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1311.5835
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Some lessons for us scientists (too) from the “Sokal affair”
Pablo Echenique-Robba
_Instituto de Química Física Rocasolano, CSIC, Spain_
_BIFI, ZCAM, DFTUZ, University of Zaragoza, Spain_
[email protected]
http://www.pabloecheniquerobba.com
###### Abstract
In this little non-technical piece, I argue that some of the lessons that can
be learnt from the bold action carried out in 1996 by the physicist Alan Sokal
and typically known as the “Sokal affair” not only apply to some sector of the
humanities (which was the original target of the hoax), but also (with much
less intensity, but still) to the hardest sciences.
The reader probably knows about the famous “Sokal affair”. This refers to an
illuminating action designed and carried out by Alan Sokal in 1996. The
physics professor at NYU submitted an article entitled “Transgressing the
boundaries: Towards a transformative hermeneutics of quantum gravity” to
_Social Text_ , a high-impact, well known academic journal of postmodern
cultural studies. In Sokal’s own words, what he wanted to test was this:
“Would a leading North American journal of cultural studies —whose editorial
collective includes such luminaries as Fredric Jameson and Andrew Ross—
publish an article liberally salted with nonsense if (a) it sounded good and
(b) it flattered the editors’ ideological preconceptions?” (Sokal, 1996a).
That is, he deliberately sent and absurd article which was “a pastiche of
left-wing cant, fawning references, grandiose quotations, and outright
nonsense…structured around the silliest quotations [by postmodernist
academics] he could find about mathematics and physics” (Wikipedia, 2013), an
article that he wrote “so that any competent physicist or mathematician (or
undergraduate physics or math major) would realize that it is a spoof” (Sokal,
1996a).
The answer to Sokal’s question was (unfortunately for our trust in the
collective intelligence of humankind) _yes_. The article got published in
_Social Text_ (Sokal, 1996b), and he soon denounced it was a hoax in the
journal _Lingua Franca_ (Sokal, 1996a).
The whole business is very interesting and several considerations enter the
mix:
First, it is important to remark that Sokal is a declared “leftist” (whatever
this 1-dimensional classification of political tendencies may mean in these
times), and one of his objectives was to denounce the anti-scientific, anti-
rationalistic attitude of a large part of the left. This is important, it is
also very sad (specially for rationalistic “leftists”), it is as valid now as
it was in 1996, but I will not focus on it here.
Another lesson that the Sokal affair teaches us is that believing in things
that make us feel good can be dangerous (to say the least). This is explicit
in his second point, “(b) it flattered the editors’ ideological
preconceptions”. Why? Because, for most people, confirming preconceptions
feels good and contradicting them feels bad.
The lesson is in fact more general than this, since confirming preconceptions
is by no means the only way of producing nice warm feelings out of beliefs and
intellectual conclusions. The sources are multiple: believing that there is
life after death, believing that medicines (such as homeopathy) exist with no
secondary effects and capable of curing virtually everything, believing that
you are right about a point and most people is wrong (Neil Armstrong didn’t go
the Moon), and many more, all make people feel good for obvious reasons.
Another way of putting it is due to David Albert. In a great interview in
which he tries to control the damage of having been inadvertently talked into
participating in the shameful film “What the bleep do we know?”, he explains
that the main difference between the views which science helps us to arrive to
and those defended by the Vatican or by the producers and fans of the film is
that the second are (and must be!) “therapeutic”, while the views suggested by
science do not have to be (and typically are not) (Albert, 2012). Science
forces us to be honest to ourselves (when it works well), and this includes
not letting warm feelings lead us to “therapeutic” but false conclusions about
the world.
Of course, these blatantly obvious concessions to one’s feelings are nowhere
to be found among successful scientists in the hard sciences, but I think that
something more subtle and related to this _is_ in place. No serious scientist
will let herself be influenced by not wanting to die, or by the desire of
having a cure-it-all medicine; that is too childish. But it is also clear that
some pressure exists to arrive to conclusions that, say, confirm what was said
in previous publications by the same scientist, that are consistent with the
achievements that were promised in the last funding grant, or that do not go
too much against the usual way of understanding things in the corresponding
field (thus making the peer-review process “smoother”). Depending on the
personality of the scientist, these pressures will be enough to lead the
discourse to wrong (but convenient) conclusions…or not. After all, confirming
and thus increasing the importance of one’s past results, getting nice grants,
and not having to struggle too much with referees suspicious of our heterodoxy
_does_ feel good. And scientists are human —despite many opinions on the
contrary.
A very nice example is one that Dennett (2009) likes to use. It seems that
when “The origin of species” was published a Robert Beverley MacKenzie
answered Darwin with a long criticism containing the following paragraph:
> But in the Theory with which we have to deal, Absolute Ignorance is the
> artificer, so that we may enunciate as the fundamental principle of the
> whole system, that in order TO MAKE A PERFECT AND BEAUTIFUL MACHINE IT IS
> NOT REQUISITE TO KNOW HOW TO MAKE IT [capital (outraged) letters in the
> original]. This proposition will be found, on a careful examination, to
> express in a condensed form the essential purport of the Theory, and to
> express in a few words all Mr Darwin’s meaning; who, by a strange inversion
> of reasoning, seems to think Absolute Ignorance fully qualified to take the
> place of Absolute Wisdom in all the achievements of creative skill.
As Dennett says, “Exactly!” This piece of text is one of the most accurate,
distilled and insightful descriptions of what Darwin had achieved, thus
proving that MacKenzie was a clever fellow who had read the whole treatise and
who had understood it thoroughly. However, he not only disagreed, he hated
Darwin’s conclusion. Why? Because it went against one of the beliefs that he
held dearest and which made him good and warm inside: that an intelligent
creator was behind life in general and humans in particular.
When doing science, it is convenient to remember Feynman’s famous aphorism:
“The first principle is that you must not fool yourself, and you are the
easiest person to fool” (Feynman, 1999, chap. 10) —which is, of course, also
applicable to me.
An example much closer to my line of work pertains some analyses of hybrid
quantum-classical models in chemical physics. I will not go so far as to state
that the authors of the corresponding papers are guilty of “fooling
themselves” with respect to their quantitative conclusions (after all, the
conclusions tend to be numerically validated, and rigorously so). But I cannot
help realizing the uncritical way in which some ill-defined and even false
statements are repeated and (it seems) carried forward from one introduction
to the next. For example, in the otherwise excellent review by (Truhlar, 2007)
(and by no means _only_ there) we can find the statement that Ehrenfest
evolution is unitary —which, being non-linear, is obviously not (Alonso et
al., 2011, 2012). I think that this should make us a bit suspicious about the
hypotheses from which these papers start, and maybe also about the
interpretation of the quantitative results. Of course, the same caution should
be exercised if we catch _ourselves_ repeating something uncritically. Nobody
is free from making this kind of mistakes.
A third lesson that we can learn from Sokal’s hoax is emphasized in the book
he later wrote together with Jean Bricmont (Sokal and Bricmont, 1998), namely,
that postmodernist writers like to misuse scientific and mathematical concepts
to support their “arguments” (e.g., a given postmodernist argued that the
famous equation $E=mc^{2}$ is a “sexed equation” because “it privileges the
speed of light over other speeds that are vitally necessary to us”).
Again, this is an extreme (and funny!) case of a much more general practice.
It is common that all kinds of thinkers use concepts from a “more fundamental”
(or just different) field to sound more clever and attach more weight to their
arguments. The trick is very simple in its workings: Since you write mostly
for your colleagues (who work in the same field as you), it is very likely
that they do not understand very well the borrowed concepts that you are
planning to use. However, they are not certain that you don’t understand them
either (hey, maybe you spent your last sabbatical reading about formal logic,
who knows). Hence, if you use the concepts with gravity and (apparent) self-
assurance, they might think that you know what you mean, and (not knowing
formal logic) they will probably assent silently. Try it, it works!
As I say, this is a common pitfall in scientific discourses and it is not
always so obvious and ridiculous as in postmodernist papers. Normally, the
discipline from which the borrowed concepts come from is very close to the one
in which the author is an expert, thus making the _bona fide_ assumption that
she knows what she means more reasonable. Also, since the borrowed concepts
_are_ in fact close, the author might misuse them, but only slightly. She is
not an expert, but she is not completely alien to them either.
I claim here that theoretical physicists (including myself) are sometimes
guilty of this kind of slight misuses related to philosophical, mathematical
or biological concepts; mathematicians borrow gaily from physics; biologists
from physics and chemistry; and theoretical chemists from quantum physics and
mathematics.
Finally, in my opinion probably the greatest warning coming from the Sokal
affair is related to the dangers of using ambiguous and vague language. One of
the points that Sokal and Bricmont (1998) discuss in their book is indeed
“manipulating words and phrases that are, in fact, meaningless” or the use of
“deliberately obscure language”, but my content is that this is not again
something circumscribed to the most absurd postmodernist texts only. This is a
practice which is all-pervading; and not only in science, but in society as a
whole. It fact, it is in science where the greatest efforts have been made to
sharpen the language, to be precise, to deal with unique meanings, to
disambiguate natural words, and I think that this is one of the main reasons
behind the enormous achievements of our scientific and technological society
(the scientific method: yes; the aforementioned honest approach to nature:
yes; the precise language: no doubt, too).
You see, if a word has three (or twenty!) possible meanings and we do not
start by declaring with care and precision which one of them we are thinking
about, it is very likely that I am using one of the meanings and you are using
a different one. If the discourse contains not only one such word but many of
them, the odds that we do not understand each other are very high. We will
very probably end talking past each other or, in the best of cases, we will
strongly disagree and we will be amazed how the other person can possibly hold
such absurd beliefs about the world. If we also include the possibility that
some of the words’ meanings have blurred boundaries (bald, tall, teenager),
that some words have no meaning at all (chakra, aura, karma, luck), or we
accept composed concepts made of words that have meaning independently but it
is destroyed upon combination (quantum healing, negative vibrations), then you
can imagine how bad the situation can get.
Many conversations are like this in everyday life and, unfortunately, also in
science (as I say, to a much lower degree, but still). Even in quantum
mechanics, one of the finest theories ever created by us humans, many
conceptual problems have survived for almost a century very likely due (in
part) to the use of ambiguous language in its very axiomatic foundations
(Bricmont, 2013, Echenique-Robba, 2013). In this case, the word “measure”
seems to be the likely culprit.
It takes a lot of work to try to be as precise as possible in every sentence,
in every word, but I think it is worth the effort. I think it is better to
write less, to publish less, but to think deeper. To stop and ask ourselves
from time to time: “What do I really mean with ‘wordX’? Am I sure that I am
using it properly? Am I sure that I can define it sharply and neatly?” I think
that being extremely careful with the meaning of words is not just being picky
and wasting others’ time, but it can serve to prove that some widely accepted
hypotheses are wrong, and to arrive to new and applicable results.
The lessons of the “Sokal affair” do not apply to cultural studies only, but
also to science.
## References
* Albert (2012) D. Z. Albert. Interview with David Albert. In _What the bleep do we know?_ 2012\. http://www.youtube.com/watch?v=K99Dic75dxg.
* Alonso et al. (2011) J. L. Alonso, A. Castro, J. Clemente-Gallardo, J. C. Cuchí, P. Echenique, and F. Falceto. Statistics and Nosé formalism for Ehrenfest dynamics. _J. Phys. A: Math. Theor._ , 44:395004, 2011. http://arxiv.org/abs/1104.2154.
* Alonso et al. (2012) J. L. Alonso, J. Clemente-Gallardo, J. C. Cuchí, P. Echenique, and F. Falceto. Ehrenfest dynamics is purity non-preserving: A necessary ingredient for decoherence. _J. Chem. Phys._ , 137:054106, 2012. http://arxiv.org/abs/1205.0885.
* Bricmont (2013) J. Bricmont. Interview with Jean Bricmont. In _Conference “Quantum Theory without Observers III” (ZiF, Bielefeld)_. 2013. http://www.youtube.com/watch?v=nQEtV5I_RNk.
* Dennett (2009) D. C. Dennett. Dennett on free will and evolution, 2009. http://www.youtube.com/watch?v=2ZhuaxZX5mc.
* Echenique-Robba (2013) P. Echenique-Robba. Shut up and let me think! Or why you should work on the foundations of quantum mechanics as much as you please, 2013. http://arxiv.org/abs/1308.5619.
* Feynman (1999) R. P. Feynman. _The pleasure of finding things out_. Basic Books, 1999.
* Sokal (1996a) A. Sokal. A physicist experiments with cultural studies. _Lingua Franca_ , May 1996, 1996a. http://bit.ly/1fEV1hF.
* Sokal (1996b) A. Sokal. Transgressing the boundaries: Towards a transformative hermeneutics of quantum gravity. _Social Text_ , 46-47:217–252, 1996b. http://bit.ly/1fEWB32.
* Sokal and Bricmont (1998) A. Sokal and J. Bricmont. _Fashionable nonsense: Postmodern intellectuals’ abuse of science_. Picador, 1998.
* Truhlar (2007) D. G. Truhlar. Decoherence in Combined Quantum Mechanical and Classical Mechanical Methods for Dynamics as Illustrated for Non-Born–Oppenheimer Trajectories. In D. A. Micha and I. Burghardt, editors, _Quantum Dynamics of Complex Molecular Systems_ , pages 227–243. Springer, Berlin, 2007.
* Wikipedia (2013) Wikipedia. Sokal affair, 2013. http://en.wikipedia.org/wiki/Sokal_affair.
|
arxiv-papers
| 2013-11-22T18:13:03 |
2024-09-04T02:49:54.112039
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pablo Echenique-Robba",
"submitter": "Pablo Echenique-Robba",
"url": "https://arxiv.org/abs/1311.5835"
}
|
1311.6004
|
# Semiclassical and quantum behavior of the Mixmaster model in the polymer
approach
Orchidea Maria Lecian [email protected] Dipartimento di Fisica (VEF), P.le A.
Moro 5 (00185) Roma, Italy Giovanni Montani
[email protected] Dipartimento di Fisica (VEF), P.le A. Moro 5
(00185) Roma, Italy ENEA-UTFUS-MAG, C.R. Frascati (Rome, Italy) Riccardo
Moriconi [email protected] Dipartimento di Fisica (VEF), P.le A. Moro 5
(00185) Roma, Italy
###### Abstract
We analyze the quantum dynamics of the Bianchi Type IX model, as described in
the so-called polymer representation of quantum mechanics, to characterize the
modifications that a discrete nature in the anisotropy variables of the
Universe induces on the morphology of the cosmological singularity. We first
perform a semiclassical analysis, to be regarded as the zeroth-order
approximation of a WKB (Wentzel-Kramers-Brillouin) approximation of the
quantum dynamics, and demonstrate how the features of polymer quantum
mechanics are able to remove the chaotic properties of the Bianchi IX
dynamics. The resulting evolution towards the cosmological singularity
overlaps the one induced, in a standard Einsteinian dynamics, by the presence
of a free scalar field. Then, we address the study of the full quantum
dynamics of this model in the polymer representation and analyze the two
cases, in which the Bianchi IX spatial curvature does not affect the wave-
packet behavior, as well as the instance, for which it plays the role of an
infinite potential confining the dynamics of the anisotropic variables. The
main development of this analysis consists of investigating how, differently
from the standard canonical quantum evolution, the high quantum number states
are not preserved arbitrarily close to the cosmological singularity. This
property emerges as a consequence, on one hand, of the no longer chaotic
features of the classical dynamics (on which the Misner analysis is grounded),
and, on the other hand, of the impossibility to remove the quantum effect due
to the spatial curvature. In the polymer picture, the quantum evolution of the
Bianchi IX model remains always significantly far from the semiclassical
behavior, as far as both the wave-packet spread and the occupation quantum
numbers are concerned. As a result, from a quantum point of view, the
Mixmaster dynamics loses any predictivity characterization for the discrete
nature of the Universe anisotropy.
###### pacs:
98.80.Qc, 04.60.Kz, 04.60.Pp
## INTRODUCTION
Even though the thermal history of the Universe is properly described by the
homogeneous and isotropic Robertson-Walker cosmological model kolbeturner , up
to the very early stage of its evolution, reliably since the inflationary
phase has been completed, the nature of the cosmological singularity is a
general property of the Einsteinian dynamics, suggesting the necessity of
relaxing the high symmetry of the geometry that characterizes the Big-Bang. A
very valuable insight on dynamical behaviors more general than the simple
isotropic case, especially in view of the classical features of the initial
singularity and of its quantum ones, is offered by the Bianchi type IX
modelbklref0 -bklref8 , essentially for the following three reasons: i) this
model is homogeneous, but its dynamics possesses the same degree of generality
as any generic inhomogeneous model ii) the canonical quantization of this
model can be performed via a minisuperspace approach, which reduces the
asymptotic evolution near the singularity to the well-defined paradigm of the
’particle in a box’ iii) the late (classical) evolution of the model is
naturally reconcilied, with or without the inflationmontani-kirillov montani-
kirillov2 , to that of a closed isotropic Universerussi .
Two years after the derivation of the Wheeler-DeWitt equation, C.W.Misner
applied this canonical quantization approach to the Bianchi IX model, which,
in the Hamiltonian version he had just provided (restating the oscillatory
regime derived by Belinski-Khalatnikov-Lifshitz), constitutes a proper
application.
The main result of the Misner quantum analysis of the Bianchi IX dynamics is
to provide a brilliant demonstration of the phenomenon for which very high
occupation numbers are preserved in the evolution towards the cosmological
singularity. This result is achieved by using the chaotic properties of the
classical Bianchi IX dynamics (the Mixmaster model) and by approximating the
potential well, in which the Universe-particle moves, by a simple square box,
instead of the equilateral triangle it indeed is.
Many subsequent studies have been pursued on such a quantum dynamics, both in
the Misner variables misnermixmaster , as well as in other frameworks (as the
Misner-Chitré schememisnercitre ,misnercitre2 ), but the original
semiclassical nature of the cosmological singularity, when considered in terms
of high occupation numbers, still remains the most striking prediction of the
canonical quantum dynamics provided by the Mixmaster model.
In the Misner variables, the dynamical problem is reduced to a two-dimensional
scheme, in which the role of the time variable is played by the Universe
volume, while the two physical degrees of freedom are represented via the
Universe anisotropies. Such a picture is elucidated by an ADM (Arnowitt-Deser-
Misner ADM ) reduction of the variational principle (based on the solution of
the superHamiltonian constraint), but its physical significance emerges in a
very transparent way, already in the direct Hamiltonian approach to the
dynamics.
Here, we address the quantum analysis of the Bianchi IX model by using the
superHamiltonian constraint, which, via the Dirac prescription, leads to the
Wheleer-DeWitt equation; this procedure is expectedly fully consistent with
the Schrödinger-like dynamics following from the ADM reduction of the Dirac
constraint. The new feature we introduce by the present study is the discrete
nature of the anisotropic degrees of freedom, by a two-dimensional
quantization approach, based on the so-called polymer representation of
quantum mechanics. The use of such a modified quantization scheme is justified
by the request that a cut-off in the spatial scale, as expected at the
Planckian level, would induce a corresponding discrete morphology in the
configuration space. The reason for retaining the isotropic Misner variables,
connected to the Universe volume, as a continuous ones, relies on the role
time plays in the dynamical scheme.
This way, we discuss first the semiclassical behavior of the model, to be
regarded to as the zeroth-order approximation of a WKB expansion of the full
quantum theory. By other words, we analyze the classical behavior of a
modified Hamiltonian dynamics of the Mixmaster model, based on the
prescriptions fixed by the classical limit of the new paradigm. In this
respect, we remark that the typical scale of the polymer discretization is not
directly related to the value of $\hbar$; the classical limit for this
quantity approaching zero still constitutes a modification of the Einsteinian
classical dynamics. Such a semiclassical study is relevant to the
interpretation of the full quantum behavior of the system, especially for the
analysis of localized wave packets. The main result of this semiclassical
analysis is the demonstration that the chaotic structure of the asymptotic
evolution of the Bianchi IX model to the cosmological singularity is naturally
removed by a dynamical mechanism very similar to the one induced on the same
dynamics by the introduction of a free massless scalar field. In the limit of
small values of the polymer lattice parameter, we calculate the modified
reflection law for the point-Universe against the potential walls, which are
due to the spatial curvature. For the general case, we provide a precise
description of how the bounce against these walls is avoided and of the
condition that the free-motion parameters of the model must satisfy for it to
take place.
The absence of chaos in the semiclassical behavior of the polymer Mixmaster
model prevents us from directly implementing the Misner procedure, which is
basic to its description of the states approaching the singularity with very
high occupation numbers. We are therefore lead to analyze separately two
cases: one in which the potential walls can be neglected in the quantum
evolution, such that we deal with free-particle wave packets, and one in which
when the potential walls play the role of a ’box’, in which the point-Universe
is confined. In both cases, the behavior of localized wave packets is analyzed
to better understand the limit up to which the Misner semiclassical feature
survives in this modified approach. The main outcome we develop here is to
identify how the presence of the walls is, sooner or later, relevant for the
wave packets evolution, such that also the states with high occupation numbers
are accordingly obliged to spread close enough to the singularity, simply
because the potential box destroys their semiclassical nature. The direct
comparison with the Misner result is not possible because of the semiclassical
behavior of the model, but the conclusion of our analysis is otherwise very
clear, because there is no chance to built up a localized state that can reach
the singularity without bouncing against the potential walls. This is because
the conditions to fix a direction in the configuration space, which would
allow for a free motion, is indeed time dependent and is, sooner or later,
violated during the evolution of the wave packets. In this scheme, there is no
possibility to retain the semiclassical features in the quantum description,
and we can therefore claim that the singularity of the Mixmaster Universe, as
viewed in the present polymer representation, can not be described by
semiclassical notion, as for the Einsteinian oscillatory regime of the
expanding and contracting independent directions, and that even its quantum
relic, i.e. the occurrence of high occupation numbers close to the singular
point, is removed in a discrete quantum picture for the anisotropy degrees of
freedom.
The relevance of this result is enforced by observing that, as investigated in
montani-kirillov , an oscillatory regime cannot exist before a real classical
limit of the Universe is reached. As a consequence, since a simple model for
the cut-off physics is able to cancel also the memory of semiclassical
features in the Planck era (as Misner argued in misner ), we are lead to
believe that the classical Mixmaster dynamics is not fully compatible with the
quantum origin of the Universe, and it is indeed a classical dynamical regime
reached by the system when the quantum effects are very small and the physical
and configuration spaces appear as bounded by continuous domains.
A certain specific interest for the implementation of a polymer approach to
the quantum dynamics of the Universe was rised by the analogy of this quantum
prescription to the main issues of Loop Quantum Cosmology, which, in the
Minisuperspace, essentially reduces to a polymer treatment of the Ashtekar-
Barbero-Immirzi variables, as adapted to the cosmological settingloop1 -loop3
.
The first Loop Quantum cosmological analysis of the Bianchi IX model was
provided in boj1 , where it was argued the non-chaotic nature of the
semiclassical dynamics. The main reason of such non-chaotic behavior of the
Bianchi IX model must be determined in the discrete nature of the Universe
volume and, in particular in its minimal (cut-off) value. In fact,
asymptotically to the singularity, the potential walls can no longer
arbitrarily growth and the point-Universe confinement is removed. This
analysis was based on the so-called inverse volume corrections and properly
accounts for the induced semiclassical implications of the Loop Quantum
Gravity theory. Nonetheless, the considered approach is based on a
regularization scheme (the so called $\mu_{0}$ one) that is under revision, in
order to provide a consistent reformulation of the Bianchi IX dynamics as done
in APS for the the isotropic Robertson-Walker geometry (the $\bar{\mu}$
regularization scheme).
A step in this direction has be pursued in wilson , where the Loop quantum
dynamics is rigorously restated by adopting the $\bar{\mu}$ regularization
scheme, demostrating that, under certain circumstances, the chaotic features
of the Bianchi IX model are removed again. However, this result holds only
when the quantum picture includes a massless scalar field, able to remove the
chaoticity even on a classical Einsteinian level. The reason of this striking
difference in the two result obtained in these two approaches, must be
individualized in the kind of semiclassical corrections discussed in wilson .
Indeed, in this analysis only the so-called Holonomy correction contributions
are considered and they are unable to induce on the dynamics the basic feature
of a volume cut-off scale. It is just this different type of quantum
corrections adopted to construct the semiclassical limit, the reliable source
of the non-generic nature of the chaoticity removal. Althought it is expected
that inverse volume corrections can remove the chaotic behavior of the Bianchi
IX model as in boj1 , nevertheless this has not been explicitly demonstrated
and it stands as a mere conjecture.
It is worth noting that a critical revision of the Loop Quantum Cosmology
picture of the primordial Universe space, was presented in cian-mon , where
the necessity of a gauge fixing in implementing the homogeneity constraint is
required. A consistent quantum reformulation of the dynamics of an homogeneous
model was then constructed in cian-ale , which allows a semiclassical limit of
the theory, in close analogy to the full Loop Quantum Gravity theory.
Despite the polymer formulation of the canonical approach to the
minisuperspace mimics very well some features of the Loop Quantum cosmology
methodology (de facto a polymer treatment of the restricted Ashtekar-Barbero-
Immirzi variables for the homogeneity constraint), however, there is a crucial
difference between our result and such recent Loop-like approaches. In fact,
we apply the polymer procedure to the anisotropic variables only (the real
degrees of freedom of the cosmological gravitational field), leaving the
Universe volume at all unaffected by the cut-off physics, in view of its time-
like behavior. The removal of the chaos, discussed here, is therefore not
related to the volume discretization and it is also difficult to characterize
its relation with the Holonomy correction approach (studying properties of the
edge morphology more than of the nodes, but non-directly reducible to the
anysotropy concept). In this respect, the present result must be regarded as
essentially an independent one with respect to the ones actually available in
Loop quantum Cosmology.
This paper is organized as follows.
In Section I, we introduce the polymer representation of quantum mechanics, by
a kynematical and a dynamical point of view. Then we analyze the continuum
limit and conclude the Section illustrating two fundamental examples of one-
dimensional systems: the polymer free particle and the particle in a box.
In Section II, we review the principal (classical and quantum) features of the
Mixmaster model, as studied by Misner inmisner .
Section III is dedicated to the study of the polymer Mixmaster model, from a
semi-classical point of view. In particular, we analyze the modified
relational motion between the Universe-particle and the walls, and we derive a
modified reflection law for one single bounce against the wall.
In Section IV, we build up the wave packets for the case when the wave
function of the Universe is related to a free polymer particle and to a
polymer particle in a square box, respectively.
Finally, Section V is devoted to the numerical integrations of the polymer
wave packets and to the analysis on the quantum numbers related to the
anisotropy.
Concluding remarks complete the paper.
## I The POLYMER REPRESENTATION OF QUANTUM MECHANICS
To apply the modified polymer approach to the Mixmaster quantum dynamics, we
briefly summarize the fundamental features of this modified quantization
scheme. In particular, after giving a general picture of the model, we
consider the two specific cases of the free particle and of the particle in a
box, which are relevant for the subseguent cosmological study.
### I.1 Kynematical properties
The Polymer representation of quantum mechanics is a non-equivalent
representation of the usual Schrödinger quantum mechanics, based on a
different kind of Canonical Commutation Rules (CCR). It is a really useful
tool to investigate the consequences of the hypothesis for which the phase
space variables are discretized.
For the definition of the kinematics of a simple one-dimensional systemcorichi
, one introduces a discrete set of kets $|\mu_{i}\rangle$, with
$\mu_{i}\in\mathbb{R}$ and $i=1,...,N$. These vectors $|\mu_{i}\rangle$ are
taken from the Hilbert space
$\mathcal{H}_{poly}=L^{2}(\mathbb{R}_{b},d\mu_{H})$, i.e. the set of square-
integrable functions defined on the Bohr compactification of the real line
$\mathbb{R}_{b}$ with a Haar measure $d\mu_{H}$. One chooses for them an inner
product with a discrete normalization
$\langle\nu|\mu\rangle=\delta_{\nu,\mu}$. The state of the system is described
by a generic linear combination of them
$|\psi\rangle=\sum\limits_{i=1}^{N}a_{i}|\mu_{i}\rangle.$ (1)
One can identify two fundamental operators in this Hilbert space: a label
operator $\widehat{\varepsilon}$ and a shift operator $\widehat{s}(\lambda)$.
They act on the kets as follows
$\widehat{\varepsilon}|\mu\rangle=\mu|\mu\rangle\quad,\quad\widehat{s}(\lambda)|\mu\rangle=|\mu+\lambda\rangle.$
(2)
To characterize our system, described by the phase space variables $p$ and
$q$, one assigns a discrete characterization to the variable $q$, and chooses
to describe the wave function of the system in the so-called $p$-polarization.
Consequently, the projection of the states on the pertinent basis vectors is
$\phi_{\mu}(p)=\langle p|\mu\rangle=e^{-i\mu p}.$ (3)
Through the introduction of two unitary operators
$U(\alpha)=e^{i\alpha\widehat{q}},V(\beta)=e^{i\beta\widehat{p}},(\alpha,\beta)\in\mathbb{R}$
which obey the Weyl Commutation Rules (WCR)
$U(\alpha)V(\beta)=e^{i\alpha\beta}V(\beta)U(\alpha)$, one sees that the label
operator is exactly the position operator, while it is not possible to define
a (differential) momentum operator, as a consequence of the discontinuity for
$\widehat{s}(\lambda)$ pointed out in Eq.(2).
### I.2 The dynamical features
For the dynamical characterization of the model, the properties of the
Hamiltonian system have to be investigated. The simplest Hamiltonian
describing a one-dimensional particle of mass $m$ in a potential $V(q)$ is
given by
$H=\frac{p^{2}}{2m}+V(q).$ (4)
In the $p$-polarization, as a consequence of the discreteness of $q$, it is
not possible to define $\widehat{p}$ as a differential operator. The standard
procedure is to define a subspace $\mathcal{H}_{\gamma_{a}}$ of
$\mathcal{H}_{poly}$ containing all vectors that live on the lattice of points
identified by the lattice spacing $a$
$\gamma_{a}=\mathcal{f}q\in\mathbb{R}|q=na,\forall n\in\mathbb{Z}\mathcal{g},$
(5)
where $a$ has the dimensions of a length.
Consequently, the basis vectors are of the form $|\mu_{n}\rangle$ (where
$\mu_{n}=an$), and the states are all of the form
$|\psi\rangle=\sum\limits_{n}b_{n}|\mu_{n}\rangle.$ (6)
The basic realization of the polymer quantization is to approximate the term
corresponding to the non-existent operator (this case $\widehat{p}$), and to
find for this approximation an appropriate and well-defined quantum operator.
The operator $\widehat{V}$ is exactly the shift operator $\widehat{s}$, in
both polarizations. Through this identification, it is possible to exploit the
properties of $\widehat{s}$ to write an approximate version of $\widehat{p}$.
For $p\ll\frac{1}{a}$, one gets
$p\simeq\frac{\sin(ap)}{a}=\frac{1}{2a}\left(e^{iap}-e^{-iap}\right)$ (7)
and then the new version of $\widehat{p}$ is
$\widehat{p}_{a}|\mu_{n}\rangle=\frac{i}{2a}\left(|\mu_{n-1}\rangle-|\mu_{n+1}\rangle\right).$
(8)
One can define an approximate version of $\widehat{p}^{2}$. For
$p\ll\frac{1}{a}$, one gets
$p^{2}\simeq\frac{2}{a^{2}}\left[1-\cos(ap)\right]=\frac{2}{a^{2}}\left[1-e^{iap}-e^{-iap}\right]$
(9)
and then the new version of $\widehat{p}^{2}$ is
$\widehat{p}_{a}^{2}|\mu_{n}\rangle=\frac{1}{a^{2}}\left[2|\mu_{n}\rangle-|\mu_{n+1}\rangle-|\mu_{n-1}\rangle\right].$
(10)
Remembering that $\widehat{q}$ is a well-defined operator as in the canonical
way, the approximate version of the starting Hamiltonian (4) is
$\widehat{H}_{a}=\frac{1}{2m}\widehat{p}_{a}^{2}+V(\widehat{q}).$ (11)
The hamiltonian operator $\widehat{H}_{a}$ is a well-defined and simmetric
operator belonging to $\mathcal{H}_{\gamma_{a}}$.
### I.3 The continuum Limit
The polymer representation of quantum mechanics is related with Schrödinger
representation can now be analyzed.
Starting from a Hilbert space $\mathcal{H}_{poly}$, one needs to verify a
limit operation to demonstrate that the space is isomorphic to the Hilbert
space $\mathcal{H}_{S}=L^{2}(\mathbb{R},dq)$ 111$L^{2}$ is the set of square-
integrable functions defined on the real line $\mathbb{R}$ with a Lebesgue
measure $dq$.
The natural way to proceed is to start from a lattice
$\gamma_{0}=\mathcal{f}q_{k}\in\mathbb{R}|q_{k}=ka_{0},\forall
k\in\mathbb{Z}\mathcal{g}$ and subdivide each interval $a_{0}$ in $2^{n}$
intervals of length $a_{n}=\frac{a_{0}}{2^{n}}$. Unfortunately, this is not
possible because, when densifying the lattice, the elements of
$\mathcal{H}_{poly}$ have a norm that tend to infinity. This is because
$\mathcal{H}_{S}$ and its states cannot be included in $\mathcal{H}_{poly}$.
However, it is possible to realize a different procedure. From a continuous
wave function, one has to find the best wave function defined on the lattice
that approximates it, in the limit when the lattice becomes denser. The
strategy to properly implement this approach is the introduction of a scale
$C_{n}$, which, in our case, is the subdivision of the real line into disjoint
intervals of the form $\alpha_{i}=[ia_{n},(i+1)a_{n})$, where the extrema of
the range are the lattice points. On this level, one approximates continuous
functions with constant intermediate states belonging to $\mathcal{H}_{S}$. So
for, one has a whole series of effective theories, depending on the scale
$C_{n}$, that approximate much and much better the continuous functions and
that have a well defined Hamiltonian. As in corichidue , by introducing a cut-
off for each Hamiltonian defined on the intervals, making the operation of
coarse graining and entering a normalization factor in the internal product,
one verifies that the existence of continous limit is equivalent to the
description of the energy spectrum (relative to the Hamiltonian defined after
the cut-off) as tending to the continuous spectrum, such that a complete set
of normalized eigenfunctions exists. The space obtained this way is isomorphic
to the space $\mathcal{H}_{S}$.
### I.4 The Free Polymer particle
In this sub-section, we analize the simplest one-dimensional system in the
presence of a discrete structure of the space variable $q$, i.e. the free
polymer particletaub . When the free polymer particle problem is taken into
account, the potential term in Eq.(11) is negligible. Therefore, in the
$p$-polarization, the quantum state of the system is described by the wave
function $\psi(p)$ via the eigenvalue problem
$\left[\frac{1}{ma^{2}}\left(1-\cos(ap)\right)-E_{a}\right]\psi(p)=0.$ (12)
Here, $E_{a}$ is an eigenvalue depending on the scale $a$, and one has
$E_{a}=\frac{1}{ma^{2}}\left[1-\cos(ap)\right]\leq\frac{2}{ma^{2}}=E_{a}^{max}.$
(13)
From Eq.(13), one sees that, for each scale $a$, there is a bounded and
continous eigenvalue. In the limit $a\rightarrow 0$, i.e. switching the
polymer effect off, one obtains the unbounded eigenvalue $E=\frac{p^{2}}{2m}$,
typical for a free particle. It is easy to verify that the solution $\psi(p)$
for the eigenvalue problem (12 has the form
$\psi(p)=A\delta(p-P_{a})+B\delta(p+P_{a}),$ (14)
where $A,B$ are integration constants and
$P_{a}=\frac{1}{a}\arccos(1-ma^{2}E_{a})$ (15)
induces modified dispersion relation in the presence of a polymer structure.
For an (inverse) Fourier transform for the eigenfunction (14), one obtains the
eigenfunction in the $q$-polarization $\psi(q)$ as
$\psi(q)=\int\psi(p)e^{ipq}=Ae^{iqP_{a}}+Be^{-iqP_{a}}.$ (16)
The eigenfunction in the $q$-polarization becomes a modified wave plane, due
to the dispersion relation (15), which are valid at each scale.
### I.5 The polymer particle in a box
In this subsection, we will analyze the dynamical features of a one-
dimensional particle in a box, within the framework of the polymer
representation of quantum mechanics. For a one-dimensional box (i.e. a
segment) of length $L=na,n\in\mathbb{N}$, the potential $V(q)=V(na)$ reads
$V(q)=\begin{cases}\infty,&x>L,x<0\\\ 0,&0<x<L\end{cases},$ (17)
i.e. in the case of a potential limited by infinite walls. In this case, the
particle behaves as a free particle within the segment, and proper boundary
conditions for eigenfunction (16) have to be imposed. In particular
$\psi(0)=\psi(L)=0\longrightarrow\begin{cases}&A=-B\\\
&LP_{a}=n\pi\end{cases},$ (18)
for which the eigenfunctions $\psi(q)$ in the $q$ polarization are obtained
$\psi(q)=2A\sin\left(\frac{n\pi q}{L}\right).$ (19)
The corresponding energy spectrum $E_{a,n}$ is a function of both the lattice
constant $a$ and the quantum number $n$, such that
$E_{a,n}=\frac{1}{ma^{2}}\left[1-\cos\left(\frac{an\pi}{L}\right)\right].$
(20)
In the limit $a\rightarrow 0$ one gets the energy spectrum of the standard
case.
## II THE MIXMASTER MODEL: CLASSICAL AND QUANTUM FEATURES
In this section, we provide a complete description of the most relevant
achievements obtained for the dynamics of the Bianchi IX cosmological model,
both in the classical and the quantum regime towards the cosmological
singularity, as they are depicted in the two pioneering worksmisnermixmaster
,misner .
### II.1 The classical dynamics
Homogeneous spaces are an important class of cosmological models. These spaces
are characterized by the preservation of the space line element under a
specific group of symmetry, and are collected in the so-called Bianchi
classificationlandau . The most general homogeneous model is the Bianchi IX
model. As demonstrated by Belinski, Khalatnikov and Lifshitz (BKL)cinque ,
when a generic inhomogeneous space approaches the singularity, it behaves as
an ensemble of Bianchi IX independent models in each point of space222Also the
Bianchi VIII as the same degree of generality but it does not admit an
isotropic limit. Following the Misner parametrizationmisnermixmaster , the
line element for the Bianchi IX model is
$ds^{2}=N(t)^{2}dt^{2}-\eta_{ab}\omega^{a}\omega^{b},$ (21)
where $\omega^{a}=\omega^{a}_{\alpha}dx^{\alpha}$ is a set of three invariant
differential forms, $N(t)$ is the lapse function and $\eta_{ab}$ is defined as
$\eta_{ab}=e^{2\alpha}(e^{2\beta})_{ab}$. In the Misner picture, $\alpha$
expresses the isotropic volume of the universe (for
$\alpha\rightarrow-\infty$, the initial singularity is reached.), while the
matrix
$\beta_{ab}=diag(\beta_{+}+\sqrt{3}\beta_{-},\beta_{+}-\sqrt{3}\beta_{-},-2\beta_{+})$
accounts for the anisotropy of this model. The introduction of the Misner
variables allows one to rewrite the super Hamiltonian constraint (written
following the ADM formalismADM ) in this simple way
$\mathcal{H}_{IX}=-p_{\alpha}^{2}+p_{+}^{2}+p_{-}^{2}+\frac{3(4\pi)^{4}}{k^{2}}e^{4\alpha}V(\beta_{\pm})=0,$
(22)
where the ($p_{\alpha},p_{\pm}$) are the conjugated momenta to
($\alpha,\beta_{\pm}$) respectively, $k=8\pi G$, and $V(\beta_{\pm})$ is the
potential term depending only on $\beta_{\pm}$, i.e. the anisotropies.
$V(\beta_{\pm})=e^{-8\beta_{+}}-4e^{-2\beta_{+}}\cosh(2\sqrt{3}\beta_{-})+\\\
+2e^{4\beta_{+}}\left[\cosh(4\sqrt{3}\beta_{-})-1\right].$ (23)
Figure 1: ”(Color online)”.Equipotential lines of Bianchi IX model in
($\beta_{+},\beta_{-}$) planemisner .
Let us execute now the ADM reduction of the dynamicsADM2 by solving the super
Hamiltonian constraint with respect to a specific conjugated momenta and then
by identifing a time-variable for the phase space. For the purposes of this
investigation, we choose to solve (22) with respect to $p_{\alpha}$ and
identify $\alpha$ as a time-variable. This choice is justified because, if we
choose a time gauge $\dot{\alpha}=1$ in the synchronous reference system
($N(t)=1$), the isotropic volume $\alpha$ depends on the synchronous time $t$
by the relation $\alpha=\frac{1}{3}\ln t$. Then one obtains
$-p_{\alpha}=\mathcal{H}_{ADM}\equiv\sqrt{p_{+}^{2}+p_{-}^{2}+\frac{3(4\pi)^{4}}{k^{2}}e^{4\alpha}V(\beta_{\pm})},$
(24)
i.e. the so-called reduced Hamiltonian of our problem. From relation (24), one
recognizes that, as studied by C.W.Misnermisner , the dynamics of the Universe
towards the singularity is mapped to the description of the motion of a
particle that lives on a plane inside a closed domain. This way, we can study
how the anisotropies $\beta_{\pm}$ change with respect to the time variable
$\alpha$ through equations of motion related to the reduced Hamiltonian
$\begin{split}&\beta^{\prime}_{\pm}=\frac{d\beta_{\pm}}{d\alpha}=\frac{p_{\pm}}{\mathcal{H}_{ADM}},\\\
&p_{\pm}^{\prime}=\frac{dp_{\pm}}{d\alpha}=\frac{3(4\pi)^{4}}{2k\mathcal{H}_{ADM}}e^{4\alpha}\frac{\partial
V(\beta_{\pm})}{\partial\beta_{\pm}}.\end{split}$ (25)
Studying the two opposite approximations of $V(\beta_{\pm})$, i.e. far from
the walls ($V\simeq 0$) and close to the walls
($V\simeq\frac{1}{3}e^{-8\beta_{+}}$), we can obtain the relative motion
between the particle and the potential wall. It is possible to obtain, for
$V\simeq 0$, the behavior of $\beta_{\pm}$ as a function of time $\alpha$ via
a simple integration of the first equation of motion. This way, one gets
$\beta_{\pm}\propto\frac{p_{\pm}}{\sqrt{p_{+}^{2}+p_{-}^{2}}}\alpha.$ (26)
Moreover, the anisotropy velocity of the particle far from the walls is
defined as
$\beta^{{}^{\prime}}=\sqrt{\left(\frac{d\beta_{+}}{d\alpha}\right)^{2}+\left(\frac{d\beta_{-}}{d\alpha}\right)^{2}}=1$
(27)
for each value of $p_{\pm}$. On the other hand, the investigation on the
motion of one of the equivalent sides allow one to understand that the walls
move towards the ’outer’ directionou with velocity
$|\beta^{\prime}_{w}|=\frac{1}{2}$. The particle always collides against the
wall and bounces from one to another. This chaotic dynamics is the analogue of
the oscillatory regime described by BKL in tre .
It is worth noting that the regime under which $V\simeq 0$ corresponds to the
Bianchi I case of the Bianchi classification, the so-called Kasner regime, in
which the particle moves as being free and the two constraints
$\begin{split}&p_{1}+p_{2}+p_{3}=1,\\\
&p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1,\end{split}$ (28)
are satisfied. Here $p_{1},p_{2},p_{3}$ are the Kasner indices, i.e. three
real numbers that express the anisotropy of the model. Writing the spatial
metric of the Bianchi I model in the synchronous reference system, i.e.
$dl^{2}=t^{2p_{1}}dx^{1}+t^{2p_{2}}dx^{2}+t^{2p_{3}}dx^{3},$ (29)
the presence of the Kasner indices inside the spatial metric is understood to
imply a different behavior along the different directions, which define the
anisotropic directions.
On the other hand, when the particle is close the wall
($V\simeq\frac{1}{3}e^{-8\beta_{+}}$), the Bianchi II model, i.e. the model
descrbing one single bounce against infinite wall potentialmontanireview , is
considered. The system (LABEL:eqHam) can be studied close to a potential wall,
and is possible to identify two constants of motion
$\begin{split}&p_{-}=cost,\\\
&K=\frac{1}{2}p_{+}+\mathcal{H}_{ADM}=cost.\end{split}$ (30)
These relations have been obtained for the ’vertical’ potential wall in
Fig.(1); it is however necessary to stress that the bounces against the
potential walls are all equivalent as far as the dynamics of the system is
concerned, as the potential walls can be obtained one from the other, by
taking into account the symmetries of the model, as analyzed in indicikasner .
A description of this regime is illustrated in Fig.(1). The anisotropies can
be parameterized as functions of both the incidence angle and of the
reflection one, $\theta_{i}$ and $\theta_{f}$, respectively. This way,
$\begin{split}&(\beta_{-}^{\prime})_{i}=\sin\theta_{i},\\\
&(\beta_{+}^{\prime})_{i}=-\cos\theta_{i},\\\
&(\beta_{-}^{\prime})_{f}=\sin\theta_{f},\\\
&(\beta_{+}^{\prime})_{f}=\cos\theta_{f}.\end{split}$ (31)
The relations (LABEL:pmeno) are used to obtain a reflection law for a generic
single bounce
$\sin\theta_{f}-\sin\theta_{i}=\frac{1}{2}\sin(\theta_{i}+\theta_{f}).$ (32)
However, there is a maximum angle $\theta_{max}$ after which no bounce occurs.
For the occurrence of a bounce, the longitudinal component of the velocity
$\beta^{\prime}_{+}$ must be greater than the wall velocity
$\beta^{\prime}_{w}$. This condition is expressed as
$|\theta_{i}|<|\theta_{max}|=\arccos\left(\frac{\beta^{\prime}_{w}}{\beta^{\prime}_{+}}\right)=\frac{\pi}{3}.$
(33)
As a result, the particle,sooner or later, will assume all the possible
directions, regardless of the initial condition. Following the convenience
choice used by C.W. Misner in misner , and taking advantage of the geometric
properties of this scheme, in the limit close to the singularity
($\alpha\rightarrow-\infty$) one finds a conservation law of the form
$<\mathcal{H}_{ADM}\alpha>=cost.$ (34)
For two successive bounces (the $i$-th and the $(i+1)$-th of the sequence),
$\alpha^{i}$ expresses the time at which the $i$-th bounce occurs and
$\mathcal{H}_{ADM}^{i}$ the value of reduced Hamiltonian (24) just before the
$i$-th bounce: relation (34) states that
$\mathcal{H}_{ADM}^{i}\alpha^{i}=\mathcal{H}_{ADM}^{i+1}\alpha^{i+1}.$ (35)
In other words, the quantity $\mathcal{H}_{ADM}\alpha$ acquires the same
costant value as just before each bounce towards the singularity.
### II.2 The quantum behavior
The canonical quantization of the system consists of the commutation relations
$[\widehat{q}_{a},\widehat{p}_{b}]=i\delta_{ab},$ (36)
which are satisfied for $\widehat{p_{a}}=-i\frac{\partial}{\partial
q_{a}}=-i\partial_{a}$ where $a,b=\alpha,\beta_{+},\beta_{-}$. By replacing
the canonical variables with the corresponding operators, the quantum behavior
of the Universe is given by the quantum version of the superhamiltonian
constrain (22), i.e. the Wheeler-deWitt equation(WDW) for the Bianchi IX model
$\widehat{\mathcal{H}}_{IX}\Psi(\alpha,\beta_{\pm})=\\\
=\left[\partial_{\alpha}^{2}-\partial_{+}^{2}-\partial_{+}^{2}+\frac{3(4\pi)^{4}}{k^{2}}e^{4\alpha}V(\beta_{\pm})\right]\Psi(\alpha,\beta_{\pm}),$
(37)
where $\Psi(\alpha,\beta_{\pm})$ is the wave function of the Universe which
provides information about the physical state of the Universe. A solution of
Eq.(37) can be looked for in the form
$\Psi=\sum_{n}\chi_{n}(\alpha)\phi_{n}(\alpha,\beta).$ (38)
The adiabatic approximation consists in requiring that the $\alpha$-evolution
be principally contained in the $\chi_{n}(\alpha)$ coefficients, while the
functions $\phi_{n}(\alpha,\beta)$ depend on $\alpha$ parametrically only. The
adiabatic approximation is therefore expressed by the condition
$|\partial_{\alpha}\chi_{n}(\alpha)|\gg|\partial_{\alpha}\phi_{n}(\alpha,\beta)|.$
(39)
By applying condition (39), the WDW Eq.(37) reduces to an eigenvalue problem
related to the reduced hamiltonian $\mathcal{H}_{ADM}$ via
$\widehat{\mathcal{H}}_{ADM}^{2}\phi_{n}=E^{2}_{n}(\alpha)\phi_{n}=\\\
=\left[-\partial_{+}^{2}-\partial_{-}^{2}+\frac{3(4\pi)^{4}}{k^{2}}e^{4\alpha}V(\widehat{\beta_{\pm}})\right]\phi_{n}.$
(40)
However, even without finding the exact expression of the eigenfunctions, one
may gain important information about the system from a quantum point of view
near the initial singularity. From Fig.(1), one can see how the potential (23)
can be modelized as an infinitely steep potential well with a triangular base.
In misner , the strong hypothesis to replace the triangular box with a squared
box having the same area $L^{2}$ is proposed. This way, the probelm describing
a two-dimensional particle in a squared box with infinite walls is recovered.
In this case, the eigenvalue problem becomes
$\widehat{\mathcal{H}}_{ADM}^{2}\phi_{n,m}=\frac{\pi^{2}(m^{2}+n^{2})}{L^{2}(\alpha)}\phi_{n,m},$
(41)
where $m,n\in\mathbb{N}$ are the quantum numbers associated to
($\beta_{+},\beta_{-}$). By a direct calculation, we can derive
$L^{2}(\alpha)=\frac{3\sqrt{3}}{4}\alpha^{2}$, such that the eigenvalue is
$E_{n,m}=\frac{2\pi}{3^{3/4}\alpha}\sqrt{m^{2}+n^{2}}.$ (42)
As demonstrated in soluzione , substituting the eigenvalue expression (42) in
the Eq.(37), the self-consistence of adiabatic approximation is ensured. Let
us use (42) with (34) to estimate the quantum numbers behavior towards the
singularity. One can see in Eq.(42) that the eigenvalue spectrum is unlimited
from above, such that, for sufficiently high occupation numbers, the replacing
$\mathcal{H}_{ADM}\simeq E_{n,m}$ is a good approximation. This way, for
$\alpha\rightarrow-\infty$, Eq.(34) becomes
$<\mathcal{H}_{ADM}\alpha>\xrightarrow[\alpha\rightarrow-\infty]{}<\sqrt{m^{2}+n^{2}}>=cost.$
(43)
Being the current state of the Universe anisotropy characterized by a
classical nature, i.e. $\sqrt{m^{2}+n^{2}}>>1$, we can say, by Eq.(43), that
this quantity is constant approaching the singularity. This way, the quantum
state of the Universe related to the anisotropies remains classical for all
the backwards history until the singularity.
## III SEMICLASSICAL Polymer approach to the MIXMASTER MODEL
The aim of the present Section is to discuss how to apply the polymer approach
of Sec.I to the Bianchi IX model at a semiclassical level and to verify if and
how the nature of the cosmological singularity is modified. Here,
“semiclassical” means that we are working with a modified super Hamiltonian
constraint obtained as the lowest order term of a WKB expansion for
$\hbar\rightarrow 0$. At this level, the modified theory is subject to a
deterministic dynamics. Following the procedure in Sec.I.2, one can choose,
with a precise physical interpretation, to define the anisotropies of the
Universe $(\beta_{+},\beta_{-})$ as discrete variables leaving the
characterization of the isotropic variable $\alpha$ unchanged, which here
plays the role of time. This procedure formally consists in the replacement
$p_{\pm}^{2}\rightarrow\frac{2}{a^{2}}\left[1-\cos(ap_{\pm})\right].$ (44)
The superhamiltonian constraint (22) becomes
$-p_{\alpha}^{2}+\frac{2}{a^{2}}\left[2-\cos(ap_{+})-\cos(ap_{-})\right]+\frac{3(4\pi)^{4}e^{4\alpha}}{k^{2}}V(\beta_{\pm})=0.$
(45)
We define $-p_{\alpha}\equiv H_{poly}$ as the reduced Hamiltonian, such that
one gets
$-p_{\alpha}\equiv H_{poly}=\\\
=\sqrt{\frac{2}{a^{2}}\left[2-\cos(ap_{+})-\cos(ap_{-})\right]+\frac{3(4\pi)^{4}e^{4\alpha}}{k^{2}}V(\beta_{\pm})}.$
(46)
Starting from the new hamiltonian formulation (46), we can get the following
set of the hamiltonian equations as
$\begin{split}&\beta^{\prime}_{\pm}=\frac{d\beta_{\pm}}{d\alpha}=\frac{\sin(ap_{\pm})}{aH_{poly}},\\\
&p_{\pm}^{\prime}=\frac{dp_{\pm}}{d\alpha}=\frac{3(4\pi)^{4}}{2kH_{poly}}e^{4\alpha}\frac{\partial
V(\beta_{\pm})}{\partial\beta_{\pm}}.\end{split}$ (47)
This modification leaves the potential $V(\beta_{\pm})$ and the isotropic
variable $\alpha$ unchanged. Therefore, even in the modified theory, the walls
move in the ’outer’ direction with velocity $|\beta^{\prime}_{w}|=\frac{1}{2}$
and the initial singularity is not expected to be removed.
Let us start by analyzing the system far from the wall, i.e. with $V\simeq 0$.
As one can see in (LABEL:eqHampoly) when $V\simeq 0$, the anisotropy velocity
is modified if compared to the standard case. In particular, the behavior of
$\beta_{\pm}$ is proportional to the time $\alpha$, as in the standard theory,
but with a different coefficient, i.e.
$\beta_{\pm}\propto\frac{\sin(ap_{\pm})}{\sqrt{4-2[\cos(ap_{+})+\cos(ap_{-})]}}\alpha.$
(48)
In particular, by the definition of the anisotropy velocity, Eq.(27), one
obtains
$\beta^{\prime}=\sqrt{\frac{\sin(ap_{+})^{2}+\sin(ap_{-})^{2}}{4-2[\cos(ap_{+})+\cos(ap_{-})]}}=r(a,p_{\pm}).$
(49)
It is worth noting that $r(a,p_{\pm})$ is a bounded function ($r\in[0,1]$) of
parameters that remains constant between one bounce and the following one.
From Eq.(48), we have a Bianchi I model modified by the polymer substitution.
As a consequence of this feature, also in the modified theory, the
anisotropies behaves respect to $\alpha$ in a proportional way. The first
important semiclassical result is the relative motion between wall and
particle. From (49), one can observe the existence of allowed values of
$(ap_{+},ap_{-})$, such that the particle velocity is smaller than the wall
velocity $\beta^{\prime}_{w}$. Therefore, the condition for a bounce is
$\beta^{\prime}=\sqrt{\frac{\sin(ap_{+})^{2}+\sin(ap_{-})^{2}}{4-2[\cos(ap_{+})+\cos(ap_{-})]}}>\frac{1}{2}=\beta^{\prime}_{w}.$
(50)
It means that the infinite sequence of bounces against the walls, typical of
the Mixmaster Model, takes place until condition (50) is valid. When
$r<\frac{1}{2}$, the particle becomes slower than the potential wall and
reaches the singularity without no other bounces.
This feature is confirmed by the analysis on the Kasner relations
(LABEL:somma_indici). The first Kasner relation is still valid in the deformed
approach, because the sum of the Kasner indices is linked by the Misner
variables just trough the isotropic variable $\alpha$. Instead, the second
Kasner relation is directly related to the anisotropy velocity montanireview
and it results modified into
$p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=\frac{1}{3}+\frac{2}{3}\left[(\beta_{+}^{\prime})^{2}+(\beta_{-}^{\prime})^{2}\right]=\\\
=1-\frac{2}{3}(1-r^{2})=1-q^{2},$ (51)
where $q^{2}=\frac{2}{3}(1-r^{2})$. We introduce $q^{2}$ in (51) because
$\frac{2}{3}(1-r^{2})\geq 0$ for any values of $(ap_{+},ap_{-})$. The
introduction of the polymer structure for the anisotropies acts the same way
as a massless scalar field in Bianchi IX modelberger ,intersezione . For this
reason, we can choose to describe the Kasner indices with the same
parametrization due to Belinski and Khalatnikov in scalar field . It is
realized through the introduction of two parameters $(u,q)$333In the standard
case (absence of the scalar field or polymer modification, i.e. $q^{2}=0$)
$p_{1},p_{2},p_{3}$ and $u$ are related this wayindicikasner :
$p_{1}(u)=-\frac{u}{1+u+u^{2}},p_{2}(u)=\frac{1+u}{1+u+u^{2}},p_{3}(u)=\frac{u(1+u)}{1+u+u^{2}}$.
In this case $0<u<1$. and it allows to represent the all possible values of
the Kasner indices. One gets
$\begin{split}&p_{1}=\frac{-u}{1+u+u^{2}},\\\
&p_{2}=\frac{1+u}{1+u+u^{2}}\left[u-\frac{u-1}{2}(1-\sqrt{1-\gamma^{2}})\right],\\\
&p_{3}=\frac{1+u}{1+u+u^{2}}\left[1+\frac{u-1}{2}(1-\sqrt{1-\gamma^{2}})\right],\\\
&\gamma^{2}=\frac{2(1+u+u^{2})q^{2}}{(u^{2}-1)^{2}}.\end{split}$ (52)
Here, $-1<u<+1$ and $-\sqrt{\frac{2}{3}}<q<\sqrt{\frac{2}{3}}$. The presence
of $\gamma^{2}$ inside Eq.’s(LABEL:paramscala) means that not all values of
$u,q$ are allowed. The $u,q$ allowed values are those which respect the
condition $\gamma^{2}<1$. Inside this region of permitted values, there are
two fundamental areas where all Kasner indices are simultaneously positive,
i.e. for $q>\frac{1}{\sqrt{2}}$ and $q<-\frac{1}{\sqrt{2}}$. When it happens,
remembering that the spatial Kasner metric is
$dl^{2}=t^{2p_{1}}dx^{1}+t^{2p_{2}}dx^{2}+t^{2p_{3}}dx^{3}$, the distances
contract along all the spatial direction approaching the singularity
($t\rightarrow 0$). It means that the system behaves as a stable Kasner regime
and the oscillatory regime is suppressed.
Furthemore, the relation (35) remains valid until $r<\frac{1}{2}$ or rather
when the particle become slower than the potential wall. When it happens,
approching the singularity, $\alpha\rightarrow-\infty$ while $H_{poly}$
remains costant without changes. In this sense, when the outgoing momenta
configuration of the $j$-th bounce is such that $r<\frac{1}{2}$, the quantity
$H_{poly}^{j}\alpha^{j}$ is no longer a constant of motion.
As in the standard case, we can introduce a parametrization for the particle
velocity components, before and after a single bounce
$\begin{split}&(\beta_{-}^{\prime})_{i}=r_{i}\sin\theta_{i},\\\
&(\beta_{+}^{\prime})_{i}=-r_{i}\cos\theta_{i},\\\
&(\beta_{-}^{\prime})_{f}=r_{f}\sin\theta_{f},\\\
&(\beta_{+}^{\prime})_{f}=r_{f}\cos\theta_{f}.\end{split}$ (53)
where $(\theta_{i},\theta_{f})$ are the incidence and the reflection angles
and $(r_{i},r_{f})$ are the anisotropy velocities before and after the bounce.
Eq. (33) states the existence of a maximum angle $\theta_{max}=\frac{\pi}{3}$
for a bounce to occur.
In the modified model, the condition for a bounce to take place is
$\theta_{i}<\theta_{max}^{poly}=\arccos(\frac{1}{2r_{i}})\leq\arccos(\frac{1}{2})=\theta_{max}=\frac{\pi}{3}.$
(54)
Figure 2: ”(Color online)”.The maximum angle for have a bounce
$\theta_{max}^{poly}$ as a function of $r$. In the $r\rightarrow 1$ limit, the
standard case is restored. This treatment make sense only for a configuration
in which the particle velocity is higher than walls velocity, i.e. for
$r>\frac{1}{2}$.
The new maximum angle $\theta_{max}^{poly}$ coincides with $\theta_{max}$ just
for $r=1$, i.e. when the standard case is restored (Fig. (2)). The last
semiclassical result is the modified reflection law for a single bounce: as in
the standard case, we can identify two constants of motion by studying the
system near the potential wall. In particular, one has
$\begin{split}&p_{-}=cost,\\\ &K=\frac{1}{2}p_{+}+H_{poly}=cost.\end{split}$
(55)
The expression of $p_{+}$ as function of $\beta^{{}^{\prime}}$ can be obtained
from (LABEL:eqHampoly):
$p_{+}=\frac{1}{a}\arcsin(a\beta_{+}^{\prime}H_{poly}).$ (56)
This way, by a substitution of Eq.(56) in Eq.(LABEL:kappapol), remembering
$\arcsin(-x)=-\arcsin(x)$ and using the parametrization (LABEL:parpoly), one
obtains
$\frac{1}{2a}\arcsin(-ar_{i}H_{poly}^{i}\cos\theta_{i})+H_{poly}^{i}=\\\
=\frac{1}{2a}\arcsin(ar_{f}H_{poly}^{f}\cos\theta_{f})+H_{poly}^{f}.$ (57)
Now we express $r$ and $H_{poly}$ as functions of $a,p_{+},p_{-}$:
$\frac{1}{2}[\arcsin(\sqrt{\sin(ap_{+}^{i})^{2}+\sin(ap_{-}^{i})^{2}}\cos\theta_{i})+\\\
+\arcsin(\sqrt{\sin(ap_{+}^{i})^{2}+\sin(ap_{-}^{i})^{2}}\frac{\cos\theta_{f}\sin\theta_{i}}{\sin\theta_{f}})]=\\\
=\sqrt{4-2(\cos(ap_{+}^{i})+\cos(ap_{-}^{i})}-\frac{\sin\theta_{i}}{\sin\theta_{f}}\times\\\
\times\sqrt{\frac{\sin(ap_{+}^{i})^{2}+\sin(ap_{-}^{i})^{2}}{\sin(ap_{+}^{f})^{2}+\sin(ap_{-}^{f})^{2}}[4-2(\cos(ap_{+}^{f}+\cos(ap_{-}^{f})]}.$
(58)
To perform a direct comparison with the standard case, a Taylor expansion up
to second order for $ap_{\pm}<<1$ for Eq.(58) is needed. This way, after
standard manipulation, the reflection law rewrites
$\frac{1}{2}\sin(\theta_{i}+\theta_{f})=\sin\theta_{f}\sqrt{1+\frac{a^{2}}{4}\frac{(p_{+}^{i})^{4}+(p_{-}^{i})^{4}}{(p_{+}^{i})^{2}+(p_{-}^{i})^{2}}}-\\\
-\sin\theta_{i}\sqrt{1+\frac{a^{2}}{4}\frac{(p_{+}^{f})^{4}+(p_{-}^{f})^{4}}{(p_{+}^{f})^{2}+(p_{-}^{f})^{2}}}.$
(59)
Defining $R=\frac{a^{2}}{4}\frac{p_{+}^{4}+p_{-}^{4}}{p_{+}^{2}+p_{-}^{2}}$,
one has
$\frac{1}{2}\sin(\theta_{i}+\theta_{f})=\sin\theta_{f}\sqrt{1+R_{i}}-\sin\theta_{i}\sqrt{1+R_{f}}.$
(60)
We obtain for $ap_{\pm}<<1$ a modified reflection law that, differently from
the standard case, depends on two parameters $(R,\theta)$. Obviously, in the
limit $ap_{\pm}\rightarrow 0$, i.e. switching off the polymer modification,
the standard reflection law (32) is recovered.
## IV POLYMER APPROACH TO THE QUANTUM MIXMASTER MODEL
We now analyze the quantum properties of the polymer Mixmaster model. As in
Sec.II.2, one searches a solution for the wave function of the form
$\Psi(p_{\pm},\alpha)=\chi(\alpha)\psi(\alpha,p_{\pm}).$ (61)
In this case, one can choose to describe the $\chi(\alpha)$ component of the
wave function in the $q$-polarization and the $\psi(\alpha,p_{\pm})$ component
of the wave function in the $p$-polarization. As in the semiclassical model,
we choose to discretized the anisotropies ($\beta_{+},\beta_{-}$) leaving
unchanged the characterization of the isotropic variable $\alpha$. Therefore,
as in Sec.I, one applies the formal substitution
$\widehat{p}_{\pm}^{2}\rightarrow\frac{2}{a^{2}}\left[1-\cos(ap_{\pm})\right]$.
Of course, the conjugated momenta $p_{\alpha}$ have a well-defined operator of
the form $\widehat{p}_{\alpha}=-i\partial_{\alpha}$. This way, we can obtain
the WDW equation for the polymer Mixmaster model writing the quantum version
of superHamiltonian in (45), that is
$[-\partial^{2}_{\alpha}+\frac{2}{a^{2}}\left(1-\cos(ap_{+})\right)+\frac{2}{a^{2}}\left(1-\cos(ap_{-})\right)+\\\
+\frac{3(4\pi)^{4}}{k^{2}}e^{4\alpha}V(\beta_{\pm})]\Psi(p_{\pm},\alpha)=0.$
(62)
The conservation of quantum numbers associated to the anisotropies, as
obtained by C.W.Misner in the standard quantum theory (see Eq.(43)), is
essentially based on a fundamental propriety of the Mixmaster Model: the
presence of chaos. Nevertheless, as in Sec.III, the chaos is removed for
discretized anisotropies of the Universe. This way, one cannot obtain for the
modified theory a conservation law towards the singularity as in the standard
case. For a quantum description, the polymer wavepackets for the theory are
needed. By a semiclassical analysis of the relational motion between the wall
and the particle, as in Sec.III, the polymer modification implies for the
particle different condition for the reach of the potential wall. This way, it
behaves as a free particle (no potential case $V=0$) or as a particle in a box
(infinitely steep potential well case). In this Section, we make use of the
the adiabatic approximation (39) as in the standard case. Following the same
procedure of Sec.II.2, the polymer WDW equation reduces to an eigenvalue
problem associated to the ADM Hamiltonian.
### IV.1 The free motion
In the free particle case, the potential term $V(\beta_{\pm})$ is negligible
in the WDW equation. As in Sec.II.2, condition (39) is applied to Eq.(62), and
the following free-particle eigenvalue problem is obtained
$\widehat{H}^{2}_{poly}\psi(p_{\pm})=k^{2}\psi(p_{\pm})=\\\
=\left[\frac{2}{a^{2}}(2-\cos(ap_{+})-\cos(ap_{-}))\right]\psi(p_{\pm}).$ (63)
From the structure of the eigenvalue problem (63), one can write
$\widehat{H^{2}}_{poly}=\widehat{H^{2}}_{+}+\widehat{H^{2}}_{-}$. As a
consequence, it is possible to describe the anisotropic wave function as
$\psi(p_{\pm})=\psi_{+}(p_{+})\psi_{-}(p_{-})$. This way, one obtains the two
independent eigenvalue problems
$\begin{split}&(\widehat{H_{+}^{2}}-k_{+}^{2})\psi_{+}(p)=\left[\frac{2}{a^{2}}[1-\cos(ap_{+})]-k_{+}^{2}\right]\psi_{+}(p)=0,\\\
&(\widehat{H_{-}^{2}}-k_{-}^{2})\psi_{-}(p)=\left[\frac{2}{a^{2}}[1-\cos(ap_{-})]-k_{-}^{2}\right]\psi_{-}(p)=0.\end{split}$
(64)
where $k^{2}=k_{+}^{2}+k_{-}^{2}$. These eigenvalue problems can be treated as
in Sec.I.4 and, by a similar procedure, one can easily verify that the
momentum wave functions $\psi_{+}(p)$ and $\psi_{-}(p)$ have the form
$\begin{split}&\psi_{+}(p_{+})=A\delta(p_{+}-p_{a}^{+})+B\delta(p_{+}+p_{a}^{+}),\\\
&\psi_{-}(p_{-})=C\delta(p_{-}-p_{a}^{-})+D\delta(p_{-}+p_{a}^{-}),\end{split}$
(65)
where $A,B,C,D$ are integration constants and $p_{a}^{+}$,$p_{a}^{-}$ are
defined as
$\begin{split}&p_{a}^{+}=\frac{1}{a}\arccos\left(1-\frac{k_{+}^{2}a^{2}}{2}\right),\\\
&p_{a}^{-}=\frac{1}{a}\arccos\left(1-\frac{k_{-}^{2}a^{2}}{2}\right).\end{split}$
(66)
From Eq.’s (LABEL:EFPR), the eigenvalue $k^{2}$ is given by
$k^{2}=k_{+}^{2}+k_{-}^{2}=\\\
=\frac{2}{a^{2}}\left[2-\cos(ap_{+})-\cos(ap_{-})\right]\leq
k^{2}_{max}=\frac{8}{a^{2}},$ (67)
i.e. a bounded and continous eigenvalue is found.
Now one can obtain $\psi(\beta_{\pm})$ by performing a Fourier trasform for
$\psi(p_{\pm})=\psi_{+}(p_{+})\psi_{-}(p_{-})$, such that
$\psi_{k}(\beta_{\pm})=\int\int
dp_{+}dp_{-}\psi(p_{\pm})e^{ip_{+}\beta_{+}}e^{ip_{-}\beta_{-}}=\\\
=C_{1}e^{ip_{a}^{+}\beta_{+}}e^{ip_{a}^{-}\beta_{-}}+C_{2}e^{ip_{a}^{+}\beta_{+}}e^{-ip_{a}^{-}\beta_{-}}+\\\
+C_{3}e^{-ip_{a}^{+}\beta_{+}}e^{ip_{a}^{-}\beta_{-}}+C_{4}e^{-ip_{a}^{+}\beta_{+}}e^{-ip_{a}^{-}\beta_{-}},$
(68)
where $C_{1}=AC$, $C_{2}=AD$, $C_{3}=BC$, $C_{4}=BD$. We are now able to build
up the polymer wave packet for the wave function of the Universe. We choose to
integrate the packet on the energies $k_{+},k_{-}$. As a consequence of the
modified dispersion relations (LABEL:reldis+), the energies eigenvalues
$k_{+},k_{-}$ can only take values within the interval
$[-\frac{2}{a},+\frac{2}{a}]$. Therefore, we have
$\Psi(\beta_{\pm},\alpha)=\iint_{-\frac{2}{a}}^{\frac{2}{a}}dk_{\pm}A(k_{\pm})\psi_{k_{\pm}}(\beta_{\pm})\chi(\alpha),$
(69)
where
$A(k_{+},k_{-})=e^{-\frac{(k_{+}-k_{+}^{0})^{2}}{2\sigma_{+}^{2}}}e^{-\frac{(k_{-}-k_{-}^{0})^{2}}{2\sigma_{-}^{2}}}$
is a Gaussian weighting function, $\sigma^{2}_{\pm}$ are the variances along
the two directions ($\beta_{+}$,$\beta_{-}$) and $k_{\pm}^{0}$ are the
energies eigenvalues around which we build up the wave packet. Let us note
from Eq.(69) that the polymer structure modifies the standard wave packet
related to the plane wave in terms of the anisotropies component as a
consequence of Eq.’s(LABEL:reldis+), i.e. the modified dispersion relations.
The shape for the isotropic component of the wave function in the free
particle case is
$\chi(\alpha)=e^{-i\int_{0}^{\alpha}kdt}=e^{-i\sqrt{k_{+}^{2}+k_{-}^{2}}\alpha}$.
This shape is a solution of the WDW equation
$\partial^{2}\chi(\alpha)+k^{2}\chi(\alpha)=0$ obtained by the application of
the adiabatic approximation (39). Furthermore, the self-consistence of this
approximation is ensured.
Figure 3: ”(Color online)”. The evolution of the polymer wave packet
$|\Psi(\alpha,\beta_{\pm})|$(upper row) and its full width at half maximum
(lower row) for the free particle case respectively for the values of
$|\alpha|=0,50,150$. The numerical integration is done for this choice of
parameters: $a=0.07,k_{+}=k_{-}=25,\sigma_{+}=\sigma_{-}=0.7$. They select an
initial semiclassical condition of a particle with a velocity smaller than the
wall velocity. It is worth noting that the particular choice of the parameters
couple ($a,\sigma_{\pm}$) is done because this way the condition
$a<<\frac{1}{\sigma_{\pm}}$ is valid. It is referred to the condition that the
typical polymer scale $a$ be much smaller than the characteristic width of the
wave packet $\frac{1}{\sigma_{\pm}}$.
Figure 4: ”(Color online)”. The solid line in the first graph represents the
polymer semiclassical trajectory identified by the choice of the initial
conditions. The dashed line represents the classical trajectory followed by a
wave packet build up in the same way of Sec.IV.1 but starting from classical
superHamiltonian constrain (22). The points in the second graph represent the
evolution of the spread $d$ as a function of $|\alpha|$. The solid line
represents the best fit for the points while the dashed line represents the
evolution of the wall position $|\beta_{w}|=\frac{1}{2}|\alpha|$.
### IV.2 Particle in a box
We analyze the problem of a particle in a box according to the Misner
hypothesys about the substitution of the triangular box by a square domain
having the same area $L^{2}$, as in Sec.II.2. Furthermore, following the
semiclassical results in Sec.III, one takes into account the outside wall
velocity defining the side of square box $L$ as
$L(\alpha)=L_{0}+|\alpha|,$ (70)
where $L_{0}$ is the side of the square box when $\alpha=0$. Proceding in the
same way as in Sec.I.5, the potential has the well-known form
$V(\beta_{\pm})=\begin{cases}\infty,&\beta_{\pm}>\frac{L(\alpha)}{2}\quad,\quad\beta_{\pm}<-\frac{L(\alpha)}{2}\\\
0,&-\frac{L(\alpha)}{2}<\beta_{\pm}<\frac{L(\alpha)}{2}\end{cases}.$ (71)
We can obtain a solution for $\psi(\beta_{\pm})$ in the same way of Sec.IV.1,
recalling that the potential form (71) implies this kind of boundary
conditions for $\psi(\beta_{\pm})$ along the two directions
$\psi_{\pm}\left(-\frac{L_{0}}{2}-\frac{\alpha}{2}\right)=\psi_{\pm}\left(+\frac{L_{0}}{2}+\frac{\alpha}{2}\right)=0.$
(72)
When one applies the conditions (72) separately along the two directions
$(\beta_{+},\beta_{-})$, one obtains
$\begin{split}&\psi_{+}(\beta_{+})=A\left[e^{\frac{in\pi\beta_{+}}{L_{0}+\alpha}}-e^{\frac{-in\pi\beta_{+}}{L_{0}+\alpha}}e^{-in\pi}\right],\\\
&\psi_{-}(\beta_{-})=B\left[e^{\frac{im\pi\beta_{-}}{L_{0}+\alpha}}-e^{\frac{-im\pi\beta_{-}}{L_{0}+\alpha}}e^{-im\pi}\right].\end{split}$
(73)
This way, $\psi(\beta_{\pm})$ is the product of the two separate wave
functions $\psi_{+}(\beta_{+})$ and $\psi_{-}(\beta_{-})$. Thus, one gets444It
is possible to evaluate the costant $AB$ by requesting that
$|\psi_{n,m}(\beta_{\pm})|^{2}=1$ over all the square box. This way,
$AB=\frac{1}{2(L_{0}+\alpha)}$ is obtained.
$\psi_{n,m}(\beta_{\pm},\alpha)=\psi_{+}(\beta_{+})\psi_{-}(\beta_{-})=\\\
=\frac{1}{2(L_{0}+\alpha)}\left[e^{\frac{in\pi\beta_{+}}{L_{0}+\alpha}}-e^{\frac{-in\pi\beta_{+}}{L_{0}+\alpha}}e^{-in\pi}\right]\times\\\
\times\left[e^{\frac{im\pi\beta_{-}}{L_{0}+\alpha}}-e^{\frac{-im\pi\beta_{-}}{L_{0}+\alpha}}e^{-im\pi}\right],$
(74)
where $A,B$ are integration constants and $(n,m)\in\mathbb{Z}$ are quantum
numbers associated anisotropy degrees of freedom. Due to the presence of the
integers quantum numbers $(n,m)$, a bounded and discrete eigenvalue spectrum
$k^{2}=k^{2}_{+}+k^{2}_{-}=\\\
=\frac{2}{a^{2}}\left[2-\cos\left(\frac{an\pi}{L_{0}+\alpha}\right)-\cos\left(\frac{am\pi}{L_{0}+\alpha}\right)\right]$
(75)
is obtained.
As in the free particle case, one builds the polymer wave packet. However, in
this case, one cannot integrate on a limited domain of energies $k_{\pm}$, and
a sum over all quantum numbers $n,m$ between $-\infty$ and $\infty$ i
necessary. This way,
$\Psi(\beta_{\pm},\alpha)=\sum_{n,m=-\infty}^{+\infty}B(n,m)\psi_{n,m}(\beta_{\pm},\alpha)\times\\\
\times
e^{-i\begin{matrix}\int_{0}^{\alpha}\sqrt{\frac{2}{a^{2}}\left[2-\cos\left(\frac{an\pi}{L_{0}+t}\right)-\cos\left(\frac{am\pi}{L_{0}+t}\right)\right]}dt,\end{matrix}}$
(76)
where
$B(n,m)=e^{-\frac{(n-n^{*})^{2}}{2\sigma_{+}^{2}}}e^{-\frac{(m-m^{*})^{2}}{2\sigma_{-}^{2}}}$
is a Gaussian weighting function and $n^{*},m^{*}$ are the quantum numbers
around which we build up the wave packet.
Let us note that, differently from the free particle case, the presence of the
polymer structure modifies the standard wave packet related to a particle in a
box in terms of the isotropic components. It happens because, in the wave
packet (76), the energies $k_{\pm}$ are expressd through ($n,m$), namely the
quantum numbers associated to the anisotropies.
As from Eq.(76), one chooses a shape for the isotropic component
$\chi(\alpha)=e^{-i\int_{0}^{\alpha}k(t)dt}=e^{-i\int_{0}^{\alpha}\sqrt{\frac{2}{a^{2}}\left[2-\cos\left(\frac{an\pi}{L_{0}+t}\right)-\cos\left(\frac{am\pi}{L_{0}+t}\right)\right]}dt}.$
(77)
In this case, Eq.(77) is a solution of the WDW equation
$\partial^{2}\chi(\alpha)+k(\alpha)^{2}\chi(\alpha)=0$ obtained by means of
the adiabatic approximation (39) in the asymptotic limit
$\alpha\rightarrow-\infty$. In this limit, the self-consistence of the
adiabatic approximation is ensured. The form of the isotropic component of the
wave function (77) is also an exact solution for the Schrödinger equation
associated to the ADM reduction.
Figure 5: ”(Color online)”. The evolution of the polymer wave packet
$|\Psi(\alpha,\beta_{\pm})|$(the first row) and its full width at half maximum
(the second row) for the particle in a box case respectively for
$|\alpha|=0,20,200$. The numerical integration is done for this choice of
parameters: $a=0.014,n^{*}=m^{*}=3000,\sigma_{+}=\sigma_{-}=50,L_{0}=52$. They
select an initial condition of a particle inside a square box with velocity
smaller than the wall velocity. This time, the particular choice of the
parameters ($a,\sigma_{\pm},L_{0}$) it is done because this way the condition
$a<<\frac{L(\alpha)}{\sigma_{\pm}}$ is valid. It concerns the condition that
the typical polymer scale $a$ is very smaller than
$\frac{L(\alpha)}{\sigma_{\pm}}$, i.e. the correct dimensional quantity
related with the width of the wave packet.
## V NUMERICAL ANALYSIS OF POLYMER WAVE PACKETS
We dedicate this section to the discussion of the polymer wave packet for the
Mixmaster towards the cosmological singularity. Both in the case of a free
particle (69) and in the one of a particle in a box (76), it is not possible
to perform an analytic integration for the wave packets. This way, in order to
obtain the quantum behavior of the wave packets near the cosmological
singularity, we evaluate them via numerical integrations.
### V.1 behavior of the free particle
In the case of a free particle, we perform the numerical integration choosing
the parameters which select semiclassical initial conditions concerning a
particle with velocity smaller than the wall one ($r<\frac{1}{2}$).
One appreciates, in the first row of the Fig.(3), the behavior towards the
singularity (formally for $|\alpha|\rightarrow\infty$) of the absolute value
of the wave packet $|\Psi(\alpha,\beta_{\pm})|$ in Eq.(69) while, in the
second row, the behavior towards the singularity of the full width at half
maximum width. It is interesting to study the evolution of $\beta_{\pm}^{m}$,
i.e. the wave packet maximum position. This way, we can see which trajectory
the wave packet follows towards the singularity. As we can see in the first
graph in Fig.(4), the behavior of the maximum position is completely
overlapping the semiclassical trajectory selected by our choice of the initial
conditions. In this sense, the polymer wave packet follows the semiclassical
trajectory until the singularity. This feature is not undermined by the spread
$d$ of the wave packet, i.e. the delocalization of the wave packet, as
expressed by the distance between the maximum position of the wave packet and
the edge of the region identified by the full width at half maximum.
Obviously, one expects that the spread velocity is really smaller than the
wall velocity. Otherwise, it would be possible for that the wave packet to
reach the potential wall. In that case, the description of the quantum system
with the wave packets for the free particle would not be correct. The second
graph in Fig.(4) represents the spread evolution, and we can see it follows a
linear behavior (solid line) with a slope much smaller than
$|\beta^{\prime}_{w}|=\frac{1}{2}$, i.e. the one related to the behavior of
the wall position (dashed line). This assures that the quantum representation
of the system near the singularity for the free particle case is well
described by the wave packet representation.
### V.2 behavior of the Particle in a box
The numerical integration related to the polymer wave packet (76) has to face
a significant technical difficulty. As a consequence of Eq.(75), the
conjugated momenta $p_{\pm}$ turn into a discretized variables. Therefore, we
select for the particle in a box the initial semiclassical condition
considering the substitution
$ap_{+}\rightarrow\frac{an\pi}{L_{0}+\alpha}\quad,\quad
ap_{-}\rightarrow\frac{am\pi}{L_{0}+\alpha}.$ (78)
It is worth noting that the initial condition of the particle depends on
$\alpha$, such that one deals with a time-dependent condition. In this
subsection, the influence of quantum numbers $n,m$ on the dynamics is
investigated. For this reason, one introduces six data sets with different
values of quantum numbers ($n^{*},m^{*}$) and box side $L_{0}$
$\begin{split}&\begin{cases}a=0.014\\\ n_{0}=1000\\\ m_{0}=1000\\\ L_{0}=17\\\
\sigma_{+}=50\\\ \sigma_{-}=50\end{cases}\begin{cases}a=0.014\\\ n_{1}=2000\\\
m_{1}=2000\\\ L_{1}=34\\\ \sigma_{+}=50\\\
\sigma_{-}=50\end{cases}\begin{cases}a=0.014\\\ n_{3}=3000\\\ m_{3}=3000\\\
L_{3}=52\\\ \sigma_{+}=50\\\ \sigma_{-}=50\end{cases}\\\
&\begin{cases}a=0.014\\\ n_{4}=6000\\\ m_{4}=6000\\\ L_{4}=103\\\
\sigma_{+}=50\\\ \sigma_{-}=50\end{cases}\begin{cases}a=0.014\\\ n_{4}=8000\\\
m_{4}=8000\\\ L_{4}=137\\\ \sigma_{+}=50\\\
\sigma_{-}=50\end{cases}\begin{cases}a=0.014\\\ n_{5}=10000\\\ m_{5}=10000\\\
L_{5}=172\\\ \sigma_{+}=50\\\ \sigma_{-}=50\end{cases}.\end{split}$ (79)
They select the same initial condition of a particle slower than potential
wall ($r<\frac{1}{2}$) and we show in Fig.(5) the evolution of
$|\Psi(\alpha,\beta_{\pm})|$ and its full width at half maximum for the first
data set. As in the free particle case, the wave packet spreads with $\alpha$,
i.e. it delocalizes until it disappears in a finite $\alpha$ time. The real
difference between free particle case and particle in a box case is the
trajectory followed by the wave packet. If we study the evolution of the wave
packet maximum position $\beta_{\pm}^{m}$ for the all data sets, we observe
that the wave packet trajectories move away from the polymer semiclassical
trajectory identified by the initial condition, as we can see in the first
graph of Fig.(6). The separation from the polymer semiclassical trajectory
depends on the quantum numbers $n^{*},m^{*}$. In particular, the larger
$n^{*},m^{*}$, the longer the semiclassical trajectory is followed. Anyway, no
matter how large they are, in a finite time $\alpha$, the wave packetstops
following the semiclassical trajectory, is directed to the potential wall and
reaches it. As in Fig.(7), this behavior is repeated for every unexpected
bounce against the wall. This way, it is not possible to chose an initial
semiclassical state (i.e. large $n^{*},m^{*}$) conserved until the
singularity.
Figure 6: ”(Color online)”. The points in the first graph represent the
evolution of the wave packet maximum position $\beta_{\pm}^{m}$ as a function
of $|\alpha|$ for all data sets. The solid line represents the polymer
semiclassical trajectory identified by the choice of the initial conditions.
The points in the second graph represent the evolution of the spread $d$ as a
function of $|\alpha|$ for all data sets. The solid line represents the
evolution of the wall position $|\beta_{w}|=\frac{1}{2}|\alpha|$. As in the
free particle case, the spread evolution follows a linear trend for all data
sets and the slopes are really smaller than the one related to the trend of
the wall position. Figure 7: ”(Color online)”. The points represent the
evolution of the wave packet maximum position $\beta_{\pm}^{m}$ as a function
of $|\alpha|$ for
$a=00.14,n^{*}=m^{*}=3000,\sigma_{+}=\sigma_{-}=50,L_{0}=32$. The two solid
lines represent the $\alpha$-evolution of the position of two opposite wall of
the square box. At last, the dashed lines represent the polymer semiclassical
trajectory identified by the choice of the initial conditions that the
wavepacket follow after each bounce for a finite $\alpha$-time.
This result is opposite respect the one in Eq.(34), where in the standard
theory the state remains classical until the singularity. It happens because
we have a time-dependent initial condition (as in Eq.(78, it depends on
$\alpha$) that changes the particle velocity. This behavior is explained if
one considers the two different data sets
$\begin{cases}a_{1}=0.014\\\ n^{*}_{1}=m^{*}_{1}=3000\\\ L_{1}=26\\\
\sigma_{+}=\sigma_{-}=50\end{cases}\quad\begin{cases}a_{2}=0.014\\\
n^{*}_{2}=m^{*}_{2}=400\\\ L_{2}=26\\\ \sigma_{+}=\sigma_{-}=50\end{cases}.$
(80)
They respectively select a particle with initial velocity $r<\frac{1}{2}$ and
with $r>\frac{1}{2}$. The first one is related to a particle in a box which
semiclassically cannot reach the potential wall, while the second one is
related to a particle in a box which semiclassically reaches the potential
wall. For our purposes, we take two data sets with same values of
$a,\sigma_{\pm},L_{0}$ but with different $n^{*}$ and $m^{*}$.
Figure 8: ”(Color online)”. The red(grey) points represent the evolution of
the distance $d$ between the wave packet maximum position and the potential
wall for $r<\frac{1}{2}$. The black points represent the evolution of the
distance $d$ between the wave packet maximum position and the potential wall
for $r>\frac{1}{2}$.
In Fig.(8), the evolution of the distance $d$ between the wave packet maximum
position and the potential wall in the two cases towards the singularity is
described.
When the first one is still traveling, the second one has already bounced on
the wall and it is travelling again. The red (light grey) points indicate the
(expected) velocity change due to the dynamical initial condition (78).
Finally, it is interesting to study the spread for the two wave pckets near
the potential wall. In Fig.(9), the two wave packets and the full width at
half maximum are sketched. Since the first wave packet should not reach the
wall, one would expect a high rate of delocalization near the wall. Instead,
as from the second line in Fig.(9), the two wave packets near the potential
wall have a comparable delocalization. Thus, we can conclude that, when the
potential is taken into account as an infinite well, any notion of a free
semiclassical wave packet is lost.
Figure 9: ”(Color online)”. The wave packets and the full width half maximum
near the potential wall for the two case with initial condition (80). The
first case is evaluated for $|\alpha|=85$, while the second case is evaluated
for $|\alpha|=45$.
## VI CONCLUDING REMARKS
The Mixmaster model, interpreted as the most general dynamics allowed by the
homogeneity constraint, constitutes a valuable prototype of the behavior of a
generic inhomogeneous model near the cosmological singularity, when referred
to sufficiently small space regions, having roughly the causal size.
Therefore, the characterization of its classical and quantum dynamics has a
very relevant value in understanding the general features of the Universe
birth.
The present work is aimed at generalizing misner , in which the classical
Mixmaster Hamiltonian dynamics is reduced to the motion of a two-dimensional
point-particle in a closed triangular-like potential and the corresponding
quantum behavior is reconducted to the one of a point-particle in a box. The
main result of the classical picture is the neverending bouncing of the
particle against the potential walls (resulting into a chaotic evolution),
while, in the quantum regime, the surprising feature emerges, of states having
very high occupation numbers which can approach the initial singularity.
This generalization is the reformulation of the quantum Mixmaster dynamics in
the polymer quantum approach. We have applied this procedure to the physical
degrees only, i.e. the Universe anisotropies, while the Universe volume has
been kept in its standard interpretation as a time variable for the system
evolution.
The semi-classical behavior of the Mixmaster model, i.e. the classical
modified dynamics by means of the polymer features, results as chaos-free, in
formal analogy with the case in which a massless scalar field is introduced in
the Einsteinian dynamics. As a consequence, the quantum regime loses its
property to admit very high occupation numbers asimptotically to the
singularity. Actually, we demonstrated that the absence of a chaotic behavior
prevents to construct the classical constant of the motion that Misner used to
infer the quantum properties for high occupation numbers. Thus, the most
impressive property of the quantum Mixmaster, i.e. its “classicality” across
the Planckian era, is no longer well-grounded.
In the polymer framework, such impossibility to recover a quasi-classical
behavior near the singularity, is enforced by noting that it is impossible to
construct wave-packets peaked around the classical trajectoreies that do not
impact against the potential. Such packets can follow the classical trajectory
for a finite time interval, after which the bounce of the wave packet against
the potential walls takes place. We showed that this fact is a direct
consequence of the time dependence of the potential well, resulting in a
condition on the free motion of the wave packets which is correspondingly time
dependent and, soon or later, is violated.
We can conclude that the polymer features of the Mixmaster model, i.e. the
implications of this particular cut-off physics on the anisotropic degrees of
freedom, enforces the relevance of the quantum nature of the model near the
cosmological singularity, since they introduce a non-local effect of the
potential walls on the behavior of wave packets, localized around classical
trajectories. This result suggests that, to better focus on the behavior of a
Mixmaster model model near the cosmological singularity, it is necessary to
implement a more rigurous semiclassical interpretation of the wavefunction
toward the cosmological singularity in presence of cut-off induced effects,
and a full quantum picture in the disscretized picture.
###### Acknowledgements.
This work has been partially developed within the framework of the CGW
collaboration (http:// www.cgwcollaboration.it).
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|
arxiv-papers
| 2013-11-23T15:09:55 |
2024-09-04T02:49:54.123106
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Orchidea Maria Lecian and Giovanni Montani and Riccardo Moriconi",
"submitter": "Riccardo Moriconi",
"url": "https://arxiv.org/abs/1311.6004"
}
|
1311.6044
|
###### Abstract.
In this article we study existence of boundary blow up solutions for some
fractional elliptic equations including
$\displaystyle(-\Delta)^{\alpha}u+u^{p}$ $\displaystyle=$ $\displaystyle f\ \
\hbox{in}\ \ \Omega,$ $\displaystyle u$ $\displaystyle=$ $\displaystyle g\ \
\hbox{on}\ \ \Omega^{c},$
$\displaystyle\lim_{x\in\Omega,x\to\partial\Omega}u(x)$ $\displaystyle=$
$\displaystyle\infty,$
where $\Omega$ is a bounded domain of class $C^{2}$, $\alpha\in(0,1)$ and the
functions $f:\Omega\to\mathbb{R}$ and $g:\bar{\Omega}^{c}\to\mathbb{R}$ are
continuous….
We prove existence and uniqueness results fro
ddd
Large solutions to elliptic equations involving fractional Laplacian
Huyuan Chen, Patricio Felmer
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático
UMR2071 CNRS-UChile, Universidad de Chile
Casilla 170 Correo 3, Santiago, Chile.
([email protected], [email protected])
and
Alexander Quaas
Departamento de Matemática, Universidad Técnica Federico Santa María
Casilla: V-110, Avda. España 1680, Valparaíso, Chile
([email protected])
## 1\. Introduction
In their pioneering work, Keller [22] and Osserman [27] studied the existence
of solutions to the nonlinear reaction diffusion equation
(1.1) $\left\\{\begin{array}[]{lll}-\Delta
u+h(u)=0,&\mbox{in}&\Omega,\\\\[5.69054pt]
u=+\infty,&\mbox{on}&\partial\Omega,\end{array}\right.$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq 2$, and $h$ is
a nondecreasing positive function. They independently proved that this
equation admits a solution if and only if $h$ satisfies
(1.2) $\int_{1}^{+\infty}\frac{ds}{\sqrt{H(s)}}<+\infty,$
where $H(s)=\int_{0}^{s}h(t)dt$, that in the case of $h(u)=u^{p}$ means $p>1$.
This integral condition on the non-linearity is known as the Keller-Osserman
criteria. The solution of (1.1) found in [22] and [27] exists as a consequence
of the interaction between the reaction and the difussion term, without the
influence of an external source that blows up at the boundary. Solutions
exploding at the boundary are usually called boundary blow up solutions or
large solutions. From then on, more general boundary blow-up problem:
(1.3) $\left\\{\begin{array}[]{lll}-\Delta
u(x)+h(x,u)=f(x),&x\in\Omega,\\\\[5.69054pt] \lim_{x\in\Omega,\
x\to\partial\Omega}u(x)=+\infty\end{array}\right.$
has been extensively studied, see [1, 2, 3, 10, 11, 12, 13, 19, 24, 25, 26,
29]. It has being extended in various ways, weakened the assumptions on the
domain and the nonlinear terms, extended to more general class of equations
and obtained more information on the uniqueness and the asymptotic behavior of
solution at the boundary.
During the last years there has been a renewed and increasing interest in the
study of linear and nonlinear integral operators, especially, the fractional
Laplacian, motivated by great applications and by important advances on the
theory of nonlinear partial differential equations, see [4, 6, 7, 9, 14, 16,
17, 18, 28, 31] for details.
In a recent work, Felmer and Quaas [14] considered an analog of (1.1) where
the laplacian is replaced by the fractional laplacian
(1.4)
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u=f(x),&\mbox{ in
}\quad\Omega,\\\\[5.69054pt] u(x)=g(x),&\mbox{ in
}\quad\bar{\Omega}^{c},\\\\[5.69054pt] \lim_{x\in\Omega,\
x\to\partial\Omega}u(x)=+\infty,\end{array}\right.$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq 2$, with
boundary $\partial\Omega$ of class $C^{2}$, $p>1$ and the fractional Laplacian
operator is defined as
$(-\Delta)^{\alpha}u(x)=-\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\delta(u,x,y)}{|y|^{N+2\alpha}}dy,\
\ x\in\Omega,$
with $\alpha\in(0,1)$ and $\delta(u,x,y)=u(x+y)+u(x-y)-2u(x)$. The authors
proved the existence of a solution to (1.4) provided that $g$ explodes at the
boundary and satisfies other technical conditions. In case the function $g$
blows up with an explosion rate as $d(x)^{\beta}$, with
$\beta\in(-\frac{2\alpha}{p-1},0)$ and $d(x)=dist(x,\partial\Omega)$, the
solution satisfies
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\beta}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$
In [14] the explosion is driven by the function $g$. The external source $f$
has a secondary role, not intervening in the explosive character of the
solution. $f$ may be bounded or unbounded, in latter case the explosion rate
has to be controlled by $d(x)^{-2\alpha p/(p-1)}$.
One interesting question not answered in [14] is the existence of a boundary
blow up solution without external source, that is assuming $g=0$ in
$\bar{\Omega}^{c}$ and $f=0$ in $\Omega$, thus extending the original result
by Keller and Osserman, where solutions exists due to the pure interaction
between the reaction and the diffusion terms. It is the purpose of this
article to answer positively this question and to better understand how the
non-local character influences the large solutions of (1.4) and what is the
structure of the large solutions of (1.4) with or without sources. Comparing
with the Laplacian case, where well possedness holds for (1.4), a much richer
structure for the solution set appears for the non-local case, depending on
the parameters and the data $f$ and $g$. In particular, Theorem 1.1 shows that
existence, uniqueness, non-existence and infinite existence may occur at
different values of $p$ and $\alpha$.
Our first result is on the existence of blowing up solutions driven by the
sole interaction between the diffusion and reaction term, assuming the
external value $g$ vanishes. Thus we will be considering the equation
$\displaystyle(-\Delta)^{\alpha}u+|u|^{p-1}u$ $\displaystyle=$ $\displaystyle
f\ \ \hbox{in}\ \ \Omega,$ (1.5) $\displaystyle u$ $\displaystyle=$
$\displaystyle 0\ \ \hbox{in}\ \ \Omega^{c},$
$\displaystyle\lim_{x\in\Omega,x\to\partial\Omega}u(x)$ $\displaystyle=$
$\displaystyle+\infty.$
On the external source $f$ we will assume the following hypotheses
* (H1)
The external source $f:\Omega\to\mathbb{R}$ is a $C^{\beta}_{loc}(\Omega)$,
for some $\beta>0$.
* (H2)
Defining $f_{-}(x)=\max\\{-f(x),0\\}$ and $f_{+}(x)=\max\\{f(x),0\\}$ we have
$\limsup_{x\in\Omega,x\to\partial\Omega}f_{+}(x)d(x)^{\frac{2\alpha
p}{p-1}}<+\infty\quad\mbox{and}\quad\lim_{x\in\Omega,x\to\partial\Omega}f_{-}(x)d(x)^{\frac{2\alpha
p}{p-1}}=0.$
A related condition that we need for non-existence results
* (H2∗)
The function $f$ satisfies
$\limsup_{x\in\Omega,x\to\partial\Omega}|f(x)|d(x)^{2\alpha}<+\infty.$
Now we are in a position to state our first theorem
###### Theorem 1.1.
Assume that $\Omega$ is an open, bounded and connected domain of class $C^{2}$
and $\alpha\in(0,1)$. Then we have:
Existence: Assume that $f$ satisfies (H1) and (H2), then there exists
$\tau_{0}(\alpha)\in(-1,0)$ such that for every $p$ satisfying
(1.6) $1+2\alpha<p<1-\frac{2\alpha}{\tau_{0}(\alpha)},$
the equation (1) possesses at least one solution $u$ satisfying
(1.7)
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$
Uniqueness: If $f$ further satisfies $f\geq 0$ in $\Omega$, then $u>0$ in
$\Omega$ and $u$ is the unique solution of (1) satisfying (1.7).
Nonexistence: If $f$ satisfies (H1) and (H2∗), then in the following three
cases:
* i)
For any $\tau\in(-1,0)\setminus\\{-\frac{2\alpha}{p-1},\ \tau_{0}(\alpha)\\}$
and $p$ satisfying (1.6) or
* ii)
For any $\tau\in(-1,0)$ and
(1.8) $p\geq 1-\frac{2\alpha}{\tau_{0}(\alpha)}\mbox{ or}$
* iii)
For any $\tau\in(-1,0)\setminus\\{\tau_{0}(\alpha)\\}$ and
(1.9) $1<p\leq 1+2\alpha,$
equation (1) does not have a solution $u$ satisfying
(1.10)
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}<+\infty.$
Special existence for $\tau=\tau_{0}(\alpha)$. Assume $f(x)\equiv 0,\
x\in\Omega$ and that
(1.11)
$\max\\{1-\frac{2\alpha}{\tau_{0}(\alpha)}+\frac{\tau_{0}(\alpha)+1}{\tau_{0}(\alpha)},1\\}<p<1-\frac{2\alpha}{\tau_{0}(\alpha)}.$
Then, there exist constants $C_{1}\geq 0$ and $C_{2}>0$, such that for any
$t>0$ there is a positive solution $u$ of equation (1) satisfying
(1.12) $C_{1}d(x)^{\min\\{\tau_{0}(\alpha)p+2\alpha,0\\}}\leq
td(x)^{\tau_{0}(\alpha)}-u(x)\leq
C_{2}d(x)^{\min\\{\tau_{0}(\alpha)p+2\alpha,0\\}}.$
###### Remark 1.1.
We remark that hypothesis (H2) and ($\rm{H2^{*}}$) are satisfied when $f\equiv
0$, so this theorem answer the question on existence rised in [14]. We also
observe that a function $f$ satisfying (H2) may also satisfy
$\lim_{x\in\Omega,x\in\partial\Omega}f(x)=-\infty,$
what matters is that the rate of explosion is smaller than $\frac{2\alpha
p}{p-1}$.
For proving the existence part of this theorem we will construct appropriate
super and sub-solutions. This construction involves the one dimensional
truncated laplacian of power functions given by
(1.13)
$C(\tau)=\int^{+\infty}_{0}\frac{\chi_{(0,1)}(t)|1-t|^{\tau}+(1+t)^{\tau}-2}{t^{1+2\alpha}}dt,$
for $\tau\in(-1,0)$ and where $\chi_{(0,1)}$ is the characteristic function of
the interval $(0,1)$. The number $\tau_{0}(\alpha)$ appearing in the statement
of our theorems is precisely the unique $\tau\in(-1,0)$ satisfying
$C(\tau)=0$. See Proposition 3.1 for details.
###### Remark 1.2.
For the uniqueness, we would like to mention that, by using iteration
technique, Kim in [23] has proved the uniqueness of solution to the problem
(1.14) $\left\\{\begin{array}[]{lll}-\Delta
u+u_{+}^{p}=0,&\mbox{in}&\Omega,\\\\[5.69054pt]
u=+\infty,&\mbox{in}&\partial\Omega,\end{array}\right.$
where $u_{+}=\max\\{u,0\\}$, under the hypotheses that $p>1$ and $\Omega$ is
bounded and satisfying $\partial\Omega=\partial\bar{\Omega}$. García-Melián in
[19, 20] introduced some improved iteration technique to obtain the uniqueness
for problem (1.14) with replacing nonlinear term by $a(x)u^{p}$. However,
there is a big difficulty for us to extend the iteration technique to our
problem (1.4) involving fractional Laplacian, which is caused by the nonlocal
character.
In the second part, we are also interested in considering the existence of
blowing up solutions driven by external source $f$ on which we assume the
following hypothesis
* (H3)
There exists $\gamma\in(-1-2\alpha,0)$ such that
$0<\liminf_{x\in\Omega,x\to\partial\Omega}f(x)d(x)^{-\gamma}\leq\limsup_{x\in\Omega,x\to\partial\Omega}f(x)d(x)^{-\gamma}<+\infty.$
Depending on the size of $\gamma$ we will say that the external source is weak
or strong. In order to gain in clarity, in this case we will state separately
the existence, uniqueness and non-existence theorem in this source-driven
case.
###### Theorem 1.2 (Existence).
Assume that $\Omega$ is an open, bounded and connected domain of class
$C^{2}$. Assume that $f$ satisfies (H1) and let $\alpha\in(0,1)$ then we have:
$(i)$ (weak source) If $f$ satisfies (H3) with
(1.15) $-2\alpha-\frac{2\alpha}{p-1}\leq\gamma<-2\alpha,$
then, for every $p$ such that (1.8) holds, equation (1.5) possesses at least
one solution $u$, with asymptotic behavior near the boundary given by
(1.16)
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\gamma-2\alpha}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\gamma-2\alpha}<+\infty.$
$(ii)$ (strong source) If $f$ satisfies (H3) with
(1.17) $-1-2\alpha<\gamma<-2\alpha-\frac{2\alpha}{p-1}$
then, for every $p$ such that
(1.18) $p>1+2\alpha,$
equation (1.5) possesses at least one solution $u$, with asymptotic behavior
near the boundary given by
(1.19)
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\frac{\gamma}{p}}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\frac{\gamma}{p}}<+\infty.$
As we already mentioned, in Theorem 1.1 the existence of blowing up solutions
results from the interaction between the reaction $u^{p}$ and the diffusion
term $(-\Delta)^{\alpha}$, while the role of the external source $f$ is
secondary. In contrast, in Theorem 1.2 the existence of blowing up solutions
results on the interaction between the external source, and the diffusion term
in case of weak source and the interaction between the external source and the
reaction term in case of strong source.
Regarding uniqueness result for solutions of (1.5), as in Theorem 1.1 we will
assume that $f$ is non-negative, hypothesis that we need for technical
reasons. We have
###### Theorem 1.3 (Uniqueness).
Assume that $\Omega$ is an open, bounded and connected domain of class
$C^{2}$, $\alpha\in(0,1)$ and $f$ satisfies (H1) and $f\geq 0$. Then we have
* i)
(weak source) the solution of (1.5) satisfying (1.16) is positive and unique,
and
* ii)
(strong source) the solution of (1.5) satisfying (1.19) is positive and
unique.
We complete our theorems with a non-existence result for solution with a
previously defined asymptotic behavior, as we saw in Theorem 1.1. We have
###### Theorem 1.4 (Non-existence).
Assume that $\Omega$ is an open, bounded and connected domain of class
$C^{2}$, $\alpha\in(0,1)$ and $f$ satisfies $(H1)$, $(H3)$ and $f\geq 0$. Then
we have
* i)
(weak source) Suppose that $p$ satisfies (1.8), $\gamma$ satisfies (1.15) and
$\tau\in(-1,0)\setminus\\{\gamma+2\alpha\\}$. Then equation (1) does not have
a solution $u$ satisfying (1.10).
* ii)
(strong source) Suppose that $p$ satisfies (1.18), $\gamma$ satisfies (1.17)
and $\tau\in(-1,0)\setminus\\{\frac{\gamma}{p}\\}$. Then, equation (1) does
not have a solution $u$ satisfying (1.10).
All theorems stated so far deal with equation (1.4) in the case $g\equiv 0$,
but they may also be applied when $g\not\equiv 0$ and, in particular, these
result improve those given in [14]. In what follows we describe how to obtain
this. We start with some notation, we consider
$L^{1}_{\omega}(\bar{\Omega}^{c})$ the weighted $L^{1}$ space in
$\bar{\Omega}^{c}$ with weight
$\omega(y)=\frac{1}{1+|y|^{N+2\alpha}},\quad\mbox{for all
}y\in\mathbb{R}^{N}.$
Our hypothesis on the external values $g$ is the following
* $(H4)\ $
1. The function $g:\bar{\Omega}^{c}\to\mathbb{R}$ is measurable and $g\in L^{1}_{\omega}(\bar{\Omega}^{c})$.
Given $g$ satisfying $(H4)$, we define
(1.20)
$G(x)=\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\tilde{g}(x+y)}{|y|^{N+2\alpha}}dy,\
\ x\in\Omega,$
where
(1.21)
$\tilde{g}(x)=\left\\{\begin{array}[]{lll}0,&x\in\bar{\Omega},\\\\[5.69054pt]
g(x),&x\in\bar{\Omega}^{c}.\end{array}\right.$
We observe that
$G(x)=-(-\Delta)^{\alpha}\tilde{g}(x),\ \ x\in\Omega.$
Hypothesis $(H4)$ implies that $G$ is continuous in $\Omega$ as seen in Lemma
2.1 and has an explosion of order $d(x)^{\beta-2\alpha}$ towards the boundary
$\partial\Omega$, if $g$ has an explosion of order $d(x)^{\beta}$ for some
$\beta\in(-1,0)$, as we shall see in Proposition 3.3. We observe that under
the hypothesis $(H4)$, if $u$ is a solution of equation (1.4), then
$u-\tilde{g}$ is the solution of
(1.22)
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=f(x)+G(x),&x\in\Omega,\\\\[5.69054pt]
u(x)=0,&x\in\bar{\Omega}^{c},\\\\[5.69054pt] \lim_{x\in\Omega,\
x\to\partial\Omega}u(x)=+\infty\end{array}\right.$
and vice versa, if $v$ is a solution of (1.22), then $v+\tilde{g}$ is a
solution of (1.4).
Thus, using Theorem 1.1-1.4, we can state the corresponding results of
existence, uniqueness and non-existence for (1.4), combining $f$ with $g$ to
define a new external source
(1.23) $F(x)=G(x)+f(x),\ \ \ x\in\Omega.$
With this we can state appropriate hypothesis for $g$ and thus we can write
theorems, one corresponding to each of Theorem 1.1, 1.2, 1.3 and 1.4. Even
though, at first sight we need that $G(x)$ is $C^{\beta}_{loc}(\Omega)$,
actually continuity of $g$ is sufficient, as we discuss Remark 4.1.
Moreover, in Remark 4.2 we explain how our results in this paper allows to
give a different proof of those obtained by Felmer and Quaas in [14],
generalizing them.
This article is organized as follows. In Section §2 we present some
preliminaries to introduce the notion of viscosity solutions, comparison and
stability theorems in case of explosion at the boundary. Then we prove an
existence theorem for the nonlinear problem with blow up at the boundary,
assuming the existence of ordered. Section §3 is devoted to obtain crucial
estimates used to construct super and sub-solutions. In Section §4 we prove
the existence of solution to (1) in Theorem 1.1 and Theorem 1.2. In section
§5, we give the proof of the uniqueness of solution to (1) in Theorem 1.1 and
Theorem 1.3. Finally, the nonexistence related to Theorem 1.1 and Theorem 1.4
are shown in Section §6.
## 2\. Preliminaries and existence theorem
The purpose of this section is to introduce some preliminaries and prove an
existence theorem for blow-up solutions assuming the existence of ordered
super-solution and sub-solution which blow up at the boundary. In order to
prove this theorem we adapt the theory of viscosity to allow for boundary blow
up.
We start this section by defining the notion of viscosity solution for non-
local equation, allowing blow up at the boundary, see for example [7]. We
consider the equation of the form:
(2.1) $(-\Delta)^{\alpha}u=h(x,u)\quad\mbox{in}\quad\Omega,\quad
u=g\quad\mbox{in}\quad\Omega^{c}.$
###### Definition 2.1.
We say that a function $u:(\partial\Omega)^{c}\to\mathbb{R}$, continuous in
$\Omega$ and in $L^{1}_{\omega}(\mathbb{R}^{N})$ is a viscosity super-solution
(sub-solution) of (2.1) if
$u\geq g\ (\mbox{resp.}\ u\leq g)\ \mbox{in}\ \bar{\Omega}^{c}$
and for every point $x_{0}\in\Omega$ and some neighborhood $V$ of $x_{0}$ with
$\bar{V}\subset\Omega$ and for any $\phi\in C^{2}(\bar{V})$ such that
$u(x_{0})=\phi(x_{0})$ and
$u(x)\geq\phi(x)\ (\mbox{resp.}\ u(x)\leq\phi(x))\ \mbox{for\ all}\ x\in V,$
defining
$\displaystyle\tilde{u}=\left\\{\begin{array}[]{lll}\phi&\mbox{in}&V,\\\\[5.69054pt]
u&\mbox{in}&V^{c},\end{array}\right.$
we have
$(-\Delta)^{\alpha}\tilde{u}(x_{0})\geq h(x_{0},u(x_{0}))\
(\mbox{resp.}(-\Delta)^{\alpha}\tilde{u}(x_{0})\leq h(x_{0},u(x_{0})).$
We say that $u$ is a viscosity solution of (2.1) if it is a viscosity super-
solution and also a viscosity sub-solution of (2.1).
It will be convenient for us to have also a notion of classical solution.
###### Definition 2.2.
We say that a function $u:(\partial\Omega)^{c}\to\mathbb{R}$, continuous in
$\Omega$ and in $L^{1}_{\omega}(\mathbb{R}^{N})$ is a classical solution of
(2.1) if $(-\Delta)^{\alpha}u(x)$ is well defined for all $x\in\Omega$,
$(-\Delta)^{\alpha}u(x)=h(x,u(x)),\quad\mbox{for all }x\in\Omega$
and $u(x)=g(x)$ a.e. in $\overline{\Omega}^{c}$. Classical super and sub-
solutions are defined similarly.
Next we have our first regularity theorem.
###### Theorem 2.1.
Let $g\in L^{1}_{\omega}(\mathbb{R}^{N})$ and $f\in C^{\beta}_{loc}(\Omega)$,
with $\beta\in(0,1)$, and $u$ be a viscosity solution of
$(-\Delta^{\alpha}u)=f\quad\mbox{ in }\quad\Omega,\quad
u=g\quad\mbox{in}\quad\Omega^{c},$
then there exists $\gamma>0$ such that $u\in C^{2\alpha+\gamma}_{loc}(\Omega)$
Proof. Suppose without loss of generality that $B_{1}\subset\Omega$ and $f\in
C^{\beta}(B_{1})$. Let $\eta$ be a non-negative, smooth function with support
in $B_{1}$, such that $\eta=1$ in $B_{1/2}$. Now we look at the equation
$-\Delta w=\eta f\quad\mbox{ in }\quad\ \mathbb{R}^{N}.$
By Hölder regularity theory for the Laplacian we find $w\in C^{2,\beta}$, so
that $(-\Delta)^{1-\alpha}w\in C^{2\alpha+\beta}$, see [32] or Theorem 3.1 in
[15]. Then, since
$(-\Delta)^{\alpha}(u-(-\Delta)^{1-\alpha}w)=0\quad\mbox{ in }\quad B_{1/2},$
we can use Theorem 1.1 and Remark 9.4 of [8] (see also Theorem 4.1 there), to
obtain that there exist $\tilde{\beta}$ such that $u-(-\Delta)^{1-\alpha}w\in
C^{2\alpha+\tilde{\beta}}(B_{1/2})$, from where we conclude. $\Box$
The Maximum and the Comparison Principles are key tools in the analysis, we
present them here for completitude.
###### Theorem 2.2.
(Maximum principle) Let ${\mathcal{O}}$ be an open and bounded domain of
$\mathbb{R}^{N}$ and $u$ be a classical solution of
(2.3) $(-\Delta)^{\alpha}u\leq 0\ \ \ \mbox{in}\ \ \ {\mathcal{O}},$
continuous in $\bar{{\mathcal{O}}}$ and bounded from above in
$\mathbb{R}^{N}$. Then $u(x)\leq M,$ for all $x\in{\mathcal{O}},$ where
$M=\sup_{x\in{\mathcal{O}}^{c}}u(x)<+\infty.$
Proof. If the conclusion is false, then there exists
$x^{\prime}\in{\mathcal{O}}$ such that $u(x^{\prime})>M$. By continuity of
$u$, there exists $x_{0}\in{\mathcal{O}}$ such that
$u(x_{0})=\max_{x\in{\mathcal{O}}}u(x)=\max_{x\in\mathbb{R}^{N}}u(x)$
and then $(-\Delta)^{\alpha}u(x_{0})>0$, which contradicts (2.3). $\Box$
###### Theorem 2.3.
(Comparison Principle) Let $u$ and $v$ be classical super-solution and sub-
solution of
$(-\Delta)^{\alpha}u+h(u)=f\ \ \mbox{in}\ \ {\mathcal{O}},$
respectively, where ${\mathcal{O}}$ is an open, bounded domain, the functions
$f:{\mathcal{O}}\to\mathbb{R}$ is continuous and $h:\mathbb{R}\to\mathbb{R}$
is increasing. Suppose further that $u$ and $v$ are continuous in
$\bar{\mathcal{O}}$ and $v(x)\leq u(x)$ for all $x\in{\mathcal{O}}^{c}$. Then
$u(x)\geq v(x),\ x\in{\mathcal{O}}.$
Proof. Suppose by contradiction that $w=u-v$ has a negative minimum in
$x_{0}\in{\mathcal{O}}$, then $(-\Delta)^{\alpha}w(x_{0})<0$ and so, by
assumptions on $u$ and $v$, $h(u(x_{0}))>h(v(x_{0}))$, which contradicts the
monotonicity of $h$. $\Box$
We devote the rest of the section to the proof of the existence theorem
through super and sub-solutions. We prove the theorem by an approximation
procedure for which we need some preliminary steps. We need to deal with a
Dirichlet problem involving fractional laplacian operator and with exterior
data which blows up away from the boundary. Precisely, on the exterior data
$g$, we assume the following hypothesis, given an open, bounded set
${\mathcal{O}}$ in $\mathbb{R}^{N}$ with $C^{2}$ boundary:
* $(G)\ $
$g:{\mathcal{O}^{c}}\to\mathbb{R}$ is in $L^{1}_{\omega}({\mathcal{O}}^{c})$
and it is of class $C^{2}$ in
$\\{z\in{\mathcal{O}}^{c},dist(z,\partial{\mathcal{O}})\leq\delta\\}$, where
$\delta>0$.
In studying the nonlocal problem (1.4) with explosive exterior source, we have
to adapt the stability theorem and the existence theorem for the linear
Dirichlet problem. The following lemma is important in this direction.
###### Lemma 2.1.
Assume that ${\mathcal{O}}$ is an open, bounded domain in $\mathbb{R}^{N}$
with $C^{2}$ boundary. Let $w:\mathbb{R}^{N}\to\mathbb{R}$:
$(i)$ If $w\in L^{1}_{\omega}(\mathbb{R}^{N})$ and $w$ is of class $C^{2}$ in
$\\{z\in\mathbb{R}^{N},d(z,\mathcal{O})\leq\delta\\}$ for some $\delta>0$,
then $(-\Delta)^{\alpha}w$ is continuous in $\bar{{\mathcal{O}}}$.
$(ii)$ If $w\in L^{1}_{\omega}(\mathbb{R}^{N})$ and $w$ is of class $C^{2}$ in
${\mathcal{O}}$, then $(-\Delta)^{\alpha}w$ is continuous in ${\mathcal{O}}$.
$(iii)$ If $w\in L^{1}_{\omega}(\mathbb{R}^{N})$ and $w\equiv 0$ in
${\mathcal{O}}$, then $(-\Delta)^{\alpha}w$ is continuous in ${\mathcal{O}}$.
Proof. We first prove (ii). Let $x\in\Omega$ and $\eta>0$ such that
$B(x,2\eta)\subset\Omega$. Then we consider
$(-\Delta)^{\alpha}u(x)=L_{1}(x)+L_{2}(x),$
where
$L_{1}(x)=\int_{B(0,\eta)}\frac{\delta(u,x,y)}{|y|^{N+2\alpha}}dy\quad\mbox{and}\quad
L_{2}(x)=\int_{B(0,\eta)^{c}}\frac{\delta(u,x,y)}{|y|^{N+2\alpha}}dy.$
Since $w$ is of class $C^{2}$ in $\mathcal{O}$, we may write $L_{1}$ as
$L_{1}(x)=\int_{0}^{\eta}\left\\{\int_{S^{N-1}}\int_{-1}^{1}\int_{1}^{1}t\omega^{t}D^{2}w(x+str\omega)\omega
dtdsd\omega\right\\}r^{1-\alpha}dr,$
where the term inside the brackets is uniformly continuous in $(x,r)$, so the
resulting function $L_{1}$ is continuous. On the other hand we may write
$L_{2}$ as
$L_{2}(x)=-2w(x)\int_{B(0,\eta)^{c}}\frac{dy}{|y|^{N+2\alpha}}-2\int_{B(x,\eta)^{c}}\frac{w(z)dz}{|z-x|^{N+2\alpha}},$
from where $L_{2}$ is also continuous. The proof of (i) and (iii) are similar.
$\Box$
The next theorem gives the stability property for viscosity solutions in our
setting.
###### Theorem 2.4.
Suppose that ${\mathcal{O}}$ is an open, bounded and $C^{2}$ domain and
$h:\mathbb{R}\to\mathbb{R}$ is continuous. Assume that $(u_{n})$,
$n\in\mathbb{N}$ is a sequence of functions, bounded in
$L^{1}_{\omega}({\mathcal{O}}^{c})$ and $f_{n}$ and $f$ are continuous in
${\mathcal{O}}$ such that:
$(-\Delta)^{\alpha}u_{n}+h(u_{n})\geq f_{n}\ (\mbox{resp.}\
(-\Delta)^{\alpha}u_{n}+h(u_{n})\leq f_{n})$ in ${\mathcal{O}}$ in viscosity
sense,
$u_{n}\to u$ locally uniformly in ${\mathcal{O}}$,
$u_{n}\to u$ in $L^{1}_{\omega}(\mathbb{R}^{N})$, and
$f_{n}\to f$ locally uniformly in ${\mathcal{O}}$.
Then, $(-\Delta)^{\alpha}u+h(u)\geq f\ (\mbox{resp.}\
(-\Delta)^{\alpha}u+h(u)\leq f)$ in ${\mathcal{O}}$ in viscosity sense.
Proof. If $|u_{n}|\leq C$ in ${\mathcal{O}}$ then we use Lemma 4.3 of [7]. If
$u_{n}$ is unbounded in ${\mathcal{O}}$, then $u_{n}$ is bounded in
${\mathcal{O}}_{k}=\\{x\in{\mathcal{O}},dist(x,\partial{\mathcal{O}})\geq\frac{1}{k}\\}$,
since $u_{n}$ is continuous in ${\mathcal{O}}$, and then by Lemma 4.3 of [7],
$u$ is a viscosity solution of $(-\Delta)^{\alpha}u+h(u)\geq f$ in
${\mathcal{O}}_{k}$ for any $k$. Thus $u$ is a viscosity solution of
$(-\Delta)^{\alpha}u+h(u)\geq f$ in ${\mathcal{O}}$ and the proof is
completed. $\Box$
An existence result for the Dirichlet problem is given as follows:
###### Theorem 2.5.
Suppose that ${\mathcal{O}}$ is an open, bounded and $C^{2}$ domain,
$g:{\mathcal{O}}^{c}\to\mathbb{R}$ satisfies $(G)$,
$f:\bar{{\mathcal{O}}}\to\mathbb{R}$ is continuous, $f\in
C^{\beta}_{loc}({\mathcal{O}})$, with $\beta\in(0,1)$, and $p>1$. Then there
exists a classical solution $u$ of
(2.4)
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=f(x),&x\in{\mathcal{O}},\\\\[5.69054pt]
u(x)=g(x),&x\in{\mathcal{O}}^{c},\end{array}\right.$
which is continuous in $\bar{{\mathcal{O}}}$.
In proving Theorem 2.5, we will use the following lemma:
###### Lemma 2.2.
Suppose that ${\mathcal{O}}$ is an open, bounded and $C^{2}$ domain,
$f:\bar{{\mathcal{O}}}\to\mathbb{R}$ is continuous and $C>0$. Then there exist
a classical solution of
(2.7)
$\displaystyle\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+Cu(x)=f(x),&x\in{\mathcal{O}},\\\\[5.69054pt]
u(x)=0,&x\in{\mathcal{O}}^{c},\end{array}\right.$
which is continuous in $\bar{{\mathcal{O}}}$.
Proof. For the existence of a viscosity solution $u$ of (2.7), that is
continuous in $\bar{{\mathcal{O}}}$, we refers to Theorem 3.1 in [14]. Now we
apply Theorem 2.6 of [7] to conclude that $u$ is
$C^{\theta}_{loc}({\mathcal{O}})$, with $\theta>0$, and then we use Theorem
2.1 to conclude that the solution is classical (see also Proposition 1.1 and
1.4 in [30]). $\Box$
Using Lemma 2.2, we find $\bar{V}$, a classical solution of
(2.8)
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}\bar{V}(x)=-1,&x\in{\mathcal{O}},\\\\[5.69054pt]
\bar{V}(x)=0,&x\in{\mathcal{O}}^{c},\end{array}\right.$
which is continuous in $\bar{{\mathcal{O}}}$ and negative in ${\mathcal{O}}$.
it is classical since we apply Theorem 2.6 of [7] to conclude that $u$ is
$C^{\theta}_{loc}({\mathcal{O}})$, with $\theta>0$, and then we use Theorem
2.1 to conclude that the solution is classical (see also Proposition 1.1 and
1.4 in [30]).
Now we prove Theorem 2.5.
Proof of Theorem 2.5. Under assumption $(G)$ and in view of the hypothesis on
$\mathcal{O}$, we may extend $g$ to $\bar{g}$ in $\mathbb{R}^{N}$ as a $C^{2}$
function in $\\{z\in\mathbb{R}^{N},d(z,\mathcal{O})\leq\delta\\}$. We
certainly have $\bar{g}\in L^{1}_{\omega}(\mathbb{R}^{N})$ and, by Lemma 2.1
$(-\Delta)^{\alpha}\bar{g}$ is continuous in $\bar{{\mathcal{O}}}$. Next we
use Lemma 2.2 to find a solution $v$ of equation (2.7) with $f(x)$ replaced by
$f(x)-(-\Delta)^{\alpha}\bar{g}(x)-C\bar{g}(x)$, where $C>0$. Then we define
$u=v+\bar{g}$ and we see that $u$ is continuous in $\bar{\mathcal{O}}$ and it
satisfies in the viscosity sense
$\displaystyle\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+Cu(x)=f(x),&x\in{\mathcal{O}},\\\\[5.69054pt]
u(x)=g(x),&x\in{\mathcal{O}}^{c}.\end{array}\right.$
Now we use Theorem Theorem 2.6 in [7] and then Theorem 2.1 to conclude that
$u$ is a classical solution. Continuing the proof, we find super and sub-
solutions for (2.4). We define
$u_{\lambda}(x)=\lambda\bar{V}(x)+\bar{g}(x),\ x\in\mathbb{R}^{N},$
where $\lambda\in\mathbb{R}$ and $\bar{V}$ is given in (2.8). We see that
$u_{\lambda}(x)=g(x)$ in ${\mathcal{O}}^{c}$ for any $\lambda$ and for
$\lambda$ large (negative), $u_{\lambda}$ satisfies
$\displaystyle(-\Delta)^{\alpha}u_{\lambda}(x)+|u_{\lambda}(x)|^{p-1}u_{\lambda}(x)-f(x)\geq(-\Delta)^{\alpha}\bar{g}(x)-\lambda-f(x)-|\bar{g}(x)|^{p},$
for $x\in{\mathcal{O}}$. Since $(-\Delta)^{\alpha}\bar{g}$, $\bar{g}$ and $f$
are bounded in $\bar{\mathcal{O}}$, choosing $\lambda_{1}<0$ large enough we
find that $u_{\lambda_{1}}\geq 0$ is a super-solution of (2.4) with
$u_{\lambda_{1}}=g$ in ${\mathcal{O}}^{c}$.
On the other hand, for $\lambda>0$ we have
$\displaystyle(-\Delta)^{\alpha}u_{\lambda}(x)+|u_{\lambda}|^{p-1}u_{\lambda}(x)-f(x)\leq(-\Delta)^{\alpha}\bar{g}(x)-\lambda+|\bar{g}|^{p-1}\bar{g}(x)-f(x).$
As before, there is $\lambda_{2}>0$ large enough, so that $u_{\lambda_{2}}$ is
a sub-solution of (2.4) with $u_{\lambda_{2}}=g$ in ${\mathcal{O}}^{c}$.
Moreover, we have that $u_{\lambda_{2}}<u_{\lambda_{1}}$ in ${\mathcal{O}}$
and $u_{\lambda_{2}}=u_{\lambda_{1}}=g$ in ${\mathcal{O}}^{c}$.
Let $u_{0}=u_{\lambda_{2}}$ and define iteratively, using the above argument,
the sequence of functions $u_{n}\ (n\geq 1)$ as the classical solutions of
$\begin{array}[]{lll}(-\Delta)^{\alpha}u_{n}(x)+Cu_{n}(x)=f(x)+Cu_{n-1}(x)-|u_{n-1}|^{p-1}u_{n-1}(x),&x\in{\mathcal{O}},\\\
\hskip 79.6678ptu_{n}(x)=g(x),\quad x\in{\mathcal{O}}^{c},&{}\end{array}$
where $C>0$ is so that the function $r(t)=Ct-|t|^{p-1}t$ is increasing in the
interval
$[\min_{x\in\bar{\mathcal{O}}}u_{\lambda_{2}}(x),\max_{x\in\bar{\mathcal{O}}}u_{\lambda_{1}}(x)]$.
Next, using Theorem 2.3 we get
$u_{\lambda_{2}}\leq u_{n}\leq u_{n+1}\leq u_{\lambda_{1}}\ \ \mbox{in}\
{\mathcal{O}},\quad\mbox{for all }n\in\mathbb{N}.$
Then we define $u(x)=\lim_{n\to+\infty}u_{n}(x),$ for $x\in{\mathcal{O}}$ and
$u(x)=g(x),$ for $x\in{\mathcal{O}}^{c}$ and we have
(2.10) $u_{\lambda_{2}}\leq u\leq u_{\lambda_{1}}\ \ \mbox{in}\ \
{\mathcal{O}}.$
Moreover, $u_{\lambda_{1}},u_{\lambda_{2}}\in L^{1}_{\omega}(\mathbb{R}^{N})$
so that $u_{n}\to u$ in $L^{1}_{\omega}(\mathbb{R}^{N}),$ as $n\to\infty$.
By interior estimates as given in [6], for any compact set $K$ of
${\mathcal{O}}$, we have that $u_{n}$ has uniformly bounded $C^{\theta}(K)$
norm. So, by Ascoli-Arzela Theorem we have that $u$ is continuous in $K$ and
$u_{n}\to u$ uniformly in $K$. Taking a sequence of compact sets
$K_{n}=\\{z\in{\mathcal{O}},d(z,\partial{\mathcal{O}})\geq\frac{1}{n}\\}$, and
${\mathcal{O}}=\cup^{+\infty}_{n=1}K_{n},$ we find that $u$ is continuous in
${\mathcal{O}}$ and, by Theorem 2.4, $u$ is a viscosity solution of (2.4). Now
we apply Theorem 2.6 of [7] to find that u is
$C^{\theta}_{loc}({\mathcal{O}})$, and then we use Theorem 2.1 con conclude
that $u$ is a classical solution. In addition, $u$ is continuous up to the
boundary by (2.10). $\Box$
Now we are in a position to prove the main theorem of this section. We prove
the existence of a blow-up solution of (1) assuming the existence of suitable
super and sub-solutions.
###### Theorem 2.6.
Assume that $\Omega$ is an open, bounded domain of class $C^{2}$, $p>1$ and
$f$ satisfy $(H1)$. Suppose there exists a super-solution $\bar{U}$ and a sub-
solution $\underline{U}$ of (1) such that $\bar{U}$ and $\underline{U}$ are of
class $C^{2}$ in $\Omega$, $\underline{U}$, $\bar{U}\in
L^{1}_{\omega}(\mathbb{R}^{N})$,
$\bar{U}\geq\underline{U}\ \ \mbox{in}\ \Omega,\ \
\liminf_{x\in\Omega,x\to\partial\Omega}\underline{U}(x)=+\infty\ \ \mbox{and}\
\ \bar{U}=\underline{U}=0\ \ \mbox{in}\ \bar{\Omega}^{c}.$
Then there exists at least one solution $u$ of (1) in the viscosity sense and
$\underline{U}\leq u\leq\bar{U}\ \ \mbox{in}\ \ \Omega.$
Additionally, if $f\geq 0$ in $\Omega,$ then $u>0$ in $\Omega$.
Proof. Let us consider $\Omega_{n}=\\{x\in\Omega:d(x)>1/n\\}$ and use Theorem
2.5 to find a solution $u_{n}$ of
(2.11)
$\left\\{\begin{array}[]{ll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=f(x),&x\in\Omega_{n},\\\\[5.69054pt]
u(x)=\underline{U}(x),&x\in\Omega_{n}^{c},\end{array}\right.$
We just replace ${\mathcal{O}}$ by $\Omega_{n}$ and define
$\delta=\frac{1}{4n}$, so that $\underline{U}(x)$ satisfies assumption $(G)$.
We notice that $\Omega_{n}$ is of class $C^{2}$ for $n\geq N_{0}$, for certain
$N_{0}$ large. Next we show that $u_{n}$ is a sub-solution of (2.11) in
$\Omega_{n+1}$. In fact, since $u_{n}$ is the solution of (2.11) in
$\Omega_{n}$ and $\underline{U}$ is a sub-solution of (2.11) in $\Omega_{n}$,
by Theorem 2.3,
$u_{n}\geq\underline{U}\ \ \mbox{in}\ \ \Omega_{n}.$
Additionally, $u_{n}=\underline{U}\ \ \mbox{in}\ \ \Omega_{n}^{c}$. Then, for
$x\in\Omega_{{n+1}}\setminus\Omega_{n}$, we have
$\displaystyle(-\Delta)^{\alpha}u_{n}(x)=-\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\delta(u_{n},x,y)}{|y|^{N+2\alpha}}dy\leq(-\Delta)^{\alpha}\underline{U}(x),$
so that $u_{n}$ is a sub-solution of (2.11) in $\Omega_{n+1}$. From here and
since $u_{n+1}$ is the solution of (2.11) in $\Omega_{n+1}$ and $\bar{U}$ is a
super-solution of (2.11) in $\Omega_{n+1}$, by Theorem 2.3, we have $u_{n}\leq
u_{n+1}\leq\bar{U}$ in $\Omega_{n+1}.$ Therefore, for any $n\geq N_{0}$,
$\underline{U}\leq u_{n}\leq u_{n+1}\leq\bar{U}\ \ \mbox{in}\ \ \Omega.$
Then we can define the function $u$ as
$u(x)=\lim_{n\to+\infty}u_{n}(x),\ x\in\Omega\ \ \mbox{and}\ \ u(x)=0,\
x\in\bar{\Omega}^{c}$
and we have
$\underline{U}(x)\leq u(x)\leq\bar{U}(x),\ x\in\Omega.$
Since $\underline{U}$ and $\bar{U}$ belong to
$L^{1}_{\omega}(\mathbb{R}^{N})$, we see that $u_{n}\to u\ \ \mbox{in}\ \
L^{1}_{\omega}(\mathbb{R}^{N}),$ as $n\to\infty$. Now we repeat the arguments
of the proof of Theorem 2.5 to find that u is a classical solution of (1).
Finally, if $f$ is positive we easily find that $u$ is positive, again by a
contradiction argument. $\Box$
## 3\. Some estimates
In order to prove our existence threorems we will use Theorem 2.6, so that it
is crucial to have available super and sub-solutions to (1.4). In this section
we provide the basic estimates that will allow to obtain in the next section
the necessary super and sub-solutions.
To this end, we use appropriate powers of the distance function $d$ and the
main result in this section are the estimates given in Proposition 3.2, that
provides the asymptotic behavior of the fractional operator applied to $d$.
But before going to this estimates, we describe the behavior of the function
$C$ defined in (1.13), which is a $C^{2}$ defined in $(-1,2\alpha)$. We have:
###### Proposition 3.1.
For every $\alpha\in(0,1)$ there exists a unique $\tau_{0}(\alpha)\in(-1,0)$
such that $C(\tau_{0}(\alpha))=0$ and
(3.1) $C(\tau)(\tau-\tau_{0}(\alpha))<0,\quad\mbox{for
all}\,\,\tau\in(-1,0)\setminus\\{\tau_{0}(\alpha)\\}.$
Moreover, the function $\tau_{0}$ satisfies
(3.2) $\lim_{\alpha\to
1^{-}}\tau_{0}(\alpha)=0\quad\mbox{and}\quad\lim_{\alpha\to
0^{+}}\tau_{0}(\alpha)=-1.$
Proof. We first observe that $C(0)<0$ since the integrand in (1.13) is zero in
$(0,1)$ and negative in $(1,+\infty)$. Next easily see that
(3.3) $\lim_{\tau\to-1^{+}}C(\tau)=+\infty,$
since, as $\tau$ approaches $-1$, the integrand loses integrability at $0$.
Next we see that $C(\cdot)$ is strictly convex in $(-1,0)$, since
$C^{\prime}(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}\chi_{(0,1)}(t)\log|1-t|+(1+t)^{\tau}\log(1+t)}{t^{1+2\alpha}}dt\
$
and
$C^{\prime\prime}(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}[\chi_{(0,1)}(t)\log|1-t|]^{2}+(1+t)^{\tau}[\log(1+t)]^{2}}{t^{1+2\alpha}}dt>0.$
The convexity $C(\cdot)$, $C(0)<0$ and (3.3) allow to conclude the existence
and uniqueness of $\tau_{0}(\alpha)\in(-1,0)$ such that (3.1) holds. To prove
the first limit in (3.2), we proceed by contradiction, assuming that for
$\\{\alpha_{n}\\}$ converging to $1$ and $\tau_{0}\in(-1,0)$ such that
$\tau_{0}(\alpha_{n})\leq\tau_{0}<0.$
Then, for a constant $c_{1}>0$ we have
$\lim_{\alpha_{n}\to
1^{-}}\int^{\frac{1}{2}}_{0}\frac{(1-t)^{\tau_{0}(\alpha_{n})}+(1+t)^{\tau_{0}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}dt\geq
c_{1}\lim_{\alpha_{n}\to
1^{-}}\int_{0}^{\frac{1}{2}}t^{1-2\alpha_{n}}dt=+\infty$
and, for a constant $c_{2}$ independent of $n$, we have
$\displaystyle\int_{\frac{1}{2}}^{+\infty}|\frac{\chi_{(0,1)}(t)(1-t)^{\tau_{0}(\alpha_{n})}+(1+t)^{\tau_{0}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}|dt$
$\displaystyle\leq$ $\displaystyle c_{2},$
contradicting the fact that $C(\tau_{0}(\alpha_{n}))=0.$ For the second limit
in (3.2), we proceed similarly, assuming that for $\\{\alpha_{n}\\}$
converging to $0$ and $\bar{\tau}_{0}\in(-1,0)$ such that
$\tau_{0}(\alpha_{n})\geq\bar{\tau}_{0}>-1.$
There are positive constants $c_{1}$ and $c_{2}$ we have such that
$\displaystyle\int^{2}_{0}|\frac{\chi_{0,1}(t)(1-t)^{\tau_{0}(\alpha_{n})}+(1+t)^{\tau_{0}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}|dt\leq
c_{1}$
and
$\displaystyle\lim_{n\to\infty}\int_{2}^{+\infty}\frac{(1+t)^{\tau_{0}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}dt$
$\displaystyle\leq$ $\displaystyle-
c_{2}\lim_{n\to\infty}\int_{2}^{+\infty}\frac{1}{t^{1+2\alpha_{n}}}dt=-\infty,$
contradicting again that $C(\tau_{0}(\alpha_{n}))=0.$ $\Box$
Next we prove our main result in this section. We assume that $\delta>0$ is
such that the distance function $d(\cdot)$ is of class $C^{2}$ in
$A_{\delta}=\\{x\in\Omega,d(x)<\delta\\}$ and we define
(3.4) $V_{\tau}(x)=\left\\{\begin{array}[]{lll}l(x),&x\in\Omega\setminus
A_{\delta},\\\\[5.69054pt] d(x)^{\tau},&x\in A_{\delta},\\\\[5.69054pt]
0,&x\in\Omega^{c},\end{array}\right.$
where $\tau$ is a parameter in $(-1,0)$ and the function $l$ is positive such
that $V_{\tau}$ is $C^{2}$ in $\Omega$. We have the following
###### Proposition 3.2.
Assume $\Omega$ is a bounded, open subset of $\mathbb{R}^{N}$ with a $C^{2}$
boundary and let $\alpha\in(0,1)$. Then there exists $\delta_{1}\in(0,\delta)$
and a constant $C>1$ such that:
$(i)$ If $\tau\in(-1,\tau_{0}(\alpha))$, then
$\frac{1}{C}d(x)^{\tau-2\alpha}\leq-(-\Delta)^{\alpha}V_{\tau}(x)\leq
Cd(x)^{\tau-2\alpha},\ \ \mbox{for all}\,\,x\in A_{\delta_{1}}.$
$(ii)$ If $\tau\in(\tau_{0}(\alpha),0)$, then
$\frac{1}{C}d(x)^{\tau-2\alpha}\leq(-\Delta)^{\alpha}V_{\tau}(x)\leq
Cd(x)^{\tau-2\alpha},\ \ \mbox{for all}\,\,x\in A_{\delta_{1}}.$
$(iii)$ If $\tau=\tau_{0}(\alpha)$, then
$|(-\Delta)^{\alpha}V_{\tau}(x)|\leq
Cd(x)^{\min\\{\tau_{0}(\alpha),2\tau_{0}(\alpha)-2\alpha+1\\}},\ \ \mbox{for
all}\,\,x\in A_{\delta_{1}}.$
Proof. By compactness we prove that the corresponding inequality holds in a
neighborhood of any point $\bar{x}\in\partial\Omega$ and without loss of
generality we may assume that $\bar{x}=0$. For a given $0<\eta\leq\delta$, we
define
$Q_{\eta}=\\{z=(z_{1},z^{\prime})\in\mathbb{R}\times\mathbb{R}^{N-1},|z_{1}|<\eta,|z^{\prime}|<\eta\\}$
and $Q_{\eta}^{+}=\\{z\in Q_{\eta},z_{1}>0\\}$. Let
$\varphi:\mathbb{R}^{N-1}\to\mathbb{R}$ be a $C^{2}$ function such that
$(z_{1},z^{\prime})\in\Omega\cap Q_{\eta}$ if and only if
$z_{1}\in(\varphi(z^{\prime}),\eta)$ and moreover,
$(\varphi(z^{\prime}),z^{\prime})\in\partial\Omega$ for all
$|z^{\prime}|<\eta$. We further assume that $(-1,0,\cdot\cdot\cdot,0)$ is the
outer normal vector of $\Omega$ at $\bar{x}$.
In the proof of our inequalities, we let $x=(x_{1},0)$, with
$x_{1}\in(0,\eta/4)$, be then a generic point in $A_{\eta/4}$. We observe that
$|x-\bar{x}|=d(x)=x_{1}$. By definition we have
(3.5)
$\displaystyle-(-\Delta)^{\alpha}V_{\tau}(x)=\frac{1}{2}\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy+\frac{1}{2}\int_{\mathbb{R}^{N}\setminus
Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy$
and we see that
(3.6) $|\int_{\mathbb{R}^{N}\setminus
Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy|\leq c|x|^{\tau},$
where the constant $c$ is independent of $x$. Thus we only need to study the
asymptotic behavior of the first integral, that from now on we denote by
$E(x_{1})$.
Our first goal is to get a lower bound for $E(x_{1})$. For that purpose we
first notice that, since $\tau\in(-1,0)$, we have that
(3.7) $d(z)^{\tau}\geq|z_{1}-\varphi(z^{\prime})|^{\tau},\quad\mbox{for
all}\quad z\in Q_{\delta}\cap\Omega.$
Now we assume that $0<\eta\leq\delta/2$, then for all $y\in Q_{\eta}$ we have
$x\pm y\in Q_{\delta}$. Thus $x\pm y\in\Omega\cap Q_{\delta}$ if and only if
$\varphi(\pm y^{\prime})<x_{1}\pm y_{1}<\delta$ and $|y^{\prime}|<\delta$.
Then, by (3.7), we have that
(3.8) $\displaystyle\quad
V_{\tau}(x+y)=d(x+y)^{\tau}\geq[x_{1}+y_{1}-\varphi(y^{\prime})]^{\tau},\quad
x+y\in Q_{\delta}\cap\Omega$
and
(3.9) $\displaystyle\quad
V_{\tau}(x-y)=d(x-y)^{\tau}\geq[x_{1}-y_{1}-\varphi(-y^{\prime})]^{\tau},\quad
x-y\in Q_{\delta}\cap\Omega.$
On the other side, for $y\in Q_{\eta}$ we have that if $x\pm y\in
Q_{\delta}\cap\Omega^{c}$ then, by definition of $V_{\tau}$, we have
$V_{\tau}(x\pm y)=0.$ Now, for $y\in Q_{\eta}$ we define the intervals
(3.10) $I_{+}=(\varphi(y^{\prime})-x_{1},\eta-x_{1})\quad\mbox{and}\quad
I_{-}=(x_{1}-\eta,x_{1}-\varphi(-y^{\prime}))$
and the functions
$\displaystyle I(y)$ $\displaystyle=$
$\displaystyle\chi_{I_{+}}(y_{1})|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}+\chi_{I_{-}}(y_{1})|x_{1}-y_{1}-\varphi(-y^{\prime})|^{\tau}-2x_{1}^{\tau},~{}~{}~{}~{}~{}$
$\displaystyle J(y_{1})$ $\displaystyle=$
$\displaystyle\chi_{(x_{1}-\eta,x_{1})}(y_{1})|x_{1}-y_{1}|^{\tau}+\chi_{(-x_{1},\eta-
x_{1})}(y_{1})|x_{1}+y_{1}|^{\tau}-2x_{1}^{\tau},$ $\displaystyle I_{1}(y)$
$\displaystyle=$ $\displaystyle\\{\chi_{I_{+}}(y_{1})-\chi_{(-x_{1},\eta-
x_{1})}(y_{1})\\}|x_{1}+y_{1}|^{\tau},$ $\displaystyle I_{2}(y)$
$\displaystyle=$
$\displaystyle\chi_{I_{+}}(y_{1})(|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}\\}),$
where $\chi_{A}$ denotes the characteristic function of the set $A$. Then,
using these definitions and inequalities (3.8) and (3.9), we have that
(3.11) $\displaystyle\quad
E(x_{1})\geq\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy=\int_{Q_{\eta}}\frac{J(y_{1})}{|y|^{N+2\alpha}}dy+E_{1}(x_{1})+E_{2}(x_{1}),$
where
(3.12)
$E_{i}(x_{1})=\int_{Q_{\eta}}\frac{I_{i}(y)+I_{-i}(y)}{|y|^{N+2\alpha}}dy,\quad
i=1,2.$
Here we have considered that
$I_{-1}(y)=\\{\chi_{I_{-}}(y_{1})-\chi_{(x_{1}-\eta,x_{1})}(y_{1})\\}|x_{1}-y_{1}|^{\tau}$
and
$I_{-2}(y)=\chi_{I_{-}}(y_{1})(|x_{1}-y_{1}-\varphi(-y^{\prime})|^{\tau}-|x_{1}-y_{1}|^{\tau}\\}),$
for $y=(y_{1},y^{\prime})\in\mathbb{R}^{N}$. We start studying the first
integral in the right hand side in (3.11). Changing variables we see that
$\int_{Q_{\eta}}\frac{J(y_{1})}{|y|^{N+2\alpha}}dy=x_{1}^{\tau-2\alpha}\int_{Q_{\frac{\eta}{x_{1}}}}\frac{J(x_{1}z_{1})x_{1}^{-\tau}}{|z|^{N+2\alpha}}dz=2x_{1}^{\tau-2\alpha}(R_{1}-R_{2}),$
where
$R_{1}=\int_{Q_{\frac{\eta}{x_{1}}}^{+}}\frac{\chi_{(0,1)}(z_{1})|1-z_{1}|^{\tau}+(1+z_{1})^{\tau}-2}{|z|^{N+2\alpha}}dz$
and
$R_{2}=\int_{Q_{\frac{\eta}{x_{1}}}^{+}}\frac{\chi_{(\frac{\eta}{x_{1}}-1,\frac{\eta}{x_{1}})}(z_{1})(1+z_{1})^{\tau}}{|z|^{N+2\alpha}}dz.$
Next we estimate these last two integrals. For $R_{1}$ we see that, for
appropriate positive constants $c_{1}$ and $c_{2}$
$\displaystyle\int_{\mathbb{R}^{N}_{+}}\frac{\chi_{(0,1)}(z_{1})|1-z_{1}|^{\tau}+(1+z_{1})^{\tau}-2}{|z|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\frac{\chi_{(0,1)}(z_{1})|1-z_{1}|^{\tau}+(1+z_{1})^{\tau}-2}{z_{1}^{1+2\alpha}}dz_{1}\int_{\mathbb{R}^{N-1}}\frac{1}{(|z^{\prime}|^{2}+1)^{\frac{N+2\alpha}{2}}}dz^{\prime}$
$\displaystyle=$ $\displaystyle c_{1}\,C(\tau)$
and
$\displaystyle\int_{(Q_{\frac{\eta}{x_{1}}}^{+})^{c}}\frac{\chi_{(0,1)}(z_{1})|1-z_{1}|^{\tau}+(1+z_{1})^{\tau}-2}{|z|^{N+2\alpha}}dz=-c_{2}\,x_{1}^{2\alpha}(1+o(1)).$
Consequently we have, for some constant $c$ that
(3.13) $\displaystyle
R_{1}=c_{1}(C(\tau)+cx_{1}^{2\alpha}+o(x_{1}^{2\alpha})).$
For $R_{2}$ we have that
(3.14) $\displaystyle\quad R_{2}$ $\displaystyle=$
$\displaystyle\int_{\frac{\eta}{x_{1}}-1}^{\frac{\eta}{x_{1}}}\frac{(1+z_{1})^{\tau}}{z_{1}^{1+2\alpha}}\int_{B_{\frac{\eta}{x_{1}}}}\frac{1}{(1+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz^{\prime}dz_{1}\leq
c_{3}x_{1}^{2\alpha-\tau+1},$
where $c_{3}>0$. Here and in what follows we denote by $B_{\sigma}$ the ball
of radius $\sigma$ in $\mathbb{R}^{N-1}$. From (3.13) and (3.14) we then
conclude that
(3.15)
$\displaystyle\int_{Q_{\eta}}\frac{J(y_{1})}{|y|^{N+2\alpha}}dy=c_{1}x_{1}^{\tau-2\alpha}(C(\tau)+cx_{1}^{2\alpha}+o(x_{1}^{2\alpha})).$
Continuing with our analysis we estimate $E_{1}(x_{1})$. We only consider the
term $I_{1}(y)$, since the estimate for $I_{1}(-y)$ is similar. We have
$\displaystyle\int_{Q_{\eta}}\frac{I_{1}(y)}{|y|^{N+2\alpha}}dy=-\int_{B_{\eta}}\int^{\varphi(y^{\prime})-x_{1}}_{-x_{1}}\frac{|x_{1}+y_{1}|^{\tau}}{|y|^{N+2\alpha}}dy_{1}dy^{\prime}=-x_{1}^{\tau-2\alpha}F_{1}(x_{1}),$
where
(3.16) $\displaystyle
F_{1}(x_{1})=\int_{B_{\frac{\eta}{x_{1}}}}\int^{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}}_{0}\frac{|z_{1}|^{\tau}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}.$
In what follows we write
$\varphi_{-}(y^{\prime})=\min\\{\varphi(y^{\prime}),0\\}$ and
$\varphi_{+}(y^{\prime})=\varphi(y^{\prime})-\varphi_{-}(y^{\prime})$. Next we
see that assuming that $0\leq\varphi_{+}(y^{\prime})\leq C|y^{\prime}|^{2}$
for $|y^{\prime}|\leq\eta$, for given $(z_{1},z^{\prime})$ satisfying $0\leq
z_{1}\leq\frac{\varphi_{+}(x_{1}z^{\prime})}{x_{1}}$ and
$|z^{\prime}|\leq\frac{\eta}{x_{1}}$ then
(3.17)
$\displaystyle(1-z_{1})^{2}+|z^{\prime}|^{2}\geq\frac{1}{4}(1+|z^{\prime}|^{2}),$
if we assume $\eta$ small enough. Thus
$\displaystyle F_{1}(x_{1})$ $\displaystyle\leq$ $\displaystyle
C\int_{B_{\frac{\eta}{x_{1}}}}\int^{\frac{\varphi_{+}(x_{1}z^{\prime})}{x_{1}}}_{0}\frac{|z_{1}|^{\tau}}{(1+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
$\displaystyle\leq$ $\displaystyle
Cx_{1}^{\tau+1}\int_{B_{\frac{\eta}{x_{1}}}}\frac{|z^{\prime}|^{2(\tau+1)}}{(1+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz^{\prime}$
$\displaystyle\leq$ $\displaystyle
Cx_{1}^{\tau+1}(x_{1}^{-2\tau+2\alpha-1}+1)\leq
Cx_{1}^{\min\\{\tau+1,2\alpha-\tau\\}}.$
Thus we have obtained
(3.18) $E_{1}(x_{1})\geq-
Cx_{1}^{\tau-2\alpha}x_{1}^{\min\\{\tau+1,2\alpha-\tau\\}}.$
We continue with the estimate of $E_{2}(x_{1})$. As before we only consider
the term $I_{2}(y)$,
(3.19) $\displaystyle\int_{Q_{\eta}}\frac{I_{2}(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int_{\varphi(y^{\prime})-x_{1}}^{\eta-
x_{1}}\frac{|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle\int_{B_{\eta}}\int_{\varphi_{-}(y^{\prime})-x_{1}}^{\eta-
x_{1}}\frac{|x_{1}+y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int_{\varphi_{-}(y^{\prime})}^{\eta}\frac{|z_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|z_{1}|^{\tau}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz_{1}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle\int_{B_{\eta}}\int_{0}^{\eta}\frac{|z_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|z_{1}|^{\tau}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int_{\varphi_{-}(y^{\prime})}^{0}\frac{-|z_{1}|^{\tau}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle E_{21}(x_{1})+E_{22}(x_{1}).$
We observe that $E_{22}(x_{1})$ is similar to $F_{1}(x_{1})$. In order to
estimate $E_{21}(x_{1})$ we use integration by parts
$\displaystyle E_{21}(x_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}\left\\{\frac{(\eta-\varphi_{-}(y^{\prime}))^{\tau+1}-\eta^{\tau+1}}{((\eta-
x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}-\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}\right\\}dy^{\prime}$
$\displaystyle+$
$\displaystyle\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int_{0}^{\eta}\frac{(z_{1}-\varphi_{-}(y^{\prime}))^{\tau+1}-(z_{1})^{\tau+1}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-x_{1})dz_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle A_{1}+A_{2}.$
For the first integral we have
$\displaystyle A_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}\left\\{\frac{-\eta^{\tau+1}}{((\eta-
x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}-\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}\right\\}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle-C(\eta)-C\int_{B_{\eta}}\frac{|y^{\prime}|^{2\tau+2}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}\geq-
Cx_{1}^{\tau-2\alpha+\tau+1}-C.$
For the second integral, since $\tau\in(-1,0)$ and
$(z_{1}-\varphi_{-}(y^{\prime}))^{\tau+1}-|z_{1}|^{\tau+1}>0$, we have that
(3.20) $\displaystyle A_{2}$ $\displaystyle\geq$
$\displaystyle\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{(z_{1}-\varphi_{-}(y^{\prime}))^{\tau+1}-|z_{1}|^{\tau+1}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-x_{1})dz_{1}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{N+2\alpha}{(\tau+1)^{2}}\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{-\varphi_{-}(y^{\prime})z_{1}^{\tau}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-x_{1})dz_{1}dy^{\prime}$
$\displaystyle\geq$ $\displaystyle
C_{3}x_{1}^{2\tau-2\alpha+1}\int_{B_{\eta/x_{1}}}\int^{1}_{0}\frac{|z^{\prime}|^{2}z_{1}^{\tau}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-1)dz_{1}dz^{\prime}$
$\displaystyle\geq$ $\displaystyle-C_{4}x_{1}^{2\tau-2\alpha+1},$
where $C_{3},C_{4}>0$ independent of $x_{1}$ and the second inequality used
$a=z_{1}$ and $b=-\varphi_{-}(y^{\prime})$ in the fact that
$(a+b)^{\tau+1}-a^{\tau+1}\leq\frac{a^{\tau}b}{\tau+1}$ for $a>0,b\geq 0$.
Thus, we have obtained
(3.21) $E_{2}(x_{1})\geq-
Cx_{1}^{\tau-2\alpha}x_{1}^{\min\\{\tau+1,2\alpha-\tau\\}}.$
The next step is to obtain the other inequality for $E(x_{1})$. By choosing
$\delta$ smaller if necessary, we can prove that
###### Lemma 3.1.
Under the regularity conditions on the boundary and with the arrangements
given at the beginning of the proof, there is $\eta>0$ and $C>0$ such that
$d(z)\geq(z_{1}-\varphi(z^{\prime}))(1-C|z^{\prime}|^{2})\quad\mbox{for all
}(z_{1},z^{\prime})\in\Omega\cap Q_{\eta}.$
Proof. Since $\varphi$ is $C^{2}$ and $\nabla\varphi(0)=0$, there exist
$\eta_{1}\in(0,1/8)$ small and $C_{1}>0$ such that $C_{1}\eta_{1}<1/4$ and
(3.22)
$|\varphi(y^{\prime})|<C_{1}|y^{\prime}|^{2},\quad|\nabla\varphi(y^{\prime})|\leq
C_{1}|y^{\prime}|,\quad\forall\ y^{\prime}\in B_{\eta_{1}}.$
Choosing $\eta_{2}\in(0,\eta_{1})$ such that for any $z=(z_{1},z^{\prime})\in
Q_{\eta_{2}}\cap\Omega$, there exists $y^{\prime}$ satisfying
$(\varphi(y^{\prime}),y^{\prime})\in\partial\Omega\cap Q_{\eta_{1}}$ and
$d(z)=|z-(\varphi(y^{\prime}),y^{\prime})|$.
We observe that $y^{\prime}$ mentioned above, is the minimizer of
$H(z^{\prime})=(z_{1}-\varphi(z^{\prime}))^{2}+|z^{\prime}-y^{\prime}|^{2},\quad|z^{\prime}|<\eta_{1},$
then
$-(z_{1}-\varphi(y^{\prime}))\nabla\varphi(y^{\prime})+(z^{\prime}-y^{\prime})=0,$
which, together with (3.22) implies that
$\displaystyle|y^{\prime}|-|z^{\prime}|$ $\displaystyle\leq$
$\displaystyle|z^{\prime}-y^{\prime}|=|(z_{1}-\varphi(y^{\prime}))\nabla\varphi(y^{\prime})|\leq(|z_{1}|+C_{1}|y^{\prime}|^{2})|\nabla\varphi(y^{\prime})|$
$\displaystyle\leq$ $\displaystyle
C_{1}(\eta_{2}+C_{1}\eta_{1}^{2})|y^{\prime}|\leq
2C_{1}\eta_{1}|y^{\prime}|<\frac{1}{2}|y^{\prime}|.$
Then
(3.23) $|y^{\prime}|\leq 2|z^{\prime}|.$
Denote the points
$z,(\varphi(y^{\prime}),y^{\prime}),(\varphi(z^{\prime}),z^{\prime})$ by
$A,B,C$, respectively, and let $\theta$ be the angle between the segment $BC$
and the hyper plane with normal vector $e_{1}=(1,0,...,0)$ and containing $C$.
Then the angle $\angle C=\frac{\pi}{2}-\theta.$ Denotes the arc from $B$ to
$C$ in the plane $ABC$ by arc$(BC)$. By the geometry, there exists some point
$x=(\varphi(x^{\prime}),x^{\prime})\in\mbox{arc}(BC)$ such that line $BC$
parallels the tangent line of arc$(BC)$ at point $x$. Then, from (3.23) we
have $|x^{\prime}|\leq\max\\{|z^{\prime}|,|y^{\prime}|\\}\leq 2|z^{\prime}|$
and so, from (3.22) we obtain
$\displaystyle\tan(\theta)=|\frac{y^{\prime}-z^{\prime}}{|y^{\prime}-z^{\prime}|}\cdot\nabla\varphi(x^{\prime})|\leq|\nabla\varphi(x^{\prime})|\leq
C_{1}|x^{\prime}|\leq 2C_{1}|z^{\prime}|,$
which implies that for some $C>0$,
(3.24) $\cos(\theta)\geq 1-C|z^{\prime}|^{2}.$
Then we complete the proof using Sine Theorem and (3.24)
$\displaystyle d(z)$ $\displaystyle=$ $\displaystyle\frac{\sin(\angle
C)}{\sin(\angle
B)}(z_{1}-\varphi(z^{\prime}))\geq(z_{1}-\varphi(z^{\prime}))\sin(\frac{\pi}{2}-\theta)$
$\displaystyle=$
$\displaystyle(z_{1}-\varphi(z^{\prime}))\cos(\theta)\geq(z_{1}-\varphi(z^{\prime}))(1-C|z^{\prime}|^{2}).\qquad\Box$
From this lemma, by making $C$ and $\eta$ smaller if necessary we obtain that
(3.25)
$d^{\tau}(z)\leq(z_{1}-\varphi(z^{\prime}))^{\tau}(1+C|z^{\prime}|^{2})\quad\mbox{for
all }z\in\Omega\cap Q_{\eta}.$
With $x=(x_{1},0)$ satisfying $x_{1}\in(0,\eta/4)$ as at the beginning of the
proof, we have that $d(x)=x_{1}$ and for any $y\in Q_{\eta}$ we see that $x\pm
y\in Q_{\delta}.$ We also see that $x\pm y\in\Omega\cap Q_{\delta}$ if and
only if $\varphi(\pm y^{\prime})<x_{1}\pm y_{1}<\delta$ and
$|y^{\prime}|<\delta$. Then, for $x\pm y\in\Omega\cap Q_{\eta}$, by (3.25) we
have,
(3.26) $\displaystyle V_{\tau}(x\pm y)=d(x\pm y)^{\tau}\leq(x_{1}\pm
y_{1}-\varphi(\pm y^{\prime}))^{\tau}(1+C|y^{\prime}|^{2}).$
For $y\in Q_{\eta}$, we define
$I_{3}(y)=C|y^{\prime}|^{2}\chi_{I_{+}}(y_{1})|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}$
and
$I_{3}(-y)=C|y^{\prime}|^{2}\chi_{I_{-}}(y_{1})|x_{1}-y_{1}-\varphi(-y^{\prime})|^{\tau},$
where $I_{+}$ and $I_{-}$ were defined in (3.10). Using (3.26) as in (3.11) we
find
(3.27) $\displaystyle E(x_{1})$ $\displaystyle=$
$\displaystyle\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy\leq\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy+E_{3}(x_{1})$
$\displaystyle=$
$\displaystyle\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy+E_{1}(x_{1})+E_{2}(x_{1})+E_{3}(x_{1}),$
where $E_{1}$ and $E_{2}$ were defined in (3.12) and
(3.28)
$E_{3}(x_{1})=\int_{Q_{\eta}}\frac{I_{3}(y)+I_{3}(-y)}{|y|^{N+2\alpha}}dy.$
We estimate $E_{3}(x_{1})$ and for that we observe that it is enough to
estimate the integral with one of the terms in (3.28) (the other is similar),
say
$\displaystyle\int_{Q_{\eta}}\frac{I_{3}(y)}{|y|^{N+2\alpha}}dy=\int_{B_{\eta}}\int_{\varphi(y^{\prime})-x_{1}}^{\eta-
x_{1}}\frac{C|y^{\prime}|^{2}|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}}{|y|^{N+2\alpha}}dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle
Cx_{1}^{\tau-2\alpha+2}\int_{B_{\frac{\eta}{x_{1}}}}\int_{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}}^{\frac{\eta}{x_{1}}}\frac{|z^{\prime}|^{2}|z_{1}-\frac{\varphi(x_{1}z^{\prime})}{x_{1}}|^{\tau}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
(3.29) $\displaystyle=$ $\displaystyle Cx_{1}^{\tau-2\alpha+2}(A_{1}+A_{2}),$
where $A_{1}$ and $A_{2}$ are integrals over properly chosen subdomains,
estimated separately.
(3.30) $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\int_{B_{\frac{\eta}{x_{1}}}}\int_{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}}^{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}+\frac{1}{2}}\frac{|z^{\prime}|^{2}|z_{1}-\frac{\varphi(x_{1}z^{\prime})}{x_{1}}|^{\tau}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
$\displaystyle\leq$
$\displaystyle\frac{c}{(\tau+1)2^{\tau+1}}\int_{B_{\frac{\eta}{x_{1}}}}\frac{|z^{\prime}|^{2}}{(1+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz^{\prime}$
(3.31) $\displaystyle\leq$ $\displaystyle
c^{\prime}\left(\frac{\eta}{x_{1}}\right)^{-2\alpha+1}.$
The inequality in (3.30) is obtained noticing that the ball $B((1,0),1/2)$ in
$R^{N}$ does not touch the band
$\\{(z_{1},z^{\prime})\,/\,|z^{\prime}|\leq\eta,\frac{\varphi(x_{1}z^{\prime})}{x_{1}}\leq
z_{1}\leq\frac{\varphi(x_{1}z^{\prime})}{x_{1}}+1/2\\}$
if $x_{1}$ is small enough, and so
$(z_{1}-1)^{2}+|z^{\prime}|^{2}\geq\frac{1}{8}+\frac{1}{2}|z^{\prime}|^{2}$.
Then simple integration gives the next term. Next we estimate $A_{2}$
(3.32) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\int_{B_{\frac{\eta}{x_{1}}}}\int^{\frac{\eta}{x_{1}}}_{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}+\frac{1}{2}}\frac{|z^{\prime}|^{2}|z_{1}-\frac{\varphi(x_{1}z^{\prime})}{x_{1}}|^{\tau}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2^{\tau}}\int_{B_{\frac{\eta}{x_{1}}}}\int^{\frac{\eta}{x_{1}}}_{\frac{\varphi(x_{1}z^{\prime})}{x_{1}}+\frac{1}{2}}\frac{|z^{\prime}|^{2}}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
$\displaystyle\leq$ $\displaystyle
c^{\prime}\left(\frac{\eta}{x_{1}}\right)^{-2\alpha+2}.$
Putting together (3.29), (3.31), (3.32) and (3.28) we obtain
(3.33)
$E_{3}(x_{1})=\int_{Q_{\eta}}\frac{(I_{3}(y)+I_{3}(-y))}{|y|^{N+2\alpha}}dy\leq
cx_{1}^{\tau}.$
From (3), but using the other inequality for $F_{1}$, that is,
$F_{1}(x_{1})\geq
C\int_{B_{\frac{\eta}{x_{1}}}}\int^{\frac{\varphi_{-}(x_{1}z^{\prime})}{x_{1}}}_{0}\frac{|z_{1}|^{\tau}}{(1+|z^{\prime}|^{2})^{(N+2\alpha)/2}}dz_{1}dz^{\prime}$
and arguing similarly we obtain as in (3.18)
(3.34) $E_{1}(x_{1})\leq
Cx_{1}^{\tau-2\alpha}x_{1}^{\min\\{\tau+1,2\alpha\\}}.$
Then we look at $E_{2}(x_{1})$ and, as in (3.19), we only consider the term
$I_{2}(y)$:
$\displaystyle\int_{Q_{\eta}}\frac{I_{2}(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle\leq$
$\displaystyle\int_{B_{\eta}}\int_{\varphi_{+}(y^{\prime})}^{\eta}\frac{|z_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|z_{1}|^{\tau}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz_{1}dy^{\prime}=\tilde{E}_{21}(x_{1}).$
In order to estimate $\tilde{E}_{21}(x_{1})$ we use integration by parts
$\displaystyle\tilde{E}_{21}(x_{1})=$
$\displaystyle\\!\\!\\!\\!\frac{1}{\tau+1}\int_{B_{\eta}}\left\\{\frac{(\eta-\varphi_{+}(y^{\prime}))^{\tau+1}-\eta^{\tau+1}}{((\eta-
x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}-\frac{(\varphi_{+}(y^{\prime}))^{\tau+1}}{((\varphi_{+}(y^{\prime})-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}\right\\}dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int_{\varphi_{+}(y^{\prime})}^{\eta}\frac{(z_{1}-\varphi_{+}(y^{\prime}))^{\tau+1}-z_{1}^{\tau+1}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-x_{1})dz_{1}dy^{\prime}$
$\displaystyle\leq\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int_{\min\\{\varphi_{+}(y^{\prime}),x_{1}\\}}^{x_{1}}\frac{(z_{1}-\varphi_{+}(y^{\prime}))^{\tau+1}-z_{1}^{\tau+1}}{((z_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(z_{1}-x_{1})dz_{1}dy^{\prime}.$
This integral can be estimated in a similar way as $E_{21}$, see (3.20) and
the estimates given before. We then obtain
(3.35) $E_{2}(x_{1})\leq Cx_{1}^{2\tau-2\alpha+1}.$
Then we conclude from (3.5), (3.11), (3.15), (3.18), (3.21), (3.27), (3.33),
(3.34) and (3.35) that
(3.36)
$\displaystyle-(-\Delta)^{\alpha}V_{\tau}(x)=Cx_{1}^{\tau-2\alpha}(C(\tau)+O(x_{1}^{\min\\{\tau+1,2\alpha\\}})),$
where there exists a constant $c>0$ so that
$|O(x_{1}^{\min\\{\tau+1,2\alpha\\}})|\leq
cx_{1}^{\min\\{\tau+1,2\alpha\\}},\qquad\mbox{for all small }x_{1}>0.$
From here, depending on the value of $\tau\in(-1,0)$, conditions (i), (ii) and
(iii) follows and the proof of the proposition is complete.$\hfill\Box$
We end this section with an estimate we need when dealing with equation (1.4)
when the external value $g$ is not zero. We have the following proposition
###### Proposition 3.3.
Assume that $\Omega$ is a bounded, open and $C^{2}$ domain in
$\mathbb{R}^{N}$. Assume that $g\in L^{1}_{\omega}(\Omega^{c})$. Assume
further that there are numbers $\beta\in(-1,0)$, $\eta>0$ and $c>1$ such that
$\frac{1}{c}\leq g(x)d(x)^{-\beta}\leq c,\ \ x\in\bar{\Omega}^{c}\ \mbox{and}\
d(x)\leq\eta.$
Then there exist $\eta_{1}>0$ and $C>1$ such that $G$, defined in (1.20),
satisfies
(3.37) $\frac{1}{C}d(x)^{\beta-2\alpha}\leq G(x)\leq Cd(x)^{\beta-2\alpha},\ \
x\in A_{\eta_{1}}.$
Proof. The proof of this proposition requires estimates similar to those in
the proof of Proposition 3.2 so we omit it. However, the function $C$ used
there and defined in (1.13), needs to be replaced here by
$\tilde{C}:(-1,0)\to\mathbb{R}$ given by
$\tilde{C}(\beta)=\int_{1}^{\infty}\frac{|t-1|^{\beta}}{t^{1+2\alpha}}dt.$
We observe that this function is always positive. $\Box$
## 4\. Proof of existence results
In this section, we will give the proof of existence of large solution to (1).
By Theorem 2.6 we only need to find ordered super and sub-solution, denoted by
$U$ and $W$, for (1) under our various assumptions. We begin with a simple
lemma that reduce the problem to find them only in $A_{\delta}$.
###### Lemma 4.1.
Let $U$ and $W$ be classical ordered super and sub-solution of (1) in the sub-
domain $A_{\delta}$. Then there exists $\lambda$ large such that
$U_{\lambda}=U-\lambda\bar{V}$ and $W_{\lambda}=W+\lambda\bar{V}$, where
$\bar{V}$ is the solution of (2.8), with ${\mathcal{O}}=\Omega$, are ordered
super and sub-solution of (1).
Proof. Notice that by negativity $\bar{V}$ in $\Omega$, we have that
$U_{\lambda}\geq U$ and $W_{\lambda}\leq W$, so they are still ordered in
$A_{\delta}$. In addition $U_{\lambda}$ satisfies
$(-\Delta)^{\alpha}U_{\lambda}+|U_{\lambda}|^{p-1}U_{\lambda}-f(x)\geq(-\Delta)^{\alpha}U+|U|^{p-1}U-f(x)+\lambda>0,\quad\mbox{in}\quad\Omega.$
This inequality holds because of our assumption in $A_{\delta}$, the fact that
$(-\Delta)^{\alpha}U+|U|^{p-1}U-f(x)$ is continuous in
$\Omega\setminus{A_{\delta}}$ and by taking $\lambda$ large enough.
By the same type of arguments we find the $W_{\lambda}$ is a sub-solution of
the first equation in (1) and we complete the proof. $\Box$
Now we are in position to prove our existence results that we already reduced
to find ordered super and sub-solution of (1) with the first equation in
$A_{\delta}$ with the desired asymptotic behavior.
Proof of Theorem 1.1 (Existence). Define
(4.1) $U_{\mu}(x)=\mu V_{\tau}(x)\quad\mbox{and}\quad\ W_{\mu}(x)=\mu
V_{\tau}(x),$
with $\tau=-\frac{2\alpha}{p-1}$. We observe that
$\tau=-\frac{2\alpha}{p-1}\in(-1,\tau_{0}(\alpha))$ and $\tau p=\tau-2\alpha$,
Then by Proposition 3.2 and $(H2)$ we find that for $x\in A_{\delta}$ and
$\delta>0$ small
$\displaystyle(-\Delta)^{\alpha}U_{\mu}(x)+U^{p}_{\mu}(x)-f(x)\geq-C\mu
d(x)^{\tau-2\alpha}+\mu^{p}d(x)^{\tau p}-Cd(x)^{\tau p},$
for some $C>0$. Then there exists a large $\mu>0$ such that $U_{\mu}$ is a
super-solution of (1) with the first equation in $A_{\delta}$ with the desired
asymptotic behavior. Now by Proposition 3.2 we have that for $x\in A_{\delta}$
and $\delta>0$ small
$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+W^{p}_{\mu}(x)-f(x)\leq-\frac{\mu}{C}d(x)^{\tau-2\alpha}+\mu^{p}d(x)^{\tau
p}-f(x)\leq 0,$
in the last inequality we have used $(H2)$ and $\mu>0$ small. Then, by Theorem
2.6 there exists a solution, with the desired asymptotic behavior. $\Box$
Proof of Theorem 1.1 (Special case $\tau=\tau_{0}(\alpha)$). We define for
$t>0$,
(4.2) $U_{\mu}(x)=tV_{\tau_{0}(\alpha)}(x)-\mu V_{\tau_{1}}(x)\quad\mbox{
and}\quad W_{\mu}(x)=tV_{\tau_{0}(\alpha)}(x)-\mu V_{\tau_{1}}(x),$
where $\tau_{1}=\min\\{\tau_{0}(\alpha)p+2\alpha,0\\}$. If $\tau_{1}=0$, we
write $V_{0}=\chi_{\Omega}$ and we have
$\displaystyle(-\Delta)^{\alpha}V_{0}(x)=\int_{\mathbb{R}^{N}\setminus\Omega}\frac{1}{|z-x|^{N+2\alpha}}dz,\quad
x\in\Omega.$
By direct computation, there exists $C>1$ such that
(4.3) $\frac{1}{C}d(x)^{-2\alpha}\leq(-\Delta)^{\alpha}V_{0}(x)\leq
Cd(x)^{-2\alpha},\quad x\in\Omega.$
We see that $\tau_{1}\in(\tau_{0}(\alpha),0]$ and, if $\tau_{1}<0$, we have
$\tau_{1}-2\alpha=\tau_{0}(\alpha)p$ and
$\tau_{1}-2\alpha<\min\\{\tau_{0}(\alpha),\tau_{0}(\alpha)-2\alpha+\tau_{0}(\alpha)+1\\}.$
, u Then, by Proposition 3.2 and (4.3), for $x\in A_{\delta}$, it follows that
$\displaystyle(-\Delta)^{\alpha}U_{\mu}(x)+|U_{\mu}(x)|^{p-1}U_{\mu}(x)$
$\displaystyle\geq$ $\displaystyle-
Ctd(x)^{\min\\{\tau_{0}(\alpha),\tau_{0}(\alpha)-2\alpha+\tau_{0}(\alpha)+1\\}}$
$\displaystyle-C\mu d(x)^{\tau_{1}-2\alpha}+t^{p}d(x)^{\tau_{0}(\alpha)p}.$
Thus, letting $\mu=t^{p}/(2C)$ if $\tau_{1}<0$ and $\mu=0$ if $\tau_{1}=0$,
for a possible smaller $\delta>0$, we obtain
$(-\Delta)^{\alpha}U_{\mu}(x)+|U_{\mu}(x)|^{p-1}U_{\mu}(x)\geq 0,\quad x\in
A_{\delta}.$
For the sub-solution, by Proposition 3.2 and (4.3), for $x\in A_{\delta}$, we
have
$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)$
$\displaystyle\leq$ $\displaystyle
Ctd(x)^{\min\\{\tau_{0}(\alpha),\tau_{0}(\alpha)-2\alpha+\tau_{0}(\alpha)+1\\}}$
$\displaystyle-\frac{\mu}{C}d(x)^{\tau_{1}-2\alpha}+t^{p}d(x)^{\tau_{0}(\alpha)p},$
where $C>1$. Then, for $\mu\geq 2Ct^{p}$ and a possibly smaller $\delta>0$
$(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)\leq 0,\ x\in
A_{\delta},$
completing the proof. $\Box$
Proof of Theorem 1.2. We define $U_{\mu}$ and $W_{\mu}$ as in (4.1). In the
case of a weak source, we take $\tau=\gamma+2\alpha$ and we observe that
$\gamma+2\alpha\geq-\frac{2\alpha}{p-1}>\tau_{0}(\alpha)$ and
$p(\gamma+2\alpha)\geq\gamma$. Using Proposition 3.2 and $(H3)$ we find that
$U_{\mu}$ is a super-solution for $\mu>0$ large (resp. $W_{\mu}$ is a sub-
solution for $\mu>0$ small) of (1) with the first equation in $A_{\delta}$ for
$\delta>0$ small. In the case of a strong source, we take
$\tau=\frac{\gamma}{p}$ and observe that $\gamma<\frac{\gamma}{p}-2\alpha$.
Using Proposition 3.2 we find
$|(-\Delta)^{\alpha}U_{m}u|,|(-\Delta)^{\alpha}U_{m}u|\leq
Cd(x)^{\frac{\gamma}{p}-2\alpha}.$
By $(H3)$ we find that $U_{\mu}$ is a super-solution for $\mu$ large (resp.
$W_{\mu}$ is a sub-solution for $\mu$ small) of (1) with the first equation in
$A_{\delta}$ for $\delta$ small. $\Box$
###### Remark 4.1.
In order to obtain the above existence results for classical solution to
(1.4), that is when $g$ is not necessarily zero, we only need use them with
$F$ as a right hand side as given in (1.23). Here we only need to assume that
$g$ satisfies $(H4)$. In fact, as above we find super and sub-solutions for
(1), with $f$ replaced by $F$. Then, as in the proof of Theorem 2.6, we find a
viscosity solution of (1) and then $v=u+\tilde{g}$ is a viscosity solution of
(1.4). Next we use Theorem 2.6 in [7] and then we use Theorem 2.1 to obtain
that $v$ is a classical solution of (1.4).
###### Remark 4.2.
Now we compare Theorem 1.1 with the result in [14]. Let us assume that $f$ and
$g$ satisfies hypothesis (F0)-(F2) and (G0)-(G3), respectively, given in [14].
We first observe that the function $F$, as defined above, satisfies $(H1)$
thanks to (G0), (G3) and (F0). Next we see that $F$ satisfies $(H2)$, since
(G2), (F1) and (F2) holds. Here we have to use Proposition 3.3. In the range
of $p$ given by (1.6), we then may apply Theorem 1.1 to obtain existence of a
blow-up solution as given in Theorem 1.1 in [14]. We see that the existence is
proved here, without assuming hypothesis $(G1)$, thus we generalized this
earlier result. Moreover, here we obtain a uniqueness and non existence of
blow-up solution, if we further assume hypotheses on $f$ and $g$, guaranteeing
hypothesis $(H2^{*})$ in Theorem 1.1. The complementary range of $p$ is
obtained using Theorem 1.2 for the existence of solutions as given in Theorem
1.1 in [14] and uniqueness and non-existence as in Theorem 1.3 and 1.4 are
truly new results. The hypotheses needed on $g$ to obtain $(H3)$ for the
function $F$ are a bit stronger, since we are requiring in $(H3)$ that the
explosion rate is the same from above and from below, while in (G2) and (G4)
they may be different.
## 5\. Proof of uniqueness results
In this section we prove our uniqueness results, which are given in Theorem
1.1 and Theorem 1.3. These results are for positive solutions, so we assume
that the external source $f$ is non-negative. We assume that there are two
positive solutions $u$ and $v$ of (1) and then define the set
(5.1) $\mathcal{A}=\\{x\in\Omega,\ u(x)>v(x)\\}.$
This set is open, $\mathcal{A}\subset\Omega$ and we only need to prove that
$\mathcal{A}=\O,$ to obtain that $u=v$, by interchanging the roles of $u$ and
$v$.
We will distinguish three cases, depending on the conditions satisfying $u$
and $v$: Case a) $u$ and $v$ satisfy (1.6) and (1.7) (uniqueness part of
Theorem 1.1), Case b) $u$ and $v$ $(\ref{gamma1})$ and (1.16) (weak source in
Theorem 1.3) and Case c) $u$ and $v$ with $(\ref{gamma2})$ -(1.19) (strong
source in Theorem 1.3).
We start our proof considering an auxiliary function
(5.2) $V(x)=\left\\{\begin{array}[]{lll}c(1-|x|^{2})^{3},&x\in
B_{1}(0),\\\\[5.69054pt] 0,&x\in B_{1}^{c}(0),\end{array}\right.$
where the constant $c$ may be chosen so that $V$ satisfies
(5.3) $(-\Delta)^{\alpha}V(x)\leq 1\quad\mbox{and}\quad
0<V(0)=\max_{x\in\mathbb{R}^{N}}V(x).$
In order to prove the uniqueness result in the three cases, we need first some
preliminary lemmas.
###### Lemma 5.1.
If $\mathcal{A}_{k}=\\{x\in\Omega,u(x)-kv(x)>0\\}\not=\O,$ for $k>1$. Then,
(5.4) $\partial\mathcal{A}_{k}\cap\partial\Omega\not=\O.$
Proof. If (5.4) is not true, there exists $\bar{x}\in\Omega$ such that
$u(\bar{x})-kv(\bar{x})=\max_{x\in\mathbb{R}^{N}}(u-kv)(x)>0,$
Then, we have
$(-\Delta)^{\alpha}(u-kv)(\bar{x})\geq 0,$
which contradicts
$\displaystyle(-\Delta)^{\alpha}(u-kv)(\bar{x})$ $\displaystyle=$
$\displaystyle-u^{p}(\bar{x})+kv^{p}(\bar{x})-(k-1)f(\bar{x})$
$\displaystyle\leq$ $\displaystyle-(k^{p}-k)v^{p}(\bar{x})<0.\hfill\Box$
###### Lemma 5.2.
If $\mathcal{A}_{k}\not=\O$, for $k>1$, then
(5.5) $\sup_{x\in\Omega}(u-kv)(x)=+\infty.$
Proof. Assume that $\bar{M}=\sup_{x\in\Omega}(u-kv)(x)<+\infty.$ We see that
$\bar{M}>0$ and there is no point $\bar{x}\in\Omega$ achieving the supreme of
$u-kv$, by the same argument given above. Let us consider
$x_{0}\in\mathcal{A}_{k}$, $r=d(x_{0})/2$ and define
(5.6) $w_{k}=u-kv\ \ \mbox{in}\ \ \mathbb{R}^{N}.$
Under the conditions of Case a) and b) (resp. Case c)), for all $x\in
B_{r}(x_{0})\cap\mathcal{A}_{k}$ we have
(5.7) $(-\Delta)^{\alpha}w_{k}(x)=-u^{p}(x)+kv^{p}(x)+(1-k)f(x)\leq-
K_{1}r^{\tau-2\alpha},$
(resp. $\leq-K_{1}r^{\gamma}$). Here we have used that $\tau=-2\alpha/(p-1)$
and, in Case a) (1.7) for $v$, in Case b) $(H3)$ and (1.15) and in Case c)
$(H3)$. Moreover, in Case a) we have considered $K_{1}=C(k^{p}-k)$ and in
Cases b) and c) $K_{1}=C(k-1)$ for some constant $C$. Now we define
$w(x)=\frac{2\bar{M}}{V(0)}V\left(\frac{x-x_{0}}{r}\right)$
for $x\in\mathbb{R}^{N},$ where $V$ is given in (5.2), and we see that
(5.8) $w(x_{0})=2\bar{M}$
and
(5.9) $(-\Delta)^{\alpha}w\leq\frac{2\bar{M}}{V(0)}r^{-2\alpha},\ \ \ \
\mbox{in}\ \ B_{r}(x_{0}).$
Since $\tau<0$ ($\gamma<-2\alpha$ in the Case c)), by Lemma 5.1 we can take
$x_{0}\in\mathcal{A}_{k}$ close to $\partial\Omega$, so that
$\frac{2\bar{M}}{V(0)}\leq K_{1}r^{\tau}\ \ \ (\ \frac{2\bar{M}}{V(0)}\leq
K_{1}r^{\gamma+2\alpha},\quad\mbox{ in Case c))}.$
From here, combining (5.7) with (5.9), we have that
$\displaystyle(-\Delta)^{\alpha}(w_{k}+w)(x)\leq 0,\ \ \ x\in
B_{r}(x_{0})\cap\mathcal{A}_{k}.$
Then, by the Maximum Principle, we obtain
(5.10) $w_{k}(x_{0})+w(x_{0})\leq\max\\{\bar{M},\sup_{x\in
B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}(w_{k}+w)\\}.$
In case we have
(5.11) $\bar{M}<\sup_{x\in B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}(w_{k}+w),$
then
(5.12) $\displaystyle w(x_{0})<(w_{k}+w)(x_{0})$ $\displaystyle\leq$
$\displaystyle\sup_{x\in B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}(w_{k}+w)(x)$
$\displaystyle\leq$ $\displaystyle\sup_{x\in
B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}w(x)=w(x_{0}),$
which is impossible. So that (5.11) is false and then, from (5.10) we get
$w(x_{0})<w_{k}(x_{0})+w(x_{0})\leq\bar{M},$
which is impossible in view of (5.8), completing the proof. $\Box$
###### Lemma 5.3.
There exists a sequence $\\{C_{n}\\}$, with $C_{n}>0$, satisfying
(5.13) $\lim_{n\to+\infty}C_{n}=0$
and such that for all $x_{0}\in\mathcal{A}_{k}$ and $k>1$ we have
$\displaystyle 0<\int_{Q_{n}}\frac{w_{k}(z)-M_{n}}{|z-x|^{N+2\alpha}}dz\leq
C_{n}r^{\tau-2\alpha},\ \ \forall x\in B_{r}(x_{0}),$
where we consider $r=d(x_{0})/2$, $Q_{n}=\\{z\in
A_{r/n}\,/\,w_{k}(z)>M_{n}\\}$ and $M_{n}=\max_{x\in\Omega\setminus
A_{r/n}}w_{k}(x).$
Proof. In Case a): we see that $Q_{n}\subset A_{r/n}$ and
$\lim_{n\to+\infty}|Q_{n}|=0$, so that using (1.10) we directly obtain
$\displaystyle\int_{Q_{n}}\frac{w_{k}(z)-M_{n}}{|z-x|^{N+2\alpha}}dz$
$\displaystyle\leq$ $\displaystyle
C_{0}r^{-N-2\alpha}\int_{A_{r/n}}d(z)^{\tau}dz$ $\displaystyle\leq$
$\displaystyle
Cr^{-N-2\alpha}\int_{0}^{r/n}t^{\tau}t^{N-1}dt\leq\frac{C}{n^{N+\tau}}r^{\tau-2\alpha},$
where $C$ depends on $C_{0}$ and $\partial\Omega$. We complete the proof
defining $C_{n}=\frac{C}{n^{N+\tau}}$.
In Case b) we argue similarly using (1.16) and define $C_{n}$ as before, while
in Case c) we argue similarly using (1.19), but defining
$C_{n}=\frac{C}{n^{N+\gamma/p}}$. $\Box$
Now we are in a position to prove our non-existence results.
Proof of uniqueness results in Cases a), b) and c). We assume that
${\mathcal{A}}\not=\O$, then there exists $k>1$ such that
${\mathcal{A}}_{k}\not=\O$. By Lemma 5.2 there exists
$x_{0}\in{\mathcal{A}}_{k}$ such that
$w_{k}(x_{0})=\max\\{w_{k}(x)\,/\,x\in\Omega\setminus A_{d(x_{0})}\\}.$
Proceeding as in Lemma 5.2 with the function
$w(x)=\frac{K_{1}}{2}r^{\tau}V(\frac{x-x_{0}}{r})$ $\mbox{and}\quad
w(x)=\frac{K_{1}}{2}r^{\gamma+2\alpha}V(\frac{x-x_{0}}{r}),\mbox{ in Case
c)},$
we see that
(5.14) $(-\Delta)^{\alpha}(w_{k}+w)(x)\leq-\frac{K_{1}}{2}r^{\tau-2\alpha},\ \
\ x\in B_{r}(x_{0})\cap\mathcal{A}_{k}.$ (5.15)
$\mbox{and}\quad(-\Delta)^{\alpha}(w_{k}+w)(x)\leq-\frac{K_{1}}{2}r^{\gamma},\mbox{
in Case c).}$
With $M_{n}$, as given in Lemma 5.3, we define
(5.16) $\bar{w}_{n}(x)=\left\\{\begin{array}[]{ll}(w_{k}+w)(x),&\mbox{if}\ \
w_{k}(x)\leq M_{n},\\\\[5.69054pt] M_{n},&\mbox{if}\ \
w_{k}(x)>M_{n},\end{array}\right.$
for $n>1$. By Lemma 5.3 we find $n_{0}$ such that
$\displaystyle(-\Delta)^{\alpha}\bar{w}_{n_{0}}(x)$ $\displaystyle=$
$\displaystyle(-\Delta)^{\alpha}(w_{k}+w)(x)+2\int_{Q_{n_{0}}}\frac{w_{k}(z)-M_{n_{0}}}{|z-x|^{N+2\alpha}}dz$
$\displaystyle\leq$ $\displaystyle 0,\quad\mbox{in}\quad
B_{r}(x_{0})\cap\mathcal{A}_{k}.$
In Case b) we have use (1.15) and in Case c) we have use (1.17), to get
similar conclusion. Then, by the Maximum Principle, we get
$\bar{w}_{n_{0}}(x_{0})\leq\max\\{M_{n_{0}},\sup_{x\in
B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}(w_{k_{0}}+w)\\}.$
Using the same argument as in (5.12), we conclude that
$\sup_{x\in B_{r}(x_{0})\cap\mathcal{A}_{k}^{c}}(w_{k_{0}}+w)>M_{n_{0}}$
does not hold and therefore
(5.17) $\bar{w}_{n_{0}}(x_{0})=w_{k}(x_{0})+w(x_{0})\leq M_{n_{0}}.$
Next, by the definition of $M_{n}$, we choose $x_{1}\in\Omega\setminus
A_{r/n_{0}}$ such that $w_{k}(x_{1})=M_{n_{0}}$. But then we have
$w_{k}(x_{0})+w(x_{0})\geq w(x_{0})=\frac{K_{1}}{2}V(0)r^{\tau}\quad\mbox{in
Case a) and b)}$ $\quad\mbox{and }\quad w_{k}(x_{0})+w(x_{0})\geq
w(x_{0})=\frac{K_{1}}{2}V(0)r^{\gamma+2\alpha}\quad\mbox{in Case c)}.$
Thus, by the asymptotic behavior of $v$, (1.6) in Case a), (1.15) in Case b)
and (1.17) in Case c), we have
$r^{\tau}\geq n_{0}^{\tau}Cv(x_{1})\quad\mbox{and}\quad r^{\gamma+2\alpha}\geq
r^{\gamma/p}\geq n_{0}^{\gamma/p}Cv(x_{1})\quad\mbox{in Case c)}.$
We recall that in Case a) $K_{1}=C(k^{p}-k)$, so from (5.17)
(5.18) $u(x_{1})>(1+c_{0})kv(x_{1}),$
where $c_{0}>0$ is a constant, not depending on $x_{0}$ and increasing in $k$.
Now we repeat this process above initiating by $x_{1}$ and $k_{1}=k(1+c_{0})$.
Proceeding inductively, we can find a sequence $\\{x_{m}\\}\subset\mathcal{A}$
such that
$u(x_{m})>(1+c_{0})^{m}kv(x_{m}),$
which contradicts the common asymptotic behavior of $u$ and $v$.
In the Case b) and c) recall that $K_{1}=C(k-1)$ and, as before, we can
proceed inductively to find a sequence $\\{x_{m}\\}\subset\mathcal{A}$ such
that
$u(x_{m})>(k+mc_{0})v(x_{m}),$
which again contradicts the common asymptotic behavior of $u$ and $v$. $\Box$
## 6\. Proof of our non-existence results
In this section we prove our non-existence results. Our arguments are based on
the construction of some special super and sub-solutions and some ideas used
in Section §5. The main portion of our proof is based on the following
proposition that we state and prove next.
###### Proposition 6.1.
Assume that $\Omega$ is an open, bounded and connected domain of class
$C^{2}$, $\alpha\in(0,1)$, $p>1$ and $f$ is nonnegative. Suppose that $U$ is a
sub or super-solution of (1) satisfying $U=0$ in $\Omega^{c}$ and (1.10) for
some $\tau\in(-1,0)$. Moreover, if $\tau>-\frac{2\alpha}{p-1}$, assume there
are numbers $\epsilon>0$ and $\delta>0$ such that, in case $U$ is a sub-
solution of (1),
(6.1) $(-\Delta)^{\alpha}U(x)\leq-\epsilon
d(x)^{\tau-2\alpha}\quad\mbox{or}\quad f(x)\geq\epsilon
d(x)^{\tau-2\alpha},\quad\mbox{for }x\in A_{\delta},$
and in case $U$ is a super-solution of (1),
(6.2) $(-\Delta)^{\alpha}U(x)\geq\epsilon d(x)^{\tau-2\alpha}\ \ \mbox{and}\ \
f(x)\leq\frac{\epsilon}{2}d(x)^{\tau-2\alpha},\quad\mbox{for }x\in
A_{\delta}.$
Then there is no solution $u$ of (1) such that, in case $U$ is a sub-solution,
(6.3) $\displaystyle 0<\liminf_{x\in\Omega,\
x\to\partial\Omega}u(x)d(x)^{-\tau}$ $\displaystyle\leq$
$\displaystyle\limsup_{x\in\Omega,\ x\to\partial\Omega}u(x)d(x)^{-\tau}$
$\displaystyle<$ $\displaystyle\liminf_{x\in\Omega,\
x\to\partial\Omega}U(x)d(x)^{-\tau}$
or in case $U$ is a super-solution,
(6.4) $\displaystyle 0<\limsup_{x\in\Omega,\
x\to\partial\Omega}U(x)d(x)^{-\tau}$ $\displaystyle<$
$\displaystyle\liminf_{x\in\Omega,\ x\to\partial\Omega}u(x)d(x)^{-\tau}$
$\displaystyle\leq$ $\displaystyle\limsup_{x\in\Omega,\
x\to\partial\Omega}u(x)d(x)^{-\tau}<\infty.$
We prove this proposition by a contradiction argument, so we assume that $u$
is a solution of (1) satisfying (6.3) or (6.4), depending on the fact that $U$
is a sub-solution or a super-solution. Since $f$ is non-negative we have that
$u>0$ in $\Omega$ and by our assumptions on $U$, there is a constant
$C_{0}\geq 1$ so that, in case $U$ is a sub-solution
(6.5) $C_{0}^{-1}\leq u(x)d(x)^{-\tau}<U(x)d(x)^{-\tau}\leq C_{0},\ \ x\in
A_{\delta}\ \ $
and, in case $U$ is a super-solution
(6.6) $C_{0}^{-1}\leq U(x)d(x)^{-\tau}<u(x)d(x)^{-\tau}\leq C_{0},\ \ x\in
A_{\delta}.$
Here $\delta$ is decreased if necessary so that (6.1), (6.2), (6.5) and (6.6)
hold. We define
(6.7) $\pi_{k}(x)=\left\\{\begin{array}[]{lll}U(x)-ku(x),&\mbox{in\ case
}U\mbox{ is a sub-solution},\\\\[5.69054pt] u(x)-kU(x),&\mbox{in\ case
}U\mbox{ is a super-solution},\end{array}\right.$
where $k\geq 0$. In order to prove Proposition 6.1, we need the following two
preliminary lemmas.
###### Lemma 6.1.
Under the hypotheses of Proposition 6.1. If
$\mathcal{A}_{k}=\\{x\in\Omega\,/\,\pi_{k}(x)>0\\}\not=\O,$ for $k>1$. Then,
(6.8) $\partial\mathcal{A}_{k}\cap\partial\Omega\not=\O.$
The proof of this lemma follows the same arguments as the proof of Lemma 5.1
so we omit it.
###### Lemma 6.2.
Under the hypotheses of Proposition 6.1. If $\mathcal{A}_{k}\not=\O,$ for
$k>1$, then
(6.9) $\sup_{x\in\Omega}\pi_{k}(x)=+\infty.$
Proof. If (6.9) fails, then we have $M=\sup_{x\in\Omega}\pi_{k}(x)<+\infty.$
We see that $M>0$ and, as in Lemma 5.2, there is no point $\bar{x}\in\Omega$
achieving $M$. By Lemma 6.1 we may choose $x_{0}\in\mathcal{A}_{k}$ and
$r=d(x_{0})/4$ such that $B_{r}(x_{0})\subset A_{\delta}$, where $r$ could be
chosen as small as we want. Here $\delta$ is as in (6.1) and (6.2).
In what follows we consider $x\in B_{r}(x_{0})\cap\mathcal{A}_{k}$ and we
notice that $3r<d(x)<5r$. We first analyze the case $U$ is a sub-solution and
$\tau\leq-\frac{2\alpha}{p-1}$. We have
$\displaystyle(-\Delta)^{\alpha}\pi_{k}(x)$ $\displaystyle\leq$
$\displaystyle-U^{p}(x)+ku^{p}(x)-(k-1)f(x)$ $\displaystyle\leq$
$\displaystyle-(k^{p-1}-1)ku^{p}(x)$ $\displaystyle\leq$
$\displaystyle-C^{-p}_{0}(k^{p-1}-1)kd(x)^{\tau p}\leq-K_{1}r^{\tau-2\alpha},$
where we have used $f\geq 0$, $k>1$, (6.5),
$K_{1}=5^{\tau-2\alpha}C^{-p}_{0}(k^{p-1}-1)k>0$ and $C_{0}$ is taken from
(6.5). Next we consider the case $U$ is a sub-solution and
$\tau>-\frac{2\alpha}{p-1}$. By the first inequality in (6.1), we have
$\displaystyle(-\Delta)^{\alpha}\pi_{k}(x)$ $\displaystyle\leq$
$\displaystyle-\epsilon d(x)^{\tau-2\alpha}+ku^{p}(x)-kf(x)$
$\displaystyle\leq$ $\displaystyle-(\epsilon-kC^{p}_{0}r^{2\alpha-\tau+\tau
p})d(x)^{\tau-2\alpha}\leq-K_{1}r^{\tau-2\alpha},$
where the last inequality is achieved by choosing $r$ small enough so that
$(\epsilon-kC^{p}_{0}r^{2\alpha-\tau+\tau p})\geq\frac{\epsilon}{2}$ and
$K_{1}=5^{\tau-2\alpha}\frac{\epsilon}{2}$. On the other hand, if the second
inequality in (6.1) holds, we have
$\displaystyle(-\Delta)^{\alpha}\pi_{k}(x)$ $\displaystyle\leq$ $\displaystyle
ku^{p}(x)-(k-1)\epsilon d(x)^{\tau-2\alpha}$ $\displaystyle\leq$
$\displaystyle-((k-1)\epsilon-kC^{p}_{0}r^{2\alpha-\tau+\tau
p})d(x)^{\tau-2\alpha}\leq-K_{1}r^{\tau-2\alpha},$
where $r$ satisfies $(k-1)\epsilon-kC^{p}_{0}r^{2\alpha-\tau+\tau
p}\geq\frac{k-1}{2}\epsilon$ and
$K_{1}=5^{\tau-2\alpha}\frac{k-1}{2}\epsilon$.
In case $U$ is a super-solution and $\tau\leq-\frac{2\alpha}{p-1}$, we argue
similarly to obtain
$\displaystyle(-\Delta)^{\alpha}\pi_{k}(x)$ $\displaystyle\leq$
$\displaystyle-u^{p}(x)+kU_{1}^{p}(x)-(k-1)f(x)\leq-K_{1}r^{\tau-2\alpha},$
where $K_{1}=5^{\tau-2\alpha}C^{-p}_{0}(k^{p-1}-1)k>0$. Finally, in case $U$
is a super-solution and $\tau>-\frac{2\alpha}{p-1}$, using (6.2) we find
$\displaystyle(-\Delta)^{\alpha}\pi_{k}(x)$ $\displaystyle\leq$
$\displaystyle-u^{p}(x)-k\epsilon d(x)^{\tau-2\alpha}+f(x)\leq-
K_{1}r^{\tau-2\alpha},$
with $K_{1}=5^{\tau-2\alpha}\frac{k}{2}\epsilon>0$. Thus, in all cases we have
obtained
(6.10) $(-\Delta)^{\alpha}\pi_{k}(x)\leq-K_{1}r^{\tau-2\alpha},\ \ x\in
B_{r}(x_{0})\cap\mathcal{A}_{k},$
for some $K_{1}=K_{1}(k)>0$ non-decreasing with $k$. From here we can argue as
in Lemma 5.2 to get a contradiction $\Box$
Now proof of Proposition 6.1 is easy.
Proof of Proposition 6.1. From (6.10), recalling that $K_{1}$ non-decreasing
with $k$, we can argue as in the proof of uniqueness result in Case b) to get
a sequence $(x_{m})$ in $A_{\delta}$ such that, for some $k_{0}>1$ and
$\bar{k}>0$, in case $U$ is a sub-solution we have
$U(x_{m})>(k_{0}+m\bar{k})u(x_{m})\ $
and, in case $U$ is a super-solution we have
$u(x_{m})>(k_{0}+m\bar{k})U(x_{m}).$
From here we obtain a contradiction with (6.5) or (6.6), for $m$ large. $\Box$
Proof of non-existence part of Theorem 1.1. For any $t>0$ we construct a sub-
solution or super-solution $U$ of (1) such that
(6.11) $\lim_{x\in\Omega,x\to\partial\Omega}U(x)d(x)^{-\tau}=t,$
and $U$ satisfies the assumption of Proposition 6.1, for different
combinations of the parameters $p$ and $\tau$. For $t>0$ and
$\mu\in\mathbb{R}$ we define
(6.12) $U_{\mu,t}=tV_{\tau}+\mu V_{0}\ \ \mbox{in}\ \ \mathbb{R}^{N},$
where $V_{0}=\chi_{\Omega}$ is the characteristic function of $\Omega$ and
$V_{\tau}$ is defined in (3.4). It is obvious that (6.11) holds for
$U_{\mu,t}$ for any $\mu\in\mathbb{R}$. To complete proof we show that for any
$t>0$, there is $\mu(t)$ such that $U_{\mu(t),t}$ is a sub-solution or super-
solution of (1), depending on the zone to which $(p,\tau)$ belongs.
Zone 1: We consider $p>1$ and $\tau\in(\tau_{0}(\alpha),0)$. By Proposition
3.2 $(ii)$, there exist $\delta_{1}>0$ and $C_{1}>0$ such that
(6.13) $(-\Delta)^{\alpha}V_{\tau}(x)>C_{1}d(x)^{\tau-2\alpha},\ \ x\in
A_{\delta_{1}}.$
Combining with $(H2^{*})$, for any $\mu>0$, there exists $\delta_{1}>0$
depending on $t$ such that
$(-\Delta)^{\alpha}U_{\mu,t}(x)+U_{\mu,t}^{p}(x)-f(x)>C_{1}td(x)^{\tau-2\alpha}-Cd(x)^{-2\alpha}\geq
0,\ \ x\in A_{\delta_{1}}.$
On the other hand, since $V_{\tau}$ is of class $C^{2}$, $f$ is continuous in
$\Omega$ and $\Omega\setminus A_{\delta_{1}}$ is compact, there exists
$C_{2}>0$ such that
(6.14) $|f|,\ |(-\Delta)^{\alpha}V_{\tau}(x)|\leq C_{2},\ \
x\in\Omega\setminus A_{\delta_{1}}.$
Then, using (4.3), there exists $\mu>0$ such that
(6.15)
$(-\Delta)^{\alpha}U_{\mu,t}(x)+U_{\mu,t}^{p}(x)-f(x)>-2C_{2}+C_{0}\mu\geq 0,\
\ x\in\Omega\setminus A_{\delta_{1}}.$
We conclude that for any $t>0$, there exists $\mu(t)>0$ such that
$U_{\mu(t),t}$ is a super-solution of (1) and, by $(H2^{*})$ and (6.13), it
satisfies (6.2).
Zone 2: We consider $p>1+2\alpha$ and $\tau\in(-1,-\frac{2\alpha}{p-1})$. By
Proposition 3.2 $(i)$ and $(ii)$, there exists $\delta_{1}>0$ depending on $t$
such that
(6.16) $(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)-f(x)\geq-
C_{1}td(x)^{\tau-2\alpha}+t^{p}d(x)^{\tau p}-Cd(x)^{-2\alpha}\geq 0,$
for $x\in A_{\delta_{1}}$ and for any $\mu>0$, where we used that
$0>\tau-2\alpha>\tau p$. On the other hand, for $x\in\Omega\setminus
A_{\delta_{1}}$, (6.15) holds for some $\mu>0$ and so we have constructed a
super-solution of (1).
Zone 3: We consider $1+2\alpha<p\leq 1-\frac{2\alpha}{\tau_{0}(\alpha)}$ and
$\tau\in(-\frac{2\alpha}{p-1},\tau_{0}(\alpha))$, which implies that $\tau
p>\tau-2\alpha$. By Proposition 3.2 $(i)$ and $f\geq 0$ in $\Omega$, there
exists $\delta_{1}>0$ so that for all $\mu\leq 0$
(6.17) $(-\Delta)^{\alpha}U_{\mu,t}(x)+U_{\mu,t}^{p}(x)-f(x)\leq-
C_{1}td(x)^{\tau-2\alpha}+t^{p}d(x)^{\tau p}\leq 0,$
for $x\in A_{\delta_{1}}$. Then, using (4.3) and (6.14), there exists
$\mu=\mu(t)<0$ such that
(6.18)
$(-\Delta)^{\alpha}U_{\mu,t}(x)+U_{\mu,t}^{p}(x)-f(x)<2C_{2}+C_{0}\mu\leq 0,\
\ x\in\Omega\setminus A_{\delta_{1}}.$
We conclude that for any $t>0$, there exists $\mu(t)<0$ such that
$U_{\mu(t),t}$ is a sub-solution of (1) and it satisfies (6.1).
We see that Zone 1, 2 and 3 cover the range of parameters in part $(i)$ of
Theorem 1.1, completing the proof in the case.
Zone 4: To cover part (ii) of Theorem 1.1 we only need to consider
$p=1-\frac{2\alpha}{\tau_{0}(\alpha)}$ with
$\tau=\tau_{0}(\alpha)=-\frac{2\alpha}{p-1}$, which implies that $\tau
p=\tau-2\alpha<\min\\{\tau-2\alpha+\tau+1,\tau\\}$. By Proposition 3.2
$(iii)$, there exists $\delta_{1}>0$ depending on $t$ such that
$\displaystyle(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)-f(x)$
$\displaystyle\geq$ $\displaystyle-
C_{1}td(x)^{\min\\{\tau-2\alpha+\tau+1,\tau\\}}+t^{p}d(x)^{\tau p}$
$\displaystyle-Cd(x)^{-2\alpha}\geq 0,\ \ \ x\in A_{\delta_{1}}$
for any $\mu>0$. For $x\in\Omega\setminus A_{\delta_{1}}$, (6.15) holds for
some $\mu>0$, so we have constructed a super-solution of (1).
We see that Zones 1, 2 and 4 cover the parameters in part $(ii)$ of Theorem
1.1, so the proof is complete in this case too.
Zone 5: We consider $1<p\leq 1+2\alpha$ and $\tau\in(-1,\tau_{0}(\alpha))$,
which implies that $\tau p>\tau-2\alpha$. By Proposition 3.2 $(i)$ and $f\geq
0$ in $\Omega$, there exists $\delta_{1}>0$ such that for all $\mu\leq 0$ and
$x\in A_{\delta_{1}}$, inequality (6.17) holds. Then, using (4.3) and (6.14),
there exists $\mu=\mu(t)<0$ such that (6.18) holds and we conclude that for
any $t>0$, there exists $\mu(t)<0$ such that $U_{\mu(t),t}$ satisfies the
first inequality of (6.1) and it is a sub-solution of (1).
We see that Zones 1 and 5 cover the parameters in part $(iii)$ of Theorem 1.1.
This completes the proof. $\Box$
Proof of Theorem 1.4. Here again we construct sub or super-solutions
satisfying Proposition 6.1 to prove the theorem. In the case of a weak source,
that is, part $(i)$ of Theorem 1.4, we have $p\geq
1-\frac{2\alpha}{\tau_{0}(\alpha)}$ and
$-2\alpha-\frac{2\alpha}{p-1}\leq\gamma<-2\alpha$, which implies that
$-1<\tau_{0}(\alpha)\leq-\frac{2\alpha}{p-1}\leq\gamma+2\alpha<0$. We consider
two zones depending on $\tau$.
Zone 1: we consider $\tau\in(\gamma+2\alpha,0)$, so we have $\gamma<\tau p$
and $\gamma<\tau-2\alpha$. By Proposition 3.2 $(ii)$ and $(H3)$, we have that,
for any $t>0$ there exist $\delta_{1}>0$, $C_{1}>0$ and $C_{2}>0$ such that
(6.19) $(-\Delta)^{\alpha}U_{\mu,t}(x)+U_{\mu,t}^{p}(x)-f(x)\leq
C_{1}td(x)^{\tau-2\alpha}+t^{p}d(x)^{\tau p}-C_{2}d(x)^{\gamma}\leq 0,$
for $x\in A_{\delta_{1}}$ and any $\mu\leq 0$. On the other hand, using (4.3)
and (6.14) we find $\mu=\mu(t)<0$ such that (6.18) holds for
$x\in\Omega\setminus A_{\delta_{1}}.$ We conclude that for any $t>0$, there
exists $\mu(t)<0$ such that $U_{\mu(t),t}$ is is a sub-solution of (1) and by
$(H3)$, it satisfies (6.1).
Zone 2: we consider $\tau\in(-1,\gamma+2\alpha)$. For
$\tau\in(\tau_{0}(\alpha),\gamma+2\alpha)$ in case
$\tau_{0}(\alpha)<\gamma+2\alpha$, by Proposition 3.2 $(i)$ there exists
$\delta_{1}>0$, depending on $t$, such that
(6.20) $(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)-f(x)\geq
C_{1}td(x)^{\tau-2\alpha}-C_{2}d(x)^{\gamma}\geq 0,$
for $x\in A_{\delta_{1}}$ and any $\mu\geq 0$. For
$\tau\in(-1,\tau_{0}(\alpha)]\cap(-1,\gamma+2\alpha)$, we have $\tau p<\gamma$
and $\tau p<\tau-2\alpha$, so by Proposition 3.2 $(ii)$ and $(iii)$, there
exists $\delta_{1}>0$ dependent of $t$ such that (6.16) holds for any $\mu\geq
0$, while for $x\in\Omega\setminus A_{\delta_{1}}$, (6.15) holds for some
$\mu>0$. We conclude that for any $t>0$, there exists $\mu(t)>0$ such that
$U_{\mu(t),t}$ is a super-solution of (1) and by $(H3)$ it satisfies (6.2),
completing the proof in the weak source case.
Next we consider the case of strong source, that is part $(ii)$ of Theorem
1.4. Here we have that
$-1<\frac{\gamma}{p}<-\frac{2\alpha}{p-1}<0.$
Here again we have two zones, depending on the parameter $\tau$.
Zone 1: we consider $\tau\in(\frac{\gamma}{p},0)$, in which case we have
$\tau-2\alpha>\gamma$ and $\tau p>\gamma$. Then there exist $\delta_{1}>0$,
$C_{1}>0$ and $C_{2}>0$ such that (6.19) holds for any $\mu\leq 0$ and using
(4.3) and (6.14), there exists $\mu=\mu(t)<0$ such that (6.18) holds for
$x\in\Omega\setminus A_{\delta_{1}}.$ Thus, for any $t>0$ there exists
$\mu(t)<0$ such that $U_{\mu(t),t}$ is a sub-solution of (1) and $(H3)$
implies the first inequality of (6.1).
Zone 2: we consider $\tau\in(-1,\frac{\gamma}{p})$, in which case we have
$\tau p<\tau-2\alpha$ and $\tau p<\gamma$. Then there exist $\delta_{1}>0$,
$C_{1}>0$ and $C_{2}>0$ such that (6.20) holds for $x\in A_{\delta_{1}}$ and
$\mu\geq 0$. We see also that for $x\in\Omega\setminus A_{\delta_{1}}$,
inequality (6.15) holds for some $\mu>0$and so for any $t>0$, there exists
$\mu(t)>0$ such that $U_{\mu(t),t}$ is a super-solution of (1).
This completes the proof of the theorem. $\Box$
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huyuan chen, Patricio Felmer, Alexander Quaas",
"submitter": "Huyuan Chen",
"url": "https://arxiv.org/abs/1311.6044"
}
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1311.6118
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# A Large Scale Pattern from Optical Quasar Polarization Vectors
Richard Shurtleff affiliation and mailing address: Department of Science,
Wentworth Institute of Technology, 550 Huntington Avenue, Boston, MA, 02115,
USA, telephone: (617) 989-4338, FAX: 617-989-4010, e-mail: [email protected]
###### Abstract
Based on a published catalog of 355 quasars with significant optical linear
polarization, it is shown here that the distribution of polarization
directions is skewed, preferentially toward one location in the sky and away
from a second. To show this, we calculate the average polarization angle as a
function of position. The function makes a clean quadrupole on the sky
offering the opportunity to apply multipoles including their spherical
harmonic, Maxwell vector and symmetric tensor representations. The evidence
suggests that observed polarization directions of optical quasars are not
independent over very large angular scales, thereby confirming similar
conclusions by others.
Keywords: Quasars: general Polarization Large scale structure
PACS: 98.54.-h, 42.25.Ja
## 1 Introduction
Given a quasar at position $Q$ on the sky and some other position $H$ on the
sky, the direction ‘toward $H$’ at $Q$ is along the great circle connecting
$Q$ to $H.$ A polarization vector $V$ at $Q$ makes an angle $\eta$ with the
direction toward $H.$ The angle $\eta$ is the polarization angle referenced to
position $H.$ Given a collection of polarized quasars, we can see if there is
a tendency for their polarization vectors to skew toward $H$ by averaging the
angles $\eta.$
We use a published catalog of $N$ = 355 significantly polarized optical
quasars (QSOs).[1, 2] Each quasar is listed with its known position $Q_{i}$
and polarization angle referenced to North $\theta_{pi},$ $i\in$
$\\{1,...,355\\}.$ For any given position $H,$ we can calculate the angles
$\eta_{i}$ that the polarization vectors make with respect to the direction
toward $H.$ The function we investigate is the average polarization angle
$\eta_{\mathrm{355}}(H),$ averaged over all 355 of the QSOs in the catalog. We
find that the average polarization angle function $\eta_{\mathrm{355}}(H)$ has
both maxima and minima at various positions $H$ in the sky; the polarization
vectors skewing toward some positions and away from others.
As described in Ref. 1 and 2, the catalog was compiled over the course of
three papers investigating possible nearest-neighbor alignments of optical
polarization vectors. The QSOs selected are located at high-latitudes
$>30^{\circ}$ in galactic coordinates and have significant $>0.6\%$ linear
polarization with well-determined polarization directions with uncertainties
of less that $14^{\circ}.$ “If we assume that the field star polarization
correctly represents the interstellar polarization affectingmore distant
objects, then interstellar polarization in our Galaxy was shown to have little
effect on the polarization angle distribution of significantly polarized
$(p>0.6\%)$ quasars.”[1, 3]
The researchers found interesting activity in roughly 20% of the sky, they dub
regions “A1” and “A3”, which led them to emphasize those regions. The result
is a catalog with half the objects covering regions A1 and A3, about 20% of
the sky. A more extensive collection would better suit our purposes here.
Motivation for the calculations in this article started with reports of
unexpected alignments of the quadrupole, octupole and other low multipoles of
the Cosmic Microwave Background temperature field.[4, 5, 6] These multipoles
have an uncanny affinity for the Ecliptic.
QSOs are far away, though not as far as the CMB sources, and their
polarization vectors point in specific directions. One can toy with the
hypothesize that the polarization vectors of QSOs could favor the direction of
the Ecliptic. To test this, the function $\eta_{\mathrm{355}}(H)$ was
constructed that measures the average polarization angle toward a given
direction. The test successfully finds preferred directions that show
interesting alignments, but with the Equatorial coordinate system, not with
the Ecliptic plane.
We find that the deflection of polarization vector effect is dictated by the
QSOs in regions A1 and A3 where previous researchers found mutual alignments.
The sparsely covered other $80\%$ of the sky with less than half the QSOs in
the catalog produces a pattern that has some indications of an effect, but the
pattern is too close to random results to be significant. More data over wider
regions of the sky is needed to see if polarization vectors are skewed in any
direction over extremely large scales.
The function $\eta_{{\mathrm{355}}}(H)$ forms a neat quadrupole pattern
superimposed on a constant average value $\bar{\eta}_{{\mathrm{355}}},$ the
monopole. The function is symmetric about the origin by construction, implying
that there can be no contributions from a dipole or octupole since these are
odd functions of $H.$ In addition to the spherical harmonics,[8] we evaluate
the pattern with Maxwell vectors[9, 10, 11] and in terms of a symmetric,
traceless, second-rank tensor.[12]
One can understand more about the pattern by looking at the distributions of
the 355 polarization angles at various positions $H.$
For an ordinary position $H$ away from both the maximum and the minimum
regions, the distribution of the 355 angles $\eta_{i}$ is close to the uniform
distribution of 355 evenly spaced angles from $0^{\circ}$ to $90^{\circ}.$ See
Fig. 4. So, away from the extrema of the function, the distributions are
consistent with a random distribution of polarization directions.
At $H_{\mathrm{min}}$ and its diametrically opposite position, where the
polarization function is a minimum, one finds the greatest deviation from the
straight-line uniform distribution. The deflection is not hap-hazard, but
distorts the straight line into a parabolic curve. See Fig. 5. Similarly, at
the function’s maxima, $H_{\mathrm{max}}$ and $-H_{\mathrm{max}}$, the
distribution arches above the straight line of the uniform distribution. The
parabolic shape itself suggests that there is a physical explanation.
In Ref. 1, the alignment of nearest neighbor QSOs was investigated. They found
evidence for a large scale mechanism affecting the polarization in transit. In
this article, the quadrupole pattern of the function
$\eta_{{\mathrm{355}}}(H)$ also suggests some large scale mechanism is
influencing the polarization directions. However Ref. 1 focused on subsets
consisting of neighboring QSOs, while the investigation here is catalog-wide
and sky-wide. Different approaches yield similar results.
In Section 2, we discuss how the 355 polarization vectors located at 355 QSOs
form the polarization angle function. In Sec. 3, we analyze the multipole
expansion of the polarization angle function. We calculate the parameters
needed to represent the function by spherical harmonics, by Maxwell vectors,
and as a symmetric traceless tensor. All three representations simplify in a
preferred coordinate system. In Sec. 4, we describe the distribution of
polarization angles at positions in the sky where function has a near-average
value, as well as at the positions where the function has minimum and maximum
values. In Sec. 5, we see that the sharpest pattern originates with the 183
QSOs in the favored regions A1 and A3. The pattern from the 172 other QSOs
hovers around random. More data from the sky outside of A1 and A3 is needed to
determine whether there is a large scale effect in that part of the sky.
## 2 The polarization angle function
For each QSO in the catalog, the listed polarization angle $\theta$ is the
angle counterclockwise from local North to the polarization direction. North
is just one position in the sky. We can calculate the polarization angle
referenced to any position $H.$ Then averaging over all QSOs gives a function
of position in the sky that we can use to see if the polarization directions
favor any particular region of the sky.
At the QSO the direction toward some position $H$ is along the great circle
connecting the QSO with $H.$ See fig. 1. We determine the angles
$\eta_{i}(H),$ $i\in$ $\\{1,...,N\\},$ between the polarization direction and
the direction toward $H$ for every QSO in the catalog. See Fig. 2. The average
of the 355 angles forms the function $\eta_{{\mathrm{355}}}(H),$ a measure of
the tendency of the polarization vectors to point toward the position $H$ in
the sky.
Because the great circle that contains $H$ also contains its diametrically
opposed position, the function $\eta_{{\mathrm{355}}}(H)$ is symmetric about
the origin by construction, $\eta_{{\mathrm{355}}}(H)$ =
$\eta_{{\mathrm{355}}}(-H)$.
We outline the calculation for clarity. Denote the direction from the origin
(Earth) to a position on the sky by a unit 3-vector in rectangular equatorial
coordinates,
$\hat{r}=\hat{r}(\alpha,\delta)=\\{\hat{r}_{x},\hat{r}_{y},\hat{r}_{z}\\}=\\{\cos{\alpha}\cos{\delta},\sin{\alpha}\cos{\delta},\sin{\delta}\\}\;,$
(1)
here $\alpha$ and $\delta$ are the Right Ascension and Declination in the
Equatorial coordinate system.
Let the position $H$ in the sky be the unit vector $\hat{r}_{H}$ =
$\hat{r}(\alpha_{H},\delta_{H})$ and let the $i$th QSO be in the direction of
the unit vector $\hat{r}_{i}$ = $\hat{r}(\alpha_{i},\delta_{i}).$ Denote by
$\phi_{Hi}$ the angle between the two directions $\hat{r}_{H}$ and
$\hat{r}_{i};$ we have $\cos{\phi_{Hi}}$ = ${\hat{r}_{H}\cdot\hat{r}_{i}}.$ It
follows that at the $i$th QSO on the sky the unit tangent vector
$\hat{s}_{Hi}$ along the great circle toward $H$ is
$\hat{s}_{Hi}=\frac{1}{\sin{\phi_{Hi}}}\,\hat{r}_{H}-\frac{1}{\tan{\phi_{Hi}}}\,\hat{r}_{i}\;.$
(2)
We can show this quickly: Clearly, as a sum over $\hat{r}_{H}$ and
$\hat{r}_{i},$ $\hat{s}_{Hi}$ is in the plane of the great circle. The scalar
product $\hat{s}_{Hi}\cdot\hat{r}_{i}$ = 0, so $\hat{s}_{Hi}$ is perpendicular
to $\hat{r}_{i}.$ Finally, a short calculation shows that
$\hat{s}_{Hi}\cdot\hat{s}_{Hi}$ = 1 and so $\hat{s}_{Hi}$ is a unit vector.
Local North at the $i$th QSO is the vector $\hat{s}_{Ni}$ in (2) with North
the direction $\\{0,0,1\\}$ and the angle between the QSO and North is
$\phi_{Ni}$ = $90^{\circ}-\delta_{i},$ where $\delta_{i}$ is the declination
for the $i$th QSO.
Given local North and the tangent vector $\hat{s}_{Hi}$ along the great circle
toward $H$ by (2), we can obtain the angle $\theta_{Hi}$ between the local
North and the tangent vector. Thus
$\cos{\theta_{Hi}}={\hat{s}_{Hi}\cdot\hat{s}_{Ni}}\;.$ (3)
Many angles have the same cosine. Let $\theta_{Hi}$ be the angle between
$0^{\circ}$ and $180^{\circ}$ measured clockwise from the local North
direction with East to the right. This matches the way polarization angles are
specified in the catalog.
Both the polarization direction at the $i$th QSO and the tangent to the great
circle toward $H$ are ‘non-oriented’ bidirectional straight lines that
intersect at the QSO. We take $\eta_{i}(H)$ to be the acute angle,
$0^{\circ}\leq$ $\eta_{i}(H)$ $\leq 90^{\circ},$ from one straight line to the
other. See fig. 2. For the case with $\theta_{Hi}$ larger than the
polarization angle $\theta_{pi}$ and with the difference less than
$90^{\circ},$ we have $\eta_{i}(H)$ = $\theta_{Hi}-\theta_{pi}\,.$ More
generally, we use
$\eta_{i}(H)\equiv\min{\\{\mid\theta_{Hi}-\theta_{pi}\mid,180^{\circ}-\mid\theta_{Hi}-\theta_{pi}\mid\\}}\;,$
(4)
which is the minimum of the two positive angles and therefore the acute angle.
A smaller value of $\eta_{i}(H)$ means the polarization and the tangent are
more nearly parallel.
Each position $H$ on the sky determines 355 angles $\eta_{i}(H),$ one for each
QSO in the catalog. We calculate the average of these 355 angles,
$\eta_{{\mathrm{355}}}(H),$
$\eta_{{\mathrm{355}}}(H)=\eta_{{\mathrm{355}}}(\alpha_{H},\delta_{H})\equiv\frac{1}{N}\sum_{i=1}^{N}\eta_{i}(H)\;,$
(5)
where the sum is over all $N$ = 355 QSOs in the catalog. The angle
$\eta_{{\mathrm{355}}}(H)$ measures how much the polarization angles at the
QSOs differ from the local direction toward $H,$ averaged over the catalog.
In practice, the value of $\eta_{{\mathrm{355}}}(\alpha,\delta)$ was
calculated every $2^{\circ}$ in $\delta$ and every
$2^{\circ}/(\cos{\delta}+0.01)$ in $\alpha.$ The factor $(\cos{\delta}+0.01)$
keeps the intervals at about $2^{\circ}$ in longitude for any latitude with a
small number 0.01 included to avoid infinities at the poles, where
$\cos{\delta}$ = $0.$ A linear interpolation of the table was used as the
function $\eta_{{\mathrm{355}}}(\alpha,\delta).$ [7]
We determine uncertainties in the calculated results by using the tabulated
uncertainties of polarization angles in the catalogue. The uncertainties in
position of the QSOs in the sky and any other sources of error are ignored.
The calculations were run 16 times in order to obtain uncertainties for the
various numerical results. In each run the 355 polarization angles
$\theta_{i}$ were varied by adding a random number $R,$ $-1\leq R\leq+1,$
times the tabulated uncertainty $\sigma_{\theta\,i},$ $\theta_{i}$ =
$\theta_{{\mathrm{best}}}+R\sigma_{\theta i}.$ The uncertainty in any
calculated result is the standard deviation of the values for the 16 runs. The
displayed value is the best value calculated using the polarization angles
listed in the catalog plus or minus the uncertainty. For example, the Right
Ascension of $H_{\mathrm{min}}$ is presented as $\alpha_{H_{\mathrm{min}}}$ =
$142.2^{\circ}\pm 6.4^{\circ},$ where $6.4^{\circ}$ is the standard deviation
of the RA in the sixteen runs: $\\{142.213,$ $142.871,$ $142.213,$ $142.871,$
$145.414,$ $166.566,$ $138.001,$ $148.532,$ $142.213,$ $148.641,$ $142.871,$
$150.547,$ $140.281,$ $142.871,$ $142.871,$ $150.547\\}.$ The value
$142.2^{\circ}$ is the best value, not the mean of the sixteen values, which
is $145.6^{\circ}.$ When the uncertainty is not written, assume the
uncertainty to be at most plus or minus one-half the least digit.
For the function $\eta_{{\mathrm{355}}}(H),$ the uncertainty
$\sigma_{\eta}(H)$ can be taken to be
$\sigma_{\eta}(H)=\left[\sum_{i=1}^{N}\left(\frac{\partial\eta_{{\mathrm{355}}}}{\partial\theta_{pi}}\right)^{2}\sigma^{2}_{pi}\right]^{1/2}=\left[\sum_{i=1}^{N}\sigma^{2}_{pi}\right]^{1/2}=0.417^{\,\circ}\;,$
(6)
the same value for any position $H$ over the whole sky.
In addition, we analyzed fake data to see what random angles would produce. In
16 runs, the observed angles $\theta_{i}$ were replaced by random values
between $0^{\circ}$ and $180^{\circ}$ keeping the 355 QSO positions fixed. The
results are needed in Sec. 5.
## 3 The Quadrupole
It is apparent from the plot of the function $\eta_{{\mathrm{355}}}(H)$ on the
sky in Fig. 3 that there is a pattern. The function has two below-average-
value regions and two above-average-value regions. Since the great circle that
contains $H$ also contains $-H,$ the function is symmetric about the origin,
$\eta_{{\mathrm{355}}}(H)$ = $\eta_{{\mathrm{355}}}(-H).$ The minima occur at
$H_{\mathrm{min}}$ and $-H_{\mathrm{min}}$ and the maxima occur at
$H_{\mathrm{max}}$ and $-H_{\mathrm{max}},$ where
$H_{\mathrm{min}}:\quad\\{{\mathrm{RA,dec}}\\}=\\{142.2^{\circ}\pm
6.4^{\circ},-31.8^{\circ}\pm 2.0\\}$ (7)
$H_{\mathrm{max}}:\quad\\{{\mathrm{RA,dec}}\\}=\\{107.6^{\circ}\pm
7.7^{\circ},49.8^{\circ}\pm 1.5^{\circ}\\}\;.$
In Fig. 3, these are indicated by minus ‘$-$’ signs and plus ‘$+$’ signs,
respectively.
The max and min values of the function $\eta_{{\mathrm{355}}}(H)$ are found to
be
${\eta_{{\mathrm{355}}}}_{\mathrm{min}}=39.72^{\circ}\pm
0.19^{\circ}\quad;\quad{\eta_{{\mathrm{355}}}}_{\mathrm{max}}=49.47^{\circ}\pm
0.18^{\circ}\;.$ (8)
The peak-to-peak range of the function is much larger than the uncertainty.
The max-min difference of the function
${\eta_{{\mathrm{355}}}}_{\mathrm{max}}-{\eta_{{\mathrm{355}}}}_{\mathrm{min}}$
is $49.47^{\circ}-39.72^{\circ}$ = $9.75^{\circ}\approx$ $20\,\sigma_{\eta}.$
Thus, the peak-to-peak range of the function is twenty times the uncertainty
in the function.
We analyze the pattern with a multipole expansion. By using great circles in
the construction of the function $\eta_{{\mathrm{355}}}(H),$ one guarantees
that the function is even in $\hat{r},$ $\eta_{{\mathrm{355}}}(\hat{r}_{H})$ =
$\eta_{{\mathrm{355}}}(-\hat{r}_{H}).$ The $l$th multipole is a homogeneous
polynomial of degree $l$ in components of $\hat{r}$, so only even $l$
multipoles should contribute. The multipoles can be represented by spherical
harmonics, by Maxwell vectors, and by symmetric traceless tensors.
Spherical Harmonics. To avoid complex numbers, the real form of the spherical
harmonics is used, denoted ‘$Y_{{\mathrm{real}}\,m}^{l}$’ and known as
‘tesseral spherical harmonics’. Essentially, sines and cosines replace the
phase factors $\exp{(im\alpha)}$ in the conventional complex-valued version,
i.e. sines for $m<$ 0 and cosines for $m>$ 0\. The five real-valued spherical
harmonics, for the quadrupole $l$ = 2,[8] are defined by
$Y_{{\mathrm{real}}\,-2}^{l=2}=-k\sin{\left(2\alpha\right)}\,\cos^{2}{\left(\delta\right)}=-2k\hat{r}_{x}\hat{r}_{y}$
(9)
$Y_{{\mathrm{real}}\,-1}^{l=2}=-k\sin{\left(\alpha\right)}\cos{\left(2\delta\right)}=-2k\hat{r}_{y}\hat{r}_{z}$
$Y_{{\mathrm{real}}\,0}^{l=2}=\frac{k}{\sqrt{3}}\left(3\sin^{2}{\left(\delta\right)}-1\right)=\frac{k}{\sqrt{3}}\left(3\hat{r}_{z}^{2}-1\right)$
$Y_{{\mathrm{real}}\,+1}^{l=2}=-k\cos{\left(\alpha\right)}\cos{\left(2\delta\right)}=-2k\hat{r}_{x}\hat{r}_{z}$
$Y_{{\mathrm{real}}\,+2}^{l=2}=+k\cos{\left(2\alpha\right)}\cos^{2}{\left(\delta\right)}=k\left(\hat{r}_{x}^{2}-\hat{r}_{y}^{2}\right)\;,$
where $k=\sqrt{15/\left(16\pi\right)}$ and we use (1) to introduce the
$\hat{r}(\alpha,\delta)$ components
$\\{\hat{r}_{x},\hat{r}_{y},\hat{r}_{z}\\}.$ These spherical harmonics are
orthogonal to all non-quadrupole poles, $l\neq 2,$ and are orthonormal among
themselves. With $m^{\prime},m\in$ $\\{-2,-1,0,1,2\\},$ we have
$\int_{\alpha=-180^{\circ}}^{180^{\circ}}\int_{\delta=-90^{\circ}}^{90^{\circ}}{Y_{{\mathrm{real}}\,m^{\prime}}^{l=2}Y_{{\mathrm{real}}\,m}^{l=2}\cos{\left(\delta\right)d\alpha
d\delta}}=\delta^{m^{\prime}m}\;,$ (10)
where the Kronecker delta $\delta^{m^{\prime}m}$ is one when the $m^{\prime}$
and $m$ are equal and zero otherwise.
Expanding the function $\eta_{{\mathrm{355}}}(H)$ =
$\eta_{{\mathrm{355}}}(\alpha,\delta)$ for the position $H(\alpha,\delta)$ as
a sum of spherical harmonics, we get
$\eta_{{\mathrm{355}}}(H)=\eta_{{\mathrm{355}}}(\alpha,\delta)=\bar{\eta}_{{\mathrm{355}}}+\sum_{m=-2}^{2}a_{2}^{m}Y_{{\mathrm{real}}\,m}^{l=2}(\alpha,\delta)+\epsilon\;,$
(11)
where we show only the monopole ($l$ = 0) and quadrupole ($l$ = 2) and lump
all other multipoles in the remainder $\epsilon$ . The remainder $\epsilon$
has a root mean square value taken over all $H$ of $\epsilon_{{\mathrm{rms}}}$
= $0.61^{\circ}$ which is just a little more than the uncertainty in
$\eta_{{\mathrm{355}}}(H),$ $\sigma_{\eta}$ = $0.42^{\circ}.$ Also, the
remainder is tiny compared with the peak-to-peak amplitude of the pattern,
$\epsilon_{{\mathrm{rms}}}$ = $0.61^{\circ}\ll$ $9.7^{\circ}.$ We ignore
$\epsilon$ in what follows.
By (10) and (11), we can determine the coefficients $a_{2}^{m}$ given the
function $\eta_{{\mathrm{355}}}(\alpha,\delta).$ We have
$a_{2}^{m}=\int_{\alpha=-180^{\circ}}^{180^{\circ}}\int_{\delta=-90^{\circ}}^{90^{\circ}}{\eta_{{\mathrm{355}}}Y_{{\mathrm{real}}\,m}^{l=2}\cos{\left(\delta\right)d\alpha
d\delta}}\;,$ (12)
which gives
$a_{2}^{-2}=-1.59^{\circ}\pm 0.20^{\circ}\;,\;a_{2}^{-1}=-7.33^{\circ}\pm
0.36^{\circ}\;,\;a_{2}^{0}=1.79^{\circ}\pm 0.34^{\circ}\;,$
$a_{2}^{+1}=3.65^{\circ}\pm 0.29^{\circ}\;,\;a_{2}^{+2}=0.46^{\circ}\pm
0.16^{\circ}\;.$ (13)
Similarly, one can calculate the constant term $\bar{\eta}_{{\mathrm{355}}},$
$\bar{\eta}_{{\mathrm{355}}}=45.0190^{\circ}\pm 0.0004^{\circ}\;.$ (14)
The five coefficients (13) and the monopole (14) determine the main features
of the function $\eta_{{\mathrm{355}}}(H)$ very accurately.
The relative importance of the quadrupole and monopole can be inferred by
computing the ‘power spectrum’ $P(l)\equiv$
$(\sum_{m}{a_{l}^{m}}^{2})/(2l+1).$ For the monopole, the average value
$\bar{\eta}_{{\mathrm{355}}}$ is proportional to $a_{0}^{0},$ $a_{0}^{0}$ =
$2\sqrt{\pi}\bar{\eta}_{{\mathrm{355}}}$ = $159.588^{\circ}.$ The coefficients
in (13) give $P(2).$ We get
$P(0)=25468.4\pm 0.5\quad;\quad P(2)=14.6\pm 1.1\;,$ (15)
both in units of degrees squared. The next even harmonic, $l$ = 4, is part of
the remainder $\epsilon$ that we ignore. It has a power $P(4)$ = 0.4 degrees
squared, so $P(4)$ is small compared to the quadrupole $P(2).$ Thus the
pattern in Fig. 3 is well approximated by a quadrupole superimposed on a
constant monopole.
Symmetric Tensor. It is well known that the $l$th multipole in a multipole
expansion determines a symmetric $l$-rank tensor that is traceless over any
two indices.[12] A quadrupole, $l$ = 2, determines a second rank symmetric
traceless tensor $T^{ij},$
$\sum_{m}{a_{2}^{m}}Y_{{\mathrm{real}}\,m}^{l=2}=\sum_{i,j}\hat{r}_{i}T^{ij}\hat{r}_{j}\;.$
(16)
The values of the $a_{2}^{m}$ are known from (13) and, the dependence on
components of $\hat{r}$ is known from the right-most version of the spherical
harmonics $Y_{{\mathrm{real}}\,m}^{l=2}$ in (9). Thus we get an expression for
$\sum{a_{2}^{m}}Y_{{\mathrm{real}}\,m}^{l=2}$ that is second order in the
components of $\hat{r}$ with numerical coefficients. Rearranging the
expression to fit the form $\sum\hat{r}_{i}T^{ij}\hat{r}_{j}$ on the right
side of (16), we find the components of the tensor $T^{ij},$
$T=\pmatrix{{-0.31\;+0.87\;-1.99}\cr{+0.87\;-0.82\;+4.00}\cr{\;-1.99\;+4.00\;+1.12}}\pm\pmatrix{{0.10\;\;0.11\;\;0.16}\cr{0.11\;\;0.17\;\;0.20}\cr{0.16\;\;0.20\;\;0.21}}\;.$
(17)
We have made the tensor symmetric; it is automatically traceless. The
determinant, $\det{T}$ = $-6.2\pm 1.9,$ is invariant under rotations. With the
tensor representation of the quadrupole, rotations act on a vector $\hat{r}$
and on a tensor $T$ as in (16), simplifying transformations to different
coordinate systems.
Maxwell vectors. These vectors represent multipoles as a sequence of monopole
($l$ = 0), dipole ($l$ = 1), two dipoles ($l$ = 2), three dipoles ($l$ = 3),
and so on.[9, 10, 11] Dipoles are vectors and working with vectors is a
convenience with abundant mathematical resources.
For the quadrupole term in (11) there are two Maxwell vectors $u_{1}$ and
$u_{2},$
$\eta_{{\mathrm{355}}}(\alpha,\delta)=\bar{\eta}_{{\mathrm{355}}}+A\left[u_{1}\cdot\vec{\bigtriangledown}\left(u_{2}\cdot\vec{\bigtriangledown}\frac{1}{r}\right)\right]_{\mathrm{S}}+\epsilon\;,$
$\eta_{{\mathrm{355}}}(\alpha,\delta)=\bar{\eta}_{{\mathrm{355}}}+A\left[3u_{1}\cdot\hat{r}\,u_{2}\cdot\hat{r}-\left(\hat{r}_{x}^{2}+\hat{r}_{y}^{2}+\hat{r}_{z}^{2}\right)u_{1}\cdot
u_{2}\right]+\epsilon\;,$ (18)
where $A$ is a constant, $\hat{r}$ is the position unit vector
$\hat{r}(\alpha,\delta)$ in (1). By adjusting the sign of $A$ if needed, we
can multiply the components of $u_{1}$ or $u_{2}$ or both by $-1.$ Thus
$u_{1}$ and $u_{2}$ are non-oriented, bidirectional.
It is important to note that the divergences $\vec{\bigtriangledown}$ in (18)
are three dimensional including contributions obtained by changing the radius
$r.$ Then the result is restricted to the $r$ = 1 unit sphere S. We choose to
leave the factor
$\left(\hat{r}_{x}^{2}+\hat{r}_{y}^{2}+\hat{r}_{z}^{2}\right)$ = 1 in the
expression, so that the expression is second order in components of $\hat{r},$
to match the other quadratic expressions
$\sum{a_{2}^{m}}Y_{{\mathrm{real}}\,m}^{l=2}$ and
$\sum\hat{r}_{i}T^{ij}\hat{r}_{j}.$
Comparing the expressions for $\eta_{{\mathrm{355}}}(\alpha,\delta)$ in (11)
and (18) we see by (16) that
$\sum{a_{2}^{m}}Y_{{\mathrm{real}}\,m}^{l=2}=A\left[3u_{1}\cdot\hat{r}u_{2}\cdot\hat{r}-\left(\hat{r}_{x}^{2}+\hat{r}_{y}^{2}+\hat{r}_{z}^{2}\right)u_{1}\cdot
u_{2}\right]=\hat{r}_{i}T^{ij}\hat{r}_{j}\;.$ (19)
It follows that the tensor components are given in terms of the vectors
$u_{1}$ and $u_{2}$ by
$T^{ij}=A\left[\frac{3}{2}\left(u_{1\,i}u_{2\,j}+u_{1\,j}u_{2\,i}\right)-\delta^{ij}u_{1}\cdot
u_{2}\right]\;.$ (20)
Comparing the expressions in (20) with numerical values of the components in
(17), one can deduce values for the components of the Maxwell vectors $u_{1}$
and $u_{2}$ and the factor $A.$ We find that $A=3.11\pm 0.11$ and
$u_{1}:\quad\\{{\mathrm{RA,dec}}\\}=\\{+136.9^{\circ}\pm
6.1^{\circ}\,,-78.8^{\circ}\pm 1.4^{\circ}\\}$ (21)
$u_{2}:\quad\\{{\mathrm{RA,dec}}\\}=\\{-63.2^{\circ}\pm
2.3^{\circ}\,,-5.3^{\circ}\pm 1.6^{\circ}\\}\;.$
From the dot product, $u_{1}\cdot u_{2}$ = $\cos{\theta_{12}},$ one finds that
$u_{1}$ and $u_{2}$ are nearly perpendicular, differing in direction by an
angle of $\theta_{12}$ = $95.3^{\circ}\pm 1.8^{\circ}.$ This quadrupole
approximates the pattern of two perpendicular dipoles, a ‘lateral quadrupole’.
Preferred Coordinate System. In a ‘preferred coordinate system’, all three
ways of describing the quadrupole simplify.
The preferred coordinate system is a rectangular coordinate system determined
by three mutually orthogonal unit vectors
${x}^{\prime},\hat{y}^{\prime},\hat{z}^{\prime}$ that are combinations of the
Maxwell vectors $u_{1}$ and $u_{2}.$ We have
$\\{a\hat{x}^{\prime},b\hat{y}^{\prime},c\hat{z}^{\prime}\\}=\\{u_{1}-u_{2},\;u_{1}+u_{2},\;\pm\,u_{1}\times
u_{2}\\}\;,$ (22) $a=\|u_{1}-u_{2}\|\quad;\quad b=\|u_{1}+u_{2}\|\quad;\quad
c=\|u_{1}\times u_{2}\|=ab/2\;.$
where the $\pm$ sign determines whether the coordinates are left- or right-
handed and $\|v\|$ is the magnitude of vector $v$. Given $a,$ we get $b$ and
$c,$ within signs, because $a^{2}+b^{2}$ = 4.
Clearly, $u_{1}$ = $(a\hat{x}^{\prime}+b\hat{y}^{\prime})/2$ and $u_{2}$ =
$(-a\hat{x}^{\prime}+b\hat{y}^{\prime})/2.$ Since the coefficients here are
the coordinates of $u_{1}$ and $u_{2}$ in the preferred coordinate system, the
Maxwell vectors are
$u_{1}^{\prime}=\frac{1}{2}\\{a,b,0\\}\quad;\quad
u_{2}^{\prime}=\frac{1}{2}\\{-a,b,0\\}\;.$ (23)
Substituting these components in (20), we get $T^{\prime},$ the traceless,
symmetric tensor in the preferred coordinate system,
$T^{\prime}=\frac{A}{4}\pmatrix{-a^{2}-4&&\phantom{a^{2}}0\phantom{2b^{2}}&&\phantom{a^{2}}0\phantom{-b^{2}}\cr\phantom{-2a^{2}}0\phantom{+b^{2}}&&b^{2}+4&&\phantom{a^{2}}0\phantom{-b^{2}}\cr\phantom{-2a^{2}}0\phantom{+b^{2}}&&\phantom{a^{2}}0\phantom{2b^{2}}&&a^{2}-b^{2}}\;.$
(24)
Note that $T^{\prime}$ is both diagonal and traceless, with determinant
$(A^{3}/64)[(a^{2}-b^{2})^{3}-3(a^{6}-b^{6})]$. Next, since the tensor
$T^{\prime}$ is diagonal, the quadratic expression in (16) only has terms with
the squares
${\hat{x}^{\prime\,2}},{\hat{y}^{\prime\,2}},{\hat{z}^{\prime\,2}}.$ By the
far-right expressions in (9), only the coefficients $a_{2}^{2\,\prime}$ and
$a_{2}^{0\,\prime},$ can be nonzero. We get
${a_{2}^{0}}^{\prime}=\frac{A\sqrt{3}}{8k}\left(a^{2}-b^{2}\right)\quad;\quad{a_{2}^{2}}^{\prime}=-\frac{3A}{2k}\;,$
(25)
where, as previously, $k=\sqrt{15/\left(16\pi\right)}.$
In the preferred coordinate system the Maxwell vectors, the tensor, and the
spherical harmonic coefficients are all simple functions of the vector
magnitudes $a$ = $\|u_{1}-u_{2}\|$ and $b$ = $\|u_{1}+u_{2}\|.$ The formulas
(22) to (25) are valid in general for quadrupoles.
For the quadrupole determined by the function $\eta_{{\mathrm{355}}}(H),$ we
can calculate values for the various quantities in the preferred coordinate
system. From the {RA,dec} values of $u_{1}$ and $u_{2}$ in (21) we get
equatorial components by (1). These give the vector magnitudes $a,b,c$ in
(22). We get
$a=1.478\pm 0.021\quad;\quad b=1.348\pm 0.023\quad;\quad c=0.9956\pm
0.0025\;,$ (26)
from which we can get $u_{1}^{\prime}$ and $u_{2}^{\prime}$ in the preferred
system by (23). Knowing $a,b,c$ also gives the tensor $T^{\prime},$ by (24),
$T^{\prime}=\frac{A}{4}\pmatrix{-6.20&&\phantom{5.}0&&\phantom{0.}0\cr\phantom{-6.}0&&5.82&&\phantom{0.}0\cr\phantom{-6.}0&&\phantom{5.}0&&0.38}\pm\frac{A}{4}\pmatrix{0.06&&\phantom{5.}0&&\phantom{0.}0\cr\phantom{.}0&&0.06&&\phantom{0.}0\cr\phantom{.}0&&\phantom{5.}0&&0.12}\;,$
(27)
where $A=3.11\pm 0.12,$ as noted previously. Finally, the non-zero quadrupole
coefficients, $a_{l=2}^{0\,\prime}$ and $a_{l=2}^{2\,\prime},$ have the values
${a_{2}^{0}}^{\prime}=0.45\pm 0.15\quad;\quad{a_{2}^{2}}^{\prime}=-8.53\pm
0.32\;.$ (28)
One can check that the trace and determinant of $T$ and the quadrupole power
$P(2)$ are equal in Equatorial and preferred coordinates.
## 4 Distributions at various positions
At each position $H$, the 355 polarization angles referenced to $H$ form a
distribution with values ranging from $0^{\circ}$ to $90^{\circ}.$ In this
section we look at the distributions at an ordinary position with an average
$\eta_{{\mathrm{355}}}(H)$ and at the positions with maximum and minimum
$\eta_{{\mathrm{355}}}(H)$.
As an ordinary position with a near-average value of the function, let
$H_{op}$ be $\\{$RA,dec$\\}$ = $\\{50^{\circ}\,,15^{\circ}\\},$ where one
finds that $\eta_{{\mathrm{355}}}(H_{op})$ = $46.1^{\circ}$ which is a little
more than $1^{\circ}$ above the average $\bar{\eta}_{{\mathrm{355}}}$ of
$45.0^{\circ}.$
It is clear from the graph, fig. 4, that the distribution of angles
$\eta_{i}(H_{op})$ is nearly a straight line. This is typical for the ordinary
positions I have looked at. At ordinary positions, the distributions of angles
$\eta_{i}(H)$ approximates closely the uniform distribution.
The uniform distribution $\eta^{{\mathrm{U}}}_{i},$ superscript U, has 355
evenly spaced angles from $0^{\circ}$ to $90^{\circ},$
$\eta^{{\mathrm{U}}}_{i}=\frac{i}{N}\,90^{\circ}\;,$ (29)
where $N$ = 355 is the number of QSOs in the catalog. One supposes that the
uniform distribution $\eta^{{\mathrm{U}}}_{i}$ is likely for the polarization
angles of independent QSO sources.
Thus, on the sky at positions where the function $\eta_{{\mathrm{355}}}$ is
close to its average value, the distributions of the observed polarization
vectors of the 355 QSOs are consistent with independent non-interacting
sources.
We turn now to the observed distributions of polarization angles $\eta_{i}(H)$
at the max and min positions labeled ‘$+$’ and ‘$-$’ in Fig. 3. At these
positions, the distributions of the angles $\eta_{i}(H)$ differ the most from
the uniform distribution.
The observed distributions toward $H_{{\mathrm{min}}}$ and
$H_{{\mathrm{max}}}$ are plotted in fig. 5 with the uniform distributions
plotted for comparison. Note that the angles $\eta_{i}$ are sorted. For
$H_{{\mathrm{min}}}$ the angles increase from $0^{\circ}$ to $90^{\circ}$
while they decrease with increasing $i$ for $H_{{\mathrm{max}}}.$ So the $i$th
QSO for one distribution is not the $i$th QSO in the other distribution, and
both differ from the $i$th QSO in the catalog, in general.
It is clear from the graph that the deviation from the uniform distribution is
maximized at mid range angles $\eta_{i}\approx$ $45^{\circ}.$ See fig. 5. The
arc-like distributions can be approximated by smooth quadratic functions. For
the distribution at $H_{{\mathrm{min}}}$ we have
$\eta_{-}=\left[\left(90^{\circ}-\mu\right)\frac{i}{N}+\mu\,\left({\frac{i}{N}}\right)^{2}\right]=\left[\left(1-\frac{\mu}{90^{\circ}}\right)\eta^{{\mathrm{U}}}_{i}+\,\mu\left(\frac{{\eta^{{\mathrm{U}}}_{i}}}{90^{\circ}}\right)^{2}\right]\;,$
(30)
where the negative subscript $\eta_{-}$ indicates the $i$th polarization angle
is less than the $i$th value in the uniform distribution
$\eta^{{\mathrm{U}}}.$ At best fit $\mu$ = $31.2^{\circ}\pm 1.0^{\circ}.$
For the distribution at $H_{{\mathrm{max}}}$ we have
$\eta_{+}=90^{\circ}-\left[\left(90^{\circ}-\nu\right)\frac{i}{N}+\nu\,\left({\frac{i}{N}}\right)^{2}\right]=90^{\circ}-\left[\left(1-\frac{\nu}{90^{\circ}}\right)\eta^{{\mathrm{U}}}_{i}+\,\nu\left(\frac{{\eta^{{\mathrm{U}}}_{i}}}{90^{\circ}}\right)^{2}\right]\;,$
(31)
where the positive sign in the subscript indicates $\eta_{+\;i}\geq$
$\eta_{i}^{{\mathrm{U}}}$ for any $i.$ Here the best-fit is found to have
$\nu$ = $26.8^{\circ}\pm 0.9^{\circ}.$
The distributions $\eta_{-}$ and $\eta_{+}$ at $H_{{\mathrm{min}}}$ and
$H_{{\mathrm{max}}},$ respectively, form smooth arcs in Fig. 5 that can be
approximated by the quadratic functions in (30) and (31). The smooth arcs
formed by the observed polarization angles of the QSOs suggests that some
large-scale mechanism exists to shape these distribution curves.
## 5 Breaking the catalog into regions
Motivated by alignments in the CMB temperature field, we search in this
article for large scale deflections of QSO polarization vectors toward a
particular direction. It is clear from the previous sections that the
catalogued polarization vectors skew toward $H_{{\mathrm{min}}}.$ However, the
catalog favors the regions A1 and A3 defined in Ref. 1 as A1:
$168^{\circ}\leq\alpha\leq 218^{\circ}$ and
$-40^{\circ}\leq\delta\leq+50^{\circ}$ and A3: $-40^{\circ}\leq\alpha\leq
0^{\circ}$ and $-60^{\circ}\leq\delta\leq+30^{\circ}.$
Thus 183 of the 355 QSOs in the catalog reside in A1A3’s 20% of the sky. Since
these are regions where QSO polarization vectors tend to align, we need to
check if the catalog-wide effect found above is global or is it related to the
alignments of the QSOs in regions A1 and A3. And we would like to know if any
effect remains once the 183 A1-A3 QSOs are excluded.
In this section we split the catalog into 183 A1-A3 QSOs and the rest, the 172
QSOs not in A1 or A3. The calculations of the previous sections are applied
with the 183 QSOs in regions A1 and A3. Then we process the 172 QSOs that are
not in A1 or A3.
We get two additional quadrupole patterns, one for the 183 QSO subset and one
for the 172 QSO subset. The quadrupole patterns are faithfully rendered by the
Maxwell vectors $u_{1}$ and $u_{2}$ with constant $A.$ This information is
collected in Table 1. We see that, compared to the full catalog, the 183 QSO
sample shifts $u_{1}$ and $u_{2}$ toward the Equatorial coordinate axes, with
$u_{1}$ coincident with the negative $z$-direction and $u_{2}$ coincident with
negative $y.$ The so called ‘preferred direction’ along $u_{1}\times u_{2}$
aligns closely with negative $x.$
Thus the $u_{1}$ and $u_{2}$ Maxwell vectors of the 183 QSO sample determine a
near-Equatorial coordinate system which has directions determined by the plane
of the Earth’s equator. Other planets have other equatorial planes, so the
coincidence suggests a local deflection of polarization vectors, which without
convincing corroboration must be deemed unlikely. The interesting outcome is
that a preferred coordinate system is determined by QSO polarization vectors.
It is difficult to say what such alignments could mean. Before wondering about
that, we should check to see if the patterns are significant and compare their
strengths with random polarization angles.
To judge the strength of the patterns, we compare the quadrupole powers $P(2)$
for the three samples. See Table 2. The 183 QSO sample has the best quadrupole
power $P(2)$ = 29.5, with the 172 QSO sample worst at 11.8.
To help judge the effect of sample size and to get quantitative information on
what random polarization angles would give, we replace the measured
polarization angles with random values between $0^{\circ}$ and $180^{\circ}$
oriented clockwise from North with East to the right at each QSO. QSO
locations are not changed. The quadrupole power of the resulting patterns is
called $P_{\mathrm{Ran}}(2)$ and listed in Table 2. All three samples have
$P_{\mathrm{Ran}}(2)$ about one sigma away from zero, as one would expect.
Also as expected, the entire 355 QSO sample has the best statistics with the
lowest $P_{\mathrm{Ran}}(2)$ = $3.7\pm 3.4,$ with the 183 and 172 samples much
larger at about 6 and 10, respectively.
Since both the 183 QSO sample and the full 355 QSO sample have powers $P(2)$
that exceed random by a factor of about five, both samples generate
significant patterns. The 172 QSO sample of QSOs outside of A1A3 lags in both
statistics and power. The 172 sample is so weak that its quadrupole power,
$P(2)$ = 11.8, is as likely as 172 QSOs in the same locations but with random
polarization directions, $P_{\mathrm{Ran}}(2)$ = $9.7\pm 6.8\;.$
It is reasonable to conclude that the 183 QSO sample drives the pattern found
for the 355 QSO catalog discussed in the previous sections of this paper.
Thus the method here differs from the analysis of Ref 1, but yields much the
same results. This may be expected since we use their catalogued data. Here we
find that QSOs in regions A1 and A3 have polarization vectors skewed toward a
particular direction by a few degrees on average, whereas Ref. 1 found mutual
alignments of neighboring QSOs in A1 and A3. However the limited data
available for QSOs outside of regions A1 and A3, some 80% of the sky, preclude
any conclusion about effects there. Since the goal here is to find a global
CMB-like effect, we wait for more data to be developed. A survey of
significantly polarized optical QSOs in the higher latitudes of the Galaxy
would be welcome.
## References
* [1] Hutsemekers, D. et al. 2005, “Mapping extreme-scale alignments of quasar polarization vectors”, Astron. Astrophys. 441 915-930.
* [2] The catalog is available in electronic form at the CDS Centre de Donn es astronomiques de Strasbourg or http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
* [3] Sluse, D., Hutsem ekers, D., Lamy, H., Cabanac, R., Quintana, H. 2005,“New optical polarization measurements of quasi-stellar objects. The data”, A&A, 433, 757
* [4] See, for example, Planck Collaboration 2013, “Planck 2013 results. XXIII. Isotropy and Statistics of the CMB”, Submitted to Astronomy & Astrophysics on March 22, 2013.
* [5] See, for example, Copi, Craig J. et al 2010, “Large angle anomalies in the CMB”, Adv.Astron. 2010, Article ID 847541; DOI: 10.1155/2010/847541.
* [6] See, for example, Hanson, D. and Lewis, A. 2009, “Estimators for CMB statistical anisotropy”, Phys Rev D 80(6), 063004-1. DOI: 10.1103/PhysRevD.80.063004
* [7] Calculations aided by computer software: Wolfram Research, Inc. 2010, Mathematica Edition: Version 8.0, Wolfram Research, Inc., Champaign, Illinois.
* [8] See, for example, T. Whittaker and G. N. Watson 1927, A Course of Modern Analysis, Cambridge UP, Cambridge UK, 4th edition.
* [9] See, for example, Land, K. and Magueijo,J. 2005, “The Multipole Vectors of WMAP, and their frames and invariants” , Mon.Not.Roy.Astron.Soc. 362:838-846.
* [10] See, for example, Weeks, J. 2004, “Maxwell’s Multipole Vectors and the CMB,” arXiv:astro-ph/0412231v2 .
* [11] See, for example, Dennis, M. R., 2004, “Canonical representation of spherical functions: Sylvester’s theorem, Maxwell’s multipoles and Majorana’s sphere”, J. Phys. A: Math. Gen., 37, 9487 .
* [12] See, for example, Guth, A. 2012, “Lecture Notes 9: Traceless Symmetric Tensor Approach to Legendre Polynomials and Spherical Harmonics”, M.I.T. Lecture note series, http://web.mit.edu/8.07/www/lecnotes/ln09-807f12.pdf
## 6 Tables
Sample | $u_{1}$: {RA,dec} | $u_{2}$: {RA,dec} | $A$
---|---|---|---
A1, A3 (183 QSOs) | $\\{-124(10),-82.7(1.8)\\}$ | $\\{-79.0(1.3),-2.9(1.7)\\}$ | 4.41(0.13)
All 355 QSOs | $\\{136.9(6.1),-78.7(1.4)\\}$ | $\\{-63.1(2.3),-5.3(1.6)\\}$ | 3.11(0.12)
not A1,A3 (172 QSOs) | $\\{84.0(6.2),-60.2(3.5)\\}$ | $\\{-37.0(3.1),-19.2(2.8)\\}$ | 2.79(0.19)
Table 1. The Maxwell representation of the quadrupole for three samples of
QSOs. The value of the average polarization angle
$\eta_{{\mathrm{N}}}(\hat{r})$ referenced to a position $\hat{r}$ for the
various samples, $N$ = 183, 355, 172 respectively, can be reconstructed from
the unit vectors $u_{1}$ and $u_{2}$ and the constant $A:$
$\eta_{{\mathrm{N}}}(\hat{r})$ = $45^{\circ}$ \+
$A\left[3u_{1}\cdot\hat{r}u_{2}\cdot\hat{r}-\left(\hat{r}_{x}^{2}+\hat{r}_{y}^{2}+\hat{r}_{z}^{2}\right)u_{1}\cdot
u_{2}\right].$ All angles are in degrees (∘). Uncertainties are in
parenthesis: $A$ = $4.41(0.13)$ means $A$ = $4.41\pm 0.13\,.$
Sample | $P(2)$ | $P_{\mathrm{Ran}}(2)$: Random angles
---|---|---
A1, A3 (183 QSOs) | 29.5(1.7) | 6.2(5.3)
All 355 QSOs | 14.6(1.1) | 3.7(3.4)
not A1,A3 (172 QSOs) | 11.8(1.6) | 9.7(6.8)
Table 2. Quadrupole powers $P(2)$ for the three samples. The 183 QSO sample
and the 355 QSO sample have significant quadrupole patterns because their
quadrupole powers $P(2)$ exceed random by a factor of 4 or 5, with
$P(2)/P_{\mathrm{Ran}}(2)$ = $29.5/6.2\approx$ 5 and
$P(2)/P_{\mathrm{Ran}}(2)$ = $14.6/3.7\approx$ 4, respectively. However, the
pattern for the 172 QSOs is not significant because the ratio
$P(2)/P_{\mathrm{Ran}}(2)\approx$ 1.2 and the quadrupole power 11.8 is well
within the plus/minus value of random, $11.8<9.7+6.8\;.$
## 7 Figures
Figure 1: (Color online) Two quasars and their polarization angles $\eta$
referenced to a given position $H.$ The polarization vectors for the two
quasars (QSOs) make angles $\eta_{1}$ and $\eta_{2}$ with respect to the
directions toward $H.$ The angles $\eta_{1}$ and $\eta_{2}$ are acute, i.e.
between $0^{\circ}$ and $90^{\circ}.$ The sphere represents the celestial
sphere in Equatorial coordinates with North upward and East to the right in
the hemisphere shown. Figure 2: (Color online) Determining the polarization
angle from catalog data. In the catalog,[1, 2] the direction of the
polarization vector of the $i$th QSO is given as the ‘polarization position
angle’ $\theta_{pi}$ from local North, measured clockwise with East to the
right. The equatorial coordinates of the QSO are also listed in the catalog,
so we can draw the great circle to $H$ and calculate the angle $\theta_{Hi}$
between the great circle and North, measured clockwise as shown. In cases like
this sketch, $\theta_{Hi}$ is larger than $\theta_{pi}$ but less than
$90^{\circ}$ larger, so the polarization angle $\eta_{i}(H)$ is the difference
of the two angles, $\eta_{i}(H)$ = $\theta_{Hi}-\theta_{pi}.$ Figure 3: (Color
online) The function $\eta_{{\mathrm{355}}}(H)$ mapped on the celestial sphere
At each position $H$ on the celestial sphere we calculate the arithmetic
average of the 355 polarization angles $\eta_{i}(H).$ The function forms a
pattern with two diametrically opposite maxima, indicated with ‘$+$’, and two
diametrically opposite minima at the positions ‘$-$’. The plot is an Aitoff
projection. Figure 4: (Color online) The distribution of polarization angles
with respect to an ordinary position. For an ordinary position $H_{op}$ with
a near-average polarization function, $\eta_{{\mathrm{355}}}(H_{op})\approx$
$45^{\circ},$ the sorted polarization angles hug the straight line uniform
distribution. The distributions at ordinary positions have small blips above,
as here, and below the straight line. Such distributions are consistent with
independent polarization vectors. Figure 5: (Color online) The distributions
of polarization angles $\eta_{i}$ at $H_{\mathrm{min}}$ and
$H_{\mathrm{max}}$. The distributions of polarization angles $\eta_{-}$ and
$\eta_{+}$ with respect to positions $H_{\mathrm{min}}$ and $H_{\mathrm{max}}$
deviate most from the uniform distributions, the straight lines. The fit of
the observed polarization angles to parabolic arcs suggests that some large
scale mechanism exists that skews the distributions toward $H_{\mathrm{min}}$
and away from $H_{\mathrm{max}}$. What mechanism(s) could accomplish this?
|
arxiv-papers
| 2013-11-24T13:27:37 |
2024-09-04T02:49:54.154197
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Richard Shurtleff",
"submitter": "Richard Shurtleff",
"url": "https://arxiv.org/abs/1311.6118"
}
|
1311.6161
|
# Current induced torques between ferromagnets and compensated
antiferromagnets: symmetry and phase coherence effects
Karthik Prakya1, Adrian Popescu2,3, and Paul M. Haney2 1\. The MITRE
Corporation, Bedford, MA 01730
2\. Center for Nanoscale Science and Technology, National Institute of
Standards and Technology, Gaithersburg, MD 20899
3\. Maryland NanoCenter, University of Maryland, College Park, MD 20742, USA
###### Abstract
It is shown that the current-induced torques between a ferromagnetic layer and
an antiferromagnetic layer with a compensated interface vanish when the
ferromagnet is aligned with an axis of spin-rotation symmetry of the
antiferromagnet. For properly chosen geometries this implies that the current
induced torque can stabilize the out-of-plane (or hard axis) orientation of
the ferromagnetic layer. This current-induced torque relies on phase coherent
transport, and we calculate the robustness of this torque to phase breaking
scattering. From this it is shown that the torque is not linearly dependent on
applied current, but has an absolute maximum.
###### pacs:
85.35.-p, 72.25.-b,
## I Introduction
Current-induced torques result from the interaction between conduction
electron spins and the magnetization of a sample when current flows through
it. This torque is generally present when the magnetization is spatially
nonuniform, and has been extensively studied in the context of magnetic domain
walls and spin valve structures. Since its theoretical predictionslonc ;
berger , extensive studies have led to a theoretical framework of current-
induced torque in ferromagnets that describes experimental results with
quantitative success.stiles:jmmm It has been proposed that current-induced
torques also exist in antiferromagnetic systems.nunez ; haney:jmmm Previous
theoretical studies considered systems composed entirely of antiferromagnetic
layers nunez ; duine:prb ; haney:prb as well as experimental wei and
theoreticalhaney:prl ; loktev ; loktev2 systems with both ferromagnetic and
antiferromagnetic layers. Theoretical work has also focused on antiferromagnet
textures.brataas1 ; brataas2 ; duine:2011 ; niu Experiments have demonstrated
current-induced torque in materials with other types of complex magnetic
ordering, such as skyrmion lattices. Recent theoryshick and
experimentjungwirth have shown that antiferromagnets exhibit anisotropic
magnetoresistance, demonstrating a coupling between magnetic order and charge
transport these materials.
Antiferromagnets exhibit an array of magnetic ordering, such as spin density
waves that are commensurate or incommensurate with the lattice, and
configurations with multiple spin density waves. As shown in Ref. haney:prl, ,
the symmetry properties of the antiferromagnetic layer can lead to torques in
multilayers with qualitatively different properties than conventional spin
valves. In particular, a collinear compensated antiferromagnetic layer
interface (with each spin in the $\pm{\hat{z}}$ direction, which we call a 1Q
spin structure) leads to a torque which vanishes when the ferromagnet is
perpendicular to the $\hat{z}$ direction. This torque can stabilize the hard-
axis orientation of the ferromagnet in systems where the antiferromagnet is
pinned. Here we treat similar systems (see Fig. 1a), and compute the current-
induced torque on the ferromagnetic layer. (Previous works have investigated
the current-induced torque on the antiferromagnetic layer in such
systems.loktev ; loktev2 )
In this work we consider a system where the antiferromagnetic layer has a 3Q
spin structure (see Fig. 1b). This is qualitatively different than the
previously studied 1Q antiferromagnet because the 3Q structure has only a
single axis of spin rotational symmetry (3-fold in this case), whereas for the
1Q antiferromagnet all directions perpendicular to the $\hat{z}$ direction are
axes of 2-fold spin rotational symmetry. We show that an important consequence
of the reduced symmetry of the 3Q antiferromagnet is that the current-induced
torque stabilizes the out-of-plane magnetic orientation only when the
magnetization is initialized nearby this orientation (in contrast, the 1Q
antiferromagnet drives any initial orientation out-of-plane). In this work we
additionally determine the effects of phase breaking scattering: The current-
induced torque relies on phase coherence, and quantifying the robustness with
respect to scattering is important to gauge the feasibility of observing these
effects in real systems.
Our results are easily generalized to multilayers composed of a free
ferromagnet layer, and a fixed magnetic layer whose spin configuration has an
axis of $n$-fold rotational symmetry. The key property of the torque is that:
if the ferromagnetic layer is aligned with an axis of spin-rotational symmetry
of the fixed layer, then the current-induced torque (in fact, all torques)
must vanish. This is seen by recognizing that, by assumption, the system is
invariant with respect to spin rotations about the ferromagnet orientation by
some angle $\phi_{n}$, and any nonzero torque (which must be perpendicular to
the ferromagnet orientation) does not respect this symmetry. For conventional
spin valves, this statement implies the well known fact that the current-
induced torque vanish when the ferromagnet layers are aligned or anti-aligned.
Identifying the points where the current-induced torque vanishes is important
because the torque may drive the magnetization to these fixed points. For
properly designed antiferromagnet-ferromagnet multilayers this property of the
torque can stabilize the out-of-plane magnetic orientation.haney:prl This is
because this orientation, being a maximum of the magnetic free energy,
represents a fixed point for the conventional micromagnetic torques. In the
absence of current-induced torques, this out-of-plane configuration is an
unstable fixed point; however if the current-induced torque drives the
ferromagnet to this orientation and exceeds the damping torque, it can
stabilize this configuration, as shown by micromagnetic simulations in Ref.
haney:prl, .
## II Method
To calculate the current-induced torques, we use the nonequilibrium Green’s
function technique within a tight binding representation. This is a well
established approach to calculating the transport properties of magnetic thin
films. We highlight the most important details here. The system is taken to
consist of two semi-infinite electrodes, with a scattering region placed
between them. There is a difference $V_{\rm app}$ in the electrochemical
potential of the two electrodes. The central quantity is the density matrix
$\rho$:
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{-\infty}^{E_{\rm F}-V_{\rm
app/2}}\left[G^{r}\left(E\right)-G^{a}\left(E\right)\right]dE+$ (1)
$\displaystyle~{}~{}~{}\int_{E_{\rm F}-V_{\rm app}/2}^{E_{\rm F}+V_{\rm
app/2}}G^{r}\left(E\right)\Gamma_{L}\left(E\right)G^{a}\left(E\right)dE.$
where
$G^{r,a}\left(E\right)=\left(E-H_{C}-\Sigma^{r,a}_{L}\left(E\right)-\Sigma^{r,a}_{R}\left(E\right)\right)^{-1}$.
$H_{C}$ is the scattering region Hamiltonian, and $\Sigma^{r}_{L}$ is the self
energy which describes the electronic coupling between the scattering region
and the semi-infinite left lead; it is given by
$\Sigma^{r}_{L}=V_{C,L}^{\dagger}g_{0,L}^{r}\left(E\right)V_{C,L}$, where
$V_{C,L}$ is the coupling matrix element between the left lead and central
region, and $g_{0,L}$ is the surface Green’s function of the isolated semi-
infinite left lead. The same form of self energy holds for the right lead.
As noted in previous works,duine:prb phase coherence plays a central role in
a number of the antiferromagnetic systems studied so far. To explore the
robustness of the torques in this system, we include an additional self energy
$\Sigma_{S}$ in Green’s function which describes elastic, phase breaking
scattering. Its form is:
$\displaystyle\Sigma_{S}\left(E\right)=iD\left(G^{r}\left(E\right)-G^{a}\left(E\right)\right)$
(2)
where $D$ parameterizes the scattering. A discussion of the parameter $D$ in
terms of real material properties and temperature is given in Sec. (III).
We assume the spin-orbit coupling is negligible, so that the current-induced
torque on the ferromagnet layer is simply given by the transverse component of
incoming spin current flux. For our geometry, the net spin current has real
space velocity in the $\hat{y}$ direction. The spin current operator
$\vec{J}\left(y\right)$ is then:
$\displaystyle\hat{\vec{J}}\left(y\right)=\sum_{\begin{subarray}{c}j\in
R\left(y\right)\\\ k\in L\left(y\right)\\\
s,s^{\prime}\end{subarray}}i\left[c^{\dagger}_{j,s}\vec{\sigma}_{s,s^{\prime}}c_{k,s^{\prime}}t_{j,k}-{\rm
h.c.}\right],$ (3)
where $R\left(y\right)$ are the set of sites with coordinate $y^{\prime}$
greater than $y$, and $L\left(y\right)$ are the set of sites with coordinate
$y^{\prime}$ less than $y$. $\vec{\sigma}$ sigma is the vector of Pauli
matrices, and we take the hopping $t_{j,k}$ between all sites $j$ and $k$ to
be spin independent. We present results in terms of torque per current (units
of ${\mu_{B}}/e$), which represents the spin torque efficiency. The absolute
value of this quantity determines the critical current needed to drive
magnetic dynamics.
As discussed in Refs. haney2007, and nikolic, , it is sometimes necessary to
compute the entire energy integral (both terms in Eq. 1) in order to find the
current-induced torques. This is particularly the case when the torques in
question are present in equilibrium (which is itself dependent on the
symmetries of the system, as discussed in Ref. haney2007, ). We checked
explicitly that the current-induced torque in question for these systems are
dominated by the nonequilibrium contribution to the density matrix (the second
term of Eq. 1), and present only this contribution in the results (the
remaining “energy integral” contribution is several orders of magnitude
smaller in all the cases we checked). We take the Fermi energy to be
$E_{F}=3.75~{}t$, and use a dense $k$-point mesh to converge the transport
integrals, up to $1000^{2}$ k-points for a unit cell having 4 atoms per layer.
Figure 1: (a) Overall system geometry (b) The crystal and spin structure for
the 3Q state. The spins at the corners of the interior box all point inward.
(c) The spin on the [111] interface of the lattice from (b). The small black,
medium red, and large gray dots represent atoms in different layers (i.e with
different y-values). The spin of dots without an arrow is completely in the
$\hat{y}$ direction, while other spins are partially canted in the ${\hat{y}}$
direction. (d) Spherical coordinate system used to describe the torques on the
ferromagnet. The blue (dark) spins in the $x-z$ plane represent the 3-fold
symmetric spins of the antiferromagnetic layer, and the skinnier red arrow
represents the orientation of the ferromagnet layer.
A schematic of the overall system is shown in Fig. 1a. It consists of semi-
infinite ferromagnetic and antiferromagnetic layers, separated by a
nonmagnetic spacer which is 3 atomic layers thick. The layers are fcc, with
interfaces in the [111] direction. We use two different spin structures for
the antiferromagnet. One is a 3Q spin structure, depicted in Fig. 1b. The spin
structure in the $(111)$ planes is shown in Fig. 1c, which shows the 3-fold
symmetry of the spin in the $x-z$ plane. Each spin also has a component along
the $y$ axis (into or out of the page). Atoms with no arrow in the figure have
a spin fully aligned in the $+\hat{y}$ direction, while other atoms’ spins are
partially canted in the $-\hat{y}$ direction, so that the net bulk spin
vanishes. Common antiferromagnetic materials such as FeMn are predicted to
have a 3Q ground state,schulthess ; footnote2 ; stocks consistent with
measurementskawarazaki ; kennedy , although there is not complete consensus
between all the experimental data. To further explore the consequences of the
antiferromagnet symmetry, we also consider a system where the $y$ component of
the spins are set to 0. This artificial system retains the 3-fold symmetry in
the $x-z$ plane, but is also symmetric under $s_{y}\leftrightarrow-s_{y}$. We
refer to this as the “no-canting” antiferromagnet. We emphasize that our
primary results generalize to any antiferromagnet for which there is an axis
of $n$-fold spin rotational symmetry, as explained in the introduction.
We present the angular variation of the torque on the ferromagnet layer in
terms of spherical coordinates, as shown in Fig. 1d. The $\hat{y}$ direction
is the hard axis of the F, which is taken to coincide with the axis of 3-fold
symmetry of the antiferromagnet. As explained in the introduction, this
alignment of hard axis and the antiferromagnet axis of spin rotational
symmetry is crucial for the out-of-plane orientation to be stabilized by the
current-induced torque. The $\hat{z}$ direction is along one of the spins of
the antiferromagnetic layer. We utilize similar schematics as Fig. 1d in the
next section to show the relative orientation of the ferromagnet layer with
the spins of the antiferromagnet.
## III Results
The current-induced torque on the ferromagnetic layer for a no-canting
antiferromagnetic system is shown in Fig. 2. Unlike the current-induced torque
in a conventional spin valve, whose magnitude has a simple $\sin(\theta)$
dependence, we find a more complex angular dependence for the torque. We first
fix $\phi=0^{\circ}$ and vary the ferromagnet orientation from $\theta=0$ to
$360^{\circ}$. These orientations are in the easy plane. The torques conform
to the 3-fold symmetry, varying approximately as $\sin\left(3\theta\right)$,
as shown in Fig. 2a. For fixed $\phi=90^{\circ}$, sweeping the polar angle
$\theta$ takes the magnetization out of the easy plane, through the hard axis
direction. The torques in this case are shown in Fig. 2c. The torques vary as
$\sin\left(2\theta\right)$, again as required by symmetry. For fixed
$\phi=45^{\circ}$, varying $\theta$ takes the ferromagnet on an “off-axis”
orbit, and the torque exhibits more complex angular dependence.
Figure 3 shows similar results for the 3Q spin structure for the same set of
magnetic orientations. The reduction in symmetry due to the inequivalence of
$s_{y}$ and $-s_{y}$ leads to more complex behavior of the torque. For
$\phi=0^{\circ}$, we note the invariance of the torque under
$\theta\rightarrow\theta+120^{\circ}$. Key data points are shown in Fig. 3c by
the black arrows. As argued in the introduction, when the ferromagnet layer is
aligned to the axis of 3-fold symmetry, the current-induced torque vanishes.
Figure 2: The angular dependence of the current-induced torque (CIT) on the
ferromagnet for the system with no antiferromagnetic canting in the
$y$-direction (the “no canting” system). The black dashed line is the torque
in the $\hat{\phi}$ direction, and the gray line with markers is the torque in
the $\hat{\theta}$ direction. (a) shows the torque as when the ferromagnet is
coplanar with the antiferromagnet spins, which shows a $\sin(3\theta)$
dependence. (b) shows an intermediate angle, and (c) shows the torque as the
ferromagnet orientation is normal to the plane of the antiferromagnet spins.
In this case, the torque varies as $\sin\left(2\theta\right)$. The diagrams to
the right of the plots show the direction of antiferromagnet spins in the x-z
plane, and with a circle representing the angles of the ferromagnet layer in
the plot. Figure 3: The angular dependence of the torque on the ferromagnet
for the system with no 3Q spin ordering of the antiferromagnet. The black
dashed line is the torque in the $\hat{\phi}$ direction, and the gray line
with markers is the torque in the $\hat{\theta}$ direction. (a) shows that the
torque again varies as $\sin(3\theta)$ when the ferromagnet layer is confined
to the $x-z$ plane (easy plane). (b) shows complex angular dependence for the
ferromagnet layer oriented along an axis of low symmetry. (c) shows that the
torque vanishes when the ferromagnet is aligned to the axis of 3-fold symmetry
of the antiferromagnet (arrows indicate these points).
To gain a fuller view of the current-induced torque near the out-of-plane
fixed point, we show the torque in the vicinity of these points in Fig. 4. For
the no-canting system, the $+\hat{y}$ and $-\hat{y}$ fixed point are
equivalent. For electrons flowing from the antiferromagnet to the ferromagnet,
these are stable fixed points. For the 3Q antiferromagnet, on the other hand,
the $+\hat{y}$ and $-\hat{y}$ fixed points are inequivalent. In this case, we
find the $+\hat{y}$ is a stable attractor, while the $-\hat{y}$ is an elliptic
fixed point. The nature of the fixed point (stable, unstable, elliptic, etc.)
is parameter dependent, making it difficult to make general statements about
the prevalence of different fixed points.
For antiferromagnetic systems it is also important to distinguish between
stable fixed points to which any initial magnetization vector is driven
(global attractors), and those fixed points for which only an initial
magnetization vector nearby is driven (local attractors). Inspection of Fig.
2a shows that, if the magnetization is in the $x-z$ plane, the torque driving
it to the out-of-plane direction is quite weak (in this case, the relevant
torque is in the $\hat{\phi}$ direction). On the other hand, if the
magnetization is near the $z-y$ plane (Fig. 2c), the torque driving it to the
out-of-plane orientation ($\Gamma_{\theta}$) is much stronger. Rather than
characterizing the flow of the current-induced torque field for any particular
system in detail (which is highly parameter dependent), we simply emphasize
that an experiment is more likely to observe these torques if the
magnetization is initially in the out-of-plane before the current is applied.
Application of a current can stabilize this configuration, so that subsequent
removal of the applied field does not result in the magnetization returning to
the easy plane.
Figure 4: A zoom-in view of the torques on the ferromagnet layer near the
fixed point of the current-induced torque. (a) shows the result for the “no
canting” system, where the $\pm y$ fixed points are equivalent. The red dot on
the sphere on the right represents the magnetic orientation shown in the left
panel. The dark blue arrows represent the orientation of the antiferromagnet
spins. (b) shows the result for the 3Q system. The torques indicate that the
$+{\hat{y}}$ orientation is a stable fixed point. (c) shows that the
$-{\hat{y}}$ orientation is an elliptic fixed point. (The three blue (dark)
arrows of (a) have no $\hat{y}$ component, while for (b) and (c), the three
similar blue (dark) arrows are canted, acquiring a small positive $\hat{y}$
component.)
In contrast to the current-induced torque in noncollinear ferromagnets, the
current-induced torque in many antiferromagnet systems rely on phase
coherence.duine:prb This is because the eigenstates of the bulk
antiferromagnet are degenerate Kramer’s doublets with opposite spins. A
distribution of these eigenstates carries no net spin current. However, spin-
dependent reflection at the ferromagnet interface leads to a superposition of
these degenerate states, which results in a nonzero spin polarization of the
current in the antiferromagnet. The component of this spin current
perpendicular to the ferromagnet is responsible for the torque on the F, and
vanishes as the coherence between the states is destroyed. The requirement of
ballistic (or quasi-ballistic) transport imposes more stringent requirements
on the existence of current-induced torques in antiferromagnets than in
ferromagnets. Materials should be nearly single crystal, and scattering (from
e.g. phonons) should be minimized. In order to estimate the acceptable limits
of electron-phonon scattering, we add an elastic scattering channel to the
Green’s function self-energy as described in Sec. II. Fig. 5 shows how
increased scattering decreases current-induced torque near the out-of-plane
fixed point of the no-canting system. Here the scattering parameter $D$ is
normalized by the square of the hopping matrix element $t$.
Figure 5: (a) The magnitude of current-induced torque near the fixed point of
the “no-canting” system as a function of the elastic scattering parameter
${D/t^{2}}$. The parameters used in the curve fit are:
$\Gamma_{0}=0.0478~{}\left(\mu_{B}/e\right),~{}A=670~{}t^{-2}$. (b) The same
plot for the 3Q system. The fit applies only to (a).
To place the result of Fig. 5 in context, we write $D$ in terms of material
properties. For simplicity, we focus on just one phase breaking process:
elastic acoustic phonon scattering. Our aim is to explicitly show that the
current-induced torque, as a function of the applied current, has a maximum
absolute value. Depending on materials properties and temperature, other
scattering processes may be more important. In any event, for acoustic phonon
scattering, $D$ takes the form:lundstrom
$\displaystyle D=\frac{E_{a}^{2}k_{\rm B}T}{\rho v^{2}a^{3}}\equiv D_{0}T$ (4)
where $E_{a}$ is the elastic deformation potential, $\rho$ is the material
density, $v$ is the speed of sound, $a$ is the lattice spacing, and $T$ is the
temperature. The linear $T$ dependence reflects the increased thermal
population of phonons with increasing temperature. Other scattering process
(e.g. electron-electron scattering, inelastic phonon scattering) depend on $T$
differently; generally $D\propto T^{p}$ where $p$ varies from 0.5 to 3 (see
Ref. mohanty, and references within).
Joule heating may increase the importance of thermal effects: for current
density $J$ flowing through a material with resistivity $\Omega$, thermal
conductivity $\kappa$, and length $L$ along the current direction (in this
case, the $\hat{y}$-direction), the spatially averaged temperature increases
by a factor on the order of $J^{2}L^{2}\Omega/\kappa$. To stabilize the out-
of-plane magnetic orientation requires a current density of $\alpha\gamma
M_{s}t_{\rm F}/2g$footnote1 , where $g$ is the current-induced torque per
current, $\alpha$ is the damping, $\gamma$ is the gyromagnetic ratio, $M_{s}$
is the saturation magnetization of the ferromagnet layer, and $t_{\rm F}$ is
the thickness of the ferromagnet layer. For the no-canting system, the
current-induced torque per current is $g=0.05~{}\mu_{\rm B}/e$. According to
this estimate and typical material parameters, this leads to a critical
current density on the order of $10^{12}~{}{\rm A/m^{2}}$. Taking
$\rho=10^{-7}~{}{\rm\Omega\cdot m},~{}\kappa=50~{}{\rm W/\left(m\cdot
K\right)},L=50~{}{\rm nm}$ leads to only a modest increase in temperature,
less than $10~{}{\rm K}$. The other parameters of Eq. 4 for metals are
typically $E_{a}=10~{}{\rm eV},~{}\rho=10^{4}~{}{\rm
kg/m^{3}},~{}v=5000~{}{\rm m/s},~{}a=0.35~{}{\rm nm}$. In total, we find a $D$
parameter on the order of $10^{-5}~{}{\rm eV}^{2}$ to $10^{-4}~{}{\rm
eV}^{2}$. In light of Fig. 5, this implies that elastic phonon scattering does
not immediately destroy the current-induced torque for the no-canting system.
On the other hand, the much weaker current-induced torque per current of the
3Q system ($g=4\times 10^{-4}~{}\mu_{\rm B}/e$) requires a 100-fold increase
in the current to stabilize the out-of-plane orientation, a current density
which exceeds the maximum these systems can accommodate.
We’ve observed that the current-induced torque decays as $1/D$ for the no-
canting system. This is not universal behavior. Indeed, the current-induced
torque in the 3Q system is nonmonotonic with scattering parameter
$D$.footnote3 Despite its non-universality, we find it instructive to assume
such a dependence in order to derive closed form expressions for the maximum
current-induced torque as a function of applied current density. Recalling
that $D$ is proportional to $T$, we find the absolute current-induced torque
$\Gamma_{\rm abs}$ (units of torque) varies with current as:
$\displaystyle\Gamma_{\rm
abs}\left(J\right)=\frac{\Gamma_{0}J}{1+AD_{0}\left(T_{0}+BJ^{2}\right)},$ (5)
where $\Gamma_{0}$ is the current-induced torque in the absence of scattering
(recall $\Gamma_{0}$ has units of torque per current), $T_{0}$ is the sample
temperature in the absence of current, $B=L^{2}\rho/\kappa$ describes the
system’s susceptibility to current-induced heating, and $D_{0}$ is defined in
Eq. 4.footnote2 The absolute current-induced torque has a maximum - for
current densities that are too large, the magnitude of the current-induced
torque decreases due to increased scattering from Joule heating. The maximum
absolute current-induced torque is given by:
$\displaystyle\Gamma_{\rm abs}^{\rm
max}=\frac{\Gamma_{0}}{3L}\sqrt{\frac{2\kappa}{\rho
D_{0}A\left(1+D_{0}T_{0}A\right)}},$ (6)
The parameters $\Gamma_{0}$ and $A$ are entirely system specific, and related
to the spin dependent transport properties of a system, and their robustness
with respect to scattering. If the above maximum torque exceeds the damping
torque $\alpha\gamma M_{s}t_{\rm F}/2g$, then the out-of-plane configuration
can be stabilized by the current-induced torque. Intuitively, it’s
advantageous to use a low $M_{s}$ material in order to reduce the critical
current, and a thin multilayer to reduce heating. For scattering processes
with different functional dependence on $T$, a similar line of reasoning
applies, although the specific form of the maximum absolute current-induced
torque will differ. It’s straightforward to show that a $T^{p}$ dependence of
$D$ results in a maximum current-induced torque expression similar to Eq. 6,
where the expression inside the square root is taken to the power $1/2p$.
## IV Conclusion
This work demonstrates the role of symmetry and phase coherence effects in the
current-induced torque present between ferromagnet and antiferromagnetic
layers with a compensated interface. Basic symmetry arguments identify the
fixed points of the current-induced torque. We demonstrate that for an
antiferromagnetic layer with a 3Q spin structure, the current-induced torque
has a complex angular dependence, and the fixed points for the current-induced
torque are generally only local attractors. This is important because
experiments designed to drive the ferromagnet to these fixed points must
initialize the ferromagnet sufficiently nearby. We also show via explicit
calculations the primary role played by phase coherence for these torques, and
show an inverse relationship between the magnitude of the current-induced
torque and the phase breaking scattering parameter. In the antiferromagnetic
system with planar spins (the no-canted system), we find the current-induced
torque to be sufficiently robust to scattering to stabilize the out-of-plane
magnetic orientation, while for the 3Q ordered antiferromagnet, the current-
induced torque is too weak to stabilize this orientation. We expect that the
robustness of this torque to scattering should be system specific, determined
by which scattering processes are dominant and the system electronic
structure.
## V Acknowledgements
A.P. acknowledges support under the Cooperative Research Agreement between the
University of Maryland and the National Institute of Standards and Technology
Center for Nanoscale Science and Technology, Award 70NANB10H193, through the
University of Maryland.
## References
* (1) L. Berger, Phys. Rev. B 54, 9353 (1996).
* (2) J. Slonczewki, J. Magn. Magn. Mater. 159, L1 (1996).
* (3) D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).
* (4) A.S. Núñez, R.A. Duine, Paul Haney, A.H. MacDonald, Phys. Rev. B 73 214426 (2006).
* (5) P.M. Haney, R.A. Duine, A.S. N ez, A.H. MacDonald, J. Magn. Magn. Mat 320, 1300 (2008).
* (6) P. M. Haney, D. Waldron, R. A. Duine, A. S. Núñez, H. Guo, and A. H. MacDonald, Phys. Rev. B 75, 174428 (2007)
* (7) R.A. Duine, P.M. Haney, A.S. Núñez, A.H. MacDonald, Phys. Rev.B 75 014433 (2007).
* (8) Z. Wei, A. Sharma, A.S. Nu nez, P.M. Haney, R.A. Duine, J. Bass, A.H. MacDonald, M. Tsoi, Phys. Rev. Lett. 98 116603 (2007).
* (9) Paul M. Haney and A. H. MacDonald, Phys. Rev. Lett 100, 196801 (2008).
* (10) H. V. Gomonay, R. V. Kunitsyn, and V. M. Loktev, Phys. Rev. B, 85, 134446 (2012).
* (11) H. V. Gomonay and V. M. Loktev, Phys. Rev. B, 81, 144427 (2010).
* (12) K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys. Rev. Lett. 106, 107206 (2011).
* (13) E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and A. Brataas, Phys. Rev. Lett. 110, 127208 (2013).
* (14) A. C. Swaving and R. A. Duine, Phys. Rev. B 83, 054428 (2011).
* (15) R. Cheng and Q. Niu, Phys. Rev. B 86, 245118 (2012).
* (16) A. B. Shick, S. Khmelevskyi, O. N. Mryasov, J. Wunderlich, and T. Jungwirth, Phys. Rev. B 81, 212409 (2010).
* (17) B. G. Park, J. Wunderlich, X. Martí, V. Holý, Y. Kurosaki, M. Yamada, H. Yamamoto, A. Nishide, J. Hayakawa, H. Takahashi, A. B. Shick, and T. Jungwirth, Nat. Mat. 10, 347 (2011).
* (18) P. M. Haney, C. Heiliger, and M. D. Stiles, Phys. Rev. B 79, 054405 (2009).
* (19) F. Mahfouzi and B. K. Nikolic, ArXiv:1202.6602 (2012).
* (20) F. Jonietz, S. M hlbauer, C. Pfleiderer, A. Neubauer, W. Münzer, A. Bauer, T. Adams, R. Georgii,P. Böni, R. A. Duine, K. Everschor, M. Garst, A. Rosch, Science 330, 6011 (2010).
* (21) T. C. Schulthess, W. H. Butler, G. M. Stocks, S. Maat, and G. J. Mankey, J. Appl. Phys. 85, 4842 (1999).
* (22) We note that the true ground state may differ slightly from the 3Q configuration in the case of FeMn, as shown in Ref. stocks, . We expect that fluctuations or deviations from ordered configurations will decrease the effectiveness of symmetry-based torques.
* (23) G. Malcolm Stocks, W. A. Shelton, Thomas C. Schulthess, Balazs jfalussy, W. H. Butler, and A. Canning, J. Appl. Phys. 91, 7355 (2002).
* (24) S. Kawarazaki, Y. Sasaki, K. Yasuda, T. Mizusaki and A. Hirai, J. Phys.: Condens. Matter 2, 5747 (1990).
* (25) S. J. Kennedy and T. J. Hicksm, J. Phys. F: Met. Phys. 17, 1599 (1987).
* (26) M. Lundstrom, Fundamentals of Carrier Transport 2nd ed. Cambridge University Press (2000).
* (27) P. Mohanty, E. M. Q. Jariwala, and R. A. Webb, Phys. Rev. Lett. 78, 3366 (1997).
* (28) This expression for the current density required to stabilize the out-of-plane orientation assumes that there is no applied magnetic field, or other sources magnetic anisotropy. In this case, the damping torque from the hard-axis anisotropy for a magnetization with small tilt angle $\beta$ away from the hard-axis is $\gamma\alpha M_{s}\beta$, while the current-induced torque is $2gJ\beta/t_{\rm F}$. Equating these two leads to the form of current density given in the text.
* (29) We assume that the material paramters in Eq. 4 are weakly temperature dependent in this analysis.
* (30) For the 3Q system, the current-induced torque decays monotonically with scattering parameter $D$ for each state (i.e each $\bf k$-point) individually. However the sign of the current-induced torque varies by state, so that there is partial cancellation when summing over all states. The increase of the total current-induced torque at small $D$ is the result of less cancellation as the states’ torque, as each state’s contribution changes slightly.
|
arxiv-papers
| 2013-11-24T19:47:22 |
2024-09-04T02:49:54.165240
|
{
"license": "Public Domain",
"authors": "Karthik Prakya, Adrian Popescu, and Paul M. Haney",
"submitter": "Paul Haney Mr.",
"url": "https://arxiv.org/abs/1311.6161"
}
|
1311.6500
|
# Stitched Panoramas from Toy Airborne Video Cameras
###### Abstract
Effective panoramic photographs are taken from vantage points that are high.
High vantage points have recently become easier to reach as the cost of
quadrotor helicopters has dropped to nearly disposable levels.111 Disposal is
trickier than it sounds. I have reclaimed such aircraft undamaged after
extended periods on roofs, in trees, among cattle, and under fast-moving cars.
Although cameras carried by such aircraft weigh only a few grams, their low-
quality video can be converted into panoramas of high quality and high
resolution. Also, the small size of these aircraft vastly reduces the risks
inherent to flight.
## 1 Introduction
High-quality panoramic photographs can now be acquired from aircraft under 100
grams. This is desirable because these ``toys'' pose a far smaller risk than
aircraft carrying a camera that itself weighs more than 100 g. (Quality
cameras are so heavy because of their glass lenses. This is unlikely to change
soon.) The risk reduction can be quantified by estimating their reduced
gravitational potential energy ($0.05\times$ mass, $0.3\times$ height:
$0.015\times$ net), kinetic energy (mass as before, $0.4\times$ airspeed:
$0.008\times$), rotor kinetic energy (about $0.05\times$), and battery energy
in mWh ($0.05\times$). Financial risk due to aircraft damage or loss is also
reduced about twentyfold. Photography from places too confined or too risky
for larger aircraft becomes possible. Also, an aircraft small and light enough
to always keep with you encourages impromptu photography: these days, SLR
cameras take far fewer photos than mobile phones do.
Capturing video from sub-100 g aircraft is common [6], but no reports have
been published about capturing still photographs. This document's novel
contribution is a complete set of techniques for acquiring high-quality
panoramas from these aircraft: how to maneuver effectively, cope with wind,
extract still frames from a video recording, robustly and automatically cull
frames to avoid motion parallax and motion blur, suppress artifacts due to the
camera's poor quality, and record simultaneously from multiple cameras. These
techniques are all simple and inexpensive, as they should be for a toy.
Fig. 1: Two videocamera-equipped quadcopters, with a shared radio-control
transmitter.
### 1.1 Quadcopters
From 1990 to 2000, electric power for radio-controlled aircraft developed from
a curiosity to a commonplace, as batteries and motors improved to match the
sheer power of piston engines. Erasing that performance deficit left the
electric drivetrain with only advantages, notably reliability, less vibration,
and mechanical simplicity—often only one moving part.
During the next decade, electric and electronic technology continued to
improve, while consumer preference for mechanical simplicity remained high.
This technological progress then produced another commonplace: the quadrotor
helicopter, or quadcopter. Because differential thrust controlled pitch, roll,
and yaw, neither servomotors nor swashplates were needed, leaving the entire
aircraft with only four moving parts. Accelerometers and gyroscopes made
flight easy to learn. Pushing all of the aircraft's complexity into software
made it inexpensive to manufacture, maintenance-free, easy to repair, and
crash-resistant—if only because it weighed less than a gerbil. The price of
camera-equipped quadcopters, such as those in fig. 1, has fallen to USD 45
[20]. (The larger quadcopters that record sporting events are quite the
opposite: many moving parts, fussy maintenance, high fragility, and a price in
the thousands of dollars.)
Fig. 2: Full $360^{\circ}$ panorama. Trenton, Ontario, 2013-08-19.
Sub-100 g aircraft are inconspicuous and quiet: using one I photographed a
family wedding's outdoor reception, without anybody noticing. Quadcopters much
lighter than 100 g are now available, but would have been uncontrollable in
the gusty 10 knot winds that day (this 75 g one just managed). Although flying
animals much smaller than that shrug off such winds, a hummingbird's
performance won't soon be matched by consumer goods. A flying weight near 100
g will likely remain optimal for a few years. This small size also permits
unplanned opportunities to be exploited, such as fig. 2, captured in
midmorning while waiting ten minutes for the beer store to open. As the saying
goes, the best camera is the one you have with you.
## 2 Converting Video to a Panoramic Photo
The inexpensive videocamera commonly used for stealth or light weight has no
official name. Vendors call it a keychain camera; hobbyists call it an ``808''
[18]. Its attributes have changed monthly for some years, but are roughly:
weight 8 g, pixel resolution $640\times 480$ to $1280\times 800$, microSD card
storage, 30 or 60 frames per second, fixed focus, and depth of field 10
cm–$\infty$. Its 2 mm diameter lens performs poorly in low light.
The camera saves a video file in motion JPEG format, which is just a
soundtrack combined with individual JPEG images [17]. Because this format does
not exploit inter-frame redundancy, it produces files 3 to 10 times larger
than those made with the modern H.264 codec. This size is acceptable, though,
because it does not constrain recording—an 8 GB card easily stores a dozen
5-minute flights. In fact, were the camera's CPU advanced enough to compress
video better, its increased power consumption would deplete the battery
faster, paradoxically decreasing the duration of both a flight and its
recording.
Individual frames from the video file can be extracted with the open-source
software FFmpeg [11]:
ffmpeg -i in.avi -vsync 0 -vcodec png -f image2 %04d.png
This command produces images named 0001.png, 0002.png, …, 1138.png.222
Alternatively, the original JPEG frames can be very quickly extracted: ffmpeg
-i in.avi -vsync 0 -vcodec copy -f image2 %04d.jpg. (The option -vcodec jpg
should be avoided, because it transcodes and further degrades each frame
instead of just extracting it.) This shortcut is convenient for video good
enough to need no improvement with the tools listed in sections 4 and 5.1.
Dropped or missing frames occur with some camera–card combinations, or when
the camera's CPU is momentarily too slow. Naïve extraction of frames
``reconstructs'' these missing frames by repeatedly duplicating the previous
frame; this duplication would slow down image stitching.333 Proper
interpolation, which analyzes frame-to-frame motion, has been implemented for
some keychain cameras [25], but this interpolation improves only video, not
stitched panoramas. Many of these consecutive duplicate frames are removed
with FFmpeg's option -vsync 0. Removing _all_ duplicate frames requires a
duplicate-file finder, such as the Linux command fdupes --delete --noprompt
*.png. Because these finders use file size as a quick first test for
duplication, they are much slower with formats that give every frame the same
file size, such as .bmp and .ppm. The .png format does not suffer from this.
These image files are sent to an automatic image stitcher, such as the free
programs AutoStitch [4, 5] and Image Composite Editor [19]. The stitcher then
produces a single panoramic image (figs. 2 and 8).
Fig. 3: Mis-stitching due to camera movement. UIUC Arboretum, Urbana,
Illinois, 2013-05-16.
## 3 Flight Path
An image stitcher must assume that the photos it is given were taken from a
single viewpoint. Because a quadcopter is hardly a stationary tripod,
stitching the video recording of an entire flight spectacularly violates this
assumption (fig. 3). For a coherent panorama, only a subinterval of the
recording should be stitched.
A convenient way to record a stitchable subinterval is to yaw (pirouette) the
quadcopter while hovering. Some drifting is tolerable if the subject is not
very nearby, and if the pirouette is less than a complete circle. Stitching is
improved when frames have more overlap, which happens with slower yaw. The
slowest practical yaw for a 100 g quadcopter is about 0.4 rad/s (16 s for a
full pirouette); this is rarely slow enough to introduce other problems like
ghosting [8, 23].
Fig. 4: Different magenta-cyan moiré patterns on three identically corrugated
roofs. The roofs differ only in their distance from the camera. UIUC Dairy
Cattle Research Unit, 2013-08-01.
### 3.1 Choosing a Stitchable Subinterval
After landing, the video is viewed on a computer to find an interval where the
desired subject is visible. To maximize the panorama's coverage, the
interval's endpoints are extended, with two constraints (often identical):
exclusion of non-yaw flight and exclusion of different viewpoints of the
subject.
These constraints generally restrict the interval to a single monotonic yaw
maneuver. One might think that several back-and-forth pans cover the subject
more widely and give more information to the stitcher; but in practice each
pan is from a slightly different location, introducing seams like the one in
the fence in fig. 8.
The position of the interval's endpoints and the duration of the entire video
determine the endpoints as a fraction of the video's duration. These fractions
then approximate the filenames. For example, consider a video lasting 100 s,
with a stitched interval starting at 50 s and ending at 60 s, and filenames
0000.png to 3000.png. The interval's filenames will be approximately 1500.png
to 1600.png. Because of duplicated frames, the numbers 1500 and 1600 are only
approximate; manual verification is needed.
### 3.2 Video Downlink
After a few flights, most pilots develop an intuition for what the
quadcopter's camera is seeing. But if the quadcopter's height exceeds that
used to capture fig. 2, about 30 m, it becomes almost too small to see, let
alone aim its camera. If this is a concern, a live video downlink can be
added. But this expense is substantial compared to the stock quadcopter,
because many parts must be removed or replaced with lighter ones to compensate
for the extra payload [1]. Also, such first-person view (FPV) flight is
riskier because of its many single points of failure. Even for a 100 g
aircraft, never mind a 5 kg one, the prudent FPV pilot keeps the aircraft near
enough for line-of-sight control, and asks an assistant ``spotter'' to
maintain situational awareness.
## 4 Suppressing Camera Artifacts
A keychain camera's poor image quality may be evident in several ways: varying
brightness and color, rolling shutter, moiré bands, and JPEG compression
blockiness. Fortunately, these artifacts can be suppressed or even eliminated.
### 4.1 Varying Brightness and Color
Variations in brightness and color are due to the camera's automatic exposure
compensation and automatic white balance [23]. When the view changes suddenly
from, say, bright cumulus clouds to tree-shaded terrain, the camera takes a
second or two to correct its exposure. Similarly, when a view of only grass
suddenly tilts up to include sky, it takes a second for the white-balanced
grayish grass to become bright green. Frames from such transitions may not be
usable.
Reducing such variations requires slower aircraft rotation. After flight it
may be too late to correct the transitional frames if color is out of gamut,
or if shadows or highlights are clipped (lost detail, in pure black shadows or
pure white highlights).
### 4.2 Rolling Shutter
Rolling shutter is a motion artifact common to small cameras: the image is
captured one scanline at a time, instead of all at once. In other words,
different parts of the image correspond to different instants in time.
Therefore, moving the camera relative to the subject produces visible warp and
skew. (This can be demonstrated by waving one's hand in front of a
photocopier's scanner as it slides along.) As with varying brightness, slower
aircraft rotation is the first cure. Also, balancing the propellers with
flecks of adhesive tape reduces mechanical vibration, which causes what
hobbyists call ``jello'' in video [13].
Unlike varying brightness, though, rolling shutter can also be suppressed
after flying [2, 14]. Rolling shutter repair is included in commercial video
software such as Adobe Premiere Pro and Adobe After Effects, and in free video
software such as the Deshaker [22] plug-in for VirtualDub [15]. However, these
tools specialize in inter-frame smoothness, which panorama stitching does not
need. Worse, they may crop the image (which reduces the panorama's coverage),
or add a black border (which confuses the stitcher). If the border's color can
be made transparent, however, commercial stitchers such as Adobe's Photomerge
may succeed. Better yet, Deshaker can fill the border with pixels from
previous or successive frames, or, when those are unavailable, with colors
extrapolated from the current frame.
These tools require the individual frames to be re-encoded as a video file, to
give the detection-and-removal algorithm more material to work with: several
successive frames of the same subject, not just a single frame.
### 4.3 Moiré
The artifact called a moiré pattern consists of undesired bands of hue or
brightness (fig. 4), seen in a subject with repetitive detail, such as
stripes, that exceeds the camera's resolution. (The pattern is due to foldover
at the camera's Nyquist frequency. Non-toy cameras suppress these patterns
with anti-alias filters.) If the stitcher tries to match these bands, which
shift from frame to frame as the camera moves, stitching quality is reduced.
This is particularly so for stitchers that match image features by hue as well
as by brightness, because a camera sensor's Bayer filter mosiac easily
produces hue bands.
Avoiding moiré patterns requires such subjects to be either very distant, or
so close that each stripe is at least two pixels wide (for a keychain camera,
at most a few hundred stripes visible at once).
### 4.4 JPEG Compression Blockiness
Some JPEG frames are compressed so strongly that a grid appears at the
boundary between $8\times 8$ blocks of pixels. As with moiré patterns, this
noise varies from frame to frame, slightly distracting the stitcher from
finding common elements across frames. It also looks ugly in the final
panorama.
This artifact is suppressed by the UnBlock algorithm [10, 12], which smooths
over the boundaries between blocks, but only aggressively enough to reach the
same distribution of discrepancies across the block boundaries as is found in
the block interiors (fig. 5). This approach prevents worse artifacts from
being introduced as a side effect. The algorithm also needs no tuning.
## 5 Kites
In winds too strong for a lightweight quadcopter, it can nevertheless be given
a high vantage point by hanging it from a toy delta-wing kite (span 1.3 m,
cost USD 5). Even with its four booms removed to prevent its propellers from
getting tangled in the kite's tether, it may still operate as a power source
and remote control for the camera (fig. 6).
An elaborate Picavet suspension [3, 21] for the camera is not in the spirit of
cheap, simple hardware. On the other hand, just dangling the camera from the
tether can cause so much camera shake that fewer than one frame in a hundred
is usable for stitching (fig. 7). Happily, the shaking can be dampened by
hanging the camera from not one but two points on the tether, at the bottom of
a `V.' Then one frame in ten has acceptably low motion blur.
### 5.1 Motion Blur
Manual culling of frames blurred by camera motion is impractical. To automate
this, one can measure how blurred each frame is, and then sort the frames by
blurriness with a Schwartzian transform. Blurriness can be measured simply and
thus robustly by re-saving the frame in JPEG format, with and without first
applying a Gaussian blur. The smaller the ratio of the sizes of the two
resulting files, the less difference the Gaussian blur made, and thus the
blurrier the original frame. (A more elaborate method, culling any frame that
has few sharp edges compared to its neighboring frames [7], fails in the
presence of the duplicate frames mentioned in section 2).
Fig. 5: Detail ($160\times 160$ pixels) from top right of fig. 7. Left:
original. Right: processed by the UnBlock algorithm.
Fig. 6: Kite hoisting a rotorless quadcopter-camera (a ``nullicopter''), while
capturing fig. 7.
Fig. 7: Strong motion blur from a kite-suspended camera. Evergreens 5 to 15 m
tall, Okanogan-Wenatchee National Forest, 2013-05-22.
This algorithm is implemented by the Ruby script in listing 1. It uses the
ImageMagick program convert to read, blur, and save files. Because the
script's performance is strongly dominated by the blur computation,
downsampling precedes the blur to speed it up sixteenfold. The downsampling
also attenuates the sharp pixel-block boundaries described in section 4.4
(fig. 5, left). This is desirable because these sharp boundaries reduce how
well the Gaussian blur approximates the original motion blur—they hide the
smooth motion blur behind artificial crisp edges. Finally, each file is given
a symbolic link from a new directory, so the new directory contains filenames
sorted by blurriness rather than by time, for convenient manual inspection.
Listing 1: Ruby script to sort frames by blurriness.
⬇
#!/usr/bin/env ruby
$src = "/my_dir/frames_from_video"
$dst = "/my_dir/frames_sorted_by_blur"
‘rm -rf #$dst; mkdir -p #$dst‘
$tmp = "/run/shm/tmp" # fast ramdisk
$a = "#$tmp/a.jpg"
$b = "#$tmp/b.jpg"
‘mkdir -p #$tmp‘
pairs = []
Dir.glob($src + "/*.jpg") {|filename|
‘convert #{filename} -resize 25% -quality 50 #$a‘
‘convert #{filename} -resize 25% -gaussian-blur 4 -quality 50 #$b‘
blur = File.size($b).to_f / File.size($a) rescue 0.0
pairs << [filename,blur]
}
pairs.sort_by! {|filename,blur| blur}
pairs.each_with_index {|(oldname,blur),i|
newname = (’%05d’ % i) + ".jpg"
‘ln -s #{oldname} #$dst/#{newname}‘
}
Of course, a Gaussian blur only approximates a motion blur. But the exact
motion blur is a combination of axial rotation and panning, which is too
expensive to measure for this quick first pass that culls almost all of the
frames. Later passes can use advanced algorithms [9, 16], which can not only
detect but even remove mild blur by estimating camera motion from consecutive
frames—although these again fail for duplicate frames. This advanced
deblurring can also improve non-kite video.
## 6 Multiple Cameras
A quadcopter may have enough thrust to carry more than one camera. If each
camera points in a slightly different direction, the panorama gets more
coverage (fig. 8). This has been proposed for Parrot's AR.Drone quadcopter
(400 g, USD 400) [7], but no implementations to date have used sub-100 g
aircraft. More typical is DARPA's ARGUS-IS cluster of several hundred cameras
[24].
Fig. 8: Top: panorama stitched from one camera's frames. Bottom: second
camera's frames added. UIUC Large Animal Clinic, 2013-10-09.
If the quadcopter's maneuverability suffers with the extra payload of more
cameras, another novel solution is to laterally combine two or more
quadcopters into an octocopter (fig. 9), dodecacopter, or hexadecacopter.444
Owning several quadcopters is not unusual: it is an inexpensive way to buy
spare parts, because a significant part of a quadcopter’s mail-order cost is
shipping. Bamboo skewers make good struts, being cheap, lighter than even a
keychain camera, and almost as stiff as carbon fiber. (The transmitter in fig.
1 is unaware that it is controlling more than one quadcopter.) The composite
aircraft is slightly less maneuverable because the stabilizers in each
quadcopter fight each other, and because roll authority is reduced. But the
more important controls—pitch, yaw, and overall thrust—have no reduced
authority. As with multiple cameras on one quadcopter, each camera points at a
different angle.
Fig. 9: Two-camera octocopter, just before capturing fig. 8.
## 7 Future Work
Multiple cameras can record stereoscopic video, especially when mounted far
apart (large interpupillary distance) on an octocopter. Sound recorded with
each camera's rudimentary microphone helps to synchronize the individual
recordings.
Stereoscopic stitched panoramas can be made with only one camera, recording
two partial pirouettes from nearby locations (half pirouette left, scoot
forward a few seconds, then half pirouette right).
An objective measure for the quality of image processing pipelines could be
constructed. The challenge, for both synthetic imagery and hundred-frame
excerpts from actual flights, would be the continually changing attributes of
keychain cameras.
## 8 Conclusion
High-quality panoramic photos can be captured with a videocamera-equipped
quadcopter of startlingly small size, low cost, and low quality, thanks to
multiple stages of software post-processing. These stages can be applied to
whichever aspects of a particular panorama need improving.
Basic piloting skill is needed, but the more fundamental skill is choosing
where to fly and when not to fly. Even without these skills, though, loss of
flight control presents a hazard hardly greater than that of a stray Frisbee.
The same cannot be said of an aircraft powerful enough to carry a 100 g
camera.555 However, a secondary hazard can be posed by flying a 100 g
quadcopter in public. Because non-aeromodelers often lump together the risks
of _all_ aircraft too small to actually sit in, pilots should avoid misleading
bystanders into thinking that a larger aircraft in that situation would pose
no greater hazard.
For help in preparing this manuscript, I thank Kevyn Collins-Thompson, James
A. Crowell, Audrey Fisher, Farouk Gaffoor, David Gee, Michel Goudeseune, and
David Schilling.
## References
* [1] Gaffoor, F. _Micro FPV HD quad—WLtoys v929/949/959 FPV HD._ www.rcgroups.com/forums/showthread.php?t=1809332, 2013.
* [2] Baker, S., Bennett, E., Kang, S. B., and Szeliski, R. ``Removing rolling shutter wobble,'' in _Proc. IEEE Computer Vision and Pattern Recognition_ , pp. 2392-2399, 2010. http://dx.doi.org/10.1109/CVPR.2010.5539932
* [3] Beutnagel, R., Bieck, W., and Böhnke, O. ``Picavet—past and present,'' in _The Aerial Eye_ 1(4), p. 6, 1995.
* [4] Brown, M. _AutoStitch._ www.cs.bath.ac.uk/brown/autostitch/autostitch.html, 2013.
* [5] Brown, M., and Lowe, D. ``Automatic panoramic image stitching using invariant features,'' in _Intl. J. Computer Vision_ 74(1), pp. 59–73. 2007. http://dx.doi.org/10.1007/s11263-006-0002-3
* [6] Chen, J. _The Micro RTF Quadcopters Thread._ www.rcgroups.com/forums/showthread.php?t=1701910, 2013.
* [7] Chen, J., and Huang, C. ``Ghosting elimination with A* seam optimization in image stitching,'' in _Proc. Intl. Conf. on Information Security and Intelligence Control_ , pp. 214–217, 2012. http://dx.doi.org/10.1109/ISIC.2012.6449744
* [8] Chen, J., and Huang, C. ``Image stitching on the unmanned air vehicle in the indoor environment,'' in _Proc. Soc. Instrument and Control Engineers_ , pp. 402–406, 2012.
* [9] Cho, S., Wang, J, and Lee, S. ``Video deblurring for hand-held cameras using patch-based synthesis,'' in _ACM Trans. Graph._ 31(4), July 2012. http://dx.doi.org/10.1145/2185520.2185560
* [10] Costella, J. _The UnBlock algorithm._ http://johncostella.webs.com/unblock/unblock_paper.pdf, 2006.
* [11] FFmpeg. _FFmpeg._ www.ffmpeg.org, 2013.
* [12] Goudeseune, C. _UnBlock._ https://github.com/camilleg/unblock, 2013.
* [13] Graham, J. ``Multirotors for the beginner, part two,'' in _Model Aviation_ 39(5), pp. 77-79, May 2013.
* [14] Grundmann, M., Kwatra, V., Castro, D., and Essa, I. ``Calibration-free rolling shutter removal,'' in _Proc. IEEE Conf. Computational Photography_ , pp. 1–8, 2012. http://dx.doi.org/10.1109/ICCPhot.2012.6215213
* [15] Lee, A. _VirtualDub._ http://virtualdub.org, 2013.
* [16] Li, Y., Kang, S., Joshi, N., Seitz, S., and Huttenlocher, D. ``Generating sharp panoramas from motion-blurred videos,'' in _Proc. IEEE Computer Vision and Pattern Recognition_ , pp. 2424–2431, 2010. http://dx.doi.org/10.1109/CVPR.2010.5539938
* [17] Library of Congress. _Motion JPEG 2000 File Format._ www.digitalpreservation.gov/formats/fdd/fdd000127.shtml, 2013.
* [18] Lohr, C. _808 Car keys micro camera, micro video recorder, review._ www.chucklohr.com/808, 2013.
* [19] Microsoft Research. _Image composite editor._ http://research.microsoft.com/ivm/ice, 2013.
* [20] My RC Mart. _WL Toys V959 4ch 2.4Ghz 4-axis RTF quadcopter (built w/ camera)._ www.myrcmart.com, WL-V959-RTF, 2013.
* [21] Picavet, P. ``La photographie aérienne: suspension pendulaire elliptique,'' in _La Revue du Cerf-Volant,_ Nov. 1912.
* [22] Thalin, G. _Deshaker._ www.guthspot.se/video/deshaker.htm, 2013.
* [23] Uyttendaele, M., Eden, A., and Szeliski, R. ``Eliminating ghosting and exposure artifacts in image mosaics,'' in _Proc. IEEE Computer Vision and Pattern Recognition_ , pp. II-509–II-516, 2001. http://dx.doi.org/10.1109/CVPR.2001.991005
* [24] Vaidya, S. ``From video to knowledge,'' in _Science and Technology Review._ https://str.llnl.gov/AprMay11/pdfs/4.11.1.pdf, pp. 8–11, April/May 2011.
* [25] Y, Robert. _Get rid of your dropped frames._ www.rcgroups.com/forums/showpost.php?p=17691753, 2010.
|
arxiv-papers
| 2013-11-11T20:32:50 |
2024-09-04T02:49:54.182671
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Camille Goudeseune",
"submitter": "Camille Goudeseune",
"url": "https://arxiv.org/abs/1311.6500"
}
|
1311.6556
|
11institutetext: Data Mining Lab, GE Global Research, JFWTC, Whitefield,
Bangalore-560066,
(11email: [email protected], [email protected]), 22institutetext:
Bidgely, Bangalore (22email: [email protected]), 33institutetext: Sabre
Airline Solutions, Bangalore (33email: [email protected])
# Double Ramp Loss Based Reject Option Classifier
Naresh Manwani 11 Kalpit Desai 22 Sanand Sasidharan 11 Ramasubramanian
Sundararajan 33
###### Abstract
We consider the problem of learning reject option classifiers. The goodness of
a reject option classifier is quantified using $0-d-1$ loss function wherein a
loss $d\in(0,.5)$ is assigned for rejection. In this paper, we propose double
ramp loss function which gives a continuous upper bound for $(0-d-1)$ loss.
Our approach is based on minimizing regularized risk under the double ramp
loss using difference of convex (DC) programming. We show the effectiveness of
our approach through experiments on synthetic and benchmark datasets. Our
approach performs better than the state of the art reject option
classification approaches.
## 1 Introduction
The primary focus of classification problems has been on algorithms that
return a prediction on every example. However, in many real life situations,
it may be prudent to reject an example rather than run the risk of a costly
potential mis-classification. Consider, for instance, a physician who has to
return a diagnosis for a patient based on the observed symptoms and a
preliminary examination. If the symptoms are either ambiguous, or rare enough
to be unexplainable without further investigation, then the physician might
choose not to risk misdiagnosing the patient (which might lead to further
complications). He might instead ask for further medical tests to be
performed, or refer the case to an appropriate specialist. Similarly, a
banker, when faced with a loan application from a customer, may choose not to
decide on the basis of the available information, and ask for a credit bureau
score. While the follow-up actions might vary (asking for more features to
describe the example, or using a different classifier), the principal response
in these cases is to “reject” the example. This paper focuses on the manner in
which this principal response is decided, i.e., which examples should a
classifier reject, and why? From a geometric standpoint, we can view the
classifier as being possessed of a decision surface (which separates points of
different classes) as well as a rejection surface. The size of the rejection
region impacts the proportion of cases that are likely to be rejected by the
classifier, as well as the proportion of predicted cases that are likely to be
correctly classified. A well-optimized classifier with a reject option is the
one which minimizes the rejection rate as well as the mis-classification rate
on the predicted examples.
Let $\mathbf{x}\in\mathbb{R}^{p}$ is the feature vector and $y\in\\{-1,+1\\}$
is the class label. Let $\mathcal{D}(\mathbf{x},y)$ be the joint distribution
of $\mathbf{x}$ and $y$. A typical reject option classifier is defined using a
bandwidth parameter ($\rho$) and a separating surface ($f(\mathbf{x})=0$).
$\rho$ is the parameter which determines the rejection region. Then a reject
option classifier $h(f(\mathbf{x}),\rho)$ is formed as:
$\displaystyle h(f(\mathbf{x}),\rho)=\begin{cases}1&\text{if
}f(\mathbf{x})>\rho\\\ 0&\text{if }|f(\mathbf{x})|\leq\rho\\\ -1&\text{if
}f(\mathbf{x})<-\rho\end{cases}$ (1)
The reject option classifier can be viewed as two parallel surfaces with the
rejection area in between. The goal is to determine $f(\mathbf{x})$ as well as
$\rho$ simultaneously. The performance of this classifier is evaluated using
$L_{0-d-1}$ [13, 9] which is
$\displaystyle L_{0-d-1}(f(\mathbf{x}),y,\rho)=\begin{cases}1,&\text{if
}yf(\mathbf{x})<-\rho\\\ d,&\text{if }|f(\mathbf{x})|\leq\rho\\\
0,&\text{otherwise}\end{cases}$ (2)
In the above loss, $d$ is the cost of rejection. If $d=0$, then we will always
reject. When $d>.5$, then we will never reject (because expected loss of
random labeling is 0.5). Thus, we always take $d\in(0,.5)$.
To learn a reject option classifier, the expectation of $L_{0-d-1}(.,.,.)$
with respect to $\mathcal{D}(\mathbf{x},y)$ (risk) is minimized. Since
$\mathcal{D}(\mathbf{x},y)$ is fixed but unknown, the empirical risk
minimization principle is used. The risk under $L_{0-d-1}$ is minimized by
generalized Bayes discriminant [9, 4], which is as below:
$\displaystyle f_{d}^{*}(\mathbf{x})=\begin{cases}-1,&\text{if
}P(y=1|\mathbf{x})<d\\\ 0,&\text{if }d\leq P(y=1|\mathbf{x})\leq 1-d\\\
1,&\text{if }P(y=1|\mathbf{x})>1-d\end{cases}$ (3)
$h(f(\mathbf{x}),\rho)$ (equation (1)) is shown to be infinite sample
consistent with respect to the generalized Bayes classifier
$f^{*}_{d}(\mathbf{x})$ described in equation (3) [15].
Loss Function | Definition
---|---
Generalized Hinge | $L_{\text{GH}}(f(\mathbf{x}),y)=\begin{cases}1-\frac{1-d}{d}yf(\mathbf{x}),&\text{if }yf(\mathbf{x})<0\\\ 1-yf(\mathbf{x}),&\text{if }0\leq yf(\mathbf{x})<1\\\ 0,&\text{otherwise}\end{cases}$
Double Hinge | $L_{\text{DH}}(f(\mathbf{x}),y)=\max[-y(1-d)f(\mathbf{x})+H(d),-ydf(\mathbf{x})+H(d),0]$
| where $H(d)=-d\log(d)-(1-d)\log(1-d)$
Table 1: Convex surrogates for $L_{0-d-1}$.
Since minimizing the risk under $L_{0-d-1}$ is computationally cumbersome,
convex surrogates for $L_{0-d-1}$ have been proposed. Generalized hinge loss
$L_{\text{GH}}$ (see Table 1) is a convex surrogate for $L_{0-d-1}$ [13, 14,
3]. It is shown that a minimizer of risk under $L_{\text{GH}}$ is consistent
to the generalized Bayes classifier [3]. Double hinge loss $L_{\text{DH}}$
(see Table 1) is another convex surrogate for $L_{0-d-1}$ [7]. Minimizer of
the risk under $L_{\text{DH}}$ is shown to be strongly universally consistent
to the generalized Bayes classifier [7].
We observe that these convex loss functions have some limitations. For
example, $L_{\text{GH}}$ is a convex upper bound to $L_{0-d-1}$ provided
$\rho<1-d$ and $L_{\text{DH}}$ forms an upper bound to $L_{0-d-1}$ provided
$\rho\in(\frac{1-H(d)}{1-d},\frac{H(d)-d}{d})$ (see Fig. 1). Also, both
$L_{\text{GH}}$ and $L_{\text{DH}}$ increase linearly in the rejection region
instead of remaining constant. These convex losses can become unbounded for
misclassified examples with the scaling of parameters of $f$. Moreover,
limited experimental results are shown to validate the practical significance
of these losses [13, 14, 3, 7]. A non-convex formulation for learning reject
option classifier is proposed in [5]. However, theoretical guarantees for the
approach proposed in [5] are not known. While learning a reject option
classifier, one has to deal with the overlapping class regions as well as the
presence of outliers. SVM and other convex loss based approaches are less
robust to label noise and outliers in the data [11]. It is shown that ramp
loss based risk minimization is more robust to noise [6].
|
---|---
(a) | (b)
Figure 1: $L_{\text{GH}}$ and $L_{\text{DH}}$ for $d=0.2$. (a) For $\rho=0.7$,
both the losses upper bound the $L_{0-d-1}$. For $\rho=2$, both the losses
fail to upper bound $L_{0-d-1}$. $L_{\text{GH}}$ and $L_{\text{DH}}$ both
increase linearly even in the rejection region than being flat.
Motivated from this, we propose double ramp loss $(L_{\text{DR}})$ which
incorporates a different loss value for rejection. $L_{\text{DR}}$ forms a
continuous nonconvex upper bound for $L_{0-d-1}$ and overcomes many of the
issues of convex surrogates of $L_{0-d-1}$. To learn a reject option
classifier, we minimize the regularized risk under $L_{\text{DR}}$ which
becomes an instance of difference of convex (DC) functions. To minimize such a
DC function, we use difference of convex programming approach [1], which
essentially solves a sequence of convex programs. The proposed method has
following advantages over the existing approaches: (1) the proposed loss
function $L_{\text{DR}}$ gives a tighter upper bound to the $L_{0-d-1}$, (2)
$L_{\text{DR}}$ requires no constraint on $\rho$ unlike $L_{\text{GH}}$ and
$L_{\text{DH}}$, (3) our approach can be easily kernelized for dealing with
nonlinear problems.
The rest of the paper is organized as follows. In Section 2 we define the
double ramp loss function $(L_{\text{DR}})$ and discuss its properties. Then
we discussed the proposed formulation based on risk minimization under
$L_{\text{DR}}$. In Section 3 we derive the algorithm for learning reject
option classifier based on regularized risk minimization under
$(L_{\text{DR}})$ using DC programming. We present experimental results in
Section 4. We conclude the paper with the discussion in Section 5.
## 2 Proposed Approach
Our approach for learning classifier with reject option is based on minimizing
regularized risk under $L_{\text{DR}}$ (double ramp loss).
### 2.1 Double Ramp Loss
We define double ramp loss function as a continuous upper bound for
$L_{0-d-1}$. This loss function is defined as a sum of two ramp loss functions
as follows:
$\displaystyle L_{\text{DR}}(f(\mathbf{x}),y,\rho)$ $\displaystyle=$
$\displaystyle\frac{d}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})+\rho\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})+\rho\big{]}_{+}\Big{]}$
(4) $\displaystyle+\frac{(1-d)}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})-\rho\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})-\rho\big{]}_{+}\Big{]}$
Figure 2: $L_{\text{DR}}$ and $L_{0-d-1}$ : $\forall\mu\geq 0,\rho\geq 0$,
$L_{\text{DR}}$ is an upper bound for $L_{0-d-1}$.
where $[a]_{+}=\max(0,a)$. $\mu\in(0,1]$ defines the slope of ramps in the
loss function. $d\in(0,.5)$ is the cost of rejection and $\rho\geq 0$ is the
parameter which defines the size of the rejection region around the
classification boundary $f(\mathbf{x})=0$.111While $L_{\text{DR}}$ is
parametrized by $\mu$ and $d$ as well, we omit them for the sake of notational
consistency. As in $L_{0-d-1}$, $L_{\text{DR}}$ also considers the region
$[-\rho,\rho]$ as rejection region. Fig. 2 shows $L_{\text{DR}}$ for
$d=0.2,\rho=2$ with different values of $\mu$.
###### Theorem 2.1
(1) $L_{\text{DR}}\geq L_{0-d-1},\;\forall\mu>0,\rho\geq 0$. (2)
$\lim_{\mu\rightarrow
0}L_{\text{DR}}(f(\mathbf{x}),\rho,y)=L_{0-d-1}(f(\mathbf{x}),\rho,y)$. (3) In
the rejection region $yf(\mathbf{x})\in(\rho-\mu^{2},-\rho+\mu)$, the loss
remains constant, that is $L_{\text{DR}}(f(\mathbf{x}),y,\rho)=d(1+\mu)$. (4)
For $\mu>0$, $L_{\text{DR}}\leq(1+\mu),\;\forall\rho\geq 0,\;\forall d\geq 0$.
(5) When $\rho=0$, $L_{\text{DR}}$ is same as $\mu$-ramp loss ([12])used for
classification problems without rejection option. (6) $L_{\text{DR}}$ is a
non-convex function of $(yf(\mathbf{x}),\rho)$.
The proof of Theorem 2.1 is provided in Appendix 0.A. We see that
$L_{\text{DR}}$ does not put any restriction on $\rho$ for it to be an upper
bound of $L_{0-d-1}$. Thus, $L_{\text{DR}}$ is a general ramp loss function
which also allows rejection option.
### 2.2 Risk Formulation Using $L_{\text{DR}}$
Let $\mathcal{S}=\\{(\mathbf{x}_{n},y_{n}),\;n=1\ldots N\\}$ be the training
dataset, where
$\mathbf{x}_{n}\in\mathbb{R}^{p},\;y_{n}\in\\{-1,+1\\},\;\forall n$. As
discussed, we minimize regularized risk under $L_{\text{DR}}$ to find a reject
option classifier. In this paper, we use $l_{2}$ regularization. Let
$\Theta=[\mathbf{w}^{T}\;\;\;b\;\;\;\rho]^{T}$. Thus, for
$f(\mathbf{x})=(\mathbf{w}^{T}\phi(\mathbf{x})+b)$, regularized risk under
double ramp loss is
$\displaystyle R(\Theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+C\sum_{n=1}^{N}L_{\text{DR}}(y_{n},\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)$
$\displaystyle=$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\Big{\\{}d\big{[}\mu-
y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}-d\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}$
$\displaystyle+(1-d)\big{[}\mu-
y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}-(1-d)\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}\Big{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\Big{\\{}d\big{[}\mu-
y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}+(1-d)\big{[}\mu-
y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}$
$\displaystyle-d\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}-(1-d)\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}\Big{\\}}$
where $C$ is regularization parameter. While minimizing $R(\Theta)$, no non-
negativity condition on $\rho$ is required due to the following lemma.
###### Lemma 1
At the minimum of $R(\Theta)$, $\rho$ must be non-negative.
Prood of the above lemma is provided in Appendix 0.B.
## 3 Solution methodology
$R(\Theta)$ (equation (2.2)) is a nonconvex function of $\Theta$. However,
$R(\Theta)$ can be written as $R(\Theta)=R_{1}(\Theta)-R_{2}(\Theta)$, where
$R_{1}(\Theta)$ and $R_{2}(\Theta)$ are convex functions of $\Theta$.
$\displaystyle R_{1}(\Theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\Big{[}d\big{[}\mu-
y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}+(1-d)\big{[}\mu-
y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}\Big{]}$ $\displaystyle R_{2}(\Theta)$
$\displaystyle=$
$\displaystyle\frac{C}{\mu}\sum_{n=1}^{N}\Big{[}d\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}+(1-d)\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}\Big{]}$
In this case, DC programming guarantees to find a local optima of $R(\Theta)$
[1]. In the simplified DC algorithm [1], an upper bound of $R(\Theta)$ is
found using the convexity property of $R_{2}(\Theta)$ as follows.
$\displaystyle R(\Theta)\leq
R_{1}(\Theta)-R_{2}(\Theta^{(l)})-(\Theta-\Theta^{(l)})^{T}\nabla
R_{2}(\Theta^{(l)})=:ub(\Theta,\Theta^{(l)})$ (5)
where $\Theta^{(l)}$ is the parameter vector after $(l)^{th}$ iteration,
$\nabla R_{2}(\Theta^{(l)})$ is a sub-gradient of $R_{2}$ at $\Theta^{(l)}$.
$\Theta^{(l+1)}$ is found by minimizing $ub(\Theta,\Theta^{(l)})$. Thus,
$R(\Theta^{(l+1)})\leq ub(\Theta^{(l+1)},\Theta^{(l)})\leq
ub(\Theta^{(l)},\Theta^{(l)})=R(\Theta^{(l)})$. Which means, in every
iteration, the DC program reduces the value of $R(\Theta)$.
### 3.1 Learning Reject Option Classifier Using DC Programming
In this section, we will derive a DC algorithm for minimizing $R(\Theta)$. We
initialize with $\Theta=\Theta^{(0)}$. For any $l\geq 0$, we find
$ub(\Theta,\Theta^{(l)})$ as an upper bound for $R(\Theta)$ (see equation (5))
as follows:
$ub(\Theta,\Theta^{(l)})=R_{1}(\Theta)-R_{2}(\Theta^{(l)})-(\Theta-\Theta^{(l)})^{T}\nabla
R_{2}(\Theta^{(l)})$
Given $\Theta^{(l)}$, we find $\Theta^{(l+1)}$ by minimizing the upper bound
$ub(\Theta,\Theta^{(l)})$. Thus,
$\displaystyle\Theta^{(l+1)}\in\arg\min_{\Theta}\;ub(\Theta,\Theta^{(l)})=\arg\min_{\Theta}\;R_{1}(\Theta)-\Theta^{T}\nabla
R_{2}(\Theta^{(l)})$ (6)
where $\nabla R_{2}(\Theta^{(l)})$ is the subgradient of $R_{2}(\Theta)$ at
$\Theta^{(l)}$. We choose $\nabla R_{2}(\Theta^{(l)})$ as:
$\displaystyle\nabla
R_{2}(\Theta^{(l)})=\sum_{n=1}^{N}\beta_{n}^{\prime(l)}[-y_{n}\phi(\mathbf{x}_{n})^{T}\;\;-y_{n}\;\;1]^{T}+\sum_{n=1}^{N}\beta_{n}^{\prime\prime(l)}[-y_{n}\phi(\mathbf{x}_{n})^{T}\;\;-y_{n}\;\;-1]^{T}$
where
$\displaystyle\begin{cases}\beta_{n}^{\prime(l)}=\frac{Cd}{\mu}\mathbb{I}_{\\{y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l)}+b^{(l)})-\rho^{(l)}<-\mu^{2}\\}}\\\
\beta_{n}^{\prime\prime(l)}=\frac{C(1-d)}{\mu}\mathbb{I}_{\\{y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l)}+b^{(l)})+\rho^{(l)}<-\mu^{2}\\}}\end{cases}$
(7)
For $f(\mathbf{x})=(\mathbf{w}^{T}\phi(\mathbf{x})+b$, we rewrite the upper
bound minimization problem described in equation (6) as follows,
$\displaystyle P^{(l+1)}$ $\displaystyle=\min_{\Theta}$ $\displaystyle
R_{1}(\Theta)-\Theta^{T}\nabla R_{2}(\Theta^{(l)})$
$\displaystyle=\smash{\displaystyle\min_{\mathbf{w},b,\rho}}$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\Big{[}d\big{[}\mu-
y_{n}f(\mathbf{x}_{n})+\rho\big{]}_{+}+(1-d)\big{[}\mu-
y_{n}f(\mathbf{x}_{n})-\rho\big{]}_{+}\Big{]}$
$\displaystyle+\sum_{n=1}^{N}\beta_{n}^{\prime(l)}[y_{n}f(\mathbf{x}_{n})-\rho]+\sum_{n=1}^{N}\beta_{n}^{\prime\prime(l)}[y_{n}f(\mathbf{x}_{n})+\rho]$
Note that $P^{(l+1)}$ is a convex optimization problem where the optimization
variables are $(\mathbf{w},b,\rho)$. We rewrite $P^{(l+1)}$ as
$\displaystyle P^{(l+1)}=$
$\displaystyle\smash{\displaystyle\min_{\mathbf{w},b,\mbox{\boldmath$\xi$}^{\prime},\mbox{\boldmath$\xi$}^{\prime\prime},\rho}}$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\big{[}d\xi_{n}^{\prime}+(1-d)\xi_{n}^{\prime\prime}\big{]}+\sum_{n=1}^{N}\beta_{n}^{\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\rho]$
$\displaystyle+\sum_{n=1}^{N}\beta_{n}^{\prime\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]$
$\displaystyle s.t.$ $\displaystyle
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq\rho+\mu-\xi_{n}^{\prime},\;\;\;\xi_{n}^{\prime}\geq
0,\;\;\;n=1\ldots N$ $\displaystyle
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq-\rho+\mu-\xi_{n}^{\prime\prime},\;\;\;\xi_{n}^{\prime\prime}\geq
0\;\;\;n=1\ldots N$
where
$\mbox{\boldmath$\xi$}^{\prime}=[\xi_{1}^{\prime}\;\;\xi_{2}^{\prime}\ldots\xi_{N}^{\prime}]^{T}$
and
$\mbox{\boldmath$\xi$}^{\prime\prime}=[\xi_{1}^{\prime\prime}\;\;\xi_{2}^{\prime\prime}\ldots\xi_{N}^{\prime\prime}]^{T}$.
The dual optimization problem $D^{(l+1)}$ of $P^{(l+1)}$ is as follows.
$\displaystyle D^{(l+1)}=$
$\displaystyle\smash{\displaystyle\min_{\mbox{\boldmath$\gamma$}^{\prime},\mbox{\boldmath$\gamma$}^{\prime\prime}}}$
$\displaystyle\frac{1}{2}\sum_{n=1}^{N}\sum_{m=1}^{N}y_{n}y_{m}(\gamma^{\prime}_{n}+\gamma_{n}^{\prime\prime})(\gamma^{\prime}_{m}+\gamma_{m}^{\prime\prime})k(\mathbf{x}_{n},\mathbf{x}_{m})-\mu\sum_{n=1}^{N}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})$
$\displaystyle s.t.$
$\displaystyle\begin{cases}-\beta_{n}^{\prime(l)}\leq\gamma_{n}^{\prime}\leq\frac{Cd}{\mu}-\beta_{n}^{\prime(l)}&n=1\ldots
N\\\
-\beta_{n}^{\prime\prime(l)}\leq\gamma_{n}^{\prime\prime}\leq\frac{C(1-d)}{\mu}-\beta_{n}^{\prime\prime(l)}&n=1\ldots
N\\\
\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})=0\;\;\sum_{n=1}^{N}(\gamma_{n}^{\prime}-\gamma_{n}^{\prime\prime})=0&\end{cases}$
where
$\mbox{\boldmath$\gamma$}^{\prime}=[\gamma_{1}^{\prime}\;\;\gamma_{2}^{\prime}\ldots\ldots\gamma_{n}^{\prime}]^{T}$
and
$\mbox{\boldmath$\gamma$}^{\prime\prime}=[\gamma_{1}^{\prime\prime}\;\;\gamma_{2}^{\prime\prime}\ldots\ldots\gamma_{n}^{\prime\prime}]^{T}$
are dual variables. The derivation of dual $D^{(l+1)}$ can be seen in Appendix
0.C. At the optimality of $P^{(l+1)}$, $\mathbf{w}$ can be found as
$\mathbf{w}=\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})\phi(\mathbf{x}_{n})$.
Since $P^{(l+1)}$ has quadratic objective and linear constraints, it holds
strong duality with $D^{(l+1)}$. Solving $D^{(l+1)}$ is more useful as it can
be easily kernelized for non-linear problems. Behavior of
$\gamma_{n}^{\prime}$ and $\gamma_{n}^{\prime\prime}$ under different cases is
as follows.
$\displaystyle\begin{cases}y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\mu>\rho&\Rightarrow\gamma_{n}^{\prime}=-\beta_{n}^{\prime(l)};\;\;\gamma_{n}^{\prime\prime}=-\beta_{n}^{\prime\prime(l)}\\\
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\mu=\rho&\Rightarrow\gamma_{n}^{\prime}\in\big{(}-\beta_{n}^{\prime(l)},\frac{Cd}{\mu}-\beta_{n}^{\prime(l)}\big{)};\;\;\gamma_{n}^{\prime\prime}=-\beta_{n}^{\prime\prime(l)}\\\
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\mu\in(-\rho,\rho)&\Rightarrow\gamma_{n}^{\prime}=\frac{Cd}{\mu}-\beta_{n}^{\prime(l)};\;\;\gamma_{n}^{\prime\prime}=-\beta_{n}^{\prime\prime(l)}\\\
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\mu=-\rho&\Rightarrow\gamma_{n}^{\prime}=\frac{Cd}{\mu}-\beta_{n}^{\prime(l)};\;\;\gamma_{n}^{\prime\prime}\in\big{(}-\beta_{n}^{\prime\prime(l)},\frac{C(1-d)}{\mu}-\beta_{n}^{\prime\prime(l)}\big{)}\\\
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\mu<-\rho&\Rightarrow\gamma_{n}^{\prime}=\frac{Cd}{\mu}-\beta_{n}^{\prime(l)};\;\;\gamma_{n}^{\prime\prime}=\frac{C(1-d)}{\mu}-\beta_{n}^{\prime\prime(l)}\end{cases}$
### 3.2 Finding $b^{(l+1)}$ and $\rho^{(l+1)}$
The dual optimization problem above gives dual variables
$\mbox{\boldmath$\gamma$}^{\prime(l+1)}$ and
$\mbox{\boldmath$\gamma$}^{\prime\prime(l+1)}$ using which the normal vector
is found as
$\mathbf{w}^{(l+1)}=\sum_{n=1}^{N}(\gamma_{n}^{\prime(l+1)}+\gamma_{n}^{\prime\prime(l+1)})y_{n}\phi(\mathbf{x}_{n})$.
To find $b^{(l+1)}$ and $\rho^{(l+1)}$, we consider
$\mathbf{x}_{n}\in\text{SV}^{\prime(l+1)}\cup\text{SV}^{\prime\prime(l+1)}$,
where
$\displaystyle\text{SV}^{\prime(l+1)}$ $\displaystyle=$
$\displaystyle\\{\mathbf{x}_{n}\;|\;y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l+1)}+b^{(l+1)})=\rho^{(l+1)}+\mu\\}$
$\displaystyle\text{SV}^{\prime\prime(l+1)}$ $\displaystyle=$
$\displaystyle\\{\mathbf{x}_{n}\;|\;y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l+1)}+b^{(l+1)})=-\rho^{(l+1)}+\mu\\}$
We already saw that
1. 1.
If $\mathbf{x}_{n}\in\text{SV}^{\prime(l+1)}$, then
$\gamma_{n}^{\prime(l+1)}\in\big{(}-\beta_{n}^{\prime(l)},\frac{Cd}{\mu}-\beta_{n}^{\prime}{(l)}\big{)}$
and $\gamma_{n}^{\prime\prime(l+1)}=-\beta_{n}^{\prime\prime(l)}$
2. 2.
If $\mathbf{x}_{n}\in\text{SV}^{\prime\prime(l+1)}$, then
$\gamma_{n}^{\prime(l+1)}=\frac{Cd}{\mu}-\beta_{n}^{\prime(l)}$ and
$\gamma_{n}^{\prime\prime(l+1)}\in\big{(}-\beta_{n}^{\prime\prime(l)},\frac{C(1-d)}{\mu}-\beta_{n}^{\prime\prime(l)}\big{)}$
We solve the system of linear equations corresponding to sets
$\text{SV}^{\prime(l+1)}$ and $\text{SV}^{\prime\prime(l+1)}$ for identifying
$b^{(l+1)}$ and $\rho^{(l+1)}$.
### 3.3 Summary of the Algorithm
We fix $d\in[0,.5]$, $\mu\in(0,1]$ and $C$ and initialize the parameter vector
$\Theta$ as $\Theta^{(0)}$. In any iteration $(l)$, we find
$\beta_{n}^{\prime(l)},\beta_{n}^{\prime\prime(l)},\;n=1\ldots N$ (see
equation (7))using $\Theta^{(l)}$. We use
$\beta_{n}^{\prime(l)},\beta_{n}^{\prime\prime(l)},\;n=1\ldots N$ and solve
$D^{(l+1)}$ to find
$\mbox{\boldmath$\gamma$}^{\prime(l+1)},\mbox{\boldmath$\gamma$}^{\prime\prime(l+1)}$.
$\mathbf{w}^{(l+1)}$ is found as
$\mathbf{w}^{(l+1)}=\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime(l+1)}+\gamma_{n}^{\prime\prime(l+1)})\phi(\mathbf{x}_{n})$.
We find $b^{(l+1)}$ and $\rho^{(l+1)}$ as described in Section 3.2. Thus, we
have found $\Theta^{(l+1)}$. Using $\Theta^{(l+1)}$, we now find
$\beta_{n}^{\prime(l+1)},\beta_{n}^{\prime\prime(l+1)},\;n=1\ldots N$. We
repeat the above two steps until the parameter vector $\Theta$ changes
significantly. More formal description of our algorithm is provided in
Algorithm 1.
Algorithm 1 Learning Reject Option Classifier by Minimizing $R(\Theta)$
Input : $d\in[0,.5],\;\mu\in(0,1],\;C>0$, $\mathcal{S}$
Output : $\mathbf{w}^{*},b^{*},\rho^{*}$
Initialize $\mathbf{w}^{(0)},b^{(0)},\rho^{(0)}$, $l=0$
repeat
Compute
$\beta_{n}^{\prime(l)}=\frac{Cd}{\mu}\mathbb{I}_{\\{y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l)}+b^{(l)})-\rho^{(l)}<-\mu^{2}\\}}$
$\beta_{n}^{\prime\prime(l)}=\frac{C(1-d)}{\mu}\mathbb{I}_{\\{y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l)}+b^{(l)})+\rho^{(l)}<-\mu^{2}\\}}$
Find
$\mbox{\boldmath$\gamma$}^{\prime(l+1)},\mbox{\boldmath$\gamma$}^{\prime\prime(l+1)}$
by solving $D^{(l+1)}$ described in equation (3.1)
Find
$\mathbf{w}^{(l+1)}=\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime(l+1)}+\gamma_{n}^{\prime\prime(l+1)})\phi(\mathbf{x}_{n})$
Find $b^{(l+1)}$ and $\rho^{(l+1)}$ by solving the system of linear equations
corresponding to sets $\text{SV}_{1}^{(l+1)}$ and $\text{SV}_{2}^{(l+1)}$,
where $\displaystyle\text{SV}^{\prime(l+1)}$ $\displaystyle=$
$\displaystyle\\{\mathbf{x}_{n}\;|\;y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l+1)}+b^{(l+1)})=\rho^{(l+1)}+\mu\\}$
$\displaystyle\text{SV}^{\prime\prime(l+1)}$ $\displaystyle=$
$\displaystyle\\{\mathbf{x}_{n}\;|\;y_{n}(\phi(\mathbf{x}_{n})^{T}\mathbf{w}^{(l+1)}+b^{(l+1)})=-\rho^{(l+1)}+\mu\\}$
until convergence of $\Theta^{(l)}$
### 3.4 $\mbox{\boldmath$\gamma$}^{\prime}$ and
$\mbox{\boldmath$\gamma$}^{\prime\prime}$ at the Convergence of Algorithm 1
At the convergence of Algorithm 1, let
$\gamma_{n}^{\prime*},\gamma_{n}^{\prime\prime*},\;n=1\ldots N$ become the
values of the dual variables. The behavior of $\gamma_{n}^{\prime*}$ and
$\gamma_{n}^{\prime\prime*}$ is described in Table 2. For any
$\mathbf{x}_{n}$, only one of $\gamma_{n}^{\prime*}$ and
$\gamma_{n}^{\prime\prime*}$ can be nonzero. We observe that parameters
$\mathbf{w},b$ and $\rho$ are determined by the points whose margin
($yf(\mathbf{x})$) is in the range
$[\rho-\mu^{2},\rho+\mu]\cup[-\rho-\mu^{2},-\rho+\mu]$. We call these points
as support vectors. We also see that $\mathbf{x}_{n}$ for which
$y_{n}f(\mathbf{x}_{n})\in(\rho+\mu,\infty)\cup(-\rho+\mu,\rho-\mu^{2})\cup(-\infty,-\rho-\mu^{2})$,
both $\gamma_{n}^{\prime*},\gamma_{n}^{\prime\prime*}=0$. Thus, points which
are correctly classified with margin at least $(\rho+\mu)$, points falling
close to the decision boundary with margin in the interval
$(-\rho+\mu,\rho-\mu^{2})$ and points misclassified with a high negative
margin (less than $-\rho-\mu^{2}$), are ignored in the final classifier. Thus,
our approach not only rejects points falling in the overlapping region of
classes, it also ignores potential outliers. We illustrate these insights
through experiments on a synthetic dataset as shown in Fig. 3. 400 points are
uniformly sampled from the square region $[0\;\;1]\times[0\;\;1]$. We consider
the diagonal passing through the origin as the separating surface and assign
labels $\\{-1,+1\\}$ to all the points using it. We changed the labels of 80
points inside the band (width=0.225) around the separating surface.
Condition | $\gamma_{n}^{\prime*}\in$ | $\gamma_{n}^{\prime\prime*}\in$
---|---|---
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\in(\rho+\mu,\infty)$ | 0 | 0
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)=\rho+\mu$ | $(0,\frac{Cd}{\mu})$ | 0
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\in[\rho-\mu^{2},\rho+\mu)$ | $\frac{Cd}{\mu}$ | 0
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\in(-\rho+\mu,\rho-\mu^{2})$ | 0 | 0
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)=-\rho+\mu$ | 0 | $(0,\frac{C(1-d)}{\mu})$
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\in[-\rho-\mu^{2},-\rho+\mu)$ | 0 | $\frac{C(1-d)}{\mu}$
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\in(-\infty,-\rho-\mu^{2})$ | 0 | 0
Table 2: Behavior of $\mbox{\boldmath$\gamma$}^{\prime*}$ and $\mbox{\boldmath$\gamma$}^{\prime\prime*}$ |
---|---
Figure 3: Figure on left shows that label noise affects points near the true
classification boundary. Classes are represented using empty circles and
triangles. Figure on right shows reject option classifier learnt using the
proposed $L_{\text{DR}}$ based approach ($C=100$, $\mu=1$, $d=.2$). Filled
circles and triangles represent the support vectors.
Fig. 3 shows the reject option classifier learnt using the proposed method. We
see that the proposed approach learns the rejection region accurately. We also
observe that all of the support vectors are near the two parallel hyperplanes.
## 4 Experimental Results
We show the effectiveness of our approach by showing its performance on
several datasets. We also compare our approach with the approach proposed in
[7].
### 4.1 Dataset Description
We report experimental results on 1 synthetic datasets and 2 datasets taken
from UCI ML repository [2].
1. 1.
Synthetic Dataset 1 : Let $f_{1}$ and $f_{2}$ be two mixture density functions
in $\mathbb{R}^{2}$ defined as follows:
$\displaystyle
f_{1}(\mathbf{x})=0.45\mathcal{U}([1,0]\times[1,1])+0.5\mathcal{U}([4,3]\times[0,1])+0.05\mathcal{U}([10,0]\times[5,5])$
$\displaystyle
f_{2}(\mathbf{x})=0.45\mathcal{U}([0,1]\times[1,1])+0.5\mathcal{U}([9,10]\times[1,0])+0.05\mathcal{U}([0,10]\times[5,5])$
where $\mathcal{U}(A)$ denotes the uniform density function with support set
$A$. We sample 150 points independently each from $f_{1}$ and $f_{2}$. We
label these points using the hyperplane with $\mathbf{w}=[1\;\;\;0]^{T}$ and
$b=0$. We choose 10% of these points uniformly at random and flip their
labels.
2. 2.
Synthetic Dataset 2 [8] : $\mathbf{m}_{k1},k=1,\ldots,10$ were drawn from
$\mathcal{N}((1,0)^{T},I)$ and labeled as class $C_{1}$. Similarly,
$\mathbf{m}_{k2},\;k=1,\ldots,10$ were drawn from $\mathcal{N}((0,1)^{T},I)$
and labeled as class $C_{2}$. For each class, 100 observations were drawn from
the following mixture distributions:
$f(\mathbf{x}|C_{i})=\sum_{k=1}^{10}\frac{1}{10}\mathcal{N}(\mathbf{m}_{ki},I/5),\;\;\;i=1,2$
3. 3.
Ionosphere Dataset [2] : This dataset describes the problem of discriminating
good versus bad radars based on whether they send some useful information
about the Ionosphere. There are 34 variables and 351 observations.
4. 4.
Parkinsons Disease Dataset [2] : This dataset is used to discriminate people
with Parkinsons disease from the healthy people. There are 22 features which
are comprised of a range of biomedical voice measurements from individuals.
There are 195 such feature vectors.
### 4.2 Experimental Setup
In the proposed $L_{\textbf{DR}}$ based approach, for solving the dual
$D^{(l)}$ at every iteration, we have used the kernlab package [10] in R. We
thank the authors of $L_{\text{DH}}$ based method [7] for providing the codes
for their approach. For nonlinear problems, we use RBF kernel. In our
approach, we set $\mu=1$. $C$ and $\sigma$ (width parameter for RBF kernel)
are chosen using 10-fold cross validation.
### 4.3 Simulation Results
For every dataset, we report results for values of $d$ in the interval
$[0.05\;\;\;.5]$ with the step size of 0.05. For every value of $d$, we find
the cross validation risk (under $L_{0-d-1}$), % accuracy on the non-rejected
examples (Acc) and % rejection rate (RR). The results provided are based on 10
repetitions of 10-fold cross validation (CV). We show the average values and
standard deviation (computed over the 10 repetitions).
We now discuss the experimental results. Fig. 4(a) shows the Synthetic dataset
and the true classification boundary. This dataset has some mislabeled points
creating noise around the classification surface. Fig. 4(b) and (c) show the
classifiers learnt using $L_{\text{DR}}$ and $L_{\text{DH}}$ based approaches
respectively for $d=0.2$. We see that $L_{\text{DR}}$ based approach
accurately finds the true classification boundary as oppose to $L_{\text{DH}}$
based approach. Also, the reject region found by $L_{\text{DR}}$ based
approach is covers the most ambiguous region unlike $L_{\text{DH}}$ based
approach which rejects almost all the points.
| |
---|---|---
(a) | (b) | (c)
Figure 4: (a) Synthetic Dataset and the true classification boundary. Reject
option classifiers learnt using (b) proposed $L_{DR}$ based approach for
$d=0.2$, (c) $L_{DH}$ based approach for $d=0.2$.
Table 3-6 show the experimental results on all the datasets. We observe the
following:
1. 1.
We see that the proposed $L_{\text{DR}}$ based method outperforms
$L_{\text{DH}}$ based approach in terms of the risk (expectation of
$L_{0-d-1}$). For Synthetic dataset 1, except for $d=0.05$ and $0.1$,
$L_{\text{DR}}$ based method has lower CV risk. For Synthetic dataset 2, both
the approaches perform comparable to each other. For Ionosphere dataset,
except for $d=0.2,0.25$ and $0.3$, $L_{\text{DR}}$ based method has lower CV
risk. For Parkinsons dataset, $L_{\text{DR}}$ based method has lower CV risk
except for $d=0.35$.
2. 2.
We also observe that $L_{\text{DR}}$ based method outputs classifiers with
significantly lesser rejection rate for all the datasets and for all values of
$d$.
Thus, for most of the cases, the proposed $L_{\text{DR}}$ based approach
outputs classifiers with lesser risk. Moreover, the learnt classifier has
always lesser rejection rate compared to the $L_{\text{DH}}$ based approach.
d | $L_{\text{DR}}$ ($C=2$) | $L_{\text{DH}}$ ($C=32$)
---|---|---
| Risk | RR | Acc on unrejected | Risk | RR | Acc on unrejected
0.05 | 0.068$\pm$0.015 | 90.87$\pm$5.79 | 75.87$\pm$7.95 | 0.05 | 100 | NA
0.1 | 0.138$\pm$0.023 | 70.35$\pm$12.18 | 79.05$\pm$6.87 | 0.105$\pm$0.002 | 95.53$\pm$1.69 | 77.20$\pm$6.06
0.15 | 0.135$\pm$0.003 | 65.41$\pm$5.06 | 89.66$\pm$0.90 | 0.136 | 72.77$\pm$0.23 | 90.56$\pm$0.66
0.2 | 0.155$\pm$0.006 | 43.18$\pm$4.31 | 88.56$\pm$0.75 | 0.17 | 72.67 | 90.36$\pm$1.44
0.25 | 0.164$\pm$0.014 | 32.13$\pm$8.43 | 87.97$\pm$1.42 | 0.204$\pm$0.003 | 66.5$\pm$1.7 | 91$\pm$0.74
0.3 | 0.148$\pm$0.012 | 13.23$\pm$7.52 | 87.67$\pm$0.69 | 0.197 | 46.73$\pm$0.14 | 89.37$\pm$0.32
0.35 | 0.134$\pm$0.005 | 4.57$\pm$1.80 | 87.68$\pm$0.23 | 0.21$\pm$0.002 | 43.33$\pm$0.65 | 90.02$\pm$0.38
0.4 | 0.131$\pm$0.003 | 1.51$\pm$0.56 | 87.29$\pm$0.30 | 0.21$\pm$0.006 | 31.17$\pm$1.26 | 87.41$\pm$0.55
0.45 | 0.128$\pm$0.002 | 0.86$\pm$0.45 | 87.45$\pm$0.25 | 0.265$\pm$0.008 | 9.13$\pm$1.1 | 75.58$\pm$0.98
0.5 | 0.136$\pm$0.01 | 0 | 86.41$\pm$0.99 | 0.297$\pm$0.004 | 0 | 70.27$\pm$0.44
Table 3: Comparison results on Synthetic Dataset 1 (linear classifiers for both the approaches). d | $L_{\text{DR}}$ ($C=64,\;\gamma=0.25$) | $L_{\text{DH}}$ ($C=64,\;\gamma=0.25$)
---|---|---
| Risk | RR | Acc on unrejected | Risk | RR | Acc on unrejected
0.05 | 0.046$\pm$0.006 | 79.5$\pm$1.47 | 97.56$\pm$2.92 | 0.046$\pm$0.004 | 86.5$\pm$0.82 | 97.26$\pm$3.8
0.1 | 0.096$\pm$0.006 | 75.45$\pm$1.12 | 92.80$\pm$2.35 | 0.1$\pm$0.005 | 76.35$\pm$1.13 | 91.65$\pm$2.0
0.15 | 0.15$\pm$0.012 | 64.3$\pm$2.32 | 86.40$\pm$2.35 | 0.139$\pm$0.01 | 52.3$\pm$2.02 | 87.6$\pm$2.4
0.2 | 0.182$\pm$0.01 | 51.2$\pm$1.90 | 84.79$\pm$1.99 | 0.162$\pm$0.007 | 40.35$\pm$1.68 | 86.75$\pm$1.22
0.25 | 0.193$\pm$0.008 | 30.3$\pm$1.01 | 83.56$\pm$1.33 | 0.18$\pm$0.008 | 31.25$\pm$1.65 | 85.74$\pm$1.47
0.3 | 0.190$\pm$0.005 | 16.4$\pm$1.74 | 83.47$\pm$0.75 | 0.183$\pm$0.013 | 18.35$\pm$2.85 | 84.4$\pm$1.2
0.35 | 0.178$\pm$0.006 | 6.85$\pm$1.43 | 83.49$\pm$0.69 | 0.178$\pm$0.008 | 10.65$\pm$1.42 | 84.21$\pm$0.80
0.4 | 0.171$\pm$0.012 | 2.6$\pm$1.26 | 83.51$\pm$1.2 | 0.177$\pm$0.006 | 5.75$\pm$0.68 | 83.75$\pm$0.76
0.45 | 0.168$\pm$0.011 | 0.65$\pm$0.41 | 83.42$\pm$1.06 | 0.182$\pm$0.008 | 2.95$\pm$0.9 | 82.61$\pm$0.87
0.5 | 0.178$\pm$0.014 | 0 | 82.2$\pm$1.36 | 0.184$\pm$0.009 | 0 | 81.65$\pm$0.88
Table 4: Comparison Results on Synthetic Dataset 2 (nonlinear classifiers using RBF kernel for both the approaches). d | $L_{\text{DR}}$ ($C=2,\;\gamma=0.125$) | $L_{\text{DH}}$ ($C=16,\;\gamma=0.125$)
---|---|---
| Risk | RR | Acc on unrejected | Risk | RR | Acc on unrejected
0.05 | 0.025$\pm$0.002 | 34.84$\pm$0.92 | 98.94$\pm$0.31 | 0.029 | 52.61$\pm$0.73 | 99.47$\pm$0.06
0.1 | 0.027$\pm$0.003 | 8.81$\pm$0.32 | 97.99$\pm$0.33 | 0.047$\pm$0.002 | 43.44$\pm$0.85 | 99.46$\pm$0.17
0.15 | 0.039$\pm$0.003 | 5.78$\pm$0.57 | 96.81$\pm$0.29 | 0.042$\pm$0.003 | 24.02$\pm$1.62 | 99.3$\pm$0.37
0.2 | 0.044$\pm$0.001 | 3.46$\pm$0.51 | 96.18$\pm$0.15 | 0.04$\pm$0.002 | 17.43$\pm$0.59 | 99.42$\pm$0.25
0.25 | 0.047$\pm$0.002 | 1.76$\pm$0.41 | 95.68$\pm$0.23 | 0.046$\pm$0.001 | 14.47$\pm$0.79 | 98.9$\pm$0.16
0.3 | 0.052$\pm$0.003 | 0.92$\pm$0.46 | 95.08$\pm$0.35 | 0.051$\pm$0.003 | 12.57$\pm$0.75 | 98.56$\pm$0.31
0.35 | 0.051$\pm$0.003 | 0.03$\pm$0.09 | 94.88$\pm$0.29 | 0.054$\pm$0.002 | 9.33$\pm$0.59 | 97.72$\pm$0.21
0.4 | 0.051$\pm$0.002 | 0 | 94.95$\pm$0.24 | 0.054$\pm$0.003 | 6.72$\pm$0.86 | 97.09$\pm$0.35
0.45 | 0.054$\pm$0.002 | 0 | 94.64$\pm$0.21 | 0.055$\pm$0.003 | 3.53$\pm$0.41 | 95.97$\pm$0.36
0.5 | 0.054$\pm$0.001 | 0 | 94.62$\pm$0.13 | 0.055$\pm$0.005 | 0 | 94.55$\pm$0.47
Table 5: Comparison results on Ionosphere dataset (nonlinear classifiers using RBF kernel for both the approaches). d | $L_{\text{DR}}$ ($C=32$) | $L_{\text{DH}}$ ($C=32$)
---|---|---
| Risk | RR | Acc on unrejected | Risk | RR | Acc on unrejected
0.05 | 0.031$\pm$0.002 | 43.88$\pm$0.80 | 98.33$\pm$0.49 | 0.043$\pm$0.001 | 86.38$\pm$0.92 | 100
0.1 | 0.051$\pm$0.004 | 41.79$\pm$0.77 | 98.07$\pm$1.03 | 0.061$\pm$0.002 | 53.76$\pm$1.64 | 98.61$\pm$0.62
0.15 | 0.071$\pm$0.002 | 40.08$\pm$1.21 | 98.14$\pm$0.48 | 0.086$\pm$0.004 | 39.56$\pm$1.13 | 95.8$\pm$0.72
0.2 | 0.095$\pm$0.004 | 37.67$\pm$1.04 | 96.99$\pm$0.55 | 0.125$\pm$0.008 | 29.78$\pm$2.06 | 90.86$\pm$1.5
0.25 | 0.133$\pm$0.009 | 20.46$\pm$2.79 | 90.26$\pm$1.30 | 0.142$\pm$0.004 | 22.3$\pm$1.95 | 89.02$\pm$0.73
0.3 | 0.129$\pm$0.01 | 4.06$\pm$2.06 | 87.83$\pm$1.15 | 0.131$\pm$0.009 | 14.19$\pm$1.05 | 89.76$\pm$1.01
0.35 | 0.134$\pm$0.007 | 2.49$\pm$1.04 | 87.19$\pm$0.76 | 0.133$\pm$0.004 | 9.97$\pm$1.18 | 89.10$\pm$0.57
0.4 | 0.131$\pm$0.008 | 0.56$\pm$0.44 | 87.06$\pm$0.75 | 0.133$\pm$0.006 | 6.10$\pm$1.62 | 88.53$\pm$0.92
0.45 | 0.133$\pm$0.013 | 0.05$\pm$0.17 | 86.72$\pm$1.28 | 0.14$\pm$0.009 | 2.92$\pm$1.09 | 86.96$\pm$1.05
0.5 | 0.133$\pm$0.009 | 0 | 86.65$\pm$0.94 | 0.139$\pm$0.008 | 0 | 86.06$\pm$0.76
Table 6: Comparison results on Parkinsons Disease dataset (linear classifiers
for both the approaches).
## 5 Conclusion and Future Work
In this paper, we have proposed a new loss function $L_{\text{DR}}$ (double
ramp loss) for learning the reject option classifier. $L_{\text{DR}}$ gives
tighter upper bound for $L_{0-d-1}$ compared to convex losses $L_{\text{DH}}$
and $L_{\text{GH}}$. Our approach learns the classifier by minimizing the
regularized risk under the double ramp loss function which becomes an instance
of DC optimization problem. Our approach can also learn nonlinear classifiers
by using appropriate kernel function. Experimentally we have shown that our
approach works superior to $L_{\text{DH}}$ based approach for learning reject
option classifier.
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## Appendix 0.A Proof of Theorem 2.1
$\displaystyle L_{\text{DR}}(f(\mathbf{x}),\rho,y)$ $\displaystyle=$
$\displaystyle\frac{d}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})+\rho\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})+\rho\big{]}_{+}\Big{]}$
$\displaystyle+\frac{(1-d)}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})-\rho\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})-\rho\big{]}_{+}\Big{]}$
1. 1.
Table 7 shows that $L_{\text{DR}}\geq L_{0-d-1},\;\forall\mu>0,\rho\geq 0$.
Interval | $L_{\text{DR}}$ | $L_{0-d-1}$
---|---|---
$yf(\mathbf{x})\in[\rho+\mu,\infty)$ | 0 | 0
$yf(\mathbf{x})\in(\rho,\rho+\mu)$ | $\in(0,d)$ | 0
$yf(\mathbf{x})\in(\rho-\mu^{2},\rho]$ | $\in[d,(1+\mu)d)$ | $d$
$yf(\mathbf{x})\in[-\rho+\mu,\rho-\mu^{2}]$ | $(1+\mu)d$ | $d$
$yf(\mathbf{x})\in[-\rho,-\rho+\mu)$ | $\in((1+\mu)d,(1+\mu)d+(1-d)]$ | $d$
$yf(\mathbf{x})\in(-\rho-\mu^{2},-\rho)$ | $\in((1+\mu)d+(1-d),(1+\mu))$ | 1
$yf(\mathbf{x})\in(-\infty,-\rho-\mu^{2}]$ | $1+\mu$ | 1
Table 7: Proof for Theorem 1.(1).
2. 2.
We need to show that $\lim_{\mu\rightarrow
0}L_{\text{DR}}(f(\mathbf{x}),\rho,y)=L_{0-d-1}(f(\mathbf{x}),\rho,y)$. We
first see the values that $L_{\text{DR}}$ take for different values of
$yf(\mathbf{x})$. Table 8 shows how $L_{\text{DR}}$ changes as a function of
$yf(\mathbf{x})$.
Interval | $L_{\text{DR}}$
---|---
$yf(\mathbf{x})\in(\rho+\mu,\infty)$ | 0
$yf(\mathbf{x})\in[\rho-\mu^{2},\rho+\mu]$ | $\frac{d}{\mu}(\mu-yf(\mathbf{x})+\rho)$
$yf(\mathbf{x})\in(-\rho+\mu,\rho-\mu^{2})$ | $(1+\mu)d$
$yf(\mathbf{x})\in[-\rho-\mu^{2},-\rho+\mu]$ | $(1+\mu)d+\frac{(1-d)}{\mu}(\mu-yf(\mathbf{x})-\rho)$
$yf(\mathbf{x})\in(-\infty,-\rho-\mu^{2})$ | $1+\mu$
Table 8: $L_{\text{DR}}$ in different intervals (Proof for Theorem 1.(iii))
Now we take the limit $\mu\rightarrow 0$, which is shown in Table 9. We see
that $\lim_{\mu\rightarrow 0}L_{\text{DR}}=L_{0-d-1}$.
Interval | $\lim_{\mu\rightarrow 0}L_{\text{DR}}$ | $L_{0-d-1}$
---|---|---
$yf(\mathbf{x})\in(\rho,\infty)$ | 0 | 0
$yf(\mathbf{x})=\rho$ | $d$ | $d$
$yf(\mathbf{x})\in(-\rho,\rho)$ | $d$ | $d$
$yf(\mathbf{x})=-\rho$ | $1$ | $1$
$yf(\mathbf{x})\in(-\infty,-\rho)$ | $1$ | $1$
Table 9: $\lim_{\mu\rightarrow 0}L_{\text{DR}}$ in different intervals (Proof
for Theorem 1.(iii))
3. 3.
In the rejection region $yf(\mathbf{x})\in(\rho-\mu^{2},-\rho+\mu)$, the loss
remains constant, that is $L_{\text{DR}}(f(\mathbf{x}),\rho,y)=d(1+\mu)$. This
can be seen in Table 8.
4. 4.
For $\mu>0$, $L_{\text{DR}}\leq(1+\mu),\;\forall\rho\geq 0,\;\forall d\geq 0$.
This can be seen in Table 8.
5. 5.
When $\rho=0$, $L_{\text{DR}}$ becomes
$\displaystyle L_{\text{DR}}(f(\mathbf{x}),0,y)$ $\displaystyle=$
$\displaystyle\frac{d}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})\big{]}_{+}\Big{]}+\frac{(1-d)}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})-\big{]}_{+}$
$\displaystyle-\big{[}-\mu^{2}-yf(\mathbf{x})\big{]}_{+}\Big{]}$
$\displaystyle=$ $\displaystyle\frac{1}{\mu}\Big{[}\big{[}\mu-
yf(\mathbf{x})\big{]}_{+}-\big{[}-\mu^{2}-yf(\mathbf{x})\big{]}_{+}\Big{]}$
which is same as the $\mu$-ramp loss function used for classification problems
without rejection option.
6. 6.
We have to show that $L_{\text{DR}}$ is non-convex function of
$(yf(\mathbf{x}),\rho)$. From (iv), we know that $L_{\text{DR}}\leq(1+\mu)$.
That is, $L_{\text{DR}}$ is bounded above. We show non-convexity of
$L_{\text{DR}}$ by contradiction.
Let $L_{\text{DR}}$ be convex function of $(yf(\mathbf{x}),\rho)$. Let
$\mathbf{z}=(yf(\mathbf{x}),\rho)$. We also rewrite
$L_{\text{DR}}(f(\mathbf{x}),\rho,y)$ as $L_{\text{DR}}(\mathbf{z})$. We
choose two points $\mathbf{z}_{1},\mathbf{z}_{2}$ such that
$L_{\text{DR}}(\mathbf{z}_{1})>L_{\text{DR}}(\mathbf{z}_{2})$. Thus, from the
definition of convexity, we have
$\displaystyle L_{\text{DR}}(\mathbf{z}_{1})\leq\lambda
L_{\text{DR}}(\frac{\mathbf{z}_{1}-(1-\lambda)\mathbf{z}_{2}}{\lambda})+(1-\lambda)L_{\text{DR}}(\mathbf{z}_{2})\;\;\;\forall\lambda\in(0,1)$
Hence,
$\frac{L_{\text{DR}}(\mathbf{z}_{1})-(1-\lambda)L_{\text{DR}}(\mathbf{z}_{2})}{\lambda}\leq
L_{\text{DR}}(\frac{\mathbf{z}_{1}-(1-\lambda)\mathbf{z}_{2}}{\lambda})$
Now, since $L_{\text{DR}}(\mathbf{z}_{1})>L_{\text{DR}}(\mathbf{z}_{2})$,
$\frac{L_{\text{DR}}(\mathbf{z}_{1})-(1-\lambda)L_{\text{DR}}(\mathbf{z}_{2})}{\lambda}=\frac{L_{\text{DR}}(\mathbf{z}_{1})-L_{\text{DR}}(\mathbf{z}_{2})}{\lambda}+L_{\text{DR}}(\mathbf{z}_{2})\rightarrow\infty\;\;\;as\;\;\;\lambda\rightarrow
0^{+}$
Thus $\lim_{\lambda\rightarrow
0^{+}}L_{\text{DR}}(\frac{\mathbf{z}_{1}-(1-\lambda)\mathbf{z}_{2}}{\lambda})=\infty$.
But $L_{\text{DR}}$ is upper bounded by $(1+\mu)d$. This contradicts that
$L_{\text{DR}}$ is convex.
## Appendix 0.B Proof of Lemma 1
Let $\Theta^{\prime}=(\mathbf{w}^{\prime},b^{\prime},\rho^{\prime})$ minimizes
$R(\Theta)$, where $\rho^{\prime}<0$. Thus $-\rho^{\prime}>0$. Consider
$\Theta^{\prime\prime}=(\mathbf{w}^{\prime},b^{\prime},-\rho^{\prime})$ as
another point.
$\displaystyle R(\Theta^{\prime})-R(\Theta^{\prime\prime})$ $\displaystyle=$
$\displaystyle\frac{C(1-2d)}{\mu}\sum_{n=1}^{N}\Big{\\{}-\big{[}\mu-
y_{n}f(\mathbf{x}_{n})+\rho^{\prime}\big{]}_{+}+\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})+\rho^{\prime}\big{]}_{+}$
$\displaystyle+\big{[}\mu-
y_{n}f(\mathbf{x}_{n})-\rho^{\prime}\big{]}_{+}-\big{[}-\mu^{2}-y_{n}f(\mathbf{x}_{n})-\rho^{\prime}\big{]}_{+}\Big{\\}}$
$\displaystyle=$ $\displaystyle
C(1-2d)\sum_{n=1}^{N}\Big{\\{}L_{ramp}(y_{n}f(\mathbf{x}_{n})+\rho^{\prime})-L_{ramp}(y_{n}f(\mathbf{x}_{n})-\rho^{\prime})\Big{\\}}$
where $L_{ramp}(t)=\frac{1}{\mu}([\mu-t]_{+}-[-\mu^{2}-t]_{+})$ is a
monotonically non-increasing function of $t$ [12]. Since $\rho^{\prime}<0$,
thus,
$y_{n}f(\mathbf{x}_{n})+\rho^{\prime}<y_{n}f(\mathbf{x}_{n})-\rho^{\prime},\;\forall
n$. This implies $L_{ramp}(y_{n}f(\mathbf{x}_{n})+\rho^{\prime})\geq
L_{ramp}(y_{n}f(\mathbf{x}_{n})-\rho^{\prime}),\;\forall n$. Also $(1-2d)\geq
0$, since $0\leq d\leq 0.5$. Thus
$R(\Theta^{\prime})-R(\Theta^{\prime\prime})\geq 0$, which contradicts that
$\Theta^{\prime}$ minimizes $R(\Theta)$. Thus, at the minimum of $R(\Theta)$,
$\rho$ must be non-negative.
## Appendix 0.C Derivation of Dual Optimization Problem $\mathcal{D}^{(l+1)}$
$\displaystyle\mathcal{P}^{(l+1)}:$
$\displaystyle\smash{\displaystyle\min_{\mathbf{w},b,\mbox{\boldmath$\xi$}^{\prime},\mbox{\boldmath$\xi$}^{\prime\prime},\rho}}$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\big{[}d\xi_{n}^{\prime}+(1-d)\xi_{n}^{\prime\prime}\big{]}+\sum_{n=1}^{N}\beta_{n}^{\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\rho]$
$\displaystyle+\sum_{n=1}^{N}\beta_{n}^{\prime\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]$
$\displaystyle s.t.$ $\displaystyle
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq\rho+\mu-\xi_{n}^{\prime},\;\;\;\xi_{n}^{\prime}\geq
0,\;\;\;n=1\ldots N$ $\displaystyle
y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq-\rho+\mu-\xi_{n}^{\prime\prime},\;\;\;\xi_{n}^{\prime\prime}\geq
0\;\;\;n=1\ldots N$
The Lagrangian for above problem will be:
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}||\mathbf{w}||^{2}+\frac{C}{\mu}\sum_{n=1}^{N}\big{[}d\xi_{n}^{\prime}+(1-d)\xi_{n}^{\prime\prime}\big{]}+\sum_{n=1}^{N}\beta_{n}^{\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\rho]+$
$\displaystyle\sum_{n=1}^{N}\beta_{n}^{\prime\prime(l)}[y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]+\sum_{n=1}^{N}\alpha_{n}^{\prime}[\rho+\mu-\xi_{n}^{\prime}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)]-\sum_{n=1}^{N}\eta_{n}^{\prime}\xi_{n}^{\prime}$
$\displaystyle+\sum_{n=1}^{N}\alpha_{n}^{\prime\prime}[-\rho+\mu-\xi_{n}^{\prime\prime}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)]-\sum_{n=1}^{N}\eta_{n}^{\prime\prime}\xi_{n}^{\prime\prime}$
where $\alpha_{n}^{\prime}$ is dual variable corresponding to constraint
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq\rho+\mu-\xi_{n}^{\prime}$,
$\alpha^{\prime\prime}_{n}$ is dual variable corresponding to
$y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)\geq-\rho+\mu-\xi_{n}^{\prime}$,
$\eta_{n}^{\prime}$ is dual variable corresponding to $\xi_{n}^{\prime}\geq
0$, $\eta_{n}^{\prime\prime}$ is dual variable corresponding to
$\xi_{n}^{\prime\prime}\geq 0$. We take the gradient of Lagrangian with
respect to the primal variables. By equating the gradient to zero, we get the
KKT conditions of optimality for this optimization problem.
$\displaystyle\begin{cases}\mathbf{w}=\sum_{n=1}^{N}y_{n}[\alpha_{n}^{\prime}+\alpha_{n}^{\prime\prime}-\beta_{n}^{\prime(l)}-\beta_{n}^{\prime\prime}{(l)}]\phi(\mathbf{x}_{n})&\\\
\sum_{n=1}^{N}y_{n}[\alpha_{n}^{\prime}+\alpha_{n}^{\prime\prime}-\beta_{n}^{\prime(l)}-\beta_{n}^{\prime\prime}{(l)}]&\\\
\eta_{n}^{\prime}+\alpha_{n}^{\prime}=\frac{Cd}{\mu}&n=1\ldots N\\\
\eta_{n}^{\prime\prime}+\alpha_{n}^{\prime\prime}=\frac{C(1-d)}{\mu}&n=1\ldots
N\\\
\sum_{n=1}^{N}[\alpha_{n}^{\prime}-\alpha_{n}^{\prime\prime}-\beta_{n}^{\prime(l)}+\beta_{n}^{\prime\prime}{(l)}]=0&\\\
\eta_{n}^{\prime}\xi_{n}^{\prime}=0,\;\;\eta_{n}^{\prime}\geq 0&n=1\ldots N\\\
\eta_{n}^{\prime\prime}\xi_{n}^{\prime\prime}=0,\;\;\eta_{n}^{\prime\prime}\geq
0&n=1\ldots N\\\
\alpha_{n}^{\prime}[\mu-\xi^{\prime}_{n}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]=0,\;\;\alpha_{n}^{\prime}\geq
0&n=1\ldots N\\\
\alpha_{n}^{\prime\prime}[\mu-\xi^{\prime\prime}_{n}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)-\rho]=0,\;\;\alpha^{\prime\prime}_{n}\geq
0&n=1\ldots N\end{cases}$
We make the dual optimization problem simpler by changing the variables in
following way:
$\displaystyle\begin{cases}\gamma_{n}^{\prime}=\alpha_{n}^{\prime}-\beta_{n}^{\prime(l)},\;\;\;n=1\ldots
N\\\
\gamma_{n}^{\prime\prime}=\alpha_{n}^{\prime\prime}-\beta_{n}^{\prime\prime(l)},\;\;\;n=1\ldots
N\end{cases}$
By changing these variables, the new KKT conditions in terms of
$\mbox{\boldmath$\gamma$}^{\prime}$ and
$\mbox{\boldmath$\gamma$}^{\prime\prime}$ are
$\displaystyle\begin{cases}\mathbf{w}=\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})\phi(\mathbf{x}_{n})&\\\
\sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})=0&\\\
\eta_{n}^{\prime}+\gamma_{n}^{\prime}+\beta_{n}^{\prime(l)}=\frac{Cd}{\mu}&n=1\ldots
N\\\
\eta_{n}^{\prime\prime}+\gamma_{n}^{\prime\prime}+\beta_{n}^{\prime\prime(l)}=\frac{C(1-d)}{\mu}&n=1\ldots
N\\\ \sum_{n=1}^{N}(\gamma_{n}^{\prime}-\gamma_{n}^{\prime\prime})=0&\\\
\eta_{n}^{\prime}\xi_{n}^{\prime}=0,\;\;\eta_{n}^{\prime}\geq 0&n=1\ldots N\\\
\eta_{n}^{\prime\prime}\xi_{n}^{\prime\prime}=0,\;\;\eta_{n}^{\prime\prime}\geq
0&n=1\ldots N\\\
(\gamma_{n}^{\prime}+\beta_{n}^{\prime(l)})[\mu-\xi^{\prime}_{n}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]=0,\;\;\gamma_{n}^{\prime}+\beta_{n}^{\prime(l)}\geq
0&n=1\ldots N\\\
(\gamma_{n}^{\prime\prime}+\beta_{n}^{\prime\prime(l)})[\mu-\xi^{\prime\prime}_{n}-y_{n}(\mathbf{w}^{T}\phi(\mathbf{x}_{n})+b)+\rho]=0,\;\;\gamma_{n}^{\prime\prime}+\beta_{n}^{\prime\prime(l)}\geq
0&n=1\ldots N\end{cases}$
Using the KKT conditions in the Langarangian, we replace the primal variables
$(\mathbf{w},b,\rho,\mbox{\boldmath$\xi$}^{\prime},\mbox{\boldmath$\xi$}^{\prime\prime})$
in terms of the dual variables
$(\mbox{\boldmath$\gamma$}^{\prime},\mbox{\boldmath$\gamma$}^{\prime\prime})$.
The dual optimization problem $\mathcal{D}^{(l+1)}$ will become:
$\displaystyle\mathcal{D}^{(l+1)}=$
$\displaystyle\smash{\displaystyle\min_{\mbox{\boldmath$\gamma$}^{\prime},\mbox{\boldmath$\gamma$}^{\prime\prime}}}$
$\displaystyle\frac{1}{2}\sum_{n=1}^{N}\sum_{m=1}^{N}y_{n}y_{m}(\gamma^{\prime}_{n}+\gamma_{n}^{\prime\prime})(\gamma^{\prime}_{m}+\gamma_{m}^{\prime\prime})k(\mathbf{x}_{n},\mathbf{x}_{m})-\mu\sum_{n=1}^{N}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})$
$\displaystyle s.t.$
$\displaystyle\begin{cases}-\beta_{n}^{\prime(l)}\leq\gamma_{n}^{\prime}\leq\frac{Cd}{\mu}-\beta_{n}^{\prime(l)}&n=1\ldots
N\\\
-\beta_{n}^{\prime\prime(l)}\leq\gamma_{n}^{\prime\prime}\leq\frac{C(1-d)}{\mu}-\beta_{n}^{\prime\prime(l)}&n=1\ldots
N\\\ \sum_{n=1}^{N}y_{n}(\gamma_{n}^{\prime}+\gamma_{n}^{\prime\prime})=0&\\\
\sum_{n=1}^{N}(\gamma_{n}^{\prime}-\gamma_{n}^{\prime\prime})=0&\end{cases}$
where
$\mbox{\boldmath$\gamma$}^{\prime}=[\gamma_{1}^{\prime}\;\;\gamma_{2}^{\prime}\ldots\ldots\gamma_{n}^{\prime}]^{T}$
and
$\mbox{\boldmath$\gamma$}^{\prime\prime}=[\gamma_{1}^{\prime\prime}\;\;\gamma_{2}^{\prime\prime}\ldots\ldots\gamma_{n}^{\prime\prime}]^{T}$.
|
arxiv-papers
| 2013-11-26T05:13:18 |
2024-09-04T02:49:54.192502
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Naresh Manwani, Kalpit Desai, Sanand Sasidharan, Ramasubramanian\n Sundararajan",
"submitter": "Naresh Manwani",
"url": "https://arxiv.org/abs/1311.6556"
}
|
1311.6603
|
# contact structure on 2-step nilpotent lie groups
babak hasanzadeye seyedi young researchers and elit club islamic azad
university tabriz branch, iran [email protected]
###### Abstract.
In this paper we study contact structure on 2-step nilpotent Lie groups. We
conside properties of normal subgroups and center of Lie groups while
cosymplectic and Sasakian structure defined on Lie group.
###### Key words and phrases:
contact structure, Lie group, nonsingular, skew adjoint
###### 1991 Mathematics Subject Classification:
Primary 53B40 , Secondary 53C60
## 1\. Introduction
let $\tilde{M}$ be an odd dimentional Riemannian manifold with a Riemannian
metric $g$ and Riemannian connection $\tilde{\nabla}$. Denote by $T\tilde{M}$
the Lie algebra of vector fields on $\tilde{M}$. Then $\tilde{M}$ is said to
be an almost contact metric manifold if there exist on $\tilde{M}$ a tensor
$\phi$ of type $(1,1)$, a vector field $\xi$ called structure vector field and
$\eta$, the dual 1-form of $\xi$ satisfying the following
(1.1) $\displaystyle\phi^{2}X=-X+\eta(X)\xi,g(X,\xi)=\eta(X)$ (1.2)
$\displaystyle\eta(\xi)=1,\phi(\xi)=0,\eta\circ\phi=0$ (1.3) $\displaystyle
g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y)$
For any $X,Y\in T\tilde{M}$. In this case
(1.4) $\displaystyle g(\phi X,Y)=-g(X,\phi Y)$
Now, let $M$ be a submanifold immersed in $\tilde{M}$. A normal almost contact
manifold is called a cosymplectic manidfold if
(1.5) $\displaystyle(\tilde{\nabla}_{X}\phi)(Y)=0,\tilde{\nabla}_{X}\xi=0$
Theorem 1.1 An almost contact metric struture $(\phi,\xi,\eta,g)$ is Sasakian
if and only if
(1.6) $\displaystyle(\nabla_{X}\phi)Y=g(X,Y)\xi-\eta(Y)X$
The Riemannian metric induced on $M$ is denoted by the same symbol $g$. Let
$TM$ and $T^{\perp}M$ be the Lie algebras of vector fields tangential and
normal to $M$ respectively, and $\nabla$ be the induced Levi-Civita connection
on $M$, then the Gauss and Weingarten formulas are given by
(1.7) $\displaystyle\tilde{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)$ (1.8)
$\displaystyle\tilde{\nabla}_{X}V=-A_{V}X+\nabla^{\perp}_{X}V$
for any $X,Y\in TM$ and $V\in T^{\perp}M$. Where $\nabla^{\perp}$ is the
connection on the normal bundle $T^{\perp}M$, $h$ is the second fundamental
form and $A_{V}$ is the Weingarten map associated with $V$ as
(1.9) $\displaystyle g(A_{V}X,Y)=g(h(X,Y),V)$
For any $x\in M$ and $X\in T_{x}M$, we write
(1.10) $\displaystyle\phi X=\Psi X+\Gamma X$
where $\Psi X\in T_{x}M$ and $\Gamma X\in T^{\perp}_{x}M$. Similary, for $V\in
T^{\perp}_{x}M$, we have
(1.11) $\displaystyle\phi V=\psi V+\gamma V$
where $\psi V$ (resp. $\gamma$v ) is the tangential componenet (resp. normal
component) of $\phi V$. From (1.4) and (1.9), it is easy to observe that for
each $x\in M$, and $X,Y\in T_{x}M$
(1.12) $\displaystyle g(\Psi X,Y)=-g(X,\Psi Y)$
and therefore $g(\Psi^{2}X,Y)=g(X,\Psi^{2}Y)$ which implies that the
endomorphism $\Psi^{2}=Q$ is self adjoint. Moreover, it can be seen that the
eignvalues of $Q$ belong to $\left[-1,0\right]$ and that each non-vanishing
eigenvalue of $Q$ has even multiplicity. We define $\nabla\Psi$,$\nabla Q$ and
$\nabla N$ by
(1.13) $\displaystyle(\nabla_{X}\Psi)Y=\nabla_{X}\Psi Y-\Psi\nabla_{X}Y$
(1.14) $\displaystyle(\nabla_{X}Q)Y=\nabla_{X}QY-Q\nabla_{X}Y$ (1.15)
$\displaystyle(\nabla_{X}N)Y=\nabla^{\perp}_{X}NY-N\nabla_{X}Y$
for any $X,Y\in TM$.
## 2\. preliminaries
Difination 1.1. A Lie group G is a smooth manifold with group structure such
that
(2.1) $\displaystyle(i)G\times G\mapsto G\quad(ii)G\mapsto G$
$\displaystyle(x,y)\mapsto xy\quad x\mapsto x^{-1}$
Are smooth.
Difination 1.2. Let G is a Lie group and $a\in G$, thus
(2.2) $\displaystyle L_{a}:G\rightarrow G$ $\displaystyle x\mapsto ax$
is called left translation with a. Also map
(2.3) $\displaystyle R_{a}:G\rightarrow G$ $\displaystyle x\mapsto xa$
Is called right translation with a.Let G is a Lie group. Vector field $X$ on G
is left invariant if
(2.4) $\displaystyle L_{a}(X)=X\quad\forall a\in G$
And that is right invariant if
(2.5) $\displaystyle R_{a}(X)=X\quad\forall a\in G$
Difination 1.3. A Lie group H of a Lir group G is a subgroup which is also a
submanifold.
Difination 1.4. Here $F=\mathcal{R}$ or $\mathcal{C}$. A Lie algebra over F is
pair $(\mathfrak{g},[.,.])$, where $\mathfrak{g}$ is a vector space over F and
(2.6)
$\displaystyle[.,.]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$
is an F-bilinear map satisfying the following properties
(2.7) $\displaystyle[X,Y]=-[Y,X]$ (2.8)
$\displaystyle[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0$
The latter is the Jacobi identity. In this paper for any $X,Y\in\mathfrak{g}$
we have
(2.9) $\displaystyle[X,Y]=\nabla_{X}Y-\nabla_{Y}X$
A Lie subalgebra of a Lie algebra is a vector space that is closed under
bracket.
Theorem 1.1. Let G is a Lie group and $\mathfrak{g}$ is a set of left
invariant vector field on G. We have
(i) $\mathfrak{g}$ is a vector space and map
$\displaystyle E:\mathfrak{g}\rightarrow T_{e}G$ $\displaystyle X\mapsto
X_{e}$
Is a linear isomorphism and therefore $dim\mathfrak{g}=dimT_{e}G=dimG$.(e is
identity element)
(ii) Left invariant vectro fields necessity are differentiable.
(iii) $(\mathfrak{g},[.,.])$ is Lie algebra.[6]
Difination 5.1. Lie algebra made of left invariant vector field on Lie group G
is called Lie algebras Lie group G. This Lie algebra is isomorphism with
$T_{e}G$, and we have
(2.10) $\displaystyle[X_{e},Y_{e}]=[X,Y]_{e}$
X and Y are unique left invariant vector field.
Theorem 2.1. Let G be a Lie group.[6]
(a) If H is a Lie subgroup of G, then $\mathfrak{h}\simeq T_{e}H\subset
T_{e}G\simeq\mathfrak{g}$ g is a Lie subalgebra.
(b) If $\mathfrak{h}\subset\mathfrak{g}$ a Lie subalgebra, there exists a
unique connected Lie subgroup $H\subset G$ with Lie algebra $\mathfrak{h}$.
For each nonzero vector field $X\in\mathfrak{g}$ , the angle $\theta(X)$;
$0\leq\theta(X)\leq\dfrac{\pi}{2}$ ; between $\phi X$ and $\mathfrak{g}$ is
called the Wirtinger angle of X. If the Wirtinger angle $\theta$ is a constant
its called slant angle of $\mathfrak{g}$. Let $H\subset G$ is a Lie subgroup
and $\mathfrak{h}$ is their Lie algebra, thus $\mathfrak{h}$ is Lie subalgebra
of $\mathfrak{g}$ and if $\mathfrak{h}$ is a slant Lie subalgebra, H is called
slant Lie subgroup.
If (G,g) be a Lie group equipted by Riemannian metric, ad is skew adjoint if
for $X,Y,Z\in\mathfrak{g}$, if
$\displaystyle g(adX(Y),Z)=-g(Y,adX(Z))$
If $X\in Z(\mathfrak{g})$ and $Y,z\in\mathfrak{g}$ we have
$\displaystyle X\langle Y,Z\rangle=\langle\nabla_{Y+Z}X,Y\rangle$
Theorem 2.3. Let g be a left invariant metric on a connected Lie group G. This
metric will also be right invariant if and only if ad(X) is skew-adjoint for
every $X\in\mathfrak{g}$.[2]
Definition 3.1. A nilpotent Lie group is a Lie group G which is connected and
whose Lie algebras is nilpotent Lie algebra $\mathfrak{g}$, that is, its Lie
algebra have a sequence of ideals of $\mathfrak{g}$ by
$\mathfrak{g}^{0}=\mathfrak{g}$,
$\mathfrak{g}^{1}=[\mathfrak{g},\mathfrak{g}]$,$\mathfrak{g}^{2}=[\mathfrak{g},\mathfrak{g}^{1}]$,…,
$\mathfrak{g}^{i}=[\mathfrak{g},\mathfrak{g}^{i-1}]$. $\mathfrak{g}$ is called
nilpotent if $\mathfrak{g}^{n}=0$ for some n.
Proposition 3.1.[3] Let $\mathfrak{g}$ be a Lie algebra.
(i)If $\mathfrak{g}$ is a nilpotent, then so are all subalgebras and
homomorphic images of $\mathfrak{g}$.
(ii) If $\dfrac{\mathfrak{g}}{Z(\mathfrak{g})}$ is nilpotent, then so is
$\mathfrak{g}$.
(iii)If $\mathfrak{g}$ is nilpotent and nonzero, then $Z(\mathfrak{g})\neq 0$.
Definition 3.2. A finite dimensional Lie algebra $\mathfrak{g}$ is 2-step
nilpotent if $\mathfrak{g}$ is not abelian and
$[\mathfrak{g},[\mathfrak{g},\mathfrak{g}]]=0$. A Lie group G is 2-step
nilpotent if its Lie algebra $\mathfrak{g}$ is 2-step nilpotent. In the other
word A Lie algebra $\mathfrak{g}$ is 2-step nilpotent if
$[\mathfrak{g},\mathfrak{g}]$ is non zero and Lies in the center of
$\mathfrak{g}$.
We may identify an element of $\mathfrak{g}$ witha left invariant vector field
on G since $T_{e}G$ may be identified with $\mathfrak{g}$. If X,y are left
invariant vector field on G, then $\nabla_{X}Y$ is left invariant also. for
$X,Y\in Z^{\perp}(\mathfrak{g})$ we have following formula:[7]
(2.11) $\displaystyle\nabla_{X}Y=\dfrac{1}{2}[X,Y]$
Definition 3.3. A 2-step nilpotent Lie algebra $\mathfrak{g}$ is nonsingular
if $adX:\mathfrak{g}\rightarrow Z(\mathfrak{g})$ is surjective for each $X\in
Z^{\perp}(\mathfrak{g})$. A 2-step nilpotent Lie group G is nonsingular if its
Lie algebra $\mathfrak{g}$ is nonsingular.
Let G denote a simply connected, 2-step nilpotent Lie group with a left
invariant metric $\langle.,.\rangle$ and let $\mathfrak{g}$ denote the Lie
algebra of G. Write $\mathfrak{g}=Z(\mathfrak{g})\oplus
Z^{\perp}(\mathfrak{g})$ where $Z^{\perp}(\mathfrak{g})$ its orthogonal
complement of center $Z(\mathfrak{g})$.
## 3\. cosymplectic and sasakian
In this section we study cosymplectic structure on 2-step nilpotent Lie
groups. Let G is a 2-step nilpotent Lie group with cosymplectic structure,
from (1.5) for any $X,Y\in\mathfrak{g}$ we have
(3.1) $\displaystyle[\phi X,Y]=[X,\phi Y]=\phi[X,Y]$
and
(3.2) $\displaystyle[X,\xi]=0$
Let G be a 2-step nilpotent nonsingular cosymplectic Lie group. if ad is skew
adjoint for any $X,Y\in\mathfrak{g}$ we have
(3.3) $\displaystyle\eta([X,Y])=g([X,Y],\xi)=-g([X,\xi],Y)$
From() we conclude $\eta(Z(\mathfrak{g}))=0$, thus $\xi$ is normal to
$Z(\mathfrak{g})$ and $\xi\in Z^{\perp}(\mathfrak{g})$, therefore
$Z(\mathfrak{g})$ is integral Lie subgroup and
(3.4) $\displaystyle\phi^{2}([X,Y])=[Y,X]$
## References
* [1] David E. blair, _Riemannian Geometry of contact and symplectic manifolds._ department of mathematics Michigan state University, USA.
* [2] J. Milnor, _Curvatures of Left Invariant Metrics on Lie Groups._ Advances in Math. 21 (1976),293-329.
* [3] James. E Humphreys, _Introduction to Lie algebras and representation theory._ Springer-Verlag New york Hiedlberg Berlin (1972)
* [4] Patrick Eberlein, _Geometry of 2-step nilpotent groups with a left invariant metric._ Ann. Sci. Ecole Norm. Sup. No. 27, p. 611-660(1994).
* [5] S. Helgason, _Differential geometry and symmetric spaces._ Academic press, New York, (1962)
* [6] Wolfgang Ziller, _Lie Groups. Representation Theory and Symmetric Spaces_ University of Pennsylvania, Fall 2010
* [7] Patrick Eberlein, _Geometry of 2-step nilpotent groups with a left invariant metric II._ Transaction of the American mathematical society, volume 343, 805-828,(1994)
* [8] Yu. Khakimdjanov, M. Gozea, A. Medinab _Symplectic or contact structures on Lie groups_ Differential Geometry and its Applications 21 (2004) 41–54
* [9] André Diatta, _Left invariant contact structures on Lie groups_ Differential Geometry and its Applications 26 (2008) 544–552
* [10] Brendan J. Foreman, _K-contact Lie groups of dimension five or greater._ arXiv:1006.1531v1
* [11] Luis A. Cordero, Phillip E. Parker. _pseudo Riemannian 2-step nilpotent Lie groups._ arXiv:math/9905188v1
* [12] André Diatta, _Riemannian geometry on contact Lie groups._ Geom Dedicata (2008) 133:83–94
|
arxiv-papers
| 2013-11-26T09:49:04 |
2024-09-04T02:49:54.202526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Babak Hasanzadeh",
"submitter": "Babak Hassanzadeh",
"url": "https://arxiv.org/abs/1311.6603"
}
|
1311.6607
|
Self-generated interior blow-up solutions in fractional elliptic equation with
absorption
Huyuan Chen, Patricio Felmer
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático
UMR2071 CNRS-UChile, Universidad de Chile
Casilla 170 Correo 3, Santiago, Chile.
([email protected], [email protected])
and
Alexander Quaas
Departamento de Matemática, Universidad Técnica Federico Santa María
Casilla: V-110, Avda. España 1680, Valparaíso, Chile
([email protected])
###### Abstract
In this paper we study positive solutions to problem involving the fractional
Laplacian
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega\setminus\mathcal{C},\\\\[5.69054pt]
u(x)=0,&x\in\Omega^{c},\\\\[5.69054pt] \lim_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}u(x)=+\infty,\end{array}\right.$ (0.1)
where $p>1$ and $\Omega$ is an open bounded $C^{2}$ domain in
$\mathbb{R}^{N}$, $\mathcal{C}\subset\Omega$ is a compact $C^{2}$ manifold
with $N-1$ multiples dimensions and without boundary, the operator
$(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is the fractional Laplacian.
We consider the existence of positive solutions for problem (0.1). Moreover,
we further analyze uniqueness, asymptotic behaviour and nonexistence.
Key words: Fractional Laplacian, Existence, Uniqueness, Asymptotic behavior,
Blow-up solution.
## 1 Introduction
In 1957, a fundamental contribution due to Keller in [11] and Osserman in [19]
is the study of boundary blow-up solutions for the non-linear elliptic
equation
$\left\\{\begin{array}[]{lll}-\Delta u+h(u)=0\ \ \rm{in}\ \
\Omega,\\\\[5.69054pt]
\lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty.\end{array}\right.$ (1.1)
They proved the existence of solutions to (1.1) when
$h:\mathbb{R}\to[0,+\infty)$ is a locally Lipschitz continuous function which
is nondecreasing and satisfies the so called Keller-Osserman condition. From
then on, the result of Keller and Osserman has been extended by numerous
mathematicians in various ways, weakening the assumptions on the domain,
generalizing the differential operator and the nonlinear term for equations
and systems. The case of $h(u)=u_{+}^{p}$ with $p=\frac{N+2}{N-2}$ is studied
by Loewner and Nirenberg [15], where in particular uniqueness and asymptotic
behavior were obtained. After that, Bandle and Marcus [2] obtained uniqueness
and asymptotic for more general non-linearties $h$. Later, Le Gall in [9]
established a uniqueness result of problem (1.1) in the domain whose boundary
is non-smooth when $h(u)=u_{+}^{2}$. Marcus and Véron [16, 18] extended the
uniqueness of blow-up solution for (1.1) in general domains whose boundary is
locally represented as a graph of a continuous function when $h(u)=u_{+}^{p}$
for $p>1$. Under this special assumption on $h$, Kim [12] studied the
existence and uniqueness of boundary blow-up solution to (1.1) in bounded
domains $\Omega$ satisfying $\partial\Omega=\partial\bar{\Omega}$. For another
interesting contributions to boundary blow-up solutions see for example
Kondratev, Nikishkin [13], Lazer, McKenna [14], Arrieta and Rodríguez-Bernal
[1], Chuaqui, Cortázar, Elgueta and J. García-Melián [4], del Pino and
Letelier [5], Díaz and Letelier [6], Du and Huang [7], García-Melián [10],
Véron [20], and the reference therein.
In a recent work, Felmer and Quaas [8] considered a version of Keller and
Osserman problem for a class of non-local operator. Being more precise, they
considered as a particular case the fractional elliptic problem
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=f(x),&x\in\Omega,\\\\[5.69054pt]
u(x)=g(x),&x\in\bar{\Omega}^{c},\\\\[5.69054pt] \lim_{x\in\Omega,\
x\to\partial\Omega}u(x)=+\infty,\end{array}\right.$ (1.2)
where $p>1$, $f$ and $g$ are appropriate functions and $\Omega$ is a bounded
domain with $C^{2}$ boundary. The operator $(-\Delta)^{\alpha}$ is the
fractional Laplacian which is defined as
$(-\Delta)^{\alpha}u(x)=-\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\delta(u,x,y)}{|y|^{N+2\alpha}}dy,\
\ x\in\Omega,$ (1.3)
with $\alpha\in(0,1)$ and $\delta(u,x,y)=u(x+y)+u(x-y)-2u(x)$.
In [8] the authors proved the existence of a solution to (1.2) provided that
$g$ explodes at the boundary and satisfies other technical conditions. In case
the function $g$ blows up with an explosion rate as $d(x)^{\beta}$, with
$\beta\in[-\frac{2\alpha}{p-1},0)$ and $d(x)=dist(x,\partial\Omega)$, it is
shown that the solution satisfies
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\beta}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$
Here the explosion is driven by the external value $g$ and the external source
$f$ has a secondary role, not intervening in the explosive character of the
solution.
More recently, Chen, Felmer and Quaas [3] extended the results in [8] studying
existence, uniqueness and non-existence of boundary blow-up solutions when the
function $g$ vanishes and the explosion on the boundary is driven by the
external source $f$, with weak or strong explosion rate. Moreover, the results
are extended even to the case where the boundary blow-up solutions in driven
internally, when the external source and value, $f$ and $g$, vanish.
Existence, uniqueness, asymptotic behavior and non-existence results for blow-
up solutions of (1.2) are considered in [3]. In the analysis developed in [3],
a key role is played by the function $C:(-1,0]\to\mathbb{R},$ that governs the
behavior of the solution near the boundary. The function $C$ is defined as
$C(\tau)=\int^{+\infty}_{0}\frac{\chi_{(0,1)}(t)|1-t|^{\tau}+(1+t)^{\tau}-2}{t^{1+2\alpha}}dt$
(1.4)
and it possess exactly one zero in $(-1,0)$ and we call it $\tau_{0}(\alpha)$.
In what follows we explain with more details the results in the case of
vanishing external source and values, that is $f=0$ in $\Omega$ and $g=0$ in
$\bar{\Omega}^{c}$, which is the case we will consider in this paper. In
Theorem 1.1 in [3], we proved that whenever
$1+2\alpha<p<1-\frac{2\alpha}{\tau_{0}(\alpha)},$
then problem (1.2) admits a unique positive solution $u$ such that
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$
On the other hand, we proved that when $p\geq 1,$ then problem (1.2) does not
admit any solution $u$ such that
$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}<+\infty,$
(1.5)
for any $\tau\in(-1,0)\setminus\\{\tau_{0}(\alpha),-\frac{2\alpha}{p-1}\\}$.
We observe that the non-existence result does not include the case when $u$
has an asymptotic behavior of the form $d(x)^{\tau_{0}(\alpha)}$, where
$\tau_{0}(\alpha)$ is precisely where $C$ vanishes. We have a a special
existence result in this case, precisely if
$\max\\{1-\frac{2\alpha}{\tau_{0}(\alpha)}+\frac{\tau_{0}(\alpha)+1}{\tau_{0}(\alpha)},1\\}<p<1-\frac{2\alpha}{\tau_{0}(\alpha)},$
then, for any $t>0$, problem (1.2) admits a positive solution $u$ such that
$\lim_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau_{0}(\alpha)}=t.$
Motivated by these results and in view of the non-local character of the
fractional Laplacian we are interested in another class of blow-up solutions.
We want to study solutions that vanish at the boundary of the domain $\Omega$
but that explodes at the interior of the domain, near a prescribed embedded
manifold. From now on, we assume that $\Omega$ is an open bounded domain in
$\mathbb{R}^{N}$ with $C^{2}$ boundary, and that there is a $C^{2}$,
$(N-1)$-dimensional manifold $\mathcal{C}$ without boundary, embedded in
$\Omega$, such that, it separates $\Omega\setminus\mathcal{C}$ in exactly two
connected components. We denote by $\Omega_{1}$ the inner component and by
$\Omega_{2}$ the external component, that is
$\bar{\Omega}_{1}\cap\partial\Omega=\emptyset$ and
$\bar{\Omega}_{2}\cap\partial\Omega=\partial\Omega.$ Throughout the paper we
will consider the distance function
$D:\Omega\setminus\mathcal{C}\to\mathbb{R}_{+},\quad D(x)={\rm
dist}(x,\mathcal{C}).$ (1.6)
Let us consider the equations, for $i=1,2$,
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega_{i},\\\\[5.69054pt]
u(x)=0,&x\in\bar{\Omega}_{i}^{c},\\\\[5.69054pt] \lim_{x\in\Omega_{i},\
x\to\partial\Omega_{i}}u(x)=+\infty,\end{array}\right.$ (1.7)
which have solutions $u_{1}$ and $u_{2}$, for $i=1,2$ respectively, in the
appropriate range of the parameters. In the local case, that is, $\alpha=1$,
these two solutions certainly do not interact among each other, but when
$\alpha\in(0,1)$, due to the non-local character of the fractional Laplacian
and the non-linear character of the equation the solutions on each side of
$\Omega$ interact and it is precisely the purpose of this paper to study their
existence, uniqueness and non-existence.
In precise terms we consider the equation
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega\setminus\mathcal{C},\\\\[5.69054pt]
u(x)=0,&x\in\Omega^{c},\\\\[5.69054pt] \lim_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}u(x)=+\infty,\end{array}\right.$ (1.8)
where $p>1$, $\Omega$ and $\mathcal{C}\subset\Omega$ are as described above.
The explosion of the solution near $\mathcal{C}$ is governed by a function
$c:(-1,0]\to\mathbb{R},$ defined as
$c(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}+(1+t)^{\tau}-2}{t^{1+2\alpha}}dt.$
(1.9)
This function plays the role of the function $C$ used in [3], but it has
certain differences. In Section §2 we prove the existence of a number
$\alpha_{0}\in(0,1)$ such that $\alpha\in[\alpha_{0},1)$ the function $c$ is
always positive in $(-1,0)$, while if $\alpha\in(0,\alpha_{0})$ then there
exists exists a unique $\tau_{1}(\alpha)\in(-1,0)$ such that
$c(\tau_{1}(\alpha))=0$ and $c(\tau)>0$ in $(-1,\tau_{1}(\alpha))$ and
$c(\tau)<0$ in $(\tau_{1}(\alpha),0)$, see Proposition 2.1. We notice here
that $\tau_{1}(\alpha)>\tau_{0}(\alpha)$ if $\alpha\in(0,\alpha_{0})$.
Now we are ready to state our main theorems on the existence uniqueness and
asymptotic behavior of interior blow-up solutions to equation (1.8). These
theorems deal separately the case $\alpha\in(0,\alpha_{0})$ and
$\alpha\in[\alpha_{0},1)$.
###### Theorem 1.1
Assume that $\alpha\in(0,\alpha_{0})$ and the assumptions on $\Omega$ and
$\mathcal{C}$. Then we have:
$(i)$ If
$1+2\alpha<p<1-\frac{2\alpha}{\tau_{1}(\alpha)},$ (1.10)
then problem (1.8) admits a unique positive solution $u$ satisfying
$0<\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{\frac{2\alpha}{p-1}}\leq\limsup_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{\frac{2\alpha}{p-1}}<+\infty.$
(1.11)
$(ii)$ If
$\max\\{1-\frac{2\alpha}{\tau_{1}(\alpha)}+\frac{\tau_{1}(\alpha)+1}{\tau_{1}(\alpha)},1\\}<p<1-\frac{2\alpha}{\tau_{1}(\alpha)}.$
(1.12)
Then, for any $t>0$, there is a positive solution $u$ of problem (1.8)
satisfying
$\lim_{x\in\Omega\setminus{\mathcal{C}},x\to{\mathcal{C}}}u(x)D(x)^{-\tau_{1}(\alpha)}=t.$
(1.13)
$(iii)$ If one of the following three conditions holds
* a)
$1<p\leq 1+2\alpha$ and $\tau\in(-1,0)\setminus\\{\tau_{1}(\alpha)\\}$,
* b)
$1+2\alpha<p<1-\frac{2\alpha}{\tau_{1}(\alpha)}$ and
$\tau\in(-1,0)\setminus\\{\tau_{1}(\alpha),-\frac{2\alpha}{p-1}\\}$ or
* c)
$p\geq 1-\frac{2\alpha}{\tau_{1}(\alpha)}$ and $\tau\in(-1,0)$,
then problem (1.8) does not admit any solution $u$ satisfying
$0<\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{-\tau}<+\infty.$
(1.14)
We observe that this theorem is similar to Theorem 1.1 in [3], where the role
of $\tau_{0}(\alpha)$ is played here by $\tau_{1}(\alpha)$. A quite different
situation occurs when $\alpha\in[\alpha_{0},1)$ and the function $c$ never
vanishes in $(-1,0)$. Precisely, we have
###### Theorem 1.2
Assume that $\alpha\in[\alpha_{0},1)$ and the assumptions on $\Omega$ and
$\mathcal{C}$. Then we have:
$(i)$ If $p>1+2\alpha$, then problem (1.8) admits a unique positive solution
$u$ satisfying (1.11).
$(ii)$ If $p>1,$ then problem (1.8) does not admit any solution $u$ satisfying
(1.14) for any $\tau\in(-1,0)\setminus\\{-\frac{2\alpha}{p-1}\\}$.
Comparing Theorem 1.1 with Theorem 1.2 we see that the range of existence for
the absorption term is quite larger for the second one, no constraint from
above. The main difference with the results in [3], Theorem 1.1, with
vanishing $f$ and $g$ occurs when $\alpha$ is large and the function $c$ does
not vanish, allowing thus for existence for all $p$ large. This difference
comes from the fact that the fractional Laplacian is a non-local operator so
that in the interior blow-up, in each of the domains $\Omega_{1}$ and
$\Omega_{2}$ there is a non-zero external value, the solutions itself acting
on the other side of $\mathcal{C}$.
The proof of our theorems is obtained through the use of super and sub-
solutions as in [3]. The main difficulty here is to find the appropriate super
and sub-solutions to apply the iteration technique to fractional elliptic
problem (1.8). Here we make use of some precise estimates based on the
function $c$ and the distance function $D$ near $\mathcal{C}$.
This article is organized as follows. In section §2, we introduce some
preliminaries and we prove the main estimates of the behavior of the
fractional Laplacian when applied to suitable powers of the function $D$. In
section §3 we prove the existence of solution to problem (1.8) as given in
Theorem 1.1 and Theorem 1.2. Finally, in Section §4 we prove the uniqueness
and nonexistence statements of these theorems.
## 2 Preliminaries
In this section, we recall some basic results from [3] and obtain some useful
estimate, which will be used in constructing super and sub-solutions of
problem (1.8). The first result states as:
###### Theorem 2.1
Assume that $p>1$ and there are super-solution $\bar{U}$ and sub-solution
$\underline{U}$ of problem (1.8) such that
$\bar{U}\geq\underline{U}\ \ {\rm{in}}\
\Omega\setminus\mathcal{C},\quad\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}\underline{U}(x)=+\infty,\quad\bar{U}=\underline{U}=0\
\ {\rm{in}}\ \Omega^{c}.$
Then problem (1.8) admits at least one positive solution $u$ such that
$\underline{U}\leq u\leq\bar{U}\ \ {\rm{in}}\ \Omega\setminus\mathcal{C}.$
Proof. The procedure is similar to the proof of Theorem 2.6 in [3], here we
give the main differences.
Let us define $\Omega_{n}:=\\{x\in\Omega\,|\,D(x)>1/n\\}$ then we solve
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u_{n}(x)+|u_{n}|^{p-1}u_{n}(x)=0,&x\in\Omega_{n},\\\\[5.69054pt]
u_{n}(x)=\underline{U},&x\in\Omega_{n}^{c}.\\\\[5.69054pt] \end{array}\right.$
(2.1)
To find these solutions of (2.1) we observe that for fix $n$ the method of
section 3 of [8] applies even if the domain is not connected since the
estimate of Lemma 3.2 holds with $\delta<1/2n$ (see also Proposition 3.2 part
ii) in [3]), form here sub and super-solution can be construct for the
Dirichlet problem and then existence holds for (2.1) by an iteration technique
(see also section 2 of [3] for that procedure). Then as in Theorem 2.6 in [3]
we have
$\underline{U}\leq u_{n}\leq u_{n+1}\leq\bar{U}\ \ \rm{in}\ \ \Omega.$
By monotonicity of $u_{n}$, we can define
$u(x):=\lim_{n\to+\infty}u_{n}(x),\ x\in\Omega\ \ {\rm{and}}\ \ u(x):=0,\
x\in\Omega^{c}.$
Which, by a stability property, is a solution of problem (1.8) with the
desired properties. $\Box$
In order to prove our existence result, it is crucial to have available super
and sub-solutions to problem (1.8). To this end, we start describing the
properties of $c(\tau)$ defined in (1.9), which is a $C^{2}$ function in
$(-1,0)$.
###### Proposition 2.1
There exists a unique $\alpha_{0}\in(0,1)$ such that
$(i)$ For $\alpha\in[\alpha_{0},1)$, we have $c(\tau)>0,$ for all
$\tau\in(-1,0);$
$(ii)$ For any $\alpha\in(0,\alpha_{0})$, there exists unique
$\tau_{1}(\alpha)\in(-1,0)$ satisfying
$c(\tau)\
\left\\{\begin{array}[]{lll}>0,&\rm{if}\quad\tau\in(-1,\tau_{1}(\alpha)),\\\\[5.69054pt]
=0,&\rm{if}\quad\tau=\tau_{1}(\alpha),\\\\[5.69054pt]
<0,&\rm{if}\quad\tau\in(\tau_{1}(\alpha),0)\end{array}\right.$ (2.2)
and
$\lim_{\alpha\to\alpha_{0}^{-}}\tau_{1}(\alpha)=0\quad\rm{and}\quad\lim_{\alpha\to
0^{+}}\tau_{1}(\alpha)=-1.$ (2.3)
Moreover, $\tau_{1}(\alpha)>\tau_{0}(\alpha)$, for all
$\alpha\in(0,\alpha_{0})$, where $\tau_{0}(\alpha)\in(-1,0)$ is the unique
zero of $C(\tau)$, defined in (1.4).
Proof. From (1.9), differentiating twice we find that
$c^{\prime\prime}(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}(\log|1-t|)^{2}+(1+t)^{\tau}(\log(1+t))^{2}}{t^{1+2\alpha}}dt>0,$
(2.4)
so that $c$ is strictly convex in $(-1,0)$. We also see easily that
$c(0)=0\quad\mbox{and}\quad\lim_{\tau\to-1^{+}}c(\tau)=\infty.$ (2.5)
Thus, if $c^{\prime}(0)\leq 0$ then $c(\tau)>0$ for $\tau\in(-1,0)$ and if
$c^{\prime}(0)>0$, then there exists $\tau_{1}(\alpha)\in(-1,0)$ such that
$c(\tau)>0$ for $\tau\in(-1,\tau_{1}(\alpha))$, $c(\tau)<0$ for
$\tau\in(\tau_{1}(\alpha),0)$ and $c(\tau_{1}(\alpha))=0$. In order to
complete our proof, we have to analyze the sign of $c^{\prime}(0)$, which
depends on $\alpha$ and to make this dependence explicit, we write
$c^{\prime}(0)=T(\alpha)$. We compute $T(\alpha)$ from (1.9), differentiating
and evaluating in $\tau=0$
$T(\alpha)=\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt.$ (2.6)
We have to prove that $T$ possesses a unique zero in the interval $(0,1)$. For
this purpose we start proving that
$\lim_{\alpha\to 1^{-}}T(\alpha)=-\infty\quad\mbox{and}\quad\lim_{\alpha\to
0^{+}}T(\alpha)=+\infty.$ (2.7)
The first limit follows from the fact that $\log(1-s)\leq-s,$ for all
$s\in[0,1/4]$, and so
$\displaystyle\lim_{\alpha\to
1^{-}}\int_{0}^{\frac{1}{2}}\frac{\log(1-t^{2})}{t^{1+2\alpha}}dt\leq-\lim_{\alpha\to
1^{-}}\int_{0}^{\frac{1}{2}}t^{1-2\alpha}dt=-\infty$
and the fact that exists a constant $t_{0}$ such that
$\int_{\frac{1}{2}}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\leq
t_{0},\qquad\mbox{for all }\alpha\in(1/2,1).$
The second limit in (2.7) follows from
$\displaystyle\lim_{\alpha\to
0^{+}}\int_{2}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\geq\log
3\lim_{\alpha\to 0^{+}}\int_{2}^{+\infty}t^{-1-2\alpha}dt=+\infty$
and the fact that there exists a constant $t_{1}$ such that
$\int_{0}^{2}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\leq t_{1},\quad\mbox{for
all }\alpha\in(0,1/2).$
On the other hand we claim that
$T^{\prime}(\alpha)=-2\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}\log
tdt<0,\ \ \ \alpha\in(0,1).$ (2.8)
In fact, since ${\log|1-t^{2}|}\log t$ is negative only for
$t\in(1,\sqrt{2})$, we have
$\displaystyle\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}\log tdt$
$\displaystyle>$
$\displaystyle\int_{0}^{\sqrt{2}-1}\frac{\log(1-t^{2})}{t^{1+2\alpha}}\log
tdt+\int_{1}^{\sqrt{2}}\log(t^{2}-1)\log tdt$ $\displaystyle\geq$
$\displaystyle\int_{0}^{\sqrt{2}-1}\frac{-t^{2}}{t^{1+2\alpha}}\log
tdt+\int_{1}^{\sqrt{2}}\log(t-1)\log tdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{\sqrt{2}-1}t^{1-2\alpha}\log
tdt+\int_{0}^{\sqrt{2}-1}\log(1+t)\log tdt$ $\displaystyle\geq$
$\displaystyle-\int_{0}^{\sqrt{2}-1}t^{1-2\alpha}\log
tdt+\int_{0}^{\sqrt{2}-1}t\log tdt>0.$
Then, (2.7) and (2.8) the existence of the desired $\alpha_{0}\in(0,1)$ with
the required properties follows, completing $(i)$ and (2.2) in $(ii)$.
To continue with the proof of our proposition, we study the first limit in
(2.3). We assume that there exist a sequence $\alpha_{n}\in(0,\alpha_{0})$ and
$\tilde{\tau}\in(-1,0)$ such that
$\lim_{n\to+\infty}\alpha_{n}=\alpha_{0}\quad\mbox{
and}\quad\lim_{n\to+\infty}\tau_{1}(\alpha_{n})=\tilde{\tau}$
and so $c(\tilde{\tau})=0$. Moreover $c(0)=0$ and
$c^{\prime}(0)=T(\alpha_{0})=0$, contradicting the strict convexity of $c$
given by (2.4). Next we prove the second limit in (2.3). We proceed by
contradiction, assuming that there exist a sequence
$\\{\alpha_{n}\\}\subset(0,1)$ and $\bar{\tau}\in(-1,0)$ such that
$\lim_{n\to+\infty}\alpha_{n}=0\quad\mbox{and}\quad\tau_{1}(\alpha_{n})\geq\bar{\tau}>-1,\quad\mbox{for
all }n\in\mathbb{N}.$
Then there exist $C_{1},C_{2}>0$, depending on $\bar{\tau}$, such that
$\displaystyle\int^{2}_{0}|\frac{|1-t|^{\tau_{1}(\alpha_{n})}+(1+t)^{\tau_{1}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}|dt\leq
C_{1}$
and
$\displaystyle\lim_{n\to\infty}\int_{2}^{+\infty}\frac{|1-t|^{\tau_{1}(\alpha_{n})}+(1+t)^{\tau_{1}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}dt\leq-
C_{2}\lim_{n\to\infty}\int_{2}^{+\infty}\frac{1}{t^{1+2\alpha_{n}}}dt=-\infty.$
Then $c(\tau_{1}(\alpha_{n}))\to-\infty$ as $n\to+\infty$, which is impossible
since $c(\tau_{1}(\alpha_{n}))=0.$
We finally prove the last statement of the proposition. Since
$\tau_{0}(\alpha)\in(-1,0)$ is such that $C(\tau_{0}(\alpha))=0$ and we have,
by definition, that
$c(\tau)=C(\tau)+\int_{1}^{+\infty}\frac{(t-1)^{\tau}}{t^{1+2\alpha}}dt,$
we find that $c(\tau_{0}(\alpha))>0$, which together with (2.2), implies that
$\tau_{0}(\alpha)\in(-1,\tau_{1}(\alpha)).$ $\Box$
Next we prove the main proposition in this section, which is on the basis of
the construction of super and sub-solutions. By hypothesis on the domain
$\Omega$ and the manifold $\mathcal{C}$, there exists $\delta>0$ such that the
distance functions $d(\cdot)$, to $\partial\Omega$, and $D(\cdot)$, to
$\mathcal{C}$, are of class $C^{2}$ in $B_{\delta}$ and $A_{\delta}$,
respectively, and $dist(A_{\delta},B_{\delta})>0$, where
$A_{\delta}=\\{x\in\Omega\ |\ D(x)<\delta\\}$ and $B_{\delta}=\\{x\in\Omega\
|\ d(x)<\delta\\}$. Now we define the basic function $V_{\tau}$ as follows
$V_{\tau}(x):=\left\\{\begin{array}[]{lll}D(x)^{\tau},&x\in
A_{\delta}\setminus\mathcal{C},\\\\[5.69054pt] d(x)^{2},&x\in
B_{\delta},\\\\[5.69054pt] l(x),&x\in\Omega\setminus(A_{\delta}\cup
B_{\delta}),\\\\[5.69054pt] 0,&x\in\Omega^{c},\end{array}\right.$ (2.9)
where $\tau$ is a parameter in $(-1,0)$ and the function $l$ is positive such
that $V_{\tau}$ is of class $C^{2}$ in $\mathbb{R}^{N}\setminus\mathcal{C}$.
###### Proposition 2.2
Let $\alpha_{0}$ and $\tau_{1}(\alpha)$ be as in Proposition 2.1.
$(i)$ If $(\alpha,\tau)\in[\alpha_{0},1)\times(-1,0)$ or
$(\alpha,\tau)\in(0,\alpha_{0})\times(-1,\tau_{1}(\alpha)),$ then there exist
$\delta_{1}\in(0,\delta]$ and $C>1$ such that
$\frac{1}{C}D(x)^{\tau-2\alpha}\leq-(-\Delta)^{\alpha}V_{\tau}(x)\leq
CD(x)^{\tau-2\alpha},\ \ x\in A_{\delta_{1}}\setminus\mathcal{C}.$
$(ii)$ If $(\alpha,\tau)\in(0,\alpha_{0})\times(\tau_{1}(\alpha),0),$ then
there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that
$\frac{1}{C}D(x)^{\tau-2\alpha}\leq(-\Delta)^{\alpha}V_{\tau}(x)\leq
CD(x)^{\tau-2\alpha},\ \ x\in A_{\delta_{1}}\setminus\mathcal{C}.$
$(iii)$ If $(\alpha,\tau)\in(0,\alpha_{0})\times\\{\tau_{1}(\alpha)\\},$ then
there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that
$|(-\Delta)^{\alpha}V_{\tau}(x)|\leq CD(x)^{\min\\{\tau,2\tau-2\alpha+1\\}},\
\ x\in A_{\delta_{1}}\setminus\mathcal{C}.$
This proposition and its proof has many similarities with Proposition 3.2 in
[3], but it has also important differences so we give a complete proof of it.
Proof. By compactness of $\mathcal{C}$, we just need to prove that the
corresponding inequality holds in a neighborhood of any point
$\bar{x}\in\mathcal{C}$ and, without loss of generality, we may assume
$\bar{x}=0$. For a given $0<\eta\leq\delta$, we define
$Q_{\eta}=(-\eta,\eta)\times B_{\eta}\subset\mathbb{R}\times\mathbb{R}^{N-1},$
where $B_{\eta}$ denotes the ball centered at the origin and with radius
$\eta$ in $\mathbb{R}^{N-1}$. We observe that $Q_{\eta}\subset\Omega.$ Let
$\varphi:\mathbb{R}^{N-1}\to\mathbb{R}$ be a $C^{2}$ function such that
$(z_{1},z^{\prime})\in\mathcal{C}\cap Q_{\delta}$ if and only if
$z_{1}=\varphi(z^{\prime})$. We further assume that $e_{1}$ is normal to
$\mathcal{C}$ at $\bar{x}$ and then there exists $C>0$ such that
$|\varphi(z^{\prime})|\leq C|z^{\prime}|^{2}$ for $|z^{\prime}|\leq\delta$.
Thus, choosing $\eta>0$ smaller if necessary we may assume that
$|\varphi(z^{\prime})|<\frac{\eta}{2}$ for $|z^{\prime}|\leq\eta$. In the
proof of our inequalities, we will consider a generic point along the normal
$x=(x_{1},0)\in A_{\eta/4}$, with $0<|x_{1}|<\eta/4$. We observe that
$|x-\bar{x}|=D(x)=|x_{1}|$. By definition we have
$-(-\Delta)^{\alpha}V_{\tau}(x)=\frac{1}{2}\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy+\frac{1}{2}\int_{\mathbb{R}^{N}\setminus
Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy.$ (2.10)
It is not difficult to see that the second integral is bounded by
$Cx_{1}^{\tau}$, for an appropriate constant $C>0$, so that we only need to
study the first integral, that from now on we denote by $\frac{1}{2}E(x_{1})$.
Our first goal is to obtain positive constants $c_{1},c_{2}$ so that lower
bound for $E(x_{1})$
$E(x_{1})\geq
c_{1}c(\tau)|x_{1}|^{\tau-2\alpha}-c_{2}|x_{1}|^{\min\\{\tau,2\tau-2\alpha+1\\}}$
(2.11)
holds, for all $|x_{1}|\leq\eta/4$. For this purpose we assume that
$0<\eta\leq\delta/2$, then for all $y=(y_{1},y^{\prime})\in Q_{\eta}$ we have
that $x\pm y\in Q_{\delta}$, so that
$D(x\pm y)\leq|x_{1}\pm y_{1}-\varphi(\pm y^{\prime})|,\quad\mbox{for\ all}\ \
y\in Q_{\eta}.$
From here and the fact that $\tau\in(-1,0)$, we have that
$E(x_{1})=\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy\geq\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy+\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy,$
(2.12)
where the functions $I$ and $J$ are defined, for $y\in Q_{\eta}$, as
$I(y)=|x_{1}-y_{1}|^{\tau}+|x_{1}+y_{1}|^{\tau}-2x_{1}^{\tau}$ (2.13)
and
$J(y)=|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}.$ (2.14)
In what follows we assume $x_{1}>0$ (the case $x_{1}<0$ is similar). For the
first term of the right hand side in (2.12), we have
$\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy=x_{1}^{\tau-2\alpha}\int_{Q_{\frac{\eta}{x_{1}}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz.$
On one hand we have that, for a constant $c_{1}$, we have
$\displaystyle\int_{\mathbb{R}^{N}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz=2c(\tau)\int_{\mathbb{R}^{N-1}}\frac{1}{(|z^{\prime}|^{2}+1)^{\frac{N+2\alpha}{2}}}dz^{\prime}=c_{1}c(\tau),$
and, on the other hand, for constants $C_{2}$ and $C_{3}$ we have
$\displaystyle|\int_{-\frac{\eta}{x_{1}}}^{\frac{\eta}{x_{1}}}\int_{|z^{\prime}|\geq\frac{\eta}{x_{1}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz|$
$\displaystyle\leq$
$\displaystyle\int_{-\frac{\eta}{x_{1}}}^{\frac{\eta}{x_{1}}}(|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}+2)dz_{1}\int_{|z^{\prime}|\geq\frac{\eta}{x_{1}}}\frac{dz^{\prime}}{|z^{\prime}|^{N+2\alpha}}\leq
C_{2}x_{1}^{2\alpha}$
and
$\displaystyle|\int_{|z_{1}|\geq\frac{\eta}{x_{1}}}\int_{\mathbb{R}^{N-1}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz|$
$\displaystyle\leq$ $\displaystyle
2\int_{\frac{\eta}{x_{1}}}^{+\infty}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}+2}{z_{1}^{1+2\alpha}}dz_{1}\int_{\mathbb{R}^{N-1}}\frac{1}{(1+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz^{\prime}\leq
C_{3}x_{1}^{2\alpha}.$
Consequently, for an appropriate constant $c_{2}$
$|\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy-
c_{1}c(\tau)x_{1}^{\tau-2\alpha}|\leq c_{2}x_{1}^{\tau}.$ (2.15)
Next we estimate the second term of the right hand side in (2.12). Since
$\int_{Q_{\eta}}\frac{J(-y)}{|y|^{N+2\alpha}}dy=\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy,$
we only need to estimate
$\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy=\int_{B_{\eta}}\int^{\eta}_{-\eta}\frac{|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}.$
(2.16)
We notice that $|x_{1}+y_{1}-\varphi(y^{\prime})|\geq|x_{1}+y_{1}|$ if and
only if
$\varphi(y^{\prime})(x_{1}+y_{1}-\frac{\varphi(y^{\prime})}{2})\leq 0.$
From here and (2.16), we have
$\displaystyle\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle\geq$
$\displaystyle\int_{B_{\eta}}\int^{-x_{1}+\frac{\varphi_{+}(y^{\prime})}{2}}_{-\eta}\frac{|x_{1}+y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\eta}_{-x_{1}+\frac{\varphi_{-}(y^{\prime})}{2}}\frac{|x_{1}+y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle E_{1}(x_{1})+E_{2}(x_{1}),$
where $\varphi_{+}(y^{\prime})=\max\\{\varphi(y^{\prime}),0\\}$ and
$\varphi_{-}(y^{\prime})=\min\\{\varphi(y^{\prime}),0\\}$. We only estimate
$E_{1}(x_{1})$ ($E_{2}(x_{1})$ is similar). Using integration by parts, we
obtain
$\displaystyle E_{1}(x_{1})$ (2.17) $\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{x_{1}-\eta}\frac{|y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|y_{1}|^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{0}_{x_{1}-\eta}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau}-y_{1}^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{-\varphi_{+}(y^{\prime})^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{(\eta-
x_{1}+\varphi_{+}(y^{\prime}))^{\tau+1}-(\eta-
x_{1})^{\tau+1}}{(\eta^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle-\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{0}_{x_{1}-\eta}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}-(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle+\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{-2^{-\tau}\varphi_{+}(y^{\prime})^{\tau+1}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{\varphi_{+}(y^{\prime})^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}+y_{1}^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{-2^{-\tau}}{\tau+1}\int_{B_{\eta}}\frac{\varphi_{+}(y^{\prime})^{\tau+1}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\min\\{\frac{\varphi_{+}(y^{\prime})}{2},x_{1}\\}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}+y_{1}^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle A_{1}(x_{1})+A_{2}(x_{1}).$
In order to estimate $A(x_{1})$, we split $B_{\eta}$ in $O=\\{y^{\prime}\in
B_{\eta}:|\frac{\varphi_{+}(y^{\prime})}{2}-x_{1}|\geq\frac{x_{1}}{2}\\}$ and
$B_{\eta}\setminus O$. On one hand we have
$\displaystyle\int_{O}\frac{|y^{\prime}|^{2\tau+2}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle
x_{1}^{2\tau-2\alpha+1}\int_{B_{\eta/{x_{1}}}}\frac{|z^{\prime}|^{2\tau+2}}{(1/4+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz^{\prime}$
$\displaystyle\leq$ $\displaystyle C(x_{1}^{2\tau-2\alpha+1}+x_{1}^{\tau}).$
On the other hand, for $y^{\prime}\in B_{\eta}\setminus O$ we have that
$|y^{\prime}|\geq c_{1}\sqrt{x_{1}}$, for some constant $c_{1}$, and then
$\displaystyle\int_{B_{\eta}\setminus
O}\frac{|y^{\prime}|^{2\tau+2}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle\int_{B_{\eta}\setminus
B_{c_{1}\sqrt{x_{1}}}}|y^{\prime}|^{2\tau+2-N-2\alpha}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle C(x_{1}^{\tau-\alpha+\frac{1}{2}}+1).$
Thus, for some $C>0$,
$\displaystyle A_{1}(x_{1})\geq-Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
(2.18)
Next we estimate $A_{2}(x_{1})$:
$\displaystyle A_{2}(x_{1})$ $\displaystyle\geq$
$\displaystyle\frac{2(N+2\alpha)}{\tau+1}\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{\varphi_{+}(y^{\prime})^{\tau+1}(y_{1}-x_{1})}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dy_{1}dy^{\prime}$
$\displaystyle\geq$ $\displaystyle
C\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{|y^{\prime}|^{2\tau+2}(y_{1}-x_{1})}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dy_{1}dy^{\prime}$
$\displaystyle\geq$ $\displaystyle
Cx_{1}^{2\tau-2\alpha+1}\int_{B_{\eta/{x_{1}}}}\int^{1}_{0}\frac{|z^{\prime}|^{2\tau+2}(z_{1}-1)}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dz_{1}dz^{\prime}$
$\displaystyle\geq$ $\displaystyle-
C_{1}x_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}},$
for some $C,C_{1}>0$. From here, (2.17) and (2.18) we obtain, for some $C>0$
$E_{1}(x_{1})\geq-Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
Using the similar estimate for $E_{2}(x_{1})$, we obtain
$\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy\geq-
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.19)
Thus, from (2.12), (2.15), (2.19) and noticing that these inequalities also
hold with $x_{1}<0$ with the obvious changes, we conclude the lower bound for
$E(x_{1})$ we gave in (2.11). Our second goal is to get an upper bound for
$E(x_{1})$ and for this, we first recall Lemma 3.1 in [3] to obtain
$D(x\pm y)^{\tau}\leq(x_{1}\pm
y_{1}-\varphi(y^{\prime}))^{\tau}(1+C|y^{\prime}|^{2}),\,\,\mbox{for
all}\quad|x_{1}|\leq\eta/4,y=(y_{1},y^{\prime})\in Q_{\eta}.$
From here we see that
$\displaystyle E(x_{1})$ $\displaystyle\leq$
$\displaystyle\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy+\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy$
(2.20)
$\displaystyle+C\int_{Q_{\eta}}\frac{I(y)+J(y)+J(-y)}{|y|^{N+2\alpha}}|y^{\prime}|^{2}dy.$
We denote by $E_{3}(x_{1})$ the third integral above. The first integral was
studied in (2.15), so we study the second integral and that we only need to
consider the term $J(y)$, since the other is completely analogous. We see that
$|x_{1}+y_{1}-\varphi(y^{\prime})|\leq|x_{1}+y_{1}|$ if and only if
$\varphi(y^{\prime})(x_{1}+y_{1}-\frac{\varphi(y^{\prime})}{2})\geq 0.$
As before, we will consider only the case $x_{1}>0$, since the other one is
analogous. From (2.16) we have
$\displaystyle\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle\leq$
$\displaystyle\int_{B_{\eta}}\int^{-x_{1}+\frac{\varphi_{-}(y^{\prime})}{2}}_{-\eta}\frac{|x_{1}+y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\eta}_{-x_{1}+\frac{\varphi_{+}(y^{\prime})}{2}}\frac{|x_{1}+y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle F_{1}(x_{1})+F_{2}(x_{1}).$
Next we estimate $F_{1}(x_{1})$ ($F_{2}(x_{1})$ is similar), using integration
by parts
$\displaystyle F_{1}(x_{1})$ $\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int_{x_{1}-\eta}^{\frac{\varphi_{-}(y^{\prime})}{2}}\frac{|y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|y_{1}|^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{\varphi_{-}(y^{\prime})}_{x_{1}-\eta}\frac{(\varphi_{-}(y^{\prime})-y_{1})^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\frac{\varphi_{-}(y^{\prime})}{2}}_{\varphi_{-}(y^{\prime})}\frac{(y_{1}-\varphi_{-}(y^{\prime}))^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((x_{1}-\varphi_{-}(y^{\prime}))^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{(\eta-
x_{1}+\varphi_{-}(y^{\prime}))^{\tau+1}-(\eta-
x_{1})^{\tau+1}}{(\eta^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle-\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\varphi_{-}(y^{\prime})}_{x_{1}-\eta}\frac{(\varphi_{-}(y^{\prime})-y_{1})^{\tau+1}-(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle+\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{2^{-\tau}(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{-(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((x_{1}-\varphi_{-}(y^{\prime}))^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\frac{\varphi_{-}(y^{\prime})}{2}}_{\varphi_{-}(y^{\prime})}\frac{(y_{1}-\varphi_{-}(y^{\prime}))^{\tau+1}+(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}\frac{2^{-\tau}(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}=B(x_{1}).$
Since $(\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}\geq x_{1}^{2}$, we have
$\displaystyle B(x_{1})$ $\displaystyle\leq$
$\displaystyle\frac{2^{-\tau}}{\tau+1}\int_{B_{\eta}}\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle
C\int_{B_{\eta}}\frac{|y^{\prime}|^{2\tau+2}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}},$
for some $C>0$ independent of $x_{1}$. Thus we have obtained that
$F_{1}(x_{1})\leq Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.21)
Similarly, we can get an analogous estimate for $F_{2}(x_{1})$ and these two
estimates imply
$\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.22)
Finally we obtain
$\displaystyle\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}|y^{\prime}|^{2}dy$
$\displaystyle=$ $\displaystyle
x_{1}^{\tau-2\alpha+2}\int_{Q_{\frac{\eta}{x_{1}}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}|z^{\prime}|^{2}dz$
$\displaystyle\leq$ $\displaystyle Cx_{1}^{\min\\{\tau,\tau-2\alpha+2\\}}$
and, in a similar way,
$\int_{Q_{\eta}}\frac{J(y)|y^{\prime}|^{2}}{|y|^{N+2\alpha}}dy\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
From the last two inequalities we obtain
$E_{3}(x_{1})\leq Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.23)
Then, taking into account (2.20), (2.15), (2.22), (2.23) and considering also
the case $x_{1}<0$, we obtain
$E(x_{1})\leq
c_{1}c(\tau)|x_{1}|^{\tau-2\alpha}+c_{2}|x_{1}|^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
(2.24)
From inequalities (2.11), (2.24) and Proposition 2.1 the result follows.
$\Box$
## 3 Existence of large solution
This section is devoted to use Proposition 2.2 to prove the existence of
solution of problem (1.8). To this purpose, our main goal is to construct
appropriate sub-solution and super-solution of problem (1.8) under the
hypotheses of Theorem 1.1 $(i)$, $(ii)$ and Theorem 1.2 $(i)$.
We begin with a simple lemma that reduces the problem to find them only in
$A_{\delta}\setminus\mathcal{C}$.
###### Lemma 3.1
Let $U$ and $W$ be classical ordered super and sub-solution of (1.8) in the
sub-domain $A_{\delta}\setminus\mathcal{C}$. Then there exists $\lambda$ large
such that $U_{\lambda}=U+\lambda\bar{V}$ and $W_{\lambda}=W-\lambda\bar{V}$,
are ordered super and sub-solution of (1.8), where $\bar{V}$ is the solution
of
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}\bar{V}(x)=1,&x\in\Omega,\\\\[5.69054pt]
\bar{V}(x)=0,&x\in\Omega^{c}.\end{array}\right.$ (3.1)
###### Remark 3.1
Here $U,W:I\\!\\!R^{N}\to\mathbb{R}$ are classical ordered of super and sub-
solution of (1.8) in the sub-domain $A_{\delta}\setminus\mathcal{C}$ if $U$
satisfies
$(-\Delta)^{\alpha}U+|U|^{p-1}U\geq 0\quad\mbox{in}\quad
A_{\delta}\setminus\mathcal{C}$
and $W$ satisfies the reverse inequality. Moreover, they satisfy
$U\geq W\ \ {\rm{in}}\
\Omega\setminus\mathcal{C},\quad\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}W(x)=+\infty,\quad
U=W=0\ \ {\rm{in}}\ \Omega^{c}.$
Proof. Notice that by the maximum principle $\bar{V}$ is nonnegative in
$\Omega$, therefore $U_{\lambda}\geq U$ and $W_{\lambda}\leq W$, so they are
still ordered. In addition $U_{\lambda}$ satisfies
$(-\Delta)^{\alpha}U_{\lambda}+|U_{\lambda}|^{p-1}U_{\lambda}\geq(-\Delta)^{\alpha}U+|U|^{p-1}U+\lambda>0,\quad\mbox{in}\quad\Omega\setminus\mathcal{C}.$
This inequality holds because of our assumption in
$A_{\delta}\setminus\mathcal{C}$ and the fact that
$(-\Delta)^{\alpha}U+|U|^{p-1}U$ is continuous in
$\Omega\setminus{A_{\delta}}$ and by taking $\lambda$ large enough.
By the same type of arguments we find that $W_{\lambda}$ is a sub-solution.
$\Box$
Proof of existence results in Theorem 1.1 $(i)$ and Theorem 1.2 $(i)$. We
define
$U_{\mu}(x)=\mu V_{\tau}(x)\ {\rm{and}}\ \ W_{\mu}(x)=\mu V_{\tau}(x),\
x\in\mathbb{R}^{N}\setminus\mathcal{C},$ (3.2)
where $V_{\tau}$ is defined in (2.9) with $\tau=-\frac{2\alpha}{q-1}$
1\. $U_{\mu}$ is Super-solution. By hypotheses of Theorem 1.1 $(i)$ and
Theorem 1.2 $(i)$, we notice that
$\tau\in(-1,0),\quad\rm{for}\ \alpha\in[\alpha_{0},1),$
$\tau\in(-1,\tau_{1}(\alpha)),\quad\rm{for}\ \alpha\in(0,\alpha_{0})$
and $\tau p=\tau-2\alpha$, then we use Proposition 2.2 part $(i)$ to obtain
that there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that
$\displaystyle(-\Delta)^{\alpha}U_{\mu}(x)+U^{p}_{\mu}(x)\geq-C\mu
D(x)^{\tau-2\alpha}+\mu^{p}D(x)^{\tau p},\quad x\in
A_{\delta_{1}}\setminus\mathcal{C}.$
Then there exist $\mu_{1}>1$ such that for $\mu\geq\mu_{1}$, we have
$(-\Delta)^{\alpha}U_{\mu}(x)+U^{p}_{\mu}(x)\geq 0,\ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$
2\. $W_{\mu}$ is Sub-solution. We use Proposition 2.2 part $(i)$ to obtain
that there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that for $x\in
A_{\delta_{1}}\setminus\mathcal{C}$, we have
$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)$
$\displaystyle\leq$
$\displaystyle-\frac{\mu}{C}D(x)^{\tau-2\alpha}+\mu^{p}D(x)^{\tau p}$
$\displaystyle\leq$
$\displaystyle(-\frac{\mu}{C}+\mu^{p})D(x)^{\tau-2\alpha}.$
Then there exists $\mu_{3}\in(0,1)$ such that for all $\mu\in(0,\mu_{3})$, it
has
$(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)\leq 0,\ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$
To conclude the proof we use Lemma 3.1 and Proposition 2.2. $\Box$
Proof of Theorem 1.1 $(ii)$. For any given $t>0$, we denote
$U(x)=tV_{\tau_{1}(\alpha)}(x),\quad x\in\mathbb{R}^{N}\setminus\mathcal{C},$
$W_{\mu}(x)=tV_{\tau_{1}(\alpha)}(x)-\mu V_{\bar{\tau}}(x),\quad
x\in\mathbb{R}^{N}\setminus\mathcal{C}$
where
$\bar{\tau}=\min\\{\tau_{1}(\alpha)p+2\alpha,\frac{1}{2}\tau_{1}(\alpha)\\}<0$.
By (1.12), we have
$\bar{\tau}\in(\tau_{1}(\alpha),0),\
\bar{\tau}-2\alpha<\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}\
\rm{and}\ \bar{\tau}-2\alpha<\tau_{1}(\alpha)p.$ (3.3)
1\. $U$ is Super-solution. We use Proposition 2.2 $(iii)$ to obtain that for
any $x\in A_{\delta_{1}}\setminus\mathcal{C}$,
$\displaystyle(-\Delta)^{\alpha}U(x)+U^{p}(x)\geq-
CtD(x)^{\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}}+t^{p}D(x)^{\tau_{1}(\alpha)p},$
together with
$\tau_{1}(\alpha)p<\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}$,
then there exists $\delta_{2}\in(0,\delta_{1}]$ such that
$(-\Delta)^{\alpha}U(x)+U^{p}(x)\geq 0,\quad x\in
A_{\delta_{2}}\setminus\mathcal{C}.$
2\. $W_{\mu}$ is Sub-solution. We use Proposition 2.2 $(ii)$ and $(iii)$ to
obtain that for $x\in A_{\delta_{1}}\setminus\mathcal{C}$,
$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)$
$\displaystyle\leq$ $\displaystyle
CtD(x)^{\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}}$
$\displaystyle-\frac{\mu}{C}D(x)^{\bar{\tau}-2\alpha}+t^{p}D(x)^{\tau_{1}(\alpha)p}.$
Then there exists $\delta_{2}\in(0,\delta_{1}]$ such that for any $\mu\geq 1$,
we have
$(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)\leq 0,\ x\in
A_{\delta_{2}}\setminus\mathcal{C}.$
To conclude the proof we use Lemma 3.1 and Proposition 2.2. $\Box$
## 4 Uniqueness and nonexistence
We prove the uniqueness statement by contradiction. Assume that $u$ and $v$
are solutions of problem (1.8) satisfying (1.11). Then there exist $C_{0}\geq
1$ and $\bar{\delta}\in(0,\delta)$ such that
$\frac{1}{C_{0}}\leq v(x)D(x)^{-\tau},\ u(x)D(x)^{-\tau}\leq C_{0},\ \ \forall
x\in A_{\bar{\delta}}\setminus\mathcal{C},$ (4.4)
where $\tau=-\frac{2\alpha}{p-1}$. We denote
$\mathcal{A}=\\{x\in\Omega\setminus\mathcal{C}\ |\ u(x)>v(x)\\}.$ (4.5)
Then $\mathcal{A}$ is open and $\mathcal{A}\subset\Omega$. Then the uniqueness
in Theorem 1.2 $(i)$ and Theorem 1.1 $(i)$ is a consequence of the following
result:
###### Proposition 4.1
Under the hypotheses of Theorem 1.2 $(i)$ and Theorem 1.1 $(i)$, we have
$\mathcal{A}=\O.$
Proof. The procedure of proof is similar as Section§5 in [3], noting that we
need to replace $d(x)$ by $D(x)$ and $\partial\Omega$ by $\mathcal{C}$ .
$\Box$
From Proposition 4.1, we can prove uniqueness part in Theorem 1.1 $(i)$ and
Theorem 1.2 $(i)$ .
The final goal in this note is to consider the nonexistence of solutions of
problem (1.8) under the hypotheses of Theorem 1.1 $(iii)$ and Theorem 1.2
$(ii)$.
###### Proposition 4.2
Under the hypotheses of Theorem 1.1 $(iii)$ and Theorem 1.2 $(ii)$, we assume
that $U_{1}$ and $U_{2}$ are both sub-solutions (or both super-solutions) of
(1.8) satisfying that $U_{1}=U_{2}=0$ in $\Omega^{c}$ and
$\displaystyle 0$ $\displaystyle<$
$\displaystyle\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}$ $\displaystyle<$
$\displaystyle\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}<+\infty,$
for $\tau\in(-1,0)$. For the case $\tau p>\tau-2\alpha$, we further assume
that
$(i)$ if $U_{1},U_{2}$ are sub-solutions, there exist $C>0$ and
$\tilde{\delta}>0$,
$(-\Delta)^{\alpha}U_{2}(x)\leq-CD(x)^{\tau-2\alpha},\quad x\in
A_{\tilde{\delta}}\setminus\mathcal{C};$ (4.6)
or
$(ii)$ if $U_{1},U_{2}$ are super-solutions, there exist $C>0$ and
$\tilde{\delta}>0$,
$(-\Delta)^{\alpha}U_{1}(x)\geq CD(x)^{\tau-2\alpha},\quad x\in
A_{\tilde{\delta}}\setminus\mathcal{C}.$ (4.7)
Then there doesn’t exist any solution $u$ of (1.8) such that
$\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}\frac{U_{1}(x)}{u(x)}<1<\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}\frac{U_{2}(x)}{u(x)}.$ (4.8)
Proof. The proof is similar as Proposition 6.1 in [3], noting again that we
need to replace $d(x)$ by $D(x)$ and $\partial\Omega$ by $\mathcal{C}$ .
$\Box$
With the help of Proposition 2.2, for given $t_{1}>t_{2}>0$, we construct two
sub-solutions (or both super-solutions) $U_{1}$ and $U_{2}$ of (1.8) such that
$\lim_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}=t_{1},\
\lim_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}=t_{2}.$
So what we have to do is to prove that for any $t>0$, we can construct super-
solution (sub-solution) of problem (1.8).
Proof of Theorem 1.1 $(iii)$ and Theorem 1.2 $(ii)$. We divide our proof of
the nonexistence results into several cases under the assumption $p>1$.
Zone 1: We consider $\tau\in(\tau_{1}(\alpha),0)$ and
$\alpha\in(0,\alpha_{0}).$ By Proposition 2.2 $(ii)$, there exists
$\delta_{1}>0$ such that
$(-\Delta)^{\alpha}V_{\tau}(x)\geq\frac{1}{C}D(x)^{\tau-2\alpha},\ \ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$ (4.9)
Since $V_{\tau}$ is $C^{2}$ in $\Omega\setminus\mathcal{C}$, then there exists
$C>0$ such that
$|(-\Delta)^{\alpha}V_{\tau}(x)|\leq C,\ \ x\in\Omega\setminus
A_{\delta_{1}}.$ (4.10)
Let $\bar{U}:=V_{\tau}+C\bar{V}\quad\rm{in}\ \
\mathbb{R}^{N}\setminus\mathcal{C}$, then we have $\bar{U}>0$ in
$\Omega\setminus\mathcal{C}$,
$(-\Delta)^{\alpha}\bar{U}\geq 0\ \ {\rm{in}}\ \
\Omega\setminus\mathcal{C}\quad{\rm{and}}\ \
(-\Delta)^{\alpha}\bar{U}(x)\geq\frac{1}{C}D(x)^{\tau-2\alpha},\ \ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$
Then, we have that $t\bar{U}$ is super-solution of (1.8) for any $t>0$. Using
Proposition 4.2, we see that there is no solution of (1.8) satisfying (1.14).
Zone 2: We consider $\tau-2\alpha<\tau p$ and
$\tau\in\left\\{\begin{array}[]{lll}(-1,0),&\alpha\in[\alpha_{0},1),\\\\[5.69054pt]
(-1,\tau_{1}(\alpha)),&\alpha\in(0,\alpha_{0}).\end{array}\right.$
Let us define
$W_{\mu,t}=tV_{\tau}-\mu\bar{V}\quad{\rm{in}}\ \
\mathbb{R}^{N}\setminus\mathcal{C},$
where $t,\mu>0$. By Proposition 2.2 $(i)$, for $x\in
A_{\delta_{1}}\setminus\mathcal{C}$,
$\displaystyle(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq-\frac{t}{C}D(x)^{\tau-2\alpha}+t^{p}D(x)^{\tau
p}.$
For any fixed $t>0$, there exists $\delta_{2}\in(0,\delta_{1}]$, for all
$\mu\geq 0$,
$(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq 0,\quad
A_{\delta_{2}}\setminus\mathcal{C}.$ (4.11)
To consider $x\in\Omega\setminus A_{\delta_{2}}$, in fact, there exists
$C_{1}>0$ such that
$t|(-\Delta)^{\alpha}V_{\tau}(x)|+t^{p}V_{\tau}^{p}(x)\leq C_{1},\quad
x\in\Omega\setminus A_{\delta_{2}}$
and
$\displaystyle(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq
C_{1}t-\mu,\quad x\in\Omega\setminus A_{\delta_{2}}$
For given $t>0$, there exists $\mu(t)>0$ such that
$(-\Delta)^{\alpha}W_{\mu(t),t}(x)+|W_{\mu,t}|^{p-1}W_{\mu(t),t}(x)\leq 0,\ \
x\in\Omega\setminus A_{\delta_{2}}.$ (4.12)
Therefore, together with (4.11) and (4.12), for any given $t>0$, there sub-
solutions $W_{\mu(t),t}$ of problem (1.8) and by Proposition 4.2, we see that
there is no solution $u$ of (1.8) satisfying (1.14).
Zone 3: We consider $\tau-2\alpha>\tau p$ and
$\tau\in\left\\{\begin{array}[]{lll}(-1,0),&\alpha\in[\alpha_{0},1),\\\\[5.69054pt]
(-1,\tau_{1}(\alpha)),&\alpha\in(0,\alpha_{0}).\end{array}\right.$
We denote that
$U_{\mu,t}=tV_{\tau}+\mu\bar{V}\quad\rm{in}\ \
\mathbb{R}^{N}\setminus\mathcal{C},$
where $t,\mu>0$. Here $U_{\mu,t}>0$ in $\Omega\setminus\mathcal{C}$. By
Proposition 2.2 $(i)$,
$\displaystyle(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq-
CtD(x)^{\tau-2\alpha}+t^{p}D(x)^{\tau p},\quad x\in
A_{\delta_{1}}\setminus\mathcal{C}.$
For any fixed $t>0$, there exists $\delta_{2}\in(0,\delta_{1}]$, for all
$\mu\geq 0$,
$(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq 0,\quad x\in
A_{\delta_{2}}\setminus\mathcal{C}.$ (4.13)
For $x\in\Omega\setminus A_{\delta_{2}}$, we see that
$(-\Delta)^{\alpha}V_{\tau}$ is bounded and
$\displaystyle(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq-Ct+\mu.$
For given $t>0$, there exists $\mu(t)>0$ such that
$(-\Delta)^{\alpha}U_{\mu(t),t}(x)+U^{p}_{\mu(t),t}(x)\geq 0,\ \
x\in\Omega\setminus A_{\delta_{2}}.$ (4.14)
Combining with (4.13) and (4.14), we have that for any $t>0$, there exists
$\mu(t)>0$ such that
$(-\Delta)^{\alpha}U_{\mu(t),t}(x)+U^{p}_{\mu(t),t}(x)\geq 0,\ \ \
x\in\Omega\setminus\mathcal{C}.$
Therefore, for any given $t>0$, there is a super-solution $U_{\mu(t),t}$ of
problem (1.8) and by Proposition 4.2, we see that there is no solution of
(1.8) satisfying (1.14).
We see that Zones 1 and 2 cover Theorem 1.1 part $(iii)$ a) since
$\tau>-2\alpha/(p-1).$ From Zones 1, 2 and 3 we cover Theorem 1.1 part $(iii)$
b) since $\tau_{1}(\alpha)>2\alpha/(p-1)$. Moreover, from Zone 1 to Zone 3, we
cover the parameters in part $(iii)$ c) of Theorem 1.1, since
$\tau_{1}(\alpha)<2\alpha/(p-1)$. Finally Theorem 1.2 part ii) can be obtained
from Zone 2 and Zone 3. This complete the proof. $\Box$
Acknowledgements. The authors thanks Peter Bates for proposing the problem.
H.C. was partially supported by Conicyt Ph.D. scholarship. P.F. was partially
supported by Fondecyt # 1110291 and Programa BASAL-CMM U. de Chile. A.Q. was
partially supported by Fondecyt # 1110210 and Programa BASAL-CMM U. de Chile.
## References
* [1] J. M. Arrieta and A. Rodríguez-Bernal, Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions, Comm. Partial Diff. Eqns., 29, 1127-1148, 2004\.
* [2] C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math., 58, 9-24, 1992.
* [3] H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian, Preprint.
* [4] M. Chuaqui, C. Cortázar, M. Elgueta and J. García-Melián, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Comm. Pure Appl. Anal., 3, 653-662, 2004.
* [5] M. del Pino and R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Analysis: Theory, Methods & Applications, 48(6), 897-904, 2002.
* [6] G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Analysis: Theory, Methods & Applications, 20(2), 97-125, 1993.
* [7] Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31, 1-18, 1999.
* [8] P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations. Asymptotic Analysis, Volume 78 (3), 123-144, 2012.
* [9] J. F. Le Gall, A path-valued Markov process and its connections with parital differential equations. In Proc. First European Congress of Mathematics, Vol. II (A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds.) 185-212, 1994. Birkhaüser, Boston.
* [10] J. García-Melián, Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. Diff. Eqns., 223(1), 208-227, 2006.
* [11] J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10, 503-510, 1957.
* [12] S. Kim, A note on boundary blow-up problem of $\Delta u=u^{p}$, IMA preprint No., 18-20, 2002.
* [13] V. A. Kondratev, V. A. Nikishkin, Asymptotics near the boundary, of a solution of a singular boundary value problem for a semilinear elliptic equation, Differential Equations 26 (1990), 345-348.
* [14] A. C. Lazer, P. J. McKenna, Asymptotic behaviour of solutions of boundary blow-up problems, Differential Integral Equations 7 (1994), 1001-1019.
* [15] C. Loewner and L. Nirenberg, Parital differential equations invariant under conformal or projective transformations, In Contributions to analysis, Academic Press, New York, 245-272, 1974.
* [16] M. Marcus and L. Véron, Uniqueness of solutions with blow up at the boundary for a class of nonlinear elliptic equations, C. R. Acad. Sci. Paris sér. I Math. 317(6), 559-563, 1993.
* [17] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equation, J. Evol. Equ. 3, 637-652, 2003\.
* [18] M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré 14(2), 237-274, 1997.
* [19] R. Osserman, On the inequality $\Delta u=f(u)$, Pac. J. Math. 7, 1641-1647, 1957.
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|
arxiv-papers
| 2013-11-26T10:02:22 |
2024-09-04T02:49:54.209095
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huyuan Chen, Patricio Felmer, Alexander Quaas",
"submitter": "Huyuan Chen",
"url": "https://arxiv.org/abs/1311.6607"
}
|
1311.6672
|
UNIVERSITÀ DEGLI STUDI DI TRENTO
Facoltà di Scienze Matematiche, Fisiche e Naturali
Dipartimento di Fisica
Tesi di Dottorato di Ricerca in Fisica
Ph.D. Thesis in Physics
From Hypernuclei to Hypermatter:
a Quantum Monte Carlo Study of Strangeness
in Nuclear Structure and Nuclear Astrophysics
> _Tutta la materia di cui siamo fatti noi l’hanno costruita le stelle, tutti
> gli elementi dall’idrogeno all’uranio sono stati fatti nelle reazioni
> nucleari che avvengono nelle supernove, cioè queste stelle molto più grosse
> del Sole che alla fine della loro vita esplodono e sparpagliano nello spazio
> il risultato di tutte le reazioni nucleari avvenute al loro interno. Per cui
> noi siamo veramente figli delle stelle._
> _Il computer non è una macchina intelligente che aiuta le persone stupide,
> anzi, è una macchina stupida che funziona solo nelle mani delle persone
> intelligenti._
###### Contents
1. Introduction
2. 1 Strangeness in nuclear systems
1. 1.1 Hyperons in finite nuclei
2. 1.2 Hyperons in neutron stars
3. 2 Hamiltonians
1. 2.1 Interactions: nucleons
1. 2.1.1 Two-body $NN$ potential
2. 2.1.2 Three-body $NNN$ potential
2. 2.2 Interactions: hyperons and nucleons
1. 2.2.1 Two-body $\Lambda N$ potential
2. 2.2.2 Three-body $\Lambda NN$ potential
3. 2.2.3 Two-body $\Lambda\Lambda$ potential
4. 3 Method
1. 3.1 Diffusion Monte Carlo
1. 3.1.1 Importance Sampling
2. 3.1.2 Sign Problem
3. 3.1.3 Spin-isospin degrees of freedom
2. 3.2 Auxiliary Field Diffusion Monte Carlo
1. 3.2.1 Propagator for nucleons: $\bm{\sigma}$, $\bm{\sigma}\cdot\bm{\tau}$ and $\bm{\tau}$ terms
2. 3.2.2 Propagator for neutrons: spin-orbit terms
3. 3.2.3 Propagator for neutrons: three-body terms
4. 3.2.4 Wave functions
5. 3.2.5 Propagator for hypernuclear systems
5. 4 Results: finite systems
1. 4.1 Nuclei
2. 4.2 Single $\Lambda$ hypernuclei
1. 4.2.1 Hyperon separation energies
2. 4.2.2 Single particle densities and radii
3. 4.3 Double $\Lambda$ hypernuclei
1. 4.3.1 Hyperon separation energies
2. 4.3.2 Single particle densities and radii
6. 5 Results: infinite systems
1. 5.1 Neutron matter
2. 5.2 $\Lambda$ neutron matter
1. 5.2.1 Test: fixed $\Lambda$ fraction
2. 5.2.2 $\Lambda$ threshold density and the equation of state
3. 5.2.3 Mass-radius relation and the maximum mass
7. 6 Conclusions
8. A AFDMC wave functions
1. A.1 Derivatives of the wave function: CM corrections
2. A.2 Derivatives of a Slater determinant
9. B $\Lambda N$ space exchange potential
10. C Acknowledgements
###### List of Figures
1. _i_.1 Neutron star structure
2. 1.1 Strangeness producing reactions
3. 1.2 $\Lambda$ hypernuclei accessible via different experimental reactions
4. 1.3 $\Lambda$ hypernuclear chart
5. 1.4 Hyperon and nucleon chemical potentials
6. 1.5 Neutron star mass-radius relation: Schulze 2011
7. 1.6 Neutron star mass-radius relation: Massot 2012
8. 1.7 Neutron star mass-radius relation: Miyatsu 2012
9. 1.8 Neutron star mass-radius relation: Bednarek 2012
10. 2.1 Two-pion exchange processes in the $NNN$ force
11. 2.2 Three-pion exchange processes in the $NNN$ force
12. 2.3 Short-range contribution in the $NNN$ force
13. 2.4 Meson exchange processes in the $\Lambda N$ force
14. 2.5 Two-pion exchange processes in the $\Lambda NN$ force
15. 4.1 Binding energies: $E$ vs. $d\tau$ for 4He, Argonnne V4’
16. 4.2 Binding energies: $E$ vs. $d\tau$ for 4He, Argonnne V6’
17. 4.3 Binding energies: $E$ vs. mixing parameter for 6He
18. 4.4 $\Lambda$ separation energy vs. $A$: closed shell hypernuclei
19. 4.5 $\Lambda$ separation energy for ${}^{5}_{\Lambda}$He vs. $W_{D}-C_{P}$: 3D plot
20. 4.6 $\Lambda$ separation energy for ${}^{5}_{\Lambda}$He vs. $W_{D}-C_{P}$: 2D plot
21. 4.7 $\Lambda$ separation energy vs. $A$
22. 4.8 $\Lambda$ separation energy vs. $A^{-2/3}$
23. 4.9 Single particle densities: $N$ and $\Lambda$ in 4He and ${}^{5}_{\Lambda}$He
24. 4.10 Single particle densities: $\Lambda$ in hypernuclei for $3\leq A\leq 91$
25. 4.11 Single particle densities: $N$ and $\Lambda$ in 4He, ${}^{5}_{\Lambda}$He and ${}^{\;\;\,6}_{\Lambda\Lambda}$He
26. 5.1 Energy per particle vs. baryon density at fixed $\Lambda$ fraction
27. 5.2 Pair correlation functions at fixed $\Lambda$ fraction: $\rho_{b}=0.16\leavevmode\nobreak\ \text{fm}^{-3}$
28. 5.3 Pair correlation functions at fixed $\Lambda$ fraction: $\rho_{b}=0.40\leavevmode\nobreak\ \text{fm}^{-3}$
29. 5.4 YNM and PNM energy difference vs. $\Lambda$ fraction
30. 5.5 Hyperon symmetry energy vs. baryon density
31. 5.6 $x_{\Lambda}(\rho_{b})$ function and $\Lambda$ threshold density
32. 5.7 YNM equation of state
33. 5.8 YNM mass-radius relation
34. 5.9 YNM mass-central density relation
###### List of Tables
1. 1.1 Nucleon and hyperon properties
2. 2.1 Parameters of the $\Lambda N$ and $\Lambda NN$ interaction
3. 2.2 Parameters of the $\Lambda\Lambda$ interaction
4. 4.1 Binding energies: nuclei, $2\leq A-1\leq 90$
5. 4.2 $\Lambda$ separation energies: $\Lambda N+\Lambda NN$ set (I) for ${}^{5}_{\Lambda}$He and ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O
6. 4.3 $\Lambda$ separation energies: $\Lambda$ hypernuclei, $3\leq A\leq 91$
7. 4.4 $\Lambda$ separation energies: $A=4$ mirror hypernuclei
8. 4.5 $\Lambda$ separation energies: effect of the CSB potential
9. 4.6 $\Lambda$ separation energies: effect of the $\Lambda N$ exchange potential
10. 4.7 Nucleon and hyperon radii in hypernuclei for $3\leq A\leq 49$
11. 4.8 $\Lambda$ separation energies: ${}^{\;\;\,6}_{\Lambda\Lambda}$He
12. 4.9 Nucleon and hyperon radii for ${}^{\;\;\,6}_{\Lambda\Lambda}$He
13. 5.1 Energy per particle: neutron matter
14. 5.2 Energy per particle: $\Lambda$ neutron matter
15. 5.3 Baryon number and $\Lambda$ fraction
16. 5.4 Coefficients of the hyperon symmetry energy fit
Empty page
## Introduction
Neutron stars (NS) are among the densest objects in the Universe, with central
densities several times larger than the nuclear saturation density
$\rho_{0}=0.16\leavevmode\nobreak\ \text{fm}^{-3}$. As soon as the density
significantly exceeds this value, the structure and composition of the NS core
become uncertain. Moving from the surface towards the interior of the star,
the stellar matter undergoes a number of transitions, Fig. _i_.1. From
electrons and neutron rich ions in the outer envelopes, the composition is
supposed to change to the $npe\mu$ matter in the outer core, a degenerate gas
of neutrons, protons, electrons and muons. At densities larger than $\sim
2\rho_{0}$ the $npe\mu$ assumption can be invalid due to the appearance of new
hadronic degrees of freedom or exotic phases.
Figure _i_.1: Schematic structure of a neutron star. Stellar parameters
strongly depend on the equation of state of the core. Figure taken from Ref.
[1]
In the pioneering work of 1960 [2], Ambartsumyan and Saakyan reported the
first theoretical evidence of hyperons in the core of a NS. Contrary to
terrestrial conditions, where hyperons are unstable and decay into nucleons
through the weak interaction, the equilibrium conditions in a NS can make the
inverse process happen. At densities of the order $2\div 3\rho_{0}$, the
nucleon chemical potential is large enough to make the conversion of nucleons
into hyperons energetically favorable. This conversion reduces the Fermi
pressure exerted by the baryons, and makes the equation of state (EoS) softer.
As a consequence, the maximum mass of the star is typically reduced.
Nowadays many different approaches of hyperonic matter are available, but
there is no general agreement among the predicted results for the EoS and the
maximum mass of a NS including hyperons. Some classes of methods extended to
the hyperonic sector predict that the appearance of hyperons at around $2\div
3\rho_{0}$ leads to a strong softening of EoS and consequently to a large
reduction of the maximum mass. Other approaches, instead, indicate much weaker
effects as a consequence of the presence of strange baryons in the core of the
star.
The situation has recently become even more controversial as a result of the
latest astrophysical observations. Until 2010, the value of $1.4\div
1.5M_{\odot}$ for the maximum mass of a NS, inferred from precise neutron star
mass determinations [4], was considered the canonical limit. First neutron
star matter calculations with the inclusion of hyperons seemed to better agree
with this value compared to the case of pure nucleonic EoS, that predicts
relatively large maximum masses ($1.8\div 2.4M_{\odot}$) [5]. The recent
measurements of unusually high masses of the millisecond pulsars PSR
J1614-2230 ($1.97(4)M_{\odot}$) [6] and PSR J1903+0327 ($2.01(4)M_{\odot}$)
[7], rule out almost all these results, making uncertain the appearance of
strange baryons in high-density matter. However, in the last three years new
models compatible with the recent observations have been proposed, but many
inconsistency still remain. The solution of this problem, known as _hyperon
puzzle_ , is far to be understood.
The difficulty of correctly describe the effect of strange baryons in the
nuclear medium, is that one needs a precise solution of a many-body problem
for a very dense system with strong and complicated interactions which are
often poorly known.
The determination of a realistic interaction among hyperons and nucleons
capable to reconcile the terrestrial measurements on hypernuclei and the NS
observations is still an unsolved question. The amount of data available for
nucleon-nucleon scattering and binding energies is enough to build
satisfactory models of nuclear forces, either purely phenomenological or built
on the basis of an effective field theory. Same approaches have been used to
derive potentials for the hyperon-nucleon and hyperon-hyperon interaction, but
the accuracy of these models is far from that of the non strange counterparts.
The main reason of this is the lack of experimental information due the
impossibility to collect hyperon-neutron and hyperon-hyperon scattering data.
This implies that interaction models must be fitted mostly on binding energies
(and possibly excitations) of hypernuclei. In the last years several
measurements of the energy of hypernuclei became available. These can be used
to validate or to constrain the hyperon-nucleon interactions within the
framework of many-body systems. The ultimate goal is then to constrain these
forces by reproducing at best the experimental energies of hypernuclei from
light systems made of few particles up to heavier systems.
The method used to accurately solve the many-body Schrödinger equation
represents the second part of the problem. Accurate calculations are indeed
limited to very few nucleons. The exact Faddeev-Yakubovsky equation approach
has been applied up to four particle systems [8]. Few nucleon systems can be
accurately described by means of techniques based on shell models calculations
like the No-Core Shell Model [9], on the Hyperspherical Harmonics approach
[10, 11, 12, 13, 14] or on QuantumMonte Carlo methods, like the Variational
Monte Carlo [15, 16] or Green Function Monte Carlo [17, 18, 19, 20]. These
methods have been proven to solve the nuclear Schrödinger equation in good
agreement with the Faddeev-Yakubovsky method [21]. For heavier nuclei,
Correlated Basis Function theory [22], Cluster Variational Monte Carlo [23,
24] and Coupled Cluster Expansion [25, 26] are typically adopted. In addition,
the class of method which includes the Brueckner-Goldstone [27] and the
Hartree-Fock [28] algorithms is widely used, also for nuclear matter
calculations. The drawback of these many-body methods is that they modify the
original Hamiltonian to a more manageable form, often introducing uncontrolled
approximations in the algorithm. In absence of an exact method for solving the
many-body Schrödinger equation for a large number of nucleons, the derivation
of model interactions and their applicability in different regimes is subject
to an unpleasant degree of arbitrariness.
In this work we address the problem of the hyperon-nucleon interaction from a
Quantum Monte Carlo point of view. We discuss the application of the Auxiliary
Field Diffusion Monte Carlo (AFDMC) algorithm to study a non relativistic
Hamiltonian based on a phenomenological hyperon-nucleon interaction with
explicit two- and three-body components. The method was originally developed
for nuclear systems [29] and it has been successfully applied to the study of
nuclei [30, 31, 32], neutron drops [33, 20, 34], nuclear matter [35, 36] and
neutron matter [37, 38, 39, 40]. We have extended this ab-initio algorithm in
order to include the lightest of the strange baryons, the $\Lambda$ particle.
By studying the ground state properties of single and double $\Lambda$
hypernuclei, information about the employed microscopic hyperon-nucleon
interaction are deduced.
The main outcome of the study on finite strange systems is that only the
inclusion of explicit $\Lambda NN$ terms provides the necessary repulsion to
realistically describe the separation energy of a $\Lambda$ hyperon in
hypernuclei of intermediate masses [41, 42, 43]. The analysis of single
particle densities confirms the importance of the inclusion of the $\Lambda
NN$ contribution. On the ground of this observation, the three-body hyperon-
nucleon interaction has been studied in detail. By refitting the coefficients
in the potential, it has been possible to reproduce at the same time the
available experimental data accessible with AFDMC calculations in a medium-
heavy mass range [43]. Other details of the hypernuclear force, like the
charge symmetry breaking contribution and the effect of a $\Lambda\Lambda$
interaction, have been successfully analyzed. The AFDMC study of $\Lambda$
hypernuclei results thus in a realistic phenomenological hyperon-nucleon
interaction accurate in describing the ground state physics of medium-heavy
mass hypernuclei.
The large repulsive contribution induced by the three-body $\Lambda NN$ term,
makes very clear the fact that the lack of an accurate Hamiltonian might be
responsible for the unrealistic predictions of the EoS, that would tend to
rule out the appearance of strange baryons in high-density matter. We
speculate that the application of the developed hyperon-nucleon interaction to
the study of the homogeneous medium would lead to a stiffer EoS for the
$\Lambda$ neutron matter. This fact might eventually reconcile the physically
expected onset of hyperons in the inner core of a NS with the observed masses
of order $2M_{\odot}$.
First steps in this direction have been taken. The study of $\Lambda$ neutron
matter at fixed $\Lambda$ fraction shows that the repulsive nature of the
three-body hyperon-nucleon interaction is still active and relevant at
densities larger than the saturation density. The density threshold for the
appearance of $\Lambda$ hyperons has then been derived and the EoS has been
computed. Very preliminary results suggest a rather stiff EoS even in the
presence of hyperons, implying a maximum mass above the observational limit.
The study of hypermatter is still work in progress.
The present work is organized as follows:
Chapter 1:
a general overview about strangeness in nuclear systems, from hypernuclei to
neutron stars, is reported with reference to the terrestrial experiments and
astronomical observations.
Chapter 2:
a description of nuclear and hypernuclear non-relativistic Hamiltonians is
presented, with particular attention to the hyperon-nucleon sector in the two-
and three-body channels.
Chapter 3:
the Auxiliary Field Diffusion Monte Carlo method is discussed in its original
form for nuclear systems and in the newly developed version with the inclusion
of strange degrees of freedom, both for finite and infinite systems.
Chapter 4:
the analysis and set up of a realistic hyperon-nucleon interaction are
reported in connection with the AFDMC results for the hyperon separation
energy. Qualitative information are also deduced from single particle
densities and root mean square radii for single and double $\Lambda$
hypernuclei.
Chapter 5:
using the interaction developed for finite strange systems, first Quantum
Monte Carlo calculations on $\Lambda$ neutron matter are presented and the
implications of the obtained results for the properties of neutron stars are
explored.
Chapter 6:
the achievements of this work are finally summarized and future perspective
are discussed.
Empty page
## Chapter 1 Strangeness in nuclear systems
Hyperons are baryons containing one or more strange quarks. They have masses
larger than nucleons and lifetimes characteristic of the weak decay. The
$\Lambda$ and $\Omega$ hyperons belong to an isospin singlet, the $\Sigma$s to
an isospin triplet and the $\Xi$ particles to an isospin doublet. In Tab. 1.1
we report the list of hyperons (excluding resonances and unnatural parity
states [44]), with their main properties. The isospin doublet of nucleons is
also shown for comparison.
Baryon | qqq | $S$ | $I$ | $m$ [MeV] | $\tau$ [$10^{-10}$ s] | Decay mode
---|---|---|---|---|---|---
$p$ | uud | $0$ | $\displaystyle\frac{1}{2}$ | $938.272\,05(2)$ | $\sim 10^{32}$ y | many
$n$ | udd | $939.565\,38(2)$ | 808(1) s | $p\,e\,\bar{\nu}_{e}$
$\Lambda$ | uds | $-1$ | 0 | $1115.683(6)$ | $2.63(2)$ | $p\,\pi^{-},n\,\pi^{0}$
$\Sigma^{+}$ | uus | $-1$ | 1 | $1189.37(7)$ | $0.802(3)$ | $p\,\pi^{0},n\,\pi^{+}$
$\Sigma^{0}$ | uds | $1192.64(2)$ | $7.4(7)$$\times 10^{-10}$ | $\Lambda\,\gamma$
$\Sigma^{-}$ | dds | $1197.45(3)$ | $1.48(1)$ | $n\,\pi^{-}$
$\Xi^{0}$ | uss | $-2$ | $\displaystyle\frac{1}{2}$ | $1314.9(2)$ | $2.90(9)$ | $\Lambda\,\pi^{0}$
$\Xi^{-}$ | dss | $1321.71(7)$ | $1.64(2)$ | $\Lambda\,\pi^{-}$
$\Omega^{-}$ | sss | $-3$ | 0 | $1672.5(3)$ | $0.82(1)$ | $\Lambda\,K^{-},\Xi^{0}\,\pi^{-},\Xi^{-}\,\pi^{0}$
Table 1.1: Nucleon and hyperon properties: quark components, strangeness,
isospin, mass, mean life and principal decay modes [44].
In the non strange nuclear sector many information are available for nucleon-
nucleon scattering. The Nijmegen $NN$ scattering database [45, 46] includes
1787 $pp$ and 2514 $np$ data in the range $0\div 350$ MeV. Due to the
instability of hyperons in the vacuum and the impossibility to collect
hyperon-neutron and hyperon-hyperon scattering data, the available information
in the strange nuclear sector are instead very limited. Although many events
have been reported both in the low and high energy regimes [47], the standard
set employed in the modern hyperon-nucleon interactions (see for example Ref.
[48]) comprises 35 selected $\Lambda p$ low energy scattering data [49] and
some $\Lambda N$ and $\Sigma N$ data at higher energies [50]. In addition
there are the recently measured $\Sigma^{+}p$ cross sections of the KEK-PS
E289 experiment [51], for a total of 52 $YN$ scattering data.
The very limited experimental possibilities of exploring hyperon-nucleon and
hyperon-hyperon interactions in elementary scattering experiments, makes the
detailed study of hypernuclei essential to understand the physics in the
strange sector. In the next, we will present a summary of the available
hypernuclei experimental data. These information are the key ingredient to
develop realistic hyperon-nucleon and hyperon-hyperon interactions, as
described in the next chapters. The theoretical evidence of the appearance of
hyperons in the core of a NS and the problem of the hyperon puzzle will then
be discussed, following the results of many-body calculations for the
available models of hypermatter.
### 1.1 Hyperons in finite nuclei
In high-energy nuclear reactions strange hadrons are produced abundantly, and
they are strongly involved in the reaction process. When hyperons are captured
by nuclei, hypernuclei are formed, which can live long enough in comparison
with nuclear reaction times. Extensive efforts have been devoted to the study
of hypernuclei. Among many strange nuclear systems, the single $\Lambda$
hypernucleus is the most investigated one [52].
The history of hypernuclear experimental research (see Refs. [53, 54, 52] for
a complete review) celebrates this year the sixtieth anniversary, since the
publication of the discovery of hypernuclei by Danysz and Pniewski in 1953
[55]. Their first event was an example of ${}^{3}_{\Lambda}$H decaying via
${}^{3}_{\Lambda}\text{H}\longrightarrow\,^{3}\text{He}+\pi^{-}\;,$ (1.1)
confirming that the bound particle was a $\Lambda$ hyperon. The event was
observed in an emulsion stack as a consequence of nuclear multifragmentation
induced by cosmic rays. This first evidence opened the study of light
$\Lambda$ hypernuclei ($A<16$) by emulsion experiments, by means of cosmic ray
observations at the beginning and then through proton and pion beams, although
the production rates were low and there was much background. In the early
70’s, the advent of kaon beam at CERN and later at Brookhaven National
Laboratory (BNL), opened the possibility of spectroscopic studies of
hypernuclei, including excited states, by means of the $(K^{-},\pi^{-})$
reaction (see Fig. 1.1). A third stage, which featured the use of the
$(\pi^{+},K^{+})$ reaction, began in the mid 1980’s at the Alternating
Gradient Synchrotron (AGS) of BNL first, and then at the proton synchrotron
(PS) of the High Energy Accelerator Organization (KEK) in Japan. Here, the
superconducting kaon spectrometer (SKS) played a key role in exploring
$\Lambda$ hypernuclear spectroscopy by the $(\pi^{+},K^{+})$ reaction.
$\gamma$-ray spectroscopy developed reaching unprecedented resolution through
the use of a germanium detector array, the Hyperball, and the high quality and
high intensity electron beams available at the Thomas Jefferson National
Accelerator Facility (JLab). This permitted the first successful
$(e,e^{\prime}K^{+})$ hypernuclear spectroscopy measurement (an historical
review of hypernuclear spectroscopy with electron beams can be found in Ref.
[56]. The detailed analysis of $\Lambda$ hypernuclei spectroscopy is reported
in Ref. [52]).
Figure 1.1: Schematic presentation of three strangeness producing reactions
used in the study of $\Lambda$ hypernuclei.
With the development of new facilities, like the japanese J-PARC (Proton
Accelerator Research Complex), other reaction channels for the production of
neutron rich $\Lambda$ hypernuclei became available. The candidates are the
single charge exchange (SCX) reactions $(K^{-},\pi^{0})$ and
$(\pi^{-},K^{0})$, and double charge exchange (DCX) reactions
$(\pi^{-},K^{+})$ and $(K^{-},\pi^{+})$. Fig. 1.2 nicely illustrates the
complementarity of the various production mechanisms and thus the need to
study hypernuclei with different reactions. Moreover, during the last 20 years
of research, great progress has been made in the investigation of
multifragmentation reactions associated with heavy ion collisions (see for
instance [57] and reference therein). This gives the opportunity to apply the
same reactions for the production of hypernuclei too [58, 59]. On the other
hand, it was noticed that the absorption of hyperons in spectator regions of
peripheral relativistic ion collisions is a promising way to produce
hypernuclei [60, 61]. Also, central collisions of relativistic heavy ions can
lead to the production of light hypernuclei [62]. Recent experiments have
confirmed observations of hypernuclei in such reactions, in both peripheral
[63, 64] and central collisions [65].
Figure 1.2: $\Lambda$ hypernuclei accessible by experiments for different
production channels. The boundaries at the neutron and proton rich side mark
the predicted drip lines by a nuclear mass formula extended to strange nuclei.
Figure taken from Ref. [66].
At the time of writing, many laboratories are making extensive efforts in the
study of $\Lambda$ hypernuclei. The status of the art together with future
prospects can be found in Refs. [67, 68, 69] for the J-PARC facility and in
Ref. [70] for the ALICE (A Large Ion Collider Experiment) experiment at the
LHC. Ref. [71] reports the status of the JLab’s Hall A program. In Ref. [72]
future prospects for the the PANDA (antiProton ANihilation at DArmstadt)
project at FAIR (Facility for Antiproton ad Ion Research) and the hypernuclear
experiments using the KAOS spectrometer at MAMI (Mainz Microtron) can be
found. Last results from the FINUDA (FIsica NUcleare a DA$\Phi$NE)
collaboration at DA$\Phi$NE, Italy, are reported in Ref. [73]. Recent interest
has been also focused on the $S=-2$ sector with the study of double $\Lambda$
hypernuclei [74] and the $S=-3$ sector with the search for $\Omega$
hypernuclei [75].
So far, there is no evidence for $\Lambda p$ and ${}^{3}_{\Lambda}$He bound
states. Only very recently the possible evidence of the three-body system
$\Lambda nn$ has been reported [76]. The first well established weakly bound
systems is ${}^{3}_{\Lambda}$H, with hyperon separation energy $B_{\Lambda}$
(the energy difference between the $A-1$ nucleus and the $A$ hypernucleus,
being $A$ the total number of baryons) of $0.13(5)$ MeV [77]. Besides the very
old experimental results [77, 78, 79], several measurements of single
$\Lambda$ hypernuclei became available in the last years trough the many
techniques described above [80, 81, 82, 83, 84, 85, 86, 73]. The update
determination of the lifetime of ${}_{\Lambda}^{3}$H and ${}_{\Lambda}^{4}$H
has been recently reported [87] and new proposals for the search of exotic
$\Lambda$ hypernuclei are constantly discussed (see for example the search for
${}_{\Lambda}^{9}$He [88]). One of the results of this investigation is the
compilation of the $\Lambda$ hypernuclear chart reported in Fig. 1.3. Although
the extensive experimental studies in the $S=-1$ strangeness sector, the
availability of information for hypernuclei is still far from the abundance of
data for the non strange sector.
Figure 1.3: $\Lambda$ hypernuclear chart presented at the XI International
Conference on Hypernuclear and Strange Particle Physics (HYP2012), October
2012, Spain. The figure has been updated from Ref. [52].
It is interesting to observe that with the increase of $A$, there is an
orderly increase of $B_{\Lambda}$ with the number of particles, of the order
of 1 MeV/nucleon (see Tab. 4.3 or the mentioned experimental references). Many
stable hypernuclei with unstable cores appears, as for example
${}^{6}_{\Lambda}$He, ${}^{8}_{\Lambda}$He, ${}^{7}_{\Lambda}$Be and
${}^{9}_{\Lambda}$Be. These evidences testify that the presence of a $\Lambda$
particle inside a nucleus has a glue like effect, increasing the binding
energy and stability of the system. This should be reflected by the attractive
behavior of the $\Lambda$-nucleon interaction, at least in the low density
regime of hypernuclei.
For $\Sigma$ hypernuclei, the situation is quite different. Up to now, only
one bound $\Sigma$ hypernucleus, ${}^{4}_{\Sigma}$He, was detected [89],
despite extensive searches. The analysis of experimental data suggests a
dominant $\Sigma$-nucleus repulsion inside the nuclear surface and a weak
attraction outside the nucleus. In the case of $\Xi$ hypernuclei, although
there is no definitive data for any $\Xi$ hypernucleus at present, several
experimental results suggest that $\Xi$-nucleus interactions are weakly
attractive [90]. No experimental indication exists for $\Omega$ hypernuclei.
It is a challenge to naturally explain the net attraction in $\Lambda$\- and
$\Xi$-nucleus potentials and at the same time the dominant repulsion in
$\Sigma$-nucleus potentials.
In addition to single hyperon nuclei, the binding energies of few double
$\Lambda$ hypernuclei (${}^{\;\;\,6}_{\Lambda\Lambda}$He [91, 92, 93],
${}^{\;10}_{\Lambda\Lambda}$Be, ${}^{\;12}_{\Lambda\Lambda}$Be and
${}^{\;12}_{\Lambda\Lambda}$Be [94, 92], ${}^{\;13}_{\Lambda\Lambda}$B [92])
have been measured. The indication is that of a weakly attractive
$\Lambda\Lambda$ interaction, which reinforces the glue like role of $\Lambda$
hyperons inside nuclei.
From the presented picture it is clear that experimental hypernuclear physics
has become a very active field of research. However there is still lack of
information, even in the most investigated sector of $\Lambda$ hypernuclei.
Due to the technical difficulties in performing scattering experiments
involving hyperons and nucleons, the present main goal is the extension of the
$\Lambda$ hypernuclear chart to the proton and neutron drip lines and for
heavier systems. Parallel studies on $\Sigma$, $\Xi$ and double $\Lambda$
hypernuclei have been and will we be funded in order to try to complete the
scheme. This will hopefully provide the necessary information for the
development of realistic hyperon-nucleon and hyperon-hyperon interactions.
### 1.2 Hyperons in neutron stars
The matter in the outer core of a NS is supposed to be composed by a
degenerate gas of neutrons, protons, electrons and muons, the $npe\mu$ matter,
under $\beta$ equilibrium. Given the energy density
$\displaystyle\mathcal{E}(\rho_{n},\rho_{p},\rho_{e},\rho_{\mu})=\mathcal{E}_{N}(\rho_{n},\rho_{p})+\mathcal{E}_{e}(\rho_{e})+\mathcal{E}_{\mu}(\rho_{\mu})\;,$
(1.2)
where $\mathcal{E}_{N}$ is the nucleon contribution, the equilibrium condition
at a given baryon density $\rho_{b}$ corresponds to the minimum of
$\mathcal{E}$ under the constraints
fixed baryon density: $\displaystyle\rho_{n}+\rho_{p}-\rho_{b}=0\;,$ (1.3a)
electrical neutrality: $\displaystyle\rho_{e}+\rho_{\mu}-\rho_{p}=0\;.$ (1.3b)
The result is the set of conditions
$\displaystyle\mu_{n}$ $\displaystyle=\mu_{p}+\mu_{e}\;,$ (1.4a)
$\displaystyle\mu_{\mu}$ $\displaystyle=\mu_{e}\;,$ (1.4b)
where $\mu_{j}=\partial\mathcal{E}/\partial\rho_{j}$ with $j=n,p,e,\mu$ are
the chemical potentials. These relations express the equilibrium with respect
to the weak interaction processes
$\displaystyle\begin{array}[]{rclrcl}n&\longrightarrow&p+e+\bar{\nu}_{e}\;,&p+e&\longrightarrow&n+\nu_{e}\;,\\\\[5.0pt]
n&\longrightarrow&p+\mu+\bar{\nu}_{\mu}\;,&p+\mu&\longrightarrow&n+\nu_{\mu}\;.\end{array}$
(1.7)
(Neutrino do not affect the matter thermodynamics so their chemical potential
is set to zero). Eqs. (1.4) supplemented by the constraints (1.3) form a
closed system which determines the equilibrium composition of the $npe\mu$
matter. Once the equilibrium is set, the energy and pressure as a function of
the baryon density can be derived and thus the EoS is obtained.
Given the EoS, the structure of a non rotating NS can be fully determined by
solving the Tolman-Oppenheimer-Volkoff (TOV) equations [95, 96]
$\displaystyle\frac{dP(r)}{dr}$
$\displaystyle=-G\frac{\Bigl{[}\mathcal{E}(r)+P(r)\Bigr{]}\Bigl{[}m(r)+4\pi
r^{3}P(r)\Bigr{]}}{r^{2}\Bigl{[}1-\frac{2Gm(r)}{r}\Bigr{]}}\;,$ (1.8a)
$\displaystyle\frac{dm(r)}{dr}$ $\displaystyle=4\pi r^{2}\mathcal{E}(r)\;,$
(1.8b)
which describe the hydrostatic equilibrium of a static spherically symmetric
star. $\mathcal{E}(r)$ and $P(r)$ are the energy density and the pressure of
the matter, $m(r)$ is the gravitational mass enclosed within a radius $r$, and
$G$ is the Gravitational constant. In the stellar interior $P>0$ and
$dP/dr<0$. The condition $P(R)=0$ fixes the stellar radius $R$. Outside the
star for $r>R$, we have $P=0$ and $\mathcal{E}=0$. Eq. (1.8b) gives thus
$m(r>R)=M=const$, which is total gravitational mass. Starting with a central
energy density $\mathcal{E}_{c}=\mathcal{E}(r=0)$ and using the above
conditions, the TOV equations can be numerically solved and the mass-radius
relation $M=M(R)$ is obtained. It can be shown [1], that the relativistic
corrections to the Newtonian law $dP(r)/dr=-Gm\mathcal{E}(r)/r^{2}$ included
in Eq. (1.8a) give an upperbound to the $M(R)$ relation, i.e. there exists a
maximum mass for a NS in hydrostatic equilibrium. It is important to note
that, given the EoS, the mass-radius relation is univocally determined. Any
modification made on the EoS will lead to a change in the $M(R)$ curve and
thus in the allowed maximum mass.
For $\rho_{b}\gtrsim 2\rho_{0}$, the inner core is thought to have the same
$npe\mu$ composition of the outer core. However, since at high densities the
nucleon gas will be highly degenerate, hyperons with energies lower than a
threshold value will become stable, because the nucleon arising from their
decay cannot find a place in phase space in accordance to the Pauli principle
[2]. Thus, beyond a density threshold we have to take into account the
contribution of hyperons to the $\beta$ equilibrium. Eq. (1.2) becomes a
function of $\rho_{b}$ (baryons: nucleons and hyperons) and $\rho_{l}$
(leptons: electrons and muons). Given the baryon density and imposing
electrical neutrality conditions, the equilibrium equations now read:
$\displaystyle Q_{b}=-1\,:$ $\displaystyle\mu_{b^{-}}$
$\displaystyle=\mu_{n}+\mu_{e}$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{\Omega^{-}}$
$\displaystyle=\mu_{\Xi^{-}}=\mu_{\Sigma^{-}}=\mu_{n}+\mu_{e}\;,$ (1.9a)
$\displaystyle Q_{b}=\phantom{+}0\,:$ $\displaystyle\mu_{b^{0}}$
$\displaystyle=\mu_{n}$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{\Xi^{0}}$
$\displaystyle=\mu_{\Sigma^{0}}=\mu_{\Lambda}=\mu_{n}\;,$ (1.9b)
$\displaystyle Q_{b}=+1\,:$ $\displaystyle\mu_{b^{+}}$
$\displaystyle=\mu_{n}-\mu_{e}$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{\Sigma^{+}}$ $\displaystyle=\mu_{p}=\mu_{n}-\mu_{e}\;,$
(1.9c)
where $Q_{b}$ is the electric charge of a baryon. As soon as the neutron
chemical potential becomes sufficiently large, energetic neutrons can decay
via weak strangeness nonconserving reactions into $\Lambda$ hyperons, leading
to a $\Lambda$ Fermi sea.
We can derive the hyperons threshold densities $\rho_{Y}$ by calculating the
minimum increase of the energy of the matter produced by adding a single
strange particle at a fixed pressure. This can be done by considering the
energy of the matter with an admixture of given hyperons and by calculating
numerically the limit of the derivative
$\displaystyle\lim_{\rho_{Y}\rightarrow
0}\,\frac{\partial\mathcal{E}}{\partial\rho_{Y}}\bigg{|}_{eq}\\!=\mu_{Y}^{0}\;.$
(1.10)
Consider for example the lightest $\Lambda$ hyperon. As long as
$\mu_{\Lambda}^{0}>\mu_{n}$, the strange baryon cannot survive because the
system will lower its energy via an exothermic reaction
$\Lambda+N\longrightarrow n+N$. However, $\mu_{n}$ increases with growing
$\rho_{b}$ and the functions $\mu_{\Lambda}^{0}(\rho_{b})$ and
$\mu_{n}^{0}(\rho_{b})$ intersect at some $\rho_{b}=\rho_{\Lambda}^{th}$ (the
left panel in Fig. 1.4). For $\rho_{b}>\rho_{\Lambda}^{th}$ the $\Lambda$
hyperons become stable in dense matter because their decay is blocked by the
Pauli principle.
Figure 1.4: Threshold chemical potentials of neutral hyperons and neutron
(left panel), and of negatively charged hyperons and the sum $\mu_{n}+\mu_{e}$
(right panel) versus baryon density. Vertical dotted lines mark the thresholds
for the creation of new hyperons. Dashed lines show the minimum chemical
potential $\mu_{Y}^{0}$ of unstable hyperons before the thresholds. Figure
taken from Ref. [1].
Although the $\Lambda$ particle is the lightest among hyperons, one expects
the $\Sigma^{-}$ to appear via
$\displaystyle n+e^{-}\longrightarrow\Sigma^{-}+\nu_{e}$ (1.11)
at densities lower than the $\Lambda$ threshold, even thought the $\Sigma^{-}$
is more massive. This is because the negatively charged hyperons appear in the
ground state of matter when their masses equal $\mu_{n}+\mu_{e}$, while the
neutral hyperon $\Lambda$ appears when its mass equals $\mu_{n}$. Since the
electron chemical potential in matter is typically larger (ultrarelativistic
degenerate electrons $\mu_{e}\sim E_{F_{e}}\sim\hbar
c(3\pi^{2}\rho_{e})^{1/3}>120\leavevmode\nobreak\
\text{MeV\leavevmode\nobreak\ for\leavevmode\nobreak\ }\rho_{e}\sim
5\%\rho_{0}$) than the mass difference
$m_{\Sigma^{-}}-m_{\Lambda}=81.76\leavevmode\nobreak\ \mbox{MeV}$, the
$\Sigma^{-}$ will appear at lower densities. However, in typical neutron
matter calculations with the inclusion of strange degrees of freedom, only
$\Lambda$, $\Sigma^{0}$ and $\Xi^{0}$ hyperons are taken into account due to
charge conservation.
The formation of hyperons softens the EoS because high energy neutrons are
replaced by more massive low energy hyperons which can be accommodated in
lower momentum states. There is thus a decrease in the kinetic energy that
produces lower pressure. The softening of the EoS of the inner core of a NS
induced by the presence of hyperons is generic effect. However, its magnitude
is strongly model dependent.
Calculations based on the extension to the hyperonic sector of the Hartree-
Fock (HF) [97, 98] and Brueckner-Hartree-Fock (BHF) [99, 100] methods, do all
agree that the appearance of hyperons around $2\div 3\rho_{0}$ leads to a
strong softening of the EoS. Consequently, the expected maximum mass is
largely reduced, as shown for instance in Fig. 1.5 and Fig. 1.6. The addition
of the hyperon-nucleon force to the pure nucleonic Hamiltonian, lowers the
maximum mass of a value between $0.4M_{\odot}$ and more than $1M_{\odot}$.
From the pure nucleonic case of $M_{\max}>1.8M_{\odot}$, the limit for
hypernuclear matter is thus reduced to the range
$1.4M_{\odot}<M_{\max}<1.6M_{\odot}$. These results, although compatible with
the canonical limit of $1.4\div 1.5M_{\odot}$, cannot be consistent with the
recent observations of $2M_{\odot}$ millisecond pulsars [6, 7].
Figure 1.5: Mass-radius and mass-central density relations for different NS
EoS obtained in Brueckner-Hartree-Fock calculations of hypernuclear matter.
V18+TBF and V18+UIX’ refer to purely nuclear matter EoS built starting from
two- and three-body nucleon-nucleon potentials (see § 2.1). The other curves
are obtained adding two different hyperon-nucleon forces among the Nijmegen
models to the previous nucleonic EoS. For more details see the original paper
[99]. Figure 1.6: Neutron star mass as a function of the circumferential
radius. QMC700 and MC _i_ -H(F)/N refer to EoS based on quark-meson coupling
model and chiral model in the Hartee(Fock) approximation without hyperons. In
the MC _i_ -H(F)/NY models also hyperons are taken into account. The canonical
maximum mass limit of $\sim 1.45M_{\odot}$ and the mass of the two heavy
millisecond pulsars PSR J1903+0327 ($1.67(2)M_{\odot}$) and PSR J1614-2230
($1.97(4)M_{\odot}$) are shown. Details on the potentials and method adopted
can be found in Ref. [98].
It is interesting to note that the hyperonic $M_{\max}$ weakly depends on the
details of the employed nucleon-nucleon interaction and even less on the
hypernuclear forces. In Ref. [97] the interaction used for the nuclear sector
is an analytic parametrization fitted to energy of symmetric matter obtained
from variational calculations with the Argonne V18 nucleon-nucleon interaction
(see § 2.1) including three-body forces and relativistic boost corrections.
Refs. [99] and [100] adopted the bare $NN$ Argonne V18 supplemented with
explicit three-nucleon forces or phenomenological density-dependent contact
terms that account for the effect of nucleonic and hyperonic three-body
interactions. The hypernuclear forces employed in these work belong to the
class of Nijmegen potentials (see § 2). Finally, in Ref. [98] chiral
Lagrangian and quark-meson coupling models of hyperon matter have been
employed. Despite the differences in the potentials used in the strange and
non strange sectors, the outcomes of these works give the same qualitative and
almost quantitative picture about the reduction of $M_{\max}$ due to the
inclusions of strange baryons. Therefore, the (B)HF results seem to be rather
robust and thus, many doubts arise about the real appearance of hyperons in
the inner core of NSs.
Other approaches, such as relativistic Hartree-Fock [101, 102, 103], standard,
density-dependent and nonlinear Relativistic Mean Field models [104, 105, 106,
107, 108] and Relativistic Density Functional Theory with density-dependent
couplings [109], indicate much weaker effects as a consequence of the presence
of strange baryons in the core of NSs, as shown for example in Fig. 1.7 and
Fig. 1.8. In all these works, it was possible to find a description of
hypernuclear matter, within the models analyzed, that produces stiff EoS,
supporting a $2M_{\odot}$ neutron star. Same conclusion has been reported in
Ref. [110] where the EoS of matter including hyperons and deconfined quark
matter has been constructed on the basis of relativistic mean-field nuclear
functional at low densities and effective Nambu-Jona-Lasinio model of quark
matter. The results of this class of calculations seem to reconcile the onset
of hyperons in the inner core of a NS with the observed masses of order
$2M_{\odot}$.
Figure 1.7: Neutron star mass-radius relations in Hartree (left panel) and
Hartree-Fock (right panel) calculations. CQMC, QMC and QHD+NL denote the
chiral quark-meson coupling, quark-meson coupling and non linear quantum
hadrodynamics employed potentials, with (npY) and without hyperons (np). For
details see Ref. [101]. Figure 1.8: Stellar mass versus circumferential radius
in non linear relativistic mean field model. The purely nucleon case is
denoted with N, the nucleon+hyperon case with NH. In the inset, the effect of
rotation at $f=317$ Hz on the mass-radius relation near $M_{\max}$. The dashed
region refers to the mass of the pulsar PSR J1614-2230. All the details are
reported in Ref. [104].
This inconsistency among different calculations and between the theoretical
results and the observational constraints, at present is still an open
question. For example, given the theoretical evidence about the appearance of
hyperons in the inner core of a NS, the results of all available (B)HF
calculations seem to be in contradiction with the picture drawn by the
relativistic mean field models. On one hand there should be uncontrolled
approximations on the method used to solve the many-body Hamiltonian. On the
other hand the employed hypernuclear interactions might not be accurate enough
in describing the physics of the infinite nuclear medium with strange degrees
of freedom. For instance, as reported in Refs. [111, 106], one of the possible
solutions to improve the hyperon-nucleon interactions might be the inclusion
of explicit three-body forces in the models. These should involve one or more
hyperons (i.e., hyperon-nucleon-nucleon, hyperon-hyperon-nucleon or hyperon-
hyperon-hyperon interactions) and they could eventually provide the additional
repulsion needed to make the EoS stiffer and, therefore the maximum mass
compatible with the current observational limits. On the grounds of this
observation, we decided to revisit the problem focusing on a systematic
construction of a realistic, though phenomenological hyperon-nucleon
interaction with explicit two- and three-body components (§ 2) by means of
Quantum Monte Carlo calculations (§ 3).
Empty page
## Chapter 2 Hamiltonians
The properties of nuclear systems arise from the interactions between the
individual constituents. In order to understand these properties, the starting
point is the determination of the Hamiltonian to be used in the description of
such systems. In principle the nuclear Hamiltonian should be directly derived
from Quantum Chromodynamics (QCD). Many efforts have been done in the last
years [112, 113, 114], but this goal is still far to be achieved.
The problem with such derivation is that QCD is non perturbative in the low-
temperature regime characteristic of nuclear physics, which makes direct
solutions very difficult. Moving from the real theory to effective models, the
structure of a nuclear Hamiltonian can be determined phenomenologically and
then fitted to exactly reproduce the properties of few-nucleon systems. In
this picture, the degrees of freedom are the baryons, which are considered as
non relativistic point-like particles interacting by means of phenomenological
potentials. These potentials describe both short and the long range
interactions, typically via one-boson and two-meson exchanges, and they have
been fitted to exactly reproduce the properties of few-nucleon systems [115].
In more details, different two-body phenomenological forms have been proposed
and fitted on the nucleon-nucleon ($NN$) scattering data of the Nijmegen
database [45, 46] with a $\chi^{2}/N_{data}\simeq 1$. The more diffuse are the
Nijmegen models [116], the Argonne models [117, 118] and the CD-Bonn [119].
Although reproducing the $NN$ scattering data, all these two-nucleon
interactions underestimate the triton binding energy, suggesting that the
contribution of a three-nucleon ($NNN$) interaction (TNI) is essential to
reproduce the physics of nuclei. The TNI is mainly attributed to the
possibility of nucleon excitation in a $\Delta$ resonance and it can be
written as different effective three-nucleon interactions which have been
fitted on light nuclei [120, 121] and on saturation properties of nuclear
matter [122]. The TNIs typically depend on the choice of the two-body $NN$
potential [123], but the final result with the total Hamiltonian should be
independent of the choice.
A different approach to the problem is the realization that low-energy QCD is
equivalent to an Effective Field Theory (EFT) which allows for a perturbative
expansion that is known as chiral perturbation theory. In the last years
modern nucleon-nucleon interaction directly derived from Chiral Effective
Field Theory ($\chi$-EFT) have been proposed, at next-to-next-to-next-to-
leading order (N3LO) in the chiral expansion [124, 125] and recently at
optimized next-to-next-to-leading order (N2LO) [126] (see Ref. [127] for a
complete review). All these potentials are able to reproduce the Nijmegen
phase shifts with $\chi/N_{data}^{2}\simeq 1$. TNIs enter naturally at N2LO in
this scheme, and they play again a pivotal role in nuclear structure
calculations [128]. The contributions of TNIs at N3LO have also been worked
out [129, 130, 131]. The $\chi$-EFT interactions are typically developed in
momentum space, preventing their straightforward application within the
Quantum Monte Carlo (QMC) framework. However, a local version of the
$\chi$-EFT potentials in coordinate space up to N2LO has been very recently
proposed and employed in QMC calculations [132].
Nuclear phenomenological Hamiltonians have been widely used to study finite
and infinite nuclear systems within different approaches. From now on, we will
focus on the Argonne $NN$ potentials and the corresponding TNIs, the Urbana IX
(UIX) [122] and the modern Illinois (ILx) [121] forms. These potentials have
been used to study nuclei, neutron drops, neutron and nuclear matter in
Quantum Monte Carlo (QMC) calculations, such as Variational Monte Carlo (VMC)
[15, 16, 24], Green Function Monte Carlo (GFMC) [133, 118, 134, 17, 135, 136,
19, 20] and Auxiliary Field Diffusion Monte Carlo (AFDMC) [37, 33, 30, 31, 20,
35, 38]. Same bare interactions have been also employed in the Fermi Hyper-
Netted Chain (FHNC) approach [22, 137], both for nuclei and nuclear matter.
With a projection of the interaction onto the model space, these Hamiltonians
are used in Effective Interaction Hyperspherical Harmonics (EIHH) [10, 13] and
Non-Symmetrized Hyperspherical Harmonics (NSHH) [14] calculations. Finally,
same potentials can be also used in Brueckner Hartree Fock (BHF) [138], Shell-
Model (SM) [139], No-Core-Shell-Model (NCSM) [9] and Coupled Cluster (CC) [26]
calculations by means of appropriate techniques to handle the short-range
repulsion of the nucleon-nucleon force, such as Brueckner $G$-matrix approach
[140, 141], $V_{low-k}$ reduction [142, 143, 144], Unitary Correlation
Operator Method (UCOM) [145] or Similarity Renormalization Group (SRG)
evolution [146, 147]. The list of methods that can handle in a successful way
the Argonne+TNIs potentials demonstrates the versatility and reliability of
this class of phenomenological nuclear Hamiltonians.
Moving from the non-strange nuclear sector, where nucleons are the only
baryonic degrees of freedom, to the strange nuclear sector, where also
hyperons enter the game, the picture becomes much less clear. There exists
only a very limited amount of scattering data from which one could construct
high-quality hyperon-nucleon ($YN$) potentials. Data on hypernuclei binding
energies and hyperon separation energies are rather scarce and can only
partially complete the scheme.
After the pioneering work reported in Ref. [148], several models have been
proposed to describe the $YN$ interaction. The more diffuse are the Nijmegen
soft-core models (like NSC89 and NSC97x) [149, 150, 151, 152, 153, 154, 155]
and the Jülic potential (J04) [156, 157, 158]. A recent review of these
interactions, together with Hartree-Fock (HF) calculations have been published
by Ðapo _et al._ in Ref. [159]. In the same framework, extended soft-core
Nijmegen potentials for strangeness $S=-2$ have been also developed [160,
161]. Very recently, the extended soft-core 08 (ESC08) model has been
completed, which represents the first unified theoretical framework involving
hyperon-nucleon, hyperon-hyperon ($YY$) and also nucleon-nucleon sectors [48].
This class of interaction has been used in different calculations for
hypernuclei [162, 163, 164, 165, 166, 159, 48] and hypermatter [159, 97, 99,
100] within different methods, but the existing data do not constrain the
potentials sufficiently. For example, six different parameterizations of the
Nijmegen $YN$ potentials fit equally well the scattering data but produce very
different scattering lengths, as reported for instance in Ref. [152]. In
addition, these potentials are not found to yield the correct spectrum of
hypernuclear binding energies. For example, the study [166] of
${}^{4}_{\Lambda}$H and ${}^{4}_{\Lambda}$He that uses Nijmegen models, does
not predict all experimental separation energies. Similar conclusions for
single- and double-$\Lambda$ hypernuclei have also been drawn in a study
employing a different many-body technique [165]. Even the most recent ESC08
model produces some overbinding of single-$\Lambda$ hypernuclei and a weakly
repulsive incremental $\Lambda\Lambda$ energy [48], not consistent with the
observed weak $\Lambda\Lambda$ attraction in
${}^{\;\;\,6}_{\Lambda\Lambda}$He.
In analogy with the nucleon-nucleon sector, a $\chi$-EFT approach for the
hyperon-nucleon interaction has been also developed. The first attempt was
proposed by Polinder and collaborators in 2006 [167], resulting in a leading
order (LO) expansion. Only recently the picture has been improved going to
next-to-leading order (NLO) [168, 169, 170]. The $YN$ $\chi$-EFT model is
still far away from the theoretical accuracy obtained in the non-strange
sector, but it is any case good enough to describe the limited available $YN$
scattering data.
As an alternative, a cluster model with phenomenological interactions has been
proposed by Hiyama and collaborators to study light hypernuclei [171, 172,
173, 174, 175, 176]. Interesting results on $\Lambda$ hypernuclei have also
been obtained within a $\Lambda$-nucleus potential model, in which the need of
a functional with a more than linear density dependence was shown, suggesting
the importance of a many-body interaction [177]. While studying $s$-shell
hypernuclei, the $\Lambda N\rightarrow\Sigma N$ coupling as a three-body
$\Lambda NN$ force has been investigated by many authors [166, 178, 179, 180].
Having strong tensor dependence it is found to play an important role,
comparable to the TNI effect in non-strange nuclei.
Finally, starting in the 1980s, a class of Argonne-like interactions has been
developed by Bodmer, Usmani and Carlson on the grounds of quantum Monte Carlo
calculations to describe the $\Lambda$-nucleon force. These phenomenological
interactions are written in coordinates space and they include two- and three-
body hyperon-nucleon components, mainly coming from two-pion exchange
processes and shorter range effects. They have been used in different forms
mostly in variational Monte Carlo calculations for single $\Lambda$
hypernuclei (${}^{3}_{\Lambda}$H [181, 182], ${}^{4}_{\Lambda}$H and
${}^{4}_{\Lambda}$He [183, 181, 182, 184], ${}^{5}_{\Lambda}$He [181, 182,
185, 186, 184, 187, 188, 189], ${}^{9}_{\Lambda}$Be [190, 191],
${}^{13}_{\leavevmode\nobreak\ \Lambda}$C [190],
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O [192, 185]), double $\Lambda$
hypernuclei (${}^{\;\;\,4}_{\Lambda\Lambda}$H,
${}^{\;\;\,5}_{\Lambda\Lambda}$H, ${}^{\;\;\,5}_{\Lambda\Lambda}$He [193] and
${}^{\;\;\,6}_{\Lambda\Lambda}$He [193, 194, 195]) and in the framework of
correlated basis function theory for $\Lambda$ hypernuclei [196], typically in
connection with the Argonne $NN$ potential.
Within the phenomenological interaction scheme, a generic nuclear system
including nucleons and hyperons, can be described by the non relativistic
phenomenological Hamiltonian
$H=H_{N}+H_{Y}+H_{YN}\;,$ (2.1)
where $H_{N}$ and $H_{Y}$ are the pure nucleonic and hyperonic Hamiltonians
and $H_{YN}$ represents the interaction Hamiltonian connecting the two
distinguishable types of baryon:
$\displaystyle H_{N}$
$\displaystyle=\frac{\hbar^{2}}{2m_{N}}\sum_{i}\nabla_{i}^{2}\;+\sum_{i<j}v_{ij}\;\,+\sum_{i<j<k}v_{ijk}\;\;\,+\,\ldots\;,$
(2.2) $\displaystyle H_{Y}$
$\displaystyle=\frac{\hbar^{2}}{2m_{\Lambda}}\sum_{\lambda}\nabla_{\lambda}^{2}\;+\sum_{\lambda<\mu}v_{\lambda\mu}\,+\sum_{\lambda<\mu<\nu}v_{\lambda\mu\nu}\;+\,\ldots\;,$
(2.3) $\displaystyle H_{YN}$ $\displaystyle=\sum_{\lambda i}v_{\lambda
i}\,+\sum_{\lambda,i<j}v_{\lambda ij}\,+\sum_{\lambda<\mu,i}v_{\lambda\mu
i}\,+\,\ldots\;.$ (2.4)
In this context, $A$ is the total number of baryons,
$A=\mathcal{N}_{N}+\mathcal{N}_{Y}$. Latin indices
$i,j,k=1,\ldots,\mathcal{N}_{N}$ label nucleons and Greek symbols
$\lambda,\mu,\nu=1,\ldots,\mathcal{N}_{Y}$ are used for the hyperons. The
Hamiltonians (2.2) and (2.3) contain the kinetic energy operator and two- and
three-body interactions for nucleons and hyperons separately. In principles
they could include higher order many-body forces that however are expected to
be less important. The Hamiltonian (2.4) describes the interaction between
nucleons and hyperons, and it involves two-body ($YN$) and three-body ($YNN$
and $YYN$) forces. At present there is no evidence for higher order terms in
the hyperon-nucleon sector.
As reported in the previous chapter, experimental data are mainly available
for $\Lambda p$ scattering and $\Lambda$ hypernuclei and present experimental
efforts are still mostly concentrated in the study of the $S=-1$ hypernuclear
sector. Information on heavier hyperon-nucleon scattering and on $\Sigma$ or
more exotic hypernuclei are very limited. For these reasons, from now on we
will focus on the phenomenological interactions involving just the $\Lambda$
hyperon. We adopt the class of Argonne-like $\Lambda$-nucleon interaction for
the strange sector and the nucleon-nucleon Argonne force with the
corresponding TNIs (UIX and ILx) for the non-strange sector. An effective
$\Lambda\Lambda$ interaction has been also employed.
### 2.1 Interactions: nucleons
We report the details of the $NN$ Argonne potential [117, 118] and the
corresponding TNIs, the Urbana IX (UIX) [122] and the Illinois (ILx) [121].
These interactions are written in coordinate space and they include different
range components coming from meson (mostly pion) exchange and phenomenological
higher order contributions.
#### 2.1.1 Two-body $NN$ potential
The nucleon-nucleon potential Argonne V18 (AV18) [117] contains a complete
electromagnetic (EM) interaction and a strong interaction part which is
written as a sum of a long-range component $v_{ij}^{\pi}$ due to one-pion
exchange (OPE) and a phenomenological intermediate- and short-range part
$v_{ij}^{R}$ :
$v_{ij}=v_{ij}^{\pi}+v_{ij}^{R}\;.$ (2.5)
Ignoring isospin breaking terms, the long-range OPE is given by
$\displaystyle v_{ij}^{\pi}=\frac{f_{\pi
NN}^{2}}{4\pi}\frac{m_{\pi}}{3}\,X_{ij}\,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\;,$
(2.6)
where $\tfrac{f_{\pi NN}^{2}}{4\pi}=0.075$ is the pion-nucleon coupling
constant [197] and
$\displaystyle
X_{ij}=Y_{\pi}(r_{ij})\,\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}+T_{\pi}(r_{ij})\,S_{ij}\;.$
(2.7)
$\bm{\sigma}_{i}$ and $\bm{\tau}_{i}$ are Pauli matrices acting on the spin or
isospin of nucleons and $S_{ij}$ is the tensor operator
$\displaystyle
S_{ij}=3\left(\bm{\sigma}_{i}\cdot\hat{\bm{r}}_{ij}\right)\left(\bm{\sigma}_{j}\cdot\hat{\bm{r}}_{ij}\right)-\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\;.$
(2.8)
The pion radial functions associated with the spin-spin (Yukawa potential) and
tensor (OPE tensor potential) parts are
$\displaystyle Y_{\pi}(r)$
$\displaystyle=\frac{\operatorname{e}^{-\mu_{\pi}r}}{\mu_{\pi}r}\xi_{Y}(r)\;,$
(2.9) $\displaystyle T_{\pi}(r)$
$\displaystyle=\left[1+\frac{3}{\mu_{\pi}r}+\frac{3}{(\mu_{\pi}r)^{2}}\right]\frac{\operatorname{e}^{-\mu_{\pi}r}}{\mu_{\pi}r}\xi_{T}(r)\;,$
(2.10)
where $\mu_{\pi}$ is the pion reduced mass
$\displaystyle\mu_{\pi}=\frac{m_{\pi}}{\hbar}=\frac{1}{\hbar}\frac{m_{\pi^{0}}+2\,m_{\pi^{\pm}}}{3}\quad\quad\frac{1}{\mu_{\pi}}\simeq
1.4\leavevmode\nobreak\ \text{fm}\;,$ (2.11)
and $\xi_{Y}(r)$ and $\xi_{T}(r)$ are the short-range cutoff functions defined
by
$\displaystyle\xi_{Y}(r)=\xi_{T}^{1/2}(r)=1-\operatorname{e}^{-cr^{2}}\quad\quad
c=2.1\leavevmode\nobreak\ \text{fm}^{-2}\;.$ (2.12)
It is important to note that since $T_{\pi}(r)\gg Y_{\pi}(r)$ in the important
region where $r\lesssim 2$ fm, the OPE is dominated by the tensor part.
The remaining intermediate- and short-range part of the potential is expressed
as a sum of central, $L^{2}$, tensor, spin-orbit and quadratic spin-orbit
terms (respectively labelled as $c$, $l2$, $t$, $ls$, $ls2$) in different $S$,
$T$ and $T_{z}$ states:
$\displaystyle\\!v_{NN}^{R}=v_{NN}^{c}(r)+v_{NN}^{l2}(r)\bm{L}^{2}+v_{NN}^{t}(r)S_{12}+v_{NN}^{ls}(r)\bm{L}\\!\cdot\\!\bm{S}+v_{NN}^{ls2}(r)(\bm{L}\\!\cdot\\!\bm{S})^{2}\;,$
(2.13)
with the radial functions $v_{NN}^{k}(r)$ written in the general form
$\displaystyle
v_{NN}^{k}(r)=I_{NN}^{k}\,T_{\pi}^{2}(r)+\bigg{[}P_{NN}^{k}+(\mu_{\pi}r)\,Q_{NN}^{k}+(\mu_{\pi}r)^{2}\,R_{NN}^{k}\bigg{]}\,W(r)\;,$
(2.14)
where the $T_{\pi}^{2}(r)$ has the range of a two-pion exchange (TPE) force
and $W(r)$ is a Wood-Saxon function which provides the short-range core:
$\displaystyle
W(r)=\Bigl{(}1+\operatorname{e}^{\frac{r-\bar{r}}{a}}\Bigr{)}^{-1}\quad\quad\bar{r}=0.5\leavevmode\nobreak\
\text{fm},\quad a=0.2\leavevmode\nobreak\ \text{fm}\;.$ (2.15)
By imposing a regularization condition at the origin, it is possible to reduce
the number of free parameters by one for each $v_{NN}^{k}(r)$. All the
parameters in the $\xi(r)$ short-range cutoff functions as well as the other
phenomenological constants are fitted on the $NN$ Nijmegen scattering data
[45, 46].
The two-body nucleon potential described above can be projected from $S$, $T$,
$T_{z}$ states into an operator format with 18 terms
$\displaystyle v_{ij}=\sum_{p=1,18}v_{p}(r_{ij})\,\mathcal{O}_{ij}^{\,p}\;.$
(2.16)
The first 14 operators are charge independent and they are the ones included
in the Argonne V14 potential (AV14):
$\displaystyle\mathcal{O}_{ij}^{\,p=1,8}$
$\displaystyle=\Bigl{\\{}1,\bm{\sigma}_{i}\cdot\bm{\sigma}_{j},S_{ij},\bm{L}_{ij}\cdot\bm{S}_{ij}\Bigr{\\}}\otimes\Bigl{\\{}1,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\Bigr{\\}}\;,$
(2.17) $\displaystyle\mathcal{O}_{ij}^{\,p=9,14}$
$\displaystyle=\Bigl{\\{}\bm{L}_{ij}^{2},\bm{L}_{ij}^{2}\;\bm{\sigma}_{i}\cdot\bm{\sigma}_{j},\left(\bm{L}_{ij}\cdot\bm{S}_{ij}\right)^{2}\Bigr{\\}}\otimes\Bigl{\\{}1,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\Bigr{\\}}\;.$
(2.18)
The first eight terms give the higher contribution to the $NN$ interaction and
they are the standard ones required to fit $S$ and $P$ wave data in both
triplet and singlet isospin states. The first six of them come from the long-
range part of OPE and the last two depend on the velocity of nucleons and give
the spin-orbit contribution. In the above expressions, $\bm{L}_{ij}$ is the
relative angular momentum of a couple $ij$
$\displaystyle\bm{L}_{ij}=\frac{1}{2i}({\bf r}_{i}-{\bf
r}_{j})\times(\bm{\nabla}_{i}-\bm{\nabla}_{j})\;,$ (2.19)
and $\bm{S}_{ij}$ the total spin of the pair
$\displaystyle\bm{S}_{ij}=\frac{1}{2}(\bm{\sigma}_{i}+\bm{\sigma}_{j})\;.$
(2.20)
Operators from 9 to 14 are included to better describe the Nijmegen higher
partial waves phase shifts and the splitting of state with different $J$
values. However, the contribution of these operators is small compared to the
total potential energy.
The four last additional operators of the AV18 potential account for the
charge symmetry breaking effect, mainly due to the different masses of charged
and neutral pions, and they are given by
$\displaystyle\mathcal{O}_{ij}^{\,p=15,18}=\Bigl{\\{}T_{ij},(\bm{\sigma}_{i}\cdot\bm{\sigma}_{j})\,T_{ij},S_{ij}\,T_{ij},\tau_{i}^{z}+\tau_{j}^{z}\Bigr{\\}}\;,$
(2.21)
where $T_{ij}$ is the isotensor operator defined in analogy with $S_{ij}$ as
$\displaystyle
T_{ij}=3\,\tau_{i}^{z}\tau_{j}^{z}-\bm{\tau}_{i}\cdot\bm{\tau}_{j}\;.$ (2.22)
The contribution to the total energy given by these four operators is however
rather small.
In QMC calculations reduced versions of the original AV18 potential are often
employed. The most used one is the Argonne V8’ (AV8’) [118] that contains only
the first eight operators and it is not a simple truncation of AV18 but also a
reprojection, which preserves the isoscalar part in all $S$ and $P$ partial
waves as well as in the ${}^{3}D_{1}$ wave and its coupling to ${}^{3}S_{1}$.
AV8’ is about $0.2\div 0.3$ MeV per nucleon more attractive than Argonne V18
in light nuclei [121, 118, 198], but its contribution is very similar to AV18
in neutron drops, where the difference is about 0.06 MeV per neutron [121].
Other common solutions are the Argonne V6’ (AV6’) and V4’ (AV4’) potentials
[118]. AV6’ is obtained by deleting the spin-orbit terms from AV8’ and
adjusting the potential to preserve the deuteron binding. The spin-orbit terms
do not contribute to $S$-wave and ${}^{1}P_{1}$ channel of the $NN$ scattering
and are the smallest contributors to the energy of 4He [21], but they are
important in differentiating between the ${}^{3}P_{0,1,2}$ channels. The AV4’
potential eliminates the tensor terms. As a result, the ${}^{1}S_{0}$ and
${}^{1}P_{1}$ potentials are unaffected, but the coupling between
${}^{3}S_{1}$ and ${}^{3}D_{1}$ channels is gone and the ${}^{3}P_{0,1,2}$
channels deteriorate further. The Fortran code for the AV18 and AVn’
potentials is available at the webpage [199].
#### 2.1.2 Three-body $NNN$ potential
The Urbana IX three-body force was originally proposed in combination with the
Argonne AV18 and AV8’ [122]. Although it slightly underbinds the energy of
light nuclei, it has been extensively used to study the equation of state of
nuclear and neutron matter [5, 37, 200, 38, 39, 40]. The Illinois forces
[121], the most recent of which is the Illinois-7 (IL7) [201], have been
introduced to improve the description of both ground- and excited-states of
light nuclei, showing an excellent accuracy [121, 17], but they produce an
unphysical overbinding in pure neutron systems [34].
The three-body Illinois potential consists of two- and three-pion exchange and
a phenomenological short-range component (the UIX force does not include the
three-pion rings):
$\displaystyle V_{ijk}=V_{ijk}^{2\pi}+V_{ijk}^{3\pi}+V_{ijk}^{R}\;.$ (2.23)
The two-pion term, as shown in Fig. 2.1, contains $P$\- and $S$-wave $\pi N$
scattering terms (respectively in Fig. 2.1(a) and Fig. 2.1(b)):
$\displaystyle V_{ijk}^{2\pi}=V_{ijk}^{2\pi,P}+V_{ijk}^{2\pi,S}\;.$ (2.24)
(a) (b)
Figure 2.1: Two-pion exchange processes in the $NNN$ force. 2.1(a) is the
Fujita-Miyazawa $P$-wave term and 2.1(b) the Tucson-Melbourne $S$-wave term.
The $P$-wave component, originally introduced by Fujita-Miyazawa [202],
describes an intermediate excited $\Delta$ resonance produced by the exchange
of two pions between nucleons $i$-$j$ and $j$-$k$, as shown in Fig. 2.1(a),
and it can be written as
$\displaystyle V_{ijk}^{2\pi,P}=A_{2\pi}^{P}\,\mathcal{O}_{ijk}^{2\pi,P}\;,$
(2.25)
where
$\displaystyle A_{2\pi}^{P}$ $\displaystyle=-\frac{2}{81}\frac{f_{\pi
NN}^{2}}{4\pi}\frac{f_{\pi\Delta
N}^{2}}{4\pi}\frac{m_{\pi}^{2}}{m_{\Delta}-m_{N}}\;,$ (2.26a)
$\displaystyle\mathcal{O}_{ijk}^{2\pi,P}$
$\displaystyle=\sum_{cyclic}\left(\phantom{\frac{1}{4}}\\!\\!\\!\\!\Bigl{\\{}X_{ij},X_{jk}\Bigr{\\}}\Bigl{\\{}\bm{\tau}_{i}\cdot\bm{\tau}_{j},\bm{\tau}_{j}\cdot\bm{\tau}_{k}\Bigr{\\}}+\frac{1}{4}\Bigl{[}X_{ij},X_{jk}\Bigr{]}\Bigl{[}\bm{\tau}_{i}\cdot\bm{\tau}_{j},\bm{\tau}_{j}\cdot\bm{\tau}_{k}\Bigr{]}\right)\;,$
(2.26b)
and the $X_{ij}$ operator is the same of Eq. (2.7). The constant
$A_{2\pi}^{P}$ is fitted to reproduce the ground state of light nuclei and
properties of nuclear matter. The $P$-wave TPE term is the longest-ranged
nuclear $NNN$ contribution and it is attractive in all nuclei and nuclear
matter. However it is very small or even slightly repulsive in pure neutron
systems.
The $S$-wave component of TPE three-nucleon force is a simplified form of the
original Tucson-Melbourne model [203], and it involves the $\pi N$ scattering
in the $S$-wave as shown in Fig. 2.1(b). It has the following form:
$\displaystyle V_{ijk}^{2\pi,S}=A_{2\pi}^{S}\,\mathcal{O}_{ijk}^{2\pi,S}\;,$
(2.27)
where
$\displaystyle A_{2\pi}^{S}$ $\displaystyle=\left(\frac{f_{\pi
NN}}{4\pi}\right)^{2}a^{\prime}m_{\pi}^{2}\;,$ (2.28a)
$\displaystyle\mathcal{O}_{ijk}^{2\pi,S}$
$\displaystyle=\sum_{cyclic}Z_{\pi}(r_{ij})Z_{\pi}(r_{jk})\,\bm{\sigma}_{i}\cdot\hat{\bm{r}}_{ij}\,\bm{\sigma}_{k}\cdot\hat{\bm{r}}_{kj}\,\bm{\tau}_{i}\cdot\bm{\tau}_{k}\;,$
(2.28b)
and the $Z_{\pi}(r)$ function is defined as
$\displaystyle
Z_{\pi}(r)=\frac{\mu_{\pi}r}{3}\Bigl{[}Y_{\pi}(r)-T_{\pi}(r)\Bigr{]}\;.$
(2.29)
The $S$-wave TPE term is required by chiral perturbation theory but in
practice its contribution is only 3%–4% of $V_{ijk}^{2\pi,P}$ in light nuclei.
The three-pion term (Fig. 2.2) was introduced in the Illinois potentials. It
consists of the subset of three-pion rings that contain only one $\Delta$ mass
in the energy denominators.
(a) (b)
Figure 2.2: Three-pion exchange processes in the $NNN$ force.
As discussed in Ref. [121], these diagrams result in a large number of terms,
the most important of which are the ones independent of cyclic permutations of
$ijk$:
$\displaystyle V_{ijk}^{3\pi}=A_{3\pi}\,\mathcal{O}_{ijk}^{3\pi}\;,$ (2.30)
where
$\displaystyle A_{3\pi}$ $\displaystyle=\left(\frac{f^{2}_{\pi
NN}}{4\pi}\frac{m_{\pi}}{3}\right)^{3}\frac{f^{2}_{\pi N\Delta}}{f^{2}_{\pi
NN}}\frac{1}{(m_{\Delta}-m_{N})^{2}}\;,$ (2.31a)
$\displaystyle\mathcal{O}_{ijk}^{3\pi}$
$\displaystyle\simeq\frac{50}{3}S_{ijk}^{\tau}\,S_{ijk}^{\sigma}+\frac{26}{3}A_{ijk}^{\tau}\,A_{ijk}^{\sigma}\;.$
(2.31b)
The letters $S$ and $A$ denote operators that are symmetric and antisymmetric
under the exchange of $j$ with $k$. Superscripts $\tau$ and $\sigma$ label
operators containing isospin and spin-space parts, respectively. The isospin
operators are
$\displaystyle S_{ijk}^{\tau}$
$\displaystyle=2+\frac{2}{3}\left(\bm{\tau}_{i}\cdot\bm{\tau}_{j}+\bm{\tau}_{j}\cdot\bm{\tau}_{k}+\bm{\tau}_{k}\cdot\bm{\tau}_{i}\right)=4\,P_{T=3/2}\;,$
(2.32a) $\displaystyle A_{ijk}^{\tau}$
$\displaystyle=\frac{1}{3}\,i\,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\times\bm{\tau}_{k}=-\frac{1}{6}\Bigl{[}\bm{\tau}_{i}\cdot\bm{\tau}_{j},\bm{\tau}_{j}\cdot\bm{\tau}_{k}\Bigr{]}\;,$
(2.32b)
where $S_{ijk}^{\tau}$ is a projector onto isospin 3/2 triples and
$A_{ijk}^{\tau}$ has the same isospin structure as the commutator part of
$V_{ijk}^{2\pi,P}$. The spin-space operators have many terms and they are
listed in the Appendix of Ref. [121]. An important aspect of this structure is
that there is a significant attractive term which acts only in $T=3/2$
triples, so the net effect of $V_{ijk}^{3\pi}$ is slight repulsion in
$S$-shell nuclei and larger attraction in $P$-shell nuclei. However, in most
light nuclei the contribution of this term is rather small, $\langle
V_{ijk}^{3\pi}\rangle\lesssim 0.1\langle V_{ijk}^{2\pi}\rangle$.
The last term of Eq. (2.23) was introduced to compensate the overbinding in
nuclei and the large equilibrium density of nuclear matter given by the
previous operators. It is strictly phenomenological and purely central and
repulsive, and it describes the modification of the contribution of the TPE
$\Delta$-box diagrams to $v_{ij}$ due to the presence of the third nucleon $k$
(Fig. 2.3). It takes the form:
$\displaystyle
V_{ijk}^{R}=A_{R}\,\mathcal{O}^{R}_{ijk}=A_{R}\sum_{cyclic}T_{\pi}^{2}(r_{ij})\,T_{\pi}^{2}(r_{jk})\;,$
(2.33)
where $T_{\pi}(r)$ is the OPE tensor potential defined in Eq. (2.10).
Figure 2.3: Repulsive short-range contribution included in the $NNN$ force.
Finally, the Illinois (Urbana IX) TNI can be written as a sum of four
different terms:
$\displaystyle
V_{ijk}=A_{2\pi}^{P}\,\mathcal{O}^{2\pi,P}_{ijk}+A_{2\pi}^{S}\,\mathcal{O}^{2\pi,S}_{ijk}+A_{3\pi}\,\mathcal{O}^{3\pi}_{ijk}+A_{R}\,\mathcal{O}^{R}_{ijk}\;.$
(2.34)
### 2.2 Interactions: hyperons and nucleons
We present a detailed description of the $\Lambda N$ and $\Lambda NN$
interaction as developed by Bodmer, Usmani and Carlson following the scheme of
the Argonne potentials [190, 183, 181, 192, 185, 191, 186, 184, 187, 188,
189]. The interaction is written in coordinates space and it includes two- and
three-body hyperon nucleon components with an explicit hard-core repulsion
between baryons and a charge symmetry breaking term. We introduce also an
effective $\Lambda\Lambda$ interaction mainly used in variational [194, 195]
and cluster model [171, 173] calculations for double $\Lambda$ hypernuclei.
#### 2.2.1 Two-body $\Lambda N$ potential
##### $\Lambda N$ charge symmetric potential
The $\Lambda$ particle has isospin $I=0$, so there is no OPE term, being the
strong $\Lambda\Lambda\pi$ vertex forbidden due to isospin conservation. The
$\Lambda$ hyperon can thus exchange a pion only with a $\Lambda\pi\Sigma$
vertex. The lowest order $\Lambda N$ coupling must therefore involve the
exchange of two pions, with the formation of a virtual $\Sigma$ hyperon, as
illustrated in Figs. 2.4(a) and 2.4(b). The TPE interaction is intermediate
range with respect to the long range part of $NN$ force. One meson exchange
processes can only occur through the exchange of a $K,K^{*}$ kaon pair, that
contributes in exchanging the strangeness between the two baryons, as shown in
Fig. 2.4(c). The $K,K^{*}$ potential is short-range and contributes to the
space-exchange and $\Lambda N$ tensor potential. The latter is expected to be
quite weak because the $K$ and $K^{*}$ tensor contributions have opposite sign
[204].
(a) (b) (c)
Figure 2.4: Meson exchange processes in the $\Lambda N$ force. 2.4(a) and
2.4(b) are the TPE diagrams. 2.4(c) represents the kaon exchange channel.
The $\Lambda N$ interaction has been modeled with an Urbana-type potential
[205] with spin-spin and space-exchange components and a TPE tail which is
consistent with the available $\Lambda p$ scattering data below the $\Sigma$
threshold:
$\displaystyle v_{\lambda i}=v_{0}(r_{\lambda
i})(1-\varepsilon+\varepsilon\,\mathcal{P}_{x})+\frac{1}{4}v_{\sigma}T^{2}_{\pi}(r_{\lambda
i})\,{\bm{\sigma}}_{\lambda}\cdot{\bm{\sigma}}_{i}\;,$ (2.35)
where
$\displaystyle v_{0}(r_{\lambda i})=v_{c}(r_{\lambda
i})-\bar{v}\,T^{2}_{\pi}(r_{\lambda i})\;.$ (2.36)
Here,
$\displaystyle
v_{c}(r)=W_{c}\Bigl{(}1+\operatorname{e}^{\frac{r-\bar{r}}{a}}\Bigr{)}^{-1}$
(2.37)
is a Wood-Saxon repulsive potential introduced, similarly to the Argonne $NN$
interaction, in order to include all the short-range contributions and
$T_{\pi}(r)$ is the regularized OPE tensor operator defined in Eq. (2.10). The
term $\bar{v}\,T^{2}_{\pi}(r_{\lambda i})$ corresponds to a TPE mechanism due
to OPE transition potentials $\left(\Lambda N\leftrightarrow\Sigma
N,\Sigma\Delta\right)$ dominated by their tensor components. The $\Lambda p$
scattering at low energies is well fitted with $\bar{v}=6.15(5)$ MeV. The
terms $\bar{v}=(v_{s}+3v_{t})/4$ and $v_{\sigma}=v_{s}-v_{t}$ are the spin-
average and spin-dependent strengths, where $v_{s}$ and $v_{t}$ denote
singlet- and triplet-state strengths, respectively. $\mathcal{P}_{x}$ is the
$\Lambda N$ space-exchange operator and $\varepsilon$ the corresponding
exchange parameter, which is quite poorly determined from the $\Lambda p$
forward-backward asymmetry to be $\varepsilon\simeq 0.1\div 0.38$. All the
parameters defining the $\Lambda N$ potential can be found in Tab. 2.1.
##### $\Lambda N$ charge symmetry breaking potential
The $\Lambda$-nucleon interaction should distinguish between the nucleon
isospin channels $\Lambda p$ and $\Lambda n$. The mirror pair of hypernuclei
${}^{4}_{\Lambda}$H and ${}^{4}_{\Lambda}$He is the main source of information
about the charge symmetry breaking (CSB) $\Lambda N$ interaction. The
experimental data for $A=4$ $\Lambda$ hypernuclei [77], show indeed a clear
difference in the $\Lambda$ separation energies for the $(0^{+})$ ground state
$\displaystyle B_{\Lambda}\left({}^{4}_{\Lambda}\text{H}\right)$
$\displaystyle=2.04(4)\leavevmode\nobreak\ \text{MeV}\;,$ (2.38a)
$\displaystyle B_{\Lambda}\left({}^{4}_{\Lambda}\text{He}\right)$
$\displaystyle=2.39(3)\leavevmode\nobreak\ \text{MeV}\;,$ (2.38b)
and for the $(1^{+})$ excited state
$\displaystyle B_{\Lambda}^{*}\left({}^{4}_{\Lambda}\text{H}\right)$
$\displaystyle=1.00(6)\leavevmode\nobreak\ \text{MeV}\;,$ (2.39a)
$\displaystyle B_{\Lambda}^{*}\left({}^{4}_{\Lambda}\text{He}\right)$
$\displaystyle=1.24(6)\leavevmode\nobreak\ \text{MeV}\;.$ (2.39b)
The differences in the hyperon separation energies are:
$\displaystyle\Delta B_{\Lambda}$ $\displaystyle=0.35(6)\leavevmode\nobreak\
\text{MeV}\;,$ (2.40a) $\displaystyle\Delta B_{\Lambda}^{*}$
$\displaystyle=0.24(6)\leavevmode\nobreak\ \text{MeV}\;.$ (2.40b)
However, the experimental values $\Delta B_{\Lambda}$ must be corrected to
include the difference $\Delta B_{c}$ due to the Coulomb interaction in order
to obtain the values to be attributed to CSB effects. By means of a
variational calculation, Bodmer and Usmani [183] estimated the Coulomb
contribution to be rather small
$\displaystyle|\Delta B_{c}|$ $\displaystyle=0.05(2)\leavevmode\nobreak\
\text{MeV}\;,$ (2.41a) $\displaystyle|\Delta B_{c}^{*}|$
$\displaystyle=0.025(15)\leavevmode\nobreak\ \text{MeV}\;,$ (2.41b)
and they were able to reproduce the differences in the $\Lambda$ separation
energies by means of a phenomenological spin dependent CSB potential. It was
found that the CSB interaction is effectively spin independent and can be
simply expressed (as subsequently reported in Ref. [186]) by
$\displaystyle v_{\lambda i}^{CSB}=C_{\tau}\,T_{\pi}^{2}\left(r_{\lambda
i}\right)\tau_{i}^{z}\quad\quad C_{\tau}=-0.050(5)\leavevmode\nobreak\
\text{MeV}\;.$ (2.42)
Being $C_{\tau}$ negative, the $\Lambda p$ channel becomes attractive while
the $\Lambda n$ channel is repulsive, consistently with the experimental
results for ${}^{4}_{\Lambda}$H and ${}^{4}_{\Lambda}$He. The contribution of
CSB is expected to be very small in symmetric hypernuclei (if Coulomb is
neglected) but could have a significant effect in hypernuclei with an neutron
(or proton) excess.
#### 2.2.2 Three-body $\Lambda NN$ potential
The $\Lambda N$ force as obtained by fitting the $\Lambda p$ scattering does
not provide a good account of the experimental binding energies, as in the
case of nuclei with the bare $NN$ interaction. A three-body $\Lambda NN$ force
is required in this scheme to solve the overbinding. The $\Lambda NN$
potential is at the same TPE order of the $\Lambda N$ force and it includes
diagrams involving two nucleons and one hyperon, as reported in Fig. 2.5.
(a) (b) (c)
Figure 2.5: Two-pion exchange processes in three-body $\Lambda NN$ force.
2.5(a) and 2.5(b) are, respectively, the $P$\- and $S$-wave TPE contributions.
2.5(c) is the phenomenological dispersive term.
The diagrams in Fig. 2.5(a) and Fig. 2.5(b) correspond respectively to the
$P$-wave and $S$-wave TPE
$\displaystyle v^{2\pi}_{\lambda ij}=v^{2\pi,P}_{\lambda
ij}+v^{2\pi,S}_{\lambda ij}\;,$ (2.43)
that can be written in the following form:
$\displaystyle v_{\lambda ij}^{2\pi,P}$
$\displaystyle=\widetilde{C}_{P}\,\mathcal{O}_{\lambda ij}^{2\pi,P}$
$\displaystyle=-\frac{C_{P}}{6}\Bigl{\\{}X_{i\lambda}\,,X_{\lambda
j}\Bigr{\\}}\,{\bm{\tau}}_{i}\cdot{\bm{\tau}}_{j}\;,$ (2.44) $\displaystyle
v_{\lambda ij}^{2\pi,S}$ $\displaystyle=C_{S}\,O_{\lambda ij}^{2\pi,S}$
$\displaystyle=C_{S}\,Z\left(r_{\lambda i}\right)Z\left(r_{\lambda
j}\right)\,{\bm{\sigma}}_{i}\cdot\hat{\bm{r}}_{i\lambda}\,{\bm{\sigma}}_{j}\cdot\hat{\bm{r}}_{j\lambda}\,{\bm{\tau}}_{i}\cdot{\bm{\tau}}_{j}\;.$
(2.45)
The structure of $V_{\lambda ij}^{2\pi}$ is very close to the Fujita-Miyazawa
$P$-wave term and the Tucson-Melbourne $S$-wave term of the nuclear
$V_{ijk}^{2\pi}$ (see Eqs. (2.26) and (2.28)). In the hypernuclear sector,
however, there are simplifications because only two nucleons at a time enter
the picture, so there are no cyclic summations, and the $\Lambda$ particle has
isospin zero, thus there is no $\bm{\tau}_{\lambda}$ operator involved. As
reported in Ref. [121], the strength of $V_{ijk}^{2\pi,S}$ is
$\left|A_{2\pi}^{S}\right|\simeq 0.8$ MeV. However, in other references it is
assumed to have a value of 1.0 MeV. Comparing the Tucson-Melbourne $NNN$ model
with Eq. (2.45) for the $\Lambda NN$ potential, one may write an identical
structure for both $S$-wave $\Lambda NN$ and $NNN$ potentials as follows:
$\displaystyle C_{S}\,\mathcal{O}_{\lambda
ij}^{2\pi,S}=A_{S}^{2\pi}\,\mathcal{O}_{ijk}^{2\pi,S}\;.$ (2.46)
This directly relates $C_{S}$ in the strange sector to $A_{2\pi}^{S}$ in the
non-strange sector. Since the $\Sigma$-$\Lambda$ mass difference is small
compared to the $\Delta$-$N$ mass difference, the $2\pi$ $\Lambda NN$
potential of $S=-1$ sector is stronger than the non-strange $NNN$ potential of
$S=0$ sector. This provides stronger strengths in the case of $\Lambda NN$
potential compared to the $NNN$ potential. It is therefore expected that the
value of $C_{S}$ would be more than 1.0 MeV, and is taken to be 1.5 MeV [189].
However, the $S$-wave component is expected to be quite weak, at least in
spin-zero core hypernuclei, and indeed it has been neglected in variational
calculations for ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O and
${}^{5}_{\Lambda}$He [192, 187, 194].
The last diagram (Fig. 2.5(c)) represents the dispersive contribution
associated with the medium modifications of the intermediate state potentials
for the $\Sigma$, $N$, $\Delta$ due to the presence of the second nucleon.
This term describes all the short-range contributions and it is expected to be
repulsive due to the suppression mechanism associated with the $\Lambda
N$-$\Sigma N$ coupling [206, 207]. The interaction of the intermediate states
$\Sigma$, $N$, $\Delta$ with a nucleon of the medium will be predominantly
through a TPE potential, proportional to $T_{\pi}^{2}(r)$, with an explicit
spin dependence (negligible for spin-zero core hypernuclei):
$\displaystyle v_{\lambda ij}^{D}=W_{D}\,\mathcal{O}_{\lambda
ij}^{D}=W_{D}\,T_{\pi}^{2}\left(r_{\lambda
i}\right)T^{2}_{\pi}\left(r_{\lambda
j}\right)\\!\\!\bigg{[}1+\frac{1}{6}{\bm{\sigma}}_{\lambda}\\!\cdot\\!\left({\bm{\sigma}}_{i}+{\bm{\sigma}}_{j}\right)\bigg{]}\;.$
(2.47)
The radial functions $T_{\pi}(r)$ and $Z_{\pi}(r)$ are the same of the nuclear
potential, see Eq. (2.10) and Eq. (2.29). The operator $X_{\lambda i}$ is the
same of Eq. (2.7), in which the first nucleon is replaced by the $\Lambda$
particle.
It is important to note that the three-body $\Lambda NN$ interaction have been
investigated in variational calculations for ${}_{\Lambda}^{5}$He [185, 187,
189], ${}_{\Lambda\Lambda}^{\;\;\,6}$He [194, 195] and
${}_{\leavevmode\nobreak\ \Lambda}^{17}$O [192, 185], resulting in a range of
values for the $C_{P}$ and $W_{D}$ parameters (see Tab. 2.1) that gives good
description of the properties of the studied hypernuclei. A unique set of
parameters that reproduces all the available experimental energies for single
(and double) $\Lambda$ hypernuclei has not been set yet.
A second crucial observation is that, differently to the nucleon sector, both
two- and three-body lambda-nucleon interactions are at the same TPE order. In
addition, the mass difference between the $\Lambda$ particle and its
excitation $\Sigma$ is much smaller than the mass difference between the
nucleon and the $\Delta$ resonance. Thus, the $\Lambda NN$ interaction can not
be neglected in this framework but it is a key ingredient in addition to the
$\Lambda N$ force for any consistent theoretical calculation involving
$\Lambda$ hyperons.
Constant | Value | Unit
---|---|---
$W_{c}$ | $2137$ | MeV
$\bar{r}$ | $0.5$ | fm
$a$ | $0.2$ | fm
$v_{s}$ | $6.33,6.28$ | MeV
$v_{t}$ | $6.09,6.04$ | MeV
$\bar{v}$ | $6.15(5)$ | MeV
$v_{\sigma}$ | $0.24$ | MeV
$c$ | $2.0$ | fm-2
$\varepsilon$ | $0.1\div 0.38$ | —
$C_{\tau}$ | -0.050(5) | MeV
$C_{P}$ | $0.5\div 2.5$ | MeV
$C_{S}$ | $\simeq 1.5$ | MeV
$W_{D}$ | $0.002\div 0.058$ | MeV
Table 2.1: Parameters of the $\Lambda N$ and $\Lambda NN$ interaction (see
[189] and reference therein). For $C_{P}$ and $W_{D}$ the variational allowed
range is shown. The value of the charge symmetry breaking parameter $C_{\tau}$
is from Ref. [186].
#### 2.2.3 Two-body $\Lambda\Lambda$ potential
Due to the impossibility to collect $\Lambda\Lambda$ scattering data,
experimental information about the $\Lambda\Lambda$ interaction can be
obtained only from the $\Lambda\Lambda$ separation energy of the observed
double $\Lambda$ hypernuclei, ${}^{\;\;\,6}_{\Lambda\Lambda}$He [91, 92, 93],
${}^{\;13}_{\Lambda\Lambda}$B [92] and the isotopes of
${}^{\;10}_{\Lambda\Lambda}$Be ($A=10\div 12$) [94, 92]. Evidence for the
production of ${}_{\Lambda\Lambda}^{\;\;\,4}$H has been reported in Ref.
[208], but no information about the $\Lambda\Lambda$ separation energy was
found. On the other hand, there is a theoretical indication for the one-boson
exchange (OBE) part of the $\Lambda\Lambda$ interaction coming from the
$SU(3)$-invariance of coupling constants, but the $\Lambda\Lambda$ force is
still far to be settled.
In the next, we follow the guide line adopted in the three- and four-body
cluster models for double $\Lambda$ hypernuclei [171, 173], which was also
used in Faddeev-Yakubovsky calculations for light double $\Lambda$ hypernuclei
[209] and in variational calculations on ${}^{\;\;\,4}_{\Lambda\Lambda}$H
[193, 210], ${}^{\;\;\,5}_{\Lambda\Lambda}$H and
${}^{\;\;\,5}_{\Lambda\Lambda}$He [193, 211] and
${}^{\;\;\,6}_{\Lambda\Lambda}$He [194, 195, 193, 211], with different
parametrizations. The employed OBE-simulating $\Lambda\Lambda$ effective
interaction is a low-energy phase equivalent Nijmegen interaction represented
by a sum of three Gaussians:
$\displaystyle
v_{\lambda\mu}=\sum_{k=1}^{3}\left(v_{0}^{(k)}+v_{\sigma}^{(k)}\,{\bm{\sigma}}_{\lambda}\cdot{\bm{\sigma}}_{\mu}\right)\operatorname{e}^{-\mu^{(k)}r_{\lambda\mu}^{2}}\;.$
(2.48)
The most recent parametrization of the potential (see Tab. 2.2), was fitted in
order to simulate the $\Lambda\Lambda$ sector of the Nijmegen F (NF)
interaction [150, 151, 152]. The NF is the simplest among the Nijmegen models
with a scalar nonet, which seems to be more appropriate than the versions
including only a scalar singlet in order to reproduce the weak binding energy
indicated by the NAGARA event [91]. The components $k=1,2$ of the above
Gaussian potential are determined so as to simulate the $\Lambda\Lambda$
sector of NF and the strength of the part for $k=3$ is adjusted so as to
reproduce the ${}^{\;\;\,6}_{\Lambda\Lambda}$He NAGARA experimental double
$\Lambda$ separation energy of $7.25\pm 0.19^{+0.18}_{-0.11}$ MeV. In 2010,
Nakazawa reported a new, more precise determination of
$B_{\Lambda\Lambda}=6.93\pm 0.16$ MeV for ${}^{\;\;\,6}_{\Lambda\Lambda}$He
[92], obtained via the $\Xi^{-}$ hyperon capture at rest reaction in a hybrid
emulsion. This value has been recently revised to $B_{\Lambda\Lambda}=6.91\pm
0.16$ MeV by the E373 (KEK-PS) Collaboration [93]. No references were found
about the refitting of the $\Lambda\Lambda$ Gaussian potential on the more
recent experimental result, which is in any case compatible with the NAGARA
event. We therefore consider the original parametrization of Ref. [173].
$\mu^{(k)}$ | $0.555$ | $1.656$ | $8.163$
---|---|---|---
$v_{0}^{(k)}$ | $-10.67$ | $-93.51$ | $4884$
$v_{\sigma}^{(k)}$ | $0.0966$ | $16.08$ | $915.8$
Table 2.2: Parameters of the the $\Lambda\Lambda$ interaction. The size
parameters $\mu^{(k)}$ are in fm-2 and the strengths $v_{0}^{(k)}$ and
$v_{\sigma}^{(k)}$ are in MeV [173].
## Chapter 3 Method
In nuclear physics, many-body calculations are used to understand the nuclear
systems in the non-relativistic regime. When interested in low energy
phenomena, a nucleus (or an extensive nucleonic system) can be described as a
collection of particles interacting via a potential that depends on positions,
momenta, spin and isospin. The properties of the system can be determined by
solving a many-body Schrödinger equation. Such calculations can study, for
example, binding energies, excitation spectra, densities, reactions and many
other aspects of nuclei. The equation of state, masses, radii and other
properties are obtained by describing astrophysical objects as a nuclear
infinite medium.
The two main problems related to microscopic few- and many-body calculations
in nuclear physics are the determination of the Hamiltonian and the method
used to accurately solve the Schrödinger equation. In the previous chapter, we
have already seen how to build a realistic nuclear Hamiltonian, including also
strange degrees of freedom. In the next we will focus on the methodological
part presenting a class of Quantum Monte Carlo algorithms, the Diffusion Monte
Carlo (DMC) and, more in detail, the Auxiliary Field Diffusion Monte Carlo
(AFDMC). Such methods are based on evolving a trial wave function in imaginary
time to yield the ground state of the system. The DMC method sums explicitly
over spin and isospin states and can use very sophisticated wave functions.
However, it is limited to small systems. In the AFDMC, in addition to the
coordinates, also the spin and isospin degrees of freedom are sampled. It can
thus treat larger systems but there are some limitations on the trial wave
function and the nuclear potentials that can be handled.
Strangeness can be included in AFDMC calculations by adding hyperons to the
standard nucleons. The interaction between hyperons and nucleons presented in
the previous chapter is written in a suitable form to be treated within this
algorithm. By extending the AFDMC nuclear wave function to the hyperonic
sector, it is possible to study both hypernuclei and hypermatter. A new QMC
approach to strange physics is thus now available.
### 3.1 Diffusion Monte Carlo
The Diffusion Monte Carlo method [212, 136, 213, 214] projects the ground-
state out of a stationary trial wave function $|\psi_{T}\rangle$ not
orthogonal to the true ground state. Consider the many-body time dependent
Schrödinger equation with its formal solution
$\displaystyle i\hbar\frac{\partial}{\partial
t}|\psi(t)\rangle=(H-E_{T})|\psi(t)\rangle\quad\Rightarrow\quad|\psi(t+dt)\rangle=\operatorname{e}^{-\frac{i}{\hbar}(H-E_{T})dt}|\psi(t)\rangle\;,$
(3.1)
and let move to the imaginary time $\tau=it/\hbar$111with this definition
$\tau$ has the dimensions of the inverse of an energy.:
$\displaystyle-\frac{\partial}{\partial\tau}|\psi(\tau)\rangle=(H-E_{T})|\psi(\tau)\rangle\quad\Rightarrow\quad|\psi(\tau+d\tau)\rangle=\operatorname{e}^{-(H-E_{T})d\tau}|\psi(\tau)\rangle\;.$
(3.2)
The stationary states $|\psi(0)\rangle=|\psi_{T}\rangle$ are the same for both
normal and imaginary time Schrödinger equations and we can expand them on a
complete orthonormal set of eigenvectors $|\varphi_{n}\rangle$ of the
Hamiltonian $H$:
$\displaystyle|\psi_{T}\rangle=\sum_{n=0}^{\infty}c_{n}|\varphi_{n}\rangle\;.$
(3.3)
Supposing that the $|\psi_{T}\rangle$ is not orthogonal to the true ground
state, i.e. $c_{0}\neq 0$, and that at least the ground state is non
degenerate, i.e. $E_{n}\geq E_{n-1}>E_{0}$, where $E_{n}$ are the eigenvalues
of $H$ related to $|\varphi_{n}\rangle$, the imaginary time evolution of
$|\psi_{T}\rangle$ is given by
$\displaystyle|\psi(\tau)\rangle$
$\displaystyle=\sum_{n=0}^{\infty}c_{n}\operatorname{e}^{-(E_{n}-E_{T})\tau}|\varphi_{n}\rangle\;,$
$\displaystyle=c_{0}\operatorname{e}^{-(E_{0}-E_{T})\tau}|\varphi_{0}\rangle+\sum_{n=1}^{\infty}c_{n}\operatorname{e}^{-(E_{n}-E_{T})\tau}|\varphi_{n}\rangle\;.$
(3.4)
If the energy offset $E_{T}$ is the exact ground state energy $E_{0}$, in the
limit $\tau\rightarrow\infty$ the components of Eq. (3.4) for $n>0$ vanish and
we are left with
$\displaystyle\lim_{\tau\rightarrow\infty}|\psi(\tau)\rangle=c_{0}|\varphi_{0}\rangle\;.$
(3.5)
Starting from a generic initial trial wave function $|\psi_{T}\rangle$ not
orthogonal to the ground state, and adjusting the energy offset $E_{T}$ to be
as close as possible to $E_{0}$, in the limit of infinite imaginary time, one
can project out the exact ground state $c_{0}|\varphi_{0}\rangle$ giving
access to the lowest energy properties of the system.
Consider the imaginary time propagation of Eq. (3.2) and insert a completeness
on the orthonormal basis $|R^{\prime}\rangle$, where $R$ represents a
configuration $\\{\bm{r}_{1},\ldots,\bm{r}_{\mathcal{N}}\\}$ of the
$\mathcal{N}$ particle system with all its degrees of freedom:
$\displaystyle|\psi(\tau+d\tau)\rangle$
$\displaystyle=\operatorname{e}^{-(H-E_{T})d\tau}|\psi(\tau)\rangle\;,$
$\displaystyle=\int
dR^{\prime}\operatorname{e}^{-(H-E_{T})d\tau}|R^{\prime}\rangle\langle
R^{\prime}|\psi(\tau)\rangle\;.$ (3.6)
Projecting on the coordinates $\langle R|$ leads to
$\displaystyle\langle R|\psi(\tau+d\tau)\rangle=\int dR^{\prime}\,\langle
R|\operatorname{e}^{-(H-E_{T})d\tau}|R^{\prime}\rangle\langle
R^{\prime}|\psi(\tau)\rangle\;,$ (3.7)
where $\langle
R|\operatorname{e}^{-(H-E_{T})d\tau}|R^{\prime}\rangle=G(R,R^{\prime},d\tau)$
is the Green’s function of the operator
$(H-E_{T})+\frac{\partial}{\partial\tau}$. Recalling that $\langle
R|\psi(\tau)\rangle=\psi(R,\tau)$, we can write Eq. (3.2) as
$\displaystyle-\frac{\partial}{\partial\tau}\psi(R,\tau)$
$\displaystyle=(H-E_{T})\psi(R,\tau)\;,$ (3.8)
$\displaystyle\psi(R,\tau+d\tau)$ $\displaystyle=\int
dR^{\prime}\,G(R,R^{\prime},d\tau)\,\psi(R^{\prime},\tau)\;.$ (3.9)
If we consider a non-interacting many-body system, i.e. the Hamiltonian is
given by the pure kinetic term
$\displaystyle
H_{0}=T=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{\mathcal{N}}\nabla_{i}^{2}\;,$ (3.10)
the Schrödinger equation (3.8) becomes a $3\mathcal{N}$-dimensional diffusion
equation. By writing the Green’s function of Eq. (3.9) in momentum space by
means of the Fourier transform, it is possible to show that $G_{0}$ is a
Gaussian with variance proportional to $\tau$
$\displaystyle G_{0}(R,R^{\prime},d\tau)=\left(\frac{1}{4\pi
Dd\tau}\right)^{\frac{3\mathcal{N}}{2}}\\!\operatorname{e}^{-\frac{(R-R^{\prime})^{2}}{4Dd\tau}}\;,$
(3.11)
where $D=\hbar^{2}/2m$ is the diffusion constant of a set of particles in
Brownian motion with a dynamic governed by random collisions. This
interpretation can be implemented by representing the wave function
$\psi(R,\tau)$ by a set of discrete sampling points, called _walkers_
$\displaystyle\psi(R,\tau)=\sum_{k}\delta(R-R_{k})\;,$ (3.12)
and evolving this discrete distribution for an imaginary time $d\tau$ by means
of Eq. (3.9):
$\displaystyle\psi(R,\tau+d\tau)=\sum_{k}G_{0}(R,R_{k},d\tau)\;.$ (3.13)
The result is a set of Gaussians that in the infinite imaginary time limit
represents a distribution of walkers according to the lowest state of the
Hamiltonian, that can be used to calculate the ground state properties of the
system.
Let now consider the full Hamiltonian where the interaction is described by a
central potential in coordinate space:
$\displaystyle
H=T+V=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{\mathcal{N}}\nabla_{i}^{2}+V(R)\;.$
(3.14)
Because $T$ and $V$ in general do not commute, it is not possible to directly
split the propagator in a kinetic and a potential part
$\displaystyle\operatorname{e}^{-(H-E_{T})d\tau}\neq\operatorname{e}^{-Td\tau}\operatorname{e}^{-(V-E_{T})d\tau}\;,$
(3.15)
and thus the analytic solution of the Green’s function $\langle
R|\operatorname{e}^{-(T+V-E_{T})d\tau}|R^{\prime}\rangle$ is not known in most
of the cases. However, by means of the Trotter-Suzuki formula to order
$d\tau^{3}$
$\displaystyle\operatorname{e}^{-(A+B)d\tau}=\operatorname{e}^{-A\frac{d\tau}{2}}\operatorname{e}^{-Bd\tau}\operatorname{e}^{-A\frac{d\tau}{2}}\,+\operatorname{o}\left(d\tau^{3}\right)\;,$
(3.16)
which is an improvement of the standard
$\displaystyle\operatorname{e}^{-(A+B)d\tau}=\operatorname{e}^{-Ad\tau}\operatorname{e}^{-Bd\tau}\,+\operatorname{o}\left(d\tau^{2}\right)\;,$
(3.17)
in the limit of small imaginary time step $d\tau$ it is possible to write an
approximate solution for $\psi(R,\tau+d\tau)$:
$\displaystyle\psi(R,\tau+d\tau)$ $\displaystyle\simeq\int dR^{\prime}\langle
R|\operatorname{e}^{-V\frac{d\tau}{2}}\operatorname{e}^{-Td\tau}\operatorname{e}^{-V\frac{d\tau}{2}}\operatorname{e}^{E_{T}d\tau}|R^{\prime}\rangle\,\psi(R^{\prime},\tau)\;,$
$\displaystyle\simeq\int dR^{\prime}\underbrace{\langle
R|\operatorname{e}^{-Td\tau}|R^{\prime}\rangle}_{G_{0}(R,R^{\prime},d\tau)}\underbrace{\phantom{\langle}\\!\\!\operatorname{e}^{-\left(\frac{V(R)+V(R^{\prime})}{2}-E_{T}\right)d\tau}}_{G_{V}(R,R^{\prime},d\tau)}\psi(R^{\prime},\tau)\;,$
$\displaystyle\simeq\left(\frac{1}{4\pi
Dd\tau}\right)^{\frac{3\mathcal{N}}{2}}\\!\\!\int
dR^{\prime}\operatorname{e}^{-\frac{(R-R^{\prime})^{2}}{4Dd\tau}}\operatorname{e}^{-\left(\frac{V(R)+V(R^{\prime})}{2}-E_{T}\right)d\tau}\psi(R^{\prime},\tau)\;,$
(3.18)
which is the same of Eq. (3.9) with the full Green’s function given by
$\displaystyle G(R,R^{\prime},d\tau)\simeq
G_{0}(R,R^{\prime},d\tau)\,G_{V}(R,R^{\prime},d\tau)\;.$ (3.19)
According to the interacting Hamiltonian, the propagation of $\psi(R,\tau)$
for $d\tau\rightarrow 0$ is thus described by the integral (3.18) and the long
imaginary time evolution necessary to project out the ground state component
of the wave function is realized by iteration until convergence is reached.
The steps of this process, that constitute the Diffusion Monte Carlo
algorithm, can be summarized as follows:
1. 1.
An initial distribution of walkers $w_{i}$ with $i=1,\ldots,\mathcal{N}_{w}$
is sampled from the trial wave function $\langle
R|\psi_{T}\rangle=\psi_{T}(R)$ and the starting trial energy $E_{T}$ is chosen
(for instance from a variational calculation or close to the expected result).
2. 2.
The spacial degrees of freedom are propagated for small imaginary time step
$d\tau$ with probability density $G_{0}(R,R^{\prime},d\tau)$, i.e. the
coordinates of the walkers are diffused by means of a Brownian motion
$\displaystyle R=R^{\prime}+\xi\;,$ (3.20)
where $\xi$ is a stochastic variable distributed according to a Gaussian
probability density with $\sigma^{2}=2Dd\tau$ and zero average.
3. 3.
For each walker, a weight
$\displaystyle\omega_{i}=G_{V}(R,R^{\prime},d\tau)=\operatorname{e}^{-\left(\frac{V(R)+V(R^{\prime})}{2}-E_{T}\right)d\tau}\;,$
(3.21)
is assigned. The estimator contributions (kinetic energy, potential energy,
root mean square radii, densities, …) are evaluated on the imaginary time
propagated configurations, weighting the results according to $\omega_{i}$.
4. 4.
The _branching_ process is applied to the propagated walkers. $\omega_{i}$
represents the probability of a configuration to multiply at the next step
according to the normalization. This process is realized by generating from
each $w_{i}$ a number of walker copies
$\displaystyle n_{i}=[\omega_{i}+\eta_{i}]\;,$ (3.22)
where $\eta_{i}$ is a random number uniformly distributed in the interval
$[0,1]$ and $[x]$ means integer part of $x$. In such a way, depending on the
potential $V(R)$ and the trial energy $E_{T}$, some configurations will
disappear and some other will replicate, resulting in the evolution of walker
population which is now made of
$\widetilde{\mathcal{N}}_{w}=\sum_{i=1}^{\mathcal{N}_{w}}n_{i}$ walkers. A
simple solution in order to control the fluctuations of walker population is
to multiply the weight $\omega_{i}$ by a factor
$\mathcal{N}_{w}/\widetilde{\mathcal{N}}_{w}$, adjusting thus the branching
process at each time step. This solution is not efficient if the potential
diverges. The corrections applied run-time could generate a lot of copies from
just few good parent walkers and the population will be stabilized but not
correctly represented. A better sampling technique is described in § 3.1.1.
5. 5.
Iterate from 2 to 4 as long as necessary until convergence is reached, i.e.
for large enough $\tau$ to reach the infinite limit of Eq. (3.5). In this
limit, the configurations $\\{R\\}$ are distributed according to the lowest
energy state $\psi_{0}(R,\tau)$. Therefore, we can compute the ground state
expectation values of observables that commute with the Hamiltonian
$\displaystyle\\!\\!\langle\mathcal{O}\rangle=\frac{\langle\psi_{0}|\mathcal{O}|\psi_{0}\rangle}{\langle\psi_{0}|\psi_{0}\rangle}=\\!\lim_{\tau\rightarrow\infty}\\!\frac{\langle\psi_{T}|\mathcal{O}|\psi(\tau)\rangle}{\langle\psi_{T}|\psi(\tau)\rangle}=\\!\lim_{\tau\rightarrow\infty}\int\\!\\!dR\frac{\langle\psi_{T}|\mathcal{O}|R\rangle\psi(R,\tau)}{\psi_{T}(R)\psi(R,\tau)}\;,$
(3.23)
by means of
$\displaystyle\langle\mathcal{O}\rangle=\frac{\sum_{\\{R\\}}\langle
R|\mathcal{O}|\psi_{T}\rangle}{\sum_{\\{R\\}}\langle
R|\psi_{T}\rangle}=\frac{\sum_{\\{R\\}}\mathcal{O}\psi_{T}(R)}{\sum_{\\{R\\}}\psi_{T}(R)}\;.$
(3.24)
Statistical error bars on expectation values are then estimated by means of
block averages and the analysis of auto-correlations on data blocks. The
direct calculation of the expectation value (3.24) gives an exact result only
when $\mathcal{O}$ is the Hamiltonian $H$ or commutes with $H$, otherwise only
“mixed” matrix elements
$\langle\mathcal{O}\rangle_{m}\neq\langle\mathcal{O}\rangle$ can be obtained.
Among the different methods to calculate expectation values for operators that
do not commute with $H$, the extrapolation method [136] is the most widely
used. Following this method, one has a better approximation to the “pure”
(exact) value by means of a linear extrapolation
$\displaystyle\langle\mathcal{O}\rangle_{p}\simeq
2\,\frac{\langle\psi_{0}|\mathcal{O}|\psi_{T}\rangle}{\langle\psi_{0}|\psi_{T}\rangle}-\frac{\langle\psi_{T}|\mathcal{O}|\psi_{T}\rangle}{\langle\psi_{T}|\psi_{T}\rangle}=2\,\langle\mathcal{O}\rangle_{m}-\langle\mathcal{O}\rangle_{v}\;,$
(3.25)
or, if the operator $\mathcal{O}$ is positive defined, by means of
$\displaystyle\langle\mathcal{O}\rangle_{p}$
$\displaystyle\simeq\frac{\left(\frac{\langle\psi_{0}|\mathcal{O}|\psi_{T}\rangle}{\langle\psi_{0}|\psi_{T}\rangle}\right)^{2}}{\frac{\langle\psi_{T}|\mathcal{O}|\psi_{T}\rangle}{\langle\psi_{T}|\psi_{T}\rangle}}=\frac{\langle\mathcal{O}\rangle_{m}^{2}}{\langle\mathcal{O}\rangle_{v}}\;,$
(3.26)
where $\langle\mathcal{O}\rangle_{v}$ is the variational estimator. The
accuracy of the extrapolation method is closely related to the trial wave
function used in the variational calculation and on the accuracy of the DMC
sampling technique.
For a many-body system, if no constraint is imposed, $H$ has both symmetric
and antisymmetric eigenstates with respect to particle exchange. It can be
proven [215] that the lowest energy solution, and hence the state projected by
imaginary time propagation, is always symmetric. Moreover, in the DMC
algorithm, the walkers distribution is sampled through the wave function, that
must be positive defined in the whole configuration space for the
probabilistic interpretation to be applicable. The projection algorithm
described above is thus referred to Boson systems only. The extension for
Fermion systems is reported in § 3.1.2.
#### 3.1.1 Importance Sampling
As discussed in the previous section, the basic version of the DMC algorithm
is rather inefficient because the weight term of Eq. (3.21) could suffer of
very large fluctuations. Indeed, because the Brownian diffusive process
ignores the shape of the potential, there is nothing that prevents two
particles from moving very close to each other, even in presence of an hard-
core repulsive potential.
The _importance function_ techniques [212, 213, 214] mitigates this problem by
using an appropriate importance function $\psi_{I}(R)$ (which is often, but
not necessarily, the same $\psi_{T}(R)$ used for the projection) to guide the
diffusive process. The idea is to multiply Eq. (3.9) by $\psi_{I}(R)$
$\displaystyle\psi_{I}(R)\psi(R,\tau+d\tau)=\int
dR^{\prime}\,G(R,R^{\prime},d\tau)\frac{\psi_{I}(R)}{\psi_{I}(R^{\prime})}\psi_{I}(R^{\prime})\psi(R^{\prime},\tau)\;,$
(3.27)
and define a new propagator
$\displaystyle\widetilde{G}(R,R^{\prime},d\tau)=G(R,R^{\prime},d\tau)\frac{\psi_{I}(R)}{\psi_{I}(R^{\prime})}\;,$
(3.28)
and a new function
$\displaystyle f(R,\tau)=\psi_{I}(R)\psi(R,\tau)\;,$ (3.29)
such that
$\displaystyle f(R,\tau+d\tau)=\int
dR^{\prime}\,\widetilde{G}(R,R^{\prime},d\tau)\,f(R^{\prime},\tau)\;.$ (3.30)
$f(R,\tau)$ represents the new probability density from which sample the
walker distribution. If $\psi_{I}(R)$ is suitably chosen, for example to be
small in the region where the potential presents the hard-core, then
$f(R,\tau)$ contains more information than the original $\psi(R,\tau)$, being
correlated to the potential by construction, and thus there is an improvement
in the quality of the DMC sampling and a reduction of the fluctuations of the
weight (3.21).
By inserting the new propagator $\widetilde{G}(R,R^{\prime},d\tau)$ in Eq.
(3.18) and expanding near $R^{\prime}$, it is possible to show (see for
instance Refs. [214]) that the integration gives an additional drift term in
$G_{0}(R,R^{\prime},d\tau)$
$\displaystyle
G_{0}(R,R^{\prime},d\tau)\rightarrow\widetilde{G}_{0}(R,R^{\prime},d\tau)=\left(\frac{1}{4\pi
Dd\tau}\right)^{\frac{3\mathcal{N}}{2}}\operatorname{e}^{-\frac{(R-R^{\prime}-v_{d}(R^{\prime})Dd\tau)^{2}}{4Dd\tau}}\;,$
(3.31)
where
$\displaystyle\bm{v}_{d}(R)=2\frac{\bm{\nabla}\psi_{I}(R)}{\psi_{I}(R)}\;,$
(3.32)
is a $3\mathcal{N}$ dimensional _drift velocity_ that drives the free
diffusion. The branching factor of Eq. (3.21) modifies in
$\displaystyle\omega_{i}\rightarrow\widetilde{\omega}_{i}=\operatorname{e}^{-\left(\frac{E_{L}(R)+E_{L}(R^{\prime})}{2}-E_{T}\right)d\tau}\;,$
(3.33)
where the potential energy is replaced by the _local energy_
$\displaystyle E_{L}(R)=\frac{H\psi_{I}(R)}{\psi_{I}(R)}\;.$ (3.34)
If the importance function is sufficiently accurate, the local energy remains
close to the ground-state energy throughout the imaginary time evolution and
the population of walkers is not subject to large fluctuations.
Going back to the imaginary time dependent Schrödinger equation, it is
possible to show (details can be found in Refs. [214]) that by multiplying Eq.
(3.8) by $\psi_{I}(R)$ we obtain a non homogenous Fokker-Plank equation for
$f(R,\tau)$
$\displaystyle\\!\\!-\frac{\partial}{\partial\tau}f(R,\tau)=$
$\displaystyle-\frac{\hbar^{2}}{2m}\nabla^{2}f(R,\tau)+\frac{\hbar^{2}}{2m}\bm{\nabla}\\!\cdot\\!\Bigl{[}\bm{v}_{d}(R)f(R,\tau)\Bigr{]}+E_{L}(R)f(R,\tau)\;,$
(3.35)
for which the corresponding Green’s function is given by the two terms of Eqs.
(3.31) and (3.33).
The DMC algorithm including the importance sampling procedure is still the
same described in § 3.1, where now the coordinates of the walkers are diffused
by the Brownian motion and guided by the drift velocity
$\displaystyle R=R^{\prime}+\xi+\bm{v}_{d}Dd\tau\;,$ (3.36)
and the branching process is given by the local energy through the weight
(3.33). The expectation values are still calculated by means of Eq. (3.24) but
now the sampling function $\psi(R,\tau)$ is replaced by $f(R,\tau)$.
#### 3.1.2 Sign Problem
As discussed in § 3.1.1, the standard DMC algorithm applies to positive
defined wave function and the result of the imaginary time projection is a
nodeless function. The ground state of a Fermionic system is instead described
by an antisymmetric wave function, to which a probability distribution
interpretation cannot be given. Moreover, the search for an antisymmetric
ground state $|\psi_{0}^{A}\rangle$ corresponds to the search for an excited
state of the many-body Hamiltonian with eigenvalue
$\displaystyle E_{0}^{A}>E_{0}^{S}\;,$ (3.37)
where $E_{0}^{S}$ and $E_{0}^{A}$ are the ground state energies for the
Bosonic and the Fermionic system.
If no constraint is imposed, the Hamiltonian has both eigenstates that are
symmetric and antisymmetric with respect to particle exchange. We can thus
rewrite Eq. (3.4) by separating Bosonic and Fermionic components:
$\displaystyle|\psi(\tau)\rangle=\sum_{n=0}^{\infty}c_{n}^{S}\operatorname{e}^{-(E_{n}^{S}-E_{T})\tau}|\varphi_{n}^{S}\rangle+\sum_{n=0}^{\infty}c_{n}^{A}\operatorname{e}^{-(E_{n}^{A}-E_{T})\tau}|\varphi_{n}^{A}\rangle\;.$
(3.38)
If we want to naively apply the standard DMC algorithm to project out the
Fermionic ground state, we need to propagate the trial wave function for long
imaginary time taking $E_{0}^{A}$ as energy reference. If the Fermionic ground
state is not degenerate, i.e. $E_{n}^{A}\geq E_{n-1}^{A}>E_{0}^{A}$, in the
limit $\tau\rightarrow\infty$ we have
$\displaystyle\lim_{\tau\rightarrow\infty}|\psi(\tau)\rangle=\lim_{\tau\rightarrow\infty}\sum_{n}c_{n}^{S}\operatorname{e}^{-(E_{n}^{S}-E_{0}^{A})\tau}|\varphi_{n}\rangle+c_{0}^{A}|\varphi_{0}^{A}\rangle\;,$
(3.39)
where at least for $E_{0}^{S}$ the Bosonic part diverges due to the condition
(3.37). However, the exponentially growing component along the symmetric
ground state does not affect the expectation of the Hamiltonian. Indeed,
during the evaluation of the integral (3.23) on an antisymmetric trial wave
function $\psi_{T}^{A}(R)$, the symmetric components of $\psi(R,\tau)$ vanish
by orthogonality and in the limit of infinite imaginary time the energy
converges to exact eigenvalue $E_{0}^{A}$. However, the orthogonality
cancellation of the Bosonic terms does not apply to the calculation of the DMC
variance for the antisymmetric energy expectation value $\langle
E_{0}^{A}\rangle$
$\displaystyle\sigma^{2}_{E_{0}^{A}}=\left|\langle
H\rangle_{\psi_{T}^{A}}^{2}-\langle H^{2}\rangle_{\psi_{T}^{A}}\right|\;,$
(3.40)
where the second term diverges. We are left thus with an exact eigenvalue
affected by an exponentially growing statistical error. The signal to noise
ratio exponentially decays. This is the well known _sign problem_ and it
represents the main limit to the straightforward application of the DMC
algorithm to Fermion systems.
In order to extend the DMC method to systems described by antisymmetric wave
functions, it is possible to artificially split the configuration space in
regions where the trial wave function does not change sign. The multi
dimensional surface where the trial wave function vanishes, denoted as _nodal
surface_ , can be used to constrain the diffusion of the walkers: whenever a
walker crosses the nodal surface it is dropped from the calculation. In such a
way only the configurations that diffuse according to region of the wave
function with definite sign are taken into account. The problem reduces thus
to a standard DMC in the subsets of the configuration space delimited by the
nodal surface. This approximate algorithm is called _fixed-node_ [216, 212,
217] and it can be proven that it always provides an upper bound to the true
Fermionic ground state.
The sign problem appears for both real and complex antisymmetric wave
functions. The latter is the case of nuclear Hamiltonians. As proposed by
Zhang _et al._ [218, 219, 220], the _constrained path_ approximation can be
used to deal with the sign problem for complex wave functions. The general
idea is to constraining the path of walkers to regions where the real part of
the overlap with the wave function is positive. If we consider a complex
importance function $\psi_{I}(R)$, in order to keep real the coordinates space
of the system, the drift term in Eq. (3.31) must be real. A suitable choice
for the drift velocity is thus:
$\displaystyle\bm{v}_{d}(R)=2\frac{\bm{\nabla}\operatorname{Re}\left[\psi_{I}(R)\right]}{\operatorname{Re}\left[\psi_{I}(R)\right]}\;.$
(3.41)
Consistently, a way to eliminate the decay of the signal to noise ratio
consists in requiring that the real part of the overlap of each walker with
the importance function must keep the same sign
$\displaystyle\frac{\operatorname{Re}\left[\psi_{I}(R)\right]}{\operatorname{Re}\left[\psi_{I}(R^{\prime})\right]}>0\;,$
(3.42)
where $R$ and $R^{\prime}$ denote the coordinates of the system after and
before the diffusion of a time step. When this condition is violate, i.e. when
the overlap between the importance function and the walker after a diffusive
step changes sign, the walker is dropped. In these scheme, the ground state
expectation value of an observable $\mathcal{O}$ (Eq. (3.24)) is given by
$\displaystyle\langle\mathcal{O}\rangle=\frac{\sum_{\\{R\\}}\mathcal{O}\operatorname{Re}\left[\psi_{T}(R)\right]}{\sum_{\\{R\\}}\operatorname{Re}\left[\psi_{T}(R)\right]}\;.$
(3.43)
Another approach to deal with the complex sign problem is the _fixed phase_
approximation, originally introduced for systems whose Hamiltonian contains a
magnetic field [221]. Let write a complex wave function as
$\displaystyle\psi(R)=\left|\psi(R)\right|\operatorname{e}^{i\phi(R)}\;,$
(3.44)
where $\phi(R)$ is the phase of $\psi(R)$, and rewrite the drift velocity as
$\displaystyle\bm{v}_{d}(R)=2\frac{\bm{\nabla}\left|\psi_{I}(R)\right|}{\left|\psi_{I}(R)\right|}=2\operatorname{Re}\left[\frac{\bm{\nabla}\psi_{I}(R)}{\psi_{I}(R)}\right]\;.$
(3.45)
With this choice, the weight for the branching process becomes
$\displaystyle\widetilde{\omega}_{i}$
$\displaystyle=\exp\Bigg{\\{}-\Bigg{[}\frac{1}{2}\Bigg{(}-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}|\psi_{I}(R)|}{|\psi_{I}(R)|}+\frac{V\psi_{I}(R)}{\psi_{I}(R)}$
$\displaystyle\quad-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}|\psi_{I}(R^{\prime})|}{|\psi_{I}(R^{\prime})|}+\frac{V\psi_{I}(R^{\prime})}{\psi_{I}(R^{\prime})}\Bigg{)}-E_{T}\Bigg{]}d\tau\Bigg{\\}}\times\frac{\left|\psi_{I}(R^{\prime})\right|}{\left|\psi_{I}(R)\right|}\frac{\psi_{I}(R)}{\psi_{I}(R^{\prime})}\;,$
(3.46)
which is the usual importance sampling factor as in Eq. (3.33) multiplied by
an additional factor that corrects for the particular choice of the drift.
Using Eq. (3.44), the last term of the previous equation can be rewritten as
$\displaystyle\frac{\left|\psi_{I}(R^{\prime})\right|}{\left|\psi_{I}(R)\right|}\frac{\psi_{I}(R)}{\psi_{I}(R^{\prime})}=\operatorname{e}^{i[\phi_{I}(R)-\phi_{I}(R^{\prime})]}\;.$
(3.47)
The so called “fixed phase” approximation is then realized by constraining the
walkers to have the same phase as the importance function $\psi_{I}(R)$. It
can be applied by keeping the real part of the last expression. In order to
preserve the normalization, one has to consider an additional term in the
Green’s function due to the phase, that must be added to the weight:
$\displaystyle\exp\Bigg{[}-\frac{\hbar^{2}}{2m}\Bigl{(}\bm{\nabla}\phi(R)\Bigr{)}^{2}d\tau\Bigg{]}\;.$
(3.48)
This factor can be included directly in $\widetilde{\omega}_{i}$ considering
the following relation:
$\displaystyle\operatorname{Re}\left[\frac{\nabla^{2}\psi_{I}(R)}{\psi_{I}(R)}\right]=\frac{\nabla^{2}\left|\psi_{I}(R)\right|}{\left|\psi_{I}(R)\right|}-\Bigl{(}\bm{\nabla}\phi(R)\Bigr{)}^{2}\;.$
(3.49)
Thus, by keeping the real part of the kinetic energy in Eq. (3.46), the
additional weight term given by the fixed phase approximation is automatically
included. The calculation of expectation values is given now by
$\displaystyle\langle\mathcal{O}\rangle=\sum_{\\{R\\}}\operatorname{Re}\left[\frac{\mathcal{O}\psi_{T}(R)}{\psi_{T}(R)}\right]\;,$
(3.50)
i.e. by the evaluation of the real part of a local operator. This is of
particular interest for the technical implementation of the DMC algorithm. As
we will see in § 3.2.4, when dealing with Fermions the wave function can be
written as a Slater determinant of single particle states. It can be shown
(see Appendix A.2) that the evaluation of local operators acting on Slater
determinants can be efficiently implemented by means of the inverse matrix of
the determinant. The fixed phase approximation allows thus to deal with the
Fermion sign problem and also provides a natural scheme to implement the DMC
method. Moreover, the above derivation can be extended to operators other than
the kinetic energy. For example, when dealing with nuclear Hamiltonians like
(2.2), spin and isospin expectation values can be evaluated by taking the real
part of local spin and isospin operators calculated on the Slater determinant.
This is actually the standard way to treat the spin-isospin dependent
components of the nuclear Hamiltonian in the Auxiliary Field Diffusion Monte
Carlo (see § 3.2).
The constrained path and the fixed phase prescriptions are both approximations
introduced to deal with the sign problem for complex wave functions. In
principle they should yield similar results if the importance function is
close enough to the real ground state of the system. Accurate $\psi_{I}(R)$
are thus needed. An additional important observation is that the DMC algorithm
with the constrained path approximation does not necessarily provide an upper
bound in the calculation of energy [222, 223]. Moreover, it has not been
proven that the fixed phase approximation gives an upper bound to the real
energy. Thus, the extension of the DMC algorithm to Fermion systems described
by complex wave functions does not obey to the Rayleigh-Ritz variational
principle. Further details on the fixed node, constrained path and fixed phase
approximations can be found in the original papers and an exhaustive
discussion is reported in the Ph.D. thesis of Armani [224].
#### 3.1.3 Spin-isospin degrees of freedom
If we want to study a nuclear many-body system described by the Hamiltonian
(2.2), we need to include also the spin-isospin degrees of freedom in the
picture. In order to simplify the notation, in the next with $A$ we will refer
to the number of nucleons. Starting from § 3.2.4 we will restore the
convention $A=\mathcal{N}_{N}+\mathcal{N}_{\Lambda}$. The typical trial many-
body wave function used in DMC calculation for nuclear systems takes the form
[136, 223]
$\displaystyle|\psi_{T}\rangle=\mathcal{S}\left[\prod_{i<j}\left(1+U_{ij}+\sum_{k}U_{ijk}\right)\right]\prod_{i<j}f_{c}(r_{ij})|\Phi_{A}\rangle\;,$
(3.51)
where $f_{c}(r_{ij})$ is a central (mostly short ranged repulsion)
correlation, $U_{ij}$ are non commuting two-body correlations induced by
$v_{ij}$ (that typically takes the same form of Eq. (2.16) for $p=2,\ldots,6$)
and $U_{ijk}$ is a simplified three-body correlation from $v_{ijk}$.
$|\Phi_{A}\rangle$ is the one-body part of the trial wave function that
determines the quantum numbers of the states and it is fully antisymmetric.
The central correlation is symmetric with respect to particle exchange and the
symmetrization operator $\mathcal{S}$ acts on the operatorial correlation part
of $|\psi_{T}\rangle$ in order to make the complete trial wave function
antisymmetric. The best trial wave function from which (3.51) is derived,
includes also spin-orbit and the full three-body correlations and it is used
in VMC calculations. See Refs. [225, 223].
Given $A$ nucleons ($Z$ protons, $A-Z$ neutrons), the trial wave function is a
complex vector in spin-isospin space with dimension
$\mathcal{N}_{S}\times\mathcal{N}_{T}$, where $\mathcal{N}_{S}$ is the number
of spin states and $\mathcal{N}_{T}$ the number of isospin states:
$\displaystyle\mathcal{N}_{S}=2^{A}\quad\quad\quad\mathcal{N}_{T}=\left(\begin{array}[]{c}A\\\
Z\end{array}\right)=\frac{A!}{Z!(A-Z)!}\;.$ (3.54)
For example, the wave function of an $A=3$ system has 8 spin components and,
considering the physical systems for $Z=1$ $\left({}^{3}\text{H}\right)$ or
$Z=2$ $\left({}^{3}\text{He}\right)$, 3 isospin states, thus a spin-isospin
structure with 24 entries. Using the notation of Ref. [136], we can write the
spin part of an $A=3$ wave function as a complex 8-vector (ignore
antisymmetrization)
$\displaystyle|\Phi_{A=3}\rangle=\left(\begin{array}[]{c}a_{\uparrow\uparrow\uparrow}\\\
a_{\uparrow\uparrow\downarrow}\\\ a_{\uparrow\downarrow\uparrow}\\\
a_{\uparrow\downarrow\downarrow}\\\ a_{\downarrow\uparrow\uparrow}\\\
a_{\downarrow\uparrow\downarrow}\\\ a_{\downarrow\downarrow\uparrow}\\\
a_{\downarrow\downarrow\downarrow}\end{array}\right)\quad\quad\text{with}\quad
a_{\uparrow\downarrow\uparrow}=\langle\uparrow\downarrow\uparrow|\Phi_{A=3}\rangle\;.$
(3.63)
The potentials ($v_{ij}$, $v_{ijk}$) and correlations ($U_{ij}$, $U_{ijk}$)
involve repeated operations on $|\psi_{T}\rangle$ but the many-body spin-
isospin space is closed under the action of the operators contained in the
Hamiltonian. As an example, consider the term
$\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$:
$\displaystyle\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$
$\displaystyle=2\left(\sigma_{i}^{+}\sigma_{j}^{-}+\sigma_{i}^{-}\sigma_{j}^{+}\right)+\sigma_{i}^{z}\sigma_{j}^{z}\;,$
$\displaystyle=2\,\mathcal{P}_{ij}^{\sigma}-1\;,$
$\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&2&0\\\ 0&2&-1&0\\\
0&0&0&1\end{array}\right)\quad\text{acting
on}\quad\left(\begin{array}[]{c}\uparrow\uparrow\\\ \uparrow\downarrow\\\
\downarrow\uparrow\\\ \downarrow\downarrow\end{array}\right)\;.$ (3.72)
The $\mathcal{P}_{ij}^{\sigma}$ exchanges the spin $i$ and $j$, so the
operator $\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ does not mix different isospin
components and acts on different, non contiguous, 4-element blocks of
$|\Phi_{A=3}\rangle$. For $i=2$ and $j=3$ we have for example:
$\displaystyle\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}\,|\Phi_{A=3}\rangle=\left(\begin{array}[]{c}a_{\uparrow\uparrow\uparrow}\\\
2a_{\uparrow\downarrow\uparrow}-a_{\uparrow\uparrow\downarrow}\\\
2a_{\uparrow\uparrow\downarrow}-a_{\uparrow\downarrow\uparrow}\\\
a_{\uparrow\downarrow\downarrow}\\\ a_{\downarrow\uparrow\uparrow}\\\
2a_{\downarrow\downarrow\uparrow}-a_{\downarrow\uparrow\downarrow}\\\
2a_{\downarrow\uparrow\downarrow}-a_{\downarrow\downarrow\uparrow}\\\
a_{\downarrow\downarrow\downarrow}\end{array}\right)\;.$ (3.81)
The action of pair operators on $|\psi_{T}\rangle$, that are the most
computationally expensive, results thus in a sparse matrix of (non contiguous)
$4\times 4$ blocks in the $A$-body problem.
In the Green Function Monte Carlo, which slightly differs from the DMC in the
way the propagator is treated, each of the $2^{A}\frac{A!}{Z!(A-Z)!}$ spin-
isospin configurations undergoes to the imaginary time evolution of Eq. (3.9).
The propagation is now acting on the component $a_{\alpha}$, being $\alpha$
the spin-isospin index,
$\displaystyle a_{\alpha}(R,\tau+d\tau)=\sum_{\beta}\int
dR^{\prime}\,G_{\alpha\beta}(R,R^{\prime},d\tau)\,a_{\beta}(R^{\prime},\tau)\;,$
(3.82)
where the Green’s function is a matrix function of $R$ and $R^{\prime}$ in
spin-isospin space, defined as
$\displaystyle G_{\alpha\beta}(R,R^{\prime},d\tau)=\langle
R,\alpha|\operatorname{e}^{-(H-E_{T})d\tau}|R^{\prime},\beta\rangle\;.$ (3.83)
Due to the the factorial growth in the number of components of the wave
function, GFMC cannot deal with systems having a large number of nucleons,
like medium-heavy nuclei or nuclear matter. Standard GFMC calculations are
indeed limited up to 12 nucleons [17, 18, 19] or 16 neutrons [20].
### 3.2 Auxiliary Field Diffusion Monte Carlo
The AFDMC algorithm was originally introduced by Schmidt and Fantoni [29] in
order to deal in an efficient way with spin-dependent Hamiltonians. Many
details on the AFDMC method can be found in Refs. [37, 33, 226, 30, 31, 38,
224, 227, 214]. The main idea is to move from the many particle wave function
of the DMC or GFMC to a single particle wave function. In this representation,
going back to the example of the previous section, the spin part of an $A=3$
wave function becomes a tensor product of 3 single particle spin states
(ignore antisymmetrization):
$\displaystyle|\Phi_{A=3}\rangle$
$\displaystyle=\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{2\uparrow}\\\
a_{2\downarrow}\end{array}\right)_{2}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{3\uparrow}\\\
a_{3\downarrow}\end{array}\right)_{3}\quad\text{with}\quad
a_{k\uparrow}=\,_{{}_{k}\,}\\!\langle\uparrow|\Phi_{A=3}\rangle\;.$ (3.90)
Taking also into account the isospin degrees of freedom, each single particle
state becomes a complex 4-vector and the total number of entries for
$|\Phi_{A=3}\rangle$ is thus 12, half of the number for the full DMC function
of Eq. (3.63). In the general case, the dimension of the multicomponent vector
describing a system with $A$ nucleons scale as $4A$. So, in this picture, the
computational cost for the evaluation of the wave function is drastically
reduced compared to the DMC-GFMC method when the number of particles becomes
large.
The problem of the single particle representation is that it is not closed
with respect to the application of quadratic spin (isospin) operators. As done
in the previous section (Eq. (3.81)), consider the operator
$\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}=2\,\mathcal{P}_{23}^{\sigma}-1$ acting on
$|\Phi_{A=3}\rangle$:
$\displaystyle\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}\,|\Phi_{A=3}\rangle$
$\displaystyle=2\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{3\uparrow}\\\
a_{3\downarrow}\end{array}\right)_{2}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{2\uparrow}\\\
a_{2\downarrow}\end{array}\right)_{3}$ (3.97)
$\displaystyle\phantom{=}-\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{2\uparrow}\\\
a_{2\downarrow}\end{array}\right)_{2}\\!\\!\otimes\\!\left(\begin{array}[]{c}a_{3\uparrow}\\\
a_{3\downarrow}\end{array}\right)_{3}\;.$ (3.104)
There is no way to express the result as a single particle wave function of
the form (3.90). At each time step, the straightforward application of the DMC
algorithm generates a sum of single particle wave functions. The number of
these functions will grows very quickly during the imaginary time evolution,
destroying the gain in computational time obtained using a smaller
multicomponent trial wave function.
In order to keep the single particle wave function representation and overcome
this problem, the AFDMC makes use of the Hubbard-Stratonovich transformation
$\displaystyle\operatorname{e}^{-\frac{1}{2}\lambda\mathcal{O}^{2}}=\frac{1}{\sqrt{2\pi}}\int\\!dx\operatorname{e}^{-\frac{x^{2}}{2}+\sqrt{-\lambda}\,x\mathcal{O}}\;,$
(3.105)
to linearize the quadratic dependence on the spin-isospin operators by adding
the integration over a new variable $x$ called _auxiliary field_. It is indeed
possible to show that the single particle wave function is closed with respect
to the application of a propagator containing linear spin-isospin operators at
most:
$\displaystyle\operatorname{e}^{-\mathcal{O}_{j}d\tau}|\Phi_{A}\rangle$
$\displaystyle=\operatorname{e}^{-\mathcal{O}_{j}d\tau}\bigotimes_{i}\left(\begin{array}[]{c}a_{i\uparrow}\\\
a_{i\downarrow}\end{array}\right)_{i}\;,$ (3.108)
$\displaystyle=\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\otimes\cdots\otimes\,\operatorname{e}^{-\mathcal{O}_{j}d\tau}\left(\begin{array}[]{c}a_{j\uparrow}\\\
a_{j\downarrow}\end{array}\right)_{j}\\!\otimes\cdots\otimes\left(\begin{array}[]{c}a_{A\uparrow}\\\
a_{A\downarrow}\end{array}\right)_{A}\;,\quad\quad$ (3.115)
$\displaystyle=\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\otimes\cdots\otimes\left(\begin{array}[]{c}\widetilde{a}_{j\uparrow}\\\
\widetilde{a}_{j\downarrow}\end{array}\right)_{j}\\!\otimes\cdots\otimes\left(\begin{array}[]{c}a_{A\uparrow}\\\
a_{A\downarrow}\end{array}\right)_{A}\;,$ (3.122)
where, working on 2-component spinors, $\mathcal{O}_{j}$ can be a $2\times 2$
spin or isospin matrix. If we are dealing with the full 4-component spinor,
$\mathcal{O}_{j}$ can be an extended $4\times 4$ spin, isospin or
isospin$\,\otimes\,$spin matrix. To get this result we have used the fact that
the operator $\mathcal{O}_{j}$ is the representation in the $A$-body tensor
product space of a one-body operator:
$\displaystyle\mathcal{O}_{j}\equiv\mathbb{I}_{1}\otimes\cdots\otimes\mathcal{O}_{j}\otimes\cdots\otimes\mathbb{I}_{A}\;.$
(3.123)
Limiting the study to quadratic spin-isospin operators and making use of the
Hubbard-Stratonovich transformation, it is thus possible to keep the single
particle wave function representation over all the imaginary time evolution.
This results in a reduced computational time for the propagation of the wave
function compared to GFMC, that allows us to simulate larger systems, from
medium-heavy nuclei to the infinite matter. In the next we will see in detail
how the AFDMC works on the Argonne V6 like potentials (§ 3.2.1), and how it is
possible to include also spin-orbit (§ 3.2.2) and three-body (§ 3.2.3) terms
for neutron systems. Finally the extension of AFDMC for hypernuclear systems
(§ 3.2.5) is presented.
#### 3.2.1 Propagator for nucleons: $\bm{\sigma}$,
$\bm{\sigma}\cdot\bm{\tau}$ and $\bm{\tau}$ terms
Consider the first six components of the Argonne $NN$ potential of Eq. (2.16).
They can be conveniently rewritten as a sum of a spin-isospin independent and
a spin-isospin dependent term
$\displaystyle
V_{NN}=\sum_{i<j}\sum_{p=1,6}v_{p}(r_{ij})\,\mathcal{O}_{ij}^{\,p}=V_{NN}^{SI}+V_{NN}^{SD}\;,$
(3.124)
where
$\displaystyle V_{NN}^{SI}$ $\displaystyle=\sum_{i<j}v_{1}(r_{ij})\;,$ (3.125)
and
$\displaystyle V_{NN}^{SD}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\,A_{i\alpha,j\beta}^{[\sigma]}\,\sigma_{j\beta}+\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta\gamma}\tau_{i\gamma}\,\sigma_{i\alpha}\,A_{i\alpha,j\beta}^{[\sigma\tau]}\,\sigma_{j\beta}\,\tau_{j\gamma}\;$
$\displaystyle\,+\frac{1}{2}\sum_{i\neq
j}\sum_{\gamma}\tau_{i\gamma}\,A_{ij}^{[\tau]}\,\tau_{j\gamma}\;.$ (3.126)
The $3A\times 3A$ matrices $A^{[\sigma]}$, $A^{[\sigma\tau]}$ and the $A\times
A$ matrix $A^{[\tau]}$ are real and symmetric under Cartesian component
interchange $\alpha\leftrightarrow\beta$, under particle exchange
$i\leftrightarrow j$ and fully symmetric with respect to the exchange
$i\alpha\leftrightarrow j\beta$. They have zero diagonal (no self interaction)
and contain proper combinations of the components of AV6 (Latin indices are
used for the nucleons, Greek ones refer to the Cartesian components of the
operators):
$\displaystyle A_{ij}^{[\tau]}$ $\displaystyle=v_{2}\left(r_{ij}\right)\;,$
$\displaystyle A_{i\alpha,j\beta}^{[\sigma]}$
$\displaystyle=v_{3}\left(r_{ij}\right)\delta_{\alpha\beta}+v_{5}\left(r_{ij}\right)\left(3\,\hat{r}_{ij}^{\alpha}\,\hat{r}_{ij}^{\beta}-\delta_{\alpha\beta}\right)\;,$
(3.127) $\displaystyle A_{i\alpha,j\beta}^{[\sigma\tau]}$
$\displaystyle=v_{4}\left(r_{ij}\right)\delta_{\alpha\beta}+v_{6}\left(r_{ij}\right)\left(3\,\hat{r}_{ij}^{\alpha}\,\hat{r}_{ij}^{\beta}-\delta_{\alpha\beta}\right)\;,$
that come from the decomposition of the operators in Cartesian coordinates:
$\displaystyle\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$
$\displaystyle=\sum_{\alpha\beta}\sigma_{i\alpha}\,\sigma_{j\beta}\,\delta_{\alpha\beta}\;,$
(3.128) $\displaystyle S_{ij}$
$\displaystyle=\sum_{\alpha\beta}\left(3\,\sigma_{i\alpha}\,\hat{r}_{ij}^{\alpha}\,\sigma_{j\beta}\,\hat{r}_{ij}^{\beta}-\sigma_{i\alpha}\,\sigma_{j\beta}\,\delta_{\alpha\beta}\right)\;.$
(3.129)
Being real and symmetric, the $A$ matrices have real eigenvalues and real
orthogonal eigenstates
$\displaystyle\sum_{j\beta}A_{i\alpha,j\beta}^{[\sigma]}\,\psi_{n,j\beta}^{[\sigma]}$
$\displaystyle=\lambda_{n}^{[\sigma]}\,\psi_{n,i\alpha}^{[\sigma]}\;,$
$\displaystyle\sum_{j\beta}A_{i\alpha,j\beta}^{[\sigma\tau]}\,\psi_{n,j\beta}^{[\sigma\tau]}$
$\displaystyle=\lambda_{n}^{[\sigma\tau]}\,\psi_{n,i\alpha}^{[\sigma\tau]}\;,$
(3.130) $\displaystyle\sum_{j}A_{ij}^{[\tau]}\,\psi_{n,j}^{[\tau]}$
$\displaystyle=\lambda_{n}^{[\tau]}\,\psi_{n,i}^{[\tau]}\;.$
Let us expand $\sigma_{i\alpha}$ on the complete set of eigenvectors
$\psi_{n,i\alpha}^{[\sigma]}$ of the matrix $A_{i\alpha,j\beta}^{[\sigma]}$ :
$\displaystyle\sigma_{i\alpha}=\sum_{n}c_{n}^{[\sigma]}\,\psi_{n,i\alpha}^{[\sigma]}=\sum_{n}\left(\sum_{j\beta}\psi_{n,j\beta}^{[\sigma]}\,\sigma_{j\beta}\right)\psi_{n,i\alpha}^{[\sigma]}\;,$
(3.131)
where we have used the orthogonality condition
$\displaystyle\sum_{i\alpha}\psi_{n,i\alpha}^{[\mathcal{O}]}\,\psi_{m,i\alpha}^{[\mathcal{O}]}=\delta_{nm}\;.$
(3.132)
Using Eq. (3.131) we can recast the first term of Eq. (3.126) in the following
form:
$\displaystyle\frac{1}{2}\sum_{i\alpha,j\beta}\sigma_{i\alpha}\,A_{i\alpha,j\beta}^{[\sigma]}\,\sigma_{j\beta}=$
$\displaystyle\,=\frac{1}{2}\sum_{i\alpha,j\beta}\left\\{\left[\sum_{n}\left(\sum_{k\gamma}\sigma_{k\gamma}\,\psi_{n,k\gamma}^{[\sigma]}\right)\psi_{n,i\alpha}^{[\sigma]}\right]A_{i\alpha,j\beta}^{[\sigma]}\left[\sum_{m}\left(\sum_{k\gamma}\sigma_{k\gamma}\,\psi_{m,k\gamma}^{[\sigma]}\right)\psi_{m,j\beta}^{[\sigma]}\right]\right\\}\;,$
$\displaystyle\,=\frac{1}{2}\sum_{i\alpha}\left\\{\left[\sum_{n}\left(\sum_{k\gamma}\sigma_{k\gamma}\,\psi_{n,k\gamma}^{[\sigma]}\right)\psi_{n,i\alpha}^{[\sigma]}\right]\left[\sum_{m}\lambda_{m}^{[\sigma]}\left(\sum_{k\gamma}\sigma_{k\gamma}\,\psi_{m,k\gamma}^{[\sigma]}\right)\psi_{m,i\alpha}^{[\sigma]}\right]\right\\}\;,$
$\displaystyle\,=\frac{1}{2}\sum_{n}\left(\sum_{k\gamma}\sigma_{k\gamma}\,\psi_{n,k\gamma}^{[\sigma]}\right)^{2}\\!\lambda_{n}^{[\sigma]}\;.$
(3.133)
Similar derivation can be written for the terms
$\tau_{i\gamma}\,\sigma_{i\alpha}$ and $\tau_{i\gamma}$ and we can define a
new set of operators expressed in terms of the eigenvectors of the matrices
$A$:
$\displaystyle\mathcal{O}_{n}^{[\sigma]}$
$\displaystyle=\sum_{j\beta}\sigma_{j\beta}\,\psi_{n,j\beta}^{[\sigma]}\;,$
$\displaystyle\mathcal{O}_{n,\alpha}^{[\sigma\tau]}$
$\displaystyle=\sum_{j\beta}\tau_{j\alpha}\,\sigma_{j\beta}\,\psi_{n,j\beta}^{[\sigma\tau]}\;,$
(3.134) $\displaystyle\mathcal{O}_{n,\alpha}^{[\tau]}$
$\displaystyle=\sum_{j}\tau_{j\alpha}\,\psi_{n,j}^{[\tau]}\;.$
The spin dependent part of the $NN$ interaction can be thus expressed as
follows:
$\displaystyle\\!\\!\\!\\!V_{NN}^{SD}=\frac{1}{2}\sum_{n=1}^{3A}\lambda_{n}^{[\sigma]}\\!\left(\mathcal{O}_{n}^{[\sigma]}\right)^{2}\\!+\frac{1}{2}\sum_{n=1}^{3A}\sum_{\alpha=1}^{3}\lambda_{n}^{[\sigma\tau]}\\!\left(\mathcal{O}_{n\alpha}^{[\sigma\tau]}\right)^{2}\\!+\frac{1}{2}\sum_{n=1}^{A}\sum_{\alpha=1}^{3}\lambda_{n}^{[\tau]}\\!\left(\mathcal{O}_{n\alpha}^{[\tau]}\right)^{2}\,.\\!$
(3.135)
$V_{NN}^{SD}$ is now written in a suitable form for the application of the
Hubbard-Stratonovich transformation of Eq. (3.105). The propagator
$\operatorname{e}^{-V_{NN}^{SD}\,d\tau}$ can be finally recast as:
$\displaystyle\operatorname{e}^{-\frac{1}{2}\sum_{n}\lambda_{n}(\mathcal{O}_{n})^{2}d\tau}$
$\displaystyle=\prod_{n}\operatorname{e}^{-\frac{1}{2}\lambda_{n}(\mathcal{O}_{n})^{2}d\tau}\,+\operatorname{o}\left(d\tau^{2}\right)\;,$
$\displaystyle\simeq\prod_{n}\frac{1}{\sqrt{2\pi}}\int\\!dx_{n}\operatorname{e}^{\frac{-x_{n}^{2}}{2}+\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}}\;,$
(3.136)
where we have used the compact notation $\mathcal{O}_{n}$ to denote the $3A$
$\mathcal{O}_{n}^{[\sigma]}$, the $9A$ $O_{n,\alpha}^{[\sigma\tau]}$ and the
$3A$ $\mathcal{O}_{n,\alpha}^{[\tau]}$ operators including the summation over
the coordinate index $\alpha$ where needed. The first step of the above
equation comes to the fact that in general the operators $\mathcal{O}_{n}$ do
not commute and so, due to Eq. (3.17), the equality is correct only at order
$d\tau^{2}$.
The standard DMC imaginary time propagation of Eq. (3.9) needs to be extended
to the spin-isospin space, as done in the GFMC algorithm via the projection of
Eqs. (3.82) and (3.83). In the AFDMC method, spin-isospin coordinates
$\\{S\\}$ are added to the spacial coordinates $\\{R\\}$, defining a set of
walkers which represents the single-particle wave function to be evolved in
imaginary time:
$\displaystyle\psi(R,S,\tau+d\tau)=\int
dR^{\prime}dS^{\prime}\,G(R,S,R^{\prime},S^{\prime},d\tau)\,\psi(R^{\prime},S^{\prime},\tau)\;.$
(3.137)
Including the integration over the Hubbard-Stratonovich auxiliary fields, the
Auxiliary Field DMC Green’s function reads (recall Eqs. (3.18) and (3.19)):
$\displaystyle G(R,S,R^{\prime},S^{\prime},d\tau)$ $\displaystyle=\langle
R,S|\operatorname{e}^{-(T+V-E_{T})d\tau}|R^{\prime},S^{\prime}\rangle\;,$
$\displaystyle\simeq\left(\frac{1}{4\pi
Dd\tau}\right)^{\frac{3\mathcal{N}}{2}}\\!\operatorname{e}^{-\frac{(R-R^{\prime})^{2}}{4Dd\tau}}\operatorname{e}^{-\left(\frac{V_{NN}^{SI}(R)+V_{NN}^{SI}(R^{\prime})}{2}-E_{T}\right)d\tau}\times$
$\displaystyle\quad\,\times\langle
S|\prod_{n=1}^{15A}\frac{1}{\sqrt{2\pi}}\int\\!dx_{n}\operatorname{e}^{\frac{-x_{n}^{2}}{2}+\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}}|S^{\prime}\rangle\;,$
(3.138)
Each operator $\mathcal{O}_{n}$ involves the sum over the particle index $j$,
as shown in Eq. (3.134). However, in the $A$-body tensor product space, each
$j$ sub-operator is a one-body operator acting on a different single particle
spin-isospin states, as in Eq. (3.123). Therefore the $j$-dependent terms
commute and we can represent the exponential of the sum over $j$ as a tensor
product of exponentials. The result is that the propagation of a spin-isospin
state $|S^{\prime}\rangle$ turns into a product of independent rotations, one
for each spin-isospin state. Considering just a spin wave function we have for
example:
$\displaystyle\operatorname{e}^{\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}^{[\sigma]}}|S^{\prime}\rangle=$
$\displaystyle\,=\operatorname{e}^{\sqrt{-\lambda_{n}d\tau}\,x_{n}\sum_{\beta}\sigma_{1\beta}\,\psi_{n,1\beta}^{[\sigma]}}\left(\begin{array}[]{c}a_{1\uparrow}\\\
a_{1\downarrow}\end{array}\right)_{1}\\!\otimes\cdots\otimes\operatorname{e}^{\sqrt{-\lambda_{n}d\tau}\,x_{n}\sum_{\beta}\sigma_{A\beta}\,\psi_{n,A\beta}^{[\sigma]}}\left(\begin{array}[]{c}a_{A\uparrow}\\\
a_{A\downarrow}\end{array}\right)_{A}\;,$ (3.143)
$\displaystyle\,=\left(\begin{array}[]{c}\widetilde{a}_{1\uparrow}\\\
\widetilde{a}_{1\downarrow}\end{array}\right)_{1}\\!\otimes\cdots\otimes\left(\begin{array}[]{c}\widetilde{a}_{A\uparrow}\\\
\widetilde{a}_{A\downarrow}\end{array}\right)_{A}\;.$ (3.148)
We can thus propagate spin-isospin dependent wave functions remaining inside
the space of single particle states.
The new coefficients $\widetilde{a}_{j\uparrow}$ and
$\widetilde{a}_{j\downarrow}$ are calculated at each time step for each
$\mathcal{O}_{n}$ operator. For neutron systems, i.e. for two-component
spinors for which only the operator $\mathcal{O}_{n}^{[\sigma]}$ is active,
there exists an explicit expression for these coefficients. Consider the
Landau relations
$\displaystyle\operatorname{e}^{i\,\vec{b}\cdot\vec{\sigma}}$
$\displaystyle=\cos(|\vec{b}|)+i\sin(|\vec{b}|)\frac{\vec{b}\cdot\vec{\sigma}}{|\vec{b}|}\;,$
(3.149) $\displaystyle\operatorname{e}^{\vec{b}\cdot\vec{\sigma}}$
$\displaystyle=\cosh(|\vec{b}|)+\sinh(|\vec{b}|)\frac{\vec{b}\cdot\vec{\sigma}}{|\vec{b}|}\;,$
(3.150)
and identify the $\vec{b}$ vector with
$\displaystyle\vec{b}=\sqrt{|\lambda_{n}|d\tau}\,x_{n}\vec{\psi}_{n,j}^{\,[\sigma]}\quad\quad\quad
b_{\beta}=\sqrt{|\lambda_{n}|d\tau}\,x_{n}\psi_{n,j\beta}^{\,[\sigma]}\;.$
(3.151)
The following expressions for the coefficients of the rotated spinors can be
then written, distinguishing the case $\lambda_{n}<0$ (Eq. (3.154)) and the
case $\lambda_{n}>0$ (Eq. (3.157)):
$\displaystyle\begin{array}[]{l}\widetilde{a}_{j\uparrow}\\!=\\!\\!\Biggl{[}\cosh(|\vec{b}|)+\sinh(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\frac{\psi_{n,jz}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\uparrow}\\!+\sinh(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\\!\\!\Biggl{[}\frac{\psi_{n,jx}^{[\sigma]}-i\,\psi_{n,jy}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\downarrow}\;,\\!\\!\\\\[13.00005pt]
\widetilde{a}_{j\downarrow}\\!=\\!\\!\Biggl{[}\cosh(|\vec{b}|)-\sinh(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\frac{\psi_{n,jz}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\downarrow}\\!+\sinh(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\\!\\!\Biggl{[}\frac{\psi_{n,jx}^{[\sigma]}+i\,\psi_{n,jy}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\uparrow}\;,\\!\\!\end{array}$
(3.154)
$\displaystyle\begin{array}[]{l}\widetilde{a}_{j\uparrow}\\!=\\!\\!\Biggl{[}\cos(|\vec{b}|)+i\,\sin(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\frac{\psi_{n,jz}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\uparrow}\\!+\sin(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\\!\\!\Biggl{[}\frac{i\,\psi_{n,jx}^{[\sigma]}+\psi_{n,jy}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\downarrow}\;,\\!\\!\\\\[13.00005pt]
\widetilde{a}_{j\downarrow}\\!=\\!\\!\Biggl{[}\cos(|\vec{b}|)+i\,\sin(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\frac{\psi_{n,jz}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\downarrow}\\!+\sin(|\vec{b}|)\operatorname{sgn}\\!\left(x_{n}\right)\\!\\!\Biggl{[}\frac{i\,\psi_{n,jx}^{[\sigma]}-\psi_{n,jy}^{[\sigma]}}{|\vec{\psi}_{n,j}^{\,[\sigma]}|}\Biggr{]}a_{j\uparrow}\;.\\!\\!\end{array}$
(3.157)
When we are dealing with the full 4-dimension single particle spinors, the
four coefficients $\widetilde{a}$ do not have an explicit expression. The
exponential of the $\mathcal{O}_{n}$ operators acting on the spinors is
calculated via a diagonalization procedure. Consider the general $4\times 4$
rotation matrix $B_{j}$ and its eigenvectors $\Psi_{m,j}\neq 0$ and
eigenvalues $\mu_{m,j}$:
$\displaystyle
B_{j}\,\Psi_{m,j}=\mu_{m,j}\,\Psi_{m,j}\quad\Rightarrow\quad\Psi_{m,j}^{-1}\,B_{j}\,\Psi_{m,j}=\mu_{m,j}\quad\quad
m=1,\ldots,4\;.$ (3.158)
Using the formal notation $\vec{\Psi}_{j}$ and $\vec{\mu}_{j}$ to denote the
$4\times 4$ matrix of eigenvectors and the 4-dimension vector of eigenvalues,
it is possible to write the action of $\operatorname{e}^{B_{j}}$ on a
4-dimensional single particle spinor $|S^{\prime}\rangle_{j}$ as follows:
$\displaystyle\operatorname{e}^{B_{j}}|S^{\prime}\rangle_{j}$
$\displaystyle=\vec{\Psi}_{j}\vec{\Psi}_{j}^{-1}\operatorname{e}^{B_{j}}\vec{\Psi}_{j}\vec{\Psi}_{j}^{-1}|S^{\prime}\rangle_{j}\;,$
$\displaystyle=\vec{\Psi}_{j}\operatorname{e}^{\vec{\Psi}_{j}^{-1}B_{j}\,\vec{\Psi}_{j}}\vec{\Psi}_{j}^{-1}|S^{\prime}\rangle_{j}\;,$
$\displaystyle=\vec{\Psi}_{j}\operatorname{e}^{\text{diag}\left(\vec{\mu}_{j}\right)}\vec{\Psi}_{j}^{-1}|S^{\prime}\rangle_{j}\;,$
$\displaystyle=\vec{\Psi}_{j}\,\text{diag}\left(\operatorname{e}^{\vec{\mu}_{j}}\right)\vec{\Psi}_{j}^{-1}|S^{\prime}\rangle_{j}\;.$
(3.159)
Each component of the rotated spinor $|\widetilde{S}^{\prime}\rangle_{j}$ is
thus derived from the eigenvectors and eigenvalues of the rotation matrix
$B_{j}$, which is built starting from the $\mathcal{O}_{n}$ operators. Moving
from neutrons to nucleons, i.e. adding the isospin degrees of freedom to the
system, the computational time spent to rotate each single particle spin-
isospin state during the propagation is increased by the time for the
diagonalization of the $4\times 4$ Hubbard-Stratonovich rotation matrices.
However, the total time for the propagation of the wave function as $A$
becomes large, is dominated by the diagonalization of the potential matrices.
Since the cost of this operation goes as the cube of the number of matrix rows
(columns), the AFDMC computational time is proportional to $A^{3}$, which is
much slower than the scaling factor $A!$ of GFMC.
In addition to the diagonalization of the AV6 potential matrices and the
spinor rotation matrices, we have to deal with the evaluation of the integral
over the auxiliary fields $x_{n}$. The easiest way, in the spirit of Monte
Carlo, is to sample the auxiliary fields from the Gaussian of Eq. (3.138),
which is interpreted as a probability distribution. The sampled values are
then used to determine the action of the operators on the spin-isospin part of
the wave function as described above. The integral over all the spin-isospin
rotations induced by the auxiliary fields eventually recovers the action of
the quadratic spin-isospin operators on a trial wave function containing all
the possible good spin-isospin states, as the GFMC one.
In this scheme, the integration over the auxiliary fields is performed jointly
with the integration over the coordinates. This generally leads to a large
variance. The integral of Eq. (3.138) should be indeed evaluated for each
sampled position and not simply estimated “on the fly”. A more refined
algorithm, in which for each sampled configuration the integral over $x_{n}$
is calculated by sampling more than one auxiliary variable, has been tested.
The energy values at convergence are the same for both approaches. However, in
the latter case the variance is much reduced, although the computational time
for each move is increased due to the iteration over the newly sampled
auxiliary points.
As done in the DMC method, see § 3.1.1, we can introduce an importance
function to guide the diffusion in the coordinate space also in the AFDMC
algorithm. The drift term (3.32) is added to the $R-R^{\prime}$ Gaussian
distribution of Eq. (3.138) and the branching weight $\widetilde{\omega}_{i}$
is given by the local energy as in Eq. (3.33). The idea of the importance
sampling can be applied to guide the rotation of the spin-isospin states in
the Hubbard-Stratonovich transformation. This can be done by properly shifting
the Gaussian over the auxiliary fields of Eq. (3.138) by means of a drift term
$\bar{x}_{n}$:
$\displaystyle\operatorname{e}^{-\frac{x_{n}^{2}}{2}+\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}}=\operatorname{e}^{-\frac{(x_{n}-\bar{x}_{n})^{2}}{2}}\operatorname{e}^{\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}}\operatorname{e}^{-\bar{x}_{n}\left(x_{n}-\frac{\bar{x}_{n}}{2}\right)}\;,$
(3.160)
where
$\displaystyle\bar{x}_{n}=\operatorname{Re}\left[\sqrt{-\lambda_{n}d\tau}\langle\mathcal{O}_{n}\rangle_{m}\right]\;,$
(3.161)
and $\langle\mathcal{O}_{n}\rangle_{m}$ is the mixed expectation value of
$\mathcal{O}_{n}$ (Eq. (3.24)) calculated on the old spin-isospin
configurations. The mixed estimator is introduced in order to guide the
rotations, by maximizing the overlap between the walker and the trial
function, which is not generally picked around $x_{n}=0$.
The last factor of Eq. (3.160) can be interpreted as an additional weight term
that has to be included in the total weight. By combining diffusion, rotation
and all the additional factors we can derive two different algorithms.
* _v1_
In the first one, the ratio between the importance functions in the new and
old configurations (see Eq. (3.28)) is kept explicit. However the drifted
Gaussian $\widetilde{G}_{0}(R,R^{\prime},d\tau)$ of Eq. (3.31) is used for the
diffusion in the coordinate space and the drifted Gaussian of Eq. (3.160) for
the sampling of auxiliary fields. The weight for the branching process
$\omega_{i}$ defined in Eq. (3.21) takes then an overall factor
$\displaystyle\frac{\langle\psi_{I}|RS\rangle}{\langle\psi_{I}|R^{\prime}S^{\prime}\rangle}\operatorname{e}^{-\frac{d(R^{\prime})\left[d(R^{\prime})+2(R-R^{\prime})\right]}{4Dd\tau}}\prod_{n}\operatorname{e}^{-\bar{x}_{n}\left(x_{n}-\frac{\bar{x}_{n}}{2}\right)}\;,$
(3.162)
due to the counter terms coming from the coordinate drift
$d(R)=\bm{v}_{d}(R)Dd\tau$ added in the original $G_{0}(R,R^{\prime},d\tau)$
and from the auxiliary field shift $\bar{x}_{n}$.
* _v2_
The second algorithm corresponds the local energy scheme described in § 3.1.1.
Again the coordinates are diffused via the drifted Gaussian
$\widetilde{G}_{0}(R,R^{\prime},d\tau)$ of Eq. (3.31) and the auxiliary fields
are sampled from the shifted Gaussian of Eq. (3.160). The branching weight
$\widetilde{w}_{i}$ is instead given by the local energy as in Eq. (3.33). The
counter terms related to $\bar{x}_{n}$ are automatically included in the
weight because the local energy
$E_{L}(R,S)=\frac{H\psi_{I}(R,S)}{\psi_{I}(R,S)}$ takes now contributions from
all the spin-isospin operators of the full potential $V_{NN}$. Actually, the
term $\operatorname{e}^{-\bar{x}_{n}x_{n}}$ vanishes during the auxiliary
field integration because $x_{n}$ can take positive and negative values. The
term $\frac{\bar{x}_{n}^{2}}{2}$ is nothing but the
$-\frac{1}{2}\lambda_{n}\langle\mathcal{O}_{n}\rangle_{m}^{2}d\tau$
contribution already included in the weight via $E_{L}(R,S)$.
Given the same choice for the drift term, that depends, for example, on the
constraint applied to deal with the sign problem, the two algorithms are
equivalent and should sample the same Green’s function.
In both versions, the steps that constitute the AFDMC algorithm are almost the
same of the DMC one, reported in § 3.1. The starting point is the initial
distribution of walkers, step 1. In step 2 the diffusion of the coordinates is
performed including the drift factor. Now also the spin-isospin degrees of
freedom are propagated, by means of the Hubbard-Stratonovich rotations and the
integral over the auxiliary fields. As in step 3, a weight is assigned to each
walker, choosing one of the two equivalent solutions proposed above (explicit
$\psi_{I}$ ratio or local energy). Both propagation and weight depend on the
prescription adopted in order to keep under control the sign problem. Usually
the fixed phase approximation (see § 3.1.2) is applied with the evaluation of
local operators. The branching process follows then the DMC version described
in step 4 and the procedure is iterated in the same way with the computation
of expectation values at convergence.
#### 3.2.2 Propagator for neutrons: spin-orbit terms
In the previous section we have seen how to deal in an efficient way with a
propagator containing the first six components of the Argonne two-body
potential. Next terms in Eq. (2.16) are the spin-orbit contributions for
$p=7,8$. Although an attempt to treat the spin-orbit terms for nucleon systems
has been reported by Armani in his Ph.D. thesis [224] (together with a
possible $\bm{L}_{ij}^{2}$ inclusion for $p=9$), at present the
$\bm{L}_{ij}\cdot\bm{S}_{ij}$ operator is consistently employed in the AFDMC
algorithm only for neutron systems. No other terms of the $NN$ interaction are
included in the full propagator, neither for nucleons nor for neutrons,
although a perturbative treatment of the remaining terms of AV18 is also
possible [136]. The full derivation of the neutron spin-orbit propagator is
reported in Ref. [37]. Here we want just to sketch the idea behind the
treatment of this non local term for which the corresponding Green’s function
is not trivial to be derived.
Consider the spin-orbit potential for neutrons:
$\displaystyle
v_{ij}^{LS}=v_{LS}(r_{ij})\,\bm{L}_{ij}\cdot\bm{S}_{ij}=v_{LS}(r_{ij})\left(\bm{L}\cdot\bm{S}\right)_{ij}\;,$
(3.163)
where
$\displaystyle v_{LS}(r_{ij})=v_{7}(r_{ij})+v_{8}(r_{ij})\;,$ (3.164)
and $\bm{L}_{ij}$ and $\bm{S}_{ij}$ are defined respectively by Eqs. (2.19)
and (2.20). As reported in Ref. [228], one way to evaluate the propagator for
$\bm{L}\cdot\bm{S}$ is to consider the expansion at first order in $d\tau$
$\displaystyle\operatorname{e}^{-v_{LS}(r_{ij})\left(\bm{L}\cdot\bm{S}\right)_{ij}d\tau}\simeq\left[1-v_{LS}(r_{ij})\left(\bm{L}\cdot\bm{S}\right)_{ij}d\tau\right]\;,$
(3.165)
acting on the free propagator $G_{0}(R,R^{\prime},d\tau)$ of Eq. (3.11). The
derivatives terms of the above expression give
$\displaystyle\left(\bm{\nabla}_{i}-\bm{\nabla}_{j}\right)G_{0}(R,R^{\prime},d\tau)=-\frac{1}{2Dd\tau}\left(\Delta\bm{r}_{i}-\Delta\bm{r}_{j}\right)G_{0}(R,R^{\prime},d\tau)\;,$
(3.166)
where $\Delta\bm{r}_{i}=\bm{r}_{i}-\bm{r}^{\prime}_{i}$ . We can then write:
$\displaystyle\left(\bm{L}\cdot\bm{S}\right)_{ij}G_{0}(R,R^{\prime},d\tau)=$
$\displaystyle\quad\quad=-\frac{1}{4i}\frac{1}{2Dd\tau}\left(\bm{r}_{i}-\bm{r}_{j}\right)\times\left(\Delta\bm{r}_{i}-\Delta\bm{r}_{j}\right)\cdot\left(\bm{\sigma}_{i}+\bm{\sigma}_{j}\right)G_{0}(R,R^{\prime},d\tau)\;,$
$\displaystyle\quad\quad=-\frac{1}{4i}\frac{1}{2Dd\tau}\left(\bm{\Sigma}_{ij}\times\bm{r}_{ij}\right)\cdot\left(\Delta\bm{r}_{i}-\Delta\bm{r}_{j}\right)G_{0}(R,R^{\prime},d\tau)\;,$
(3.167)
where $\bm{\Sigma}_{ij}=\bm{\sigma}_{i}+\bm{\sigma}_{j}$ and
$\bm{r}_{ij}=\bm{r}_{i}-\bm{r}_{j}$, and the relation
$\bm{a}\cdot\left(\bm{b}\times\bm{c}\right)=\bm{c}\cdot\left(\bm{a}\times\bm{b}\right)$
has been used.
By inserting the last expression in Eq. (3.165) and re-exponentiating,
including also the omitted sum over particle indices $i$ and $j$, the
following propagator is obtained:
$\displaystyle\mathcal{P}_{LS}\simeq\operatorname{e}^{\sum_{i\neq
j}\frac{v_{LS}(r_{ij})}{8iD}\left(\bm{\Sigma}_{ij}\times\bm{r}_{ij}\right)\cdot\left(\Delta\bm{r}_{i}-\Delta\bm{r}_{j}\right)}\;.$
(3.168)
The effect of $\mathcal{P}_{LS}$ can be studied starting from the formal
solution
$\displaystyle\psi(R,S,\tau+d\tau)\stackrel{{\scriptstyle LS}}{{\simeq}}\int
dR^{\prime}dS^{\prime}\,G_{0}(R,R^{\prime},d\tau)\,\mathcal{P}_{LS}\,\psi(R^{\prime},S^{\prime},\tau)\;,$
(3.169)
and expanding the propagator to the second order and the wave function
$\psi(R^{\prime},S^{\prime},\tau)$ to the first order in $R-R^{\prime}$. It is
possible to show (see Ref. [37] for the details) that the spin-orbit
contribution of the propagator takes a simple form, but two- and three-body
extra corrections appear. However, in the case of neutrons these additional
terms contain quadratic spin operators and so they can be handled by the
Hubbard-Stratonovich transformation and the rotations over new auxiliary
fields.
#### 3.2.3 Propagator for neutrons: three-body terms
As reported in § 2.1.2, the Illinois (Urbana IX) TNI can be written as a sum
of four different terms:
$\displaystyle
V_{ijk}=A_{2\pi}^{P}\,\mathcal{O}^{2\pi,P}_{ijk}+A_{2\pi}^{S}\,\mathcal{O}^{2\pi,S}_{ijk}+A_{3\pi}\,\mathcal{O}^{3\pi}_{ijk}+A_{R}\,\mathcal{O}^{R}_{ijk}\;.$
(3.170)
For neutron systems, being $\bm{\tau}_{i}\cdot\bm{\tau}_{j}=1$, the operator
structure simplify in such a way that $V_{ijk}$ can be recast as a sum of two-
body terms only [37, 33]. We can therefore handle also the TNI in the AFDMC
propagator by means of the Hubbard-Stratonovich transformation. Let analyze
how each term of the above relation can be conveniently rewritten for neutron
systems.
* •
$\mathcal{O}^{2\pi,P}_{ijk}$ _term_. The $P$-wave 2$\pi$ exchange term (and
also the 3$\pi$ exchange one) of Eq. (2.26) includes the OPE operator
$X_{ij}$, defined in Eq. (2.7). $X_{ij}$ involves the
$\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ and the $S_{ij}$ operators that can be
decomposed via Eqs. (3.128) and (3.129) in order to define a $3A\times 3A$
matrix $X_{i\alpha,j\beta}$ analogous to the $A_{i\alpha,j\beta}^{[\sigma]}$
of Eq. (3.127), where $v_{3}(r_{ij})\\!\rightarrow Y_{\pi}(r_{ij})$ and
$v_{5}(r_{ij})\\!\rightarrow T_{\pi}(r_{ij})$. The OPE operator can be thus
expressed as
$\displaystyle
X_{ij}=\sigma_{i\alpha}\,X_{i\alpha,j\beta}\,\sigma_{j\beta}\;,$ (3.171)
where the matrix $X_{i\alpha,j\beta}$ is real with zero diagonal and has the
same symmetries of $A_{i\alpha,j\beta}^{[\sigma]}$. The commutator over the
$\bm{\tau}_{i}$ operators vanishes, while the anticommutator gives simply a
factor 2. Recalling that $X_{ij}=X_{ji}$ we can derive the following relation:
$\displaystyle\sum_{i<j<k}\mathcal{O}^{2\pi,P}_{ijk}$
$\displaystyle=\frac{1}{3!}\sum_{i\neq j\neq
k}\sum_{cyclic}2\phantom{\frac{1}{4}}\\!\\!\\!\\!\Bigl{\\{}X_{ij},X_{jk}\Bigr{\\}}\;,$
$\displaystyle=2\sum_{i\neq j\neq k}X_{ik}X_{kj}\;,$
$\displaystyle=2\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\Biggl{(}\sum_{k\gamma}X_{i\alpha,k\gamma}\,X_{k\gamma,j\beta}\Biggr{)}\sigma_{j\beta}\;,$
$\displaystyle=2\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\,X^{2}_{i\alpha,j\beta}\,\sigma_{j\beta}\;.$
(3.172)
* •
$\mathcal{O}^{2\pi,S}_{ijk}$ _term_. In the $S$-wave TPE term the isospin
operators do not contribute and we are left with
$\displaystyle\sum_{i<j<k}\mathcal{O}_{ijk}^{2\pi,S}$
$\displaystyle=\frac{1}{3!}\sum_{i\neq j\neq
k}\sum_{cyclic}Z_{\pi}(r_{ij})Z_{\pi}(r_{jk})\,\bm{\sigma}_{i}\cdot\hat{\bm{r}}_{ij}\,\bm{\sigma}_{k}\cdot\hat{\bm{r}}_{kj}\;,$
$\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\Biggl{[}\sum_{k}Z_{\pi}(r_{ik})\,\hat{r}_{ik}^{\alpha}\,Z_{\pi}(r_{jk})\,\hat{r}_{jk}^{\beta}\Biggr{]}\sigma_{j\beta}\;,$
$\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\,Z_{i\alpha,j\beta}\,\sigma_{j\beta}\;.$
(3.173)
* •
$\mathcal{O}^{3\pi}_{ijk}$ _term_. The 3$\pi$ exchange term, even with the
isospin reduction for neutrons, still keeps a very complicated operator
structure. As reported in Ref. [33], this factor can be conveniently written
as a sum of a spin independent and a spin dependent components
$\displaystyle\sum_{i<j<k}\mathcal{O}_{ijk}^{3\pi}=V_{c}^{3\pi}+V_{\sigma}^{3\pi}\;,$
(3.174)
with
$\displaystyle V_{c}^{3\pi}$ $\displaystyle=\frac{400}{18}\sum_{i\neq
j}X_{i\alpha,j\beta}^{2}\,X_{i\alpha,j\beta}\;,$ (3.175) $\displaystyle
V_{\sigma}^{3\pi}$ $\displaystyle=\frac{200}{54}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\Biggl{(}\sum_{\gamma\delta\mu\nu}X_{i\gamma,j\mu}^{2}\,X_{i\delta,j\nu}\,\varepsilon_{\alpha\gamma\delta}\,\varepsilon_{\beta\mu\nu}\Biggr{)}\sigma_{j\beta}\;,$
$\displaystyle=\frac{200}{54}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\,W_{i\alpha,j\beta}\,\sigma_{j\beta}\;,$
(3.176)
where $\varepsilon_{\alpha\beta\gamma}$ is the full antisymmetric tensor.
* •
$\mathcal{O}^{R}_{ijk}$ _term_. The last spin independent term can be recast
as a two body operator as follows
$\displaystyle\sum_{i<j<k}\mathcal{O}^{R}_{ijk}=G_{0}^{R}+\frac{1}{2}\sum_{i}\left(G_{i}^{R}\right)^{2}\;,$
(3.177)
with
$\displaystyle G_{0}^{R}$ $\displaystyle=-\sum_{i<j}T_{\pi}^{4}(r_{ij})\;,$
(3.178) $\displaystyle G_{i}^{R}$ $\displaystyle=\sum_{k\neq
i}T_{\pi}^{2}(r_{ik})\;.$ (3.179)
Finally, for neutron systems we can still write the spin dependent part of the
$NN$ potential in the form of Eq. (3.126), with the inclusion of TNI
contributions:
$\displaystyle V_{NN}^{SD}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta}\sigma_{i\alpha}\,A_{i\alpha,j\beta}^{[\sigma]}\,\sigma_{j\beta}\;,$
(3.180)
where now
$\displaystyle A_{i\alpha,j\beta}^{[\sigma]}\longrightarrow
A_{i\alpha,j\beta}^{[\sigma]}+2A_{2\pi}^{P}\,X^{2}_{i\alpha,j\beta}+\frac{1}{2}A_{2\pi}^{S}\,Z_{i\alpha,j\beta}+\frac{200}{54}\,A_{3\pi}\,W_{i\alpha,j\beta}\;.$
(3.181)
The central term of the two-body potential of Eq. (3.125) keeps also
contributions from the TNI 3$\pi$ exchange term and from the phenomenological
term, and it reads now:
$\displaystyle V_{NN}^{SI}\longrightarrow
V_{NN}^{SI}+A_{3\pi}V_{c}^{3\pi}+A_{R}\left[G_{0}^{R}+\frac{1}{2}\sum_{i}\left(G_{i}^{R}\right)^{2}\right]\;.$
(3.182)
#### 3.2.4 Wave functions
In this section the trial wave functions used in AFDMC calculations for
nuclear and hypernuclear systems will be presented, distinguishing between the
case of finite and infinite systems. Restoring the convention of Chapter 2,
which is commonly used in the literature for hypernuclear systems, $A$ will
refer to the total number of baryons, $\mathcal{N}_{N}$ nucleons plus
$\mathcal{N}_{\Lambda}$ lambda particles. Latin indices will be used for the
nucleons, Greek $\lambda$, $\mu$ and $\nu$ indices for the lambda particles.
Finally, the first letters of the Greek alphabet
($\alpha,\beta,\gamma,\delta,\ldots$) used as indices will refer to the
Cartesian components of the operators.
##### Non strange finite and infinite systems
As already sketched in § 3.2, the AFDMC wave function is written in the single
particle state representation. The trial wave function for nuclear systems,
which is used both as projection and importance function
$|\psi_{I}\rangle\equiv|\psi_{T}\rangle$, is assumed of the form [31, 38]
$\displaystyle\psi_{T}^{N}(R_{N},S_{N})=\prod_{i<j}f_{c}^{NN}(r_{ij})\,\Phi_{N}(R_{N},S_{N})\;,$
(3.183)
where $R_{N}=\\{\bm{r}_{1},\ldots,\bm{r}_{\mathcal{N}_{N}}\\}$ are the
Cartesian coordinates and $S_{N}=\\{s_{1},\ldots,s_{\mathcal{N}_{N}}\\}$ the
spin-isospin coordinates, represented as complex 4- or 2-component vectors:
nucleons: $\displaystyle\quad s_{i}=\left(\begin{array}[]{c}a_{i}\\\ b_{i}\\\
c_{i}\\\
d_{i}\end{array}\right)_{i}\\!=a_{i}|p\uparrow\rangle_{i}+b_{i}|p\downarrow\rangle_{i}+c_{i}|n\uparrow\rangle_{i}+d_{i}|n\downarrow\rangle_{i}\;,$
(3.188) neutrons: $\displaystyle\quad s_{i}=\left(\begin{array}[]{c}a_{i}\\\
b_{i}\end{array}\right)_{i}\\!=a_{i}|n\uparrow\rangle_{i}+b_{i}|n\downarrow\rangle_{i}\;,$
(3.191)
with
$\left\\{|p\uparrow\rangle,|p\downarrow\rangle,|n\uparrow\rangle,|n\downarrow\rangle\right\\}$
the proton-neutron-up-down basis.
The function $f_{c}^{NN}(r)$ is a symmetric and spin independent Jastrow
correlation function, solution of the Schrödinger-like equation for
$f_{c}^{NN}(r<d)$
$\displaystyle-\frac{\hbar^{2}}{2\mu_{NN}}\nabla^{2}f_{c}^{NN}(r)+\eta\,v_{c}^{NN}(r)f_{c}^{NN}(r)=\xi
f_{c}^{NN}(r)\;,$ (3.192)
where $v_{c}^{NN}(r)$ is the spin independent part of the two-body $NN$
interaction, $\mu_{NN}=m_{N}/2$ the reduced mass of the nucleon pair and
$\eta$ and the healing distance $d$ are variational parameters. For distances
$r\geq d$ we impose $f_{c}^{NN}(r)=1$. The role of the Jastrow function is to
include the short-range correlations in the trial wave function. In the AFDMC
algorithm the effect is simply a reduction of the overlap between pairs of
particles, with the reduction of the energy variance. Since there is no change
in the phase of the wave function, the $f_{c}^{NN}$ function does not
influence the computed energy value in the long imaginary time projection.
The antisymmetric part $\Phi_{N}(R_{N},S_{N})$ of the trial wave function
depends on the system to be studied (finite or infinite). As already seen, it
is generally built starting from single particle states
$\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})$, where $\epsilon$ is the set of
quantum numbers describing the state and $\bm{r}_{i}$, $s_{i}$ the single
particle nucleon coordinates. The antisymmetry property is then realized by
taking the Slater determinant of the $\varphi_{\epsilon}^{N}$:
$\displaystyle\Phi_{N}(R_{N},S_{N})=\mathcal{A}\Bigg{[}\prod_{i=1}^{\mathcal{N}_{N}}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigg{]}=\det\Bigl{\\{}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}\;.$
(3.193)
For nuclei and neutron drops [31] a good set of quantum number is given by
$\epsilon=\\{n,j,m_{j}\\}$. The single particle states are written as:
$\displaystyle\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})=R_{n,j}^{N}(r_{i})\Bigl{[}Y_{l,m_{l}}^{N}(\Omega)\,\chi_{s,m_{s}}^{N}(s_{i})\Bigr{]}_{j,m_{j}}\;,$
(3.194)
where $R_{n,j}^{N}$ is a radial function, $Y_{l,m_{l}}^{N}$ the spherical
harmonics depending on the solid angle $\Omega$ and $\chi_{s,m_{s}}^{N}$ the
spinors in the proton-neutron-up-down basis. The angular functions are coupled
to the spinors using the Clebsh-Gordan coefficients to have orbitals in the
$\\{n,j,m_{j}\\}$ basis according to the usual shell model classification of
the nuclear single particle spectrum. For finite systems, in order to make the
wave function translationally invariant, the single particle orbitals have to
be defined with respect to the center of mass (CM) of the system. We have
thus:
$\displaystyle\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\longrightarrow\varphi_{\epsilon}^{N}(\bm{r}_{i}-\bm{r}_{CM},s_{i})\quad\quad\text{with}\quad\bm{r}_{CM}=\frac{1}{\mathcal{N}_{N}}\sum_{i=1}^{\mathcal{N}_{N}}\bm{r}_{i}\;.$
(3.195)
In order to deal with new shifted coordinates, we need to correct all the
first and second derivatives of trial wave function with respect to
$\bm{r}_{i}$. The derivation of such corrections is reported in Appendix A.
The choice of the radial functions $R_{n,j}^{N}$ depends on the system studied
and, typically, solutions of the self-consistent Hartree-Fock problem with
Skyrme interactions are adopted. For nuclei the Skyrme effective interactions
of Ref. [229] are commonly used. For neutron drops, the Skyrme SKM force of
Ref. [230] has been considered.
An additional aspect to take care when dealing with finite systems, is the
symmetry of the wave function. Because the AFDMC projects out the lowest
energy state not orthogonal to the starting trial wave function, it is
possible to study a state with given symmetry imposing to the trial wave
function the total angular momentum $J$ experimentally observed. This can be
achieved by taking a sum over a different set of determinants
$\displaystyle\det\Bigl{\\{}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}\longrightarrow\left[\sum_{\kappa}c_{\kappa}\,\text{det}_{\kappa}\Bigl{\\{}\varphi_{\epsilon_{\kappa}}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}\right]_{J,M_{J}}\;,$
(3.196)
where the $c_{\kappa}$ coefficients are determined in order to have the
eigenstate of total angular momentum $J=j_{1}+\ldots+j_{\mathcal{N}_{N}}$.
For nuclear and neutron matter [38], the antisymmetric part of the wave
function is given by the ground state of the Fermi gas, built from a set of
plane waves. The infinite uniform system at a given density is simulated with
$\mathcal{N}_{N}$ nucleons in a cubic box of volume $L^{3}$ replicated into
the space by means of periodic boundary conditions (PBC):
$\displaystyle\varphi_{\epsilon}^{N}(\bm{r}_{1}+L\hat{\bm{r}},\bm{r}_{2},\ldots,s_{i})=\varphi_{\epsilon}^{N}(\bm{r}_{1},\bm{r}_{2},\ldots,s_{i})\;.$
(3.197)
Working in a discrete space, the momentum vectors are quantized and can be
expressed as
$\displaystyle\bm{k}_{\epsilon}=\frac{2\pi}{L}\left(n_{x},n_{y},n_{z}\right)_{\epsilon}\;,$
(3.198)
where $\epsilon$ labels the quantum state and $n_{x}$, $n_{y}$ and $n_{z}$ are
integer numbers labelling the momentum shell. The single particle states are
then given by
$\displaystyle\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})=\operatorname{e}^{-i\bm{k}_{\epsilon}\cdot\bm{r}_{i}}\chi_{s,m_{s}}^{N}(s_{i})\;.$
(3.199)
In order to meet the requirement of homogeneity and isotropy, the shell
structure of the system must be closed. The total number of Fermions in a
particular spin-isospin configuration that can be correctly simulated in a box
corresponds to the closure of one of the
$\left(n_{x},n_{y},n_{z}\right)_{\epsilon}$ shells. The list of the first
closure numbers is
$\displaystyle\mathcal{N}_{c}=1,7,19,27,33,57,81,93\ldots\;.$ (3.200)
Given a closure number $\mathcal{N}_{c}^{N}$, we can thus simulate an infinite
system by means of a periodic box with $2\,\mathcal{N}_{c}^{N}$ neutrons (up
and down spin) or $4\,\mathcal{N}_{c}^{N}$ nucleons (up and down spin and
isospin). Although the use of PBC should reduce the finite-size effects, in
general there are still sizable errors in the kinetic energy arising from
shell effects in filling the plane wave orbitals, even at the closed shell
filling in momentum space. However, in the thermodynamical limit
$\mathcal{N}_{c}^{N}\rightarrow\infty$, exact results should be obtained. For
symmetric nuclear matter (SNM), 28, 76 and 108 nucleons have been used [35],
resulting in comparable results for the energy per particle at a given
density. In the case of pure neutron matter (PNM), finite-size effects are
much more evident [38] and the thermodynamical limit is not reached
monotonically. Typically, PNM is simulated using 66 neutrons, which was found
to give the closest kinetic energy compared to the Fermi gas in the range of
$\mathcal{N}_{c}^{N}$ corresponding to feasible computational times.
As reported in Ref. [231], twist-averaged boundary conditions (TABC) can be
imposed on the trial wave function to reduce the finite-size effects. One can
allow particles to pick up a phase $\theta$ when they wrap around the periodic
boundaries:
$\displaystyle\varphi_{\epsilon}^{N}(\bm{r}_{1}+L\hat{\bm{r}},\bm{r}_{2},\ldots,s_{i})=\operatorname{e}^{i\theta}\varphi_{\epsilon}^{N}(\bm{r}_{1},\bm{r}_{2},\ldots,s_{i})\;.$
(3.201)
The boundary condition $\theta=0$ corresponds to the PBC, $\theta\neq 0$ to
the TABC. It has been shown that if the twist phase is integrated over, the
finite size effects are substantially reduced. TABC has been used in PNM
calculations [38], showing a small discrepancy in the energy per particle for
38, 45, 66 and 80 neutrons at fixed density. A remarkable result is that the
PNM energy for 66 neutrons using PBC is very close to the extrapolated result
obtained employing the TABC, validating then the standard AFDMC calculation
for 66 particles. Compare to PBC, employing the TABC results in a more
computational time and they have not been used in this work.
##### Strange finite and infinite systems
The $\Lambda$ hyperon, having isospin zero, does not participate to the
isospin doublet of nucleons. Referring to hypernuclear systems, we can
therefore treat the additional strange baryons as distinguishable particles
writing a trial wave function of the form
$\displaystyle\psi_{T}(R,S)=\prod_{\lambda i}f_{c}^{\Lambda N}(r_{\lambda
i})\,\psi_{T}^{N}(R_{N},S_{N})\,\psi_{T}^{\Lambda}(R_{\Lambda},S_{\Lambda})\;,$
(3.202)
where
$R=\\{\bm{r}_{1},\ldots,\bm{r}_{\mathcal{N}_{N}},\bm{r}_{1},\ldots,\bm{r}_{\mathcal{N}_{\Lambda}}\\}$
and
$S=\\{s_{1},\ldots,s_{\mathcal{N}_{N}},s_{1},\ldots,s_{\mathcal{N}_{\Lambda}}\\}$
refer to the coordinates of all the baryons and $\psi_{T}^{N}(R_{N},S_{N})$ is
the nucleon single particle wave function of Eq. (3.183).
$\psi_{T}^{\Lambda}(R_{\Lambda},S_{\Lambda})$ is the lambda single particle
wave function that takes the same structure of the nucleon one:
$\displaystyle\psi_{T}^{\Lambda}(R_{\Lambda},S_{\Lambda})=\prod_{\lambda<\mu}f_{c}^{\Lambda\Lambda}(r_{\lambda\mu})\,\Phi_{\Lambda}(R_{\Lambda},S_{\Lambda})\;,$
(3.203)
with
$\displaystyle\Phi_{\Lambda}(R_{\Lambda},S_{\Lambda})=\mathcal{A}\Bigg{[}\prod_{\lambda=1}^{\mathcal{N}_{\Lambda}}\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})\Bigg{]}=\det\Bigl{\\{}\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})\Bigr{\\}}\;.$
(3.204)
$R_{\Lambda}=\\{\bm{r}_{1},\ldots,\bm{r}_{\mathcal{N}_{\Lambda}}\\}$ are the
hyperon Cartesian coordinates and
$S_{\Lambda}=\\{s_{1},\ldots,s_{\mathcal{N}_{\Lambda}}\\}$ the hyperon spin
coordinates, represented by the 2-dimension spinor in the lambda-up-down
basis:
$\displaystyle s_{\lambda}=\left(\begin{array}[]{c}u_{\lambda}\\\
v_{\lambda}\end{array}\right)_{\lambda}\\!=u_{\lambda}|\Lambda\uparrow\rangle_{\lambda}+v_{\lambda}|\Lambda\downarrow\rangle_{\lambda}\;.$
(3.207)
The $\Lambda\Lambda$ Jastrow correlation function $f_{c}^{\Lambda\Lambda}(r)$
is calculated by means of Eq. (3.192) for the hyperon-hyperon pair using the
central channel of the $\Lambda\Lambda$ potential of Eq. (2.48). Eq. (3.192)
is also used to calculate the hyperon-nucleon correlation function
$f_{c}^{\Lambda N}(r)$ of the hypernuclear wave function (3.202) by
considering the pure central term of the $\Lambda N$ potential of Eq. (2.35)
and using the reduced mass
$\displaystyle\mu_{\Lambda N}=\frac{m_{\Lambda}\,m_{N}}{m_{\Lambda}+m_{N}}\;.$
(3.208)
For $\Lambda$ hypernuclei (and $\Lambda$ neutron drops) the hyperon single
particle states take the same structure as the nuclear case, and they read:
$\displaystyle\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})=R_{n,j}^{\Lambda}(r_{\lambda})\Bigl{[}Y_{l,m_{l}}^{\Lambda}(\Omega)\,\chi_{s,m_{s}}^{\Lambda}(s_{\lambda})\Bigr{]}_{j,m_{j}}\;.$
(3.209)
Although the AFDMC code for hypernuclei is set up for an arbitrary number of
hyperons, we focused on single and double $\Lambda$ hypernuclei. Having just
two hyperons to deal with, only one radial function $R_{n,j}^{\Lambda}$ is
needed. Being the mass difference between the neutron and the $\Lambda$
particle small, we used the neutron $1s_{1/2}$ radial function also for the
hyperon.
Dealing with finite systems, the coordinates of all the baryons must be
related to the CM, that now is given by the coordinates of particles with
different mass. Nucleon and hyperon single particle orbitals are thus defined
as:
$\displaystyle\begin{array}[]{rcl}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\\!\\!\\!&\longrightarrow&\\!\\!\varphi_{\epsilon}^{N}(\bm{r}_{i}-\bm{r}_{CM},s_{i})\\\\[5.0pt]
\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})\\!\\!\\!&\longrightarrow&\\!\\!\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda}-\bm{r}_{CM},s_{\lambda})\end{array}$
(3.212)
where
$\displaystyle\bm{r}_{CM}=\frac{1}{M}\left(m_{N}\sum_{i=1}^{\mathcal{N}_{N}}\bm{r}_{i}+m_{\Lambda}\sum_{\lambda=1}^{\mathcal{N}_{\Lambda}}\bm{r}_{\lambda}\right)\quad\text{with}\quad
M=\mathcal{N}_{N}\,m_{N}+\mathcal{N}_{\Lambda}\,m_{\Lambda}\;.$ (3.213)
As in the nuclear case, the use of relative coordinates introduces corrections
in the calculation of the derivatives of the trial wave function. For
hypernuclei such corrections, and in general the evaluation of derivatives,
are more complicated than for nuclei. This is because we have to deal with two
set of spacial coordinates ($R_{N}$ and $R_{\Lambda}$) and the Jastrow
function $f_{c}^{\Lambda N}$ depends on both. The derivatives of the trial
wave function including CM corrections are reported in Appendix A.
For $\Lambda$ neutron (nuclear) matter the antisymmetric part of the hyperon
wave function is given by the ground state of the Fermi gas, as for nucleons.
We are thus dealing with two Slater determinants of plane waves with
$\bm{k}_{\epsilon}$ vectors quantized in the same $L^{3}$ box (see Eq.
(3.198)). The dimension of the simulation box, and thus the quantization of
the $\bm{k}_{\epsilon}$ vectors, is given by the total numeric baryon density
$\displaystyle\rho_{b}=\frac{\mathcal{N}_{b}}{L^{3}}=\frac{\mathcal{N}_{N}+\mathcal{N}_{\Lambda}}{L^{3}}=\rho_{N}+\rho_{\Lambda}\;,$
(3.214)
and the number of nucleons and lambda particles. The hyperon single particle
states correspond then to
$\displaystyle\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})=\operatorname{e}^{-i\bm{k}_{\epsilon}\cdot\bm{r}_{\lambda}}\chi_{s,m_{s}}^{\Lambda}(s_{\lambda})\;,$
(3.215)
where the $\bm{k}_{\epsilon}$ structure derived from $\rho_{b}$ is used also
for the the nuclear part. The requirements of homogeneity and isotropy
discussed in the previous section still apply and so the lambda plane waves
have to close their own momentum shell structure. Given the list of closure
numbers (3.200), we can add $2\,\mathcal{N}_{c}^{\Lambda}$ hyperons (up and
down spin) to the $2\,\mathcal{N}_{c}^{N}$ neutrons or
$4\,\mathcal{N}_{c}^{N}$ nucleons in the periodic box.
The wave functions described so far are appropriate only if we consider
nucleons and hyperons as distinct particles. In this way, it is not possible
to include the $\Lambda N$ exchange term of Eq. (2.35) directly in the
propagator, because it mixes hyperon and nucleon states. The complete
treatment of this factor would require a drastic change in the AFDMC code
and/or a different kind of nuclear-hypernuclear interactions, as briefly
discussed in Appendix B. A perturbative analysis of the $v_{0}(r_{\lambda
i})\,\varepsilon\,\mathcal{P}_{x}$ term is however possible and it is reported
in the next section.
#### 3.2.5 Propagator for hypernuclear systems
Consider a many-body system composed by nucleons and hyperons, interacting via
the full Hamiltonian (2.1) and described by the trial wave function (3.202).
Suppose to switch off all the spin-isospin interactions in all the channels
and keep only the central terms:
$\displaystyle
H=T_{N}+T_{\Lambda}+V_{NN}^{c}+V_{\Lambda\Lambda}^{c}+V_{\Lambda N}^{c}\;,$
(3.216)
where also the central contributions from the three-body interactions are
included. Neglecting the spin-isospin structure of the trial wave function we
can follow the idea of the standard DMC described in § 3.1 and write the
analog of Eq. (3.18):
$\displaystyle\psi(R_{N},R_{\Lambda},\tau+d\tau)\simeq\int
dR^{\prime}_{N}\,dR^{\prime}_{\Lambda}\langle
R_{N},R_{\Lambda}|\operatorname{e}^{-\left(V_{NN}^{c}+V_{\Lambda\Lambda}^{c}+V_{\Lambda
N}^{c}\right)\frac{d\tau}{2}}\operatorname{e}^{-T_{N}d\tau}\operatorname{e}^{-T_{\Lambda}d\tau}\times$
$\displaystyle\hskip
14.22636pt\times\operatorname{e}^{-\left(V_{NN}^{c}+V_{\Lambda\Lambda}^{c}+V_{\Lambda
N}^{c}\right)\frac{d\tau}{2}}\operatorname{e}^{E_{T}d\tau}|R^{\prime}_{N},R^{\prime}_{\Lambda}\rangle\,\psi(R^{\prime}_{N},R^{\prime}_{\Lambda},\tau)\;,$
$\displaystyle\simeq\int
dR^{\prime}_{N}\,dR^{\prime}_{\Lambda}\underbrace{\langle
R_{N}|\operatorname{e}^{-T_{N}d\tau}|R^{\prime}_{N}\rangle}_{G_{0}^{N}(R_{N},R^{\prime}_{N},d\tau)}\underbrace{\langle
R_{\Lambda}|\operatorname{e}^{-T_{\Lambda}d\tau}|R^{\prime}_{\Lambda}\rangle}_{G_{0}^{\Lambda}(R_{\Lambda},R^{\prime}_{\Lambda},d\tau)}\times$
$\displaystyle\hskip
14.22636pt\times\underbrace{\phantom{\langle}\\!\\!\operatorname{e}^{-\left(\widetilde{V}_{NN}^{c}(R_{N},R^{\prime}_{N})+\widetilde{V}_{\Lambda\Lambda}^{c}(R_{\Lambda},R^{\prime}_{\Lambda})+\widetilde{V}_{\Lambda
N}^{c}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda})-E_{T}\right)d\tau}}_{G_{V}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda},d\tau)}\psi(R^{\prime}_{N},R^{\prime}_{\Lambda},\tau)\;,$
$\displaystyle\simeq\left(\frac{1}{4\pi
D_{N}d\tau}\right)^{\frac{3\mathcal{N}_{N}}{2}}\\!\\!\left(\frac{1}{4\pi
D_{\Lambda}d\tau}\right)^{\frac{3\mathcal{N}_{\Lambda}}{2}}\\!\\!\int
dR^{\prime}_{N}\,dR^{\prime}_{\Lambda}\operatorname{e}^{-\frac{(R_{N}-R^{\prime}_{N})^{2}}{4D_{N}d\tau}}\operatorname{e}^{-\frac{(R_{\Lambda}-R^{\prime}_{\Lambda})^{2}}{4D_{\Lambda}d\tau}}\times$
$\displaystyle\hskip
14.22636pt\times\operatorname{e}^{-\left(\widetilde{V}_{NN}^{c}(R_{N},R^{\prime}_{N})+\widetilde{V}_{\Lambda\Lambda}^{c}(R_{\Lambda},R^{\prime}_{\Lambda})+\widetilde{V}_{\Lambda
N}^{c}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda})-E_{T}\right)d\tau}\psi(R^{\prime}_{N},R^{\prime}_{\Lambda},\tau)\;,$
(3.217)
where
$\displaystyle\widetilde{V}_{NN}^{c}(R_{N},R^{\prime}_{N})$
$\displaystyle=\frac{1}{2}\Bigl{[}V_{NN}^{c}(R_{N})+V_{NN}^{c}(R^{\prime}_{N})\Bigr{]}\;,$
$\displaystyle\widetilde{V}_{\Lambda\Lambda}^{c}(R_{\Lambda},R^{\prime}_{\Lambda})$
$\displaystyle=\frac{1}{2}\Bigl{[}V_{\Lambda\Lambda}^{c}(R_{\Lambda})+V_{\Lambda\Lambda}^{c}(R^{\prime}_{\Lambda})\Bigr{]}\;,$
(3.218) $\displaystyle\widetilde{V}_{\Lambda
N}^{c}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda})$
$\displaystyle=\frac{1}{2}\Bigl{[}V_{\Lambda
N}^{c}(R_{N},R_{\Lambda})+V_{\Lambda
N}^{c}(R^{\prime}_{N},R^{\prime}_{\Lambda})\Bigr{]}\;,$
and $D_{N}=\hbar^{2}/2m_{N}$ and $D_{\Lambda}=\hbar^{2}/2m_{\Lambda}$ are the
diffusion constants of the Brownian motion of nucleons and lambda particles.
The evolution in imaginary time is thus performed in the same way of the
standard DMC algorithm. A set of walkers, which now contains nucleon and
hyperon coordinates, is diffused according to
$G_{0}^{N}(R_{N},R^{\prime}_{N},d\tau)$ and
$G_{0}^{\Lambda}(R_{\Lambda},R^{\prime}_{\Lambda},d\tau)$. A weight
$\omega_{i}=G_{V}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda},d\tau)$
is assigned to each waker and it is used for the estimator contributions and
the branching process. We can also apply the importance function technique,
the result of which is the inclusion of a drift term in the diffusion of each
type of baryon and the use of the local energy for the branching weight. The
drift velocities take the same form of Eq. (3.32), but now the derivatives are
calculated with respect to nucleon or hyperon coordinates, including all the
possible CM (for finite systems) and Jastrow corrections, as reported in
Appendix A.
Reintroduce now the spin-isospin structure in the wave function and consider
then the spin-isospin dependent interactions. For the nuclear part, we can
still deal with AV6 like potentials for nucleon systems by means of the
Hubbard-Stratonovich transformation, as discussed in § 3.2.1. In the case of
pure neutron systems, we can also include spin-orbit and three-body
contributions as reported in § 3.2.2 and § 3.2.3. In the next we will discuss
how to deal with the spin-isospin dependent part of the hypernuclear
potentials, in both two- and three-body channels.
##### Two-body terms
Consider the full two-body $\Lambda N$ interaction described in the previous
chapter:
$\displaystyle V_{\Lambda N}$ $\displaystyle=\sum_{\lambda i}\left(v_{\lambda
i}+v_{\lambda i}^{CSB}\right)\;,$ $\displaystyle=\sum_{\lambda
i}v_{0}(r_{\lambda i})(1-\varepsilon)+\sum_{\lambda i}v_{0}(r_{\lambda
i})\,\varepsilon\,\mathcal{P}_{x}+\sum_{\lambda
i}\frac{1}{4}v_{\sigma}T^{2}_{\pi}(r_{\lambda
i})\,\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$
$\displaystyle\quad+\sum_{\lambda i}C_{\tau}\,T_{\pi}^{2}\left(r_{\lambda
i}\right)\tau_{i}^{z}\;,$ $\displaystyle=\sum_{\lambda i}v_{0}(r_{\lambda
i})(1-\varepsilon)+\sum_{\lambda i}B_{\lambda
i}^{[\mathcal{P}_{x}]}\,\mathcal{P}_{x}+\sum_{\lambda
i}\sum_{\alpha}\sigma_{\lambda\alpha}\,B_{\lambda
i}^{[\sigma]}\,\sigma_{i\alpha}+\sum_{i}B_{i}^{[\tau]}\,\tau_{i}^{z}\;,$
(3.219)
where
$\displaystyle B_{\lambda i}^{[\mathcal{P}_{x}]}=v_{0}(r_{\lambda
i})\,\varepsilon\quad\quad B_{\lambda
i}^{[\sigma]}=\frac{1}{4}v_{\sigma}T^{2}_{\pi}(r_{\lambda i})\quad\quad
B_{i}^{[\tau]}=\sum_{\lambda}C_{\tau}\,T_{\pi}^{2}\left(r_{\lambda
i}\right)\;.$ (3.220)
The first term of Eq. (3.219) is simply a spin independent factor and can be
included in the $V_{\Lambda N}^{c}$ contribution of Eq. (3.217). The remaining
terms involve operators acting on nucleons and hyperons and need to be
discussed separately.
* •
$\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$ _term_. The quadratic spin-spin
term of the $\Lambda N$ interaction is written in same form of the nucleon-
nucleon one of Eq. (3.126). However, in general the matrix $B_{\lambda
i}^{[\sigma]}$ is not a square matrix ($\dim B_{\lambda
i}^{[\sigma]}=\mathcal{N}_{\Lambda}\times\mathcal{N}_{N}$) and so we can not
follow the derivation of § 3.2.1. Recalling that we are working with single
particle wave functions and that each spin-isospin operator is the
representation in the $A$-body tensor product space of a one-body operator as
in Eq. (3.123), we can write
$\displaystyle\\!\\!\\!\\!\sum_{\alpha}\sigma_{\lambda\alpha}\otimes\sigma_{i\alpha}=\frac{1}{2}\sum_{\alpha}\left[\left(\sigma_{\lambda\alpha}\oplus\sigma_{i\alpha}\right)^{2}-\left(\sigma_{\lambda\alpha}\otimes\mathbb{I}_{i\alpha}\right)^{2}-\left(\mathbb{I}_{\lambda\alpha}\otimes\sigma_{i\alpha}\right)^{2}\right]\;.\\!$
(3.221)
The square of the Pauli matrices of the last two terms gives the identity with
respect to the single particle state $\lambda$ or $i$, so that they can be
simply written as a spin independent contribution
$\displaystyle\sum_{\alpha}\sigma_{\lambda\alpha}\otimes\sigma_{i\alpha}=-3+\frac{1}{2}\sum_{\alpha}\left(\mathcal{O}_{\lambda
i,\alpha}^{[\sigma_{\Lambda N}]}\right)^{2}\;,$ (3.222)
where we have defined a new spin-spin operator
$\displaystyle\mathcal{O}_{\lambda i,\alpha}^{[\sigma_{\Lambda
N}]}=\sigma_{\lambda\alpha}\oplus\sigma_{i\alpha}\;.$ (3.223)
We can now write the the $\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$ term as
follows
$\displaystyle V_{\Lambda N}^{\sigma\sigma}$ $\displaystyle=\sum_{\lambda
i}\sum_{\alpha}\sigma_{\lambda\alpha}\,B_{\lambda
i}^{[\sigma]}\,\sigma_{i\alpha}\;,$ $\displaystyle=-3\sum_{\lambda
i}B_{\lambda i}^{[\sigma]}+\frac{1}{2}\sum_{\lambda i}\sum_{\alpha}B_{\lambda
i}^{[\sigma]}\left(\mathcal{O}_{\lambda i,\alpha}^{[\sigma_{\Lambda
N}]}\right)^{2}$ (3.224)
The first term is a central factor that can be included in $V_{\Lambda
N}^{c}$. The second term is written in the same way of the spin-isospin
dependent part of the nuclear interaction of Eq. (3.135). With a little abuse
of notation
$n=\\{\lambda,i\\}={1,\ldots,\mathcal{N}_{N}\,\mathcal{N}_{\Lambda}}$, the
spin dependent part of the propagator for the
$\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$ takes a suitable form for the
application of the Hubbard-Stratonovich transformation:
$\displaystyle\operatorname{e}^{-\sum_{\lambda
i}\sum_{\alpha}\sigma_{\lambda\alpha}\,B_{\lambda
i}^{[\sigma]}\,\sigma_{i\alpha}\,d\tau}$
$\displaystyle=\operatorname{e}^{-3\sum_{n}B_{n}^{[\sigma]}-\frac{1}{2}\sum_{n\alpha}B_{n}^{[\sigma]}\left(\mathcal{O}_{n,\alpha}^{[\sigma_{\Lambda
N}]}\right)^{2}d\tau}\;,$ $\displaystyle=\operatorname{e}^{-V_{\Lambda
N}^{c\,[\sigma]}}\prod_{n\alpha}\operatorname{e}^{-\frac{1}{2}B_{n}^{[\sigma]}\left(\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda
N}]}\right)^{2}d\tau}\,+\operatorname{o}\\!\left(d\tau^{2}\right)\;,$
$\displaystyle\simeq\operatorname{e}^{-V_{\Lambda
N}^{c\,[\sigma]}}\prod_{n\alpha}\frac{1}{\sqrt{2\pi}}\int\\!dx_{n\alpha}\operatorname{e}^{\frac{-x_{n\alpha}^{2}}{2}+\sqrt{-B_{n}^{[\sigma]}d\tau}\,x_{n\alpha}\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda
N}]}}\;.$ (3.225)
Recalling Eq. (3.138), we can write the hyperon spin dependent part of the
AFDMC propagator for hypernuclear systems as
$\displaystyle\langle
S_{N}S_{\Lambda}|\prod_{n\alpha=1}^{3\mathcal{N}_{N}\mathcal{N}_{\Lambda}}\\!\frac{1}{\sqrt{2\pi}}\int\\!dx_{n\alpha}\operatorname{e}^{\frac{-x_{n\alpha}^{2}}{2}+\sqrt{-B_{n}^{[\sigma]}d\tau}\,x_{n\alpha}\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda
N}]}}|S^{\prime}_{N}S^{\prime}_{\Lambda}\rangle\;.$ (3.226)
By the definition of Eq. (3.223), it comes out that the action of the operator
$\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda N}]}$ on the spinor
$|S^{\prime}_{N},S^{\prime}_{\Lambda}\rangle$ factorizes in a
$\sigma_{i\alpha}$ rotation for the nucleon spinor $|S_{N}\rangle$ and a
$\sigma_{\lambda\alpha}$ rotation for the $\Lambda$ spinor
$|S_{\Lambda}\rangle$, coupled by the same coefficient
$\sqrt{-B_{n}^{[\sigma]}d\tau}\,x_{n\alpha}$.
* •
$\tau_{i}^{z}$ _term_. As already seen in § 3.2, the single particle wave
function is closed with respect to the application of a propagator containing
linear spin-isospin operators. The action of the CSB potential corresponds to
the propagator
$\displaystyle\operatorname{e}^{-\sum_{i}B_{i}^{[\tau]}\,\tau_{i}^{z}\,d\tau}=\prod_{i}\operatorname{e}^{-B_{i}^{[\tau]}\,\tau_{i}^{z}\,d\tau}\,+\operatorname{o}\left(d\tau^{2}\right)\;,$
(3.227)
that, acting on the trial wave function, simply produces a rotation of the
nucleon spinors, as in Eq. (3.148). Being the CSB term already linear in
$\tau_{i}^{z}$, there is no need for Hubbard-Stratonovich transformation. The
$\tau_{i}^{z}$ rotations can be applied after the integration over auxiliary
fields on the spinors modified by the Hubbard-Stratonovich rotations. In the
$\psi_{I}$ ratio AFDMC algorithm (_v1_) there are no additional terms in the
weight coming from the CSB rotations. If we use the local energy version of
the algorithm (_v2_), we need to subtract the CSB contribution to $E_{L}(R)$
(Eq. (3.34)) because there are no counter terms coming from the importance
sampling on auxiliary fields. Note that in neutron systems, $\tau_{i}^{z}$
gives simply a factor $-1$, so the CSB becomes a positive central contribution
($C_{\tau}<0$) to be added in $V_{\Lambda N}^{c}$.
* •
$\mathcal{P}_{x}$ _term_. As discussed in § 3.2.4, the structure of our trial
wave function for hypernuclear systems prevents the straightforward inclusion
of the $\Lambda N$ space exchange operator in the AFDMC propagator. We can try
to treat this contribution perturbatively: $\mathcal{P}_{x}$ is not included
in the propagator but its effect is calculated as a standard estimator on the
propagated wave function. The action of $\mathcal{P}_{x}$ is to exchange the
position of one nucleon and one hyperon, modifying thus the CM of the whole
system due to the mass difference between the baryons. To compute this
potential contribution we have thus to sum over all the hyperon-nucleon pairs.
For each exchanged pair, all the positions are referred to the new CM and the
wave function is evaluated and accumulated. Then, particles are moved back to
the original positions and a new pair is processed. At the end of the sum the
contribution $\sum_{\lambda i}\mathcal{P}_{x}\,\psi$ is obtained. As reported
in Refs. [191, 188, 195, 189], the $\Lambda N$ space exchange operator induces
strong correlations and thus a perturbative approach might not be appropriate.
A possible non perturbative extension of the AFDMC code for the space exchange
operator is outlined in Appendix B.
In the two body hypernuclear sector a $\Lambda\Lambda$ interaction is also
employed, as reported in § 2.2.3. The potential described in Eq. (2.48) can be
recast as
$\displaystyle V_{\Lambda\Lambda}$
$\displaystyle=\sum_{\lambda<\mu}\sum_{k=1}^{3}\left(v_{0}^{(k)}+v_{\sigma}^{(k)}\,{\bm{\sigma}}_{\lambda}\cdot{\bm{\sigma}}_{\mu}\right)\operatorname{e}^{-\mu^{(k)}r_{\lambda\mu}^{2}}\;,$
$\displaystyle=\sum_{\lambda<\mu}\sum_{k=1}^{3}v_{0}^{(k)}\operatorname{e}^{-\mu^{(k)}r_{\lambda\mu}^{2}}+\frac{1}{2}\sum_{\lambda\neq\mu}\sum_{\alpha}\sigma_{\lambda\alpha}\,C_{\lambda\mu}^{[\sigma]}\,\sigma_{\mu\alpha}\;,$
(3.228)
where
$\displaystyle
C_{\lambda\mu}^{[\sigma]}=\sum_{k=1}^{3}v_{\sigma}^{(k)}\operatorname{e}^{-\mu^{(k)}r_{\lambda\mu}^{2}}\;.$
(3.229)
The first term of $V_{\Lambda\Lambda}$ is a pure central factor to be included
in $V_{\Lambda\Lambda}^{c}$, while the second part has exactly the same form
of the isospin component of Eq. (3.126). We can thus diagonalize the $C$
matrix and define a new operator $\mathcal{O}_{n,\alpha}^{[\sigma_{\Lambda}]}$
starting from the eigenvectors $\psi_{n,\lambda}^{[\sigma_{\Lambda}]}$:
$\displaystyle\mathcal{O}_{n,\alpha}^{[\sigma_{\Lambda}]}$
$\displaystyle=\sum_{\lambda}\sigma_{\lambda\alpha}\,\psi_{n,\lambda}^{[\sigma_{\Lambda}]}\;.$
(3.230)
In this way, the spin dependent part of the hyperon-hyperon interaction
becomes
$\displaystyle
V_{\Lambda\Lambda}^{SD}=\frac{1}{2}\sum_{n=1}^{\mathcal{N}_{\Lambda}}\sum_{\alpha=1}^{3}\lambda_{n}^{[\sigma_{\Lambda}]}\\!\left(\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda}]}\right)^{2}\;,$
(3.231)
and we can apply the Hubbard-Stratonovich transformation to linearize the
square dependence of $\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda}]}$ introducing
the integration over $3\,\mathcal{N}_{\Lambda}$ new auxiliary fields and the
relative $|S^{\prime}_{\Lambda}\rangle$ rotations.
At the end, using the diagonalization of the potential matrices and the
derivation reported in this section, the spin-isospin dependent part of the
nuclear and hypernuclear two-body potentials (but spin-orbit term for
simplicity) reads:
$\displaystyle V_{NN}^{SD}+V_{\Lambda N}^{SD}$
$\displaystyle=\frac{1}{2}\sum_{n=1}^{3\mathcal{N}_{N}}\lambda_{n}^{[\sigma]}\\!\left(\mathcal{O}_{n}^{[\sigma]}\right)^{2}\\!+\frac{1}{2}\sum_{n=1}^{3\mathcal{N}_{N}}\sum_{\alpha=1}^{3}\lambda_{n}^{[\sigma\tau]}\\!\left(\mathcal{O}_{n\alpha}^{[\sigma\tau]}\right)^{2}\\!+\frac{1}{2}\sum_{n=1}^{\mathcal{N}_{N}}\sum_{\alpha=1}^{3}\lambda_{n}^{[\tau]}\\!\left(\mathcal{O}_{n\alpha}^{[\tau]}\right)^{2}$
$\displaystyle\,+\frac{1}{2}\sum_{n=1}^{\mathcal{N}_{\Lambda}}\sum_{\alpha=1}^{3}\lambda_{n}^{[\sigma_{\Lambda}]}\\!\left(\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda}]}\right)^{2}\\!+\frac{1}{2}\sum_{n=1}^{\mathcal{N}_{N}\mathcal{N}_{\Lambda}}\\!\sum_{\alpha=1}^{3}B_{n}^{[\sigma]}\\!\left(\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda
N}]}\right)^{2}\\!+\sum_{i=1}^{\mathcal{N}_{N}}B_{i}^{[\tau]}\,\tau_{i}^{z}\;.$
(3.232)
Using a compact notation, the AFDMC propagator for hypernuclear systems of Eq.
(3.217) with the inclusion of spin-isospin degrees of freedom becomes:
$\displaystyle G(R,S,R^{\prime},S^{\prime},d\tau)=\langle
R,S|\operatorname{e}^{-(T_{N}+T_{\Lambda}+V_{NN}+V_{\Lambda\Lambda}+V_{\Lambda
N}-E_{T})d\tau}|R^{\prime},S^{\prime}\rangle\;,$ $\displaystyle\hskip
3.99994pt\simeq\left(\frac{1}{4\pi
D_{N}d\tau}\right)^{\frac{3\mathcal{N}_{N}}{2}}\\!\\!\left(\frac{1}{4\pi
D_{\Lambda}d\tau}\right)^{\frac{3\mathcal{N}_{\Lambda}}{2}}\\!\\!\operatorname{e}^{-\frac{(R_{N}-R^{\prime}_{N})^{2}}{4D_{N}d\tau}}\operatorname{e}^{-\frac{(R_{\Lambda}-R^{\prime}_{\Lambda})^{2}}{4D_{\Lambda}d\tau}}\times$
$\displaystyle\hskip
17.07182pt\times\operatorname{e}^{-\left(\widetilde{V}_{NN}^{c}(R_{N},R^{\prime}_{N})+\widetilde{V}_{\Lambda\Lambda}^{c}(R_{\Lambda},R^{\prime}_{\Lambda})+\widetilde{V}_{\Lambda
N}^{c}(R_{N},R_{\Lambda},R^{\prime}_{N},R^{\prime}_{\Lambda})-E_{T}\right)d\tau}$
$\displaystyle\hskip 17.07182pt\times\langle
S_{N},S_{\Lambda}|\prod_{i=1}^{\mathcal{N}_{N}}\operatorname{e}^{-B_{i}^{[\tau]}\,\tau_{i}^{z}\,d\tau}\prod_{n=1}^{\mathcal{M}}\frac{1}{\sqrt{2\pi}}\int\\!dx_{n}\operatorname{e}^{\frac{-x_{n}^{2}}{2}+\sqrt{-\lambda_{n}d\tau}\,x_{n}\mathcal{O}_{n}}|S^{\prime}_{N},S^{\prime}_{\Lambda}\rangle\;,$
(3.233)
where $|R,S\rangle\equiv|R_{N},R_{\Lambda},S_{N},S_{\Lambda}\rangle$ is the
state containing all the coordinates of the baryons and
$\widetilde{V}_{NN}^{c}$, $\widetilde{V}_{\Lambda\Lambda}^{c}$ and
$\widetilde{V}_{\Lambda N}^{c}$ defined in Eqs. (3.218) contain all the
possible central factors. Formally,
$\mathcal{M}=15\,\mathcal{N}_{N}+3\,\mathcal{N}_{\Lambda}+3\,\mathcal{N}_{N}\mathcal{N}_{\Lambda}$
and $\mathcal{O}_{n}$ stays for the various operators of Eq. (3.232), which
have a different action on the spinors
$|S^{\prime}_{N},S^{\prime}_{\Lambda}\rangle$. The
$\mathcal{O}_{n}^{[\sigma]}$, $\mathcal{O}_{n\alpha}^{[\sigma\tau]}$ and
$\mathcal{O}_{n\alpha}^{[\tau]}$ act on the nucleon spinor
$|S^{\prime}_{N}\rangle$. The $\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda}]}$
rotates the lambda spinor $|S^{\prime}_{\Lambda}\rangle$.
$\mathcal{O}_{n\alpha}^{[\sigma_{\Lambda N}]}$ acts on both baryon spinors
with a separate rotation for nucleons and hyperons coupled by the same
coefficient $(-B_{n}^{[\sigma]}d\tau)^{1/2}\,x_{n}$ (recall Eq. (3.223)). The
algorithm follows then the nuclear version (§ 3.2.1) with the sampling of the
nucleon and hyperon coordinates and of the auxiliary fields, one for each
linearized operator. The application of the propagator of Eq. (3.233) has the
effect to rotate the spinors of the baryons. The weight for each walker is
then calculate starting from the central part of the interaction with possible
counter terms coming from the importance sampling on spacial coordinates and
on auxiliary fields (algorithm _v1_), or by means of the local energy
(algorithm _v2_). Fixed phase approximation, branching process and
expectation values are the same discussed in § 3.1.
##### Three-body terms
We have already shown in § 3.2.3 that for neutron systems the three-body
nucleon force can be recast as a sum of two-body terms only. In the case of
the three-body $\Lambda NN$ interaction it is possible to verify that the same
reduction applies both for nucleon and neutron systems. Let consider the full
potential of Eqs. (2.43) and (2.47)
$\displaystyle V_{\Lambda NN}=\sum_{\lambda,i<j}\left(v_{\lambda
ij}^{2\pi,P}+v_{\lambda ij}^{2\pi,S}+v_{\lambda ij}^{D}\right)\;,$ (3.234)
and assume the notations:
$\displaystyle T_{\lambda i}=T_{\pi}(r_{\lambda i})\quad\quad Y_{\lambda
i}=Y_{\pi}(r_{\lambda i})\quad\quad Q_{\lambda i}=Y_{\lambda i}-T_{\lambda
i}\;.$ (3.235)
By expanding the operators over the Cartesian components as done in Eqs.
(3.128) and (3.129), it is possible to derive the following relations:
$\displaystyle V_{\Lambda NN}^{2\pi,S}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta\gamma}\tau_{i\gamma}\,\sigma_{i\alpha}\left(-\frac{C_{P}}{3}\sum_{\lambda}\sum_{\delta}\Theta_{\lambda
i}^{\alpha\delta}\,\Theta_{\lambda
j}^{\beta\delta}\right)\sigma_{j\beta}\,\tau_{j\gamma}\;,$ (3.236)
$\displaystyle V_{\Lambda NN}^{2\pi,P}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta\gamma}\tau_{i\gamma}\,\sigma_{i\alpha}\,\Xi_{i\alpha,j\beta}\,\sigma_{j\beta}\,\tau_{j\gamma}\;,$
(3.237) $\displaystyle V_{\Lambda NN}^{D}$
$\displaystyle=W_{D}\\!\sum_{\lambda,i<j}T^{2}_{\lambda i}\,T^{2}_{\lambda
j}+\frac{1}{2}\sum_{\lambda i}\sum_{\alpha}\sigma_{\lambda\alpha}\,D_{\lambda
i}^{[\sigma]}\,\sigma_{i\alpha}\;,$ (3.238)
where
$\displaystyle\Theta_{\lambda i}^{\alpha\beta}$ $\displaystyle=Q_{\lambda
i}\,\delta^{\alpha\beta}+3\,T_{\lambda i}\hat{r}_{\lambda
i}^{\alpha}\,\hat{r}_{\lambda i}^{\beta}\;,$ (3.239)
$\displaystyle\Xi_{i\alpha,j\beta}$
$\displaystyle=\frac{1}{9}C_{S}\,\mu_{\pi}^{2}\sum_{\lambda}\,Q_{i\lambda}\,Q_{\lambda
j}\,|r_{i\lambda}||r_{j\lambda}|\,\hat{r}_{i\lambda}^{\alpha}\,\hat{r}_{j\lambda}^{\beta}\;,$
(3.240) $\displaystyle D_{\lambda i}^{[\sigma]}$
$\displaystyle=\frac{1}{3}W_{D}\\!\sum_{j,j\neq i}T^{2}_{\lambda
i}\,T^{2}_{\lambda j}\;.$ (3.241)
By combining then Eq. (3.239) and Eq. (3.240), the TPE term of the three-body
hyperon-nucleon interaction can be recast as:
$\displaystyle V_{\Lambda NN}^{2\pi}=\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta\gamma}\tau_{i\gamma}\,\sigma_{i\alpha}\,D_{i\alpha,j\beta}^{[\sigma\tau]}\,\sigma_{j\beta}\,\tau_{j\gamma}\;,$
(3.242)
where
$\displaystyle D_{i\alpha,j\beta}^{[\sigma\tau]}$
$\displaystyle=\sum_{\lambda}\Bigg{\\{}\\!-\frac{1}{3}C_{P}Q_{\lambda
i}Q_{\lambda j}\delta_{\alpha\beta}-C_{P}Q_{\lambda i}T_{\lambda
j}\,\hat{r}_{j\lambda}^{\,\alpha}\,\hat{r}_{j\lambda}^{\,\beta}-C_{P}Q_{\lambda
j}T_{\lambda i}\,\hat{r}_{i\lambda}^{\,\alpha}\,\hat{r}_{i\lambda}^{\,\beta}$
$\displaystyle\quad+\left[-3\,C_{P}T_{\lambda i}T_{\lambda
j}\left({\sum_{\delta}}\,\hat{r}_{i\lambda}^{\,\delta}\,\hat{r}_{j\lambda}^{\,\delta}\right)+\frac{1}{9}C_{S}\mu_{\pi}^{2}Q_{\lambda
i}Q_{\lambda
j}\,|r_{i\lambda}||r_{j\lambda}|\right]\,\hat{r}_{i\lambda}^{\,\alpha}\,\hat{r}_{j\lambda}^{\,\beta}\Bigg{\\}}\;.$
(3.243)
Finally, the $\Lambda NN$ interaction takes the following form:
$\displaystyle V_{\Lambda NN}$
$\displaystyle=W_{D}\\!\sum_{\lambda,i<j}T^{2}_{\lambda i}\,T^{2}_{\lambda
j}+\frac{1}{2}\sum_{\lambda i}\sum_{\alpha}\sigma_{\lambda\alpha}\,D_{\lambda
i}^{[\sigma]}\,\sigma_{i\alpha}$ $\displaystyle\quad\,+\frac{1}{2}\sum_{i\neq
j}\sum_{\alpha\beta\gamma}\tau_{i\gamma}\,\sigma_{i\alpha}\,D_{i\alpha,j\beta}^{[\sigma\tau]}\,\sigma_{j\beta}\,\tau_{j\gamma}\;.$
(3.244)
The first term is a pure central factor that can be included in $V_{\Lambda
N}^{c}$. The second factor is analogous to the
$\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$ term (3.219) of the two body
hyperon-nucleon interaction. The last term acts only on the spin-isospin of
the two nucleons $i$ and $j$ and has the same structure of the nuclear
$\bm{\sigma}\cdot\bm{\tau}$ contribution described by the matrix
$A_{i\alpha,j\beta}^{[\sigma\tau]}$. The three-body hyperon-nucleon
interaction is then written as a sum of two-body operators only, of the same
form of the ones already discussed for the $NN$ and $\Lambda N$ potentials. We
can therefore include also these contributions in the AFDMC propagator of Eq.
(3.233) by simply redefining the following matrices:
$\displaystyle B_{\lambda i}^{[\sigma]}$ $\displaystyle\longrightarrow
B_{\lambda i}^{[\sigma]}+D_{\lambda i}^{[\sigma]}\;,$ (3.245) $\displaystyle
A_{i\alpha,j\beta}^{[\sigma\tau]}$ $\displaystyle\longrightarrow
A_{i\alpha,j\beta}^{[\sigma\tau]}+D_{i\alpha,j\beta}^{[\sigma\tau]}\;.$
(3.246)
The algorithm follows then the steps already discussed in the previous
section. Note that in the case of pure neutron systems, the last term of Eq.
(3.244) simply reduces to a $\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ contribution
that is included in the propagator by redefining the nuclear matrix
$A_{i\alpha,j\beta}^{[\sigma]}$.
With the AFDMC method extended to the hypernuclear sector, we can study finite
and infinite lambda-nucleon and lambda-neutron systems. In the first case we
can treat Hamiltonians that include the full hyperon-nucleon, hyperon-nucleon-
nucleon and hyperon-hyperon interaction of Chapter 2, but we are limited to
the Argonne V6 like potentials for the nuclear sector. However it has been
shown that this approach gives good results for finite nuclei [31] and nuclear
matter [35, 36]. In the latter case, instead, we can also add the nucleon
spin-orbit contribution, so AV8 like potentials, and the three-neutron force.
The neutron version of the AFDMC code has been extensively and successfully
applied to study the energy differences of oxygen [30] and calcium [32]
isotopes, the properties of neutron drops [33, 20, 34] and the properties of
neutron matter in connection with astrophysical observables [37, 38, 39, 40].
Very recently, the AFDMC algorithm has been also used to perform calculations
for neutron matter using chiral effective field theory interactions [132].
## Chapter 4 Results: finite systems
This chapter reports on the analysis of finite systems, nuclei and
hypernuclei. For single $\Lambda$ hypernuclei a direct comparison of energy
calculations with experimental results is given for the $\Lambda$ separation
energy, defined as:
$\displaystyle
B_{\Lambda}\left(\,{}^{A}_{\Lambda}\text{Z}\,\right)=E\left(\,{}^{A-1}\text{Z}\,\right)-E\left(\,{}^{A}_{\Lambda}\text{Z}\,\right)\;,$
(4.1)
where, using the notation of the previous chapters, ${}^{A}_{\Lambda}\text{Z}$
refers to the hypernucleus and ${}^{A-1}\text{Z}$ to the corresponding
nucleus. $E$ is the binding energy of the system, i.e. the expectation value
of the Hamiltonian on the ground state wave function
$\displaystyle
E(\kappa)=\frac{\langle\psi_{0,\kappa}|H_{\kappa}|\psi_{0,\kappa}\rangle}{\langle\psi_{0,\kappa}|\psi_{0,\kappa}\rangle}\;,\quad\quad\kappa=\text{nuc},\text{hyp}\;,$
(4.2)
that we can compute by means of the AFDMC method. In the case of double
$\Lambda$ hypernuclei, the interesting experimental observables we can have
access are the double $\Lambda$ separation energy
$\displaystyle B_{\Lambda\Lambda}\left(\,{}^{\leavevmode\nobreak\
\,A}_{\Lambda\Lambda}\text{Z}\,\right)=E\left(\,{}^{A-2}\text{Z}\,\right)-E\left(\,{}^{\leavevmode\nobreak\
\,A}_{\Lambda\Lambda}\text{Z}\,\right)\;,$ (4.3)
and the incremental $\Lambda\Lambda$ energy
$\displaystyle\Delta B_{\Lambda\Lambda}\left(\,{}^{\leavevmode\nobreak\
\,A}_{\Lambda\Lambda}\text{Z}\,\right)=B_{\Lambda\Lambda}\left(\,{}^{\leavevmode\nobreak\
\,A}_{\Lambda\Lambda}\text{Z}\,\right)-2B_{\Lambda}\left(\,{}^{A-1}_{\quad\;\Lambda}\text{Z}\,\right)\;.$
(4.4)
The calculation of these quantities proceeds thus with the computation of the
binding energies for both strange and non strange systems. Moreover it is
interesting to compare other observables among the systems with strangeness
$0$, $-1$ and $-2$, such as the single particle densities. By looking at the
densities in the original nucleus and in the one modified by the addition of
the lambda particles, information about the hyperon-nucleon interaction can be
deduced.
As reported in Ref. [31], the ground state energies of 4He, 8He, 16O and 40Ca
have been computed using the AV6’ potential (§ 2.1.1). Due to the limitations
in the potential used, the results cannot reproduce the experimental energies
and all the nuclei result less bound than expected. However, given the same
simplified interaction, the published AFDMC energies are close to the GFMC
results, where available. AFDMC has also been used to compute the energy
differences between oxygen [30] and calcium [32] isotopes, by studying the
external neutrons with respect to a nuclear core obtained from Hartree-Fock
calculations using Skyrme forces. In this case the results are close to the
experimental ones.
The idea behind the AFDMC analysis of $\Lambda$ hypernuclei follows in some
sense the one assumed in the study of oxygen and calcium isotopes by the
analysis of energy differences. The two-body nucleon interaction employed is
limited to the first six operators of AV18. However, if we use the same
potential for the nucleus and the core of the corresponding hypernucleus, and
take the difference between the binding energies of the two systems, the
uncertainties in the $NN$ interaction largely cancel out. We shall see that
this assumption, already used in other works [148, 181], is indeed consistent
with our results, thereby confirming that the specific choice of the nucleon
Hamiltonian does not significantly affect the results on $B_{\Lambda}$. On the
grounds of this observation, we can focus on the interaction between hyperons
and nucleons, performing QMC simulations with microscopic interactions in a
wide mass range.
### 4.1 Nuclei
Let us start with the AFDMC study of finite nuclei. In the previous chapter,
we have seen that two versions of the AFDMC algorithm, that should give the
same results, are available (_v1_ and _v2_). Before including the strange
degrees of freedom, we decided to test the stability and accuracy of the two
algorithms, within the fixed phase approximation, by performing some test
simulations on 4He. The result of $-27.13(10)$ MeV for the AV6’ potential
reported in Ref. [31], was obtained employing the algorithm _v2_ using single
particle Skyrme orbitals and a particular choice of the parameters for the
solution of the Jastrow correlation equation (see § 3.2.4). In order to check
the AFDMC projection process, we tried to modify the starting trial wave
function:
* •
we changed the healing distance $d$ and the quencher parameter $\eta$ for the
Jastrow function $f_{c}^{NN}$;
* •
we used a different set of radial functions, labelled as HF-B1 [232], coming
from Hartree-Fock calculations for the effective interaction B1 of Ref. [233].
The B1 is a phenomenological two-body nucleon-nucleon potential fitted in
order to reproduce the binding energies and densities of various light nuclei
and of nuclear matter in the HF approximation.
Although a central correlation function should not affect the computed energy
value, in the version _v2_ of the algorithm an unpleasant dependence on
$f_{c}^{NN}$ was found, and in particular as a function of the quencher
parameter $\eta$. This dependence is active for both the AV4’ and the AV6’
potentials, regardless of the choice of the single particle orbitals. The time
step extrapolation ($d\tau\rightarrow 0$) of the energy does not solve the
issue. Energy differences are still more than 1 MeV among different setups for
the trial wave function. By varying the parameter $\eta$ from zero (no Jastrow
at all) to one (full central channel of the $NN$ potential used for the
solution of Eq. (3.192)), the energies increase almost linearly. For example,
in the case of AV4’ for the Skyrme orbital functions, the energy of 4He goes
from $-31.3(2)$ MeV for $\eta=0$, to $-27.2(2)$ MeV for $\eta=1$. Same effect
is found for the HF-B1 orbitals with energies going from $-32.5(2)$ MeV to
$-28.4(2)$ MeV. The inclusion of the pure central Jastrow introduces thus
strong biases in the evaluation of the total energy. Moreover, there is also a
dependence on the choice of the single particle orbitals, as shown from the
results for AV4’. Same conclusions follow for the AV6’ potential.
On the grounds of these observations we moved from the AFDMC local energy
scheme to the importance function ratio scheme (version _v1_), with no
importance sampling on auxiliary fields. In this case the bias introduced by
the Jastrow correlation function is still present but reduced to $0.3\div 0.4$
MeV for AV4’ and $0.1\div 0.2$ MeV for AV6’. In spite of the improvement with
respect to the previous case, we decided to remove this source of uncertainty
from the trial wave function and proceed with the test of the _v1_ algorithm
with no Jastrow. It has to be mentioned that a new sampling procedure, for
both coordinates and auxiliary fields, capable to reduce the dependence on
central correlations is being studied.
As shown in Figs. 4.1 and 4.2, the _v1_ extrapolated energies obtained using
different single particle orbitals are consistent within the Monte Carlo
statistical errors, both for the AV4’ and the AV6’ potentials.
Figure 4.1: Binding energy of 4He as a function of the Monte Carlo imaginary
time step. Results are obtained using the AV4’ $NN$ potential. Red dots are
the AFDMC results for the Skyrme radial orbitals. Blue triangles the ones for
the HF-B1 orbitals. For comparison, the GFMC result of Ref. [198], corrected
by the Coulomb contribution (see text for details), is reported with the green
band. Figure 4.2: Binding energy of 4He as a function of the Monte Carlo
imaginary time step. Results are obtained using the AV6’ $NN$ potential. As in
Fig. 4.1, red dots refers to the AFDMC results for the Skyrme radial functions
and blue triangles for the HF-B1 orbitals. The green arrow points to the GFMC
result.
For $\mathcal{N}_{N}=4$ we can compare the AFDMC results with the GFMC ones.
In our calculations the Coulomb interaction is not included. A precise VMC
estimate, that should be representative also for the GFMC estimate, of the
Coulomb expectation value for 4He is $0.785(2)$ MeV [234]. The AFDMC values of
$-32.67(8)$ MeV (Skyrme) and $-32.7(1)$ MeV (HF-B1) for 4He with AV4’ are thus
very close to the GFMC $-32.11(2)$ MeV of Ref. [198] for the same potential
once the Coulomb contribution is subtracted. Our results are still $\sim
0.1\div 0.2$ MeV above the GFMC one, most likely due to the removal of the
sign problem constraint applied at the end of the GFMC runs (release node
procedure [235]).
Although AFDMC and GFMC energies for 4He described by the AV4’ potential are
consistent, a clear problem appears using the AV6’ interaction (Fig. 4.2).
With the two sets of radial functions, the energies are $-19.59(8)$ MeV
(Skyrme) and $-19.53(13)$ MeV (HF-B1) and thus the AFDMC actually projects out
the same ground state. However, the GFMC estimate is $-26.15(2)$ MeV minus the
Coulomb contribution. This large difference in the energies cannot be
attributed to the GFMC release node procedure. The difference in using AV4’
and AV6’ is the inclusion of the tensor term $S_{ij}$ of Eq. (2.8). The
Hamiltonian moves then from real to complex and this might result in a phase
problem during the imaginary time propagation. There might be some issues with
the fixed phase approximation or with the too poor trial wave function (or
both), which does not include operatorial correlations. This is still an
unsolved question but many ideas are being tested. According to the lack of
control on the AFDMC simulations for the AV6’ potential, from now on we will
limit the study to AV4’. As we shall see, this choice does not affect the
result on energy differences as the hyperon separation energy, which is the
main observable of this study for finite systems.
In order to complete the check of the accuracy of the algorithm _v1_ for 4He,
we performed simulations using the Minnesota potential of Ref. [236]. This
two-nucleon interaction has the same operator structure of AV4’ but much
softer cores. Our AFDMC result for the energy is $-30.69(7)$ MeV. It has to be
compared with the $-29.937$ MeV ($-30.722$ MeV with the Coulomb subtraction)
obtained with the Stochastic Variational Method (SVM) [237], that has been
proven to give consistent results with the GFMC algorithm for 4He [21]. The
agreement of the results is remarkable.
Moreover, we tested the consistency of the _v1_ algorithm for the AV4’
potential by studying the deuteron, tritium and oxygen nuclei.
* •
The AFDMC binding energy for 2H is $-2.22(5)$ MeV, in agreement with the
experimental $-2.225$ MeV. The result is significant because, although the
Argonne V4’ was exactly fitted in order to reproduce the deuteron energy, our
starting trial wave function is just a Slater determinant of single particle
orbitals, with no correlations.
* •
The result for 3H is $-8.74(4)$ MeV, close to the GFMC $-8.99(1)$ MeV of Ref.
[198]. As for 4He, the small difference in the energies is probably due to the
release node procedure in GFMC. Without the Coulomb contribution, we obtained
the same energy $-8.75(4)$ MeV also for 3He. In AV4’ there are no charge
symmetry breaking terms. Therefore, this result can be seen as a consistency
test on the correct treatment of the spin-isospin operators acting on the wave
function during the Hubbard-Stratonovich rotations.
* •
For 16O we found the energy values of $-176.8(5)$ MeV for the Skyrme orbitals
and $-174.3(8)$ MeV for the HF-B1 radial functions. The energy difference is
of order 1% even for a medium mass nucleus. The projection mechanism is
working accurately regardless the starting trial function. GFMC results are
limited to 12 nucleons [17, 18, 19], so we cannot compare the two methods for
$\mathcal{N}_{N}=16$. The binding energy cannot be compared with the
experimental data due to the poor employed Hamiltonian. However the AFDMC
results are consistent with the overbinding predicted by the available GFMC
energies for AV4’ [198] and the nucleus results stable under alpha particle
break down, as expected.
On the grounds of the results of these consistency checks, in the present work
we adopt the version _v1_ of the AFDMC algorithm employing the nuclear
potential AV4’ for both nuclei and hypernuclei.
System | $E_{\text{AFDMC}}$ | $E_{\text{GFMC}}$ | $E_{\text{exp}}$ | $E_{\text{AFDMC}}/\mathcal{N}_{N}$ | $E_{\text{exp}}/\mathcal{N}_{N}$
---|---|---|---|---|---
2H | -2.22(5) | — | -2.225 | -1.11 | -1.11
3H | -8.74(4) | -8.99(1) | -8.482 | -2.91 | -2.83
3He | -8.75(4) | — | -7.718 | -2.92 | -2.57
4He | -32.67(8) | -32.90(3) | -28.296 | -8.17 | -7.07
5He | -27.96(13) | -31.26(4) | -27.406 | -5.59 | -5.48
6He | -29.87(14) | -33.00(5) | -29.271 | -4.98 | -4.88
12C | -77.31(25)* | — | -92.162 | -6.44 | -7.68
15O | -144.9(4) | — | -111.955 | -9.66 | -7.46
16O | -176.8(5) | — | -127.619 | -11.05 | -7.98
17O | -177.0(6) | — | -131.762 | -10.41 | -7.75
40Ca | -597(3) | — | -342.052 | -14.93 | -8.55
48Ca | -645(3) | — | -416.001 | -13.44 | -8.67
90Zr | -1457(6) | — | -783.899 | -16.19 | -8.71
Table 4.1: Binding energies (in MeV) for different nuclei. AFDMC and GFMC
results are obtained using the the AV4’ $NN$ potential. The GFMC data are from
Ref. [198] corrected by the Coulomb contribution (see text for details). In
the fourth column the experimental results are from Ref. [238]. Errors are
less than 0.1 KeV. In the last two columns the calculated and experimental
binding energies per particle. For the note * on 12C see the text.
As reported in Tab. 4.1, the resulting absolute binding energies using AV4’
are not comparable with experimental ones, as expected, due to the lack of
information about the nucleon interaction in the Hamiltonian. With the
increase of the number of particles, the simulated nuclei become more an more
bound until the limit case of 90Zr, for which the estimated binding energy is
almost twice the experimental one. Looking at the results for helium isotopes,
we can see that for $\mathcal{N}_{N}=3$ and $4$ the energies are compatible
with GFMC calculations, once the Coulomb contribution is removed. For 5He and
6He instead, we obtained discrepancies between the results for the two
methods. However this is an expected result. When moving to open shell
systems, as 5He and 6He with one or two neutrons out of the first $s$ shell,
the structure of the wave function becomes more complicated and results are
more dependent on the employed $\psi_{T}$. For example, in the case of 6He, in
order to have total angular momentum zero, the two external neutrons can
occupy the $1p_{3/2}$ or the $1p_{1/2}$ orbitals of the nuclear shell model
classification. By using just one of the two $p$ shells, one gets the
unphysical result $E(^{5}\text{He})<E(^{6}\text{He})$. The reported binding
energy has been instead obtained by considering the linear combination of the
Slater determinants giving $J=0$
$\displaystyle\Phi(R_{N},S_{N})=(1-c)\det\Bigl{\\{}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}_{1p_{3/2}}\\!\\!+c\,\det\Bigl{\\{}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}_{1p_{1/2}}\;,$
(4.5)
and minimizing the energy with respect to the mixing parameter $c$, as shown
in Fig. 4.3. However the final result is still far from the GFMC data. This is
a clear indication that better wave functions are needed for open shell
systems. A confirmation of that is the non physical result obtained for the
12C nucleus (marked in Tab. 4.1 with *), which is even less bound than
expected, although the employed AV4’ potential, resulting thus unstable under
$\alpha$ break down. In the case of $\mathcal{N}_{N}=12$ indeed, the 8
additional neutrons and protons to the alpha core have just been placed in the
$1p_{3/2}$ shell without any linear combination of the other possible setups
giving zero total angular momentum. This result will be useful in the hyperon
separation energy estimate anyway. In fact, we shall see in the next section
that regardless the total binding energies, by using the same nucleon
potential to describe nuclei and the core of hypernuclei, the obtained hyperon
separation energy is in any case realistic.
Last comment on a technical detail regarding the computation of AFDMC
observables. As shown in Fig. 4.1 and 4.2, the extrapolation of the energy
values in the limit $d\tau\rightarrow 0$ is linear. This is consistent with
the application of the Trotter-Suzuki formula of Eq. (3.17) in the Hubbard-
Stratonovich transformation (3.136), that is thus correct at order
$\sqrt{d\tau}^{\,2}$. Focusing on the AV4’ case, for 4He the time step
extrapolation is almost flat. The differences between the final results and
the energies computed at large $d\tau$ are less than $0.5\%$ and almost within
the statistical errors of the Monte Carlo run. The situation dramatically
changes with the increase of the particle number. For $\mathcal{N}_{N}=16$
this difference is around $2\%$. For 40 and 48 particles, large time step
values and the extrapolated ones are, respectively, $6\%$ and $8.5\%$
different. Therefore, the binding energies must always be carefully studied by
varying the time step of the AFDMC run. The same behavior has been found for
observables other than the total energy (single particle densities and radii).
Each reported result in this chapter has been thus obtained by means of a
computationally expensive procedure of imaginary time extrapolation.
Figure 4.3: 6He binding energy as a function of the mixing parameter $c$ of
Eq. (4.5). The arrows point to the results for the pure $1p_{3/2}$
($-27.65(8)$ MeV) and $1p_{1/2}$ ($-25.98(8)$ MeV) configurations used for the
two external neutrons. The green line is the GFMC result of Ref. [198]
corrected by the VMC Coulomb expectation contribution $0.776(2)$ MeV [234].
### 4.2 Single $\Lambda$ hypernuclei
When a single $\Lambda$ particle is added to a core nucleus, the wave function
of Eq. (3.202) is given by
$\displaystyle\psi_{T}(R,S)=\prod_{i}f_{c}^{\Lambda N}(r_{\Lambda
i})\,\psi_{T}^{N}(R_{N},S_{N})\,\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\Lambda},s_{\Lambda})\;.$
(4.6)
The structure of the nucleon trial wave function is the same of Eq. (3.183),
used in the AFDMC calculations for nuclei. The hyperon Slater determinant is
simply replaced by the single particle state $\varphi_{\epsilon}^{\Lambda}$ of
Eq. (3.209), assumed to be the neutron $1s_{1/2}$ radial function, as already
described in the previous chapter. In order to be consistent with the
calculations for nuclei, we neglected the Jastrow $\Lambda N$ correlation
function which was found to produce a similar but smaller bias on the total
energy. As radial functions we used the same Skyrme set employed in the
calculations for the nuclei of Tab. 4.1.
The $\Lambda$ separation energies defined in Eq. (4.1), are calculated by
taking the difference between the nuclei binding energies presented in the
previous section, and the AFDMC energies for hypernuclei, given the same
nucleon potential. By looking at energy differences, we studied the
contribution of the $\Lambda N$ and $\Lambda NN$ terms defined in Chapter 2.
By comparing AFDMC results with the expected hyperon separation energies,
information about the hyperon-nucleon interaction are deduced. Some
qualitative properties have been also obtained by studying the nucleon and
hyperon single particle densities and the root mean square radii.
#### 4.2.1 Hyperon separation energies
We begin the study of $\Lambda$ hypernuclei with the analysis of closed shell
hypernuclei, in particular ${}^{5}_{\Lambda}$He and
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O. We have seen in the previous
section that the AFDMC algorithm is most accurate in describing closed shell
nuclei. Results for 4He and 16O with the AV4’ potential are indeed consistent
and under control. This give us the possibility to realistically describe the
hyperon separation energy for such systems and deduce some general properties
of the employed hyperon-nucleon force.
The step zero of this study was the inclusion in the Hamiltonian of the $NN$
AV4’ interaction and the two-body $\Lambda N$ charge symmetric potential of
Eq. (2.35). The employed parameters $\bar{v}$ and $v_{\sigma}$ are reported in
Tab 2.1. The exchange parameter $\varepsilon$ has been initially set to zero
due to the impossibility of including the space exchange operator directly in
the AFDMC propagator (see § 3.2.4). As reported in Tab. 4.2, the AV4’
$\Lambda$ separation energy for ${}^{5}_{\Lambda}$He is more than twice the
expected value. For the heavier ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O the
discrepancy is even larger. Actually, this is an expected result. As firstly
pointed out by Dalitz [148], $\Lambda N$ potentials, parameterized to account
for the low-energy $\Lambda N$ scattering data and the binding energy of the
$A=3,4$ hypernuclei, overbind ${}^{5}_{\Lambda}$He by $2\div 3$ MeV. That is,
the calculated $A=5$ $\Lambda$ separation energy is about a factor of 2 too
large. This fact is usually reported as _$A=5$ anomaly_. With only a $\Lambda
N$ potential fitted to $\Lambda p$ scattering, the heavier hypernuclei result
then strongly overbound.
$NN$ potential | ${}^{5}_{\Lambda}$He | ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O
---|---|---
$V_{\Lambda N}$ | $V_{\Lambda N}$+$V_{\Lambda NN}$ | $V_{\Lambda N}$ | $V_{\Lambda N}$+$V_{\Lambda NN}$
Argonne V4’ | 7.1(1) | 5.1(1) | 43(1) | 19(1)
Argonne V6’ | 6.3(1) | 5.2(1) | 34(1) | 21(1)
Minnesota | 7.4(1) | 5.2(1) | 50(1) | 17(2)
Expt. | 3.12(2) | 13.0(4)
Table 4.2: $\Lambda$ separation energies (in MeV) for ${}^{5}_{\Lambda}$He and
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O obtained using different nucleon
potentials (AV4’, AV6’, Minnesota) and different hyperon-nucleon interaction
(two-body alone and two-body plus three-body, set of parameters (I)) [41]. In
the last line the experimental $B_{\Lambda}$ for ${}^{5}_{\Lambda}$He is from
Ref. [77]. Since no experimental data for ${}^{17}_{\leavevmode\nobreak\
\Lambda}$O exists, the reference separation energy is the semiempirical value
reported in Ref. [192].
As suggested by the same Dalitz [148] and successively by Bodmer and Usmani
[181], the inclusion of a $\Lambda$-nucleon-nucleon potential may solve the
overbinding problem. This is indeed the case, as reported for instance in
Refs. [192, 184, 189]. Therefore, in our AFDMC calculations we included the
three-body $\Lambda NN$ interaction developed by Bodmer, Usmani and Carlson
and described in § 2.2.2. Among the available parametrizations coming from
different VMC studies of light hypernuclei, the set of parameters for the
$\Lambda NN$ potential has been originally taken from Ref. [185], being the
choice that made the variational $B_{\Lambda}$ for ${}_{\Lambda}^{5}$He and
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O compatible with the expected
results. It reads:
$\hypertarget{par_I}{(\text{I})}\phantom{I}\quad\left\\{\begin{array}[]{rcll}C_{P}&\\!=\\!&0.60&\\!\text{MeV}\\\
C_{S}&\\!=\\!&0.00&\\!\text{MeV}\\\
W_{D}&\\!=\\!&0.015&\\!\text{MeV}\end{array}\right.$
The inclusion of the $\Lambda NN$ force reduces the overbinding and thus the
hyperon separation energies, as reported in Tab. 4.2. Although the results are
still not compatible with the experimental ones, the gain in energy due to the
inclusion of the three-body hypernuclear force is considerable.
It has to be pointed out that this result might in principle depend on the
particular choice of the $NN$ interaction used to describe both nucleus and
hypernucleus. One of the main mechanisms that might generate this dependence
might be due to the different environment experienced by the hyperon in the
hypernucleus because of the different nucleon densities and correlations
generated by each $NN$ potential. To discuss this possible dependence, we
performed calculations with different $NN$ interactions having very different
saturation properties. As it can be seen from Tab. 4.2, for
${}^{5}_{\Lambda}$He the extrapolated $B_{\Lambda}$ values with the two-body
$\Lambda N$ interaction alone are about 10% off and well outside statistical
errors. In contrast, the inclusion of the three-body $\Lambda NN$ force gives
a similar $\Lambda$ binding energy independently to the choice of the $NN$
force. On the grounds of this observation, we feel confident that the use of
AV4’, for which AFDMC calculations for nuclei are under control, will in any
case return realistic estimates of $B_{\Lambda}$ for larger masses when
including the $\Lambda NN$ interaction. We checked this assumption performing
simulations in ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O, where the
discrepancy between the $\Lambda$ separation energy computed using the
different $NN$ interactions and the full $\Lambda N$+$\Lambda NN$ force is
less than few per cent (last column of Tab.4.2). The various $NN$ forces
considered here are quite different. The AV6’ includes a tensor force, while
AV4’ and Minnesota have a simpler structure with a similar operator structure
but very different intermediate- and short-range correlations. The fact that
the inclusion of the $\Lambda NN$ force does not depend too much on the
nuclear Hamiltonian is quite remarkable, because the different $NN$ forces
produce a quite different saturation point for the nuclear matter EoS,
suggesting that our results are pretty robust.
Figure 4.4: $\Lambda$ separation energy as a function of $A$ for closed shell
hypernuclei, adapted from Ref. [41]. Solid green dots (dashed curve) are the
available $B_{\Lambda}$ experimental or semiempirical values. Empty red dots
(upper banded curve) refer to the AFDMC results for the two-body $\Lambda N$
interaction alone. Empty blue diamonds (middle banded curve) are the results
with the inclusion also of the three-body hyperon-nucleon force in the
parametrization (I).
For ${}^{5}_{\Lambda}$He the hyperon separation energy with the inclusion of
the $\Lambda NN$ force with the set of parameters (I) reduces of a factor
$\sim 1.4$. For ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O the variation is
around $40\div 50\%$. In order to check the effect of the three-body force
with increasing the particle number, we performed simulations for the next
heavier closed or semi-closed shell $\Lambda$ hypernuclei,
${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca and ${}^{91}_{\leavevmode\nobreak\
\Lambda}$Zr. The $\Lambda$ separation energies for all the studied closed
shell hypernuclei are shown in Fig. 4.4. While the results for lighter
hypernuclei might be inconclusive in terms of the physical consistency of the
$\Lambda NN$ contribution to the hyperon binding energy in AFDMC calculations,
the computations for ${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca and
${}^{91}_{\leavevmode\nobreak\ \Lambda}$Zr reveal a completely different
picture. The saturation binding energy provided by the $\Lambda N$ force alone
is completely unrealistic, while the inclusion of the $\Lambda NN$ force gives
results that are much closer to the experimental behavior. Therefore, the
$\Lambda$-nucleon-nucleon force gives a very important repulsive contribution
towards a realistic description of the saturation of medium-heavy hypernuclei
[41]. However, with the given parametrization, only a qualitative agreement
wiht the expected separation energies is reproduced. A refitting procedure for
the three-body hyperon-nucleon interaction might thus improve the quality of
the results.
As already discussed in § 2.2.2, the $C_{S}$ parameter can be estimated by
comparing the $S$-wave term of the $\Lambda NN$ force with the Tucson-
Melbourne component of the $NNN$ interaction. We take the suggested
$C_{S}=1.50$ MeV value [189], in order to reduce the number of fitting
parameters. This choice is justified because the $S$-wave component of the
three-body $\Lambda NN$ interaction is sub-leading. We indeed verified that a
change in the $C_{S}$ value yields a variation of the total energy within
statistical error bars, and definitely much smaller than the variation in
energy due to a change of the $W_{D}$ parameter.
Figure 4.5: $\Lambda$ separation energy for ${}^{5}_{\Lambda}$He as a function
of strengths $W_{D}$ and $C_{P}$ of the three-body $\Lambda NN$ interaction
[43]. The red grid represents the experimental $B_{\Lambda}=3.12(2)$ MeV [77].
The dashed yellow curve is the interception between the expected result and
the $B_{\Lambda}$ surface in the $W_{D}-C_{P}$ parameter space. Statistical
error bars on AFDMC results (solid black dots) are of the order of $0.10\div
0.15$ MeV. Figure 4.6: Projection of Fig. 4.5 on the $W_{D}-C_{P}$ plane [43].
Error bars come from a realistic conservative estimate of the uncertainty in
the determination of the parameters due to the statistical errors of the Monte
Carlo calculations. Blue and green dashed, long-dashed and dot-dashed lines
(lower curves) are the variational results of Ref. [189] for different
$\varepsilon$ and $\bar{v}$ (two-body $\Lambda N$ potential). The dashed box
corresponds to the parameter domain of Fig. 4.5. Black empty dots and the red
band (upper curve) are the projected interception describing the possible set
of parameters reproducing the experimental $B_{\Lambda}$.
In Fig. 4.5 we report the systematic study of the $\Lambda$ separation energy
of ${}_{\Lambda}^{5}$He as a function of both $W_{D}$ and $C_{P}$. Solid black
dots are the AFDMC results. The red grid represents the experimental
$B_{\Lambda}=3.12(2)$ MeV [77]. The dashed yellow curve follows the set of
parameters reproducing the expected $\Lambda$ separation energy. The same
curve is also reported in Fig. 4.6 (red banded curve with black empty dots and
error bars), that is a projection of Fig. 4.5 on the $W_{D}-C_{P}$ plane. The
dashed box represents the $W_{D}$ and $C_{P}$ domain of the previous picture.
For comparison, also the variational results of Ref. [189] are reported. Green
curves are the results for $\bar{v}=6.15$ MeV and $v_{\sigma}=0.24$ MeV, blue
ones for $\bar{v}=6.10$ MeV and $v_{\sigma}=0.24$ MeV. Dashed, long-dashed and
dot-dashed lines correspond respectively to $\varepsilon=0.1$, $0.2$ and
$0.3$. In our calculations we have not considered different combinations for
the parameters of the two-body $\Lambda N$ interaction, focusing on the three-
body part. We have thus kept fixed $\bar{v}$ and $v_{\sigma}$ to the same
values of the green curves of Fig. 4.6 which are the same reported in Tab.
2.1. Moreover, we have set $\varepsilon=0$ for all the hypernuclei studied due
to the impossibility of exactly including the $\mathcal{P}_{x}$ exchange
operator in the propagator. A perturbative analysis of the effect of the
$v_{0}(r)\varepsilon(\mathcal{P}_{x}-1)$ term on the hyperon separation energy
is reported in § 4.2.1.
As it can be seen from Fig. 4.5, $B_{\Lambda}$ significantly increases with
the increase in $C_{P}$, while it decreases with $W_{D}$. This result is
consistent with the attractive nature of $V_{\Lambda NN}^{2\pi,P}$ and the
repulsion effect induced by $V_{\Lambda NN}^{D}$. It is also in agreement with
all the variational estimates on ${}^{5}_{\Lambda}$He (see for instance Refs.
[184, 189]). Starting from the analysis of the results in the $W_{D}-C_{P}$
space for ${}_{\Lambda}^{5}$He, we performed simulations for the next closed
shell hypernucleus ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O. Using the
parameters in the red band of Fig. 4.6 we identified a parametrization able to
reproduce the experimental $B_{\Lambda}$ for both ${}_{\Lambda}^{5}$He and
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O at the same time within the AFDMC
framework:
$\hypertarget{par_II}{(\text{II})}\quad\left\\{\begin{array}[]{rcll}C_{P}&\\!=\\!&1.00&\\!\text{MeV}\\\
C_{S}&\\!=\\!&1.50&\\!\text{MeV}\\\
W_{D}&\\!=\\!&0.035&\\!\text{MeV}\end{array}\right.$
Given the set (II), the $\Lambda$ separation energy of the closed shell
hypernuclei reported in Fig. 4.4 has been re-calculated. We have seen that
$B_{\Lambda}$ is not sensitive neither to the details of the $NN$ interaction,
nor to the total binding energies of nuclei and hypernuclei, as verified by
the good results in Tab. 4.2 even for the problematic case of AV6’ (see §
4.1). On the grounds of this observation, we tried to simulate also open shell
hypernuclei, using the $\Lambda N$, $\Lambda NN$ set (I) and $\Lambda NN$ set
(II) potentials. The binding energies for these systems might not be accurate,
as in the case of the corresponding nuclei. The hyperon separation energy is
expected to be in any case realistic. All the results obtained so far in the
mass range $3\leq A\leq 91$ are summarized in Fig. 4.7 and Fig. 4.8.
Figure 4.7: $\Lambda$ separation energy as a function of $A$. Solid green dots
(dashed curve) are the available $B_{\Lambda}$ experimental or semiempirical
values. Empty red dots (upper banded curve) refer to the AFDMC results for the
two-body $\Lambda N$ interaction alone. Empty blue diamonds (middle banded
curve) and empty black triangles (lower banded curve) are the results with the
inclusion also of the three-body hyperon-nucleon force, respectively for the
parametrizations (I) and (II). Figure 4.8: $\Lambda$ separation energy as a
function of $A^{-2/3}$, adapted from Ref. [43]. The key is the same of Fig.
4.7.
We report $B_{\Lambda}$ as a function of $A$ and $A^{-2/3}$, which is an
approximation of the $A$ dependence of the kinetic term of the Hamiltonian.
Solid green dots are the available experimental data, empty symbols the AFDMC
results. The red curve is obtained using only the two-body hyperon-nucleon
interaction in addition to the nuclear AV4’ potential. The blue curve refers
to the results for the same systems when also the three-body $\Lambda NN$
interaction with the old set of parameters (I) is included. The black lower
curve shows the results obtained by including the three-body hyperon-nucleon
interaction described by the new parametrization (II). A detailed comparison
between numerical and experimental results for the hyperon-separation energy
is given in Tab. 4.3.
System | $E$ | $B_{\Lambda}$ | Expt. $B_{\Lambda}$
---|---|---|---
${}^{3}_{\Lambda}$H | -1.00(14) | -1.22(15) | 0.13(5) [77]
${}^{4}_{\Lambda}$H | -9.69(8) | 0.95(9) | 2.04(4) [77]
${}^{4}_{\Lambda}$He | -9.97(8) | 1.22(9) | 2.39(3) [77]
${}^{5}_{\Lambda}$He | -35.89(12) | 3.22(14) | 3.12(2) [77]
${}^{6}_{\Lambda}$He | -32.72(15) | 4.76(20) | 4.25(10) [77]
${}^{7}_{\Lambda}$He | -35.82(15) | 5.95(25) | 5.68(28) [86]
${}^{13}_{\leavevmode\nobreak\ \Lambda}$C | -88.5(26)* | 11.2(4) | 11.69(12) [78]
${}^{16}_{\leavevmode\nobreak\ \Lambda}$O | -157.5(6) | 12.6(7) | 12.50(35) [80]
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O | -189.2(4) | 12.4(6) | 13.0(4) [192]
${}^{18}_{\leavevmode\nobreak\ \Lambda}$O | -189.7(6) | 12.7(9) | —
${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca | -616(3) | 19(4) | 19.24(0) [181]
${}^{49}_{\leavevmode\nobreak\ \Lambda}$Ca | -665(4) | 20(5) | —
${}^{91}_{\leavevmode\nobreak\ \Lambda}$Zr | -1478(7) | 21(9) | 23.33(0) [181]
Table 4.3: Binding energies and $\Lambda$ separation energies (in MeV)
obtained using the two-body plus three-body hyperon-nucleon interaction with
the set of parameters (II) [43]. The results already include the CSB
contribution. The effect is evident only for light systems, as discussed in
the next section. In the last column, the expected $B_{\Lambda}$ values. Since
no experimental data for $A=17,41,91$ exists, the reference separation
energies are semiempirical values.
From Fig. 4.7 and Fig. 4.8 we can see that the new parametrization for the
three-body hyperon-nucleon interaction correctly reproduces the experimental
saturation property of the $\Lambda$ separation energy. All the separation
energies for $A\geq 5$ are compatible or very close to the expected results,
where available, as reported in Tab. 4.3. Since for
${}^{18}_{\leavevmode\nobreak\ \Lambda}$O and ${}^{49}_{\leavevmode\nobreak\
\Lambda}$Ca no experimental data have been found, the values of $12.7(9)$ MeV
and $20(5)$ MeV are AFDMC predictions, that follows the general trend of the
experimental curve. Although for $A\geq 41$ the Monte Carlo statistical error
bars become rather large, the extrapolation of the $\Lambda$ binding energy
for $A\rightarrow\infty$ points to the correct region for the expected value
$D_{\Lambda}\sim 30$ MeV of $s_{\Lambda}$ states in nuclear matter.
We can find the same problems discussed in the case of nuclei (§ 4.1) in the
analysis of the total hypernuclear binding energies for $A\geq 5$. For
instance, the binding energy of ${}^{13}_{\leavevmode\nobreak\ \Lambda}$C is
non physical, as for the energy of the core nucleus 12C. However, the energy
difference is consistent with the expected result. Moreover, for the core wave
function of ${}^{7}_{\Lambda}$He we have used the same mixing parameter
adopted in the description of 6He (see Eq. (4.5) and Fig. 4.3), in order to
have at least the correct ordering in the hypernuclear energy spectrum.
However, the same hyperon separation energy can be found by just using the
$1p_{3/2}$ shell for the outer neutrons for both strange and non strange
nucleus. Our working hypothesis regarding the computation of the hyperon
separation energy is thus correct, at least for medium-heavy hypernuclei.
For $A<5$ our results are more than 1 MeV off from experimental data. For
${}^{3}_{\Lambda}$H, the $\Lambda$ separation energy is even negative, meaning
that the hypernucleus is less bound than the corresponding nucleus 2H. We can
ascribe this discrepancy to the lack of accuracy of our wave function for few-
body systems. Since the $\Lambda$ hyperon does not suffer from Pauli blocking
by the other nucleons, it can penetrate into the nuclear interior and form
deeply bound hypernuclear states. For heavy systems the $\Lambda$ particle can
be seen as an impurity that does not drastically alter the nuclear wave
function. Therefore, the trial wave function of Eq. (4.6) with the single
particle state $\varphi_{\epsilon}^{\Lambda}$ described by the $1s_{1/2}$
neutron orbital, is accurate enough as starting point for the imaginary time
propagation. For very light hypernuclei, for which the first nucleonic $s$
shell is not closed, this might not be the case. In order to have a correct
projection onto the ground state, the single particle orbitals of both
nucleons and lambda might need to be changed when the hyperon is added to the
nucleus. Moreover, in very light hypernuclei, the neglected nucleon-nucleon
and hyperon-nucleon correlations, might result in non negligible contributions
to the $\Lambda$ binding energy. A study of these systems within a few-body
method or a different projection algorithm like the GFMC, might solve this
issue.
##### Effect of the charge symmetry breaking term
The effect of the CSB potential has been studied for the $A=4$ mirror
hypernuclei. As reported in Tab. 4.4, without the CSB term there is no
difference in the $\Lambda$ binding energy of ${}^{4}_{\Lambda}$H and
${}^{4}_{\Lambda}$He. When CSB is active, a splitting appears due to the
different behavior of the $\Lambda p$ and $\Lambda n$ channels. The strength
of the difference $\Delta B_{\Lambda}^{CSB}$ is independent on the parameters
of the three-body $\Lambda NN$ interaction and it is compatible with the
experimental result [77], although the $\Lambda$ separation energies are not
accurate.
Parameters | System | $B_{\Lambda}^{sym}$ | $B_{\Lambda}^{CSB}$ | $\Delta B_{\Lambda}^{CSB}$
---|---|---|---|---
Set (I) | ${}^{4}_{\Lambda}$H | 1.97(11) | 1.89(9) | 0.24(12)
${}^{4}_{\Lambda}$He | 2.02(10) | 2.13(8)
Set (II) | ${}^{4}_{\Lambda}$H | 1.07(8) | 0.95(9) | 0.27(13)
${}^{4}_{\Lambda}$He | 1.07(9) | 1.22(9)
Expt. [77] | ${}^{4}_{\Lambda}$H | — | 2.04(4) | 0.35(5)0
${}^{4}_{\Lambda}$He | — | 2.39(3)
Table 4.4: $\Lambda$ separation energies (in MeV) for the $A=4$ mirror $\Lambda$ hypernuclei with (fourth column) and without (third column) the inclusion of the charge symmetry breaking term [43]. In the last column the difference in the separation energy induced by the CSB interaction. First and second rows refer to different set of parameters for the $\Lambda NN$ interaction, while the last row is the experimental result. System | $p$ | $n$ | $\Delta_{np}$ | $\Delta B_{\Lambda}$
---|---|---|---|---
${}^{4}_{\Lambda}$H | 1 | 2 | $+1$ | $-0.12(8)$
${}^{4}_{\Lambda}$He | 2 | 1 | $-1$ | $+0.15(9)$
${}^{5}_{\Lambda}$He | 2 | 2 | $0$ | $+0.02(9)$
${}^{6}_{\Lambda}$He | 2 | 3 | $+1$ | $-0.06(8)$
${}^{7}_{\Lambda}$He | 2 | 4 | $+2$ | $-0.18(8)$
${}^{16}_{\leavevmode\nobreak\ \Lambda}$O | 8 | 7 | $-1$ | $+0.27(35)$
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O | 8 | 8 | $0$ | $+0.15(35)$
${}^{18}_{\leavevmode\nobreak\ \Lambda}$O | 8 | 9 | $+1$ | $-0.74(49)$
Table 4.5: Difference (in MeV) in the hyperon separation energies induced by
the CSB term (Eq. (2.42)) for different hypernuclei [43]. The fourth column
reports the difference between the number of neutrons and protons. Results are
obtained with the full two- plus three-body (set (II)) hyperon-nucleon
interaction. In order to reduce the errors, $\Delta B_{\Lambda}$ has been
calculated by taking the difference between total hypernuclear binding
energies, instead of the hyperon separation energies.
The same CSB potential of Eq. (2.42) has been included in the study of
hypernuclei for $A>4$. In Tab. 4.5 the difference in the hyperon separation
energies $\Delta B_{\Lambda}=B_{\Lambda}^{CSB}-B_{\Lambda}^{sym}$ is reported
for different hypernuclei up to $A=18$. The fourth column shows the difference
between the number of neutrons and protons
$\Delta_{np}=\mathcal{N}_{n}-\mathcal{N}_{p}$. For the symmetric hypernuclei
${}^{5}_{\Lambda}$He and ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O the CSB
interaction has no effect, being this difference zero. In the systems with
neutron excess ($\Delta_{np}>0$), the effect of the CSB consists in decreasing
the hyperon separation energy compared to the charge symmetric case. When
$\Delta_{np}$ becomes negative, $\Delta B_{\Lambda}>0$ due to the attraction
induced by the CSB potential in the $\Lambda p$ channel, that produces more
bound hypernuclei. Being $\Delta_{np}$ small, these effects are in any case
rather small and they become almost negligible compared to the statistical
errors on $B_{\Lambda}$ when the number of baryons becomes large enough
($A>16$). However, in the case of $\Lambda$ neutron matter, the CSB term might
have a relevant effect for large enough $\Lambda$ fraction.
##### Effect of the hyperon-nucleon space-exchange term
As already mentioned in the previous chapter, the inclusion of the $\Lambda N$
space exchange operator of Eq. (2.35) in the AFDMC propagator is not yet
possible. In § 3.2.5 we presented a possible perturbative approach for the
treatment of such term. In Tab. 4.6 we report the results of this analysis.
All the results for ${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca are consistent
within the statistical errors. On the contrary, for lighter systems the
$\Lambda$ separation energy seems rather sensitive to the value of the
exchange parameter $\varepsilon$. Considering larger values for $\varepsilon$,
$B_{\Lambda}$ generally increases. This trend is opposite to what is found for
instance in Ref. [188]. We recall that only the computation of the Hamiltonian
expectation value by means of Eq. (3.24) gives exact results. For other
operators, like the space exchange $\mathcal{P}_{x}$, the pure estimators have
to be calculated with the extrapolation method via the two relations (3.25) or
(3.26). The variational estimate $\langle\mathcal{P}_{x}\rangle_{v}$ is thus
needed. In the mentioned reference, the importance of space exchange
correlations for variational estimates is discussed. Being these correlations
neglected in this work, our perturbative treatment of the $\mathcal{P}_{x}$
contribution might not be accurate. Moreover, the evidence of the importance
of space exchange correlations might invalid the perturbative approach itself.
An effective but more consistent treatment of this term could consist in a
slight change in the strength of the central $\Lambda N$ potential. However,
due to the very limited information about the space exchange parameter and its
effect on single $\Lambda$ hypernuclei heavier than ${}^{5}_{\Lambda}$He, this
approach has not been considered in the present work. Recent calculations of
many hadron systems within an EFT treatment at NLO for the full $SU(3)$
hadronic spectrum confirmed indeed that exchange terms are sub-leading [170].
System | $\varepsilon=0.0$ | $\varepsilon=0.1$ | $\varepsilon=0.3$
---|---|---|---
${}^{5}_{\Lambda}$He | 3.22(14) | 3.89(15) | 4.67(25)
${}^{17}_{\leavevmode\nobreak\ \Lambda}$O | 12.4(6) | 12.9(9) | 14.0(9)
${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca | 19(4) | 21(5) | 25(7)
Table 4.6: Variation of the $\Lambda$ separation energy as a consequence of
the exchange potential $v_{0}(r)\varepsilon(\mathcal{P}_{x}-1)$ in the
$\Lambda N$ interaction of Eq. (2.35). The contribution of $\mathcal{P}_{x}$
is treated perturbatively for different value of the parameter $\varepsilon$.
The interaction used is the full AV4’+$\Lambda N$+$\Lambda NN$ set (II).
Results are expressed in MeV.
#### 4.2.2 Single particle densities and radii
Single particle densities can be easily computed in Monte Carlo calculations
by considering the expectation value of the density operator
$\displaystyle\hat{\rho}_{\kappa}(r)=\sum_{i}\delta(r-r_{i})\quad\quad\kappa=N,\Lambda\;,$
(4.7)
where $i$ is the single particle index running over nucleons for
$\rho_{N}=\langle\hat{\rho}_{N}\rangle$ or hyperons for
$\rho_{\Lambda}=\langle\hat{\rho}_{\Lambda}\rangle$. The normalization is
given by
$\displaystyle\int dr4\pi r^{2}\rho_{\kappa}(r)=1\;.$ (4.8)
Root mean square radii $\langle r_{\kappa}^{2}\rangle^{1/2}$ are simply
calculated starting from the Cartesian coordinates of nucleons and hyperons. A
consistency check between AFDMC densities and radii is then taken by verifying
the relation
$\displaystyle\langle r_{\kappa}^{2}\rangle=\int dr4\pi
r^{4}\rho_{\kappa}(r)\;.$ (4.9)
Before reporting the results we recall that also for densities and radii the
AFDMC calculation can only lead to mixed estimators. The pure estimators are
thus approximated by using Eq. (3.25) or Eq. (3.26). The two relations should
lead to consistent results. This is the case for the nucleon and hyperon
radii. In computing the densities instead, the low statistics for
$r\rightarrow 0$ generates differences in the two approaches. For nucleons
these discrepancies are almost within the statistical errors. For hyperons,
the much reduced statistics (1 over $A-1$ for single $\Lambda$ hypernuclei)
and the fact that typically the $\Lambda$ density is not peaked in $r=0$,
create some uncertainties in the region for small $r$, in particular for the
first estimator. We therefore chose to adopt the pure estimator of Eq. (3.26)
to have at least a positive definite estimate. Finally, it has to be pointed
out that the pure extrapolated results are sensitive to the quality of the
variational wave function and the accuracy of the projection sampling
technique. Although we successfully tested the AFDMC propagation, we are
limited in the choice of the VMC wave function. In order to be consistent with
the mixed estimators coming from AFDMC calculations, we considered the same
trial wave functions also for the variational runs. This might introduce some
biases in the evaluation of pure estimators. Therefore, the results presented
in the following have to be considered as a qualitative study on the general
effect of the hypernuclear forces on the nucleon and hyperon distributions.
In Fig. 4.9 we report the results for the single particle densities for 4He
and ${}^{5}_{\Lambda}$He. The green curves are the densities of nucleons in
the nucleus, while the red and blue curves are, respectively, the density of
nucleons and of the lambda particle in the hypernucleus. In the left panel the
results are obtained using AV4’ for the nuclear part and the two-body $\Lambda
N$ interaction alone for the hypernuclear component. In the right panel the
densities are calculated with the full two- plus three-body (set (II))
hyperon-nucleon interaction.
Figure 4.9: Single particle densities for nucleons in 4He [green, upper banded
curve] and for nucleons [red, middle banded curve] and the lambda particle
[blue, lower banded curve] in ${}^{5}_{\Lambda}$He [43]. In the left panel the
results for the two-body $\Lambda N$ interaction alone. In the right panel the
results with the inclusion also of the three-body hyperon-nucleon force in the
parametrization (II). The AV4’ potential has been used for the nuclear core.
Figure 4.10: Single particle densities for the $\Lambda$ particle in different
hypernuclei [43]. Top panel reports the results for the two-body $\Lambda N$
interaction alone. Bottom panel shows the results when the three-body hyperon-
nucleon interaction with the set of parameters (II) is also included. The
nuclear core is described by the AV4’ potential.
The addition of the $\Lambda$ particle to the nuclear core of 4He has the
effect to slightly reduce the nucleon density in the center. The $\Lambda$
particle tries to localize close to $r=0$, enlarging therefore the nucleon
distribution. When the three-body $\Lambda NN$ interaction is turned on (right
panel of Fig. 4.9), the repulsion moves the nucleons to large distances but
the main effect is that the hyperon is pushed away from the center of the
system. As can be seen from Fig. 4.10, this effect is much more evident for
large $A$. When the hypernucleus is described by the $\Lambda N$ interaction
alone, the $\Lambda$ particle is localized near the center, in the range $r<2$
fm (left panel of Fig. 4.10). The inclusion of the three-body $\Lambda NN$
potential forces the hyperon to move from the center, in a region that roughly
correspond to the skin of nucleons (see Tab. 4.7). Although these densities
are strictly dependent to the nuclear interaction, by using the AV6’ potential
we found the same qualitative effects on the $\Lambda$ particle, confirming
the importance of the three-body hyperon-nucleon interaction and its repulsive
nature. Due to the limitations discussed above and the use of too simplified
interactions for the nucleon-nucleon force, the comparison with the available
VMC density profiles [187, 192] is difficult.
System | nucleus | hypernucleus
---|---|---
$r_{N}^{\text{exp}}$ | $r_{N}$ | $r_{N}$ | $r_{\Lambda}$
2H - ${}^{3}_{\Lambda}$H | 2.142 | 1.48(8) | 1.9(1) | 2.00(16)
3H - ${}^{4}_{\Lambda}$H | 1.759 | 1.5(1) | 1.77(9) | 2.12(15)
3He - ${}^{4}_{\Lambda}$He | 1.966 | 1.5(1) | 1.77(9) | 2.10(14)
4He - ${}^{5}_{\Lambda}$He | 1.676 | 1.57(9) | 1.58(7) | 2.2(2)
5He - ${}^{6}_{\Lambda}$He | — | 2.02(16) | 2.16(17) | 2.43(17)
6He - ${}^{7}_{\Lambda}$He | 2.065 | 2.3(2) | 2.4(2) | 2.5(2)
15O - ${}^{16}_{\leavevmode\nobreak\ \Lambda}$O | — | 2.20(12) | 2.3(1) | 3.2(3)
16O - ${}^{17}_{\leavevmode\nobreak\ \Lambda}$O | 2.699 | 2.16(12) | 2.23(11) | 3.3(3)
17O - ${}^{18}_{\leavevmode\nobreak\ \Lambda}$O | 2.693 | 2.26(13) | 2.32(14) | 3.3(3)
40Ca - ${}^{41}_{\leavevmode\nobreak\ \Lambda}$Ca | 3.478 | 2.8(2) | 2.8(2) | 4.2(5)
48Ca - ${}^{49}_{\leavevmode\nobreak\ \Lambda}$Ca | 3.477 | 3.1(2) | 3.1(2) | 4.3(5)
Table 4.7: Nucleon and hyperon root mean square radii (in fm) for nuclei and
corresponding $\Lambda$ hypernuclei. The employed nucleon-nucleon potential is
AV4’. For the strange sector we used the full two- plus three-body hyperon-
nucleon force in the parametrization (II). The experimental nuclear charge
radii are from Ref. [239]. Errors are on the fourth digit.
In Tab. 4.7 we report the nucleon and hyperon root mean square radii for
nuclei and hypernuclei. The experimental nuclear charge radii are reported as
a reference. AFDMC $r_{N}$, that do not distinguish among protons and
neutrons, are typically smaller than the corresponding experimental results.
This can be understood as a consequence of the employed AV4’ $NN$ interaction
that overbinds nuclei. The main qualitative information is that the hyperon
radii are systematically larger than the nucleon ones, as expected by looking
at the single particle densities. Starting from $A=5$, the nucleon radii in
the nucleus and the corresponding hypernucleus do not change, although the
differences in the nucleon densities for $r\rightarrow 0$. This is due to the
small contribution to the integral (4.9) given by the density for $r$ close to
zero. For the hypernuclei with $A<5$, AFDMC calculations predict larger
$r_{N}$ when the hyperon is added to the core nucleus. This is inconsistent
with the results of Ref. [184], where a shrinking of the core nuclei due to
the presence of the $\Lambda$ particle in $A\leq 5$ hypernuclei is found. We
need to emphasize once more that the results presented in this section are
most likely strictly connected to the employed nucleon-nucleon potential. For
instance, the shrinkage of hypernuclei has been investigated experimentally by
$\gamma$-ray spectroscopy [52, 240]. In the experiment of Ref. [240], by
looking at the electric quadrupole transition probability from the excited
$5/2^{+}$ state to the ground state in ${}^{7}_{\Lambda}$Li, a $19\%$
shrinkage of the intercluster distance was inferred, assuming the two-body
cluster structure core+deuteron. Therefore, the AFDMC study of densities and
radii, differently from the analysis of $\Lambda$ separation energies, cannot
lead to accurate results at this level. It has to be considered as a first
explorative attempt to get hypernuclear structure information from Diffusion
Monte Carlo simulations.
### 4.3 Double $\Lambda$ hypernuclei
In the single particle wave function representation, two $\Lambda$ particles
with antiparallel spin can be added to a core nucleus filling the first
hyperon $s$ shell, assumed to be the neutron $1s_{1/2}$ Skyrme radial function
as in the case of single $\Lambda$ hypernuclei. The complete hypernuclear wave
function is given by Eq. (3.202), where the nucleon trial wave function is the
same used in the AFDMC calculations for nuclei and in this case also the
hyperon Slater determinant is employed. Although the effect on the total
energy introduced by a $\Lambda\Lambda$ correlation function is found to be
negligible, for consistency with the calculations for nuclei and single
$\Lambda$ hypernuclei we neglected the central Jastrow correlations.
The double $\Lambda$ separation energy and the incremental $\Lambda\Lambda$
energy of Eqs. (4.3) and (4.4) are calculated starting from the energy of the
nucleus and the corresponding single and double $\Lambda$ hypernuclei
described by the same $NN$ AV4’ potential. Due to the difficulties in treating
open shell nuclei and the limited amount of data about double $\Lambda$
hypernuclei, we performed the AFDMC study for just the lightest
$\Lambda\Lambda$ hypernucleus for which energy experimental information are
available, ${}^{\;\;\,6}_{\Lambda\Lambda}$He.
#### 4.3.1 Hyperon separation energies
In Tab. 4.8 we report the total binding energies for 4He, ${}^{5}_{\Lambda}$He
and ${}^{\;\;\,6}_{\Lambda\Lambda}$He in the second column, the single or
double hyperon separation energies in the third and the incremental binding
energy in the last column. The value of $B_{\Lambda\Lambda}$ confirms the weak
attractive nature of the $\Lambda\Lambda$ interaction [173, 150, 151, 152].
Starting from 4He and adding two hyperons with $B_{\Lambda}=3.22(14)$ MeV, the
energy of ${}^{\;\;\,6}_{\Lambda\Lambda}$He would be $1.0\div 1.5$ MeV less
than the actual AFDMC result. Therefore the $\Lambda\Lambda$ potential of Eq.
(2.48) induces a net attraction between hyperons, at least at this density.
System | $E$ | $B_{\Lambda(\Lambda)}$ | $\Delta B_{\Lambda\Lambda}$
---|---|---|---
4He | -32.67(8) | — | —
${}^{5}_{\Lambda}$He | -35.89(12) | 3.22(14) | —
${}^{\;\;\,6}_{\Lambda\Lambda}$He | -40.6(3) | 7.9(3) | 1.5(4)
${}^{\;\;\,6}_{\Lambda\Lambda}$He | Expt. [91] | $7.25\pm 0.19^{+0.18}_{-0.11}$ | $1.01\pm 0.20^{+0.18}_{-0.11}$
Table 4.8: Comparison between 4He and the corresponding single and double
$\Lambda$ hypernuclei [43]. In the second column the total binding energies
are reported. The third column shows the single or double $\Lambda$ separation
energies. In the last column the incremental binding energy $\Delta
B_{\Lambda\Lambda}$ is reported. All the results are obtained using the
complete two- plus three-body (set (II)) hyperon-nucleon interaction with the
addition of the $\Lambda\Lambda$ force of Eq. (2.48). The results are
expressed in MeV.
Our $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$ are very close to the
expected results for which the potential has originally been fitted within the
cluster model. The latest data $B_{\Lambda\Lambda}=6.91(0.16)$ MeV and $\Delta
B_{\Lambda\Lambda}=0.67(0.17)$ MeV of Ref. [93] suggest a weaker attractive
force between the two hyperons. A refit of the interaction of the form
proposed in Eq. (2.48) would be required. It would be interesting to study
more double $\Lambda$ hypernuclei within the AFDMC framework with the $\Lambda
N$, $\Lambda NN$ and $\Lambda\Lambda$ interaction proposed. Some experimental
data are available in the range $A=7\div 13$, but there are uncertainties in
the identification of the produced double $\Lambda$ hypernuclei, reflecting in
inconsistencies about the sign of the $\Lambda\Lambda$ interaction [241, 242].
An ab-initio analysis of these systems might put some constraints on the
hyperon-hyperon force, which at present is still poorly known, and give
information on its density dependence. Also the inclusion of the
$\Lambda\Lambda N$ force would be important.
#### 4.3.2 Single particle densities and radii
For the sake of completeness, we also report the results for the single
particle densities (Fig. 4.11) and root mean square radii (Tab. 4.9) for the
double $\Lambda$ hypernucleus ${}^{\;\;\,6}_{\Lambda\Lambda}$He. By looking at
the densities profiles, when a second hyperon is added to
${}^{5}_{\Lambda}$He, the nucleon density at the center reduces further. The
hyperon density, instead, seems to move a bit toward $r=0$ consistently with
weak attractive behavior of the employed $\Lambda\Lambda$ interaction.
However, the nucleon and hyperon radii are almost the same of
${}^{5}_{\Lambda}$He. These conclusions are thus rather speculative,
particularly recalling the discussion on single particle densities of § 4.2.2.
System | $r_{N}$ | $r_{\Lambda}$
---|---|---
4He | 1.57(9) | —
${}^{5}_{\Lambda}$He | 1.58(7) | 2.2(2)
${}^{\;\;\,6}_{\Lambda\Lambda}$He | 1.7(2) | 2.3(2)
Table 4.9: Nucleon and hyperon root mean square radii (in fm) for 4He and the
corresponding single and double $\Lambda$ hypernuclei. The employed
interactions are the $NN$ AV4’ plus the full two- and three-body hyperon-
nucleon force (set (II)). Figure 4.11: Single particle densities for nucleons
in 4He [green banded curve], ${}^{5}_{\Lambda}$He [red banded curve] and
${}^{\;\;\,6}_{\Lambda\Lambda}$He [light blue banded curve], and for the
$\Lambda$ particle in ${}^{5}_{\Lambda}$He [blue banded curve] and
${}^{\;\;\,6}_{\Lambda\Lambda}$He [brown banded curve]. The results are
obtained using the AV4’ potential for nucleons and the two- plus three-body
hyperon-nucleon force (II). In the case of ${}^{\;\;\,6}_{\Lambda\Lambda}$He,
the $\Lambda\Lambda$ interaction of Eq. (2.48) is also employed.
Empty page
## Chapter 5 Results: infinite systems
Neutron matter has been deeply investigated in previous works using the
Auxiliary Field DMC algorithm. The EoS at zero temperature has been derived in
both constrained path [37] and fixed phase [38] approximations. In the low
density regime, the ${}^{1\\!}S_{0}$ superfluid energy gap has also been
studied [39]. In the high density regime, the connections between three-body
forces, nuclear symmetry energy and the neutron star maximum mass are
extensively discussed in Refs. [40, 243].
In this chapter we will review some details of the AFDMC simulations for pure
neutron matter (PNM). They will be useful to extend the calculations for the
inclusion of strange degrees of freedom. We will then focus on the hyperon
neutron matter (YNM), firstly with the test of the AFDMC algorithm extended to
the strange sector in connection with the developed hyperon-nucleon
interactions. Starting from the derivation of the threshold density for the
appearance of $\Lambda$ hyperons, a first attempt to construct a realistic EoS
for YNM will be presented. The corresponding limit for the maximum mass will
be finally discussed.
### 5.1 Neutron matter
As already described in Chapter 3, due to the simplification in the potentials
for neutron only systems, PNM can investigated by means of AFDMC calculations
using the Argonne V8’ two-body potential and including three-body forces. The
contribution of terms in the Argonne potential beyond spin-orbit are usually
very small in nuclei and in low density nuclear and neutron matter. It may
become significative only for very large densities [38]. Predicted maximum
masses of a NS for the two Argonne potentials are very close and both below
$1.8M_{\odot}$, as a consequence of the softness of the corresponding EoS [5,
40]. Being the present observational limit for $M_{\max}$ around $2M_{\odot}$
[6, 7], three-neutron forces must be repulsive at high densities. As reported
in Ref. [34], the Illinois 7 TNI is attractive and produces a too soft EoS.
The Urbana IX interaction instead provides a strong repulsive contribution to
the total energy. The inclusion of the UIX force in addition to the two-body
AV8’ interaction in AFDMC calculations for PNM generates a rather stiff EoS.
The predicted maximum mass is around $2.4M_{\odot}$ [40], in agreement with
the result coming from the AV18+UIX calculation of Akmal, Pandharipande and
Ravenhall [5]. It follows that the AFDMC method to solve the AV8’+UIX nuclear
Hamiltonian is a valuable tool for the investigation of neutron matter
properties and neutron stars observables. This is the starting point for the
study of $\Lambda$ neutron matter.
All the AFDMC results for PNM have been obtained using the version _v2_ of the
algorithm. Simulations are typically performed at fixed imaginary time step
$d\tau=2\cdot 10^{-5}\leavevmode\nobreak\ \text{MeV}^{-1}$, that should be
small enough to provide a good approximation of the extrapolated result [37].
The wave function of Eq. (3.183) includes a Jastrow correlation function among
neutrons and a Slater determinant of plane waves coupled with two-component
spinors. For infinite neutron systems, AFDMC calculations do not depend on the
Jastrow functions. Moreover by changing the algorithm to version _v1_ ,
results are less than 1% different. This is because the employed trial wave
function is already a good approximation of the real ground state wave
function. Moreover the interaction is simplified with respect to the case of
finite nucleon systems due to absence of the $\bm{\tau}_{i}\cdot\bm{\tau}_{j}$
contributions.
In Chapter 3 we have seen that finite size effects appear because of the
dependence of the Fermi gas kinetic energy to the number of particles. The
kinetic energy oscillations of $\mathcal{N}_{F}$ free Fermions imply that the
energy of $\mathcal{N}_{F}=38$ is lower than either $\mathcal{N}_{F}=14$ or
$\mathcal{N}_{F}=66$. This is reflected in the energy of PNM for different
number of neutrons with PBC conditions (Eq. (3.197)). At each density it
follows that $E(38)<E(14)<E(66)$ [38]. However, as already discussed in §
3.2.4, the results for 66 neutrons are remarkably close to the extrapolated
TABC energy. 66 is thus the typical number of particle employed in AFDMC
calculations for PNM.
Finite size effects could appear also from the potential, in particular at
high density, depending on the range of the interaction. Monte Carlo
calculations are generally performed in a finite periodic box with size $L$
and all inter-particle distances are truncated within the sphere of radius
$L/2$. Usually, tail corrections due to this truncation are estimated with an
integration of the two-body interaction from $L/2$ up to infinity. However,
this is possible only for spin independent terms. As originally reported in
Ref. [37], in order to correctly treat all the tail corrections to the
potential, it is possible to include the contributions given by neighboring
cells to the simulation box. Each two-body contribution to the potential is
given by
$\displaystyle v_{p}(r)\equiv
v_{p}(|x,y,z|)\longrightarrow\sum_{i_{x},i_{y},i_{z}}v_{p}\Bigl{(}\big{|}(x+i_{x}L)\hat{x}+(y+i_{y}L)\hat{y}+(z+i_{z}L)\hat{z}\big{|}\Bigr{)}\;,$
(5.1)
where $v_{p}(r)$ are the potential functions of Eq. (2.16) and
$i_{x},i_{y},i_{z}$ are $0,\pm 1,\pm 2,\ldots$ depending on the number of the
boxes considered. The inclusion of the first 26 additional neighbor cells,
that corresponds to $i_{x},i_{y},i_{z}$ taking the values $-1$, $0$ and $1$,
is enough to extend the calculation for inter-particle distances larger than
the range of the potential [37, 38]. Finite-size corrections due to three-body
forces can be included in the same way as for the nucleon-nucleon interaction,
although their contribution is very small compared to the potential energy.
Their effect is appreciable only for a small number of particles and at large
density, i.e., if the size of the simulation box is small. We will see that
these corrections are actually non negligible for the correct computation of
energy differences in $\Lambda$ neutron matter. By looking at the results
reported in the mentioned references, for PNM we can estimate that the finite-
size errors in AFDMC calculations, due to both kinetic and potential energies,
do not exceed 2% of the asymptotic value of the energy calculated by using
TABC.
It was found [38, 40] that the EoS of PNM can be accurately parametrized using
the following polytrope functional form:
$\displaystyle
E(\rho_{n})=a\left(\frac{\rho_{n}}{\rho_{0}}\right)^{\alpha}+b\left(\frac{\rho_{n}}{\rho_{0}}\right)^{\beta}\;,$
(5.2)
where $E(\rho_{n})$ is the energy per neutron as a function of the neutron
density $\rho_{n}$, and the parameters $a$, $\alpha$, $b$, and $\beta$ are
obtained by fitting the QMC results. $\rho_{0}=0.16\leavevmode\nobreak\
\text{fm}^{-3}$ is the nuclear saturation density. AFDMC energies per particle
as a function of the neutron density, together with the fitted parameters for
both AV8’ and the full AV8’+UIX Hamiltonians, are reported in Tab. 5.1. The
plots of the EoS are shown in the next section, Fig. 5.1.
$\rho_{n}$ | AV8’ | AV8’+UIX
---|---|---
0.08 | 9.47(1) | 10.49(1)
0.16 | 14.47(2) | 19.10(2)
0.24 | 19.98(3) | 31.85(3)
0.32 | 26.45(3) | 49.86(5)
0.40 | 34.06(5) | 74.19(5)
0.48 | 42.99(8) | 105.9(1)
0.56 | — | 145.3(1)
0.60 | 58.24(8) | 168.1(2)
0.70 | 73.3(1) | —
$\begin{aligned} &\phantom{a=2.04(7)}\\\ &\text{polytrope}\\\ &\text{parameters}\\\ &\phantom{\beta=0.47(1)}\end{aligned}$ | $\begin{aligned} a&=2.04(7)\\\ \alpha&=2.15(2)\\\ b&=12.47(47)\\\ \beta&=0.47(1)\end{aligned}$ | $\begin{aligned} a&=5.66(3)\\\ \alpha&=2.44(1)\\\ b&=13.47(3)\\\ \beta&=0.51(1)\end{aligned}$
Table 5.1: Energy per particle in neutron matter for selected densities [34,
243]. $a$, $\alpha$, $b$ and $\beta$ are the fitted polytrope coefficients of
Eq. (5.2).
### 5.2 $\Lambda$ neutron matter
The study of $\Lambda$ neutron matter follows straightforwardly from PNM
calculations with the extension of the wave function (Eq. (3.202)) and the
inclusion of the strange part of the Hamiltonian (Eqs. (2.3) and (2.4)), in
analogy with the simulations for finite strange systems. In addition to the
Slater determinant of plane waves for neutrons, there is now the determinant
for the $\Lambda$ particles. Both sets of plane waves have quantized
$\bm{k}_{\epsilon}$ vectors given by Eq. (3.198), and each type of baryon
fills its own momentum shell. As discussed in § 3.2.4, the requirement of
homogeneity and isotropy implies the closure of the momentum shell structure,
both for neutrons and hyperons. The consequence is that in AFDMC calculations
we are limited in the possible choices for the $\Lambda$ fraction, defined as
$\displaystyle
x_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{b}}=\frac{\mathcal{N}_{\Lambda}}{\mathcal{N}_{n}+\mathcal{N}_{\Lambda}}\;,$
(5.3)
where $\rho_{\Lambda}$ is the hyperon density and $\rho_{b}$ the total baryon
density of Eq. (3.214). Employing the TABC (Eq. (3.201)) would allow to
consider a number of particles corresponding to open shells, providing more
freedom in the choice of $x_{\Lambda}$. However, this approach has not been
investigate in this work.
As soon as the hyperons appear in the bulk of neutrons, i.e. above a $\Lambda$
threshold density $\rho_{\Lambda}^{th}$, the EoS becomes a function of both
baryon density and $\Lambda$ fraction, which are connected by the equilibrium
condition $\mu_{\Lambda}=\mu_{n}$ (see § 1.2). The $\Lambda$ threshold density
and the function $x_{\Lambda}(\rho_{b})$ are key ingredients to understand the
high density properties of hypermatter and thus to predict the maximum mass.
We will start the discussion with the test analysis of $\Lambda$ neutron
matter at a fixed $\Lambda$ fraction. We will then move to the realistic case
of variable $x_{\Lambda}$.
#### 5.2.1 Test: fixed $\Lambda$ fraction
In order to test the feasibility of AFDMC calculations for hypermatter, we
considered the limiting case of small $\Lambda$ fraction, in order to look at
the hyperon as a small perturbation in the neutron medium. We filled the
simulation box with 66 neutrons and just one $\Lambda$ particle, i.e.
$x_{\Lambda}=0.0149$. Although the first momentum shell for the strange
baryons is not completely filled (for $\mathcal{N}_{c}=1$ the occupation
number is 2, spin up and spin down $\Lambda$ particles), the requirement of
homogeneity and isotropy is still verified. The first $\bm{k}_{\epsilon}$
vector, indeed, is $\frac{2\pi}{L}(0,0,0)$ and thus the corresponding plane
wave is just a constant, giving no contribution to the kinetic energy. In
order to keep the $\Lambda$ fraction small we are allowed to use one or two
hyperons in the box (next close shell is for 14 particles) and, possibly,
change the number of neutrons, as we will see. Using just one lambda hyperon
there is no need to include the $\Lambda\Lambda$ interaction. The closest
hyperon will be in the next neighboring cell at distances larger than the
range of the hyperon-hyperon force, at least for non extremely high densities.
Therefore, we proceeded with the inclusion of the AV8’+UIX potentials for
neutrons, adding the $\Lambda N$+$\Lambda NN$ interactions in both
parametrizations (I) and (II).
In Tab. 5.2 we report the energy as a function of the baryon density for
different combinations of the employed potentials. The parameters of the
polytrope function of Eq. (5.2) that fits the AFDMC results are also shown.
The plot of the fits, for both PNM and YNM are reported in Fig. 5.1.
By looking at the dashed lines, corresponding to calculations without the
neutron TNI, it is evident the softness of the PNM EoS (green) discussed in
the previous section. The addition of the hyperon-nucleon two-body interaction
(blue) implies, as expected (see § 1.2), a further reduction of the energy per
particle, even for the small and constant $\Lambda$ fraction. The inclusion of
the three-body $\Lambda NN$ interaction (red), instead, makes the EoS stiffer
at high density, even stiffer than the PNM one for the set of parameters (II).
This result is rather interesting because it means that the hyperon-nucleon
force used has a strong repulsive component that is effective also at
densities larger than nuclear saturation density, where the interaction was
originally fitted on medium-heavy hypernuclei.
When the Urbana IX TNI is employed (solid lines), the PNM EoS (green) becomes
stiff. As in the previous case, the inclusion of the two-body $\Lambda N$
interaction softens the EoS (blue), although the effect is not dramatic for
the small $x_{\Lambda}$ considered. The three-body hyperon-nucleon force gives
a repulsive contribution to the total energy (red). The effect is more evident
for the parametrization (II), for which the PNM and YNM EoS are almost on top
of each other. The small constant fraction of hyperons in the neutron medium
induces very small modifications in the energy per particle. This is due to
the repulsive contribution of the $\Lambda NN$ interaction still active at
high densities.
These results do not describe the realistic EoS for $\Lambda$ neutron matter,
because they are computed at a fixed $\Lambda$ fraction for each baryon
density. However, the high density part of the curves gives us some indication
about the behavior of the hyperon-nucleon interaction in the infinite medium.
The fundamental observation is that the $\Lambda NN$ force is repulsive,
confirming our expectations. By varying the $\Lambda$ fraction, for example
considering two hyperons over 66 neutrons, the qualitative picture drawn in
Fig. 5.1 is the same, but a small reasonable increase in the softening of the
EoS is found. This is consistent with the theoretical prediction related to
the appearance of strange baryons in NS matter and gives us the possibility to
quantitatively predict the entity of the softening in a Quantum Monte Carlo
framework.
$\rho_{b}$ | AV8’ | AV8’ | AV8’
---|---|---|---
$\Lambda N$ | $\Lambda N$+$\Lambda NN$ (I) | $\Lambda N$+$\Lambda NN$ (II)
0.08 | 8.71(1) | 8.84(1) | 8.92(1)
0.16 | 13.11(3) | 13.44(2) | 13.76(1)
0.24 | 17.96(2) | 18.71(2) | 19.31(3)
0.32 | 23.81(4) | 25.02(4) | 26.09(3)
0.40 | 30.72(4) | 32.75(6) | 34.20(6)
0.48 | 38.84(6) | 42.03(6) | 43.99(4)
0.56 | 48.37(7) | 52.30(8) | 55.18(8)
0.60 | 53.24(7) | 57.9(1) | 61.42(7)
0.70 | 67.1(1) | 74.0(1) | 78.7(1)
0.80 | 83.1(1) | 91.7(1) | 98.0(1)
$\begin{aligned} &\phantom{a=2.54(13)}\\\ &\text{polytrope}\\\ &\text{parameters}\\\ &\phantom{\beta=0.38(2)}\end{aligned}$ | $\begin{aligned} a&=2.54(13)\\\ \alpha&=2.00(3)\\\ b&=10.52(15)\\\ \beta&=0.38(2)\end{aligned}$ | $\begin{aligned} a&=2.80(13)\\\ \alpha&=2.02(3)\\\ b&=10.60(16)\\\ \beta&=0.38(2)\end{aligned}$ | $\begin{aligned} a&=2.75(9)\\\ \alpha&=2.07(2)\\\ b&=10.98(11)\\\ \beta&=0.41(2)\end{aligned}$
$\rho_{b}$ | AV8’+UIX | AV8’+UIX | AV8’+UIX
$\Lambda N$ | $\Lambda N$+$\Lambda NN$ (I) | $\Lambda N$+$\Lambda NN$ (II)
0.08 | 9.72(2) | 9.77(1) | 9.87(1)
0.16 | 17.53(2) | 17.88(2) | 18.16(1)
0.24 | 29.29(5) | 29.93(2) | 30.57(2)
0.32 | 46.17(7) | 47.38(5) | 48.55(4)
0.40 | 68.86(8) | 71.08(7) | 72.87(7)
0.48 | 98.71(8) | 101.7(1) | 104.68(9)
0.56 | 135.9(1) | 140.19(9) | 144.(1)
0.60 | 157.0(1) | 162.3(1) | 167.0(1)
$\begin{aligned} &\phantom{a=5.48(12)}\\\ &\text{polytrope}\\\ &\text{parameters}\\\ &\phantom{\beta=0.47(1)}\end{aligned}$ | $\begin{aligned} a&=5.48(12)\\\ \alpha&=2.42(1)\\\ b&=12.06(14)\\\ \beta&=0.47(1)\end{aligned}$ | $\begin{aligned} a&=5.55(5)\\\ \alpha&=2.44(1)\\\ b&=12.32(6)\\\ \beta&=0.49(1)\end{aligned}$ | $\begin{aligned} a&=5.76(7)\\\ \alpha&=2.43(1)\\\ b&=12.39(8)\\\ \beta&=0.49(1)\end{aligned}$
Table 5.2: Energy per particle in $\Lambda$ neutron matter as a function of
the baryon density. The $\Lambda$ fraction is fixed at $x_{\Lambda}=0.0149$.
Different columns correspond to different nucleon-nucleon and hyperon-nucleon
potentials. $a$, $\alpha$, $b$ and $\beta$ are the fitted polytrope
coefficients (Eq. (5.2)). The curves are reported in Fig. 5.1. Figure 5.1:
Energy per particle as a function of the baryon density for $\Lambda$ neutron
matter at fixed $\Lambda$ fraction $x_{\Lambda}=0.0149$. Green curves refer to
the PNM EoS, blue and red to the YNM EoS with the inclusion of the two-body
and two- plus three-body hyperon nucleon force. In the upper panel the results
are for the $\Lambda NN$ parametrization (I). In the lower panel the set (II)
has been used. Dashed lines are obtained using the AV8’ nucleon-nucleon
potential. Solid lines represent the results with the inclusion of the $NNN$
Urbana IX potential. Figure 5.2: $nn$ (dashed lines) and $\Lambda n$ (solid
lines) pair correlation functions in $\Lambda$ neutron matter for
$\rho_{b}=0.16\leavevmode\nobreak\ \text{fm}^{-3}$ and $x_{\Lambda}=0.0149$.
The nucelon-nucleon potential is AV8’+UIX. In the upper panel only the two-
body hyperon-nucleon potential has been used. In the lower panel also the
three body $\Lambda NN$ force in the parametrization (II) has been considered.
The subscript $u$ ($d$) refers to the neutron or lambda spin up (down). Figure
5.3: Same of Fig. 5.2 but for the baryon density
$\rho_{b}=0.40\leavevmode\nobreak\ \text{fm}^{-3}$.
Before moving to the derivation of the $\Lambda$ threshold density and the
hypermatter EoS, let us analyze the pair correlation functions calculated for
$\Lambda$ neutron matter at fixed $\Lambda$ fraction $x_{\Lambda}=0.0149$.
Figs. 5.2 and 5.3 report the neutron-neutron and lambda-neutron pair
correlation functions $g(r)$ for different baryon density, $\rho_{b}=\rho_{0}$
and $\rho_{b}=0.40\leavevmode\nobreak\ \text{fm}^{-3}$. Dashed lines refer to
$g_{nn}(r)$ in the central (black), spin singlet (light blue) and spin triplet
(brown) channels. Solid lines to $g_{\Lambda n}(r)$ in the central (blue),
$\Lambda$ spin up - $n$ spin down (red) and $\Lambda$ spin up - $n$ spin up
(green) channels respectively. In the upper panels results obtained using the
two-body $\Lambda N$ interaction only are displayed. In the lower panels the
three-body $\Lambda NN$ force in the parametrization (II) is also included.
The main information we can obtain from the plots is the non negligible effect
on inter-particle distances due to the inclusion of the three-body $\Lambda
NN$ force. Without TNI among hyperons and neutrons, the central $\Lambda n$
correlation function presents a maximum around $1.0\div 1.2$ fm, depending on
the density. This is a consequence of the attractive $\Lambda N$ force that
tends to create a shell of neutrons surrounding the hyperon impurity. The
effect is also visible at high density, although reduced. When the $\Lambda
NN$ is considered, the shell effect disappears and the $g_{\Lambda n}(r)$
resembles the neutron-neutron one, particularly at high density. The inclusion
of the repulsive three-body force avoids the clustering of $\Lambda$ particles
in favor of a more homogenous lambda-neutron medium. The use of a $\Lambda n$
central correlation, has the only effect of reducing the value of $g_{\Lambda
n}(r)$ in the origin, moving the central functions close to the PNM ones. For
the small $\Lambda$ fraction considered here, the neutron-neutron correlation
functions are not sensitive to the presence of the hyperon. Indeed, similar
results can be obtained for PNM.
It is interesting to observe the projection of the pair correlation functions
in the spin channels. For neutrons the Pauli principle tends to suppress the
presence of close pairs of particles with parallel spin. For the $\Lambda$-$n$
pair, theoretically there is no Pauli effect because the two particles belong
to different isospin spaces. However, the employed hyperon-nucleon interaction
involves a $\bm{\sigma}_{\lambda}\cdot\bm{\sigma}_{i}$ contribution (recall
Eqs. (2.35) and (2.47)). This is almost negligible in the case of the $\Lambda
N$ potential alone (upper panels of Figs. 5.2 and 5.3). It has instead a
sizable effect in the dominant three-body force, for which the channel
$\Lambda$ spin up - $n$ spin down separates from the $\Lambda$ spin up - $n$
spin up, revealing a (weak) net repulsion between parallel configurations.
Same effect can be found for $\Lambda$ reversed spin.
#### 5.2.2 $\Lambda$ threshold density and the equation of state
In order to address the problem of $\Lambda$ neutron matter, we make use of a
formal analogy with the study of two components Fermi gas used in the analysis
of asymmetric nuclear matter. When protons are added to the bulk of neutrons,
the energy per baryon can be expressed in terms of the isospin asymmetry
$\displaystyle\delta_{I}=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}=1-2x_{p}\quad\quad
x_{p}=\frac{\rho_{p}}{\rho_{b}}\;,$ (5.4)
as a sum of even powers of $x_{p}$
$\displaystyle
E_{pn}(\rho_{b},x_{p})=E_{pn}(\rho_{b},1/2)+S_{pn}^{(2)}(\rho_{b})(1-2x_{p})^{2}+S_{pn}^{(4)}(\rho_{b})(1-2x_{p})^{4}+\ldots\;,$
(5.5)
where $x_{p}$ is the proton fraction and $S_{pn}^{(2i)}(\rho_{b})$ with
$i=1,2,\ldots$ are the nuclear symmetry energies. Typically, higher order
corrections for $i>1$ are ignored. The nuclear symmetry energy
$S_{pn}(\rho_{b})\equiv S_{pn}^{(2)}(\rho_{b})$ is then defined as the
difference between the energy per baryon of PNM
$E_{\text{PNM}}(\rho_{b})=E_{pn}(\rho_{b},0)$ and the energy per baryon of
symmetric nuclear matter (SNM)
$E_{\text{SNM}}(\rho_{b})=E_{pn}(\rho_{b},1/2)$.
$E_{pn}(\rho_{b},x_{p})$ can be rewritten in terms of the PNM energy:
$\displaystyle E_{pn}(\rho_{b},x_{p})$
$\displaystyle=E_{\text{SNM}}(\rho_{b})+S_{pn}(\rho_{b})\Bigl{(}1-2x_{p}\Bigr{)}^{2}\;,$
$\displaystyle=E_{\text{SNM}}(\rho_{b})+\Bigl{[}E_{\text{PNM}}(\rho_{b})-E_{\text{SNM}}(\rho_{b})\Bigr{]}\Bigl{(}1-2x_{p}\Bigr{)}^{2}\;,$
$\displaystyle=E_{\text{PNM}}(\rho_{b})+S_{pn}(\rho_{b})\Bigl{(}-4x_{p}+4x_{p}^{2}\Bigr{)}\;.$
(5.6)
In AFDMC calculations the Coulomb interaction is typically neglected. The
difference between PNM and asymmetric nuclear matter is thus related to the
isospin dependent terms of the nucleon-nucleon interactions. The effect of
these components of the potential is parametrized by means of a function of
the proton fraction and a function of the baryon density.
We can try to make an analogy between asymmetric nuclear matter and
hypermatter, by replacing the protons with the $\Lambda$ particles. In this
case the difference with the PNM case is given by the “strangeness asymmetry”
$\displaystyle\delta_{S}=\frac{\rho_{n}-\rho_{\Lambda}}{\rho_{n}+\rho_{\Lambda}}=1-2x_{\Lambda}\;,$
(5.7)
and the effect on the energy per particle is related to the hyperon-nucleon
interactions and the difference in mass between neutron and $\Lambda$. In the
case of $\Lambda$ neutron matter, the analog of Eq. (5.5) should contain also
odd powers of $\delta_{S}$. These contributions are negligible for asymmetric
nuclear matter due to the smallness of the charge symmetry breaking in $NN$
interaction. Being the $\Lambda$ particles distinguishable from neutrons,
there are no theoretical arguments to neglect the linear term in
$(1-2x_{\Lambda})$. However, we can try to express the energy per particle of
$\Lambda$ neutron matter as an expansion over the $\Lambda$ fraction, by
introducing an “hyperon symmetry energy” $S_{\Lambda n}(\rho_{b})$ such that
$\displaystyle E_{\Lambda
n}(\rho_{b},x_{\Lambda})=E_{\text{PNM}}(\rho_{b})+S_{\Lambda
n}(\rho_{b})\Bigl{(}-x_{\Lambda}+x_{\Lambda}^{2}\Bigr{)}\;.$ (5.8)
The expression for the energy difference directly follows from Eq. (5.8):
$\displaystyle\Delta E_{\Lambda n}(\rho_{b},x_{\Lambda})=E_{\Lambda
n}(\rho_{b},x_{\Lambda})-E_{\text{PNM}}(\rho_{b})=S_{\Lambda
n}(\rho_{b})\Bigl{(}-x_{\Lambda}+x_{\Lambda}^{2}\Bigr{)}\;.$ (5.9)
The idea is then to perform simulations for different $\Lambda$ fraction in
order to fit the hyperon symmetry energy $S_{\Lambda n}(\rho_{b})$. The main
problem in this procedure is the limitation in the values of the hyperon
fraction we can consider. In order to keep $x_{\Lambda}$ small we can use up
to 2 lambdas in the first momentum shell and try to vary the number of
neutrons from 66 to 14, as reported in Tab. 5.3. In fact, moving to the next
$\Lambda$ shell implies a total of 14 strange baryons and a number of neutrons
that is computationally demanding. Moreover, we cannot neglect the
$\Lambda\Lambda$ interaction for 14 hyperons in a box, even at low density.
The inclusion of the hyperon-hyperon force would lead to additional
uncertainties in the calculation and it has not been taken into account at
this point.
$\mathcal{N}_{n}$ | $\mathcal{N}_{\Lambda}$ | $\mathcal{N}_{b}$ | $x_{\Lambda}$ | $x_{\Lambda}\leavevmode\nobreak\ \%$
---|---|---|---|---
66 | 0 | 66 | 0.0000 | 0.0%
66 | 1 | 67 | 0.0149 | 1.5%
54 | 1 | 55 | 0.0182 | 1.8%
38 | 1 | 39 | 0.0256 | 2.6%
66 | 2 | 68 | 0.0294 | 2.9%
54 | 2 | 56 | 0.0357 | 3.6%
38 | 2 | 40 | 0.0500 | 5.0%
14 | 1 | 15 | 0.0667 | 6.7%
Table 5.3: Neutron, lambda and total baryon number with the corresponding
$\Lambda$ fraction for $\Lambda$ matter calculations.
Because of finite size effects, we have to be careful in calculating the
difference $\Delta E_{\Lambda n}$. Being the $\Lambda$ fraction small, we can
suppose that these effects on the total energy are mainly due to neutrons. By
taking the difference between YNM and PNM energies for the same number of
neutrons, the finite size effects should cancel out. We can see the problem
from a different equivalent point of view. The starting point is the energy of
PNM obtained with 66 neutrons in the box. If we consider the $\Lambda$ matter
described by $66n+1\Lambda$ or $66n+2\Lambda$ there are no problems in
evaluating $\Delta E_{\Lambda n}$. When moving to a different $\Lambda$
fraction, the number of neutrons $\mathcal{M}$ in the strange matter has to be
changed. In order to take care of the modified neutron shell, a reasonable
approach is to correct the YNM energy by the contribution given by the PNM
“core” computed with 66 and $\mathcal{M}$ neutrons:
$\displaystyle E_{\Lambda n}^{corr}(\rho_{b},x_{\Lambda})$
$\displaystyle=E_{\Lambda
n}^{\mathcal{M}}(\rho_{b},x_{\Lambda})+\Bigl{[}E_{\text{PNM}}^{66}(\rho_{b})-E_{\text{PNM}}^{\mathcal{M}}(\rho_{b})\Bigr{]}$
$\displaystyle=E_{\text{PNM}}^{66}(\rho_{b})+S_{\Lambda
n}(\rho_{b})\Bigl{(}-x_{\Lambda}+x_{\Lambda}^{2}\Bigr{)}\;.$ (5.10)
In this way we obtain
$\displaystyle\Delta E_{\Lambda n}(\rho_{b},x_{\Lambda})=S_{\Lambda
n}(\rho_{b})\Bigl{(}-x_{\Lambda}+x_{\Lambda}^{2}\Bigr{)}=E_{\Lambda
n}^{\mathcal{M}}(\rho_{b},x_{\Lambda})-E_{\text{PNM}}^{\mathcal{M}}(\rho_{b})\;,$
(5.11)
that exactly corresponds to the result of Eq. (5.9).
We verified that energy oscillations for different number of particles keep
the same ordering and relative magnitude around the value for 66 neutrons when
the density is changed. Actually this is true only when finite size effects
due to the truncation of the interaction are also considered. The effect of
tail corrections due to the potential is indeed severe, because it depends on
both the number of particles and the density, getting worst for few particles
and at high densities. In order to control these effects, we performed
simulations for PNM and YNM with different number of neutrons including tail
corrections for the $NN$ potential and also for the $NNN$, $\Lambda N$ and
$\Lambda NN$ forces which are all at the same TPE order and thus have similar
interaction range. The result is that, once all the finite size effects are
correctly taken into account, the $\Delta E_{\Lambda n}$ values for different
densities and number of particles, thus hyperon fraction, can actually be
compared.
The result of this analysis is reported in Fig. 5.4. The values of the
difference $\Delta E_{\Lambda n}$ are shown as a function of the $\Lambda$
fraction for different baryon densities up to
$\rho_{b}=0.40\leavevmode\nobreak\ \text{fm}^{-3}$. As expected, the energy
difference is almost linear in $x_{\Lambda}$, at least for the range of
$\Lambda$ fraction that has been possible to investigate. For
$x_{\Lambda}=0.0294,0.0357,0.05$ two hyperons are involved in the calculation.
For these cases, we also tried to include the hyperon-hyperon interaction in
addition to the AV8’+UIX+$\Lambda N$+$\Lambda NN$ potentials. The
$\Lambda\Lambda$ contribution is negligible up to $\rho_{b}\sim 2.5\rho_{0}$,
where some very small effects are found, although compatible with the previous
results within the statistical error bars. For densities higher than
$\rho_{b}=0.40\leavevmode\nobreak\ \text{fm}^{-3}$, finite size effects become
harder to correct. Although the distribution of energy values generally
follows the trend of the lower density data, the approximations used to
compute $\Delta E_{\Lambda n}$ might not be accurate enough. A more refined
procedure to reduce the dependence on shell closure, for example involving the
twist-averaged boundary conditions, it is possibly needed.
Figure 5.4: YNM and PNM energy difference as a function of the $\Lambda$
fraction for different baryon densities. The employed potential is the full
AV8’+UIX+$\Lambda N$+$\Lambda NN$ parametrization (II). Dashed lines
correspond to the quadratic fit $\Delta E_{\Lambda n}(x_{\Lambda})=S_{\Lambda
n}(-x_{\Lambda}+x_{\Lambda}^{2})$. In the range of $\Lambda$ fraction shown,
$\Delta E_{\Lambda n}$ is essentially given by the linear term in
$x_{\Lambda}$.
We used the quadratic function $\Delta E_{\Lambda n}(x_{\Lambda})=S_{\Lambda
n}(-x_{\Lambda}+x_{\Lambda}^{2})$ to fit the $\Delta E_{\Lambda n}$ values of
Fig. 5.4. For each density the coefficient $S_{\Lambda n}$ has been plotted as
a function of the baryon density, as shown in Fig. 5.5. In the case of
asymmetric nuclear matter, close to the saturation density the nuclear
symmetry energy is parametrized with a linear function of the density [40].
The data in Fig. 5.5 actually manifest a linear behavior for
$\rho_{b}\sim\rho_{0}$ but the trend deviates for large density. We can try to
fit the $S_{\Lambda n}$ points including the second order term in the
expansion over $\rho_{b}-\rho_{0}$:
$\displaystyle S_{\Lambda n}(\rho_{b})=S_{\Lambda n}^{(0)}+S_{\Lambda
n}^{(1)}\left(\frac{\rho_{b}-\rho_{0}}{\rho_{0}}\right)+S_{\Lambda
n}^{(2)}\left(\frac{\rho_{b}-\rho_{0}}{\rho_{0}}\right)^{2}\;.$ (5.12)
The results are shown in Fig. 5.5 with the dashed line. The three parameters
of the $S_{\Lambda n}(\rho_{b})$ function are reported in Tab. 5.4.
$S_{\Lambda n}^{(0)}$ | $S_{\Lambda n}^{(1)}$ | $S_{\Lambda n}^{(2)}$
---|---|---
65.6(3) | 46.4(1.6) | -10.2(1.3)
Table 5.4: Coefficients (in MeV) of the hyperon symmetry energy function of
Eq. (5.12). The parameters are obtained from the quadratic fit on the $\Delta
E_{\Lambda n}$ results reported in Fig. 5.4. Figure 5.5: Hyperon symmetry
energy as a function of the baryon density. Red dots are the points obtained
by the the quadratic fit $\Delta E_{\Lambda n}(x_{\Lambda})=S_{\Lambda
n}(-x_{\Lambda}+x_{\Lambda}^{2})$ on the data of Fig. 5.4. The dashed line is
the $S_{\Lambda n}(\rho_{b})$ fitted curve of Eq. (5.12).
After fitting the hyperon symmetry energy we have a complete parametrization
for the EoS of $\Lambda$ neutron matter depending on both baryon density and
$\Lambda$ fraction (Eq. (5.8)). For $x_{\Lambda}=0$ the relation reduces to
the EoS of PNM parametrized by the polytrope of Eq. (5.2) whose coefficients
are reported in Tab. 5.1. For $x_{\Lambda}>0$ the presence of hyperons
modifies the PNM EoS through the hyperon symmetry energy and the quadratic
term in $x_{\Lambda}$. The derivation of $S_{\Lambda n}$ has been performed
for small $x_{\Lambda}$ ($\sim 10\%$), corresponding to a baryon density up to
$\sim 3\rho_{0}$. However, this should be enough to derive at least the
$\Lambda$ threshold density by imposing the chemical potentials equilibrium
condition $\mu_{\Lambda}=\mu_{n}$.
Let us start defining the energy density $\mathcal{E}$ for the $\Lambda$
neutron matter as
$\displaystyle\mathcal{E}_{\Lambda n}(\rho_{b},x_{\Lambda})$
$\displaystyle=\rho_{b}E_{\Lambda
n}(\rho_{b},x_{\Lambda})+\rho_{n}m_{n}+\rho_{\Lambda}m_{\Lambda}\;,$
$\displaystyle=\rho_{b}\Bigl{[}E_{\Lambda
n}(\rho_{b},x_{\Lambda})+m_{n}+x_{\Lambda}\Delta m\Bigr{]}\;,$ (5.13)
where
$\displaystyle\rho_{n}=(1-x_{\Lambda})\rho_{b}\quad\quad\quad\rho_{\Lambda}=x_{\Lambda}\rho_{b}\;,$
(5.14)
and $\Delta m=m_{\Lambda}-m_{n}$. For $x_{\Lambda}=0$ the relation corresponds
to the PNM case. The chemical potential is generally defined as the derivative
of the energy density with respect to the number density, evaluated at fixed
volume:
$\displaystyle\mu=\frac{\partial\mathcal{E}}{\partial\rho}\Bigg{|}_{V}\;.$
(5.15)
In AFDMC calculations, because of the requirement of the momentum shell
closure, the number of particles has to be fixed. The density is increased by
changing the volume, i.e. reducing the size of the simulation box. Therefore,
Eq. (5.15) must include a volume correction of the form
$\displaystyle\mu=\frac{\partial\mathcal{E}}{\partial\rho}+\rho\frac{\partial
E}{\partial\rho}\;.$ (5.16)
Our chemical potentials are thus given by
$\displaystyle\mu_{\kappa}(\rho_{b},x_{\Lambda})=\frac{\partial\mathcal{E}_{\Lambda
n}(\rho_{b},x_{\Lambda})}{\partial\rho_{\kappa}}+\rho_{\kappa}\frac{\partial
E_{\Lambda n}(\rho_{b},x_{\Lambda})}{\partial\rho_{\kappa}}\;,$ (5.17)
where $\kappa=n,\Lambda$ and the derivatives of the energy per particle and
energy density must be calculated with respect to $\rho_{b}$ and
$x_{\Lambda}$:
$\displaystyle\frac{\partial\mathcal{F}_{\Lambda
n}(\rho_{b},x_{\Lambda})}{\partial\rho_{\kappa}}$
$\displaystyle=\frac{\partial\mathcal{F}_{\Lambda
n}(\rho_{b},x_{\Lambda})}{\partial\rho_{b}}\frac{\partial\rho_{b}}{\partial\rho_{\kappa}}+\frac{\partial\mathcal{F}_{\Lambda
n}(\rho_{b},x_{\Lambda})}{\partial x_{\Lambda}}\frac{\partial
x_{\Lambda}}{\partial\rho_{\kappa}}\;.$ (5.18)
Recalling Eq. (5.14) we have
$\displaystyle\frac{\partial\rho_{b}}{\partial\rho_{n}}=1\quad\quad\frac{\partial\rho_{b}}{\partial\rho_{\Lambda}}=1\quad\quad\frac{\partial
x_{\Lambda}}{\partial\rho_{n}}=-\frac{x_{\Lambda}}{\rho_{b}}\quad\quad\frac{\partial
x_{\Lambda}}{\partial\rho_{\Lambda}}=\frac{1-x_{\Lambda}}{\rho_{b}}\;,$ (5.19)
and thus the neutron and lambda chemical potentials take the form:
$\displaystyle\mu_{n}(\rho_{b},x_{\Lambda})$
$\displaystyle=\frac{\partial\mathcal{E}_{\Lambda
n}}{\partial\rho_{b}}-\frac{x_{\Lambda}}{\rho_{b}}\frac{\partial\mathcal{E}_{\Lambda
n}}{\partial x_{\Lambda}}+(1-x_{\Lambda})\rho_{b}\frac{\partial E_{\Lambda
n}}{\partial\rho_{b}}-x_{\Lambda}(1-x_{\Lambda})\frac{\partial E_{\Lambda
n}}{\partial x_{\Lambda}}\;,$ (5.20)
$\displaystyle\mu_{\Lambda}(\rho_{b},x_{\Lambda})$
$\displaystyle=\frac{\partial\mathcal{E}_{\Lambda
n}}{\partial\rho_{b}}+\frac{1-x_{\Lambda}}{\rho_{b}}\frac{\partial\mathcal{E}_{\Lambda
n}}{\partial x_{\Lambda}}+x_{\Lambda}\rho_{b}\frac{\partial E_{\Lambda
n}}{\partial\rho_{b}}+x_{\Lambda}(1-x_{\Lambda})\frac{\partial E_{\Lambda
n}}{\partial x_{\Lambda}}\;.$ (5.21)
The two $\mu_{n}$ and $\mu_{\Lambda}$ surfaces in the $\rho_{b}-x_{\Lambda}$
space cross each other defining the curve $x_{\Lambda}(\rho_{b})$ reported in
Fig. 5.6. This curve describes the equilibrium condition
$\mu_{\Lambda}=\mu_{n}$. It thus defines the $\Lambda$ threshold density
$x_{\Lambda}(\rho_{\Lambda}^{th})=0$ and provides the equilibrium $\Lambda$
fraction for each density. For the given parametrization of the hyperon
symmetry energy, the threshold density is placed around $1.9\rho_{0}$, which
is consistent with the theoretical indication about the onset of strange
baryons in the core of a NS. Once the $\Lambda$ particles appear, the hyperon
fraction rapidly increases due to the decrease of the energy and pressure that
favors the $n\rightarrow\Lambda$ transition (see § 1.2). However, there is a
saturation effect induced by the repulsive nature of the hyperon-nucleon
interaction that slows down the production of $\Lambda$ particle at higher
density.
Figure 5.6: $\Lambda$ fraction as a function of the baryon density. The curve
describes the equilibrium condition $\mu_{\Lambda}=\mu_{n}$. The red line is
the result for the quadratic fit on the $\Delta E_{\Lambda n}$ data of Fig.
5.4. The blue dotted vertical line indicates the $\Lambda$ threshold densities
$\rho_{\Lambda}^{th}$ such that $x_{\Lambda}(\rho_{\Lambda}^{th})=0$.
By using the $\Lambda$ threshold density $\rho_{\Lambda}^{th}$ and the
equilibrium $\Lambda$ fraction values $x_{\Lambda}(\rho_{b})$ in Eq. (5.8), we
can finally address the $\Lambda$ neutron matter EoS. The result is reported
in Fig. 5.7. The green dashed line is the PNM EoS for AV8’, the green solid
line the one for AV8’+UIX. Red curve is instead the YNM EoS coming from the
AV8’+UIX+$\Lambda N$+$\Lambda NN$ (II) potentials. At the threshold density
there is a strong softening of the EoS induced by the rapid production of
hyperons. However the EoS becomes soon almost as stiff as the PNM EoS due to
hyperon saturation and the effect of the repulsion among hyperons and
neutrons. In $\rho_{b}=\rho_{\Lambda}^{th}$ there is a phase transition
between PNM and YNM. For densities close to the threshold density, the
pressure becomes negative. This is a non physical finite size effect due to
the small number of particles considered in the simulations, not large enough
for the correct description of a phase transition. However, in the
thermodynamical limit the effect should disappear. We could mitigate this
effect by using a Maxwell construction between the PNM and the YNM EoS. The
details of the density dependence of the energy per baryon at the hyperon
threshold are however not relevant for the derivation of the maximum mass.
The derived model for the EoS of $\Lambda$ neutron matter should be a good
approximation up to $\rho_{b}\sim 3\rho_{0}$. The behavior of the energy per
baryon after this limit depends on density and $\Lambda$ fraction to which we
do not have controlled access with the present AFDMC calculations. Moreover,
starting from $\rho_{b}>0.6\leavevmode\nobreak\ \text{fm}^{-3}$, $\Sigma^{0}$
hyperons could be formed, as shown in Fig. 1.4. The behavior of the energy
curve should thus be different. However, there are already strong indications
for a weak softening of the EoS induced by the presence of hyperons in the
neutron bulk when the hyperon-nucleon potentials employed for hypernuclei are
used.
Figure 5.7: Equation of state for $\Lambda$ neutron matter. Green solid
(dashed) curves refer to the PNM EoS calculated with the AV8’+UIX (AV8’)
potential. Red line is the EoS for YNM corresponding to the quadratic fit on
the $\Delta E$ data of Fig. 5.4. The employed hyperon-nucleon potential is the
full two- plus three-body in the parametrization (II). The $\Lambda$ threshold
density is displayed with the blue dotted vertical line.
#### 5.2.3 Mass-radius relation and the maximum mass
In Chapter 1 we have seen that, given the EoS, the mass-radius relation and
the predicted maximum mass are univocally determined. The $M(R)$ curves are
the solutions of the TOV equations (1.8), which involve the energy density
$\mathcal{E}$ and the pressure $P$. For YNM the energy density is given by Eq.
(5.8) supplemented by the hyperon threshold density and the
$x_{\Lambda}(\rho_{b})$ curve. For the pressure we can simply use the relation
$\displaystyle P_{\Lambda n}(\rho_{b},x_{\Lambda})=\rho_{b}^{2}\frac{\partial
E_{\Lambda n}(\rho_{b},x_{\Lambda})}{\partial\rho_{b}}\;,$ (5.22)
where the additional term due to density dependence of the $\Lambda$ fraction
vanishes once the equilibrium condition $\mu_{\Lambda}=\mu_{n}$ is given.
Fig. 5.8 reports the $M(R)$ curves solution of the TOV equations for the EoS
reported in Fig. 5.7. Green curves are the PNM relations for AV8’ (dashed) and
AV8’+UIX (solid). Red one is the result for the $\Lambda$ neutron matter
described by the full nucleon-nucleon and hyperon-nucleon interaction in the
parametrization (II). The shaded region corresponds to the excluded region by
the causality condition [244]
$\displaystyle
M\lesssim\beta\frac{c^{2}}{G}R\quad\quad\quad\beta=\frac{1}{2.94}\;,$ (5.23)
where $G$ is the gravitational constant and $c$ the speed of light. The curves
with the inclusion of the TNI partially enter the forbidden region. This is
due to the behavior of our EoS that evaluated for very high densities becomes
superluminal. A connection to the maximally stiff EoS given by the condition
$P<1/3\,\mathcal{E}$ should be needed. However, we can estimate the effect on
the maximum mass to be rather small, not changing the general picture.
Figure 5.8: Mass-radius relation for $\Lambda$ neutron matter. Green solid
(dashed) curves refer to the PNM calculation with the AV8’+UIX (AV8’)
potential. Red line is the result for the YNM corresponding to the quadratic
fit on the $\Delta E_{\Lambda n}$ data of Fig. 5.4. The light blue and brown
bands correspond to the masses of the millisecond pulsars PSR J1614-2230
($1.97(4)M_{\odot}$) [6] and PSR J1903+0327 ($2.01(4)M_{\odot}$) [7]. The gray
shaded region is the excluded part of the plot according to causality.
The maximum mass for PNM obtained using the Argonne V8’ and Urbana IX
potentials is reduced from $\sim 2.45M_{\odot}$ to $\sim 2.40M_{\odot}$ by the
inclusion of $\Lambda$ hyperons. This small reduction follows by the stiffness
of the YNM EoS for densities larger than $\rho_{b}\sim 3\rho_{0}$, up to which
our model gives a good description of the strange system. However, by limiting
the construction of the $M(R)$ relation in the range of validity of the
employed YNM model, the mass of the star is already at $\sim 1.81M_{\odot}$
around $R=12.5$ km, and at $\sim 1.98M_{\odot}$ if we extend the range up to
$\rho_{b}=0.55\leavevmode\nobreak\ \text{fm}^{-3}$. These values are larger
than the predicted maximum mass for hypermatter in all (B)HF calculations (see
§ 1.2).
Regardless of the details of the real behavior of the EoS for
$\rho_{b}>3\rho_{0}$, we can speculate that a maximum mass of $2M_{\odot}$ can
be supported by the $\Lambda$ neutron matter described by means of the
realistic AV8’+UIX potentials plus the here developed two- and three-body
hyperon-nucleon interactions. The key ingredient of the picture is the
inclusion of the repulsive $\Lambda NN$ force that has been proven to give a
fundamental contribution in the realistic description of $\Lambda$
hypernuclei. Although very preliminary, our first AFDMC calculations for
hypermatter suggest that a $2M_{\odot}$ neutron star including hyperons can
actually exist.
Figure 5.9: Stellar mass versus central density for $\Lambda$ neutron matter.
The key is the same of Fig. 5.8. The vertical blue dotted line represents the
maximum central density for the stability of the star when TNI forces are
considered.
The solution of the TOV equations provides additional information on the
central density $\rho_{c}$ of the star. The behavior of the star mass as a
function of the central density determines the stability condition of the NS
trough the relation $dM(\rho_{c})/d\rho_{c}>0$. For non rotating neutron
stars, configurations that violate this condition are unstable and will
collapse into black holes [1]. As can be seen from Fig. 5.9 where the mass-
central density relation is reported, the maximum mass also determines the
maximum central density for stable NSs. Within our model, $\rho_{c}^{\max}$ is
around $5.7\rho_{0}$ for both PNM and YNM when the three-nucleon force is
considered in the calculation. Given the fact the inter-particle distance
scale as $\rho_{c}^{-1/3}$, we can estimate that for the given
$\rho_{c}^{\max}$ baryons are not extremely packed. The baryon-baryon
distances are of the order of few fermi, comparable to the range of the hard
core of the nucleon-nucleon and hyperon-nucleon interactions considered.
Therefore, in this framework there is no evidence for the appearance of exotic
phases like quark matter. Our YNM EoS is stiff enough to realistically
describe the infinite medium supporting a $2M_{\odot}$ NS without requiring
other additional degrees of freedom for the inner core.
Empty page
## Chapter 6 Conclusions
In this work the recent developments in Quantum Monte Carlo calculations for
nuclear systems including strange degrees of freedom have been reported. The
Auxiliary Field Diffusion Monte Carlo algorithm has been extended to the
strange sector by the inclusion of the lightest among the hyperons, the
$\Lambda$ particle. This gave us the chance to perform detailed calculations
for $\Lambda$ hypernuclei, providing a microscopic framework for the study of
the hyperon-nucleon interaction in connection with the available experimental
information. The extension of the method for strange neutron matter, put the
basis for the first Diffusion Monte Carlo analysis of the hypernuclear medium,
with the derivation of neutron star observables of great astrophysical
interest.
The main outcome of the study of $\Lambda$ hypernuclei, is that, within the
employed phenomenological model for hyperon-nucleon forces, the inclusion of a
three-body $\Lambda NN$ interaction is fundamental to reproduce the ground
state physics of medium-heavy hypernuclei, in particular the observed
saturation property of the hyperon binding energy. By accurately refitting the
three-body hyperon-nucleon interaction, we obtain a substantial agreement with
the experimental separation energies, that are strongly overestimated by the
use of a bare $\Lambda N$ interaction. The result is of particular interest
because with the employed algorithm, heavy hypernuclei up to 91 particles have
been investigated within the same theoretical framework, providing a realistic
description able to reproduce the extrapolation of the hyperon binding energy
in the infinite medium. By employing an effective hyperon-hyperon interaction,
first steps in the study of $S=-2$ $\Lambda$ hypernuclei have also been taken.
The interest in these systems is motivated by the controversial results coming
from both theoretical and experimental studies.
Preliminary AFDMC results on hypermatter indicate that the hyperon-nucleon
interaction fitted on finite strange nuclei leads to a stiff equation of state
for the strange infinite medium. Within our model, $\Lambda$ particles start
to appear in the neutron bulk around twice the saturation density,
consistently with different theoretical previsions. However, the predicted
softening of the equation of state seems not to be dramatic, due to the
strongly repulsive nature of the employed three-body hyperon-nucleon force.
This fact helps to understand how the necessary appearance of hyperons at some
value of the nucleon density in the inner core of a neutron star might
eventually be compatible with the observed neutron star masses of order
$2M_{\odot}$.
Both works on hypernuclei and hypermatter represent the first Diffusion Monte
Carlo study of finite and infinite strange nuclear systems, and thus are
subject to further improvements. The algorithm for (hyper)nuclei should be
refined in order to become more independent from the starting trial wave
function that should include also correlations other than the pure central.
Together with the accurate treatment of the tensor (and spin-orbit) potential
term and, possibly, with the inclusion of the density dependent nucleon-
nucleon interaction developed in the framework of correlated basis function
[245], the algorithm might become a powerful tool for the precise
investigation not only of energy differences but also of other structural
ground state properties such as density and radii. From the methodological
point of view, the algorithm for infinite strange systems could benefit from
the inclusion of twist-averaged boundary conditions, that would allow for a
more refined study of the equation of state of the hypernuclear medium and
thus the derivation of the maximum mass.
It would be interesting to perform benchmark calculations with the employed
hyperon-nucleon force by means of few-body methods. This would reduce the
uncertainties on the fitted interaction, providing more insight on the
structure of the phenomenological potential for light hypernuclei. On the
other hand, by projecting the three-body interaction on the triplet and
singlet isospin channels, it would be possible to fit the experimental data
for large hypernuclei in order to better capture the features of the
interaction that are relevant for the neutron star physics without
significantly change the compatibility of the results with the lighter strange
nuclei. This could definitely determine a stiff equation of state for the
hyperon neutron matter supporting a $2M_{\odot}$ star.
In the same contest, the study of asymmetric nuclear matter with the inclusion
of hyperon degrees of freedom is very welcome. At present this project has not
started yet and so the goal is far to be achieved. However this is one of the
more promising direction in order to describe the properties of stellar matter
at high densities by means of accurate microscopic calculations with realistic
interactions.
The very recent indication of a bound $\Lambda nn$ three-body system [76],
might motivate the AFDMC investigation of hyper neutron drops. Weakly bound
systems are typically not easily accessible by means of standard AFDMC method
for finite systems. The study of neutron systems confined by an external
potential with the inclusion of one or more hyperons, could give fundamental
information about the hyperon-neutron and hyperon-hyperon interaction in
connection with the experimental evidence of light neutron rich hypernuclei,
such as ${}^{6}_{\Lambda}$H [85], or the theoretical speculation of exotic
neutron systems, as the bound $\Lambda\Lambda nn$ system.
## Appendix A AFDMC wave functions
### A.1 Derivatives of the wave function: CM corrections
As seen in § 3.2.4, for finite systems the single particle orbitals must be
referred to the CM of the system:
$\bm{r}_{p}\rightarrow\bm{r}_{p}-\bm{r}_{CM}$. Each derivative with respect to
nucleon or hyperon coordinates has thus to be calculated including CM
corrections. Let Call $\bm{\rho}_{i}$ the relative coordinates and
$\bm{r}_{i}$ the absolute ones for nucleons, and $\bm{\rho}_{\Lambda}$,
$\bm{r}_{\lambda}$ the analogues for the hyperons. Then
$\displaystyle\bm{\rho}_{i}=\bm{r}_{i}-\bm{\rho}_{CM}\qquad\bm{\rho}_{\lambda}=\bm{r}_{\lambda}-\bm{\rho}_{CM}\;,$
(A.1)
with
$\displaystyle\bm{\rho}_{CM}=\frac{1}{M}\left(m_{N}\sum_{k}\bm{r}_{k}+m_{\Lambda}\sum_{\nu}\bm{r}_{\nu}\right)\qquad
M=\mathcal{N}_{N}\,m_{N}+\mathcal{N}_{\Lambda}\,m_{N}m_{\Lambda}\;.$ (A.2)
In order to simplify the notation, in the next we will use $r_{p}$ instead of
$\bm{r}_{p}$. The equations for the first derivatives will be valid for the
Cartesian component of the position vectors. In the relations for the second
derivatives implicit sums over Cartesian components will be involved.
Consider a function of the relative nucleon and hyperon coordinates:
$\displaystyle f(\rho_{N},\rho_{\Lambda})\equiv
f(\rho_{1},\ldots,\rho_{\mathcal{N}_{N}},\rho_{1},\ldots,\rho_{\mathcal{N}_{\Lambda}})\;,$
(A.3)
In order to calculate the derivatives of $f(\rho_{N},\rho_{\Lambda})$ with
respect to $r_{p}$, we need to change variable. Recalling that now all the
coordinates (nucleons and hyperons) are connected together via the CM, we have
$\displaystyle\frac{\partial}{\partial r_{i}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\sum_{j}\frac{\partial\rho_{j}}{\partial
r_{i}}\frac{\partial}{\partial\rho_{j}}f(\rho_{N},\rho_{\Lambda})+\sum_{\mu}\frac{\partial\rho_{\mu}}{\partial
r_{i}}\frac{\partial}{\partial\rho_{\mu}}f(\rho_{N},\rho_{\Lambda})\;,$ (A.4)
$\displaystyle\frac{\partial}{\partial r_{\lambda}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\sum_{\mu}\frac{\partial\rho_{\mu}}{\partial
r_{\lambda}}\frac{\partial}{\partial\rho_{\mu}}f(\rho_{N},\rho_{\Lambda})+\sum_{j}\frac{\partial\rho_{j}}{\partial
r_{\lambda}}\frac{\partial}{\partial\rho_{j}}f(\rho_{N},\rho_{\Lambda})\;,$
(A.5)
where
$\displaystyle\frac{\partial\rho_{j}}{\partial
r_{i}}=\delta_{ij}-\frac{m_{N}}{M}\,,\quad\;\frac{\partial\rho_{\mu}}{\partial
r_{i}}=-\frac{m_{N}}{M}\,,\quad\;\frac{\partial\rho_{\mu}}{\partial
r_{\lambda}}=\delta_{\lambda\mu}-\frac{m_{\Lambda}}{M}\,,\quad\;\frac{\partial\rho_{j}}{\partial
r_{\lambda}}=-\frac{m_{\Lambda}}{M}\;.$ (A.6)
The CM corrected first derivates take then the form:
$\displaystyle\frac{\partial}{\partial r_{i}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\left[\frac{\partial}{\partial\rho_{i}}-\frac{m_{N}}{M}\left(\sum_{j}\frac{\partial}{\partial\rho_{j}}+\sum_{\mu}\frac{\partial}{\partial\rho_{\mu}}\right)\right]f(\rho_{N},\rho_{\Lambda})\;,$
(A.7) $\displaystyle\frac{\partial}{\partial
r_{\lambda}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\left[\frac{\partial}{\partial\rho_{\lambda}}-\frac{m_{\Lambda}}{M}\left(\sum_{j}\frac{\partial}{\partial\rho_{j}}+\sum_{\mu}\frac{\partial}{\partial\rho_{\mu}}\right)\right]f(\rho_{N},\rho_{\Lambda})\;.$
(A.8)
For the second derivatives we have:
$\displaystyle\frac{\partial^{2}}{\partial
r_{i}^{2}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\left[\frac{\partial^{2}}{\partial\rho_{i}^{2}}-2\frac{m_{N}}{M}\left(\sum_{j}\frac{\partial^{2}}{\partial\rho_{i}\partial\rho_{j}}+\sum_{\mu}\frac{\partial^{2}}{\partial\rho_{i}\partial\rho_{\mu}}\right)\right.$
$\displaystyle+\left.\frac{m_{N}^{2}}{M^{2}}\left(\sum_{jk}\frac{\partial^{2}}{\partial\rho_{j}\partial\rho_{k}}+\sum_{\mu\nu}\frac{\partial^{2}}{\partial\rho_{\mu}\partial\rho_{\nu}}+2\sum_{j\mu}\frac{\partial^{2}}{\partial\rho_{j}\partial\rho_{\mu}}\right)\right]f(\rho_{N},\rho_{\Lambda})\;,$
(A.9) $\displaystyle\frac{\partial^{2}}{\partial
r_{\lambda}^{2}}f(\rho_{N},\rho_{\Lambda})$
$\displaystyle=\left[\frac{\partial^{2}}{\partial\rho_{\lambda}^{2}}-2\frac{m_{\Lambda}}{M}\left(\sum_{\mu}\frac{\partial^{2}}{\partial\rho_{\lambda}\partial\rho_{\mu}}+\sum_{j}\frac{\partial^{2}}{\partial\rho_{\lambda}\partial\rho_{j}}\right)\right.$
$\displaystyle\left.+\frac{m_{\Lambda}^{2}}{M^{2}}\left(\sum_{\mu\nu}\frac{\partial^{2}}{\partial\rho_{\mu}\partial\rho_{\nu}}+\sum_{jk}\frac{\partial^{2}}{\partial\rho_{j}\partial\rho_{k}}+2\sum_{\mu
j}\frac{\partial^{2}}{\partial\rho_{\mu}\partial\rho_{j}}\right)\right]f(\rho_{N},\rho_{\Lambda})\;.$
(A.10)
Consider now the hypernuclear wave function of Eq. (3.202) and assume the
compact notation:
$\displaystyle\psi_{T}$ $\displaystyle=\prod_{\lambda i}f_{c}^{\Lambda
N}(r_{\lambda
i})\,\psi_{T}^{N}(R_{N},S_{N})\,\psi_{T}^{\Lambda}(R_{\Lambda},S_{\Lambda})\;,$
$\displaystyle=\prod_{\lambda i}f_{c}^{\Lambda N}(r_{\lambda
i})\prod_{i<j}f_{c}^{NN}(r_{ij})\prod_{\lambda<\mu}f_{c}^{\Lambda\Lambda}(r_{\lambda\mu})\det\Bigl{\\{}\varphi_{\epsilon}^{N}(\bm{r}_{i},s_{i})\Bigr{\\}}\det\Bigl{\\{}\varphi_{\epsilon}^{\Lambda}(\bm{r}_{\lambda},s_{\lambda})\Bigr{\\}}\;,$
$\displaystyle=J_{\Lambda
N}\,J_{NN}\,J_{\Lambda\Lambda}\,\text{det}_{N}\,\text{det}_{\Lambda}\;.$
(A.11)
The trial wave function is written in the single particle representation and
thus it should be possible to factorize the calculation of the derivatives on
each component. However, when we use the relative coordinates with respect to
the CM, the antisymmetric part of the wave function
$\text{det}_{N}\,\text{det}_{\Lambda}$ has to be treated as a function of both
nucleon and hyperon coordinates, like the function
$f(\rho_{N},\rho_{\Lambda})$ used above. The Jastrow correlation functions
instead, being functions of the distances between two particles, are not
affected by the CM corrections. It is then possible to obtain in a simple way
the derivatives with respect to the nucleon and hyperon coordinates by
calculating the local derivatives:
$\displaystyle\frac{\partial_{p}\psi_{T}}{\psi_{T}}=\frac{\frac{\partial}{\partial
R_{p}}\psi_{T}}{\psi_{T}}\quad\quad\text{with}\quad p=N,\Lambda\;,$ (A.12)
which are of particular interest in the AFDMC code for the calculation of the
drift velocity of Eq. (3.32) and the local energy of Eq. (3.34). The first
local derivatives read
$\displaystyle\frac{\partial_{N}\psi_{T}}{\psi_{T}}$
$\displaystyle=\frac{\partial_{N}J_{NN}}{J_{NN}}+\frac{\partial_{N}J_{\Lambda
N}}{J_{\Lambda
N}}+\frac{\partial_{N}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}\;,$
(A.13) $\displaystyle\frac{\partial_{\Lambda}\psi_{T}}{\psi_{T}}$
$\displaystyle=\frac{\partial_{\Lambda}J_{\Lambda\Lambda}}{J_{\Lambda\Lambda}}+\frac{\partial_{\Lambda}J_{\Lambda
N}}{J_{\Lambda
N}}+\frac{\partial_{\Lambda}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}\;,$
(A.14)
while the second local derivatives take the form
$\displaystyle\frac{\partial_{N}^{2}\psi_{T}}{\psi_{T}}$
$\displaystyle=\frac{\partial_{N}^{2}J_{NN}}{J_{NN}}+\frac{\partial_{N}^{2}J_{\Lambda
N}}{J_{\Lambda
N}}+\frac{\partial_{N}^{2}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}+2\frac{\partial_{N}J_{NN}}{J_{NN}}\frac{\partial_{N}J_{\Lambda
N}}{J_{\Lambda N}}$
$\displaystyle\quad\,+2\frac{\partial_{N}J_{NN}}{J_{NN}}\frac{\partial_{N}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}+2\frac{\partial_{N}J_{\Lambda
N}}{J_{\Lambda
N}}\frac{\partial_{N}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}\;,$
(A.15) $\displaystyle\frac{\partial_{\Lambda}^{2}\psi_{T}}{\psi_{T}}$
$\displaystyle=\frac{\partial_{\Lambda}^{2}J_{\Lambda\Lambda}}{J_{\Lambda\Lambda}}+\frac{\partial_{\Lambda}^{2}J_{\Lambda
N}}{J_{\Lambda
N}}+\frac{\partial_{\Lambda}^{2}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}+2\frac{\partial_{\Lambda}J_{\Lambda\Lambda}}{J_{\Lambda\Lambda}}\frac{\partial_{\Lambda}J_{\Lambda
N}}{J_{\Lambda N}}$
$\displaystyle\quad\,+2\frac{\partial_{\Lambda}J_{\Lambda\Lambda}}{J_{\Lambda\Lambda}}\frac{\partial_{\Lambda}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}+2\frac{\partial_{\Lambda}J_{\Lambda
N}}{J_{\Lambda
N}}\frac{\partial_{\Lambda}\left(\text{det}_{N}\text{det}_{\Lambda}\right)}{\text{det}_{N}\text{det}_{\Lambda}}\;.$
(A.16)
The derivatives of Jastrow correlation functions require a standard
calculations, while for the derivatives of the Slater determinant (SD) we need
to include CM corrections as in Eqs. (A.7), (A.8), (A.9) and (A.10). Moreover,
the derivative of a SD is typically rather computationally expensive and in
the above relations many terms, also with mixed derivatives, are involved. An
efficiently way to deal with derivatives of a SD is described in the next
section.
### A.2 Derivatives of a Slater determinant
Consider a Slater determinant $|\mathcal{A}|$. Let us define
$A_{ij}=f_{i}(j)$, so that $\partial_{j}A_{ij}=f^{\prime}_{i}(j)$. Assume
${}^{i}B$ a matrix equal to $A$ but with the column $i$ replaced by the
derivative of $f$: ${}^{i}B_{ki}=f^{\prime}_{k}(i)$ and
${}^{i}B_{kj}=f_{k}(j)$ for $j\neq i$. Consider then the trivial identity
$\displaystyle|Q|=|Q|\sum_{i}Q_{ij}Q_{ji}^{-1}=\sum_{i}Q_{ij}(Q_{ji}^{-1}|Q|)\;,$
(A.17)
and the following relation
$\displaystyle Q_{ji}^{-1}|Q|=(-1)^{i+j}|Q^{(ij)}|\;,$ (A.18)
where the minor $Q^{(ij)}$ is, by definition, $j$-independent. The first
derivative of a SD takes the form
$\displaystyle\partial_{j}|A|=|A|\sum_{i}A_{ji}^{-1}(\partial_{j}A_{ij})=|A|\sum_{i}A_{ji}^{-1}f^{\prime}_{i}(j)\;,$
(A.19)
and the second derivative reads:
$\displaystyle\partial_{j}^{2}|A|=|A|\sum_{i}A_{ji}^{-1}(\partial^{2}_{j}A_{ij})=|A|\sum_{i}A_{ji}^{-1}f^{\prime\prime}_{i}(j)\;.$
(A.20)
An efficient way to compute the second mixed derivative of a SD
$\partial_{j}\partial_{i}|A|$ is to write the first derivative as
$|^{j}B|=\partial_{j}|A|$, i.e.
$\displaystyle|^{j}B|=|A|\sum_{i}A_{ji}^{-1}f^{\prime}_{i}(j)\;.$ (A.21)
Using the relation (A.19) for $|^{i}B|$, we can write
$\displaystyle\partial_{j}\partial_{i}|A|=\partial_{j}|^{i}B|=|^{i}B|\sum_{k}(^{i}B)_{jk}^{-1}(\partial_{j}{{}^{i}B_{kj}})\;.$
(A.22)
Choosing $j\neq i$ we have that
$(\partial_{j}{{}^{i}B_{kj}})=(\partial_{j}A_{kj})=f^{\prime}_{k}(j)$ and,
using (A.21), it is possible to rewrite the previous equation as:
$\displaystyle\partial_{j}\partial_{i}|A|=|A|\left(\sum_{k}(^{i}B)_{jk}^{-1}f^{\prime}_{k}(j)\right)\left(\sum_{k}A_{ik}^{-1}f^{\prime}_{k}(i)\right)\;.$
(A.23)
Consider now the Sherman-Morrison formula
$\displaystyle(A+\bm{u}\,\bm{v}^{T})^{-1}=A^{-1}-\frac{(A^{-1}\bm{u}\,\bm{v}^{T}A^{-1})}{1+\bm{v}^{T}A^{-1}\bm{u}}\;,$
(A.24)
with $\bm{u},\bm{v}$ vectors. If we choose $(A+\bm{u}\,\bm{v}^{T})={{}^{i}B}$,
i.e.
$\displaystyle
u_{k}=f^{\prime}_{k}(i)-f_{k}(i)\qquad\left\\{\begin{array}[]{ll}v_{k}=0&k\neq
i\\\ v_{k}=1&k=i\end{array}\right.$ (A.27)
we can use the Sherman-Morrison relation to to compute $(^{i}B)^{-1}$:
$\displaystyle(^{i}B)_{jk}^{-1}=A_{jk}^{-1}-A_{ik}^{-1}\frac{\displaystyle\left(\sum_{k}A_{jk}^{-1}f^{\prime}_{k}(i)\right)-\left(\sum_{k}A_{jk}^{-1}f_{k}(i)\right)}{\displaystyle
1+\left(\sum_{k}A_{ik}^{-1}f^{\prime}_{k}(i)\right)-\left(\sum_{k}A_{ik}^{-1}f_{k}(i)\right)}\;.$
(A.28)
Recalling that $f_{k}(i)=A_{ki}$ and assuming $j\neq i$ we have
$\displaystyle(^{i}B)_{jk}^{-1}=A_{jk}^{-1}-A_{ik}^{-1}\frac{\displaystyle\left(\sum_{k}A_{jk}^{-1}f^{\prime}_{k}(i)\right)-\cancelto{0}{\left(\sum_{k}A_{jk}^{-1}A_{ki}\right)}}{\displaystyle\cancel{1}+\left(\sum_{k}A_{ik}^{-1}f^{\prime}_{k}(i)\right)-\cancel{\left(\sum_{k}A_{ik}^{-1}A_{ki}\right)}}\;.$
(A.29)
Finally the second mixed derivative ($j\neq i$) of a SD results:
$\displaystyle\partial_{j}\partial_{i}|A|$
$\displaystyle=|A|\\!\left\\{\left[\sum_{k}A_{ik}^{-1}f^{\prime}_{k}(i)\right]\\!\\!\left[\sum_{k}A_{jk}^{-1}f^{\prime}_{k}(j)\right]\\!-\\!\left[\sum_{k}A_{ik}^{-1}f^{\prime}_{k}(j)\right]\\!\\!\left[\sum_{k}A_{jk}^{-1}f^{\prime}_{k}(i)\right]\right\\}\,.$
(A.30)
Eqs. (A.19), (A.20) and (A.30) are used to calculate the derivatives with all
the CM corrections of the Slater determinant
$f(\rho_{N},\rho_{\Lambda})=\text{det}_{N}\text{det}_{\Lambda}$. The
derivation of these equations is actually valid for any single particle
operator $\mathcal{O}_{j}$. Eqs. (A.19), (A.20) and (A.30) can be thus used to
describe the linear or quadratic action of a single particle operator on a SD,
that can be expressed as a local operator:
$\displaystyle\frac{\mathcal{O}_{j}|A|}{|A|}$
$\displaystyle=\sum_{i}A_{ji}^{-1}(\mathcal{O}_{j}A_{ij})\;,$ (A.31)
$\displaystyle\frac{\mathcal{O}_{j}^{2}|A|}{|A|}$
$\displaystyle=\sum_{i}A_{ji}^{-1}(\mathcal{O}^{2}_{j}A_{ij})\;,$ (A.32)
$\displaystyle\frac{\mathcal{O}_{j}\mathcal{O}_{i}|A|}{|A|}$
$\displaystyle=\left\\{\left[\sum_{k}A_{ik}^{-1}(\mathcal{O}_{i}A_{ki})\right]\\!\\!\left[\sum_{k}A_{jk}^{-1}(\mathcal{O}_{j}A_{kj})\right]\right.$
$\displaystyle\hskip
9.10509pt-\left.\left[\sum_{k}A_{ik}^{-1}(\mathcal{O}_{j}A_{kj})\right]\\!\\!\left[\sum_{k}A_{jk}^{-1}(\mathcal{O}_{i}A_{ki})\right]\right\\}\;.$
(A.33)
For example, considering the spin term of Eq. (3.126) we have:
$\displaystyle\frac{\sigma_{i\alpha}\,\sigma_{j\beta}|A|}{|A|}$
$\displaystyle=\left\\{\left[\sum_{k}A_{jk}^{-1}\sigma_{j\beta}A_{kj}\right]\\!\\!\left[\sum_{k}A_{ik}^{-1}\sigma_{i\alpha}A_{ki}\right]\right.$
$\displaystyle\hskip
9.10509pt-\left.\left[\sum_{k}A_{jk}^{-1}\sigma_{i\alpha}A_{ki}\right]\\!\\!\left[\sum_{k}A_{ik}^{-1}\sigma_{j\beta}A_{kj}\right]\right\\}\;,$
(A.34)
where $|A|$ could be again the SD $\text{det}_{N}\text{det}_{\Lambda}$ of the
trial wave function.
## Appendix B $\Lambda N$ space exchange potential
As proposed by Armani in his Ph.D. thesis [224], the inclusion of the
$\mathcal{P}_{x}$ operator in the AFDMC propagator can be possibly realized by
a mathematical extension of the isospin of nucleons
$\displaystyle\left(\begin{array}[]{c}p\\\
n\end{array}\right)\otimes\Bigl{(}\Lambda\Bigr{)}\quad\longrightarrow\quad\left(\begin{array}[]{c}p\\\
n\\\ \Lambda\end{array}\right)\;,$ (B.6)
such that in the wave function hyperon and nucleon states can be mixed,
referring now to indistinguishable particles. An antisymmetric wave function
with respect to particle exchange must be an eigenstate of the pair exchange
operator $\mathcal{P}_{pair}$ with eigenvalue $-1$:
$\displaystyle-1=\mathcal{P}_{pair}=\mathcal{P}_{x}\,\mathcal{P}_{\sigma}\,\mathcal{P}_{\tau}\quad\Rightarrow\quad\mathcal{P}_{x}=-\mathcal{P}_{\sigma}\,\mathcal{P}_{\tau}\;,$
(B.7)
where $\mathcal{P}_{x}$ exchanges the coordinates of the pair,
$\mathcal{P}_{\sigma}$ the spins and $\mathcal{P}_{\tau}$ the extended
isospins:
$\displaystyle\mathcal{P}_{\sigma}(i\longleftrightarrow j)$
$\displaystyle=\frac{1}{2}\left(1+\sum_{\alpha=1}^{3}\sigma_{i\alpha}\,\sigma_{j\alpha}\right)\;,$
(B.8) $\displaystyle\mathcal{P}_{\tau}(i\longleftrightarrow j)$
$\displaystyle=\frac{1}{2}\left(\frac{2}{3}+\sum_{\alpha=1}^{8}\lambda_{i\alpha}\,\lambda_{j\alpha}\right)\;.$
(B.9)
The particle indices $i$ and $j$ run over nucleons and hyperons and the
$\lambda_{i\alpha}$ are the eight Gell-Mann matrices. $\mathcal{P}_{x}$ takes
now a suitable form (square operators) for the implementation in the AFDMC
propagator. The technical difficulty in such approach is that we need to
deeply modify the structure of the code. The hypernuclear wave function has to
be written as a single Slater determinant including nucleons and hyperons
states, matched with the new 3-component isospinor and 2-component spinors, so
a global 6-component vector. All the potential operators must be represented
as $6\times 6$ matrices and the ones acting on nucleons and hyperons
separately must be projected on the correct extended isospin states:
$\displaystyle\mathcal{P}_{N}$
$\displaystyle=\frac{2+\sqrt{3}\,\lambda_{8}}{3}\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&0\end{array}\right)\;,$ (B.13)
$\displaystyle\mathcal{P}_{\Lambda}$
$\displaystyle=\frac{1-\sqrt{3}\,\lambda_{8}}{3}\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&1\end{array}\right)\;.$ (B.17)
In addition, due to the non negligible mass difference between nucleons and
hyperons, also the kinetic operator must be splitted for states with different
mass:
$\displaystyle\operatorname{e}^{-d\tau\frac{\hbar^{2}}{2}\sum_{i}\mathcal{O}_{m_{i}}\nabla_{i}^{2}}\quad\quad\text{with}\quad\mathcal{O}_{m_{i}}=\left(\begin{array}[]{ccc}1/m_{N}&0&0\\\
0&1/m_{N}&0\\\ 0&0&1/m_{\Lambda}\end{array}\right)\;.$ (B.21)
Finally, it is not even clear if all the operators of the two- and three-body
hyperon-nucleon interaction will be still written in a suitable form for the
application of the the Hubbard-Stratonovich transformation. For pure neutron
systems this approach might simply reduce to an analog of the nucleonic case.
The extended spin-isospin vector will have four components and all the
operators will be represented as $4\times 4$ matrices coupled with the
$\mathcal{P}_{N}$ and $\mathcal{P}_{\Lambda}$ on the reduced space. The
$\mathcal{O}_{m_{i}}$ operator will have just two diagonal elements with the
mass of the neutron and the hyperon. Although this purely mathematical
approach could be applied, many questions arise from the physical point of
view. By considering an extended isospin vector, states with different
strangeness (0 for nucleons and $-1$ for the $\Lambda$ particle) will mix
during the imaginary time evolution. This violates the conservation of
strangeness that should be instead verified by the strong interaction. The
picture becomes even less clear if we consider the $\Lambda\Lambda$
interaction of Eq. (2.48), because strangeness will be distributed among all
the particles but the potential is explicitly developed for hyperon-hyperon
pairs. Thus, for the phenomenological interactions introduced in Chapter 2,
this mathematical approach is not feasible and it has not been investigated in
this work.
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## Appendix C Acknowledgements
Se scrivere una tesi di fisica in inglese è già di per sé un arduo compito,
tentare di riportare i ringraziamenti in lingua anglofona è un’impresa
impossibile, almeno per un veronese quadratico medio come me. Seguirà dunque
uno sproloquio in quella che più si avvicina alla mia lingua madre, nel quale
mi auguro di non dimenticare nessuno, anche se so che sarà inevitabile,
abbiate pazienza.
Inizio col ringraziare la mia famiglia, mamma Graziella e papà Franco in
primis. Nonostante abbia fatto di tutto per rendermi odioso e insopportabile,
soprattutto in periodi di scadenze e consegne, mi hanno sempre sostenuto e
incoraggiato, spingendomi ad andare avanti. Sempre pronti ad ascoltarmi,
continuamente mi chiedevano “Come va a Trento? I tuoi studi?”. E nonostante
poi avessero le idee ancora più confuse di prima al sentire le mie fumose
spiegazioni su iperoni e stelle di neutroni, ogni volta tornavano a informarsi
sul mio lavoro per avere anche solo una vaga idea di quello che facevo per
portare a casa quei quattro euro della borsa di dottorato. Ringrazio Simone e
Sara: sarà la lontananza, sarà che superata la soglia degli “enta” uno inizia
anche a maturare (ahahah), sarà lo snowboard o il downhill ma negli ultimi
anni ci siamo ri-avvicinati parecchio, finendo addirittura in vacanza a
Dublino assieme! Nonostante non sia più un ragazzetto sbarbatello (no speta,
quello lo sono ancora), mi è stato utile avere il supporto, i consigli e la
complicità del fratello maggiore che, anche se non lo ammetterà mai
pubblicamente, so che mi vuole bene. E quindi, grazie! Meritano altrettanti
ringraziamenti i nonni, gli zii e cugini di Caprino e dintorni, e quelli più
geograficamente “lontani” di Verona, che non hanno mai smesso di credere in
me. E perché per i libri, le trasferte, i soggiorni all’estero c’è MasterCard,
ma sapere che nel paesello vengo pubblicizzato con espressioni del tipo “Varda
che me neòdo l’è ’n sciensiato!” non ha prezzo!
Accademicamente parlando non posso non esser grato a Francesco, che mi ha
seguito in questi tre anni di dottorato (e ancor prima durante la laurea
magistrale), con spiegazioni, discussioni, consigli tecnici o anche solo
chiacchierate, soprattutto in questi ultimi mesi parecchio impegnativi sotto
tutti i punti di vista. Nonostante i suoi mille impegni e viaggi, è sempre
stato un punto di riferimento. Aggiungiamo lo Stefano, senza l’aiuto del quale
probabilmente avrei dovuto trovare lavoro come operatore ecologico in quel di
Verona. A parte le mille questioni di fisica o le discussioni su quel cavolo
di codice, lo ringrazio per la vagonata di consigli in generale, per
l’ospitalità, le battute del piffero, le (forse troppe) birrette e le partite
a biliardo super professionali… Assieme a lui è d’obbligo ringraziare la
mitica Serena, che diciamolo, è la persona che porta i pantaloni in quella
famiglia e detto questo ho già detto tutto! Un grazie anche alle due belvette,
che mi hanno fatto un sacco ridere finché ero ospite in casa (e che casa!)
Gandolfi. Tornando un po’ indietro nel tempo devo sicuramente ringraziare il
buon Paolo “Ormoni”, che mi iniziò all’AFDMC e mise le basi per quello che
sarebbe stato poi il mio progetto sui sistemi “strani”. Senza di lui credo non
avrei mai potuto affrontare quel codice e la Bash in generale. Per chiudere la
parte accademica ringrazio poi tutti i LISCers, che hanno contribuito a creare
un ambiente di lavoro intellettualmente stimolante, e tutti coloro con i quali
ho avuto modo di parlare di fisica, Kevin, Steve, Bob, Ben, Abhi, i due
Alessandro e i colleghi di ufficio, i quali però meritano un paragrafo a
parte. Ah sì, non posso certo dimenticare l’infinita pazienza di Micaela che
con la fisica centra poco, ma in merito a burocrazia e organizzazione è
insuperabile.
Veniamo dunque al reparto amicizie: qui potrei dilungarmi fin troppo ma ho
scelto di limitarmi un po’, dividendo il campione in due sottoinsiemi
geografici, quello trentino (in senso lato) e quello più storico veronese,
seguendo un percorso un po’ random (deformazione professionale).
Iniziamo con la completa e incontrollabile degenerazione del mio ufficio,
dall’insostituibile (e dico sul serio $\heartsuit$) Roberto allo shallissimo
Alessandro, dallo “svizzerooooo” Elia al “miserabile” Paolo (con nostro grande
divertimento in perenne lotta per il titolo di maschio omega). E l’ormai santa
donna Giorgia che ha recentemente installato una serie di filtri per escludere
le nostre impertinenti voci. Non dimentichiamo coloro che in principio
colonizzarano l’open space al LISC: il canterino Emmanuel, il già citato Paolo
“Ormoni” e il mitico Enrico (che quando leggerà queste righe inizierà a
riprodurre senza sosta una delle parodie dei prodotti Apple). Aggiungiamo i
colleghi di FBK naturalizzati LISC, quali il Mostarda, l’Amadori, il Fossati
con la fortissima Saini al seguito (ho volutamente messo i cognomi per
subrazzarvi un po’), i personaggi di “passaggio” come Marco e gli adottati da
altri atenei come l’Alessandro (Lovato). Quest’ultimo (eccellente) fisico
merita un ringraziamento particolare (oltre ad una già preventivata cena in
quel di Chicago) per l’estrema ospitalità e il supporto che mi ha dato (e che
spero continuerà a darmi) oltreoceano, non solo per questioni di fisica. In
realtà ognuna delle persone qui citate meriterebbe un grazie su misura, ma non
è facile (e probabilmente nemmeno opportuno) riportare tutto su queste pagine.
Chi mi è stato particolarmente vicino sa già che gli sono grato per tutto, non
servono molte parole…
Uscendo dall’ufficio la cosa si complica perché il numero di persone da
ringraziare cresce di molto. E quindi un caloroso grazie a Giuseppe, Paolo e
Chiara, Alessia, Nicolò, Irena e Nicolò, Sergio, Mattia, Roberta, Nicola,
Cinzia, Giada, Marco, Giovanni, Sebastiano, Fernando, Eleonora, Letizia,
Nikolina, David, Eleonora, Federica, Beatrice, Marta, Fata e a questo punto
sono costretto a mettere un politico _et al._ , non abbiatene a male. Con
alcune di queste persone ho convissuto, con altre si usciva a fare festa,
altre ancora erano e sono “semplicemente” amici, ma tutti hanno contribuito in
qualche modo a farmi trascorrere momenti fantastici in questi tre anni.
Essendo l’autore di questo lavoro mi riservo il diritto di ringraziare in
separata sede Marianna e Gemma: nonostante ci sarebbero molte cose da dire in
merito, mi limiterò ad un semplice ma profondo “grazie!”. Per lo stesso motivo
della precedente proposizione, estendo temporalmente e geograficamente un
ringraziamento anche a Francesco a al Bazza, che col mio dottorato non
centrano un tubo ma che sono stati elementi portanti della mia lunga avventura
trentina e la coda (in termini probabilistici) della loro influenza si fa
tuttora sentire.
Nelle lande veronesi è d’obbligo citare tutti gli amici storici e meno
storici, che nell’ultimo periodo ho avuto modo di vedere più spesso perché,
sarà la moda del momento o qualche virus contagioso, ma qui si stanno sposando
tutti! E dunque grazie ad Andrea, Marco e Jessica, Alice e Francesco, Davide
ed Elisa, Roberta e Alberto, Matteo, Erika, Letizia, Daniela, Mirko, Silvia e
tutti gli altri con cui ho bevuto $n$ birrette (con $n$ spesso troppo grande)
in quel di Caprino e dintorni. Sono particolarmente grato all’IIIIIIIIIING.
Giacomo e alla gnocca Giulia: nell’ultimo periodo non c’è più stato modo di
vedersi spesso ma le serate passate in vostra compagnia mi accompagneranno
sempre col sorriso. Infine, non certo per ordine di importanza, devo
ringraziare di cuore Alessandra (e con lei tutta la famiglia), che per molti
anni è stata al mio fianco sostenendomi, sopportandomi, incoraggiandomi,
facendomi arrabbiare e divertire allo stesso tempo, ma che il destino (o
chi/cosa per esso) ha voluto le nostre strade prendessero due direzioni
diverse, ma nulla o nessuno potrà mai cancellare tutto ciò che di bello e
buono c’è stato. Per cui grazie!
Eccoci dunque alla fine del mio sproloquio. Non mi resta che ringraziare tutte
quelle cose che, pur essendo inanimate, mi hanno fatto penare ma al tempo
stesso esaltare non poco, fra cui meritano un posto di eccellenza Gnuplot,
LaTeX e gli script Bash. Chiudo (stavolta sul serio) ringraziando questo pazzo
2013 che mi ha portato immense soddisfazioni e altrettante sofferenze, ma che
con il suo carico di grandi (a volte fin troppo) novità mi ha stupito e mi ha
spinto a reagire con coraggio facendomi sentire veramente vivo…
> _Meglio aggiungere vita ai giorni che non giorni alla vita._
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|
arxiv-papers
| 2013-11-26T14:01:09 |
2024-09-04T02:49:54.227951
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Diego Lonardoni",
"submitter": "Diego Lonardoni",
"url": "https://arxiv.org/abs/1311.6672"
}
|
1311.6704
|
# BCS-BEC crossover in relativistic Fermi systems
Lianyi He1, Shijun Mao2 and Pengfei Zhuang3 1 Frankfurt Institute for Advanced
Studies, 60438 Frankfurt am Main, Germany
2 School of Science, Xi an Jiaotong University, Xi an 710049, China
3 Physics Department, Tsinghua University and Collaborative Innovation Center
of Quantum Matter, Beijing 100084, China
###### Abstract
We review the BCS-BEC crossover in relativistic Fermi systems, including the
QCD matter at finite density. In the first part we study the BCS-BEC crossover
in a relativistic four-fermion interaction model and show how the relativistic
effect affects the BCS-BEC crossover. In the second part, we investigate both
two-color QCD at finite baryon density and pion superfluid at finite isospin
density, by using an effective Nambu–Jona-Lasinio model. We will show how the
model describes the weakly interacting diquark and pion condensates at low
density and the BEC-BCS crossover at high density.
Keywords: BCS-BEC crossover, relativistic systems, two-color QCD, pion
superfluid
###### pacs:
03.75.Hh, 11.10.Wx, 12.38.-t, 74.20.Fg
## I Introduction
It is generally believed that, by tuning the attractive strength in a Fermi
system, one can realize a smooth crossover from the Bardeen–Cooper–Schrieffer
(BCS) superfluidity at weak attraction to Bose–Einstein condensation (BEC) of
tightly bound difermion molecules at strong attraction Eagles ; Leggett ;
BCSBEC1 ; BCSBEC2 ; BCSBEC3 ; BCSBEC4 ; BCSBEC5 ; BCSBEC6 ; BCSBEC7 . The
typical system is a dilute atomic Fermi gas in three dimensions, where the
effective range $r_{0}$ of the short-range attractive interaction is much
smaller than the inter-particle distance. Therefore, the system can be
characterized by a dimensionless parameter $1/(k_{\rm f}a_{s})$, where $a_{s}$
is the $s$-wave scattering length of the short-range interaction and $k_{\rm
f}$ is the Fermi momentum in the absence of interaction. The BCS-BEC crossover
occurs when the parameter $1/(k_{\rm f}a_{s})$ is tuned from negative to
positive values, and the BCS and BEC limits correspond to the cases $1/(k_{\rm
f}a_{s})\rightarrow-\infty$ and $1/(k_{\rm f}a_{s})\rightarrow+\infty$,
respectively.
This BCS-BEC crossover phenomenon has been successfully demonstrated in
ultracold fermionic atoms, where the $s$-wave scattering length and hence the
parameter $1/(k_{\rm f}a_{s})$ are tuned by means of the Feshbach resonance
BCSBECexp1 ; BCSBECexp2 ; BCSBECexp3 . At the resonant point or the so-called
unitary point with $a_{s}\rightarrow\infty$, the only length scale of the
system is the inter-particle distance ($\sim k_{\rm f}^{-1}$). Therefore, the
properties of the system at the unitary point $1/(k_{\rm f}a_{s})=0$ become
universal, i.e., independent of the details of the interactions. All physical
quantities, scaled by their counterparts for the non-interacting Fermi gases,
become universal constants. Determining these universal constants has been one
of the most intriguing topics in the research of cold Fermi gases Unitary ;
Unitary1 ; Unitary11 ; Unitary2 ; Unitary21 .
The BCS-BEC crossover has become an interesting and important issue for the
studies of dense and strongly interacting matter, i.e., nuclear or quark
matter which may exist in the core of compact stars BCSBECNM ; BCSBECNM1 ;
BCSBECNM2 ; BCSBECNM3 ; pairsize ; kitazawa ; kitazawa1 ; kitazawa2 ;
kitazawa3 ; Abuki ; RBCSBEC ; RBCSBEC1 ; RBCSBEC2 ; RBCSBEC3 ; RBCSBEC4 ;
RBCSBEC5 ; RBCSBEC6 ; RBCSBEC7 ; RBCSBEC8 ; RBCSBEC9 ; RBCSBEC10 ; BCSBECQCD ;
BCSBECQCD1 ; BCSBECQCD2 ; BCSBECQCD3 ; BCSBECQCD4 ; BCSBECQCD5 ; BCSBECQCD6 ;
BCSBECQCD7 ; BCSBECQCD8 ; BCSBECQCD9 . By analogy with the usual superfluid,
the BCS-BEC crossover in dense quark matter can be theoretically described
pion2 ; pion21 ; pion22 ; pion1 ; pion11 by the quark chemical potential which
is positive in BCS and negative in BEC, the size of the Cooper pair which is
large in BCS and small in BEC, and the scaled pair condensate which is small
in BCS and large in BEC. However, unlike the fermion-fermion scattering in
cold atom systems, quarks are unobservable degrees of freedom. The quark-quark
scattering can not be measured or used to experimentally identify the BCS-BEC
crossover, and its function to characterize the BCS-BEC crossover at quark
level is replaced by the difermion molecules scattering maozh ; detmold . In
the BCS quark superconductor/superfluid, the large and overlapped pairs lead
to a large pair-pair cross section, and the small and individual pairs in the
BEC superconductor/superfluid interact weakly with small cross section.
A good knowledge of Quantum Chromodynamics (QCD) at finite temperature and
density kapusta is significant for us to understand a wide range of physical
phenomena in thermal and dense nuclear matter and quark matter. To understand
the evolution of the early universe in the first few seconds after the big
bang, we need the nature of the QCD phase transitions at temperature $T\sim
170$ MeV and nearly vanishing density. On the other hand, to understand the
physics of compact stars we need the knowledge of the equation of state and
dynamics of QCD matter at high density and low temperature.
Color superconductivity in dense quark matter is due to the attractive
interaction in certain diquark channels CSCearly ; CSCbegin ; CSCbegin1 ;
CSCreview ; CSCreview1 ; CSCreview2 ; CSCreview3 ; CSCreview4 ; CSCreview5 ;
CSCreview6 . Taking into account only the screened (color) electric
interaction which is weakened at the Debye mass scale $g\mu$ ($g$ is the QCD
coupling constant and $\mu$ the quark chemical potential), the early studies
CSCearly predicted a rather small pairing gap $\Delta\sim 1$ MeV at moderate
density with $\mu\sim\Lambda_{\text{QCD}}$ ($\sim 300$ MeV as the QCD energy
scale). The breakthrough in the field was made in CSCbegin ; CSCbegin1 where
it was observed that the pairing gap is about 2 orders of magnitude larger
than the previous prediction, by using the instanton-induced interactions and
the phenomenological four-fermion interactions. On the other hand, it was
first pointed out by Son that, at asymptotic high density the unscreened
magnetic interaction becomes dominant CSCgap . This leads to a non-BCS gap
$\Delta\sim\mu g^{-5}\exp{\left(-c/g\right)}$ pQCD ; pQCD1 ; pQCD2 ; pQCD3 ;
pQCD4 ; pQCD5 ; pQCD6 with $c=3\pi^{2}/\sqrt{2}$, which matches the large
magnitude of $\Delta$ at moderate density. The same phenomena happen in pion
superfluid at moderate isospin density ISO .
Such gaps at moderate density are so large that they may fall outside of
applicability range of the usual BCS-like mean field theory. It was estimated
that the size of the Cooper pairs or the superconducting coherence length
$\xi_{c}$ becomes comparable to the averaged inter-quark distance $d$ pairsize
at moderate density with $\mu\sim\Lambda_{\text{QCD}}$. This feature is quite
different from the standard BCS superconductivity in metals with $\xi_{c}\gg
d$. Qualitatively, we can examine the ratio of the superconducting transition
temperature $T_{c}$ to the Fermi energy $E_{\text{f}}$,
$\kappa=T_{c}/E_{\text{f}}$ TC . There are $\kappa\sim 10^{-5}$ for ordinary
BCS superconductors and $\kappa\sim 10^{-2}$ for high temperature
superconductors. For quark matter at moderate density, taking
$E_{\text{f}}\simeq 400$ MeV and $T_{c}\simeq 50$ MeV TCQ , we find that
$\kappa$ is even higher, $\kappa\sim 10^{-1}$, which is close to that for the
resonant superfluidity in strongly interacting atomic Fermi gases TC . This
indicates that the color superconductor and pion superfluid at moderate
density are likely in the strongly coupled region or the BCS-BEC crossover
region. It is known that the pairing fluctuation effects play important role
in the BCS-BEC crossover BCSBEC5 . The effects of the pairing fluctuations on
the quark spectral properties, including possible pseudogap formation in
heated quark matter (above the color superconducting transition temperature),
was elucidated by Kitazawa, Koide, Kunihiro, and Nemoto kitazawa ; kitazawa1 ;
kitazawa2 ; kitazawa3 .
Since the dense quark matter is relativistic, it is interesting to study how
the relativistic effect affects the BCS-BEC crossover. This is recently
investigated Abuki ; RBCSBEC ; RBCSBEC1 ; RBCSBEC2 ; RBCSBEC3 ; RBCSBEC4 ;
RBCSBEC5 ; RBCSBEC6 ; RBCSBEC7 ; RBCSBEC8 ; RBCSBEC9 ; RBCSBEC10 in the
Nozieres-Schmitt-Rink (NSR) theory above the critical temperature, the boson-
fermion model, and the BCS-Leggett mean field theory at zero temperature. It
is shown that, not only the BCS superfluidity and the nonrelativistic BEC for
heavy molecules but also the nonrelativistic and relativistic BEC for nearly
massless molecules can be smoothly connected. In the first part of this
article, we will review the studies on the BCS-BEC crossover in a relativistic
four-fermion interaction model. By using the generalized mean field theory or
the so-called pseudogap theory at finite temperature, we are able to predict
the size of the pseudogap energy in color superconducting quark matter at
moderate baryon density.
Lattice simulation of QCD at finite temperature and vanishing density has been
successfully performed. However, at large baryon density the lattice
simulation has not yet been successfully done due to the sign problem Lreview
; Lreview1 : the fermion determinant is not positively definite in the
presence of a nonzero baryon chemical potential $\mu_{\text{B}}$. To study the
nature of QCD matter at finite density, we first study some special theories
which possess positively definite fermion determinant and can be simulated on
the lattice. One case is the so-called QCD-like theories at finite $\mu_{\rm
B}$ QC2D ; QC2D1 ; QC2D2 ; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ; QL03 ; ratti ;
2CNJL04 where quarks are in a real or pseudo-real representation of the gauge
group, including two-color QCD with quarks in the fundamental representation
and QCD with quarks in the adjoint representation. While these theories are
not real QCD, they can be simulated on the lattice LQC2D ; LQC2D1 ; LQC2D2 ;
LQC2D3 ; LQC2D4 ; LBECBCS and may give us some information of real QCD at
finite baryon density. Notice that the lightest baryon state in two-color QCD
is just the scalar diquarks. Another interesting case is real QCD at finite
isospin chemical potential $\mu_{\text{I}}$ ISO , where the chemical
potentials for light $u$ and $d$ quarks have opposite signs and hence the
fermion determinant is positively definite. For both two-color QCD at finite
$\mu_{\rm B}$ and QCD at finite $\mu_{\rm I}$, chiral perturbation theory and
other effective models predict a continuous quantum phase transition from the
vacuum to the matter phase when $\mu_{\rm B}$ or $\mu_{\text{I}}$ is equal to
the pion mass $m_{\pi}$ QC2D ; QC2D1 ; QC2D2 ; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ;
QL03 ; ratti ; 2CNJL04 ; ISO ; ISOother01 ; ISOother011 ; ISOother012 ;
ISOother013 ; ISOother014 ; ISOother015 ; ISOother016 ; ISOother017 ;
ISOother018 ; ISOother019 ; ISOother0110 ; ISOother0111 ; ISOother0112 ;
ISOother0113 ; ISOother0114 ; ISOother0115 ; boser ; ISOother02 ; ISOother021
, in contrast to real QCD at finite $\mu_{\rm B}$ where the phase transition
takes place when $\mu_{\text{B}}$ is approximately equal to the nucleon mass
$m_{\rm N}$. This transition has also been verified by lattice simulations
LQC2D ; LQC2D1 ; LQC2D2 ; LQC2D3 ; LQC2D4 ; LBECBCS ; Liso ; Liso1 ; Liso2 ;
Liso3 . The resulting matter near the quantum phase transition is expected to
be a dilute Bose condensate with weakly repulsive interactions Bose01 .
The Bose-Einstein condensation (BEC) phenomenon is believed to widely exist in
dense and strongly interacting matter. For instance, pions or Kaons can
condense in neutron stars if the electron chemical potential exceeds the
effective mass of pions or Kaons PiC ; PiC1 ; PiC2 ; PiC3 ; PiC4 . However,
the condensation of pions and Kaons in neutron stars is rather complicated due
to the meson-nucleon interactions in dense nuclear medium. On the other hand,
at asymptotically high density, perturbative QCD calculations show that the
ground state of dense QCD is a weakly coupled BCS superfluid with condensation
of overlapping Cooper pairs pQCD ; pQCD1 ; pQCD2 ; pQCD3 ; pQCD4 ; pQCD5 ;
pQCD6 ; ISO . For two-color QCD at finite baryon density or QCD at finite
isospin density, the BCS superfluid state at high density and the Bose
condensate of diquarks or pions have the same symmetry breaking pattern and
thus are smoothly connected with one another ISO . The BCS and BEC state are
both characterized by the nonzero expectation value $\langle qq\rangle\neq 0$
for two-color QCD at finite baryon density or
$\langle\bar{u}i\gamma_{5}d\rangle\neq 0$ for QCD at finite isospin density.
This phenomenon is just the BCS-BEC crossover discussed by Eagles Eagles and
Leggett Leggett in condensed matter physics.
While lattice simulations of two-color QCD at finite baryon chemical potential
LQC2D ; LQC2D1 ; LQC2D2 ; LQC2D3 ; LQC2D4 ; LBECBCS and QCD at finite isospin
chemical potential Liso ; Liso1 ; Liso2 ; Liso3 can be performed, it is still
interesting to employ some effective models to describe the crossover from the
Bose condensate at low density to the BCS superfluidity at high density. The
chiral perturbation theories as well as the linear sigma models QC2D ; QC2D1 ;
QC2D2 ; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ; QL03 ; ratti ; 2CNJL04 ; ISO ;
ISOother01 ; ISOother011 ; ISOother012 ; ISOother013 ; ISOother014 ;
ISOother015 ; ISOother016 ; ISOother017 ; ISOother018 ; ISOother019 ;
ISOother0110 ; ISOother0111 ; ISOother0112 ; ISOother0113 ; ISOother0114 ;
ISOother0115 ; boser ; ISOother02 ; ISOother021 , which describe only the
physics of Bose condensate, does not meet our purpose. The Nambu–Jona-Lasinio
(NJL) model NJL with quarks as elementary blocks, which describes well the
chiral symmetry breaking and low energy phenomenology of the QCD vacuum, is
generally believed to work at low and moderate temperatures and densities
NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3 . In the second part of this
article, we present our study of the BEC-BCS crossover in two-color QCD at
finite baryon density in the frame of NJL model. Pion superfluid and the BEC-
BCS crossover at finite isospin density are discussed in this model too.
The article is organized as follows. In Section II we present our study on the
BCS-BEC crossover in relativistic Fermi systems by using a four-fermion
interaction model. Both zero temperature and finite temperature cases are
studied. In Section III the two-color QCD at finite baryon density and pion
superfluid at finite isospin density are studied in the NJL model. We
summarize in Section IV. Throughout the article, we use
$K\equiv(i\omega_{n},{\bf k})$ and $Q\equiv(i\nu_{m},{\bf q})$ to denote the
four momenta for fermions and bosons, respectively, where
$\omega_{n}=(2n+1)\pi T$ and $\nu_{m}=2m\pi T$ ($m,n$ integer) are the
Matsubara frequencies. We will use the following notation for the frequency
summation and momentum integration,
$\sum_{P}=T\sum_{l}\sum_{\bf p},\ \ \ \ \sum_{\bf p}=\int\frac{d^{3}{\bf
p}}{(2\pi)^{3}},\ \ \ P=K,Q,\ \ l=n,m.$ (1)
## II BCS-BEC crossover with relativistic fermions
The motivation of studying BCS-BEC crossover with relativistic fermions is
mostly due to the study of dense and hot quark matter which may exist in
compact stars and can be created in heavy ion collisions. However, we shall
point out that a relativistic theory is also necessary for non-relativistic
systems when the attractive interaction strength becomes super strong. To this
end, let us first review the nonrelativistic theory of BCS-BEC crossover in
dilute Fermi gases with $s$-wave interaction.
The Leggett mean field theory Leggett is successful to describe the BCS-BEC
crossover at zero temperature in dilute Fermi gases with short-range $s$-wave
interaction. The BCS-BEC crossover can be realized in a dilute two-component
Fermi gas with fixed total density $n=k_{\rm f}^{3}/(3\pi^{2})$ ($k_{\rm f}$
is the Fermi momentum) by tuning the $s$-wave scattering length $a_{s}$ from
negative to positive. Theoretically, the BCS-BEC crossover can be seen if we
self-consistently solve the gap and number equations for the pairing gap
$\Delta_{0}$ and the fermion chemical potential $\mu_{\rm n}$ BCSBEC3 ,
$\displaystyle-\frac{m}{4\pi a_{s}}$ $\displaystyle=$ $\displaystyle\sum_{\bf
k}\left({\frac{1}{2\sqrt{\xi_{\bf k}^{2}+\Delta_{0}^{2}}}-\frac{m}{{\bf
k}^{2}}}\right),$ $\displaystyle\frac{k_{\rm f}^{3}}{3\pi^{2}}$
$\displaystyle=$ $\displaystyle\sum_{\bf k}\left(1-\frac{\xi_{\bf
k}}{\sqrt{\xi_{\bf k}^{2}+\Delta_{0}^{2}}}\right)$ (2)
with $\xi_{\bf k}={\bf k}^{2}/(2m)-\mu_{\rm n}$ and $a_{s}$ being the s-wave
scattering length. The fermion mass $m$ plays a trivial role here, since the
BCS-BEC crossover depends only on a dimensionless parameter $\eta=1/(k_{\rm
f}a_{s})$. This is the so-called universality for such a nonrelativistic
syetem. The BCS-BEC crossover can be characterized by the behavior of the
chemical potential $\mu_{\rm n}$: it coincides with the Fermi energy
$\epsilon_{\rm f}=k_{\rm f}^{2}/(2m)$ in the BCS limit
$\eta\rightarrow-\infty$, but becomes negative in the BEC region. In the BEC
limit $\eta\rightarrow+\infty$, we have $\mu_{\rm n}\rightarrow-E_{b}/2$ with
$E_{b}=1/(ma_{s}^{2})$ being the molecular binding energy. Therefore, in the
nonrelativistic theory the chemical potential will tend to be negatively
infinity in the strong coupling BEC limit.
Then a problem arises if we look into the physics of the BEC limit from a
relativistic point of view. In the relativistic description, the fermion
dispersion (without pairing) becomes $\xi_{\bf k}^{\pm}=\sqrt{{\bf
k}^{2}+m^{2}}\pm\mu$, where $\mp$ correspond to fermion and anti-fermion
degrees of freedom, and $\mu$ is the chemical potential in relativistic theory
kapusta . In the non-relativistic limit $|{\bf k}|\ll m$, if $|\mu-m|\ll m$,
we can neglect the anti-fermion degrees of freedom and recover the
nonrelativistic dispersion $\xi_{\bf k}^{-}\simeq{\bf k}^{2}/(2m)-(\mu-m)$.
Therefore, the quantity $\mu-m$ plays the role of the chemical potential
$\mu_{\rm n}$ in nonrelativistic theory. While $\mu_{\rm n}$ can be
arbitrarily negative in the nonrelativistic theory, $\mu$ is under some
physical constraint in the relativistic theory. Since the molecule binding
energy $E_{b}$ can not exceed two times the constituent mass $m$, even at
super strong coupling the absolute value of the nonrelativistic chemical
potential $\mu_{\rm n}=\mu-m\simeq-E_{b}/2$ can not exceed the fermion mass
$m$, and the relativistic chemical potential $\mu$ should be always positive.
If the system is dilute, i.e., the Fermi momentum satisfies $k_{\rm f}\ll m$,
we expect that the non-relativistic theory works well when the attraction is
not very strong (the binding energy $E_{b}\ll 2m$). However, if the attraction
is strong enough to ensure $E_{b}\sim 2m$, relativistic effects will appear.
From $E_{b}\sim 1/(ma_{s}^{2})$, we can roughly estimate that the
nonrelativistic theory becomes unphysical for $a_{s}\sim 1/m$, corresponding
to super strong attraction. This can be understood if we consider $1/m$ as the
Compton wavelength $\lambda_{c}$ of a particle with mass $m$.
What will then happen in the strong attraction limit in an attractive Fermi
gas? If the attraction is so strong that $E_{b}\rightarrow 2m$ and
$\mu\rightarrow 0$, the excitation spectra $\xi_{\bf k}^{-}$ and $\xi_{\bf
k}^{+}$ for fermions and anti-fermions become nearly degenerate, and non-
relativistic limit cannot be reached even though $k_{\rm f}\ll m$. This means
that the anti-fermion pairs can be excited by strong attraction and the
condensed bosons and anti-bosons become both nearly massless. Therefore,
without any model dependent calculation, we observe an important relativistic
effect on the BCS-BEC crossover: there exists a relativistic BEC (RBEC) state
Abuki ; RBCSBEC ; RBCSBEC1 ; RBCSBEC2 ; RBCSBEC3 ; RBCSBEC4 ; RBCSBEC5 ;
RBCSBEC6 ; RBCSBEC7 ; RBCSBEC8 ; RBCSBEC9 ; RBCSBEC10 which is smoothly
connected to the nonrelativistic BEC (NBEC) state. The RBEC is not a specific
phenomenon for relativistic Fermi systems, it should appear in any Fermi
system if the attraction can be strong enough, even though the initial non-
interacting gas satisfies the non-relativistic condition $k_{\rm f}\ll m$.
We now study the BCS-BEC crossover in a relativistic four-fermion interaction
model, which we expect to recover the nonrelativistic theory in a proper
limit. The Lagrangian density of the model is given by
${\cal L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi+{\cal L}_{\rm
I}(\bar{\psi},\psi),$ (3)
where $\psi,\bar{\psi}$ denote the Dirac fields with mass $m$ and ${\cal
L}_{\rm I}$ describes the attractive interaction among the fermions. For the
sake of simplicity, we consider only the dominant interaction in the scalar
$J^{P}=0^{+}$ channel RBCS ; RBCS1 ; RBCS2 , which in the nonrelativistic
limit recovers the $s$-wave interaction in the nonrelativistic theory. The
interaction Lagrangian ${\cal L}_{\rm I}$ can be modeled by a contact
interaction Abuki ; RBCSBEC ; RBCSBEC1 ; RBCSBEC2 ; RBCSBEC3 ; RBCSBEC4 ;
RBCSBEC5 ; RBCSBEC6 ; RBCSBEC7 ; RBCSBEC8 ; RBCSBEC9 ; RBCSBEC10
${\cal L}_{\rm
I}=\frac{g}{4}\left(\bar{\psi}i\gamma_{5}C\bar{\psi}^{\text{T}}\right)\left(\psi^{\text{T}}Ci\gamma_{5}\psi\right),$
(4)
where $g>0$ is the coupling constant and $C=i\gamma_{0}\gamma_{2}$ is the
charge conjugation matrix. Generally, by increasing the attractive coupling
$g$, the crossover from condensation of spin-zero Cooper pairs at weak
coupling to the Bose-Einstein condensation of bound bosons at strong coupling
can be realized.
We start our calculation from the partition function ${\cal Z}$ in the
imaginary time formalism,
${\cal Z}=\int[d\bar{\psi}][d\psi]e^{\int dx({\cal
L}+\mu\bar{\psi}\gamma_{0}\psi)}$ (5)
with $\int dx\equiv\int_{0}^{1/T}d\tau\int d^{3}{\bf r}$, $x=(\tau,{\bf r})$
and $\mu$ being the chemical potential conjugating to the charge density
operator $\psi^{\dagger}\psi=\bar{\psi}\gamma_{0}\psi$. Similar to the method
in the study of superconductivity, we introduce the Nambu-Gor’kov spinors nam
$\Psi=\left(\begin{array}[]{cc}\psi\\\
C\bar{\psi}^{\text{T}}\end{array}\right),\ \ \
\bar{\Psi}=\left(\begin{array}[]{cc}\bar{\psi}&\psi^{\text{T}}C\end{array}\right)$
(6)
and the auxiliary pair field
$\Delta(x)=(g/2)\psi^{\text{T}}(x)Ci\gamma_{5}\psi(x)$. After performing the
Hubbard-Stratonovich transformation hs ; hs1 , we rewrite the partition
function as
${\cal Z}=\int[d\bar{\Psi}][d\Psi][d\Delta][d\Delta^{*}]e^{-{\cal A}_{\rm
eff}}$ (7)
with
${\cal A}_{\rm eff}=\int dx\frac{|\Delta(x)|^{2}}{g}-\frac{1}{2}\int dx\int
dx^{\prime}\bar{\Psi}(x){\bf G}^{-1}(x,x^{\prime})\Psi(x^{\prime})$ (8)
and the inverse fermion propagator ${\bf G}^{-1}$
$\displaystyle{\bf
G}^{-1}=\left(\begin{array}[]{cc}i\gamma^{\mu}\partial_{\mu}-m+\mu\gamma_{0}&i\gamma_{5}\Delta(x)\\\
i\gamma_{5}\Delta^{*}(x)&i\gamma^{\mu}\partial_{\mu}-m-\mu\gamma_{0}\end{array}\right)\delta(x-x^{\prime}).$
(11)
Integrating out the fermion fields, we obtain
${\cal Z}=\int[d\Delta][d\Delta^{*}]e^{-{\cal S}_{\rm
eff}\left[\Delta,\Delta^{*}\right]}$ (12)
with the bosonized effective action
$\displaystyle{\cal S}_{\rm eff}=\int
dx\frac{|\Delta(x)|^{2}}{g}-\frac{1}{2}\text{Tr}\ln\left[{\bf
G}^{-1}(x,x^{\prime})\right].$ (13)
### II.1 Zero temperature analysis
While in general case the pairing fluctuations are important BCSBEC1 ; BCSBEC2
; BCSBEC5 , the Leggett mean field theory is already good to describe
qualitatively the BCS-BEC crossover at zero temperature. This can be naively
seen from the fact that the dominant contribution of fluctuations to the
thermodynamic potential at low temperature is from the massless Goldstone
modes gold ; gold1 and is therefore proportional to $T^{4}$ BCSBEC3 . At zero
temperature it vanishes. However, since the quantum fluctuations are not taken
into account, the mean field theory cannot predict quantitatively the
universal constants in the unitary limit and the boson-boson scattering length
in the BEC limit. Since our goal in this paper is to study the BCS-BEC
crossover with relativistic fermions on a qualitative level, we shall take the
mean field approximation for the study of the zero temperature case.
In the mean field approximation, we consider the uniform and static saddle
point $\Delta(x)=\Delta_{0}$ which serves as the order parameter of the
fermionic superfluidity. Due to the U$(1)$ symmetry of the Lagrangian, the
phase of the order parameter can be chosen arbitrarily and we therefore set
$\Delta_{0}$ to be real without loss of generality. The thermodynamic
potential density $\Omega=T{\cal S}_{\rm eff}(\Delta_{0})/V$ at the saddle
point can be evaluated askapusta
$\displaystyle\Omega={\Delta_{0}^{2}\over g}-\sum_{\bf k}\left(E_{\bf
k}^{-}+E_{\bf k}^{+}-\xi_{\bf k}^{-}-\xi_{\bf k}^{+}\right),$ (14)
where the relativistic BCS-like excitation spectra read $E_{\bf
k}^{\pm}=\sqrt{(\xi_{\bf k}^{\pm})^{2}+\Delta_{0}^{2}}$ and $\xi_{\bf
k}^{\pm}=\epsilon_{\bf k}\pm\mu$ with $\epsilon_{\bf k}=\sqrt{{\bf
k}^{2}+m^{2}}$, and the superscripts - and + correspond to the contributions
from fermions and anti-fermions, respectively. Minimizing $\Omega$ with
respect to $\Delta_{0}$, we obtain the gap equation to determine the physical
$\Delta_{0}$,
$\frac{1}{g}={1\over 2}\sum_{\bf k}\left(\frac{1}{E_{\bf
k}^{-}}+\frac{1}{E_{\bf k}^{+}}\right).$ (15)
From the thermodynamic relation we also obtain the number equation for the
fermion density $n=k_{f}^{3}/(3\pi^{2})$ bcs ,
$n=\sum_{\bf k}\left[\left(1-\frac{\xi_{\bf k}^{-}}{E_{\bf
k}^{-}}\right)-\left(1-\frac{\xi_{\bf k}^{+}}{E_{\bf k}^{+}}\right)\right].$
(16)
The relativistic model with contact four-fermion interaction is non-
renormalizable and a proper regularization is needed. In this study we
introduce a momentum cutoff $\Lambda$ to regularize the divergent momentum
integrals. The cutoff $\Lambda$ then serves as a model parameter. For a more
realistic model such as fermions interact via exchange of bosons (Yukawa
coupling), the cutoff does not appear in principle but the calculation becomes
much more complicated. To compare with the nonrelativistic theory, we also
replace the coupling constant $g$ by a “renormalized” coupling $U$, which is
given by
$-\frac{1}{U}=\frac{1}{g}-\frac{1}{2}\sum_{\bf k}\left(\frac{1}{\epsilon_{\bf
k}-m}+\frac{1}{\epsilon_{\bf k}+m}\right).$ (17)
This corresponds to subtracting the right hand side of the gap equation (15)
at $\Delta_{0}=0$ and $\mu=m$. Such a subtraction is consistent with the
formula derived from the relativistic two-body scattering matrix Abuki . The
effective $s$-wave scattering length $a_{s}$ can be defined as $U=4\pi
a_{s}/m$. Actually, by defining the new coupling constant $U$, we find that
$a_{s}$ recovers the definition of the $s$-wave scattering length in the
nonrelativistic limit. While this is a natural extension of the coupling
constant renormalization in nonrelativistic theory, we keep in mind that the
ultraviolet divergence cannot be completely removed, and the momentum cutoff
$\Lambda$ still exists in the relativistic theory. For our convenience, we
define the relativistic Fermi energy $E_{\rm f}$ as $E_{\rm f}=\sqrt{k_{\rm
f}^{2}+m^{2}}$, which recovers the Fermi kinetic energy $E_{\rm
f}-m=\epsilon_{\rm f}\simeq k_{\rm f}^{2}/(2m)$ in nonrelativistic limit.
In the nonrelativistic theory for BCS-BEC crossover in dilute Fermi gases,
there are only two characteristic lengths, i.e., $k_{\rm f}^{-1}$ and $a_{s}$.
The BCS-BEC crossover then shows the universality: after a proper scaling, all
physical quantities depend only on the dimensionless coupling $\eta=1/(k_{\rm
f}a_{s})$. Especially, in the unitary limit $a_{s}\rightarrow\infty$, all
scaled physical quantities become universal constants. However, unlike in the
nonrelativistic theory where the fermion mass $m$ plays a trivial role, in the
relativistic theory a new length scale, namely the Compton wavelength
$\lambda_{c}=m^{-1}$ appears. As a consequence, the BCS-BEC crossover should
depend on not only the dimensionless coupling $\eta$, but also the
relativistic parameter $\zeta=k_{\rm f}/m=k_{\rm f}\lambda_{c}$. Since the
cutoff $\Lambda$ is needed, the result also depends on $\Lambda/m$ or
$\Lambda/k_{\rm f}$. By scaling all energies by $\epsilon_{f}$ and momenta by
$k_{f}$, the gap and number equations (15) and (16) become dimensionless,
$\displaystyle-\frac{\pi}{2}\eta$ $\displaystyle=$
$\displaystyle\int_{0}^{z}x^{2}dx\left[\left(\frac{1}{E_{x}^{-}}-\frac{1}{\epsilon_{x}-2\zeta^{-2}}\right)+\left(\frac{1}{E_{x}^{+}}-\frac{1}{\epsilon_{x}+2\zeta^{-2}}\right)\right],$
$\displaystyle\frac{2}{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{z}x^{2}dx\left[\left(1-\frac{\xi_{x}^{-}}{E_{x}^{-}}\right)-\left(1-\frac{\xi_{x}^{+}}{E_{x}^{+}}\right)\right],$
(18)
with
$E_{x}^{\pm}=\sqrt{\left(\xi_{x}^{\pm}\right)^{2}+\left(\Delta_{0}/\epsilon_{\rm
f}\right)^{2}}$, $\xi_{x}^{\pm}=\epsilon_{x}\pm\mu/\epsilon_{\rm f}$,
$\epsilon_{x}=2\zeta^{-1}\sqrt{x^{2}+\zeta^{-2}}$, and
$z=\zeta^{-1}\Lambda/m=\Lambda/k_{\rm f}$. It becomes now clear that the BCS-
BEC crossover in such a relativistic system is characterized by three
dimensionless parameters, $\eta,\zeta$, and $\Lambda/m$.
We now study in what condition we can recover the nonrelativistic theory in
the limit $\zeta\ll 1$. Expanding $\epsilon_{x}$ in powers of $\zeta$,
$\epsilon_{x}=x^{2}+2\zeta^{-2}+O(\zeta^{2})$, we can recover the
nonrelativistic version of $\xi_{x}$. However, we cannot simply neglect the
terms corresponding to anti-fermions, namely the second terms on the right
hand sides of Eq(II.1). Such terms can be neglected only when $|\mu-m|\ll m$
and $\Delta_{0}\ll m$. When the coupling is very strong, we expect that
$\mu\rightarrow 0$, these conditions cannot be met and the contribution from
anti-fermions becomes significant. Therefore, we find that the so-called
nonrelativistic condition $\zeta\ll 1$ for free Fermi gas is not sufficient to
recover the nonrelativistic limit of the BCS-BEC crossover. Actually, another
important condition $a_{s}\ll 1/m$ should be imposed to guarantee the molecule
binding energy $E_{b}\ll 2m$. With the dimensionless coupling $\eta$, this
condition becomes
$\displaystyle\eta=1/(k_{\rm f}a_{s})=(m/k_{\rm f})(1/ma_{s})\ll m/k_{\rm
f}=\zeta^{-1}.$ (19)
Therefore, the complete condition for the nonrelativistic limit of the BCS-BEC
crossover can be expressed as
$\displaystyle\zeta\ll 1\ \ \ \ {\rm and}\ \ \ \ \ \eta\ll\zeta^{-1}.$ (20)
To confirm the above conclusion we solve the gap and number equations
numerically. In Fig.1 we show the condensate $\Delta_{0}$ and the
nonrelativistic chemical potential $\mu-m$ as functions of $\eta$ in the
region $-1<\eta<1$ for several values of $\zeta<1$. In this region the cutoff
$\Lambda$ dependence is weak and can be neglected. For sufficiently small
$\zeta$, we really recover the Leggett result of BCS-BEC crossover in
nonrelativistic dilute Fermi gases. With increasing $\zeta$, however, the
universality is broken and the deviation becomes more and more remarkable. On
the other hand, when we increase the coupling $\eta$, especially for
$\eta\sim\zeta^{-1}$, the difference between our calculation at any fixed
$\zeta$ and the Leggett result becomes larger due to relativistic effects.
This means that even for the case $\zeta\ll 1$ we cannot recover the
nonrelativistic result when the coupling $\eta$ becomes of order of
$\zeta^{-1}$. This can be seen clearly from the $\eta$ dependence of
$\Delta_{0}$ and $\mu$ in a wider $\eta$ region, shown in Fig.2. We find the
critical coupling $\eta_{c}\simeq 2\zeta^{-1}$ in our numerical calculations
with the cutoff $\Lambda/m=10$, which is consistent with the above estimation.
Beyond this critical coupling, the behavior of the chemical potential $\mu$
changes characteristically and approaches zero, and the condensate
$\Delta_{0}$ becomes of order of the relativistic Fermi energy $E_{\rm f}$ or
the fermion mass $m$ rapidly. In the region $\eta\sim\eta_{c}$, the
relativistic effect already becomes important, even though the initial
noninteracting Fermi gas satisfies the nonrelativistic condition $\zeta\ll 1$.
On the other hand, the smaller the parameter $\zeta$, the stronger the
coupling to exhibit the relativistic effects. For the experiments of ultra
cold atomic Fermi gases, it seems that it is very hard to show this effect,
since the system is very dilute and we cannot reach such strong attraction.
Figure 1: The condensate $\Delta_{0}$ and nonrelativistic chemical potential
$\mu-m$, scaled by the nonrelativistic Fermi energy $\epsilon_{\rm f}$, as
functions of $\eta$ in the region $-1<\eta<1$ for several values of $\zeta$.
In the calculations we set $\Lambda/m=10$.
Figure 2: The condensate $\Delta_{0}$ and chemical potential $\mu$, scaled by
the relativistic Fermi energy $E_{\rm f}$, as functions of $\eta$ in a wide
region $-1<\eta<20$ for several values of $\zeta$. In the calculations we set
$\Lambda/m=10$.
We can derive an analytical expression for the critical coupling $\eta_{c}$ or
$U_{c}$ for the RBEC state. At the critical coupling, we can take $\mu\simeq
0$ and $\Delta_{0}\ll m$ approximately, and the gap equation becomes
$\frac{1}{U_{c}}\simeq\frac{m^{2}}{2\pi^{2}}\int_{0}^{\Lambda}\frac{dk}{\sqrt{k^{2}+m^{2}}}.$
(21)
Completing the integral we obtain
$\displaystyle U_{c}^{-1}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}U_{0}^{-1}f(\Lambda/m),$ $\displaystyle\eta_{c}$
$\displaystyle=$ $\displaystyle\frac{2}{\pi}\zeta^{-1}f(\Lambda/m)$ (22)
with $f(x)=\ln(x+\sqrt{x^{2}+1})$ and $U_{0}=4\pi/m^{2}=4\pi\lambda_{c}^{2}$.
While in the nonrelativistic region with $\eta\ll\zeta^{-1}$ the result is
almost cutoff independent, in the RBEC region $\eta\sim\zeta^{-1}$ the
solution becomes sensitive to the cutoff $\Lambda$. For a realistic model such
as Yukawa coupling model, the solution at super strong coupling should be
sensitive to the model parameters.
To see what happens in the region with $\eta\sim\zeta^{-1}$, we first discuss
the fermion and anti-fermion momentum distributions $n_{-}({\bf k})$ and
$n_{+}({\bf k})$. From the number equation we have
$n_{\pm}({\bf k})=\frac{1}{2}\left(1-\frac{\xi_{\bf k}^{\pm}}{E_{\bf
k}^{\pm}}\right).$ (23)
In the nonrelativistic BCS and BEC regions with $\eta\ll\zeta^{-1}$, we have
$\Delta_{0}\ll E_{\rm f}$, and the anti-fermion degree of freedom can be
safely neglected, i.e., $n_{+}({\bf k})\simeq 0$. In the weak coupling BCS
limit we have $\Delta_{0}\ll\epsilon_{\rm f}$. Therefore $n_{-}({\bf k})$
deviates slightly from the standard Fermi distribution at the Fermi surface,
especially we have $n_{-}({\bf 0})\simeq 1$. In the deep NBEC region, we have
$\Delta_{0}\sim\eta\epsilon_{\rm f}$ and $|\mu-m|\sim\eta^{2}\epsilon_{\rm
f}$. From $|\mu-m|\gg\Delta_{0}$, we find that $n_{-}({\bf 0})\ll 1$ and
$n_{-}({\bf k})$ become very smooth in the momentum space. However, in the
RBEC region with $\eta\sim\zeta^{-1}$, $\mu$ approaches zero and the anti-
fermions become nearly degenerate with the fermions. We have
$n_{-}({\bf k})\simeq n_{+}({\bf k})=\frac{1}{2}\left(1-\frac{\epsilon_{\bf
k}}{\sqrt{\epsilon_{\bf k}^{2}+\Delta_{0}^{2}}}\right).$ (24)
Therefore, unlike the NBEC case, $n_{\pm}({\bf 0})$ here can be large as long
as $\Delta_{0}$ is of order of $m$.
In the nonrelativistic BCS and BEC regions, the total net density
$n=n_{-}-n_{+}$ is approximately $n\simeq n_{-}=\sum_{\bf k}n_{-}({\bf k})$,
and the contribution from the anti-fermions can be neglected, i.e.,
$n_{+}=\sum_{\bf k}n_{+}({\bf k})\simeq 0$. However, when we approach the RBEC
region, the contributions from fermions and anti-fermions are almost equally
important. Near the onset of the RBEC region with $\Delta_{0}<m$ we can
estimate
$n_{-}\simeq
n_{+}\simeq\frac{\Delta_{0}^{2}}{8\pi^{2}}\int_{0}^{\Lambda}dk\frac{k^{2}}{k^{2}+m^{2}}=\frac{\Delta_{0}^{2}\Lambda}{8\pi^{2}}\left(1-\frac{m}{\Lambda}\arctan{\frac{\Lambda}{m}}\right).$
(25)
For $m\ll\Lambda$, the second term in the bracket can be omitted, and we get
$n_{-}\simeq n_{+}\simeq\Delta_{0}^{2}\Lambda/(8\pi^{2})$. Therefore, in the
RBEC region, the system in fact becomes very dense even though the net density
$n$ is dilute: the densities of fermions and anti-fermions are both much
larger than $n$ but their difference produces a small net density $n$.
In the nonrelativistic theory of BCS-BEC crossover in dilute Fermi gases, the
attraction strength and number density are reflected in the theory in a
compact way through the combined dimensionless quantity $\eta=1/(k_{\rm
f}a_{s})$, and therefore changing the density of the system cannot induce a
BCS-BEC crossover. However, if the universality is broken, there would exist
an extra density dependence which may induce a BCS-BEC crossover. In
nonrelativistic Fermi systems, the breaking of the universality can be induced
by a finite-range interaction density . In the relativistic theory, the
universality is naturally broken by the $\zeta=k_{\rm f}/m$ dependence which
leads to the extra density effect.
To study this phenomenon, we calculate the “phase diagram” in the
$U_{0}/U-k_{\rm f}/m$ plane where $U_{0}/U$ reflects the pure coupling
constant effect and $k_{\rm f}/m$ reflects the pure density effect. The reason
why we do not present the phase diagram in the $\eta-\zeta$ plane is that both
$\eta=1/(k_{\rm f}a_{s})$ and $\zeta=k_{\rm f}/m$ include the density effect.
To identify the BCS-like and BEC-like phases, we take a look at the lower
branch of the excitation spectra, i.e., $E_{\bf k}^{-}=\sqrt{(\epsilon_{\bf
k}-\mu)^{2}+\Delta_{0}^{2}}$. This excitation spectrum is qualitatively
different for $\mu>m$ and $\mu<m$. Actually, the minimum of the dispersion is
located at nonzero momentum $|{\bf k}|=\sqrt{\mu^{2}-m^{2}}$ for $\mu>m$ (BCS-
like) and located at zero momentum $|{\bf k}|=0$ for $\mu<m$ (BEC-like).
Therefore, the BCS state and the BEC state can be separated by the line
determined by the condition $\mu=m$. The phase diagram is shown in Fig.3. The
BEC state is below the line $\mu=m$ and the BCS-like state is above the line.
We see clearly two ways to realize the BCS-BEC crossover, by changing the
attraction strength at some fixed density and changing the density at some
fixed attraction strength. Note that we only plot the line which separates the
BCS-like region and the BEC-like region. Above and close to the line $\mu=m$
there should exist a crossover region, like the phase diagram given in density
.
Figure 3: The phase diagram in the $U_{0}/U-k_{\rm f}/m$ plane. The line which
separates the BCS-like region and the BEC-like region is defined as $\mu=m$.
The density induced BCS-BEC crossover can be realized in dense QCD or QCD-like
theories, such as QCD at finite isospin density ISO ; ISOother01 ; ISOother011
; ISOother012 ; ISOother013 ; ISOother014 ; ISOother015 ; ISOother016 ;
ISOother017 ; ISOother018 ; ISOother019 ; ISOother0110 ; ISOother0111 ;
ISOother0112 ; ISOother0113 ; ISOother0114 ; ISOother0115 ; boser ; ISOother02
; ISOother021 and two-color QCD at finite baryon density QC2D ; QC2D1 ; QC2D2
; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ; QL03 ; ratti ; 2CNJL04 . The new feature in
QCD and QCD-like theories is that the effective quark mass $m$ decreases with
increasing density due to the effect of chiral symmetry restoration at finite
density, which would lower the BCS-BEC crossover line in the phase diagram.
To study the evolution of the collective modes in the BCS-BEC crossover, we
investigate the fluctuations around the saddle point $\Delta(x)=\Delta_{0}$.
To this end, we write $\Delta(x)=\Delta_{0}+\phi(x)$ and expand the effective
action ${\cal S}_{\rm eff}$ to the quadratic terms in $\phi$ (Gaussian
fluctuations). We obtain
${\cal S}_{\rm Gauss}[\phi]={\cal S}_{\rm
eff}[\Delta_{0}]+\frac{1}{2}\sum_{Q}\Phi^{\dagger}(Q){\bf M}(Q)\Phi(Q),$ (26)
where $\Phi$ is defined as $\Phi^{\dagger}(Q)=(\phi^{*}(Q),\phi(-Q))$. The
matrix ${\bf M}(Q)$ then determines the spectra of the collective bosonic
excitations.
The inverse propagator ${\bf M}$ of the collective modes is a $2\times 2$
matrix. The elements are given by
$\displaystyle{\bf M}_{11}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{g}+\frac{1}{2}\sum_{K}\text{Tr}\left[i\gamma_{5}{\cal
G}_{11}(K+Q)i\gamma_{5}{\cal G}_{22}(K)\right],$ $\displaystyle{\bf
M}_{12}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[i\gamma_{5}{\cal
G}_{12}(K+Q)i\gamma_{5}{\cal G}_{12}(K)\right],$ $\displaystyle{\bf
M}_{21}(Q)$ $\displaystyle=$ $\displaystyle{\bf M}_{12}(-Q),$
$\displaystyle{\bf M}_{22}(Q)$ $\displaystyle=$ $\displaystyle{\bf
M}_{11}(-Q),$ (27)
where ${\cal G}_{ij}(i,j=1,2)$ are the elements of the fermion propagator
${\cal G}={\bf G}[\Delta_{0}]$ in the Nambu-Gor’kov space. The explicit form
of the fermion propagator is given by
$\displaystyle{\cal G}_{11}$ $\displaystyle=$
$\displaystyle{i\omega_{n}+\xi_{\bf k}^{-}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{+}\gamma_{0}+{i\omega_{n}-\xi_{\bf
k}^{+}\over(i\omega_{n})^{2}-(E_{\bf k}^{+})^{2}}\Lambda_{-}\gamma_{0},$
$\displaystyle{\cal G}_{12}$ $\displaystyle=$
$\displaystyle{i\Delta_{0}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{+}\gamma_{5}+{i\Delta_{0}\over(i\omega_{n})^{2}-(E_{\bf
k}^{+})^{2}}\Lambda_{-}\gamma_{5},$ $\displaystyle{\cal G}_{22}$
$\displaystyle=$ $\displaystyle{\cal G}_{11}(\mu\rightarrow-\mu),$
$\displaystyle{\cal G}_{21}$ $\displaystyle=$ $\displaystyle{\cal
G}_{12}(\mu\rightarrow-\mu),$ (28)
where the energy projectors are defined as
$\Lambda_{\pm}({\bf k})={1\over
2}\left[1\pm{\gamma_{0}\left(\mbox{\boldmath{$\gamma$}}\cdot{\bf
k}+m\right)\over\epsilon_{\bf k}}\right].$ (29)
At zero temperature, ${\bf M}_{11}$ and ${\bf M}_{12}$ can be evaluated as
$\displaystyle{\bf M}_{11}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{g}+\sum_{\bf k}\Bigg{[}\left(\frac{(\upsilon_{\bf
k}^{-})^{2}(\upsilon_{{\bf k}+{\bf q}}^{-})^{2}}{i\nu_{m}-E_{\bf
k}^{-}-E_{{\bf k}+{\bf q}}^{-}}-\frac{(u_{\bf k}^{-})^{2}(u_{{\bf k}+{\bf
q}}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{{\bf k}+{\bf q}}^{-}}\right){\cal
T}_{+}+\left(\frac{(u_{\bf k}^{+})^{2}(u_{{\bf k}+{\bf
q}}^{+})^{2}}{i\nu_{m}-E_{\bf k}^{+}-E_{{\bf k}+{\bf
q}}^{+}}-\frac{(\upsilon_{\bf k}^{+})^{2}(\upsilon_{{\bf k}+{\bf
q}}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{{\bf k}+{\bf q}}^{+}}\right){\cal
T}_{+}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ +\left(\frac{(\upsilon_{\bf
k}^{-})^{2}(u_{{\bf k}+{\bf q}}^{+})^{2}}{i\nu_{m}-E_{\bf k}^{-}-E_{{\bf
k}+{\bf q}}^{+}}-\frac{(u_{\bf k}^{-})^{2}(\upsilon_{{\bf k}+{\bf
q}}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{{\bf k}+{\bf q}}^{+}}\right){\cal
T}_{-}+\left(\frac{(u_{\bf k}^{+})^{2}(\upsilon_{{\bf k}+{\bf
q}}^{-})^{2}}{i\nu_{m}-E_{\bf k}^{+}-E_{{\bf k}+{\bf
q}}^{-}}-\frac{(\upsilon_{\bf k}^{+})^{2}(u_{{\bf k}+{\bf
q}}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{{\bf k}+{\bf q}}^{-}}\right){\cal
T}_{-}\Bigg{]},$ $\displaystyle{\bf M}_{12}(Q)$ $\displaystyle=$
$\displaystyle\sum_{\bf k}\Bigg{[}\left(\frac{u_{\bf k}^{-}\upsilon_{\bf
k}^{-}u_{{\bf k}+{\bf q}}^{-}\upsilon_{{\bf k}+{\bf q}}^{-}}{i\nu_{m}+E_{\bf
k}^{-}+E_{{\bf k}+{\bf q}}^{-}}-\frac{u_{\bf k}^{-}\upsilon_{\bf k}^{-}u_{{\bf
k}+{\bf q}}^{-}\upsilon_{{\bf k}+{\bf q}}^{-}}{i\nu_{m}-E_{\bf k}^{-}-E_{{\bf
k}+{\bf q}}^{-}}\right){\cal T}_{+}+\left(\frac{u_{\bf k}^{+}\upsilon_{\bf
k}^{+}u_{{\bf k}+{\bf q}}^{+}\upsilon_{{\bf k}+{\bf q}}^{+}}{i\nu_{m}+E_{\bf
k}^{+}+E_{{\bf k}+{\bf q}}^{+}}-\frac{u_{\bf k}^{+}\upsilon_{\bf k}^{+}u_{{\bf
k}+{\bf q}}^{+}\upsilon_{{\bf k}+{\bf q}}^{+}}{i\nu_{m}-E_{\bf k}^{+}-E_{{\bf
k}+{\bf q}}^{+}}\right){\cal T}_{+}$ (30) $\displaystyle\ \ \ \ \
+\left(\frac{u_{\bf k}^{-}\upsilon_{\bf k}^{-}u_{{\bf k}+{\bf
q}}^{+}\upsilon_{{\bf k}+{\bf q}}^{+}}{i\nu_{m}+E_{\bf k}^{-}+E_{{\bf k}+{\bf
q}}^{+}}-\frac{u_{\bf k}^{-}\upsilon_{\bf k}^{-}u_{{\bf k}+{\bf
q}}^{+}\upsilon_{{\bf k}+{\bf q}}^{+}}{i\nu_{m}-E_{\bf k}^{-}-E_{{\bf k}+{\bf
q}}^{+}}\right){\cal T}_{-}+\left(\frac{u_{\bf k}^{+}\upsilon_{\bf
k}^{+}u_{{\bf k}+{\bf q}}^{-}\upsilon_{{\bf k}+{\bf q}}^{-}}{i\nu_{m}+E_{\bf
k}^{+}+E_{{\bf k}+{\bf q}}^{-}}-\frac{u_{\bf k}^{+}\upsilon_{\bf k}^{+}u_{{\bf
k}+{\bf q}}^{-}\upsilon_{{\bf k}+{\bf q}}^{-}}{i\nu_{m}-E_{\bf k}^{+}-E_{{\bf
k}+{\bf q}}^{-}}\right){\cal T}_{-}\Bigg{]},$
where $(u_{\bf k}^{\pm})^{2}=(1+\xi_{\bf k}^{\pm}/E_{\bf k}^{\pm})/2$ and
$(\upsilon_{\bf k}^{\pm})^{2}=(1-\xi_{\bf k}^{\pm}/E_{\bf k}^{\pm})/2$ are the
BCS distributions and ${\cal T}_{\pm}=1/2\pm({\bf k}\cdot{\bf q}+\epsilon_{\bf
k}^{2})/(2\epsilon_{\bf k}\epsilon_{{\bf k}+{\bf q}})$. At ${\bf q}=0$, we
have ${\cal T}_{+}=1$ and ${\cal T}_{-}=0$.
Taking the analytical continuation $i\nu_{m}\rightarrow\omega+i0^{+}$, the
excitation spectrum $\omega=\omega({\bf q})$ of the collective mode is
obtained by solving the equation
$\det{{\bf M}[{\bf q},\omega({\bf q})]}=0.$ (31)
To make the result more physical, we decompose the complex fluctuation field
$\phi(x)$ into its amplitude mode $\lambda(x)$ and phase mode $\theta(x)$,
i.e., $\phi(x)=(\lambda(x)+i\theta(x))/\sqrt{2}$. Then in terms of the phase
and amplitude fields the Gaussian part of the effective action takes the form
$\left(\begin{array}[]{cc}\lambda^{*}&\theta^{*}\end{array}\right)\left(\begin{array}[]{cc}{\bf
M}_{11}^{+}+{\bf M}_{12}&i{\bf M}_{11}^{-}\\\ -i{\bf M}_{11}^{-}&{\bf
M}_{11}^{+}-{\bf M}_{12}\end{array}\right)\left(\begin{array}[]{c}\lambda\\\
\theta\end{array}\right)$ (32)
with ${\bf M}_{11}^{\pm}({\bf q},\omega)=({\bf M}_{11}({\bf q},\omega)\pm{\bf
M}_{11}({\bf q},-\omega))/2$. Note that ${\bf M}_{11}^{+}({\bf q},\omega)$ and
${\bf M}_{11}^{-}({\bf q},\omega)$ are even and odd functions of $\omega$,
respectively. Considering ${\bf M}_{11}^{-}({\bf q},0)=0$, the amplitude and
phase modes decouple exactly at $\omega=0$. Furthermore, using the gap
equation for $\Delta_{0}$ we find ${\bf M}_{11}^{+}(0,0)={\bf M}_{12}(0,0)$,
which ensures the gapless phase mode at ${\bf q}=0$, i.e., the Goldstone mode
corresponding to the spontaneous breaking of the global $U(1)$ symmetry of the
Lagrangian density (4).
We now determine the excitation spectrum of the Goldstone mode at low momentum
and frequency, i.e., $\omega,|{\bf q}|\ll\text{min}_{\bf k}\\{E_{\bf
k}^{\pm}\\}$. In this region, the dispersion takes the linear form
$\omega({\bf q})=c|{\bf q}|$. The behavior of the Goldstone mode velocity $c$
in the BCS-BEC crossover is most interesting since it determines the low
temperature behavior of the thermodynamic quantities. To calculate the
velocity $c$, we make a small ${\bf q}$ and $\omega$ expansion of the
effective action, that is,
$\displaystyle{\bf M}_{11}^{+}+{\bf M}_{12}$ $\displaystyle=$ $\displaystyle
A+C|{\bf q}|^{2}-D\omega^{2}+\cdots,$ $\displaystyle{\bf M}_{11}^{+}-{\bf
M}_{12}$ $\displaystyle=$ $\displaystyle H|{\bf q}|^{2}-R\omega^{2}+\cdots,$
$\displaystyle{\bf M}_{11}^{-}$ $\displaystyle=$ $\displaystyle
B\omega+\cdots.$ (33)
The Goldstone mode velocity $c$ can be shown to be
$c=\sqrt{H\over B^{2}/A+R}.$ (34)
The corresponding eigenvector of ${\bf M}$ is $(\lambda,\theta)=(-ic|{\bf
q}|B/A,1)$, which is a pure phase mode at ${\bf q}=0$ but has an admixture of
the amplitude mode controlled by $B$ at finite ${\bf q}$. The explicit forms
of $A,\ B,\ R$, and $H$ can be calculated as
$\displaystyle A$ $\displaystyle=$ $\displaystyle 4\Delta_{0}^{2}R,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle{1\over 4}\sum_{\bf
k}\left({\xi_{\bf k}^{-}\over E_{\bf k}^{-3}}-{\xi_{\bf k}^{+}\over E_{\bf
k}^{+3}}\right),$ $\displaystyle R$ $\displaystyle=$ $\displaystyle{1\over
8}\sum_{\bf k}\left({1\over E_{\bf k}^{-3}}+{1\over E_{\bf k}^{+3}}\right),$
$\displaystyle H$ $\displaystyle=$ $\displaystyle{1\over 16}\sum_{\bf
k}\Bigg{[}\frac{1}{E_{\bf k}^{-3}}\left(\frac{\xi_{\bf k}^{-}}{\epsilon_{\bf
k}}+\left(\frac{\Delta_{0}^{2}}{E_{\bf k}^{-2}}-\frac{\xi_{\bf
k}^{-}}{3\epsilon_{\bf k}}\right)\frac{{\bf k}^{2}}{\epsilon_{\bf
k}^{2}}\right)$ (35) $\displaystyle+\frac{1}{E_{\bf
k}^{+3}}\left(\frac{\xi_{\bf k}^{+}}{\epsilon_{\bf
k}}+\left(\frac{\Delta_{0}^{2}}{E_{\bf k}^{+2}}-\frac{\xi_{\bf
k}^{+}}{3\epsilon_{\bf k}}\right)\frac{{\bf k}^{2}}{\epsilon_{\bf
k}^{2}}\right)$ $\displaystyle+2\left(\frac{1}{E_{\bf k}^{-}}+\frac{1}{E_{\bf
k}^{+}}-2\frac{E_{\bf k}^{-}E_{\bf k}^{+}-\xi_{\bf k}^{-}\xi_{\bf
k}^{+}+\Delta_{0}^{2}}{E_{\bf k}^{-}E_{\bf k}^{+}(E_{\bf k}^{-}+E_{\bf
k}^{+})}\right)$ $\displaystyle\times\frac{1}{\epsilon_{\bf
k}^{2}}\left(1-\frac{{\bf k}^{2}}{3\epsilon_{\bf k}^{2}}\right)\Bigg{]}.$
In the nonrelativistic limit with $\zeta\ll 1$ and $\eta\ll\zeta^{-1}$, we
have $|\mu-m|,\Delta_{0}\ll m$, all the terms that include anti-fermion energy
can be neglected. The fermion dispersion relation $\xi_{\bf k}^{-}$ and
$E_{\bf k}^{-}$ can be well approximated as $\xi_{\bf k}={\bf
k}^{2}/(2m)-(\mu-m)$ and $E_{\bf k}=\sqrt{\xi_{\bf k}^{2}+\Delta_{0}^{2}}$,
and we can take ${\cal T}_{+}\simeq 1$. In this limit the functions ${\bf
M}_{11}$ and ${\bf M}_{12}$ are the same as those obtained in nonrelativistic
theory BCSBEC3 . In this case, we have
$\displaystyle B$ $\displaystyle=$ $\displaystyle{1\over 2}\sum_{\bf
k}{\xi_{\bf k}\over E_{\bf k}^{3}},$ $\displaystyle R$ $\displaystyle=$
$\displaystyle{1\over 8}\sum_{\bf k}{1\over E_{\bf k}^{3}},$ $\displaystyle H$
$\displaystyle=$ $\displaystyle{1\over 16}\sum_{\bf k}{1\over E_{\bf
k}^{3}}\left({\xi_{\bf k}\over m}+{\Delta_{0}^{2}\over E_{\bf k}^{2}}{{\bf
k}^{2}\over m^{2}}\right).$ (36)
In the weak coupling BCS limit, all the integrated functions peak near the
Fermi surface, we have $B=0$ and $c=\sqrt{H/R}$. Working out the integrals we
recover the well known result $c=\zeta/\sqrt{3}$ for nonrelativistic fermionic
superfluidity in the weak coupling limit. In the NBEC region, the Fermi
surface does not exist and $B$ becomes nonzero. An explicit calculation shows
$c=\zeta/\sqrt{3\pi\eta}\ll\zeta$ BCSBEC3 . This result can be rewritten as
$c^{2}=4\pi n_{B}a_{B}/m_{B}^{2}$, where $m_{B}=2m,\ a_{B}=2a_{s}$, and
$n_{B}=n/2$ are the corresponding mass, scattering length and density of
bosons. This recovers the result for weakly interacting Bose condensate Bose01
.
In the RBEC region with $\eta\sim\zeta^{-1}$, the chemical potential
$\mu\rightarrow 0$. Therefore, the terms include anti-fermion energy become
nearly degenerate with the fermion terms and cannot be neglected. Notice that
$B$ is an odd function of $\mu$, it vanishes for $\mu\rightarrow 0$. Taking
$\mu=0$ we obtain
$\displaystyle R$ $\displaystyle=$ $\displaystyle{1\over 4}\sum_{\bf k}{1\over
E_{\bf k}^{3}},$ $\displaystyle H$ $\displaystyle=$ $\displaystyle{1\over
8}\sum_{\bf k}{1\over E_{\bf k}^{3}}\left(3-{{\bf k}^{2}\over\epsilon_{\bf
k}^{2}}+{\Delta_{0}^{2}\over E_{\bf k}^{2}}{{\bf k}^{2}\over\epsilon_{\bf
k}^{2}}\right),$ (37)
where $E_{\bf k}=\sqrt{\epsilon_{\bf k}^{2}+\Delta_{0}^{2}}$ is now the
degenerate dispersion relation in the limit $\mu\rightarrow 0$. In the RBEC
region the Goldstone mode velocity can be well approximated as
$c=\lim_{\Lambda\rightarrow\infty}\sqrt{H/R}$. Therefore, we find
$c\rightarrow 1$ in this region. This is consistent with the Goldstone boson
velocity for relativistic Bose-Einstein condensation boser .
On the other hand, in the ultra relativistic BCS state with $k_{\rm f}\gg m$
and $\Delta_{0}\ll\mu\simeq E_{\rm f}$, all the terms that include anti-
fermion energy can be neglected again. This case corresponds to color
superconductivity in high density quark matter. In this case we have $B\simeq
0$ and
$\displaystyle R$ $\displaystyle\simeq$ $\displaystyle{\mu^{2}\over
16\pi^{2}}\int_{0}^{\infty}dk{1\over\left[(k-\mu)^{2}+\Delta_{0}^{2}\right]^{3/2}},$
$\displaystyle H$ $\displaystyle\simeq$ $\displaystyle{\mu^{2}\over
32\pi^{2}}\int_{0}^{\infty}dk{\Delta_{0}^{2}\over\left[(k-\mu)^{2}+\Delta_{0}^{2}\right]^{5/2}}.$
(38)
For weak coupling, we have $\Delta_{0}\ll\mu$, and a simple algebra shows
$H/R=3$. Therefore we recover the well-known result $c=1/\sqrt{3}$ for BCS
superfluidity in ultra relativistic Fermi gases.
The mixing of amplitude and phase modes undergoes characteristic changes in
the BCS-NBEC-RBEC crossover. In the weak coupling BCS region, all the
integrands peak near the Fermi surface and we have $B=0$ due to the particle-
hole symmetry. In this region the amplitude and phase modes decouple exactly.
In the NBEC region where $\eta\ll\zeta^{-1}$, while the anti-fermion term in
$B$ can be neglected, we have $B\neq 0$ since the particle-hole symmetry is
lost, which induces strong phase-amplitude mixing. In the RBEC region, while
both particle-hole and anti-particle–anti-hole symmetries are lost, they
cancel each other and we have again $B=0$. This can be seen from the fact that
for $\mu\rightarrow 0$ the first and second terms in $B$ cancel each other.
Thus in the RBEC region, the amplitude and phase modes decouple again. The
above observation for the phase-amplitude mixing in NBEC and RBEC regions can
also be explained in the frame of the bosonic field theory for Bose-Einstein
condensation. In the nonrelativistic field theory, the off-diagonal elements
of the inverse boson propagator are proportional to $i\omega$ nao , which
induces a strong phase-amplitude mixing. However, in the relativistic field
theory, the off-diagonal elements are proportional to $i\mu\omega$ kapusta ;
boser . Therefore the phase-amplitude mixing is weak for the Bose-Einstein
condensation of nearly massless bosons ($\mu\rightarrow 0$).
### II.2 Finite temperature analysis
In this subsection we turn to the finite temperature case. First, we consider
the BCS mean field theory at finite temperature. In the mean field
approximation, we consider a uniform and static saddle point
$\Delta(x)=\Delta_{\text{sc}}$. In this part we denote the superfluid order
parameter by $\Delta_{\text{sc}}$ for convenience. The thermodynamic potential
in the mean field approximation can be evaluated as
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle{\Delta_{\text{sc}}^{2}\over g}-\sum_{\bf
k}\Bigg{\\{}\left(E_{\bf k}^{+}+E_{\bf k}^{-}-\xi_{\bf k}^{+}-\xi_{\bf
k}^{-}\right)$ (39) $\displaystyle-{T}\left[\ln(1+e^{-E_{\bf
k}^{+}/T})+\ln(1+e^{-E_{\bf k}^{-}/T})\right]\Bigg{\\}},$
where $E_{\bf k}^{\pm}$ now reads $E_{\bf k}^{\pm}=\sqrt{(\xi_{\bf
k}^{\pm})^{2}+\Delta_{\text{sc}}^{2}}$. Minimizing $\Omega$ with respect to
$\Delta_{\text{sc}}$, we get the gap equation at finite temperature,
$\frac{1}{g}=\sum_{\bf k}\left[\frac{1-2f(E_{\bf k}^{-})}{2E_{\bf
k}^{-}}+\frac{1-2f(E_{\bf k}^{+})}{2E_{\bf k}^{+}}\right],$ (40)
where $f(x)=1/(e^{x/T}+1)$ is the Fermi-Dirac distribution function.
Meanwhile, the number equation at finite temperature can be expressed as
$n=\sum_{\bf k}\left\\{\left[1-\frac{\xi_{\bf k}^{-}}{E_{\bf
k}^{-}}(1-2f(E_{\bf k}^{-}))\right]-\left[1-\frac{\xi_{\bf k}^{+}}{E_{\bf
k}^{+}}(1-2f(E_{\bf k}^{+}))\right]\right\\}.$ (41)
We note that the first and second terms in the square bracket on the right
hand sides of equations (40) and (41) correspond to fermion and anti-fermion
degrees of freedom, respectively.
Generally, we expect that the order parameter $\Delta_{\rm sc}$ vanishes at
some critical temperature due to thermal excitation of fermionic
quasiparticles. At weak coupling, the BCS mean field theory is enough to
predict quantitatively the critical temperature. However, it fails to recover
the correct critical temperature for Bose-Einstein condensation at strong
coupling add1 ; add12 . Therefore, to study the BCS-BEC crossover at finite
temperature, the effects of pairing fluctuations should be considered. There
exist many methods to treat pair fluctuations at finite temperature. In the
NSR theory BCSBEC1 , which is also called $G_{0}G_{0}$ theory, the pair
fluctuations enter only the number equation, but the fermion loops which
appear in the pair propagator are constructed by bare Green function $G_{0}$.
As a consequence, such a theory is, in principle, approximately valid only at
$T\geq T_{c}$. For the study of BCS-BEC crossover, one needs a theory which is
valid not only above the critical temperature but also in the symmetry
breaking phase. While such a strict theory has not been reached so far, some
T-matrix approaches are developed, see, for instance BCSBEC5 ; add1 ; add12 .
An often used treatment for the pair fluctuations in these approaches is the
asymmetric pair approximation or the so-called $G_{0}G$ scheme BCSBEC5 ; G0G ;
G0G1 ; G0G2 . The effect of the pair fluctuations in the $G_{0}G$ method is
treated as a fermion pseudogap which has been widely discussed in high
temperature superconductivity. In contrast to the $G_{0}G_{0}$ scheme, the
$G_{0}G$ scheme keeps the Ward identity BCSBEC5 .
To generalize the mean field theory to including the effects of uncondensed
pairs, we first reexpress the BCS mean field theory by using the $G_{0}G$
formalism BCSBEC5 . Such a formalism is convenient for us to go beyond the BCS
and include uncondensed pairs at finite temperature.
Let us start from the fermion propagator ${\cal G}$ in the superfluid phase.
The inverse fermion propagator reads
${\cal
G}^{-1}(K)=\left(\begin{array}[]{cc}G_{0}^{-1}(K;\mu)&i\gamma_{5}\Delta_{\text{sc}}\\\
i\gamma_{5}\Delta_{\text{sc}}&G_{0}^{-1}(K;-\mu)\end{array}\right)$ (42)
with the inverse free fermion propagator given by
$G_{0}^{-1}(K;\mu)=(i\omega_{n}+\mu)\gamma_{0}-\mbox{\boldmath{$\gamma$}}\cdot{\bf
k}-m.$ (43)
The fermion propagator can be formally expressed as
${\cal G}(K)=\left(\begin{array}[]{cc}G(K;\mu)&F(K;\mu)\\\
F(K;-\mu)&G(K;-\mu)\end{array}\right).$ (44)
The normal and anomalous Green’s functions can be explicitly expressed as
$\displaystyle G(K;\mu)={i\omega_{n}+\xi_{\bf
k}^{-}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{+}\gamma_{0}+{i\omega_{n}-\xi_{\bf
k}^{+}\over(i\omega_{n})^{2}-(E_{\bf k}^{+})^{2}}\Lambda_{-}\gamma_{0},$
$\displaystyle F(K;\mu)={i\Delta_{\text{sc}}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{+}\gamma_{5}+{i\Delta_{\text{sc}}\over(i\omega_{n})^{2}-(E_{\bf
k}^{+})^{2}}\Lambda_{-}\gamma_{5}.$ (45)
On the other hand, the normal and anomalous Green’s functions can also be
expressed as
$\displaystyle G(K;\mu)$ $\displaystyle=$
$\displaystyle\left[G_{0}^{-1}(K;\mu)-\Sigma_{\text{sc}}(K)\right]^{-1},$
$\displaystyle F(K;\mu)$ $\displaystyle=$
$\displaystyle-G(K;\mu)i\gamma_{5}\Delta_{\text{sc}}G_{0}(K;-\mu),$ (46)
where the fermion self-energy $\Sigma_{\text{sc}}(K)$ is given by
$\Sigma_{\text{sc}}(K)=i\gamma_{5}\Delta_{\text{sc}}G_{0}(K;-\mu)i\gamma_{5}\Delta_{\text{sc}}=-\Delta_{\text{sc}}^{2}G_{0}(-K;\mu).$
(47)
According to the Green’s function relations, the gap equation can be expressed
as
$\displaystyle\Delta_{\text{sc}}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\sum_{K}\text{Tr}\left[i\gamma_{5}F(K;\mu)\right]$
(48) $\displaystyle=$
$\displaystyle\frac{g}{2}\Delta_{\text{sc}}\sum_{K}\text{Tr}\left[G(K;\mu)G_{0}(-K;\mu)\right].$
The number equation reads
$n=\sum_{K}\text{Tr}\left[\gamma_{0}G(K;\mu)\right].$ (49)
Completing the Matsubara frequency summation, we obtain the gap equation (40)
and the number equation (41).
There are two lessons from the above formalism. First, in the BCS mean field
theory, fermion–fermion pairs and anti-fermion–anti-fermion pairs explicitly
enter the theory below $T_{c}$ only through the condensate
$\Delta_{\text{sc}}$. In the $G_{0}G$ formalism, the fermion self-energy can
equivalently be expressed as
$\Sigma_{\text{sc}}(K)=\sum_{Q}t_{\text{sc}}(Q)G_{0}(Q-K;\mu)$ (50)
associated with a condensed pair propagator given by
$t_{\text{sc}}(Q)=-\frac{\Delta_{\text{sc}}^{2}}{T}\delta(Q).$ (51)
Second, the BCS mean field theory can be associated with a specific pair
susceptibility $\chi(Q)$ defined by
$\chi_{\text{BCS}}(Q)=\frac{1}{2}\sum_{K}\text{Tr}\left[G_{0}(Q-K;\mu)G(K;\mu)\right].$
(52)
With this susceptibility, the gap equation can also be expressed as
$1-g\chi_{\text{BCS}}(Q=0)=0.$ (53)
This implies that the uncondensed pair propagator takes the form
$t(Q)=\frac{-g}{1-g\chi_{\text{BCS}}(Q)},$ (54)
and $t^{-1}(Q=0)$ is proportional to the pair chemical potential
$\mu_{\text{pair}}$. Therefore, the fact that in the superfluid phase the pair
chemical potential vanishes leads to the BEC-like condition
$t^{-1}(Q=0)=0.$ (55)
While the uncondensed pairs do not play any real role in the BCS mean field
theory, such a specific choice of the pair susceptibility and the BEC-like
condition tell us how to go beyond the BCS mean field theory and include the
effects of uncondensed pairs.
We now go beyond the BCS mean field approximation and include the effects of
uncondensed pairs by using the $G_{0}G$ formalism. It is clear that, in the
BCS mean field approximation, the fermion self-energy $\Sigma_{\text{sc}}(K)$
includes contribution only from the condensed pairs. At finite temperature,
the uncondensed pairs with nonzero momentum can be thermally excited, and the
total pair propagator should contain both the condensed (sc) and uncondensed
or “pseudogap”-associated (pg) contributions. Then we write BCSBEC5
$\displaystyle t(Q)$ $\displaystyle=$ $\displaystyle
t_{\text{pg}}(Q)+t_{\text{sc}}(Q),$ $\displaystyle t_{\text{pg}}(Q)$
$\displaystyle=$ $\displaystyle\frac{-g}{1-g\chi(Q)},\ \ \ Q\neq 0,$
$\displaystyle t_{\text{sc}}(Q)$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\text{sc}}^{2}}{T}\delta(Q).$ (56)
Then the fermion self-energy reads
$\Sigma(K)=\sum_{Q}t(Q)G_{0}(Q-K;\mu)=\Sigma_{\text{sc}}(K)+\Sigma_{\text{pg}}(K),$
(57)
where the BCS part is
$\Sigma_{\text{sc}}(K)=\sum_{Q}t_{\text{sc}}(Q)G_{0}(Q-K;\mu)$ (58)
and the pseudogap part reads
$\Sigma_{\text{pg}}(K)=\sum_{Q}t_{\text{pg}}(Q)G_{0}(Q-K;\mu).$ (59)
The pair susceptibility $\chi(Q)$ is still given by the $G_{0}G$ form,
$\chi(Q)=\frac{1}{2}\sum_{K}\text{Tr}\left[G_{0}(Q-K;\mu)G(K;\mu)\right],$
(60)
where the full fermion propagator now becomes
$G(K;\mu)=\left[G_{0}^{-1}(K;\mu)-\Sigma(K)\right]^{-1}.$ (61)
The $G_{0}G$ formalism used here can be diagrammatically illustrated in Fig.4.
The order parameter $\Delta_{\text{sc}}$ and the chemical potential $\mu$ are
in principle determined by the BEC condition $t_{\text{pg}}^{-1}(0)=0$ and the
number equation $n=\sum_{K}\text{Tr}\left[\gamma_{0}G(K;\mu)\right]$.
Figure 4: Diagramatic representation of the propagator $t_{\text{pg}}$ for the
uncondensed pairs and the fermion self-energy BCSBEC5 . The total fermion
self-energy contains contributions from condensed ($\Sigma_{\text{sc}}$) and
uncondensed ($\Sigma_{\text{pg}}$) pairs. The dashed, thin solid and thick
solid lines in $t_{\text{pg}}$ represent, respectively, the coupling constant
$g/2$, bare propagator ${G}_{0}$ and full propagator ${G}$.
However, since the explicit form of the full propagator $G(K)$ is not known _a
priori_ , the above equations are no long simple algebra equations and become
hard to handle analytically. In the superfluid phase $T\leq T_{c}$, the BEC
condition $t_{\text{pg}}^{-1}(0)=0$ implies that $t_{\text{pg}}(Q)$ is
strongly peaked around $Q=0$. This allows us to take the approximation
$\Sigma(K)\simeq-\Delta^{2}G_{0}(-K;\mu),$ (62)
where $\Delta^{2}$ contains contributions from both the condensed and
uncondensed pairs. We have
$\Delta^{2}=\Delta_{\text{sc}}^{2}+\Delta_{\text{pg}}^{2},$ (63)
where the pseudogap energy $\Delta_{\text{pg}}$ is defined as
$\Delta_{\text{pg}}^{2}=-\sum_{Q\neq 0}t_{\text{pg}}(Q).$ (64)
We should point out that, well above the critical temperature $T_{c}$ such an
approximation is no longer good, since the BEC condition is not valid in
normal phase and $t_{\text{pg}}(Q)$ becomes no longer peaked around $Q=0$.
Like in the nonrelativistic theory BCSBEC5 ; G0G ; G0G1 ; G0G2 , we can show
that $\Delta_{\text{pg}}^{2}$ physically corresponds to the classical
fluctuations of the order parameter field $\Delta(x)$, i.e.,
$\Delta_{\text{pg}}^{2}\simeq\langle|\Delta|^{2}\rangle-\langle|\Delta|\rangle^{2}.$
(65)
Therefore, the effects of the quantum fluctuations are not included in such a
theory. In fact, at zero temperature $\Delta_{\rm pg}$ vanishes and the theory
recovers exactly the BCS mean field theory. Such a theory may be called a
generalized mean field theory. But as we will show below that, such a theory
is already good to describe the BCS-NBEC-RBEC crossover in relativistic Fermi
systems.
Under the approximation (62), the full Green’s function $G(K;\mu)$ can be
evaluated explicitly as
$\displaystyle G(K;\mu)={i\omega_{n}+\xi_{\bf
k}^{-}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{+}\gamma_{0}+{i\omega_{n}-\xi_{\bf
k}^{+}\over(i\omega_{n})^{2}-(E_{\bf k}^{+})^{2}}\Lambda_{-}\gamma_{0},$ (66)
where the excitation spectra becomes
$\displaystyle E_{\bf k}^{\pm}=\sqrt{(\xi_{\bf
k}^{\pm})^{2}+\Delta^{2}}=\sqrt{(\xi_{\bf
k}^{\pm})^{2}+\Delta_{\text{sc}}^{2}+\Delta_{\text{pg}}^{2}}.$ (67)
We see clearly that the excitation gap at finite temperature becomes $\Delta$
rather than the superfluid order parameter $\Delta_{\text{sc}}$. With the
explicit form of the full Green’s function $G(K;\mu)$, the pairing
susceptibility $\chi(Q)$ can be evaluated as
$\displaystyle\chi(Q)$ $\displaystyle=$ $\displaystyle\sum_{\bf
k}\Bigg{\\{}\left[\frac{1-f(E_{\bf k}^{-})-f(\xi_{{\bf q}-{\bf
k}}^{-})}{E_{\bf k}^{-}+\xi_{{\bf q}-{\bf k}}^{-}-q_{0}}\frac{E_{\bf
k}^{-}+\xi_{\bf k}^{-}}{2E_{\bf k}^{-}}-\frac{f(E_{\bf k}^{-})-f(\xi_{{\bf
q}-{\bf k}}^{-})}{E_{\bf k}^{-}-\xi_{{\bf q}-{\bf k}}^{-}+q_{0}}\frac{E_{\bf
k}^{-}-\xi_{\bf k}^{-}}{2E_{\bf
k}^{-}}\right]\left(\frac{1}{2}+\frac{\epsilon_{\bf k}^{2}-{\bf k}\cdot{\bf
q}}{2\epsilon_{\bf k}\epsilon_{{\bf q}-{\bf k}}}\right)$ (68)
$\displaystyle+\left[\frac{1-f(E_{\bf k}^{-})-f(\xi_{{\bf q}-{\bf
k}}^{+})}{E_{\bf k}^{-}+\xi_{{\bf q}-{\bf k}}^{+}+q_{0}}\frac{E_{\bf
k}^{-}-\xi_{\bf k}^{-}}{2E_{\bf k}^{-}}-\frac{f(E_{\bf k}^{-})-f(\xi_{{\bf
q}-{\bf k}}^{+})}{E_{\bf k}^{-}-\xi_{{\bf q}-{\bf k}}^{+}-q_{0}}\frac{E_{\bf
k}^{-}+\xi_{\bf k}^{-}}{2E_{\bf
k}^{-}}\right]\left(\frac{1}{2}-\frac{\epsilon_{\bf k}^{2}-{\bf k}\cdot{\bf
q}}{2\epsilon_{\bf k}\epsilon_{{\bf q}-{\bf k}}}\right)\Bigg{\\}}$
$\displaystyle+\left(E_{\bf k}^{\pm},\xi_{\bf k}^{\pm},q_{0}\rightarrow E_{\bf
k}^{\mp},\xi_{\bf k}^{\mp},-q_{0}\right)$
with $q_{0}=i\nu_{m}$. Using the BEC condition $t_{\text{pg}}^{-1}(0)=0$, we
obtain the gap equation
$\frac{1}{g}=\sum_{\bf k}\left[\frac{1-2f(E_{\bf k}^{-})}{2E_{\bf
k}^{-}}+\frac{1-2f(E_{\bf k}^{+})}{2E_{\bf k}^{+}}\right].$ (69)
The number equation $n=\sum_{K}\text{Tr}\left[\gamma_{0}G(K;\mu)\right]$ now
becomes
$n=\sum_{\bf k}\left\\{\left[1-\frac{\xi_{\bf k}^{-}}{E_{\bf
k}^{-}}(1-2f(E_{\bf k}^{-}))\right]-\left[1-\frac{\xi_{\bf k}^{+}}{E_{\bf
k}^{+}}(1-2f(E_{\bf k}^{+}))\right]\right\\}.$ (70)
While they take the same forms as those in the BCS mean field theory, the
excitation gap is replaced by $\Delta$ which contains the contribution from
the uncondensed pairs. The order parameter $\Delta_{\rm sc}$, the pseudogap
energy $\Delta_{\rm pg}$, and the chemical potential $\mu$ are determined by
solving together the gap equation, the number equation, and Eq. (64).
In the nonrelativistic limit with $|\mu-m|,\Delta\ll m$, all the terms
including anti-fermion dispersion relations can be safely neglected, and the
fermion dispersion relations are well approximated as $\xi_{\bf k}={\bf
k}^{2}/(2m)-(\mu-m)$ and $E_{\bf k}=\sqrt{\xi_{\bf k}^{2}+\Delta^{2}}$. Taking
into account $|{\bf q}|\ll m$, we obtain
$\displaystyle\chi_{\text{NR}}(Q)$ $\displaystyle=$ $\displaystyle\sum_{\bf
k}\Bigg{[}\frac{1-f(E_{\bf k})-f(\xi_{{\bf q}-{\bf k}})}{E_{\bf k}+\xi_{{\bf
q}-{\bf k}}-q_{0}}\frac{E_{\bf k}+\xi_{\bf k}}{2E_{\bf k}}$ (71)
$\displaystyle\ -\frac{f(E_{\bf k})-f(\xi_{{\bf q}-{\bf k}})}{E_{\bf
k}-\xi_{{\bf q}-{\bf k}}+q_{0}}\frac{E_{\bf k}-\xi_{\bf k}}{2E_{\bf
k}}\Bigg{]},$
which is just the same as the pair susceptibility obtained in the
nonrelativistic theory BCSBEC5 ; G0G ; G0G1 ; G0G2 .
However, solving Eq. (64) together with the gap and number equations is still
complicated. Fortunately, the BEC condition allows us to do further
approximations for the pair propagator $t_{\text{pg}}(Q)$. Using the BEC
condition $1-g\chi(0)=0$ for the superfluid phase, we have
$t_{\text{pg}}(Q)=\frac{1}{\chi(Q)-\chi(0)}.$ (72)
The pseudogap contribution is dominated by the gapless pair dispersion in low
energy domain. Then we can expand the susceptibility around $Q=0$ and obtain
$t_{\text{pg}}(Q)\simeq\frac{1}{Z_{1}q_{0}+Z_{2}q_{0}^{2}-\xi^{2}{\bf
q}^{2}},$ (73)
where the coefficients $Z_{1},Z_{2}$ and $\xi^{2}$ are given by
$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial\chi(Q)}{\partial q_{0}}\Bigg{|}_{Q=0},$
$\displaystyle Z_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial^{2}\chi(Q)}{\partial
q_{0}^{2}}\Bigg{|}_{Q=0},$ $\displaystyle\xi^{2}$ $\displaystyle=$
$\displaystyle-{1\over 2}\frac{\partial^{2}\chi(Q)}{\partial{\bf
q}^{2}}\Bigg{|}_{Q=0}.$ (74)
Taking the first and second order derivatives with respect to $q_{0}$, we
obtain
$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle\sum_{\bf
k}\frac{1}{2E_{\bf k}^{-}}\left[\frac{1-f(E_{\bf k}^{-})-f(\xi_{{\bf
k}}^{-})}{E_{\bf k}^{-}+\xi_{\bf k}^{-}}+\frac{f(E_{\bf k}^{-})-f(\xi_{\bf
k}^{-})}{E_{\bf k}^{-}-\xi_{\bf k}^{-}}\right]$ $\displaystyle+\left(E_{\bf
k}^{\pm},\xi_{\bf k}^{\pm}\rightarrow E_{\bf k}^{\mp},\xi_{\bf
k}^{\mp}\right),$ $\displaystyle Z_{2}$ $\displaystyle=$
$\displaystyle\sum_{\bf k}\frac{1}{2E_{\bf k}^{-}}\left[\frac{1-f(E_{\bf
k}^{-})-f(\xi_{{\bf k}}^{-})}{(E_{\bf k}^{-}+\xi_{\bf
k}^{-})^{2}}-\frac{f(E_{\bf k}^{-})-f(\xi_{\bf k}^{-})}{(E_{\bf
k}^{-}-\xi_{\bf k}^{-})^{2}}\right]$ (75) $\displaystyle+\left(E_{\bf
k}^{\pm},\xi_{\bf k}^{\pm}\rightarrow E_{\bf k}^{\mp},\xi_{\bf
k}^{\mp}\right).$
Using the identity $(E_{\bf k}^{\pm})^{2}-(\xi_{\bf k}^{\pm})^{2}=\Delta^{2}$,
they can be rewritten as
$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2\Delta^{2}}\left[n-2\sum_{\bf k}\left(f(\xi_{\bf
k}^{-})-f(\xi_{\bf k}^{+})\right)\right],$ $\displaystyle Z_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2\Delta^{4}}\sum_{\bf
k}\left[\frac{(E_{\bf k}^{-})^{2}+(\xi_{\bf k}^{-})^{2}}{E_{\bf
k}^{-}}\left(1-2f(E_{\bf k}^{-})\right)-2\xi_{\bf k}^{-}\left(1-2f(\xi_{\bf
k}^{-})\right)\right]+\left(E_{\bf k}^{\pm},\xi_{\bf k}^{\pm}\rightarrow
E_{\bf k}^{\mp},\xi_{\bf k}^{\mp}\right).$ (76)
Taking the second order derivative with respect to ${\bf q}$, we get
$\displaystyle\xi^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\bf
k}\Bigg{\\{}\frac{1}{2E_{\bf k}^{-}}\left[\frac{1-f(E_{\bf k}^{-})-f(\xi_{{\bf
k}}^{-})}{E_{\bf k}^{-}+\xi_{\bf k}^{-}}+\frac{f(E_{\bf k}^{-})-f(\xi_{\bf
k}^{-})}{E_{\bf k}^{-}-\xi_{\bf k}^{-}}\right]\frac{\epsilon_{\bf k}^{2}-{\bf
k}^{2}x^{2}}{\epsilon_{\bf k}^{3}}$ (77) $\displaystyle-\left[\frac{1}{E_{\bf
k}^{-}}\left(\frac{1-f(E_{\bf k}^{-})-f(\xi_{{\bf k}}^{-})}{(E_{\bf
k}^{-}+\xi_{\bf k}^{-})^{2}}-\frac{f(E_{\bf k}^{-})-f(\xi_{\bf
k}^{-})}{(E_{\bf k}^{-}-\xi_{\bf
k}^{-})^{2}}\right)+\frac{2f^{\prime}(\xi_{\bf
k}^{-})}{\Delta^{2}}\right]\frac{{\bf k}^{2}x^{2}}{\epsilon_{\bf k}^{2}}$
$\displaystyle-\left[\frac{1-f(E_{\bf k}^{-})-f(\xi_{\bf k}^{+})}{E_{\bf
k}^{-}+\xi_{\bf k}^{+}}\frac{E_{\bf k}^{-}-\xi_{\bf k}^{-}}{2E_{\bf
k}^{-}}-\frac{f(E_{\bf k}^{-})-f(\xi_{\bf k}^{+})}{E_{\bf k}^{-}-\xi_{\bf
k}^{+}}\frac{E_{\bf k}^{-}+\xi_{\bf k}^{-}}{2E_{\bf k}^{-}}-\frac{1-2f(E_{\bf
k}^{-})}{2E_{\bf k}^{-}}\right]\frac{\epsilon_{\bf k}^{2}-{\bf
k}^{2}x^{2}}{2\epsilon_{\bf k}^{4}}\Bigg{\\}}+\left(E_{\bf k}^{\pm},\xi_{\bf
k}^{\pm}\rightarrow E_{\bf k}^{\mp},\xi_{\bf k}^{\mp}\right)$
with $x=\cos\theta$ and $f^{\prime}(x)=df(x)/dx$. In the superfluid phase the
equation (64) becomes
$\Delta_{\text{pg}}^{2}=\frac{1}{Z_{2}}\sum_{\bf q}\frac{b(\omega_{\bf
q}-\nu)+b(\omega_{\bf q}+\nu)}{2\omega_{\bf q}},$ (78)
where $b(x)=1/(e^{x/T}-1)$ is the Bose-Einstein distribution function, and
$\omega_{\bf q}$ and $\nu$ are defined as
$\omega_{\bf q}=\sqrt{\nu^{2}+c^{2}{\bf q}^{2}},\ \ \
\nu=\frac{Z_{1}}{2Z_{2}},\ \ c^{2}=\frac{\xi^{2}}{Z_{2}}.$ (79)
Without numerical calculations we have the following observations from the
above equations.
1) At zero temperature, the pseudogap $\Delta_{\text{pg}}$ vanishes
automatically and the theory reduces to the BCS mean field theory Abuki ;
RBCSBEC ; RBCSBEC1 ; RBCSBEC2 ; RBCSBEC3 ; RBCSBEC4 ; RBCSBEC5 ; RBCSBEC6 ;
RBCSBEC7 ; RBCSBEC8 ; RBCSBEC9 ; RBCSBEC10 . Therefore such a theory can be
called a generalized mean field theory at finite temperature.
2) For dilute systems with $k_{f}\ll m$ or $n\ll m^{3}$, if the coupling is
not strong enough, i.e., the molecule binding energy $E_{b}\ll 2m$, the theory
reduces to its nonrelativistic version BCSBEC6 .
3) If $Z_{1}q_{0}$ dominates the propagator $t_{\text{pg}}$, the pair
dispersion is quadratic in $|{\bf q}|$, and therefore the pseudogap behaves as
$\Delta_{\text{pg}}\propto T^{3/4}$ at low temperature. On the other hand, if
$Z_{2}q_{0}^{2}$ is the dominant term, the pair dispersion is linear in $|{\bf
q}|$, and the pseudogap behaves as $\Delta_{\text{pg}}\propto T$ at low
temperature. In the following, we will show that the first case occurs in the
NBEC region and the second case occurs in the RBEC region.
4)From the explicit expression of $Z_{1}$ in Eq.(76), we find that the
quantity in the square brackets can be identified as the total number density
$n_{\text{B}}$ of the bound pairs (bosons). Therefore, we have
$n_{\text{B}}=Z_{1}\Delta^{2}.$ (80)
From the relation $\Delta^{2}=\Delta_{\text{sc}}^{2}+\Delta_{\text{pg}}^{2}$,
$n_{\rm B}$ can be decomposed into the condensed pair density $n_{\text{sc}}$
and the uncondensed pair density $n_{\text{pg}}$, i.e.,
$n_{\text{sc}}=Z_{1}\Delta_{\text{sc}}^{2},\ \ \ \
n_{\text{pg}}=Z_{1}\Delta_{\text{pg}}^{2}.$ (81)
The fraction of the condensed pairs can be defined by
$P_{c}=\frac{n_{\text{sc}}}{n/2}=\frac{2Z_{1}\Delta_{\text{sc}}^{2}}{n}.$ (82)
5) In the weak coupling BCS region, the density $n$ can be well approximated
as
$n\simeq 2\sum_{\bf k}\left(f(\xi_{\bf k}^{-})-f(\xi_{\bf k}^{+})\right),$
(83)
which leads to consistently $n_{\text{B}}=0$ in this region. In the strong
coupling BEC region, however, almost all fermions form two-body bound states
which results in $n_{\text{B}}\simeq n/2$. At $T=0$, we have
$\Delta_{\text{pg}}=0$, $n_{\text{B}}=n_{\text{sc}}$, and $P_{c}\simeq 1$. At
the critical temperature $T=T_{c}$, the order parameter $\Delta_{\text{sc}}$
vanishes and the uncondensed pair density $n_{\text{pg}}$ becomes dominant.
At the critical temperature $T_{c}$, the order parameter $\Delta_{\text{sc}}$
vanishes but the pseudogap $\Delta_{\text{pg}}$ in general does not vanish.
The transition temperature $T_{c}$ can be determined by solving the gap and
number equations together with Eq.(78). In general there also exists a limit
temperature $T^{*}$ where the pseudogap becomes small enough. The region
$T_{c}<T<T^{*}$ is the so-called pseudogap phase. Above the critical
temperature $T_{c}$, the order parameter $\Delta_{\text{sc}}$ vanishes, and
the BEC condition is no longer valid. As a consequence, the propagator $t_{\rm
pg}(Q)$ can be expressed as
$t_{\text{pg}}(Q)=\frac{1}{\chi(Q)-\chi(0)-Z_{0}}$ (84)
with $Z_{0}=1/g-\chi(0)\neq 0$. As an estimation of $\Delta_{\text{pg}}$ above
$T_{c}$, we still perform the low energy expansion for the susceptibility,
$t_{\text{pg}}(Q)\simeq\frac{1}{Z_{1}q_{0}+Z_{2}q_{0}^{2}-\xi^{2}|{\bf
q}|^{2}-Z_{0}}.$ (85)
Therefore above $T_{c}$ the pseudogap equation becomes
$\Delta_{\text{pg}}^{2}=\frac{1}{Z_{2}}\sum_{\bf
q}\frac{b(\omega^{\prime}_{\bf q}-\nu)+b(\omega^{\prime}_{\bf
q}+\nu)}{2\omega^{\prime}_{\bf q}}$ (86)
with
$\omega^{\prime}_{\bf q}=\sqrt{\nu^{2}+\lambda^{2}+c^{2}{\bf q}^{2}},\ \ \
\lambda^{2}=Z_{0}/Z_{2}.$ (87)
The equation (86) together with the number equation determines the pseudogap
$\Delta_{\text{pg}}$ and the chemical potential $\mu$ above $T_{c}$. Since the
pair dispersion is no longer gapless, we expect that $Z_{0}$ increases with
increasing temperature and therefore $\Delta_{\text{pg}}$ drops down and
approaches zero at some dissociation temperature $T^{*}$.
In the end of this part, we discuss the thermodynamics of the system. The BCS
mean field theory does not include the contribution from the uncondensed
bosons which dominate the thermodynamics at strong coupling. In the
generalized mean field theory, the total thermodynamic potential $\Omega$
contains both the fermionic and bosonic contributions,
$\Omega=\Omega_{\text{cond}}+\Omega_{\text{fermion}}+\Omega_{\text{boson}},$
(88)
where $\Omega_{\text{cond}}=\Delta_{\text{sc}}^{2}/g$ is the condensation
energy, $\Omega_{\text{fermion}}$ is the fermionic contribution,
$\displaystyle\Omega_{\text{fermion}}$ $\displaystyle=$
$\displaystyle\sum_{\bf k}\Bigg{\\{}\left(\xi_{\bf k}^{+}+\xi_{\bf
k}^{-}-E_{\bf k}^{+}-E_{\bf k}^{-}\right)$
$\displaystyle-T\left[\ln{(1+e^{-E_{\bf k}^{+}/T})}+\ln{(1+e^{-E_{\bf
k}^{-}/T})}\right]\Bigg{\\}},$
and $\Omega_{\text{boson}}$ is the contribution from uncondensed pairs,
$\Omega_{\text{boson}}=\sum_{Q}\ln[1-g\chi(Q)].$ (90)
Under the approximation (73) for the pair propagator, the bosonic contribution
in the superfluid phase can be evaluated as
$\Omega_{\text{boson}}=T\sum_{\bf q}\left[\ln{(1-e^{-\omega_{\bf
q}^{+}/T})}+\ln{(1-e^{-\omega_{\bf q}^{-}/T})}\right]$ (91)
with $\omega_{\bf q}^{\pm}=\omega_{\bf q}\pm\nu$.
There exist two limiting cases for the bosonic contribution. If $Z_{1}q_{0}$
dominates the pair propagator, the pair dispersion is quadratic in $|{\bf
q}|$. In this case $\Omega_{\text{boson}}$ recovers the thermodynamic
potential of a nonrelativistic boson gas,
$\Omega_{\text{boson}}^{\text{NR}}=T\sum_{\bf q}\ln\left[1-e^{-{\bf
q}^{2}/(2m_{\text{B}}T)}\right].$ (92)
On the other hand, if $Z_{2}q_{0}^{2}$ dominates, the pair dispersion is
linear in $|{\bf q}|$. In the case we obtain the thermodynamic potential for
an ultra relativistic boson gas
$\Omega_{\text{boson}}^{\text{UR}}=2T\sum_{\bf q}\ln\left(1-e^{-c|{\bf
q}|/T}\right)$ (93)
with $c\rightarrow 1$ for the RBEC region. As we will see below, the former
and latter cases correspond to the NBEC and RBEC regions, respectively. The
bosons and fermions behave differently in thermodynamics. As is well known,
the specific heat $C$ of an ideal boson gas at low temperature is proportional
to $T^{\alpha}$ with $\alpha=3/2$ for nonrelativistic case and $\alpha=3$ for
ultra relativistic case. However the BCS mean field theory only predicts an
exponential law $C\propto e^{-\Delta_{0}/T}$ at low temperature.
We now apply the generalized mean field theory to study the BCS-BEC crossover
with massive relativistic fermions. We assume here that the density $n$
satisfies $n<m^{3}$ or $\zeta<1$. In this case the system is not ultra
relativistic and can even be treated nonrelativistically in some parameter
region.
From the study in the previous subsection at $T=0$, if the dimensionless
coupling $\eta$ varies from $-\infty$ to $+\infty$, the system undergoes two
crossovers, the crossover from the BCS state to the NBEC state around
$\eta\sim 0$ and the crossover from the NBEC state to the RBEC state around
$\eta\sim\zeta^{-1}$. The NBEC state and the RBEC state can be characterized
by the molecule binding energy $E_{b}$. We have $E_{b}\ll 2m$ in the NBEC
state and $E_{b}\sim 2m$ in the RBEC state.
1) BCS region. In the weak coupling BCS region, there exist no bound pairs in
the system. In this case, $Z_{1}$ is small enough and $Z_{2}$ dominates the
pair dispersion BCSBEC5 . We have $\Delta_{\text{pg}}^{2}\propto
1/(Z_{2}c^{3})$, where $c$ can be proven to be approximately equal to the
Fermi velocity BCSBEC5 . Since $\Delta$ is small in the weak coupling region,
we can show that the pseudogap $\Delta_{\rm pg}$ is much smaller than the zero
temperature gap $\Delta_{0}$ and therefore can be safely neglected in this
region. Therefore, the BCS mean field theory is good enough at any
temperature, and the critical temperature satisfies the well known relation
$T_{c}\simeq 0.57\Delta_{0}$. In the nonrelativistic limit with $\zeta\ll 1$,
the anti-fermion degrees of freedom can be ignored and the critical
temperature reads bcs
$T_{c}=\frac{8e^{\gamma-2}}{\pi}\epsilon_{\rm f}e^{2\eta/\pi},$ (94)
where $\gamma$ is Euler’s constant. Since the bosonic contribution can be
neglected, the specific heat at low temperature behaves as $C\propto
e^{-\Delta_{0}/T}$.
2) NBEC region. In the NBEC region we have $\eta>1$ and $\eta\ll\zeta^{-1}$.
The molecule binding energy $E_{b}\ll 2m$ and $|\mu-m|\ll m$. The system is a
nonrelativistical boson gas with effective boson mass $2m$, if $\zeta\ll 1$.
In this case, the anti-fermion degree of freedom can be neglected, and we
recover the nonrelativistic result BCSBEC6 . In this region, the gap $\Delta$
becomes of order of the Fermi kenetic energy $\epsilon_{f}$. From
$Z_{1}\propto 1/\Delta^{2}$ and $Z_{2}\propto 1/\Delta^{4}$, $Z_{1}q_{0}$ is
the dominant term and the pair dispersion becomes quadratic in $|{\bf q}|$.
Therefore, the propagator of the uncondensed pairs can be well approximated by
$t_{\text{pg}}(q)\simeq\frac{Z_{1}^{-1}}{q_{0}-|{\bf
q}|^{2}/\left(2m_{\text{B}}\right)},$ (95)
where the pair mass $m_{\text{B}}$ is given by
$m_{\text{B}}=Z_{1}/(2\xi^{2})$. Then we obtain
$Z_{1}\Delta_{\text{pg}}^{2}=\sum_{\bf q}b\left(\frac{|{\bf
q}|^{2}}{2m_{\text{B}}}\right)=\left(\frac{m_{\text{B}}T}{2\pi}\right)^{3/2}\zeta\left(\frac{3}{2}\right).$
(96)
Since $Z_{1}\Delta_{\text{pg}}^{2}$ equals the total boson density
$n_{\text{B}}$ at $T=T_{c}$, we obtain the critical temperature for Bose-
Einstein condensation in nonrelativistic boson gas kapusta ,
$T_{c}=\frac{2\pi}{m_{\text{B}}}\left(\frac{n_{\text{B}}}{\zeta(\frac{3}{2})}\right)^{2/3}.$
(97)
The boson mass $m_{\text{B}}$ is generally expected to be equal to the boson
chemical potential $\mu_{\text{B}}=2\mu$. In the nonrelativistic limit
$\zeta\ll 1$, we find $m_{\text{B}}\simeq 2m$ and $n_{\rm B}\simeq n/2$. The
critical temperature becomes $T_{c}=0.218\epsilon_{\rm f}$. Since $Z_{1}$
dominates the pair dispersion, we have $\Delta_{\rm pg}\propto T^{3/4}$ and
$C\propto T^{3/2}$ at low temperature.
3) RBEC region. In this region the molecule binding energy $E_{b}\rightarrow
2m$ and the chemical potential $\mu\rightarrow 0$. Nonrelativistic limit
cannot be reached even for $\zeta\ll 1$. Since the bosons with their mass
$m_{\text{B}}=2\mu$ become nearly massless in this region, anti-bosons can be
easily excited. At $T=T_{c}$ we have
$n_{\text{B}}=n_{\text{b}}-n_{\bar{\text{b}}}=Z_{1}\Delta_{\text{pg}}^{2},$
(98)
where $n_{\text{b}}$ and $n_{\bar{\text{b}}}$ are the densities for boson and
anti-boson, respectively. Note that $n_{\text{b}}$ and $n_{\bar{\text{b}}}$
are both large, and their difference produces a small density
$n_{\text{B}}\simeq n/2$. On the other hand, for $\mu\rightarrow 0$ we can
expand $Z_{1}$ in powers of $\mu$,
$Z_{1}\simeq R\mu+O(\mu^{3})=\frac{R}{2}m_{\text{B}}+O(\mu^{3})$ (99)
and hence $Z_{2}$ dominates the pair dispersion. In this case, the propagator
of the uncondensed pairs can be approximated as
$t_{\text{pg}}(Q)\simeq\frac{Z_{2}^{-1}}{q_{0}^{2}-c^{2}|{\bf q}|^{2}}.$ (100)
Therefore we obtain
$Z_{2}\Delta_{\text{pg}}^{2}\simeq\sum_{\bf q}\frac{b\left(c|{\bf
q}|\right)}{c|{\bf q}|}=\frac{T^{2}}{12c^{3}}.$ (101)
At low temperature we have $\Delta_{\rm pg}\propto T$. Combining the above
equations, we obtain the expression for $T_{c}$,
$T_{c}=\left(\frac{24c^{3}Z_{2}}{R}\frac{n_{\text{B}}}{m_{\text{B}}}\right)^{1/2}.$
(102)
In the RBEC limit $\mu\rightarrow 0$, we find that the above result recovers
the critical temperature for ultra relativistic Bose-Einstein condensation
kapusta ; haber ; haber1 ,
$T_{c}=\left(\frac{3n_{\text{B}}}{m_{\text{B}}}\right)^{1/2}.$ (103)
The specific heat at low temperature behaves as to $C\propto T^{3}$.
Now turn to numerical results. In Fig.5 we show numerical results for the
critical temperature $T_{c}$, the chemical potential $\mu(T_{c})$, and the
pseudogap energy $\Delta_{\text{pg}}(T_{c})$ as functions of the dimensionless
coupling parameter $\eta$. In the calculations we have set $\Lambda/m=10$ and
$k_{\rm f}/m=0.5$. The BCS-NBEC-RBEC crossover can be seen directly from the
behavior of the chemical potential $\mu$. In the BCS region
$-\infty<\eta<0.5$, $\mu$ is larger than the fermion mass $m$ and approaches
to the Fermi energy $E_{\rm f}$ in the weak coupling limit
$\eta\rightarrow-\infty$. The NBEC region roughly corresponds to $-0.5<\eta<4$
and the NBEC-RBEC crossover occurs at about $\eta\simeq 4$. The critical
coupling $\eta\simeq 4$ for the NBEC-RBEC crossover is consistent the previous
analytical result (II.1).
Figure 5: The critical temperature $T_{c}$ (a), chemical potential
$\mu(T_{c})$ (b) and pseudogap $\Delta_{\text{pg}}(T_{c})$ (c) as functions of
coupling $\eta$ at $\Lambda/m=10$ and $k_{\rm f}/m=0.5$. $T_{c},\mu$ and
$\Delta_{\text{pg}}$ are all scaled by the Fermi energy $E_{\rm f}$. The
dashed line is the standard critical temperature for the ideal boson gas in
(a) and stands for the position $\mu=m$ in (b), and the dotted line in (a) is
the limit temperature $T^{*}$ where the pseudogap starts to disappear. Figure
6: The boson number fraction $r_{\text{B}}$ and the fermion number fraction
$r_{\text{F}}$ at the critical temperature $T_{c}$ as functions of the
coupling $\eta$ at $\Lambda/m=10$ and $k_{f}/m=0.5$.
The critical temperature $T_{c}$, plotted as the solid line in Fig.5a, shows
significant change from the weak to strong coupling. To compare it with the
critical temperature for Bose-Einstein condensation, we solve the equation
kapusta
$\int\frac{d^{3}{\bf q}}{(2\pi)^{3}}\left[b\left(\epsilon_{\bf
q}^{\text{B}}-\mu_{\text{B}}\right)-b\left(\epsilon_{\bf
q}^{\text{B}}+\mu_{\text{B}}\right)\right]\Big{|}_{\mu_{\text{B}}=m_{\text{B}}}=n_{\text{B}}$
(104)
with $\epsilon_{\bf q}^{\text{B}}=\sqrt{{\bf q}^{2}+m_{\text{B}}^{2}}$, boson
mass $m_{\text{B}}=2\mu$, and boson density $n_{\text{B}}=n/2$. The critical
temperature obtained from this equation is also shown in Fig.5a as a dashed
line. In the weak coupling region $T_{c}$ is very small and agrees with the
BCS mean field theory. In the NBEC region $T_{c}$ changes smoothly and there
is no remarkable difference between the solid and dashed lines. Around the
coupling $\eta_{c}=4$, $T_{c}$ increases rapidly and then varies smoothly
again. In the RBEC region, the critical temperature deviates significantly
from the critical temperature for ideal boson gas (dashed line). Note that,
$T_{c}$ is of the order of the Fermi kinetic energy $\epsilon_{\rm f}\simeq
k_{\rm f}^{2}/(2m)$ in the NBEC region but becomes as large as the Fermi
energy $E_{f}$ in the RBEC region. The pseudogap $\Delta_{\text{pg}}$ at
$T=T_{c}$, shown in Fig.5c, behaves the same as the critical temperature. To
see clearly the pseudogap phase, we also show in Fig.5a the limit temperature
$T^{*}$ as a dotted line. The pseudogap exists in the region between the solid
and dotted lines and becomes small enough above the dotted line.
To explain why the critical temperature in the RBEC region deviates remarkably
from the result for ideal boson gas, we calculate the boson number fraction
$r_{\text{B}}=n_{\text{B}}/(n/2)$ and the fermion number fraction
$r_{\text{F}}=1-r_{\text{B}}$ at $T=T_{c}$ and show them as functions of the
coupling $\eta$ in Fig.6. While $r_{\rm B}\simeq 1$ in the NBEC region,
$r_{\rm B}$ is obviously less than $1$ in the RBEC region. This conclusion is
consistent with the results from the NSR theory Abuki . In the NBEC region,
the binding energy of the molecules is $E_{b}\simeq
1/ma_{s}^{2}=2\eta^{2}\epsilon_{\rm f}$, which is much larger than the
critical temperature $T_{c}\simeq 0.2\epsilon_{\rm f}$. In this case the
molecules can be safely regarded as point bosons at temperature near
$T=T_{c}$. However, the critical temperature in the RBEC region becomes as
large as the Fermi energy $E_{\rm f}$, which is of the order of the molecule
binding energy $E_{b}\simeq 2m$. Due to the competition between the
condensation and dissociation of composite bosons in hot medium, the bosons in
the RBEC region cannot be regarded as point particles and the critical
temperature should deviates from the result for ideal boson gas. This may be a
general feature of a composite boson system, especially for a system where the
condensation temperature $T_{c}$ is of the order of the molecule binding
energy. This phenomenon can also be explained by the competition between free
energy and entropy Abuki : in terms of entropy a two-fermion state is more
favorable than a one-boson state, but in terms of free energy it is less
favorable. Since the condensation temperature $T_{c}$ in the RBEC region is of
the order of $\sqrt{n_{\text{B}}/m_{\text{B}}}\sim\sqrt{n/\mu}$, we conclude
that only for a system with sufficiently small value of $k_{\rm f}/m$, the
critical temperature for relativistic boson gas can be reached and is much
smaller than $2m$.
We now apply the generalized mean field theory to study strong coupling
superfluidity/superconductivity in ultra relativistic Fermi systems. A
possible ultra relativistic superfluid/superconductor is color superconducting
quark matter which may exist in the core of compact stars. The high density
quark matter corresponds to the ultra relativistic case $n\gg m_{0}^{3}$,
where $m_{0}$ is the current quark mass. For light $u$ and $d$ quarks,
$m_{0}\simeq 5$MeV. At moderate baryon density with the quark chemical
potential $\mu\sim 400$ MeV, the quark energy gap $\Delta$ due to color
superconductivity is of the order of 100 MeV. Since $\Delta/\mu$ is of order
$0.1$, the color superconductor is not located in the weak coupling region. As
a result, the pseudogap effect is expected to be significant near the critical
temperature. To study color superconductivity at $\mu\sim 400$ MeV where
perturbative method does not work, we employ the generalized NJL model with
four-fermion interaction in the scalar diquark channel. Since the strange
quark degree of freedom has no effect for $\mu\sim 400$MeV, we restrict us to
the two-flavor case. The Lagrangian density is given by
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{q}(i\gamma^{\mu}\partial_{\mu}-m_{0})q+G_{\text{s}}\left[\left(\bar{q}q\right)^{2}+\left(\bar{q}i\gamma_{5}\tau
q\right)^{2}\right]$
$\displaystyle+G_{\text{d}}\sum_{a=2,5,7}\left(\bar{q}i\gamma^{5}\tau_{2}\lambda_{a}C\bar{q}^{\text{T}}\right)\left(q^{\text{T}}Ci\gamma^{5}\tau_{2}\lambda_{a}q\right),$
where $q$ and $\bar{q}$ denote the two-flavor quark fields, $\tau_{i}$
$(i=1,2,3)$ are the Pauli matrices in flavor space and $\lambda_{a}$
$(a=1,2,...,8)$ are the Gell-Mann matrices in color space, and $G_{\text{s}}$
and $G_{\text{d}}$ are coupling constants for meson and diquark channels.
At $\mu\sim 400$MeV, the chiral symmetry gets restored and we do not need to
consider the possibility of nonzero chiral condensate $\langle\bar{q}q\rangle$
which generates an effective quark mass $M\gg m$. The order parameter field
for color superconductivity is defined as
$\Phi_{a}=-2G_{\text{d}}q^{\text{T}}Ci\gamma^{5}\tau_{2}\lambda_{a}q.$ (106)
Nonzero expectation values of $\Phi_{a}$ spontaneously breaks the color SU(3)
symmetry down to a SU(2) subgroup. Due to the color SU(3) symmetry of the
Lagrangian, the effective potential depends only on the combination
$\Delta_{2}^{2}+\Delta_{5}^{2}+\Delta_{7}^{2}$ with
$\Delta_{a}=\langle\Phi_{a}\rangle$. Therefore we can choose a specific gauge
$\Delta_{\text{sc}}=\Delta_{2}\neq 0,\Delta_{5}=\Delta_{7}=0$ without loss of
generality. In this gauge, the red and green quarks participate in the pairing
and condensation, leaving the blue quarks unpaired.
Since the blue quarks do not participate pairing, the generalized mean field
theory cannot be directly applied to the color superconducting phase
$T<T_{c}$. The difficulty here is due to the complicated pairing fluctuations
from $\Phi_{5}$ and $\Phi_{7}$. However, we can still apply the theory to
study the critical temperature $T_{c}$ and the pseudogap $\Delta_{\rm pg}$ at
$T=T_{c}$. At and above the critical temperature, the order parameter
$\Delta_{\rm sc}$ vanished and the broken color SU$(3)$ symmetry gets
restored. Therefore, all three colors becomes degenerate. At $T=T_{c}$, we
have $\Delta=\Delta_{\text{pg}}$. The pair susceptibility $\chi(Q)$ can be
derived. It takes the same form as that for the toy U$(1)$ model but there
exists a prefactor $N_{f}(N_{c}-1)$ ($N_{f}=2$ and $N_{c}=3$ are the numbers
of flavor and color) due to the existence of flavor and color degrees of
freedom. The gap equation can be obtained from the BEC-like condition
$1-4G_{\rm d}\chi(0)=0$.
Figure 7: The critical temperature $T_{c}$ for two-flavor color superconductor
as a function of the pairing gap $\Delta_{0}$ at zero temperature in the BCS
mean field theory (dashed line) and in the generalized mean field theory
(solid line). Figure 8: The pseudogap $\Delta_{\text{pg}}$ in two-flavor
color superconductor at $T=T_{c}$ as a function of $\Delta_{0}$.
For numerical calculations, we take the current quark mass $m_{0}=5$ MeV, the
momentum cutoff $\Lambda=650$ MeV, and the quark chemical potential $\mu=400$
MeV. For convenience, we use the pairing gap $\Delta_{0}$ at zero temperature
(obtained by solving the BCS gap equation at $T=0$) to characterize the
diquark coupling strength $G_{\text{d}}$. It is generally believed the pairing
gap $\Delta_{0}$ at $\mu\sim 400$MeV is of order $100$MeV.
In Fig.7 we show the critical temperature $T_{c}$ as a function of
$\Delta_{0}$ in the generalized mean field theory and in the BCS mean field
theory. The critical temperature is not strongly modified by the pairing
fluctuations in a wide range of $\Delta_{0}$. The difference between the two
can reach about $20\%$ for the strong coupling case $\Delta_{0}\simeq 200$
MeV. In Fig.8, we show the pseudogap $\Delta_{\text{pg}}$ at the critical
temperature $T=T_{c}$. In a wide range of coupling strength, the pseudogap is
of the order of the zero temperature gap $\Delta_{0}$. Such a behavior means
that the two-flavor color superconductivity at moderate density ($\mu\sim
400$MeV) is likely in the BCS-BEC crossover region and is similar to the
behavior of the pseudogap in cuprates BCSBEC5 ; G0G ; G0G1 ; G0G2 .
## III BEC-BCS crossover in two-color QCD and Pion superfluid
In Section II we have studied the BCS-BEC crossover with relativistic fermions
with a constant mass $m$. In QCD, however, the effective quark mass $M$
generally varies with the temperature and density. For two-flavor QCD, the
current masses of light $u$ and $d$ quarks are very small, about $5$MeV. Their
effective masses are generated by the nonzero chiral condensate
$\langle\bar{q}q\rangle$ which breaks the chiral symmetry of QCD. In the
superfluid state, the lightest fermionic quasiparticle has the spectrum
$E({\bf k})=\sqrt{(\sqrt{{\bf k}^{2}+M^{2}}-\mu)^{2}+\Delta^{2}}.$ (107)
The BEC-BCS crossover occurs when the chemical potential $\mu$ equals the
effective mass $M$. However, since $M$ is dynamically generated by the chiral
condensate $\langle\bar{q}q\rangle$, it varies with the chemical potential or
density. Therefore, there exists interesting interplay between the chiral
symmetry restoration and the BEC-BCS crossover. The decreasing of the
effective mass $M$ with increasing density lowers the crossover density, and
we expect that the BEC-BCS crossover occurs in the nonperturbative region
($\mu\sim\Lambda_{\rm QCD}$) where perturbative QCD does not work. In this
section, we study BEC-BCS crossover in two-color QCD (number of color
$N_{c}=2$) at finite baryon chemical potential $\mu_{\text{B}}$ and in real
QCD at finite isospin chemical potential $\mu_{\text{I}}$ by using the NJL
model.
### III.1 Effective action of two-color QCD at finite T and $\mu_{\text{B}}$
For vanishing current quark mass $m_{0}$, two-color QCD possesses an enlarged
flavor symmetry SU$(2N_{f})$ ($N_{f}$ is the number of flavors), the so-called
Pauli-Gursey symmetry gur which connects quarks and antiquarks QC2D ; QC2D1 ;
QC2D2 ; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ; QL03 ; ratti ; 2CNJL04 . For
$N_{f}=2$, the flavor symmetry SU$(2N_{f})$ is spontaneously broken down to
Sp$(2N_{f})$ driven by a nonzero quark condensate $\langle\bar{q}q\rangle$ and
there arise five Goldstone bosons: three pions and two scalar diquarks. The
scalar diquarks are the lightest baryons that carry baryon number. For
nonvanishing current quark mass, the flavor symmetry is explicitly broken,
resulting in five degenerate pseudo-Goldstone bosons with a small mass
$m_{\pi}$. At finite baryon chemical potential $\mu_{\text{B}}$, the flavor
symmetry SU$(2N_{f})$ is explicitly broken down to
SU${}_{\text{L}}(N_{f})\otimes$SU${}_{\text{R}}(N_{f})\otimes$U${}_{\text{B}}(1)$.
Further, a nonzero diquark condensate $\langle qq\rangle$ can form at large
chemical potential and breaks spontaneously the U${}_{\text{B}}(1)$ symmetry.
In two-color QCD, the scalar diquarks are in fact the lightest “baryons”.
Therefore, we expect a baryon superfluid phase with $\langle qq\rangle\neq 0$
for $|\mu_{\text{B}}|>m_{\pi}$.
First, we construct a NJL model for two-color two-flavor QCD with the above
flavor symmetry. We consider a contact current-current interaction
$\displaystyle{\cal
L}_{\text{int}}=G_{\text{c}}\sum_{{\text{a}}=1}^{3}(\bar{q}\gamma_{\mu}t_{\text{a}}q)(\bar{q}\gamma^{\mu}t_{\text{a}}q)$
(108)
inspired by QCD. Here $t_{\text{a}}$ (${\text{a}}=1,2,3$) are the generators
of color SU${}_{\text{c}}(2)$ and $G_{\text{c}}$ is a phenomenological
coupling constant. After the Fierz transformation we can obtain an effective
NJL Lagrangian density with scalar mesons and color singlet scalar diquarks
ratti
$\displaystyle{\cal L}_{\text{NJL}}$ $\displaystyle=$
$\displaystyle\bar{q}(i\gamma^{\mu}\partial_{\mu}-m_{0})q+G\left[(\bar{q}q)^{2}+(\bar{q}i\gamma_{5}\mbox{\boldmath{$\tau$}}q)^{2}\right]$
(109)
$\displaystyle+G(\bar{q}i\gamma_{5}\tau_{2}t_{2}q_{c})(\bar{q}_{c}i\gamma_{5}\tau_{2}t_{2}q),$
where $q_{c}={\cal C}\bar{q}^{\text{T}}$ and $\bar{q}_{c}=q^{\text{T}}{\cal
C}$ are the charge conjugate spinors with ${\cal C}=i\gamma_{0}\gamma_{2}$ and
$\tau_{\text{i}}$ (${\text{i}}=1,2,3$) are the Pauli matrices in the flavor
space. The four-fermion coupling constants for the scalar mesons and diquarks
are the same, $G=3G_{\text{c}}/4$ ratti , which ensures the enlarged flavor
symmetry SU$(2N_{f})$ of two-color QCD in the chiral limit $m_{0}=0$. One can
show explicitly that there are five Goldstone bosons (three pions and two
diquarks) driven by a nonzero quark condensate $\langle\bar{q}q\rangle$. With
explicit chiral symmetry broken $m_{0}\neq 0$, pions and diquarks are also
degenerate, and their mass $m_{\pi}$ can be determined by the standard method
for the NJL model NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3 .
The partition function of the two-color NJL model (109) at finite temperature
$T$ and baryon chemical potential $\mu_{\text{B}}$ is
$\displaystyle Z_{\text{NJL}}=\int[d\bar{q}][dq]e^{\int dx\left({\cal
L}_{\text{NJL}}+\frac{\mu_{\text{B}}}{2}\bar{q}\gamma_{0}q\right)},$ (110)
The partition function can be bosonized after introducing the auxiliary boson
fields
$\displaystyle\sigma(x)=-2G\bar{q}(x)q(x),\ \ \
\mbox{\boldmath{$\pi$}}(x)=-2G\bar{q}(x)i\gamma_{5}\mbox{\boldmath{$\tau$}}q(x)$
(111)
for mesons and
$\displaystyle\phi(x)=-2G\bar{q}_{c}(x)i\gamma_{5}\tau_{2}t_{2}q(x)$ (112)
for diquarks. With the help of the Nambu-Gor’kov representation
$\bar{\Psi}=\left(\begin{array}[]{cc}\bar{q}&\bar{q}_{c}\end{array}\right)$,
the partition function can be written as
$\displaystyle{\cal
Z}_{\text{NJL}}=\int[d\bar{\Psi}][d\Psi][d\sigma][d\mbox{\boldmath{$\pi$}}][d\phi^{\dagger}][d\phi]e^{-{\cal
A_{\text{eff}}}},$ (113)
where the action ${\cal A}_{\text{eff}}$ is given by
$\displaystyle{\cal A_{\text{eff}}}=\int
dx\frac{\sigma^{2}(x)+\mbox{\boldmath{$\pi$}}^{2}(x)+|\phi(x)|^{2}}{4G}-\frac{1}{2}\int
dx\int dx^{\prime}\bar{\Psi}(x){\bf G}^{-1}(x,x^{\prime})\Psi(x^{\prime})$
(114)
with the inverse quark propagator defined as
$\displaystyle{\bf
G}^{-1}(x,x^{\prime})=\left(\begin{array}[]{cc}\gamma^{0}(-\partial_{\tau}+\frac{\mu_{\text{B}}}{2})+i\mbox{\boldmath{$\gamma$}}\cdot\mbox{\boldmath{$\nabla$}}-\mathcal{M}(x)&-i\gamma_{5}\phi(x)\tau_{2}t_{2}\\\
-i\gamma_{5}\phi^{\dagger}(x)\tau_{2}t_{2}&\gamma^{0}(-\partial_{\tau}-\frac{\mu_{\text{B}}}{2})+i\mbox{\boldmath{$\gamma$}}\cdot\mbox{\boldmath{$\nabla$}}-\mathcal{M}^{\text{T}}(x)\end{array}\right)\delta(x-x^{\prime})$
(117)
and
$\mathcal{M}(x)=m_{0}+\sigma(x)+i\gamma_{5}\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\pi$}}(x)$.
After integrating out the quarks, we can reduce the partition function to
$\displaystyle{\cal Z}_{\text{NJL}}$ $\displaystyle=$
$\displaystyle\int[d\sigma][d\mbox{\boldmath{$\pi$}}][d\phi^{\dagger}][d\phi]e^{-{\cal
S}_{\text{eff}}[\sigma,\mbox{\boldmath{$\pi$}},\phi^{\dagger},\phi]},$ (118)
where the bosonized effective action ${\cal S}_{\text{eff}}$ is given by
$\displaystyle{\cal
S}_{\text{eff}}[\sigma,\mbox{\boldmath{$\pi$}},\phi^{\dagger},\phi]$
$\displaystyle=$ $\displaystyle\int
dx\frac{\sigma^{2}(x)+\mbox{\boldmath{$\pi$}}^{2}(x)+|\phi(x)|^{2}}{4G}$ (119)
$\displaystyle-\frac{1}{2}\text{Tr}\ln{\bf G}^{-1}(x,x^{\prime}).$
Here the trace Tr is taken in color, flavor, spin, Nambu-Gor’kov and
coordinate ($x$ and $x^{\prime}$) spaces. The thermodynamic potential density
of the system is given by
$\Omega(T,\mu_{\text{B}})=-\lim_{V\rightarrow\infty}(T/V)\ln Z_{\text{NJL}}$.
The effective action ${\cal S}_{\text{eff}}$ as well as the thermodynamic
potential $\Omega$ cannot be evaluated exactly in the $3+1$ dimensional case.
In this work, we firstly consider the saddle point approximation, i.e., the
mean-field approximation. Then we investigate the fluctuations around the mean
field.
In the mean field approximation, all bosonic auxiliary fields are replaced by
their expectation values. Therefore, we set
$\langle\sigma(x)\rangle=\upsilon$, $\langle\phi(x)\rangle=\Delta$, and
$\langle\mbox{\boldmath{$\pi$}}(x)\rangle=0$. While $\Delta$ can be set to be
real due to the U${}_{\rm B}(1)$ symmetry, we do not do this in the formalism.
We will show in the following that all physical results depend only on
$|\Delta|^{2}$. The zeroth order or mean-field effective action reads
${\cal
S}_{\text{eff}}^{(0)}=\frac{V}{T}\left[\frac{\upsilon^{2}+|\Delta|^{2}}{4G}-\frac{1}{2}\sum_{K}\text{Trln}{\cal
G}^{-1}(K)\right],$ (120)
where the inverse of the Nambu-Gor’kov quark propagator ${\cal G}^{-1}(K)$ is
given by
$\displaystyle\left(\begin{array}[]{cc}(i\omega_{n}+\frac{\mu_{\text{B}}}{2})\gamma^{0}-\mbox{\boldmath{$\gamma$}}\cdot{\bf
k}-M&-i\gamma_{5}\Delta\tau_{2}t_{2}\\\
-i\gamma_{5}\Delta^{\dagger}\tau_{2}t_{2}&(i\omega_{n}-\frac{\mu_{\text{B}}}{2})\gamma^{0}-\mbox{\boldmath{$\gamma$}}\cdot{\bf
k}-M\end{array}\right)\ $ (123)
with the effective quark mass defined as $M=m_{0}+\upsilon$. The mean-field
thermodynamic potential $\Omega_{0}=(T/V){\cal S}_{\text{eff}}^{(0)}$ can be
evaluated as
$\displaystyle\Omega_{0}=\frac{\upsilon^{2}+|\Delta|^{2}}{4G}-2N_{c}N_{f}\sum_{\bf
k}\left[\mathcal{W}(E_{\bf k}^{+})+\mathcal{W}(E_{\bf k}^{-})\right]$ (124)
with the function $\mathcal{W}(E)=E/2+T\ln{(1+e^{-E/T})}$ and the BCS-like
quasiparticle dispersion relations $E_{\bf k}^{\pm}=\sqrt{(E_{\bf
k}\pm\mu_{\text{B}}/2)^{2}+|\Delta|^{2}}$ and $E_{\bf k}=\sqrt{{\bf
k}^{2}+M^{2}}$. The signs $\mp$ correspond to quasiquark and quasi-antiquark
excitations, respectively. The integral over the quark momentum ${\bf k}$ is
divergent, and some regularization scheme should be adopted. In this work, we
employ a hard three-momentum cutoff $\Lambda$.
The physical values of the variational parameters $M$ (or $\upsilon$) and
$\Delta$ should be determined by the saddle point condition
$\displaystyle\frac{\delta{\cal
S}_{\text{eff}}^{(0)}[\upsilon,\Delta]}{\delta\upsilon}=0,\ \ \ \ \
\frac{\delta{\cal S}_{\text{eff}}^{(0)}[\upsilon,\Delta]}{\delta\Delta}=0,$
(125)
which minimizes the mean-field effective action ${\cal S}_{\text{eff}}^{(0)}$.
One can show that the saddle point condition is equivalent to the following
Green’s function relations
$\displaystyle\langle\bar{q}q\rangle$ $\displaystyle=$
$\displaystyle\sum_{K}\text{Tr}{\cal G}_{11}(K)\ ,$
$\displaystyle\langle\bar{q}_{c}i\gamma_{5}\tau_{2}t_{2}q\rangle$
$\displaystyle=$ $\displaystyle\sum_{K}\text{Tr}\left[{\cal
G}_{12}(K)i\gamma_{5}\tau_{2}t_{2}\right]\ ,$ (126)
where the matrix elements of ${\cal G}$ are explicitly given by
$\displaystyle{\cal G}_{11}(K)$ $\displaystyle=$
$\displaystyle{i\omega_{n}+\xi_{\bf k}^{-}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{\bf k}^{+}\gamma_{0}+{i\omega_{n}-\xi_{\bf
k}^{+}\over(i\omega_{n})^{2}-(E_{\bf k}^{+})^{2}}\Lambda_{\bf
k}^{-}\gamma_{0}\ ,$ $\displaystyle{\cal G}_{22}(K)$ $\displaystyle=$
$\displaystyle{i\omega_{n}-\xi_{\bf k}^{-}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{\bf k}^{-}\gamma_{0}+{i\omega_{n}+\xi_{\bf
k}^{+}\over(i\omega_{n})^{2}-(E_{\bf k}^{+})^{2}}\Lambda_{\bf
k}^{+}\gamma_{0}\ ,$ $\displaystyle{\cal G}_{12}(K)$ $\displaystyle=$
$\displaystyle{-i\Delta\tau_{2}t_{2}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{\bf
k}^{+}\gamma_{5}+{-i\Delta\tau_{2}t_{2}\over(i\omega_{n})^{2}-(E_{\bf
k}^{+})^{2}}\Lambda_{\bf k}^{-}\gamma_{5}\ ,$ $\displaystyle{\cal G}_{21}(K)$
$\displaystyle=$
$\displaystyle{-i\Delta^{\dagger}\tau_{2}t_{2}\over(i\omega_{n})^{2}-(E_{\bf
k}^{-})^{2}}\Lambda_{\bf
k}^{-}\gamma_{5}+{-i\Delta^{\dagger}\tau_{2}t_{2}\over(i\omega_{n})^{2}-(E_{\bf
k}^{+})^{2}}\Lambda_{\bf k}^{+}\gamma_{5},$
with the massive energy projectors
$\Lambda_{\bf k}^{\pm}={1\over
2}\left[1\pm{\gamma_{0}\left(\mbox{\boldmath{$\gamma$}}\cdot{\bf
k}+M\right)\over E_{\bf k}}\right]\ .$ (127)
Here we have defined the notation $\xi_{\bf k}^{\pm}=E_{\bf
k}\pm\mu_{\text{B}}/2$.
Next, we consider the fluctuations around the mean field, corresponding to the
collective bosonic excitations. Making the field shifts for the auxiliary
fields,
$\displaystyle\sigma(x)\rightarrow\upsilon+\sigma(x),\ \
\mbox{\boldmath{$\pi$}}(x)\rightarrow 0+\mbox{\boldmath{$\pi$}}(x),$
$\displaystyle\phi(x)\rightarrow\Delta+\phi(x),\ \
\phi^{\dagger}(x)\rightarrow\Delta^{\dagger}+\phi^{\dagger}(x),$ (128)
we can express the total effective action as
$\displaystyle{\cal S}_{\text{eff}}$ $\displaystyle=$ $\displaystyle{\cal
S}_{\text{eff}}^{(0)}+\int
dx\left(\frac{\sigma^{2}+\mbox{\boldmath{$\pi$}}^{2}+|\phi|^{2}}{4G}+\frac{\upsilon\sigma+\Delta\phi^{\dagger}+\Delta^{\dagger}\phi}{2G}\right)$
(129) $\displaystyle-$
$\displaystyle\frac{1}{2}\text{Tr}\ln{\left[\mathbbold{1}+\int dx_{1}{\cal
G}(x,x_{1})\Sigma(x_{1},x^{\prime})\right]}.$
Here ${\cal G}(x,x^{\prime})$ is the Fourier transformation of ${\cal G}(K)$,
and $\Sigma(x,x^{\prime})$ is defined as
$\displaystyle\Sigma(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\sigma(x)-i\gamma_{5}\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\pi$}}(x)&-i\gamma_{5}\phi(x)\tau_{2}t_{2}\\\
-i\gamma_{5}\phi^{\dagger}(x)\tau_{2}t_{2}&-\sigma(x)-i\gamma_{5}\mbox{\boldmath{$\tau$}}^{\text{T}}\cdot\mbox{\boldmath{$\pi$}}(x)\end{array}\right)$
(132) $\displaystyle\times$ $\displaystyle\delta(x-x^{\prime}).$ (133)
With the help of the derivative expansion
$\displaystyle\text{Tr}\ln{\left[\mathbbold{1}+{\cal
G}\Sigma\right]}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\text{Tr}[{\cal
G}\Sigma]^{n},$ (134)
we can calculate the effective action in powers of the fluctuations
$\sigma(x),\mbox{\boldmath{$\pi$}}(x),\phi(x),\phi^{\dagger}(x)$.
The first-order effective action ${\cal S}_{\text{eff}}^{(1)}$ which includes
linear terms of the fluctuations should vanish exactly, since the expectation
value of the fluctuations should be exactly zero. In fact, ${\cal
S}_{\text{eff}}^{(1)}$ can be evaluated as
$\displaystyle{\cal S}_{\text{eff}}^{(1)}$ $\displaystyle=$ $\displaystyle\int
dx\Bigg{\\{}\left[\frac{\upsilon}{2G}+\frac{1}{2}\text{Tr}\left({\cal
G}_{11}+{\cal G}_{22}\right)\right]\sigma(x)$ (135) $\displaystyle+$
$\displaystyle\frac{1}{2}\text{Tr}\left[i\gamma_{5}\left({\cal
G}_{11}\mbox{\boldmath{$\tau$}}+{\cal
G}_{22}\mbox{\boldmath{$\tau$}}^{\text{T}}\right)\right]\cdot\mbox{\boldmath{$\pi$}}(x)$
$\displaystyle+$
$\displaystyle\left[\frac{\Delta}{2G}+\frac{1}{2}\text{Tr}\left(i\gamma_{5}\tau_{2}t_{2}{\cal
G}_{12}\right)\right]\phi^{\dagger}(x)$ $\displaystyle+$
$\displaystyle\left[\frac{\Delta^{\dagger}}{2G}+\frac{1}{2}\text{Tr}\left(i\gamma_{5}\tau_{2}t_{2}{\cal
G}_{21}\right)\right]\phi(x)\Bigg{\\}}.$
We observe that the coefficient of $\mbox{\boldmath{$\pi$}}(x)$ is
automatically zero after taking the trace in Dirac spin space. The
coefficients of $\phi(x),\phi^{\dagger}(x)$ and $\sigma(x)$ vanish once the
quark propagator takes the mean-field form and $M,\Delta$ take the physical
values that satisfies the saddle point condition. Therefore, in the present
approach, the saddle point condition plays a crucial role in having vanishing
linear terms in the expansion.
The quadratic term ${\cal S}_{\text{eff}}^{(2)}$ or the Gaussian fluctuation
corresponds to collective bosonic excitations. Working in the momentum space
is convenient. It can be expressed as
$\displaystyle{\cal S}_{\text{eff}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{Q}\Bigg{\\{}\frac{|\sigma(Q)|^{2}+|\mbox{\boldmath{$\pi$}}(Q)|^{2}+|\phi(Q)|^{2}}{2G}$
(136) $\displaystyle+\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal
G}(K)\Sigma(-Q){\cal G}(K+Q)\Sigma(Q)\right]\Bigg{\\}}.$
Here $A(Q)$ is the Fourier transformation of the field $A(x)$, and $\Sigma(Q)$
is defined as
$\Sigma(Q)=\left(\begin{array}[]{cc}-\sigma(Q)-i\gamma_{5}\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\pi$}}(Q)&-i\gamma_{5}\phi(Q)\tau_{2}t_{2}\\\
-i\gamma_{5}\phi^{\dagger}(-Q)\tau_{2}t_{2}&-\sigma(Q)-i\gamma_{5}\mbox{\boldmath{$\tau$}}^{\text{T}}\cdot\mbox{\boldmath{$\pi$}}(Q)\end{array}\right).$
(137)
After taking the trace in the Nambu-Gor’kov space, we find that ${\cal
S}_{\text{eff}}^{(2)}$ can be written in the following bilinear form
$\displaystyle{\cal S}_{\text{eff}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{Q}\left(\begin{array}[]{ccc}\phi^{\dagger}(Q)&\phi(-Q)&\sigma^{\dagger}(Q)\end{array}\right){\bf
M}(Q)\left(\begin{array}[]{cc}\phi(Q)\\\ \phi^{\dagger}(-Q)\\\
\sigma(Q)\end{array}\right)$ (147)
$\displaystyle+\frac{1}{2}\sum_{Q}\left(\begin{array}[]{ccc}\pi_{1}^{\dagger}(Q)&\pi_{2}^{\dagger}(Q)&\pi_{3}^{\dagger}(Q)\end{array}\right){\bf
N}(Q)\left(\begin{array}[]{cc}\pi_{1}(Q)\\\ \pi_{2}(Q)\\\
\pi_{3}(Q)\end{array}\right).$
For $\Delta\neq 0$, the matrix ${\bf M}$ is non-diagonal and can be expressed
as
${\bf
M}(Q)=\left(\begin{array}[]{ccc}\frac{1}{4G}+\Pi_{11}(Q)&\Pi_{12}(Q)&\Pi_{13}(Q)\\\
\Pi_{21}(Q)&\frac{1}{4G}+\Pi_{22}(Q)&\Pi_{23}(Q)\\\
\Pi_{31}(Q)&\Pi_{32}(Q)&\frac{1}{2G}+\Pi_{33}(Q)\end{array}\right).$ (149)
The polarization functions $\Pi_{\text{ij}}(Q)$
(${\text{i}},{\text{j}}=1,2,3$) are one-loop susceptibilities composed of the
Nambu-Gor’kov quark propagator. They can be expressed as
$\displaystyle\Pi_{11}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{22}(K)\Gamma{\cal
G}_{11}(P)\Gamma\right],$ $\displaystyle\Pi_{12}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{12}(K)\Gamma{\cal
G}_{12}(P)\Gamma\right],$ $\displaystyle\Pi_{13}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{12}(K)\Gamma{\cal
G}_{11}(P)+{\cal G}_{22}(K)\Gamma{\cal G}_{12}(P)\right],$
$\displaystyle\Pi_{21}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{21}(K)\Gamma{\cal
G}_{21}(P)\Gamma\right],$ $\displaystyle\Pi_{22}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{11}(K)\Gamma{\cal
G}_{22}(P)\Gamma\right],$ $\displaystyle\Pi_{23}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{11}(K)\Gamma{\cal
G}_{21}(P)+{\cal G}_{21}(K)\Gamma{\cal G}_{22}(P)\right],$
$\displaystyle\Pi_{31}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{21}(K){\cal
G}_{11}(P)\Gamma+{\cal G}_{22}(K){\cal G}_{21}(P)\Gamma\right],$
$\displaystyle\Pi_{32}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\left[{\cal G}_{11}(K){\cal
G}_{12}(P)\Gamma+{\cal G}_{12}(K){\cal G}_{22}(P)\Gamma\right],$
$\displaystyle\Pi_{33}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\big{[}{\cal G}_{11}(K){\cal
G}_{11}(P)+{\cal G}_{22}(K){\cal G}_{22}(P)$ (150) $\displaystyle+{\cal
G}_{12}(K){\cal G}_{21}(P)+{\cal G}_{21}(K){\cal G}_{12}(P)\big{]},$
with $P=K+Q$, $\Gamma=i\gamma_{5}\tau_{2}t_{2}$, where the trace is taken in
color, flavor and spin spaces. Using the fact ${\cal
G}_{22}(K,\mu_{\text{B}})={\cal G}_{11}(K,-\mu_{\text{B}})$ and ${\cal
G}_{21}(K,\mu_{\text{B}})={\cal G}_{12}^{\dagger}(K,-\mu_{\text{B}})$, we can
show
$\displaystyle\Pi_{22}(Q)=\Pi_{11}(-Q),\ \ \ \
\Pi_{12}(Q)=\Pi_{21}^{\dagger}(Q),$
$\displaystyle\Pi_{13}(Q)=\Pi_{31}^{\dagger}(Q)=\Pi_{23}^{\dagger}(-Q)=\Pi_{32}(-Q).$
(151)
Therefore, only five of the polarization functions are independent. At $T=0$,
their explicit form is shown in Appendix A. For general case, we can show
$\Pi_{12}\propto\Delta^{2}$ and $\Pi_{13}\propto M\Delta$. Therefore, in the
normal phase with $\Delta=0$, the matrix ${\bf M}$ recovers the diagonal form.
The off-diagonal elements $\Pi_{13}$ and $\Pi_{23}$ represents the mixing
between the sigma meson and the diquarks. At large chemical potentials where
the chiral symmetry is approximately restored, $M\rightarrow m_{0}$, this
mixing effect can be safely neglected.
On the other hand, the matrix ${\bf N}$ of the pion sector is diagonal and
proportional to the identity matrix, i.e.,
${\bf
N}_{\text{ij}}(Q)=\delta_{\text{ij}}\left[\frac{1}{2G}+\Pi_{\pi}(Q)\right],\ \
\ \text{i,j}=1,2,3.$ (152)
This means that pions are eigen mesonic excitations even in the superfluid
phase. The polarization function $\Pi_{\pi}(Q)$ is given by
$\displaystyle\Pi_{\pi}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K}\text{Tr}\bigg{[}{\cal
G}_{11}(K)i\gamma_{5}{\cal G}_{11}(P)i\gamma_{5}+{\cal
G}_{22}(K)i\gamma_{5}{\cal G}_{22}(P)i\gamma_{5}$ (153) $\displaystyle-{\cal
G}_{12}(K)i\gamma_{5}{\cal G}_{21}(P)i\gamma_{5}-{\cal
G}_{21}(K)i\gamma_{5}{\cal G}_{12}(P)i\gamma_{5}\bigg{]}.$
Its explicit form at $T=0$ is given in Appendix A. We find that $\Pi_{\pi}(Q)$
and $\Pi_{33}(Q)$ is different only up to a term proportional to $M^{2}$.
Therefore, at high density with $\langle\bar{q}q\rangle\rightarrow 0$, the
spectra of pions and sigma meson become nearly degenerate, which represents
the approximate restoration of chiral symmetry.
The U${}_{\text{B}}(1)$ baryon number symmetry is spontaneously broken by the
nonzero diquark condensate $\langle qq\rangle$ in the superfluid phase (even
for $m_{0}\neq 0$), resulting in one Goldstone boson. In our model, this is
ensured by the condition $\det{\bf M}(Q=0)=0$. From the explicit form of the
polarization functions given in Appendix A, we find that this condition holds
if and only if the saddle point condition (125) for $\upsilon$ and $\Delta$ is
satisfied.
### III.2 Vacuum and model parameter fixing
For a better understanding of our derivation in the following, it is useful to
review the vacuum state at $T=\mu_{\text{B}}=0$. In the vacuum, it is evident
that $\Delta=0$ and the mean-field effective potential $\Omega_{\text{vac}}$
can be evaluated as
$\displaystyle\Omega_{\text{vac}}(M)=\frac{(M-m_{0})^{2}}{4G}-2N_{c}N_{f}\sum_{\bf
k}E_{\bf k}.$ (154)
The physical value of $M$, denoted by $M_{*}$, satisfies the saddle point
condition $\partial\Omega_{\text{vac}}/\partial M=0$ and minimizes
$\Omega_{\text{vac}}$.
The meson and diquark excitations can be obtained from ${\cal
S}_{\text{eff}}^{(2)}$, which in the vacuum can be expressed as
$\displaystyle{\cal
S}_{\text{eff}}^{(2)}=-\frac{1}{2}\int\frac{d^{4}Q}{(2\pi)^{4}}\bigg{[}\sigma(-Q){\cal
D}_{\sigma}^{*-1}(Q)\sigma(Q)$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\sum_{i=1}^{3}\pi_{i}(-Q){\cal D}_{\pi}^{*-1}(Q)\pi_{i}(Q)$ $\displaystyle\ \
\ \ \ \ \ \ \ \ \ +\sum_{i=1}^{2}\phi_{i}(-Q){\cal
D}_{\phi}^{*-1}(Q)\phi_{i}(Q)\bigg{]},$ (155)
where $\phi_{1},\phi_{2}$ are the real and imaginary parts of $\phi$,
respectively. The inverse propagators in vacuum can be expressed in a
symmetrical form NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3
$\displaystyle{\cal D}^{*-1}_{l}(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2G}+\Pi_{l}^{*}(Q),\ \ \ \ l=\sigma,\pi,\phi$
$\displaystyle\Pi_{l}^{*}(Q)$ $\displaystyle=$ $\displaystyle
2iN_{c}N_{f}(Q^{2}-\epsilon_{l}^{2})I(Q^{2})$ (156)
$\displaystyle-4iN_{c}N_{f}\int\frac{d^{4}K}{(2\pi)^{4}}\frac{1}{K^{2}-M_{*}^{2}}$
with $\epsilon_{\sigma}=2M_{*}$ and $\epsilon_{\pi}=\epsilon_{\phi}=0$, where
the function $I(Q^{2})$ is defined as
$\displaystyle
I(Q^{2})=\int\frac{d^{4}K}{(2\pi)^{4}}\frac{1}{(K_{+}^{2}-M_{*}^{2})(K_{-}^{2}-M_{*}^{2})}$
(157)
with $K_{\pm}=K\pm Q/2$. Keeping in mind that $M_{*}$ satisfies the saddle
point condition, we find that the pions and diquarks are Goldstone bosons in
the chiral limit, corresponding to the symmetry breaking pattern
SU$(4)\rightarrow$Sp$(4)$. Using the gap equation of $M_{*}$, we find that the
masses of mesons and diquarks can be determined by the equation
$\displaystyle
m_{l}^{2}=-\frac{m_{0}}{M_{*}}\frac{1}{4iGN_{c}N_{f}I(m_{l}^{2})}+\epsilon_{l}^{2}.$
(158)
Since the $Q^{2}$ dependence of the function $I(Q^{2})$ is very weak, we find
$m_{\pi}^{2}\sim m_{0}$ and $m_{\sigma}^{2}\simeq 4M_{*}^{2}+m_{\pi}^{2}$.
Since pions and diquarks are deep bound states, their propagators can be well
approximated by ${\cal D}_{\pi}^{*}(Q)\simeq-g_{\pi
qq}^{2}/(Q^{2}-m_{\pi}^{2})$ with $g_{\pi qq}^{-2}\simeq-2iN_{c}N_{f}I(0)$
NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3 . The pion decay constant
$f_{\pi}$ can be determined by the matrix element of the axial current,
$\displaystyle iQ_{\mu}f_{\pi}\delta_{\text{ij}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{Tr}\int\frac{d^{4}K}{(2\pi)^{4}}\left[\gamma_{\mu}\gamma_{5}\tau_{\text{i}}{\cal
G}(K_{+})g_{\pi qq}\gamma_{5}\tau_{\text{j}}{\cal G}(K_{-})\right]$ (159)
$\displaystyle=$ $\displaystyle 2N_{c}N_{f}g_{\pi
qq}M_{*}Q_{\mu}I(Q^{2})\delta_{\text{ij}}.$
Here ${\cal G}(K)=(\gamma^{\mu}K_{\mu}-M_{*})^{-1}$. Therefore, the pion decay
constant can be expressed as
$\displaystyle f_{\pi}^{2}\approx-2iN_{c}N_{f}M_{*}^{2}I(0).$ (160)
Finally, together with (158) and (160), we recover the well-known Gell-
Mann–Oakes–Renner relation
$m_{\pi}^{2}f_{\pi}^{2}=-m_{0}\langle\bar{q}q\rangle_{0}$ gell .
Set | $\Lambda$ | $G\Lambda^{2}$ | $m_{0}$ | $\langle\bar{u}u\rangle_{0}^{1/3}$ | $M_{*}$ | $m_{\pi}$
---|---|---|---|---|---|---
1 | 657.9 | 3.105 | 4.90 | -217.4 | 300 | 133.6
2 | 583.6 | 3.676 | 5.53 | -209.1 | 400 | 134.0
3 | 565.8 | 4.238 | 5.43 | -210.6 | 500 | 134.2
4 | 565.4 | 4.776 | 5.11 | -215.1 | 600 | 134.4
Table 1: Model parameters (3-momentum cutoff $\Lambda$, coupling constant $G$,
and current quark mass $m_{0}$) and related quantities (quark condensate
$\langle\bar{u}u\rangle_{0}$, effective quark mass $M_{*}$ and pion mass
$m_{\pi}$ in units of MeV) for the two-flavor two-color NJL model (109). The
pion decay constant is fixed to be $f_{\pi}=75$ MeV.
There are three parameters in our model, the current quark mass $m_{0}$, the
coupling constant $G$ and the cutoff $\Lambda$. In principle they should be
determined from the known values of the pion mass $m_{\pi}$, the pion decay
constant $f_{\pi}$ and the quark condensate $\langle\bar{q}q\rangle_{0}$
NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3 . Since two-color QCD does
not correspond to our real world, we get the above values from the empirical
values $f_{\pi}\simeq 93$MeV,
$\langle\bar{u}{u}\rangle_{0}\simeq(250$MeV$)^{3}$ in the $N_{c}=3$ case,
according to the relation $f_{\pi}^{2},\langle\bar{q}{q}\rangle_{0}\sim
N_{c}$. To obtain the model parameters, we fix the values of the pion decay
constant $f_{\pi}$ and slightly vary the values of the chiral condensate
$\langle\bar{q}q\rangle_{0}$ and the pion mass $m_{\pi}$. Thus, we can obtain
different sets of model parameters corresponding to different values of
effective quark mass $M_{*}$ and hence different values of the sigma meson
mass $m_{\sigma}$. Four sets of model parameters are shown in Table. 1. As we
will show in the following that, the physics near the quantum phase transition
point $\mu_{\text{B}}=m_{\pi}$ is not sensitive to different model parameter
sets, since the low energy dynamics is dominated by the pseudo-Goldstone
bosons (i.e., the diquarks). However, at high density, the physics becomes
sensitive to different model parameter sets corresponding to different sigma
meson masses. The predictions by the chiral perturbation theory should be
recovered in the limit $m_{\sigma}/m_{\pi}\rightarrow\infty$.
### III.3 Quantum phase transition and diquark Bose condensation
Now we begin to study the properties of two-color matter at finite baryon
density. Without loss of generality, we set $\mu_{\text{B}}>0$. In this
section, we study the two-color baryonic matter in the dilute limit, which
forms near the quantum phase transition point $\mu_{\text{B}}=m_{\pi}$. Since
the diquark condensate is vanishingly small near the quantum phase transition
point, we can make a Ginzburg-Landau expansion for the effective action. As we
will see below, this corresponds to the mean-field theory of weakly
interacting dilute Bose condensates.
Since the diquark condensate $\Delta$ is vanishingly small near the quantum
phase transition, we can derive the Ginzburg-Landau free energy functional
$V_{\text{GL}}[\Delta(x)]$ at $T=0$ for the order parameter field
$\Delta(x)=\langle\phi(x)\rangle$ in the static and long-wavelength limit. The
general form of $V_{\text{GL}}[\Delta(x)]$ can be written as
$\displaystyle V_{\text{GL}}[\Delta(x)]=\int
dx\Bigg{[}\Delta^{\dagger}(x)\left(-\delta\frac{\partial^{2}}{\partial\tau^{2}}+\kappa\frac{\partial}{\partial\tau}-\gamma\mbox{\boldmath{$\nabla$}}^{2}\right)\Delta(x)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\alpha|\Delta(x)|^{2}+\frac{1}{2}\beta|\Delta(x)|^{4}\Bigg{]},$ (161)
where the coefficients $\alpha,\beta,\gamma,\delta,\kappa$ should be low
energy constants which depend only on the vacuum properties. The calculation
is somewhat similar to the derivation of Ginzburg-Landau free energy of a
superconductor from the microscopic BCS theory nao , but for our case there is
a difference in that we have another variational parameter, i.e., the
effective quark mass $M$ which should be a function of $|\Delta|^{2}$
determined by the saddle point condition.
In the static and long-wavelength limit, the coefficients $\alpha,\beta$ of
the potential terms can be obtained from the effective action ${\cal
S}_{\text{eff}}$ in the mean-field approximation. At $T=0$, the mean-field
effective action reads ${\cal S}_{\text{eff}}^{(0)}=\int dx\Omega_{0}$, where
the mean-field thermodynamic potential is given by
$\displaystyle\Omega_{0}(|\Delta|^{2},M)=\frac{(M-m_{0})^{2}+|\Delta|^{2}}{4G}-N_{c}N_{f}\sum_{\bf
k}(E_{\bf k}^{+}+E_{\bf k}^{-}).$ (162)
The Ginzburg-Landau coefficients $\alpha,\beta$ can be obtained via a Taylor
expansion of $\Omega_{0}$ in terms of $|\Delta|^{2}$,
$\Omega_{0}=\Omega_{\text{vac}}(M_{*})+\alpha|\Delta|^{2}+\frac{1}{2}\beta|\Delta|^{4}+O(|\Delta|^{6}),$
(163)
where $\Omega_{\text{vac}}(M_{*})$ is the vacuum contribution which should be
subtracted. One should keep in mind that the effective quark mass $M$ is not a
fixed parameter, but an implicit function of $|\Delta|^{2}$ determined by the
gap equation $\partial\Omega_{0}/\partial M=0$.
For convenience, we define $y\equiv|\Delta|^{2}$. The Ginzburg-Landau
coefficient $\alpha$ is defined as
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{d\Omega_{0}(y,M)}{dy}\Bigg{|}_{y=0}$ (164)
$\displaystyle=$ $\displaystyle\frac{\partial\Omega_{0}(y,M)}{\partial
y}\Bigg{|}_{y=0}+\frac{\partial\Omega_{0}(y,M)}{\partial
M}\frac{dM}{dy}\Bigg{|}_{y=0}$ $\displaystyle=$
$\displaystyle\frac{\partial\Omega_{0}(y,M)}{\partial y}\Bigg{|}_{y=0},$
where the indirect derivative term vanishes due to the saddle point condition
for $M$. After some simple algebra, we get
$\alpha=\frac{1}{4G}-N_{c}N_{f}\sum_{\bf k}\frac{E_{\bf k}^{*}}{E_{\bf
k}^{*2}-\mu_{\text{B}}^{2}/4}$ (165)
with $E_{\bf k}^{*}=\sqrt{{\bf k}^{2}+M_{*}^{2}}$. We can make the above
expression more meaningful using the pion mass equation in the same three-
momentum regularization schemeNJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3
,
$\frac{1}{4G}-N_{c}N_{f}\sum_{\bf k}\frac{E_{\bf k}^{*}}{E_{\bf
k}^{*2}-m_{\pi}^{2}/4}=0.$ (166)
We therefore obtain a $G$-independent result
$\alpha=\frac{1}{4}N_{c}N_{f}(m_{\pi}^{2}-\mu_{\text{B}}^{2})\sum_{\bf
k}\frac{E_{\bf k}^{*}}{(E_{\bf k}^{*2}-m_{\pi}^{2}/4)(E_{\bf
k}^{*2}-\mu_{\text{B}}^{2}/4)}.$ (167)
From the fact that $m_{\pi}\ll 2M_{*}$ and $\beta>0$ (see below), we see
clearly that a second order quantum phase transition takes place exactly at
$\mu_{\text{B}}=m_{\pi}$. Therefore, the Ginzburg-Landau free energy is
meaningful only near the quantum phase transition point, i.e.,
$|\mu_{\text{B}}-m_{\pi}|\ll m_{\pi}$, and $\alpha$ can be further simplified
as
$\alpha\simeq(m_{\pi}^{2}-\mu_{\text{B}}^{2}){\cal J},$ (168)
where the factor ${\cal J}$ is defined as
$\displaystyle{\cal J}=\frac{1}{4}N_{c}N_{f}\sum_{\bf k}\frac{E_{\bf
k}^{*}}{(E_{\bf k}^{*2}-m_{\pi}^{2}/4)^{2}}.$ (169)
The coefficient $\beta$ of the quartic term can be evaluated from the
definition
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{d^{2}\Omega_{0}(y,M)}{dy^{2}}\Bigg{|}_{y=0}$ (170)
$\displaystyle=$ $\displaystyle\frac{\partial^{2}\Omega_{0}(y,M)}{\partial
y^{2}}\Bigg{|}_{y=0}+\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M\partial
y}\frac{dM}{dy}\Bigg{|}_{y=0}.$
Notice that the last indirect derivative term does not vanish here and will be
important for us to obtain a correct diquark-diquark scattering length. The
derivative $dM/dy$ can be analytically derived from the gap equation for $M$.
From the fact that $\partial\Omega_{0}/\partial M=0$, we obtain
$\frac{\partial}{\partial y}\left(\frac{\partial\Omega_{0}(y,M)}{\partial
M}\right)+\frac{\partial}{\partial
M}\left(\frac{\partial\Omega_{0}(y,M)}{\partial M}\right)\frac{dM}{dy}=0.$
(171)
Then we get
$\frac{dM}{dy}=-\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M\partial
y}\left(\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M^{2}}\right)^{-1}.$ (172)
Then the practical expression for $\beta$ can be written as
$\displaystyle\beta=\beta_{1}+\beta_{2},$ (173)
where $\beta_{1}$ is the direct derivative term
$\beta_{1}=\frac{\partial^{2}\Omega_{0}(y,M)}{\partial y^{2}}\Bigg{|}_{y=0},$
(174)
and $\beta_{2}$ is the indirect term
$\beta_{2}=-\left(\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M\partial
y}\right)^{2}\left(\frac{\partial^{2}\Omega_{0}(y,M)}{\partial
M^{2}}\right)^{-1}\Bigg{|}_{y=0}.$ (175)
Near the quantum phase transition, we can set $\mu_{\text{B}}=m_{\pi}$ in
$\beta$. After a simple algebra, the explicit forms of $\beta_{1}$ and
$\beta_{2}$ can be evaluated as
$\displaystyle\beta_{1}=\frac{1}{4}N_{c}N_{f}\sum_{e=\pm}\sum_{\bf
k}\frac{1}{(E_{\bf k}^{*}-em_{\pi}/2)^{3}}$ (176)
and
$\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle-\left\\{\frac{1}{2}N_{c}N_{f}\sum_{e=\pm}\sum_{\bf
k}\frac{M_{*}}{E_{\bf k}^{*}}\frac{1}{(E_{\bf
k}^{*}-em_{\pi}/2)^{2}}\right\\}^{2}$ (177)
$\displaystyle\times\left(\frac{m_{0}}{2GM_{*}}+2N_{c}N_{f}\sum_{\bf
k}\frac{M_{*}^{2}}{E_{\bf k}^{*3}}\right)^{-1}.$
The $G$-dependent term $m_{0}/(2GM_{*})$ in (177) can be approximated as
$m_{\pi}^{2}f_{\pi}^{2}/M_{*}^{2}$ by using the relation
$m_{\pi}^{2}f_{\pi}^{2}=-m_{0}\langle\bar{q}{q}\rangle_{0}$.
The kinetic terms in the Ginzburg-Landau free energy can be derived from the
inverse of the diquark propagator. In the general case with $\Delta\neq 0$,
there exists mixing between the diquarks and the sigma meson. However,
approaching the quantum phase transition point, $\Delta\rightarrow 0$, the
problem is simplified. After the analytical continuation
$i\nu_{m}\rightarrow\omega+i0^{+}$, the inverse of the diquark propagator at
$\mu_{\text{B}}=m_{\pi}+0^{+}$ can be evaluated as
$\displaystyle{\cal D}_{\text{d}}^{-1}(\omega,{\bf
q})=\frac{1}{4G}+\Pi_{\text{d}}(\omega,{\bf q}),$ (178)
where the polarization function $\Pi_{\text{d}}(\omega,{\bf q})$ is given by
$\displaystyle\Pi_{\text{d}}(\omega,{\bf q})=N_{c}N_{f}\sum_{\bf
k}\frac{E^{*}_{{\bf k}}+E^{*}_{{\bf k}+{\bf
q}}}{(\omega+\mu_{\text{B}})^{2}-(E^{*}_{{\bf k}}+E^{*}_{{\bf k}+{\bf
q}})^{2}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left(1+\frac{{\bf k}\cdot({\bf k}+{\bf q})+M_{*}^{2}}{E^{*}_{{\bf
k}}E^{*}_{{\bf k}+{\bf q}}}\right).$ (179)
In the static and long-wavelength limit ($\omega,|{\bf q}|\rightarrow 0$), the
coefficients $\kappa,\delta,\gamma$ can be determined by the Taylor expansion
${\cal D}_{\text{d}}^{-1}(\omega,{\bf q})={\cal D}_{\text{d}}^{-1}(0,{\bf
0})-\delta\omega^{2}-\kappa\omega+\gamma{\bf q}^{2}$. Notice that $\alpha$ is
identical to ${\cal D}_{\text{d}}^{-1}(0,{\bf 0})$, which is in fact the
Thouless criterion for the superfluid transition. On the other hand, ${\cal
D}_{\text{d}}^{-1}(\omega,{\bf q})$ can be related to the pion propagator in
the vacuum, i.e., ${\cal D}_{\text{d}}^{-1}(\omega,{\bf q})=(1/2){\cal
D}_{\pi}^{*-1}(\omega+\mu_{\text{B}},{\bf q})$. In the static and long-
wavelength limit and for $\mu_{\text{B}}\rightarrow m_{\pi}\ll 2M_{*}$ it can
be well approximated asNJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3
$\displaystyle{\cal D}_{\text{d}}^{-1}(\omega,{\bf q})\simeq-{\cal
J}\left[(\omega+\mu_{\text{B}})^{2}-{\bf q}^{2}-m_{\pi}^{2}\right],$ (180)
where ${\cal J}$ is the factor defined in (168) which describes the coupling
strength of the quark-quark-diquark interaction ${\cal J}\simeq g_{\pi
qq}^{-2}/2$ and with which we obtain $\delta\simeq\gamma\simeq{\cal J}$ and
$\kappa\simeq 2\mu_{\text{B}}{\cal J}$.
Set | 1 | 2 | 3 | 4
---|---|---|---|---
$a_{\text{dd}}$ from Eq. (184) | 0.0631 | 0.0635 | 0.0637 | 0.0639
$a_{\text{dd}}$ from Eq. (187) | 0.0624 | 0.0628 | 0.0630 | 0.0633
Table 2: The values of diquark-diquark scattering length $a_{\text{dd}}$ (in
units of $m_{\pi}^{-1}$) for different model parameter sets.
We now show how the Ginzburg-Landau free energy can be reduced to the theory
describing weakly repulsive Bose condensates, i.e., the Gross-Pitaevskii free
energy GP01 ; GP02 .
First, since the Bose condensed matter is indeed dilute, let us consider the
nonrelativistic version with $\omega\ll m_{\pi}$ and neglecting the kinetic
term $\propto\partial^{2}/\partial\tau^{2}$. To this end, we define the
nonrelativistic chemical potential $\mu_{\text{d}}$ for diquarks,
$\mu_{\text{d}}=\mu_{\text{B}}-m_{\pi}$. The coefficient $\alpha$ can be
further simplified as
$\displaystyle\alpha$ $\displaystyle\simeq$
$\displaystyle-\mu_{\text{d}}(2m_{\pi}{\cal J}).$ (181)
Then the Ginzburg-Landau free energy can be reduced to the Gross-Pitaevskii
free energy of a dilute repulsive Bose gas, if we define a new condensate wave
function $\psiup(x)$ as
$\displaystyle\psiup(x)=\sqrt{2m_{\pi}{\cal J}}\Delta(x).$ (182)
The resulting Gross-Pitaevskii free energy is given by
$\displaystyle V_{\text{GP}}[\psiup(x)]$ $\displaystyle=$ $\displaystyle\int
dx\bigg{[}\psiup^{\dagger}(x)\left(\frac{\partial}{\partial\tau}-\frac{\mbox{\boldmath{$\nabla$}}^{2}}{2m_{\pi}}\right)\psiup(x)$
(183)
$\displaystyle-\mu_{\text{d}}|\psiup(x)|^{2}+\frac{1}{2}g_{0}|\psiup(x)|^{4}\bigg{]}$
with $g_{0}=4\pi a_{\text{dd}}/m_{\pi}$. The repulsive diquark-diquark
interaction is characterized by a positive scattering length $a_{\text{dd}}$
defined as
$\displaystyle a_{\text{dd}}=\frac{\beta}{16\pi m_{\pi}}{\cal J}^{-2}.$ (184)
Keep in mind that the scattering length obtained here is at the mean-field
level. We will discuss the possible beyond-mean-field corrections later.
Therefore, near the quantum phase transition where the density $n$ satisfies
$na_{\text{dd}}^{3}\ll 1$, the system is indeed a weakly interacting Bose
condensate Bose01 .
Even though we have shown that the Ginzburg-Landau free energy is indeed a
Gross-Pitaevskii version near the quantum phase transition, a key problem is
whether the obtained diquark-diquark scattering length is quantitatively
correct. The numerical calculation for (184) is straightforward. The obtained
values of $a_{\text{dd}}$ for the four model parameter sets are shown in Table
2. We can also give an analytical expression based on the formula of the pion
decay constant in the three-momentum cutoff scheme,
$\displaystyle f_{\pi}^{2}=N_{c}M_{*}^{2}\sum_{\bf k}\frac{1}{E_{\bf
k}^{*3}}.$ (185)
According to the fact $m_{\pi}\ll 2M_{*}$, $\beta$ and ${\cal J}$ can be well
approximated as
$\displaystyle\beta\simeq\frac{f_{\pi}^{2}}{M_{*}^{2}}-\frac{(2f_{\pi}^{2}/M_{*})^{2}}{m_{\pi}^{2}f_{\pi}^{2}/M_{*}^{2}+4f_{\pi}^{2}}\simeq\frac{f_{\pi}^{2}m_{\pi}^{2}}{4M_{*}^{4}},$
$\displaystyle{\cal J}\simeq\frac{f_{\pi}^{2}}{2M_{*}^{2}}.$ (186)
There, the diquark-diquark scattering length $a_{\text{dd}}$ in the limit
$m_{\pi}/(2M_{*})\rightarrow 0$ is only related to the pion mass and decay
constant,
$\displaystyle a_{\text{dd}}=\frac{m_{\pi}}{16\pi f_{\pi}^{2}}.$ (187)
The values of $a_{\text{dd}}$ for the four model parameter sets according to
the above expression are also listed in Table 2. The errors are always about
$1\%$ comparing with the exact numerical results, which means that the
expression (187) is a good approximation for the diquark-diquark scattering
length. The error should come from the finite value of $m_{\pi}/(2M^{*})$. We
can obtain a correction in powers of $m_{\pi}/(2M^{*})$ schulze , but it is
obviously small, and its explicit form is not shown here.
The result $a_{\text{dd}}\propto m_{\pi}$ is universal for the scattering
lengths of the pseudo-Goldstone bosons. Eventhough the SU$(4)$ flavor symmetry
is explicitly broken in presence of a nonzero quark mass, a descrete symmetry
$\phi_{1},\phi_{2}\leftrightarrow\pi_{1},\pi_{2}$ holds exactly for arbitrary
quark mass. This also means that the partition function of two-color QCD has a
descrete symmetry $\mu_{\text{B}}\leftrightarrow\mu_{\text{I}}$ QL03 . Because
of this descrete symmetry of two-color QCD, the analytical expression (187) of
$a_{\text{dd}}$ (which is in fact the diquark-diquark scattering length in the
$B=2$ channel) should be identical to the pion-pion scattering length at tree
level in the $I=2$ channel which was first obtained by Weinberg many years ago
pipi . Therefore, the mean-field theory can describe not only the quantum
phase transition to a dilute diquark condensate but also the effect of
repulsive diquark-diquark interaction.
Figure 9: The baryon chemical potential $\mu_{\text{B}}$ and the pressure $P$
as functions of the baryon density $n$ for different model parameter sets. The
solid lines correspond to the direct mean-field calculation and the dashed
lines are given by (III.3).
The mean-field equations of state of the dilute diquark condensate are thus
determined by the Gross-Pitaevskii free energy (183). Minimizing
$V_{\text{GP}}[\psiup(x)]$ with respect to a uniform condensate $\psiup$, we
find that the physical minimum is given by
$\displaystyle|\psiup_{0}|^{2}=\frac{\mu_{\text{d}}}{g_{0}},$ (188)
and the baryon density is $n=|\psiup_{0}|^{2}$. Using the thermodynamic
relations, we therefore get the well-known results at $T=0$ for the pressure
$P$, the energy density ${\cal E}$ and the chemical potential $\mu_{\text{B}}$
in terms of the baryon density $n$ Bose01 ,
$\displaystyle P(n)=\frac{2\pi a_{\text{dd}}}{m_{\pi}}n^{2},$
$\displaystyle{\cal E}(n)=m_{\pi}n+\frac{2\pi a_{\text{dd}}}{m_{\pi}}n^{2},$
$\displaystyle\mu_{\text{B}}(n)=m_{\pi}+\frac{4\pi a_{\text{dd}}}{m_{\pi}}n.$
(189)
We can examine the above results through a direct numerical calculation with
the mean-field thermodynamic potential. Since all the thermodynamic quantities
are relative to the vacuum, we subtract the vacuum contribution from the
pressure, $P=-(\Omega_{0}(n)-\Omega_{0}(0))$, and the baryon density reads
$n=-\partial\Omega_{0}/\partial\mu_{\text{B}}$. In Fig.9 we show the numerical
results for the pressure and the chemical potential as functions of the
density for the four model parameter sets. At low enough density, the
equations of state are indeed consistent with the results (III.3) with the
scattering length given by (184). It is evident that the results at low
density are not sensitive to different model parameter sets, since the physics
at low density should be dominated by the pseudo-Goldstone bosons.
Further, since our treatment is only at the mean-field level, the Lee-Huang-
Yang corrections Bose03 which are proportional to $(na_{\text{dd}}^{3})^{1/2}$
are absent in the equations of state. To obtain such corrections, it is
necessary to go beyond the mean field, and there is also a beyond-mean-field
correction to the scattering length $a_{\text{dd}}$ Unitary ; HU ; Diener .
We can also consider a relativistic version of the Gross-Pitaevskii free
energy via defining the condensate wave function
$\displaystyle\Phi(x)=\sqrt{{\cal J}}\Delta(x).$ (190)
In this case, the Ginzburg-Landau free energy is reduced to a relativistic
version of the Gross-Pitaevskii free energy,
$\displaystyle V_{\text{RGP}}[\Phi(x)]$ $\displaystyle=$ $\displaystyle\int
dx\bigg{[}\Phi^{\dagger}(x)\left(-\frac{\partial^{2}}{\partial\tau^{2}}+2\mu_{\text{B}}\frac{\partial}{\partial\tau}-\mbox{\boldmath{$\nabla$}}^{2}\right)\Phi(x)$
(191)
$\displaystyle+(m_{\pi}^{2}-\mu_{\text{B}}^{2})|\Phi(x)|^{2}+\frac{\lambda}{2}|\Phi(x)|^{4}\bigg{]}.$
The self-interacting coupling $\lambda=\beta{\cal J}^{-2}$ is now
dimensionless and can be approximated by $\lambda\simeq
m_{\pi}^{2}/f_{\pi}^{2}$. For realistic values of $m_{\pi}$ and $f_{\pi}$, we
find $\lambda\sim O(1)$. In this sense, the Bose condensate is not weakly
interacting, except for the low density limit $na_{\text{dd}}^{3}\ll 1$. One
should keep in mind that this result cannot be applied to high density, since
it is valid only near the quantum phase transition point.
An ideal Bose-Einstein condensate is not a superfluid. In presence of weak
repulsive interactions among the bosons, a Goldstone mode which has a linear
dispersion at low energy appears, and the condensate becomes a superfluid
according to Landau’s criterion $\min_{\bf q}[\omega({\bf q})/|{\bf q}|]>0$.
The Goldstone mode which is also called the Bogoliubov mode here should have a
dispersion given by Bose01
$\displaystyle\omega({\bf q})=\sqrt{\frac{{\bf
q}^{2}}{2m_{\pi}}\left(\frac{{\bf q}^{2}}{2m_{\pi}}+\frac{8\pi
a_{\text{dd}}n}{m_{\pi}}\right)},\ \ \ \ |{\bf q}|\ll m_{\pi}.$ (192)
Since the Gross-Pitaevskii free energy obtained above is at the classical
level, to study the bosonic collective excitations we should consider the
fluctuations around the mean field. The propagator of the bosonic collective
modes is given by ${\bf M}^{-1}(Q)$ and ${\bf N}^{-1}(Q)$. The Bogoliubov mode
corresponds to the lowest excitation obtained from the equation $\det{\bf
M}(\omega,{\bf q})=0$. With the explicit form of the matrix elements of ${\bf
M}$ in the superfluid phase, we can analytically show $\det{\bf M}(0,{\bf
0})=0$ which ensures the Goldstone’s theorem. In fact, for $(\omega,{\bf
q})=(0,{\bf 0})$, we find $\det{\bf M}=({\bf M}_{11}^{2}-|{\bf
M}_{12}|^{2}){\bf M}_{33}+2|{\bf M}_{13}|^{2}(|{\bf M}_{12}|-{\bf M}_{11})$.
Using the saddle point condition for $\Delta$, we can show that ${\bf
M}_{11}(0,{\bf 0})=|{\bf M}_{12}(0,{\bf 0})|$ and hence the Goldstone’s
theorem holds in the superfluid phase. Further, we may obtain an analytical
expression of the velocity of the Bogoliubov mode via a Taylor expansion for
${\bf M}(\omega,{\bf q})$ around $(\omega,{\bf q})=(0,{\bf 0})$ like those
done in BCSBEC3 . Such a calculation for our case is more complicated due to
the mixing between the sigma meson and diquarks, and it cannot give the full
dispersion (192).
On the other hand, considering $\Delta\rightarrow 0$ near the quantum phase
transition point, we can expand the matrix elements of ${\bf M}$ in powers of
$|\Delta|^{2}$. The advantage of such an expansion is that it can not only
give the full dispersion (192) but also link the meson properties in the
vacuum. Formally, we can write down the following expansions,
$\displaystyle{\bf M}_{11}(\omega,{\bf q})={\cal
D}_{\text{d}}^{-1}(\omega,{\bf q})+|\Delta|^{2}A(\omega,{\bf
q})+O(|\Delta|^{4}),$ $\displaystyle{\bf M}_{22}(\omega,{\bf q})={\cal
D}_{\text{d}}^{-1}(-\omega,{\bf q})+|\Delta|^{2}A(-\omega,{\bf
q})+O(|\Delta|^{4}),$ $\displaystyle{\bf M}_{12}(\omega,{\bf q})={\bf
M}_{21}^{\dagger}(\omega,{\bf q})=\Delta^{2}B(\omega,{\bf
q})+O(|\Delta|^{4}),$ $\displaystyle{\bf M}_{13}(\omega,{\bf q})={\bf
M}_{31}^{\dagger}(\omega,{\bf q})=\Delta H(\omega,{\bf q})+O(|\Delta|^{3}),$
$\displaystyle{\bf M}_{23}(\omega,{\bf q})={\bf M}_{32}^{\dagger}(\omega,{\bf
q})=\Delta^{\dagger}H(-\omega,{\bf q})+O(|\Delta|^{3}),$ $\displaystyle{\bf
M}_{33}(\omega,{\bf q})={\cal D}_{\sigma}^{*-1}(\omega,{\bf
q})+O(|\Delta|^{2}).$ (193)
Notice that the effective quark mass $M$ is regarded as a function of
$|\Delta|^{2}$ as we have done in deriving the Ginzburg-Landau free energy.
Since we are interested in the dispersion in the low energy limit, i.e.,
$\omega,|{\bf q}|\ll m_{\pi}$, we can approximate the coefficients of the
leading order terms as their values at $(\omega,{\bf q})=(0,{\bf 0})$,
$\displaystyle A(\omega,{\bf q})\simeq A(-\omega,{\bf q})\simeq A(0,{\bf
0})\equiv A_{0},$ $\displaystyle B(\omega,{\bf q})\simeq B(0,{\bf 0})\equiv
B_{0},$ $\displaystyle H(\omega,{\bf q})\simeq H(-\omega,{\bf q})\simeq
H(0,{\bf 0})\equiv H_{0}.$ (194)
Further, from $m_{\sigma}\gg m_{\pi}$, we can approximate the inverse sigma
propagator ${\cal D}_{\sigma}^{*-1}(\omega,{\bf q})$ as its value at
$(\omega,{\bf q})=(0,{\bf 0})$. Therefore, the dispersion of the Goldstone
mode in the low energy limit can be determined by the following equation,
$\displaystyle\det\left(\begin{array}[]{ccc}{\cal
D}_{\text{d}}^{-1}(\omega,{\bf q})+|\Delta|^{2}A_{0}&\Delta^{2}B_{0}&\Delta
H_{0}\\\ \Delta^{\dagger 2}B_{0}&{\cal D}_{\text{d}}^{-1}(-\omega,{\bf
q})+|\Delta|^{2}A_{0}&\Delta^{\dagger}H_{0}\\\ \Delta^{\dagger}H_{0}&\Delta
H_{0}&{\cal D}_{\sigma}^{*-1}(0,{\bf 0})\end{array}\right)$ (198)
$\displaystyle=0.$ (199)
Now we can link the coefficients $A_{0},B_{0},H_{0}$ and ${\cal
D}_{\sigma}^{-1}(0,{\bf 0})$ to the derivatives of the mean-field
thermodynamic potential $\Omega_{0}$ and its Ginzburg-Landau coefficients.
Firstly, using the explicit form of ${\bf M}_{12}$, we find
$\displaystyle|{\bf M}_{12}(0,{\bf 0})|=|\Delta|^{2}\beta_{1}\Longrightarrow
B_{0}=\beta_{1}.$ (200)
Secondly, using the fact that
$\displaystyle{\bf M}_{11}(0,{\bf 0})-|{\bf M}_{12}(0,{\bf
0})|=\frac{\partial\Omega_{0}}{\partial|\Delta|^{2}},$ (201)
and together with the definition for $A(\omega,{\bf q})$,
$\displaystyle A(\omega,{\bf q})$ $\displaystyle=$ $\displaystyle\frac{d{\bf
M}_{11}(y,M)}{dy}\Bigg{|}_{y=0}$ (202) $\displaystyle=$
$\displaystyle\frac{\partial{\bf M}_{11}(y,M)}{\partial
y}\Bigg{|}_{y=0}+\frac{\partial{\bf M}_{11}(y,M)}{\partial
M}\frac{dM}{dy}\Bigg{|}_{y=0},$
we find the following exact relation:
$\displaystyle A_{0}=\beta+B_{0}=\beta+\beta_{1}.$ (203)
On the other hand, we have the following relations for $H_{0}$ and ${\cal
D}_{\sigma}^{*-1}(0,{\bf 0})$,
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M\partial
y}\Bigg{|}_{y=0},$ $\displaystyle{\cal D}_{\sigma}^{*-1}(0,{\bf 0})$
$\displaystyle=$ $\displaystyle\frac{\partial^{2}\Omega_{0}(y,M)}{\partial
M^{2}}\Bigg{|}_{y=0}.$ (204)
One can check the above results from the explicit forms of ${\bf M}_{13}$ and
${\bf M}_{33}$ in Appendix A directly. Thus we have
$\displaystyle-\frac{H_{0}^{2}}{{\cal D}_{\sigma}^{*-1}(0,{\bf
0})}=\beta_{2}.$ (205)
According to the above relations, Eq. (198) can be reduced to
$\displaystyle 3\beta^{2}|\Delta|^{4}+2\beta|\Delta|^{2}[{\cal
D}_{\text{d}}^{-1}(\omega,{\bf q})+{\cal D}_{\text{d}}^{-1}(-\omega,{\bf q})]$
$\displaystyle+$ $\displaystyle{\cal D}_{\text{d}}^{-1}(\omega,{\bf q}){\cal
D}_{\text{d}}^{-1}(-\omega,{\bf q})=0.$ (206)
It is evident that only the coefficient $\beta$ appears in the final equation.
Further, in the nonrelativistic limit $\omega,|{\bf q}|\ll m_{\pi}$ and near
the quantum phase transition point, ${\cal D}_{\text{d}}^{-1}(\omega,{\bf q})$
can be approximated as
$\displaystyle{\cal D}_{\text{d}}^{-1}(\omega,{\bf q})\simeq-2m_{\pi}{\cal
J}\left(\omega-\frac{{\bf q}^{2}}{2m_{\pi}}+\mu_{\text{d}}\right).$ (207)
Together with the mean-field results for the chemical potential
$\mu_{\text{d}}=g_{0}|\psiup_{0}|^{2}=\beta|\Delta|^{2}/(2m_{\pi}{\cal J})$
and for the baryon density $n=|\psiup_{0}|^{2}$, we finally get the Bogoliubov
dispersion (192).
We here emphasize that the mixing between the sigma meson and the diquarks,
denoted by the terms $\Delta H_{0}$ and $\Delta^{\dagger}H_{0}$, plays an
important role in recovering the correct Bogoliubov dispersion. Even though we
do get this dispersion, we find that the procedure is quite different to the
standard theory of weakly interacting Bose gas Bose01 ; GP01 ; GP02 . There,
the elementary excitation is given only by the diquark-diquark sectors, i.e.,
$\det\left(\begin{array}[]{cc}{\bf M}_{11}(Q)&{\bf M}_{12}(Q)\\\ {\bf
M}_{21}(Q)&{\bf M}_{22}(Q)\end{array}\right)=0\ \ \ \ \ \ \ \Longrightarrow$
$\displaystyle\det\left(\begin{array}[]{cc}-\omega+\frac{{\bf
q}^{2}}{2m_{\pi}}-\mu_{\text{d}}+2g_{0}|\psiup_{0}|^{2}&g_{0}|\psiup_{0}|^{2}\\\
g_{0}|\psiup_{0}|^{2}&\omega+\frac{{\bf
q}^{2}}{2m_{\pi}}-\mu_{\text{d}}+2g_{0}|\psiup_{0}|^{2}\end{array}\right)$
(210) $\displaystyle=0.$ (211)
But in our case, we cannot get the correct Bogoliubov excitation if we simply
set $H_{0}=0$ and consider only the diquark-diquark sector. In fact, this
requires $A_{0}=2B_{0}=2\beta$ which is not true in our case.
One can also check how the momentum dependence of $A,B,H$ and ${\cal
D}_{\sigma}^{*-1}$ modifies the dispersion. This needs direct numerical
solution of the equation $\det{\bf M}(\omega,{\bf q})=0$. We have examined
that for $|\mu_{\text{B}}-m_{\pi}|$ up to $0.01m_{\pi}$, the numerical result
agrees well with the Bogoliubov formula (192). However, at higher density, a
significant deviation is observed. This is in fact a signature of BEC-BCS
crossover which will be discussed later.
Up to now we have studied the properties of the dilute Bose condensate induced
by a small diquark condensate $\langle qq\rangle$. The chiral condensate
$\langle\bar{q}{q}\rangle$ will also be modified in the medium. In such a
dilute Bose condensate, we can study the response of the chiral condensate to
the baryon density $n$.
To this end, we expand the effective quark mass $M$ in terms of
$y=|\Delta|^{2}$. We have
$M-M_{*}=\frac{dM}{dy}\bigg{|}_{y=0}y+O(y^{2})$ (212)
The expansion coefficient can be approximated as
$\displaystyle\frac{dM}{dy}\bigg{|}_{y=0}$ $\displaystyle\simeq$
$\displaystyle-\frac{2f_{\pi}^{2}/M_{*}}{m_{\pi}^{2}f_{\pi}^{2}/M_{*}^{2}+4f_{\pi}^{2}}$
(213) $\displaystyle=$
$\displaystyle-\frac{1}{2M_{*}}\left[1+O\left(\frac{m_{\pi}^{2}}{4M_{*}^{2}}\right)\right].$
Using the definition of the effective quark mass,
$M=m_{0}-2G\langle\bar{q}{q}\rangle$, we find
$\frac{\langle\bar{q}{q}\rangle_{n}}{\langle\bar{q}{q}\rangle_{0}}=1-\frac{|\Delta|^{2}}{4G\langle\bar{q}{q}\rangle_{0}M_{*}}\simeq
1-\frac{|\Delta|^{2}}{2M_{*}^{2}}.$ (214)
Since the baryon number density reads $n=|\psiup_{0}|^{2}=2m_{\pi}{\cal
J}|\Delta|^{2}$, using the fact that ${\cal J}\simeq
f_{\pi}^{2}/(2M_{*}^{2})$, we obtain to leading order
$\frac{\langle\bar{q}{q}\rangle_{n}}{\langle\bar{q}{q}\rangle_{0}}\simeq
1-\frac{n}{2f_{\pi}^{2}m_{\pi}}.$ (215)
This formula is in fact a two-color analogue of the density dependence of the
chiral condensate in the $N_{c}=3$ case, where we have cohen ; cohen2
$\frac{\langle\bar{q}{q}\rangle_{n}}{\langle\bar{q}{q}\rangle_{0}}\simeq
1-\frac{\Sigma_{\pi{\text{N}}}}{f_{\pi}^{2}m_{\pi}^{2}}n$ (216)
with $\Sigma_{\pi\text{N}}$ being the pion-nucleon sigma term. In Fig.10, we
show the numerical results via solving the mean-field gap equations. One finds
that the chiral condensate has a perfect linear behavior at low density. For
large value of $M_{*}$ ( and hence the sigma meson mass $m_{\sigma}$), the
linear behavior persists even at higher density.
Figure 10: The ratio
$R_{n}=\langle\bar{q}{q}\rangle_{n}/\langle\bar{q}{q}\rangle_{0}$ as a
function of $n/(f_{\pi}^{2}m_{\pi})$ for different model parameter sets. The
dashed line is the linear behavior given by (215).
In fact, the Eq. (215) can be obtained in a model independent way. Applying
the Hellmann-Feynman theorem to a dilute diquark gas with energy density
${\cal E}(n)$ given by (III.3), we can obtain (215) directly. According to the
Hellmann-Feynman theorem, we have
$2m_{0}(\langle\bar{q}{q}\rangle_{n}-\langle\bar{q}{q}\rangle_{0})=m_{0}\frac{d{\cal
E}}{dm_{0}}.$ (217)
The derivative $d{\cal E}/dm_{0}$ can be evaluated via the chain rule $d{\cal
E}/dm_{0}=(d{\cal E}/dm_{\pi})(dm_{\pi}/dm_{0})$. Together with the Gell-
Mann–Oakes–Renner relation
$m_{\pi}^{2}f_{\pi}^{2}=-m_{0}\langle\bar{q}{q}\rangle_{0}$ and the fact that
$da_{\text{dd}}/dm_{\pi}\simeq a_{\text{dd}}/m_{\pi}$, we can obtain to
leading order Eq. (215). Beyond the leading order, we find that the correction
of order $O(n^{2})$ vanishes. Thus, the next-to-leading order correction
should be $O(n^{5/2})$ coming from the Lee-Huang-Yang correction to the
equation of state HFiso .
Finally, we can show analytically that the “chiral rotation” behavior QC2D ;
QC2D1 ; QC2D2 ; QC2D3 ; QC2D4 ; QC2D5 ; QC2D6 ; QL03 ; ratti ; 2CNJL04 ; ISO
predicted by the chiral perturbation theories is valid in the NJL model near
the quantum phase transition. In the chiral perturbation theories, the
chemical potential dependence of the chiral and diquark condensates can be
analytically expressed as
$\frac{\langle\bar{q}{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}=\frac{m_{\pi}^{2}}{\mu_{\text{B}}^{2}},\
\ \ \ \frac{\langle
q{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}=\sqrt{1-\frac{m_{\pi}^{4}}{\mu_{\text{B}}^{4}}}.$
(218)
Near the phase transition point, we can expand the above formula in powers of
$\mu_{\text{d}}=\mu_{\text{B}}-m_{\pi}$. To leading order, we have
$\frac{\langle\bar{q}{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}\simeq
1-\frac{2\mu_{\text{d}}}{m_{\pi}},\ \ \ \ \frac{\langle
q{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}\simeq
2\sqrt{\frac{\mu_{\text{d}}}{m_{\pi}}}.$ (219)
Using the mean-field result (188) for the chemical potential $\mu_{\text{d}}$,
one can easily check that the above relations are also valid in our NJL model.
In the above studies we focused on the “physical point” with $m_{0}\neq 0$. In
the final part of this section, we briefly discuss the chiral limit with
$m_{0}=0$.
We may naively expect that the results at $m_{0}\neq 0$ can be directly
generalized to the chiral limit via setting $m_{\pi}=0$. The ground state is a
noninteracting Bose condensate of massless diquarks, due to $m_{\pi}=0$ and
$a_{\text{dd}}=0$. However, this cannot be true since many divergences develop
due to the vanishing pion mass. In fact, the conclusion of second order phase
transition is not correct since the Ginzburg-Landau coefficient $\beta$
vanishes. Instead, the superfluid phase transition is of strongly first order
in the chiral limit ratti ; ISOother02 ; ISOother021 .
Figure 11: The chiral and diquark condensates (in units of
$\langle\bar{q}{q}\rangle_{0}$) as functions of the baryon chemical
potential(in units of $m_{\pi}$) for different model parameter sets.
In the chiral limit, the effective action in the vacuum should depend only on
the combination $\sigma^{2}+\mbox{\boldmath{$\pi$}}^{2}+|\phi|^{2}$ due to the
exact flavor symmetry SU$(4)\simeq$ SO$(6)$. The vacuum is chosen to be
associated with a nonzero chiral condensate $\langle\sigma\rangle$ without
loss of generality. At zero and at finite chemical potential, the
thermodynamic potential $\Omega_{0}(M,|\Delta|)$ has two minima locating at
$(M,|\Delta|)=(a,0)$ and $(M,|\Delta|)=(0,b)$. At zero chemical potential,
these two minima are degenerate due to the exact flavor symmetry. However, at
nonzero chemical potential (even arbitrarily small), the minimum $(0,b)$ has
the lowest free energy. Analytically, we can show $b\rightarrow M_{*}$ at
$\mu_{\text{B}}=0^{+}$. This means that the superfluid phase transition in the
chiral limit is of strongly first order, and takes place at arbitrarily small
chemical potential. Since the effective quark mass $M$ keeps vanishing in the
superfluid phase, a low density Bose condensate does not exist in the chiral
limit.
### III.4 Matter at High Density: BEC-BCS crossover and Mott Transition
The investigations in the previous subsection are restricted near the quantum
phase transition point $\mu_{\text{B}}=m_{\pi}$. Generally the state of matter
at high density should not be a relativistic Bose condensate described by
(191). In fact, perturbative QCD calculations show that the matter is a weakly
coupled BCS superfluid at asymptotic density pQCD ; pQCD1 ; pQCD2 ; pQCD3 ;
pQCD4 ; pQCD5 ; pQCD6 . In this section, we will discuss the evolution of the
superfluid matter as the baryon density increases from the NJL model point of
view.
The numerical results for the chiral condensate $\langle\bar{q}q\rangle$ and
diquark condensate $\langle qq\rangle$ are shown in Fig.11. As a comparison,
we also show the analytical result (218) predicted by the chiral perturbation
theories (dashed lines). Since both the NJL model and chiral perturbation
theories are equivalent realization of chiral symmetry as an effective low-
energy theory of QCD, the behavior of the chiral condensate is almost the same
in the two cases. However, for the diquark condensate, there is quantitative
difference at large chemical potential. To understand this deviation, we
compare the linear sigma model and its limit of infinite sigma mass. The
former is similar to the NJL model with a finite sigma mass, and the latter
corresponds to the chiral perturbation theories. We consider the O$(6)$ linear
sigma model 2CNJL04
${\cal
L}_{\text{LSM}}=\frac{1}{2}(\partial_{\mu}\mbox{\boldmath{$\varphi$}})^{2}-\frac{1}{2}m^{2}\mbox{\boldmath{$\varphi$}}^{2}+\frac{1}{4}\lambda\mbox{\boldmath{$\varphi$}}^{4}-H\sigma$
(220)
with
$\mbox{\boldmath{$\varphi$}}=(\sigma,\mbox{\boldmath{$\pi$}},\phi_{1},\phi_{2})$
and $m^{2}<0$. The model parameters $m^{2},\lambda,H$ can be determined from
the vacuum phenomenology. In this model, we can show that the chiral and
diquark condensates are given by
$\displaystyle\frac{\langle\bar{q}{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}=\frac{m_{\pi}^{2}}{\mu_{\text{B}}^{2}},$
$\displaystyle\frac{\langle
q{q}\rangle_{\mu_{\text{B}}}}{\langle\bar{q}{q}\rangle_{0}}=\sqrt{1-\frac{m_{\pi}^{4}}{\mu_{\text{B}}^{4}}+2\frac{\mu_{\text{B}}^{2}-m_{\pi}^{2}}{m_{\sigma}^{2}-m_{\pi}^{2}}}.$
(221)
In the limit $m_{\sigma}\rightarrow\infty$, the above results are indeed
reduced to the result (218) of chiral perturbation theories. However, for
finite values of $m_{\sigma}$, the difference between the two might be
remarkable at large chemical potential.
While the Ginzburg-Landau free energy can be reduced to the Gross-Pitaevskii
free energy near the quantum phase transition point, it is not the case at
arbitrary $\mu_{\text{B}}$.
When $\mu_{\text{B}}$ increases, we find that the fermionic excitation spectra
$E_{\bf k}^{\pm}$ undergo a characteristic change. Near the quantum phase
transition $\mu_{\text{B}}=m_{\pi}$, they are nearly degenerate, due to
$m_{\pi}\ll 2M_{*}$ and their minima located at $|{\bf k}|=0$. However, at
very large $\mu_{\text{B}}$, the minimum of $E_{\bf k}^{-}$ moves to $|{\bf
k}|\simeq\mu_{\text{B}}/2$ from $M\rightarrow m_{0}$. Meanwhile the excitation
energy of the anti-fermion excitation become much larger than that of the
fermion excitation and can be neglected. This characteristic change of the
fermionic excitation spectra takes place when the minimum of the lowest band
excitation $E_{\bf k}^{-}$ moves from $|{\bf k}|=0$ to $|{\bf k}|\neq 0$,
i.e., $\mu_{\text{B}}/2=M(\mu_{\text{B}})$. A schematic plot of this
characteristic change is shown in Fig.12. The equation
$\mu_{\text{B}}/2=M(\mu_{\text{B}})$ defines the so-called crossover point
$\mu_{\text{B}}=\mu_{0}$ which can be numerically determined by the mean-field
gap equations. The numerical results of the crossover chemical potential
$\mu_{0}$ for the four model parameter sets are shown in Table.3. For
reasonable parameter sets, the crossover chemical potential is in the range
$(1.6-2)m_{\pi}$. We notice that this crossover chemical potential agrees with
the result from lattice simulation LBECBCS .
Set | 1 | 2 | 3 | 4
---|---|---|---|---
chemical potential $\mu_{0}$ | 1.65 | 1.81 | 1.95 | 2.07
Table 3: The crossover chemical potential $\mu_{0}$ (in units of $m_{\pi}$)
for different model parameter sets. Figure 12: A schematic plot of the
fermionic excitation spectrum in the BEC state (left) and the BCS state
(right).
In fact, an analytical expression for $\mu_{0}$ can be achieved according to
the fact that the chiral rotation behavior
$\langle\bar{q}q\rangle_{\mu_{\text{B}}}/\langle\bar{q}q\rangle_{0}\simeq
m_{\pi}^{2}/\mu_{\text{B}}^{2}$ is still valid in the NJL model at large
chemical potentials as shown in Fig.11. We obtain
$\displaystyle\frac{\mu_{0}}{2}\simeq\frac{m_{\pi}^{2}}{\mu_{0}^{2}}M_{*}\ \
\Longrightarrow\ \ \mu_{0}\simeq(2M_{*}m_{\pi}^{2})^{1/3}.$ (222)
Using the fact that $m_{\sigma}\simeq 2M_{*}$, we find that $\mu_{0}$ can be
expressed as
$\displaystyle\frac{\mu_{0}}{m_{\pi}}\simeq\left(\frac{m_{\sigma}}{m_{\pi}}\right)^{1/3}.$
(223)
Thus, in the nonlinear sigma model limit
$m_{\sigma}/m_{\pi}\rightarrow\infty$, there should be no BEC-BCS crossover.
On the other hand, this means that the physical prediction power of the chiral
perturbation theories is restricted near the quantum phase transition point.
The fermionic excitation gap $\Delta_{\text{ex}}$ (as shown in Fig.12),
defined as the minimum of the fermionic excitation energy, i.e.,
$\Delta_{\text{ex}}=\min_{\bf k}\\{E_{\bf k}^{-},E_{\bf k}^{+}\\}$, can be
evaluated as
$\Delta_{\text{ex}}=\left\\{\begin{array}[]{r@{\quad,\quad}l}\sqrt{(M-\frac{\mu_{\text{B}}}{2})^{2}+|\Delta|^{2}}&\mu_{\text{B}}<\mu_{0}\\\
|\Delta|&\mu_{\text{B}}>\mu_{0}.\end{array}\right.$ (224)
It is evident that the fermionic excitation gap is equal to the superfluid
order parameter only in the BCS regime. This is similar to the BEC-BCS
crossover in nonrelativistic systems BCSBEC3 , and we find that the
corresponding fermion chemical potential $\mu_{\rm n}$ can be defined as
$\mu_{\rm n}=\mu_{\text{B}}/2-M$. The numerical results of the fermionic
excitation gap $\Delta_{\text{ex}}$ for different model parameter sets are
shown in Fig.13. We find that for a wide range of the baryon chemical
potential, it is of order $O(M_{*})$. The fermionic excitation gap is equal to
the pairing gap $|\Delta|$ only at the BCS side of the crossover, and exhibits
a minimum at the quantum phase transition point.
On the other hand, the momentum distributions of quarks (denoted by $n({\bf
k})$) and antiquarks (denoted by $\bar{n}({\bf k})$) can be evaluated using
the quark Green function ${\cal G}_{11}(K)$. We obtain
$\displaystyle n({\bf k})=\frac{1}{2}\left(1-\frac{\xi_{\bf k}^{-}}{E_{\bf
k}^{-}}\right),\ \ \ \ \text{for quarks},$ $\displaystyle\bar{n}({\bf
k})=\frac{1}{2}\left(1-\frac{\xi_{\bf k}^{+}}{E_{\bf k}^{+}}\right),\ \ \ \
\text{for antiquarks}.$ (225)
The numerical results for $n({\bf k})$ and $\bar{n}({\bf k})$ (for model
parameter set 1) are shown in Fig.14. Near the quantum phase transition point,
the quark momentum distribution $n({\bf k})$ is a very smooth function in the
whole momentum space. In the opposite limit, i.e., at large chemical
potentials, it approaches unity at $|{\bf k}|=0$ and decreases rapidly around
the effective “Fermi surface” at $|{\bf k}|\simeq|\mu|$. For the antiquarks,
we find that the momentum distribution $\bar{n}({\bf k})$ exhibits a
nonmonotonous behavior: it is suppressed at both low and high densities and is
visible only at moderate chemical potentials. However, even at very large
chemical potentials, e.g., $\mu_{\text{B}}=10m_{\pi}$, the momentum
distribution $n({\bf k})$ does not approach the standard BCS behavior, which
means that the dense matter is not a weakly coupled BCS superfluid for a wide
range of the baryon chemical potential. Actually, at $\mu_{\text{B}}\simeq
10m_{\pi}$, the ratio $|\Delta|/\mu$ is about $0.5$, which means that the
dense matter is still a strongly coupled superfluid.
Figure 13: The fermionic excitation gap $\Delta_{\text{ex}}$ (in units of
$M_{*}$) as a function of the baryon chemical potential (in units of
$m_{\pi}$) for different model parameter sets. The effective quark mass $M$
and the pairing gap $|\Delta|$ are also shown by dashed and dash-dotted lines,
respectively.
Figure 14: The momentum distributions for quarks (upper panel) and antiquarks
(lower panel) for various values of $\mu_{\text{B}}$. The momentum is scaled
by $|\mu|=|\mu_{\text{B}}/2-M|$.
The Goldstone mode also undergoes a characteristic change in the BEC-BCS
crossover. Near the quantum phase transition point, i.e., in the dilute limit,
the Goldstone mode recovers the Bogoliubov excitation of weakly interacting
Bose condensates. In the opposite limit, we expect the Goldstone mode
approaches the Anderson-Bogoliubov mode of a weakly coupled BCS superfluid,
which takes a dispersion $\omega({\bf q})=|{\bf q}|/\sqrt{3}$ up to the two-
particle continuum $\omega\simeq 2|\Delta|$. In fact, at large chemical
potentials, we can safely neglect the mixing between the sigma meson and
diquarks. The Goldstone boson dispersion is thus determined by the equation
$\det\left(\begin{array}[]{cc}{\bf M}_{11}(Q)&{\bf M}_{12}(Q)\\\ {\bf
M}_{21}(Q)&{\bf M}_{22}(Q)\end{array}\right)=0.$ (226)
Therefore, at very large chemical potentials where $|\Delta|/\mu$ becomes
small enough, the Goldstone mode recovers the Anderson-Bogoliubov mode of a
weakly coupled BCS superfluid.
Finally, we should emphasize that the existence of a smooth crossover from the
Bose condensate to the BCS superfluid depends on whether there exists a
deconfinement phase transition at finite $\mu_{\text{B}}$ LBECBCS ; decon and
where it takes place. Recent lattice calculation predicts a deconfinement
crossover which occurs at a baryon chemical potential larger than that of the
BEC-BCS crossover LBECBCS .
Figure 15: The mass spectra of mesons and diquarks (in units of $m_{\pi}$) as
functions of the baryon chemical potential (in units of $m_{\pi}$) for model
parameter set 1. For other model parameter sets, the mass of the heaviest mode
is changed but others are almost the same.
As in real QCD with two quark flavors, we expect the chiral symmetry is
restored and the spectra of sigma meson and pions become degenerate at high
density note2 . For the two-flavor case and with vanishing $m_{0}$, the
residue
SU${}_{\text{L}}(2)\otimes$SU${}_{\text{R}}(2)\otimes$U${}_{\text{B}}(1)$
symmetry group at $\mu_{\text{B}}\neq 0$ is spontaneously broken down to
Sp${}_{\text{L}}(2)\otimes$Sp${}_{\text{R}}(2)$ in the superfluid medium with
nonzero $\langle qq\rangle$, resulting in one Goldstone boson. For small
nonzero $m_{0}$, we expect the spectra of sigma meson and pions become
approximately degenerate when the in-medium chiral condensate
$\langle\bar{q}{q}\rangle$ becomes small enough.
In fact, according to the result
$\langle\bar{q}{q}\rangle_{n}/\langle\bar{q}{q}\rangle_{0}\simeq
1-n/(2f_{\pi}^{2}m_{\pi})$ at low density, we can roughly expect that the
chiral symmetry is approximately restored at $n\sim 2f_{\pi}^{2}m_{\pi}$. From
the chemical potential dependence of the chiral condensate
$\langle\bar{q}{q}\rangle$ shown in Fig.11, we find that it becomes smaller
and smaller as the density increases. As a result, we should have nearly
degenerate spectra for the sigma meson and pions. To show this we need the
explicit form of the matrix ${\bf M}(Q)$ and ${\bf N}(Q)$ given in Appendix A.
From ${\bf M}_{13},{\bf M}_{32}\propto M\Delta$ at high density with
$\langle\bar{q}{q}\rangle\rightarrow 0$, they can be safely neglected and the
sigma meson decouples from the diquarks. The propagator of the sigma meson is
then given by ${\bf M}_{33}^{-1}(Q)$. From the explicit form of the
polarization functions $\Pi_{\sigma}(Q)=\Pi_{33}(Q)$ and $\Pi_{\pi}(Q)$, we
can see that the inverse propagators of the sigma meson and pions differ from
each other in a term proportional to $M^{2}$. Thus at high density their
spectra are nearly degenerate, and their masses are given by the equation
$1-2G\Pi_{\pi}(\omega,{\bf 0})=0.$ (227)
Using the mean-field gap equation for $\Delta$, we find that the solution is
$\omega=\mu_{\text{B}}$, which means that the meson masses are equal to
$\mu_{\text{B}}$ at large chemical potentials. In Fig.15, we show the chemical
potential dependence of the meson and diquark masses determined at zero
momentum. We find that the chiral symmetry is approximately restored at
$\mu_{\text{B}}\simeq 3m_{\pi}$, corresponding to $n\simeq
3.5f_{\pi}^{2}m_{\pi}$, where the $\pi$ and ”anti-d” start to be degenerate.
In the normal phase with $\mu_{\text{B}}<m_{\pi}$, diquark, anti-diquark,
$\pi$ and $\sigma$ themselves are eigen modes of the collective excitation of
the system, but in the symmetry breaking phase with $\mu_{\text{B}}>m_{\pi}$,
except for $\pi$ which is still an eigen mode, diquark, anti-diquark and
$\sigma$ are no longer eigen modes ratti ; tomas . However, if we neglect the
mixing between $\sigma$ and anti-diquark in the symmetry breaking phase hao ,
$\sigma$ is still the eigen mode and it becomes degenerate with anti-diquark
and $\pi$ at high enough chemical potential. This is clearly shown in Fig.15
of Ref. ISOother03 .
Figure 16: The two-particle continua $\omega_{\bar{q}q}$ and $\omega_{qq}$ (in
units of $M_{*}$) as functions of the baryon chemical potential (in units of
$m_{\pi}$) for different model parameter sets. The degenerate mass of pions
and sigma meson is shown by dashed line.
Even though the deconfinement transition or crossover which corresponds to the
gauge field sector cannot be described in the NJL model, we can on the other
hand study the meson Mott transition associated with the chiral restoration
mott ; mott1 ; mott2 ; precursor ; precursor1 ; precursor2 ; precursor3 ;
note4 . The meson Mott transition is defined as the point where the meson
energy becomes larger than the two-particle continuum $\omega_{\bar{q}q}$ for
the decay process $\pi\rightarrow\bar{q}q$ at zero momentum, which means that
the mesons are no longer bound states. The two-particle continuum
$\omega_{\bar{q}q}$ is different at the BEC and the BCS sides. From the
explicit form of $\Pi_{\pi}(Q)$, we find
$\omega_{\bar{q}q}=\left\\{\begin{array}[]{r@{\quad,\quad}l}\sqrt{(M-\frac{\mu_{\text{B}}}{2})^{2}+|\Delta|^{2}}+\sqrt{(M+\frac{\mu_{\text{B}}}{2})^{2}+|\Delta|^{2}}&\mu_{\text{B}}<\mu_{0}\\\
|\Delta|+\sqrt{(M+\frac{\mu_{\text{B}}}{2})^{2}+|\Delta|^{2}}&\mu_{\text{B}}>\mu_{0}.\end{array}\right.$
(228)
Thus the pions and the sigma meson will undergo a Mott transition when their
masses become larger than the two-particle continuum $\omega_{\bar{q}q}$,
i.e., $\mu_{\text{B}}>\omega_{\bar{q}q}$. Using the mean-field results for
$\Delta$ and $M$, we can calculate the two-particle continuum
$\omega_{\bar{q}q}$ as a function of $\mu_{\text{B}}$, which is shown in
Fig.16. We find that the Mott transition does occur at a chemical potential
$\mu_{\text{B}}=\mu_{\text{M}1}$ which is sensitive to the value of $M_{*}$.
The values of $\mu_{\text{M}1}$ for the four model parameter sets are shown in
Table.4. For reasonable model parameter sets, the value of $\mu_{\text{M}1}$
is in the range $(7-10)m_{\pi}$. Above this chemical potential, the mesons are
no longer stable bound states and can decay into quark-antiquark pairs even at
zero momentum. We note that the Mott transition takes place well above the
chiral restoration, in contrast to the pure finite temperature case where the
mesons are dissociated once the chiral symmetry is restored mott ; mott1 ;
mott2 ; precursor ; precursor1 ; precursor2 ; precursor3 .
Set | 1 | 2 | 3 | 4
---|---|---|---|---
$\mu_{\text{M}1}$ | 7.22 | 7.76 | 8.63 | 9.62
$\mu_{\text{M}2}$ | 5.29 | 6.06 | 6.96 | 7.92
Table 4: The chemical potentials $\mu_{\text{M}1}$ and $\mu_{\text{M}2}$ (in
units of $m_{\pi}$) for different model parameter sets.
On the other hand, we find from the explicit forms of the meson propagators in
Appendix A that the decay process $\pi\rightarrow qq$ is also possible at
${\bf q}\neq 0$ (even though $|{\bf q}|$ is small) due to the presence of
superfluidity. Thus, we have another unusual Mott transition in the superfluid
phase. Notice that this process is not in contradiction to the baryon number
conservation law, since the U${}_{\text{B}}(1)$ baryon number symmetry is
spontaneously broken in the superfluid phase. Quantitatively, this transition
occurs when the meson mass becomes larger than the two-particle continuum
$\omega_{qq}$ for the decay process $\pi\rightarrow qq$ at ${\bf q}=0^{+}$. In
this case, we have
$\omega_{qq}=\left\\{\begin{array}[]{r@{\quad,\quad}l}2\sqrt{(M-\frac{\mu_{\text{B}}}{2})^{2}+|\Delta|^{2}}&\mu_{\text{B}}<\mu_{0}\\\
2|\Delta|&\mu_{\text{B}}>\mu_{0}.\end{array}\right.$ (229)
The two-particle continuum $\omega_{qq}$ is also shown in Fig.16. We find that
the unusual Mott transition does occur at another chemical potential
$\mu_{\text{B}}=\mu_{\text{M}2}$ which is also sensitive to the value of
$M_{*}$. The values of $\mu_{\text{M}2}$ for the four model parameter sets are
also shown in Table.4. For reasonable model parameter sets, this value is in
the range $(5-8)m_{\pi}$. This process can also occur in the 2SC phase of
quark matter in the $N_{c}=3$ case ebert . In the 2SC phase, the symmetry
breaking pattern is
SU${}_{\text{c}}(3)\otimes$U${}_{\text{B}}(1)\rightarrow$SU${}_{\text{c}}(2)\otimes\tilde{\text{U}}_{\text{B}}(1)$
where the generator of the residue baryon number symmetry
$\tilde{\text{U}}_{\text{B}}(1)$ is
$\tilde{\text{B}}={\text{B}}-2T_{8}/\sqrt{3}=\text{diag}(0,0,1)$ corresponding
to the unpaired blue quarks. Thus the baryon number symmetry for the paired
red and green quarks are broken and our results can be applied. To show this
explicitly, we write down the explicit form of the polarization function for
pions in the 2SC phase ebert
$\Pi_{\pi}^{2\text{SC}}(Q)=\Pi_{\pi}^{\text{2-color}}(Q)+\sum_{K}\text{Tr}[{\cal
G}_{0}(K)i\gamma_{5}{\cal G}_{0}(P)i\gamma_{5}],$ (230)
where ${\cal G}_{0}(K)$ is the propagator for the unpaired blue quarks. Here
$\Pi_{\pi}^{\text{2-color}}(Q)$ is given by (153) (the effective quarks mass
$M$ and the pairing gap $\Delta$ should be given by the $N_{c}=3$ case of
course) and corresponds to the contribution from the paired red and green
sectors. The second term is the contribution from the unpaired blue quarks.
Therefore, the unusual decay process is only available for the paired quarks.
### III.5 Beyond-Mean-Field Corrections
The previous investigations are restricted to the mean-field approximation,
even though the bosonic collective excitations are studied. In this part, we
will include the Gaussian fluctuations in the thermodynamic potential, and
thus really go beyond the mean field. The scheme of going beyond the mean
field is somewhat like those done in the study of finite temperature
thermodynamics of the NJL model zhuang01 ; zhuang02 ; however, in this paper
we will focus on the beyond-mean-field corrections at zero temperature, i.e.,
the pure quantum fluctuations. We will first derive the thermodynamic
potential beyond the mean field which is valid at arbitrary chemical potential
and temperature, and then briefly discuss the beyond-mean-field corrections
near the quantum phase transition. The numerical calculations are deferred for
future studies.
In Gaussian approximation, the partition function can be expressed as
$\displaystyle Z_{\text{NJL}}\simeq\exp{\left(-{\cal
S}_{\text{eff}}^{(0)}\right)}\int[d\sigma][d\mbox{\boldmath{$\pi$}}][d\phi^{\dagger}][d\phi]e^{-{\cal
S}_{\text{eff}}^{(2)}}$ (231)
Integrating out the Gaussian fluctuations, we can express the total
thermodynamic potential as
$\Omega(T,\mu_{\text{B}})=\Omega_{0}(T,\mu_{\text{B}})+\Omega_{\text{fl}}(T,\mu_{\text{B}}),$
(232)
where the contribution from the Gaussian fluctuations can be written as
$\displaystyle\Omega_{\text{fl}}=\frac{1}{2}\sum_{Q}\big{[}\ln\det{\bf
M}(Q)+\ln\det{\bf N}(Q)\big{]}.$ (233)
However, there is a problem with the above expression, since it is actually
ill-defined: the sum over the boson Matsubara frequency is divergent and we
need appropriate convergent factors to make it meaningful. In the simpler case
without superfluidity, the convergent factor is simply given by
$e^{i\nu_{m}0^{+}}$ zhuang01 ; zhuang02 . In our case, the situation is
somewhat different due to the introduction of the Nambu-Gor’kov spinors. Keep
in mind that in the equal time limit, there are additional factors
$e^{i\omega_{n}0^{+}}$ for ${\cal G}_{11}(K)$ and $e^{-i\omega_{n}0^{+}}$ for
${\cal G}_{22}(K)$. Therefore, to get the proper convergent factors for
$\Omega_{\text{fl}}$, we should keep these factors when we make the sum over
the fermion Matsubara frequency $\omega_{n}$ in evaluating the polarization
functions $\Pi_{\text{ij}}(Q)$ and $\Pi_{\pi}(Q)$.
The problem in the expression of $\Omega_{\text{fl}}$ is thus from the
opposite convergent factors for ${\bf M}_{11}$ and ${\bf M}_{22}$. From the
above arguments, we find that there is a factor $e^{i\nu_{m}0^{+}}$ for ${\bf
M}_{11}$ and $e^{-i\nu_{m}0^{+}}$ for ${\bf M}_{22}$. Keep in mind that the
Matsubara sum $\sum_{m}$ is converted to a standard contour integral
($i\nu_{m}\rightarrow z$). The convergence for $z\rightarrow+\infty$ is
automatically guaranteed by the Bose distribution function
$b(z)=1/(e^{z/T}-1)$, we thus should treat only the problem for
$z\rightarrow-\infty$. To this end, we write the first term of
$\Omega_{\text{fl}}$ as
$\displaystyle\sum_{Q}\ln\det{\bf M}(Q)$ $\displaystyle=$
$\displaystyle\sum_{Q}\bigg{[}\ln{\bf M}_{11}e^{i\nu_{m}0^{+}}+\ln{\bf
M}_{22}e^{-i\nu_{m}0^{+}}$ (234) $\displaystyle+\ln\left(\frac{\det{\bf
M}}{{\bf M}_{11}{\bf M}_{22}}\right)e^{i\nu_{m}0^{+}}\bigg{]}.$
Using the fact that ${\bf M}_{22}(Q)={\bf M}_{11}(-Q)$, we obtain
$\displaystyle\sum_{Q}\ln\det{\bf M}(Q)=\sum_{Q}\ln\left[\frac{{\bf
M}_{11}(Q)}{{\bf M}_{22}(Q)}\det{\bf M}(Q)\right]e^{i\nu_{m}0^{+}}.$ (235)
Therefore, the well-defined form of $\Omega_{\text{fl}}$ is given by the above
formula together with the other term $\sum_{Q}\ln\det{\bf N}(Q)$ associated
with a factor $e^{i\nu_{m}0^{+}}$.
The Matsubara sum can be written as the contour integral via the theorem
$\sum_{m}g(i\nu_{m})=\oint_{\text{C}}dz/(2\pi i)b(z)g(z)$, where C runs on
either side of the imaginary $z$ axis, enclosing it counterclockwise.
Distorting the contour to run above and below the real axis, we obtain
$\displaystyle\Omega_{\text{fl}}$ $\displaystyle=$ $\displaystyle\sum_{\bf
q}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}b(\omega)\big{[}\delta_{\text{M}}(\omega,{\bf
q})+\delta_{11}(\omega,{\bf q})$ (236) $\displaystyle-\delta_{22}(\omega,{\bf
q})+3\delta_{\pi}(\omega,{\bf q})\big{]},$
where the scattering phases are defined as
$\displaystyle\delta_{\text{M}}(\omega,{\bf q})=\text{Im}\ln\det{\bf
M}(\omega+i0^{+},{\bf q}),$ $\displaystyle\delta_{11}(\omega,{\bf
q})=\text{Im}\ln{\bf M}_{11}(\omega+i0^{+},{\bf q}),$
$\displaystyle\delta_{22}(\omega,{\bf q})=\text{Im}\ln{\bf
M}_{22}(\omega+i0^{+},{\bf q}),$ $\displaystyle\delta_{\pi}(\omega,{\bf
q})=\text{Im}\ln\left[(2G)^{-1}+\Pi_{\pi}(\omega+i0^{+},{\bf q})\right].$
(237)
Keep in mind the pressure of the vacuum should be zero, the physical
thermodynamic potential at finite temperature and chemical potential should be
defined as
$\Omega_{\text{phy}}(T,\mu_{\text{B}})=\Omega(T,\mu_{\text{B}})-\Omega(0,0).$
(238)
As we have shown in the mean-field theory, at $T=0$, the vacuum state is
restricted in the region $|\mu_{\text{B}}|<m_{\pi}$. In this region, all
thermodynamic quantities should keep zero, no matter how large the value of
$\mu_{\text{B}}$ is. While this should be an obvious physical conclusion, it
is important to check whether our beyond-mean-field theory satisfies this
condition.
Notice that the physical thermodynamic potential is defined as
$\Omega_{\text{phy}}(\mu_{\text{B}})=\Omega(\mu_{\text{B}})-\Omega(0)$, we
therefore should prove that the thermodynamic potential
$\Omega(\mu_{\text{B}})$ stays constant in the region
$|\mu_{\text{B}}|<m_{\pi}$. For the mean-field part $\Omega_{0}$, the proof is
quite easy. Because of the fact that $M_{*}>m_{\pi}/2$, the solution for $M$
is always given by $M=M_{*}$. Thus $\Omega_{0}$ keeps its value at
$\mu_{\text{B}}=0$ in the region $|\mu_{\text{B}}|<m_{\pi}$.
Now we turn to the complicated part $\Omega_{\text{fl}}$. From $\Delta=0$, all
the off-diagonal elements of ${\bf M}$ vanishes, $\Omega_{\text{fl}}$ is
reduced to
$\displaystyle\Omega_{\text{fl}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{Q}\ln\left[\frac{1}{2G}+\Pi_{\sigma}(Q)\right]e^{i\nu_{m}0^{+}}$
(239)
$\displaystyle+\frac{3}{2}\sum_{Q}\ln\left[\frac{1}{2G}+\Pi_{\pi}(Q)\right]e^{i\nu_{m}0^{+}}$
$\displaystyle+\sum_{Q}\ln\left[\frac{1}{4G}+\Pi_{\text{d}}(Q)\right]e^{i\nu_{m}0^{+}}$
with $\Pi_{\sigma}(Q)=\Pi_{33}(Q)$, and we should set $\Delta=0$ and $M=M_{*}$
in evaluating the polarization functions. First, we can easily show that the
contributions from the sigma meson and pions do not have explicit
$\mu_{\text{B}}$ dependence and thus keep the same values as those at
$\mu_{\text{B}}=0$. In fact, since the effective quark mass $M$ keeps its
vacuum value $M_{*}$ guaranteed by the mean-field part, all the
$\mu_{\text{B}}$ dependence in $\Pi_{\sigma,\pi}(Q)$ is included in the Fermi
distribution functions $f(E\pm\mu_{\text{B}}/2)$. From
$M_{*}>\mu_{\text{B}}/2$, they vanish automatically at $T=0$. In fact, from
the explicit expressions for $\Pi_{\sigma,\pi}(Q)$ in Appendix A, we can check
that there is no $\mu_{\text{B}}$ independence in $\Pi_{\sigma,\pi}(Q)$.
The diquark contribution, however, has an explicit $\mu_{\text{B}}$ dependence
through the combination $i\nu_{m}+\mu_{\text{B}}$ in the polarization function
$\Pi_{\text{d}}(Q)$. The diquark contribution (at $T=0$) can be written as
$\displaystyle\Omega_{\text{d}}$ $\displaystyle=$ $\displaystyle-\sum_{\bf
q}\int_{-\infty}^{0}\frac{d\omega}{\pi}\delta_{\text{d}}(\omega,{\bf q}),$
$\displaystyle\delta_{\text{d}}(\omega,{\bf q})$ $\displaystyle=$
$\displaystyle\text{Im}\ln\left[(4G)^{-1}+\Pi_{\text{d}}(\omega+i0^{+},{\bf
q})\right].$ (240)
Making a shift $\omega\rightarrow\omega-\mu_{\text{B}}$, and noticing that
fact $\Pi_{\text{d}}(\omega-\mu_{\text{B}},{\bf q})=\Pi_{\pi}(\omega,{\bf
q})/2$, we obtain
$\displaystyle\Omega_{\text{d}}=-\sum_{\bf
q}\int_{-\infty}^{-\mu_{\text{B}}}\frac{d\omega}{\pi}\delta_{\pi}(\omega,{\bf
q}).$ (241)
To show that the above quantity is $\mu_{\text{B}}$ independent, we separate
it into a pole part and a continuum part. There is a well-defined two-particle
continuum $E_{c}({\bf q})$ for pions at arbitrary momentum ${\bf q}$,
$\displaystyle E_{c}({\bf q})=\text{min}_{\bf k}\left(E_{\bf k}^{*}+E_{\bf
k+q}^{*}\right).$ (242)
The pion propagator has two symmetric poles $\pm\omega_{\pi}({\bf q})$ when
${\bf q}$ satisfies $\omega_{\pi}({\bf q})<E_{c}({\bf q})$. Thus in the region
$|\omega|<E_{c}({\bf q})$, the scattering phase $\delta_{\pi}$ can be
analytically evaluated as
$\displaystyle\delta_{\pi}(\omega,{\bf
q})=\pi\left[\Theta\left(-\omega-\omega_{\pi}({\bf
q})\right)-\Theta\left(\omega-\omega_{\pi}({\bf q})\right)\right].$ (243)
From $E_{c}({\bf q})>\omega_{\pi}({\bf q})>m_{\pi}>\mu_{\text{B}}$, the
thermodynamic potential $\Omega_{\text{d}}$ can be separated as
$\displaystyle\Omega_{\text{d}}=\sum_{\bf q}\left[\omega_{\pi}({\bf
q})-E_{c}({\bf q})\right]-\sum_{\bf q}\int_{-\infty}^{-E_{c}({\bf
q})}\frac{d\omega}{\pi}\delta_{\pi}(\omega,{\bf q}),$ (244)
which is indeed $\mu_{\text{B}}$ independent. Notice that in the first term
the integral over ${\bf q}$ is restricted in the region $|{\bf q}|<q_{c}$
where $q_{c}$ is defined as $\omega_{\pi}(q_{c})=E_{c}(q_{c})$.
In conclusion, we have shown that the thermodynamic potential $\Omega$ in the
Gaussian approximation stays constant in the vacuum state, i.e., at
$|\mu_{\text{B}}|<m_{\pi}$ and at $T=0$. All other thermodynamic quantities
such as the baryon number density keep zero in the vacuum. The subtraction
term $\Omega(0,0)$ in the Gaussian approximation can be expressed as
$\displaystyle\Omega(0,0)$ $\displaystyle=$
$\displaystyle\Omega_{\text{vac}}(M_{*})+\frac{5}{2}\sum_{\bf
q}\left[\omega_{\pi}({\bf q})-E_{c}({\bf q})\right]$ (245)
$\displaystyle-\sum_{\bf q}\int_{-\infty}^{-E_{c}({\bf
q})}\frac{d\omega}{2\pi}\left[\delta_{\sigma}(\omega,{\bf
q})+5\delta_{\pi}(\omega,{\bf q})\right].$
Now we consider the beyond-mean-field corrections near the quantum phase
transition point $\mu_{\text{B}}=m_{\pi}$. Notice that the effective quark
mass $M$ and the diquark condensate $\Delta$ are determined at the mean-field
level, and the beyond-mean-field corrections are possible only through the
equations of state.
Formally, the Gaussian contribution to the thermodynamic potential
$\Omega_{\text{fl}}$ is a function of $\mu_{\text{B}},M$ and $y=|\Delta|^{2}$,
i.e., $\Omega_{\text{fl}}=\Omega_{\text{fl}}(\mu_{\text{B}},y,M)$. In the
superfluid phase, the total baryon density including the Gaussian contribution
can be evaluated as
$\displaystyle
n(\mu_{\text{B}})=n_{0}(\mu_{\text{B}})+n_{\text{fl}}(\mu_{\text{B}}),$ (246)
where the mean-field part is simply given by
$n_{0}(\mu_{\text{B}})=-\partial\Omega_{0}/\partial\mu_{\text{B}}$ and the
Gaussian contribution can be expressed as
$\displaystyle
n_{\text{fl}}(\mu_{\text{B}})=-\frac{\partial\Omega_{\text{fl}}}{\partial\mu_{\text{B}}}-\frac{\partial\Omega_{\text{fl}}}{\partial
y}\frac{dy}{d\mu_{\text{B}}}-\frac{\partial\Omega_{\text{fl}}}{\partial
M}\frac{dM}{d\mu_{\text{B}}}.$ (247)
The physical values of $M$ and $|\Delta|^{2}$ should be determined by their
mean-field gap equations. In fact, from the gap equations
$\partial\Omega_{0}/\partial M=0$ and $\partial\Omega_{0}/\partial y=0$, we
obtain
$\displaystyle\frac{\partial^{2}\Omega_{0}}{\partial\mu_{\text{B}}\partial
M}+\frac{\partial^{2}\Omega_{0}}{\partial y\partial
M}\frac{dy}{d\mu_{\text{B}}}+\frac{\partial^{2}\Omega_{0}}{\partial
M^{2}}\frac{dM}{d\mu_{\text{B}}}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{\partial^{2}\Omega_{0}}{\partial\mu_{\text{B}}\partial
y}+\frac{\partial^{2}\Omega_{0}}{\partial
y^{2}}\frac{dy}{d\mu_{\text{B}}}+\frac{\partial^{2}\Omega_{0}}{\partial
M\partial y}\frac{dM}{d\mu_{\text{B}}}$ $\displaystyle=$ $\displaystyle 0.$
(248)
Thus, we can obtain the derivatives $dM/d\mu_{\text{B}}$ and
$dy/d\mu_{\text{B}}$ analytically. Finally, $n_{\text{fl}}(\mu_{\text{B}})$ is
a continuous function of $\mu_{\text{B}}$ guaranteed by the properties of
second order phase transition, and we have $n_{\text{fl}}(m_{\pi})=0$.
Next we focus on the beyond-mean-field corrections near the quantum phase
transition. Since the diquark condensate $\Delta$ is vanishingly small, we can
expand the Gaussian part $\Omega_{\text{fl}}$ in powers of $|\Delta|^{2}$.
Notice that $\mu_{\text{B}}$ and $M$ can be evaluated as functions of
$|\Delta|^{2}$ from the Ginzburg-Landau potential and mean-field gap
equations. Thus to order $O(|\Delta|^{2})$, the expansion takes the form
$\displaystyle\Omega_{\text{fl}}\simeq\eta|\Delta|^{2},$ (249)
where the expansion coefficient $\eta$ is defined as
$\displaystyle\eta=\left(\frac{\partial\Omega_{\text{fl}}}{\partial
y}+\frac{\partial\Omega_{\text{fl}}}{\partial\mu_{\text{B}}}\frac{d\mu_{\text{B}}}{dy}+\frac{\partial\Omega_{\text{fl}}}{\partial
M}\frac{dM}{dy}\right)\Bigg{|}_{\mu_{\text{B}}=m_{\pi},y=0,M=M_{*}}.$ (250)
Using the definition of $n_{\text{fl}}$, we find that $\eta$ can be related to
$n_{\text{fl}}$ by
$\displaystyle\eta=n_{\text{fl}}(m_{\pi})\frac{d\mu_{\text{B}}}{dy}\bigg{|}_{y=0}.$
(251)
Therefore, the coefficient $\eta$ vanishes, and the leading order of the
expansion should be $O(|\Delta|^{4})$.
As shown above, to leading order, the expansion of $\Omega_{\text{fl}}$ can be
formally expressed as
$\displaystyle\Omega_{\text{fl}}\simeq-\frac{\zeta}{2}\beta|\Delta|^{4}.$
(252)
The explicit form of $\zeta$ is quite complicated and we do not show it here.
Notice that the factor $\zeta$ is in fact $\mu_{\text{B}}$ independent, thus
the total baryon density to leading order is
$\displaystyle
n=n_{0}+\zeta\beta|\Delta|^{2}\frac{d|\Delta|^{2}}{d\mu_{\text{B}}}\bigg{|}_{\mu_{\text{B}}=m_{\pi}}.$
(253)
Near the quantum phase transition point, the mean-field contribution is
$n_{0}=|\psiup_{0}|^{2}=2m_{\pi}{\cal J}|\Delta|^{2}$ from the Gross-
Pitaevskii free energy. The last term can be evaluated using the analytical
result
$\displaystyle|\psiup_{0}|^{2}=\frac{\mu_{\text{d}}}{g_{0}}\Longrightarrow|\Delta|^{2}=\frac{2m_{\pi}{\cal
J}}{\beta}\mu_{\text{d}},$ (254)
which is in fact the solution of the mean-field gap equations. Therefore, to
leading order, the total baryon density reads
$\displaystyle n=(1+\zeta)2m_{\pi}{\cal J}|\Delta|^{2}.$ (255)
On the other hand, the total pressure $P$ can be expressed as
$\displaystyle P=(1+\zeta)\frac{\beta}{2}|\Delta|^{4}.$ (256)
Thus we find that the leading order quantum corrections are totally included
in the numerical factor $\zeta$. Setting $\zeta=0$, we recover the mean-field
results obtained previously.
Including the quantum fluctuations, the equations of state shown in (III.3)
are modified to be
$\displaystyle P(n)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\zeta}\frac{2\pi a_{\text{dd}}}{m_{\pi}}n^{2},$
$\displaystyle\mu_{\text{B}}(n)$ $\displaystyle=$ $\displaystyle
m_{\pi}+\frac{1}{1+\zeta}\frac{4\pi a_{\text{dd}}}{m_{\pi}}n.$ (257)
This means that, to leading order, the effect of quantum fluctuations is
giving a correction to the diquark-diquark scattering length. The renormalized
scattering length is
$\displaystyle a_{\text{dd}}^{\prime}=\frac{a_{\text{dd}}}{1+\zeta}.$ (258)
Generally, we have $\zeta>0$ and the renormalized scattering length is smaller
than the mean-field result.
An exact calculation of the numerical factor $\zeta$ can be performed. In this
work we will give an analytical estimation of $\zeta$ based on the fact that
the quantum fluctuations are dominated by the gapless Goldstone mode. To this
end, we approximate the Gaussian contribution $\Omega_{\text{fl}}$ as
$\displaystyle\Omega_{\text{fl}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\sum_{Q}\ln\bigg{[}{\cal D}_{\text{d}}^{-1}(Q){\cal
D}_{\text{d}}^{-1}(-Q)+3\beta^{2}|\Delta|^{4}$ (259)
$\displaystyle+2\beta|\Delta|^{2}\left({\cal D}_{\text{d}}^{-1}(Q)+{\cal
D}_{\text{d}}^{-1}(-Q)\right)\bigg{]},$
where ${\cal D}_{\text{d}}^{-1}(Q)$ is given by (178) and can be approximated
by (180). Subtracting the value of $\Omega_{\text{fl}}$ at
$\mu_{\text{B}}=m_{\pi}$ with $\Delta=0$, and using the result
$\mu_{\text{B}}=m_{\pi}+g_{0}|\psiup_{0}|^{2}$ from the Gross-Pitaevskii
equation, we find that $\zeta$ can be evaluated as
$\displaystyle\zeta=\frac{\beta}{{\cal
J}^{2}}\left(I_{1}+I_{2}\right)\simeq\frac{m_{\pi}^{2}}{f_{\pi}^{2}}\left(I_{1}+I_{2}\right),$
(260)
where the numerical factors $I_{1}$ and $I_{2}$ are given by
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{m}\sum_{\bf X}\frac{Z_{m}^{2}+{\bf
X}^{2}}{(Z_{m}^{2}-{\bf X}^{2})^{2}-4Z_{m}^{2}},$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle 4\sum_{m}\sum_{\bf X}\frac{(3Z_{m}^{2}-{\bf
X}^{2})^{2}}{\left[(Z_{m}^{2}-{\bf X}^{2})^{2}-4Z_{m}^{2}\right]^{2}}.$ (261)
Here the dimensionless notations $Z_{m}$ and ${\bf X}$ are defined as
$Z_{m}=i\nu_{m}/m_{\pi}$ and ${\bf X}={\bf q}/m_{\pi}$ respectively. Notice
that the integral over ${\bf X}$ is divergent and hence such an estimation has
no prediction power due to the fact that the NJL model is nonrenormalizable.
However, regardless of the numerical factor $I_{1}+I_{2}$, we find
$\zeta\propto m_{\pi}^{2}/f_{\pi}^{2}$. Thus, the correction should be small
for the case $m_{\pi}\ll 2M_{*}$.
While the effect of the Gaussian fluctuations at zero temperature is to give a
small correction to the diquark-diquark scattering length and the equations of
state, it can be significant at finite temperature. In fact, as the
temperature approaches the critical value of superfluidity, the Gaussian
fluctuations should dominate. In this part, we will show that to get a correct
critical temperature in terms of the baryon density $n$, we must go beyond the
mean field. The situation is analogous to the Nozieres–Schmitt-Rink treatment
of molecular condensation in strongly interacting Fermi gases BCSBEC1 ;
BCSBEC2 .
The transition temperature $T_{c}$ is determined by the Thouless criterion
${\cal D}_{\text{d}}^{-1}(0,{\bf 0})=0$ which can be shown to be consistent
with the saddle point condition $\delta{\cal
S}_{\text{eff}}/\delta\phi|_{\phi=0}=0$. Its explicit form is a BCS-type gap
equation
$\displaystyle\frac{1}{4G}=N_{c}N_{f}\sum_{e=\pm}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\frac{1-2f(\xi_{\bf k}^{e})}{2\xi_{\bf k}^{e}}.$ (262)
Meanwhile, the effective quark mass $M$ satisfies the mean-field gap equation
$\displaystyle\frac{M-m_{0}}{2GM}=N_{c}N_{f}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}{1-f(\xi_{\bf k}^{-})-f(\xi_{\bf k}^{+})\over E_{\bf k}}.$
(263)
To obtain the transition temperature as a function of $n$, we need the so-
called number equation given by $n=-\partial\Omega/\partial\mu_{\text{B}}$,
which includes both the mean-field contribution
$n_{0}(\mu_{\text{B}},T)=2N_{f}\sum_{\bf k}\left[f(\xi_{\bf k}^{-})-f(\xi_{\bf
k}^{+})\right]$ and the Gaussian contribution
$n_{\text{fl}}(\mu_{\text{B}},T)=-\partial\Omega_{\text{fl}}/\partial\mu_{\text{B}}$.
At the transition temperature with $\Delta=0$, $\Omega_{\text{fl}}$ can be
expressed as
$\displaystyle\Omega_{\text{fl}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}b(\omega)$ (264)
$\displaystyle\times\ [2\delta_{\text{d}}(\omega,{\bf
q})+\delta_{\sigma}(\omega,{\bf q})+3\delta_{\pi}(\omega,{\bf q})],$
where the scattering phases are defined as $\delta_{\text{d}}(\omega,{\bf
q})=\text{Im}\ln[1/(4G)+\Pi_{\text{d}}(\omega+i0^{+},{\bf q})]$ for the
diquarks, $\delta_{\sigma}(\omega,{\bf
q})=\text{Im}\ln[1/(2G)+\Pi_{\sigma}(\omega+i0^{+},{\bf q})]$ for the sigma
meson and $\delta_{\pi}(\omega,{\bf
q})=\text{Im}\ln[1/(2G)+\Pi_{\pi}(\omega+i0^{+},{\bf q})]$ for the pions.
Obviously, the polarization functions should take their forms at finite
temperature in the normal phase.
The transition temperature $T_{c}$ at arbitrary baryon number density $n$ can
be determined numerically via solving simultaneously the gap and number
equations. However, in the dilute limit $n\rightarrow 0$ which we are
interested in this section, analytical result can be achieved. Keep in mind
that $T_{c}\rightarrow 0$ when $n\rightarrow 0$, we find that the Fermi
distribution functions $f(\xi_{\bf k}^{\pm})$ vanish exponentially (from
$M_{*}-m_{\pi}/2\gg T_{c}$) and we obtain $\mu_{\text{B}}=m_{\pi}$ and
$M=M_{*}$ from the gap Eqs. (262) and (263), respectively. Meanwhile the mean-
field contribution of the density $n_{0}$ can be neglected and the total
density $n$ is thus dominated by the Gaussian part $n_{\text{fl}}$. When
$T\rightarrow 0$ we can show that $\Pi_{\sigma}(\omega,{\bf q})$ and
$\Pi_{\pi}(\omega,{\bf q})$ are independent of $\mu_{\text{B}}$, and the
number equation is reduced to
$\displaystyle n=-\sum_{\bf
q}\int_{-\infty}^{\infty}\frac{d\omega}{\pi}b(\omega)\frac{\partial\delta_{\text{d}}(\omega,{\bf
q})}{\partial\mu_{\text{B}}}.$ (265)
From $T_{c}\rightarrow 0$, the inverse diquark propagator can be reduced to
${\cal D}_{\text{d}}^{-1}(\omega,{\bf q})$ in (178). Thus the scattering phase
$\delta_{\text{d}}$ can be well approximated by $\delta_{\text{d}}(\omega,{\bf
q})=\pi[\Theta(\mu_{\text{B}}-\epsilon_{\bf
q}-\omega)-\Theta(\omega-\mu_{\text{B}}-\epsilon_{\bf q})]$ with
$\epsilon_{\bf q}=\sqrt{{\bf q}^{2}+m_{\pi}^{2}}$. Therefore, the number
equation can be further reduced to the well-known equation for ideal Bose-
Einstein condensation,
$\displaystyle n=\sum_{\bf q}\left[b(\epsilon_{\bf
q}-\mu_{\text{B}})-b(\epsilon_{\bf
q}+\mu_{\text{B}})\right]\bigg{|}_{\mu_{\text{B}}=m_{\pi}}.$ (266)
Since the above equation is valid only in the low density limit $n\rightarrow
0$, the critical temperature is thus given by the nonrelativistic result
$\displaystyle
T_{c}=\frac{2\pi}{m_{\pi}}\left[\frac{n}{\xi(3/2)}\right]^{2/3}.$ (267)
At finite density but $na_{\text{dd}}^{3}\ll 1$, there exists a correction to
$T_{c}$ which is proportional to $n^{1/3}a_{\text{dd}}$ Bose01 . Such a
correction is hard to handle analytically in our model since we should
consider simultaneously the corrections to $M$ and $\mu_{\text{B}}$, as well
as the contributions from the sigma meson and pions.
### III.6 BCS-BEC crossover in Pion superfluid
The other situation, which can be simulated by lattice QCD, is quark matter at
finite isospin density. The physical motivation to study QCD at finite isospin
density and the corresponding pion superfluid is related to the investigation
of compact stars, isospin asymmetric nuclear matter and heavy ion collisions
at intermediate energies. In early studies on dense nuclear matter and compact
stars, it has been suggested that charged pions are condensed at sufficiently
high density PiC ; PiC1 ; PiC2 ; PiC3 ; PiC4 ; he36 ; he361 . The QCD phase
structure at finite isospin chemical potential is recently investigated in
many low energy effective models, such as chiral perturbation theory, linear
sigma model, NJL model, random matrix method, and ladder QCD ISO ; ISOother01
; ISOother011 ; ISOother012 ; ISOother013 ; ISOother014 ; ISOother015 ;
ISOother016 ; ISOother017 ; ISOother018 ; ISOother019 ; ISOother0110 ;
ISOother0111 ; ISOother0112 ; ISOother0113 ; ISOother0114 ; ISOother0115 ;
boser ; ISOother02 ; ISOother021 ; ISOother03 ; Liso ; Liso1 ; Liso2 ; Liso3 ;
ran212 ; ran2121 ; lad23 . In this subsection, we review the pion superfluid
and the corresponding BCS-BEC crossover in the frame of two-flavor NJL model.
Since the isospin chemical potential which triggers the pion condensation is
large, $\mu_{I}\geq m_{\pi}$, we neglect the diquark condensation which is
favored at large baryon chemical potential and small isospin chemical
potential.
The Lagrangian density of the two-flavor NJL model at quark level is defined
as NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3
${\cal
L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m_{0}+\gamma_{0}\mu\right)\psi+G\Big{[}\left(\bar{\psi}\psi\right)^{2}+\left(\bar{\psi}i\gamma_{5}\tau_{i}\psi\right)^{2}\Big{]}$
(268)
with scalar and pseudoscalar interactions corresponding to $\sigma$ and $\pi$
excitations, where $m_{0}$ is the current quark mass, $G$ is the four-quark
coupling constant with dimension GeV-2, $\tau_{i}\ (i=1,2,3)$ are the Pauli
matrices in flavor space, and
$\mu=diag\left(\mu_{u},\mu_{d}\right)=diag\left(\mu_{B}/3+\mu_{I}/2,\mu_{B}/3-\mu_{I}/2\right)$
is the quark chemical potential matrix with $\mu_{u}$ and $\mu_{d}$ being the
$u$\- and $d$-quark chemical potentials and $\mu_{B}$ and $\mu_{I}$ the baryon
and isospin chemical potentials.
At zero isospin chemical potential, the Lagrangian density has the symmetry of
$U_{B}(1)\bigotimes SU_{L}(2)\bigotimes SU_{R}(2)$ corresponding to baryon
number symmetry, isospin symmetry and chiral symmetry. At finite isospin
chemical potential, the symmetries $SU_{L}(2)\bigotimes SU_{R}(2)$ are firstly
explicitly broken down to $U_{L}(1)\bigotimes U_{R}(1)$, and then the nonzero
pion condensate leads to the spontaneous breaking of $U_{I=L+R}(1)$, with
pions as the corresponding Goldstone modes. At $\mu_{B}=0$, the Fermi surface
of $u(d)$ and anti-$d(u)$ quarks coincide and hence the condensate of $u$ and
anti-$d$ is favored at $\mu_{I}>0$ and the condensate of $d$ and anti-$u$
quarks is favored at $\mu_{I}<0$. Finite $\mu_{B}$ provides a mismatch between
the two Fermi surfaces and will reduce the pion condensation.
Introducing the chiral and pion condensates
$\sigma=\langle\bar{\psi}\psi\rangle,\ \ \ \ \
\pi=\langle\bar{\psi}i\gamma_{5}\tau_{1}\psi\rangle$ (269)
and taking them to be real, the quark propagator ${\cal S}$ in mean field
approximation can be expressed as a matrix in the flavor space
${\cal
S}^{-1}(k)=\left(\begin{array}[]{cc}\gamma^{\mu}k_{\mu}+\mu_{u}\gamma_{0}-M_{q}&2iG\pi\gamma_{5}\\\
2iG\pi\gamma_{5}&\gamma^{\mu}k_{\mu}+\mu_{d}\gamma_{0}-M_{q}\end{array}\right)$
(270)
with the effective quark mass $M_{q}=m_{0}-2G\sigma$ generated by the chiral
symmetry breaking. By diagonalizing the propagator, the thermodynamic
potential $\Omega(T,\mu_{B},\mu_{I},M_{q},\pi)$ can be simply expressed as a
condensation part plus a summation part of four quasiparticle contributions
ISOother02 ; ISOother021 ; ISOother03 . The gap equations to determine the
condensates $\sigma$ (or effective quark mass $M_{q}$) and $\pi$ can be
obtained by the minimum of the thermodynamic potential,
${\partial\Omega\over\partial M_{q}}=0,\ \ \
{\partial\Omega\over\partial\pi}=0.$ (271)
In the NJL model, the meson modes are regarded as quantum fluctuations above
the mean field. The two quark scattering via meson exchange can be effectively
expressed at quark level in terms of quark bubble summation in Random Phase
Approximation (RPA) NJLreview ; NJLreview1 ; NJLreview2 ; NJLreview3 . The
quark bubbles are defined as
$\Pi_{mn}(q)=i\int{d^{4}k\over(2\pi)^{4}}Tr\left(\Gamma_{m}^{*}{\cal
S}(k+q)\Gamma_{n}{\cal S}(k)\right)$ (272)
with indices $m,n=\sigma,\pi_{+},\pi_{-},\pi_{0}$, where the trace
$Tr=Tr_{C}Tr_{F}Tr_{D}$ is taken in color, flavor and Dirac spaces, the four
momentum integration is defined as $\int d^{4}k/(2\pi)^{4}=iT\sum_{j}\int
d^{3}{\bf k}/(2\pi)^{3}$ with fermion frequency $k_{0}=i\omega_{j}=i(2j+1)\pi
T\ (j=0,\pm 1,\pm 2,\cdots)$ at finite temperature $T$, and the meson vertices
are from the Lagrangian density (268),
$\Gamma_{m}=\left\\{\begin{array}[]{ll}1&m=\sigma\\\
i\gamma_{5}\tau_{+}&m=\pi_{+}\\\ i\gamma_{5}\tau_{-}&m=\pi_{-}\\\
i\gamma_{5}\tau_{3}&m=\pi_{0}\ ,\end{array}\right.\ \
\Gamma_{m}^{*}=\left\\{\begin{array}[]{ll}1&m=\sigma\\\
i\gamma_{5}\tau_{-}&m=\pi_{+}\\\ i\gamma_{5}\tau_{+}&m=\pi_{-}\\\
i\gamma_{5}\tau_{3}&m=\pi_{0}\ .\\\ \end{array}\right.$ (273)
Since the quark propagator ${\cal S}$ contains off-diagonal elements, we must
consider all possible channels in the bubble summation in RPA. Using matrix
notation for the meson polarization function $\Pi(q)$ in the $4\times 4$ meson
space, the meson propagator can be expressed as hao
${\cal
D}(q)={2G\over{1-2G\Pi(q)}}={2G\over\text{det}\left[1-2G\Pi(q)\right]}{\cal
M}(q).$ (274)
Since the isospin symmetry is spontaneously broken in the pion superfluid, the
original meson modes $\sigma,\pi_{+},\pi_{-},\pi_{0}$ with definite isospin
quantum number are no longer the eigen modes of the Hamiltonian of the system,
the new eigen modes
$\overline{\sigma},\overline{\pi}_{+},\overline{\pi}_{-},\overline{\pi}_{0}$
are linear combinations of the old ones, their masses
$M_{i}(i=\overline{\sigma},\overline{\pi}_{+},\overline{\pi}_{-},\overline{\pi}_{0})$
are determined by poles of the meson propagator at $q_{0}=M_{i}$ and ${\bf
q=0}$,
$\text{det}\left[1-2G\Pi(M_{i},{\bf 0})\right]=0,$ (275)
and their coupling constants are defined as the residues of the propagator at
the poles,
$g^{2}_{iq\overline{q}}={2G\sum_{m}{\cal M}_{mm}(M_{i},{\bf 0})\over
d\text{det}\left[1-2G\Pi(q_{0},{\bf
0})\right]/dq_{0}^{2}{\big{|}}_{q_{0}=M_{i}}}.$ (276)
Since the NJL model is non-renormalizable, we can employ a hard three momentum
cutoff $\Lambda$ to regularize the gap equations for quarks and pole equations
for mesons. In the following numerical calculations, we take the current quark
mass $m_{0}=5$ MeV, the coupling constant $G=4.93$ GeV-2 and the cutoff
$\Lambda=653$ MeVzhuang02 . This group of parameters correspond to the pion
mass $m_{\pi}=134$ MeV, the pion decay constant $f_{\pi}=93$ MeV and the
effective quark mass $M_{q}=310$ MeV in the vacuum.
Numerically solving the minimum of thermodynamic potential, the order
parameters look like the case in Fig.11, with replacing $\langle qq\rangle$ by
pion condensate and $\mu_{B}$ by $\mu_{I}$. The conclusion, which pion
superfluid phase transition at zero $\mu_{B}$ occurs at finite isospin density
$\mu_{I}=m_{\pi}$, can be clearly seen by comparing the gap equation and the
polarization function $\Pi_{\pi_{+}\pi_{+}}$. Namely, the phase transition
line from normal state to pion superfluid on $T-\mu_{I}$ plane at $\mu_{B}=0$
is determined by condition that $1-2G\Pi_{\pi_{+}\pi_{+}}(q_{0}=0,{\bf 0})$,
and the mass of $\pi_{+}$ is always zero at phase boundary.
Figure 17: (upper) Meson spectra and effective quark mass at $T=\mu_{B}=0$ as
a function of isospin chemical potential $\mu_{I}$. In normal phase, the meson
eigenmode is $\sigma,\pi_{0},\pi_{+},\pi_{-}$, and in pion superfluid state,
they are denoted by
$\overline{\sigma},\pi_{0},\overline{\pi}_{+},\overline{\pi}_{-}$. ISOother03
(lower) The coupling constants for $\sigma,\pi_{0},\pi_{+},\pi_{-}$ in the
normal phase and
$\overline{\sigma},\pi_{0},\overline{\pi}_{+},\overline{\pi}_{-}$ in the pion
superfluid phase as functions of $\mu_{I}$ at $T=\mu_{B}=0$. hao
The meson mass and the coupling constant $g_{iq\overline{q}}$ at zero
temperature and finite isospin chemical potential are shown in Fig.17. Note
that the meson mass spectrum is similar to what shown in Fig.15, the ”anti-d”
and ”d” there correspond to the $\pi_{+}$ and $\pi_{-}$ here. The condition
for a meson to decay into a $q$ and a $\overline{q}$ is that its mass lies
above the $q-\overline{q}$ threshold. From the pole equation (275), the
heaviest mode in the pion superfluid is $\overline{\sigma}$ and its mass is
beyond the threshold value. As a result, there exists no $\overline{\sigma}$
meson in the pion superfluid, and the coupling constant
$g_{\overline{\sigma}q\overline{q}}$ drops down to zero at the critical point
$\mu^{c}_{I}$ and keeps zero at $\mu_{I}>\mu^{c}_{I}$ hao .
As discussed in the previous section, there are some characteristic quantities
to describe the BCS-BEC crossover in superfluid or superconductor pion2 ;
pion21 ; pion22 ; pion1 ; pion11 , which are difficult to be experimentally
measured but can be used to confirm the BCS-BEC crossover picture in pion
superfluid. Here we calculate the scaled binding energy $\epsilon/\mu_{I}$ as
a function of $\mu_{I}$ in pion superfluid at $T=0$ and $T=100$ MeV, shown in
Fig.18. The binding energy of $\overline{\pi}_{+}$ is defined as the the mass
difference between $\overline{\pi}_{+}$ and the two quarks,
$\epsilon=M_{\overline{\pi}_{+}}-M_{u}-M_{\overline{d}}$. With decreasing
isospin chemical potential, the binding energy increases, indicating a BCS-BEC
crossover in the pion superfluid. When the medium becomes hot, the condensate
melts and the pairs are gradually dissociated.
Figure 18: The scaled $\overline{\pi}_{+}$ binding energy $\epsilon/\mu_{I}$
as a function of isospin chemical potential $\mu_{I}$ in the pion superfluid.
The solid and dashed lines are for $T=0$ and $100$ MeV, and the baryon
chemical potential keeps zero $\mu_{B}=0$.
In pion superfluid, the pairs themselves, namely the pion mesons, are
observable objects. We propose to measure the $\pi-\pi$ scattering to probe
the properties of the pion condensate and in turn the BCS-BEC crossover. On
one hand, since pions are Goldstone modes corresponding to the chiral symmetry
spontaneous breaking, the $\pi-\pi$ scattering provides a direct way to link
chiral theories and experimental data and has been widely studied in many
chiral models chiralpi1 ; chiralpi11 ; schulze ; quack ; huang . Note that
pions are also the Goldstone modes of the isospin symmetry spontaneous
breaking, the $\pi-\pi$ scattering should be a sensitive signature of the pion
superfluid phase transition. On the other hand, the $\pi-\pi$ scattering
behaves different according to the BCS-BEC crossover picture. In the BCS quark
superfluid, the large and overlapped pairs lead to large pair-pair cross
section, but the small and individual pairs in the BEC superfluid interact
weakly.
Figure 19: The lowest order diagrams for $\pi-\pi$ scattering in the pion
superfluid. The solid and dashed lines are respectively quarks and mesons
(pions or $\sigma$), and the dots denote meson-quark vertices.
We now study $\pi-\pi$ scattering at finite isospin chemical potential. To the
lowest order in $1/N_{c}$ expansion, where $N_{c}$ is the number of colors,
the invariant amplitude ${\cal T}$ is calculated from the diagrams shown in
Fig.19 for the $s$ channel. Note that the $s$-wave $\pi-\pi$ scattering
calculated schulze ; quack ; huang in NJL model of $1/N_{c}$ order is
consistent with the Weinberg limit pipi and the experimental data pocanic in
vacuum. Different from the calculation in normal state schulze ; quack ; huang
where both the box and $\sigma$-exchange diagrams contribute, the
$\sigma$-exchange diagrams vanish in the pion superfluid due to the
disappearance of the $\overline{\sigma}$ meson. This greatly simplifies the
calculation in the pion superfluid.
For the calculation in normal matter at $\mu_{I}=0$, people are interested in
the $\pi$-$\pi$ scattering amplitude with definite isospin, ${\cal
T}_{I=0,1,2}$, which can be measured in experiments due to isospin symmetry.
However, the nonzero isospin chemical potential breaks down the isospin
symmetry and makes the scattering amplitude ${\cal T}_{I=0,1,2}$ not well
defined. In fact, the new meson modes in the pion superfluid do not carry
definite isospin quantum numbers. Unlike the chiral dynamics in normal matter,
where the three degenerated pions are all the Goldstone modes corresponding to
the chiral symmetry spontaneous breaking, the pion mass splitting at finite
$\mu_{I}$ results in only one Goldstone mode $\overline{\pi}_{+}$ in the pion
superfluid.
The scattering amplitude for any channel of the box diagrams can be expressed
as
$i{\cal
T}_{s,t,u}(q)=-2g_{\overline{\pi}q\overline{q}}^{4}\int{d^{4}k\over(2\pi)^{4}}Tr\prod_{l=1}^{4}\left[\gamma_{5}\tau{\cal
S}_{l}\right]$ (277)
with the quark propagators ${\cal S}_{1}={\cal S}_{3}={\cal S}(k)$, ${\cal
S}_{2}={\cal S}(k+q)$, and ${\cal S}_{4}={\cal S}(k-q)$ for the $s$ and $t$
channels and ${\cal S}_{1}={\cal S}_{3}={\cal S}(k+q)$ and ${\cal S}_{2}={\cal
S}_{4}={\cal S}(k)$ for the $u$ channel. To simplify the numerical
calculation, we consider in the following the limit of the scattering at
threshold $\sqrt{s}=2M_{\overline{\pi}}$ and $t=u=0$, where $s,t$ and $u$ are
the Mandelstam variables. In this limit, the amplitude approaches to the
scattering length. Note that the threshold condition can be fulfilled by a
simple choice of the pion momenta, $q_{a}=q_{b}=q_{c}=q_{d}=q$ and
$q^{2}=M_{\overline{\pi}}^{2}=s/4$, which facilitates a straightforward
computation of the diagrams. Doing the fermion frequency summation over the
internal quark lines, the scattering amplitude for the process of
$\overline{\pi}_{+}\ +\ \overline{\pi}_{+}\rightarrow\overline{\pi}_{+}\ +\
\overline{\pi}_{+}$ in the pion superfluid is simplified as
$\displaystyle{\cal T}_{+}$ $\displaystyle=$ $\displaystyle
18g_{\overline{\pi}_{+}q\overline{q}}^{4}\int{d^{3}{\bf
k}\over(2\pi)^{3}}\Bigg{\\{}{1\over
E_{+}^{3}}\Big{[}\Big{(}f(E_{+}^{-})-f(-E_{+}^{+})\Big{)}$ (278)
$\displaystyle-
E_{+}\Big{(}f^{\prime}(E_{+}^{-})+f^{\prime}(-E_{+}^{+})\Big{)}\Big{]}$
$\displaystyle+{1\over
E_{-}^{3}}\Big{[}\Big{(}f(E_{-}^{-})-f(-E_{-}^{+})\Big{)}$ $\displaystyle-
E_{-}\Big{(}f^{\prime}(E_{-}^{-})+f^{\prime}(-E_{-}^{+})\Big{)}\Big{]}\Bigg{\\}},$
where $E_{\pm}^{\mp}=E_{\pm}\mp\mu_{B}/3$ are the energies of the four
quasiparticles with
$E_{\pm}=\sqrt{\left(E\pm\mu_{I}/2\right)^{2}+4G^{2}\pi^{2}}$ and
$E=\sqrt{{\bf k}^{2}+M_{q}^{2}}$, $f(x)$ is the Fermi-Dirac distribution
function $f(x)=\left(e^{x/T}+1\right)^{-1}$, and $f^{\prime}(x)=df/dx$ is the
first order derivative of $f$. For the scattering amplitude outside the pion
superfluid, one should consider both the box and $\sigma$-exchange diagrams.
The calculation is straightforward.
Figure 20: (Color online) The scaled scattering amplitude ${\cal T}_{+}$ as a
function of isospin chemical potential $\mu_{I}$ at two values of temperature
$T$.
In Fig.20, we plot the scattering amplitude $|{\cal T}_{+}|$ as a function of
isospin chemical potential $\mu_{I}$ at two temperatures $T=0$ and $T=100$
MeV, keeping baryon chemical potential $\mu_{B}=0$. The normal matter with
$\mu_{I}<\mu_{I}^{c}$ is dominated by the explicit isospin symmetry breaking
and spontaneous chiral symmetry breaking, and the pion superfluid with
$\mu_{I}>\mu_{I}^{c}$ and the corresponding BEC-BCS crossover is controlled by
the spontaneous isospin symmetry breaking and chiral symmetry restoration.
From (277), the scattering amplitude is governed by the meson coupling
constant, ${\cal T}_{+}\sim g_{\overline{\pi}_{+}q\overline{q}}^{4}$. From
Fig.17, the meson mode $\overline{\pi}_{+}$ in the pion superfluid phase is
always a bound state, its coupling to quarks drops down with decreasing
$\mu_{I}$, and therefore the scattering amplitude $\left|{\cal T}_{+}\right|$
decreases when the system approaches to the phase transition and reaches zero
at the critical point $\mu^{c}_{I}$, due to
$g_{\overline{\pi}_{+}q\overline{q}}=0$ at this point. The critical isospin
chemical potential is $\mu^{c}_{I}=m_{\pi}=134$ MeV at $T=0$ and $142$ MeV at
$T=100$ MeV. After crossing the border of the phase transition, the coupling
constant changes its trend and starts to go up with decreasing isospin
chemical potential in the normal matter, and the scattering amplitude smoothly
increases and finally approaches its vacuum value at $\mu_{I}\to 0$.
The above $\mu_{I}$-dependence of the meson-meson scattering amplitude in the
pion superfluid with $\mu_{I}>\mu_{I}^{c}$ can be understood well from the
point of view of BCS-BEC crossover. We recall that the BCS and BEC states are
defined in the sense of the degree of overlapping among the pair wave
functions. The large pairs in BCS state overlap each other, and the small
pairs in BEC state are individual objects. Therefore, the cross section
between two pairs should be large in the BCS state and approach zero in the
limit of BEC. From our calculation shown in Fig.20, the $\pi-\pi$ scattering
amplitude is a characteristic quantity for the BCS-BEC crossover in pion
superfluid. The overlapped quark-antiquark pairs in the BCS state at higher
isospin density have large scattering amplitude, while in the BEC state at
lower isospin density with separable pairs, the scattering amplitude becomes
small. This provides a sensitive observable for the BCS-BEC crossover at quark
level, analogous to the fermion scattering in cold atom systems.
Figure 21: (Color online) The scattering amplitude ${\cal T}_{+}$ as a
function of temperature $T$ at two values of isospin chemical potential
$\mu_{I}$ in the pion superfluid.
The minimum of the scattering amplitude at the critical point can generally be
understood in terms of the interaction between the two quarks. A strong
interaction means a tightly bound state with small meson size and small meson-
meson cross section, and a weak interaction means a loosely bound state with
large meson size and large meson-meson cross section. Therefore, the minimum
of the meson scattering amplitude at the critical point indicates the most
strong quark interaction at the phase transition. This result is consistent
with theoretical calculations for the ratio $\eta/s$ kovtun ; csernai of shear
viscosity to entropy density and for the quark potential mu2 ; jiang , which
show a strongly interacting quark matter around the phase transition.
With increasing temperature, the pairs will gradually melt and the coupling
constant $g_{\overline{\pi}q\overline{q}}$ drops down in the hot medium,
leading to a smaller scattering amplitude at $T=100$ MeV in the pion
superfluid, in comparison with the case at $T=0$, as shown in Fig.20. To see
the continuous temperature effect on the scattering amplitude in the BCS and
BEC states, we plot in Fig.21 $\left|{\cal T}_{+}\right|$ as a function of $T$
at $\mu_{I}=160$ and $\mu_{I}=400$ MeV, still keeping $\mu_{B}=0$. While the
temperature dependence is similar in both cases, the involved physics is
different. In the BCS state at $\mu_{I}=400$ MeV, $\left|{\cal T}_{+}\right|$
is large and drops down with increasing temperature and finally vanishes at
the critical temperature $T_{c}=188$ MeV. Above $T_{c}$ the system becomes a
fermion gas with weak coupling and without any pair. In the BEC state at
$\mu_{I}=160$ MeV, the scattering amplitude becomes much smaller (multiplied
by a factor of $10$ in Fig.21). At a lower critical temperature $T_{c}=136$
MeV, the condensate melts but the still strong coupling between quarks makes
the system be a gas of free pairs.
The meson scattering amplitude $\left|{\cal T}_{+}\right|$ shown in Figs.20
and 21 are obtained in a particular model, the NJL model, which has proven to
be rather reliable in the study on chiral, color and isospin condensates at
low temperature. Since there is no confinement in the model, one may ask the
question to what degree the conclusions obtained here can be trusted. From the
general picture for BCS and BEC states, the feature that the meson scattering
amplitude approaches to zero in the process of BCS-BEC crossover can be
geometrically understood in terms of the degree of overlapping between the two
pairs. Therefore, the qualitative conclusion of taking meson scattering as a
probe of BCS-BEC crossover at quark level may survive any model dependence.
Our result that the molecular scattering amplitude approaches to zero in the
BEC limit is consistent with our previous result in Eq.(187) and the recent
work for a general fermion gas he2 . Different from a system with finite
baryon density where the fermion sign problem Lreview ; Lreview1 makes it
difficult to simulate QCD on lattice, there is in principle no problem to do
lattice QCD calculations at finite isospin density Liso ; Liso1 ; Liso2 ;
Liso3 . From the recent lattice QCD results detmold at nonzero isospin
chemical potential in a canonical approach, the scattering length in the pion
superfluid increases with increasing isospin density, which qualitatively
supports our conclusion here.
## IV Summary
In summary, we have presented in the article the studies of BCS-BEC crossover
in relativistic Fermi systems, especially in QCD matter at finite density.
We studied the BCS-BEC crossover in a relativistic four-fermion interaction
model. The relativistic effect is significant: A crossover from
nonrelativistic BEC to ultra relativistic BEC is possible, if the attraction
can be strong enough. In the relativistic theory, changing the density of the
system can naturally induce a BCS-BEC crossover from high density to low
density. The mean field theory is generalized to including the contribution
from uncondensed pairs. Applying the generalized mean field theory to color
superconducting quark matter at moderate density, the role of pairing
fluctuations becomes important: The size of the pseudogap at $\mu\sim 400$MeV
can reach the order of $100$ MeV at the critical temperature.
We investigated two-color QCD at finite baryon density in the frame of NJL
model. We can describe the weakly interacting diquark condensate at low
density and the BEC-BCS crossover at high density. The baryon chemical
potential for the predicted crossover is consistent with the lattice
simulations of two-color QCD at finite $\mu_{\rm B}$. The study is directly
applied to real QCD at finite isospin density. We proposed the meson-meson
scattering in pion superfluid as a sensitive probe of the BCS-BEC crossover.
Acknowledgement: LH is supported by the Helmholtz International Center for
FAIR within the framework of the LOEWE program launched by the State of Hesse,
and SM and PZ are supported by the NSFC and MOST under grant Nos. 11335005,
2013CB922000 and 2014CB845400.
## Appendix A The One-Loop Susceptibilities
In this appendix, we evaluate the explicit forms of the one-loop
susceptibilities $\Pi_{\text{ij}}(Q)$ (${\text{i}},{\text{j}}=1,2,3$) and
$\Pi_{\pi}(Q)$. At arbitrary temperature, their expressions are rather huge.
However, at $T=0$, they can be written in rather compact forms. For
convenience, we define $\Delta=|\Delta|e^{i\theta}$ in this appendix.
First, the polarization functions $\Pi_{11}(Q)$ and $\Pi_{12}(Q)$ can be
evaluated as
$\displaystyle\Pi_{11}(Q)$ $\displaystyle=$ $\displaystyle N_{c}N_{f}\sum_{\bf
k}\Bigg{[}\left(\frac{(u_{\bf k}^{-})^{2}(u_{\bf p}^{-})^{2}}{i\nu_{m}-E_{\bf
k}^{-}-E_{\bf p}^{-}}-\frac{(v_{\bf k}^{-})^{2}(v_{\bf
p}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{{\bf p}}^{-}}-\frac{(u_{\bf
k}^{+})^{2}(u_{\bf p}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf
p}^{+}}+\frac{(v_{\bf k}^{+})^{2}(v_{\bf p}^{+})^{2}}{i\nu_{m}-E_{\bf
k}^{+}-E_{\bf p}^{+}}\right){\cal T}_{+}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\ \ \ +\left(\frac{(u_{\bf k}^{-})^{2}(v_{\bf p}^{+})^{2}}{i\nu_{m}-E_{\bf
k}^{-}-E_{\bf p}^{+}}-\frac{(v_{\bf k}^{-})^{2}(u_{\bf
p}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{+}}-\frac{(u_{\bf
k}^{+})^{2}(v_{\bf p}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf
p}^{-}}+\frac{(v_{\bf k}^{+})^{2}(u_{\bf p}^{-})^{2}}{i\nu_{m}-E_{\bf
k}^{+}-E_{\bf p}^{-}}\right){\cal T}_{-}\Bigg{]},$ $\displaystyle\Pi_{12}(Q)$
$\displaystyle=$ $\displaystyle N_{c}N_{f}\sum_{\bf
k}\Bigg{[}\left(\frac{u_{\bf k}^{-}v_{\bf k}^{-}u_{\bf p}^{-}v_{\bf
p}^{-}}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{-}}-\frac{u_{\bf k}^{-}v_{\bf
k}^{-}u_{\bf p}^{-}v_{\bf p}^{-}}{i\nu_{m}-E_{\bf k}^{-}-E_{\bf
p}^{-}}+\frac{u_{\bf k}^{+}v_{\bf k}^{+}u_{\bf p}^{+}v_{\bf
p}^{+}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf p}^{+}}-\frac{u_{\bf k}^{+}v_{\bf
k}^{+}u_{\bf p}^{+}v_{\bf p}^{+}}{i\nu_{m}-E_{\bf k}^{+}-E_{\bf
p}^{+}}\right){\cal T}_{+}$ (279) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\left(\frac{u_{\bf k}^{-}v_{\bf k}^{-}u_{\bf p}^{+}v_{\bf
p}^{+}}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{+}}-\frac{u_{\bf k}^{-}v_{\bf
k}^{-}u_{\bf p}^{+}v_{\bf p}^{+}}{i\nu_{m}-E_{\bf k}^{-}-E_{\bf
p}^{+}}+\frac{u_{\bf k}^{+}v_{\bf k}^{+}u_{\bf p}^{-}v_{\bf
p}^{-}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf p}^{-}}-\frac{u_{\bf k}^{+}v_{\bf
k}^{+}u_{\bf p}^{-}v_{\bf p}^{-}}{i\nu_{m}-E_{\bf k}^{+}-E_{\bf
p}^{-}}\right){\cal T}_{-}\Bigg{]}e^{2i\theta},$
where ${\bf p}={\bf k}+{\bf q}$. Here ${\cal T}_{\pm}$ are factors arising
from the trace in spin space,
${\cal T}_{\pm}=\frac{1}{2}\pm\frac{{\bf k}\cdot{\bf p}+M^{2}}{2E_{\bf
k}E_{\bf p}},$ (280)
and $u_{\bf k}^{\pm},v_{\bf k}^{\pm}$ are the BCS distribution functions
defined as
$(u_{\bf k}^{\pm})^{2}=\frac{1}{2}\left(1+\frac{\xi_{\bf k}^{\pm}}{E_{\bf
k}^{\pm}}\right),\ \ \ \ \ (v_{\bf
k}^{\pm})^{2}=\frac{1}{2}\left(1-\frac{\xi_{\bf k}^{\pm}}{E_{\bf
k}^{\pm}}\right).$ (281)
At $Q=0$, we find
$\Pi_{12}(0)=\Delta^{2}\frac{1}{4}N_{c}N_{f}\sum_{\bf k}\left[\frac{1}{(E_{\bf
k}^{-})^{3}}+\frac{1}{(E_{\bf k}^{+})^{3}}\right].$ (282)
Thus, near the quantum phase transition point, we have
$\Pi_{12}(0)=\Delta^{2}\beta_{1}+O(|\Delta|^{4})$. On the other hand, a simple
algebra shows
$\frac{1}{4G}+\Pi_{11}(0)-|\Pi_{12}(0)|=\frac{\partial\Omega_{0}}{\partial|\Delta|^{2}}.$
(283)
Therefore, the mean-field gap equation for $\Delta$ ensures the Goldstone’s
theorem in the superfluid phase.
The term $\Pi_{13}$ standing for the mixing between the sigma meson and the
diquarks reads
$\displaystyle\Pi_{13}(Q)$ $\displaystyle=$ $\displaystyle N_{c}N_{f}\sum_{\bf
k}\Bigg{[}\left(\frac{u_{\bf k}^{+}v_{\bf k}^{+}(v_{\bf p}^{+})^{2}+u_{\bf
p}^{+}v_{\bf p}^{+}(v_{\bf k}^{+})^{2}}{i\nu_{m}-E_{\bf k}^{+}-E_{\bf
p}^{+}}+\frac{u_{\bf k}^{+}v_{\bf k}^{+}(u_{\bf p}^{+})^{2}+u_{\bf
p}^{+}v_{\bf p}^{+}(u_{\bf k}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf
p}^{+}}-\frac{u_{\bf k}^{-}v_{\bf k}^{-}(u_{\bf p}^{-})^{2}+u_{\bf
p}^{-}v_{\bf p}^{-}(u_{\bf k}^{-})^{2}}{i\nu_{m}-E_{\bf k}^{-}-E_{{\bf
p}}^{-}}-\frac{u_{\bf k}^{-}v_{\bf k}^{-}(v_{\bf p}^{-})^{2}+u_{\bf
p}^{-}v_{\bf p}^{-}(v_{\bf k}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf
p}^{-}}\right){\cal I}_{+}$ $\displaystyle+\left(\frac{u_{\bf k}^{+}v_{\bf
k}^{+}(u_{\bf p}^{-})^{2}+u_{\bf p}^{-}v_{\bf p}^{-}(v_{\bf
k}^{+})^{2}}{i\nu_{m}-E_{\bf k}^{+}-E_{{\bf p}}^{-}}+\frac{u_{\bf k}^{+}v_{\bf
k}^{+}(v_{\bf p}^{-})^{2}+u_{\bf p}^{-}v_{\bf p}^{-}(u_{\bf
k}^{+})^{2}}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf p}^{-}}-\frac{u_{\bf k}^{-}v_{\bf
k}^{-}(v_{\bf p}^{+})^{2}+u_{\bf p}^{+}v_{\bf p}^{+}(u_{\bf
k}^{-})^{2}}{i\nu_{m}-E_{\bf k}^{-}-E_{\bf p}^{+}}-\frac{u_{\bf k}^{-}v_{\bf
k}^{-}(u_{\bf p}^{+})^{2}+u_{\bf p}^{+}v_{\bf p}^{+}(v_{\bf
k}^{-})^{2}}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{+}}\right){\cal
I}_{-}\Bigg{]}e^{i\theta},$
where the factors ${\cal I}_{\pm}$ are defined as
${\cal I}_{\pm}=\frac{M}{2}\left(\frac{1}{E_{\bf k}}\pm\frac{1}{E_{\bf
p}}\right).$ (285)
One can easily find $\Pi_{13}\sim M\Delta$, thus it vanishes when $\Delta$ or
$M$ approaches zero. At $Q=0$, we have
$\Pi_{13}(0)=\Delta\frac{1}{2}N_{c}N_{f}\sum_{\bf k}\frac{M}{E_{\bf
k}}\left[\frac{\xi_{\bf k}^{-}}{(E_{\bf k}^{-})^{3}}+\frac{\xi_{\bf
k}^{+}}{(E_{\bf k}^{+})^{3}}\right].$ (286)
Thus the quantity $H_{0}$ defined in (III.3) can be evaluated as
$H_{0}=\frac{1}{2}N_{c}N_{f}\sum_{e=\pm}\sum_{\bf k}\frac{M_{*}}{E_{\bf
k}^{*}}\frac{1}{(E_{\bf
k}^{*}-em_{\pi}/2)^{2}}=\frac{\partial^{2}\Omega_{0}(y,M)}{\partial M\partial
y}\Bigg{|}_{y=0}.$ (287)
The polarization function $\Pi_{33}$ which stands for the sigma meson can be
evaluated as
$\displaystyle\Pi_{33}(Q)$ $\displaystyle=$ $\displaystyle N_{c}N_{f}\sum_{\bf
k}\Bigg{[}(v_{\bf k}^{-}u_{\bf p}^{-}+u_{\bf k}^{-}v_{\bf
p}^{-})^{2}\left(\frac{1}{i\nu_{m}-E_{\bf k}^{-}-E_{\bf
p}^{-}}-\frac{1}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{-}}\right){\cal
T}^{\prime}_{-}+(v_{\bf k}^{+}u_{\bf p}^{+}+u_{\bf k}^{+}v_{\bf
p}^{+})^{2}\left(\frac{1}{i\nu_{m}-E_{\bf k}^{+}-E_{\bf
p}^{+}}-\frac{1}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf p}^{+}}\right){\cal
T}^{\prime}_{-}$ $\displaystyle+(v_{\bf k}^{+}v_{\bf p}^{-}+u_{\bf
k}^{+}u_{\bf p}^{-})^{2}\left(\frac{1}{i\nu_{m}-E_{\bf k}^{+}-E_{\bf
p}^{-}}-\frac{1}{i\nu_{m}+E_{\bf k}^{+}+E_{\bf p}^{-}}\right){\cal
T}^{\prime}_{+}+(v_{\bf k}^{-}v_{\bf p}^{+}+u_{\bf k}^{-}u_{\bf
p}^{+})^{2}\left(\frac{1}{i\nu_{m}-E_{\bf k}^{-}-E_{\bf
p}^{+}}-\frac{1}{i\nu_{m}+E_{\bf k}^{-}+E_{\bf p}^{+}}\right){\cal
T}^{\prime}_{+}\Bigg{]},$
where the factors ${\cal T}^{\prime}_{\pm}$ are defined as
${\cal T}^{\prime}_{\pm}=\frac{1}{2}\pm\frac{{\bf k}\cdot{\bf
p}-M^{2}}{2E_{\bf k}E_{\bf p}}.$ (289)
At $Q=0$ and for $\Delta=0$, we find
$\displaystyle{\bf M}_{33}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2G}-2N_{c}N_{f}\sum_{\bf k}\frac{1}{E_{\bf
k}^{*}}+2N_{c}N_{f}\sum_{\bf k}\frac{M_{*}^{2}}{E_{\bf k}^{*3}}$ (290)
$\displaystyle=$ $\displaystyle\frac{\partial^{2}\Omega_{0}(y,M)}{\partial
M^{2}}\Bigg{|}_{y=0}.$
Finally, the polarization function $\Pi_{\pi}(Q)$ for pions can be obtained by
replacing ${\cal T}^{\prime}_{\pm}\rightarrow{\cal T}_{\pm}$. Thus, when
$M\rightarrow 0$, the sigma meson and pions become degenerate and chiral
symmetry is restored.
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|
arxiv-papers
| 2013-11-26T15:32:31 |
2024-09-04T02:49:54.271949
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lianyi He, Shijun Mao, Pengfei Zhuang",
"submitter": "Shijun Mao",
"url": "https://arxiv.org/abs/1311.6704"
}
|
1311.6706
|
# The role of local and global geometry in quantum entanglement percolation
Gerald John Lapeyre Jr ICFO–Institut de Ciències Fotòniques, Mediterranean
Technology Park, 08860 Castelldefels, Spain
###### Abstract
We prove that enhanced entanglement percolation via lattice transformation is
possible even if the new lattice is more poorly connected in that: i) the
coordination number (a local property) decreases, or ii) the classical
percolation threshold (a global property) increases. In searching for
protocols to transport entanglement across a network, it seems reasonable to
try transformations that increase connectivity. In fact, all examples that we
are aware of violate both conditions i and ii. One might therefore conjecture
that all good transformations must violate them. Here we provide a counter-
example that satisfies both conditions by introducing a new method, partial
entanglement swapping. This result shows that a transformation may not be
rejected on the basis of satisfying conditions i or ii. Both the result and
the new method constitute steps toward answering basic questions, such as
whether there is a minimum amount of local entanglement required to achieve
long-range entanglement.
entanglement, quantum networks, entanglement percolation, entanglement
distribution
###### pacs:
03.67.Bg, 03.67.Hk, 64.60.ah, 03.67.Pp
## I Introduction
Distribution of quantum entanglement on networks has been studied vigorously
over the past few years. This has been driven by the fact that entanglement is
the fundamental resource in quantum information, but it is created locally via
interaction, while it is often consumed in systems with widely separated
components. In an ideal description, each node of the network represents a
collection of qubits, and each edge or link represents entangled states of
qubits in different nodes.
But, even in the case of transporting entanglement along a chain of partially
entangled pure states, using perfect quantum operations, the resulting
entanglement decays exponentially in the number of links. Unfortunately,
technical and fundamental limits on effectively moving entanglement over even
a single link further complicate the ideal picture and have led to elaborate
protocols involving the distribution, storage, and purification of entangled
states. The most direct approach is the quantum repeater which has been
proposed to overcome these limitations on a one-dimensional chain of nodes
Briegel _et al._ (1998); Dür _et al._ (1999); Childress _et al._ (2005);
Hartmann _et al._ (2007); Sangouard _et al._ (2011); Meter _et al._ (2012).
There are examples of practical, deployed quantum networks, such as quantum
key distribution networks. But the technical challenges in implementing
quantum repeaters remain too great to be useful in contemporary quantum key
distribution networks Scarani _et al._ (2009). Typically, entanglement is
established over only a single link, while at each node information is
processed classically and re-encoded in a quantum state.
A different approach is to use the entire network, rather than a linear chain,
to distribute entanglement. The availability of multiple paths is used to
overcome the the inevitable decay of entanglement. This leads to models that
are immediately more interesting because it is not obvious how to prove which
of two protocols is better, let alone which protocol is optimal. In fact
percolation theory Stauffer and Aharony (1991); Grimmett (1999) has provided
powerful tools for evaluating protocols. The best protocols use quantum
operations to transform the initial lattice into a different lattice Acín _et
al._ (2007); Lapeyre Jr. _et al._ (2009); Perseguers _et al._ (2010).
As in the one-dimensional case, more realistic studies of multi-dimensional
networks have been done, for instance by considering mixed states and
imperfect quantum operations Cuquet and Calsamiglia (2009); Broadfoot _et
al._ (2009, 2010); Lapeyre Jr. _et al._ (2012); Cuquet and Calsamiglia
(2011). But sharp questions, say in the thermodynamic limit, are difficult to
pose in these dirtier situations because of the decay of entanglement.
Furthermore, questions about asymptotic behavior remain that are not only of
intrinsic interest, but address fundamental limits on entanglement
distribution. These are the questions that we address here.
This paper has two main goals. The first goal is to show that enhanced
entanglement percolation (defined below) via lattice transformation is
possible even if the coordination number of the transformed lattice decreases
or the classical percolation threshold increases. The second goal is to
introduce a new tool that we call partial entanglement swapping. In partial
swapping, we simply stop the swapping procedure after the first step, the
projection, and evaluate whether the output state and the new geometry may be
more profitably used in a different operation. In fact, the usefulness of the
tool is demonstrated by using it to accomplish the first goal. Although the
idea behind partial swapping is simple, it introduces a complication. In
previous entanglement percolation protocols, the Bell measurement in the
computational basis is optimal. But, the optimal basis for partial swapping is
not obvious and depends on the amount of initial entanglement.
## II Entanglement Percolation
Entanglement percolation is described in detail in several sources Acín _et
al._ (2007); Perseguers _et al._ (2008); Lapeyre Jr. _et al._ (2009);
Perseguers _et al._ (2013); Perseguers (2010). Here we give only a brief
description. We consider the following class of entanglement percolation
models. Each node consists of a collection of qubits. Each edge, or link,
consists of a partially entangled pure state between two qubits, each on a
different node. These states
$\,|\alpha\rangle\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are written in a
Schmidt basis as
$\,|\alpha\rangle=\sqrt{\alpha_{0}}\,|00\rangle+\sqrt{\alpha_{1}}\,|11\rangle,$
where the Schmidt coefficients $\alpha_{0},\alpha_{1}$ satisfy
$\alpha_{0}\geq\alpha_{1}$ and $\alpha_{0}+\alpha_{1}=1$. If
$\alpha_{0}=\alpha_{1}=1/2$, the state is maximally entangled, and is called a
Bell pair or singlet. If either $\alpha_{0}$ or $\alpha_{1}$ vanishes, then
the state is separable and is useless for quantum information tasks. The
smallest Schmidt coefficient may be used as a measure of entanglement with the
amount of entanglement increasing with $\alpha_{1}$.
Figure 1: Transformation of kagome to square lattice. Circles represent
qubits. Lines represent partially entangled bi-partite states. Left) Full
entanglement swapping is performed for each pair of links marked with a (blue)
loop. Right) The result is the square lattice, where the vertical (dashed)
links are the outcome of the swap and the horizontal links remain in the state
$\,|\alpha\rangle$. The remainder of the QEP protocol is described in the
text.
The lattice is initialized with identical states $\,|\alpha\rangle$ on each
link. So we have one free parameter $\alpha_{1}$. The goal is to design a
protocol to maximally entangle two arbitrary nodes $A$ and $B$. The utility of
the protocol is measured by the probability of success
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)$ as the distance between
$A$ and $B$ tends to infinity. We require that the protocols use only local
operations and classical communication (LOCC) Nielsen and Chuang (2000). This
means that quantum operations that include interaction between qubits on
different nodes are not allowed. But classical communication between all nodes
is allowed.
### II.1 Classical Entanglement Percolation
The simplest entanglement distribution protocol is called classical
entanglement percolation (CEP). For some lattices, better protocols have been
found, the so-called quantum entanglement percolation (QEP) protocols. The
reason for this distinction and the relation between CEP and QEP will be made
clear below. For now, we note that it is QEP that uses lattice transformation.
We introduce the CEP and QEP using the kagome lattice shown in Fig. 1, because
it allows a concise exposition. First we describe CEP. For the moment,
consider choosing fixed $A$ and $B$. In step $1$ we perform an LOCC operation
on each link, optimally converting it with probability $p=2\alpha_{1}$ to a
Bell pair, and probability $1-p$ to a separable state. This operation is
called a singlet conversion and $p$ is the singlet conversion probability
(SCP). After step $1$, we have a lattice in which each link is either open (a
Bell pair), or closed (separable). In step $2$, we search for an unbroken path
of open links between $A$ and $B$. If no such path exists, then
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=0$. If a path does exist,
then at each intermediate node we perform an entanglement swapping operation.
Because the input links are singlets, each swap succeeds with probability $1$,
so that $\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=1$. We call a
protocol that succeeds with probability $1$ deterministic. This description
corresponds exactly to classical bond percolation, with density of open bonds
$p=2\alpha_{1}$.
Lattice | $p_{c}$ for bond percolation
---|---
triangular | $2\sin(\pi/18)\approx 0.347$
square | $1/2$
kagome | $\approx 0.5244053$ MC estimate
hexagonal | $1-2\sin(\pi/18)\approx 0.653$
Table 1: $p_{c}$ for bond percolation on some lattices. All critical
densities are exactGrimmett (1999) except for $p_{c}(\text{kagome})$Ziff and
Suding (1997).
The critical bond density for the kagome lattice is
${p^{\text{kag}}_{c}}\approx 0.52$. Thus
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=0$ if
$p<{p^{\text{kag}}_{c}}$ and $\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow
B)>0$ if $p>{p^{\text{kag}}_{c}}$.
### II.2 Quantum Entanglement Percolation
A QEP scheme for the kagome lattice is shown in Fig. 1. We first perform
swapping on all pairs of qubits enclosed in loops. Each of the input states is
$\,|\alpha\rangle$, so the probability of obtaining a singlet in the resulting
vertical link is $p=2\alpha_{1}$. We then perform a singlet conversion on the
remaining horizontal bonds, resulting in a square lattice where each link is a
Bell pair with probability $p$ and is separable otherwise. Finally we perform
step $2$ of CEP (swapping with singlets) on this square lattice. This is
successful precisely when $p>{p^{\square}_{c}}$, where ${p^{\square}_{c}}$ is
the critical density for bond percolation on the square lattice. Since
${p^{\square}_{c}}=1/2$, it follows that long-distance entanglement on the
kagome lattice is possible with this QEP scheme, but not with CEP, if
$\alpha_{1}$ satisfies ${p^{\square}_{c}}<2\alpha_{1}<{p^{\text{kag}}_{c}}$.
CEP always gives an easily computable upper bound on the minimum initial
entanglement required for long-distance entanglement. Thus, CEP serves as a
benchmark to compare with any QEP protocol. Because we are not interested in
QEPs that perform worse than CEP, we will call any advantageous QEP simply a
QEP. However, the measure by which the QEP is advantageous may vary. We call
the smallest value of $\alpha_{1}$ such that long-range entanglement is
possible the lower threshold or percolation threshold $\hat{\alpha}_{c}$. We
call the smallest value of $\alpha_{1}$ such that long-range entanglement is
achieved with probability $1$ the upper threshold $\hat{\alpha}^{*}_{c}$. Note
that $\hat{\alpha}_{c}$ marks a phase transition, but $\hat{\alpha}^{*}_{c}$
does not. For every lattice, CEP gives $\hat{\alpha}^{*}_{c}=1/2$. We call a
QEP robust if it satisfies at least one of two conditions. 1) that it lowers
the percolation threshold $\hat{\alpha}_{c}$, and 2) that the upper threshold
satisfies $\hat{\alpha}^{*}_{c}<1/2$ . We are interested in isolating the
effect of the geometry of the transformed lattice on the performance of the
QEP. We therefore emphasize that we will compare the geometry of the classical
transformed lattice to that of the the initial lattice with no reference to
quantum states.
### II.3 Lattice structure and entanglement distribution
For any lattice, CEP is defined and the relevant quantities can be taken
directly from percolation theory. But there is no generic prescription for
constructing a QEP. In previous work, QEP protocols have been identified by
choosing a lattice and searching for good lattice transformations.
In the example above, the initial lattice was transformed into one new
lattice. However, in general, transformations may take the initial lattice
${\cal L}$ to multiple, decoupled lattices $\\{{\cal L}^{\prime}_{i}\\}$
Lapeyre Jr. _et al._ (2009). It is reasonable to search for $\\{{\cal
L}^{\prime}_{i}\\}$ that are more highly connected than ${\cal L}$. In fact,
in all of the examples of QEP given in refs Acín _et al._ (2007); Perseguers
_et al._ (2008); Lapeyre Jr. _et al._ (2009); Perseguers _et al._ (2010) one
${\cal L}^{\prime}_{i}$ has average coordination number greater than or equal
to that of ${\cal L}$ (condition i). Furthermore, one ${\cal L}^{\prime}_{i}$
has a classical percolation threshold that is less than or equal to that of
${\cal L}$ (condition ii). In Refs. Acín _et al._ (2007); Perseguers _et
al._ (2008); Lapeyre Jr. _et al._ (2009) this is easy to see because the
lattices involved are well-known111These are: double-bond hexagonal to
triangular; square to two square; kagome to square; bowtie to square and
triangular; asymmetric triangular to square and triangular.. The protocols in
Ref Perseguers _et al._ (2010) generate multi-partite entanglement from the
initial bi-partite states. The multi-partite swapping was explicitly designed
to increase connectivity. These protocols give the best performance to date,
and often result in less common or unclassified lattices including non-planar
graphs and lattices whose sites have different coordination numbers.
Given that all known protocols satisfy conditions i and ii, a natural question
is whether this must always be the case. Must these properties, one local and
one global, that are associated with high connectivity, be non-decreasing in
an advantageous QEP? In the following section we present a counter-example
demonstrating that the answer to this question is “no”. In fact both
conditions are violated, and the improvement is robust. To achieve this, we
introduce a new ingredient into the lattice transformation protocols.
## III QEP for the triangular lattice
CEP on the triangular lattice corresponds to classical bond percolation. With
CEP, long-range entanglement is only possible for
$\alpha_{1}>\hat{\alpha}_{c}(\text{CEP})=p_{c}^{\triangle}/2\approx 0.1736$,
and deterministic long-range entanglement is only possible for maximally
entangled initial states, i.e
$\alpha_{1}=\hat{\alpha}^{*}_{c}(\text{CEP})=1/2$. Here we present a QEP that
transforms the triangular lattice into the hexagonal lattice on which a
singlet can be created between any two nodes with probability $1$ if
$\alpha_{1}\gtrsim 0.3246$. That is, the upper threshold
$\hat{\alpha}^{*}_{c}$ is lowered. This is possible even though, classically,
the hexagonal lattice has larger critical density $p_{c}$ and smaller
coordination number than the triangular lattice.
### III.1 Partial entanglement swapping
Figure 2: Entanglement swapping. (a) a measurement is performed on qubits $1$
and $2$. (b) after the measurement, qubits $3$ and $4$ are in one of four
states $\\{{\,|\phi_{m}\rangle}\\}$, which are partially or maximally
entangled. In full entanglement swapping, a singlet conversion is performed on
the pair $(3,4)$ which results in either a maximally entangled state, or a
separable state. In partial entanglement swapping, only the measurement is
performed. (c) A distillation is then performed on the output state together
with another entangled pair, here in the state $\,|\alpha\rangle$, to produce
(d) either a more highly entangled pair, or a separable state.
In order to show the counter-example promised in the introduction, it is
enough to consider one of the outcomes from the same swapping measurement used
in previous studies on entanglement percolation. However, we consider here
more general measurements that allow us to optimize for certain figures of
merit. In this paper we consider entanglement swapping using Bell measurements
on two qubits, one from each pair, as shown in Fig. 2). For brevity, we omit
referring to any necessary local unitaries. We call the usual entanglement
swapping in any of these bases full entanglement swapping. We shall always
assume that the two input pairs are in the same state $\,|\alpha\rangle$.
Following Ref. Perseguers _et al._ (2008), we define an orthonormal basis
$\\{\,|\negmedspace\uparrow\rangle,\,|\negmedspace\downarrow\rangle\\}_{j}$
for each qubit $j=1,2$
$\begin{pmatrix}\,|\negmedspace\uparrow\rangle\\\
\,|\negmedspace\downarrow\rangle\end{pmatrix}_{j}=U_{j}\begin{pmatrix}\,|0\rangle\\\
\,|1\rangle\end{pmatrix}_{j},\quad U_{j}\in\mathcal{U}(2),$
and the Bell vectors
$\,|\Phi^{\pm}\rangle=\frac{\,|\negmedspace\uparrow\uparrow\rangle\pm\,|\negmedspace\downarrow\downarrow\rangle}{\sqrt{2}}\quad\text{and}\quad\,|\Psi^{\pm}\rangle=\frac{\,|\negmedspace\uparrow\downarrow\rangle\pm\,|\negmedspace\downarrow\uparrow\rangle}{\sqrt{2}}.$
The four measurement outcomes are
$\\{{\,|\phi_{m}\rangle}\\}\quad\text{ with probabilities }\quad\\{p_{m}\\}.$
Furthermore, $p_{\text{min}}=\min\\{p_{m}\\}$, and
$p_{\text{max}}=\max\\{p_{m}\\}$ are given by
$p_{\text{min}}=\alpha_{0}\alpha_{1}\quad\text{ and }\quad
p_{\text{max}}=\frac{1}{2}-\alpha_{0}\alpha_{1}.$
There is a bijective mapping between the probabilities $\\{p_{m}\\}$ and
$\\{U_{j}\\}$. In particular, every (orderless) choice of $\\{p_{m}\\}$
satisfying $p_{\text{min}}\leq p_{m}\leq p_{\text{max}}$ and $\sum_{m}p_{m}=1$
corresponds to a Bell measurement. The smallest Schmidt coefficients of the
output states are given by
$\lambda_{m}=\frac{1}{2}\left(1-\sqrt{1-\frac{\alpha_{0}^{2}\alpha_{1}^{2}}{p_{m}^{2}}}\right).$
In full entanglement swapping, we first perform the Bell measurement, and then
perform a singlet conversion on the output state. Since a singlet conversion
succeeds with probability equal to twice the smallest Schmidt coefficient, the
average SCP for full entanglement swapping is given by
$S_{M}=2\sum_{m}p_{m}\lambda_{m}$.
However, in partial entanglement swapping, we perform the Bell measurement
only, and not the singlet conversion. Instead of immediately doing a singlet
conversion we take advantage of the new geometry of the output state. We
attempt to distill a singlet from the output state and another entangled pair.
Although this is a simple idea, it is quite useful, and it has not been used
in previous work on entanglement distribution. From majorization theory
Nielsen (1999); Nielsen and Vidal (2001), we find that we can distill a Bell
pair from two partially entangled pairs with optimal probability
$p_{\text{distill}}=\min\\{1,2\left[1-(1-\beta_{1})(1-\gamma_{1})\right]\\},$
(1)
where $\beta_{1}$ and $\gamma_{1}$ are the smallest Schmidt coefficients of
the input states Lapeyre Jr. _et al._ (2009). In the example below, the
second state used in the distillation will be $\,|\alpha\rangle$. Thus, the
input states to the distillation have $\beta_{1}=\alpha_{1}$ and
$\gamma_{1}=\lambda_{m}$. The average SCP from combining the partial swapping
with distillation is then
$S_{M}=\sum_{m}p_{m}\min\left\\{1,2-\alpha_{0}\left(1+\sqrt{1-\frac{\alpha_{0}^{2}\alpha_{1}^{2}}{p_{m}^{2}}}\right)\right\\}.$
(2)
#### III.1.1 Swapping in ZZ basis
Suppose the measurement is in the ZZ basis, $U_{1}=U_{2}=\openone_{2}$. This
is the measurement that maximizes the average SCP in full swapping. Thus, it
is the one used in all previous entanglement percolation schemes (with a
modified version for multi-partite entanglement percolation). In this case,
$p_{1}=p_{2}=p_{\text{min}}$ and $p_{3}=p_{4}=p_{\text{max}}$, with
corresponding smallest Schmidt coefficients
$\lambda(p_{\max})=\frac{\alpha_{1}^{2}}{\alpha_{0}^{2}+\alpha_{1}^{2}},\quad\lambda(p_{\min})=\frac{1}{2}.$
Two of the outcomes are already singlets. Each of the other two may be
distilled together with $\,|\alpha\rangle$ into a singlet with probability
$p=\min\left\\{1,2\left(1-\frac{\alpha^{3}_{0}}{\alpha_{0}^{2}+\alpha_{1}^{2}}\right)\right\\},$
(3)
given by (1). The average SCP using partial swapping in the ZZ basis is then
$S_{ZZ}=\alpha_{0}\alpha_{1}+(1-2\alpha_{0}\alpha_{1})\min\left\\{1,2\left(1-\frac{\alpha^{3}_{0}}{\alpha_{0}^{2}+\alpha_{1}^{2}}\right)\right\\}.$
(4)
#### III.1.2 Swapping in XZ basis
Suppose the measurement is in the XZ basis. Then $p_{m}=1/4$ and
$\lambda_{m}=\frac{1}{2}(1-\sqrt{1-16\alpha_{0}^{2}\alpha_{1}^{2}})$ for all
$m$. The average SCP using partial swapping in the XZ basis is then
$S_{XZ}=\min\left\\{1,2-\alpha_{0}\left(1+\sqrt{1-16\alpha_{0}^{2}\alpha_{1}^{2}}\right)\right\\}.$
(5)
### III.2 The protocol
The QEP protocol proceeds as follows. Consider the triangular lattice with
each bond consisting of a single, partially entangled pure state. In step $1$,
we perform partial entanglement swappings on selected bonds as shown in Fig.
3.
Figure 3: Transformation of triangular to hexagonal lattice. a) triangular
lattice. A partial swap is applied to the dotted (red) lines. In following
frames, outcomes of partial swaps are shown as dashed (green) lines. Partial
swaps are applied to pairs of links shown as dotted lines. f) portion of
hexagonal lattice with double links. One link of each double link is in the
state $\,|\alpha\rangle$, the other link is one of the four outcomes of the
partial swap $\\{{\,|\phi_{m}\rangle}\\}$.
At the end of step $1$ we have a hexagonal lattice with double links. In each
pair, one link is the initial state $\,|\alpha\rangle$ and one link is one of
$\\{{\,|\phi_{m}\rangle}\\}$. Step $2$ consists of the following. For each
double link, if the outcome ${\,|\phi_{m}\rangle}$ is already a Bell pair,
then we do nothing. Otherwise, we attempt to distill a singlet from the two
links ${\,|\phi_{m}\rangle}$ and $\,|\alpha\rangle$.
#### III.2.1 Protocol in ZZ basis
Suppose we do the partial swap in the ZZ basis. Two outcomes are singlets and
two are partially entangled. From (3) we see that we create a singlet on every
bond of the hexagonal lattice deterministically if $\alpha_{0}$ is less than
the real root $\alpha_{0}^{*}\approx 0.6478$ of
$\alpha_{0}^{3}-\alpha_{0}^{2}+\alpha_{0}-1/2=0$. Equivalently, the condition
is $\hat{\alpha}^{*}_{c}\approx 0.3522$. The critical threshold for this
protocol is found by using (4) and solving
$S_{ZZ}(\alpha_{1})={p^{\hexagon}_{c}}$, where ${p^{\hexagon}_{c}}$ is the
classical threshold on the hexagonal lattice, with the result
$\hat{\alpha}_{c}\approx 0.1988$.
#### III.2.2 Protocol in XZ basis
Suppose we do the partial swap in the XZ basis. The smallest value of
$\alpha_{1}$ for which (5) equals $1$ is $\hat{\alpha}^{*}_{c}\approx 0.3246$.
The solution of $S_{XZ}(\alpha_{1})={p^{\hexagon}_{c}}$ is
$\hat{\alpha}_{c}\approx 0.2200$.
Protocol | $\hat{\alpha}_{c}$ | $\hat{\alpha}^{*}_{c}$
---|---|---
CEP | 0.1736 | 1/2
QEP ZZ | 0.1988 | 0.3522
QEP XZ | 0.2200 | 0.3246
QEP optimal | 0.1961 | 0.3246
Table 2: Percolation thresholds $\hat{\alpha}_{c}$ and upper thresholds
$\hat{\alpha}^{*}_{c}$ for entanglement protocols on the triangular lattice.
Figure 4: Average SCP distilled from double links on the hexagonal lattice.
Solid curve, ZZ basis. Dash-dot curve, XZ basis. Dashed curve, optimal basis.
Dotted line $p_{c}$ for bond percolation on the hexagonal lattice.
#### III.2.3 Protocol in other Bell bases
We optimized over all Bell measurements with only two distinct values of
$p_{m}$. Visual inspection showed that the optimum average SCP occurs when the
second argument to $\min$ in (5) is equal to $1$, which occurs for
$p_{1}=\alpha_{0}^{2}\alpha_{1}/\sqrt{1-2\alpha_{1}}$. Inserting this into
(2), and solving $S_{M}(\alpha_{1})={p^{\hexagon}_{c}}$, we find the lower
threshold $\hat{\alpha}_{c}\approx 0.1961$, which is a small improvement over
the ZZ basis. Optimizing for the upper threshold $\hat{\alpha}^{*}_{c}$, we
find $p_{m}=1/4$, which is the XZ basis. A numerical search for more general
Bell measurements strongly suggests that the optimum Bell protocol has only
two distinct values of $p_{m}$ for all $\alpha_{1}$.
In summary, we found that the optimal Bell basis has exactly two distinct
values of $p_{m}$, which depend on $\alpha_{1}$. At the lower threshold, the
optimal basis gives only a slight improvement over the ZZ basis. As the upper
threshold is approached, the optimal basis approaches the XZ basis. These
results are summarized in Fig. 4 and Table 2. We did not investigate non-Bell
measurements.
## IV Discussion
We have introduced a new tool, partial entanglement swapping, for entanglement
percolation via lattice transformation. This adds flexibility in optimally
combining the quantum and the geometric aspects of QEPs. We have demonstrated
the utility of partial swapping by using it to design a QEP that transforms
the triangular lattice to the hexagonal lattice. Partial entanglement swapping
allows sufficient concentration of entanglement to overcome lowered
connectivity in the transformed lattice. In particular, there is a least
initial amount of entanglement above which long-distance entanglement is
deterministic. Thus, we have proven that non-decreasing connectivity, as
measured by coordination number and percolation threshold, is not required for
QEP. However, in the present example, we find that CEP still provides the
optimal percolation threshold. It is interesting to note that the only other
known QEP for the triangular lattice uses multi-partite entanglement to
enhance the connectivity of the lattice by creating a non-planar graph
Perseguers _et al._ (2010). Thus, the question of whether a transformed
lattice with lower connectivity can give a lower critical threshold remains
open. Also unknown is whether the critical threshold of the triangular lattice
can be lowered via a transformation to a planar graph, or whether the
triangular lattice is, in a sense, a maximally connected planar graph.
In addition to answering a conjecture on the geometrical constraints on QEP,
partial swapping enlarges the toolbox for QEP. It may be combined with other
techniques to push the initial entanglement thresholds lower. However, even in
this simple example, the search becomes more complicated because we find that
the optimal measurement basis for partial swapping depends on the amount of
initial entanglement. Still more interesting than each new protocol would be a
proof, constructive or otherwise, of the existence of a minimum threshold for
a particular lattice or class of lattices.
## V Acknowledgments
The author thanks Jan Wehr for discussions and for asking a question that led
to the present work. This work was supported in part by the Spanish MICINN
(TOQATA, FIS2008-00784), by the ERC (QUAGATUA, OSYRIS), and EU projects SIQS,
EQUAM, and the Templeton Foundation.
## References
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|
arxiv-papers
| 2013-11-26T15:35:40 |
2024-09-04T02:49:54.298048
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gerald John Lapeyre Jr",
"submitter": "Gerald Lapeyre Jr.",
"url": "https://arxiv.org/abs/1311.6706"
}
|
1311.6900
|
11institutetext: 1Department of Applied Mathematics, Naval Postgraduate
School, Monterey, CA
2Institute for Computational Engineering & Sciences, The University of Texas
at Austin, Austin, TX
3Department of Aerospace Engineering & Engineering Mechanics, The University
of Texas at Austin, Austin, TX
4Departments of Mechanical Engineering and Jackson School of Geosciences, The
University of Texas at Austin, Austin, TX
# Discretely exact derivatives for hyperbolic PDE-constrained optimization
problems discretized by the discontinuous Galerkin method111This document has
been approved for public release; its distribution is unlimited.
Lucas C. Wilcox1 Georg Stadler2
Tan Bui-Thanh2,3 and Omar Ghattas2,4
###### Abstract
This paper discusses the computation of derivatives for optimization problems
governed by linear hyperbolic systems of partial differential equations (PDEs)
that are discretized by the discontinuous Galerkin (dG) method. An efficient
and accurate computation of these derivatives is important, for instance, in
inverse problems and optimal control problems. This computation is usually
based on an adjoint PDE system, and the question addressed in this paper is
how the discretization of this adjoint system should relate to the dG
discretization of the hyperbolic state equation. Adjoint-based derivatives can
either be computed before or after discretization; these two options are often
referred to as the optimize-then-discretize and discretize-then-optimize
approaches. We discuss the relation between these two options for dG
discretizations in space and Runge–Kutta time integration. The influence of
different dG formulations and of numerical quadrature is discussed. Discretely
exact discretizations for several hyperbolic optimization problems are
derived, including the advection equation, Maxwell’s equations and the coupled
elastic-acoustic wave equation. We find that the discrete adjoint equation
inherits a natural dG discretization from the discretization of the state
equation and that the expressions for the discretely exact gradient often have
to take into account contributions from element faces. For the coupled
elastic-acoustic wave equation, the correctness and accuracy of our derivative
expressions are illustrated by comparisons with finite difference gradients.
The results show that a straightforward discretization of the continuous
gradient differs from the discretely exact gradient, and thus is not
consistent with the discretized objective. This inconsistency may cause
difficulties in the convergence of gradient based algorithms for solving
optimization problems.
###### Keywords:
Discontinuous Galerkin PDE-constrained optimization Discrete adjoints Elastic
wave equation Maxwell’s equations
## 1 Introduction
Derivatives of functionals, whose evaluation depends on the solution of a
partial differential equation (PDE), are required, for instance, in inverse
problems and optimal control problems, and play a role in error analysis and a
posteriori error estimation. An efficient method to compute derivatives of
functionals that require the solution of a state PDE is through the solution
of an adjoint equation. In general, this adjoint PDE differs from the state
PDE. For instance, for a state equation that involves a first-order time
derivative, the adjoint equation must be solved backwards in time.
If the partial differential equation is hyperbolic, the discontinuous Galerkin
(dG) method is often a good choice to approximate the solution due to its
stability properties, flexibility, accuracy and ease of parallelization.
Having chosen a dG discretization for the state PDE, the question arises how
to discretize the adjoint equation and the expression for the gradient, and
whether and how their discretization should be related to the discretization
of the state equation. One approach is to discretize the adjoint equation and
the gradient independently from the state equation, possibly leading to
inaccurate derivatives as discussed below. A different approach is to derive
the discrete adjoint equation based on the discretized state PDE and a
discretization of the cost functional. Sometimes this latter approach is
called the discretize-then-optimize approach, while the former is known as
optimize-then-discretize. For standard Galerkin discretizations, these two
possibilities usually coincide; however, they can differ, for instance, for
stabilized finite element methods and for shape derivatives
CollisHeinkenschloss02 ; Gunzburger03 ; HinzePinnauUlbrichEtAl09 ; Braack09 .
In this paper, we study the interplay between these issues for derivative
computation in optimization problems and discretization by the discontinuous
Galerkin method.
While computing derivatives through adjoints on the infinite-dimensional level
and then discretizing the resulting expressions (optimize-then-discretize)
seems convenient, this approach can lead to inaccurate gradients that are not
proper derivatives of any optimization problem. This can lead to convergence
problems in optimization algorithms due to inconsistencies between the cost
functional and gradients Gunzburger03 . This inaccuracy is amplified when
inconsistent gradients are used to approximate second derivatives based on
first derivatives, as in quasi-Newton methods such as the BFGS method.
Discretizing the PDE and the cost functional first (discretize-then-optimize),
and then computing the (discrete) derivatives guarantees consistency. However,
when an advanced discretization method is used, computing the discrete
derivatives can be challenging. Thus, understanding the relation between
discretization and adjoint-based derivative computation is important. In this
paper, we compute derivatives based on the discretized equation and then study
how the resulting adjoint discretization relates to the dG discretization of
the state equation, and study the corresponding consistency issues for the
gradient.
_Related work:_ Discretely exact gradients can also be generated via
algorithmic differentiation (AD) GriewankWalther08 . While AD guarantees the
computation of exact discrete gradients, it is usually slower than hand-coded
derivatives. Moreover, applying AD to parallel implementations can be
challenging UtkeHascoetHeimbachEtAl09 . The notion of _adjoint consistency_
for dG (see Hartmann07 ; AlexeSandu10 ; HarrimanGavaghanSuli04 ;
OliverDarmofal09 ; SchutzMay13 ) is related to the discussion in this paper.
Adjoint consistency refers to the fact that the exact solution of the dual (or
adjoint) problem satisfies the discrete adjoint equation. This property is
important for dG discretizations to obtain optimal-order $L^{2}$-convergence
with respect to target functionals. The focus of this paper goes beyond
adjoint consistency to consider consistency of the gradient expressions, and
considers in what sense the discrete gradient is a discretization of the
continuous gradient. For discontinuous Galerkin discretization, the latter
aspect is called dual consistency in AlexeSandu10 . A systematic study
presented in Leykekhman12 compares different dG methods for linear-quadratic
optimal control problems subject to advection-diffusion-reaction equations. In
particular, the author targets commutative dG schemes, i.e., schemes for which
dG discretization and the gradient derivation commute. Error estimates and
numerical experiments illustrate that commutative schemes have desirable
properties for optimal control problems.
_Contributions:_ Using example problems, we illustrate that the discrete
adjoint of a dG discretization is, again, a dG discretization of the
continuous adjoint equation. In particular, an upwind numerical flux for the
hyperbolic state equation turns into a downwind flux in the adjoint, which has
to be solved backwards in time and converges at the same convergence order as
the state equation. We discuss the implications of numerical quadrature and of
the choice of the weak or strong form of the dG discretization on the adjoint
system. In our examples, we illustrate the computation of derivatives with
respect to parameter fields entering in the hyperbolic system either as a
coefficient or as forcing terms. Moreover, we show that discretely exact
gradients often involve contributions at element faces, which are likely to be
neglected in an optimize-then-discretize approach. These contributions are a
consequence of the discontinuous basis functions employed in the dG method and
since they are at the order of the discretization error, they are particularly
important for not fully resolved problems.
_Limitations:_ We restrict ourselves to problems governed by _linear_
hyperbolic systems. This allows for an explicit computation of the upwind
numerical flux in the dG method through the solution of a Riemann problem.
Linear problems usually do not require flux limiting and do not develop shocks
in the solution, which makes the computation of derivatives problematic since
numerical fluxes with limiters are often non-differentiable and defining
adjoints when the state solution involves shocks is a challenge GilesUlbrich10
; GilesUlbrich10a .
_Organization:_ Next, in Section 2, we discuss the interplay of the derivative
computation of a cost functional with the spatial and temporal discretization
of the governing hyperbolic system; moreover, we discuss the effects of
numerical quadrature. For examples of linear hyperbolic systems with
increasing complexity we derive the discrete adjoint systems and gradients in
Section 3, and we summarize important observations. In Section 4, we
numerically verify our expressions for the discretely exact gradient for a
cost functional involving the coupled acoustic-elastic wave equation by
comparing to finite differences, and finally, in Section 5, we summarize our
observations and draw conclusions.
## 2 Cost functionals subject to linear hyperbolic systems
### 2.1 Problem formulation
Let $\Omega\subset\mathbb{R}^{d}$ ($d=1,2,3$) be an open and bounded domain
with boundary $\Gamma=\partial\Omega$, and let $T>0$. We consider the linear
$n$-dimensional hyperbolic system
$\displaystyle\boldsymbol{q}_{t}+\nabla\cdot(\mathbf{F}\boldsymbol{q})$
$\displaystyle=\boldsymbol{f}\qquad$ $\displaystyle\text{\ on\
}\Omega\times(0,T),$ (1a) where, for $(\boldsymbol{x},t)\in\Omega\times(0,T)$,
$\boldsymbol{q}(\boldsymbol{x},t)\in\mathbb{R}^{n}$ is the vector of state
variables and $\boldsymbol{f}(\boldsymbol{x},t)\in\mathbb{R}^{n}$ is an
external force. The flux $\mathbf{F}\boldsymbol{q}\in\mathbb{R}^{n\times d}$
is linear in $\boldsymbol{q}$ and the divergence operator is defined as
$\nabla\cdot(\mathbf{F}\boldsymbol{q})=\sum_{i=1}^{d}{(A_{i}\boldsymbol{q})}_{x_{i}}$
with matrix functions $A_{i}:\Omega\to\mathbb{R}^{n\times n}$, where the
indices denote partial differentiation with respect to $x_{i}$. Together with
(1a), we assume the boundary and initial conditions $\displaystyle
B\boldsymbol{q}(\boldsymbol{x},t)$
$\displaystyle=\boldsymbol{g}(\boldsymbol{x},t)\qquad$
$\displaystyle\boldsymbol{x}\in\Gamma,\>t\in(0,T),$ (1b)
$\displaystyle\boldsymbol{q}(\boldsymbol{x},0)$
$\displaystyle=\boldsymbol{q}_{0}(\boldsymbol{x})\qquad$
$\displaystyle\boldsymbol{x}\in\Omega.$ (1c)
Here, $B:\Gamma\to\mathbb{R}^{l\times n}$ is a matrix function that takes into
account that boundary conditions can only be prescribed on inflow
characteristics. Under these conditions, (1) has a unique solution
$\boldsymbol{q}$ in a proper space $Q$ GustafssonKreissOliger95 .
We target problems, in which the flux, the right hand side, or the boundary or
initial condition data in (1) depend on parameters $\mathbf{c}$ from a space
$U$. These parameters can either be finite-dimensional, i.e.,
$\mathbf{c}=(c_{1},\ldots,c_{k})$ with $k\geq 1$, or infinite-dimensional,
e.g., a function $\mathbf{c}=\mathbf{c}(\boldsymbol{x})$. Examples for
functions $\mathbf{c}$ are material parameters such as the wave speed, or the
right-hand side forcing in (1a).
Our main interest are inverse and estimation problems, and optimal control
problems governed by hyperbolic systems of the form (1). This leads to
optimization problems of the form
$\min_{\mathbf{c},\boldsymbol{q}}\tilde{\mathcal{J}}(\mathbf{c},\boldsymbol{q})\quad\text{subject
to }~{}\eqref{eq:hyper},$ (2)
where $\tilde{\mathcal{J}}$ is a cost function that depends on the parameters
$\mathbf{c}$ and on the state $\boldsymbol{q}$. The parameters $\mathbf{c}$
may be restricted to an admissible set $U_{a\\!d}\subset U$ for instance to
incorporate bound constraints. If $U_{a\\!d}$ is chosen such that for each
$\mathbf{c}\in U_{a\\!d}$ the state equation (1) admits a unique solution
$\boldsymbol{q}:=\mathcal{S}(\mathbf{c})$ (where $\mathcal{S}$ is the solution
operator for the hyperbolic system), then (2) can be written as an
optimization problem in $\mathbf{c}$ only, namely
$\min_{\mathbf{c}\in
U_{a\\!d}}\mathcal{J}(\mathbf{c}):=\tilde{\mathcal{J}}(\mathbf{c},\mathcal{S}(\mathbf{c})).$
(3)
Existence and (local) uniqueness of solutions to (2) and (3) depend on the
form of the cost function $\tilde{\mathcal{J}}$, properties of the solution
and parameter spaces and of the hyperbolic system and have to be studied on a
case-to-case basis (we refer, for instance, to BorziSchulz12 ; Gunzburger03 ;
Lions85 ; Troltzsch10 ). Our main focus is not the solution of the
optimization problem (3), but the computation of derivatives of $\mathcal{J}$
with respect to $\mathbf{c}$, and the interplay of this derivative computation
with the spatial and temporal discretization of the hyperbolic system (1).
Gradients (and second derivatives) of $\mathcal{J}$ are important to solve (3)
efficiently, and can be used for studying parameter sensitivities or
quantifying the uncertainty in the solution of inverse problems Bui-
ThanhGhattasMartinEtAl13 .
### 2.2 Compatibility of boundary conditions
To ensure the existence of a solution to the adjoint equation, compatibility
conditions between boundary terms in the cost function $\tilde{\mathcal{J}}$,
the boundary operator $B$ in (1b) and the operator $\mathbf{F}$ in (1a) must
hold. We consider cost functions of the form
$\tilde{\mathcal{J}}(\mathbf{c},\boldsymbol{q})=\int_{0}^{T}\\!\\!\\!\int_{\Omega}j_{\Omega}(\boldsymbol{q})\,dx\,dt+\int_{0}^{T}\\!\\!\\!\int_{\Gamma}j_{\Gamma}(C\boldsymbol{q})\,dx\,dt+\int_{\Omega}j_{T}(\boldsymbol{q}(T))\,dx,$
(4)
where $j_{\Omega}:\mathbb{R}^{n}\to\mathbb{R}$,
$j_{\Gamma}:\mathbb{R}^{m}\to\mathbb{R}$ and
$J_{T}:\mathbb{R}^{n}\to\mathbb{R}$ are differentiable, and
$C:\Gamma\to\mathbb{R}^{m\times n}$ is a matrix-valued function. We denote the
derivatives of the functional under the integrals by
$j_{\Omega}^{\prime}(\cdot)$, $j_{\Gamma}^{\prime}(\cdot)$ and
$j_{T}^{\prime}(\cdot)$. The Fréchet derivative of $\mathcal{J}$ with respect
to $\boldsymbol{q}$ in a direction $\tilde{}\boldsymbol{q}$ is given by
$\tilde{\mathcal{J}}_{\boldsymbol{q}}(\mathbf{c},\boldsymbol{q})(\tilde{}\boldsymbol{q})=\int_{0}^{T}\\!\\!\\!\int_{\Omega}j^{\prime}_{\Omega}(\boldsymbol{q})\tilde{}\boldsymbol{q}\,dx\,dt+\int_{0}^{T}\\!\\!\\!\int_{\Gamma}j^{\prime}_{\Gamma}(C\boldsymbol{q})C\tilde{}\boldsymbol{q}\,dx\,dt\\\
+\int_{\Omega}j^{\prime}_{T}(\boldsymbol{q}(T))\tilde{}\boldsymbol{q}(T)\,dx.$
(5)
The boundary operators $B$ and $C$ must be compatible in the sense discussed
next. Denoting the outward pointing normal along the boundary $\Gamma$ by
$\boldsymbol{n}={(n_{1},\ldots,n_{d})}^{T}$, we use the decomposition
$A:=\sum_{i=1}^{d}n_{i}A_{i}=L^{T}\operatorname{diag}(\lambda_{1},\ldots,\lambda_{n})L,$
(6)
with $L\in\mathbb{R}^{n\times n}$ and $\lambda_{1}\geq\ldots\geq\lambda_{n}$.
Note that $L^{-1}=L^{T}$ if $A$ is symmetric. The positive eigenvalues
$\lambda_{1},\ldots,\lambda_{s}$, correspond to the $s\geq 0$ incoming
characteristics, and the negative eigenvalues
$\lambda_{n-m+1},\ldots,\lambda_{n}$ to the $m\geq 0$ outgoing
characteristics. Here, we allow for the zero eigenvalues
$\lambda_{s+1}=\cdots=\lambda_{n-m}=0$. To ensure well-posedness of the
hyperbolic system $\eqref{eq:hyper}$, the initial values of $\boldsymbol{q}$
can only be specified along incoming characteristics. The first $s$ rows
corresponding to incoming characteristics can be identified with the boundary
operator $B$ in (1b). To guarantee well-posedness of the adjoint equation, $C$
has to be chosen such that the cost functional $\tilde{\mathcal{J}}$ only
involves boundary measurements for outgoing characteristics. These correspond
to the rows of $L$ with negative eigenvalues and have to correspond to the
boundary operator $C$. It follows from (6) that
$A=\begin{bmatrix}B\\\ O\\\ C\end{bmatrix}^{-1}\begin{pmatrix}\lambda_{1}\\\
&\ddots\\\ &&\lambda_{n}\end{pmatrix}\begin{bmatrix}B\\\ O\\\
C\end{bmatrix}=\mathrm{bar}C^{T}B-\mathrm{bar}B^{T}C,$ (7)
where $O\in\mathbb{R}^{(n-s-m)\times n}$ and
$\mathrm{bar}B\in\mathbb{R}^{l\times n}$, $\mathrm{bar}C\in\mathbb{R}^{m\times
n}$ are derived properly. If $A$ is symmetric, then
$\mathrm{bar}C^{T}=B^{T}\operatorname{diag}(\lambda_{1},\ldots,\lambda_{S})$,
and
$\mathrm{bar}B^{T}=-C^{T}\operatorname{diag}(\lambda_{n-m+1},\ldots,\lambda_{n})$.
As will be shown in the next section, the matrix $\mathrm{bar}B$ is the
boundary condition matrix for the adjoint equation. For a discussion of
compatibility between the boundary term in a cost functional and hyperbolic
systems in a more general context we refer to GilesPierce97 ; Hartmann07 ;
AlexeSandu10 . In the next section, we formally derive the infinite-
dimensional adjoint system and derivatives of the cost functional
$\tilde{\mathcal{J}}$.
### 2.3 Infinite-dimensional derivatives
For simplicity, we assume that only the flux $\mathbf{F}$ (i.e., the matrices
$A_{1},A_{2},A_{3}$) depend on $\mathbf{c}$, but $B$, $C$, $\boldsymbol{f}$
and $\boldsymbol{q}_{0}$ do not depend on the parameters $\mathbf{c}$. We use
the formal Lagrangian method Troltzsch10 ; BorziSchulz12 , in which we
consider $\mathbf{c}$ and $\boldsymbol{q}$ as independent variables and
introduce the Lagrangian function
$\displaystyle\mathscr{L}(\mathbf{c},\boldsymbol{q},\boldsymbol{p}):=\tilde{\mathcal{J}}(\mathbf{c},\boldsymbol{q})+\int_{0}^{T}\\!\\!\int_{\Omega}{\left(\boldsymbol{q}_{t}+\nabla\cdot(\mathbf{F}\boldsymbol{q})-\boldsymbol{f},\boldsymbol{p}\right)}_{W}\,dx\,dt$
(8)
with $\boldsymbol{p},\boldsymbol{q}\in Q$, where $\boldsymbol{q}$ satisfies
the boundary and initial conditions (1b) and (1c), $\boldsymbol{p}$ satisfies
homogeneous versions of these conditions, and $\mathbf{c}\in U_{a\\!d}$. Here,
${(\cdot\,,\cdot)}_{W}$ denotes a $W$-weighted inner product in
$\mathbb{R}^{n}$, with a symmetric positive definite matrix
$W\in\mathbb{R}^{n\times n}$ (which may depend on $\boldsymbol{x}$). The
matrix $W$ can be used to make a hyperbolic system symmetric with respect to
the $W$-weighted inner product, as for instance in the acoustic and coupled
elastic-acoustic wave examples discussed in Sections 3.2 and 3.4. In
particular, this gives the adjoint equation a form very similar to the state
equation. If boundary conditions depend on the parameters $\mathbf{c}$, they
must be enforced weakly through a Lagrange multiplier in the Lagrangian
function and cannot be added in the definition of the solution space for
$\boldsymbol{q}$. For instance, if the boundary operator $B=B(\mathbf{c})$
depends on $\mathbf{c}$, the boundary condition (1b) must be enforced weakly
through a Lagrange multiplier, amounting to an additional term in the
Lagrangian functional (8).
Following the Lagrangian approach Troltzsch10 ; BorziSchulz12 , the gradient
of $\mathcal{J}$ coincides with the gradient of $\mathscr{L}$ with respect to
$\mathbf{c}$, provided all variations of $\mathscr{L}$ with respect to
$\boldsymbol{q}$ and $\boldsymbol{p}$ vanish. Requiring that variations with
respect to $\boldsymbol{p}$ vanish, we recover the state equation. Variations
with respect to $\boldsymbol{q}$ in directions $\tilde{}\boldsymbol{q}$, that
satisfy homogeneous versions of the initial and boundary conditions (1b) and
(1c), result in
$\displaystyle\mathscr{L_{\boldsymbol{q}}(\mathbf{c},\boldsymbol{q},\boldsymbol{p})}(\tilde{}\boldsymbol{q})$
$\displaystyle=\tilde{\mathcal{J}}_{\boldsymbol{q}}(\mathbf{c},\boldsymbol{q})(\tilde{}\boldsymbol{q})-\int_{0}^{T}\\!\\!\int_{\Omega}\left(\boldsymbol{p}_{t},W\tilde{}\boldsymbol{q}\right)+\left(\mathbf{F}\tilde{}\boldsymbol{q},\nabla(W\boldsymbol{p})\right)\,dx\,dt$
$\displaystyle+\int_{\Omega}\left(\tilde{}\boldsymbol{q}(T),W\boldsymbol{p}(T)\right)+\int_{0}^{T}\\!\\!\int_{\Gamma}\left(\boldsymbol{n}\cdot\mathbf{F}\tilde{}\boldsymbol{q},W\boldsymbol{p}\right)\,dx\,dt,$
where we have used integration by parts in time and space. As will be
discussed in Section 2.5, integration by parts can be problematic when
integrals are approximated using numerical quadrature and should be avoided to
guarantee exact computation of discrete derivatives. In this section, we
assume continuous functions $\boldsymbol{q},\boldsymbol{p}$ and exact
computation of integrals. Since $\boldsymbol{n}\cdot\mathbf{F}=A$, (7) implies
that
$\int_{0}^{T}\\!\\!\int_{\Gamma}(\boldsymbol{n}\cdot\mathbf{F}\tilde{}\boldsymbol{q},W\boldsymbol{p})\,dx\,dt=\int_{0}^{T}\\!\\!\int_{\Gamma}(C\tilde{}\boldsymbol{q},\mathrm{bar}BW\boldsymbol{p})-(B\tilde{}\boldsymbol{q},\mathrm{bar}CW\boldsymbol{p})\,dx\,dt.$
(9)
Using the explicit form of the cost given in (5), and that all variations with
respect to arbitrary $\tilde{}\boldsymbol{q}$ that satisfy
$B\tilde{}\boldsymbol{q}=0$ must vanish, we obtain
$\displaystyle W\boldsymbol{p}_{t}+\mathbf{F}^{\star}\nabla(W\boldsymbol{p})$
$\displaystyle=j^{\prime}_{\Omega}(\boldsymbol{q})\qquad$
$\displaystyle\text{\ on\ }\Omega\times(0,T)$ (10a)
$\displaystyle\mathrm{bar}BW\boldsymbol{p}(\boldsymbol{x},t)$
$\displaystyle=-j^{\prime}_{\Gamma}(C\boldsymbol{q}(\boldsymbol{x},t))\qquad$
$\displaystyle\boldsymbol{x}\in\Gamma,\>t\in(0,T),$ (10b) $\displaystyle
W\boldsymbol{p}(\boldsymbol{x},T)$
$\displaystyle=-j^{\prime}_{T}(\boldsymbol{q}(\boldsymbol{x},T))\qquad$
$\displaystyle\boldsymbol{x}\in\Omega.$ (10c)
Here, $\mathbf{F}^{\star}$ is the adjoint of $\mathbf{F}$ with respect to the
Euclidean inner product. Note that the adjoint system (10) is a final value
problem and thus is usually solved backwards in time. Note that, differently
from the state system, the adjoint system is not in conservative form.
Next we compute variations of $\mathscr{L}$ with respect to the parameters
$\mathbf{c}$ and obtain for variations $\tilde{}\mathbf{c}$ that
$\displaystyle\mathscr{L}_{\mathbf{c}}(\mathbf{c},\boldsymbol{q},\boldsymbol{p})(\tilde{}\mathbf{c})$
$\displaystyle=\int_{0}^{T}\\!\\!\\!\int_{\Omega}{\left(\nabla\cdot(\mathbf{F}_{\mathbf{c}}(\tilde{}\mathbf{c})\boldsymbol{q}),\boldsymbol{p}\right)}_{W}=\int_{0}^{T}\\!\\!\\!\int_{\Omega}\sum_{i=1}^{d}{\left({({A_{i}}_{\mathbf{c}}(\tilde{}\mathbf{c})\boldsymbol{q})}_{x_{i}},\boldsymbol{p}\right)}_{W}$
(11a)
Since $\boldsymbol{q}$ and $\boldsymbol{p}$ are assumed to solve the state and
adjoint system,
$\mathcal{J}(\mathbf{c})(\tilde{}\mathbf{c})=\mathscr{L}_{\mathbf{c}}(\mathbf{c},\boldsymbol{q},\boldsymbol{p})(\tilde{}\mathbf{c}),\
\text{where}\ \boldsymbol{q}\ \text{solves~{}\eqref{eq:hyper}}\ \text{and}\
\boldsymbol{p}\ \text{solves~{}\eqref{eq:adjhyper}}.$ (12)
Next, we present the dG discretization of the hyperbolic system (1) and
discuss the interaction between discretization and the computation of
derivatives.
### 2.4 Discontinuous Galerkin discretization
For the spatial discretization of hyperbolic systems such as (1), the
discontinuous Galerkin (dG) method has proven to be a favorable choice. In the
dG method, we divide the domain $\Omega$ into disjoint elements
${\Omega^{e}}$, and use polynomials to approximate $\boldsymbol{q}$ on each
element ${\Omega^{e}}$. The resulting approximation space is denoted by
$Q^{h}$, and elements $\boldsymbol{q}_{h}\in Q^{h}$ are polynomial on each
element, and discontinuous across elements. Using test functions
$\boldsymbol{p}_{h}\in Q^{h}$, dG discretization in space implies that for
each element ${\Omega^{e}}$
$\begin{split}\int_{{\Omega^{e}}}(\frac{\partial}{\partial
t}\boldsymbol{q}_{h},W\boldsymbol{p}_{h})&-(\mathbf{F}\boldsymbol{q}_{h},\nabla(W\boldsymbol{p}_{h}))\,d\boldsymbol{x}+\\\
&\int_{\Gamma^{e}}\boldsymbol{n}^{-}\cdot({(\mathbf{F}\boldsymbol{q}_{h})}^{\dagger},W\boldsymbol{p}_{h}^{-})\,d\boldsymbol{x}=\int_{{\Omega^{e}}}(\boldsymbol{f},W\boldsymbol{p}_{h})\,d\boldsymbol{x}\end{split}$
(13)
for all times $t\in(0,T)$. Here, $(\cdot\,,\cdot)$ is the inner product in
$\mathbb{R}^{n}$ and $\mathbb{R}^{d\times n}$ and the symmetric and positive
definite matrix $W$ acts as a weighting matrix in this inner product.
Furthermore, ${(\mathbf{F}\boldsymbol{q}_{h})}^{\dagger}$ is the numerical
flux, which connects adjacent elements. The superscript “$-$” denotes that the
inward values are chosen on $\Gamma^{e}$, i.e., the values of the
approximation on $\Omega^{e}$; the superscript “$+$” denotes that the outwards
values are chosen, i.e., the values of an element $\Omega^{e^{\prime}}$ that
is adjacent to $\Omega^{e}$ along the shared boundary $\Gamma^{e}$. Here,
$\boldsymbol{n}^{-}$ is the outward pointing normal on element $\Omega^{e}$.
The formulation (13) is often referred to as the _weak form_ of the dG
discretization HesthavenWarburton08 ; Kopriva09 . The corresponding _strong
form_ dG discretization is obtained by element-wise integration by parts in
space in (13), resulting in
$\begin{split}\int_{{\Omega^{e}}}(\frac{\partial}{\partial
t}\boldsymbol{q}_{h},W\boldsymbol{p}_{h})&+(\nabla\cdot\mathbf{F}\boldsymbol{q}_{h},W\boldsymbol{p}_{h})\,d\boldsymbol{x}-\\\
&\int_{\Gamma^{e}}\boldsymbol{n}^{-}\cdot(\mathbf{F}\boldsymbol{q}_{h}^{-}-{(\mathbf{F}\boldsymbol{q}_{h})}^{\dagger},W\boldsymbol{p}_{h}^{-})\,d\boldsymbol{x}=\int_{{\Omega^{e}}}(\boldsymbol{f},W\boldsymbol{p}_{h})\,d\boldsymbol{x}\end{split}$
(14)
for all $t\in(0,T)$. To find a solution to the optimization problem (3),
derivatives of $\mathcal{J}$ with respect to the parameters $\mathbf{c}$ must
be computed. There are two choices for computing derivatives, namely deriving
expressions for the derivatives of the continuous problem (3), and then
discretizing these equations, or first discretizing the problem and then
computing derivatives of this fully discrete problem. If the latter approach
is taken, i.e., the discrete adjoints are computed, the question arises
weather the discrete adjoint equation is an approximation of the continuous
adjoint and if the discrete adjoint equation is again a dG discretization.
Moreover, what are the consequences of choosing the weak or the strong form
(13) or (14)? A sketch for the different combinations of discretization and
computation of derivatives is also shown in Figure 1.
### 2.5 Influence of numerical quadrature
In a numerical implementation, integrals are often approximated using
numerical quadrature. A fully discrete approach has to take into account the
resulting quadrature error; in particular, integration by parts can incur an
error in combination with numerical quadrature. Below, we first discuss
implications of numerical quadrature in space and then comment on numerical
integration in time. In our example problems in Section 3, we use integral
symbols to denote integration in space and time, but do not assume exact
integration. In particular, we avoid integration by parts or highlight when
integration by parts is used.
${\mathcal{J}}$${\mathcal{J}^{dG}_{h}}$${\mathcal{J}^{dG}_{h,k}}$${\mathcal{J}^{\prime}}$${\mathcal{J}^{dG^{\prime}}_{h}}$${\mathcal{J}^{dG^{\prime}}_{h,k}}$gradgradhgradh,kdG
in spacetime-discretizationdiscretize in spacetime-discretization Figure 1:
Sketch to illustrate the relation between discretization of the problem
(horizontal arrows, upper row), computation of the gradient with respect to
the parameters $\mathbf{c}$ (vertical arrows) and discretization of the
gradient (horizontal arrows, lower row). The problem discretization (upper
row) requires discretization of the state equation and of the cost functional
in space (upper left horizontal arrow) and in time (upper right horizontal
arrow). The vertical arrows represent the Frèchet derivatives of $\mathcal{J}$
(left), of the semidiscrete cost $\mathcal{J}^{dG}_{h}$ (middle) and the fully
discrete cost $\mathcal{J}^{dG}_{h,k}$ (right). The discretization of the
gradient (bottom row) requires space (left arrow) and time (right arrow)
discretization of the state equation, the adjoint equation and the expression
for the gradient. Most of our derivations follow the fully discrete approach,
i.e., the upper row and right arrows; The resulting discrete expressions are
then interpreted as discretizations of the corresponding continuous equations
derived by following the vertical left arrow.
#### 2.5.1 Numerical integration in space
The weak form (13) and the strong form (14) of dG are equivalent provided
integrals are computed exactly and, as a consequence, integration by parts
does not result in numerical error. If numerical quadrature is used, these
forms are only numerically equivalent under certain conditions
KoprivaGassner10 ; in general, they are different. To compute fully discrete
gradients, we thus avoid integration by parts in space whenever possible. As a
consequence, if the weak form for the state equation is used, the adjoint
equation is in strong form; this is illustrated and further discussed in
Section 3.
#### 2.5.2 Numerical integration in time
We use a method-of-lines approach, that is, from the dG discretization in
space we obtain a continuous-in-time system of ordinary differential equations
(ODEs), which is then discretized by a Runge–Kutta method. Discrete adjoint
equations and the convergence to their continuous counterparts for systems of
ODEs discretized by Runge–Kutta methods have been studied, for instance in
Hager00 ; Walther07 . For the time-discretization we build on these results.
An alternative approach to discretize in time is using a finite element method
for the time discretization, which allows a fully variational formulation of
the problem in space-time; we refer, for instance to BeckerMeidnerVexler07
for this approach applied to parabolic optimization problems.
In both approaches, the computation of derivatives requires the entire time
history of both the state and the adjoint solutions. For realistic application
problems, storing this entire time history is infeasible, and storage
reduction techniques, also known as checkpointing strategies have to be
employed. These methods allow to trade storage against computation time by
storing the state solution only at certain time instances, and then
recomputing it as needed when solving the adjoint equation and computing the
gradient BeckerMeidnerVexler07 ; GriewankWalther00 ; GriewankWalther08 .
## 3 Example problems
The purpose of this section is to illustrate the issues discussed in the
previous sections on example problems. We present examples with increasing
complexity and with parameters entering differently in the hyperbolic systems.
First, in Section 3.1, we derive expressions for derivatives of a functional
that depends on the solution of the one-dimensional advection equation. Since
linear conservation laws can be transformed in systems of advection equations,
we provide extensive details for this example. In particular, we discuss
different numerical fluxes. In Section 3.2, we compute expressions for the
derivatives of a functional with respect to the local wave speed in an
acoustic wave equation. This is followed by examples in which we compute
derivatives with respect to a boundary forcing in Maxwell’s equation (Section
3.3) and derivatives with respect to the primary and secondary wave speeds in
the coupled acoustic-elastic wave equation (Section 3.4). At the end of each
example, we summarize our observations in remarks.
Throughout this section, we use the dG discretization introduced in Section
2.4 and denote the finite dimensional dG solution spaces by $P^{h}$ and
$Q^{h}$. These spaces do not include the boundary conditions which are usually
imposed weakly through the numerical flux in the dG method, and they also do
not include initial/final time conditions, which we specify explicitly.
Functions in these dG spaces are smooth (for instance polynomials) on each
element $\Omega^{e}$ and discontinuous across the element boundaries
$\partial{\Omega^{e}}$. As before, for each element we denote the inward value
with a superscript “-” and the outward value with superscript “+”. We use the
index $h$ to denote discretized fields and denote by $[\\![\cdot]\\!]$ the
jump, by $[\cdot]$ the difference and by $\\{\\!\\!\\{\cdot\\}\\!\\!\\}$ the
mean value at an element interface $\partial{\Omega^{e}}$. To be precise, for
a scalar dG-function $u_{h}$, these are defined as
$[\\![u_{h}]\\!]=u_{h}^{-}\boldsymbol{n}^{-}+u_{h}^{+}\boldsymbol{n}^{+}$,
$[u_{h}]=u_{h}^{-}-u_{h}^{+}$ and
$\\{\\!\\!\\{u_{h}\\}\\!\\!\\}=(u_{h}^{-}+u_{h}^{+})/2$. Likewise, for a
vector $\boldsymbol{v}_{h}$ we define
$[\\![\boldsymbol{v}_{h}]\\!]=\boldsymbol{v}_{h}^{-}\cdot\boldsymbol{n}^{-}+\boldsymbol{v}_{h}^{+}\cdot\boldsymbol{n}^{+}$,
$[\boldsymbol{v}_{h}]=\boldsymbol{v}_{h}^{-}-\boldsymbol{v}_{h}^{+}$ and
$\\{\\!\\!\\{\boldsymbol{v}_{h}\\}\\!\\!\\}=(\boldsymbol{v}_{h}^{-}+\boldsymbol{v}_{h}^{+})/2$
and for a second-order tensor $\SS_{h}$ we have
$[\\![\SS_{h}]\\!]=\SS_{h}^{-}\boldsymbol{n}^{-}+\SS_{h}^{+}\boldsymbol{n}^{+}$.
The domain boundary $\partial\Omega$ is denoted by $\Gamma$.
Throughout this section, we use the usual symbol to denote integrals, but we
do not assume exact quadrature. Rather, integration can be replaced by a
numerical quadrature rule and, as a consequence, the integration by parts
formula does not hold exactly. To avoid numerical errors when using numerical
quadrature, we thus avoid integration by parts in space or point out when
integration by parts is used. Since our focus is on the spatial dG
discretization, we do not discretize the problem in time and assume exact
integration in time.
### 3.1 One-dimensional advection equation
We consider the one-dimensional advection equation on the spatial domain
$\Omega=(x_{l},x_{r})\subset\mathbb{R}$. We assume a spatially varying,
continuous positive advection velocity $a(x)\geq a_{0}>0$ for $x\in\Omega$ and
a forcing $f(x,t)$ for $(x,t)\in\Omega\times(0,T)$. The advection equation
written in conservative form is given by
$\displaystyle u_{t}+{(au)}_{x}$ $\displaystyle=f\quad$ $\displaystyle\text{\
on\ }\Omega\times(0,T),$ (15a) with the initial condition $\displaystyle
u(x,0)$ $\displaystyle=u_{0}(x)\quad$ $\displaystyle\text{\ for\ }x\in\Omega,$
(15b) and, since $a>0$, the inflow boundary is $\Gamma_{l}:=\\{x_{l}\\}$,
where we assume $\displaystyle u(x_{l},t)$ $\displaystyle=u_{l}(t)\quad$
$\displaystyle\text{\ for\ }t\in(0,T).$ (15c)
The discontinuous Galerkin (dG) method for the numerical solution of (15a) in
strong form is: Find $u_{h}\in P^{h}$ with $u_{h}(x,0)=u_{0}(x),x\in\Omega$
such that for all test functions $p_{h}\in P^{h}$ holds
$\int_{\Omega}(u_{h,t}+{(au_{h})}_{x}-f)p_{h}\,dx=\sum_{e}\int_{\partial{\Omega^{e}}}n^{-}\left(au_{h}^{-}-{(au_{h})}^{\dagger}\right)p_{h}^{-}\,dx$
(16)
for all $t\in(0,T)$. Here, $a\in U$ can be an infinite-dimensional continuous
function, or a finite element function. For $\alpha\in[0,1]$, the numerical
flux on the boundary is replaced by the numerical flux,
${(au_{h})}^{\dagger}$, given by
${(au_{h})}^{\dagger}=a\\{\\!\\!\\{u_{h}\\}\\!\\!\\}+\frac{1}{2}\left\lvert
a\right\rvert(1-\alpha)[\\![u_{h}]\\!].$ (17)
This is a central flux for $\alpha=1$, and an upwind flux for $\alpha=0$. Note
that in one spatial dimension, the outward normal $n$ is $-1$ and $+1$ on the
left and right side of ${\Omega^{e}}$, respectively. Since $a$ is assumed to
be continuous on $\Omega$, we have $a^{-}=a^{+}:=a$. Moreover, since $a$ is
positive, we neglect the absolute value in the following.
As is standard practice HesthavenWarburton08 ; Kopriva09 we incorporate the
boundary conditions weakly through the numerical flux by choosing the
“outside” values $u_{h}^{+}$ as $u_{h}^{+}=u_{l}$ for $x=x_{l}$ and
$u_{h}^{+}=u_{h}^{-}$ for $x=x_{r}$ for the computation of
${(au_{h})}^{\dagger}$. This implies that
$[\\![u_{h}]\\!]=n^{-}(u_{h}^{-}-u_{l})$ on $\Gamma_{l}$ and
$[\\![u_{h}]\\!]=0$ on the outflow boundary $\Gamma_{r}:=\\{x_{r}\\}$. For
completeness, we also provide the dG discretization of (15a) in weak form:
Find $u_{h}\in P^{h}$ with $u_{h}(x,0)=u_{0}(x),x\in\Omega$ such that for all
$p_{h}\in P^{h}$ holds
$\int_{\Omega}(u_{h,t}-f)p_{h}-au_{h}p_{h,x}\,dx=-\sum_{e}\int_{\partial{\Omega^{e}}}n^{-}{(au_{h})}^{\dagger}p_{h}^{-}\,dx$
for all $t\in(0,T)$, which is found by element-wise integration by parts in
(16).
Adding the boundary contributions in (16) from adjacent elements
${\Omega^{e}}$ and ${\Omega^{e^{\prime}}}$ to their shared point
$\partial{\Omega^{e}}\cap\partial{\Omega^{e^{\prime}}}$, we obtain
$\frac{1}{2}\left\lvert
a\right\rvert(\alpha-1)[\\![u_{h}]\\!][\\![p_{h}]\\!]+a[\\![u_{h}]\\!]\\{\\!\\!\\{p_{h}\\}\\!\\!\\}.$
Thus, (16) can also be written as
$\int_{\Omega}(u_{h,t}+{(au_{h})}_{x}-f)p_{h}\,dx=\sum_{e}\int_{\partial{\Omega^{e}}\setminus\Gamma}n^{-}\left(a\\{\\!\\!\\{p_{h}\\}\\!\\!\\}+{a}\frac{1}{2}(\alpha-1)[\\![p_{h}]\\!]\right)u_{h}^{-}\,dx\\\
+\int_{\Gamma_{l}}n^{-}\left(a\frac{1}{2}p_{h}^{-}+{a}\frac{1}{2}(\alpha-1)n^{-}p_{h}^{-}\right)(u_{h}^{-}-u_{l})\,dx.$
(18)
We consider an objective functional for the advection velocity $a\in U$ given
by
$\mathcal{J}(a):=\tilde{\mathcal{J}}(a,u_{h})=\int_{0}^{T}\\!\\!\int_{\Omega}j_{\Omega}(u_{h})\,dx\,dt+\int_{0}^{T}\\!\\!\int_{\Gamma}j_{\Gamma}(u_{h})\,dx\,dt+\int_{\Omega}r(a)\,dx,$
(19)
with differentiable functions $j_{\Omega}:\Omega\to\mathbb{R}$,
$j_{\Gamma}:\Gamma\to\mathbb{R}$ and $r:\Omega\to\mathbb{R}$. For illustration
purposes, we define the boundary term in the cost on both, the inflow and the
outflow part of the boundary, and comment on the consequences in Remark 1. To
derive the discrete gradient of $\mathcal{J}$, we use the Lagrangian function
$\mathcal{L}:U\times P^{h}\times P^{h}\rightarrow\mathbb{R}$, which combines
the cost (19) with the dG discretization (18):
$\displaystyle\mathcal{L}(a,u_{h},p_{h}):=$
$\displaystyle\tilde{\mathcal{J}}(a,u_{h})+\int_{0}^{T}\\!\\!\int_{\Omega}(u_{h,t}+{(au_{h})}_{x}-f)p_{h}\,dx\,dt$
$\displaystyle-\sum_{e}\int_{0}^{T}\\!\\!\int_{\partial{\Omega^{e}}\setminus\Gamma}n^{-}\left(a\\{\\!\\!\\{p_{h}\\}\\!\\!\\}+{a}\frac{1}{2}(\alpha-1)[\\![p_{h}]\\!]\right)u_{h}^{-}\,dx\,dt$
$\displaystyle-\int_{0}^{T}\int_{\Gamma_{l}}n^{-}\left(a\frac{1}{2}p_{h}^{-}+{a}\frac{1}{2}(\alpha-1)n^{-}p_{h}^{-}\right)(u_{h}^{-}-u_{l})\,dx\,dt.$
By requiring that all variations with respect to $p_{h}$ vanish, we recover
the state equation (18). Variations with respect to $u_{h}$ in a direction
$\tilde{u}_{h}$, which satisfies the homogeneous initial conditions
$\tilde{u}_{h}(x,0)=0$ for $x\in\Omega$ result in
$\displaystyle\\!\\!\\!\\!\mathcal{L}_{u_{h}}(a,u_{h},p_{h})(\tilde{u}_{h})$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\int_{\Omega}j^{\prime}_{\Omega}(u_{h})\tilde{u}_{h}-p_{h,t}\tilde{u}_{h}+{(a\tilde{u}_{h})}_{x}p_{h}\,dx\,dt+\int_{\Omega}p_{h}(x,T)\tilde{u}_{h}(x,T)\,dx$
$\displaystyle-\sum_{e}\int_{0}^{T}\\!\\!\int_{\partial{\Omega^{e}}\setminus\Gamma}\\!\\!\\!n^{-}\left(a\\{\\!\\!\\{p_{h}\\}\\!\\!\\}+{a}\frac{1}{2}(\alpha-1)[\\![p_{h}]\\!]\right)\tilde{u}_{h}^{-}\,dx\,dt+\int_{0}^{T}\\!\\!\int_{\Gamma}j^{\prime}_{\Gamma}(u_{h})\tilde{u}_{h}\,dx\,dt$
$\displaystyle-\int_{0}^{T}\\!\\!\int_{\Gamma_{l}}n^{-}\left(a\frac{1}{2}p_{h}^{-}+{a}\frac{1}{2}(\alpha-1)n^{-}p_{h}^{-}\right)\tilde{u}_{h}^{-}\,dx\,dt$
Since we require that arbitrary variations with respect to $u_{h}$ must
vanish, $p_{h}$ has to satisfy $p_{h}(x,T)=0$ for all $x\in\Omega$, and
$\int_{\Omega}-p_{h,t}\tilde{u}_{h}+{(a\tilde{u}_{h})}_{x}p_{h}+j_{\Omega}^{\prime}(u_{h})\tilde{u}_{h}\,dx=\sum_{e}\int_{\partial{\Omega^{e}}\setminus\Gamma_{l}}n^{-}{(ap_{h})}^{\dagger}\tilde{u}_{h}^{-}\,dx\\\
+\int_{\Gamma_{l}}n^{-}\left(\frac{a(2-\alpha)}{2}p_{h}^{-}+j_{\Gamma}^{\prime}(u_{h})\right)\tilde{u}_{h}^{-}\,dx$
(20a) for all $t\in(0,T)$ and for all $\tilde{u}_{h}$, with the adjoint flux
${(ap_{h})}^{\dagger}:=a\\{\\!\\!\\{p_{h}\\}\\!\\!\\}+\frac{1}{2}{a}(\alpha-1)[\\![p_{h}]\\!]$
(20b) and with
$p_{h}^{+}:=-\frac{j_{\Gamma}^{\prime}(u_{h})}{a(1-\frac{\alpha}{2})}-\frac{\alpha}{2-\alpha}p_{h}^{-}\quad\text{on}\
\Gamma_{r}.$ (20c)
The outside value $p_{h}^{+}$ in (20c) is computed such that
$n^{-}{(ap_{h})}^{\dagger}=j^{\prime}_{\Gamma}(u_{h})$ on $\Gamma_{r}$. The
equations (20) are the weak form of a discontinuous Galerkin discretization of
the adjoint equation, with flux given by (20b). An element-wise integration by
parts in space in (20a), results in the corresponding strong form of the
discrete adjoint equation:
$\begin{split}\int_{\Omega}\big{(}-p_{h,t}-ap_{h,x}+&j_{\Omega}^{\prime}(u_{h})\big{)}\tilde{u}_{h}\,dx=-\sum_{e}\int_{\partial{\Omega^{e}}\setminus\Gamma_{l}}n^{-}\left(ap_{h}^{-}-{(ap_{h})}^{\dagger}\right)\tilde{u}_{h}^{-}\,dx\\\
&+\int_{\Gamma_{l}}n^{-}\left(-\frac{a\alpha}{2}p_{h}^{-}+j_{\Gamma}^{\prime}(u_{h})\right)\tilde{u}_{h}^{-}\,dx\end{split}$
(21)
Note that this integration by parts is not exact if numerical quadrature is
used. It can be avoided if the dG weak form of the adjoint equation (20) is
implemented directly.
Provided $u_{h}$ and $p_{h}$ are solutions to the state and adjoint equations,
respectively, the gradient of $\mathcal{J}$ with respect to $a$ is found by
taking variations of the Lagrangian with respect to $a$ in a direction
$\tilde{a}$:
$\displaystyle\mathcal{L}_{a}(a,u_{h},p_{h})(\tilde{a})=$
$\displaystyle\int_{\Omega}r^{\prime}(a)\tilde{a}\,dx+\int_{0}^{T}\\!\\!\int_{\Omega}{(\tilde{a}u_{h})}_{x}p_{h}\,dx\,dt$
$\displaystyle-\sum_{e}\int_{0}^{T}\\!\\!\int_{\partial{\Omega^{e}}\setminus\Gamma}n^{-}\tilde{a}\left(\frac{1}{2}(\alpha-1)[\\![p_{h}]\\!]+\\{\\!\\!\\{p_{h}\\}\\!\\!\\}\right)u_{h}^{-}\,dx\,dt$
$\displaystyle-\int_{0}^{T}\\!\\!\int_{\Gamma_{l}}n^{-}\tilde{a}\left(\frac{1}{2}(\alpha-1)n^{-}p_{h}^{-}+\frac{1}{2}p_{h}^{-}\right)(u_{h}^{-}-u_{l})\,dx\,dt.$
Thus, the gradient of $\mathcal{J}$ is given by
${\mathcal{J}}^{\prime}(a)(\tilde{a})=\int_{\Omega}r^{\prime}(a)\tilde{a}\,dx+\int_{0}^{T}\\!\\!\int_{\Omega}G\tilde{a}\,dx\,dt+\sum_{\mathsf{f}}\int_{0}^{T}\\!\\!\int_{\mathsf{f}}g\tilde{a}\,dx\,dt,$
(22)
where $G$ is defined for each element ${\Omega^{e}}$, and $g$ for each inter-
element face as follows:
$\displaystyle G\tilde{a}=p_{h}{(\tilde{a}u_{h})}_{x}\quad\text{and}\quad
g=\begin{cases}\frac{1}{2}(\alpha-1)[\\![u_{h}]\\!][\\![p_{h}]\\!]+[\\![u_{h}]\\!]\\{\\!\\!\\{p_{h}\\}\\!\\!\\}&\
\text{if}\ \mathsf{f}\not\subset\Gamma,\\\ \frac{1}{2}\alpha
p_{h}^{-}(u_{h}^{-}-u_{l})&\ \text{if}\ \mathsf{f}\subset\Gamma_{l},\\\ 0&\
\text{if}\ \mathsf{f}\subset\Gamma_{r}.\end{cases}$
Note that the element boundary jump terms arising in $\mathcal{J}^{\prime}(a)$
are a consequence of using the dG method to discretize the state equation.
While in general these terms do not vanish, they become small as the
discretization resolves the continuous state and adjoint variables. However,
these terms must be taken into account to compute discretely exact gradients.
We continue with a series of remarks:
###### Remark 1
The boundary conditions for the adjoint variable $p_{h}$ that are weakly
imposed through the adjoint numerical flux are the Dirichlet condition
$ap_{h}(x_{r})=-j^{\prime}_{\Gamma}(u)$ at $\Gamma_{r}$, which, due to the
sign change for the advection term in (20) and (21), is an inflow boundary for
the adjoint equation. At the (adjoint) outflow boundary $\Gamma_{l}$, the
adjoint scheme can only be stable if ${j_{\Gamma}}|_{\Gamma_{l}}\equiv 0$.
This corresponds to the discussion from Section 2.2 on the compatibility of
boundary operators. The discrete adjoint scheme is consistent (in the sense
that the continuous adjoint variable $p$ satisfies the discrete adjoint
equation (20)) when $\alpha=0$, i.e., for a dG scheme based on an upwind
numerical flux. Thus, only dG discretizations based on upwind fluxes at the
boundary can be used in adjoint calculus. Hence, in the following we restrict
ourselves to upwind fluxes.
###### Remark 2
While the dG discretization of the state equation is in conservative form, the
discrete adjoint equation is not. Moreover, using dG method in strong form for
the state system, the adjoint system is naturally dG method in weak form (see
(20)), and element-wise integration by parts is necessary to find the adjoint
in strong form (21). Vice versa, using the weak form of dG for the state
equation, the adjoint equation is naturally in strong form. These two forms
can be numerically different if the integrals are approximated through a
quadrature rule for which integration by parts does not hold exactly. In this
case, integration by parts should be avoided to obtain exact discrete
derivatives.
###### Remark 3
The numerical fluxes (17) and (20b) differ by the sign in the upwinding term
only. Thus, an upwinding flux for the state equation becomes a downwinding
flux for the adjoint equation. This is natural since the advection velocity
for the adjoint equation is $-a$, which makes the adjoint numerical flux an
upwind flux for the adjoint equation.
### 3.2 Acoustic wave equation
Next, we derive expressions for the discrete gradient with respect to the
local wave speed in the acoustic wave equation. This is important, for
instance, in seismic inversion using full wave forms Fichtner11 ;
LekicRomanowicz11 ; EpanomeritakisAkccelikGhattasEtAl08 ;
PeterKomatitschLuoEtAl11 . If the dG method is used to discretize the wave
equation (as, e.g., in Bui-ThanhBursteddeGhattasEtAl12 ; Bui-
ThanhGhattasMartinEtAl13 ; CollisObervanBloemenWaanders10 ), the question on
the proper discretization of the adjoint equation and of the expressions for
the derivatives arises. Note that in Section 3.4 we present the discrete
derivatives with respect to the (possibly discontinuous) primary and secondary
wave speeds in the coupled acoustic-elastic wave equation, generalizing the
results presented in this section. However, for better readability we choose
to present this simpler case first and then present the results for the
coupled acoustic-elastic equation in compact form in Section 3.4.
We consider the acoustic wave equation written as first-order system as
follows:
$\displaystyle e_{t}-\nabla\cdot\boldsymbol{v}$ $\displaystyle=0\quad$
$\displaystyle\text{\ on\ }\Omega\times(0,T),$ (23a)
$\displaystyle\rho\boldsymbol{v}_{t}-\nabla(\lambda e)$
$\displaystyle=\boldsymbol{f}\quad$ $\displaystyle\text{\ on\
}\Omega\times(0,T),$ (23b) where $\boldsymbol{v}$ is the velocity, $e$ the
dilatation (trace of the strain tensor), $\rho=\rho(\boldsymbol{x})$ is the
mass density, and $\lambda=c^{2}\rho$, where $c(\boldsymbol{x})$ denotes the
wave speed. Together with (23a) and (23b), we assume the initial conditions
$\displaystyle
e(\boldsymbol{x},0)=e_{0}(\boldsymbol{x}),\>\boldsymbol{v}(\boldsymbol{x},0)=\boldsymbol{v}_{0}(\boldsymbol{x})\quad\text{\
for\ }\boldsymbol{x}\in\Omega,$ (23c) and the boundary conditions
$\displaystyle
e(\boldsymbol{x},t)=e_{\text{bc}}(\boldsymbol{x},t),\>\>\boldsymbol{v}(\boldsymbol{x},t)=\boldsymbol{v}_{\text{bc}}(\boldsymbol{x},t)\quad\text{\
for\ }(\boldsymbol{x},t)\in\Gamma\times(0,T).$ (23d)
Note that the dG method discussed below uses an upwind numerical flux, and
thus the boundary conditions (23) are automatically only imposed at inflow
boundaries. Through proper choice of $e_{\text{bc}}$ and
$\boldsymbol{v}_{\text{bc}}$ classical wave equation boundary conditions can
be imposed, e.g., WilcoxStadlerBursteddeEtAl10 ; FengTengChen07 .
The choice of the dilatation $e$ together with the velocity $\boldsymbol{v}$
in the first order system formulation is motivated from the strain-velocity
formulation used for the coupled elastic and acoustic wave equation in Section
3.4. To write (23a) and (23b) in second-order form, we define the pressure as
$p=-\lambda e$ and obtain the pressure-velocity form as
$\displaystyle p_{t}+\lambda\nabla\cdot\boldsymbol{v}$ $\displaystyle=0\quad$
$\displaystyle\text{\ on\ }\Omega\times(0,T),$
$\displaystyle\rho\boldsymbol{v}_{t}+\nabla p$
$\displaystyle=\boldsymbol{f}\quad$ $\displaystyle\text{\ on\
}\Omega\times(0,T),$
which is equivalent to the second-order formulation
$p_{tt}=\lambda\nabla\cdot\left(\frac{1}{\rho}\nabla
p\right)-\lambda\nabla\cdot\left(\frac{1}{\rho}\boldsymbol{f}\right)\quad\text{\
on\ }\Omega\times(0,T).$ (25)
The strong form dG discretization of (23) is: Find
$(e_{h},\boldsymbol{v}_{h})\in P^{h}\times Q^{h}$ satisfying the initial
conditions (23c) such that for all test functions
$(h_{h},\boldsymbol{w}_{h})\in P^{h}\times Q^{h}$ and for all $t\in(0,T)$
holds:
$\begin{split}&\int_{\Omega}(e_{t,h}-\nabla\cdot\boldsymbol{v}_{h})\lambda
h_{h}\,d\boldsymbol{x}+\int_{\Omega}(\rho\boldsymbol{v}_{t,h}-\nabla(\lambda
e_{h})-\boldsymbol{f})\cdot\boldsymbol{w}_{h}\,d\boldsymbol{x}\\\
&=-\sum_{e}\int_{\partial{\Omega^{e}}}\boldsymbol{n}^{-}\cdot\left(\boldsymbol{v}_{h}^{-}-\boldsymbol{v}_{h}^{\dagger}\right)\lambda
h_{h}^{-}+\left({(\lambda e_{h})}^{-}-{(\lambda
e_{h})}^{\dagger}\right)\boldsymbol{n}^{-}\cdot\boldsymbol{w}_{h}^{-}\,d\boldsymbol{x}.\end{split}$
(26)
Note that above, the inner product used for (23a) is weighted by $\lambda$,
which makes the first-order form of the wave equation a symmetric operator and
also allows for a natural interpretation of the adjoint variables, as shown
below. Assuming that $c$ and $\rho$ are continuous, we obtain the upwind
numerical fluxes:
$\displaystyle\boldsymbol{n}^{-}\cdot\boldsymbol{v}_{h}^{\dagger}$
$\displaystyle=\boldsymbol{n}^{-}\cdot\\{\\!\\!\\{\boldsymbol{v}_{h}\\}\\!\\!\\}-\frac{c}{2}[e_{h}],$
(27a) $\displaystyle{(\lambda e_{h})}^{\dagger}$
$\displaystyle=\lambda\\{\\!\\!\\{e_{h}\\}\\!\\!\\}-\frac{\rho
c}{2}[\\![\boldsymbol{v}_{h}]\\!].$ (27b)
Adding the boundary contributions from two adjacent elements ${\Omega^{e}}$
and ${\Omega^{e^{\prime}}}$ in (26) to a shared edge (in 2D) or face (in 3D),
one obtains
$\lambda[\\![\boldsymbol{v}_{h}]\\!]\\{\\!\\!\\{h_{h}\\}\\!\\!\\}+\frac{c\lambda}{2}[e_{h}][h_{h}]+\frac{\rho
c}{2}[\\![\boldsymbol{v}_{h}]\\!][\\![\boldsymbol{w}_{h}]\\!]+\lambda[\\![e_{h}]\\!]\cdot\\{\\!\\!\\{\boldsymbol{w}_{h}\\}\\!\\!\\}.$
(28)
We compute the discrete gradient with respect to the wave speed $c$ for a cost
functional of the form
$\tilde{\mathcal{J}}(c,\boldsymbol{v}_{h})=\int_{0}^{T}\\!\\!\int_{\Omega}j_{\Omega}(\boldsymbol{v}_{h})\,d\boldsymbol{x}\,dt+\int_{0}^{T}\\!\\!\int_{\Gamma}j_{\Gamma}(\boldsymbol{v}_{h})\,d\boldsymbol{x}\,dt+\int_{\Omega}r(c)\,d\boldsymbol{x},$
(29)
with differentiable functions $j_{\Omega}:\Omega\to\mathbb{R}$,
$j_{\Gamma}:\Gamma\to\mathbb{R}$ and $r:\Omega\to\mathbb{R}$. To ensure
compatibility as discussed in Section 2.2, the boundary term $j_{\Gamma}$ in
(29) can only involve outgoing characteristics. We introduce the Lagrangian
function, use that all its variations with respect to $\boldsymbol{v}$ and $e$
must vanish, and integrate by parts in time $t$, resulting in the following
adjoint equation: Find $(h_{h},\boldsymbol{w}_{h})\in P^{h}\times Q^{h}$
satisfying the finial time conditions $h(\boldsymbol{x},T)=0$,
$\boldsymbol{w}(\boldsymbol{x},T)=0$ for $\boldsymbol{x}\in\Omega$, such that
for all test functions $(\tilde{e}_{h},\tilde{}\boldsymbol{v}_{h})\in
P^{h}\times Q^{h}$ and for all $t\in(0,T)$ holds:
$\begin{split}&\int_{\Omega}-\tilde{e}_{h}\lambda
h_{h,t}-\nabla\cdot\tilde{}\boldsymbol{v}_{h}\lambda
h_{h}-\rho\tilde{}\boldsymbol{v}_{h}\cdot\boldsymbol{w}_{h,t}-\nabla(\lambda\tilde{e}_{h})\cdot\boldsymbol{w}_{h}+j_{\Omega}^{\prime}(\boldsymbol{v}_{h})\cdot\tilde{}\boldsymbol{v}_{h}\,d\boldsymbol{x}\\\
&=-\sum_{e}\int_{\partial{\Omega^{e}}}\left(\boldsymbol{n}^{-}\cdot\boldsymbol{w}_{h}^{\dagger}\right)\lambda\tilde{e}_{h}^{-}+{(\lambda
h_{h})}^{\dagger}\boldsymbol{n}^{-}\cdot\tilde{}\boldsymbol{v}_{h}^{-}\,d\boldsymbol{x}-\int_{\Gamma}j_{\Gamma}^{\prime}(\boldsymbol{v}_{h})\cdot\tilde{}\boldsymbol{v}_{h}\,d\boldsymbol{x},\end{split}$
(30)
where the adjoint numerical fluxes are given by
$\displaystyle\boldsymbol{n}^{-}\cdot\boldsymbol{w}_{h}^{\dagger}$
$\displaystyle=\boldsymbol{n}^{-}\cdot\\{\\!\\!\\{\boldsymbol{w}_{h}\\}\\!\\!\\}+\frac{c}{2}[h_{h}],$
(31a) $\displaystyle{(\lambda h_{h})}^{\dagger}$
$\displaystyle=\lambda\\{\\!\\!\\{h_{h}\\}\\!\\!\\}+\frac{\rho
c}{2}[\\![\boldsymbol{w}_{h}]\\!].$ (31b)
Note that (30) is the weak form of the dG discretization for an acoustic wave
equation, solved backwards in time. This is a consequence of the symmetry of
the differential operator in the acoustic wave equation, when considered in
the appropriate inner product. Comparing (31) and (27) shows that the adjoint
numerical flux (31) is the downwind flux in the adjoint variables for the
adjoint wave equation. The strong dG form corresponding to (30) can be
obtained by element-wise integration in parts in space.
Finally, we present expressions for the derivative of $\mathcal{J}$ with
respect to the wave speed $c$, which are found as variations of the Lagrangian
with respect to $c$. This results in
${\mathcal{J}}^{\prime}(c)(\tilde{c})=\int_{\Omega}r^{\prime}(c)\tilde{c}\,d\boldsymbol{x}+\int_{0}^{T}\\!\\!\int_{\Omega}G\tilde{c}\,d\boldsymbol{x}\,dt+\sum_{\mathsf{f}}\int_{0}^{T}\\!\\!\int_{\mathsf{f}}g\tilde{c}\,d\boldsymbol{x}\,dt,$
(32)
where $G$ is defined on each element ${\Omega^{e}}$, and $g$ for each inter-
element face $\mathsf{f}$ as follows:
$\displaystyle G\tilde{c}$ $\displaystyle=-2\nabla(\rho
c\tilde{c}e_{h})\cdot\boldsymbol{w}_{h}+2\rho
c\tilde{c}(e_{h,t}-\nabla\cdot\boldsymbol{v}_{h})h_{h},$ (33a) $\displaystyle
g$ $\displaystyle=2\rho
c[\\![\boldsymbol{v}_{h}]\\!]\\{\\!\\!\\{h_{h}\\}\\!\\!\\}+\frac{3}{2}\rho
c^{2}[e_{h}][h_{h}]+\frac{1}{2}\rho[\\![\boldsymbol{v}_{h}]\\!][\\![\boldsymbol{w}_{h}]\\!]+2\rho
c[\\![e_{h}]\\!]\cdot\\{\\!\\!\\{\boldsymbol{w}_{h}\\}\\!\\!\\}$ (33b)
Above, $(\boldsymbol{v}_{h},e_{h})$ is the solution of the state equation (26)
and $(\boldsymbol{w}_{h},h_{h})$ the solution of the adjoint equation (30).
Since the state equation (23) is satisfied in the dG sense, (33) simplifies
provided $\rho c\tilde{c}h_{h}\in P^{h}$, or if a quadrature method is used in
which the values of $\rho c\tilde{c}h_{h}$ at the quadrature points coincide
with the values of a function in $P^{h}$ at these points. The latter is, for
instance, always the case when the same nodes are used for the quadrature and
the nodal basis. Then, (33) simplifies to
$\displaystyle G\tilde{c}$ $\displaystyle=-2\nabla(\rho
c\tilde{c}e_{h})\cdot\boldsymbol{w}_{h},$ (34a) $\displaystyle g$
$\displaystyle=\frac{1}{2}\rho
c^{2}[e_{h}][h_{h}]+\frac{1}{2}\rho[\\![\boldsymbol{v}_{h}]\\!][\\![\boldsymbol{w}_{h}]\\!]+2\rho
c[\\![e_{h}]\\!]\cdot\\{\\!\\!\\{\boldsymbol{w}_{h}\\}\\!\\!\\}.$ (34b)
As for the one-dimensional advection problem (see Remark 3), the upwind flux
in the state equation becomes a downwind flux in the adjoint equation, and
thus an upwind flux for the adjoint equation when solved backwards in time.
###### Remark 4
As in the advection example, the discrete gradient has boundary contributions
that involve jumps of the dG variables at the element boundaries (see (33b)
and (34b)). These jumps are at the order of the dG approximation error and
thus tend to zero as the dG solution converges to the continuous solution
either through mesh refinement or improvement of the approximation on each
element.
### 3.3 Maxwell’s equations
Here we derive expressions for the discrete gradient with respect to the
current density in Maxwell’s equations (specifically boundary current density
in our case). This can be used, for instance, in the determination and
reconstruction of antennas from boundary field measurements Nicaise00 and
controlling electromagnetic fields using currents Lagnese89 ; Yousept12 . The
time-dependent Maxwell’s equations in a homogeneous isotropic dielectric
domain $\Omega\subset\mathbb{R}^{3}$ is given by:
$\displaystyle\mu\boldsymbol{H}_{t}$
$\displaystyle=-\nabla\times\boldsymbol{E}$ $\displaystyle\qquad\text{on}\
\Omega\times(0,T),$ (35a) $\displaystyle\epsilon\boldsymbol{E}_{t}$
$\displaystyle=\phantom{-}\nabla\times\boldsymbol{H}$
$\displaystyle\qquad\text{on}\ \Omega\times(0,T),$ (35b)
$\displaystyle\nabla\cdot\boldsymbol{H}$ $\displaystyle=0$
$\displaystyle\qquad\text{on}\ \Omega\times(0,T),$ (35c)
$\displaystyle\nabla\cdot\boldsymbol{E}$ $\displaystyle=0$
$\displaystyle\qquad\text{on}\ \Omega\times(0,T),$ (35d) where
$\boldsymbol{E}$ is the electric field and $\boldsymbol{H}$ is the magnetic
field. Moreover, $\mu$ is the permeability and $\epsilon$ is the permittivity,
which can both be discontinuous across element interfaces. The impedance $Z$
and the conductance $Y$ of the material are defined as
$Z=\frac{1}{Y}=\sqrt{\frac{\mu}{\epsilon}}$. Note that we follow a standard
notation for Maxwell’s equation, in which the vectors $\boldsymbol{H}$ and
$\boldsymbol{E}$ are denoted by bold capital letters. Together with equations
(35a)–(35d), we assume the initial conditions
$\displaystyle\boldsymbol{E}(\boldsymbol{x},0)=\boldsymbol{E}_{0}(\boldsymbol{x}),\>\boldsymbol{H}(\boldsymbol{x},0)=\boldsymbol{H}_{0}(\boldsymbol{x})\quad\text{on}\
\Omega,$ (35e) and boundary conditions
$\displaystyle\boldsymbol{n}\times\boldsymbol{H}=-\boldsymbol{J}_{s}\quad\text{on}\
\Gamma.$ (35f) This classic boundary condition can be converted to equivalent
inflow characteristic boundary conditions TengLinChangEtAl08 . Here,
$\boldsymbol{J}_{s}(\boldsymbol{x},t)$ is a spatially (and possibly time-
dependent) current density flowing tangentially to the boundary.
If the initial conditions satisfy the divergence conditions (35c) and (35d),
the time evolved solution will as well HesthavenWarburton02 . Thus, the
divergence conditions can be regarded as a consistency condition on the
initial conditions. We consider a dG discretization of Maxwell’s equations hat
only involves equations (35a) and (35b) explicitly. The dG solution then
satisfies the divergence conditions up to discretization error. The strong
form dG discretization of equation (35) is: Find
$(\boldsymbol{H}_{h},\boldsymbol{E}_{h})\in P^{h}\times Q^{h}$ satisfying the
initial conditions (35e), such that
$\begin{split}\int_{\Omega}(\mu\boldsymbol{H}_{h,t}+\nabla\times\boldsymbol{E}_{h})\cdot\boldsymbol{G}_{h}\,d\boldsymbol{x}+\int_{\Omega}(\epsilon\boldsymbol{E}_{h,t}-\nabla\times\boldsymbol{H}_{h})\cdot\boldsymbol{F}_{h}\,d\boldsymbol{x}=\qquad\qquad\\\
\sum_{e}\\!\int_{\partial{\Omega^{e}}}\\!\\!\\!\\!\\!-\left(\boldsymbol{n}^{-}\\!\times\\!(\boldsymbol{E}_{h}^{\dagger}-\boldsymbol{E}_{h}^{-})\right)\cdot\boldsymbol{G}_{h}\,d\boldsymbol{x}+\sum_{e}\\!\int_{\partial{\Omega^{e}}}\\!\\!\\!\left(\boldsymbol{n}^{-}\times(\boldsymbol{H}_{h}^{\dagger}-\boldsymbol{H}_{h}^{-})\right)\cdot\boldsymbol{F}_{h}\,d\boldsymbol{x}\end{split}$
(36)
for all $(\boldsymbol{G}_{h},\boldsymbol{F}_{h})\in P^{h}\times Q^{h}$, and
for all $t\in(0,T)$. The upwind numerical flux states are given such that
$\displaystyle\boldsymbol{n}^{-}\times(\boldsymbol{E}_{h}^{\dagger}-\boldsymbol{E}_{h}^{-})$
$\displaystyle=-\frac{1}{2\\{\\!\\!\\{Y\\}\\!\\!\\}}\boldsymbol{n}^{-}\times(Y^{+}[\boldsymbol{E}_{h}]+\boldsymbol{n}^{-}\times[\boldsymbol{H}_{h}]),$
(37a)
$\displaystyle\boldsymbol{n}^{-}\times(\boldsymbol{H}_{h}^{\dagger}-\boldsymbol{H}_{h}^{-})$
$\displaystyle=-\frac{1}{2\\{\\!\\!\\{Z\\}\\!\\!\\}}\boldsymbol{n}^{-}\times(Z^{+}[\boldsymbol{H}_{h}]-\boldsymbol{n}^{-}\times[\boldsymbol{E}_{h}]).$
(37b)
The boundary conditions (35f) are imposed via the upwind numerical flux by
setting exterior values on the boundary of the domain $\Gamma$ such that
$\displaystyle\boldsymbol{H}_{h}^{+}$
$\displaystyle=-\boldsymbol{H}_{h}^{-}+2\boldsymbol{J}_{s},$ (38a)
$\displaystyle\boldsymbol{E}_{h}^{+}$ $\displaystyle=\boldsymbol{E}_{h}^{-},$
(38b)
with the continuously extended material parameters $Y^{+}=Y^{-}$ and
$Z^{+}=Z^{-}$. Using the upwind numerical flux implicitly means that the
boundary conditions are only set on the incoming characteristics.
Next, we compute the discrete adjoint equation and the gradient with respect
to the boundary current density $\boldsymbol{J}_{s}$ for an objective
functional of the form
$\tilde{\mathcal{J}}(\boldsymbol{J}_{s},\boldsymbol{E}_{h})=\int_{0}^{T}\\!\\!\int_{\Omega}j_{\Omega}(\boldsymbol{E}_{h})\,d\boldsymbol{x}\,dt+\int_{0}^{T}\\!\\!\int_{\Gamma}r(\boldsymbol{J}_{s})\,d\boldsymbol{x}\,dt,$
(39)
with differentiable functions $j_{\Omega}:\Omega\to\mathbb{R}$ and
$r:\Gamma\to\mathbb{R}$. We introduce the Lagrangian function, and derive the
adjoint equation by imposing that all variations of the Lagrangian with
respect to $\boldsymbol{H}_{h}$ and $\boldsymbol{E}_{h}$ must vanish. After
integration by parts in time $t$, this results in the following adjoint
equation: Find $(\boldsymbol{G}_{h},\boldsymbol{F}_{h})\in P^{h}\times Q^{h}$
such that
$\begin{split}\int_{\Omega}\mu\boldsymbol{G}_{h,t}\cdot\tilde{}\boldsymbol{H}_{h}+\boldsymbol{F}_{h}\cdot(\nabla\times\tilde{}\boldsymbol{H}_{h})\,d\boldsymbol{x}+\int_{\Omega}\epsilon\boldsymbol{F}_{h,t}\cdot\tilde{}\boldsymbol{E}_{h}-\boldsymbol{G}_{h}\cdot(\nabla\times\tilde{}\boldsymbol{E}_{h})\,d\boldsymbol{x}=\qquad\qquad\\\
\sum_{e}\\!\int_{\partial{\Omega^{e}}}\\!\\!\\!\\!-\left(\boldsymbol{n}^{-}\times\boldsymbol{F}_{h}^{\dagger}\right)\cdot\tilde{}\boldsymbol{H}_{h}\,d\boldsymbol{x}+\sum_{e}\\!\int_{\partial{\Omega^{e}}}\left(\boldsymbol{n}^{-}\times\boldsymbol{G}_{h}^{\dagger}\right)\cdot\tilde{}\boldsymbol{E}_{h}\,d\boldsymbol{x}-\int_{\Omega}j_{\Omega}^{\prime}(\boldsymbol{E}_{h})\cdot\tilde{}\boldsymbol{E}_{h}\,d\boldsymbol{x}\end{split}$
(40)
for all $(\tilde{}\boldsymbol{H}_{h},\tilde{}\boldsymbol{E}_{h})\in
P^{h}\times Q^{h}$ and all $t\in(0,T)$, with the final time conditions
$\boldsymbol{G}_{h}(\boldsymbol{x},T)=0,\>\boldsymbol{F}_{h}(\boldsymbol{x},T)=0\quad\text{on}\
\Omega,$ (41)
and the adjoint numerical flux states
$\displaystyle\boldsymbol{F}_{h}^{\dagger}$
$\displaystyle=\frac{\\{\\!\\!\\{Y\boldsymbol{F}_{h}\\}\\!\\!\\}}{\\{\\!\\!\\{Y\\}\\!\\!\\}}+\frac{1}{2\\{\\!\\!\\{Y\\}\\!\\!\\}}\left(\boldsymbol{n}^{-}\times[\boldsymbol{G}_{h}]\right),$
(42a) $\displaystyle\boldsymbol{G}_{h}^{\dagger}$
$\displaystyle=\frac{\\{\\!\\!\\{Z\boldsymbol{G}_{h}\\}\\!\\!\\}}{\\{\\!\\!\\{Z\\}\\!\\!\\}}-\frac{1}{2\\{\\!\\!\\{Z\\}\\!\\!\\}}\left(\boldsymbol{n}^{-}\times[\boldsymbol{F}_{h}]\right),$
(42b)
with exterior values on the boundary $\Gamma$ given by
$\boldsymbol{G}_{h}^{+}=-\boldsymbol{G}_{h}^{-}$ and
$\boldsymbol{F}_{h}^{+}=\boldsymbol{F}_{h}^{-}$. These exterior states enforce
the continuous adjoint boundary condition
$\boldsymbol{n}\times\boldsymbol{G}=0\quad\text{on}\ \Gamma.$
To compare with the numerical flux (37) of the state equation, we rewrite the
adjoint numerical flux states as
$\displaystyle\boldsymbol{n}^{-}\times(\boldsymbol{F}_{h}^{\dagger}-\boldsymbol{F}_{h}^{-})$
$\displaystyle=-\frac{1}{2\\{\\!\\!\\{Y\\}\\!\\!\\}}\boldsymbol{n}^{-}\times(Y^{+}[\boldsymbol{F}_{h}]-\boldsymbol{n}^{-}\times[\boldsymbol{G}_{h}]),$
$\displaystyle\boldsymbol{n}^{-}\times(\boldsymbol{G}_{h}^{\dagger}-\boldsymbol{G}_{h}^{-})$
$\displaystyle=-\frac{1}{2\\{\\!\\!\\{Z\\}\\!\\!\\}}\boldsymbol{n}^{-}\times(Z^{+}[\boldsymbol{G}_{h}]+\boldsymbol{n}^{-}\times[\boldsymbol{F}_{h}]).$
Note that, even with discontinuities in the material parameters, (40) is the
weak form of the dG discretization for a Maxwell’s system solved backwards in
time. As in the acoustic example, the adjoint numerical flux states (42) come
from the downwind flux.
Differentiating the Lagrangian with respect to the boundary current
$\boldsymbol{J}_{s}$ yields an equation for the derivative in direction
$\tilde{}\boldsymbol{J}_{s}$, namely
${\mathcal{J}}^{\prime}(\boldsymbol{J}_{s})(\tilde{}\boldsymbol{J}_{s})=\int_{0}^{T}\\!\\!\\!\int_{\Gamma}r^{\prime}(\boldsymbol{J}_{s})\cdot\tilde{}\boldsymbol{J}_{s}\,d\boldsymbol{x}+\int_{0}^{T}\\!\\!\int_{\Gamma}\boldsymbol{g}\cdot\tilde{}\boldsymbol{J}_{s}\,d\boldsymbol{x}\,dt$
(44a) with
$\boldsymbol{g}=\boldsymbol{n}^{-}\times\boldsymbol{F}_{h}+\frac{1}{Y}\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times\boldsymbol{G}_{h}\right)\right),$
(44b)
where $(\boldsymbol{G}_{h},\boldsymbol{F}_{h})$ is the solution of the
discrete adjoint equation (40).
###### Remark 5
Since the boundary force $\boldsymbol{J}_{s}$ enters linearly in Maxwell’s
equation, the gradient expression (44) does not involve contributions from the
element boundaries as for the advection and the acoustic wave example.
### 3.4 Coupled elastic-acoustic wave equation
Finally, we present expressions for the derivatives with respect to the
primary and secondary wave speeds in the coupled acoustic-elastic wave
equation. This section generalizes Section 3.2 to the coupled acoustic-elastic
wave equation. We derive derivative expressions with respect to both wave
speeds, and allow for discontinuous wave speeds across elements. We only
present a condensed form of the derivations, and verify our results for the
gradient numerically in Section 4.
The coupled linear elastic-acoustic wave equation for isotropic material
written in first-order velocity strain form is given as
$\displaystyle\boldsymbol{E}_{t}$
$\displaystyle=\frac{1}{2}\left(\nabla\boldsymbol{v}+\nabla\boldsymbol{v}^{T}\right)$
$\displaystyle\qquad\text{\ on\ }\Omega\times(0,T)$ (45a)
$\displaystyle\rho\boldsymbol{v}_{t}$
$\displaystyle=\nabla\cdot\left(\lambda\operatorname{tr}(\boldsymbol{E})\boldsymbol{I}+2\mu\boldsymbol{E}\right)+\rho\boldsymbol{f}$
$\displaystyle\qquad\text{\ on\ }\Omega\times(0,T)$ (45b) where
$\boldsymbol{E}$ is the strain tensor, $\boldsymbol{v}$ is the displacement
velocity, $\boldsymbol{I}$ is the identity tensor, $\rho=\rho(\boldsymbol{x})$
is the mass density, $\boldsymbol{f}$ is a body force per unit mass, and
$\lambda=\lambda(\boldsymbol{x})$ and $\mu=\mu(\boldsymbol{x})$ are the Lamé
parameters. In addition to the conditions (45a) and (45b) on the body
$\Omega$, we assume the initial conditions
$\displaystyle\boldsymbol{v}(0,\boldsymbol{x})=\boldsymbol{v}_{0}(\boldsymbol{x}),\>\boldsymbol{E}(0,\boldsymbol{x})=\boldsymbol{E}_{0}(\boldsymbol{x}),\quad\text{\
for\ }\boldsymbol{x}\in\Omega,$ (45c) and the boundary conditions
$\displaystyle\SS(\boldsymbol{x},t)\boldsymbol{n}=\boldsymbol{t}^{\text{bc}}(t)\qquad\text{on}\
\Gamma.$ (45d) Here, $\boldsymbol{t}^{\text{bc}}$ is the traction on the
boundary of the body.
The stress tensor $\SS$ is related to the strain through the constitutive
relation (here, $\boldsymbol{\mathsf{C}}$ is the forth-order constitutive
tensor):
$\SS=\boldsymbol{\mathsf{C}}\boldsymbol{E}=\lambda\operatorname{tr}(\boldsymbol{E})\boldsymbol{I}+2\mu\boldsymbol{E},$
(46)
where $\operatorname{tr}(\cdot)$ is the trace operator. There are also
boundary conditions at material interfaces. For an elastic-elastic interface
$\Gamma^{\text{ee}}$ the boundary conditions are
$\displaystyle\boldsymbol{v}^{+}=\boldsymbol{v}^{-},\quad\SS^{+}\boldsymbol{n}^{-}$
$\displaystyle=\SS^{-}\boldsymbol{n}^{-}\qquad\text{on}\ \Gamma^{\text{ee}}$
and for acoustic-elastic and acoustic-acoustic interfaces
$\Gamma^{\text{ae}}$, the boundary conditions are
$\displaystyle\boldsymbol{n}\cdot\boldsymbol{v}^{+}=\boldsymbol{n}\cdot\boldsymbol{v}^{-},\quad\SS^{+}\boldsymbol{n}^{-}=\SS^{-}\boldsymbol{n}^{-}\qquad\text{on}\
\Gamma^{\text{ae}}.$
The strong form dG discretization of equation (45) is: Find
$(\boldsymbol{E}_{h},\boldsymbol{v}_{h})\in P^{h}\times Q^{h}$ such that
$\int_{\Omega}\boldsymbol{E}_{h,t}:\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}\,d\boldsymbol{x}+\int_{\Omega}\rho\boldsymbol{v}_{h,t}\cdot\boldsymbol{w}_{h}\,d\boldsymbol{x}-\int_{\Omega}\operatorname{sym}(\nabla\boldsymbol{v}_{h}):\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}\,d\boldsymbol{x}\\\
-\int_{\Omega}\left(\nabla\cdot(\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h})+\boldsymbol{f}\right)\cdot\boldsymbol{w}_{h}\,d\boldsymbol{x}=\sum_{e}\int_{\partial{\Omega^{e}}}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{v}_{h}^{\dagger}-\boldsymbol{v}_{h}^{-}\right)\right):\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}\,d\boldsymbol{x}\\\
+\sum_{e}\int_{\partial{\Omega^{e}}}\left(\left({(\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h})}^{\dagger}-{(\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h})}^{-}\right)\boldsymbol{n}^{-}\right)\cdot\boldsymbol{w}_{h}\,d\boldsymbol{x}$
(49)
for all $(\boldsymbol{H}_{h},\boldsymbol{w}_{h})\in P^{h}\times Q^{h}$ where
$\operatorname{sym}$ is the mapping to get the symmetric part of a tensor,
i.e.,
$\operatorname{sym}(\boldsymbol{A})=\frac{1}{2}\left(\boldsymbol{A}+\boldsymbol{A}^{T}\right)$.
Note that the constitutive tensor $\boldsymbol{\mathsf{C}}$ is used in the
inner product for the weak form. Here, the upwind states are given such that
$\displaystyle\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{v}_{h}^{\dagger}-\boldsymbol{v}_{h}^{-}\right)\right)$
$\displaystyle={-k_{0}\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]+\rho^{+}c_{p}^{+}[\\![\boldsymbol{v}_{h}]\\!]\right)}\left(\boldsymbol{n}^{-}\otimes\boldsymbol{n}^{-}\right)$
$\displaystyle\quad+k_{1}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]\right)\right)\right)$
$\displaystyle\quad+k_{1}\rho^{+}c_{s}^{+}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\boldsymbol{v}_{h}]\right)\right)\right),$
(50a)
$\displaystyle\left({(\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h})}^{\dagger}-{(\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h})}^{-}\right)\boldsymbol{n}^{-}$
$\displaystyle={-k_{0}\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]+\rho^{+}c_{p}^{+}[\\![\boldsymbol{v}_{h}]\\!]\right)}\rho^{-}c_{p}^{-}\boldsymbol{n}^{-}$
$\displaystyle\quad+k_{1}\rho^{-}c_{s}^{-}\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]\right)$
$\displaystyle\quad+k_{1}\rho^{+}c_{s}^{+}\rho^{-}c_{s}^{-}\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\boldsymbol{v}_{h}]\right),$
(50b) with $k_{0}=1/(\rho^{-}c_{p}^{-}+\rho^{+}c_{p}^{+})$ and $\displaystyle
k_{1}$
$\displaystyle=\begin{dcases}\frac{1}{\rho^{-}c_{s}^{-}+\rho^{+}c_{s}^{+}}&\text{when}\
\mu^{-}\neq 0,\\\ 0&\text{when}\ \mu^{-}=0,\end{dcases}$
where $c_{p}:=\sqrt{(\lambda+2\mu)/\rho}$ is the primary wave speed and
$c_{s}:=\sqrt{\mu/\rho}$ is the secondary wave speed. The traction boundary
conditions are imposed through the upwind numerical flux by setting exterior
values on $\Gamma$ to
$\displaystyle\boldsymbol{v}_{h}^{+}$ $\displaystyle=\boldsymbol{v}_{h}^{-},$
$\displaystyle\boldsymbol{\mathsf{C}}^{+}\boldsymbol{E}_{h}^{+}\boldsymbol{n}^{+}$
$\displaystyle=\begin{cases}-2\boldsymbol{t}^{\text{bc}}+\boldsymbol{\mathsf{C}}^{-}\boldsymbol{E}_{h}^{-}\boldsymbol{n}^{-}&\
\text{if $\mu^{-}\not=0$},\\\
-2\left(\boldsymbol{n}^{-}\cdot\left(\boldsymbol{t}^{\text{bc}}-\boldsymbol{\mathsf{C}}^{-}\boldsymbol{E}_{h}^{-}\boldsymbol{n}^{-}\right)\right)\boldsymbol{n}^{-}-\boldsymbol{\mathsf{C}}^{-}\boldsymbol{E}_{h}^{-}\boldsymbol{n}^{-}&\
\text{if $\mu^{+}=0$},\end{cases}$
with the continuously extended material parameters $\rho^{+}=\rho^{-}$,
$\mu^{+}=\mu^{-}$, and $\lambda^{+}=\lambda^{-}$.
We assume a cost function that depends on the primary and secondary wave
speeds $c_{p}$ and $c_{s}$ through the solution $\boldsymbol{v}_{h}$ of (49)
${\mathcal{J}}(c_{p},c_{s})=\int_{0}^{T}\\!\\!\\!\int_{\Omega}j_{\Omega}(\boldsymbol{v}_{h})\,d\boldsymbol{x}\,dt+\int_{\Omega}r_{p}(c_{p})\,d\boldsymbol{x}+\int_{\Omega}r_{s}(c_{s})\,d\boldsymbol{x}.$
(51)
By using a sum of spatial Dirac delta distributions in $j_{\Omega}(\cdot)$,
this can include seismogram data, as common in seismic inversion. Using the
Lagrangian function and integration by parts in time, we obtain the following
adjoint equation: Find $(\boldsymbol{H}_{h},\boldsymbol{w}_{h})\in P^{h}\times
Q^{h}$ such that
$\int_{\Omega}-\boldsymbol{H}_{h,t}:\boldsymbol{\mathsf{C}}\tilde{\boldsymbol{E}}_{h}\,d\boldsymbol{x}-\int_{\Omega}\rho\boldsymbol{w}_{h,t}\cdot\tilde{\boldsymbol{v}}_{h}\,d\boldsymbol{x}-\int_{\Omega}\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}:\operatorname{sym}(\nabla\tilde{\boldsymbol{v}}_{h})\,d\boldsymbol{x}\\\
-\int_{\Omega}\boldsymbol{w}_{h}\cdot\left(\nabla\cdot(\boldsymbol{\mathsf{C}}\tilde{\boldsymbol{E}}_{h})\right)\,d\boldsymbol{x}=-\sum_{e}\int_{\partial{\Omega^{e}}}\operatorname{sym}(\boldsymbol{n}^{-}\otimes\boldsymbol{w}_{h}^{\dagger}):\boldsymbol{\mathsf{C}}\tilde{\boldsymbol{E}}_{h}\,d\boldsymbol{x}\\\
-\sum_{e}\int_{\partial{\Omega^{e}}}\left({(\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h})}^{\dagger}\boldsymbol{n}^{-}\right)\cdot\tilde{\boldsymbol{v}}_{h}\,d\boldsymbol{x},-\int_{\Omega}j_{\Omega}(\boldsymbol{v}_{h})\cdot\tilde{\boldsymbol{v}}_{h}$
(52)
for all $(\tilde{\boldsymbol{E}}_{h},\tilde{\boldsymbol{v}}_{h})\in
P^{h}\times Q^{h}$ in with final conditions $\boldsymbol{w}_{h}(T)=0$ and
$\boldsymbol{H}_{h}(T)=0$ the fluxes are given by
$\displaystyle\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\boldsymbol{w}_{h}^{\dagger}\right)$
$\displaystyle=k_{0}\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]+\boldsymbol{n}^{-}\cdot\left(2\\{\\!\\!\\{\rho
c_{p}\boldsymbol{w}_{h}\\}\\!\\!\\}\right)\right)\left(\boldsymbol{n}^{-}\otimes\boldsymbol{n}^{-}\right)$
$\displaystyle\quad-
k_{1}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]\right)\right)\right)$
$\displaystyle\quad-
k_{1}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times\left(2\\{\\!\\!\\{\rho
c_{s}\boldsymbol{w}_{h}\\}\\!\\!\\}\right)\right)\right)\right),$
$\displaystyle{\left(\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}\right)}^{\dagger}\boldsymbol{n}^{-}$
$\displaystyle=k_{0}\left(\boldsymbol{n}^{-}\cdot\left(\left(\rho^{+}c_{p}^{+}\boldsymbol{\mathsf{C}}^{-}\boldsymbol{H}_{h}^{-}+\rho^{-}c_{p}^{-}\boldsymbol{\mathsf{C}}^{+}\boldsymbol{H}_{h}^{+}\right)\boldsymbol{n}^{-}\right)+\rho^{-}c_{p}^{-}\rho^{+}c_{p}^{+}[\\![\boldsymbol{w}_{h}]\\!]\right)\boldsymbol{n}^{-}$
$\displaystyle\quad-
k_{1}\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times\left(\left(\rho^{+}c_{s}^{+}\boldsymbol{\mathsf{C}}^{-}\boldsymbol{H}_{h}^{-}+\rho^{-}c_{s}^{-}\boldsymbol{\mathsf{C}}^{+}\boldsymbol{H}_{h}^{+}\right)\boldsymbol{n}^{-}\right)\right)$
$\displaystyle\quad-
k_{1}\rho^{-}c_{s}^{-}\rho^{+}c_{s}^{+}\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\boldsymbol{w}_{h}]\right).$
We can rewrite this into a form similar to the upwind states of the state
equation (50) as
$\displaystyle\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{w}_{h}^{\dagger}-\boldsymbol{w}_{h}^{-}\right)\right)$
$\displaystyle=k_{0}\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]-\rho^{+}c_{p}^{+}[\\![\boldsymbol{w}_{h}]\\!]\right)\left(\boldsymbol{n}^{-}\otimes\boldsymbol{n}^{-}\right)$
$\displaystyle\quad-
k_{1}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]\right)\right)\right)$
$\displaystyle\quad+k_{1}\rho^{+}c_{s}^{+}\operatorname{sym}\left(\boldsymbol{n}^{-}\otimes\left(\boldsymbol{n}^{-}\times\left(\boldsymbol{n}^{-}\times[\boldsymbol{w}_{h}]\right)\right)\right),$
$\displaystyle\left({(\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h})}^{\dagger}-{(\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h})}^{-}\right)\boldsymbol{n}^{-}$
$\displaystyle=k_{0}\left(-\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]+\rho^{+}c_{p}^{+}[\\![\boldsymbol{w}_{h}]\\!]\right)\rho^{-}c_{p}^{-}\boldsymbol{n}^{-}$
$\displaystyle\quad+k_{1}\rho^{-}c_{s}^{-}\boldsymbol{n}^{-}\times(\boldsymbol{n}^{-}\times[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!])$
$\displaystyle\quad-
k_{1}\rho^{-}c_{s}^{-}\rho^{+}c_{s}^{+}\boldsymbol{n}^{-}\times(\boldsymbol{n}^{-}\times[\boldsymbol{w}_{h}]).$
Here, the adjoint boundary conditions are imposed through the adjoint
numerical flux by setting exterior values on $\Gamma$ to
$\displaystyle\boldsymbol{w}_{h}^{+}$ $\displaystyle=\boldsymbol{w}_{h}^{-},$
$\displaystyle\boldsymbol{\mathsf{C}}^{+}\boldsymbol{H}_{h}^{+}\boldsymbol{n}^{+}$
$\displaystyle=\begin{cases}\phantom{-}\boldsymbol{\mathsf{C}}^{-}\boldsymbol{H}_{h}^{-}\boldsymbol{n}^{-}&\
\text{if $\mu^{-}\not=0$},\\\
-\boldsymbol{\mathsf{C}}^{-}\boldsymbol{H}_{h}^{-}\boldsymbol{n}^{-}+2\left(\boldsymbol{n}^{-}\cdot\left(\boldsymbol{\mathsf{C}}^{-}\boldsymbol{H}_{h}^{-}\boldsymbol{n}^{-}\right)\right)\boldsymbol{n}^{-}&\
\text{if $\mu^{-}=0$},\end{cases}$
with the continuously extended material parameters $\rho^{+}=\rho^{-}$,
$\mu^{+}=\mu^{-}$, and $\lambda^{+}=\lambda^{-}$.
We assume a discretization of $c_{p}$ and $c_{s}$ and a numerical quadrature
rule such that the state equation can be used to simplify the expression for
the gradient; see the discussion in Section 3.2. The discrete gradient with
respect to $c_{p}$ is then
${\mathcal{J}}_{c_{p}}(c_{p},c_{s})(\tilde{c}_{p})=\int_{\Omega}r_{p}^{\prime}(c_{p})\tilde{c}_{p}\,d\boldsymbol{x}+\int_{0}^{T}\\!\\!\int_{\Omega}G_{p}\tilde{c}_{p}\,d\boldsymbol{x}\,dt+\sum_{\partial{\Omega^{e}}}\int_{0}^{T}\\!\\!\int_{\partial{\Omega^{e}}}g_{p}\tilde{c}^{-}_{p}\,d\boldsymbol{x}\,dt,$
(53a) where $G_{p}$ is defined on each element ${\Omega^{e}}$, and $g_{p}$ for
each element boundary as follows:
$\begin{split}G_{p}\tilde{c}_{p}&=-2\left(\nabla\left(\rho{c_{p}}{\tilde{c}_{p}}\operatorname{tr}(\boldsymbol{E}_{h})\right)\right)\cdot\boldsymbol{w}_{h},\\\
g_{p}&=-k_{0}^{2}\rho^{-}\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]+k_{0}^{2}\rho^{-}{\left(\rho^{+}{c_{p}}^{+}\right)}^{2}[\\![\boldsymbol{v}_{h}]\\!][\\![\boldsymbol{w}_{h}]\\!]\\\
&\quad+k_{0}^{2}\rho^{-}\rho^{+}{c_{p}}^{+}\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!][\\![\boldsymbol{w}_{h}]\\!]-[\\![\boldsymbol{v}_{h}]\\!]\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]\right)\\\
&\quad+2k_{0}\rho^{-}{c_{p}}^{-}\operatorname{tr}(\boldsymbol{E}_{h}^{-})\left(\boldsymbol{n}^{-}\cdot[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]-\rho^{+}{c_{p}}^{+}[\\![\boldsymbol{w}_{h}]\\!]\right)\\\
&\quad+2\rho^{-}c_{p}^{-}\operatorname{tr}(\boldsymbol{E}_{h}^{-})\boldsymbol{w}_{h}^{-}\cdot\boldsymbol{n}^{-},\end{split}$
(53b)
where $(\boldsymbol{v}_{h},\boldsymbol{E}_{h})$ is the solution of the state
equation (49) and $(\boldsymbol{w}_{h},\boldsymbol{H}_{h})$ is the solution of
the adjoint equation (52). The discrete gradient of $\mathcal{J}$ with respect
to $c_{s}$ is
${\mathcal{J}}_{c_{s}}(c_{p},c_{s})(\tilde{c}_{s})=\int_{\Omega}r_{s}^{\prime}(c_{s})\tilde{c}_{s}\,d\boldsymbol{x}+\int_{0}^{T}\\!\\!\int_{\Omega}G_{s}\tilde{c}_{s}\,d\boldsymbol{x}\,dt+\sum_{\partial{\Omega^{e}}}\int_{0}^{T}\\!\\!\int_{\partial{\Omega^{e}}}g_{s}\tilde{c}^{-}_{s}\,d\boldsymbol{x}\,dt,$
(54a) where $G_{s}$ is defined on each element ${\Omega^{e}}$, and $g_{s}$ for
each element boundary as follows:
$\begin{split}G_{s}\tilde{c}_{s}&=-4\left(\nabla\cdot\left(\rho
c_{s}\tilde{c}_{s}\left(\boldsymbol{E}_{h}-\operatorname{tr}(\boldsymbol{E}_{h})\boldsymbol{I}\right)\right)\right)\cdot\boldsymbol{w}_{h},\\\
g_{s}&=-k_{1}^{2}\rho^{-}\\!\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{[\\![\boldsymbol{\mathsf{C}}\boldsymbol{E}_{h}]\\!]}\right)\right)\cdot\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{\big{(}[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]-\rho^{+}c_{s}^{+}[\boldsymbol{w}_{h}]\big{)}}\right)\right)\\\
&\quad-
k_{1}^{2}\rho^{-}\rho^{+}{c_{s}}^{+}\\!\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{[\boldsymbol{v}_{h}]}\right)\right)\cdot\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{\big{(}[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]-\rho^{+}c_{s}^{+}[\boldsymbol{w}_{h}]\big{)}}\right)\right)\\\
&\quad+4k_{0}\rho^{-}{c_{s}}^{-}\\!\left(\boldsymbol{n}^{-}\cdot\boldsymbol{E}_{h}^{-}\boldsymbol{n}^{-}-\operatorname{tr}(\boldsymbol{E}_{h}^{-})\right)\left(\boldsymbol{n}^{-}\cdot\left([\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]-\rho^{+}{c_{p}}^{+}[\boldsymbol{w}_{h}]\right)\right)\\\
&\quad+4k_{1}\rho^{-}{c_{s}}^{-}\\!\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{\big{(}\boldsymbol{E}_{h}^{-}\boldsymbol{n}^{-}\big{)}}\right)\right)\\!\cdot\\!\left(\boldsymbol{n}^{-}\\!\times\\!\left(\boldsymbol{n}^{-}\\!\times{\big{(}[\\![\boldsymbol{\mathsf{C}}\boldsymbol{H}_{h}]\\!]\\!-\\!\rho^{+}{c_{s}}^{+}[\boldsymbol{w}_{h}]\big{)}}\right)\right)\\\
&\quad+4\rho^{-}c_{s}^{-}\\!\left(\left(\boldsymbol{E}_{h}^{-}-\operatorname{tr}(\boldsymbol{E}_{h}^{-})\boldsymbol{I}\right)\boldsymbol{n}^{-}\right)\cdot\boldsymbol{w}_{h}^{-},\end{split}$
(54b)
where $(\boldsymbol{v}_{h},\boldsymbol{E}_{h})$ is the solution of the state
equation (49) and $(\boldsymbol{w}_{h},\boldsymbol{H}_{h})$ is the solution of
the adjoint equation (52). Note that since we allow for discontinuous wave
speeds $c_{p}$ and $c_{s}$ (and perturbations $\tilde{c}_{p}$ and
$\tilde{c}_{s}$), the boundary contributions to the gradients, i.e., the last
terms in (53a) and (54a) are written as sums of integrals over individual
element boundaries, i.e., each boundary face appears twice in the overall sum.
This differs from the previous examples, where we assumed continuous
parameters and thus combined contributions from adjacent elements to shared
faces $\mathsf{f}$.
###### Remark 6
The expressions for the $c_{p}$-gradient (53) reduce to the result found for
the acoustic equation (32) and (34) for continuous parameter fields
$\rho,c_{p},c_{s}$ and continuous parameter perturbations $\tilde{c}_{p}$. To
verify this, one adds contributions from adjacent elements to common
boundaries, and the terms in (53b) combine or cancel.
###### Remark 7
Above, we have derived expressions for the derivatives with respect to the
primary and secondary wave speeds. If, instead of $c_{p}$ and $c_{s}$,
derivatives with respect to an alternative pair of parameters in the stress
tensor—such as the Lamé parameters, or Poisson’s ratio and Young’s modulus—are
derived, the adjoint equations remain unchanged, but the expressions for the
derivatives change according to the chain rule.
## 4 Numerical verification for coupled elastic-acoustic wave propagation
Here, we numerically verify the expressions for the discrete gradients with
respect to the wave speeds for the elastic-acoustic wave problem derived in
Section 3.4. For this purpose, we compare directional finite differences with
directional gradients based on the discrete gradient. To emphasize the
correctness of the discrete gradient, we use coarse meshes in these
comparisons, which underresolve the wave fields. As test problem, we use the
Snell law example from Section 6.2 in WilcoxStadlerBursteddeEtAl10 with the
material parameters and the wave incident angle specified there. For our
tests, we use the simple distributed objective function
$\mathcal{J}(c_{p},c_{s}):=\int_{0}^{T}\\!\\!\int_{\Omega}\boldsymbol{v}_{h}\cdot\boldsymbol{v}_{h}\,d\boldsymbol{x}\,dt$
The discretization of the wave equation follows WilcoxStadlerBursteddeEtAl10 ,
i.e., we use spectral elements based on Gauss-Lobatto-Lagrange (GLL) points on
hexahedral meshes. The use of GLL quadrature results in underintegration even
if the elements are images of the reference element under an affine
transformation. In Figure 2, we summarize results for the directional
derivatives in the direction $\tilde{c}_{p}:=\sin(\pi x)\cos(\pi y)\cos(\pi
z)$; we compare the finite difference directional derivatives
$d_{\epsilon}^{\text{fd}}:=\frac{\mathcal{J}(c_{p}+\epsilon\tilde{c}_{p})-\mathcal{J}(c_{p})}{\epsilon}$
(55)
with the directional derivatives $d^{\text{di}}$ and $d^{\text{co}}$ defined
by
$d^{\text{di}}:=\mathcal{J}_{c_{p}}(c_{p},c_{s})(\tilde{c}_{p}),\quad
d^{\text{co}}:=\mathcal{J}^{\text{cont}}_{c_{p}}(c_{p},c_{s})(\tilde{c}_{p}),$
(56)
where $\mathcal{J}_{c_{p}}(c_{p},c_{s})$ denotes the discrete gradient (53),
and $\mathcal{J}^{\text{co}}_{c_{p}}(c_{p},c_{s})$ denotes the gradient
obtained when neglecting the jump term in the boundary contributions $g_{p}$
in (53b). These jump terms are likely to be neglected if the continuous
gradient expressions are discretized instead of following a fully discrete
approach. The resulting error is of the order of the discretization error and
thus vanishes as the discrete solutions converge. However, this error can be
significant on coarse meshes, on which the wave solution is not well resolved.
$1$$2$$3$$0.9$$1$$1.1$$1.2$$1.3$mesh leveldirectional
derivative$d^{\text{fd}}_{\epsilon},\epsilon=10^{-3}$$d^{\text{fd}}_{\epsilon},\epsilon=10^{-4}$$d^{\text{fd}}_{\epsilon},\epsilon=10^{-5}$disc.
grad. $d^{\text{di}}$cont. grad. $d^{\text{co}}$
mesh level 1
---
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-2}$ | 1.489022
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-3}$ | 1.244358
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-4}$ | 1.219841
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-5}$ | 1.217389
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-6}$ | 1.217143
$d^{\text{fd}}_{\epsilon},\epsilon=10^{-7}$ | 1.217118
$d^{\text{di}}$ | 1.217117
Figure 2: Directional derivatives computed using one-sided finite differences
(55), and the discrete and the continuous gradients (56). Left: Results on
mesh levels 1,2 and 3 corresponding to meshes with 16, 128 and 1024 finite
elements with polynomial order $N=4$. The finite difference directional
derivatives $d^{\text{fd}}_{\epsilon}$ converge to the discrete gradient
$d^{\text{di}}$ as $\epsilon$ is reduced. Note that as the mesh level is
increased, the continuous gradient $d^{\text{co}}$ converges to
$d^{\text{di}}$. Right: Convergence of finite difference directional
derivative on the coarsest mesh. Digits for which the finite difference
gradient coincides with the discrete gradient are shown in bold.
Next, we study the accuracy of the discrete gradient for pointwise
perturbations to the wave speed. Since the same discontinuous basis functions
as for the wave equation are also used for the local wave speeds, a point
perturbation in $c_{p}^{-}$ or $c_{s}^{-}$ at an element boundary face results
in a globally discontinuous perturbation direction $\tilde{c}_{p}$ and
$\tilde{c}_{s}$. In Table 1, we present the discrete directional gradient
$d^{\text{di}}$ with finite difference directional gradients
$d^{\text{fd}}_{\epsilon}$ for unit vector perturbations of both wave speeds.
Compared to in the table in Figure 2, where the directional derivatives for
smooth perturbations are reported, pointwise perturbations of the wave speeds
$c_{p}$ or $c_{s}$ result in smaller changes in the cost functional, and
numerical roundoff influences the accuracy of finite difference directional
derivatives. As a consequence, fewer digits coincide between the finite
difference directional derivatives and the discrete gradients.
Table 1: Comparison of pointwise material gradients for Snell problem from (WilcoxStadlerBursteddeEtAl10, , Section 6.2). The derivatives $d^{\text{fd}}_{\epsilon}$ and $d^{\text{di}}$ with respect to the local wave speed (either $c_{p}$ or $c_{s}$) for points with coordinates $(x,y,z)$ are reported. We use the final time $T=1$ and spectral elements of polynomial order $N=6$ in space. The meshes for level 1 and 2 consist of 16 and 128 finite elements, respectively. Digits where the finite difference approximation coincides with the discrete gradient are shown in bold. mesh level | $(x,y,z)$ | pert. | $d^{\text{di}}$ | $d^{\text{fd}}_{\epsilon}$
---|---|---|---|---
$\\#$tsteps | | field | | $\epsilon=10^{-3}$ | $\epsilon=10^{-4}$ | $\epsilon=10^{-5}$
1/101 | $(0,0,0)$ | $c_{p}$ | 1.8590e-4 | 1.8549e-4 | 1.8581e-4 | 1.8560e-4
2/202 | $(0,0,0)$ | $c_{p}$ | 2.2102e-5 | 2.2094e-5 | 2.2007e-5 | 2.1504e-5
1/101 | $(0,0,1)$ | $c_{s}$ | 1.1472e-5 | 1.1453e-5 | 1.1372e-5 | 1.0942e-5
1/101 | $(-0.5,-0.5,0.5)$ | $c_{s}$ | 2.8886e-3 | 2.8802e-3 | 2.8877e-3 | 2.8870e-3
## 5 Conclusions
Our study yields that the discretely exact adjoint PDE of a dG-discretized
linear hyperbolic equation is a proper dG discretization of the continuous
adjoint equation, provided an upwind flux is used. Thus, the adjoint PDE
converges at the same rate as the state equation. When integration by parts is
avoided to eliminate quadrature errors, a weak dG discretization of the state
PDE leads to a strong dG discretization of the adjoint PDE, and vice versa.
The expressions for the discretely exact gradient can contain contributions at
element faces and, hence, differ from a straightforward discretization of the
continuous gradient expression. These element face contributions are at the
order of the discretization order and are thus more significant for poorly
resolved state PDEs. We believe that these observations are relevant for
inverse problems and optimal control problems governed by hyperbolic PDEs
discretized by the discontinuous Galerkin method.
## Acknowledgments
We would like to thank Jeremy Kozdon and Gregor Gassner for fruitful
discussions and helpful comments, and Carsten Burstedde for his help with the
implementation of the numerical example presented in Section 4. Support for
this work was provided through the U.S. National Science Foundation (NSF)
grant CMMI-1028889, the Air Force Office of Scientific Research’s
Computational Mathematics program under the grant FA9550-12-1-0484, and
through the Mathematical Multifaceted Integrated Capability Centers (MMICCs)
effort within the Applied Mathematics activity of the U.S. Department of
Energy’s Advanced Scientific Computing Research program, under Award Number
DE-SC0009286. The views expressed in this document are those of the authors
and do not reflect the official policy or position of the Department of
Defense or the U.S. Government.
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|
arxiv-papers
| 2013-11-27T08:54:05 |
2024-09-04T02:49:54.309345
|
{
"license": "Public Domain",
"authors": "Lucas C. Wilcox, Georg Stadler, Tan Bui-Thanh, Omar Ghattas",
"submitter": "Lucas Wilcox",
"url": "https://arxiv.org/abs/1311.6900"
}
|
1311.6932
|
# A novel framework for image forgery localization
Davide Cozzolino, Diego Gragnaniello, Luisa Verdoliva
DIETI - University Federico II of Naples
Via Claudio 21, Naples - ITALY
{davide.cozzolino, diego.gragnaniello, verdoliv}@unina.it
###### Abstract
Image forgery localization is a very active and open research field for the
difficulty to handle the large variety of manipulations a malicious user can
perform by means of more and more sophisticated image editing tools. Here, we
propose a localization framework based on the fusion of three very different
tools, based, respectively, on sensor noise, patch-matching, and machine
learning. The binary masks provided by these tools are finally fused based on
some suitable reliability indexes. According to preliminary experiments on the
training set, the proposed framework provides often a very good localization
accuracy and sometimes valuable clues for visual scrutiny.
## I Introduction
This paper describes the strategy followed by the the GRIP team of the
University Federico II of Naples (Italy) to tackle the first IEEE IFS-TC Image
Forensics Challenge on image forgery localization.
In order to deal with forgeries of different nature (copy-paste from the same
or a different image, exemplar-based inpainting) we use techniques based on
Photo Response Non Uniformity (PRNU) noise, which deal uniformly with all
these attacks, and on which this team has gathered a solid experience [1, 2,
3, 4]. This approach, however, relies on some strong hypotheses, not always
satisfied. In fact, it requires the knowledge of the camera PRNU itself, or
else a sufficient number of images (at least a few dozens) to estimate it.
Therefore, we decided to complement the PRNU-based technique with two more
techniques oriented, respectively, to copy-move and splicing forgeries,
implementing eventually a simple decision fusion rule. Specifically, to
localize copy-move forgeries we developed a simple technique based on the
PatchMatch algorithm [7] for fast block matching, while to localize splicings
we propose here a a new algorithm, based on some recently proposed local
descriptors [8].
In the following of the paper we describe in detail the proposed strategy,
devoting Section II, III, and IV to forgery localization based on PRNU noise,
PatchMatch, and local descriptors, respectively. Section V describes the
fusion algorithm and shows some results obtained on the test set.
## II PRNU-based localization
The PRNU pattern, originated by imperfections in the sensor silicon wafer, is
unique for each camera and stable in time, representing therefore a sort of a
camera fingerprint, which is present in all pristine images produced by the
camera but absent in tampered areas. By detecting the presence/absence of the
camera PRNU in the image under test, one is able to make reliable decisions on
the presence of forgeries.
Let $y$ be a digital image observed at the camera output, either as a single
color band or the composition of multiple color bands. with $y_{i}$ the value
of pixel $i$. In a simplified model [6], we can write $y$ as
$y=(1+k)x+\theta=xk+x+\theta$ (1)
where $x$ is the ideal noise-free image, $k$ the camera PRNU, $\theta$ an
additive noise term which accounts for all types of disturbances, and products
between images are pixel-wise. Part of the “noise” can be removed by
subtracting from $y$ an estimate of the true image, $\widehat{x}=f(y)$,
provided by a denoising filter obtaining the so-called noise residual,
expressed after some manipulations as
$r=y-\widehat{x}=yk+n=z+n$ (2)
where all disturbances, including the denoising error, have been included in a
single zero-mean noise term $n$.
The noise residual can be used for camera identification. In fact, the
correlation index between $r$ and a given PRNU, $h$, is a random variable with
zero mean whenever $h\neq k$, and with a mean significantly different from
zero only when $h=k$, pointing to the camera that generated the image. The
same approach can be used to detect image forgeries, by computing a
correlation index field pixel-by-pixel by sliding a window of suitable size on
the image, and carrying out a local decision test. When the computed
correlation index $\rho_{i}$ is smaller than expected, a tampering of the
corresponding pixel $i$ is likely.
In the concise description above, the camera PRNU pattern was assumed to be
already available, but this is only true if we have a collection of images
taken by the camera large enough to carry out a reliable estimate. However,
this is not the case in this challenge, since we are only given a large number
of images, with no information on their origin. More precisely, $N=$1500
training images are available, 1050 of them pristine and 450 fake, while the
test set comprises 700 fake images. In principle, each of these images could
have been taken by a different camera, frustrating any attempt to use a PRNU-
based strategy. However, we rely on the reasonable conjecture that the unknown
number of cameras $M$ used to build the database is much smaller than $N$. Our
algorithm comprises the following steps, described in detail in the rest of
the Section:
* •
group the training images in $C+1$ clusters (one for left-overs), based on
their noise residuals;
* •
estimate the PRNU for the $C$ valid clusters;
* •
associate each test image with one of the clusters;
* •
localize forgeries.
### II-A Implemented method
Our first problem is to cluster the images based on their noise residuals. At
the end of the process, clusters formed by a sufficient number of images will
allow us to estimate the corresponding camera PRNU and perform forgery
detection. To carry out the clustering we use the algorithm proposed in [9]
which is a simplified version of the well-known pairwise nearest neighbor
(PNN) algorithm. In PNN, at the beginning each data vector $v_{j}$ is the
center of a cluster with just one element, $w_{j}=1$. Then, the two closest
centers, say $v^{\prime}$ and $v^{\prime\prime}$ are merged together, provided
they are closer than a given threshold, generating by weighted averaging a new
center that replaces the existing ones, in formulas
$\displaystyle v_{\rm new}$ $\displaystyle=$
$\displaystyle(w^{\prime}v^{\prime}+w^{\prime\prime}v^{\prime\prime})/(w^{\prime}+w^{\prime\prime})$
$\displaystyle w_{\rm new}$ $\displaystyle=$ $\displaystyle
w^{\prime}+w^{\prime\prime}$ (3)
By so doing, the number of centers decreases by one at a time, and the process
continues until all centers are farthest apart than the threshold, providing
the desired clustering. Even fast versions of PNN, however, are
computationally demanding, as distances among all couples of data vectors must
be computed. The algorithm proposed in [9] introduces some modifications to
reduce computation time, like picking at random couples to be compared with
the threshold, or looking for all points of a cluster before proceeding with
another one.
In our case, the data vectors are the normalized noise residuals
$r_{j}/y_{j}=k_{j}+n_{j}/y_{j}$, which represent basic estimates of the camera
PRNU that are gradually improved through merging. The distance measure is the
Peak to Correlation Energy ratio (PCE) [10], more robust than the correlation
index. We carry out the clustering on the training set using a threshold equal
to 50. By so doing we identify 44 different clusters, for a total of 746
pristine images out of the 1050 available and 315 fakes out of 450 (see
Fig.1).
Although in the clustering phase we estimate the PRNU by unweighted averaging
of the normalized noise residuals, the final estimate for the cluster $C$ is
computed as:
$\widehat{k}_{C}=\sum_{j\in C}y_{j}r_{j}/\sum_{j\in C}y_{j}^{2}$ (4)
where the weighting terms $y_{j}$ account for the fact that dark areas of the
image present an attenuated PRNU and hence should contribute less to the
overall estimate.
At this point we can try to associate the test images with one of the
estimated PRNU’s using again PCE. With a threshold equal to 100 we are able to
classify 431 of the 700 images available, about 60% of the total, shown in
Fig.1.
Figure 1: Number of images belonging to the clustered sets.
For all forged images belonging to one of the identified clusters, forgery
detection is carried out as proposed in [6] using the normalized correlation
index between $r_{{}_{W_{i}}}$ and $z_{{}_{W_{i}}}$, the restrictions of $r$
and $z$, respectively, to the 129$\times$ 129 window $W_{i}$ centered on the
target pixel. There are two main differences with respect to the original
algorithm. First, to improve the quality of the noise residuals we resort to
nonlocal denoising. This choice, as shown in [1, 4], improves the separation
between image content and PRNU, especially in textured areas. In addition, we
use an adaptive decision threshold here, which depends on the reliability of
the correlation field, measured through PCE. In fact, given the lack of
information on the camera used to take the photos, correlation fields are not
equally reliable.
It is worth underlining that the correlation might happen to be very low when
the image is dark, saturated or strongly textured, increasing the false alarm
probability in these areas. In [6] this problem is addresses by means of a
“predictor” which, based on local images features, such as texture, flatness
and intensity, computes the expected value of the correlation index under the
hypothesis that PRNU is present. In this work we do not use the predictor, as
it proves unreliable when estimated only on a few images. However, we keep
enforcing a control on saturated areas, where PRNU is totally unreliable. In
Fig.2 we show two images of the training set with the corresponding
correlation maps (low values correspond to red in this case) and detection
masks.
Figure 2: Two training fake images, correlation maps and color-coded detection
masks. Gray: genuine pixel declared genuine, red: genuine pixel declared
tampered (error), white: tampered pixel declared genuine (error), green:
tampered pixel declared tampered.
## III Copy-move forgery localization based on PatchMatch
Localization of copy-move forgeries is a very active field of research and
several papers face the problem, the majority of which based on keypoint
identification [11] followed by feature extraction and matching. This approach
works quite well for classical copy-move forgeries, where a large compact
object is copied from source to target location, with some possible
modification (rotation, resizing, and so on) [12]. Things are more difficult
when multiple small regions are copied from all over the image and combined
together to cover a large object (exemplar-based inpainting), since keypoint
identification and feature matching becomes much quite unreliable. In this
case, better results can be obtained by computing a dense motion field by some
block-matching algorithm, as done in [13, 14].
We have followed a similar line of work, resorting to PatchMatch [7], a
recently proposed editing algorithm, which provides an accurate (though
approximate) motion field much faster than exact algorithms. The main steps of
the localization algorithm are the same as in the methods based on feature
extraction [11], namely, (dense) motion field estimation, filtering and post-
processing. In particular, matching is performed directly on the RGB image,
normalized to gain robustness against changes of illumination, with
7$\times$7-pixel patches. Once the motion vector field is computed, we single
out regions with homogeneous motion by a suitable linear filtering (robust to
moderate resizing). To avoid false alarms we remove matches between spatially
close regions, and matches obtained in perfectly flat areas, as in the
presence of saturation. Then for each motion vector we compare the image with
its shifted version and compute a dense correlation map which, after
thresholding and morphological operations, provides the binary map relative to
a single copied object. Of course, we detect both the source and target
regions, associated with opposite motion vectors. To deal with rotations and
relatively large resizing, we evaluate the motion vector field for a fixed
number of rotations and resizings, taking advantage of PatchMatch speed.
Two sample results on the training set are shown in Fig.3, referring to a
classical copy-move and an exemplar-based inpainting. Note that the algorithm
is not able to distinguish the original object from the copy. However, we can
use the information coming from the PRNU-based approach (when available) on
remove this uncertainty as in the example of Fig.4. This technique, however,
is reliable only when the tested objects are relatively large and the
correlation map is sufficiently reliable (PCE$>$150), in all other cases we
declare both regions as forged.
Figure 3: Two training fake images, their ground truth, and the output of our
algorithm.
Figure 4: A training fake image, its correlation map, its PatchMatch-based
map, and the final color-coded mask.
## IV Splicing localization by local descriptors
The techniques described above work only on a fraction of all the images,
those with copy-moves forgeries and those for which the PRNU pattern could be
estimated. To integrate this information we propose a novel algorithm
effective also on splicings, namely, objects copied from different images. In
particular, given the good performance obtained in forgery detection by the
local descriptors proposed in [15] we have implemented the same procedure on a
sliding-window basis. The algorithm performs a classification step for each
block, followed by an aggregation phase driven by a suitable reliability
measure, required to merge all data available at a given pixel.
In order to perform classification a feature extraction process is required
with a successive training of a SVM classifier with linear kernel. Features
are computed on 10000 $128\times 128$-pixel blocks, 5000 pristine and 5000
fake, extracted by the training images. More precisely, in view of the
subsequent integration with a reliable copy-move detector, we focus on
performance for splicings, and train the classifier only on the 144 spliced
images found in the training set. Note that, in this context, a fake block is
not a block drawn entirely from a splicing, but rather a boundary block, since
relevant information to discover a forgery is hidden in the transition area.
More precisely, we label as fake only the blocks which, according to the
ground truth, comprise from 20% to 80% forged pixels. The high-pass filter to
compute the descriptor is the best one found in phase 1 for detection, a 3rd
order linear filter [15].
The image under test is analyzed in a sliding-window modality, with partially
overlapping $128\times 128$-pixels blocks and a 16-pixel step. For each block
we computed the distance of the corresponding feature vector from the SVM
hyperplane, the larger the distance, the more reliable the result. By
aggregating all these values for each pixel we obtain an index related to the
probability that the pixel has been tampered, named SDH (Sum of Distances from
the Hyperplane). The final binary map is obtained by thresolding this index.
An empirical analysis on the training set suggested a threshold equal to
0.25*max(SDH). Fig.5 shows some sample results.
Figure 5: Two training fake images, their SDH map and the color coded
detection mask.
Figure 6: Flow chart of the combination strategy.
## V Combination strategy
The flow-chart of Fig.6 describes our fusion strategy. A general guideline was
to keep into great account all information about reliability. In particular,
since F-measure results computed on the training set made very clear the
superior reliability of the PatchMatch-based detector, we use only its map
when available, and integrate it with the PRNU-based map only when the latter
is itself extremely reliable (PCE$>$1200). Then when no copy-move is detected,
we trust, in decreasing order, the PRNU-based map and the Local Detector map.
It is worth underlining that the latter map, although less reliable than the
previous two, is always available, and hence allows us to make a decision on
all the test images. On the training set, this strategy provided an average
F-measure equal to 0.4153, while on the test set we obtained the best result
of phase 2 of the Challenge with 0.4072. Four sample results on the test set
are shown in Fig.7.
Figure 7: Four images from the test set and their output masks.
This work confirms that no single tool is sufficient to deal with the
diversity of possible image manipulations. Although we obtained encouraging
results, there is ample space for further improvements. The PRNU-based
technique, for example, is not able to detect very small forgeries, and gives
too many false alarms in the absence of a predictor. The PatchMatch-based
technique also needs improvements to reduce the false alarm rates. The LD-
based technique is at an embryonal stage and a deep analysis is required to
optimize it. Finally, the information fusion is also rather naive, and a
smarter fusion rule could be devised as done for example in [16].
## References
* [1] G. Chierchia, S. Parrilli, G. Poggi, C. Sansone, and L. Verdoliva, “On the influence of denoising in PRNU based forgery detection,” in Proc. of the 2nd ACM workshop on Multimedia in Forensics, Security and Intelligence pp. 117–122, 2010.
* [2] G. Chierchia, S. Parrilli, G. Poggi, L. Verdoliva, and C. Sansone, “PRNU-based detection of small-size image forgeries,” International Conference on Digital Signal Processing (DSP), pp. 1–6, 2011.
* [3] G. Chierchia, G. Poggi, C. Sansone, and L. Verdoliva, “PRNU-based forgery detection by global risk minimization,” IEEE International Workshop on Multimedia Signal Processing (MMSP), 2013.
* [4] G. Chierchia, G. Poggi, C. Sansone, and L. Verdoliva, “A Bayesian-MRF approach for PRNU-based image forgery detection,” IEEE Trans. on Information Forensics and Security, submitted, 2013.
* [5] J. Lukas, J. Fridrich, and M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. of SPIE, vol. 6072, pp. 362–372, 2006.
* [6] M. Chen, J. Fridrich, M. Goljan, and J. Lukas, “Determining Image Origin and Integrity Using Sensor Noise,” IEEE Transactions on Information Forensics and Security, vol. 3, no. 1, pp. 74–90, 2008.
* [7] C. Barnes, E. Shechtman, A. Finkelstein, and D.B. Goldman, “PatchMatch: a randomized correspondence algorithm for structural image editing,” ACM Transactions on Graphics, vol. 28, no. 3, 2009.
* [8] J. Fridrich, and J. Kodovský, “Rich models for steganalysis of digital images,” IEEE Transactions on Information Forensics and Security, vol. 7, no. 3, pp. 868 -882, june 2012.
* [9] G.J. Bloy, “Blind Camera Fingerprinting and Image Clustering,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 3, pp. 532–535, Mar. 2008.
* [10] M. Goljan, and J. Fridrich, “Digital camera identification from images – Estimating false acceptance probability,” in Proc. 8th Int. Workshop Digital Watermarking, 2008\.
* [11] V. Christlein, C. Riess, J. Jordan, and E. Angelopoulou, “An Evaluation of Popular Copy-Move Forgery Detection Approaches,” IEEE Trans. on Information Forensics and Security, vol. 7, no. 6, pp. 1841 1854, 2012.
* [12] X. Pan, and S. Lyu, “Region duplication detection using image feature matching,” IEEE Trans. on Information Forensics and Security, vol. 5, no. 4, pp. 857–867, dec. 2010.
* [13] A. Langille, and M. Gong, “An efficient match-based duplication detection algorithm,” Canadian Conf. on Computer and Robot Vision, 2006\.
* [14] I.-C. Chang, J.C. Yu, and C.-C. Chang, “A forgery detection algorithm for exemplar-based inpainting images using multi-region relation,” Image and Vision Computing, vol. 31, no. 1, pp. 57 -71, Oct. 2013.
* [15] D. Cozzolino, D. Gragnaniello and L. Verdoliva, “Image forgery detection based on the fusion of machine learning and block-matching methods,” IEEE First Forensic Challenge (phase 1), 2013.
* [16] D. Cozzolino, F. Gargiulo, C. Sansone, L. Verdoliva, “Multiple Classifier Systems for Image Forgery Detection,” International Conference on Image Analysis and Processing (ICIAP), 2013\.
|
arxiv-papers
| 2013-11-27T11:06:05 |
2024-09-04T02:49:54.322671
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Davide Cozzolino and Diego Gragnaniello and Luisa Verdoliva",
"submitter": "Luisa Verdoliva",
"url": "https://arxiv.org/abs/1311.6932"
}
|
1311.6934
|
# Image forgery detection based on the fusion of machine learning and block-
matching methods
Davide Cozzolino, Diego Gragnaniello, Luisa Verdoliva
DIETI - University Federico II of Naples
Via Claudio 21, Naples - ITALY
{davide.cozzolino, diego.gragnaniello, verdoliv}@unina.it
###### Abstract
Dense local descriptors and machine learning have been used with success in
several applications, like classification of textures, steganalysis, and
forgery detection. We develop a new image forgery detector building upon some
descriptors recently proposed in the steganalysis field suitably merging some
of such descriptors, and optimizing a SVM classifier on the available training
set. Despite the very good performance, very small forgeries are hardly ever
detected because they contribute very little to the descriptors. Therefore we
also develop a simple, but extremely specific, copy-move detector based on
region matching and fuse decisions so as to reduce the missing detection rate.
Overall results appear to be extremely encouraging.
## I Introduction
This paper describes the strategy followed by the the GRIP team of the
University Federico II of Naples (Italy) to tackle phase 1 of the first IEEE
IFS-TC Image Forensics Challenge on image forgery detection.
This team has been working in recent years on the forgery detection problem,
focusing on techniques based on camera sensor noise, a.k.a. PRNU (photo
response non-uniformity) noise [1, 2, 3, 4] and on techniques based on dense
local descriptors and machine learning [5]. Therefore, we decided to follow
both these approaches for detection, on two separate lines of development,
with the aim of fusing decisions at some later time of the process. Indeed, it
is well known [6] that, given the different types of forgery encountered in
practice, and the wide availability of powerful photo-editing tools, several
detection approaches should be used at the same time and judiciously merged in
order to obtain the best possible performance. Based on this consideration, we
also followed a third line of development working on a technique for copy-move
forgery detection which, although applicable only to a fraction of the image
set, provides very reliable results.
Unfortunately it was very soon clear that the PRNU-based approach was bound to
be of little use. Lacking any information on the cameras used to take the
photos, we had to cluster the images based on their noise residuals and
estimate each camera’s PRNU based on the clustered images. However, more than
20% of the test images could not be clustered at all and in some cases the
number of images collected in a cluster was too small to obtain a reliable
estimate of the PRNU.
On the contrary, techniques based on dense local descriptors appeared from the
beginning very promising, and we pursued actively this line of development,
drawing also from the relevant literature in the steganalysis field.
Complementing such techniques with a simple copy-move detector, tuned so as to
guarantee very high specificity, lead us eventually to obtain very promising
results.
The rest of the paper comprises only two sections, one dealing with dense
local descriptors and machine learning and the other with copy-move detection.
In each Section we provide experimental results obtained on the training set.
## II Dense local descriptors for splicing detection
Several techniques have been proposed in the last decade for splicing
detection based on machine learning. Major efforts have been devoted to find
good statistical models for natural images in order to single out the features
that guarantee the highest discriminative power. Often, in order to capture
more meaningful statistics, transform-domain features have been used, as in
[7] where the image undergoes block-wise discrete cosine transform (DCT) with
various block sizes and first-order (histogram based) and higher-order
(transition probabilities) features are collected and merged. Given the good
results obtained in terms of detection accuracy, an expanded Markov-based
scheme in DCT and DWT domains is followed in [8]. Interestingly, the method
proposed [7] was inspired by prior work carried out in steganalysis which,
despite the obvious differences with respect to the forgery detection field,
pursues a very similar goal, that is, detecting seemingly invisible
alterations of the natural characteristics of an image.
The same path is followed in the forgery detection technique proposed in [9],
based on an approach proposed for steganalysis in [10, 11]. The major
contribution consists in deriving the features based on some co-occurrence
matrices computed on the thresholded prediction-error image (also called
residual image). In fact, modeling the residuals rather than the pixel values
is very sensible in these low-level methods (not based on image semantic),
since the image content does not help detecting local alterations and should
be suppressed altogether. In the context of forgery detection, in particular,
considering that splicing typically introduces sharp edges, it is reasonable
to characterize statistically some edge image, which can also be the output of
a simple high-pass filter (like a derivative of first order). As a further
advantage, the residual image has a much narrower dynamic range than the
original one, allowing for a compact and robust statistical description by
means of co-occurrences.
The processing path outlined above, already proposed in [10], can be therefore
summarized in the following steps
1. 1.
computation of the high-pass residuals;
2. 2.
truncation and quantization;
3. 3.
feature extraction based on co-occurrence matrices of selected neighbors;
4. 4.
design of a suitable classifier on the training set.
Given its compelling rationale, and some promising results obtained in the
literature, we will follow this path, here. Nonetheless, a large number of
design choices must be made, beginning from the high-pass filter, to end with
the classifier, which impact heavily on the performance and require lengthy
development and testing. Fortunately, we can rely on the precious results
described in a recent work on steganalysis [12], where a large number of
models have been considered and analyzed, and made available online to the
research community [13]. Specifically, in [12] a number of different high-pass
filters have been considered, both linear and nonlinear, with various
supports, different quantization and truncation strategies for the residues
have been implemented and, based on some preliminary experiments, the use of
some selected groups of neighbors for co-occurrence computation has been
suggested. There is no doubt, as the Authors themselves point out, that better
design choices are possible, especially when aiming at slightly different
goals, but the wealth of models they provide allow for the rapid development
and optimization of a specific processing chain, which can be then improved,
in part already in this work, under some specific respects.
### II-A Implemented method
In [12] 39 different high-pass filters are proposed, which work on the
grayscale version of the original image obtained by standard conversion. All
such filters are extremely simple, since their goal is to highlight minor
variations w.r.t. to typical behaviors. Typical example are the first order
horizontal linear and symmetric nonlinear filters defined by
$\displaystyle r_{i,j}$ $\displaystyle=$ $\displaystyle x_{i,j+1}-x_{i,j}$
$\displaystyle r_{i,j}$ $\displaystyle=$
$\displaystyle\min[(x_{i,j+1}-x_{i,j}),(x_{i+1,j}-x_{i,j})]$
Fig.1 shows the effect of applying one of such filters to a training image of
the challenge. Of course, it is not obvious by visual inspection that the
forged region (in black in the ground-truth) exhibits characteristics
different from those of the rest of the image, and such to allow the
identification of the forgery.
Figure 1: A training image with its ground truth and an example residual
image.
Residuals are in general real-valued and, although typically small, are
defined on a wide range. To enable their meaningful characterization in terms
of co-occurrence they must be quantized and truncated. Following [12] we use
$\widehat{r}_{ij}={\rm trunc}_{T}({\rm round}(r_{ij}/q))$
with $q$ the quantization step and $T$ the truncation value. We keep using
$T$=2 to limit the matrix size but consider exclusively $q$=1, partly to
reduce complexity, but mainly to limit the risk of overfitting to our training
set. Each quantized residual can eventually take on 5 values, from -2 to +2.
We then compute co-occurrences on four consecutive pixels along the same row
or column, obtaining 625 entries, which can be highly reduced thanks to
symmetries.
In the classification phase we depart significantly from the reference
technique, due to the overfitting problem mentioned before. In fact, each
individual model comprises 169 features for linear filters and 325 for non
linear ones, a number large but still adequate for a training set comprising
about 1500 images (450 fake and 1050 pristine), as in our case. Merging all
models, however, would lead to a much larger number of features probably too
large to expect a meaningful training. The Authors of [12] dealt successfully
with this problem using a suitable ensemble classifier [14]. In this
challenge, however, we have a training set about ten times smaller, which
raises serious doubts on the chances of success of this approach.
We decided therefore to test each model individually, relying heavily on cross
validation to gain a reasonable insight into their actual performance. In each
experiment, we selected at random 5/6 of the pristine images and 5/6 of the
fake ones to train a SVM classifier. The remaining images of each class were
then used to test the trained classifier. To reduce randomness, each
experiment was repeated 18 times, selecting the training and test set at
random, and results were eventually averaged. Fig.2(top) shown the results for
the 39 models considered, in terms of expected score, defined as
$S=\frac{\Pr(\widehat{F}|F)+\Pr(\widehat{P}|P)}{2}$
with $P[F]$ indicating the event “image pristine[fake]” and
$\widehat{P}[\widehat{F}]$ the event “decision pristine[fake]”, respectively.
For several models the predicted score is in the order of 94%, hence very
promising. Then we tried to merge the features of a limited number of models,
up to four, not to exceed the number of training images. Results are reported
in Tab.I in terms of score obtained before and after merging. They show a
limited improvement, if any, over the best single-model classifier, and a non-
monotonic behavior, ringing an alarm bell on stability.
To improve robustness, we considered a different measure of performance. For
each SVM classifier, we moved the separating hyperplane along the orthogonal
direction, and built the corresponding ROC. Then we computed, for each model,
the Area Under the receiver operating Curve (AUC), because a large AUC implies
not only a good performance in the best operating point, but also robustness
w.r.t. changing conditions. Fig.2(bottom) shows results. We then tried merging
the best models selected with this criterion, obtaining the results reported
in Tab.II. This time, performance improves monotonically, supporting the use
of a merged set of features selected with this latter choice.
Eventually, our SVM classifier uses the merging of all the features of models
17 31 34 and 36, and is trained over the whole phase-1 training set.
Figure 2: Scores (top) and AUC (bottom) for all models. Model | Type | Score | AUC | Score/merg.
---|---|---|---|---
3 | non linear, 1st order | 0.9429 | 0.9724 | 0.9429
4 | non linear, 1st order | 0.9403 | 0.9693 | 0.9154
12 | non linear, 2nd order | 0.9389 | 0.9685 | 0.9415
11 | non linear, 2nd order | 0.9371 | 0.9595 | 0.9163
TABLE I: Score obtained before and after merging by the top-score individual models. Model | Type | Score | AUC | Score/merg.
---|---|---|---|---
36 | linear, 3rd order | 0.9289 | 0.9765 | 0.9289
34 | linear, 1st order | 0.9316 | 0.9751 | 0.9462
17 | non linear, 3rd order | 0.9369 | 0.9736 | 0.9481
31 | non linear, square 5$\times$ 5 | 0.9371 | 0.9727 | 0.9531
TABLE II: Score obtained before and after merging by the top-AUC individual
models.
## III Copy-move detection by PatchMatch
Many algorithms have been proposed in the literature for copy-move forgery
detection, typically based on matching techniques, e.g., [15, 16, 17]. The
major source of difference between them resides in the hypotheses made on the
nature of the forgery. In particular, detection performance and algorithm
complexity depend heavily on the size of the copied region, on its content as
compared with the target region background, and on the presence/absence of
further processing on such regions, such as rotation, resizing, change of
illumination, and so on. Algorithms aiming at the detection of large copy-
moves characterized by rigid translation can be quite simple, while they grow
more and more complex as constraints are relaxed including new potential
targets.
Let us focus, for the time being, on the simplest possible problem, in which
one or more patches of the image are copied somewhere else by pure
translation. Then, a pretty general detection algorithm might comprise the
following steps
1. 1.
computation of a dense motion-vector field;
2. 2.
segmentation of the field in regions characterized by homogeneous motion
vectors;
3. 3.
elimination of candidate matching regions based on size, matching error, and
other criteria.
In the hypotheses cited above, and barring pathological cases such as
uniformly dark or saturated areas, any copy-move forgery of reasonable size
can be detected easily, and with very high confidence. Indeed, it is very
difficult to find identical regions in a pristine natural image, a chance that
becomes totally negligible as the region size grows larger.
If we abandon the strong constraints considered before, things become quickly
much more difficult. Rotation and resizing imply a non-constant motion field
in copied areas, and also an intensity mismatch due to pixel interpolation,
further increased by possible changes of illumination. Algorithms have been
proposed to deal with all these problems but, besides being more complex they
provide weaker guarantees on the absence of false alarms. For example, in a
highly textured areas, like a close up of a tree, it might be very difficult
to decide whether a certain leaf is a rotated and rescaled version of another
or not.
These considerations serve to justify some important design choices in the
development and fine-tuning of our approach. Consider, in fact, that we are
trying to optimize the performance of a composite detector obtained through
the suitable fusion with a machine-learning method. Under this perspective,
the marginal accuracy of the copy-move detector becomes immaterial w.r.t. its
contribution to the overall performance. Preliminary experiments show that the
detector described in the previous Section is characterized by an excellent
and well balanced performance on the training set, with very low missing-
detection and false-alarm rates (e.g. 0.0726 and 0.0213, respectively, for the
best score). The copy-move detector cannot reduce the overall false-alarm
rate, since its “pristine” decision means only that there is (probably) no
copy-move forgery, but a splicing could still be present. However, it can help
reducing the missing-detection rate, by revealing all those copy-move
forgeries that have escaped the previous detector, very likely because they
are too small to impact on the descriptor. To this end, it is necessary that
it be extremely specific, assuring that its “fake” decision is very reliable.
Based on these considerations, we develop an algorithm aimed basically at
detecting rigid-translation copy-move forgeries, with little tolerance for
other forms of processing, thus ensuring a very high specificity.
### III-A Implemented method
As outlined before, our first processing step is the computation of a dense
motion vector field based on block matching. Carrying out an exact search for
each block of the image, however, is exceedingly burdensome, and in fact this
step is often replaced by simpler, though less reliable methods, e.g. [15].
Here we resort to PatchMatch, an iterative algorithm recently proposed for
image editing applications [18, 19]. Patchmatch provides a very accurate and
regular motion field, but we chose it primarily for its rapid convergence,
which makes it about 100 times faster than exact methods, allowing us to
process in reasonable time a large database of images.
We use 7$\times$7 pixel patches, a size that guarantees a good compromise
among accuracy, resolution and speed. All image pixels are preliminarily
adjusted to unitary norm, in order to single out copy-moves also in the
presence of some intensity adjustments. After computing the motion vector
field, we carry out a filtering on both components to identify regions with
homogeneous motion. Choosing an appropriate filter, we can also identify
regions where motion vectors slowly increase or decrease linearly, thus
identifying also copy-moves with moderate resizing.
Once a relatively large region with uniform motion is identified, all matches
obtained in perfectly flat areas, as in presence of saturation, are removed;
in addition, very small regions are deleted automatically through
morphological filtering. Eventually, after elimination of unsuitable
candidates, the image is classified as fake if at least one duplicated region
is detected. To find also rotated copy-moves, we simply repeat the procedure
for a number of rotations of the image, taking advantage of PatchMatch speed.
Our experiments showed that a sampling step of 15 degrees guarantees accurate
detection.
Fig.3 shows three images with copy-move forgeries, the corresponding ground
truth, and the detection map output by our method. Note that the forgery is
easily detected, and the map is quite accurate, although the original and
copied regions are not distinguished from one another. Turning to results, our
method detects only 271 of the 450 fakes of the training set, most of the
other cases being splicing. However it declares fakes only 5 of the 1050
pristine images, and therefore its specificity, 99.52%, is extremely high as
was desired from the beginning. We exploit this property in the final
decision, by declaring a fake when at least one of the methods does.
Consequently, the score on the training set increases from 0.9531 to 0.9737.
Note that using this strategy we obtained the best score of phase 1A of the
Challenge with 0.9429.
Figure 3: Three training images with copy-move forgeries, their ground truth,
and detection maps output by our method.
## References
* [1] G. Chierchia, S. Parrilli, G. Poggi, C. Sansone, and L. Verdoliva, “On the influence of denoising in PRNU based forgery detection,” in Proceedings of the 2nd ACM workshop on Multimedia in Forensics, Security and Intelligence pp. 117–122, 2010.
* [2] G. Chierchia, S. Parrilli, G. Poggi, L. Verdoliva, and C. Sansone, “PRNU-based detection of small-size image forgeries,” International Conference on Digital Signal Processing (DSP), pp. 1–6, 2011.
* [3] G. Chierchia, G. Poggi, C. Sansone, and L. Verdoliva, “PRNU-based forgery detection by global risk minimization,” IEEE International Workshop on Multimedia Signal Processing (MMSP), 2013.
* [4] G. Chierchia, G. Poggi, C. Sansone, and L. Verdoliva, “A Bayesian-MRF approach for PRNU-based image forgery detection,” IEEE Transactions on Information Forensics and Security, submitted, 2013.
* [5] D. Gragnaniello, G. Poggi, C. Sansone, and L. Verdoliva, “Fingerprint Liveness Detection based on Weber Local Image Descriptor,” IEEE Workshop on Biometric Measurements and Systems for Security and Medical Applications, 2013\.
* [6] D. Cozzolino, F. Gargiulo, C. Sansone, and L. Verdoliva, “Multiple Classifier Systems for Image Forgery Detection,” International Conference on Image Analysis and Processing (ICIAP), 2013\.
* [7] Y.Q. Shi, C. Chen, and G. Xuan, “Steganalysis versus splicing detection,” International Workshop on Digital Watermarking, December 2007.
* [8] Z. He, W. Lu, W. Sun, and J. Huang, “Digital image splicing detection basedon Markov features in DCT and DWT domain,” Pattern Recognition, vol. 45, pp. 4292–4299, 2012.
* [9] W. Wang, J. Dong, and T. Tan, “Effective image splicing detection based on image chroma,” IEEE International Conference on Image Processing, pp. 1257–1260, 2009.
* [10] D. Zou, Y.Q. Shi, W. Su, and G.R. Xuan, “Steganalysis based on markov model of tresholded prediction-error image,” International Conference on Multimedia and Expo, pp. 1365–1368, 2006.
* [11] T. Pevný, P. Bas, and J. Fridrich, “Steganalysis by subtractive pixel adjacency matrix,” IEEE Transactions on Information Forensics and Security, vol. 5, no. 2, pp. 215 -224, june 2010.
* [12] J. Fridrich, and J. Kodovský, “Rich models for steganalysis of digital images,” IEEE Transactions on Information Forensics and Security, vol. 7, no. 3, pp. 868 -882, june 2012.
* [13] http://www.ws.binghamton.edu/fridrich/.
* [14] J. Fridrich, and J. Kodovský, “Ensemble classifiers for steganalysis of digital media,” IEEE Transactions on Information Forensics and Security, vol. 7, no. 2, pp. 432 -444, april 2012.
* [15] A. Langille, and M. Gong, “An efficient match-based duplication detection algorithm,” Canadian Conf. on Computer and Robot Vision, 2006\.
* [16] X. Pan, and S. Lyu, “Region duplication detection using image feature matching,” IEEE Transactions on Information Forensics and Security, vol. 5, no. 4, pp. 857–867, dec. 2010.
* [17] R. Davarzani, K. Yaghmaie, S. Mozaffari, M. Tapak, “Copy-move forgery detection using multiresolution local binary patterns,” Forensic Science International vol. 231, pp.61 72, 2013.
* [18] C. Barnes, E. Shechtman, A. Finkelstein, and D.B. Goldman, “PatchMatch: a randomized correspondence algorithm for structural image editing,” ACM Transactions on Graphics (Proc. SIGGRAPH), vol. 28, no. 3, 2009.
* [19] http://gfx.cs.princeton.edu/pubs/Barnes_2009_PAR/.
|
arxiv-papers
| 2013-11-27T11:17:55 |
2024-09-04T02:49:54.329507
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Davide Cozzolino and Diego Gragnaniello and Luisa Verdoliva",
"submitter": "Luisa Verdoliva",
"url": "https://arxiv.org/abs/1311.6934"
}
|
1311.6952
|
RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL
LAPLACIAN
Patricio Felmer and Ying Wang
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático,
UMR2071 CNRS-UChile, Universidad de Chile
( [email protected] and [email protected] )
###### Abstract
The aim of this paper is to study radial symmetry and monotonicity properties
for positive solution of elliptic equations involving the fractional
Laplacian. We first consider the semi-linear Dirichlet problem
$(-\Delta)^{\alpha}u=f(u)+g,\ \ {\rm{in}}\ \ B_{1},\quad u=0\ \ {\rm in}\ \
B_{1}^{c},$ (0.1)
where $(-\Delta)^{\alpha}$ denotes the fractional Laplacian, $\alpha\in(0,1)$,
and $B_{1}$ denotes the open unit ball centered at the origin in
$\mathbb{R}^{N}$ with $N\geq 2$. The function $f:[0,\infty)\to\mathbb{R}$ is
assumed to be locally Lipschitz continuous and $g:B_{1}\to\mathbb{R}$ is
radially symmetric and decreasing in $|x|$.
In the second place we consider radial symmetry of positive solutions for the
equation
$(-\Delta)^{\alpha}u=f(u),\ \ {\rm{in}}\ \ \mathbb{R}^{N},$ (0.2)
with $u$ decaying at infinity and $f$ satisfying some extra hypothesis, but
possibly being non-increasing.
Our third goal is to consider radial symmetry of positive solutions for system
of the form
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha_{1}}u=f_{1}(v)+g_{1},&{\rm{in}}\quad
B_{1},\\\\[5.69054pt] (-\Delta)^{\alpha_{2}}v=f_{2}(u)+g_{2},&{\rm{in}}\quad
B_{1},\\\\[5.69054pt] u=v=0,&{\rm{in}}\quad B_{1}^{c},\end{array}\right.$
(0.3)
where $\alpha_{1},\alpha_{2}\in(0,1)$, the functions $f_{1}$ and $f_{2}$ are
locally Lipschitz continuous and increasing in $[0,\infty)$, and the functions
$g_{1}$ and $g_{2}$ are radially symmetric and decreasing.
We prove our results through the method of moving planes, using the recently
proved ABP estimates for the fractional Laplacian. We use a truncation
technique to overcome the difficulty introduced by the non-local character of
the differential operator in the application of the moving planes.
Key words: Fractional Laplacian, Radial Symmetry, Moving Planes.
## 1 Introduction
The purpose of this paper is to study symmetry and monotonicity properties of
positive solutions for equations involving the fractional Laplacian through
the use of moving planes arguments. The first part of this article is devoted
to the following semi-linear Dirichlet problem
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u=f(u)+g,&{\rm in}\quad
B_{1},\\\\[5.69054pt] u=0,&{\rm in}\quad B_{1}^{c},\end{array}\right.$ (1.1)
where $B_{1}$ denotes the open unit ball centered at the origin in
$\mathbb{R}^{N},$ $N\geq 2$ and $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is
the fractional Laplacian defined as
$(-\Delta)^{\alpha}u(x)=P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2\alpha}}dy,$
(1.2)
$x\in B_{1}$. Here $P.V.$ denotes the principal value of the integral, that
for notational simplicity we omit in what follows.
During the last years, non-linear equations involving general integro-
differential operators, especially, fractional Laplacian, have been studied by
many authors. Caffarelli and Silvestre [5] gave a formulation of the
fractional Laplacian through Dirichlet-Neumann maps. Various regularity issues
for fractional elliptic equations has been studied by Cabré and Sire [2],
Caffarelli and Silvestre [6], Capella, Dávila, Dupaigne and Sire [7], Ros-Oton
and Serra [22] and Silvestre [25]. Existence and related results were studied
by Cabré and Tan [4], Dipierro, Palatucci and Valdinoci [12], Felmer, Quaas
and Tan [13], and Servadei and Valdinoci [24]. Great attention has also been
devoted to symmetry results for equations involving the fractional Laplacian
in $\mathbb{R}^{N}$, such as in the work by Li [19] and Chen, Li and Ou [8,
9], where the method of moving planes in integral form has been developed to
treat various equations and systems, see also Ma and Chen [20]. On the other
hand, using the local formulation of Caffarelli and Silvestre, Cabré and Sire
[3] applied the sliding method to obtain symmetry results for nonlinear
equations with fractional laplacian and Sire and Valdinoci [28] studied
symmetry properties for a boundary reaction problem via a geometric
inequality. Finally, in [13] the authors used the method of moving planes in
integral form to prove symmetry results for
$(-\Delta)^{\alpha}u+u=h(u)\ \ \rm{in}\ \ \mathbb{R}^{N},$ (1.3)
taking advantage of the representation formula for $u$ given by
$u(x)=\mathcal{K}\ast h(u)(x),\quad x\in\mathbb{R}^{N},$
where the kernel $\mathcal{K}$, associated to the linear part of the equation,
plays a key role in the arguments. This approach is not possible to be used
for problem (1.1), since a similar representation formula is not available in
general.
The study of radial symmetry and monotonicity of positive solutions for non-
linear elliptic equations in bounded domains using the moving planes method
based on the Maximum Principle was initiated with the work by Serrin [23] and
Gidas, Ni and Nirenberg [14], with important subsequent advances by Berestycki
and Nirenberg [1]. We refer to the review by Pacella and Ramaswamy [21] for a
more complete discussion of the method and it various applications. In [1] the
Maximum Principle for small domain, based on the Aleksandrov-Bakelman-Pucci
(ABP) estimate, was used as a tool to obtain much general results, specially
avoiding regularity hypothesis on the domain. In a recent article Guillen and
Schwab, [16], provided an ABP estimate for a class of fully non-linear
elliptic integro-differential equations. Motivated by this work, we obtain a
version of the Maximum Principle for small domain and we apply it with the
moving planes method as in [1] to prove symmetry and monotonicity properties
for positive solutions to problem (1.1) in the ball and in more general
domains.
We consider the following hypotheses on the functions $f$ and $g$:
* $(F1)\ $
The function $f:[0,\infty)\to\mathbb{R}$ is locally Lipschitz.
* $(G)\ $
The function $g:{B_{1}}\to\mathbb{R}$ is radially symmetric and decreasing in
$|x|$.
Before stating our first theorem we make precise the notion of solution that
we use in this article. We say that a continuous function
$u:\mathbb{R}^{N}\to\mathbb{R}$ is a classical solution of equation (1.1) if
the fractional Laplacian of $u$ is defined at any point of $B_{1}$, according
to the definition given in (1.2), and if $u$ satisfies the equation and the
external condition in a pointwise sense. This notion of solution is extended
in a natural way to the other equations considered in this paper.
Now we are ready for our first theorem on radial symmetry and monotonicity
properties for positive solutions of equation (1.1). It states as follows:
###### Theorem 1.1
Assume that the functions $f$ and $g$ satisfy $(F1)$ and $(G)$, respectively.
If $u$ is a positive classical solution of (1.1), then $u$ must be radially
symmetric and strictly decreasing in $r=|x|$ for $r\in(0,1)$.
The proof of Theorem 1.1 is given in Section §3, where we prove a more general
symmetry and monotonicity result for equation (1.1) on a general domain
$\Omega$, which is convex and symmetric in one direction, see Theorem 3.1.
We devote the second part of this article to study symmetry results for a non-
linear equation as (1.1), but in $\mathbb{R}^{N}$ and with $g\equiv 0$. For
the problem in $\mathbb{R}^{N}$, the moving planes procedure has to start a
different way because we cannot use the Maximum Principle for small domain. We
refer to the work by Gidas, Ni and Nirenberg [15], Berestycki and Nirenberg
[1], Li [17], and Li and Ni [18], where these results were studied assuming
some additional hypothesis on $f$, allowing for decay properties of the
solution $u$. A general result in this direction was obtained by Li [17] for
the equation
$-\Delta u=f(u),\ \ \ \mbox{in}\ \ \mathbb{R}^{N},$
with $u$ decaying at infinity and $f$ satisfying the following hypothesis:
* $(F2)\ $
1. There exists $s_{0}>0,\ \gamma>0$ and $C>0$ such that
$\frac{f(v)-f(u)}{v-u}\leq C(u+v)^{\gamma}\ \ \ \ \mbox{for all }\quad
0<u<v<s_{0}.$ (1.4)
Motivated by these results, we are interested in similar properties of
positive solutions for equations involving the fractional Laplacian under
assumption (F2). Here is our second main theorem.
###### Theorem 1.2
Assume that $\alpha\in(0,1),$ $N\geq 2$, the function $f$ satisfies
$(F1)-(F2)$ and $u$ is a positive classical solution for the equation
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u=f(u)\ \ \ in\ \
\mathbb{R}^{N},\\\\[5.69054pt] u>0\ \ in\ \ \mathbb{R}^{N},\ \
\lim_{|x|\to\infty}u(x)=0.\end{array}\right.$ (1.5)
Assume further that there exists
$m>\max\\{\frac{2\alpha}{\gamma},\frac{N}{\gamma+2}\\}$ (1.6)
such that $u$ satisfies
$u(x)=O(\frac{1}{|x|^{m}}),\quad\mbox{as}\quad|x|\to\infty,$ (1.7)
then $u$ is radially symmetric and strictly decreasing about some point in
$\mathbb{R}^{N}$.
In [13], Felmer, Quaas and Tan studied symmetry of positive solutions using
the integral form of the moving planes method, assuming that the function $f$
is such that $h(\xi)\equiv f(\xi)+\xi$ is super-linear, with sub-critical
growth at infinity and
* $(H)\ $
1. $h\in C^{1}(\mathbb{R}),$ increasing and there exists $\tau>0$ such that
$\lim_{v\to 0}\frac{h^{\prime}(v)}{v^{\tau}}=0.$
We see that Theorem 1.2 generalizes Theorem 1.3 in [13], since here we do not
assume $f$ is differentiable and we do not require $h$ to be increasing. In
Section §4 we present an extension of Theorem 1.2 to $f(\xi)=\xi^{p}-\xi^{q}$,
with $0<q<1<p$, that is not covered by the results in [13] either, see Theorem
4.1. This non-linearity was studied by Valdebenito in [27], where decay and
symmetry results were obtained using local extension as in Caffarelli and
Silvestre [5] and regular moving planes.
For the particular case $f(u)=u^{p}$, for some $p>1$, we see that (H) is not
satisfied, but that (F2) does hold. Thus, if we knew the solution of (1.5)
satisfies decay assumption (1.7) in this setting, we would have symmetry
results in these cases. See [15] and [17] for the proof of decay properties in
the case of the Laplacian.
The third part of this paper is devoted to investigate the radial symmetry of
non-negative solutions for the following system of non-linear equations
involving fractional Laplacians with different orders,
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha_{1}}u=f_{1}(v)+g_{1},&\mbox{in}\quad
B_{1},\\\\[5.69054pt] (-\Delta)^{\alpha_{2}}v=f_{2}(u)+g_{2},&\mbox{in}\quad
B_{1},\\\\[5.69054pt] u=v=0,&\mbox{in}\quad B_{1}^{c},\end{array}\right.$
(1.8)
where $\alpha_{1},\alpha_{2}\in(0,1)$. We have following results for system
(1.8):
###### Theorem 1.3
Suppose $f_{1}$ and $f_{2}$ are locally Lipschitz continuous and increasing
functions defined in $[0,\infty)$ and $g_{1}$ and $g_{2}$ satisfy (G). Assume
that $(u,v)$ are positive, classical solutions of system (1.8), then $u$ and
$v$ are radially symmetric and strictly decreasing in $r=|x|$ for $r\in(0,1)$.
We prove our theorems using the moving planes method. The main difficulty
comes from the fact that the fractional Laplacian is a non-local operator, and
consequently Maximum Principle and Comparison Results require information on
the solutions in the whole complement of the domain, not only at the boundary.
To overcome this difficulty, we introduce a new truncation technique which is
well adapted to be used with the method of moving planes.
The rest of the paper is organized as follows. In Section §2, we recall the
ABP estimate for equations involving fractional Laplacian, as proved in [16]
and we prove a form of Maximum Principle for domains with small measure. In
Section §3, we prove Theorem 1.1 by the moving planes method and we extend our
symmetry results to general domains with one dimensional convexity and
symmetry properties. In Section §4, the radial symmetry of solutions for
equation (1.5) in $\mathbb{R}^{N}$ is obtained. An extension to a non-
lipschitzian non-linearity is given. In Section §5, we complete the proof of
Theorem 1.3. And finally, Section §6 is devoted to discuss (1.1) for a non-
local operator with non-homogeneous kernel.
## 2 Preliminaries
A key tool in the use of the moving planes method is the Maximum Principle for
small domain, which is a consequence of the ABP estimate. In [16], Guillen and
Schwab showed an ABP estimate (see Theorem 9.1) for general integro-
differential operators. In this section we recall this estimate in the case of
the fractional Laplacian in any open and bounded domain. Then we obtain the
Maximum Principle for small domains.
We start with the ABP estimate for the fractional Laplacian, which is stated
as follows:
###### Proposition 2.1
Let $\Omega$ be a bounded, connected open subset of $\mathbb{R}^{N}$. Suppose
that $h:\Omega\to\mathbb{R}$ is in $L^{\infty}(\Omega)$ and $w\in
L^{\infty}(\mathbb{R}^{N})$ is a classical solution of
$\left\\{\begin{array}[]{lll}\Delta^{\alpha}w(x)\leq
h(x),&x\in\Omega,\\\\[5.69054pt] w(x)\geq
0,&x\in\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$ (2.1)
Then there exists a positive constant $C$, depending on $N$ and $\alpha$, such
that
$-\inf_{\Omega}w\leq
Cd^{\alpha}\|h^{+}\|_{L^{\infty}(\Omega)}^{1-\alpha}\|h^{+}\|_{L^{N}(\Omega)}^{\alpha},$
(2.2)
where $d=\mbox{diam}({\Omega})$ is the diameter of $\Omega$ and
$h^{+}(x)=\max\\{h(x),0\\}$.
Here and in what follows we write
$\Delta^{\alpha}w(x)=-(-\Delta)^{\alpha}w(x).$
We have the following corollary
###### Corollary 2.1
Under the assumptions of Proposition 2.1, with $\Omega$ not necessarily
connected, we have
$-\inf_{\Omega}w\leq
Cd^{\alpha}\|h^{+}\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{\alpha}{N}}.$ (2.3)
Proof. We let $w_{0}\in L^{\infty}(\mathbb{R}^{N})$ be a classical solution of
$\left\\{\begin{array}[]{lll}\Delta^{\alpha}w_{0}(x)=\|h^{+}\|_{L^{\infty}(\Omega)}\chi_{\Omega}(x),&x\in
B_{d}(x_{0}),\\\\[5.69054pt] w_{0}(x)=0,&x\in\mathbb{R}^{N}\setminus
B_{d}(x_{0}),\end{array}\right.$ (2.4)
where $x_{0}\in\Omega$ and $\Omega\subset B_{d}(x_{0})$. We observe that
$B_{d}(x_{0})$ is connected and that $w_{0}\leq 0$ in all $\mathbb{R}^{N}$. By
Comparison Principle, we see that
$\inf_{\mathbb{R}^{N}}w_{0}\leq\inf_{\mathbb{R}^{N}}w,$
where $w$ is the solution of (2.1). Then we use Proposition 2.1 to obtain that
$-\inf_{\mathbb{R}^{N}}w_{0}=-\inf_{B_{d}(x_{0})}w_{0}\leq
C(2d)^{\alpha}\|h^{+}\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{\alpha}{N}}$
and then we conclude
$-\inf_{\Omega}w=-\inf_{\mathbb{R}^{N}}w\leq
Cd^{\alpha}\|h^{+}\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{\alpha}{N}}.\qquad\Box$
###### Remark 2.1
We notice that, under a possibly different constant $C>0$, the ABP estimate
for problem (2.1) with $\alpha=1$
$\left\\{\begin{array}[]{lll}\Delta w(x)\leq h(x),&x\in\Omega,\\\\[5.69054pt]
w(x)\geq 0,&x\in\partial\Omega,\end{array}\right.$
is precisely (2.2) with $\alpha=1$.
As a consequence of the ABP estimate just recalled, we have the Maximum
Principle for small domain, which is stated as follows:
###### Proposition 2.2
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^{N}$. Suppose that
$\varphi:\Omega\to\mathbb{R}$ is in $L^{\infty}(\Omega)$ and $w\in
L^{\infty}(\mathbb{R}^{N})$ is a classical solution of
$\left\\{\begin{array}[]{lll}\Delta^{\alpha}w(x)\leq\varphi(x)w(x),&x\in\Omega,\\\\[5.69054pt]
w(x)\geq 0,&x\in\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$ (2.5)
Then there is $\delta>0$ such that whenever $|\Omega^{-}|\leq\delta$, $w$ has
to be non-negative in $\Omega$. Here $\Omega^{-}=\\{x\in\Omega\ |\ w(x)<0\\}$.
Proof. By (2.5), we observe that
$\left\\{\begin{array}[]{lll}\Delta^{\alpha}w(x)\leq\varphi(x)w(x),&x\in\Omega^{-},\\\\[5.69054pt]
w(x)\geq 0,&x\in\mathbb{R}^{N}\setminus\Omega^{-}.\end{array}\right.$
Then, using Corollary 2.1 with $h(x)=\varphi(x)w(x)$, we obtain that
$\displaystyle\|w\|_{L^{\infty}(\Omega^{-})}=-\inf_{\Omega^{-}}w$
$\displaystyle\leq$ $\displaystyle Cd_{0}^{\alpha}\|(\varphi
w)^{+}\|_{L^{\infty}(\Omega^{-})}|\Omega^{-}|^{\frac{\alpha}{N}},$
where constant $C>0$ depends on $N$ and $\alpha$. Here
$d_{0}=\mbox{diam}(\Omega^{-})$. Thus
$\|w\|_{L^{\infty}(\Omega^{-})}\leq
Cd_{0}^{\alpha}\|\varphi\|_{L^{\infty}(\Omega)}\|w\|_{L^{\infty}(\Omega^{-})}|\Omega^{-}|^{\frac{\alpha}{N}}.$
We see that, if $|\Omega^{-}|$ is such that
$Cd_{0}^{\alpha}\|\varphi\|_{L^{\infty}(\Omega)}|\Omega^{-}|^{\alpha/N}<1$,
then we must have that
$\|w\|_{L^{\infty}(\Omega^{-})}=0.$
This implies that $|\Omega^{-}|=0$ and since $\Omega^{-}$ is open, we have
$\Omega^{-}=\emptyset$, completing the proof. $\Box$
## 3 Proof of Theorem 1.1.
In this section we provide a proof of Theorem 1.1 on the radial symmetry and
monotonicity of positive solutions to equation (1.1) in the unit ball. For
this purpose we use the of moving planes method, for which we give some
preliminary notation. We define
$\Sigma_{\lambda}=\\{x=(x_{1},x^{\prime})\in B_{1}\ |\ x_{1}>\lambda\\},$
(3.1) $T_{\lambda}=\\{x=(x_{1},x^{\prime})\in\mathbb{R}^{N}\ |\
x_{1}=\lambda\\},$ (3.2) $u_{\lambda}(x)=u(x_{\lambda})\quad\mbox{and}\quad
w_{\lambda}(x)=u_{\lambda}(x)-u(x),$ (3.3)
where $\lambda\in(0,1)$ and $x_{\lambda}=(2\lambda-x_{1},x^{\prime})$ for
$x=(x_{1},x^{\prime})\in\mathbb{R}^{N}.$ For any subset $A$ of
$\mathbb{R}^{N}$, we write $A_{\lambda}=\\{x_{\lambda}:\,x\in A\\}$, the
reflection of $A$ with regard to $T_{\lambda}$.
Proof of Theorem 1.1. We divide the proof in three steps.
Step 1: We prove that if $\lambda\in(0,1)$ is close to $1$, then
$w_{\lambda}>0$ in $\Sigma_{\lambda}$. For this purpose, we start proving that
if $\lambda\in(0,1)$ is close to 1, then $w_{\lambda}\geq 0$ in
$\Sigma_{\lambda}$. If we define
$\Sigma_{\lambda}^{-}=\\{x\in\Sigma_{\lambda}\ |\ w_{\lambda}(x)<0\\},$ then
we just need to prove that if $\lambda\in(0,1)$ is close to 1 then
$\Sigma_{\lambda}^{-}=\emptyset.$ (3.4)
By contradiction, we assume (3.4) is not true, that is
$\Sigma^{-}_{\lambda}\not=\emptyset$. We denote
$w_{\lambda}^{+}(x)=\left\\{\begin{array}[]{lll}w_{\lambda}(x),&x\in\Sigma_{\lambda}^{-},\\\\[5.69054pt]
0,&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-},\end{array}\right.$ (3.5)
$w_{\lambda}^{-}(x)=\left\\{\begin{array}[]{lll}0,&x\in\Sigma_{\lambda}^{-},\\\\[5.69054pt]
w_{\lambda}(x),&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-}\end{array}\right.$
(3.6)
and we observe that $w_{\lambda}^{+}(x)=w_{\lambda}(x)-w_{\lambda}^{-}(x)$ for
all $x\in\mathbb{R}^{N}.$ Next we claim that for all $0<\lambda<1$, we have
$(-\Delta)^{\alpha}w_{\lambda}^{-}(x)\leq 0,\ \ \ \ \forall\
x\in\Sigma_{\lambda}^{-}.$ (3.7)
By direct computation, for $x\in\Sigma_{\lambda}^{-}$, we have
$\displaystyle(-\Delta)^{\alpha}w_{\lambda}^{-}(x)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\frac{w_{\lambda}^{-}(x)-w_{\lambda}^{-}(z)}{|x-z|^{N+2\alpha}}dz=-\int_{\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle-\int_{(B_{1}\setminus(B_{1})_{\lambda})\cup((B_{1})_{\lambda}\setminus
B_{1})}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle-\int_{(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})\cup(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz-\int_{(\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle=$ $\displaystyle-I_{1}-I_{2}-I_{3}.$
We look at each of these integrals separately. Since $u=0\ in\
(B_{1})_{\lambda}\setminus B_{1}$ and $u_{\lambda}=0\ in\
B_{1}\setminus(B_{1})_{\lambda}$, we have
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{(B_{1}\setminus(B_{1})_{\lambda})\cup((B_{1})_{\lambda}\setminus
B_{1})}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz$ $\displaystyle=$
$\displaystyle\int_{(B_{1})_{\lambda}\setminus
B_{1}}\frac{u_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz-\int_{B_{1}\setminus(B_{1})_{\lambda}}\frac{u(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle=$ $\displaystyle\int_{(B_{1})_{\lambda}\setminus
B_{1}}u_{\lambda}(z)(\frac{1}{|x-z|^{N+2\alpha}}-\frac{1}{|x-z_{\lambda}|^{N+2\alpha}}))dz\geq
0,$
since $u_{\lambda}\geq 0$ and $|x-z_{\lambda}|>|x-z|$ for all
$x\in\Sigma_{\lambda}^{-}$ and $z\in(B_{1})_{\lambda}\setminus B_{1}.$ In
order to study the sign of $I_{2}$ we first observe that
$w_{\lambda}(z_{\lambda})=-w_{\lambda}(z)$ for any $z\in\mathbb{R}^{N}$. Then
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})\cup(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz+\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z_{\lambda})}{|x-z_{\lambda}|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}w_{\lambda}(z)(\frac{1}{|x-z|^{N+2\alpha}}-\frac{1}{|x-z_{\lambda}|^{N+2\alpha}})dz\geq
0,$
since $w_{\lambda}\geq 0$ in $\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}$
and $|x-z_{\lambda}|>|x-z|$ for all $x\in\Sigma_{\lambda}^{-}$ and
$z\in\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}.$ Finally, since
$w_{\lambda}(z)<0$ for $z\in\Sigma_{\lambda}^{-}$, we have
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{(\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2\alpha}}dz=\int_{\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z_{\lambda})}{|x-z_{\lambda}|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle-\int_{\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z_{\lambda}|^{N+2\alpha}}dz\geq
0.$
Hence, we obtain (3.7), proving the claim. Now we apply (3.7) and linearity of
the fractional Laplacian to obtain that, for $x\in\Sigma_{\lambda}^{-},$
$(-\Delta)^{\alpha}w_{\lambda}^{+}(x)\geq(-\Delta)^{\alpha}w_{\lambda}(x)=(-\Delta)^{\alpha}u_{\lambda}(x)-(-\Delta)^{\alpha}u(x).$
(3.8)
Combining equation (1.1) with (3.8) and (3.5), for $x\in\Sigma_{\lambda}^{-}$
we have
$\displaystyle(-\Delta)^{\alpha}w_{\lambda}^{+}(x)$ $\displaystyle\geq$
$\displaystyle(-\Delta)^{\alpha}u_{\lambda}(x)-(-\Delta)^{\alpha}u(x)$
$\displaystyle=$ $\displaystyle f(u_{\lambda}(x))+g(x_{\lambda})-f(u(x))-g(x)$
$\displaystyle=$
$\displaystyle\frac{f(u_{\lambda}(x))-f(u(x))}{u_{\lambda}(x)-u(x)}w_{\lambda}^{+}(x)+g(x_{\lambda})-g(x).$
Let us define
$\varphi(x)=-({f(u_{\lambda}(x))-f(u(x))})/({u_{\lambda}(x)-u(x)})$ for
$x\in\Sigma_{\lambda}^{-}$. By assumption $(F1)$, we have that $\varphi\in
L^{\infty}(\Sigma_{\lambda}^{-})$. By assumption $(G)$, we have that
$g(x_{\lambda})\geq g(x)$, since for all $x\in\Sigma_{\lambda}^{-}$ and
$0<\lambda<1$, we have $|x|>|x_{\lambda}|$. Hence, we have
$\Delta^{\alpha}w_{\lambda}^{+}(x)\leq\varphi(x)w_{\lambda}^{+}(x),\ \
x\in\Sigma_{\lambda}^{-}$ (3.9)
and since $w_{\lambda}^{+}=0$ in $(\Sigma_{\lambda}^{-})^{c}$ we may apply
Proposition 2.2. Choosing $\lambda\in(0,1)$ close enough to $1$ we find that
$|\Sigma_{\lambda}^{-}|$ is small and then
$w_{\lambda}=w_{\lambda}^{+}\geq 0\ \ \ \ \mbox{in}\ \ \Sigma_{\lambda}^{-}.$
But this is a contradiction with our assumption so we have
$w_{\lambda}\geq 0\ \ \ in\ \ \Sigma_{\lambda}.$
In order to complete Step 1, we claim that for $0<\lambda<1$, if
$w_{\lambda}\geq 0$ and $w_{\lambda}\not\equiv 0$ in $\Sigma_{\lambda}$, then
$w_{\lambda}>0$ in $\Sigma_{\lambda}$. Assuming the claim is true, we complete
the proof, since the function $u$ is positive in $B_{1}$ and $u=0$ on
$\partial B_{1}$, so that $w_{\lambda}$ is positive in $\partial
B_{1}\cap\partial\Sigma_{\lambda}$ and then, by continuity $w_{\lambda}\not=0$
in $\Sigma_{\lambda}$.
Now we prove the claim. Assume there exists $x_{0}\in\Sigma_{\lambda}$ such
that $w_{\lambda}(x_{0})=0,$ that is, $u_{\lambda}(x_{0})=u(x_{0})$. Then we
have that
$\displaystyle(-\Delta)^{\alpha}w_{\lambda}(x_{0})$ $\displaystyle=$
$\displaystyle(-\Delta)^{\alpha}u_{\lambda}(x_{0})-(-\Delta)^{\alpha}u(x_{0})=g((x_{0})_{\lambda})-g(x_{0}).$
Since $x_{0}\in\Sigma_{\lambda}$, we have $|x_{0}|>|(x_{0})_{\lambda}|$, then
by assumption $(G)$ we have $g((x_{0})_{\lambda})\geq g(x_{0})$ and thus
$(-\Delta)^{\alpha}w_{\lambda}(x_{0})\geq 0.$ (3.10)
On the other hand, defining
$A_{\lambda}=\\{(x_{1},x^{\prime})\in\mathbb{R}^{N}\ |\ x_{1}>\lambda\\}$,
since $w_{\lambda}(z_{\lambda})=-w_{\lambda}(z)$ for any $z\in\mathbb{R}^{N}$
and $w_{\lambda}(x_{0})=0$, we find
$\displaystyle(-\Delta)^{\alpha}w_{\lambda}(x_{0})$ $\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2\alpha}}dz-\int_{\mathbb{R}^{N}\setminus
A_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2\alpha}}dz$ $\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2\alpha}}dz-\int_{A_{\lambda}}\frac{w_{\lambda}(z_{\lambda})}{|x_{0}-z_{\lambda}|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}w_{\lambda}(z)(\frac{1}{|x_{0}-z|^{N+2\alpha}}-\frac{1}{|x_{0}-z_{\lambda}|^{N+2\alpha}})dz.$
Since $|x_{0}-z_{\lambda}|>|x_{0}-z|$ for $z\in A_{\lambda}$ ,
$w_{\lambda}(z)\geq 0$ and $w_{\lambda}(z)\not\equiv 0$ in $A_{\lambda}$, from
here we get
$(-\Delta)^{\alpha}w_{\lambda}(x_{0})<0,$ (3.11)
which contradicts (3.10), completing the proof of the claim.
Step 2: We define $\lambda_{0}=\inf\\{\lambda\in(0,1)\ |\ w_{\lambda}>0\ \
\rm{in}\ \ \Sigma_{\lambda}\\}$ and we prove that $\lambda_{0}=0$. Proceeding
by contradiction, we assume that $\lambda_{0}>0$, then $w_{\lambda_{0}}\geq 0$
in $\Sigma_{\lambda_{0}}$ and $w_{\lambda_{0}}\not\equiv 0$ in
$\Sigma_{\lambda_{0}}$. Thus, by the claim just proved above, we have
$w_{\lambda_{0}}>0$ in $\Sigma_{\lambda_{0}}$.
Next we claim that if $w_{\lambda}>0$ in $\Sigma_{\lambda}$ for
$\lambda\in(0,1)$, then there exists $\epsilon\in(0,\lambda)$ such that
$w_{\lambda_{\epsilon}}>0$ in $\Sigma_{\lambda_{\epsilon}}$, where
$\lambda_{\epsilon}=\lambda-\epsilon$. This claim directly implies that
$\lambda_{0}=0$, completing Step 2.
Now we prove the claim. Let $D_{\mu}=\\{x\in\Sigma_{\lambda}\ |\
dist(x,\partial\Sigma_{\lambda})\geq\mu\\}$ for $\mu>0$ small. Since
$w_{\lambda}>0$ in $\Sigma_{\lambda}$ and $D_{\mu}$ is compact, then there
exists $\mu_{0}>0$ such that $w_{\lambda}\geq\mu_{0}$ in $D_{\mu}$. By
continuity of $w_{\lambda}(x)$, for $\epsilon>0$ small enough and denoting
$\lambda_{\epsilon}=\lambda-\epsilon,$ we have that
$w_{\lambda_{\epsilon}}(x)\geq 0\ \ \rm{in}\ \ D_{\mu}.$
As a consequence,
$\Sigma_{\lambda_{\epsilon}}^{-}\subset\Sigma_{\lambda_{\epsilon}}\setminus
D_{\mu}$
and $|\Sigma_{\lambda_{\epsilon}}^{-}|$ is small if $\epsilon$ and $\mu$ are
small. Using (3.7) and proceeding as in Step 1, we have for all
$x\in\Sigma_{\lambda_{\epsilon}}^{-}$ that
$\displaystyle(-\Delta)^{\alpha}w_{\lambda_{\epsilon}}^{+}(x)$
$\displaystyle=$
$\displaystyle(-\Delta)^{\alpha}u_{\lambda_{\epsilon}}(x)-(-\Delta)^{\alpha}u(x)-(-\Delta)^{\alpha}w_{\lambda_{\epsilon}}^{-}(x)$
$\displaystyle\geq$
$\displaystyle(-\Delta)^{\alpha}u_{\lambda_{\epsilon}}(x)-(-\Delta)^{\alpha}u(x)$
$\displaystyle=$
$\displaystyle\varphi(x)w_{\lambda_{\epsilon}}^{+}(x)+g(x_{\lambda})-g(x)\geq\varphi(x)w_{\lambda_{\epsilon}}^{+}(x),$
where
$\varphi(x)=\frac{f(u_{\lambda_{\epsilon}}(x))-f(u(x))}{u_{\lambda_{\epsilon}}(x)-u(x)}$
is bounded by assumption $(F1)$.
Since $w_{\lambda_{\epsilon}}^{+}=0$ in
$(\Sigma_{\lambda_{\epsilon}}^{-})^{c}$ and
$|\Sigma_{\lambda_{\epsilon}}^{-}|$ is small, for $\epsilon$ and $\mu$ small,
Proposition 2.2 implies that $w_{\lambda_{\epsilon}}\geq 0$ in
$\Sigma_{\lambda_{\epsilon}}$. Thus, since $\lambda_{\epsilon}>0$ and
$w_{\lambda_{\epsilon}}\not\equiv 0$ in $\Sigma_{\lambda_{\epsilon}}$, as
before we have $w_{\lambda_{\epsilon}}>0$ in $\Sigma_{\lambda_{\epsilon}}$,
completing the proof of the claim.
Step 3: By Step 2, we have $\lambda_{0}=0$, which implies that
$u(-x_{1},x^{\prime})\geq u(x_{1},x^{\prime})$ for $x_{1}\geq 0.$ Using the
same argument from the other side, we conclude that $u(-x_{1},x^{\prime})\leq
u(x_{1},x^{\prime})$ for $x_{1}\geq 0$ and then
$u(-x_{1},x^{\prime})=u(x_{1},x^{\prime})$ for $x_{1}\geq 0.$ Repeating this
procedure in all directions we obtain radial symmetry of $u$.
Finally, we prove $u(r)$ is strictly decreasing in $r\in(0,1)$. Let us
consider $0<x_{1}<\widetilde{x}_{1}<1$ and let
$\lambda=\frac{x_{1}+\widetilde{x}_{1}}{2}$. Then, as proved above we have
$w_{\lambda}(x)>0\ \ \mbox{for}\ \ x\in\Sigma_{\lambda}.$
Then
$\displaystyle 0<w_{\lambda}(\widetilde{x}_{1},0,\cdots,0)$ $\displaystyle=$
$\displaystyle
u_{\lambda}(\widetilde{x}_{1},0,\cdots,0)-u(\widetilde{x}_{1},0,\cdots,0)$
$\displaystyle=$ $\displaystyle
u(x_{1},0,\cdots,0)-u(\widetilde{x}_{1},0,\cdots,0),$
that is $u(x_{1},0,\cdots,0)>u(\widetilde{x}_{1},0,\cdots,0).$ Using the
radial symmetry of $u$, we conclude from here the monotonicty of $u$. $\Box$
The proof of Theorem 1.1 can be applied directly to prove symmetry results for
problem (1.1) in more general domains. We have the following definition
###### Definition 3.1
We say that domain $\Omega\subset\mathbb{R}^{N}$ is convex in the $x_{1}$
direction:
$(x_{1},x^{\prime}),(x_{1},y^{\prime})\in\Omega\Rightarrow(x_{1},tx^{\prime}+(1-t)y^{\prime})\in\Omega,\
\ \forall\ t\in(0,1).$
Now we state the more general theorem:
###### Theorem 3.1
Let $\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is an open and bounded set. Assume
further that $\Omega$ is convex in the $x_{1}$ direction and symmetric with
respect to the plane $x_{1}=0$. Assume that the function $f$ satisfies $(F1)$
and $g$ satisfies
* $(\widetilde{G})\ $
The function $g:\Omega\to\mathbb{R}$ is symmetric with respect to $x_{1}=0$
and decreasing in the $x_{1}$ direction, for $x=(x_{1},x^{\prime})\in\Omega$,
$x_{1}>0$.
Let $u$ be a positive classical solution of
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)=f(u(x))+g(x),&x\in\Omega,\\\\[5.69054pt]
u(x)=0,&x\in\Omega^{c}.\end{array}\right.$ (3.12)
Then $u$ is symmetric with respect to $x_{1}$ and it is strictly decreasing in
the $x_{1}$ direction for $x=(x_{1},x^{\prime})\in\Omega$, $x_{1}>0$.
## 4 Symmetry of solutions in $\mathbb{R}^{N}$
In this section we study radial symmetry results for positive solution of
equation (1.5) in $\mathbb{R}^{N}$, in particular we will provide a proof of
Theorem 1.2. In the case of the whole space, the moving planes procedure needs
to be started in a different way, because we cannot use the Maximum Principle
for small domains. We use the moving plane method as for the second order
equation as in the work by Li [17] (see also [21]).
In this section we use the notation introduced in (3.1)-(3.3) and we let $u$
be a classical positive solution of (1.5). In order to prove Theorem 1.2 we
need some preliminary lemmas.
###### Lemma 4.1
Under the assumptions of Theorem 1.2, for any $\lambda\in\mathbb{R}$, we have
$\int_{\Sigma_{\lambda}}(f(u_{\lambda})-f(u))^{+}(u_{\lambda}-u)^{+}dx<+\infty.$
Proof. By our hypothesis, for any given $\lambda\in\mathbb{R}$, we may choose
$R>1$ and some constant $c>1$ such that
$\frac{1}{c|x|^{m}}\leq u(x),u_{\lambda}(x)\leq\frac{c}{|x|^{m}}<s_{0}\ \ \
for\ all\ x\in B^{c}_{R},$
where $s_{0}$ is the constant in condition (F2).
If $u_{\lambda}(x)>u(x)$ for some $x\in\Sigma_{\lambda}\cap B^{c}_{R},$ we
have $0<u(x)<u_{\lambda}(x)<s_{0}$. Using (1.4) with $v=u_{\lambda}(x)$, then
$\frac{f(u_{\lambda}(x))-f(u(x))}{u_{\lambda}(x)-u(x)}\leq
C(u(x)+u_{\lambda}(x))^{\gamma}\leq 2^{\gamma}Cu^{\gamma}_{\lambda}(x),$
then
$\displaystyle(f(u_{\lambda}(x))-f(u(x)))^{+}(u_{\lambda}(x)-u(x))^{+}$
$\displaystyle\leq$ $\displaystyle
2^{\gamma}Cu^{\gamma}_{\lambda}(x)[(u_{\lambda}(x)-u(x))^{+}]^{2}$
$\displaystyle\leq$ $\displaystyle\tilde{C}u^{\gamma+2}_{\lambda}(x),$
for certain $\tilde{C}>0$. We observe that, if $u_{\lambda}(x)\leq u(x)$ for
some $x\in\Sigma_{\lambda}\cap B^{c}_{R},$ then inequality above is obvious.
Therefore,
$\displaystyle(f(u_{\lambda})-f(u))^{+}(u_{\lambda}-u)^{+}\leq\tilde{C}u^{\gamma+2}_{\lambda}\
\ \ in\ \ \Sigma_{\lambda}\cap B^{c}_{R}.$
Now we integrate in $\Sigma_{\lambda}\cap B^{c}_{R}$ to obtain
$\displaystyle\int_{\Sigma_{\lambda}\cap
B^{c}_{R}}(f(u_{\lambda})-f(u))^{+}(u_{\lambda}-u)^{+}dx$ $\displaystyle\leq$
$\displaystyle\tilde{C}\int_{\Sigma_{\lambda}\cap{B^{c}_{R}}}u^{\gamma+2}_{\lambda}(x)dx$
$\displaystyle\leq$ $\displaystyle
C\int_{{B^{c}_{R}}}|x|^{-m(\gamma+2)}dx<+\infty,$
where the last inequality holds by (1.6). Since $u$ and $u_{\lambda}$ are
bounded and $f$ is locally Lipschitz, we have
$\displaystyle\int_{\Sigma_{\lambda}\cap
B_{R}}(f(u_{\lambda})-f(u))^{+}(u_{\lambda}-u)^{+}dx<+\infty$
and the proof is complete. $\Box$
It will be convenient for our analysis to define the following function
$w(x)=\left\\{\begin{array}[]{lll}(u_{\lambda}-u)^{+}(x),&x\in\Sigma_{\lambda},\\\\[5.69054pt]
(u_{\lambda}-u)^{-}(x),&x\in\Sigma_{\lambda}^{c},\end{array}\right.$ (4.1)
where $(u_{\lambda}-u)^{+}(x)=\max\\{(u_{\lambda}-u)(x),\ 0\\}$,
$(u_{\lambda}-u)^{-}(x)=\min\\{(u_{\lambda}-u)(x),\ 0\\}$. We have
###### Lemma 4.2
Under the assumptions of Theorem 1.2, there exists a constant $C>0$ such that
$\int_{\Sigma_{\lambda}}(-\Delta)^{\alpha}({u_{\lambda}}-u)(u_{\lambda}-u)^{+}dx\geq
C(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}.$
(4.2)
Proof. We start observing that, given $x\in\Sigma_{\lambda}$, we have
$\displaystyle w(x_{\lambda})$ $\displaystyle=$
$\displaystyle(u_{\lambda}-u)^{-}(x_{\lambda})=\min\\{(u_{\lambda}-u)(x_{\lambda}),\
0\\}=\min\\{(u-u_{\lambda})(x),\ 0\\}$ $\displaystyle=$
$\displaystyle-\max\\{(u_{\lambda}-u)(x),\ 0\\}=-(u_{\lambda}-u)^{+}(x)=-w(x)$
and similarly $w(x)=-w(x_{\lambda})$ for $x\in\Sigma_{\lambda}^{c}$ so that
$w(x)=-w(x_{\lambda})\quad\mbox{for}\quad x\in\mathbb{R}^{N}.$ (4.3)
This implies
$\displaystyle\int_{\mathbb{R}^{N}}|w|^{\frac{2N}{N-2\alpha}}dx=\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx+\int_{\Sigma_{\lambda}^{c}}|w|^{\frac{2N}{N-2\alpha}}dx=2\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx.$
(4.4)
Next we see that for any $x\in\Sigma_{\lambda}\cap{\rm{supp}}(w)$ we have that
$w(x)=(u_{\lambda}-u)(x)$ and
$(-\Delta)^{\alpha}(u_{\lambda}-u)(x)\geq(-\Delta)^{\alpha}w(x),\quad\forall\
x\in\Sigma_{\lambda}\cap\rm{supp}(w),$
$\displaystyle(-\Delta)^{\alpha}w(x)-(-\Delta)^{\alpha}(u_{\lambda}-u)(x)=\int_{\mathbb{R}^{N}}\frac{(u_{\lambda}-u)(z)-w(z)}{|x-z|^{N+2\alpha}}dz$
(4.5) $\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\cap({\rm{supp}}(w))^{c}}\frac{(u_{\lambda}-u)(z)}{|x-z|^{N+2\alpha}}dz+\int_{\Sigma_{\lambda}^{c}\cap({\rm{supp}}(w))^{c}}\frac{(u_{\lambda}-u)(z)}{|x-z|^{N+2\alpha}}dz$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\cap({\rm{supp}}(w))^{c}}(u_{\lambda}-u)(z)(\frac{1}{|x-z|^{N+2\alpha}}-\frac{1}{|x-z_{\lambda}|^{N+2\alpha}})dz\leq
0,$
where we used that $u_{\lambda}-u\leq 0$ in
$\Sigma_{\lambda}\cap({\rm{supp}}(w))^{c}$ and $|x-z|\leq|x-z_{\lambda}|$ for
$x,z\in\Sigma_{\lambda}.$ From (4.5), using the equation and Lemma 4.1 we find
that
$\displaystyle\int_{\Sigma_{\lambda}}(-\Delta)^{{\alpha}}w\,wdx$
$\displaystyle\leq$
$\displaystyle\int_{\Sigma_{\lambda}}(-\Delta)^{{\alpha}}(u_{\lambda}-u)(u_{\lambda}-u)^{+}dx$
(4.6) $\displaystyle\leq$
$\displaystyle\int_{\Sigma_{\lambda}}(f(u_{\lambda})-f(u))^{+}(u_{\lambda}-u)^{+}dx<\infty.$
(4.7)
From here the following integrals are finite and, taking into account (4.3),
we obtain that
$\displaystyle\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx+\int_{\Sigma_{\lambda}^{c}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx$
(4.8) $\displaystyle=$ $\displaystyle
2\int_{\Sigma_{\lambda}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx.$
Now we can use the Sobolev embedding from $H^{\alpha}(\mathbb{R}^{N})$ to
$L^{\frac{2N}{N-2\alpha}}(\mathbb{R}^{N})$ to find a constant $C$ so that
$\displaystyle\int_{\Sigma_{\lambda}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx\geq
C(\int_{\mathbb{R}^{N}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}$
(4.9) $\displaystyle=$ $\displaystyle
C(2\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}.$
On the other hand, from (4.3) and (4.6) we find that
$\displaystyle\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{\alpha}{2}}w|^{2}dx$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{N}}(-\Delta)^{\alpha}w\cdot
wdx=2\int_{\Sigma_{\lambda}}(-\Delta)^{\alpha}w\cdot wdx$ (4.10)
$\displaystyle\leq$ $\displaystyle
2\int_{\Sigma_{\lambda}}(-\Delta)^{\alpha}({u_{\lambda}}-u)(u_{\lambda}-u)^{+}dx.$
From (4.9) and (4.10) the proof of the lemma is completed. $\Box$
Now we are ready to complete the
Proof of Theorem 1.2. We divide the proof into three steps.
Step 1: We show that $\lambda_{0}:=\sup\\{\lambda\ |\ u_{\lambda}\leq u\ in\
\Sigma_{\lambda}\\}$ is finite. Using $(u_{\lambda}-u)^{+}$ as a test function
in the equation for $u$ and $u_{\lambda}$, using (1.4) and Hölder inequality,
for $\lambda$ big (negative), we find that
$\displaystyle\int_{\Sigma_{\lambda}}(-\Delta)^{\alpha}({u_{\lambda}}-u)(u_{\lambda}-u)^{+}dx\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=\int_{\Sigma_{\lambda}}(f(u_{\lambda})-f(u))(u_{\lambda}-u)^{+}dx$
$\displaystyle\leq\int_{\Sigma_{\lambda}}[\frac{f(u_{\lambda})-f(u)}{u_{\lambda}-u}]^{+}[(u_{\lambda}-u)^{+}]^{2}dx$
$\displaystyle\leq
C\int_{\Sigma_{\lambda}}{u^{\gamma}_{\lambda}}w^{2}dx\leq\bar{C}\int_{\Sigma_{\lambda}}|x_{\lambda}|^{-m\gamma}w^{2}dx$
$\displaystyle\leq\bar{C}(\int_{\Sigma_{\lambda}}|x_{\lambda}|^{-\frac{Nm\gamma}{2\alpha}}dx)^{\frac{2\alpha}{N}}(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}.$
By Lemma 4.2, there exists a constant $C>0$ such that
$\displaystyle(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}$
$\displaystyle\leq$ $\displaystyle
C(\int_{\Sigma_{\lambda}}|x_{\lambda}|^{-\frac{Nm\gamma}{2\alpha}}dx)^{\frac{2\alpha}{N}}(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}},$
but we have
$\displaystyle\int_{\Sigma_{\lambda}}|x_{\lambda}|^{-\frac{Nm\gamma}{2\alpha}}dx$
$\displaystyle\leq$
$\displaystyle\int_{\Sigma^{c}_{\lambda}}|x|^{-\frac{Nm\gamma}{2\alpha}}dx\leq\int_{B^{c}_{|\lambda|}}|x|^{-\frac{Nm\gamma}{2\alpha}}dx=c{|\lambda|}^{\frac{N}{2\alpha}(2\alpha-m\gamma)},$
so that, using (1.6), we can choose $R>0$ big enough such that
$CR^{2\alpha-m\gamma}\leq\frac{1}{2}$, then we obtain
$\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx=0,\ \ \forall\
\lambda<-R.$
Thus $w=0$ in $\Sigma_{\lambda}$ and then $u_{\lambda}\leq u$ in
$\Sigma_{\lambda},$ for all $\lambda<-R,$ concluding that $\lambda_{0}\geq-R.$
On the other hand, since $u$ decays at infinity, then there exists
$\lambda_{1}$ such that $u(x)<u_{\lambda_{1}}(x)$ for some
$x\in\Sigma_{\lambda_{1}}.$ Hence $\lambda_{0}$ is finite.
Step 2: We prove that $u\equiv u_{\lambda_{0}}$ in $\Sigma_{\lambda_{0}}$.
Assuming the contrary, we have $u\neq u_{\lambda_{0}}$ and $u\geq
u_{\lambda_{0}}$ in $\Sigma_{\lambda_{0}}$. Assume next that there exists
$x_{0}\in\Sigma_{\lambda_{0}}$ such that $u_{\lambda_{0}}(x_{0})=u(x_{0}),$
then we have
$(-\Delta)^{\alpha}u_{\lambda_{0}}(x_{0})-(-\Delta)^{\alpha}u(x_{0})=f(u_{\lambda_{0}}(x_{0}))-f(u(x_{0}))=0.$
(4.11)
On the other hand,
$\displaystyle(-\Delta)^{\alpha}u_{\lambda_{0}}(x_{0})-(-\Delta)^{\alpha}u(x_{0})=-\int_{\mathbb{R}^{N}}\frac{u_{\lambda_{0}}(y)-u(y)}{|x_{0}-y|^{N+2\alpha}}dy$
$\displaystyle=$
$\displaystyle-\int_{\Sigma_{\lambda_{0}}}(u_{\lambda_{0}}(y)-u(y))(\frac{1}{|x_{0}-y|^{N+2\alpha}}-\frac{1}{|x_{0}-y_{\lambda_{0}}|^{N+2\alpha}})dy>0,$
which contradicts (4.11). As a sequence, $u>u_{\lambda_{0}}$ in
$\Sigma_{\lambda_{0}}$.
To complete Step 2, we only need to prove that $u\geq u_{\lambda}$ in
$\Sigma_{\lambda}$ continues to hold when
${\lambda_{0}}<\lambda<{\lambda_{0}}+\varepsilon$, where $\varepsilon>0$
small. Let us consider then $\varepsilon>0$, to be chosen later, and take
$\lambda\in({\lambda_{0}},{\lambda_{0}}+\varepsilon)$. Let $P=(\lambda,0)$ and
$B(P,R)$ be the ball centered at $P$ and with radius $R>1$ to be chosen later.
Define $\tilde{B}=\Sigma_{\lambda}\cap B(P,R)$ and let us consider
$(u_{\lambda}-u)^{+}$ test function in the equation for $u$ and $u_{\lambda}$
in $\Sigma_{\lambda}$, then from Lemma 4.2 we find
$\displaystyle(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}$
$\displaystyle\leq$ $\displaystyle
C\int_{\Sigma_{\lambda}}(f(u_{\lambda})-f(u))(u_{\lambda}-u)^{+}dx.$ (4.12)
We estimate the integral on the right. Since $f$ is locally Lipschitz, using
Hölder inequality, we have
$\displaystyle\int_{\tilde{B}}(f(u_{\lambda})-f(u))(u_{\lambda}-u)^{+}dx\leq
C\int_{\tilde{B}}|w|^{2}\chi_{{\rm{supp}}{(u_{\lambda}-u)^{+}}}dx$ (4.13)
$\displaystyle=$ $\displaystyle
C|\tilde{B}\cap{{\rm{supp}}{(u_{\lambda}-u)^{+}}}|^{\frac{2\alpha}{N}}(\int_{\tilde{B}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}.$
On the other hand, for the integral over
$\Sigma_{\lambda}\setminus{\tilde{B}}$, we assume $R$ and $R_{0}$ are such
that $\Sigma_{\lambda}\setminus{\tilde{B}}\subset{B^{c}(P,R)}\subset
B^{c}_{R_{0}}(0)$, proceeding as in Step 1, we have
$\displaystyle\int_{\Sigma_{\lambda}\setminus{\tilde{B}}}(f(u_{\lambda})-f(u))(u_{\lambda}-u)^{+}dx$
$\displaystyle\leq$
$\displaystyle{C}\int_{\Sigma_{\lambda}\setminus{\tilde{B}}}u^{\gamma}_{\lambda}w^{2}dx$
(4.14) $\displaystyle\leq$
$\displaystyle{C}(\int_{\Sigma_{\lambda}\setminus{\tilde{B}}}|x_{\lambda}|^{-\frac{Nm\gamma}{2\alpha}}dx)^{\frac{2\alpha}{N}}(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}$
$\displaystyle\leq$ $\displaystyle
C{R_{0}}^{2\alpha-m\gamma}(\int_{\Sigma_{\lambda}}|w|^{\frac{2N}{N-2\alpha}}dx)^{\frac{N-2\alpha}{N}}.$
Now we choose $R_{0}$ such that $C{R_{0}}^{2\alpha-m\gamma}<1/2$, then choose
$R$ so that $\Sigma_{\lambda}\setminus{\tilde{B}}\subset{B^{c}(P,R)}\subset
B^{c}_{R_{0}}(0)$ and then choose $\varepsilon>0$ so that
$C|\tilde{B}\cap{{\rm{supp}}{(u_{\lambda}-u)^{+}}}|^{\frac{2\alpha}{N}}<1/2$.
With this choice of the parameters, from (4.12), (4.13) and (4.14) it follows
that $w=0$ in $\Sigma_{\lambda}$, which is a contradiction, completeing Step
2.
Step 3: By translation, we may say that $\lambda_{0}=0.$ An repeating the
argument from the other side, we find that $u$ is symmetric about
$x_{1}$-axis. Using the same argument in any arbitrary direction, we finally
conclude that $u$ is radially symmetric.
Finally, we prove that $u(r)$ is strictly decreasing in $r>0$, by using the
same arguments as in the case of a ball. This completes the proof. $\Box$
At the end of this section we want to give a theorem on radial symmetry of
solutions for equation (1.5) in a case where $f$ is only locally Lipschitz in
$(0,\infty)$, see [11] and [10] for the case of the Laplacian. In precise
terms we have
###### Theorem 4.1
Let $u$ be a positive classical solution of
$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u=u^{p}-u^{q}\ \ \ in\ \
\mathbb{R}^{N},\\\\[5.69054pt] u>0\ \ in\ \ \mathbb{R}^{N},\ \
\lim_{|x|\to\infty}u(x)=0,\end{array}\right.$ (4.15)
satisfying
$u(x)=O(|x|^{-\frac{N+2\alpha}{q}})\ \ \ \ as\ |x|\to\infty,$ (4.16)
where $\alpha\in(0,1),$ $N\geq 2$ and $0<q<1<p$. Then $u$ is radially
symmetric and strictly decreasing about some point.
Proof. We denote $f(u)=u^{p}-u^{q}$ for $u>0$, and consider $\gamma>0$ and
$s_{0}$ small enough, then for all $u,v$ satisfying $0<u<v<s_{0}$, we have
$\frac{f(v)-f(u)}{v-u}<0\leq C(u+v)^{\gamma},$
for some constant $C>0$, so that (F2) holds. We also observe that for a
positive classical solution $u$ of (4.15), $u\geq c$ in any bounded domain
$\Omega$, for a constant $c>0$ depending on $\Omega$ and then, in (4.13) we
may use Lipschitz continuity of $f$ in the bounded interval $[c,\sup u]$. We
set $m=\frac{N+2\alpha}{q}$ and $\gamma$ may be chosen so that (1.6) holds.
The proof of Theorem 4.1 goes in the same way as that of Theorem 1.2. $\Box$
###### Remark 4.1
In a work by Valdebenito [27], the estimate (4.16) is obtained by using super
solutions and Theorem 4.1 is proved using the local extension of equation
(4.15) as given by Caffarelli and Silvestre in [5] and then using a regular
moving planes argument as developed for elliptic equations with non-linear
boundary conditions by Terracini [26].
## 5 Symmetry results for system
The aim of this section is to prove Theorem 1.3 by the moving planes method
applied to a system of equations in the unit ball $B_{1}$. Let
$\Sigma_{\lambda}$ and $T_{\lambda}$ be defined as in Section §3. For
$x=(x_{1},x^{\prime})\in\mathbb{R}^{N}$ and $\lambda\in(0,1)$ we let
$x_{\lambda}=(2\lambda-x_{1},x^{\prime})$,
$u_{\lambda}(x)=u(x_{\lambda}),\ \ \ \ w_{\lambda,u}(x)=u_{\lambda}(x)-u(x),$
$v_{\lambda}(x)=v(x_{\lambda}),\quad\mbox{and}\quad
w_{\lambda,v}(x)=v_{\lambda}(x)-v(x).$
Proof of Theorem 1.3. We will split this proof into three steps.
Step 1: We start the moving planes proving that if $\lambda$ is close to $1$,
then $w_{\lambda,u}$ and $w_{\lambda,v}$ are positive in $\Sigma_{\lambda}$.
For that purpose we define
$\Sigma_{\lambda,u}^{-}=\\{x\in\Sigma_{\lambda}\ |\
w_{\lambda,u}(x)<0\\}\quad\mbox{and}\quad\Sigma_{\lambda,v}^{-}=\\{x\in\Sigma_{\lambda}\
|\ w_{\lambda,v}(x)<0\\}.$
We show next that $\Sigma_{\lambda,u}^{-}$ is empty for $\lambda$ close to 1.
Assume, by contradiction, that $\Sigma_{\lambda,u}^{-}$ is not empty and
define
$w_{\lambda,u}^{+}(x)=\left\\{\begin{array}[]{lll}w_{\lambda,u}(x),&x\in\Sigma_{\lambda,u}^{-},\\\\[5.69054pt]
0,&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda,u}^{-}\end{array}\right.$ (5.1)
and
$w_{\lambda,u}^{-}(x)=\left\\{\begin{array}[]{lll}0,&x\in\Sigma_{\lambda,u}^{-},\\\\[5.69054pt]
w_{\lambda,u}(x),&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda,u}^{-}.\end{array}\right.$
(5.2)
Using the arguments given in Step 1 of the proof of Theorem 1.1, we get
$(-\Delta)^{\alpha_{1}}w_{\lambda,u}^{+}(x)\geq(-\Delta)^{\alpha_{1}}w_{\lambda,u}(x)\quad\mbox{and}\quad(-\Delta)^{\alpha_{1}}w_{\lambda,u}^{-}(x)\leq
0,$ (5.3)
for all $x\in\Sigma^{-}_{\lambda,u}$. From here, using equation (1.8), for
$x\in\Sigma^{-}_{\lambda,u}$ we have
$\displaystyle(-\Delta)^{\alpha_{1}}w_{\lambda,u}^{+}(x)$ $\displaystyle\geq$
$\displaystyle(-\Delta)^{\alpha_{1}}u_{\lambda}(x)-(-\Delta)^{\alpha_{1}}u(x)$
(5.4) $\displaystyle=$ $\displaystyle
f_{1}(v_{\lambda}(x))+g_{1}(x_{\lambda})-f_{1}(v(x))-g_{1}(x)$
$\displaystyle=$
$\displaystyle\varphi_{v}(x)w_{\lambda,v}(x)+g_{1}(x_{\lambda})-g_{1}(x)$
$\displaystyle\geq$ $\displaystyle\varphi_{v}(x)w_{\lambda,v}(x),$
where
$\varphi_{v}(x)=({f_{1}(v_{\lambda}(x))-f_{1}(v(x)))}/({v_{\lambda}(x)-v(x)})$
and where we used that $g_{1}$ is radially symmetric and decreasing, with
$|x|>|x_{\lambda}|$. We further observe that, since $f_{1}$ is locally
Lipschitz continuous, we have that $\varphi_{v}(\cdot)\in
L^{\infty}(\Sigma^{-}_{\lambda,u})$. Now we consider (5.4) together with
$w_{\lambda,u}^{+}=0$ in $(\Sigma_{\lambda,u}^{-})^{c}$ and
$w_{\lambda,u}^{+}<0$ in $\Sigma_{\lambda,u}^{-}$, to use Proposition 2.1 to
find a constant $C>0$, depending on $N$ and $\alpha$ only, such that
$\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})}\leq
C\|(-\varphi_{v}w_{\lambda,v})^{+}\|^{1-\alpha_{1}}_{L^{\infty}(\Sigma_{\lambda,u}^{-})}\|(-\varphi_{v}w_{\lambda,v})^{+}\|^{\alpha_{1}}_{L^{N}(\Sigma_{\lambda,u}^{-})}$
(5.5)
We observe that $diam(\Sigma_{\lambda,u}^{-})\leq 1.$ Since $f_{1}$ is
increasing, we have
$\displaystyle-\varphi_{v}w_{\lambda,v}$ $\displaystyle=$ $\displaystyle
f_{1}(v)-f_{1}(v_{\lambda})\leq 0\ \ in\
(\Sigma_{\lambda,v}^{-})^{c}\quad\mbox{and}$ (5.6)
$\displaystyle-\varphi_{v}w_{\lambda,v}$ $\displaystyle=$ $\displaystyle
f_{1}(v)-f_{1}(v_{\lambda})>0\ \ in\ \Sigma_{\lambda,v}^{-}.$ (5.7)
Denoting
$\Sigma_{\lambda}^{-}=\Sigma_{\lambda,u}^{-}\cap\Sigma_{\lambda,v}^{-},$ from
(5.5), (5.6) and (5.7), we obtain
$\displaystyle\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})}\leq
C\|(-\varphi_{v}w_{\lambda,v})^{+}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}|\Sigma_{\lambda}^{-}|^{\frac{\alpha_{1}}{N}},$
(5.8)
Similar to (5.1) and (5.2), we define
$\displaystyle
w_{\lambda,v}^{+}(x)=\left\\{\begin{array}[]{lll}w_{\lambda,v}(x),&x\in\Sigma_{\lambda,v}^{-},\\\\[5.69054pt]
0,&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda,v}^{-}\end{array}\right.$
and
$\displaystyle
w_{\lambda,v}^{-}(x)=\left\\{\begin{array}[]{lll}0,&x\in\Sigma_{\lambda,v}^{-},\\\\[5.69054pt]
w_{\lambda,v}(x),&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda,v}^{-}.\end{array}\right.$
With this definition (5.8) becomes
$\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})}\leq
C\|w_{\lambda,v}^{+}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}|\Sigma_{\lambda}^{-}|^{\frac{\alpha_{1}}{N}},$
(5.11)
where we used that $\varphi_{v}$ is bounded and we have changed the constant
$C$, if necessary. At this point we observe that if $w_{\lambda,v}^{+}=0$ then
$w_{\lambda,u}^{+}=0$ providing a contradiction. Thus we have that
$\Sigma_{\lambda,v}^{-}\not=\emptyset$ and we may argue in a completely
analogous way to obtain
$\|w_{\lambda,v}^{+}\|_{L^{\infty}(\Sigma_{\lambda,v}^{-})}\leq
C\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}|\Sigma_{\lambda}^{-}|^{\frac{\alpha_{2}}{N}},$
(5.12)
that combined with (5.11) yields
$\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})}\leq
C^{2}|\Sigma_{\lambda}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})},$
and
$\|w_{\lambda,v}^{+}\|_{L^{\infty}(\Sigma_{\lambda,v}^{-})}\leq
C^{2}|\Sigma_{\lambda}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}\|w_{\lambda,v}^{+}\|_{L^{\infty}(\Sigma_{\lambda,v}^{-})}.$
Now we just take $\lambda$ close enough to $1$ so that
$C^{2}|\Sigma_{\lambda}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}<1$ and we
conclude that
$\|w_{\lambda,u}^{+}\|_{L^{\infty}(\Sigma_{\lambda,u}^{-})}=\|w_{\lambda,v}^{+}\|_{L^{\infty}(\Sigma_{\lambda,v}^{-})}=0,$
so $|\Sigma_{\lambda,u}^{-}|=|\Sigma_{\lambda,v}^{-}|=0$ and since
$\Sigma_{\lambda,u}^{-}$ and $\Sigma_{\lambda,v}^{-}$ are open we have that
$\Sigma_{\lambda,u}^{-},\Sigma_{\lambda,v}^{-}=\O$, which is a contradiction.
Thus we have that $w_{\lambda,u}\geq 0$ in $\Sigma_{\lambda}$ when $\lambda$
is close enough to $1$. Similarly, we obtain $w_{\lambda,v}\geq 0$ in
$\Sigma_{\lambda}$ for $\lambda$ close to $1$. In order to complete Step 1 we
will prove a bit more general statement that will be useful later, that is,
given $0<\lambda<1$, if $w_{\lambda,u}\geq 0,w_{\lambda,v}\geq 0$,
$w_{\lambda,u}\not\equiv 0$ and $w_{\lambda,v}\not\equiv 0$ in
$\Sigma_{\lambda}$, then $w_{\lambda,u}>0$ and $w_{\lambda,v}>0$ in
$\Sigma_{\lambda}$. For proving this property suppose there exists
$x_{0}\in\Sigma_{\lambda}$ such that
$w_{\lambda,u}(x_{0})=0.$ (5.13)
On one hand, by using similar arguments yielding (3.11) we find that
$(-\Delta)^{\alpha_{1}}w_{\lambda,u}(x_{0})<0.$ (5.14)
On the other hand, by our assumption we have that
$w_{\lambda,v}(x_{0})=v_{\lambda}(x_{0})-v(x_{0})\geq 0$ and since
$|x_{0}|>|(x_{0})_{\lambda}|$, from the monotonicity hypothesis on $f_{1}$ and
$g_{1}$, we obtain
$f_{1}(v_{\lambda}(x_{0}))\geq f_{1}(v(x_{0})),\ \ \ \ \
g_{1}((x_{0})_{\lambda})\geq g_{1}(x_{0}).$
Thus, using (1.8), we find
$\displaystyle(-\Delta)^{\alpha_{1}}w_{\lambda,u}(x_{0})$ $\displaystyle=$
$\displaystyle
f_{1}(v_{\lambda}(x_{0}))+g_{1}((x_{0})_{\lambda})-f_{1}(v(x_{0}))-g_{1}(x_{0})\geq
0,$
which is impossible with (5.14). This completes Step 1.
Step 2: We prove that $\lambda_{0}=0$, where
$\lambda_{0}=\inf\\{\lambda\in(0,1)\ |\ w_{\lambda,u}\ ,\ w_{\lambda,v}>0\ \
\rm{in}\ \ \Sigma_{\lambda}\\}.$
If not, that is, if $\lambda_{0}>0$ we have that
$w_{\lambda_{0},u},w_{\lambda_{0},v}\geq 0$ and
$w_{\lambda_{0},u},w_{\lambda_{0},v}\not\equiv 0$ in $\Sigma_{\lambda_{0}}$.
If we use the property we just proved above, we may assume that
$w_{\lambda_{0},u}>0$ and $w_{\lambda_{0},v}>0$ in $\Sigma_{\lambda_{0}}$. In
what follows we argue that the plane can be moved to left, that is, that there
exists $\epsilon\in(0,\lambda)$ such that $w_{{\lambda_{\epsilon}},u}>0$ and
$w_{{\lambda_{\epsilon}},v}>0$ in $\Sigma_{\lambda_{\epsilon}}$, where
$\lambda_{\epsilon}=\lambda_{0}-\epsilon$, providing a contradiction with the
definition of $\lambda_{0}$.
Let us consider the set $D_{\mu}=\\{x\in\Sigma_{\lambda}\ |\
dist(x,\partial\Sigma_{\lambda})\geq\mu\\}$ for $\mu>0$ small. Since
$w_{\lambda,u},w_{\lambda,v}>0$ in $\Sigma_{\lambda}$ and $D_{\mu}$ is
compact, then there exists $\mu_{0}>0$ such that
$w_{\lambda,u},w_{\lambda,v}\geq\mu_{0}$ in $D_{\mu}$. By continuity of
$w_{\lambda,u}(x)$ and $w_{\lambda,v}(x)$, for $\epsilon>0$ small enough, we
have that
$w_{\lambda_{\epsilon},u},\ w_{\lambda_{\epsilon},v}\geq 0\ \ \rm{in}\ \
D_{\mu}$
and, as a consequence,
$\Sigma_{\lambda_{\epsilon},u}^{-},\Sigma_{\lambda_{\epsilon},v}^{-}\subset\Sigma_{\lambda_{\epsilon}}\setminus
D_{\mu},$ and $|\Sigma_{\lambda_{\epsilon},u}^{-}|$ and
$|\Sigma_{\lambda_{\epsilon},v}^{-}|$ are small if $\epsilon$ and $\mu$ are
small.
Since $f_{1}$ and $f_{2}$ are locally Lipschitz continuous and increasing,
$g_{1}$ and $g_{2}$ are radially symmetric and decreasing, we may repeat the
arguments given in Step 1 to obtain
$\|w_{\lambda_{\epsilon},u}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},u}^{-})}\leq
C^{2}|\Sigma_{\lambda_{\epsilon}}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}\|w_{\lambda_{\epsilon},u}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},u}^{-})}$
and
$\|w_{\lambda_{\epsilon},v}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},v}^{-})}\leq
C^{2}|\Sigma_{\lambda_{\epsilon}}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}\|w_{\lambda_{\epsilon},v}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},v}^{-})}$
where
$\Sigma_{\lambda_{\epsilon}}^{-}=\Sigma_{\lambda_{\epsilon},u}^{-}\cap\Sigma_{\lambda_{\epsilon},v}^{-}$.
Now we may choose $\epsilon$ and $\mu$ small such that
$C^{2}|\Sigma_{\lambda_{\epsilon}}^{-}|^{\frac{{\alpha_{1}}+{\alpha_{2}}}{N}}<1,$
then we obtain
$\|w_{\lambda_{\epsilon},u}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},u}^{-})}=\|w_{\lambda_{\epsilon},v}^{+}\|_{L^{\infty}(\Sigma_{\lambda_{\epsilon},v}^{-})}=0$.
From here we argue as in Step 1 to obtain that $w_{\lambda_{\epsilon},u}$ and
$w_{\lambda_{\epsilon},v}$ are positive in $\Sigma_{\lambda_{\epsilon}}$,
completing Step 2.
Finally, we obtain that $u$ and $v$ are radially symmetric and strictly
decreasing respect to $r=|x|$ for $r\in(0,1)$ in the same way in Step 3 in the
proof of Theorem 1.1. $\Box$
## 6 The case of a non-local operator with non-homogeneous kernel.
The main purpose of this section is to discuss radial symmetry for a problem
with a non-local operator $\mathcal{L}$ of fractional order, but with a non-
homogeneous kernel. The operator is defined as follows:
$\mathcal{L}u(x)=P.V.\int_{\mathbb{R}^{N}}(u(x)-u(y)){K_{\mu}}(x-y)dy,$ (6.1)
where the kernel ${K_{\mu}}$ satisfies that
$K_{\mu}(x)=\left\\{\begin{array}[]{lll}\frac{1}{|x|^{N+2\alpha_{1}}},&|x|<1,\\\\[5.69054pt]
\frac{\mu}{|x|^{N+2\alpha_{2}}},&|x|\geq 1\end{array}\right.$ (6.2)
with $\mu\in[0,1]$ and $\alpha_{1},\alpha_{2}\in(0,1)$. Being more precise, we
consider the equation
$\left\\{\begin{array}[]{lll}\mathcal{L}u(x)=f(u(x))+g(x),&x\in
B_{1},\\\\[5.69054pt] u(x)=0,&x\in B_{1}^{c},\end{array}\right.$ (6.3)
and our theorem states
###### Theorem 6.1
Assume that the function $f$ satisfies $(F1)$ and $g$ satisfies $(G)$. If $u$
is a positive classical solution of (6.3), then $u$ must be radially symmetric
and strictly decreasing in $r=|x|$ for $r\in(0,1)$.
The idea for Theorem 6.1 is to take advantage of the fact that the non-local
operator $\mathcal{L}$ differs from the fractional Laplacian by a zero order
operator. Using this idea, we obtain a Maximum Principle for domains with
small volume through the ABP-estimate given Proposition 2.1 and we are able to
use the moving planes method as in the case of the fractional Laplacian. We
prove first
###### Proposition 6.1
Let ${\Sigma_{\lambda}}$ and ${\Sigma_{\lambda}^{-}}$ be defined as in the
Section §3. Suppose that $\varphi\in L^{\infty}({\Sigma_{\lambda}})$ and that
$w_{\lambda}\in L^{\infty}(\mathbb{R}^{N})\cap C(\mathbb{R}^{N})$ is a
solution of
$\left\\{\begin{array}[]{lll}-\mathcal{L}w_{\lambda}(x)\leq\varphi(x)w_{\lambda}(x),&x\in{\Sigma_{\lambda}},\\\\[5.69054pt]
w_{\lambda}(x)\geq
0,&x\in\mathbb{R}^{N}\setminus{\Sigma_{\lambda}},\end{array}\right.$ (6.4)
where $\mathcal{L}$ was defined in (6.1). Then, if $|{\Sigma_{\lambda}^{-}}|$
is small enough, $w_{\lambda}$ is non-negative in ${\Sigma_{\lambda}}$, that
is,
${w_{\lambda}}\geq 0\ \ \rm{in}\ \ {\Sigma_{\lambda}}.$
Proof. We define $w^{+}_{\lambda}(x)$ as in (3.5), then we have
$\displaystyle\mathcal{L}w^{+}_{\lambda}(x)$ $\displaystyle=$
$\displaystyle\int_{B_{1}(x)}\frac{w_{\lambda}^{+}(x)-w_{\lambda}^{+}(z)}{|x-z|^{N+2\alpha_{1}}}dz+\mu\int_{\mathbb{R}^{N}\setminus
B_{1}(x)}\frac{w_{\lambda}^{+}(x)-w_{\lambda}^{+}(z)}{|x-z|^{N+2\alpha_{2}}}dz$
$\displaystyle=$ $\displaystyle(-\Delta)^{\alpha_{1}}w^{+}_{\lambda}(x)$
$\displaystyle+\int_{\mathbb{R}^{N}\setminus
B_{1}(x)}(w_{\lambda}^{+}(x)-w_{\lambda}^{+}(z))(\frac{\mu}{|x-z|^{N+2\alpha_{2}}}-\frac{1}{|x-z|^{N+2\alpha_{1}}})dz$
$\displaystyle\leq$
$\displaystyle(-\Delta)^{\alpha_{1}}w^{+}_{\lambda}(x)+2C_{0}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}\
,\ \ \ x\in{\Sigma_{\lambda}^{-}},$
where $C_{0}=\int_{\mathbb{R}^{N}\setminus
B_{1}}|\frac{\mu}{|y|^{N+2\alpha_{2}}}-\frac{1}{|y|^{N+2\alpha_{1}}}|dy$. Thus
we have
$\Delta^{\alpha_{1}}w^{+}_{\lambda}(x)\leq-\mathcal{L}w^{+}_{\lambda}(x)+2C_{0}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}\
,\ \ \ x\in{\Sigma_{\lambda}^{-}}.$ (6.5)
Since $K_{\mu}$ is radially symmetric and decreasing in $|x|$, we may repeat
the arguments used to prove (3.7) to get
$\mathcal{L}w_{\lambda}^{-}(x)\leq 0,\ \ \ \ \forall\
x\in\Sigma_{\lambda}^{-},$ (6.6)
where $0<\lambda<1$ and $w_{\lambda}^{-}$ was defined in (3.6). Using (6.5),
the linearity of $\mathcal{L}$, (6.6) and equation (6.4), for all
$x\in\Sigma_{\lambda}^{-}$, we have
$\displaystyle\Delta^{\alpha_{1}}w^{+}_{\lambda}(x)$ $\displaystyle\leq$
$\displaystyle-\mathcal{L}w_{\lambda}(x)+\mathcal{L}w^{-}_{\lambda}(x)+2C_{0}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}$
(6.7) $\displaystyle\leq$
$\displaystyle-\mathcal{L}w_{\lambda}(x)+2C_{0}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}$
$\displaystyle\leq$
$\displaystyle\varphi(x)w_{\lambda}(x)+2C_{0}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}\leq
C_{1}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})},$
where $C_{1}=\|\varphi\|_{L^{\infty}({\Sigma_{\lambda}})}+2C_{0}$ and we
notice that $w_{\lambda}=w_{\lambda}^{+}$ in $\Sigma_{\lambda}^{-}$. Hence, we
have
$\left\\{\begin{array}[]{lll}\Delta^{\alpha_{1}}w^{+}_{\lambda}(x)\leq
C_{1}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})},&x\in{\Sigma_{\lambda}^{-}},\\\\[5.69054pt]
w^{+}_{\lambda}(x)=0,&x\in\mathbb{R}^{N}\setminus{\Sigma_{\lambda}^{-}}.\end{array}\right.$
(6.8)
Then, using Proposition 2.1 with
$h(x)=C_{1}\|w^{+}_{\lambda}\|_{L^{\infty}({\Sigma_{\lambda}^{-}})}$, we
obtain a constant $C>0$ such that
$\displaystyle\|w^{+}_{\lambda}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}=-\inf_{\Sigma_{\lambda}^{-}}w^{+}_{\lambda}\leq
Cd^{\alpha_{1}}\|w^{+}_{\lambda}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}|\Sigma_{\lambda}^{-}|^{\frac{\alpha_{1}}{N}},$
where $d=diam(\Sigma_{\lambda}^{-})$. If $|\Sigma_{\lambda}^{-}|$ is small
enough we conclude that
$\|w_{\lambda}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}=\|w^{+}_{\lambda}\|_{L^{\infty}(\Sigma_{\lambda}^{-})}=0,$
from where we complete the proof. $\Box$
Now we provide a proof for Theorem 6.1.
Proof of Theorem 6.1. The proof of this theorem goes like the one for Theorem
1.1 where we use Proposition 6.1 instead of Proposition 2.1 and $\mathcal{L}$
instead of $(-\Delta)^{\alpha}$. The only place where there is a difference is
in the following property: for $0<\lambda<1$, if $w_{\lambda}\geq 0$ and
$w_{\lambda}\not\equiv 0$ in $\Sigma_{\lambda}$, then $w_{\lambda}>0$ in
$\Sigma_{\lambda}$.
For $\mu\in(0,1]$, since $K_{\mu}$ is radially symmetric and strictly
decreasing, the proof of the property is similar to that given in Theorem 1.1.
So we only need to prove it in case $\mu=0$ so the kernel $K_{0}$ vanishes
outside the unit ball $B_{1}$. Let us assume that $w_{\lambda}\geq 0$ and
$w_{\lambda}\not\equiv 0$ in $\Sigma_{\lambda}$ and, by contradiction, let us
assume $\Sigma_{0}=\\{x\in\Sigma_{\lambda}\ |\ w_{\lambda}(x)=0\\}\not=\O$. By
our assumptions on $w_{\lambda}$ we have that
$\Sigma_{\lambda}\setminus\Sigma_{0}=\\{x\in\Sigma_{\lambda}\ |\
w_{\lambda}(x)>0\\}$ is open and nonempty. Let us consider
$x_{0}\in\Sigma_{0}$ such that
$dist(x_{0},\Sigma_{\lambda}\setminus\Sigma_{0})\leq 1/2,$ (6.9)
and observe that $(\Sigma_{\lambda}\setminus\Sigma_{0})\cap B_{1}(x_{0})$ is
nonempty. Using (6.3) we have
$\displaystyle\mathcal{L}w_{\lambda}(x_{0})$ $\displaystyle=$
$\displaystyle\mathcal{L}u_{\lambda}(x_{0})-\mathcal{L}u(x_{0})$ (6.10)
$\displaystyle=$ $\displaystyle
f(u_{\lambda}(x_{0}))-f(u(x_{0}))+g((x_{0})_{\lambda})-g(x_{0})$
$\displaystyle=$ $\displaystyle g((x_{0})_{\lambda})-g(x_{0})\geq 0,$
where the last inequality holds by monotonicity assumption on $g$ and since
$|x_{0}|>|(x_{0})_{\lambda}|$. On the other hand, denoting by
$A_{\lambda}=\\{(x_{1},x^{\prime})\in\mathbb{R}^{N}\ |\ x_{1}>\lambda\\}$,
since $w_{\lambda}(x_{0})=0$ and $w_{\lambda}(z_{\lambda})=-w_{\lambda}(z)$
for any $z\in\mathbb{R}^{N}$, we have
$\displaystyle\mathcal{L}w_{\lambda}(x_{0})$ $\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}w_{\lambda}(z){K_{0}}(x_{0}-z)dz-\int_{\mathbb{R}^{N}\setminus
A_{\lambda}}w_{\lambda}(z){K_{0}}(x_{0}-z)dz$ $\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}w_{\lambda}(z){K_{0}}(x_{0}-z)dz-\int_{A_{\lambda}}w_{\lambda}(z_{\lambda}){K_{0}}(x_{0}-z_{\lambda})dz$
$\displaystyle=$
$\displaystyle-\int_{A_{\lambda}}w_{\lambda}(z)({K_{0}}(x_{0}-z)-{K_{0}}(x_{0}-z_{\lambda}))dz.$
Since $|x_{0}-z_{\lambda}|>|x_{0}-z|$ for $z\in A_{\lambda}$, by definition of
$K_{0}$, $\Sigma_{\lambda}$ and $\Sigma_{0}$, we have that
${K_{0}}(x_{0}-z)>{K_{0}}(x_{0}-z_{\lambda})\quad\mbox{and}\quad
w_{\lambda}(z)>0\quad{\rm{for}}\quad
z\in(\Sigma_{\lambda}\setminus\Sigma_{0})\cap B_{1}(x_{0}),$
and we also have that $w_{\lambda}(z)\geq 0$ and
${K_{0}}(x_{0}-z)\geq{K_{0}}(x_{0}-z_{\lambda})$ for all $z\in A_{\lambda},$
so that
$\mathcal{L}w_{\lambda}(x_{0})<0,$
contradicting (6.10). Hence $\Sigma_{0}$ is empty and then $w_{\lambda}>0$ in
$\Sigma_{\lambda}$, completing the proof of the theorem. $\Box$
###### Remark 6.1
The theorem we just proved can be extended to more general non-homogeneous
kernels in the following class
$K(x)=\left\\{\begin{array}[]{lll}|x|^{-N-2\alpha},&x\in B_{r},\\\\[5.69054pt]
\theta(x),&x\in B^{c}_{r},\end{array}\right.$ (6.11)
here $\alpha\in(0,1)$, $r>0$ and the function $\theta:B^{c}_{r}\to\mathbb{R}$
satisfies that
* $(C)\ $
$\theta\in L^{1}(B^{c}_{r})$ is nonnegative, radially symmetric and such that
the kernel $K$ is decreasing.
Acknowledgements: P.F. was partially supported by Fondecyt Grant # 1110291,
BASAL-CMM projects and CAPDE, Anillo ACT-125. Y.W. was partially supported by
Becas CMM.
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|
arxiv-papers
| 2013-11-27T12:33:30 |
2024-09-04T02:49:54.336846
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Patricio Felmer and Ying Wang",
"submitter": "Ying Wang",
"url": "https://arxiv.org/abs/1311.6952"
}
|
1311.6961
|
# Integrated Fiber-Mirror Ion Trap for Strong Ion-Cavity Coupling
B. Brandstätter [email protected] Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria A. McClung
Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck,
Austria Norman Bridge Laboratory of Physics 12-33, California Institute of
Technology, Pasadena, California 91125, USA K. Schüppert Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria B.
Casabone Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria K. Friebe Institut für Experimentalphysik, Universität
Innsbruck, 6020 Innsbruck, Austria A. Stute Institut für Experimentalphysik,
Universität Innsbruck, 6020 Innsbruck, Austria P.O. Schmidt QUEST Institute
for Experimental Quantum Metrology, Physikalisch-Technische Bundesanstalt,
38116 Braunschweig, Germany Institut für Quantenoptik, Leibniz Universität
Hannover, 30167 Hannover, Germany C. Deutsch Laboratoire Kastler Brossel,
ENS/UPMC-Paris 6/CNRS, 24 rue Lhomond, F-75005 Paris, France Menlo Systems
GmbH, Am Klopferspitz 19a, 82152 Martinsried, Germany J. Reichel Laboratoire
Kastler Brossel, ENS/UPMC-Paris 6/CNRS, 24 rue Lhomond, F-75005 Paris, France
R. Blatt Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020
Innsbruck, Austria. T.E. Northup Institut für Experimentalphysik,
Universität Innsbruck, 6020 Innsbruck, Austria
###### Abstract
We present and characterize fiber mirrors and a miniaturized ion-trap design
developed to integrate a fiber-based Fabry-Perot cavity (FFPC) with a linear
Paul trap for use in cavity-QED experiments with trapped ions. Our fiber-
mirror fabrication process not only enables the construction of FFPCs with
small mode volumes, but also allows us to minimize the influence of the
dielectric fiber mirrors on the trapped-ion pseudopotential. We discuss the
effect of clipping losses for long FFPCs and the effect of angular and lateral
displacements on the coupling efficiencies between cavity and fiber. Optical
profilometry allows us to determine the radii of curvature and ellipticities
of the fiber mirrors. From finesse measurements we infer a single-atom
cooperativity of up to $12$ for FFPCs longer than $200~{}\mu$m in length;
comparison to cavities constructed with reference substrate mirrors produced
in the same coating run indicates that our FFPCs have similar scattering
losses. We characterize the birefringence of our fiber mirrors, finding that
careful fiber-mirror selection enables us to construct FFPCs with degenerate
polarization modes. As FFPCs are novel devices, we describe procedures
developed for handling, aligning and cleaning them. We discuss experiments to
anneal fiber mirrors and explore the influence of the atmosphere under which
annealing occurs on coating losses, finding that annealing under vacuum
increases the losses for our reference substrate mirrors. X-ray photoelectron
spectroscopy measurements indicate that these losses may be attributable to
oxygen depletion in the mirror coating. Special design considerations enable
us to introduce an FFPC into a trapped ion setup. Our unique linear Paul trap
design provides clearance for such a cavity and is miniaturized to shield
trapped ions from the dielectric fiber mirrors. We numerically calculate the
trap potential in the absence of fibers. In the experiment additional
electrodes can be used to compensate distortions of the potential due to the
fibers. Home-built fiber feedthroughs connect the FFPC to external optics, and
an integrated nanopositioning system affords the possibility of retracting or
realigning the cavity without breaking vacuum.
## I Introduction
Optical cavities can enhance the interaction between matter and light. In
quantum information experiments, high-finesse cavities act as an interface
between stationary and flying qubits, where flying qubits connect
computational nodes comprised of stationary qubits. Experimentally, atoms and
photons have proven to be promising candidates for the physical implementation
of stationary and flying qubits Zoller _et al._ (2005), respectively. The
building blocks for an elementary quantum network have recently been
demonstrated using single neutral atoms in high-finesse cavities Ritter _et
al._ (2012).
Trapped ions are promising candidates for quantum information processing, as
techniques for high-fidelity quantum operations are well established Leibfried
_et al._ (2003); Häffner, Roos, and Blatt (2008). Furthermore, trapped ions
offer a range of advantages for quantum networks: ions are stably trapped in
Paul traps for up to several days and can be well localized via ground-state
cooling. This localization allows the ions to be accurately positioned inside
a cavity mode for optimized coupling Stute _et al._ (2012a). Several research
groups are currently working on the technological challenge of integrating
ions and cavities, and the coupling of ions to the field of a high-finesse
cavity has already been shown in a range of setups Guthöhrlein _et al._
(2001); Mundt _et al._ (2002); Russo _et al._ (2009); Leibrandt _et al._
(2009); Herskind _et al._ (2009); Sterk _et al._ (2012). Single photons have
been produced with a trapped ion inside a cavity Keller _et al._ (2004);
Barros _et al._ (2009), and entanglement between single ions and single
cavity photons has recently been demonstrated Stute _et al._ (2012b).
For a high-fidelity ion-photon quantum interface, the coherent coupling
strength must be larger than the ion’s rate of spontaneous decay. In order to
maximize this coupling in cavity-QED systems, both the cavity length and the
cavity waist should be minimized. Such microcavities require mirror surfaces
with small radii of curvature. Additionally, the mirror substrates should have
low surface roughness to minimize losses in the mirror coating. These low-loss
coatings at optical wavelengths are dielectric. For an ion in vacuum close to
such dielectric mirrors, image charges and charge build-up on the mirrors
potentially distort the ion’s trapping potential.
The term fiber-based Fabry-Perot cavity (FFPC) describes two opposing optical
fiber tips, each with a mirror coating. At least one of the mirror surfaces is
concave and aligned relative to the other such that a stable standing wave
forms between them. FFPCs provide both a high coupling rate and a small
dielectric cross-section and are thus a promising way to integrate ions with
cavities Hunger _et al._ (2010). A recent experiment provides the first
demonstration of an ion coupled to a fiber-cavity mode Steiner _et al._
(2013).
In Sec. II, we discuss the development and optimization of FFPCs for ion
traps. We develop solutions for specific technological challenges in Sec. III,
including the annealing and baking of fiber mirrors. Finally, in Sec. IV, we
present a novel design for a linear Paul trap integrated with an FFPC. This
experimental system should enable us to reach the strong coupling regime
Kimble (2008) with a single calcium ion inside the high-finesse FFPC.
## II Fiber-based Fabry-Perot cavities
Microfabricated optical cavities have several advantages over conventional
cavities in cavity-QED experiments. Microcavities offer access to smaller mode
volumes than have been demonstrated with macroscopic mirrors Hunger _et al._
(2010) and thus higher interaction rates of atoms or solid-state emitters with
photons. Additionally, photons that exit the cavity are directly coupled into
a fiber. Furthermore, FFPCs provide flexibility in experimental setups, where
small mirrors are often easier to implement than centimeter-scale mirrors
fabricated on superpolished substrates. In the future it may be possible to
integrate microcavity arrays in scalable systems for quantum information
processing.
This range of advantages has motivated parallel development of microcavities
using various technologies. We identify three criteria for microcavity
development: (i) surfaces with small radii of curvature, (ii) surface
roughness low enough so that it does not contribute appreciably to the mirror
losses, and (iii) surfaces to which a low-loss mirror coating can be applied,
i.e., by ion-beam sputtering. Such surfaces are produced by silicon wet-
etching Trupke _et al._ (2005), enclosing nitrogen bubbles in borosilicate
and polishing away the bubbles’ upper half Cui _et al._ (2006), or by
transferring a coating produced on a microlens onto an optical fiber Steinmetz
_et al._ (2006); Muller _et al._ (2009). All these approaches have been used
to produce cavities with moderate finesses of up to $6\times 10^{3}$. Recent
developments in the fabrication of glass microcavities by shaping surfaces
with controlled re-flow of borosilicate glassRoy and Barrett (2011) yielded
finesses of up to $3.2\times 10^{4}$. However, the best microcavity finesse of
$1.5\times 10^{5}$ has been measured recentlyMuller _et al._ (2010) with
cavities constructed from coated, concave optical-fiber facets shaped by
CO2-laser ablation Hunger _et al._ (2010).
In this process, a short pulse of focused CO2-laser light is absorbed in the
cleaved tip of a fiber and creates a depression by locally evaporating the
material. The created surface has a roughness of only $(0.2\pm 0.1)$ nm Hunger
_et al._ (2010). The parameters of the generated surface structures, such as
radius of curvature and diameter of the depression, are set by the pulse
duration, power, and beam waist of the CO2 laser. A highly reflective coating
is then applied to the shaped fiber surfaces by ion-beam sputtering in a high-
vacuum environment. FFPCs produced in this way are being used in atom-chip
setups, in which strong coupling to a BEC has been demonstrated Colombe _et
al._ (2007). Currently, implementations of FFPCs with solid-state emitters
Muller _et al._ (2010), ion traps Steiner _et al._ (2013); Wilson _et al._
(2011); VanDevender _et al._ (2010), and neutral atoms are being developed in
several groups worldwide.
In this section, we present the recent development of FFPCs suitable for
integration with ion traps. We show that we can produce cavities of length up
to $350~{}\mu$m and finesse up to $1.1\times 10^{5}$. Furthermore, we
characterize the cavity losses due to surface roughness and the cavity
birefringence, and we describe technologies for cavity alignment and fiber-
mirror cleaning.
### II.1 Development of FFPCs for ion traps
For neutral atoms, short cavities are favorable as they provide a small mode
volume and do not influence the trapping potential seen by the atom. To
implement FFPCs with ion traps, however, a sufficient separation between the
fibers and the ion is necessary so that the trapping potential is not
distorted by charges on the dielectric mirrors.
We report on the construction of FFPCs suitable for ion traps and on the
effects of increasing the cavity length, such as decreased finesse and
coupling efficiencies between fiber mode and cavity mode. Furthermore, we
present measurements of general interest when working with FFPCs: a direct
measurement of the scattering losses due to surface roughness, and a
characterization of the birefringence of fiber mirrors.
#### II.1.1 Construction of long FFPCs
Figure 1: (a) Composite microscope photo of a fiber mirror, assembled from
multiple photos with different focal length. A highly reflective mirror
coating is fabricated on the entire fiber facet. At the mirror center, a light
reflection from the curved surface can be seen. (b) Photo of an FFPC. The
fibers are copper coated for ultra-high–vacuum compatibility and have a
cladding diameter of $200~{}\mu$m. The copper is etched back about
$400~{}\mu$m from the fiber facets. The glass cladding and the gray titanium
layer around the cladding, which starts about $100~{}\mu$m behind the facet,
can be seen. The two opposing mirrors form a Fabry-Perot cavity $200~{}\mu$m
in length.
Initial development of FFPCs focused on short cavities, such as the
$38.6~{}\mu$m FFPC used to strongly couple a BEC to a cavity field Colombe
_et al._ (2007). In order to construct cavities suited for ion-trap
experiments, we have developed technologies that allow us to increase the
length of FFPCs up to $350~{}\mu$m. These technologies include fabrication of
structures using higher CO2-laser powers and wider beam waists as well as the
use of non-standard $200~{}\mu$m-diameter fibers. Fig. 1 shows a composite
microscope picture of a coated fiber tip and a photo of an FFPC in our
laboratory.
Because of their effects on the trapping potential, fibers should remain far
from the ion. As we increase the separation $L$ between the two fiber mirrors
of a cavity, the spot size of the cavity mode at each mirror increases as a
function of $L$ and the mirror’s radius of curvature $r$. If the mirror
diameter $2\rho_{\mathrm{m}}$ is not much larger than the field diameter
$2w_{\mathrm{m}}$, the cavity mode is clipped at the mirror edge, reducing the
cavity finesse. For a conventional mirror, which has a spherical curvature
over its entire surface, $2\rho_{\mathrm{m}}$ is the physical diameter of the
mirror. However, a fiber mirror can be approximated as spherical only over the
length scale of the depression created in the CO2-ablation process. Thus,
$2\rho_{\mathrm{m}}$ corresponds to this length scale, which is bounded above
by the fiber diameter but is often much smaller due to limitations in the
ablation process.
In Ref. Hunger _et al._ (2010), fiber mirrors were produced with radius of
curvature between $40~{}\mu$m and $2$ mm and $2\rho_{\mathrm{m}}$ between
$10~{}\mu$m and $45~{}\mu$m. To calculate the clipping losses associated with
a particular cavity geometry, the numerical methods of Fox and Li can be
usedFox and Li (1961); Siegman (1986). For example, if we require round-trip
clipping losses to be less than 10 ppm, then with spherical mirrors of $2$ mm
radius of curvature and $2\rho_{\mathrm{m}}=45~{}\mu$m, we are limited to
$L\leq 70~{}\mu$m. (Choosing radii of curvature in the near-confocal limit
would improve this bound but also reduce the atom-cavity coupling; the near-
planar assumption is a reasonable compromise and also robust to small
variations in cavity length.) This bound is incompatible with the target
lengths $L\gtrsim 150~{}\mu$m planned for our experimental system. (See Sec.
IV.2.1.)
One solution to minimize clipping losses is to produce larger mirror
structures on the fiber tips, that is, to modify the laser ablation process.
Specifically, we increase the beam waist at the fiber tip and use higher
CO2-laser power. Optimizing the laser ablation parameters is challenging due
to two competing processes: while the incident laser light evaporates fiber
material, mapping the Gaussian beam profile onto a concave depression, it also
induces sufficient heat to locally melt the fiber tip, producing a convex
structure due to surface tension. To avoid melting, heat needs to be conducted
away efficiently by either cooling the fiber or creating a heat sink. Instead
of standard $125~{}\mu$m-diameter fibers, we choose to use fibers of
$200~{}\mu$m diameter, where the additional glass functions as a heat sink.
However, the use of a non-standard fiber size means that fiber connectors and
tools for cleaving and splicing are more difficult to obtain.
Structures on the fiber tips are analyzed with an optical profilometerFog ,
and structure diameters $2\rho_{\mathrm{m}}$ and mirror depth $z$ are
extracted by fitting a polynomial to the profilometer data and finding its
turning points. The distance between the turning points is defined as
$2\rho_{\mathrm{m}}$. The radius of curvature $r$ of each fiber is
approximated via the fit of a circle to the surface. In Fig. 2, we show the
profilometer data as well as the fit for one fiber. The CO2-laser ablation
structures are not rotationally symmetric but have an elliptical shape due to
astigmatism of the CO2-laser beam Hunger _et al._ (2010). We determine the
degree of ellipticity by identifying major and minor axes and comparing the
two radii of curvature. Note that $r$, $2\rho_{\mathrm{m}}$, and $z$ are mean
values of the fits to both axes.
Figure 2: Fiber surface measured by optical profilometry; compare also Fig. 3
in Ref. Hunger _et al._ (2010). The depression on the fiber surface is
elliptical. Along the major and minor axes ($i=1,2$) of the structure, we fit
a polynomial and determine structure diameter $2\rho_{\mathrm{m}_{i}}$ and
structure depth $z_{i}$. $2\rho_{\mathrm{m}_{i}}$ and $z_{i}$ are defined at
the turning points of the polynomial. From the fit of a circle (note the
different axis scales of ordinate and abscissa), we extract the radius of
curvature $r_{i}$. Furthermore, we determine the ellipticity of the fiber and
the mean values of the fitted parameters, $r$, $2\rho_{\mathrm{m}}$, and $z$.
For the surface in this figure, these values are:
$2\rho_{\mathrm{m}_{i}}=(82,84)~{}\mu$m and $2\rho_{\mathrm{m}}=83~{}\mu$m,
$z_{i}=(2.6,2.8)~{}\mu$m and $z=2.7~{}\mu$m, and $r_{i}=(343,332)~{}\mu$m and
$r=338~{}\mu$m.
Using the technique described here, CO2-laser waists between $18~{}\mu$m and
$80~{}\mu$m and powers between $0.3$ W and $1.1$ W were used in Ref. Hunger
_et al._ (2010). In contrast, we modify the parameters to a beam waist of
$92~{}\mu$m and a laser power of $4.6$ W. (The pulse duration is $\sim 30$
ms.) As a result, we produce fiber-mirror structures with radius of curvature
between $180~{}\mu$m and $420~{}\mu$m and structure diameters of up to
$80~{}\mu$m.
In a single coating process, $76$ fibers produced with the CO2-laser
parameters specified above were coated with a highly reflective coating
centered around $860$ nm (ATFilms). On an alignment stage for test setups, we
construct and characterize the FFPCs. The mirrors were characterized at $844$
nm because a laser was available whose wavelength could be tuned over several
nanometers, facilitating the measurement of short cavity lengths. In Fig. 3,
we show the cavity finesse as a function of distance between the mirrors for
one FFPC. The cavity is set up with a single-mode fiber mirror as cavity input
and a multimode-fiber mirror as cavity output. The single-mode fiber mirror
has a diameter of $2\rho_{\mathrm{m}}=67~{}\mu$m and a radius of curvature of
$r=209~{}\mu$m; for the multimode fiber mirror, $2\rho_{\mathrm{m}}=80~{}\mu$m
and $r=355~{}\mu$m.
Figure 3: Fiber-cavity finesse at a wavelength of $844$ nm as a function of
the cavity length; compare also Fig. $9$ in Ref. Hunger _et al._ (2010). The
points are measurement values from the fiber mirrors specified in the text;
the finesse decreases for longer cavities. Error bars represent one standard
deviation. The solid line shows the calculated finesse due to clipping losses
from the mirrors, where the mirrors are modeled as spheres with diameter
2$\rho_{\mathrm{m}}$ and radii of curvature given in the text. The grey dashed
line gives the cavity’s stability edge for these radii of curvature. The lack
of agreement between data and calculation demonstrates that additional loss
sources play a significant role.
The measured finesse declines gradually with mirror separation, from an
initial value of 71,600 at $85~{}\mu$m to roughly half of that at
$231~{}\mu$m. If the decrease in finesse were due to clipping losses, we would
expect a constant finesse for almost all cavity lengths, with a steep drop a
few microns before the stability boundary at $r=209~{}\mu$m. The clipping
losses from numerical calculations, which assume spherical mirrors, are also
plotted in Fig. 3 and do not agree with the data.
First, the most likely source of additional losses is a non-uniform thickness
of the coating layers: for steep mirror surfaces, ion-beam-sputtered layers
may be too thin Roy and Barrett (2011), shifting the coating towards
wavelengths shorter than the target value. As the mode-field diameter at each
mirror increases with increasing $L$, the mode may enter a region where the
coating is no longer suited for the measurement wavelength and transmission
losses are higher. This effect was observed in Ref. Roy and Barrett (2011), in
which lower finesses for cavities with smaller radii of curvature were
measured.
Second, for cavities at or beyond the stability boundary, that is, the final
two data points in Fig. 3, it is surprising that a nonzero finesse is
observed. For the radii of curvature determined from optical profilometry,
this is not a stable cavity configuration. A possible explanation is that the
assumption of a spherical mirror, while useful, is only an approximation.
Since the mode size at the mirror increases near the stability boundary, the
mode extends to regions where the curvature of the mirror deviates
significantly from a spherical fit. A more realistic model would assume a
Gaussian curvature for the mirrors, but it is difficult to use such a model to
calculate the expected losses in this regime accurately; the numerical
integration does not reduce to a fast Hankel transform as it does for
spherical mirrorsSiegman (1986).
We note that Fig. $9$ of Ref. Hunger _et al._ (2010) presents finesse
measurements for short cavities that are compatible with the clipping-loss
model. However, for each cavity, only one data point with reduced finesse is
measured, and the theory curve corresponds to an average set of parameters
rather than the specific parameters of each set of mirrors. Thus, there is not
enough information to determine whether our measurements are consistent with
those of Ref. Hunger _et al._ (2010). It would be interesting to explore the
hypothesis of coating layer uniformity by repeating these measurements over a
range of mirror curvatures.
#### II.1.2 Coupling efficiencies between cavity and fiber
In this section, we show a measurement of the transmitted intensity through an
FFPC as a function of the cavity length. The transmission is the product of
four terms: the fiber in-coupling efficiency, mode matching from the fiber
into the cavity, impedance matching of the cavity, and collection efficiency
of the output fiber. For free-space cavities, only the second and third terms
are relevant, and the input mode is matched to the mode of the cavity by beam
shaping and alignment. With FFPCs, in contrast, the cavity mirrors are built
into the in-coupling and out-coupling fibers, fixing this mode-matching
coefficient to a value determined by the cavity and the fiber parameters.
Because of this key difference, it is interesting to consider mode-matching
from a single-mode fiber to an FFPC.
The mode overlap $\epsilon$ is defined as the overlap between the TEM00 mode
of the cavity and the spatial mode of the single-mode fiber. This overlap
depends on the radius of curvature of the mirror, on the core diameter of the
fiber, and on the cavity length. In addition, either an offset of the mirror
center from the fiber core or an angle between mirror and fiber core causes a
mode mismatch which cannot be corrected. Ref. Hunger _et al._ (2010) shows
that mode matching can be as high as 85% for a short FFPC but calculates that
it declines for longer cavities. Assuming that the mirror surface is
orthogonal to the fiber core ($\theta=0$) and that there is no offset between
the core and the mirror center ($d=0$), the coupling efficiency $\epsilon_{a}$
is given by Joyce and DeLoach (1984)
$\epsilon_{a}=\frac{4}{(\frac{w_{\mathrm{f}}}{w_{\mathrm{c}}}+\frac{w_{\mathrm{c}}}{w_{\mathrm{f}}})^{2}+\frac{s^{2}}{z_{\mathrm{R_{f}}}z_{\mathrm{R_{c}}}}}$
(1)
with waists $w_{\mathrm{f}}$, $w_{\mathrm{c}}$ and Rayleigh lengths
$z_{\mathrm{R_{f}}}$, $z_{\mathrm{R_{c}}}$ of the beam exiting the fiber and
of the cavity mode, respectively. The distance from the waist of the mode
exiting the fiber to the cavity waist is denoted by $s$.
A tilt of the fiber with respect to the mirror by an angle $\theta$ reduces
the overlap by a factor of $e^{-(\theta/\theta_{e})^{2}}$ for small values of
$\theta$, with the angular tolerance $\theta_{e}$. Similarly, a displacement
of the fiber core from the cavity axis reduces $\epsilon$ by a factor of
$e^{-(d/d_{e})^{2}}$, where analytic expressions for angular tolerance
$\theta_{e}$ and displacement tolerance $d_{e}$ can be found in Ref. Joyce and
DeLoach (1984). The total mode overlap is then given by
$\epsilon=\epsilon_{a}e^{-(d/d_{e})^{2}}e^{-(\theta/\theta_{e})^{2}}$.
Fig. 4(a) and Fig. 4(b) show the calculated mode overlap for a single-mode
fiber of $6~{}\mu$m core diameter and an FFPC with $r_{1}=209~{}\mu$m and
$r_{2}=355~{}\mu$m. Cavity lengths up to $209~{}\mu$m are plotted, considering
non-zero values for $\theta$ and $d$. For $\theta=0^{\circ}$ and $d=0~{}\mu$m,
the mode matching between fiber and cavity increases steeply for short cavity
lengths, has a maximum of $0.84$ at length $54~{}\mu$m, and decreases to half
that value by $200~{}\mu$m. Thus, we see that with proper alignment, it is
possible to build long cavities with reasonable mode matching. As $\theta$ and
$d$ increase, the maximum value for $\epsilon$ drops, but $\epsilon$ becomes
relatively insensitive to cavity length. The range of values for $\theta$ and
$d$ plotted in Fig. 4 reflects estimates of realistic errors in the fiber-
mirror fabrication procedure. Over this range, and for all cavity lengths
shown, these errors cause $\epsilon$ to decrease by almost an order of
magnitude.
The theory predicts a steep decrease in mode matching as the cavity length
approaches the smaller radius of curvature of the two mirrors, $209~{}\mu$m.
Therefore, when mode matching is important, cavity lengths close to the
stability boundary should be avoided. As we have seen in Sec. II.1.1, however,
the stability boundary of a fiber cavity does not correspond to a calculation
based on spherical mirror parameters.
Figure 4: (a) Coupling efficiency from a single-mode fiber to an FFPC
calculated from Eq. 1, taking into account non-zero values of $\theta$ without
displacement ($d=0~{}\mu$m). (b) Coupling efficiency for small displacements
$d$ ($\theta=0^{\circ}$). (c) Measurement of the transmission through an FFPC
as a function of the cavity length, referenced to the first data point. The
error bars represent one standard deviation.
In Fig. 4(c), we show a measurement of the transmission of the FFPC discussed
in Sec. II.1.1. Note that due to the large core diameter and acceptance angle
of multimode fibers, collection efficiency is unity for our cavity parameters
Hunger _et al._ (2010). We measure the transmission through the cavity and
the fibers normalized to the transmission of the first point as a function of
the cavity length. The transmission first decreases as a function of cavity
length and then remains constant at around $20\%$ of the initial value for
cavities longer than $150~{}\mu$m. Only the relative transmission is measured
because an absolute transmission is difficult to calibrate and does not
provide additional information on how mode overlap scales with length Hunger
_et al._ (2010); Hood, Kimble, and Ye (2001). In order to extract the mode-
matching efficiency from this transmission measurement and compare it to Figs.
4(a) and (b), we would have to determine independently whether the decrease in
transmission is due to increasing impedance mismatch or increasing mode
mismatch.
For experiments in which optimal transmission through long fiber cavities is
important, the coupling can be improved by minimizing cavity scatter and
absorption losses, therefore optimizing the impedance matching. The mode
matching can be maximized by minimizing $\theta$ and $d$. An interesting
possibility for improving the mode matching would be to tailor the single-mode
fiber mode, e.g., by expanding the fiber core at the tip.
#### II.1.3 Surface-loss measurement
We measure the scattering losses in the mirror coatings due to surface
roughness of the CO2-laser shaped fibers, and we find no such losses within a
$1$ ppm measurement error. This measurement is the first direct comparison of
the losses of fiber-mirror coatings (including losses induced by surface
roughness) with losses of identical coatings fabricated on fused silica
substrates. For this purpose, fibers and substrates were coated together in
the same fabrication run.
The losses of highly reflective mirror coatings depend critically on the
surface quality of the mirror substrate: surface roughness of the substrate
material results in scattering losses of the mirror. In order to reduce the
scattering losses to $\sim 1$ ppm at near-infrared wavelengths, the surface
roughness needs to be less than $1$ Å rms Hunger _et al._ (2010). Mirror
substrates of surface roughness less than $1$ Å rms are referred to as
superpolished. With superpolished substrates, we assume that all scattering
and absorption losses come from point defects in the coating.
As CO2-laser ablation is a novel technique for producing curved mirror
substrates, it is of great interest to determine the quality of the shaped
surfaces and the induced scattering losses. Fiber surface roughness $\sigma$
has previously been measured with an atomic force microscope and linked to the
scattering losses $S$ via $S\approx(4\pi\sigma/\lambda)^{2}$, where
$\sigma=(0.2\pm 0.1)$ nm, corresponding to $S=(10\pm 10)$ ppm for near-
infrared light at $\lambda=780$ nm Hunger _et al._ (2010).
In contrast, we compare fiber mirrors with reference mirrors produced on
superpolished substrates. We measure reference-mirror cavities to have a
finesse of $(1.10\pm 0.04)\times 10^{5}$, in comparison to a finesse of
$(1.14\pm 0.05)\times 10^{5}$ for the fiber-mirror cavities. For identical
mirrors, the total losses per mirror
$\mathcal{L}_{\mathrm{tot}}=\mathcal{T+L}$ , the sum of transmission
$\mathcal{T}$ and losses $\mathcal{L}$, are calculated from the finesse via
$\mathcal{L}_{\mathrm{tot}}=\pi/\mathcal{F}$. The reference substrates and the
fiber mirrors thus have identical total losses of
$\mathcal{L}_{\mathrm{tot}}=(28\pm 1)$ ppm. In an additional measurement (Sec.
III.1), we determine the total losses of the same reference substrates after
annealing the mirrors. This measurement of losses of $(17\pm 1)$ ppm agrees
with $15$ ppm target transmission of the coating and $2$ ppm scatter and
absorption losses from the coating.
We conclude that fiber surface roughness does not cause any additional
scattering losses. This result suggests that it may be possible to construct
FFPCs with finesses as high as those achieved with state-of-the-art
superpolished mirrors Rempe _et al._ (1992).
#### II.1.4 Birefringence of fiber mirrors
We investigate the birefringent splitting of orthogonal polarization modes in
FFPCs. We observe that FFPCs exhibit significant birefringence, whereas
cavities built with mirrors produced on superpolished substrates fabricated in
the same coating process do not exhibit measurable birefringence. Furthermore,
we observe that the birefringent splitting of the FFPCs varies as a function
of the cavity alignment.
For experiments in which quantum information is encoded in photon
polarization, it is advantageous for the modes of orthogonal linear
polarization to be degenerate in the cavity Ritter _et al._ (2012); Stute
_et al._ (2012b). Therefore, it is of interest to control the birefringence of
the cavity mirrors. Typically, the birefringence in a cavity of mirrors
fabricated on superpolished substrates is induced via stress inside the
mirrors due to the mounting Lynn (2003). In contrast, fiber mirrors are
mounted a few millimeters from the mirror surface, a length scale much greater
than the surface diameter. Therefore, we assume that the stress is intrinsic
to the coating and is created in the fabrication process of the coating.
To measure the birefringent splitting, we measure the transmission curve of an
FFPC while driving the cavity and sweeping its frequency. To calibrate the
abscissa, we modulate the driving laser to produce frequency sidebands with a
known splitting. We repeatedly rotate and realign the fibers and measure the
relative detuning between the polarization modes. The observed splittings
range from smaller than the FWHM of the cavity resonance ($\approx 30$ MHz) up
to a few gigahertz. This result is comparable to the birefringent splitting of
$200$ MHz measured in Ref. Hunger _et al._ (2010). Furthermore, when aligning
the cavity to support different TEM modes, we observe that these TEM modes
have different birefringent splittings.
To quantify this last observation, we have modified the experimental setup
because the TEM mode of the cavity is not preserved inside the output
multimode fiber. The multimode fiber is replaced by a mirror fabricated on a
superpolished substrate that was coated in the same coating run as the fiber.
The TEM mode is then imaged with a CCD camera, while a photodiode measures the
cavity transmission curve. We measure a splitting of $37$ MHz for TEM11 and
$122$ MHz for TEM01 (Fig. 5), demonstrating that the birefringent splitting is
dependent on the TEM mode and thus on the specific mirror region that the
cavity mode samples.
Figure 5: Birefringent splitting of the orthogonal polarization modes inside
the cavity described in the text. Birefringent splitting of (a) $37$ MHz for
TEM11 and of (b) $122$ MHz for TEM01. The $100$ MHz sidebands for the
frequency calibration are indicated with dashed lines. The insets show CCD-
camera images of the cavity modes.
We conclude that birefringence is not homogeneous over the mirror coating on
the fiber. Thus, simply rotating two birefringent fiber mirrors with respect
to one another will not necessarily eliminate cavity birefringence. However,
by careful selection of fibers and proper alignment, we have built cavities
which satisfy a target birefringent splitting, in our case, degenerate
polarization modes.
### II.2 Ion-cavity system
We now focus on the relevant rates of our experimental ion-cavity system
considering ${}^{40}\mathrm{Ca}^{+}$ and the FFPC parameters from the
measurements shown above. Until very recently, experimental ion-cavity systems
have used cavities constructed with superpolished-substrate mirrors
Guthöhrlein _et al._ (2001); Mundt _et al._ (2002); Russo _et al._ (2009);
Leibrandt _et al._ (2009); Herskind _et al._ (2009); in addition, an ion has
now been coupled to an FFPC Steiner _et al._ (2013). All of these systems
operate in a regime in which the coherent coupling rate $g$ between a single
ion and a photon is smaller than the rate of at least one incoherent process,
such as scattering from the ion or loss from the cavity. To increase the
coherent coupling strength $g$ between ion and photon, the cavity mode waist
$w_{0}$ and the cavity length $L$ should be minimized, as for a dipole
coupling
$g=\frac{\lambda}{\pi w_{0}}\sqrt{\frac{3c\gamma_{c}}{L}},$ (2)
where $\gamma_{c}$ is the spontaneous emission rate between the states coupled
via the cavity, and $w_{0}$ is calculated via the length $L$ and the radius of
curvature $r$ of the mirrors. For a cavity in which both mirrors have the same
radius of curvature,
$w_{0}^{2}=\frac{\lambda}{2\pi}\sqrt{L(2r-L)}.$ (3)
Laser machining of fiber mirrors produces radii of curvature that are two
orders of magnitude smaller than radii of curvature produced via
superpolishing techniques Hunger _et al._ (2010). Thus, short cavity lengths
and small cavity waists are inherent to FFPCs, allowing for high coupling
rates. Our setup is comprised of a linear Paul trap, described in Sec. IV.2.1,
and an FFPC. In the setup, the cavity axis is perpendicular to the trap axis.
Ions trapped along this axis would have a separation of around $5~{}\mu$m.
Thus, the number of ions that can be coupled to the same antinode of the
cavity depends on the size of the cavity waist. For a mode waist $w_{0}$ of
$5~{}\mu$m, it is possible to have two ions displaced symmetrically from the
maximum of the cavity mode. Both ions are then coupled to the cavity with
$88\%$ of the maximum strength.
Our cavity mirrors are chosen for optimum finesse at either the
$4P_{1/2}-3D_{3/2}$ or the $4P_{3/2}-3D_{5/2}$ transition of
${}^{40}\mathrm{Ca}^{+}$, which have wavelengths of $866$ nm and $854$ nm,
respectively. The $3D_{3/2}$ and $3D_{5/2}$ states are metastable states, and
the $3D_{5/2}$ state is used for quantum-information processing Häffner, Roos,
and Blatt (2008). Here, we calculate the system parameters, that is, the
coherent coupling rate $g$ and cavity decay rate $\kappa$, for the FFPC
characterized in Sec. II.1.1. The finesse and the cavity length have been
measured, and $w_{0}$ and $g$ are calculated from the radius of curvature
measured interferometrically and are therefore approximate values. $\kappa$ is
calculated via the finesse $\mathcal{F}$ by
$\kappa=\frac{c\pi}{2\mathcal{F}L},$ (4)
and the single-atom cooperativity $C$ is given by
$C=\frac{g^{2}}{2\kappa\gamma}.$ (5)
Table 1: Cavity-QED system parameters for various setups of neutral-atom and ion experiments using FFPCs and cavities consisting of superpolished mirror substrates. These parameters are compared to our ${}^{40}\mathrm{Ca}^{+}$–FFPC system with the FFPC characterized in Sec. II.1.1; given the measured values of cavity length $L$ and finesse $\mathcal{F}$, we calculate the mode waist $w_{0}$, cavity decay rate $\kappa$, coherent coupling rate $g$ and single-atom cooperativity $C$. experiment | $L$ ($\mu$m) | $w_{0}$ ($\mu$m) | $\mathcal{F}$ | $\kappa$
(MHz$/2\pi$) | $g$ (MHz$/2\pi$) | $\gamma$ (MHz$/2\pi$) | $C$
---|---|---|---|---|---|---|---
neutral Cs Hood _et al._ (2000) | $10.9$ | $14.0$ | $480,000$ | $14.1$ | $110$ | $2.6$ | $164$
neutral Rb \- FFPC Colombe _et al._ (2007) | $38.6$ | $3.9$ | $37,000$ | $53$ | $215$ | $3$ | $145$
$\text{Ca}^{+}$ ions Keller _et al._ (2004) | $8000.0$ | $37.0$ | $78,000$ | $1.2$ | $0.92$ | $11.2$ | $0.03$
$\text{Ca}^{+}$ ions Stute _et al._ (2012a) | $19960.0$ | $13.2$ | $77,000$ | $0.05$ | $1.43$ | $11.2$ | $1.8$
$\text{Yb}^{+}$ ions - FFPC Steiner _et al._ (2013) | $230.0$ | $6.6$ | $1,000$ | $320$ | $6$ | $2$ | $0.03$
current setup | $85.0$ | $5.1$ | $72,000$ | $12$ | $41$ | $11.2$ | $6.1$
$\text{Ca}^{+}$ ions - FFPC | $131.0$ | $5.4$ | $64,000$ | $9$ | $31$ | $11.2$ | $4.8$
| $206.0$ | $3.2$ | $45,000$ | $8$ | $41$ | $11.2$ | $9.3$
The parameters of our ion-cavity system are compared in Tab. 1 to a selection
of single-atom cavity-QED experiments. With neutral Cs and Rb atoms,
cooperativities over $10^{2}$ have been demonstrated by using short, high-
finesse cavities Hood _et al._ (2000) and by using an FFPC to obtain a small
mode waist Colombe _et al._ (2007). In contrast, ion-trap experiments have
been limited to $C\lesssim 1$, primarily because relatively long cavities have
been necessary to avoid distortion of the trapping potential Keller _et al._
(2004); Stute _et al._ (2012b). Here, we see that fiber mirrors offer a
promising route towards much shorter cavities and smaller mode waists, as in
the first demonstration of an ion-trap FFPC Steiner _et al._ (2013).
In our system, the atomic decay rates $\gamma$ of the $4P_{3/2}$ state and the
$4P_{1/2}$ state are $2\pi\times 11.4$ MHz and $2\pi\times 11.2$ MHz,
respectively, including decay channels to both $S$ and $D$ manifolds. From
Tab. 1, we see that the coherent coupling rate $g$ is larger than $\kappa$ and
$\gamma$ for the range of possible cavity lengths of this FFPC. Longer
cavities exhibit smaller $\kappa$, although the finesse decreases with
increasing cavity length (Fig. 3). Additionally, the sharp decrease in cavity
waist gives increasingly larger $g$ as the near-concentric limit is
approached. These trends contribute to higher cooperativities as the cavity
length is increased.
### II.3 Practical techniques for FFPCs
FFPCs differ from conventional cavities in several ways. The mirror is
fabricated on the fiber, so that different techniques are required to align a
cavity. Furthermore, due to the small size of both fibers and mirrors,
technologies for mounting and cleaning as well as for annealing of the fiber
mirrors are necessary. In this section, we describe new methods developed in
our work with FFPCs. Experiments annealing fiber mirrors are presented
separately in Sec. III together with measurements of annealing mirrors
fabricated on superpolished substrates.
#### II.3.1 Fiber preparation
We work with copper-coated single-mode and multimode fibers of non-standard
$200~{}\mu$m diameterIVG . Here, we summarize techniques for fiber
preparation.
Etching copper coating from fibers: Fibers need to be stripped properly before
they are cleaved or connectorized. We etch away the copper with a $25\%$
nitric acid (HNO3) solution or a $20\%$ iron(III) chloride (FeCl3) solution at
$50^{\circ}$C until the coating is no longer visible. This process takes only
a few minutes. The first method is faster but also requires more precaution in
handling the chemicals.
Removing titanium from fibers: After the copper is etched off, the glass
fibers are still coated with a thin layer of titanium, which is used as an
adhesive between glass and copper. This layer is not insulating, and if the
fibers have contact with trap electrodes, they cause a short circuit.
Furthermore, the titanium layer shifts the ion’s trapping potential when it is
brought close to the trap center. We find that the titanium can be scratched
away gently with diamond paste, which is then rinsed off thoroughly with
solvents.
Mounting fibers: To protect the fibers from dirt or damage, they need to be
mounted properly during every stage of the experimental process, e.g., in the
coating chamber, during testing or in the experimental setup. In the coating
device, the fiber holders need to be vacuum compatible. We clamp each fiber
with a screw inside an aluminum cylinder. The fiber tip protrudes $0.5$ mm
from the cylinder for the coating. We encase the fiber in a Teflon sleeve so
that the screw does not damage it. The same holders are used for fiber storage
before and after coating. However, to set up a test cavity, the fibers should
not be clamped rigidly with screws as it impairs their optical transmission.
Instead, we fix the fibers with a magnet in stainless-steel v-grooves. In the
experimental setup, the fibers are then glued with UHV epoxyEPO onto Pyrex
v-grooves.
Connectorizing fibers: In the testing process, we often switch between fibers
and thus prefer slide-on slide-off bare-fiber adapters to connectors that need
to be glued. The bare-fiber adapters are custom-made for the $200~{}\mu$m
cladding diameterBul .
Splicing non-standard $200~{}\mu$m diameter fibers to standard $125~{}\mu$m
diameter fibers: To be able to use standard fiber tools, it is useful to work
with $125~{}\mu$m diameter fibers. We find that it is possible to splice the
$200~{}\mu$m diameter fibers to standard $125~{}\mu$m diameter fibers of the
same core diameter with a commercial fiber splicerVyt with negligible losses.
In the splicing process, we account for the larger diameter of the
$200~{}\mu$m fiber by shifting the splice filament such that it preferentially
heats the larger fiber.
#### II.3.2 Alignment of long cavities
To obtain a cavity-transmission signal, it is sufficient to align the fibers
by eye to form a very short cavity of about $30~{}\mu$m in length and sweep
the length across a free spectral range. However, as the CO2-laser ablation
does not produce perfect surfaces — generally, the center of the depression is
offset from the center of the fiber facet, and the mirror surface is not
exactly orthogonal to the fiber core — the cavity must be aligned further via
optimization of the cavity transmission signal. For aligning FFPCs, one fiber
is fixed while the second fiber is mounted on a six-axis nano-positioning
systemTho . To build longer FFPCs, the distance between the mirrors is then
increased stepwise while optimizing the alignment by maximizing the
transmission signal. Using this technique, we are able to build cavities of
lengths up to $350~{}\mu$m.
We observe that cavities of up to $\sim 100~{}\mu$m in length are robust to
changes in mirror position or angle. As the fiber mirrors are separated
further, however, very small changes misalign the cavity even though the
mirrors’ radii of curvature suggest that the cavity is still far away from the
edge of the stability region. In this case, the size of the mirror is the
limiting factor for misalignment. As a consequence, for building long
cavities, care must be taken that the fibers are mounted very stably.
#### II.3.3 Cleaning fiber mirrors
To clean ultra-low loss mirrors fabricated on a superpolished substrate, the
mirror is rotated on a spin cleaner and the surface is swabbed with water,
acetone and isopropyl alcohol during rotation Northup (2008). Fiber mirrors,
however, are too delicate to swab. The high-temperature gradient of the
CO2-ablation process makes the fiber tip brittle, and we find that any stress
or pressure usually breaks the tip.
One obvious strategy has been to keep the fiber mirrors as clean as possible
and shield them from contamination. Unfortunately, even in a clean
environment, the fiber mirrors sometimes decrease in finesse as they
accumulate dust. To address this problem, we have developed a cleaning
procedure for ultra-low loss fiber mirrors. We use spectrophotometric grade
solvents, heated to $50^{\circ}$C, to clean the mirror fiber tips in an
ultrasonic bath for two minutes, first in acetone and then in methanol.
Immediately after taking the fiber out of the methanol, we use clean helium to
dry the mirror surface for at least half a minute.
There are a few cases in which this method does not recover the initial
finesse. However, we typically see full recovery of the finesse by cleaning
fiber mirrors with this procedure.
## III Annealing mirrors
Ultralow-loss mirrors at optical wavelengths are routinely employed in quantum
optics experiments. Using ion-beam sputtering, mirrors can be fabricated with
total losses (transmission, absorption, scatter) as low as $1.6$ ppm in the
near infrared Rempe _et al._ (1992). In order to achieve such low losses in
dielectric mirror coatings, it is a standard procedure to anneal the coatings
after fabrication. Annealing leads to homogenization of the oxide layers and
improves the stoichiometry of non-perfect oxides Atanassova, Dimitrova, and
Koprinarova (1995), reducing coating losses typically by $10$ ppm. This
procedure is thus a key step in the process of manufacturing ultra-low-loss
mirrors.
The recent development of fiber cavities raises the question of whether
annealing fiber mirrors is possible. Since the surface roughness of
CO2-laser–ablated fiber tips is comparable to that of superpolished mirror
substrates (Sec. II.1.3), the finesse of fiber cavities can in principle reach
the record values achieved with mirrors fabricated on substrates. To reach
this high finesse, however, annealing would be essential.
Our initial efforts to anneal fiber mirrors have been unsuccessful. We
attribute some of the difficulties to possible chemical reactions of the
fiber-coating material with oxygen in the air. For this reason, we have
investigated annealing under vacuum. Furthermore, knowledge about the effects
of baking mirrors under vacuum is essential for all experiments in which low-
loss mirrors are placed under ultra-high vacuum (UHV), which requires a vacuum
bake. The typical temperatures for a vacuum bake are lower than annealing
temperatures, but the same chemical processes are at work. In various
experiments, degradation of cavity mirrors under vacuum has been observed
(Ref. Cetina _et al._ (2013); Sterk _et al._ (2012) and references therein),
but evidence of changes in mirrors under vacuum has been mostly anecdotal.
Because the cavities are part of complex experimental systems, in which
repeated bake-outs are impractical, a careful study of these effects has not
yet been undertaken.
In order to study annealing and baking under vacuum systematically, we use
reference mirrors which have been produced in the same coating run as the
fiber mirrors (Sec. II.1). These reference mirrors are fabricated on fused
silica substrates of half-inch and $7.75$ mm diameters. They are coated with a
highly reflective coating comprised of $37$ alternating layers of Ta2O5 and
SiO2, where the inner- and outermost layers are Ta2O5. The layers are
deposited using ion-beam sputtering, and each layer has a $\lambda/4$
thickness for peak reflectivity at $\lambda=860$ nm. Using these mirrors, we
systematically measure effects from annealing under air and vacuum in a clean
and controlled system.
In this section, annealing refers to a $90$ minute ramp from room temperature
to $450^{\circ}$C, a $90$ minute bake at $450^{\circ}$C, and a $90$ minute
ramp down to room temperature. Vacuum annealing consists of placing the
mirrors in a clean stainless-steel chamber, which is then pumped to pressures
lower than $10^{-5}$ mbar by a turbo pump, after which the temperature ramp is
started. For annealing under air, the mirrors are placed inside clean glass
Petri dishes. Care is taken that the mirrors are properly cleaned before any
finesse measurement Northup (2008).
### III.1 Repeated annealing under vacuum and under air
Figure 6: Finesse after annealing at $450^{\circ}$C under alternating air and
vacuum environments. Annealing under vacuum shows repeatable losses in cavity
finesse, and annealing under air repeatable gains up to a maximum finesse of
$(1.80\pm 0.03)\times 10^{5}$ for mirror pair $1$ (points). Mirror pair $2$
(open circles) establishes the maximum finesse after repeated annealing under
air. The error bars represent one standard deviation of the measurement
uncertainty.
With a first pair of coated mirrors, we constructed a cavity with a finesse of
$(1.05\pm 0.09)\times 10^{5}$ prior to annealing. After an initial test in
which the mirror pair was annealed under vacuum, the finesse had degraded to
$(3.9\pm 0.6)\times 10^{4}$. To investigate this unexpected result, we
conducted a series of measurements, in which we alternated between annealing
under vacuum and air. Annealing these mirrors under air resulted in a recovery
of the finesse, that is, a decrease of the losses that had been induced by
vacuum annealing. In fact, the new finesse of $(1.4\pm 0.1)\times 10^{5}$ was
higher than the initial value, indicating that annealing had removed intrinsic
coating losses as expected. Two subsequent measurements showed that the losses
when annealing under vacuum and gains when annealing under air are repeatable,
and the maximum finesse for this pair of mirrors is $(1.80\pm 0.03)\times
10^{5}$; these data are summarized in Fig. 6.
With a second pair of reference mirrors, we reproduce the initial finesse of
the first pair: $(1.06\pm 0.06)\times 10^{5}$. Annealing directly under air as
the only step yields a finesse of $(1.75\pm 0.06)\times 10^{5}$, implying that
the maximum finesse of this coating is independent of previous annealing
cycles. Repeated annealing under air established the maximum finesse for this
pair of mirrors. This finesse corresponds to total losses
$\mathcal{L}_{\mathrm{tot}}$ of $17$ ppm. We attribute 2ppm Rempe _et al._
(1992) to scattering and absorption losses and $15$ ppm to transmission,
consistent with the target transmission of the coating run.
Our finding that the change in finesse depends on the annealing environment
suggests that annealing affects the chemical composition of the mirror
coating. We hypothesize that during a vacuum bake, oxygen escapes from the
outermost Ta2O5 layer, which leads to defects in the coating. Subsequent
annealing under air gives the surface the possibility to regain the oxygen,
thus removing these defects of the chemical structure. In order to test this
theory of oxygen depletion, we conducted a series of X-ray photoelectron
spectroscopy measurements.
### III.2 X-ray photoelectron spectroscopy measurements
X-ray photoelectron spectroscopy (XPS) is used to quantitatively determine the
chemical composition of the Ta2O5 layer on the surface of the mirror coatings.
We acquire the XPS data with a Thermo Multilab $2000$ utilizing monochromatic
Al K$\alpha$ radiation at $1486.6$ eV. The atomic composition of the samples
is obtained from XPS survey scans taken with an overall resolution of $2.0$
eV. The oxygen and tantalum content are determined from the O ($1$s) and the
Ta ($4$d) lines, respectively, measured with a higher resolution of $0.1$ eV.
Measured intensity ratios are converted into atomic percentages using the
theoretical photoionization cross-sections of Scofield Scofield (1976), also
taking into account the energy-dependent transmission of the electron-energy
analyzer Klauser _et al._ (2010).
#### III.2.1 XPS measurements of mirrors annealed under air and under vacuum
We acquire XPS spectra from two mirrors and compare their chemical
composition. Prior to the measurement, one mirror was annealed in air, while
the other was annealed in vacuum. To calculate the amount of oxygen and
tantalum from the XPS spectra, we subtract the background and integrate over
the O ($1$s) and Ta ($4$d$5$) photoelectron lines. When weighted by the
Scofield sensitivity factors, which represent the emission probability of an
electron, these integrals give the relative proportions of the elements in the
material. The sensitivity factor is $15.64$ for the Ta ($4$d$5$) line and
$2.93$ for the O ($1$s) line.
Using this method, we compare the chemical compositions of the two mirrors.
The oxygen concentration of the mirror annealed under vacuum is $(0.9\pm
0.7)\%$ lower than the oxygen concentration of the mirror annealed under air.
This difference would constitute a loss of every $90$th oxygen atom from the
surface of the mirror annealed under vacuum. The large error bars are due to
the relative uncertainty of $0.5\%$ between measurements on the same
apparatus. To resolve the effect more clearly, we conduct a second experiment
based on the measurement of a single mirror over time.
#### III.2.2 Continuous XPS measurement during vacuum annealing
To observe the effect of oxygen loss from the surface directly, we perform
real-time XPS measurements during the process of annealing in vacuum on a
mirror that has previously been annealed in air. The mirror is placed inside
the XPS vacuum chamber and a reference XPS measurement is taken. Between
subsequent XPS measurements, the mirror temperature is increased stepwise up
to $608^{\circ}$C over $200$ minutes. This procedure gives an exact chemical
analysis of the mirror surface at each step of the annealing process.
Integrating the area under the oxygen and tantalum peaks, we calculate the
oxygen content in the surface of the mirror. The insets of Fig. 7 show the O
($1$s) and the Ta ($4$d) lines of one of the XPS spectra which we use for this
analysis. The temperature of the mirror in this measurement is measured with a
pyrometer (IMPAC) with a measurement uncertainty of $20^{\circ}$C.
Figure 7: Oxygen content in the Ta2O5 layer on the mirror surface. The
temperature of the mirror coating is increased stepwise and XPS measurements
are taken at each temperature. The mirror has been annealed in air before the
measurement. The solid line is taken during the heating process, the dotted
line during the cool-down of the substrate. The uncertainty of the temperature
measurement is $20^{\circ}$C. The relative uncertainty of the oxygen content
is $0.5$%. The insets show sample XPS spectra of the O (1s) and the Ta (4d)
lines (solid), including the background (dashed), at one temperature setting.
Figure 7 shows the results of these measurements, in which the oxygen content
decreases as the temperature increases. The atomic percentage of oxygen of the
mirror before annealing is $78.2$%; at $405^{\circ}$C it drops to $75.7$%. The
oxygen content briefly recovers between $450^{\circ}$C and $550^{\circ}$C,
suggesting a phase transition Bansal (1994) or outgassing of oxygen. At
$608^{\circ}$C, the oxygen content reaches its lowest point of $75.7$%. The
mirror cool-down lasts $90$ minutes, during which the oxygen content neither
decreases nor increases significantly. We note that the absolute uncertainty
of the measurement is around $10$%. The entire annealing process results in a
$(2.5\pm 0.7)\%$ drop in oxygen content, supporting the hypothesis of oxygen
depletion from the Ta2O5 layer.
The discrepancy between this result and our earlier measurement of Sec.
III.2.1 might be due to oxygen reuptake when the vacuum-baked mirror was in
air before the XPS measurement. Both measurements show that we can attribute
the lower finesse of the vacuum-annealed mirrors to the loss of oxygen of the
Ta2O5-layer on the mirror surface.
### III.3 Baking under vacuum
Up to now, we have only presented measurements of mirror annealing at
temperatures of $450^{\circ}$C and higher. The depletion of oxygen observed at
these temperatures suggests that this effect also takes place — in a moderate
form — when baking mirrors at standard temperatures for a vacuum bake-out,
typically $200^{\circ}$C to $300^{\circ}$C. The XPS measurement of Sec.
III.2.2 shows a linear decrease of oxygen when heating the mirror from room
temperature up to $405^{\circ}$C in vacuum. At a temperature of $160^{\circ}$C
one percent of the oxygen is already lost from the surface, and at
$300^{\circ}$C $1.8\%$ of the oxygen is lost.
We expect that if we bake mirrors under vacuum conditions at different
temperatures, one would find decreasing mirror finesses as the temperature of
the bake increases. This measurement would presumably show the same
temperature dependence of oxygen depletion following a vacuum bake as the XPS
measurements. We can estimate the mirror losses by linearly extrapolating the
two annealing measurements from Fig. 6 to lower temperatures. The first
annealing under vacuum was performed with non-annealed mirrors, while the
second time, these mirrors had been pre-annealed under air. According to these
measurements, we would expect $32$ ppm and $8$ ppm of additional losses at a
baking temperature of $300^{\circ}$C for the non-annealed mirrors and the
annealed mirrors, respectively. To understand whether this difference in
losses can be attributed to the pre-annealing, it would be interesting to
repeat the measurements for several mirror pairs.
A pre-annealed test mirror, however, baked under vacuum conditions at
$300^{\circ}$C, showed a decrease of the finesse from $1.9\times 10^{5}$ to
$4\times 10^{4}$ after baking. These $62$ ppm of additional mirror losses are
higher than expected from the annealing results. Furthermore, the mirror
finesse could not be recovered by successive air bakes, suggesting that the
mirror was damaged in the baking process. Lacking additional undamaged test
mirrors, we did not perform further vacuum bakes at moderate temperatures.
However, a systematic study of vacuum bakes over a range of temperatures would
provide valuable information for cavity setups in UHV.
### III.4 Discussion
Annealing under vacuum decreases the mirror finesse rather than increasing it.
As a consequence, fiber mirrors should not be annealed under vacuum.
Tests of annealing fiber mirrors under air have not been successful so far.
Even when a clean annealing environment is established, the fiber mirrors seem
to get dirty after baking in air and cannot be cleaned successfully. We
suspect that contamination from the copper coating of the fiber damages the
mirror coating. A way to remove this source of contamination is the use of
chemically more inert fiber coating materials such as gold.
Furthermore, we expect the mirror losses to increase under a vacuum bake even
at moderate temperatures. The XPS measurements of Sec. III.2.2 show that the
oxygen decreases linearly with increasing baking temperature, and thus the
amount of defects in the mirror increases. Vacuum baking should therefore be
done at the lowest possible temperatures, although baking at higher
temperatures under oxygen atmosphere might be a solution.
## IV Experimental ion-trap apparatus with an FFPC
In our combined FFPC ion-trap setup, the fibers sit on UHV-compatible
positioners enabling in-vacuum cavity alignment. In addition, these
positioners allow us to pull back the fibers from the trap center, so that the
ion trap can be tested without the influence of the fibers on the ion’s
trapping potential. The ion trap is a modified version of the linear Paul trap
presented in Refs. Gulde (2003); Riebe (2005). Here, we describe in detail the
ion trap, the integration of the FFPC into the trap setup, and the underlying
design considerations.
### IV.1 Experimental design considerations
Ions are trapped quasi-permanently and well isolated from environmental
perturbations in RF Paul traps Paul (1990) under UHV conditions. Possible ion-
trap designs include surface-electrode traps Chiaverini _et al._ (2005),
segmented linear traps based on microchip technology Schulz _et al._ (2008),
‘endcap’ Schrama _et al._ (1993); Wilson _et al._ (2011) or stylus ion traps
Maiwald _et al._ (2009), and linear blade traps Gulde (2003); Riebe (2005).
Two criteria for selecting a specific design are low ion heating rate and deep
trap depth, both of which contribute to long ion lifetimes in the trap. The
heating rate increases with decreasing ion-electrode distance Turchette _et
al._ (2000), while for comparable trap dimensions and applied RF voltages, the
trap depth in three-dimensional traps is considerably deeper than in two-
dimensional traps. Linear blade traps, with heating rates as low as a few
quanta per second and trap depths on the order of tens of eV, are known to
have long ion lifetimes Rohde _et al._ (2001); Benhelm _et al._ (2008).
When considering the implementation of dielectric mirrors into an ion trap,
one should keep in mind the effects of dielectrics on the ion. Charges on
dielectrics in vacuum are quasi-permanent and distort the ion-trap potential.
They can be produced by UV light via photoelectron ionization Harlander _et
al._ (2010) in a way that is not well understood and difficult to model. The
best strategy is to avoid any charging of dielectrics and to minimize the
influence of possible charges on the ion. As a solution, dielectric mirrors
are either placed far away from the trap Mundt _et al._ (2002); Herskind _et
al._ (2009); Russo _et al._ (2009); Leibrandt _et al._ (2009); Guthöhrlein
_et al._ (2001) or the dielectric components are well shielded Steiner _et
al._ (2013); Brady _et al._ (2011); Wilson _et al._ (2011); VanDevender _et
al._ (2010); Kim, Maunz, and Kim (2011).
Therefore, when integrating an FFPC into an ion trap, the following
restrictions should be respected: the ion-trap potential should be as deep as
possible, and the trap geometry should be such that it shields the ion from
any charges on the fibers. Furthermore, exposure of the fibers to UV light
should be minimized in order to keep them from accumulating charges. In case
the fibers become charged, the trap design should be flexible enough to
compensate for those charges, i.e., through the application of compensation
voltages. Here, we describe a setup that combines these features.
### IV.2 Ion trap and vacuum chamber
#### IV.2.1 Miniaturized linear Paul trap
We choose a miniaturized linear Paul trap similar to the standard design
described in Refs. Gulde (2003); Riebe (2005); see Fig. 8. Four blade-shaped
electrodes operated with radio frequency (RF) and ground (GND) voltages
confine the ion radially, and two tip electrodes with positive voltage add
confinement along the trap axis. The trap has a deep trapping potential of
several electron volts inherent to three-dimensional traps, but in contrast to
traps of similar design Gulde (2003); Riebe (2005); Schulz _et al._ (2008),
it is miniaturized in order to make its dimensions comparable to those of the
FFPCs, thus shielding the ions from charges on the fibers. The distance
between opposing blade tips on the diagonal is $340~{}\mu$m, which means that
the minimum ion-electrode distance is only $170~{}\mu$m; in contrast, in the
design of Ref. Riebe (2005) the ion-electrode distance is $800~{}\mu$m. The
distance along the trap axis between the two tip electrodes is $2.8$ mm, about
half the length of previous designs. These axial electrodes have $300~{}\mu$m
diameter holes for optical access along the trap axis. Another significant
change is that the angle between neighboring blade electrodes is not
$90^{\circ}$. In order to provide space for the fibers, the two angles between
the blades shielding the fibers are increased to $120^{\circ}$. As a result,
the other two angles are $60^{\circ}$. This change does not alter the trap
depth significantly. The trap has four additional rod-like electrodes,
parallel to the trap axis, $1.7$ mm from the trap center, and of $200~{}\mu$m
diameter. These electrodes allow for compensation of ions’ micromotion. The
rods are supplied with independent voltages, enabling compensation for charges
on the dielectric fibers. In Sec. IV.4, we show a simulation of the ion-trap
potential.
Figure 8: (a,b) Miniaturized ion-trap design. Red: two radio-frequency (RF)
and two ground (GND) blades of the linear Paul trap; the distance between two
opposing blades on the diagonal is $340~{}\mu$m (the distances between
neighboring blades are $290~{}\mu$m and $150~{}\mu$m). Blue: endcap
electrodes, $2.8$ mm apart. Green: four compensation electrodes, $1.7$ mm from
the trap center. White: Ceramic (MACOR) mount. Yellow: Holes in which
alignment rods are temporarily inserted. The plane for optical access,
including laser cooling, manipulation, and fluorescence detection, is
perpendicular to the fiber-cavity axis. $300~{}\mu$m wide holes in the endcaps
provide additional optical access in this plane. (c) Photo of the ion-trap
center. (d) CCD camera image of a linear string of ions.
Precise machining and positioning of the trap electrodes is necessary for a
trap of such small electrode separations. The stainless-steel blade electrodes
are aligned and mounted via two precision-machined glass-ceramic (MACOR)
holders. The fabrication tolerance for the ceramic mounts is less than
$50~{}\mu$m. The dimensions of the holders are then measured after machining,
and the blade electrodes are subsequently electron-discharge machined to fit
the holders exactly. Precision alignment of the blades with respect to the
ceramic mount and to one another is done with alignment rods, which are later
removed. After mounting the electrodes, we measured the dimensions of the ion
trap using a microscope. We find that the inaccuracy in blade-to-blade
separation is less than $30~{}\mu$m.
As the ion-electrode distance is very small, we expect higher ion-heating
rates in comparison with larger traps, so that it may be difficult to work
with ions in the motional ground state. However, the advantages of this design
are manifold: the trap has a deep trapping potential, while the electrode
distances are comparable to those of microfabricated traps and to the size of
the FFPC. Small traps do not need RF voltages as high as those of large traps
and are driven by simple RF resonators Gandolfi _et al._ (2012). The blade
separation along the FFPC axis is only $150~{}\mu$m, thus shielding the ions.
The small diameter of the holes in the tip electrodes helps us to align laser
beams on the ion. We have successfully loaded strings of ions and single ions
in the ion trap (Fig. 8(d)).
#### IV.2.2 Vacuum vessel
To minimize collisions of ions with background gas, the trap needs to be
mounted under ultra-high vacuum. The chamber is designed to optimize vacuum
conditions, optical access, and stability requirements. The implementation of
the FFPC leaves only one plane of optical access available for lasers and
collection of fluorescence from the ions. Therefore, we chose an octagonal
vacuum chamber, which provides optical access from eight sides in this plane.
Fig. 9 shows a technical drawing of the experimental chamber. The axis
orthogonal to both the FFPC and the trap axes is used for fluorescence
detection. Here, within two inverted viewports, high NA objectives are
installed which allow for efficient light collection and thus fast detection
of the state of the ion with a camera and a PMT.
The FFPC fibers are fed into vacuum with a home-made fiber feedthrough
comprising stainless-steel tubes brazed into a CF-flange. The inner diameter
of the tubes is $0.5$ mm, and the fibers are glued into the tubes with vacuum
epoxyEPO . In contrast to commercial fiber feedthroughs, a home-made
feedthrough is advantageous as it is compatible with any fiber type, including
non-standard cladding diameters.
The trap is mounted together with the FFPC on the top flange of the vacuum
chamber, which also supports all electrical feedthroughs, the fiber
feedthrough, and the calcium oven. A vibration-isolating materialDuP is
sandwiched between that top flange and the trap mount. The vacuum chamber sits
within a hole in an optical breadboard, which allows us to use short mounting
posts for optical components, thus providing improved stability.
Figure 9: Technical drawing of the vacuum vessel and the ion-trap fiber-
cavity apparatus.
### IV.3 Integration of fibers
The FFPC needs to be aligned with respect to the ion trap such that the center
of the ion trap overlaps with the waist of the cavity mode. Furthermore, the
fiber mirrors need to be aligned precisely to form a Fabry-Perot resonator,
and the cavity length has to be actively stabilized.
In the FFPC experiment of Ref. Colombe _et al._ (2007), it was possible to
shift the trapping potential for neutral atoms with respect to the cavity
using currents on the chip. The alignment of the fiber mirrors was done before
the system was placed under ultra-high vacuum: one fiber was glued on a cavity
mount carefully positioned with respect to the atom chip, and the second fiber
was then aligned to form an FFPC with the first fiber and glued with UHV
epoxy. While the glue cured, the fiber was continuously aligned by optimizing
the cavity transmission signal.
We have tested the method described above to implement an FFPC into the ion
trap. We find that with short cavities of about $70~{}\mu$m in length, this
method is successful. Unfortunately, long cavities need to be aligned with
considerably higher precision, and small drifts of the fiber mirrors lead to
misalignment of the cavity. Although it is possible to cure the glue while
keeping the FFPC aligned, the cavity signal degrades over time after the glue
has set. Furthermore, the cavity signal disappears when the cavity assembly is
moved or rotated. These effects may be caused by slight temperature changes or
by tension in the cavity mount, and we expect that vacuum baking would also
contribute to misalignment. It may be possible to stabilize the fiber mirror
position passively well enough to maintain alignment, but instead we decided
on an active technique to mount the FFPC inside vacuum.
Each fiber is mounted on a three-axis nanopositioning system compatible with
ultra-high vacuum. Along the cavity axis, the fibers can be translated by up
to $8$ mmSMA (a). Along the other two axes, smaller positioners provide $4$ mm
of traveling rangeSMA (b). The positioners operate via the slip-stick
principle and have a minimum step size of $50$ nm, but can be moved with sub-
nanometer resolution by charging the piezo actuators. The three-axis system
for each fiber has dimensions of $(17\times 22\times 21)$ mm. Fig. 10(a) shows
a test setup of an FFPC aligned with the positioning system.
Figure 10: FFPC positioning system. (a) 3D nanopositioning setup for each
fiber. (b) Each fiber is glued to a glass v-groove and mounted on a shear-mode
piezoelectric crystal which allows active length stabilization of the cavity.
The piezoelectric crystals are glued to insulating MACOR spacers which are
screwed onto the nanopositioning stages.
Each fiber is glued to a Pyrex v-groove, which sits on a shear-mode
piezoelectric actuator; see Fig.10(b). The additional actuator is needed to
stabilize the cavity length actively as the bandwidth of the positioning
system is too low. These actuators are fixed to the positioners, and the whole
assembly is mounted on the same holder as the ion trap. Fig. 11 shows a photo
of the ion-trap setup together with the FFPC aligned via the micropositioning
system.
Figure 11: Photo of the linear Paul trap with an integrated FFPC. The trap
axis is horizontal, and the fiber-positioning systems are visible above and
below the trap. The calcium oven points towards the trap center from the right
side. Vacuum-compatible coaxial cables connect positioners and trap electrodes
to vacuum feedthroughs.
The in-vacuum positioning system offers a range of advantages for the setup.
First, it provides the option of realigning the cavity under vacuum in case of
misalignment due to baking or transport of the vacuum chamber. Second, it is
possible to pull the fibers back by almost a centimeter from the trapping
region, allowing the trapping of ions without dielectrics close to the
trapping region. Also, if the fibers are out of the trapping region during ion
loading, the trap blades shield the fibers, and the fiber mirrors do not get
coated with calcium. Furthermore, the fibers can be moved towards the trap
center iteratively while compensating for charges on the dielectric mirrors
via the compensation electrodes. Finally, the positioners allow us to change
the mirror separation of the FFPC inside vacuum and thus build cavities of
variable cavity length and waist, resulting in an adjustable coupling
parameter $g$.
### IV.4 Simulations
The trap potential of the Paul trap is characterized in numerical simulations
using CPOCPO and Matlab. CPO solves the electromagnetic field equations for
each electrode, which are then combined with Matlab to give the net potential
over the trapping region. The fibers are included in the simulations as
dielectric cylinders. These simulations were used to determine trap depth and
frequencies and to optimize the trap geometry.
Figure 12: (a) Simulation of the trap potential of the miniaturized linear
Paul trap. The radio-frequency (RF) and ground (GND) blades as well as the
dielectric fibers are indicated in black. The potential in the plane
perpendicular to the trap axis is plotted in eV. The fibers are separated by
$200~{}\mu$m. (b) Trap potential without fibers. (c) Trap potential of a
symmetric trap with $90^{\circ}$ angles between RF and ground blades and blade
separations of $1.6$ mm.
We calculate the trap potential of the trap described in Sec. IV.2.1 for the
following parameters: RF amplitude of $130$ V, RF frequency of $2\pi\times 35$
MHz, and tip electrodes at $200$ V. As we cannot predict the amount of charge
on the fibers, we assume it to be zero in the simulations. Fig. 12(a) shows
the potential in the radial direction. The radial trap frequencies are
$2\pi\times 9.9$ MHz and $2\pi\times 9.6$ MHz, the axial trap frequency is
$1.9$ MHz, and the trap depth is $1.3$ eV.
In Fig. 12(b), the potential of the same trap without fibers is shown. In the
absence of the dielectric fibers, the trap depth is now $2$ eV instead of
$1.3$ eV.
Changing the blade angle from the asymmetric case of $120^{\circ}$ and
$60^{\circ}$ between RF and ground blades to angles of $90^{\circ}$ results in
a increase of trap depth of the radio frequency potential by a factor of
three. At the same time, however, the influence of the DC endcap potential
decreases, and therefore the overall trap depth does not change significantly.
The radial symmetry of the trap is no longer broken and the radial trap
frequencies are degenerate.
The trap implemented in the experimental setup has blade separations that are
five times smaller than those in the trap of Ref. Riebe (2005). To maintain a
stable trapping configuration in the smaller trap, either the radio-frequency
amplitude has to be decreased or its frequency has to be increased. Both
result in a lower trap depth: the depth of the miniaturized trap is an order
of magnitude smaller. Fig. 12(c) shows the potential of the trap from Ref.
Riebe (2005), in which the blade separations along the diagonal are $1.6$ mm,
all angles between RF and ground blades are $90^{\circ}$, and the trap depth
is $20$ eV.
The simulations confirm that the trapping potential of the trap implemented in
the setup has suitable trap depth and trap frequencies. Although not as deep
as a conventional Paul trap, it is nevertheless an order of magnitude deeper
than typical planar ion traps, giving this geometry a clear advantage over
planar designs.
## V Summary and outlook
We have built an ion-trap apparatus with an integrated FFPC of tunable length
and thus of tunable coupling parameter $g$. In the process, we have set up and
tested a miniature linear Paul trap for 40Ca+. Furthermore, we have extended
the parameter regime for CO2-laser shaped fiber tips in order to build high-
finesse FFPCs of up to $350~{}\mu$m in length, compatible with the integration
of an ion trap. We have performed several experiments to characterize the
properties of FFPCs, which we expect will further the development of this new
technology. We describe several techniques for handling mirror-coated fibers
and discuss methods to improve the fiber-mirror finesse.
Our apparatus is intended for performing cavity QED experiments. In
particular, we would like to demonstrate strong coupling between an ion and a
photon, in which $g\gg\kappa,\gamma$ Kimble (2008). We have calculated that
our system enters this regime for a cavity in near-concentric configuration.
Similar to the cavity described in Ref. Stute _et al._ (2012a), the fiber
cavity can be tuned into resonance with the $P$ to $D$ transition in 40Ca+,
where $D$ is a metastable state used as a qubit state. With a drive laser
coupling the electronic ground state $S$ and the intermediate state $P$, a
vacuum-stimulated Raman transition from $S$ to $D$ produces single photons
inside the cavity. An ion coupled to an FFPC would enable a coherent
interaction rate $g$ that dominates over the ion’s spontaneous decay rate
$\gamma$, providing access to a new experimental regime for ion-trap cavity
QED experiments.
###### Acknowledgements.
The authors would like to thank Ramin Lalezari from ATFilms for valuable
discussions on mirror coatings; Frederik Klauser from the Institute of
Physical Chemistry of the University of Innsbruck for his aid with the XPS
measurements; Stefan Haslwantler, Johannes Ghetta, Ben Ames, and Jasleen
Lugani for valuable discussions and aid in the construction of the FFPCs and
the experimental apparatus; and the mechanical workshop of the Institute of
Experimental Physics at the University of Innsbruck. We gratefully acknowledge
support from the Austrian Science Fund (FWF): Project. Nos. F4003 and F4019,
the European Research Council through the CRYTERION Project, the European
Commission via the Atomic QUantum TEchnologies (AQUTE) Integrating Project,
and the Institut für Quanteninformation GmbH.
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|
arxiv-papers
| 2013-11-27T13:10:02 |
2024-09-04T02:49:54.348839
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Birgit Brandst\\\"atter, Andrew McClung, Klemens Sch\\\"uppert, Bernardo\n Casabone, Konstantin Friebe, Andreas Stute, Piet O. Schmidt, Christian\n Deutsch, Jakob Reichel, Rainer Blatt, Tracy E. Northup",
"submitter": "Birgit Brandstaetter",
"url": "https://arxiv.org/abs/1311.6961"
}
|
1311.7036
|
Testing cosmological models with the brightness profile of distant
galaxies–References
# Testing cosmological models with the brightness profile of distant galaxies
I. Olivares-Salaverri1 and M. B. Ribeiro2
1Observatório do Valongo Universidade Federal do Rio de Janeiro Brazil
2Instituto de Física Universidade Federal do Rio de Janeiro Brazil
(2013)
###### Abstract
The goal of this work is to use observed galaxy surface brightness profiles at
high redshifts to determine, among a few candidates, the cosmological model
best suited to interpret these observations. Theoretical predictions of
galactic surface brightness profiles are compared to observational data in two
cosmological models, $\Lambda$CDM and Einstein-de Sitter, to calculate the
evolutionary effects of different spacetime geometries in these profiles in
order to try to find out if the available data is capable of indicating the
cosmology that most adequately represents actual galactic brightness profiles
observations. Starting from the connection between the angular diameter
distance and the galactic surface brightness as advanced by Ellis and Perry
(1979), we derived scaling relations using data from the Virgo galactic
cluster in order to obtain theoretical predictions of the galactic surface
brightness modeled by the Sérsic profile at redshift values equal to a sample
of galaxies in the range $1.5\lesssim z\lesssim 2.3$ composed by a subset of
Szomoru’s et al. (2012) observations. We then calculated the difference
between theory and observation in order to determine the changes required in
the effective radius and effective surface brightness so that the observed
galaxies may evolve to have features similar to the Virgo cluster ones. Our
results show that within the data uncertainties of this particular subset of
galaxies it is not possible to distinguish which of the two cosmological
models used here predicts theoretical curves in better agreement with the
observed ones, that is, one cannot identify a clear and detectable difference
in galactic evolution incurred by the galaxies of our sample when applying
each cosmology. We also concluded that the Sérsic index $n$ does not seem to
play a significant effect in the evolution of these galaxies. Further
developments of the methodology employed here to test cosmological models are
also discussed.
###### keywords:
cosmology: theory - galaxies: distances and redshifts, structure, evolution
## 1 Introduction
Observational cosmology attempts to understand the large-scale matter
distribution in the universe and its geometry by basically following two
different methodologies. The first, known as the direct-manner, or data-
driven, approach, seeks to describe what is actually observed without
addressing the question of why we observe such an universe the way it is,
whereas the theory-based, or model-based, one interprets the observations
based on explanations that can produce the observed patterns (Ellis 2006).
The theory-based approach consists of assuming a model based on a spacetime
geometry and then determines the values of the free parameters by comparing
the theoretical predictions with astronomical observations of distant objects.
Currently, the most accepted cosmological model, the $\Lambda$CDM cosmology,
is based on this theory-based approach. It concludes that the universe is
almost entirely made up of dark matter and dark energy, whose compositions are
presently unknown.
The data-driven approach of observational cosmology claims that we are in
principle capable of determining the spacetime geometry on the null cone by
means of astrophysical observations, that is, using data available on the past
null cone (Kristian & Sachs 1966; Ellis et al. 1985). One specific study that
follows this direct manner methodology was advanced by Ellis & Perry (1979),
who developed a very detailed discussion connecting the galactic brightness
profiles with cosmological models. Their aim was to determine the spacetime
geometry of the universe by measuring the angular diameter distance
$d_{\scriptscriptstyle A}$, also known as area distance, of distant galaxies
through their surface brightness photometric data. Such a task was, however,
made very difficult due to lack of detailed knowledge about the structure and
evolution of galaxies. To this day this difficulty still remains.
Here we propose a method for testing cosmological models partially based on
Ellis & Perry (1979) methodology, but less ambitious than theirs. Our approach
differs from these authors in the sense that we do not aim to determine the
entire spacetime geometry from observations. Our goal is to seek consistency
between detailed astronomical observations of quantities describing actual
galactic structure as compared to their predictions made by a specific
cosmological model. To be precise, we start by assuming a cosmological model
and then discuss the consistency between the model’s predictions and actual
observations of the surface brightness of distant galaxies and other related
quantities. The idea is to obtain a glimpse of the structure and evolution of
galaxies by means of observations of our local universe, identify observing
parameters which, in principle, are cosmological-model independent, that is,
independent of the spacetime geometry, and then assume that galactic scaling
relations do not change significantly, say within $3\sigma$, with the
redshift. In this way we could select distant galaxies that form a homogeneous
class of objects, defined as a set of similar galactic properties which can be
found at different galactic evolutionary stages, such as morphology, so that
one can compare objects at different redshift values. This implies in assuming
that the variations in the scaling relations are related to the variations in
the intrinsic structural parameters of a previously selected homogeneous class
of objects (Ellis et al. 1984). By starting with a cosmological model we are
able to assess to what extent the assumed cosmology affects actual galaxy
evolution modeling carried out in extragalactic astrophysics where some
cosmological model, nowadays the $\Lambda$CDM, is implicitly assumed.
In this paper we address the first step in this approach of cosmological model
testing using actual galactic data, whose basic methodology was briefly
advanced elsewhere (Olivares-Salaverri & Ribeiro 2009, 2010). The purpose here
is to verify if sample galactic brightness profiles vary with predicted
theoretical ones when one changes the cosmological model, that is, if surface
brightness profiles are affected by the spacetime geometry. We adopt two
distinct cosmologies, the $\Lambda$CDM and, for simplicity in this initial
approach, Einstein-de Sitter (EdS) model. We then calculate photometric
scaling relations using Virgo cluster galaxies by means of the Kormendy et al.
(2009) data and use our theory to predict the galactic surface brightness at
high redshift values in the two cosmological models, assuming that these
scaling relations do not change with the redshift. We then compare these
predictions with a subsample of high redshift galactic surface brightness data
of Szomoru et al. (2012; from now on S12).
Our results show that the observed high-redshift galactic brightness profiles
differ from the theoretically predicted ones obtained by assuming that they
follow scaling relations derived from the Virgo cluster. Such a difference
occurs when the theoretical results are obtained by using both cosmological
models studied here. Therefore, for these galaxies to change their features to
the ones found in the Virgo cluster, an intrinsic evolution must take place.
Such an evolution is similar in both cosmological models. Consequently, our
results do not allow us to conclude which cosmology produces more or less
evolution or is more suitable to represent the process in which high redshift
galaxies develop into local galaxies having scaling relations similar to the
ones observed in the Virgo galactic cluster, at least as far the chosen
particular subset of S12 galaxies is concerned.
The outline of the paper is as follow. In §2 we discuss cosmological distance
measures and their connections to astrophysical observables and §3 shows how
the surface brightness of cosmological sources are connected to those distance
measures. We present in §4 the received surface brightness using the profile
due to Sérsic (1968). In §5 we calculate two photometric scaling relations of
the Virgo cluster so that the next section (§6) shows our comparison, using
the two cosmologies assumed here, between the prediction of the surface
brightness obtained by means of these scaling relations and the observations
of S12 high redshift galaxies. In §7 we calculate in both cosmological models
how galaxies whose redshift values are equal to the ones in our chosen
subsample of S12 observations would have to evolve to end up with features
similar to the galaxies in the Virgo cluster. Finally, in §8 we summarize the
results and present our conclusions.
## 2 Cosmological distances
We start by considering that source and observer are at relative motion to one
another. From the point of view of the source, the light beams that travel
along future null geodesics define a solid angle
$\mathrm{d}\Omega_{\scriptscriptstyle G}$ with the origin at the source and
have a transverse section area $\mathrm{d}\sigma_{\scriptscriptstyle G}$ at
the observer (Ellis 1971; see also Fig. 2 of Ribeiro 2005 where cosmological
distances are also discussed in some detail).
The specific radiative intensity $F_{em}$ is the emitted, or intrinsic,
radiation measured at the source in a unit 2-sphere $S_{unit}$ lying in the
locally Euclidean space at rest with the source and centered at it, also
assumed to radiate locally with spherical symmetry. It is related to the
intrinsic source luminosity $L$ by,
$L=\int_{S_{unit}}F_{em}{\mathrm{d}}\Omega_{\scriptscriptstyle G}=4\pi
F_{em}.$ (1)
Let us now define $F_{re}$ as the flux radiated by the source, but measured by
an observer located at some future time $t_{0}$ relative to the source. This
is, of course, the received flux in the area
$\mathrm{d}\sigma_{\scriptscriptstyle G}$ at rest with the observer and
implies a certain distance between source and observer, distance which is
geometrically defined along a null curve in an expanding spacetime where both
source and observer are located. Thus, the source luminosity is given by,
$L=\int_{S_{\\!\\!phys}}(1+z)^{2}F_{re}{\mathrm{d}}\sigma_{\scriptscriptstyle
G},$ (2)
where $z$ is the redshift and $S_{\\!\\!phys}$ is the physical surface
receiving the flux, e.g., a detector. The factor $(1+z)^{2}$ appears here
because both source and observer have their geometrical locus in a curved and
expanding spacetime (Ellis 1971). Now, it has long been known that the area
law establishes that the source intrinsic luminosity is independent from the
observer (Ellis 1971). Therefore, these two equations are equal, yielding,
$L=\int_{S}F_{em}\;{\mathrm{d}}\Omega_{\scriptscriptstyle
G}=\int_{S}(1+z)^{2}F_{re}\;{\mathrm{d}}\sigma_{\scriptscriptstyle G},$ (3)
$F_{em}\;{\mathrm{d}}\Omega_{\scriptscriptstyle
G}=const=(1+z)^{2}F_{re}\;{\mathrm{d}}\sigma_{\scriptscriptstyle G}.$ (4)
Considering the source’s viewpoint, we may define the galaxy area distance
$d_{\scriptscriptstyle G}$ as (Ellis 1971; see also Fig. 2 of Ribeiro 2005),
${\mathrm{d}}\sigma_{\scriptscriptstyle G}={d_{\scriptscriptstyle
G}}^{2}{\mathrm{d}}\Omega_{\scriptscriptstyle G}.$ (5)
Thus, equation (4) becomes,
$F_{re}=\frac{F_{em}}{{d_{\scriptscriptstyle
G}}^{2}(1+z)^{2}}=\frac{L}{4\pi}\frac{1}{{d_{\scriptscriptstyle
G}}^{2}(1+z)^{2}}.$ (6)
The factor $(1+z)^{2}$ can be understood as arising from (i) the energy loss
of each photon due to the redshift $z$, and (ii) the lower measured rate of
incoming photons due to time dilation (Ellis 1971). Since the galaxy area
distance $d_{\scriptscriptstyle G}$ appearing in equation (6) cannot be
measured as ${\mathrm{d}}\Omega_{\scriptscriptstyle G}$ is defined at the
source, we need to change this equation into another one containing measurable
quantities. This can be done as follows.
From the point of view of the observer, the light beams that travel along its
past null geodesics leave the source and converge to the observer, defining a
solid angle $d\Omega_{\scriptscriptstyle A}$ with the origin at the observer
and having transverse section area $d\sigma_{\scriptscriptstyle A}$ at the
source (Ellis 1971; see also Fig. 1 of Ribeiro 2005). Then we can define the
angular diameter distance $d_{\scriptscriptstyle A}$ as being given by,
${\mathrm{d}}\sigma_{\scriptscriptstyle A}={d_{\scriptscriptstyle
A}}^{2}{\mathrm{d}}\Omega_{\scriptscriptstyle A}.$ (7)
Now we can use the reciprocity theorem, due to Etherington (1933; see also
Ellis 1971, 2007), to relate $d_{\scriptscriptstyle G}$ to
$d_{\scriptscriptstyle A}$. This theorem is written as follows,
${d_{\scriptscriptstyle G}}^{2}=(1+z)^{2}{d_{\scriptscriptstyle A}}^{2}.$ (8)
Thus, it is now possible to connect the flux received by the observer and the
angular diameter distance by combining equations (6) and (8), yielding
$F_{re}=\frac{F_{em}}{{d_{\scriptscriptstyle A}}^{2}(1+z)^{4}}.$ (9)
The received flux $F_{re}$ and the redshift $z$ are astronomically measurable
quantities. So, if the angular diameter distance $d_{\scriptscriptstyle A}$ is
somehow determined astronomically, or obtained from theory as a function of
$z$, then the intrinsic flux $F_{em}$ and, therefore, the intrinsic luminosity
$L$ are both determined for all redshifts.
## 3 Connection with the surface photometry of cosmological sources
As discussed in §2, equation (9) connects the received and emitted fluxes of
sources located in a curved spacetime, but that expression is valid for point
like sources. Galaxies, however, form extended sources of light and their
characterization requires defining another quantity, the surface brightness,
better suited to describe them. The received surface brightness $B_{re}$ is
defined as the ratio between the received flux and the observed solid angle of
the galaxy,
$B_{re}\equiv\frac{F_{re}}{{\mathrm{d}}\Omega_{\scriptscriptstyle A}}.$ (10)
Considering equations (7) and (9), this expression can be rewritten as,
$B_{re}=\frac{F_{em}}{{\mathrm{d}}\sigma_{\scriptscriptstyle
A}}\frac{1}{(1+z)^{4}}.$ (11)
If we define the emitted surface brightness $B_{em}$ as the intrinsic flux of
the source $F_{em}$ per area unit in the rest frame of the source, we have
that
$B_{em}\equiv\frac{F_{em}}{{\mathrm{d}}\sigma{\scriptscriptstyle A}}.$ (12)
Thus, equation (11) in fact connects the received and emitted surface
brightness, as follows,
$B_{re}=\frac{B_{em}}{(1+z)^{4}}.$ (13)
This expression is simply the so-called Tolman surface brightness test for
cosmological sources, showing that galactic surface brightness does not depend
on the distance. This equation also shows that if there is no significant
cosmological effects, that is, if source and observer are close enough to be
considered at rest with one another and Newtonian approximation is valid, then
the source redshift is not significant. In this case there is no cosmological
contribution ($z\sim 0$) and $B_{re}=B_{em}$. This can be considered as a
consequence of the Liouville theorem (Bradt 2004). It is also worth mentioning
that several authors name the radiation measured by the observer as intensity
$I$, and the radiation emitted by the source as surface brightness $B$,
instead of terms adopted here, respectively, received surface brightness
$B_{re}$ and emitted surface brightness $B_{em}$. This is often the case in
texts where General Relativity is not considered.
Actual astronomical observations are carried out in observational bandwidths
and, therefore, the equations discussed so far should take this fact into
account. The specific received surface brightness $B_{re,\nu_{re}}$ gives the
amount of radiation received by the observer per unit solid angle measured at
the observer in the frequency range $\nu_{re}$ and
$\nu_{re}+{\mathrm{d}}\nu_{re}$. Clearly
$B_{re}=\int_{0}^{\infty}B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}$. Considering
equation (13), we have that,
$B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}=\frac{B_{em,\nu_{em}}}{(1+z)^{4}}\,{\mathrm{d}}\nu_{em},$
(14)
where we had defined the specific emitted surface brightness as follows,
$B_{em,\nu_{em}}=B_{em}J(\nu_{em}).$ (15)
Here $J(\nu)$ is the galactic spectral energy distribution (SED) giving the
proportion of radiation at each frequency, being normalized by the condition
$\int_{0}^{\infty}J(\nu)\,{\mathrm{d}}\nu=1.$ (16)
From our definitions it also follows that
$B_{em}=\int_{0}^{\infty}B_{em,\nu_{em}}{\mathrm{d}}\nu_{em}$. We need now to
relate the received and emitted frequencies. This is accomplished by the
definition of the redshift,
$\nu_{em}=\nu_{re}(1+z),$ (17)
implying that the SED of the source is observed according to
$J(\nu_{em})=J\left[\nu_{re}(1+z)\right]$ and equation (14) can be rewritten
as follows,
$B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}=\frac{B_{em,\nu_{em}}}{(1+z)^{3}}\,{\mathrm{d}}\nu_{re}.$
(18)
The variables in the equation above depend on some implicit parameters. In
order to reveal these dependencies, let us start by assuming our galaxy as
having spherical symmetry with space points defined by the radius $R$.
Furthermore, if this galaxy has a circular projection in the celestial sphere,
any angle $\alpha$ measured by the observer corresponds to the radius $R$ in
the source by means of the angular diameter distance at a given redshift.
Hence, we have that (see Ellis & Perry 1979, Fig. 1),
$R(z)=d_{\scriptscriptstyle A}(z)\;\alpha.$ (19)
This expression is in fact a simplification of equation (7) where area and
solid angle are respectively approximated to length and angle so that
$d_{\scriptscriptstyle A}$ can be estimated observationally (Ellis 1971,
Ribeiro 2005). Indeed, it is used in observational cosmology tests under the
name “angular diameter redshift relation” since the angular diameter distance
has all cosmological information. So, different values of the angular diameter
distance are related to different cosmological models. As mentioned above, in
this paper the two chosen cosmological models are EdS and $\Lambda$CDM.
The EdS cosmology has zero curvature and no cosmological constant so, in this
model the angular diameter distance may be written as below,
$d_{\scriptscriptstyle
A,EdS}(z)=\frac{2c}{H_{0}(1+z)}\biggl{[}1-\frac{1}{\sqrt{(1+z)}}\biggr{]},$
(20)
where $c$ is the light speed and $H_{0}$ is the Hubble constant. This
expression shows that $d_{\scriptscriptstyle A,EdS}$ reaches a maximum value
at $z=1.25$ and then starts decreasing, asymptotically vanishing at the big
bang singularity hypersurface.
In the case of the $\Lambda$CDM model, several tests have been carried out in
the last few years to measure the parameter values of the model leading to
such a degree of accuracy that it became known as the concordance model, being
the most accepted cosmology nowadays. Those tests involve studies of the
cosmic microwave background radiation, baryonic acoustic oscillations and type
Ia supernovae. Komatsu et al. (2009) presented values for several parameters
in this cosmology to a high degree of accuracy, such as $H_{0}=71.8$ km s-1
Mpc-1, $\Omega_{m_{0}}=0.273$ and $\Omega_{\Lambda}=0.727$. We used these
values to calculate numerically the angular diameter distance in both
cosmologies, that is, $d_{\scriptscriptstyle A,\Lambda CDM}$ and
$d_{\scriptscriptstyle A,EdS}$, where the latter is evaluated directly from
equation (20).
Returning to equation (18), other implicit parameter dependence also occurs in
both specific surface brightness. The received one depends on the observed
parameters $\alpha$, $z$ and $\nu_{re}$, so that we should write it as
$B_{re,\nu_{re}}=B_{re,\nu_{re}}(\alpha,z)$. The specific emitted surface
brightness depends on the source parameters $R$, $\nu_{em}$ and, if allowed
for the intrinsic evolution of the source, also in the $z$. Thus,
$B_{em,\nu_{em}}=B_{em}(R,z)\,J[\nu_{re}(1+z),R,z]$. With these dependencies,
equation (18) turns out to be written as the expression below (Ellis & Perry
1979),
$B_{re,\nu_{re}}(\alpha,z)=\frac{B_{em}(R,z)}{(1+z)^{3}}\,J\left[\nu_{re}(1+z),R,z\right].$
(21)
Note that this equation is completely general, i.e., valid for any
cosmological model.
Next we shall show how the surface brightness $B_{em}(R,z)$ can be
characterized by means of the Sérsic (1968) profile and obtain an explicit
expression for the received surface brightness.
## 4 Received Sérsic surface brightness
The Sérsic profile was not commonly considered among astronomers after its
proposal. Gradually, however, some authors started to claim that the Sérsic
index is not simply a parameter capable of providing a better mathematical
fit, but that it does have a physical meaning (Ciotti 1991; Caon et al. 1993;
D’Onofrio et al. 1994). Nowadays, this profile seems to be more accepted since
one can find in the recent literature several papers using it as well as
relating its parameters to other astrophysical quantities (Davies et al. 1988;
Prugniel & Simien 1997; Ciotti & Bertin 1999; Trujillo et al. 2001; Mazure &
Capelato 2002; Graham 2001, 2002; Graham & Driver 2005; La Barbera et al.
2005; Coppola et al. 2009; Chakrabarty & Jackson 2009; Laurikainen et al.
2010). In view of this, we believe that this profile is the most suitable for
the purposes of this paper.
The Sérsic profile can be presented in two slightly different parametric
formats, although both of them characterize the same brightness profile. The
difference lies in the interpretation of the parameters. The first one can be
written as,
$B_{\mathrm{S_{1}},em}(R,z)=B_{0}(z)\,\exp\left\\{-\left[\frac{R(z)}{a(z)}\right]^{1/n}\right\\},$
(22)
where $B_{0}$ is the brightness amplitude, $a$ is the scalar radius and $n$ is
the Sérsic index. Ellis & Perry (1979) implicitly used this form. The second
way of writing the Sérsic profile is given by the following expression,
$B_{\mathrm{S_{2}},em}(R,z)=B_{eff}(z)\,\exp\left\\{\displaystyle-
b_{n}\left[\left(\frac{R(z)}{R_{eff}(z)}\right)^{1/n}-1\right]\right\\},$ (23)
where $R_{eff}(z)$ is the effective radius, $B_{eff}(z)$ is the brightness at
the effective radius and $b_{n}$ is a parameter dependent on the value of $n$.
The main difference between these two equations is the nature of their
parameters. In equation (22) $B_{0}$ and $a$ do not have a clear physical
meaning, whereas the parameters $B_{\scriptscriptstyle{eff}}$ and
$R_{\scriptscriptstyle{eff}}$ appearing in equation (23) are more easily
interpreted. $R_{\scriptscriptstyle{eff}}$ is defined as the isophote that
contains half of the total luminosity and $B_{\scriptscriptstyle{eff}}$ is the
value of the brightness in that radius (Ciotti 1991; Caon et al. 1993). For
this reason we believe that the second form above is more appropriate for our
analysis and we shall use it from now on.
A relationship between the different parameters in equations (22) and (23) can
be obtained by equating these two expressions, yielding,
$B_{0}(z)=B_{\scriptscriptstyle eff}(z)\,e^{b_{n}},$ (24)
$a(z)=\frac{R_{\scriptscriptstyle eff}(z)}{{b_{n}}^{n}}.$ (25)
The evolution of the galactic structure is implicit in this profile in view of
the fact that both $B_{eff}(z)$ and $R_{eff}(z)$ are redshift dependent. In
addition, the connection of this profile to a cosmological model and,
therefore, to the underlying curved spacetime geometry and its evolution
occurs in the intrinsic radius $R(z)$ by means of equation (19). Thus,
galaxies can possibly appear to experience two simultaneous evolutionary
effects, intrinsic source evolution and cosmological, or geometrical,
evolution. Since at our current knowledge of galactic structure both effects
cannot be easily separated, if they can be separated at all, from now on we
shall assume that $R(z)$ depends only on the underlying spacetime geometry
given by a chosen cosmological model. Furthermore, at first we shall not
consider a possible intrinsic evolution of the Sérsic index in the form
$n=n(z)$, because we assume that its change is produced via galactic merger
processes (Naab & Trujillo 2006). So, the way that the evolutionary dependency
is defined in the emitted surface brightness (eq. 23) means that we are
implicitly considering galaxies belonging to a set with similar properties, or
a homogeneous class of objects. Departures from this class occur by smooth
dependency in the evolution of the intrinsic parameters such as $B_{eff}(z)$
and $R_{eff}(z)$ (Ellis et al. 1984).
Let us now return to the properties of the Sérsic profile. There are
analytical and exact expressions relating to $b_{n}$ and $n$. Analytical
expressions for $b_{n}$ were given by several authors (Capaccioli 1989; Ciotti
1991; Prugniel & Simien 1997), whereas others worked out exact values for this
parameter (Ciotti 1991; Graham & Driver 2005; Mazure & Capelato 2002). Ciotti
& Bertin (1999) analyzed the exact value form and concluded that a fourth
order expansion is enough to obtain good results, which are even better than
the values obtained from the analytical expressions. Such an expansion is
enough for the purposes of this paper and may be written as below,
$b_{n}=2n-\frac{1}{3}+\frac{4}{405n}+\frac{46}{25515n^{2}}.$ (26)
Having expressed $B_{em}$ in terms of the Sérsic profile, we can now obtain
the equation for the received surface brightness $B_{re}(\alpha,z)$ since
equation (21) gives the relationship between the emitted and received surface
brightness. Considering the emitted brightness as modeled by the second form
of the Sérsic profile (eq. 23), we then substitute the latter equation into
the former and obtain the following expression,
$\displaystyle B_{re,\nu_{re}}(\alpha,z)$ $\displaystyle=$
$\displaystyle\frac{B_{eff}(z)}{(1+z)^{3}}\;J[\nu_{re}(1+z),R,z]\times$ (27)
$\displaystyle\times\exp{\left\\{-b_{n}\left[{\left(\frac{R(z)}{R_{eff}(z)}\right)}^{1/n}-1\right]\right\\}}.$
Let us now define two auxiliary quantities required to study this problem from
an extragalactic point of view (Caon et al. 1993; Graham & Driver 2005),
$\mu_{re,\nu_{re}}(\alpha,z)\equiv-2.5\log\left(B_{re,\nu_{re}}\right),$ (28)
$\mu_{eff}(z)\equiv-2.5\log\left[B_{eff}(z)\right].$ (29)
Equation (28) is given in units of $[\mu_{re,\nu_{re}}]$ = [mag/arc sec2].
Considering these definitions, equation (27) can be rewritten as,
$\displaystyle\mu_{re,\nu_{re}}(\alpha,z)$ $\displaystyle=$
$\displaystyle\mu_{eff}(z)+7.5\log(1+z)$ (30)
$\displaystyle-2.5\log\left\\{J\left[\nu_{re}(1+z),R,z\right]\right\\}$
$\displaystyle+\left[\frac{2.5}{\ln(10)}\right]b_{n}\left\\{\left[\frac{R(z)}{R_{eff}(z)}\right]^{1/n}-1\right\\}.$
The equation above allows us to predict the galactic surface brightness in a
given redshift and in a specific cosmological model. Nevertheless, in order to
relate this expression with actual observations, we still need further
assumptions regarding galactic structure. Next we shall make use of Kormendy
et al. (2009) data to derive photometric scaling relations of the Virgo galaxy
cluster and suppose that these relations do not change drastically out to the
redshift range under study here.
## 5 Photometric scaling relations of the Virgo Cluster
In order to compare theoretical predictions of the surface brightness profiles
with observational data we require information regarding the galactic
structure, information which can be obtained after investigating scaling
relations that different kinds of galaxies follow in the local universe. The
most well-known of those relations are the ones that relate luminosity with
velocity dispersion, such as the Faber-Jackson relation (Faber & Jackson 1976)
for elliptical galaxies, and the Tully-Fisher relation (Tully & Fisher 1977)
for spiral galaxies. Another, more general, scaling relation is the
fundamental plane (Djorgovski & Davis 1987) which correlates velocity
dispersion with effective brightness and effective radius instead of the
luminosity. It is worth mentioning that the fundamental plane is a
generalization of the Kormendy relation (Kormendy 1977).
The parameters involved in these scaling relations are measured through
spectroscopic and photometric techniques. After knowing that the Sérsic index
and the velocity dispersion correlate, a fundamental plane that only uses
photometric parameters was proposed by Graham (2002). This variant, called
photometric plane, uses parameters which appear in the definition of the
Sérsic profile, i.e., the Sérsic index $n$, the effective surface brightness
$\mu_{eff}$ and the effective radius $R_{eff}$. Since the study presented in
this paper deals with photometric parameters, we shall use the photometric
plane to obtain the galactic scaling relations required in our analysis.
Specifically, our focus will be on the galaxies belonging to the Virgo cluster
as their data (Kormendy et al. 2009) form the most exhaustively studied
galactic surface brightness dataset in the local Universe.
### 5.1 Virgo cluster data
Kormendy et al. (2009) studied 42 galaxies of the Virgo cluster on the V band,
$\lambda_{V,eff}=(5450\pm 880)$ Å, whose morphological types are elliptic (E),
lenticular (S0) and spheroid (Sph). They used the Sérsic profile to fit the
observed surface brightness and, in the authors’ words, “the Sérsic functions
fit the main parts of the profiles of both elliptical and spheroidal galaxies
astonishingly well on large ranges in surface brightness.”
As the observed data of the Virgo cluster is well fitted by the Sérsic
profile, the parameters obtained through these fittings can be considered as
reliable. Thus, using the parameters involved in the photometric plane, $n$,
$\mu_{eff}$ and $R_{eff}$, we carried out linear fittings relating $\mu_{eff}$
to $n$ and $R_{eff}$ to $n$. Plots showing these fittings are presented in
Fig. 1 and the fitted parameters are as follows,
$\mu_{eff,V}=(0.38\pm 0.09)n+(20.5\pm 0.5),$ (31) $\log R_{eff,V}=(0.21\pm
0.03)n-(0.5\pm 0.1).$ (32)
The index $\scriptstyle V$ stands for the Virgo cluster.
Figure 1: Left: Plot of the logarithm of the effective radius $R_{eff,V}$ vs.
the Sérsic index $n$ of galaxies from the Virgo cluster. The straight line
represents the linear fit whose values are given in eq. (32). Right: Plot of
the effective brightness $\mu_{eff,V}$ and the Sérsic index $n$ of galaxies
belonging to the Virgo cluster. The straight line represents the linear fit
whose values are given in eq. (31). Both: Data taken from Kormendy et al.
(2009).
Analyzing the plots we can observe that most galaxies with low Sérsic index do
not correlate. These galaxies are mostly of spheroidal types and are the main
contributors to the large errors in the Y-axis. As our aim is to compare the
prediction of the surface brightness with high-redshift observational data, if
we knew the morphological types of these high-redshift galaxies we could
select specific galaxy types to obtain the scaling relation. But, this is
usually not possible because galactic morphology at high redshifts is not as
well established as in the local universe. Considering such constraint, we
resorted on using galaxies of the Virgo cluster whose scaling relations were
just derived in order to predict the surface brightness at any redshift and
compare it with high redshift galaxy observations of S12. On top of the
scaling relations, this is done by employing the two cosmological models
adopted here.
## 6 High-redshift vs. predicted galactic surface brightness profiles in
$\Lambda$CDM and EdS cosmologies
In order to analyze the possible effects that different cosmological models
can have in the estimated galactic evolution, we shall compare theoretical
predictions of the surface brightness data with high-redshift galactic
brightness profiles. An important issue concerning high redshift galactic
surface brightness is the depth in radius of these images. Observing the
universe at high redshift involves lower resolution images, so obtaining a
good sample of the brightness profiles as complete as possible in radius is
not an easy task.
### 6.1 The high redshift galactic data of Szomoru et al. (2012)
S12 were mainly interested in investigating if the interpretation of the
compactness of high redshift galaxies was due to a lack of deep images in
radius which would lead to a misinterpretation of the compactness pattern.
They observed a stellar mass limited sample of 21 quiescent galaxies in the
redshift range $1.5<z<2.5$ having ${\cal{M}}_{*}>5.10^{10}{\cal{M}}_{\odot}$
and obtained their surface brightness using the Sérsic profile with high
values in radius. We chose S12 data for our purposes mainly because of these
features.
S12 used NIR data taken with HST WFC3 as part of the CANDELS survey.
Specifically, the surface brightness profiles were obtained in the $H_{160}$
band, $\lambda_{H,eff}=(13923\pm 3840)$ Å, that is, comparable with the $V$
band rest-frame in the redshift range under consideration. From the 21
galaxies we have selected a subsample having Sérsic indexes grouped in three
sets: $n\sim 1$, $n\sim 4$ and $n\sim 5$. The last two sets are more numerous
and are located in several redshift values, especially the group with $n\sim
5$. Table LABEL:table1 shows our subsample with the identification number in
the S12 catalog and their respective Sérsic indexes and the redshifts.
Table 1: Identification (ID) number, Sérsic index $n$ and redshift of the galaxy subsample selected from S12 and used in this paper. ID | $n$ | $z$
---|---|---
2.856 | 1.20 $\pm$ 0.08 | 1.759
3.548 | 3.75 $\pm$ 0.48 | 1.500
2.531 | 4.08 $\pm$ 0.30 | 1.598
3.242 | 4.17 $\pm$ 0.45 | 1.910
3.829 | 4.24 $\pm$ 1.15 | 1.924
1.971 | 5.07 $\pm$ 0.31 | 1.608
3.119 | 5.09 $\pm$ 0.60 | 2.349
6.097 | 5.26 $\pm$ 0.56 | 1.903
1.088 | 5.50 $\pm$ 0.67 | 1.752
### 6.2 Theoretical predictions of the surface brightness
As discussed above, equation (30) allows us to calculate the surface
brightness theoretical prediction of a hypothetical galaxy in a given
redshift. We have already calculated scaling relations for local galaxies
which we assume to be maintained out to the observed S12 redshift values.
However, to see if the observed high redshift galaxies behave like the Virgo
cluster ones, i.e., if they obey the assumed scaling relations, we must
calculate the errors of the theoretical predictions in eq. (30).
Errors for $\Delta\mu_{re,\nu_{re}}$ are estimated quadratically as follows,
$\displaystyle\Delta\mu_{re,\nu_{re}}$ $\displaystyle=$
$\displaystyle\biggl{[}\biggl{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial\mu_{eff}}\Delta\mu_{eff}\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial
z}\Delta z\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial
n}\Delta n\biggr{)}^{2}$ (33)
$\displaystyle+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial R}\Delta
R\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial\log(R_{eff})}\Delta\log(R_{eff})\biggr{)}^{2}\biggl{]}^{1/2}.$
Let us analyze individually each of the uncertainty terms in this expression.
#### 6.2.1 $\Delta\mu_{eff}$
The effective surface brightness is calculated from the linear fit of the
Virgo cluster galaxy scaling relation,
$\mu_{eff}=A_{\mu_{eff}}n+B_{\mu_{eff}},$ (34)
where $A$ and $B$ are the linear fit parameters given in equation (31). Its
uncertainty yields,
$\Delta\mu_{eff}=\biggl{[}\biggl{(}\frac{\partial\mu_{eff}}{\partial
A_{\mu_{eff}}}\Delta
A_{\mu_{eff}}\biggr{)}^{2}+\biggl{(}\frac{\partial\mu_{eff}}{\partial n}\Delta
n\biggr{)}^{2}+\biggl{(}\frac{\partial\mu_{eff}}{\partial B_{\mu_{eff}}}\Delta
B_{\mu_{eff}}\biggr{)}^{2}\biggr{]}^{1/2}.$ (35)
This expression can can be rewritten as below,
$\Delta\mu_{eff}=\biggl{[}(n\Delta A_{\mu_{eff}})^{2}+(A_{\mu_{eff}}\Delta
n)^{2}+(\Delta B_{\mu_{eff}})^{2}\biggr{]}^{1/2},$ (36)
where $\Delta A$ and $\Delta B$ come from the linear fit and $\Delta n$ comes
from the observation.
#### 6.2.2 $\Delta z$
Redshift uncertainty was given by S12 only if $z$ was measured
photometrically. However, some of S12’s galaxies had their redshifts measured
spectroscopically, then the errors were not made available. Since there is a
mixture of photometric and spectroscopic redshifts in S12’s sample, we decided
to avoid including them in our calculations and have effectively assumed
$\Delta z\sim 0$.
#### 6.2.3 $\Delta n$
The value of Sérsic index and its error came directly from the observations
shown in S12 (see Table LABEL:table1).
#### 6.2.4 $\Delta R$
The projected radius is obtained by means of equation (19), whose quadratic
uncertainty may be written as follows,
$\Delta R=\biggl{[}\biggl{(}\frac{\partial
R}{\partial\alpha}\Delta\alpha\biggr{)}^{2}+\biggl{(}\frac{\partial
R}{\partial d_{A}}\Delta d_{A}\biggr{)}^{2}\biggr{]}^{1/2}.$ (37)
The angle uncertainty $\Delta\alpha$ was neglected by S12, so
$\Delta\alpha\sim 0$. Regarding the angular diameter distance, the small
uncertainties in the parameters of the cosmological models are such that once
propagated they change very little the value of $d_{A}$. So, we effectively
have $\Delta d_{A}\sim 0$.
#### 6.2.5 $\Delta\log(R_{eff})$
Similarly to $\Delta\mu_{eff}$, the uncertainty $\Delta\log(R_{eff})$ is
derived from the linear fit of the Virgo galaxy cluster,
$\log(R_{eff})=A_{\log(R_{eff})}n+B_{\log(R_{eff})},$ (38)
where $A$ and $B$ are the parameters given by equation (32). Therefore,
$\displaystyle\Delta\log(R_{eff})$ $\displaystyle=$
$\displaystyle\biggl{\\{}\biggl{[}\frac{\partial\log(R_{eff})}{\partial
A_{\log(R_{eff})}}\Delta
A_{\log(R_{eff})}\biggr{]}^{2}+\biggl{[}\frac{\partial\log(R_{eff})}{\partial
n}\Delta n\biggr{]}^{2}$ (39)
$\displaystyle+\biggl{[}\frac{\partial\log(R_{eff})}{\partial
B_{\log(R_{eff})}}\Delta B_{\log(R_{eff})}\biggr{]}^{2}\biggr{\\}}^{1/2},$
which may be rewritten as,
$\Delta\log(R_{eff})=\biggl{\\{}\bigl{[}n\Delta
A_{\log(R_{eff})}\bigr{]}^{2}+(A\Delta n)^{2}+\bigl{[}\Delta
B_{\log(R_{eff})}\bigr{]}^{2}\biggr{\\}}^{1/2}.$ (40)
Just like in $\Delta\mu_{eff}$, $\Delta A_{\log(R_{eff})}$ and $\Delta
B_{\log(R_{eff})}$ come from the linear fit and $\Delta n$ is given by
observations.
#### 6.2.6 $\Delta\mu_{re,\nu_{re}}$
Putting together all these expressions for uncertainties in equation (33) and
remembering the relationship between $b_{n}$ and $n$ (eq. 26), the uncertainty
in the theoretical prediction of the surface brightness yields,
$\displaystyle\Delta\mu_{re,\nu_{re}}$ $\displaystyle=$
$\displaystyle\Biggl{(}(n\Delta A_{\mu_{eff}})^{2}+(A_{\mu_{eff}}\Delta
n)^{2}+(\Delta B_{\mu_{eff}})^{2}+$ (41)
$\displaystyle+\biggl{[}\biggl{(}\frac{2.5}{\ln
10}\frac{{\mathrm{d}}b_{n}}{{\mathrm{d}}n}\biggl{\\{}\bigg{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}-1\biggr{\\}}-$
$\displaystyle-\frac{2.5b_{n}}{n^{2}\ln
10}\biggl{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}\log\biggl{\\{}\frac{R}{10^{\log(R_{eff})}}\biggr{\\}}\biggr{)}\Delta
n\biggr{]}^{2}+$
$\displaystyle+\biggl{\\{}\frac{2.5b_{n}}{n}\biggl{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}\biggr{\\}}^{2}\biggl{\\{}[n\Delta
A_{\log(R_{eff})}]^{2}+$ $\displaystyle+[A_{\log(R_{eff})}\Delta
n]^{2}+[\Delta B_{\log(R_{eff})}]^{2}\biggr{\\}}\Biggr{)}^{1/2}.$
#### 6.2.7 Comparing theory and observation
Before we can actually compare our S12 galaxy subsample with the theoretical
predictions of the surface brightness profile, we still need to estimate the
spectral energy distribution (SED) $J$. We proceed on this point from the very
simple working assumption of a constant value for the SED, since using the
overall energy distribution of the galaxies in the Virgo cluster in different
bandwidths is beyond the aims of this paper. Henceforth, we assume the working
value of $J=0.5$.
The scaling relations obtained from the galaxies of the Virgo cluster, eqs.
(31) and (32), allow us to calculate $\mu_{eff}$ and $R_{eff}$ for a given
Sérsic index value. So, these two equations may be rewritten as,
$\mu_{eff,V}=0.38n+20.5,$ (42)
where its error is given by,
$\Delta\mu_{eff,V}=0.09n+0.5,$ (43)
and
$\log R_{eff,V}=0.21n-0.5,$ (44)
whose uncertainty is,
$\Delta\log R_{eff,V}=0.03n+0.1.$ (45)
Graphs showing the S12 data, labeled as “Obs”, and the theoretical predictions
of the surface brightness profile, labeled as “Pre”, in each of the two
cosmological models adopted in this work, $\Lambda$CDM and Einstein-de Sitter,
are shown in Figs. 2 to 11. One can clearly see that the observed and
predicted results are very different, a result which shows that the Virgo
cluster galaxies do not behave as our S12 subsample. So, for our high redshift
galaxies to evolve into the Virgo cluster ones the scaling relations
parameters have to change in order to reflect such an evolution. One can also
see that the difference in the results when employing each of the two
cosmological models is not at all significant. Thus, our next step is to work
out the changes in the parameter to find out if the evolution these galaxies
have to sustain is strongly dependent on the assumed underlying cosmology.
Figure 2: Both: galaxy 2.856 from S12 labeled as “Obs” at $z=1.759$ with
$n=1.2\pm 0.08$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 3: Both: galaxy 2.531 galaxy from S12 labeled as “Obs” at $z=1.598$
with $n=4.08\pm 0.3$. Theoretical prediction using eqs. (30) and (41) assuming
the scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 4: Both: galaxy 3.242 from S12 labeled as “Obs” at $z=2.47$ with
$n=4.17\pm 0.45$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 5: Both: galaxy 3.548 from S12 labeled as “Obs” at $z=1.5$ with
$n=3.75\pm 0.48$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 6: Both: galaxy 3.829 from S12 labeled as “Obs” at $z=1.924$ with
$n=4.24\pm 1.15$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 7: Both: galaxy 1.088 from S12 labeled as “Obs” at $z=1.752$ with
$n=5.5\pm 0.67$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre” points. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 8: Both: galaxy 1.971 from S12 labeled as “Obs” at $z=1.608$ with
$n=5.07\pm 0.31$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 9: Both: galaxy 2.514 from S12 labeled as “Obs” at $z=1.548$ with
$n=5.73\pm 0.93$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 10: Both: galaxy 3.119 from S12 labeled as “Obs” at $z=2.349$ with
$n=5.09\pm 0.6$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
Figure 11: Both: galaxy 6.097 from S12 labeled as “Obs” at $z=1.903$ with
$n=5.26\pm 0.56$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.
## 7 Evolution to the Virgo cluster scaling parameters in $\Lambda$CDM and
EdS cosmologies
As seen above, the theoretical prediction of the surface brightness profiles
obtained through scaling relations derived from data of the Virgo galactic
cluster are very different from the observed surface brightness of S12
galaxies in both cosmological models studied here. Hence, in order to
ascertain the possible evolution that these galaxies would have to experience
such that they end up with surface brightness equal to the ones in the Virgo
cluster, we need to look carefully at the parameter evolution of the scaling
relations. Assuming that they do not evolve through the Sérsic index $n$ (see
above), we can study such an evolution via the effective brightness and the
effective radius.
Let $\mu_{eff,evo}$ and $\log R_{eff,evo}$ be, respectively, the evolution of
the effective brightness and the effective radius. We can estimate these two
quantities similarly to our previous calculation of the scaling relations. For
values of the Sérsic index of our S12 galactic subsample, we can obtain their
respective Virgo cluster galaxies effective brightness and effective radius
$\mu_{eff,V}$ and $\log R_{eff,V}$ by means of the expressions (42) to (45).
We then modify the linear fit parameters $A$ and $B$ in these expressions
($y=Ax+B$), adjusting them so that the results approximate the points of the
theoretical predictions with S12’s observed values in order to find
$\mu_{eff,Szo}$ and $\log R_{eff,Szo}$, that is, the values of effective
brightness and effective radius of our subsample of S12’s galaxies. Therefore,
both $\mu_{eff,evo}$ and $\log R_{eff,evo}$ are given as follows,
$\mu_{eff,evo}=\mu_{eff,Szo}-\mu_{eff,V},$ (46) $\log R_{eff,evo}=\log
R_{eff,Szo}-\log R_{eff,V}.$ (47)
The respective uncertainties yield,
$\Delta\mu_{eff,evo}=\Delta\mu_{eff,Szo}+\Delta\mu_{eff,V},$ (48) $\Delta\log
R_{eff,evo}=\Delta\log R_{eff,Szo}+\Delta\log R_{eff,V}.$ (49)
Table LABEL:table2 shows the effective brightness and effective radius for the
Virgo cluster galaxies obtained with Sérsic indexes $n$ equal to those in our
S12 subsample presented in Table LABEL:table1. These quantities were obtained
using the Virgo scaling relations.
Table 2: Effective brightness $\mu_{eff,V}$ and effective radius $\log R_{eff,V}$ obtained with Virgo cluster galaxies scaling relations for values of $n$ equal to those in our subsample of S12 galaxies. ID | $\mu_{eff,V}$ | $\log R_{eff,V}$
---|---|---
2.856 | 21.0 $\pm$ 0.7 | -0.3 $\pm$ 0.2
3.548 | 21.9 $\pm$ 0.9 | 0.3 $\pm$ 0.3
2.531 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
3.242 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
3.829 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
1.971 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
3.119 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
6.097 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
1.088 | 23 $\pm$ 1 | 0.6 $\pm$ 0.3
Table LABEL:table3 shows the same quantities for the adjusted parameters of
our subsample of S12 galaxies using the two cosmological models considered
here and Table LABEL:table4 presents the evolution of the effective brightness
and effective radius calculated using eqs. (46) to (49) in both cosmological
models in the V band.
Table 3: Effective brightness $\mu_{eff,Szo}$ and effective radius $\log R_{eff,Szo}$ of the adjusted parameters from the galaxies selected from S12 for values of $n$ equal to the ID galaxies in the V band in the two cosmological models considered in this paper. ID | $\mu_{eff,Szo,\Lambda CDM}$ | $\log R_{eff,Szo,\Lambda CDM}$ | $\mu_{eff,Szo,EdS}$ | $\log R_{eff,Szo,EdS}$
---|---|---|---|---
2.856 | 16.5 $\pm$ 0.6 | -0.1 $\pm$ 0.1 | 16.7 $\pm$ 0.6 | -0.1 $\pm$ 0.1
3.548 | 16.5 $\pm$ 0.8 | -0.2 $\pm$ 0.3 | 16.4 $\pm$ 0.8 | -0.3 $\pm$ 0.2
2.531 | 16.7 $\pm$ 0.9 | 0.0 $\pm$ 0.3 | 17.1 $\pm$ 0.9 | -0.1 $\pm$ 0.3
3.242 | 14.9 $\pm$ 0.9 | -0.4 $\pm$ 0.3 | 14.3 $\pm$ 0.9 | -0.6 $\pm$ 0.3
3.829 | 18.1 $\pm$ 0.9 | 0.2 $\pm$ 0.2 | 17.1 $\pm$ 0.9 | -0.1 $\pm$ 0.3
1.971 | 19.3 $\pm$ 0.9 | 0.5 $\pm$ 0.3 | 20 $\pm$ 1 | 0.5 $\pm$ 0.3
3.119 | 16 $\pm$ 1 | -0.1 $\pm$ 0.3 | 16 $\pm$ 1 | -0.3 $\pm$ 0.3
6.097 | 17 $\pm$ 1 | 0.3 $\pm$ 0.3 | 18 $\pm$ 1 | 0.3 $\pm$ 0.3
1.088 | 16 $\pm$ 1 | 17 $\pm$ 1 | 17 $\pm$ 1 | -0.2 $\pm$ 0.3
Table 4: Evolution of the effective brightness $\mu_{eff,evo}$ and effective radius $\log R_{eff,evo}$ of the galaxies selected from S12 for values of $n$ equal to the ID galaxies in the V band in the two cosmological models considered here. ID | $\mu_{eff,evo,\Lambda CDM}$ | $\mu_{eff,evo,EdS}$ | $\log R_{eff,evo,\Lambda CDM}$ | $\log R_{eff,evo,EdS}$
---|---|---|---|---
2.856 | -5 $\pm$ 2 | -4 $\pm$ 2 | 0.2 $\pm$ 0.3 | 0.2 $\pm$ 0.3
3.548 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.4 $\pm$ 0.4 | -0.6 $\pm$ 0.5
2.531 | -5 $\pm$ 2 | -5 $\pm$ 2 | -0.4 $\pm$ 0.5 | -0.5 $\pm$ 0.5
3.242 | -7 $\pm$ 2 | -7 $\pm$ 2 | -0.7 $\pm$ 0.5 | -1.0 $\pm$ 0.4
3.829 | -4 $\pm$ 2 | -5 $\pm$ 2 | -0.2 $\pm$ 0.3 | -0.5 $\pm$ 0.5
1.971 | -3 $\pm$ 2 | -2 $\pm$ 2 | 0.0 $\pm$ 0.5 | 0.0 $\pm$ 0.5
3.119 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.7 $\pm$ 0.5 | -0.8 $\pm$ 0.5
6.097 | -5 $\pm$ 2 | -4 $\pm$ 2 | -0.3 $\pm$ 0.5 | -0.2 $\pm$ 0.5
1.088 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.7 $\pm$ 0.6 | -0.8 $\pm$ 0.6
The results show that the evolution that will have to occur so that the S12
high redshift galaxies have effective brightness and effective radius equal to
the ones in the Virgo cluster is similar in both cosmological models.
Specifically, it seems that in EdS model the effective radius evolution is
higher than the one occurred in the $\Lambda$CDM model. Nevertheless, the
evolution of the effective surface brightness is almost the same in both
models. We also note that the difference in the Sérsic index values do not
appear to affect our results, which also clearly show that the uncertainties
in the measurements of both quantities we deal with here are just too high to
allow us to distinguish the underlying cosmological model that best represents
the data. Basically our methodology is limited by the uncertainties, at least
as far as the S12 subset data is concerned.
As final words, this work is based on the assumption that the galaxies we
compare belong to a group whose members share at least one common feature
regardless of the redshift, otherwise comparison among them becomes
impossible. In other words, our basic assumption is that our selected galaxies
belong to a homogeneous class of galaxies. In the analysis we carried out
above we grouped our galaxies using only the Sérsic indexes, this therefore
being our defining criterion of a homogeneous class of objects. So, in a sense
we followed Ellis & Perry (1979) and adopted morphology as our definition of a
homogeneous class.
## 8 Summary and conclusions
In this paper we have compared high redshift surface brightness observational
data with theoretical surface brightness predictions for two cosmological
models, namely the $\Lambda$CDM and Einstein-de Sitter, in order to test if
such comparison allows us to distinguish the cosmology that best fits the
observational data. We started by reviewing the expressions for the emitted
and received bolometric source brightness and then obtained their respective
specific expressions in the context of galactic surface brightness (Ellis
1971; Ellis & Perry 1979).
Using the Sérsic profile, we have obtained scaling relations between the
surface effective brightness $\mu_{eff}$ and Sérsic index, as well as between
the effective radius $R_{eff}$ and $n$, for the Virgo cluster galaxies using
Kormendy et al. (2009) data. Assuming this scaling relation, we have
calculated theoretical predictions of the surface brightness and compared them
with some of the observed surface brightness profiles of high-redshift
galaxies in a subsample of Szomoru et al. (2012) galaxies in the two
cosmological models considered here. Our results showed that although the
Sérsic profile fits well the observed brightness, the results for surface
brightness is different from the theoretical predictions. Such difference was
used to calculate the amount of evolution that the high redshift galaxies
would have to experience in order to achieve the Virgo cluster structure once
they arrive at $z\sim 0$. We concluded from our results that the cosmological
evolution is quite similar in the two models considered in this paper. We also
noted that galaxies having different Sérsic indexes do not seem to follow a
different evolutionary path.
Overall, with the data and errors available for the chosen subset of galactic
profiles used here we cannot distinguish between the two different
cosmological models assumed in this work. That is, assuming that the high
redshift galaxies will evolve to have features similar to the ones found in
the Virgo cluster, is it not possible to conclude which cosmological model
will predict theoretical surface brightness curves similar to the observed
ones due to the high uncertainties in the data used here. We also noted that
the Sérsic index does not seem to play any significant evolutionary role, as
the evolution we discussed is apparently not affected by the value of $n$.
Nevertheless, this work used only the Sérsic index to define a homogeneous
class of objects.
Summing up those results, it seems reasonable that future studies of this kind
should also select galaxies based on other features besides morphology in
order to increase the number of common properties between high and low
redshift galaxies, and, of course, using different data samples than those
adopted here. More common features such as the Sérsic index are essential for
a better definition of a homogeneous class of cosmological objects whose
observational features are possibly able to distinguish among different
cosmological scenarios. However, care should be taken to avoid features which
can possibly suffer dramatic cosmological evolution.
## Acknowledgments
I.O.-S. is grateful to the Brazilian agency CAPES for financial support.
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|
arxiv-papers
| 2013-11-27T16:50:44 |
2024-09-04T02:49:54.371734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Iker Olivares-Salaverri and Marcelo Byrro Ribeiro (Universidade\n Federal do Rio de Janeiro)",
"submitter": "Marcelo Byrro Ribeiro",
"url": "https://arxiv.org/abs/1311.7036"
}
|
1311.7053
|
# Superdiffusion of 2D Yukawa liquids due to a perpendicular magnetic field
Yan Feng [email protected] Los Alamos National Laboratory, Mail Stop E526,
Los Alamos, New Mexico 87545, USA J. Goree Bin Liu Department of Physics
and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA T. P.
Intrator M. S. Murillo Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA
###### Abstract
Stochastic transport of a two-dimensional (2D) dusty plasma liquid with a
perpendicular magnetic field is studied. Superdiffusion is found to occur
especially at higher magnetic fields with $\beta$ of order unity. Here,
$\beta=\omega_{c}/\omega_{pd}$ is the ratio of the cyclotron and plasma
frequencies for dust particles. The mean-square displacement
${\rm{MSD}}=4D_{\alpha}t^{\alpha}$ is found to have an exponent $\alpha>1$,
indicating superdiffusion, with $\alpha$ increasing monotonically to $1.1$ as
$\beta$ increases to unity. The 2D Langevin molecular dynamics simulation used
here also reveals that another indicator of random particle motion, the
velocity autocorrelation function (VACF), has a dominant peak frequency
$\omega_{peak}$ that empirically obeys
$\omega_{peak}^{2}=\omega_{c}^{2}+\omega_{pd}^{2}/4$.
###### pacs:
52.27.Gr, 52.27.Lw, 66.10.C-
## I I. Introduction
Transport of charged particles under magnetic fields is important in studying
plasma physics processes such as ion transport in tokamaks Tsypin:1998 and
the solar wind into Earth’s magnetosphere Hasegawa:2004 . An external magnetic
field complicates the motion of all charged particles, as compared with the
case without a magnetic field, so that their transport due to collisions is
changed fundamentally. Kinetic theory including the effects of cyclotron
motion Spitzer:1956 is needed to study the collisional transport of plasmas
with magnetic fields. Dusty plasmas provide an experimental and theoretical
platform to study fundamental transport concepts.
Dusty plasma Shukla:2002 ; Fortov:2005 ; Morfill:2009 ; Piel:2010 ;
Bonitz:2010 is a four-component mixture of ions, electrons, gas atoms and
electrically charged micron-sized dust particles. These dust particles are
negatively charged, and their mutual repulsion is often described by the
Yukawa or Debye-Hückel potential Konopka:2000 ,
${\phi(r)=Q^{2}{\rm exp}(-r/\lambda_{D})/4\pi\epsilon_{0}r,}$ (1)
where $Q$ is the particle charge and $\lambda_{D}$ is the screening length due
to electrons and ions. Due to their high particle charge, dust particles are
strongly coupled, so that a collection of dust particles exhibits properties
of liquids or solids. The size of dust particles allows directly imaging them
and tracking their motion, so that collisional transport phenomena can be
observed experimentally at the level of individual particles. Experiments can
be performed either with a single horizontal layer of dust (2D) or with dust
that fills a volume (3D). For the case of a 2D dusty plasma, which we will
study, the electrons and ions fill a 3D volume, while the dust is constrained
by strong dc electric fields to move only on a single plane Feng:2011 .
Superdiffusion is a type of anomalous transport where particle displacements
exhibit a scaling with time that is greater than for normal diffusion. When
the time dependence of mean-square displacement of a particle is fit to the
form
${{\rm MSD}(t)=4D_{\alpha}t^{\alpha}}$ (2)
over times long enough for multiple collisions to occur, the signatures of
normal diffusion and superdiffusion are $\alpha=1$ and $\alpha>1$,
respectively. The coefficient $D_{\alpha}$ is not truly a diffusion
coefficient if $\alpha>1$, but nevertheless it is useful for quantifying the
magnitude of random particle displacements.
For 2D systems, anomalous transport including superdiffusion has often been
reported, for various unmagnetized systems including dusty plasmas.
Indications of this kind of anomalous transport, attributed to low
dimensionality, are often found in non-converging integrals for the random
motion Alder:1967 ; Alder:1970 ; Ernst:1970 ; Dorfman:1970 ; Donko:2009 . In a
magnetized system, however, the trajectories of charged particles are
fundamentally changed from those in an unmagnetized system, so it is an open
question whether collisional particle motion is described as diffusion or
superdiffusion. In this paper we seek to answer this question.
Previous work has been reported for unmagnetized 2D dusty plasmas to assess
whether $\alpha>1$. This previous work includes experiments Nunomura:2006 ;
Liu:2008 and theoretical simulations Liu:2007 ; Hou:2009 ; Ott:2009 . Other
transport coefficients that have been studied experimentally for 2D dusty
plasmas include shear viscosity Nosenko:2004 ; Feng:2011 ; Hartmann:2011 and
thermal conductivity Nunomura:2005 ; Nosenko:2008 ; Feng:2012a ; Feng:2012b .
Simulations have also been reported for shear viscosity Liu:2005 ; Donko:2006
, longitudinal viscosity Feng:2013 , and thermal conductivity Donko:2009 ;
Hou:2009a ; Kudelis:2013 .
Our paper is motivated by the recent attention given to dusty plasma behavior
under magnetic field Wang:2002 ; Filippov:2003 ; Jiang:2007 ; Dyachkov:2009 ;
Banerjee:2010 ; Vasiliev:2011 ; Kahlert:2013 ; Kong:2013 ; Kopp:2014 . This
attention is driven by experiments, which have only recently begun. There are
at least three magnetized dusty plasma devices Schwabe:2011 ; Reichstein:2012
; Thomas:2012 , which now, or soon will be, producing experimental data. The
prospects for these experiments has motivated simulations, including
Uchida:2004 ; Hou:2009b ; Bonitz:2010b ; Ott:2011a , to study waves for 2D
Yukawa liquids and solids under a magnetic field. In this literature, the
magnetic field strength is quantified by a ratio of the cyclotron and plasma
frequencies for dust particles, $\beta=\omega_{c}/\omega_{pd}$. For transport
coefficients in magnetized strongly coupled plasmas, at the time we began
writing this paper the literature included only studies for 3D systems, such
as Ott:2011 and one paper on a 2D Coulomb liquid Ranganathan:2002 .
As we were finishing this paper, we learned of a new work, the first for
diffusion in 2D Yukawa liquids, by Ott, Löwen and Bonitz Ott:2014 . Using a
frictionless MD simulation, they determined the MSD for a wide range of time
and $\beta$. Not claiming that it represented a diffusion coefficient, they
reported a coefficient $D_{\alpha}$ evaluated at a particular time. Our
results complement those of Ott:2014 . We investigate whether motion is
superdiffusive, and we characterize a peak in the spectrum of the velocity
autocorrelation function (VACF), which is another measure of random motion.
Our simulation was not frictionless; we use a 2D Langevin MD simulation that
includes the effects of gas friction, which are present in experiments.
We find that 2D motion of dust particles in a perpendicular magnetic field is
superdiffusive when $\beta\approx 1$. We also find that the VACF has a
spectrum that is dominated by a large peak due to a combination of cyclotron
motion and bouncing of particles within the cage defined by their neighbors.
We find an empirical expression for the frequency of this peak.
## II II. Characterizing random motion
We now review the measures of random motion that we use, the MSD (mean square
displacement) and VACF (velocity autocorrelation function).
### II.1 A. MSD and superdiffusion
Mean-squared displacement (MSD) characterizes self-diffusion Einstein:1956 .
It is defined as ${\rm MSD}(t)=\langle|\bf{r}_{i}({\it t})-\bf{r}_{i}({\rm
0})|^{2}\rangle$, where $\bf{r}_{i}({\it t})$ is the position of particle $i$
at time of $t$. Here, $\langle\rangle$ denotes the ensemble average over all
particles and different initial times Vaulina:2002 .
The MSD is a time series that reveals how random particle motion has different
regimes, according to the time scale. For strongly coupled systems such as
liquids, when the time is very short a particle moves mainly inside the cage
formed by its nearest neighbors, which is called caged motion Donko:2002 , the
particle motion is termed “ballistic” Liu:2007 , and the MSD scales $\propto
t^{2}$. At longer times, when several collisions have occurred, a particle can
escape its cage and displace with a random walk described as self-diffusion.
For these longer times, if the MSD time series is a straight line in a log-log
plot, it is described by a power law, Eq. (2). In this equation, the factor of
$4$ comes from the two dimensionality of our studied system, for 3D systems it
would be $6$. Normal diffusion is characterized by $\alpha=1$; while
superdiffusion and subdiffusion are characterized by $\alpha>1$ and
$\alpha<1$, respectively. Both superdiffusion and subdiffusion are also called
anomalous diffusion.
A criterion is needed to judge whether motion is superdiffusive. Since data
from simulations and experiments will never yield a value that is exactly $1$,
some authors apply a more stringent criterion of $\alpha\geq 1.1$ for
superdiffusion Liu:2007 , instead of $\alpha>1$. Another practical
consideration is the time duration of the MSD data. Indications have been
reported Ott:2009 that after a longer time interval, superdiffusive motion in
a 2D Yukawa liquid vanishes, becoming diffusive with $\alpha=1$ at long times.
Thus, it is desirable to assess the value of $\alpha$ for various time
intervals, and to assess whether it trends to unity at long times, as we shall
do in this paper.
### II.2 B. Velocity autocorrelation function
Like the MSD, the VACF measures the temporal development of particles that are
tracked individually, as they collide with others, but it is the fluctuating
velocity rather than position that is used. The VACF is defined Schmidt:1997
as the time series $\langle\bf{v}_{i}({\it t})\cdot\bf{v}_{i}({\rm
0})\rangle$, where $\langle\rangle$ also denotes the ensemble average over all
particles and different initial times. If there are no magnetic fields, so
that the particles move only because of their own inertia and collisions, the
VACF will exhibit a damped oscillation, for strongly coupled systems such as a
solid or liquid. The oscillation reflects the caging motion Donko:2002 of a
particle due to the interaction with its nearest neighbors. In strongly
coupled systems, diffusive motion arises from the gradual escape of a particle
from a cage (the so-called decaging motion) so that a particle becomes
displaced. For normal diffusion, the VACF can be used to calculate the
diffusion coefficient Hansen:1986 ; Vaulina:2008 ; Dzhumagulova:2012 . We will
use the VACF for another purpose because we will find that the motion is
superdiffusive. In particular, we will use it to compute the vibrational
density of states, which is the modulus of the Fourier transform of the VACF
time series, plotted as a function of $\omega$ Schmidt:1997 ; Goncalves:1992 ;
Teweldeberhan:2010 . This vibration density of states will reveal any
preferred frequency for particle motion.
In this paper we add a magnetic field, and we expect that the time series for
the VACF will oscillate, due not only to random interparticle interactions but
also to cyclotron motion of individual particles. We expect that both kinds of
oscillatory motion will be revealed in the vibrational density of states.
## III III. Simulation method
We performed Langevin MD simulations, with additional Lorentz forces acting on
dust particles due to the external perpendicular magnetic field. For each
particle $i$, we integrate the Langevin equation
${m\ddot{\bf r}_{i}=Q\dot{\bf r}_{i}\times{\bf B}-\nabla\Sigma\phi_{i,j}-\nu
m\dot{\bf r}_{i}+\zeta_{i}(t),}$ (3)
with a Lorentz force $Q\dot{\bf r}_{i}\times{\bf B}$, frictional drag
Klumov:2009 $-\nu m\dot{\bf r}_{i}$ and a random force $\zeta_{i}(t)$. The
random force $\zeta_{i}(t)$ is assumed to have a Gaussian distribution with a
zero mean, according to the fluctuation-dissipation theorem Feng:2008 ;
Gunsteren:1982 . For the binary interaction potential $\phi_{i,j}$ we use the
Yukawa repulsion, Eq. (1). Note that when there is a strong magnetic field,
the dynamics of electrons and ions that account for the shielding may be
completely changed Schwabe:2011 , so that the interparticle interaction of 2D
dusty plasmas may be more complicated. In this paper, we assume that the
interparticle interaction is still the Yukawa interaction, as the zeroth order
approximation.
We consider a uniform magnetic field in the $z$ direction perpendicular to the
x-y plane in which the particles are constrained to move. We use the Langevin
integrator of Gunsteren and Berendsen Gunsteren:1982 . Time scales are
normalized by the nominal plasma frequency,
$\omega_{pd}=(Q^{2}/2\pi\epsilon_{0}ma^{3})^{1/2}$ Kalman:2004 , which is also
a time scale for interparticle collisions, in a system that is strongly
coupled. Here, $m$ is the particle mass and $a\equiv(n\pi)^{-1/2}$ is the
Wigner-Seitz radius Kalman:2004 for an areal number density $n$. The magnetic
field is characterized using the dimensionless parameter of
$\beta=\omega_{c}/\omega_{pd}$, where $\omega_{c}$ is the cyclotron frequency
of the dust particle. The time scale for gas frictional damping Liu:2003 is
chosen as $\nu=0.027\omega_{pd}$ to mimic typical experimental conditions
Feng:2008 , while the time scale for cyclotron motion is variable, by choosing
$\beta$. We vary $\beta$ from $0$ to $1$, where the upper end of this range
corresponds to an extremely strong magnetic field Bonitz:2013 . For example,
for a typical 2D dusty plasma experiment of Feng:2010 ; Feng:2011 with $8$
micron diameter particles, $\beta=1$ corresponds to a magnetic field of
$B=1.3\times 10^{4}~{}{\rm T}$. Note that under stronger magnetic fields,
plasma sources relying on capacitively coupled radio-frequency power can have
some inhomogeneities. For example, filaments or enhanced ionization that are
aligned parallel to the magnetic field lines were observed in Schwabe:2011 .
This nonuniformity of the plasma was observed to affect microparticle motion
by causing an inhomogeneous “pattern formation” Schwabe:2011 . We assume a
spatially uniform plasma in our simulations, so that comparing our results to
experiment must await a future experiment with conditions that are more
uniform than in Schwabe:2011 . It is reasonable to anticipate such results
because there are new facilities coming online that have the flexibility to
alter the operation and design of their plasma source. We integrate Eq. (3)
using a time step of $0.037\omega_{pd}^{-1}$, which we checked to be small
enough for both the collisional and cyclotron motion.
The simulation parameters are chosen so that the collection of dust particles
will behave as a liquid, according to the phase diagram of Hartmann:2005 . To
describe the dust particle charge, kinetic temperature $T$ and areal number
density, we use the dimensionless quantities
$\Gamma=Q^{2}/(4\pi\epsilon_{0}ak_{B}T)$ and $\kappa\equiv a/\lambda_{D}$. We
choose $\Gamma=200$ and $\kappa=2$ as typical liquid conditions that are
experimentally attainable using dusty plasmas.
We emphasize that for these parameters, in the absence of a magnetic field, it
has been shown Ott:2009 that motion is nearly that of normal diffusion, with
$\alpha\approx 1$. We will determine how this conclusion changes as a magnetic
field is added.
Another dimensionless parameter for magnetized dusty plasmas is the inverse
Hall parameter for the dust $R_{c}=\omega_{c}/\nu$. When this ratio is much
greater than unity, dust particles can complete circular orbits before the
trajectory is disturbed by collisions with neutral gas, which occur at a rate
$\nu$ Thomas:2012 . For the gas conditions simulated in our Langevin equation,
$R_{c}=\beta/0.027$.
Our simulation size is $N=1024$ particles constrained to planar motion in a
rectangle of dimensions $65.5a\times 56.7a$. As in Hou:2009 ; Ott:2011 , we
use periodic boundary conditions. We truncate the Yukawa potential at radii
beyond $22.9a$ with a switching function to give a smooth cutoff between
$22.9a$ and $24.8a$ to avoid an unphysical sudden force change when a particle
moves a small distance Feng:2013 . All simulation runs start from a random
configuration of 1024 particles, then run $10^{5}$ steps to reach the steady
conditions before starting recording data. After that, particle trajectories
of the next $10^{7}$ steps are saved for data analysis. Note that the total
time duration of $10^{7}\times 0.037\omega_{pd}^{-1}=3.7\times
10^{5}\omega_{pd}^{-1}$ corresponds to $\approx 3.4$ hours for a typical value
of $\omega_{pd}=30~{}{\rm s}^{-1}$ in 2D dusty plasma experiments Feng:2010 ;
Feng:2011 , much longer than experimental runs. Representative trajectories
are shown in Fig. 1. We verified that our simulations are free of any
nonuniformity such as a flow or a localized peak in number density or kinetic
temperature.
## IV IV. Results
### IV.1 A. Superdiffusion
The calculated MSD time series for different $\beta$ values are presented in
Fig. 2. As expected, displacements are reduced with an increasing magnetic
field, i.e., an increasing $\beta$. After the initial ballistic portion, the
MSD time series has its diffusive portion at longer times. For fitting the MSD
data to determine $\alpha$, we will use the range of $100<\omega_{pd}t<1000$,
which is for times later than the ballistic portion. We present the MSD curves
two ways, normalized by the plasma frequency $\omega_{pd}$ and $f_{c}$ (where
$f_{c}=\beta\omega_{pd}/2\pi$ is the cyclotron frequency) in Fig. 2(a) and
(b), respectively. In the latter we see that the oscillations in the MSD occur
at the cyclotron frequency and its harmonics, indicating the dominant role of
cyclotron motion at that time scale.
As our first main result, we present the exponent $\alpha$ in Fig. 3(a). For
these $\Gamma$ and $\kappa$ conditions, we find that a 2D Yukawa liquid
exhibits superdiffusion $\alpha=1.1$ for a large magnetic field $\beta=1$, and
weak superdiffusion $1<\alpha<1.1$ for weaker magnetic fields. Without a
magnetic field, $\beta=0$, we find nearly normal diffusion, $\alpha\approx 1$,
as was reported for previous unmagnetized simulations Ott:2009 . Figure 3(a)
shows that there is a monotonic trend for $\alpha$ to increase with $\beta$,
i.e., for superdiffusion to become stronger as the magnetic field is
increased. This result is different from the claim of Ranganathan et al. who
simulated a 2D Coulomb liquid and reported normal diffusion Ranganathan:2002 .
We find that the fitting exponent $\alpha$ depends slightly on the time range
chosen for fitting. We chose four different fitting time ranges to detect how
sensitive of the fitting exponent $\alpha$ is related to the time. From Fig.
3(a), as the time range for the fitting is longer, a clear trend that the
exponent $\alpha$ is smaller can be easily detected. In a previous study of
superdiffusion in 2D Yukawa liquids Ott:2009 , Ott and Bonitz also found that
the exponent $\alpha$ changes as they chose different time ranges to fit.
As we noted in the Introduction, the trajectories of random particle motion in
a liquid are completely different in the presence of a magnetic field, so that
before we conducted our simulations, there was no particular reason to expect
motion to be either normal diffusion or superdiffusion. Our results in Fig. 3
make it clear that adding a magnetic field does cause superdiffusion. The
motion is nearly normal diffusion in the absence of magnetic field, but then
it becomes weakly superdiffusive as a magnetic field is added with a small
value of $\beta$, with the superdiffusion becoming stronger and reaching
$\alpha=1.1$ at a high magnetic field of $\beta=1$. This superdiffusive
tendency is not as powerful as in some cases, such as the unmagnetized
simulation of Liu:2007 where $\alpha=1.3$ was reported. While the
superdiffusive tendency found here is less profound, there is no doubt that it
is present for the time intervals that we studied: our results in Fig. 3 show
very little scatter, and our fitting of the MSD curves in Fig. 2 that yielded
the data in Fig. 3 had an exceptionally high coefficient of determination.
Thus, we are confident in our empirical finding that motion is superdiffusive
when magnetic field is added to a 2D strongly coupled plasma, when modeled as
a Yukawa liquid in the presence of gas collisions, as in our simulation. We
offer some discussion of this empirical finding in Sec. IV C.
We also characterize the coefficient $D_{\alpha}$ in Fig. 3(b). As in Ott:2014
, we can see that $D_{\alpha}$ decreases monotonically as $\beta$ increases,
meaning that the perpendicular magnetic field suppresses the self-diffusion of
particles in 2D Yukawa liquids. We fit the data for $D_{\alpha}$ vs. $\beta$
for the fitting time range of $100<\omega_{pd}t<1000$ in Fig. 3(b) to an
expression
${D_{\alpha}=D_{\alpha 0}/(1+\xi\beta)^{2}.}$ (4)
We chose the form of Eq. (4) so that it has an asymptotic behavior that is a
constant $D_{\alpha 0}$ in the absence of magnetic field $\beta=0$ and
diminishes with the same scaling as classical diffusion $\propto 1/\beta^{2}$
for large magnetic field. For the range of $\beta$ that we explored, this
expression fits the data well, with empirical coefficients $D_{\alpha
0}=0.00616a^{2}\omega_{pd}$ and $\xi=1.083$. This expression fits our data
somewhat better than the expression derived from the Langevin equation by
Ranganathan et al. Ranganathan:2002
${D_{\alpha}=D_{\alpha 0}/(1+\xi\beta^{2}).}$ (5)
Fits to both expressions are shown in Fig. 3. We note that these expressions,
Eqs. (4) and (5), each have two free parameters ($D_{\alpha 0}$ and $\xi$) and
a tendency toward classical diffusion, which is different from the three-
parameter fit used in Ott:2014 which tends toward Bohm diffusion,
$D_{\alpha}\propto 1/\beta$ for strong magnetic fields. We did not extend our
simulation to large enough $\beta$ to test whether classical or Bohm diffusion
better describes the transport because experiments might not be feasible at
such a high magnetic field.
### IV.2 B. VACF peak frequency
We can seek insight into the peculiarities of thermal motion under the
partially magnetized conditions where we observed superdiffusion. To do this,
we examine the velocity autocorrelation function (VACF), which is closely
related to diffusion; its integral diverges for superdiffusion but converges
for diffusion. Transforming the VACF in Fig. 4 to yield its spectrum, Fig.
5(a), our attention is drawn to the most prominent feature: a large peak. This
peak is such a dominant feature of the VACF spectrum that it seems likely that
to gain an understanding of the thermal motion, in the presence of both
collisions and magnetic field, will require an understanding of the peak and
its tendencies as the magnetic field is changed. Therefore we wish to
characterize the frequency of the peak and its dependence on the parameters
that characterize collisions ($\omega_{pd}$) and cyclotron motion
($\omega_{c}$).
In Fig. 4, we see that oscillations occur with or without magnetic field, but
they are larger in amplitude and more persistent in time when the magnetic
field is large. When there is a magnetic field, the oscillation frequency is
close to the cyclotron frequency, as seen in Fig. 4(b) where time is
normalized by $f_{c}^{-1}$. The decay of VACF is slower for stronger magnetic
field, which is natural result of stronger cyclotron motion. Inspecting both
Fig. 1 and Fig. 2, we also notice that, within a specific time range, the
typical displacement of a particle under a stronger magnetic field is smaller.
It seems that, under a stronger magnetic field, a particle needs a longer time
to escape the cage formed by its nearest neighbors, i.e., a longer decaging
time, due to stronger cyclotron motion.
The vibrational density of states Goncalves:1992 ; Teweldeberhan:2010 is
presented in Fig. 5(a). We calculated this as the spectral power of the VACF
by a Fourier transformation of the normalized VACF time series. This
vibrational density of states describes the collective motion of the
particles. In Fig. 5(a) we see that this spectral power is not flat, but has a
dominant peak of finite width. The prominence of this peak indicates that the
thermal motion has a favored frequency.
As our second main result, in Fig. 5(b) we find that the peak frequency
$\omega_{peak}$ increases with magnetic field $\beta$ according an empirical
fit
${\omega_{peak}^{2}/\omega_{pd}^{2}=0.25+\beta^{2},}$ (6)
or equivalently
${\omega_{peak}^{2}=0.25\omega_{pd}^{2}+\omega_{c}^{2}.}$ (7)
This expression combines two kinds of motion, collective motion at
$\omega_{pd}$ and cyclotron motion of a single particle at $\omega_{c}$.
Figure 5(b) and the expression Eq. (7) illustrate how these two kinds of
motion combine, for thermal motion of a strongly coupled plasma under magnetic
field. There is a favored frequency, which is somewhat larger than the
cyclotron frequency. We note that the coefficient of $0.25$ in Eq. (7) was
determined for our conditions, $\Gamma=200$ and $\kappa=2$; we have not
determined whether it varies with those parameters. We can also express this
peak frequency using the Einstein frequency $\omega_{E}$, which is the
oscillation frequency that a charged particle’s motion would have in a cage
formed by all the other particles, if all the other particles were stationary.
From Fig. 2(b) in Kalman:2004 , the Einstein frequency in our simulation
conditions is $\omega_{E}\approx 0.35\omega_{pd}$, so that we also find that
this peak frequency can also be expressed as
$\omega_{peak}^{2}=2\omega_{E}^{2}+\omega_{c}^{2}$, which is the same as the
expression for $\omega_{1,\infty}$ in Bonitz:2010b . For comparison, in Fig.
5(b) we also show the peak frequencies obtained from the frictionless
simulation of Bonitz:2010b .
We note that our vibrational density of states is not the only way to
characterize the frequency content of thermal motion. Another way, which has
been widely used in the literature for strongly coupled plasmas, is the wave
spectrum. For a 2D Yukawa liquid under a magnetic field, it has been used for
example by Hou et al. Hou:2009b . To compare how our vibrational density of
states and the wave spectrum quantify the frequency content, we present the
wave spectrum in Fig. 6. We computed it using Eqs. (2-4) of Hou:2009b , which
use as their inputs the positions as well as velocities of particles, not just
the velocities as in the vibrational density of states. Other differences are
that the wave spectrum is resolved in both the magnitude and direction of the
wave vector $k$. The direction refers to the particle velocity, as compared to
the arbitrarily chosen direction of $k$, and it is said to be longitudinal or
transverse according to whether $v$ is parallel or perpendicular to $k$,
respectively. Examining the power spectra in Fig. 6, we see that the frequency
content favors a peak frequency, which depends on the strength of the magnetic
field, and the spectrum has a finite width about this peak frequency. The peak
frequency is generally slightly higher than the cyclotron frequency, except
when the magnetic field is absent, as was the case for our vibrational density
of states. The wave number dependence, which is measured only by the wave
spectra and not the vibrational density of states, shows how the wave becomes
optical (i.e., $\omega$ does not approach zero for zero wave number) when a
magnetic field is present. Note that, our obtained phonon spectra agree well
with the experimental and simulation results in Hartmann:2013 . As is well
known for strongly coupled plasmas, the wave spectrum also shows how the wave
starts as a forward wave, $d\omega/dk>0$, for small $k$, but can become
backward, $d\omega/dk<0$, for larger $k$ where the wavelength is on the order
of the particle spacing. Under a magnetic field, there is not only the
dominant oscillation at a frequency somewhat above $\omega_{c}$, but also a
lower-frequency mode at $\omega/\omega_{pd}\ll 0.5$. The latter mode was
remarked upon by several authors for both 2D and 3D liquids under a magnetic
field. Our Fig. 6 shows how this low frequency mode occurs more strongly for a
transverse polarization.
### IV.3 C. Conceptual discussion of diffusion
Our empirical findings, superdiffusion in the presence of a magnetic field and
a VACF spectral peak that varies with the magnetic field, both hint at the
complexity of random particle motion. This complexity arises from a
combination of two kinds of particle motion: caged motion in a liquid and
cyclotron motion in a magnetic field. To gain an appreciation for this
complexity, we review here some of the concepts for diffusion in various
physical systems, starting with some of the simplest ones. This discussion
will lead us to recognize there is no obvious intuitive reason to expect
normal diffusion, given the complex nature of the random motion for a liquid
with magnetized particle motion.
Diffusion is often described as a process of random displacements for a
specified time interval. The diffusion coefficient is estimated by dividing
the square of the typical step-size displacement by the typical time interval
between steps. Superdiffusion can happen when there is an unusual abundance of
large displacements. Lévy-flight displacements (in certain physical systems)
Sokolov:2000 are extreme examples of these large displacements, and they
result in severe superdiffusion. More subtle increases in the abundance of
large displacements will lead to a less severe superdiffusion.
For an electron or ion in a magnetized weakly-coupled plasma, there is a kind
of normal diffusion called “classical diffusion.” The intuitive estimate for
the classical diffusion coefficient is traditionally obtained by estimating
the step size as the cyclotron radius and the time interval as the inverse
Coulomb collision frequency. (There are several kinds of Coulomb collision
frequencies; the relevant one is for perpendicular momentum deflection).
For Brownian motion of an isolated dust particle in gas, there is again a
“step size” displacement between collisions, and a typical time between
collisions. For the Brownian motion, the step size is the mean free path
between collisions with gas atoms, and that is the only length scale. There is
also only one time scale for the Brownian motion: the collision frequency with
gas atoms. Displacements for a given time interval have a Gaussian
distribution, and the resulting motion is diffusive.
For an unmagnetized strongly-coupled plasma, there is again a single length
scale: the spacing between particles. (This is so unless there are modes
present, which might have a particular wavelength and add another length
scale.) There are two time scales of note: the Einstein time for oscillations
in a cage, and a decaging time for a particle, which depends on temperature
and structure. The latter time scale would lead to the diffusion. In the 3D
case random motion should be diffusive in the absence of hydrodynamic flows.
In the 2D case, however, there can be superdiffusion, which has been
attributed to long-time correlations arising possibly from hydrodynamic modes,
according to earlier literature for transport in 2D systems Ernst:1970 .
Adding a sufficiently strong magnetic field to the strongly coupled plasma,
gyration will provide an additional time and length scale. The additional
length scale means that sometimes a diffusive step size might correspond to
the cyclotron radius (as for classical diffusion in a weakly coupled plasma),
or sometimes it might correspond to the interparticle spacing (as for a
strongly coupled plasma without magnetic field). Or the step size might be
some mixture of the two. There is no longer the simplicity of a single
mechanism for random motion. This complex mixing of collective motion at
$\omega_{pd}$ (without magnetic field) and cyclotron motion at $\omega_{c}$
(due to magnetic field) can be seen in our result for the vibrational density
of state, Fig. 5(b). More than one mechanism is at play, so that there is no
compelling reason to expect that the step size of a displacement will be that
of normal diffusion. Thus, there is no definitive reason to expect that self-
diffusion will occur with the displacements increasing with time exactly as
was the case for only one mechanism. In other words, the complexity of the
random motion means that there is no simple reason for us to anticipate
intuitively whether motion will be diffusive or superdiffusive. This situation
leads us to rely on numerical simulations to provide an empirical answer.
## V V. Summary
In conclusion, we have performed Yukawa MD simulations to study the diffusion
and superdiffusion of 2D liquid dusty plasmas under a uniform perpendicular
magnetic field. We characterized the stochastic motion of using the mean-
squared displacement MSD, velocity autocorrelation function VACF, vibrational
density of states, and phonon spectra. It is expected that adding a magnetic
field will reduce the displacements of charged particles as they undergo
collisions, and this indeed occurs. However, we also find that adding the
magnetic field also changes the scaling of those displacements with respect to
time so that the MSD scales with a greater power of time and the motion
becomes superdiffusive. The vibrational density of states has only one
dominant peak for all simulated conditions, and this peak frequency can be
expressed a function of $\omega_{c}$ and $\omega_{pd}$. These conclusions may
be tested in future experiments with magnetized dusty plasmas.
We thank M. Bonitz and T. Ott for valuable discussions and providing the data
in Fig. 5(b). Work at LANL was supported by the LANL Laboratory Directed
Research and Development program and Department of Energy contract No.
W-7405-ENG-36, and work at Iowa was supported by National Science Foundation
Grant No. 1162645.
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Figure 1: (Color online). Particle trajectories in the x-y plane for
different constant perpendicular magnetic field strengths,
$\beta=\omega_{c}/\omega_{pd}=0$ (a), 0.5 (b), and 1.0 (c). The random walk
without a magnetic field changes its character, becoming more circular and
wander less as the magnetic field increases in (b) and (c). Color represents
time, and only $\approx 10\%$ of the simulated region and $\approx 0.03\%$ of
the simulation duration are shown here. Our simulation conditions are
$\Gamma=200$ and $\kappa=2$. Figure 2: (Color online). Mean-squared
displacement MSD for different magnetic fields, in the unit of $\omega_{pd}t$
(a), or $f_{c}t$ (b). At long times $100<\omega_{pd}t<1000$, well after the
ballistic portion, we can fit the MSD time series to Eq. (2). As $\beta$
increases, the MSD curves are lower and lower, indicating that the wandering
motion of particles is suppressed by magnetic field, and $D_{\alpha}$
decreases. Oscillations at shorter times are due to cyclotron motion of
individual particles, as is best seen in (b), where time is normalized by
$1/f_{c}$. Dips in the MSD time series occur around one cyclotron period, two
periods, and so on. Here, the MSD is normalized using the Wigner-Seitz radius
$a$. Our simulation conditions are $\Gamma=200$ and $\kappa=2$. Figure 3:
(Color online). (a) Indication of superdiffusion. The exponent $\alpha$ is
$>1$, especially for strong magnetic fields $\beta\approx 1$. These data are
the results of fitting the MSD time series in Fig. 2 to the exponential
scaling of Eq. (2) in the indicated time ranges. This result $\alpha>1$ for a
2D Yukawa liquid is different from that of Ranganathan et al. who reported
normal diffusion, $\alpha=1$, for a 2D Coulomb liquid Ranganathan:2002 . As
the time range for the fitting is longer, we can see a clear trend that the
exponent $\alpha$ is smaller. In (b), as $\beta$ increases from 0 to 1, the
coefficient $D_{\alpha}$ decreases monotonically more than $70\%$ as $\beta$
increases, which means that the magnetic field greatly suppresses the
wandering of particles. Note that the scatter of our data for each $\beta$
value corresponds to the error bar. Fits of our $D_{\alpha}$ data for the time
range of $100<\omega_{pd}t<1000$ only to our empirical expressions Eq. (4) and
Eq. (5) derived by Ranganathan et al. Ranganathan:2002 are shown as solid and
dashed lines, respectively. Figure 4: (Color online). Velocity
autocorrelation function, VACF, for different magnetic fields. Time is
normalized by $1/\omega_{pd}$ in (a) and $1/f_{c}$ in (b). The oscillations
decay more slowly with higher magnetic field, as seen in (a) for increasing
$\beta$. These oscillations are mainly due to the cyclotron motion, since its
frequency is nearly the same as $f_{c}$. The period of oscillation is related
to the magnetic field strength, as seen in (b) where VACF curves for two
values of $\beta$ are nearly aligned. Figure 5: (Color online). (a)
Vibrational density of states, i.e., spectral power of the normalized VACF for
different magnetic fields. The curves exhibit a dominant peak; the frequency
of this peak is plotted in (b). The peak frequency increases monotonically as
$\beta$ increases. The peak frequency fits an empirical curve,
$\omega_{peak}^{2}/\omega_{pd}^{2}=0.25+\beta^{2}$, i.e.,
$\omega_{peak}^{2}=0.25\omega_{pd}^{2}+\omega_{c}^{2}$, shown as a smooth
curve. This fit shows how the peak frequency is always greater than the
cyclotron frequency. For comparison, we also plot the data from the
frictionless simulation of Bonitz:2010b for the peak frequency of the
longitudinal waves at the wave number of $ka=5.55$. Figure 6: (Color online).
The longitudinal and transverse phonon spectra of our 2D Yukawa liquid when
$\beta=0$ (a,b), $\beta=0.5$ (c,d), $\beta=1$ (e,f). These spectra differ from
the the vibrational density in Fig. 5 because they reflect both spatial and
temporal fluctuations as characterized by a current Hou:2009b , not just the
temporal fluctuations characterized by the VACF.
|
arxiv-papers
| 2013-11-27T17:58:22 |
2024-09-04T02:49:54.382742
|
{
"license": "Public Domain",
"authors": "Yan Feng, J. Goree, Bin Liu, T. P. Intrator, M. S. Murillo",
"submitter": "Yan Feng",
"url": "https://arxiv.org/abs/1311.7053"
}
|
1311.7071
|
# Sparse Linear Dynamical System with Its Application in Multivariate Clinical
Time Series
Zitao Liu
Department of Computer Science
University of Pittsburgh
Pittsburgh, PA 15213
[email protected]
&Milos Hauskrecht
Department of Computer Science
University of Pittsburgh
Pittsburgh, PA 15213
[email protected]
###### Abstract
Linear Dynamical System (LDS) is an elegant mathematical framework for
modeling and learning multivariate time series. However, in general, it is
difficult to set the dimension of its hidden state space. A small number of
hidden states may not be able to model the complexities of a time series,
while a large number of hidden states can lead to overfitting. In this paper,
we study methods that impose an $\ell_{1}$ regularization on the transition
matrix of an LDS model to alleviate the problem of choosing the optimal number
of hidden states. We incorporate a generalized gradient descent method into
the Maximum a Posteriori (MAP) framework and use Expectation Maximization (EM)
to iteratively achieve sparsity on the transition matrix of an LDS model. We
show that our Sparse Linear Dynamical System (SLDS) improves the predictive
performance when compared to ordinary LDS on a multivariate clinical time
series dataset.
## 1 Introduction
Developing accurate models of dynamical systems is critical for their
successful applications in outcome prediction, decision support, and optimal
control. A large spectrum of models have been developed and successfully
applied for these purposes in the literature [3, 11, 20]. In this paper we
focus on a popular model for time series analysis: the Linear Dynamical System
(LDS) [15] and its application to clinical time series [18, 19]. We aim to
develop a method to learn an LDS that performs better on future value
predictions when learned from a small amount of complex multivariate time
series dataset.
LDS is a widely used model for time series analysis of real-valued sequences.
The model is Markovian and assumes the dynamic behaviour of the system is
captured well using a small set of real-valued hidden-state variables and
linear-state transitions corrupted by a Gaussian noise. The observations in
LDS, similarly to hidden states, are real-valued. Briefly, the observations at
time $t$ are linear combinations of hidden state values for the same time.
While in some LDS applications the model parameters are known a priori, in the
majority of real-world applications the model parameters are unknown, and we
need learn them from data that consists of observation sequences we assume
were generated by the LDS model. While this can be done using standard LDS
learning approaches, the problem of learning an LDS model gives rise to
numerous important questions: Given the multivariate observation sequences,
how many hidden states are needed to represent the system dynamics well?
Moreover, since transition and observation matrices depend on the number of
hidden states, how do we prevent the overfit of the model parameters when the
number of examples is small?
In this work we address the above issues by presenting a method based on the
sparse representation of LDS (SLDS) that is able to adjust (depending on the
observation sequences in the data) the number of hidden states and at the same
time prevent the overfit of the model. Our approach builds upon the
probabilistic formulation of the LDS model, and casts the optimization of its
parameters as a maximum a posteriori (MAP) estimate, where the choice of the
parameter priors biases the model towards sparse solutions.
Our SLDS approach is distinctly different from previous work [4, 6, 9]. [4]
formulates the traditional Kalman filter as a one-step update optimization
procedure and incorporates sparsity constraints to achieve sparsity in the
hidden states. [9] trains an LDS for each training example and tries to find a
sparse linear combination of coefficients in order to combine the ensemble of
models. Neither [4] nor [9] directly achieve sparsity on the parameters of the
LDS, and furthermore, the performance of their resulting models still depends
on the optimal number of hidden states. [6] introduces a Bayesian
nonparametric approach to the identification of observation-only linear
systems, where no hidden states are involved. The underlining assumption is
that the observations are obtained from linear combinations of previous
observations and some system inputs, which may be too restrictive to model
complex multivariate time series and makes the model more sensitive to noisy
observations and outliers.
We test our sparse solution on the problem of modeling the dynamics of
sequences of laboratory test results. We show that it improves the learning of
the LDS model and leads to better accuracy in predicting future time-series
values.
Our paper is organized as follows. In Section 2 we review the basics of the
linear dynamical system. In Section 3 we describe SLDS – our method of
sparsifying the LDS parameters. Inference and learning details of SLDS are
explained in Section 3. Experimental results that compare SLDS method to
ordinary LDS are presented in Section 4. In Section 5, we summarize the work
and outline possible future extensions.
## 2 Linear Dynamical System (LDS)
The Linear Dynamical System (LDS) is a real-valued time series model that
represents observation sequences indirectly with the help of hidden states.
Let $\\{{z}_{t}\\}$, $\\{{y}_{t}\\}$ define sequences of hidden states and
observations respectively. The LDS models the dynamics of these sequences in
terms of the state transition probability $p({z}_{t}|{z}_{t-1})$, and state-
observation probability $p({y}_{t}|{z}_{t})$. These probabilities are modeled
using the following equations:
${z}_{t}=A{z}_{t-1}+{e}_{t};\hskip 14.22636pt{y}_{t}=C{z}_{t}+{v}_{t},$ (1)
where ${y}_{t}$ is a $d\times 1$ observation vector made at (current time)
$t$, and ${z}_{t}$ an $l\times 1$ hidden states vector. The transitions among
the current and previous hidden states are linear and captured in terms of an
$l\times l$ transition matrix $A$. The stochastic component of the transition,
${e}_{t}$, is modeled by a zero-mean Gaussian noise
${e}_{t}\sim\mathcal{N}(0,Q)$ with an $l\times 1$ zero mean and an $l\times l$
covariance matrix _Q_. The observations sequence is derived from the hidden
states sequence. The dependencies in between the two are linear and modeled
using a $d\times l$ emission matrix _C_. A zero mean Gaussian noise
${v}_{t}\sim\mathcal{N}(0,R)$ models the stochastic relation in between the
states and observation. In addition to $A,C,Q,R$, the LDS is defined by the
initial state distribution for ${z}_{1}$ with mean $\boldsymbol{\pi}_{1}$ and
covariance matrix $V_{1}$,
${z}_{1}\sim\mathcal{N}(\boldsymbol{\pi_{1}},V_{1})$. The complete set of the
LDS parameters is $\Omega=\\{A,C,Q,R,\boldsymbol{\pi_{1}},V_{1}\\}$. The
parameters of the LDS model can be learned using either the Expectation-
Maximization (EM) algorithm [8] or spectral learning algorithms [16, 21].
## 3 Sparse Linear Dynamical System (SLDS)
In this section, we propose a sparse representation of LDS that is able to
adjust the number of hidden states and at the same time prevents the overfit
of the model. More specifically, we impose $\ell_{1}$ regularizers on every
element of the transition matrix $A_{ij}$, which leads to zero entries in the
transition matrix _A_. The zero entries in the transition matrix of LDS indeed
reduce the actual number of parameters of LDS, sparsify the hidden states, and
avoid the overfitting problem from the real data, even if we set the number of
hidden states originally picked is too large.
To achieve sparsity on the transition matrix, we introduce a Laplacian prior
to each element of _A_ , $A_{ij}$, since Laplacian priors are equivalent to
$\ell_{1}$ regularizations [5, 10, 24]. In general, the Laplacian distribution
has the following form:
$p(x|\mu,\lambda)=\frac{1}{2\lambda}\exp(-\frac{|x-\mu|}{\lambda})$,
$\lambda\geq 0$ where $\mu$ is the location parameter and $\lambda$ is the
scale parameter. Here, we assume every element $A_{ij}$ is independent to each
other and has the following Laplacian density ($\mu=0$ and $\lambda=1/\beta$),
$p(A_{ij}|\beta)=\frac{\beta}{2}\exp(-\beta|A_{ij}|)$. Hence, the prior
probability for _A_ is
$p(A|\beta)=\prod_{i=1}^{l}\prod_{j=1}^{l}p(A_{ij}|\beta)$ and the log joint
distribution for SLDS is:
$\displaystyle\log p(\mathbf{z},\mathbf{y},A)=\log p(A)+\log
p(z_{1})+\sum_{t=2}^{T}\log p(z_{t}|z_{t-1},A)+\sum_{t=1}^{T}\log
p(y_{t}|z_{t})$ (2)
where _T_ is the observation sequence length.
### 3.1 Learning
In this section we develop an EM algorithm for the MAP estimation of the SLDS.
Let $\hat{z}_{t|T}\equiv\mathbb{E}[z_{t}|\mathbf{y}]$,
$M_{t|T}\equiv\mathbb{E}[z_{t}z_{t}^{{}^{\prime}}|\mathbf{y}]$,
$M_{t,t-1|T}\equiv\mathbb{E}[z_{t}z_{t-1}^{{}^{\prime}}|\mathbf{y}]$ and
define the $\mathcal{Q}$ function as
$\mathcal{Q}=\mathbb{E}_{\mathbf{z}}\Big{[}\log
p(\mathbf{z},\mathbf{y},A)|\Omega\Big{]}$, where
$\displaystyle\mathcal{Q}=\mathbb{E}_{\mathbf{z}}\Big{[}\log
p(z_{1})\Big{]}+\log p(A)+\mathbb{E}_{\mathbf{z}}\Big{[}\sum_{t=2}^{T}\log
p(z_{t}|z_{t-1},A)\Big{]}+\mathbb{E}_{\mathbf{z}}\Big{[}\sum_{t=1}^{T}\log
p(y_{t}|z_{t})\Big{]}$ (3)
In the E-step (Inference), we follow the backward algorithm in [8] to compute
$\mathbb{E}[z_{t}|\mathbf{y}]$,
$\mathbb{E}[z_{t}z_{t}^{{}^{\prime}}|\mathbf{y}]$ and
$\mathbb{E}[z_{t}z_{t-1}^{{}^{\prime}}|\mathbf{y}]$, which are sufficient
statistics of the expected log likelihood. In the M-step (Learning), we try to
find $\Omega$ that maximizes the likelihood lower bound $\mathcal{Q}$. In the
following, we derive the M-step for gradient based optimization of the
parameters $\Omega$. We omit the explicit conditioning on $\Omega$ for
notational brevity. Since the $\mathcal{Q}$ function is non-differentiable
with respect to _A_ , but differentiable with respect to all the other
variables ($C,R,Q,\pi_{1},V_{1}$ ), we separate the optimization into two
parts.
Optimization of $A$. In each iteration in the M-step, we need to maximize
$\mathbb{E}_{\mathbf{z}}\Big{[}\sum_{t=2}^{T}\log
p(z_{t}|z_{t-1},A)\Big{]}+\log p(A)$ with respect to _A_ , which is equivalent
to minimizing a function $f(A)$ that
$f(A)=\underbrace{\frac{1}{2}\sum_{t=2}^{T}\mathbb{E}_{\mathbf{z}}\Big{[}(z_{t}-Az_{t-1})^{\prime}Q^{-1}(z_{t}-Az_{t-1})\Big{]}}_{\text{g(A)}}+\underbrace{\beta||A||_{1}}_{\text{h(A)}}$
(4)
where $||A||_{1}$ is the $\ell_{1}$ norm on every element of matrix _A_ ,
$||A||_{1}=\sum_{i=1}^{l}\sum_{j=1}^{l}||A_{ij}||_{1}$.
As we can see $f(A)$ is convex but non-differentiable and we can easily
decompose $f(A)$ into two parts: $f(A)=g(A)+h(A)$, as shown in eq.(4). Since
$g(A)$ is differentiable, we can adopt the generalized gradient descent
algorithm to minimize $f(A)$. The update rule is:
$A^{(k+1)}=\mbox{prox}_{\alpha_{k}}(A^{(k)}-\alpha_{k}\bigtriangledown
g(A^{(k)}))$ where $\alpha_{k}$ is the step size at iteration _k_ and the
proximal function $\mbox{prox}_{\alpha_{k}}(A)$ is defined as the soft-
thresholding function $S_{\beta\alpha_{k}}(A)$
$[S_{\beta\alpha_{k}}(A)]_{ij}=\begin{cases}A_{ij}-\beta\alpha_{k}&\text{if
}A_{ij}>\beta\alpha_{k}\\\ 0&\text{if }-\beta\alpha_{k}\leq
A_{ij}\leq\beta\alpha_{k}\\\ A_{ij}+\beta\alpha_{k}&\text{if
}A_{ij}<-\beta\alpha_{k}\end{cases}$
###### Theorem 1.
Generalized gradient descent with a fixed step size $\alpha\leq
1/(||Q^{-1}||_{F}\cdot||\sum_{t=2}^{T}M_{t-1|T}||_{F})$ for minimizing eq.(4)
has convergence rate $O(1/k)$, where k is the number of iterations.
###### Proof.
$g(A)$ is differentiable with respect to _A_ , and its gradient is
$\bigtriangledown
g(A)=Q^{-1}(A\sum_{t=2}^{T}M_{t-1|T}-\sum_{t=2}^{T}M_{t,t-1|T})$. Using simple
algebraic manipulation we arrive at $||\bigtriangledown g(X)-\bigtriangledown
g(Y)||_{F}=||Q^{-1}(X-Y)\sum_{t=2}^{T}M_{t-1|T}||_{F}\leq||Q^{-1}||_{F}\cdot||\sum_{t=2}^{T}M_{t-1|T}||_{F}\cdot||X-Y||_{F}$
where $||\cdot||_{F}$ is the Frobenius norm and the inequality holds because
of the sub-multiplicative property of Frobenius norm. Since we know from
eq.(4), $f(A)=g(A)+\beta||A||_{1}$, and $g(A)$ has Lipschitz continuous
gradient with constant $||Q^{-1}||_{F}\cdot||\sum_{t=2}^{T}M_{t-1|T}||_{F}$,
according to [7, 22],
$f(A^{(k)})-f(A^{*})\leq||A^{(0)}-A^{*}||^{2}_{F}/2\alpha k$, where $A^{(0)}$
is the initial value and $A^{*}$ is the optimal value for $A$; _k_ is the
number of iterations. ∎
Theorem 1 gives us a simple way to set the step size during the generalized
gradient updates and also guarantees the fast convergence rate.
Optimization of $\Omega\backslash A=\\{C,R,Q,\pi_{1},V_{1}\\}$. Each of these
parameters is estimated similarly to the approach in [8] by taking the
corresponding derivative of the eq.(3), setting it to zero, and by solving it
analytically. Update rules for $\Omega\backslash A=\\{C,R,Q,\pi_{1},V_{1}\\}$
are shown in Algorithm 1.
The M-Step for optimizing $\Omega$ is summarized in Algorithm 1 in Appendix.
## 4 Experiments
We test our approach on time series data obtained from electronic health
records of 4,486 post-surgical cardiac patients stored in PCP database [2, 13,
12, 23]. To test the performance of our prediction model, we randomly select
600 patients that have at least 10 _Complete Blood Count_ (CBC) tests 111CBC
panel is used as a broad screening test to check for such disorders as anemia,
infection, and other diseases. ordered during their hospitalizations. The
three tests used in this experiment are Mean Corpuscular Hemoglobin
Concentration (MCHC), Mean Corpuscular Hemoglobin (MCH) and Mean Corpuscular
Volume (MCV). These time series data are noisy, their signals fluctuate in
time, and observations are obtained with varied time-interval period.
In order to get regularly sampled multivariate time series dataset, we apply
an 8-hour discretization on our original multivariate time series dataset and
use linear interpolation to fill the missing gaps from discretization. We
compare our sparse LDS (SLDS) with ordinary LDS (OLDS) on the above
multivariate time series dataset.
To evaluate the performance of our SLDS approach we split our time series for
600 patients into the training and testing sets, such that 50/100 times series
form the training data, and 500 are used for testing.
Evaluation Metric. Our objective is to test the predictive performance of our
approach by its ability to predict the future value of an observation for a
patient for some future time t given a sequence of patient’s past
observations. We judge the quality of the prediction using the Average Mean
Absolute Error (AMAE) on multiple test data predictions. More specifically,
the AMAE is defined as follows:
$AMAE=m^{-1}n^{-1}\sum_{i=1}^{m}\sum_{j=1}^{n}|y_{i,j}-\hat{y}_{i,j}|$ (5)
where $y_{i,j}$ is the _j_ th true observation from time series _i_ ,
$\hat{y}_{i,j}$ is the corresponding predicted value of $y_{i,j}$. _m_ is the
number of time series and _n_ is the length of each time series.
To conduct the evaluation, we use the test dataset to generate various
prediction tasks as follows. For each patient $p$ and complete time series _i_
for that patient, we calculate the number of observations $n_{i}^{p}$ in that
time series _i_. We use $n_{i}^{p}$ to generate different pairs of indices
$(\psi,\phi)$ for that patient, such that $1\leq\psi<\phi\leq n_{i}^{p}$,
where $\psi$ is the index of the last observation assumed to be seen, and
$\phi$ is the index of the observation we would like to predict. By adding
time stamp reading to each index, the two indices help us define all possible
prediction tasks that we can formulate on that time series. For each time
series _i_ from patient _p_ , we proceed by randomly picking 5 different pairs
of indices (or 5 different prediction tasks) for the total of 2500 predictions
tasks (500 x 5 = 2500). For each method, we repeat this random sampling
predictions 10 times and we use the Average Mean Absolute Error (AMAE) on
these tasks to judge the quality of test predictions. The prediction results
are shown in Table 1, Figure 2 and Figure 2.
From Figure 2 and Figure 2, we can see that OLDS achieves its lowest
prediction error _AMAE_ when the number of hidden states is 5. By varying the
number of states, the errors for OLDS first improve (till the optimal number
of states is reached) and then increase when the number of hidden states
exceeds the optimal point. This clearly shows the overfitting problem. The
SLDS performs similarly; its performance first impoves and after that it
deteriorates. However, its errors deteriorate at slower pace which shows it is
more robust to the overfitting problem. Comparing the two methods, the SLDS
always outperfroms the OLDS, indicating that the additional sparsity term
included in optimization helps it to better fit the underlying structure of
the transition matrix.
Table 1: Average mean absolute error for OLDS and SLDS with different hidden states sizes. # of states | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 12 | 15
---|---|---|---|---|---|---|---|---|---|---
OLDS(50) | 1.1653 | 1.0553 | 1.0561 | 0.6667 | 0.7268 | 0.9209 | 0.9502 | 0.9402 | 1.2291 | 1.2993
SLDS(50) | 0.6256 | 0.5858 | 0.5821 | 0.5684 | 0.6506 | 0.6200 | 0.8854 | 0.9236 | 1.0134 | 1.0430
OLDS(100) | 1.1527 | 1.0427 | 1.0039 | 0.6406 | 0.7153 | 0.8364 | 0.9210 | 0.9327 | 1.1427 | 1.2427
SLDS(100) | 0.5709 | 0.5429 | 0.5889 | 0.6379 | 0.6897 | 0.6949 | 0.7643 | 0.7811 | 0.7874 | 0.8309
Figure 1: AMAE on 50 training examples.
Figure 2: AMAE on 100 training examples.
## 5 Conclusion
In this paper, we have presented a sparse linear dynamical system (SLDS) for
multivariate time series predictions. Comparing with the traditional linear
state-space systems, SLDS model tries to (1) prevent the overfitting problem
and (2) represent additional structure in the transition matrix. Experimental
results on real world clinical data from electronic health records systems
demonstrated that this novel model achieves errors that is statistically
significantly lower than errors of ordinary linear dynamical system. We would
like to note that the results presented in this work are preliminary and
include only three time series. Further investigation of more complex higher
dimensional time-series data is needed and will be conducted in the future. In
addition, we would like to study group lasso regularization techniques which
we believe would be able to better control the dimensionality of the hidden
state space. Finally, we plan to study extensions of our model to switching-
state and controlled dynamical systems [14, 17].
Acknowledgement: This research work was supported by grants R01LM010019 and
R01GM088224 from the National Institutes of Health. Its content is solely the
responsibility of the authors and does not necessarily represent the official
views of the NIH. We would like to thank Eric Heim and Mahdi Pakdaman for
useful discussions and comments on this work.
## References
* [1] Iyad Batal, Dmitriy Fradkin, James Harrison, Fabian Moerchen, and Milos Hauskrecht. Mining recent temporal patterns for event detection in multivariate time series data. In SIGKDD, pages 280–288, 2012.
* [2] Iyad Batal, Hamed Valizadegan, Gregory F Cooper, and Milos Hauskrecht. A pattern mining approach for classifying multivariate temporal data. In BIBM, pages 358–365. IEEE, 2011.
* [3] Iyad Batal, Hamed Valizadegan, Gregory F Cooper, and Milos Hauskrecht. A temporal pattern mining approach for classifying electronic health record data. TIST, Special Issue on Health Informatics, 2013.
* [4] Adam Charles, Muhammad Salman Asif, Justin Romberg, and Christopher Rozell. Sparsity penalties in dynamical system estimation. In Information Sciences and Systems (CISS), 2011 45th Annual Conference on, pages 1–6. IEEE, 2011.
* [5] Scott Shaobing Chen, David L Donoho, and Michael A Saunders. Atomic decomposition by basis pursuit. SIAM journal on scientific computing, 20(1):33–61, 1998.
* [6] Alessandro Chiuso and Gianluigi Pillonetto. Learning sparse dynamic linear systems using stable spline kernels and exponential hyperpriors. In NIPS, pages 397–405, 2010.
* [7] Massimo Fornasier and Holger Rauhut. Iterative thresholding algorithms. Applied and Computational Harmonic Analysis, 25(2):187–208, 2008\.
* [8] Zoubin Ghahramani and Geoffrey E Hinton. Parameter estimation for linear dynamical systems. Technical report, Technical Report CRG-TR-96-2, University of Totronto, 1996.
* [9] Bernard Ghanem and Narendra Ahuja. Sparse coding of linear dynamical systems with an application to dynamic texture recognition. In Pattern Recognition (ICPR), 2010 20th International Conference on, pages 987–990. IEEE, 2010.
* [10] Yue Guan and Jennifer G Dy. Sparse probabilistic principal component analysis. In Proceedings of AISTATS, volume 5, pages 185–192, 2009.
* [11] James Douglas Hamilton. Time series analysis, volume 2. Cambridge Univ Press, 1994.
* [12] M. Hauskrecht, M. Valko, I. Batal, G. Clermont, S. Visweswaran, and G.F. Cooper. Conditional outlier detection for clinical alerting. In AMIA Annual Symposium Proceedings, volume 2010, page 286. American Medical Informatics Association, 2010.
* [13] Milos Hauskrecht, Iyad Batal, Michal Valko, Shyam Visweswaran, Gregory F Cooper, and Gilles Clermont. Outlier detection for patient monitoring and alerting. Journal of Biomedical Informatics, 2012.
* [14] Milos Hauskrecht and Hamish Fraser. Modeling treatment of ischemic heart disease with partially observable markov decision processes. Proceedings of the AMIA Symposium, page 538, 1998.
* [15] Rudolph Emil Kalman et al. A new approach to linear filtering and prediction problems. Journal of basic Engineering, 82(1):35–45, 1960.
* [16] T. Katayama. Subspace methods for system identification. Springer, 2005.
* [17] Branislav Kveton and Milos Hauskrecht. Solving factored mdps with exponential-family transition models. In ICAPS, pages 114–120, 2006.
* [18] Zitao Liu and Milos Hauskrecht. Clinical time series prediction with a hierarchical dynamical system. In Artificial Intelligence in Medicine, pages 227–237. Springer, 2013.
* [19] Zitao Liu, Lei Wu, and Milos Hauskrecht. Modeling clinical time series using gaussian process sequences. In SIAM International Conference on Data Mining (SDM), pages 623–631, 2013.
* [20] Lennart Ljung and Torkel Glad. Modeling of dynamic systems. 1994\.
* [21] P.V. Overschee, BLD Moor, D.A. Hensher, J.M. Rose, W.H. Greene, K. Train, W. Greene, E. Krause, J. Gere, and R. Hibbeler. Subspace Identification for the Linear Systems: Theory–Implementation. Boston: Kluwer AcademicPublishers, 1996.
* [22] NZ Shor. The rate of convergence of the generalized gradient descent method. Cybernetics and Systems Analysis, 4(3):79–80, 1968.
* [23] Michal Valko and Milos Hauskrecht. Feature importance analysis for patient management decisions. In MEDINFO, 2010.
* [24] Peter M Williams. Bayesian regularization and pruning using a laplace prior. Neural computation, 7(1):117–143, 1995.
## Appendix
Algorithm 1 EM: M-step for the ($k+1$)th iteration. (We omit the explicit
superscript $(k+1)$ for notational brevity.)
INPUT:
* •
Observation sequence $y_{t}$s, $t=1,\ldots,T$.
* •
Sufficient statistics $\hat{z}_{t|T}$, $M_{t|T}$, $M_{t,t-1|T}$,
$i=1,\ldots,T$ from the ($k+1$)th iteration in E-Step.
PROCEDURE:
1: Update $\Omega\backslash A$:
$C=(\sum_{t=1}^{T}y_{t}\hat{z}_{t|T}^{{}^{\prime}})(\sum_{t=1}^{T}M_{t|T})^{-1}$,
$Q=\frac{1}{T-1}(\sum_{t=2}^{T}M_{t|T}-A\sum_{t=2}^{T}M_{t-1,t|T})$,
$R=\frac{1}{T}\sum_{t=1}^{T}(y_{t}y_{t}^{{}^{\prime}}-C\hat{z}_{t|T}y_{t}^{{}^{\prime}})$,
$\pi_{1}=\hat{z}_{1|T}$,
$V_{1}=M_{1|T}-\hat{z}_{1|T}\hat{z}_{1|T}^{{}^{\prime}}$.
2: Initialize $A$,
$A=(\sum_{t=2}^{T}M_{t,t-1|T})(\sum_{t=2}^{T}M_{t-1|T})^{-1}$.
3: repeat
4: Compute the fixed step size $\alpha$,
$\alpha=1/(||{Q}^{-1}||_{F}\cdot||\sum_{t=2}^{T}M_{t-1|T}||_{F})$.
5: Compute gradient of $g(A)$, $\bigtriangledown
g(A)={Q}^{-1}A\sum_{t=2}^{T}M_{t-1|T}-{Q}^{-1}\sum_{t=2}^{T}M_{t,t-1|T}$.
6: Update $A$, $A=S_{\beta\alpha}(A-t\bigtriangledown g(A))$.
7: until Convergence
OUTPUT: $\Omega^{(k+1)}=\\{A,C,Q,R,\boldsymbol{\pi}_{1},V_{1}\\}$.
|
arxiv-papers
| 2013-11-27T18:58:07 |
2024-09-04T02:49:54.392732
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zitao Liu and Milos Hauskrecht",
"submitter": "Zitao Liu",
"url": "https://arxiv.org/abs/1311.7071"
}
|
1311.7110
|
On the Pursuit of Generalizations for the Petrov Classification and the
Goldberg-Sachs Theorem
Carlos Batista
Doctoral Thesis
Universidade Federal de Pernambuco, Departamento de Física
Supervisor: Bruno Geraldo Carneiro da Cunha
Brazil - November - 2013
Thesis presented to the graduation program of the Physics Department of
Universidade Federal de Pernambuco as part of the duties to obtain the degree
of Doctor of Philosophy in Physics.
Examining Board:
Prof. Amilcar Rabelo de Queiroz (IF-UNB, Brazil)
Prof. Antônio Murilo Santos Macêdo (DF-UFPE, Brazil)
Prof. Bruno Geraldo Carneiro da Cunha (DF-UFPE, Brazil)
Prof. Fernando Roberto de Luna Parisio Filho (DF-UFPE, Brazil)
Prof. Jorge Antonio Zanelli Iglesias (CECs, Chile)
Abstract
The Petrov classification is an important algebraic classification for the
Weyl tensor valid in 4-dimensional space-times. In this thesis such
classification is generalized to manifolds of arbitrary dimension and
signature. This is accomplished by interpreting the Weyl tensor as a linear
operator on the bundle of $p$-forms, for any $p$, and computing the Jordan
canonical form of this operator. Throughout this work the spaces are assumed
to be complexified, so that different signatures correspond to different
reality conditions, providing a unified treatment. A higher-dimensional
generalization of the so-called self-dual manifolds is also investigated.
The most important result related to the Petrov classification is the
Goldberg-Sachs theorem. Here are presented two partial generalizations of such
theorem valid in even-dimensional manifolds. One of these generalizations
states that certain algebraic constraints on the Weyl “operator” imply the
existence of an integrable maximally isotropic distribution. The other version
of the generalized Goldberg-Sachs theorem states that these algebraic
constraints imply the existence of a null congruence whose optical scalars
obey special restrictions.
On the pursuit of these results the spinorial formalism in 6 dimensions was
developed from the very beginning, using group representation theory. Since
the spinors are full of geometric significance and are suitable tools to deal
with isotropic structures, it should not come as a surprise that they provide
a fruitful framework to investigate the issues treated on this thesis. In
particular, the generalizations of the Goldberg-Sachs theorem acquire an
elegant form in terms of the pure spinors.
Keywords: General relativity, Weyl tensor, Petrov classification,
Integrability, Isotropic distributions, Goldberg-Sachs theorem, Spinors,
Clifford algebra.
This thesis is based on the following published articles:
$\bullet$ Carlos Batista, Weyl tensor classification in four-dimensional
manifolds of all signatures, General Relativity and Gravitation 45 (2013),
785\.
$\bullet$ Carlos Batista, A generalization of the Goldberg-Sachs theorem and
its consequences, General Relativity and Gravitation 45 (2013), 1411.
$\bullet$ Carlos Batista and Bruno G. Carneiro da Cunha, Spinors and the Weyl
tensor classification in six dimensions, Journal of Mathematical Physics 54
(2013), 052502\.
$\bullet$ Carlos Batista, On the Weyl tensor classification in all dimensions
and its relation with integrability properties, Journal of Mathematical
Physics 54 (2013), 042502.
Acknowledgments
In order for such a long work, lasting almost five years, to succeed it is
unavoidable to have the aid and the support of a lot of people. In this
section I would like to sincerely thank to everybody that contributed in some
way to my doctoral course.
I want to acknowledge my supervisor, Bruno Geraldo Carneiro da Cunha, for the
sensitivity in suggesting a research project that fully matches my
professional tastes. I also thank for all advise he gave me during our
frequent meetings. It is inspiring to be supervised by such a wise scientist
as Professor Bruno. Finally I thank for the freedom and the continued support
he provided me, so that I could follow my own track. I take the chance to
acknowledge all other Professors from UFPE that contributed to my education,
particularly the Professors Antônio Murilo, Sérgio Coutinho, Henrique Araújo
and Liliana Gheorghe, whose knowledge and commitment have inspired me.
In the same vein, I thank to all the mates as well as to the staff of the
physics department. Specially, I thank to my doctorate fellow Fábio Novaes
Santos for all the times he patiently helped me, thank you very much. I also
would like to mention my friends Carolina Cerqueira, Danilo Pinheiro, Diego
Leite and Rafael Alves, who contributed for a more pleasant environment in the
physics department. I acknowledge the really qualified and efficient work of
the graduation secretary Alexsandra Melo as well as the friendship and support
of the under-graduation secretary Paula Franssinete.
Finally, and most importantly, I would like to thank for the unconditional
support of all my family. Particularly, I thank to my mother, Ana Lúcia, and
to my sister, Natália Augusta, for always encouraging me to study, since my
childhood, as well as stimulating my vocation. I also thank to my parents in
law, Guilherme e Lúcia Helena, for taking responsibility on the construction
of my house, what allowed me to proceed using my whole time to study. To
conclude, I want to effusively and repeatedly thank to my wife, Juliana. In
addition for her being my major inspiration, she supports me and encourages me
like no one else. There are no words to say how much I am glad for having her
besides me. I love you, my wife!!
During my Ph.D. I received financial support from CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior) and CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico). It is worth mentioning that I
really appreciate doing what I enjoy, study physics and mathematics, on my own
country and still be paid for this. I will do my best, as I always tried to,
in order for my work as a researcher and as a Professor, in the future, to
return this investment.
> _“The black holes of nature are the most perfect macroscopic objects there
> are in the universe: the only elements in their construction are our
> concepts of space and time. And since the general theory of relativity
> provides only a single unique family of solutions for their descriptions,
> they are the simplest objects as well.”_
>
>
> Subrahmanyan Chandrasekhar
> (The Mathematical Theory of Black Holes)
###### Contents
1. 0 Motivation and Outline
2. I Review and Classical Results
1. 1 Introducing General Relativity
1. 1.1 Gravity is Curvature
2. 1.2 Riemannian Geometry, the Formalism of Curved Spaces
3. 1.3 Geodesics
4. 1.4 Symmetries and Conserved Quantities
5. 1.5 Einstein’s Equation
6. 1.6 Differential Forms
7. 1.7 Cartan’s Structure Equations
8. 1.8 Distributions and Integrability
9. 1.9 Higher-Dimensional Spaces
2. 2 Petrov Classification, Six Different Approaches
1. 2.1 Weyl Tensor as an Operator on the Bivector Space
2. 2.2 Annihilating Weyl Scalars
3. 2.3 Boost Weight
4. 2.4 Bel-Debever and Principal Null Directions
5. 2.5 Spinors, Penrose’s Method
6. 2.6 Clifford Algebra
7. 2.7 Interpreting the PNDs
8. 2.8 Examples
9. 2.9 Other Classifications
3. 3 Some Theorems on Petrov Types
1. 3.1 Shear, Twist and Expansion
2. 3.2 Goldberg-Sachs
3. 3.3 Mariot-Robinson
4. 3.4 Peeling Property
5. 3.5 Symmetries
3. II Original Research
1. 4 Generalizing the Petrov Classification and the Goldberg-Sachs Theorem to All Signatures
1. 4.1 Null Frames
2. 4.2 Generalized Petrov Classification
1. 4.2.1 Euclidean Signature
2. 4.2.2 Lorentzian Signature
3. 4.2.3 Split Signature
4. 4.2.4 Annihilating Weyl Scalars
3. 4.3 Generalized Goldberg-Sachs Theorem
4. 4.4 Geometric Consequences of the Generalized Goldberg-Sachs Theorem
1. 4.4.1 Complex Manifolds
2. 4.4.2 General Results
3. 4.4.3 Euclidean Signature
4. 4.4.4 Lorentzian Signature
5. 4.4.5 Split Signature
2. 5 Six Dimensions Using Spinors
1. 5.1 From Vectors to Spinors
1. 5.1.1 A Null Frame
2. 5.1.2 Clifford Algebra in 6 Dimensions
3. 5.1.3 Isotropic Subspaces
2. 5.2 Other Signatures
3. 5.3 An Algebraic Classification for the Weyl Tensor
4. 5.4 Generalized Goldberg-Sachs
1. 5.4.1 Lorentzian Signature
5. 5.5 Example, Schwarzschild in 6 Dimensions
3. 6 Integrability and Weyl Tensor Classification in All Dimensions
1. 6.1 Algebraic Classification for the Weyl Tensor
1. 6.1.1 Inner Product of $p$-forms
2. 6.1.2 Even Dimensions
3. 6.1.3 An Elegant Notation
2. 6.2 Integrability of Maximally Isotropic Distributions
3. 6.3 Optical Scalars and Harmonic Forms
4. 6.4 Generalizing Mariot-Robinson and Goldberg-Sachs Theorems
4. 7 Conclusion and Perspectives
5. A Segre Classification and its Refinement
6. B Null Tetrad Frame
7. C Clifford Algebra and Spinors
8. D Group Representations
### Chapter 0 Motivation and Outline
The so called Petrov classification is an algebraic classification for the
Weyl tensor of a 4-dimensional curved space-time that played a prominent role
in the development of general relativity. Particularly, it helped on the
search of exact solutions for Einstein’s equation, the most relevant example
being the Kerr metric. Furthermore, such classification contributed for the
physical understanding of gravitational radiation. There are several theorems
concerning this classification, they associate the Petrov type of the Weyl
tensor with physical and geometric properties of the space-time. Probably the
most important of these theorems is the Goldberg-Sachs theorem, which states
that in vacuum the Weyl tensor is algebraically special if, and only if, the
space-time admits a shear-free congruence of null geodesics. It was because of
this theorem that Kinnersley was able to find all type $D$ vacuum solutions
for Einstein’s equation, an impressive result given that such equation is
highly non-linear.
Since the Petrov classification and the Goldberg-Sachs theorem have been of
major importance for the study of 4-dimensional Lorentzian spaces, it is quite
natural trying to generalize these results to manifolds of arbitrary dimension
and signature. This is the goal of the present thesis. In what follows the
Petrov classification will be extended to all dimensions and signatures in a
geometrical approach. Moreover, there will be presented few generalizations of
the Goldberg-Sachs theorem valid in even-dimensional spaces. The relevance of
this work is enforced by the increasing significance of higher-dimensional
manifolds in physics and mathematics.
This thesis was split in two parts. The part I shows the classical results
concerning the Petrov classification and its associated theorems, while part
II presents the work developed by the present author during the doctoral
course. In chapter 1 the basic tools of general relativity and differential
geometry necessary for the understanding of this thesis are reviewed. It is
shown that gravity manifests itself as the curvature of the space-time and it
is briefly discussed the relevance of higher-dimensional manifolds. Chapter 2
shows six different routes to define the Petrov classification. In addition,
the so called principal null directions are interpreted from the physical and
geometrical points of view. Chapter 3 presents some of the most important
theorems concerning the Petrov classification, as the Goldberg-Sachs, the
Mariot-Robinson and the Peeling theorems. In chapter 4 the Petrov
classification is generalized to 4-dimensional spaces of arbitrary signature
in a unified approach, with each signature being understood as a choice of
reality condition on a complex space. Moreover, it is shown that this
generalized classification is related to the existence of important geometric
structures. Chapter 5 develops the spinorial formalism in 6 dimensions with
the aim of uncovering results that are hard to perceive by means of the
standard vectorial approach. In particular, the spinorial language reveals
that the Weyl tensor can be seen as an operator on the space of 3-vectors,
which is exploited in order to classify this tensor. It is also proved an
elegant partial generalization of the Goldberg-Sachs theorem making use of the
concept of pure spinors. An algebraic classification for the Weyl tensor valid
in arbitrary dimension and signature is then developed in chapter 6, where it
is also proved two partial generalizations of the Goldberg-Sachs theorem valid
in even-dimensional manifolds. Finally, chapter 7 discuss the conclusions and
perspectives of this work.
Some background material is also presented in the appendices. Appendix A
introduces a classical algebraic classification for square matrices called the
Segre classification and defines a refinement for it. Such refined
classification is used throughout the thesis. Appendix B describes what a null
tetrad is. The formal treatment of Clifford algebra and spinors is addressed
in appendix C, where some pedagogical examples are also worked out. Finally,
appendix D introduces and give some examples of the basics concepts on group
representation theory.
## Part I Review and Classical Results
### Chapter 1 Introducing General Relativity
Right after Albert Einstein arrived at his special theory of relativity, in
1905, he noticed that the Newtonian theory of gravity needed to be modified.
Newton’s theory predict that when a gravitational system is perturbed the
effect of such perturbation is immediately felt at all points of space, in
other words the gravitational interaction propagates with infinite velocity.
This, however, is in contradiction with one of the main results of special
relativity, that no information can propagate faster than light. Moreover,
according to Einstein’s results energy and mass are equivalent, which implies
that the light must feel the gravitational attraction, in disagreement with
the Newtonian gravitational theory.
It took long 10 years for Einstein to establish a relativistic theory of
gravitation, the General Theory of Relativity. In spite of the sophisticated
mathematical background necessary to understand this theory, it turns out that
it has a beautiful geometrical interpretation. According to general
relativity, gravity shows itself as the curvature of the space-time. Such
theory has had several experimental confirmations, notably the correct
prediction of Mercury’s perihelion precession and the light deflection. In
particular, it is worth noting that the GPS technology strongly relies on the
general theory of relativity.
The aim of the present chapter is to describe the basic tools of general
relativity necessary in the rest of the thesis. Readers already familiar with
such theory are encouraged to skip this chapter. Throughout this thesis it
will be assumed that repeated indices are summed, the so-called Einstein
summation convention. The symmetrization and anti-symmetrization of indices
are respectively denoted by round and square brackets. So that, for instance,
$T_{(\mu\nu)}=\frac{1}{2}(T_{\mu\nu}+T_{\nu\mu})$ and
$L_{[\mu\nu\rho]}=\frac{1}{6}(L_{\mu\nu\rho}+L_{\nu\rho\mu}+L_{\rho\mu\nu}-L_{\nu\mu\rho}-L_{\rho\nu\mu}-L_{\mu\rho\nu})$.
#### 1.1 Gravity is Curvature
According to the special theory of relativity we live in a four-dimensional
flat space-time endowed with the metric:
$ds^{2}\,=\,\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,=\,dt^{2}\,-\,dx^{2}\,-\,dy^{2}\,-\,dz^{2}\,,$
where $\\{x^{\mu}\\}=\\{t,x,y,z\\}$ are cartesian coordinates. Note that if we
make a Poincaré transformation,
$x^{\mu}\mapsto\Lambda^{\mu}_{\phantom{\mu}\nu}x^{\nu}+a^{\mu}$, where
$a^{\mu}$ is constant and
$\eta_{\rho\sigma}\Lambda^{\rho}_{\phantom{\rho}\mu}\Lambda^{\sigma}_{\phantom{\sigma}\nu}=\eta_{\mu\nu}$,
then the metric remains invariant. Physically, performing a Poincaré
transformation means changing from one inertial frame to another, which should
not change the Physics. But, in addition to the inertial coordinates we are
free to use any coordinate system of our preference. For example, in a
particular problem it might be convenient to use spherical coordinates on the
space. The procedure of changing coordinates is simple, for example, if
$g_{\mu\nu}$ is the metric on the coordinate system $\\{x^{\mu}\\}$ then using
new coordinates, $\\{x^{\prime\mu}\\}$, we have:
$g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,=\,g_{\mu\nu}\,\left(\frac{\partial
x^{\mu}}{\partial x^{\prime\rho}}dx^{\prime\rho}\right)\,\left(\frac{\partial
x^{\nu}}{\partial
x^{\prime\sigma}}dx^{\prime\sigma}\right)\;\Rightarrow\;g^{\prime}_{\rho\sigma}\,=\,\frac{\partial
x^{\mu}}{\partial x^{\prime\rho}}\frac{\partial x^{\nu}}{\partial
x^{\prime\sigma}}\,g_{\mu\nu}\,.$
Where $g^{\prime}_{\rho\sigma}$ are the components of the metric on the
coordinates $\\{x^{\prime\mu}\\}$. In general, if
$T^{\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}$
are the components of a tensor $\boldsymbol{T}$ on the coordinate system
$\\{x^{\mu}\\}$, then its components on the coordinates $\\{x^{\prime\mu}\\}$
are:
$T^{\prime\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}\,=\,\left(\frac{\partial
x^{\prime\mu_{1}}}{\partial x^{\rho_{1}}}\ldots\frac{\partial
x^{\prime\mu_{p}}}{\partial x^{\rho_{p}}}\right)\,\left(\frac{\partial
x^{\sigma_{1}}}{\partial x^{\prime\nu_{1}}}\ldots\frac{\partial
x^{\sigma_{q}}}{\partial
x^{\prime\nu_{q}}}\right)\,T^{\rho_{1}\ldots\rho_{p}}_{\phantom{\rho_{1}\ldots\rho_{p}}\sigma_{1}\ldots\sigma_{q}}\,.$
(1.1)
So far so good. But there is one important thing whose transformation under
coordinate changes is non trivial, the derivative. Let $V^{\mu}$ be the
components of a vector on the coordinate system $\\{x^{\mu}\\}$. Then it is a
simple matter to prove that $\partial_{\nu}V^{\mu}$ does not transform as a
tensor under a general coordinate change. Nevertheless, after some algebra, it
can be proved that defining
$\Gamma^{\mu}_{\nu\rho}\,\equiv\,\frac{1}{2}\,g^{\mu\sigma}\left(\partial_{\nu}g_{\rho\sigma}+\partial_{\rho}g_{\nu\sigma}-\partial_{\sigma}g_{\nu\rho}\right)\,,$
(1.2)
with $g^{\mu\nu}$ being the inverse of $g_{\mu\nu}$ and $\partial_{\nu}$ being
the partial derivative with respect to the coordinate $x^{\nu}$, then the
combination
$\displaystyle\nabla_{\nu}\,V^{\mu}\,\equiv\,\frac{\partial V^{\mu}}{\partial
x^{\nu}}\,+\,\Gamma^{\mu}_{\nu\rho}\,V^{\rho}\,=\,\partial_{\nu}\,V^{\mu}\,+\,\Gamma^{\mu}_{\nu\rho}\,V^{\rho}$
(1.3)
does transform as a tensor. The object $\Gamma^{\mu}_{\nu\rho}$, called
Christoffel symbol (it is not a tensor), serves to correct the non-tensorial
character of the partial derivative. The operator $\nabla_{\nu}$ is called the
covariant derivative, it has the remarkable property that when acting on a
tensor it yields another tensor. Its action on a general tensor is, for
example,
$\nabla_{\nu}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\mu_{3}}\,=\,\partial_{\nu}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\mu_{3}}\,+\,\Gamma^{\mu_{1}}_{\nu\sigma}\,T^{\sigma}_{\phantom{\sigma}\mu_{2}\mu_{3}}-\Gamma^{\sigma}_{\nu\mu_{2}}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\sigma\mu_{3}}-\Gamma^{\sigma}_{\nu\mu_{3}}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\sigma}\,.$
(1.4)
Using this formula it is straightforward to prove that
$\nabla_{\rho}g_{\mu\nu}=0$, so the metric is covariantly constant. Since
coordinates are physically meaningless we should always work with tensorial
objects, because they are invariant under coordinate changes. Therefore, we
should only use covariant derivatives instead of partial derivatives. Although
they seem awkward, the covariant derivatives are, actually, quite common. For
instance, in 3-dimensional calculus it is well-known that the divergence of a
vector field in spherical coordinates looks different than in cartesian
coordinates, this happens because we are implicitly using the covariant
derivative.
Now comes a puzzle. From the physical point of view one might expect that no
reference frame is better than another, all of them are equally arbitrary. In
particular, the concept of acceleration is relative, since according to the
classical Einstein’s mental experiment (Gedankenexperiment) gravity and
acceleration are locally indistinguishable, the so-called equivalence
principle. In spite of this, Minkowski space-time has an infinite class of
privileged frames, the cartesian frames (also called inertial frames). From
the geometrical point of view these frames are special because the Christoffel
symbols, $\Gamma^{\mu}_{\nu\sigma}$, vanish identically in all points. But, as
just advocated, the existence of these preferred frames is not a reasonable
assumption. Therefore, we conclude that the space-time should not admit the
existence of a frame such that $\Gamma^{\mu}_{\nu\sigma}$ vanishes in all
points. Geometrically this implies that the space-time is curved! Somebody
could argue that the inertial frames represent non-accelerated observers and,
therefore, may exist. But our universe is full of mass everywhere, which
implies that the gravitational field is omnipresent. Using then the
equivalence principle we conclude that all objects are accelerated, so that it
is nonsense to admit the existence of globally non-accelerated frames. Now we
might wonder ourselves: If the space-time is not flat then why has special
relativity been so successful? The reason is that in every point of a curved
space-time we can always choose a reference frame such that
$g_{\mu\nu}=\eta_{\mu\nu}$ and $\Gamma^{\mu}_{\nu\sigma}=0$ at this point.
Hence, special relativity is always valid locally.
Another natural question that emerges is: What causes space-time bending? Let
us try to answer this. In special relativity a free particle moves on straight
lines, which are the geodesics of flat space-time. Analogously, on a curved
space-time the free particles shall move along the geodesics. Thus, no matter
the peculiarities of a particle, if it is free it will follow the geodesic
path compatible with its initial conditions of position and velocity. This
resembles gravity, which, due to the equality of the inertial and
gravitational masses, is such that all particles with the same initial
condition follow the same trajectory. For example, a canon-ball and a feather
both acquire the same acceleration under the gravitational field. Therefore,
it is reasonable to say that the gravity bends the space-time. There is
another path which leads us to the same conclusion. In line with Einstein’s
elevator experiment, gravity is locally equivalent to acceleration. Now
suppose we are in a reference frame such that $\Gamma^{\mu}_{\nu\sigma}=0$,
then if this referential is accelerated it is simple matter to verify that the
Christoffel symbol will be different from zero. Thus acceleration is related
to the non-vanishing of $\Gamma^{\mu}_{\nu\sigma}$. Furthermore, the lack of a
coordinate system such that $\Gamma^{\mu}_{\nu\sigma}=0$ in all points of the
space-time implies that the space-time is curved. So that we arrive at the
following relations:
$\textrm{Gravity}\;\;\longleftrightarrow\;\;\textrm{Acceleration}\;\;\longleftrightarrow\;\;\Gamma^{\mu}_{\nu\sigma}\neq
0\;\;\longleftrightarrow\;\;\textrm{Curvature}\,,$
which again leads us to the conclusion that gravity causes the curvature of
the space-time. This is the main content of the General Theory of Relativity.
In the standard model of particles the fundamental forces of nature are
transmitted by bosons: photons carry the electromagnetic force, $W$ and $Z$
bosons communicate the weak interaction and gluons transmit the strong nuclear
force. In the same vein, the gravitational interaction might be carried by a
boson, dubbed the graviton. Indeed, heuristically speaking, since the emission
of a particle of non-integer spin changes the total angular momentum of the
system111For instance, suppose that a particle has integer spin and then emits
a fermion. So, by the rule of angular momenta addition (see eq. (D.3) in
appendix D), it follows that its angular momentum after the emission is a
superposition of non-integer values. Therefore it must have changed. it
follows that interactions carried by fermions are generally incompatible with
the existence of static forces [1]. Now comes the question: What are the mass
and the spin of the graviton? Since the gravitational force has a long range
(energy goes as $1/r$) it follows that the mass must be zero, just as the mass
of the photon. Moreover, since the graviton is a boson its spin must be
integer. One can prove that it must be different from zero, since a scalar
theory of gravitation predicts that the light is not affected by gravity [2],
which contradicts the experiments and the fact that energy and mass are
equivalent. The spin should also be different from one, since the interaction
carried by a massless particle of spin one is the electromagnetic force which
can be both attractive and repulsive, whereas gravity only attracts. It turns
out that the graviton has spin 2. Indeed, in [1] it is shown how to start from
the theory of a massless spin 2 particle on flat space-time and arrive at the
general theory of relativity. For a wonderful introductory course in general
relativity see [3]. More advanced texts are available at [4, 5]. Historical
remarks and interesting philosophical thoughts can be found in [6].
#### 1.2 Riemannian Geometry, the Formalism of
Curved Spaces
In order to make calculations on general relativity it is of fundamental
importance to get acquainted with the tools of Riemannian geometry. The intent
of the present section is to briefly introduce the bare minimum concepts on
such subject necessary for the understanding of this thesis.
Roughly, an $n$-dimensional manifold $M$ is a smooth space such that locally
it looks like $\mathbb{R}^{n}$. For example, the 2-sphere is a 2-dimensional
manifold, since it is smooth and if we look very close to some patch of the
spherical surface it will look like a flat plane (the Earth surface is round,
but for its inhabitants it, locally, looks like a plane). More precisely, a
manifold of dimension $n$ is a topological set such that the neighborhood of
each point can be mapped into a patch of $\mathbb{R}^{n}$ by a coordinate
system in a way that the overlapping neighborhoods are consistently joined [4,
7]. Now imagine curves passing through a point $p$ belonging to the surface of
the 2-sphere. The possible directions that these curves can take generate a
plane, called the tangent space of $p$. Generally, associated to each point
$p\in M$ of an $n$-dimensional manifold we have a vector space of dimension
$n$, denoted by $T_{p}M$ and called the tangent space of $p$. A vector field
$\boldsymbol{V}$ is then a map that associates to every point of the manifold
a vector belonging to its tangent space. The union of the tangent spaces of
all points of a manifold $M$ is called the tangent bundle and denoted by $TM$.
A vector field is just an element of the tangent bundle.
Now, suppose that we introduce a coordinate system $\\{x^{\mu}\\}$ in the
neighborhood of $p\in M$ and let $\boldsymbol{V}$ be a vector field in this
neighborhood. Denoting by $V^{\mu}$ the components of $\boldsymbol{V}$ on such
coordinate system then it is convenient to use the following abstract
notation:
$\boldsymbol{V}\,=\,V^{\mu}\,\frac{\partial\,}{\partial
x^{\mu}}\,\equiv\,V^{\mu}\,\partial_{\mu}\,.$
This is useful because when we make a coordinate transformation,
$x^{\mu}\mapsto x^{\prime\mu}$, and use the chain rule to transform the
partial derivative we find that the components of the vector field change just
as displayed in (1.1). Therefore, the vector fields on a manifold can be
interpreted as differential operators that act on the space of functions over
the manifold. Furthermore, the partial derivatives $\\{\partial_{\mu}\\}$
provide a basis for the tangent space at each point, forming the so-called
coordinate frame. For example, on the 2-sphere we can say that
$\\{\partial_{\theta},\partial_{\phi}\\}$ is a coordinate frame, where
$\theta$ is the polar angle while $\phi$ denotes the azimuthal angle.
A metric $\boldsymbol{g}$ is a symmetric non-degenerate map that act on two
vector fields and gives a function over the manifold. In this thesis it will
always be assumed that the manifold is endowed with a metric, hence the pair
$(M,\boldsymbol{g})$ will sometimes be called the manifold. In particular,
note that the Minkowski manifold is $(\mathbb{R}^{4},\eta_{\mu\nu})$. The
components of the metric on a coordinate frame are denoted by
$g_{\mu\nu}=\boldsymbol{g}(\partial_{\mu},\partial_{\nu})$. By conveniently
choosing a coordinate frame, we can always manage to put the matrix
$g_{\mu\nu}$ in a diagonal form such that all slots are $\pm 1$ at some
arbitrary point $p\in M$, $g_{\mu\nu}\mapsto
g^{\prime}_{\mu\nu}=\operatorname{diag}(1,1,\ldots,-1,-1,\ldots)$. The modulus
of the metric trace when it is in such diagonal form is called the signature
of the metric and denoted by $s$, $s=|\Sigma_{\mu}\,g^{\prime}_{\mu\mu}|$.
Denoting by $n$ the dimension of the manifold then if $s=n$ the metric is said
to be Euclidean, for $s=(n-2)$ the signature is Lorentzian and if $s=0$ the
metric is said to have split signature. In Riemannian geometry it is customary
to low and raise indices using the metric, $g_{\mu\nu}$, and its inverse,
$g^{\mu\nu}$.
The partial derivative of a scalar function,
$\partial_{\mu}f\equiv\nabla_{\mu}f$, is a tensor. But, as discussed in the
preceding section, when acting on tensors this partial derivative must be
replaced by the covariant derivative, defined on equations (1.2) and (1.4). In
the formal jargon, this tensorial derivative is called a connection.
Particularly, the connection defined by (1.2) and (1.4) is named the Levi-
Civita connection. The covariant derivative share many properties with the
usual partial derivative, it is linear and obey the Leibniz rule. However,
these two derivatives also have a big difference: while the partial
derivatives always commute, the covariant derivatives generally do not. More
precisely it is straightforward to prove that:
$\displaystyle(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})\,V^{\rho}\,=\,R^{\rho}_{\phantom{\rho}\sigma\mu\nu}\,V^{\sigma}\,,$
(1.5) $\displaystyle
R^{\rho}_{\phantom{\rho}\sigma\mu\nu}\,\equiv\,\partial_{\mu}\Gamma^{\rho}_{\sigma\nu}-\partial_{\nu}\Gamma^{\rho}_{\sigma\mu}+\Gamma^{\rho}_{\kappa\mu}\Gamma^{\kappa}_{\sigma\nu}-\Gamma^{\rho}_{\kappa\nu}\Gamma^{\kappa}_{\sigma\mu}\,.$
(1.6)
The object $R^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ is called the Riemann
tensor. Although its definition was made in terms of the non-tensorial
Christoffel symbols, $R^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ is indeed a
tensor, as the left hand side of equation (1.5) is a tensor. The Riemann
tensor is also called the curvature tensor, because it measures the curvature
of the manifold222Actually it measures the curvature of the tangent bundle..
In particular, a manifold is flat if, and only if, the Riemann tensor
vanishes. Defining
$R_{\rho\sigma\mu\nu}=g_{\rho\kappa}R^{\kappa}_{\phantom{\kappa}\sigma\mu\nu}$
then, after some algebra, it is possible to prove that this tensor has the
following symmetries.
$R_{\rho\sigma\mu\nu}=R_{[\rho\sigma][\mu\nu]}\,\,;\;R_{\rho\sigma\mu\nu}=R_{\mu\nu\rho\sigma}\,\,;\;R_{\rho[\sigma\mu\nu]}=0\,\,;\;\nabla_{[\kappa}R_{\rho\sigma]\mu\nu}=0$
(1.7)
Particularly, the last two symmetries above are called Bianchi identities.
There are other important tensors that are constructed out of the Riemann
curvature tensor:
$\displaystyle
R_{\mu\nu}\,\equiv\,R^{\rho}_{\phantom{\rho}\mu\rho\nu}\quad;\quad
R\,\equiv\,g^{\mu\nu}R_{\mu\nu}\,=\,R^{\nu}_{\phantom{\nu}\nu}$ $\displaystyle
C_{\rho\sigma\mu\nu}\equiv
R_{\rho\sigma\mu\nu}-\frac{2}{n-2}\left(g_{\rho[\mu}R_{\nu]\sigma}-g_{\sigma[\mu}R_{\nu]\rho}\right)+\frac{2}{(n-1)(n-2)}R\,g_{\rho[\mu}g_{\nu]\sigma}\,.$
These tensors are respectively called Ricci tensor, Ricci scalar and Weyl
tensor. The Ricci tensor is symmetric, while the Weyl tensor has all the
symmetries of equation (1.7) except for the last one, the differential Bianchi
identity. The Weyl tensor will be of central importance in this piece of work,
since the main goal of this thesis is to define an algebraic classification
for this tensor and relate such classification with integrability properties.
The Weyl tensor has two landmarks: it is traceless,
$C^{\rho}_{\phantom{\rho}\sigma\rho\nu}=0$, and it is invariant under
conformal transformations, i.e., if we transform the metric as
$g_{\mu\nu}\mapsto\Omega^{2}g_{\mu\nu}$ then the tensor
$C^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ remains invariant.
#### 1.3 Geodesics
Given two points $p_{1}$ and $p_{2}$ on a manifold $(M,\boldsymbol{g})$, the
trajectory of minimum length connecting these points is called a geodesic. If
$x^{\mu}(\tau)$ is a curve joining these points, with
$x^{\mu}(\tau_{i})=p_{i}$, then its length is given by:
$\Delta(\tau_{1},\tau_{2})\,=\,\int_{\tau_{1}}^{\tau_{2}}\,\sqrt{g_{\mu\nu}\,\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}\,\,d\tau\,.$
Note that $\Delta$ is invariant under the change of parametrization of the
curve. Let us exploit this freedom adopting the arc length,
$s(\tau)\equiv\Delta(\tau_{1},\tau)$, as the curve parameter. Then performing
a standard variational calculation we find that the curve of minimum length
connecting $p_{1}$ and $p_{2}$ satisfies the following differential equation
known as the geodesic equation:
$\frac{d^{2}x^{\rho}}{ds^{2}}\,+\,\Gamma^{\rho}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}\,=\,0$
(1.8)
Note that using cartesian coordinates on the Minkowski space we have that
$\Gamma^{\rho}_{\mu\nu}=0$, so that eq. (1.8) implies that the geodesics of
flat space are straight lines, as it should be. Using equations (1.3) and
(1.8) we find that the geodesic equation can be elegantly expressed by:
$T^{\mu}\,\nabla_{\mu}\,T^{\nu}\,=\,0\,,\quad
T^{\mu}\equiv\frac{dx^{\mu}}{ds}\,.$ (1.9)
Note that the vector field $T^{\mu}$ is tangent to the curve. If instead of
the arc length parameter, $s$, we have used another parameter $\tau$, we would
have found the equation $N^{\mu}\nabla_{\mu}N^{\nu}=fN^{\nu}$, where
$N^{\mu}\equiv\frac{dx^{\mu}}{d\tau}$ and $f$ is some function. The parameters
$\tau^{\prime}$ such that $f=0$ are called affine parameters. It is simple
matter to verify that the affine parameters are all linearly related to the
arc length, $\tau^{\prime}=a\,s+b$ with $a\neq 0$ and $b$ being constants.
Physically, the arc length $s$ of a time-like curve (geodesic or not)
represents the proper time of the observer following this curve. In general
relativity, free massive particles follow time-like geodesics, whereas free
massless particles describe null geodesics. It is worth remarking that here a
particle is said to be free when the only force acting on it is the
gravitational force.
In order to gain some intuition on the formalism introduced so far, let us go
back to the example of the $2$-sphere. Let $S$ be a sphere of radius $r$
embedded on the 3-dimensional Euclidean space $\mathbb{R}^{3}$, as depicted in
figure 1.1. The metric of the 3-dimensional space is
$ds^{2}=dx^{2}+dy^{2}+dz^{2}$. Then, the points on the sphere can be locally
labeled by the coordinates $\theta$ and $\phi$ related to the cartesian
coordinates by $x=r\sin\theta\cos\phi$, $y=r\sin\theta\sin\phi$ and
$z=r\cos\theta$. Inserting these expressions in the 3-dimensional metric and
assuming that $r$ is constant we are led to the metric of the $2$-sphere,
$ds^{2}=r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$. Once we have this
metric we can compute its associated curvature by means of equation (1.6). In
particular, the Ricci scalar is found to be $R=2/r^{2}$. So, the bigger the
radius the smaller the curvature. Now, let $\boldsymbol{V}$ be a vector field
tangent to the sphere, $\boldsymbol{V}\cdot\hat{\boldsymbol{r}}=0$. Where the
dot denotes the inner product of $\mathbb{R}^{3}$. Then, the covariant
derivative of $\boldsymbol{V}$ along some curve tangent to the sphere is just
the projection of the ordinary derivative of $\boldsymbol{V}$ along this curve
onto the tangent planes of the sphere, see figure 1.1. For instance, the
covariant derivative of $\boldsymbol{V}$ along the great circle
$\theta=\frac{\pi}{2}$ is
$\nabla_{\phi}\boldsymbol{V}=\frac{d\boldsymbol{V}}{d\phi}-(\hat{\boldsymbol{r}}\cdot\frac{d\boldsymbol{V}}{d\phi})\hat{\boldsymbol{r}}$.
Particularly, one can prove that $\nabla_{\phi}\hat{\boldsymbol{\phi}}=0$,
which implies that such great circle is a geodesic curve. In general, all
great circles of the $2$-sphere are geodesic curves.
Figure 1.1: Sphere embedded in the 3-dimensional Euclidean space. The vector
fields $\hat{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\phi}}$ are tangent
to the sphere. On the right hand side it is illustrated that the covariant
derivative of a vector field tangent to the sphere is the projection of the
ordinary derivative onto the plane tangent to the spherical surface.
#### 1.4 Symmetries and Conserved Quantities
Suppose that a space-time is symmetric on the direction of the coordinate
vector $\boldsymbol{K}=\partial_{1}$, i.e., it looks the same irrespective of
the value of the coordinate $x^{1}$. This implies that in this coordinate
system we have $\partial_{1}\,g_{\mu\nu}=0$. Then, using the fact that
$K^{\mu}=\delta^{\mu}_{1}$ and the expression for the Christoffel symbol in
terms of the derivatives of the metric, we easily find that:
$\nabla_{\mu}\,K_{\nu}\,=\,\frac{1}{2}\,\left(\partial_{\mu}\,g_{\nu
1}\,-\,\partial_{\nu}\,g_{\mu
1}\right)\;\Rightarrow\;\;\nabla_{\mu}\,K_{\nu}\,+\,\nabla_{\nu}\,K_{\mu}\,=\,0\,.$
(1.10)
Conversely, if a vector field $\boldsymbol{K}$ satisfies
$\nabla_{(\mu}K_{\nu)}=0$ then it is simple matter to prove that on a
coordinate system in which $\boldsymbol{K}$ is a coordinate vector the
relation $K^{\mu}\partial_{\mu}g_{\rho\sigma}=0$ holds. The equation
$\nabla_{(\mu}K_{\nu)}=0$ is the so-called Killing equation and the vector
field $\boldsymbol{K}$ is called a Killing vector field. In general the
symmetries of a space-time are not obvious from the expression of the metric.
For example, the Minkowski space-time has 10 independent Killing vector
fields, although only 4 symmetries are obvious from the usual expression of
this metric. That is the reason why the Killing vectors are so important, they
characterize the symmetries of a manifold without explicitly using
coordinates.
From the Noether theorem it is known that continuous symmetries are associated
to conserved charges. So the Killing vector fields must be related to
conserved quantities. Indeed, if $\boldsymbol{K}$ is a Killing vector and
$\boldsymbol{T}$ is the affinely parameterized vector field tangent to a
geodesic curve then the scalar $T^{\mu}K_{\mu}$ is constant along such
geodesic,
$T^{\nu}\nabla_{\nu}(T^{\mu}K_{\mu})=T^{\nu}T^{\mu}\nabla_{(\nu}K_{\mu)}=0$.
Physically, this means that along free-falling orbits the component of the
momentum along the direction of a Killing vector is conserved. The use of
these conserved quantities are generally quite helpful to find the solutions
of the geodesic equation. For instance, since the Schwarzschild space-time has
4 independent Killing vectors it follows that the geodesic trajectories can be
found without solving the geodesic equation. But, in addition to the Killing
vectors, there are other tensors associated with the symmetries of a manifold.
For example, let $K_{\nu_{1}\nu_{2}\ldots\nu_{p}}$ be a completely symmetric
tensor obeying to the equation
$\nabla_{(\mu}\,K_{\nu_{1}\nu_{2}\ldots\nu_{p})}\,=\,0\,,$
then the scalar $K_{\nu_{1}\ldots\nu_{p}}T^{\nu_{1}}\ldots T^{\nu_{q}}$ is
conserved along the geodesic generated by $\boldsymbol{T}$. The tensor
$K_{\nu_{1}\nu_{2}\ldots\nu_{p}}$ is called a Killing tensor of order $p$.
Another important class of tensors associated to symmetries is formed by the
Killing-Yano (KY) tensors. These are skew-symmetric tensors,
$Y_{\nu_{1}\nu_{2}\ldots\nu_{p}}=Y_{[\nu_{1}\nu_{2}\ldots\nu_{p}]}$, that obey
to the equation
$\nabla_{\mu}Y_{\nu_{1}\ldots\nu_{p}}+\nabla_{\nu_{1}}Y_{\mu\ldots\nu_{p}}=0$.
If $T^{\mu}$ generates an affinely parameterized geodesic then
$Y_{\nu_{1}\nu_{2}\ldots\nu_{p}}T^{\nu_{p}}$ is covariantly constant along the
geodesic. Note also that if $Y_{\mu\nu}$ is a Killing-Yano tensor then
$K_{\mu\nu}=Y_{\mu}^{\phantom{\mu}\rho}Y_{\rho\nu}$ is a Killing tensor of
order two. Although we can always construct Killing tensors out of KY tensors,
not all Killing tensors are made from KY tensors [8]. For more details about
KY tensors see [5].
There are also tensors associated to scalars conserved only along null
geodesics. A totally symmetric tensor $\boldsymbol{L}$ is said to be a
conformal Killing tensor (CKT) when the equation
$\nabla_{(\nu}L_{\mu_{1}\ldots\mu_{p})}=g_{(\nu\mu_{1}}A_{\mu_{2}\ldots\mu_{p})}$
holds for some tensor $\boldsymbol{A}$. If $\boldsymbol{L}$ is a CKT of order
$p$ and $\boldsymbol{l}$ is tangent to an affinely parameterized null geodesic
then the scalar $L_{\mu_{1}\ldots\mu_{p}}l^{\mu_{1}}\ldots l^{\mu_{p}}$ is
constant along such geodesic. It is not so hard to prove that if
$\boldsymbol{K}$ is a Killing tensor on the manifold $(M,\boldsymbol{g})$ then
$L_{\mu_{1}\ldots\mu_{p}}=\Omega^{2p}\,K_{\mu_{1}\ldots\mu_{p}}$ is a CKT of
the manifold $(M,\tilde{\boldsymbol{g}})$ with
$\tilde{g}_{\mu\nu}=\Omega^{2}\,g_{\mu\nu}$. In the same vein, we say that a
completely skew-symmetric tensor $\boldsymbol{Z}$ is a conformal Killing-Yano
(CKY) tensor if it satisfies the equation
$\nabla_{(\nu}Z_{\mu_{1})\mu_{2}\ldots\mu_{p}}=g_{\nu[\mu_{1}}H_{\mu_{2}\ldots\mu_{p}]}+g_{\mu_{1}[\nu}H_{\mu_{2}\ldots\mu_{p}]}$
for some tensor $\boldsymbol{H}$ [5].
Generally it is highly non-trivial to guess whether a manifold possess a
Killing tensor, a KY tensor as well as its conformal versions. Therefore, such
tensors are said to represent hidden symmetries. Since the Kerr metric has
just 2 independent Killing vectors it is not possible to find the geodesic
trajectories using only these symmetries. But, in 1968, B. Carter was able to
discover another conserved quantity that enabled him to solve the geodesic
equation [9]. Two years later Walker and Penrose demonstrated that this “new”
conserved scalar is associated to a Killing tensor of order two [10].
Thereafter it has been proved that this Killing tensor is the “square” of a KY
tensor [8].
#### 1.5 Einstein’s Equation
Hopefully we already convinced ourselves that the gravitational field is
represented by the metric, $g_{\mu\nu}$, of a curved manifold
$(M,\boldsymbol{g})$. But we do not know yet how to find this metric given the
distribution of masses throughout the space-time. For example, in the
Newtonian theory the gravitational field is represented by a scalar, the
gravitational potential $\phi$, whose equation of motion is
$\nabla^{2}\phi=4\pi G\varrho$, where $G$ is the gravitational constant and
$\varrho$ is the mass density. Analogously, we need to find the equation of
motion for the metric $g_{\mu\nu}$. It can already be expected that,
differently from the Newtonian theory, the source of gravity is not just the
mass density, but the energy content as a whole, since in relativity mass and
energy are equivalent.
A wise path to find the correct field equation satisfied by $g_{\mu\nu}$ is to
guess a reasonable action representing the gravitational field and its
interaction with the other fields. Let us start analyzing how the metric
couples to the matter fields. Well, this is simple: given the action of a
field in special relativity we just need to replace the Minkowski metric by
$\boldsymbol{g}$ and substitute the partial derivatives by covariant
derivatives. There is, however, an important detail missing. In order for the
action to look the same in any coordinate system we must impose for it to be a
scalar. It is simple matter to prove that the volume element of space-time
$d^{4}x=dx^{0}dx^{1}dx^{2}dx^{3}$ is not invariant under coordinate
transformations. This can be fixed by taking $\sqrt{|g|}d^{4}x$ as the volume
element, with $g$ being the determinant of $g_{\mu\nu}$. Regarding the action
of the gravitational field, the simplest non-trivial scalar that can be
constructed out of the metric is the Ricci scalar $R$, defined in section 1.2.
Therefore we find that a reasonable action is:
$S\,=\,\frac{1}{16\pi G}\int
R\,\sqrt{|g|}d^{4}x\,+\,\int\mathcal{L}_{m}(\varphi_{i},\nabla_{\mu}\varphi_{i},g_{\mu\nu})\,\sqrt{|g|}d^{4}x\,.$
(1.11)
Where $\mathcal{L}_{m}$ is the Lagrangian density of the matter fields
$\varphi_{i}$. Then, using the least action principle, we can prove that the
equation of motion for the field $g_{\mu\nu}$ is given by the so-called
Einstein’s equation [5]:
$R_{\mu\nu}\,-\,\frac{1}{2}R\,g_{\mu\nu}\,=\,8\pi G\,T_{\mu\nu}\;;\quad
T^{\mu\nu}\,\equiv\,\frac{2}{\sqrt{|g|}}\,\frac{\delta S_{m}}{\delta
g_{\mu\nu}}\,.$ (1.12)
The symmetric tensor $T_{\mu\nu}$ is the energy-momentum tensor of the matter
fields. Particularly, in vacuum we have $T_{\mu\nu}=0$. Einstein’s equation
matches the geometry of the space-time, on the left hand side, to the energy
content, on the right hand side. Note that this equation is highly non-linear,
since the Ricci tensor and the Ricci scalar depends on the square of the
metric as well as on the inverse of the metric. This non-linearity can be
easily grasped using physical intuition. Since the graviton carries energy it
produces gravity, which then interact with this graviton and so on. In other
words, the graviton interacts with itself. This differs from classical
electrodynamics, where the photon has zero electric charge and, therefore,
generates no electromagnetic field.
As a simple and important example let us work out the case where just the
electromagnetic field is present. In relativistic theory this field is
represented by a co-vector $A_{\mu}$, the vector potential. From this field
one can construct the skew-symmetric tensor
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$. The action of the
electromagnetic field is given by:
$S_{em}\,=\,-\frac{1}{16\pi}\,\int
g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\,\sqrt{|g|}d^{4}x\,.$ (1.13)
Taking the functional derivative of this action with respect to the metric
yields the following energy-momentum tensor for the electromagnetic field:
$\mathcal{T}_{\mu\nu}\,=\,\frac{1}{4\pi}\,\left(F_{\mu\sigma}F_{\nu}^{\phantom{\nu}\sigma}-\frac{1}{4}\,g_{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\right)\,.$
(1.14)
Furthermore, computing the functional derivative of the action (1.13) with
respect to $A_{\mu}$ and equating to zero yields $\nabla^{\nu}F_{\mu\nu}=0$,
which is equivalent to Maxwell’s equations without sources. The set of
equations $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G\mathcal{T}_{\mu\nu}$,
$\nabla^{\nu}F_{\mu\nu}=0$ and $F_{\mu\nu}=2\nabla_{[\mu}A_{\nu]}$ is called
Einstein-Maxwell’s equations.
In this section we have considered that the gravitational Lagrangian is given
by the Ricci scalar $R$, which yields Einstein’s theory. Although general
relativity has had several experimental confirmations it is expected that for
really intense gravitational fields this Lagrangian shall be corrected by
higher order terms, such as $R^{2}$,
$R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$, $\partial_{\mu}R\,\partial^{\mu}R$
and so on. Indeed, string theory predicts that the gravitational action
contains terms of all orders on the curvature. In this picture the Einstein-
Hilbert action, $S=\frac{1}{16\pi G}\int R\sqrt{|g|}d^{n}x$, is just a weak
field approximation for the complete action.
#### 1.6 Differential Forms
Just as in section 1.2 it was valuable to say that the tangent space is
spanned by the differential operators $\partial_{\mu}$, it is also fruitful to
assume that the dual of this space, the space of linear functionals on
$T_{p}M$, is generated by the differentials $dx^{\mu}$. Thus if $A_{\mu}$ are
the components of a co-vector field in the coordinates $\\{x^{\mu}\\}$, then
we shall represent the abstract tensor $\boldsymbol{A}$ as follows:
$\boldsymbol{A}\,=\,A_{\mu}\,dx^{\mu}\,.$
With such definition it follows that $A_{\mu}$ will properly transform under
coordinate changes, see eq. (1.1). Therefore, an arbitrary tensor
$\boldsymbol{T}$ has the following abstract representation:
$\boldsymbol{T}\,=\,T^{\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}\,\partial_{\mu_{1}}\otimes\ldots\otimes\partial_{\mu_{p}}\otimes
dx^{\nu_{1}}\otimes\ldots\otimes dx^{\nu_{q}}\,.$
Since formally $dx^{\mu}$ is a linear functional on the space of vector
fields, its action on a vector field gives a scalar. Such action is defined by
$dx^{\mu}(\partial_{\nu})=\delta^{\mu}_{\,\nu}$, so that if $\boldsymbol{A}$
is co-vector and $\boldsymbol{V}$ is a vector then
$\boldsymbol{A}(\boldsymbol{V})=A_{\mu}V^{\mu}$.
A particularly relevant class of tensors are the so-called differential forms,
which are tensors with all indices down and totally skew-symmetric. For
instance, $F_{\mu_{1}\ldots\mu_{p}}=F_{[\mu_{1}\ldots\mu_{p}]}$ is called a
$p$-form and the vectorial space generated by all $p$-forms at some point
$x\in M$ is denoted by $\wedge^{p}M|_{x}$. A fundamental operation when
dealing with forms is the exterior product, whose definition is:
$\boldsymbol{F}\wedge\boldsymbol{H}\,=\,\frac{(p+q)!}{p!\,q!}\,F_{[\mu_{1}\ldots\mu_{p}}\,H_{\nu_{1}\ldots\nu_{q}]}\,dx^{\mu_{1}}\otimes\ldots\otimes
dx^{\mu_{p}}\otimes dx^{\nu_{1}}\otimes\ldots\otimes dx^{\nu_{q}}\,.$
Where $\boldsymbol{F}$ is a $p$-form and $\boldsymbol{H}$ is a $q$-form, so
that their exterior product yields a $(p+q)$-form. As an example note that the
following relation holds:
$\displaystyle dx^{1}\wedge dx^{2}\wedge dx^{3}\,=\,$
$\displaystyle(dx^{1}\otimes dx^{2}\otimes dx^{3}+dx^{2}\otimes dx^{3}\otimes
dx^{1}+dx^{3}\otimes dx^{1}\otimes dx^{2}+$ $\displaystyle\,-$
$\displaystyle\phantom{(}dx^{2}\otimes dx^{1}\otimes dx^{3}-dx^{3}\otimes
dx^{2}\otimes dx^{1}-dx^{1}\otimes dx^{3}\otimes dx^{2}\,)\,.$
In $n$ dimensions the set $\\{1,dx^{\mu_{1}},dx^{\mu_{1}}\wedge
dx^{\mu_{2}},\ldots,dx^{1}\wedge\ldots\wedge dx^{n}\\}$, which contains
$2^{n}$ elements, forms a basis for the space of differential forms, called
exterior bundle. In particular, a general $p$-form $\boldsymbol{F}$ can be
written as:
$\boldsymbol{F}\,=\,\frac{1}{p!}\,F_{\mu_{1}\ldots\mu_{p}}\,dx^{\mu_{1}}\wedge
dx^{\mu_{2}}\wedge\ldots\wedge dx^{\mu_{p}}\,.$
A $p$-form is called simple when it can be expressed as the exterior product
of $p$ 1-forms. For instance, every $n$-form is simple.
Another important operation involving differential forms is the interior
product, which essentially is the contraction of a differential form
$\boldsymbol{F}$ with a vector field $\boldsymbol{V}$ yielding another form
$\boldsymbol{H}\equiv\boldsymbol{V}\lrcorner\boldsymbol{F}$. If
$\boldsymbol{F}$ is a $p$-form then the interior product of $\boldsymbol{V}$
and $\boldsymbol{F}$ is the $(p-1)$-form defined by
$H_{\mu_{2}\ldots\mu_{p}}\equiv V^{\mu_{1}}F_{\mu_{1}\mu_{2}\ldots\mu_{p}}$.
When $\boldsymbol{V}\lrcorner\boldsymbol{F}=0$ we say that the differential
form $\boldsymbol{F}$ annihilates $\boldsymbol{V}$.
Suppose that $(M,\boldsymbol{g})$ is an $n$-dimensional manifold. Then we can
introduce the so-called Levi-Civita symbol
$\varepsilon_{\mu_{1}\ldots\mu_{n}}$, defined as the unique object, up to a
sign, that is totally skew-symmetric and normalized as $\varepsilon_{12\ldots
n}=\pm 1$. Although this symbol is not a tensor we can use it to define the
important tensor $\boldsymbol{\epsilon}$ called the volume-form and defined by
[11]:
$\epsilon_{\mu_{1}\ldots\mu_{n}}\,\equiv\,\sqrt{|g|}\,\varepsilon_{\mu_{1}\ldots\mu_{n}}\;\Rightarrow\quad\boldsymbol{\epsilon}\,=\,\sqrt{|g|}\,dx^{1}\wedge\ldots\wedge
dx^{n}\,,$
where $g$ denotes the determinant of the matrix $g_{\mu\nu}$. After some
algebra it can be proved that this tensor obeys to the following useful
identity [11]:
$\epsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{p+1}\ldots\nu_{n}}\,\epsilon_{\mu_{1}\ldots\mu_{p}\,\sigma_{p+1}\ldots\sigma_{n}}\;=\;p!(n-p)!\,(-1)^{\frac{n-s}{2}}\delta_{\sigma_{p+1}}^{\;[\nu_{p+1}}\ldots\delta_{\sigma_{n}}^{\;\nu_{n}]}\,.$
(1.15)
Where $s$ is the signature of the metric. Moreover, the volume-form can be
used to define an important operation called Hodge dual. The Hodge dual of a
$p$-form $\boldsymbol{F}$ is a $(n-p)$-form denoted by $\star\boldsymbol{F}$
and defined by:
$\left(\star
F\right)_{\mu_{1}\ldots\mu_{n-p}}\;=\;\frac{1}{p!}\,\epsilon^{\nu_{1}\ldots\nu_{p}}_{\phantom{\nu_{1}\ldots\nu_{p}}\mu_{1}\ldots\mu_{n-p}}\,F_{\nu_{1}\ldots\nu_{p}}\,.$
(1.16)
Finally, the last relevant operation on the space of forms is the exterior
differentiation, $d$. This differential operation maps $p$-forms into
$(p+1)$-forms as follows:
$d\boldsymbol{F}=\frac{1}{p!}\,\partial_{\nu}F_{\mu_{1}\ldots\mu_{p}}\,dx^{\nu}\wedge
dx^{\mu_{1}}\wedge\ldots\wedge dx^{\mu_{p}}\,.$
Although we have used the partial derivative, we could have used the covariant
derivative and the result would be the same, because of the symmetry
$\Gamma^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\nu\mu}$ of the Christoffel symbol.
Therefore, the term on the right hand side of the above equation is indeed a
tensor. A remarkable property of the exterior derivative is that its square is
zero, $d(d\boldsymbol{F})=0$, which stems from the commutativity of the
partial derivatives.
As an application of this formalism note that the source-free Maxwell’s
equations can be elegantly expressed in terms of differential forms. The
vector potential $A_{\mu}$ is a 1-form, $\boldsymbol{A}=A_{\mu}dx^{\mu}$. The
field strength, $F_{\mu\nu}\equiv\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, is
nothing more than the exterior derivative of $\boldsymbol{A}$,
$\boldsymbol{F}=d\boldsymbol{A}$. In particular, this implies that
$d\boldsymbol{F}=0$. The missing equation is $\nabla^{\nu}F_{\mu\nu}=0$, which
can be proved to be equivalent to $d(\star\boldsymbol{F})=0$. Hence, in the
absence of sources, the electromagnetic field is represented by a 2-form,
$\boldsymbol{F}$, obeying the equations $d\boldsymbol{F}=0$ and
$d(\star\boldsymbol{F})=0$.
#### 1.7 Cartan’s Structure Equations
Up to now we have adopted the coordinate frames $\\{\partial_{\mu}\\}$ and
$\\{dx^{\mu}\\}$ as bases for the tangent space and for its dual respectively.
Often it is convenient to use a non-coordinate frame
$\\{\boldsymbol{e}_{a}=e_{a}^{\phantom{a}\mu}\partial_{\mu}\\}$, where the
index $a$ is not a vectorial index, but rather a label for the $n$ vector
fields composing the frame. Associated to this non-coordinate vector frame is
the so-called dual frame
$\\{\boldsymbol{e}^{a}=e^{a}_{\phantom{a}\mu}dx^{\mu}\\}$, defined to be such
that $\boldsymbol{e}^{a}(\boldsymbol{e}_{b})=\delta^{a}_{\,b}$. Given a
tensor, say $T^{\mu}_{\phantom{\mu}\nu}$, its components in the frame
$\\{\boldsymbol{e}_{a}\\}$ are defined by $T^{a}_{\phantom{a}b}\equiv
T^{\mu}_{\phantom{\mu}\nu}e^{a}_{\phantom{a}\mu}e_{b}^{\phantom{b}\nu}$. In
particular, note that
$g_{ab}=\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})$. Once fixed the
frame $\\{\boldsymbol{e}_{a}\\}$, let us define the set of $n^{2}$ connection
1-forms $\boldsymbol{\omega}^{a}_{\phantom{a}b}$ by the following relation:
$V^{\mu}\nabla_{\mu}\boldsymbol{e}^{a}\,=\,-\,\boldsymbol{\omega}^{a}_{\phantom{a}b}(\boldsymbol{V})\,\boldsymbol{e}^{b}\,,\quad\forall\;\textrm{
vector field }\;\boldsymbol{V}\,.$ (1.17)
Then expanding $\boldsymbol{e}^{a}$ in a coordinate frame and using equation
(1.6) we can, after some algebra, prove the following identities [12]:
$d\boldsymbol{e}^{a}+\boldsymbol{\omega}^{a}_{\phantom{a}b}\wedge\boldsymbol{e}^{b}\,=\,0\quad;\quad\frac{1}{2}R^{a}_{\phantom{a}bcd}\,\boldsymbol{e}^{c}\wedge\boldsymbol{e}^{d}\,=\,d\boldsymbol{\omega}^{a}_{\phantom{a}b}+\boldsymbol{\omega}^{a}_{\phantom{a}c}\wedge\boldsymbol{\omega}^{c}_{\phantom{c}b}\,.$
(1.18)
Where $R^{a}_{\phantom{a}bcd}$ are the components of the Riemann tensor with
respect to the frame $\\{\boldsymbol{e}_{a}\\}$. These equations are known as
the Cartan structure equations. Moreover, defining the scalars
$\omega_{ab}^{\phantom{ab}c}\equiv\boldsymbol{\omega}^{c}_{\phantom{c}b}(\boldsymbol{e}_{a})$
we can easily prove that
$\nabla_{a}\boldsymbol{e}_{b}=\omega_{ab}^{\phantom{ab}c}\boldsymbol{e}_{c}$.
Sometimes it is of particular help to work with frames such that $g_{ab}$ is a
constant scalar. In this case the components of the connection 1-forms obey to
the constraint $\omega_{abc}=-\omega_{acb}$, where
$\omega_{abc}\equiv\omega_{ab}^{\phantom{ab}d}\,g_{dc}$. Indeed, using the
fact that the metric is covariantly constant along with the Leibniz rule
yield:
$0\,=\,\nabla_{c}\,\left[\,\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})\,\right]\,=\,\boldsymbol{g}(\nabla_{c}\boldsymbol{e}_{a},\boldsymbol{e}_{b})+\boldsymbol{g}(\boldsymbol{e}_{a},\nabla_{c}\boldsymbol{e}_{b})\,=\,\omega_{ca}^{\phantom{ca}d}\,g_{db}+\omega_{cb}^{\phantom{cb}d}\,g_{ad}\,.$
Just as the language of differential forms provides an elegant and fruitful
way to deal with Maxwell’s equations, Cartan’s structure equations do the same
in Riemannian geometry. Particularly, equation (1.18) gives, in general, the
quicker way to compute the Riemann tensor of a manifold. For applications and
geometrical insights on the meaning of these equations see [2].
From the physical point of view, the relevance of Cartan’s structure equations
stems from its relation with the formulation of general relativity as a gauge
theory. It is well-known that, except for gravity, the fundamental
interactions of nature are currently described by gauge theories, more
precisely Yang-Mills theories. Although not widely advertised, it turns out
that general relativity can also be cast in the language of gauge
theories333Actually, the most simple gauge formulation of gravity, called
Einstein-Cartan theory, is equivalent to general relativity just in the
absence of spin. In the presence of matter with spin the former theory allows
a non-zero torsion [13].. In this approach the gauge group of gravity is the
group of Lorentz transformations, $SO(3,1)$ [13]. Indeed, those acquainted
with the formalism of non-abelian gauge theory will recognize the second
identity of (1.18) as the equation defining the curvature associated to the
connection $\boldsymbol{\omega}^{a}_{\phantom{a}b}$.
#### 1.8 Distributions and Integrability
Let $(M,\boldsymbol{g})$ be an $n$-dimensional manifold, then a
$q$-dimensional distribution in $M$ is a smooth map that associates to every
point $p\in M$ a vector subspace of dimension $q$, $\Delta_{p}\subset T_{p}M$.
We say that the set of vector fields $\\{\boldsymbol{V}_{i}\\}$ generates this
distribution when they span the vector subspace $\Delta_{p}$ for every point
$p\in M$. For instance, a non-vanishing vector field generates a 1-dimensional
distribution. We say that a distribution of dimension $q$ is integrable when
there exists a smooth family of submanifolds of $M$ such that the tangent
spaces of these submanifolds are $\Delta_{p}$. This means that locally $M$
admits coordinates $\\{x^{1},\ldots,x^{q},y^{1},\ldots,y^{n-q}\\}$ such that
the vector fields $\\{\partial_{x^{i}}\\}$ generate $\Delta_{p}$. In this case
the family of submanifolds is given by the hyper-surfaces of constant
$y^{\alpha}$.
Given a set of $q$ vector fields $\\{\boldsymbol{V}_{i}\\}$ that are linearly
independent at every point then it generates a $q$-dimensional distribution
denoted by $Span\\{\boldsymbol{V}_{i}\\}$. One might then wonder, how can we
know if such distribution is integrable? Before answering this question it is
important to introduce the Lie bracket. If $\boldsymbol{V}$ and
$\boldsymbol{Z}$ are vector fields then their Lie bracket is another vector
field defined by:
$[\boldsymbol{V},\boldsymbol{Z}]\,\equiv\,V^{\mu}\nabla_{\mu}\boldsymbol{Z}-Z^{\mu}\nabla_{\mu}\boldsymbol{V}\,=\,\left(V^{\mu}\partial_{\mu}\,Z^{\nu}-Z^{\mu}\partial_{\mu}\,V^{\nu}\right)\partial_{\nu}\,.$
As a warming exercise let us work out an example on the $n$-dimensional
Euclidian space, $(\mathbb{R}^{n},\delta_{\mu\nu})$. Let $f(\boldsymbol{r})$
be some function on this manifold, then generally the surfaces of constant $f$
foliate the space, with the leafs being orthogonal to $\boldsymbol{\nabla}f$
as depicted in figure 1.2. Therefore, if $\boldsymbol{V}$ is some vector field
tangent to the foliating surfaces then
$\boldsymbol{V}\cdot\boldsymbol{\nabla}f=0$. Differentiating this last
equation we get
$\partial_{\mu}(\boldsymbol{V}\cdot\boldsymbol{\nabla}f)\,=\,0\;\;\Rightarrow\;\;(\partial_{\mu}V^{\nu})\,\partial_{\nu}f\,+\,V^{\nu}\,\partial_{\mu}\partial_{\nu}f\,=\,0\,.$
Therefore, if $\boldsymbol{Z}$ is another vector field tangent to the leafs of
constant $f$ then
$[\boldsymbol{V},\boldsymbol{Z}]\cdot\boldsymbol{\nabla}f=\left(V^{\mu}\partial_{\mu}Z^{\nu}-Z^{\mu}\partial_{\mu}V^{\nu}\right)\,\partial_{\nu}f=-V^{\mu}Z^{\nu}\partial_{\mu}\partial_{\nu}f+Z^{\mu}V^{\nu}\partial_{\mu}\partial_{\nu}f=0\,.$
This means that the Lie bracket of two vector fields tangent to the foliating
surfaces yield another vector field tangent to these surfaces. Now let
$\boldsymbol{\theta}\neq 0$ be a 1-form proportional to $df$,
$\boldsymbol{\theta}=h\,df$. Then note that a vector field $\boldsymbol{V}$ is
tangent to the leafs of constant $f$ if, and only if,
$\boldsymbol{\theta}(\boldsymbol{V})=0$. In addition, note that
$d\boldsymbol{\theta}\wedge\boldsymbol{\theta}=0$ and that
$d(\frac{1}{h}\boldsymbol{\theta})=0$.
Figure 1.2: The space is foliated by the surfaces of constant $f$. The vector
field $\boldsymbol{\nabla}f$ is orthogonal to the leafs of the foliation,
while $\boldsymbol{V}$ and $\boldsymbol{Z}$ are tangent.
The results obtained in the preceding paragraph are just a special case of a
well-known theorem called the Frobenius theorem, which states that the
distribution generated by the vector fields $\\{\boldsymbol{V}_{i}\\}$ is
integrable if, and only if, there exists a set of functions $C_{ij}^{k}$ such
that $[\boldsymbol{V}_{i},\boldsymbol{V}_{j}]=C_{ij}^{k}\,\boldsymbol{V}_{k}$.
In other words, this distribution is integrable if, and only if, the vector
fields $\boldsymbol{V}_{i}$ form a closed algebra under the Lie brackets [14].
The Frobenius theorem can be presented in a “dual” version, in terms of
differential forms. Let $\\{\boldsymbol{V}_{i}\\}$ be a set of $q$ vector
fields generating a $q$-dimensional distribution. Then we can complete this
set with more $(n-q)$ vector fields, $\\{\boldsymbol{U}_{\alpha}\\}$, so that
$\\{\boldsymbol{V}_{i},\boldsymbol{U}_{\alpha}\\}$ spans the tangent space at
every point. Associated to this frame is a dual frame of 1-forms
$\\{\boldsymbol{\omega}^{i},\boldsymbol{\theta}^{\alpha}\\}$ such that
$\boldsymbol{\omega}^{i}(\boldsymbol{V}_{j})=\delta^{i}_{\,j}$,
$\boldsymbol{\omega}^{i}(\boldsymbol{U}_{\alpha})=0$,
$\boldsymbol{\theta}^{\alpha}(\boldsymbol{V}_{i})=0$ and
$\boldsymbol{\theta}^{\alpha}(\boldsymbol{U}_{\beta})=\delta^{\alpha}_{\,\beta}$.
Note that a vector field is tangent to the distribution if, and only if, it is
annihilated by all the $(n-q)$ 1-forms $\boldsymbol{\theta}^{\alpha}$. The
dual version of the Frobenius theorem then states that the distribution
generated by $\\{\boldsymbol{V}_{i}\\}$ is integrable if, and only if,
$d\boldsymbol{\theta}^{\alpha}\wedge\boldsymbol{\theta}^{1}\wedge\boldsymbol{\theta}^{2}\wedge\ldots\wedge\boldsymbol{\theta}^{(n-q)}\,=\,0\quad\;\forall\;\;\alpha\in\\{1,\ldots,(n-q)\\}\,.$
(1.19)
Defining
$\boldsymbol{\Theta}\equiv\boldsymbol{\theta}^{1}\wedge\ldots\wedge\boldsymbol{\theta}^{(n-q)}$,
then note that a vector field $\boldsymbol{V}$ is tangent to the distribution
generated by $\\{\boldsymbol{V}_{i}\\}$ if, and only if,
$\boldsymbol{V}\lrcorner\boldsymbol{\Theta}=0$. Now suppose that there exists
a non-zero function $h$ such that $d(h\boldsymbol{\Theta})=0$, then expanding
this equation and taking the wedge product with $\boldsymbol{\theta}^{\alpha}$
we arrive at the equation (1.19). Conversely, if the distribution generated by
$\\{\boldsymbol{V}_{i}\\}$ is integrable then, by definition, one can
introduce coordinates $\\{x^{1},\ldots,x^{q},y^{1},\ldots,y^{n-q}\\}$ such
that the vector fields $\\{\partial_{x^{i}}\\}$ generate this distribution.
Since $dy^{\alpha}(\partial_{x^{i}})=0$, it follows that
$\boldsymbol{\Theta}=\frac{1}{h}(dy^{1}\wedge\ldots\wedge dy^{n-q})$ for some
non-vanishing function $h$, which implies that $d(h\boldsymbol{\Theta})=0$. We
proved, therefore, that the distribution annihilated by $\boldsymbol{\Theta}$
is integrable if, and only if, there exists some non-zero function $h$ such
that $d(h\boldsymbol{\Theta})=0$. Equivalently, it can be stated that the
distribution annihilated by a simple form $\boldsymbol{\Theta}$ is integrable
if, and only if, there exists a 1-form $\boldsymbol{\varphi}$ such that
$d\boldsymbol{\Theta}=\boldsymbol{\varphi}\wedge\boldsymbol{\Theta}$.
The integrability of distributions plays an important role in Caratheodory’s
formulation of thermodynamics. In his formalism, the equilibrium states of a
thermodynamical system form a differentiable manifold $\mathcal{M}$. In such a
manifold it is defined a global function $U$, the internal energy, and two
$1$-forms, $\boldsymbol{W}$ and $\boldsymbol{Q}$, representing the work done
and the received heat, respectively. The first law of thermodynamics is then
written as $dU=\boldsymbol{Q}-\boldsymbol{W}$. A curve in this manifold is
called adiabatic if its tangent vector field is annihilated by
$\boldsymbol{Q}$. According to Caratheodory, the second law of thermodynamics
says that in the neighborhood of every point $x\in\mathcal{M}$ there are
points $y$ such that there is no adiabatic curve joining $x$ to $y$. He was
able to prove that this formulation of the second law guarantees that the
distribution annihilated by $\boldsymbol{Q}$ is integrable. Particularly, this
implies that there exist functions $T$ and $S$ such that $\boldsymbol{Q}=TdS$.
Physically, these functions are the temperature, $T$, and the entropy, $S$.
For more details see [14] and references therein.
#### 1.9 Higher-Dimensional Spaces
Einstein’s general relativity postulates that we live in a 4-dimensional
Lorentzian manifold, which means that the space-time has 3 spatial dimensions
and one time dimension. There are, however, some theories claiming that our
space-time can have more spatial dimensions. Particularly, in order to provide
a consistent quantum theory, superstring theory requires the space-time
dimension to be 10 or 11 [15]. Which justifies the study of higher-dimensional
general relativity.
One might wonder: If these extra dimensions exist then why they have not been
perceived yet? A reasonable reason is that these dimensions can be highly
wrapped. For example, if we look at a long pipe that is far from us it will
appear that it is just a one-dimensional line. But as we get closer and closer
to the pipe we will note that it is actually a cylinder, which has two
dimensions. An instructive example for understanding the role played by a
curled dimension is to solve Schrödinger equation for a particle of mass $m$
inside an infinite well. Let the space be 2-dimensional with one of the
dimensions being a circle of radius R while the other dimension is open and
has an infinite well of size L, then the energy spectrum of this system is
easily proved to be [16]:
$E_{p,q}\,=\,\frac{\hbar^{2}\pi^{2}}{2m}\,\left(\frac{p}{\textsf{L}}\right)^{2}\,+\,\frac{\hbar^{2}}{2m}\,\left(\frac{q-1}{\textsf{R}}\right)^{2}\,,\quad
p,q\in\\{1,2,3,\ldots\\}\,.$
The first term on the right hand side of this equation is just the regular
spectrum of a 1-dimensional infinite well of size L, while the second term is
the contribution from the extra dimension. Then note that if R is very small,
$\textsf{R}\ll\textsf{L}$, then it will be necessary a lot of energy to excite
the modes with quantum number $q$. Thus in the limit $\textsf{R}\rightarrow 0$
the system will remain in a state with $q=1$, which implies that we retrieve
the spectrum of a 1-dimensional well. Thus if the extra dimensions are very
tiny the only hope to detect them is through very energetic experiments444In
closed string theory a new phenomenon emerges. Since strings can wrap around a
curled dimension there exist winding modes that need little energy to be
excited when R is much smaller than the Planck length. Furthermore, due to a
symmetry called $T$-duality, in closed string theory very small radius turns
out to be equivalent to very large radius.. Indeed, currently the LHC555LHC is
the abbreviation for Large Hadron Collider, the most energetic particle
accelerator in the world. is probing the existence of extra dimensions.
In addition to the possibility of our universe having extra dimensions and to
the obvious mathematical relevance, the study of higher-dimensional curved
spaces has other applications. For example, in classical mechanics the phase
space of a system is a $2p$-dimensional manifold endowed with a symplectic
structure, where $p$ is the number of degrees of freedom [17]. As a
consequence, higher-dimensional spaces are also of interest to thermodynamics
and statistical mechanics.
It is needless to explain the physical relevance of the Lorentzian signature.
But it is worth highlighting that other signatures are also important in
physics, let alone in mathematics. Spaces with split signature are of
relevance for the theory of integrable systems, Yang-Mills fields and for
twistor theory [19]. Moreover, the Euclidean signature emerges when we make a
Wick rotation on the time coordinate in order to make path integrals
convergent. The Euclidean curved spaces are sometimes called gravitational
instantons, although it is more appropriate to define a gravitational
instanton as a complete 4-dimensional Ricci-flat Euclidean manifold that is
asymptotically-flat and whose Weyl tensor is self-dual [18]. Analogously to
the instantons solutions of Yang-Mills theory, gravitational instantons
provide a dominant contribution to Feynman path integral, justifying its
physical interest [18]. Non-Lorentzian signatures are also of relevance for
string theory.
Given the importance of these topics, the present thesis will investigate some
properties of higher-dimensional curved spaces of arbitrary signature. The
path adopted here is to work with complexified manifolds so that the results
can be carried to any signature by judiciously choosing a reality condition
[20]. The technique of using complexified geometry with the aim of extracting
results for real spaces can be fruitful and enlightening, an approach that was
advocated by McIntosh and Hickman in a series of papers [21], where
4-dimensional general relativity was explored using complexified manifolds.
### Chapter 2 Petrov Classification, Six Different Approaches
The Petrov classification is an algebraic classification for the curvature,
more precisely for the Weyl tensor, valid in 4-dimensional Lorentzian
manifolds. It has been of invaluable relevance for the development of general
relativity, in particular it played a prominent role on the discovery of Kerr
metric [22], which is probably the most important solution of general
relativity. Furthermore, guided by such classification and a theorem due to
Goldberg and Sachs [23], Kinnersley was able to find all type $D$ vacuum
solutions [24], a really impressive accomplishment since Einstein’s equation
is non-linear. Moreover, this classification contributed for the study of
gravitational radiation [25, 26], the peeling theorem being one remarkable
example [27].
Such classification was created by the Russian mathematician Alexei
Zinovievich Petrov in 1954111Petrov obtained this classification in a previous
article published in 1951 but, as himself acknowledges in [28], the proofs in
this first work were not precise. [28] with the intent of classifying Einstein
space-times. A. Z. Petrov has worked on differential geometry and general
relativity, and he has been one of the most important scientists responsible
for the spread of Einstein’s gravitational theory inside the Soviet Union222A
short biography of A. Z. Petrov can be found in Kazan University’s website
[29].. In particular, around 1960 he has written a really remarkable book on
general relativity that certainly has been of great relevance for the
dissemination of this theory on such an isolated nation [30].
In its original form, this classification consisted only of three types, $I$,
$II$ and $III$. Few years later, in 1960, Roger Penrose developed spinorial
techniques to general relativity and, as a consequence, has found out that
these types could be further refined, adding the types $D$ and $N$ to the
classification [31]. It is worth mentioning that by the same time Robert
Debever and Louis Bel arrived at such refinement by a different path [25, 32],
in particular they have developed an alternative approach to define the Petrov
types, the so-called Bel-Debever criteria.
The route adopted by A. Z. Petrov to arrive at his classification amounts to
reinterpreting the Weyl tensor as an operator acting on the space of
bivectors. As time passed by, several other methods to attack such
classification were developed. Since these approaches look very different from
each other, it comes as a surprise that all of them are equivalent. The intent
of the present chapter is to describe six different ways to attain this
classification. As one of the goals of this thesis is to describe an
appropriate generalization for the Petrov classification valid in dimensions
greater than four, the analysis of these different approaches proves to be
important because in higher dimensions many of these methods are not
equivalent anymore. Therefore, in order to find a suitable higher-dimensional
generalization for the Petrov classification it is helpful to investigate the
benefits and flaws of each method in 4 dimensions.
Throughout this chapter it will be assumed that the space-time is a
4-dimensional manifold endowed with a metric of Lorentzian signature,
$(M,\boldsymbol{g})$. Furthermore, the tangent bundle is assumed to be endowed
with the Levi-Civita connection, hence the curvature referred here is with
respect to this connection. All calculations are assumed to be local, in a
neighborhood of an arbitrary point $p\in M$.
#### 2.1 Weyl Tensor as an Operator on the Bivector Space
In this section the so-called bivector approach will be used to define the
Petrov classification. To this end the results of appendices A and B will be
necessary, so that the reader is advised to take a look at these appendices
before proceeding.
The Weyl tensor is the trace-less part of the Riemann tensor, it has the
following symmetries (see section 1.2):
$C_{\mu\nu\rho\sigma}=C_{[\mu\nu][\rho\sigma]}=C_{\rho\sigma\mu\nu}\;;\;\;C^{\mu}_{\phantom{\mu}\nu\mu\sigma}=0\;;\;\;C_{\mu[\nu\rho\sigma]}=0\,.$
(2.1)
Skew-symmetric tensors of rank 2 are called a bivectors,
$B_{\mu\nu}=B_{[\mu\nu]}$. Since the Weyl tensor is anti-symmetric in the
first and second pairs of indices, it follows that this tensor can be
interpreted as a linear operator that maps bivectors into bivectors in the
following way:
$B_{\mu\nu}\mapsto
T_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}\;,\;\textrm{where}\;\;B_{\mu\nu}=B_{[\mu\nu]}\;,T_{\mu\nu}=T_{[\mu\nu]}\,.$
(2.2)
Studying the possible eigenbivectors of this operator we arrive at the Petrov
classification, actually this was the original path taken by A. Z. Petrov
[28]. In order to enlighten the analysis it is important to review some
properties of bivectors in four dimensions. Let us denote the volume-form of
the 4-dimensional Lorentzian manifold $(M,g_{\mu\nu})$ by
$\epsilon_{\mu\nu\rho\sigma}$. This is a totally anti-symmetric tensor,
$\epsilon_{\mu\nu\rho\sigma}=\epsilon_{[\mu\nu\rho\sigma]}$, whose non-zero
components in an orthonormal frame are $\pm 1$. It is well-known that it
satisfies the following identity [11]:
$\epsilon^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\,\epsilon_{\mu_{1}\mu_{2}\sigma_{1}\sigma_{2}}=-2\,\left(\,\delta_{\sigma_{1}}^{\nu_{1}}\,\delta_{\sigma_{2}}^{\nu_{2}}\,-\,\delta_{\sigma_{2}}^{\nu_{1}}\,\delta_{\sigma_{1}}^{\nu_{2}}\,\right)\,.$
(2.3)
By means of the volume-form we can define the Hodge dual operation that maps
bivectors into bivectors. The dual of the bivector $\boldsymbol{B}$ is defined
by
$\left(\star
B\right)_{\mu\nu}\,\equiv\,\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}B^{\rho\sigma}\,.$
(2.4)
Let us denote by $\mathfrak{B}_{\mathbb{C}}$ the complexification of the
bivector bundle. Using equation (2.3) it is easy matter to see that the double
dual of a bivector is it negative, $[\star(\star B)]_{\mu\nu}=-B_{\mu\nu}$.
This implies that the 6-dimensional space $\mathfrak{B}_{\mathbb{C}}$ can be
split into the direct sum of the two 3-dimensional eigenspaces of the dual
operation.
$\mathfrak{B}_{\mathbb{C}}=\mathfrak{D}\oplus\mathfrak{\overline{D}}$ (2.5)
$\mathfrak{D}=\\{Z_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}\,|\,\left(\star
Z\right)_{\mu\nu}=iZ_{\mu\nu}\\}\;;\;\mathfrak{\overline{D}}=\\{Y_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}\,|\,\left(\star
Y\right)_{\mu\nu}=-iY_{\mu\nu}\\}$
The elements of $\mathfrak{D}$ are called self-dual bivectors, whereas a
bivector belonging to $\mathfrak{\overline{D}}$ is dubbed anti-self-dual. By
means of the volume-form it is also possible to split the Weyl tensor into a
sum of the dual part, $C^{+}$, and the anti-dual part, $C^{-}$:
$C_{\mu\nu\rho\sigma}=C^{+}_{\mu\nu\rho\sigma}+C^{-}_{\mu\nu\rho\sigma}\;\;;\;\;C^{\pm}_{\mu\nu\rho\sigma}\equiv\frac{1}{2}\left(C_{\mu\nu\rho\sigma}\mp\frac{i}{2}\,C_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\epsilon_{\alpha\beta\rho\sigma}\right)\,.$
(2.6)
It is then immediate to verify the following relations:
$C^{+}_{\mu\nu\rho\sigma}\,Y^{\rho\sigma}\,=\,0\quad\forall\;\boldsymbol{Y}\in\overline{\mathfrak{D}}\quad;\quad
C^{-}_{\mu\nu\rho\sigma}\,Z^{\rho\sigma}\,=\,0\quad\forall\;\boldsymbol{Z}\in\mathfrak{D}\,.$
This means that in order to analyse the action of Weyl tensor on
$\mathfrak{B}_{\mathbb{C}}$ it is sufficient to study the action of $C^{+}$ in
$\mathfrak{D}$ and the action of $C^{-}$ in $\overline{\mathfrak{D}}$.
However, by the definition on eq. (2.6), $C^{-}$ is the complex conjugate of
$C^{+}$, so that it is enough to study just the operator
$C^{+}:\,\mathfrak{D}\rightarrow\mathfrak{D}$. Since this operator is trace-
less and act on a 3-dimensional space it follows that it can have the
following algebraic types according to the refined Segre classification (see
appendix A):
$\left\\{\begin{array}[]{ll}\textbf{Type O}&\rightarrow\;C^{+}\,=\,0\\\
\textbf{Type I}&\rightarrow\;C^{+}\,\textrm{ is type }\;[1,1,1|\,]\,\textrm{
or }[1,1|1]\\\ \textbf{Type D}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[(1,1),1|\,]\\\ \textbf{Type II}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[2,1|\,]\\\ \textbf{Type N}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[\,|2,1]\\\ \textbf{Type III}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[\,|\,3]\,.\end{array}\right.$ (2.7)
These are the so-called Petrov types. Therefore, in order to determine the
Petrov classification of the Weyl tensor using this approach we must follow
four steps: 1) Choose a basis for the space of self-dual bivectors
$\mathfrak{D}$; 2) Calculate the action of the operator defined by (2.2) in
this basis in order to find a $3\times 3$ matrix representation for $C^{+}$;
3) Find the eigenvalues and eigenvectors of this matrix; 4) Use this
eigenvalue structure to determine the algebraic type of such matrix according
to the refined Segre classification (appendix A) and after this use equation
(2.7).
With the aim of making connection with the forthcoming sections, let us follow
some of these steps explicitly. Once introduced a null tetrad frame
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
(see appendix B), the ten independent components of the Weyl tensor can be
written in terms of five complex scalars:
$\Psi_{0}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}l^{\rho}m^{\sigma}\;;\;\Psi_{1}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}m^{\sigma}\;;\;\Psi_{2}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}\overline{m}^{\rho}n^{\sigma}$
$\Psi_{3}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}\overline{m}^{\rho}n^{\sigma}\;;\;\Psi_{4}\equiv
C_{\mu\nu\rho\sigma}n^{\mu}\overline{m}^{\nu}n^{\rho}\overline{m}^{\sigma}\,.$
(2.8)
These are the so-called Weyl scalars. A basis to the space of self-dual
bivectors, $\mathfrak{D}$, is given by:
$Z^{1}_{\mu\nu}=2\,l_{[\mu}m_{\nu]}\;;\;Z^{2}_{\mu\nu}=2\,\overline{m}_{[\mu}n_{\nu]}\;;\;Z^{3}_{\mu\nu}=2\,n_{[\mu}l_{\nu]}+2\,m_{[\mu}\overline{m}_{\nu]}$
(2.9)
In this basis the representation of operator
$C^{+}:\,\mathfrak{D}\rightarrow\mathfrak{D}$ is
$[C^{+}]=2\left[\begin{array}[]{ccc}\Psi_{2}&\Psi_{4}&-2\Psi_{3}\\\
\Psi_{0}&\Psi_{2}&-2\Psi_{1}\\\ \Psi_{1}&\Psi_{3}&-2\Psi_{2}\\\
\end{array}\right]$ (2.10)
Note that this matrix has vanishing trace, as claimed above equation (2.7).
Thus, in order to get the Petrov type of the Weyl tensor we just have to
calculate the Weyl scalars, using eq. (2.1), plug them on the above matrix and
investigate the algebraic type of such matrix.
When the Weyl tensor is type $I$ it is said to be algebraically general,
otherwise it is called algebraically special. If the Weyl tensor is type O in
all points we say that the space-time is conformally flat, which means there
exists a coordinate system such that $g_{\mu\nu}=\Omega^{2}\eta_{\mu\nu}$.
Note that the Petrov classification is local, so that the type of the Weyl
tensor can vary from point to point on space-time. In spite of this it is
interesting that the majority of the exact solutions has the same Petrov type
in all points of the manifold. For instance, all known black holes are type
$D$ and the plane gravitational waves are type $N$.
As pointed out at the beginning of this chapter, when Petrov classification
first emerged only three types were defined, known as types I, II and III [26,
28]. With the contributions of Penrose, Debever and Bel these types were
refined as depicted below.
$\textrm{I}-\textrm{Refinement}-^{\nearrow}_{\searrow}\begin{array}[]{l}I\\\
D\end{array}\quad;\quad\textrm{II}-\textrm{Refinement}-^{\nearrow}_{\searrow}\begin{array}[]{c}II\\\
N\end{array}\quad;\quad\textrm{III}\longrightarrow III$
Indeed, from the definition of Petrov types presented on equation (2.7) it is
already clear that the type $D$ can be seen as special case of the type $I$,
while type $N$ is a specialization of type $II$333 It is worth mentioning that
in ref. [25] L. Bel has used a different convention, denoting the types $I$,
$D$, $II$, $III$ and $N$ by $I$, $II_{a}$, $II_{b}$, $III_{a}$ and $III_{b}$
respectively..
More details about the bivector method will be given in chapter 4, where this
approach will be used to classify the Weyl tensor in any signature, see also
[33]. In particular, chapter 4 advocates that the bivector approach is endowed
with an enlightening geometrical significance. A careful investigation of the
bivector method in higher dimensions was performed in [34].
#### 2.2 Annihilating Weyl Scalars
In this section a different characterization of the Petrov types will be
presented. In this approach the different types are featured by the
possibility of annihilating some Weyl tensor components using a suitable
choice of basis. As a warming up example let us investigate the type $D$.
According to eq. (2.7), in this case the algebraic type of $C^{+}$ is
$[(1,1),1|\,]$, which means that such operator can be put on the diagonal form
$\operatorname{diag}(\lambda,\lambda,\lambda^{\prime})$. But since
$\operatorname{tr}(C^{+})=0$, we must have $\lambda^{\prime}=-2\lambda$, hence
$C^{+}=\operatorname{diag}(\lambda,\lambda,-2\lambda)$. Now, looking at eq.
(2.10) we see that this is compatible with the Weyl scalars
$\Psi_{0},\Psi_{1},\Psi_{3}$ and $\Psi_{4}$ being all zero. In general, each
Petrov type enables one to find a suitable basis where some Weyl scalars can
be made to vanish.
The Lorentz transformations at point $p\in M$ is the set of linear
transformations on tangent space, $T_{p}M$, which preserves the inner
products. These transformations can be obtained by a composition of the
following three simple operations in a null tetrad frame
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$:
(i) Lorentz Boost
$\boldsymbol{l}\rightarrow\lambda\boldsymbol{l}\;;\;\;\boldsymbol{n}\rightarrow\lambda^{-1}\boldsymbol{n}\;;\;\;\boldsymbol{m}\rightarrow
e^{i\theta}\boldsymbol{m}\;;\;\;\overline{\boldsymbol{m}}\rightarrow
e^{-i\theta}\overline{\boldsymbol{m}}$ (2.11)
(ii) Null Rotation Around $\boldsymbol{l}$
$\boldsymbol{l}\rightarrow\boldsymbol{l}\,;\;\,\boldsymbol{n}\rightarrow\boldsymbol{n}+w\boldsymbol{m}+\overline{w}\,\overline{\boldsymbol{m}}+|w|^{2}\boldsymbol{l}\,;\;\,\boldsymbol{m}\rightarrow\boldsymbol{m}+\overline{w}\boldsymbol{l}\,;\;\,\overline{\boldsymbol{m}}\rightarrow\overline{\boldsymbol{m}}+w\boldsymbol{l}$
(2.12)
(iii) Null Rotation Around $\boldsymbol{n}$
$\boldsymbol{l}\rightarrow\boldsymbol{l}+\overline{z}\boldsymbol{m}+z\overline{\boldsymbol{m}}+|z|^{2}\boldsymbol{n}\,;\;\,\boldsymbol{n}\rightarrow\boldsymbol{n}\,;\;\,\boldsymbol{m}\rightarrow\boldsymbol{m}+z\boldsymbol{n}\,;\;\,\overline{\boldsymbol{m}}\rightarrow\overline{\boldsymbol{m}}+\overline{z}\boldsymbol{n}.$
(2.13)
Where $\lambda$ and $\theta$ are real numbers while $z$ and $w$ are complex,
composing a total of six real parameters. This should be expected from the
fact that the Lorentz group, in a 4-dimensional space-time, has 6 dimensions.
In order to verify that these transformations do indeed preserve the inner
products, note that the metric
$g_{\mu\nu}=2l_{(\mu}n_{\nu)}-2m_{(\mu}\overline{m}_{\nu)}$ remains invariant
by them.
Now let us try to annihilate the maximum number of Weyl scalars by
transforming the null tetrad under the Lorentz group. After performing a null
rotation around $\boldsymbol{n}$ the Weyl scalars change as follows:
$\displaystyle\Psi_{0}\rightarrow\Psi^{\prime}_{0}(z)\,=\,\Psi_{0}\,+\,4\,z\,\Psi_{1}\,$
$\displaystyle+\,6\,z^{2}\,\Psi_{2}\,+\,4\,z^{3}\,\Psi_{3}\,+\,z^{4}\,\Psi_{4}\;;$
$\displaystyle\Psi_{1}\rightarrow\Psi^{\prime}_{1}(z)\,=\,\frac{1}{4}\frac{d}{dz}\Psi^{\prime}_{0}(z)\;\;\;;$
$\displaystyle\;\;\;\Psi_{2}\rightarrow\Psi^{\prime}_{2}(z)=\frac{1}{3}\frac{d}{dz}\Psi^{\prime}_{1}(z)\;;$
(2.14)
$\displaystyle\Psi_{3}\rightarrow\Psi^{\prime}_{3}(z)\,=\,\frac{1}{2}\frac{d}{dz}\Psi^{\prime}_{2}(z)\;\;;$
$\displaystyle\;\;\Psi_{4}\rightarrow\Psi^{\prime}_{4}(z)=\frac{d}{dz}\Psi^{\prime}_{3}(z)=\Psi_{4}\,,$
which can be proved using equations (2.1) and (2.13). Now if we set
$\Psi^{\prime}_{0}=0$ we will have a fourth order polynomial in $z$ equal to
zero444Here it is being assumed that $\Psi_{4}\neq 0$, which is always allowed
if the Weyl tensor does not vanish identically. Indeed, if the Weyl tensor is
non-zero and $\Psi_{4}=0$ then by means of a null rotation around
$\boldsymbol{l}$ we can easily make $\Psi_{4}\neq 0$.. Thus, in general we
have four distinct values of the parameter $z$ which accomplish this, call
these values $\\{z_{1},z_{2},z_{3},z_{4}\\}$. Then the Petrov types can be
defined as follows:
$\left\\{\begin{array}[]{ll}\textbf{Type O}&\rightarrow\;\textrm{Weyl tensor
is zero}\\\ \textbf{Type I}&\rightarrow\;\textrm{All roots are different}\\\
\textbf{Type D}&\rightarrow\;\textrm{Two pairs of roots
coincide},\,z_{1}=z_{2}\neq z_{3}=z_{4}\\\ \textbf{Type
II}&\rightarrow\;\textrm{Two roots coincide},\,z_{1}=z_{2}\neq z_{3}\neq
z_{4}\neq z_{1}\\\ \textbf{Type III}&\rightarrow\;\textrm{Three roots
coincide},\,z_{1}=z_{2}=z_{3}\neq z_{4}\\\ \textbf{Type
N}&\rightarrow\;\textrm{All roots
coincide},\,z_{1}=z_{2}=z_{3}=z_{4}\,.\end{array}\right.$ (2.15)
These four roots define four Lorentz transformations. By means of eq. (2.13)
such transformations lead to four privileged null vector fields
$\boldsymbol{l}^{\prime}_{i}$, which are the ones obtained by performing these
transformations on the vector field $\boldsymbol{l}$ of the original null
tetrad:
$\boldsymbol{l}\,\rightarrow\,\,\boldsymbol{l}^{\prime}_{i}\,=\,\boldsymbol{l}+\overline{z_{i}}\,\boldsymbol{m}+z_{i}\,\overline{\boldsymbol{m}}+|z_{i}|^{2}\,\boldsymbol{n}\;,\quad
i\in\\{1,2,3,4\\}\,.$ (2.16)
These real null directions are called the principal null directions (PNDs) of
the Weyl tensor. Moreover, when $z_{i}$ is a degenerated root the PND
$\boldsymbol{l}^{\prime}_{i}$ is said to be a repeated PND555The concept of
repeated PND can also be extracted from the bivector formalism of section 2.1,
as proved on reference [35].. When $z_{i}$ is a root of order $q$, we say that
the associated PND has degeneracy $q$. By the above definition of Petrov
classification it then follows that the Petrov type $I$ admits four distinct
PNDs; in type $D$ there are two pairs of repeated PNDs; in type $II$ there
exists three distinct PNDs, one being repeated; in type $III$ we have two
PNDs, one of which is repeated with triple degeneracy; in type $N$ there is
only one PND, this PND in repeated and has degree of degeneracy four.
In type $I$ once we set $\Psi^{\prime}_{0}=0$, by making $z=z_{i}$, the other
Weyl scalars are all different from zero, as can be seen from equations (2.14)
and (2.15). Then performing a null rotation around $\boldsymbol{l}$, which
makes $\Psi^{\prime}_{\alpha}\rightarrow\Psi^{\prime\prime}_{\alpha}$, it is
possible to make $\Psi^{\prime\prime}_{4}$ vanish while keeping
$\Psi^{\prime\prime}_{0}=0$, no other scalars can be made to vanish. Thus in
type $I$ the Weyl scalars $\Psi_{0}$ and $\Psi_{4}$ can always be made to
vanish by a judicious choice of null tetrad. As a further example let us treat
the type $D$. In the type $D$ setting $z=z_{1}$ it follows from equations
(2.14) and (2.15) that $\Psi^{\prime}_{0}=\Psi^{\prime}_{1}=0$. After this we
can perform a null rotation around $\boldsymbol{l}$ in order to set
$\Psi^{\prime\prime}_{3}=\Psi^{\prime\prime}_{4}=0$ while keeping
$\Psi^{\prime\prime}_{0}=\Psi^{\prime\prime}_{1}=0$. The table below sums up
what can be accomplished using this kind of procedure.
Type $O$ $-$ All | Type $II$ $-$ $\Psi_{0},\Psi_{1},\Psi_{4}$ | Type $D$ $-$ $\Psi_{0},\Psi_{1},\Psi_{3},\Psi_{4}$
---|---|---
Type $I$ $-$ $\Psi_{0},\Psi_{4}$ | Type $III$ $-$ $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{4}$ | Type $N$ $-$ $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{3}$
Table 2.1: Weyl scalars that can be made to vanish, by a suitable choice of
basis, on each Petrov type.
Although the definition of the Petrov types given in the present section looks
completely different from the one given in section (2.1) it is not hard to
prove that they are actually equivalent. As an example let us work out the
type $N$ case. According to the table 2.1, if the Weyl tensor is type $N$ it
follows that it is possible to find a null tetrad on which the only non-
vanishing Weyl scalar is $\Psi_{4}$. In this basis eq. (2.10) yield that
$C^{+}$ has the following matrix representation:
$C_{N}\,=\,2\left[\begin{array}[]{ccc}0&\Psi_{4}&0\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right]\,.$
Along with appendix A this means that the algebraic type of the operator
$C_{N}$ is $[\,|2,1]$, which perfectly matches the definition of eq. (2.7).
More details about the approach adopted in this section can be found in [12].
#### 2.3 Boost Weight
In this section the boost transformations, eq. (2.11), will be used to provide
another form of expressing the Petrov types. In order to accomplish this we
first need to see how the Weyl scalars behave under Lorentz boosts. Inserting
eq. (2.11) into the definition of the Weyl scalars, eq. (2.1), we easily find
the following transformation:
$\Psi_{\alpha}\,\longrightarrow\,\gamma^{(2-\alpha)}\,\Psi_{\alpha}\
\;\;,\quad\,\gamma\equiv
e^{i\theta}\,\lambda\;,\,\;\alpha\in\\{0,1,2,3,4\\}\,.$ (2.17)
In jargon we say that the Weyl scalar $\Psi_{\alpha}$ has boost weight
$\mathfrak{b}=(2-\alpha)$. Note, particularly, that the maximum boost weight
(b.w.) that a component of the Weyl tensor can have is $\mathfrak{b}=2$, while
the minimum is $\mathfrak{b}=-2$.
Given the components of the Weyl tensor on a particular basis, we shall denote
by $\mathfrak{b}_{+}$ the b.w. of the non-vanishing Weyl scalar with maximum
boost weight. Analogously, $\mathfrak{b}_{-}$ denotes the b.w. of the non-
vanishing Weyl tensor component with minimum boost weight. For instance, using
eq. (2.17) and table (2.1) we see that if the Weyl tensor is type $III$ then
it is possible to find a null frame in which $\mathfrak{b}_{+}=-1$. In general
we can define the Petrov types using this kind of reasoning, the bottom line
is summarized below:
$\left\\{\begin{array}[]{ll}\textbf{Type I}&\rightarrow\;\textrm{There is a
frame in which}\;\mathfrak{b}_{+}=+1\\\ \textbf{Type
II}&\rightarrow\;\textrm{There is a frame in which}\;\mathfrak{b}_{+}=\,0\\\
\textbf{Type III}&\rightarrow\;\textrm{There is a frame in
which}\;\mathfrak{b}_{+}=-1\\\ \textbf{Type N}&\rightarrow\;\textrm{There is a
frame in which}\;\mathfrak{b}_{+}=-2\\\ \textbf{Type
D}&\rightarrow\;\textrm{There is a frame in
which}\;\mathfrak{b}_{+}=\;\mathfrak{b}_{-}=0\\\ \textbf{Type
O}&\rightarrow\;\textrm{Weyl tensor vanishes
identically}\,.\end{array}\right.$ (2.18)
On the boost weight approach the different Petrov types have a hierarchy: The
type $I$ is the most general, type $II$ is a special case of the type $I$,
type $III$ is a special case of type $II$ and the type $N$ is a special case
of type $III$. The type $D$ is also a special case of type $II$, in this type
all non-vanishing components of the Weyl tensor have zero boost weight.
A classification for the Weyl tensor using the boost weight method can be
naturally generalized to higher dimensions, which yields the so-called CMPP
classification [36]. The CMPP classification has been intensively investigated
in the last ten years, see, for example, [37, 38] and references therein.
#### 2.4 Bel-Debever and Principal Null Directions
Few years after the release of Petrov’s original article defining his
classification, Bel and Debever have, independently, found an equivalent, but
quite different, way to define the Petrov types [25, 32]. On such approach the
Petrov types are defined in terms of algebraic conditions involving the Weyl
tensor and the principal null directions defined in section 2.2.
Since the null tetrad frame at a point $p\in M$ forms a local basis for the
tangent space $T_{p}M$, it follows that the Weyl tensor can be expanded in
terms of the tensorial product of this basis. Because of the symmetries of
this tensor, eq. (2.1), it follows that the expansion shall be expressed in
terms of the following kind of combination:
$\langle
e,v,u,t\rangle_{\mu\nu\rho\sigma}\,\equiv\,4\,e_{[\mu}v_{\nu]}\,u_{[\rho}t_{\sigma]}\,+\,4\,u_{[\mu}t_{\nu]}\,e_{[\rho}v_{\sigma]}\,.$
Once introduced a null tetrad
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$,
the Weyl tensor can be written as the following expansion:
$\displaystyle
C_{\mu\nu\rho\sigma}=\Big{\\{}\,\frac{1}{2}(\Psi_{2}+\overline{\Psi}_{2})\big{[}\langle
l,n,l,n\rangle+\langle
m,\overline{m},m,\overline{m}\rangle\big{]}+\Psi_{0}\langle
n,\overline{m},n,\overline{m}\rangle+$ $\displaystyle+\Psi_{4}\langle
l,m,l,m\rangle-\Psi_{2}\langle
l,m,n,\overline{m}\rangle-\frac{1}{2}(\Psi_{2}-\overline{\Psi}_{2})\langle
l,n,m,\overline{m}\rangle+$ (2.19) $\displaystyle+\Psi_{1}\big{[}\langle
l,n,n,\overline{m}\rangle+\langle
n,\overline{m},\overline{m},m\rangle\big{]}+\Psi_{3}\big{[}\langle
l,m,m,\overline{m}\rangle-\langle
l,n,l,m\rangle\big{]}+\;c.c.\,\Big{\\}}_{\mu\nu\rho\sigma}\,.$
Where $c.c.$ denotes the complex conjugate of all previous terms inside the
curly bracket. In particular, note that the right hand side of the above
equation is real and has the symmetries of the Weyl tensor. We can verify that
such expansion is indeed correct by contracting equation (2.19) with the null
frame and checking that equation (2.1) is satisfied. Now, contracting equation
(2.19) with $l^{\nu}l^{\rho}$ yield:
$C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}=\left[\Psi_{1}(l_{\mu}\overline{m}_{\sigma}+\overline{m}_{\mu}l_{\sigma})+c.c.\right]-2\left(\Psi_{0}\overline{m}_{\mu}\overline{m}_{\sigma}+c.c.\right)-2\left(\Psi_{2}+\overline{\Psi}_{2}\right)l_{\mu}l_{\sigma}\,.$
The above expression, in turn, immediately implies the following identities:
$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}\,=\,\left(\Psi_{1}\,l_{[\alpha}\overline{m}_{\mu]}l_{\sigma}+c.c.\right)\,-\,2\,\left(\Psi_{0}\,l_{[\alpha}\overline{m}_{\mu]}\overline{m}_{\sigma}+c.c.\right)\,,$
(2.20)
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}\,=\,-2\,\left(\Psi_{0}\,l_{[\alpha}\overline{m}_{\mu]}\overline{m}_{[\sigma}l_{\beta]}+c.c.\right)\,.$
(2.21)
From which we conclude that the combination on the left hand side of eq.
(2.21) vanishes if, and only if, $\Psi_{0}=0$. Hence, by the definition given
in section 2.2, it follows that $\boldsymbol{l}$ is a principal null direction
if, and only if,
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0$. Analogously,
eq. (2.20) and the definition below eq. (2.16) imply that $\boldsymbol{l}$ is
a repeated PND if, and only if,
$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0$. In the same vein, the
following relations can be proved:
$\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{2}=0\;\;\Leftrightarrow\;\;$
$\displaystyle C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}\,=\,0$
$\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{2}=\Psi_{3}=0\;\;\Leftrightarrow\;\;$
$\displaystyle C_{\mu\nu\rho\sigma}l^{\rho}\,=\,0$
Using these results and table 2.1 it is then simple matter to arrive at the
following alternative definition for the Petrov types:
$\left\\{\begin{array}[]{ll}\textbf{Type I}&\rightarrow\;\textrm{ exists
$\boldsymbol{l}$ such that
}l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0\\\ \textbf{Type
II}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0\\\ \textbf{Type
III}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}=0\\\ \textbf{Type
N}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}C_{\mu\nu\rho\sigma}l^{\rho}=0\\\ \textbf{Type D}&\rightarrow\;\textrm{ exist
$\boldsymbol{l},\boldsymbol{n}$ such that
}l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0=n_{[\alpha}C_{\mu]\nu\rho\sigma}n^{\nu}n^{\rho}\\\
\textbf{Type O}&\rightarrow\;\textrm{ exist $\boldsymbol{l},\boldsymbol{n}$
such that
}C_{\mu\nu\rho\sigma}l^{\rho}=0=C_{\mu\nu\rho\sigma}n^{\rho}\,.\end{array}\right.$
Where it was assumed that $\boldsymbol{l}$ and $\boldsymbol{n}$ are real null
vectors such that $l^{\mu}\,n_{\mu}=1$. On such definition it is assumed that
the Petrov types obey the same hierarchy of the preceding section:
$O\,\subset\,N\,\subset\,III\,\subset\,II\,\subset\,I\;\textrm{ and
}\;O\,\subset\,D\,\subset\,II\,.$
These algebraic constraints involving the Weyl tensor and null directions are
called Bel-Debever conditions. In reference [39] these conditions were
investigated in higher-dimensional space-times and connections with the CMPP
classification were made.
#### 2.5 Spinors, Penrose’s Method
In this section we will take advantage of the spinorial formalism in order to
describe the Petrov classification, an approach introduced by R. Penrose [31].
Here it will be assumed that the reader is already familiar with the spinor
calculus in 4-dimensional general relativity. For those not acquainted with
this language, a short course is available in [40]. For a more thorough
treatment with diverse applications [41] is recommended. Appendix C of the
present thesis provides the general formalism of spinors in arbitrary
dimensions.
On the spinorial formalism of 4-dimensional Lorentzian manifolds we have two
types of indices, the ones associated with Weyl spinors of positive chirality,
$A,B,C,...\in\\{1,2\\}$, and the ones related to semi-spinors of negative
chirality, $\dot{A},\dot{B},\dot{C},...\in\\{1,2\\}$. It is also worth
mentioning that the complex conjugation changes the chirality of the spinorial
indices. In this language a vectorial index is equivalent to the “product” of
two spinorial indices, one of positive chirality and one of negative
chirality:
$V_{\mu}\,\sim\,V_{A\dot{A}}\,.$
The spaces of semi-spinors are endowed with skew-symmetric metrics
$\varepsilon_{AB}=\varepsilon_{[AB]}$ and
$\overline{\varepsilon}_{\dot{A}\dot{B}}=\overline{\varepsilon}_{[\dot{A}\dot{B}]}$.
This anti-symmetry implies, for instance, that
$\zeta^{A}\zeta_{A}=\zeta^{A}\varepsilon_{AB}\zeta^{B}=0$ for every spinor
$\boldsymbol{\zeta}$. These spinorial metrics are related to the space-time
metric by the relation
$g_{\mu\nu}\sim\varepsilon_{AB}\overline{\varepsilon}_{\dot{A}\dot{B}}$. In
this formalism the Weyl tensor is represented by
$C_{\mu\nu\rho\sigma}\,\sim\,\left(\,\Psi_{ABCD}\,\overline{\varepsilon}_{\dot{A}\dot{B}}\overline{\varepsilon}_{\dot{C}\dot{D}}\,+\,c.c.\,\right)\,.$
(2.22)
Where $\Psi$ is a completely symmetric object, $\Psi_{ABCD}=\Psi_{(ABCD)}$,
and $c.c.$ denotes the complex conjugate of the previous terms inside the
bracket. Since $\boldsymbol{\varepsilon}$ carry the degrees of freedom of the
space-time metric, it follows that the degrees of freedom of the Weyl tensor
are entirely contained on $\boldsymbol{\Psi}$. Therefore, classify the Weyl
tensor is then equivalent to classify $\boldsymbol{\Psi}$.
It is a well-known result in this formalism that every object with completely
symmetric chiral indices, $S_{A_{1}A_{2}\ldots A_{p}}=S_{(A_{1}A_{2}\ldots
A_{p})}$, can be decomposed as a symmetrized direct product of spinors,
$S_{A_{1}A_{2}\ldots
A_{p}}=\zeta^{1}_{(A_{1}}\zeta^{2}_{A_{2}}\ldots\zeta^{p}_{A_{p})}$ [40].
Particularly, we can always find spinors
$\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}$ and
$\boldsymbol{\chi}$ such that
$\Psi_{ABCD}\,=\,\zeta_{(A}\,\theta_{B}\,\xi_{C}\,\chi_{D)}\,.$ (2.23)
We can then easily classify the Weyl tensor according to the possibility of
the spinors $\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}$ and
$\boldsymbol{\chi}$ being proportional to each other. Denoting de
proportionality of the spinors by “$\leftrightarrow$” and the non-
proportionality by “$\nleftrightarrow$”, we shall define:
$\left\\{\begin{array}[]{ll}\textbf{Type
I}&\rightarrow\;\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}\textrm{
and }\boldsymbol{\chi}\textrm{ are non-propotional to each other}\\\
\textbf{Type II}&\rightarrow\;\textrm{One pair coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\nleftrightarrow\xi\nleftrightarrow\boldsymbol{\chi}\nleftrightarrow\boldsymbol{\zeta}\\\
\textbf{Type III}&\rightarrow\;\textrm{Three spinors coincidence,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\leftrightarrow\boldsymbol{\xi}\nleftrightarrow\boldsymbol{\chi}\\\
\textbf{Type D}&\rightarrow\;\textrm{Two pairs coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\nleftrightarrow\boldsymbol{\xi}\leftrightarrow\boldsymbol{\chi}\\\
\textbf{Type N}&\rightarrow\;\textrm{All spinors coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\leftrightarrow\boldsymbol{\xi}\leftrightarrow\boldsymbol{\chi}\\\
\textbf{Type
O}&\rightarrow\;\boldsymbol{\zeta}\,=\,\boldsymbol{\theta}\,=\,\boldsymbol{\xi}\,=\,\boldsymbol{\chi}\,=\,0\,.\end{array}\right.$
(2.24)
The spinors that appear on the decomposition of $\boldsymbol{\Psi}$ are called
the principal spinors of the Weyl tensor, since they are intimately related to
the principal null directions. Indeed, the real null vectors generated by
these spinors,
$l_{1}^{\,\mu}\,\sim\,\zeta^{A}\overline{\zeta}^{\dot{A}}\;\;;\;\;l_{2}^{\,\mu}\,\sim\,\theta^{A}\overline{\theta}^{\dot{A}}\;\;;\;\;l_{3}^{\,\mu}\,\sim\,\xi^{A}\overline{\xi}^{\dot{A}}\;\;;\;\;l_{4}^{\,\mu}\,\sim\,\chi^{A}\overline{\chi}^{\dot{A}}\,,$
point in the principal null directions of the Weyl tensor. Hence, the
coincidence of the principal spinors is equivalent to coincidence of PNDs,
which makes a bridge between the spinorial approach to the Petrov
classification and the approach adopted in section 2.2.
The spinorial formalism allows us to see quite neatly which Weyl scalars can
be made to vanish by a suitable choice of null tetrad frame on each Petrov
type. If $\\{o_{{}_{A}},\iota_{{}_{A}}\\}$ forms a spin frame,
$o_{{}_{A}}\iota^{A}=1$, then we can use them to build a null tetrad frame, as
shown in appendix B. So using equations (2.1) and (B.1) we can prove that the
Weyl scalars are given by:
$\displaystyle\Psi_{0}=\Psi_{ABCD}o^{A}o^{B}o^{C}o^{D}\;;\;\Psi_{1}=\Psi_{ABCD}o^{A}o^{B}o^{C}\iota^{D}\;;\;\Psi_{2}=\Psi_{ABCD}o^{A}o^{B}\iota^{C}\iota^{D}$
$\displaystyle\Psi_{3}=\Psi_{ABCD}o^{A}\iota^{B}\iota^{C}\iota^{D}\;;\;\Psi_{4}=\Psi_{ABCD}\iota^{A}\iota^{B}\iota^{C}\iota^{D}\,.$
(2.25)
Thus, for example, if the Weyl tensor is type $D$ according to eq. (2.24) then
there exists non-zero spinors $\boldsymbol{\zeta}$ and $\boldsymbol{\xi}$ such
that $\Psi_{ABCD}=\zeta_{(A}\zeta_{B}\xi_{C}\xi_{D)}$. Since
$\boldsymbol{\zeta}\nleftrightarrow\boldsymbol{\xi}$ it follows that
$\zeta_{A}\xi^{A}=w\neq 0$. Therefore, setting $o_{{}_{A}}=\zeta_{{}_{A}}$ and
$\iota_{{}_{A}}=w^{-1}\xi_{{}_{A}}$ it follows that
$\\{\boldsymbol{o},\boldsymbol{\iota}\\}$ forms a spin frame. Then using
equation (2.25) we easily find that in this frame
$\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0$, which agrees with table 2.1. By means
of the same reasoning it is straightforward to work out the other types and
verify that the definitions of the Petrov types presented on (2.24) perfectly
matches the table 2.1.
In the same vein, the bivector method of section 2.1 can be easily understood
on the spinorial formalism. In the spinorial language a self-dual bivector is
represented by a symmetric spinor $\phi^{AB}=\phi^{(AB)}$, so that the map
$C^{+}$ is represented by
$\phi_{AB}\mapsto\phi^{\prime}_{AB}=\Psi_{ABCD}\phi^{CD}$. Thus, for example,
if the Weyl tensor is type $N$ then we can find a spin frame
$\\{\boldsymbol{o},\boldsymbol{\iota}\\}$ such that
$\Psi_{ABCD}=o_{{}_{A}}o_{{}_{B}}o_{{}_{C}}o_{{}_{D}}$. Then defining
$\phi_{1}^{AB}=o^{A}o^{B}$, $\phi_{2}^{AB}=o^{(A}\iota^{B)}$ and
$\phi_{3}^{AB}=\iota^{A}\iota^{B}$, it follows that the action of $C^{+}$ in
this basis of self-dual bivectors yields $\phi^{\prime}_{1}=0$,
$\phi^{\prime}_{2}=0$ and $\phi^{\prime}_{3}=\phi_{1}$, which agrees with
equation (2.7).
#### 2.6 Clifford Algebra
In this section the formalism of Clifford algebra will be used to describe
another form to arrive at the Petrov classification. For those not acquainted
with the tools of geometric algebra, appendix C introduces the necessary
background. Let
$\\{\hat{\boldsymbol{e}}_{0},\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2},\hat{\boldsymbol{e}}_{3},\\}$
be a local orthonormal frame on a 4-dimensional Lorentzian manifold
$(M,\boldsymbol{g})$,
$\frac{1}{2}\left(\hat{\boldsymbol{e}}_{a}\hat{\boldsymbol{e}}_{b}\,+\,\hat{\boldsymbol{e}}_{b}\hat{\boldsymbol{e}}_{a}\right)\,=\,\boldsymbol{g}(\hat{\boldsymbol{e}}_{a},\hat{\boldsymbol{e}}_{b})\,=\,\eta_{ab}\,=\,\operatorname{diag}(1,-1,-1,-1)\,.$
Denoting by $\eta^{ab}$ the inverse matrix of $\eta_{ab}$, we shall define
$\hat{\boldsymbol{e}}^{a}=\eta^{ab}\hat{\boldsymbol{e}}_{b}$. Let us denote
the space spanned by the bivector fields by $\Gamma(\wedge^{2}M)$. Then, in
the formalism of geometric calculus [42, 43] the Weyl tensor is a linear
operator on the space of bivectors,
$\mathcal{C}:\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)$, whose action
is666All results in this thesis are local, so that it is always being assumed
that we are in the neighborhood of some point. Thus, formally, instead of
$\Gamma(\wedge^{2}M)$ we should have written $\Gamma(\wedge^{2}M)|_{N_{x}}$,
which is the restriction of the space of sections of the bivector bundle to
some neighborhood $N_{x}$ of a point $x\in M$. So we are choosing a particular
local trivialization of the bivector bundle.
$\mathcal{C}(\boldsymbol{V}\wedge\boldsymbol{U})\,=\,V^{a}\,U^{b}\,C_{abcd}\,\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d}\,,$
(2.26)
where $C_{abcd}$ are the components of the Weyl tensor on the frame
$\\{\hat{\boldsymbol{e}}_{a}\\}$. In the above equation
$\boldsymbol{V}\wedge\boldsymbol{U}$ means the anti-symmetrized part of the
Clifford product of $\boldsymbol{V}$ and $\boldsymbol{U}$,
$\boldsymbol{V}\wedge\boldsymbol{U}=\frac{1}{2}(\boldsymbol{V}\boldsymbol{U}-\boldsymbol{U}\boldsymbol{V})$.
Then using (2.26) and equation (C.4) of appendix C we find:
$\displaystyle\hat{\boldsymbol{e}}^{a}\,\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})\,$
$\displaystyle=\,C_{abcd}\,\hat{\boldsymbol{e}}^{a}\,(\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d})=C_{abcd}\,\hat{\boldsymbol{e}}^{a}\,\frac{1}{2}(\hat{\boldsymbol{e}}^{c}\hat{\boldsymbol{e}}^{d}-\hat{\boldsymbol{e}}^{d}\hat{\boldsymbol{e}}^{c})$
$\displaystyle=\,C_{abcd}\,\frac{1}{2}(2\,\eta^{ac}\,\hat{\boldsymbol{e}}^{d}-2\,\eta^{ad}\,\hat{\boldsymbol{e}}^{c}+2\,\hat{\boldsymbol{e}}^{a}\wedge\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d})$
$\displaystyle=\,2\,C_{\phantom{c}bcd}^{c}\,\hat{\boldsymbol{e}}^{d}\,-\,C_{b[acd]}\,\hat{\boldsymbol{e}}^{a}\wedge\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d}$
(2.27)
Equation (2.27) makes clear that on the Clifford algebra formalism the single
equation
$\hat{\boldsymbol{e}}^{a}\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})=0$
is equivalent to the trace-less property and the Bianchi identity satisfied by
the Weyl tensor. There are two other symmetries satisfied by this tensor, see
(2.1), which are the anti-symmetry on the first and second pairs of indices,
$C_{abcd}=C_{[ab][cd]}$ and the symmetry by the exchange of these pairs,
$C_{abcd}=C_{cdab}$. But the latter symmetry can be derived from the Bianchi
identity, while the former is encapsulated in the present formalism by the
fact that the operator $\mathcal{C}$ maps bivectors into bivectors. Thus we
conclude that on the Clifford algebra approach all the symmetries of the Weyl
tensor are encoded in the following relations:
$\mathcal{C}:\,\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)\quad;\quad\hat{\boldsymbol{e}}^{a}\,\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})\,=\,0\,.$
(2.28)
Before proceeding let us define the following bivectors:
$\boldsymbol{\sigma}_{i}\,=\,\hat{\boldsymbol{e}}_{0}\wedge\hat{\boldsymbol{e}}_{i}\;\,;\quad\boldsymbol{I}\boldsymbol{\sigma}_{i}\,=\,\frac{1}{2}\epsilon^{ijk}\,\hat{\boldsymbol{e}}_{j}\wedge\hat{\boldsymbol{e}}_{k}$
Where $i,j,k$ are indices that run from 1 to 3, $\epsilon^{ijk}$ is a totally
anti-symmetric object with $\epsilon^{123}=1$ and
$\boldsymbol{I}=\hat{\boldsymbol{e}}_{0}\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{3}$
is the pseudo-scalar defined on appendix C. In particular, using these
definitions and the Bianchi identity it is not difficult to prove that the
following equation holds:
$\mathcal{C}(\boldsymbol{\sigma}_{i})\,=\,-2\left[\,C_{0i0j}+\boldsymbol{I}\,C_{0kli}\epsilon^{klj}\,\right]\,\boldsymbol{\sigma}_{j}$
(2.29)
Also, expanding equation (2.28) we find the following explicit relations:
$\displaystyle\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{\sigma}_{2})\,+\,\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{\sigma}_{3})\,=\,0$
$\displaystyle\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{\sigma}_{1})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{2})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{3})$
$\displaystyle\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{\sigma}_{2})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{3})$
(2.30)
$\displaystyle\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{\sigma}_{3})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{2})$
Summing the last three relations above and then using the first one, we find
$\sum_{i}\boldsymbol{I}\boldsymbol{\sigma}_{i}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})=0$.
Then using this identity on the last three relations of (2.30) we conclude
that
$\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})=\boldsymbol{I}\mathcal{C}(\boldsymbol{\sigma}_{i})$.
By means of this and the identity $\boldsymbol{I}^{2}=-1$ we also find that
$\mathcal{C}(\boldsymbol{I}\,\boldsymbol{I}\boldsymbol{\sigma}_{i})=\boldsymbol{I}\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})$.
Since $\\{\boldsymbol{\sigma}_{i},\boldsymbol{I}\boldsymbol{\sigma}_{i}\\}$ is
a basis for the bivector space it follows that in general
$\mathcal{C}(\boldsymbol{I}\boldsymbol{B})\,=\,\boldsymbol{I}\,\mathcal{C}(\boldsymbol{B})\quad\;\forall\;\boldsymbol{B}\in\Gamma(\wedge^{2}M)\,.$
(2.31)
Now recall from appendix C that the pseudo-scalar $\boldsymbol{I}$ commutes
with the elements of even order, in particular it commutes with all bivectors.
Moreover, equation (2.31) guarantees that $\boldsymbol{I}$ commutes with the
Weyl operator. Therefore, when dealing with the Weyl operator acting on the
bivector space we can treat the $\boldsymbol{I}$ as if it were a scalar.
Furthermore, since $\boldsymbol{I}^{2}=-1$ we can pretend that
$\boldsymbol{I}$ is the imaginary unit, $\boldsymbol{I}\sim i=\sqrt{-1}$, so
that we can reinterpret the operator $\mathcal{C}$ as an operator on the
complexification of the real space generated by
$\\{\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2},\boldsymbol{\sigma}_{3}\\}$.
With these conventions the equation (2.29) can be written as777A similar
phenomenon happens on the Clifford algebra of the space $\mathbb{R}^{3}$. In
this case the pseudo-scalar commutes with all elements of the algebra and
obeys to the relation $\boldsymbol{I}^{2}=-1$, so that it can actually be
interpreted as the imaginary unit, $\boldsymbol{I}\sim i=\sqrt{-1}$. This is
the geometric explanation of why the complex numbers are so useful when
dealing with rotations in 3 dimensions.:
$\mathcal{C}(\boldsymbol{\sigma}_{i})\,=\,\mathcal{C}_{ij}\,\boldsymbol{\sigma}_{j}\quad;\;\mathcal{C}_{ij}\sim-2\left(C_{0i0j}+i\,C_{0kli}\epsilon^{klj}\right)$
(2.32)
Now we can easily define a classification for the Weyl tensor. Using equation
(2.32) and the symmetries of the Weyl tensor it is trivial to prove that this
matrix is trace-less, $\mathcal{C}_{ii}=0$. Therefore, the possible algebraic
types for the operator $\mathcal{C}$ are the same as the ones listed on eq.
(2.7).
Note that this classification is, in principle, different from the one shown
on subsection 2.1. While the latter uses the space of self-dual bivectors to
define a 3-dimensional operator, the operator introduced in the present
subsection acts on the space generated by
$\\{\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2},\boldsymbol{\sigma}_{3}\\}$,
which is not the space of self-dual bivectors. The remarkable thing is that
these two classifications turns out to be equivalent. This can be seen by
noting that to every eigen-bivector of $\mathcal{C}$ we can associate a self-
dual bivector that is eigen-bivector of $C^{+}$ with the “same” eigenvalue.
Indeed, if $\boldsymbol{B}$ is an eigen-bivector of the operator $\mathcal{C}$
on the Clifford algebra approach then
$\mathcal{C}(\boldsymbol{B})=(\lambda_{1}+\boldsymbol{I}\lambda_{2})\boldsymbol{B}$,
where $\lambda_{1}$ and $\lambda_{2}$ are real numbers. Then using equation
(C.6) of appendix C we see that
$\boldsymbol{B}_{+}=(1-i\boldsymbol{I})\boldsymbol{B}$ is a self-dual
bivector. Moreover, we can use equation (2.31) to prove that
$\mathcal{C}(\boldsymbol{B}_{+})=(\lambda_{1}+i\lambda_{2})\boldsymbol{B}_{+}$.
To finish the proof just note that the Weyl operator defined on (2.26) agrees
with the definition of the section 2.1, see equation (2.2). Hence we have that
$C^{+}(\boldsymbol{B}_{+})=(\lambda_{1}+i\lambda_{2})\boldsymbol{B}_{+}$. More
details about this method can be found in [43, 44]. In particular, reference
[43] has exploited the Clifford algebra formalism to find canonical forms for
the Weyl operator for each algebraic type. As an aside, it is worth mentioning
that the whole formalism of general relativity can be translated to the
Clifford algebra language with some advantages [45].
#### 2.7 Interpreting the PNDs
In the previous sections it has been proved that every space-time with non-
vanishing Weyl tensor admits some privileged null directions, four at most,
called the principal null directions (PNDs). In the present section we will
investigate the role played by these directions both from the geometrical and
physical points of view.
According to [46, 39], in 1922 Élie Cartan has pointed out that the Weyl
tensor of a general 4-dimensional space-time defined four distinguished null
directions endowed with some invariance properties under the parallel
transport over infinitesimal closed loops. It turns out that these directions
were the principal null directions of the Weyl tensor, in spite of Petrov’s
article defining his classification have appeared three decades later. Suppose
that a vector $\boldsymbol{v}$ belonging to the tangent space at a point $p\in
M$ is parallel transported along an infinitesimal parallelogram with sides
generated by $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$, as illustrated on
the figure below.
It is a well-known result of Riemannian geometry that the change on the vector
$\boldsymbol{v}$ caused by the parallel transport over the loop is given by
$\delta v^{\mu}\equiv
v^{\prime\mu}-v^{\mu}=-\epsilon\,R^{\mu}_{\phantom{\mu}\nu\rho\sigma}\,v^{\nu}\,t_{1}^{\rho}\,t_{2}^{\sigma}\,.$
(2.33)
Where $\boldsymbol{v}^{\prime}$ is the vector after the parallel transport and
$\epsilon$ is proportional to the area of the parallelogram. In vacuum, as
henceforth assumed in this section, Einstein’s equation implies that the
Riemann tensor is equal to the Weyl tensor. So that in this case one can
substitute $R^{\mu}_{\phantom{\mu}\nu\rho\sigma}$ by
$C^{\mu}_{\phantom{\mu}\nu\rho\sigma}$ in equation (2.33). Now let us search
for null directions that are preserved by this kind of parallel transport.
Let $\boldsymbol{v}=\boldsymbol{l}$ be a PND and $\boldsymbol{n}$ a null
vector such that $l^{\mu}n_{\mu}=1$. Then, from section 2.4, we have that
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0$. Contracting
this equation with $t_{2}^{\mu}n^{\beta}$ we easily find that
$C^{\sigma}_{\phantom{\sigma}\nu\rho\mu}l^{\nu}l^{\rho}t_{2}^{\mu}\propto
l^{\sigma}$ for any $\boldsymbol{t}_{2}$ orthogonal to $\boldsymbol{l}$. Thus
PNDs are the null directions with the property of being invariant by the
parallel transport around infinitesimal parallelograms generated by the PND
itself and any direction orthogonal to it. In the same vein, if
$\boldsymbol{l}$ is a repeated principal null direction then
$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0$. Contracting this last
equation with $t_{2}^{\sigma}n^{\alpha}$ we find that $\delta l^{\mu}\propto
l^{\mu}$ for any parallelogram such that one of the sides is generated by
$\boldsymbol{l}$. If $\boldsymbol{l}$ is a triply degenerated PND then
$C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}=0$, which by contraction with
$t_{1}^{\mu}t_{2}^{\nu}n^{\alpha}$ yield that $\delta l^{\mu}\propto l^{\mu}$
for any parallelogram. Finally, if $\boldsymbol{l}$ is a PND with degree of
degeneracy four then $C_{\mu\nu\rho\sigma}l^{\sigma}=0$, so that $\delta
l^{\mu}=0$ for any parallelogram. Table 2.2 summarizes these geometric
properties of the PNDs.
$q\,=\,1$ | $q\,=\,2$ | $q\,=\,3$ | $q\,=\,4$
---|---|---|---
$\boldsymbol{t}_{1}\,=\,\boldsymbol{l}$ | $\boldsymbol{t}_{1}\,=\,\boldsymbol{l}$ | $\boldsymbol{t}_{1}$ arbitrary | $\boldsymbol{t}_{1}$ arbitrary
$\;t_{2}^{\mu}\,l_{\mu}=0$ | $\boldsymbol{t}_{2}$ arbitrary | $\boldsymbol{t}_{2}$ arbitrary | $\boldsymbol{t}_{2}$ arbitrary
$\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}=0$
Table 2.2: Invariance of the PNDs under parallel transport over an
infinitesimal parallelogram with sides generated by $\boldsymbol{t}_{1}$ and
$\boldsymbol{t}_{2}$. In the first row $q$ denotes the degeneracy of the PND
$\boldsymbol{l}$.
In ref. [47] it was shown another geometric interpretation for the principal
null directions. Glossing over the subtleties, it was proved there that a null
direction is a PND when the Riemannian curvature of a 2-space generated by
this null direction and a space-like vector field $\boldsymbol{t}$ is
independent of $\boldsymbol{t}$.
One of the first physicists to investigate the physical meaning of the Petrov
types was F. Pirani. In ref. [26] he has tried to find a plausible definition
of gravitational radiation by comparing with the electromagnetic case. In this
article it has been shown that the energy-momentum tensor associated with
electromagnetic radiation admits no time-like eigenvector and one null
eigenvector at most, this null vector turned out to point in the direction of
the radiation propagation. Searching for an analogous condition in general
relativity Pirani investigated the eigenbivectors of Riemann tensor. The
intersection of the planes generated by such eigenbivectors defined what he
called Riemann principal directions (RPDs), which are not the PNDs, as they
are not necessarily null. But it turns out that the null Riemann principal
directions are repeated PNDs. Thus, mimicking the electromagnetic case, Pirani
arrived at the conclusion that if a space-time admits a time-like RPD then no
gravitational radiation should be present. Along with the results of Bel [25],
this means that no gravitational radiation is allowed on Petrov types $I$ and
$D$, which is reasonable since all static space-times are either type $I$ or
$D$. Pirani and Bel interpreted the repeated PNDs of types $II$, $III$ and $N$
as the direction of the gravitational radiation propagation [25, 26].
In order to understand the physical meaning of the PNDs, the analogy between
the electromagnetic theory and general relativity was also exploited by other
physicists. In [48, 25] L. Bel has introduced a tensor of rank four that is
quadratic on the Riemann tensor and that in vacuum has properties that
perfectly mimics the electromagnetic energy-momentum tensor. Such tensor is
now called the Bel-Robinson tensor [41]. Then Debever proved that in vacuum
this tensor is completely determined by the principal null directions of the
Weyl tensor [32], a result that can be easily verified using the spinorial
formalism. In ref. [31], Penrose has argued that the PNDs are related to the
gravitational energy density, enforcing and complementing Debever’s results.
Penrose also concluded that pure gravitational radiation should be present
only in type $N$ space-times, since only in this case the Weyl tensor
satisfies the massless wave-equation.
Finally, according to the Goldberg-Sachs theorem, the repeated PNDs in vacuum
are tangent to a congruence of null geodesics that is shear-free. This
celebrated theorem is behind the integrability of Einstein’s equation for
space-times of type $D$ [24]. This important result will be deeply exploited
on the forthcoming chapters. One of the goals of this thesis is to prove a
suitable generalization of this theorem valid in higher dimensions, which will
be accomplished in chapters 5 and 6.
#### 2.8 Examples
1) Schwarzschild space-time
Schwarzschild space-time is the unique spherically-symmetric solution of
Einstein’s equation in vacuum. In a static and spherically symmetric
coordinate system its metric is given by
$ds^{2}=f^{2}\,dt^{2}-f^{-2}\,dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,,\;\;f^{2}=1-\frac{2M}{r}\,.$
A suitable orthonormal frame and a suitable null tetrad are then,
$\displaystyle\hat{\boldsymbol{e}}_{0}=f^{-1}\,\partial_{t}\;;\;\;\hat{\boldsymbol{e}}_{1}=f\,\partial_{r}\;;\;\;\hat{\boldsymbol{e}}_{2}=\frac{1}{r}\partial_{\theta}\;;\;\;\hat{\boldsymbol{e}}_{3}=\frac{1}{r\sin\theta}\partial_{\varphi}\,;\,\textrm{
and }$
$\displaystyle\boldsymbol{l}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}_{1})\,;\;\boldsymbol{n}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}-\hat{\boldsymbol{e}}_{1})\,;\;\boldsymbol{m}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}+i\hat{\boldsymbol{e}}_{3})\,;\;\overline{\boldsymbol{m}}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}-i\hat{\boldsymbol{e}}_{3})\,.$
Since the vector field $\partial_{t}=f\hat{\boldsymbol{e}}_{0}$ is a time-like
hyper-surface orthogonal Killing vector field, the space-time is called
static. In other words this means that the above metric is invariant by the
transformations $t\rightarrow-t$ and $t\rightarrow t+\epsilon$, where
$\epsilon$ is a constant. Such symmetries imply that the Weyl tensor cannot be
of Petrov types $II$, $III$ or $N$. For instance, if some static space-time
were type $N$ it would have just one PND,
$\boldsymbol{l}=\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}$ where
$\hat{\boldsymbol{e}}$ is some space-like vector of unit norm. But using the
symmetry $t\rightarrow-t$ we conclude that the null vector
$\boldsymbol{l}^{\prime}=-\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}$
should also be a PND, which contradicts the type $N$ hypothesis. Thus the
Schwarzschild solution must be either type $I$ or $D$. Indeed, calculating the
Weyl scalars, by means of (2.1), on the above null frame we get:
$\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\;\;;\;\;\Psi_{2}=\frac{M}{r^{3}}\,.$
Then, thanks to table 2.1, we conclude that the Schwarzschild space-time has
Petrov type $D$, with $\boldsymbol{l}$ and $\boldsymbol{n}$ being repeated
PNDs. Actually, it can be proved that the whole family of Kerr-Newman
solutions is type $D$.
2) Plane Gravitational Waves
Physically, plane waves are characterized by the existence of plane wave-
fronts (equipotentials) orthogonal to the direction of propagation. Since the
graviton is a massless particle, it follows that the gravitational field
propagates along a null direction $\boldsymbol{l}$. In order for all the
points on a wave-front remain on the same phase as propagation occurs, the
null vector field $\boldsymbol{l}$ must be covariantly constant throughout the
space-time. In particular, this implies that $\boldsymbol{l}$ remains
unchanged by parallel transport, which according to table 2.2 implies that the
space-time must be type $N$ if vacuum is assumed. Therefore, a manifold that
represents the propagation of plane gravitational waves might be type $N$.
Indeed, if a space-time admits a covariantly constant null vector
$\boldsymbol{l}$ then its metric must be of the following form [49, 50]:
$ds^{2}\,=\,2dudr\,+\,2H(u,x,y)du^{2}\,-\,dx^{2}\,-\,dy^{2}\,,$
where $\boldsymbol{l}=\partial_{r}$. A manifold with such metric is called a
$pp$-wave space-time. Choosing the other vectors of the null tetrad to be
$\boldsymbol{n}=\partial_{u}-H\partial_{r}$ and
$\boldsymbol{m}=\frac{1}{\sqrt{2}}(\partial_{x}+i\partial_{y})$ it follows
that all the Weyl scalars vanish except for
$\Psi_{4}\propto(\partial_{w}\partial_{w}H)$, where $w$ is a complex
coordinate defined by $w=x+iy$. This implies that in points of space-time
where $\partial_{w}\partial_{w}H\neq 0$ the Weyl tensor is type $N$ with PND
given by $\boldsymbol{l}=\partial_{r}$. Note that in general this $pp$-wave
metric is not a vacuum solution, since its Ricci tensor generally does not
vanish,
$R_{\mu\nu}\propto(\partial_{\overline{w}}\partial_{w}H)l_{\mu}l_{\nu}$. In
order to gain some insight on the meaning of the these coordinates, note that
in the limit $H\rightarrow 0$ the above metric is just the Minkowski metric
with $u=\frac{1}{\sqrt{2}}(t+z)$ and $r=\frac{1}{\sqrt{2}}(t-z)$, where the
frame $\\{\partial_{t},\partial_{x},\partial_{y},\partial_{z}\\}$ is a global
inertial frame on the Minkowski space-time. The plane wave space-time is of
great relevance for the quantum theory of gravity because all its curvature
invariants vanish [51], so that the quantum corrections for the Einstein-
Hilbert action do not contribute [52]. There is also an interesting article by
Penrose proving that all space-times in a certain limit are $pp$-wave [53].
The $pp$-wave solution provides an illustration that the Petrov type can vary
from point to point on the manifold, it is local classification. For instance,
if $H=(x^{2}+y^{2})^{2}=ww\bar{w}\bar{w}$ then the only non-vanishing Weyl
scalar is $\Psi_{4}\propto\bar{w}\bar{w}$. Therefore, in this case the Petrov
classification is type $O$ at the points satisfying $(x^{2}+y^{2})=0$ and type
$N$ outside the 2-dimensional time-like surface $(x^{2}+y^{2})=0$.
3) Cosmological Model (FLRW)
Astronomical observations reveal that on large scales (above $10^{24}m$) the
universe looks homogeneous and isotropic on the spatial sections. This leads
us to the so-called FLRW cosmological model, whose metric is of the following
form [54]:
$ds^{2}=dt^{2}-R^{2}(t)\left[\frac{dr^{2}}{1-\kappa
r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\right]\;;\;\kappa=0,\pm
1\,.$
The metric inside the square bracket is the general metric of a 3-dimensional
homogeneous and isotropic space, the case $\kappa=0$ being the flat space,
$\kappa=1$ being the 3-sphere and $\kappa=-1$ is the hyperbolic 3-space. Now
let us see that the Petrov classification of such metric must be type $O$.
Suppose, by contradiction, that the Petrov type is different from $O$ at some
point. Then at this point the Weyl tensor would admit at least one and at most
four PNDs. If $\boldsymbol{l}$ is a PND then, as it is a null vector, it must
be of the form $\boldsymbol{l}=\lambda(\partial_{t}+\hat{\boldsymbol{e}})$,
where $\hat{\boldsymbol{e}}$ is a unit space-like vector and $\lambda\neq 0$
is a real scalar. But this distinguishes a privileged spatial direction, the
one tangent to $\hat{\boldsymbol{e}}$, which contradicts the isotropy
assumption. Homogeneity then guarantees that the same is true on the other
points of space. Thus we conclude that the FLRW space-time is type $O$.
Indeed, it is not so hard to verify that the Weyl tensor of this metric
vanishes.
#### 2.9 Other Classifications
In this chapter it was shown that a space-time can be classified using the
Petrov type of the Weyl tensor. In the next chapter it will be presented
several important theorems involving the Petrov classification, confirming its
usefulness. But this is not the only form to classify a manifold at all. In
this section three other noteworthy methods to classify a space-time will be
presented.
In section 1.4 it was said that the symmetries of a manifold are represented
by the Killing vectors. These vector fields have an important property, the
Lie bracket of any two Killing vectors is another Killing vector. Therefore,
the Killing vectors of a manifold generate a Lie group known as the group of
motions of the space-time. For instance, the group of motions of the flat
space-time is the Poincaré group. We can, thus, classify the space-times
according to the group of motions. For details and applications see [49, 30].
Let $\boldsymbol{v}$ be a vector belonging to the tangent space at a point
$p\in M$ of the 4-dimensional space-time $(M,\boldsymbol{g})$. Then if we
perform the parallel transport of such vector along a closed loop then the
final result will be another vector $\boldsymbol{v}^{\prime}$. It is easy to
see that $\boldsymbol{v}^{\prime}$ is related to $\boldsymbol{v}$ by a linear
transformation. The group formed by all such transformations, for all closed
loops, is called the Holonomy group of $p$ and denoted by $H_{p}$. Since the
metric is covariantly constant it follows that $H_{p}\subset O(1,3)$.
Moreover, the holonomy group is the same at all points of a connected domain
[55], so the holonomy provides a global classification for the space-times.
Connections between the Petrov classification and holonomy groups were studied
in [56].
Just as the Weyl tensor provides a map of bivectors into bivectors, the Ricci
tensor can be seen as an operator on the tangent space whose action is defined
by $V^{\mu}\mapsto V^{\prime\mu}=R^{\mu}_{\phantom{\mu}\nu}V^{\nu}$. Such
operator can be algebraically classified by means of the refined Segre
classification (appendix A), yielding another independent way to classify the
curvature of a manifold. For instance, in the $pp$-wave space-time (see the
preceding section) the Ricci tensor has the form $R_{\mu\nu}=\lambda
l_{\mu}l_{\nu}$ with $\boldsymbol{l}$ being a null vector field. In this case,
if $\lambda\neq 0$ the algebraic type of the Ricci tensor is $[\,|1,1,2]$.
Since Einstein’s equation (1.12) connects the Ricci tensor to the energy-
momentum tensor it turns out that classify one of these tensors is tantamount
to classify the other. Because of the latter fact it follows that the so-
called energy conditions impose restrictions over such algebraic
classification. For example, the type $[1,3|\,]$ is not compatible with the
dominant energy condition. The classification of the Ricci tensor is of
particular help when the Weyl tensor vanishes, since in this case the
curvature is entirely determined by the former tensor. More about this
classification is available in [49]. In the forthcoming chapters we will be
interested in the vacuum case, $R_{\mu\nu}=0$, so that the classification of
the Ricci tensor will play no role.
### Chapter 3 Some Theorems on Petrov Types
One could devise a lot of different forms to classify the curvature of a
space-time, but certainly many of them will be of little help both for the
Physical understanding and for solving equations. The major relevance of the
Petrov classification does not come from the algebraic classification in
itself, but from its connection with Physics and, above all, with geometry.
The Physical content behind this classification is mainly based on the
interpretation of the principal null directions, discussed in section 2.7.
Regarding the geometric content there exist several theorems relating the
Petrov classification with geometric restrictions on the space-time. The
intent of the present chapter is to show some of the most important theorems
along this line.
As a warming up for what comes, let us consider an example showing that it is
quite natural that algebraic restrictions on the curvature yield geometric
constraints on the space-time and vice versa. Let $(M,\boldsymbol{g})$ be a
4-dimensional space-time containing a covariantly constant vector field,
$\nabla_{\mu}\,K_{\nu}=0$. Then, using equation (1.5) we arrive at the
following consequence:
$R_{\mu\nu\rho}^{\phantom{\mu\nu\rho}\sigma}\,K_{\sigma}\,=\,\left(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu}\right)\,K_{\rho}\,=\,0\,.$
(3.1)
Conversely, if $R_{\mu\nu\rho}^{\phantom{\mu\nu\rho}\sigma}K_{\sigma}=0$ then
$K_{\mu}$ must be a multiple of a covariantly constant vector field. Thus we
obtained a connection between an algebraic condition,
$R_{\mu\nu\rho}^{\phantom{\mu\nu\rho}\sigma}K_{\sigma}=0$, and a geometric
restriction, the constancy of $\boldsymbol{K}$. In particular, if
$\boldsymbol{K}$ is null then equation (3.1) implies that Petrov
classification is type $N$. Note also that some geometric constraints are
quite severe. For instance, if the space-time admits four constant vector
fields that are linearly independent at every point then eq. (3.1) implies
that $R_{\mu\nu\rho}^{\phantom{\mu\nu\rho}\sigma}=0$, i.e., the manifold is
flat.
#### 3.1 Shear, Twist and Expansion
Before proceeding to the theorems on Petrov types it is important to introduce
the geodesic congruences, which is the aim of this section. In particular, it
will be shown the physical interpretation of the expansion, shear and twist
parameters. This will be of great relevance for the forthcoming sections.
Let $(M,\boldsymbol{g})$ be a $4$-dimensional Lorentzian manifold and
$N_{p}\subset M$ be the neighborhood of some point $p\in M$. A congruence of
geodesics in $N_{p}$ is a family of geodesics such that at each point of
$N_{p}$ passes one, and just one, of these geodesics. Such congruence defines
a vector field $T^{\mu}$ that is tangent to the geodesics and affinely
parameterized, $T^{\mu}\nabla_{\mu}T^{\nu}=0$. Now, suppose that the
congruence is time-like and that its tangent vector field is normalized so
that $T^{\mu}T_{\mu}=1$. It is possible to study how the geodesics on the
congruence move relative to each other by introducing a set of $3$ vector
fields $E_{i}^{\mu}$ called deviation vector fields. These vector fields are
orthogonal to the direction of propagation and they connect a fiducial
geodesic $\gamma$ on the congruence to the neighbors geodesics, as depicted on
the figure 3.1.
Figure 3.1: A congruence of geodesics, $\boldsymbol{T}$ is the tangent vector
field and $\boldsymbol{E}$ measures the relative deviation of the geodesics.
The vector fields $E_{i}^{\mu}$ are assumed to commute with $T^{\mu}$, so that
a suitable coordinate system can be introduced, with the affine parameters of
the geodesics, $\tau$, being one of the coordinates. Therefore we have
$[\boldsymbol{E}_{i},\boldsymbol{T}]=E_{i}^{\mu}\nabla_{\mu}\boldsymbol{T}-T^{\mu}\nabla_{\mu}\boldsymbol{E}_{i}=0$.
Then the relative movements of the geodesics on the congruence are measured by
the variation of $\boldsymbol{E}_{i}$ along the geodesics:
$\frac{dE_{i}^{\nu}}{d\tau}\,=\,T^{\mu}\nabla_{\mu}E_{i}^{\nu}\,=\,E_{i}^{\mu}\nabla_{\mu}T^{\nu}\,=\,M^{\nu\mu}\,E_{i\,\mu}\,\,,\;\,M^{\nu\mu}=\nabla^{\mu}T^{\nu}\,.$
(3.2)
The geodesic character of $\boldsymbol{T}$ and the constancy of its norm
easily implies that $M_{\mu\nu}T^{\nu}=0$ and $T^{\mu}M_{\mu\nu}=0$. Denoting
by $P_{\mu\nu}=g_{\mu\nu}-T_{\mu}T_{\nu}$ the projection operator on the space
generated by $\\{\boldsymbol{E}_{i}\\}$, we can split the tensor $M_{\mu\nu}$
into its irreducible parts: the trace, $\theta=M^{\mu}_{\phantom{\mu}\mu}$,
the traceless symmetric part, $\sigma_{\mu\nu}=M_{(\mu\nu)}-\frac{1}{3}\theta
P_{\mu\nu}$ and the skew-symmetric part, $\omega_{\mu\nu}=M_{[\mu\nu]}$. These
three parts of the tensor $\boldsymbol{M}$ are named the expansion, the shear
and the twist, respectively. In order to understand the origin of these names
let us work out a simple example.
Suppose that the vectors on the 3-dimensional Euclidian space,
$(\mathbb{R}^{3},\delta_{ij})$, obey the equation of motion
$\frac{d\hat{\boldsymbol{E}}}{dt}=\mathbf{M}\,\hat{\boldsymbol{E}}$, where
$\mathbf{M}$ is a $3\times 3$ matrix. Now let us split this matrix as the sum
of its trace, the trace-less symmetric part and the skew-symmetric part,
$\mathbf{M}=\frac{1}{3}\theta\mathbf{1}+\boldsymbol{\sigma}+\boldsymbol{\omega}$.
Then plugging this into the equation of motion and assuming that $\delta t$ is
an infinitesimal time interval, we get:
$\hat{\boldsymbol{E}^{\prime}}\,\equiv\,\hat{\boldsymbol{E}}(t+\delta
t)\,=\,\hat{\boldsymbol{E}}(t)\,+\,\delta
t\,\left[\frac{1}{3}\,\theta\,\mathbf{1}\,+\,\boldsymbol{\sigma}\,+\,\boldsymbol{\omega}\right]\,\hat{\boldsymbol{E}}(t)\,.$
(3.3)
Now we shall analyse the individual effect of each of the terms inside the
square bracket on the above equation. Let
$\\{\hat{\boldsymbol{E}}_{1},\hat{\boldsymbol{E}}_{2},\hat{\boldsymbol{E}}_{3}\\}$
be a cartesian frame,
$\hat{\boldsymbol{E}}_{i}\cdot\hat{\boldsymbol{E}}_{j}=\delta_{ij}$, so that
these vectors generate a cube of unit volume, see figure 3.2. Thus if
$\boldsymbol{\sigma}=\boldsymbol{\omega}=0$ then eq. (3.3) implies that the
infinitesimal evolution of these vectors is
$\hat{\boldsymbol{E}^{\prime}}_{i}=(1+\frac{1}{3}\delta
t\theta)\hat{\boldsymbol{E}}_{i}$. This says that the cube generated by the
vectors $\\{\hat{\boldsymbol{E}}_{i}\\}$ is expanded by the same amount on all
sides, so that its shape is kept invariant while its volume get multiplied by
$(1+\delta t\theta)$. Therefore, it is appropriate to call $\theta$ the
expansion parameter.
Suppose now that both $\theta$ and $\boldsymbol{\omega}$ vanish. Since
$\boldsymbol{\sigma}$ is a symmetric real matrix then it is always possible to
choose an orthonormal frame in which it takes the diagonal form. Let us
suppose that we are already on this frame,
$\boldsymbol{\sigma}=\operatorname{diag}(\lambda_{1},\lambda_{2},\lambda_{3})$.
Then eq. (3.3) yield $\hat{\boldsymbol{E}^{\prime}}_{i}=(1+\delta
t\lambda_{i})\hat{\boldsymbol{E}}_{i}$, i.e., the sides of the cube changes
their length by different amounts but keep the direction fixed. It is simple
matter to verify that after the infinitesimal evolution the volume changes by
$\delta t(\lambda_{1}+\lambda_{2}+\lambda_{3})$, which is zero since the trace
of $\boldsymbol{\sigma}$ vanishes. Thus it is reasonable to call
$\boldsymbol{\sigma}$ the shear.
Finally, setting $\theta$ and $\boldsymbol{\sigma}$ equal to zero and using
the matrix $\boldsymbol{\omega}$ define the vector
$\hat{\boldsymbol{\omega}}\equiv(\omega_{32},\omega_{13},\omega_{21})$. Then a
simple algebra reveals that eq. (3.3) yield
$\hat{\boldsymbol{E}^{\prime}}_{i}=\hat{\boldsymbol{E}}_{i}+\delta
t\,\hat{\boldsymbol{\omega}}\times\hat{\boldsymbol{E}}_{i}$, where “$\times$”
denotes the vectorial product of $\mathbb{R}^{3}$. This implies that the frame
vectors are all infinitesimally rotated around the vector
$\hat{\boldsymbol{\omega}}$ by the angle $\delta
t|\hat{\boldsymbol{\omega}}|$, which justifies calling $\boldsymbol{\omega}$
the twist. Since this is a rotation it follows that the volume of the cube
does not change. Figure 3.2 depicts the action of the expansion, the shear and
the twist.
Figure 3.2: The illustration on the left side shows a unit cube before the
infinitesimal evolution. Then the next 3 pictures display the changes caused
by an expansion, a shear and a twist, respectively. The shear and the twist
keep the volume invariant.
To analyze the relative movements of a congruence of null geodesics is a bit
trickier. The problem is that in this case the space orthogonal to the
geodesics also contains the vectors tangent to the congruence, as a null
vector is orthogonal to itself. Therefore, we must ignore the part of the
orthogonal space that is tangent to the null geodesics and work in an
effective 2-dimensional space-like subspace. Let $\boldsymbol{l}$ be a vector
field tangent to a congruence of null geodesics affinely parameterized. Thus
introducing a frame
$\\{\boldsymbol{l},\boldsymbol{n},\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2}\\}$
such that the non-zero inner products are $l^{\mu}n_{\mu}=1$ and
$\hat{e}_{i}^{\mu}\hat{e}_{j\,\mu}=-\delta_{ij}$, then the space of effective
deviation vectors is generated by $\\{\hat{\boldsymbol{e}}_{i}\\}$. So that
equation (3.2) yields:
$\frac{d\hat{\boldsymbol{e}}_{i}}{d\tau}\,=\,\hat{e}_{i}^{\,\mu}\,\nabla_{\mu}\,\boldsymbol{l}\,\equiv\,\alpha_{i}\,\boldsymbol{l}\,+\,\beta_{i}\,\boldsymbol{n}\,+\,N_{ij}\,\hat{\boldsymbol{e}}_{j}\,\;\Rightarrow\quad\frac{d\hat{\boldsymbol{e}}_{i}}{d\tau}\,\sim\,N_{ij}\,\hat{\boldsymbol{e}}_{j}\,.$
(3.4)
Where the symbol “$\sim$” means equal except for terms proportional to
$\boldsymbol{l}$ and it was used the fact that $\beta_{i}=0$, once
$l^{\mu}l_{\mu}=0$. Thus on a null congruence we say that the expansion, shear
and twist are respectively given by the trace, the trace-less symmetric part
and the skew-symmetric part of the $2\times 2$ matrix $N_{ij}$. By means of
equation (3.4) we see that the matrix $\mathbf{N}$ is defined by,
$N_{ij}=-\boldsymbol{g}(\nabla_{\hat{\boldsymbol{e}}_{i}}\boldsymbol{l},\hat{\boldsymbol{e}}_{j})$.
We can encapsulate the four real components of the matrix $\mathbf{N}$ on the
following three parameters called the optical scalars of the null congruence:
$\displaystyle\theta\,\equiv\,\frac{1}{2}\left(N_{11}+N_{22}\right)\,;\;\,\omega\,\equiv\,\frac{1}{2}\left(N_{21}-N_{12}\right)\,;$
$\displaystyle\sigma\,\equiv\,-\frac{1}{2}\left[(N_{11}-N_{22})+i(N_{12}+N_{21})\right]\,.$
The real scalars $\theta$ and $\omega$ are respectively called expansion and
twist, while the complex scalar $\sigma$ is the shear of the null geodesic
congruence. Using these definitions it is possible to split the matrix
$\mathbf{N}$ as the sum of its trace, its symmetric and trace-less part and
its skew-symmetric part as follows:
$\mathbf{N}\,=\,\theta\left[\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right]\,+\,\frac{1}{2}\left[\begin{array}[]{cc}-(\sigma+\overline{\sigma})&i(\sigma-\overline{\sigma})\\\
i(\sigma-\overline{\sigma})&(\sigma+\overline{\sigma})\\\
\end{array}\right]\,+\,\omega\left[\begin{array}[]{cc}0&-1\\\ 1&0\\\
\end{array}\right]\,.$
Now it is useful to introduce the complex vector
$\boldsymbol{m}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}+i\,\hat{\boldsymbol{e}}_{2})$,
so that
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
forms a null tetrad frame (appendix B). Then using the definitions of
$\boldsymbol{m}$ and $\mathbf{N}$ it is straightforward to prove the following
relations:
$\boldsymbol{g}(m^{\mu}\,\nabla_{\mu}\boldsymbol{l},\boldsymbol{m})\,=\,\sigma\;;\quad\boldsymbol{g}(m^{\mu}\,\nabla_{\mu}\boldsymbol{l},\overline{\boldsymbol{m}})\,=\,-(\theta\,+\,i\omega)\,.$
(3.5)
These are useful expressions that will be adopted as the definitions for the
optical scalars of a null geodesic congruence in a 4-dimensional space-time.
Some important classes of space-times are defined by means of the optical
scalars. In any dimension the Kundt class of space-times is defined as the one
possessing a congruence of null geodesics that is shear-free ($\sigma=0$),
twist-free ($\omega=0$) and with vanishing expansion ($\theta=0$), $pp$-wave
being the most important member of this class [38, 50, 57]. The Robinson-
Trautman space-times are defined, in any dimension, as the ones containing a
congruence of null geodesics that is shear-free, twist-free but with non-zero
expansion, the Schwarzschild solution being one important example [50, 58]. As
a final comment it is worth mentioning that a congruence of null orbits is
hypersurface-orthogonal ($l_{[\mu}\nabla_{\nu}l_{\rho]}=0$) if, and only if,
the orbits are geodesic and twist-free [58]. Now we are ready to go on and
study the theorems on the Petrov classification.
#### 3.2 Goldberg-Sachs
The so-called Goldberg-Sachs (GS) theorem is the most important theorem about
the Petrov classification. It was first proved by J. Goldberg and R. Sachs
[23] and its mathematical formulation is the following:
###### Theorem 1
In a non-flat vacuum space-time (vanishing Ricci tensor and non-zero Riemann
tensor) the Weyl scalars $\Psi_{0}$ and $\Psi_{1}$ vanish simultaneously if,
and only if, the null vector field $\boldsymbol{l}$ is geodesic and shear-
free.
Where in the above theorem it was used the notation introduced in section 2.1.
A relatively compact proof of this theorem can be found in ref. [12].
According to section 2.4 the condition $\Psi_{0}=\Psi_{1}=0$ is equivalent to
the relation $l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0$, which means
that $\boldsymbol{l}$ is a repeated principal null direction. An equivalent
form of stating this theorem is saying that in vacuum a null vector field is
geodesic and shear-free if, and only if, it points in a repeated PND. In
particular, algebraically special vacuum space-times must admit a shear-free
congruence of null geodesics.
A particularly interesting situation occurs in vacuum solutions of Petrov type
$D$. Since in this case the Weyl tensor admits two repeated PNDs (section 2.2)
it follows that there exist two independent null geodesic congruences that are
shear-free. This apparently inconsequential geometric restriction has enabled
the complete integration of Einstein’s field equation [24], i.e., all type $D$
vacuum solutions were analytically found. In addition, the Goldberg-Sachs
theorem has also played a prominent role on the original derivation of Kerr
solution [22]. Interestingly, all known black-holes are of type $D$.
Let us suppose that a conformal transformation is made on the space-time,
$(M,\boldsymbol{g})\mapsto(M,\tilde{\boldsymbol{g}}=\Omega^{2}\boldsymbol{g})$.
Then if
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
is a null tetrad frame in $(M,\boldsymbol{g})$ then
$\\{\widetilde{\boldsymbol{l}}=\boldsymbol{l},\widetilde{\boldsymbol{n}}=\Omega^{-2}\boldsymbol{n},\tilde{\boldsymbol{m}}=\Omega^{-1}\boldsymbol{m},\tilde{\overline{\boldsymbol{m}}}=\Omega^{-1}\overline{\boldsymbol{m}}\\}$
will be a null tetrad on $(M,\tilde{\boldsymbol{g}})$. Then defining
$V_{\mu}\equiv\partial_{\mu}\ln\Omega$ and working out the transformation of
the Christoffel symbol it is a simple matter to prove the following relation:
$\tilde{\nabla}_{\mu}\,\tilde{l}^{\nu}\,=\,\nabla_{\mu}\,l^{\nu}\,+\,\delta^{\nu}_{\,\mu}\,l^{\rho}V_{\rho}\,+\,V_{\mu}\,l^{\nu}\,-\,l_{\mu}\,g^{\nu\rho}\,V_{\rho}\,.$
From which we immediately see that if $\boldsymbol{l}$ is geodesic in
$(M,\boldsymbol{g})$ so will be $\tilde{\boldsymbol{l}}$ in
$(M,\tilde{\boldsymbol{g}})$, although not affinely parameterized in general.
Moreover, using equation (3.5) we find that $\sigma=0$ if, and only if,
$\tilde{\sigma}=0$. Therefore, on null congruences the geodesic shear-free
condition is invariant under conformal transformations. Since the Weyl tensor
is also invariant under these transformations we conclude that there exists a
kind of asymmetry on the GS theorem as stated above, as the vacuum condition
is not invariant under conformal transformations. Noting this, I. Robinson and
A. Schild have been able to generalize the GS theorem to conformally Ricci-
flat space-times [59].
Fourteen years after the appearance of the GS theorem, J. Pleblański and S.
Hacyan noticed that in vacuum the existence of a null congruence that is
geodesic and shear-free is equivalent to the existence of two integrable
distributions of isotropic planes [60]. This is of great geometric relevance
and will be exploited on the next chapter in order to generalize the GS
theorem to 4-dimensional manifolds of all signatures.
Since non-linear equations are hard to deal with, sometimes it is useful to
linearize Einstein’s equation in order to study some properties of general
relativity. But it is very important to keep in mind that many features of the
linearized model are not carried to the complete theory. Particularly, in ref.
[61] it was proved that the Goldberg-Sachs theorem is not valid in linearized
gravity. The proof consisted of presenting explicit examples of linearized
space-times admitting a null vector field that is geodesic and shear-free but
is not a repeated PND on the linearized theory.
Since the GS theorem proved to be of great relevance to 4-dimensional general
relativity, recently a lot of effort has been made in order to generalize this
theorem to higher dimensions. But this task is not trivial at all. For
instance, in [62] it was proved that in 5 dimensions a repeated PND (according
to Bel-Debever criteria) is not necessarily shear-free. Indeed, the shear-free
condition turns out to be quite restrictive in dimensions greater than 4. A
suitable higher-dimensional generalization of the PNDs are the so-called Weyl
aligned null directions (WANDs) [36]. Although the WANDs share many properties
with the 4-dimensional PNDs there are also some important differences. For
example, while in four dimensions a non-zero Weyl tensor admits at least one
and at most four PNDs, in higher dimensions a non-vanishing Weyl tensor may
admit from zero up to infinitely many WANDs [63]. Some progress towards a
higher-dimensional generalization of the GS theorem was already accomplished
using this formalism [63, 64, 65, 38]. In particular it was proved that every
space-time admitting a repeated WAND has at least one repeated WAND that is
geodesic. Moreover, in chapter 6 it will be presented a particular
generalization of this theorem valid in even dimensions.
The equivalence between the geodesic and shear-free condition and the
integrability of null planes provides another path to generalize the GS
theorem. A partial generalization of the Goldberg-Sachs theorem using this
method has been accomplished in 2011 by Taghavi-Chabert [66, 67]. He has
proved that in a Ricci-flat manifold of dimension $d=2n+\epsilon$, with
$\epsilon=0,1$, if the Weyl tensor is algebraically special but generic
otherwise then the manifold admits an integrable $n$-dimensional isotropic
distribution. Such generalisation will be exploited and reinterpreted in
chapters 5 and 6.
#### 3.3 Mariot-Robinson
We call $F_{\mu\nu}=F_{[\mu\nu]}\neq 0$ a null bivector when
$F^{\mu\nu}F_{\mu\nu}=0=F^{\mu\nu}\,\star F_{\mu\nu}$, where
$\star\boldsymbol{F}$ is the Hodge dual of $\boldsymbol{F}$, defined on
equation (2.4). It can be proved that $\boldsymbol{F}$ is a real null bivector
if, and only if, there exists some null vector $\boldsymbol{l}$ and a space-
like vector $\boldsymbol{e}$ such that:
$F_{\mu\nu}\,=\,2\,l_{[\mu}\,e_{\nu]}\;;\quad l^{\mu}\,e_{\mu}\,=\,0\,.$
The null vector $\boldsymbol{l}$ is then called the principal null vector of
$\boldsymbol{F}$. Up to a multiplicative constant, $\boldsymbol{l}$ is the
unique vector that simultaneously obeys to the algebraic relations
$F_{\mu\nu}\,l^{\nu}=0$ and $F_{[\mu\nu}\,l_{\rho]}=0$. The Mariot-Robinson
theorem is then given by [68]:
###### Theorem 2
A 4-dimensional Lorentzian manifold admits a null bivector obeying to the
source-free Maxwell’s equations if, and only if, the principal null vector of
such bivector generates a null congruence that is geodesic and shear-free.
A simple proof of this theorem using spinors is given in [40]. More
explicitly, such theorem guarantees that if
$F_{\mu\nu}=l_{\mu}e_{\nu}-e_{\mu}l_{\nu}$ obeys the equations
$\nabla^{\mu}F_{\mu\nu}=0$ and $\nabla^{\mu}\,(\star F)_{\mu\nu}=0$ then the
null vector field $\boldsymbol{l}$ must be geodesic and shear-free.
Conversely, if $\boldsymbol{l}$ generates a null congruence of shear-free
geodesics then one can always find a space-like vector field $\boldsymbol{e}$
such that $F_{\mu\nu}=l_{\mu}e_{\nu}-e_{\mu}l_{\nu}$ obeys the equations
$\nabla^{\mu}F_{\mu\nu}=0$ and $\nabla^{\mu}\,(\star F)_{\mu\nu}=0$. Using
this result and the Goldberg-Sachs theorem we immediately arrive at the
following interesting consequence:
###### Corollary 1
A vacuum space-time is algebraically special according to the Petrov
classification if, and only if, it admits a null bivector obeying to source-
free Maxwell’s equations.
In this corollary the Maxwell field, $\boldsymbol{F}$, was assumed to be a
test field, which means that its energy was assumed to be low enough to be
neglected on Einstein’s equation, so that the space-time can be assumed to be
vacuum. But, actually, this corollary remains valid if we also consider that
the electromagnetic field distorts the space-time, i.e, if the metric obeys
the equation $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G\,T_{\mu\nu}$, where
$T_{\mu\nu}$ is the energy-momentum tensor of the electromagnetic field
$\boldsymbol{F}$.
Physically, a null Maxwell field represents electromagnetic radiation. Suppose
that
$\\{\hat{\boldsymbol{e}}_{t},\hat{\boldsymbol{e}}_{x},\hat{\boldsymbol{e}}_{y},\hat{\boldsymbol{e}}_{z}\\}$
is a Lorentz frame, then a plane electromagnetic wave of frequency $\omega$
propagating on the direction $\hat{\boldsymbol{e}}_{z}$ is generated by the
electric field $\mathbf{E}=E_{0}\cos[\omega(z-t)]\,\hat{\boldsymbol{e}}_{x}$
and the magnetic field
$\mathbf{B}=E_{0}\cos[\omega(z-t)]\,\hat{\boldsymbol{e}}_{y}$. Indeed, it is
simple matter to verify that these fields are solutions of the Maxwell’s
equations without sources. The field $\boldsymbol{F}$ associated to such
electric and magnetic fields is $F_{\mu\nu}=2\,l_{[\mu}e_{\nu]}$, with
$\boldsymbol{l}=(\hat{\boldsymbol{e}}_{t}+\hat{\boldsymbol{e}}_{z})$ and
$\boldsymbol{e}=-E_{0}\cos[\omega(z-t)]\,\hat{\boldsymbol{e}}_{x}$, which is a
null bivector. The energy-momentum tensor of such field is given by
$T_{\mu\nu}=\frac{e^{\rho}e_{\rho}}{4\pi}l_{\mu}l_{\nu}$.
Given the null field $F_{\mu\nu}=2\,l_{[\mu}e_{\nu]}$ then the bivectors
$\boldsymbol{F}^{\pm}=(\boldsymbol{F}\pm i\star\boldsymbol{F})$ are given by
$F^{+}_{\mu\nu}=2\,l_{[\mu}m_{\nu]}$ and
$F^{-}_{\mu\nu}=2\,l_{[\mu}\overline{m}_{\nu]}$, where $\boldsymbol{m}$ is a
complex null vector field orthogonal to $\boldsymbol{l}$. In section 3.2 it
was commented that the existence of a shear-free congruence of null geodesics
is equivalent to the existence of two integrable distributions of isotropic
planes. Therefore, the Mariot-Robinson theorem guarantees that the existence
of a null solution for the source-free Maxwell’s equations is equivalent to
the existence of two integrable distributions of isotropic planes. These
distributions are the ones generated by $\\{\boldsymbol{l},\boldsymbol{m}\\}$
and $\\{\boldsymbol{l},\overline{\boldsymbol{m}}\\}$.
By means of the language of isotropic distributions, the Mariot-Robinson
theorem admits a generalization valid in all even dimensions and all
signatures. In [69] the proof was made using spinors, while in [70] a
simplified proof using just tensors is presented. This generalized version of
the Mariot-Robinson theorem will be discussed in chapter 6.
#### 3.4 Peeling Property
In this section it will be shown that the Weyl tensor of an asymptotically
flat space-time has a really simple fall off behaviour near the null infinity.
But before enunciating this beautiful result it is necessary to introduce the
concept of asymptotic flatness. By an asymptotically flat space-time it is
meant one that looks like Minkowski space-time as we approach the infinity.
But in order to extract any mathematical consequence of this hypothesis it is
necessary to make a rigorous definition of what “looks like Minkowski” means.
This is a bit complicated since coordinates are meaningless in general
relativity, so that it is not reasonable to say that the metric of an
asymptotically flat space-time must approach the Minkowski metric as the
spatial coordinates go to infinity.
In order to avoid taking coordinates to infinity it is interesting to perform
a conformal transformation,
$g_{\mu\nu}\mapsto\widetilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$, that brings
the points from the infinity of an asymptotically flat space-time to a finite
distance. Thus although $\int ds=\int\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$ goes
to infinity as $x^{\mu}\rightarrow\infty$ we can manage to make $\int
d\tilde{s}=\int\Omega\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$ finite by properly
making $\Omega\rightarrow 0$ as $x^{\mu}\rightarrow\infty$. So that the
infinity of the space-time $(M,\boldsymbol{g})$ is represented by the boundary
$\Omega=0$ on the space-time $(M,\widetilde{\boldsymbol{g}})$. Using this
reasoning a space-time $(M,\boldsymbol{g})$ is said to be asymptotically flat
when there exists another space-time
$(\widetilde{M},\widetilde{\boldsymbol{g}})$, called the non-physical space-
time, such that: (1) $M\subset\widetilde{M}$ and $\widetilde{M}$ has a
boundary given by $\Omega=0$ that represents the null infinity of
$(M,\boldsymbol{g})$; (2) $\widetilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$ and
$\partial_{\mu}\Omega\neq 0$ on the boundary $\Omega=0$; (3) The Ricci tensor
of $(M,\boldsymbol{g})$ vanishes on the neighborhood of $\Omega=0$. For
details and motivation of this definition see [40, 27, 4].
Since we have some freedom on the definition of $\Omega$, we can choose it to
be the affine parameter of a null geodesic on
$(\widetilde{M},\widetilde{\boldsymbol{g}})$, let
$\tilde{\boldsymbol{l}}=\frac{d\;}{d\Omega}$ be the tangent to this geodesic.
Such geodesic then defines another null geodesic on $(M,\boldsymbol{g})$ whose
tangent shall be denoted by $\boldsymbol{l}=\frac{d\,}{dr}$. Imposing $r$ to
be an affine parameter we find that $r=-\Omega^{-1}$, so that
$l^{\mu}=\Omega^{2}\widetilde{l}^{\mu}$. The non-physical manifold,
$(\widetilde{M},\widetilde{\boldsymbol{g}})$, and the vector
$\tilde{n}_{\mu}=\partial_{\mu}\Omega$ are assumed to be completely regular on
the boundary $\Omega=0$. Using this and the transformation rule of the Ricci
scalar under conformal transformations we find that the vector field
$\tilde{\boldsymbol{n}}$ becomes null, according to $\tilde{\boldsymbol{g}}$,
as we approach the boundary of $\widetilde{M}$. Note also that
$\tilde{l}^{\mu}\tilde{n}_{\mu}=1$, hence we can find a complex vector
$\tilde{\boldsymbol{m}}$ so that, near the boundary,
$\\{\tilde{\boldsymbol{l}},\tilde{\boldsymbol{n}},\tilde{\boldsymbol{m}},\overline{\tilde{\boldsymbol{m}}}\\}$
forms a null tetrad of $(\widetilde{M},\widetilde{\boldsymbol{g}})$. Since
$\boldsymbol{l}=\Omega^{2}\widetilde{\boldsymbol{l}}$ and
$\boldsymbol{g}=\Omega^{-2}\widetilde{\boldsymbol{g}}$ we find that the
corresponding null tetrad of $(M,\boldsymbol{g})$ is such that
$\boldsymbol{n}=\tilde{\boldsymbol{n}}$ and
$\boldsymbol{m}=\Omega\tilde{\boldsymbol{m}}$.
Since $(\widetilde{M},\widetilde{\boldsymbol{g}})$ is regular at $\Omega=0$ it
is expected that the Weyl scalars of the non-physical space-time are all non-
vanishing and of the same order on the boundary. However, it can be proved
that the Weyl scalars of $(\widetilde{M},\widetilde{\boldsymbol{g}})$ are
generally of order $\Omega$ [40], $\widetilde{\Psi}_{\alpha}\sim O(\Omega)$.
Using this fact along with equation (2.1) and the transformation of the null
tetrad frame, we can easily find the behaviour of the Weyl scalars of
$(M,\boldsymbol{g})$. For example,
$\displaystyle\Psi_{0}\,=$
$\displaystyle\,C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}l^{\rho}m^{\sigma}\,=\,\Omega^{-2}\,\widetilde{C}_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}l^{\rho}m^{\sigma}$
$\displaystyle=$
$\displaystyle\,\Omega^{-2}\,\widetilde{C}_{\mu\nu\rho\sigma}\,\Omega^{2}\tilde{l}^{\mu}\,\Omega\tilde{m}^{\nu}\,\Omega^{2}\tilde{l}^{\rho}\,\Omega\tilde{m}^{\sigma}\,=\,\Omega^{4}\,\widetilde{\Psi}_{0}\,\sim\,O(\Omega^{5})\,.$
Where it was used the fact that $C^{\mu}_{\phantom{\mu}\nu\rho\sigma}$ is
invariant by conformal transformations, which implies that
$C_{\mu\nu\rho\sigma}=g_{\mu\kappa}C^{\kappa}_{\phantom{\kappa}\nu\rho\sigma}=\Omega^{-2}\widetilde{C}_{\mu\nu\rho\sigma}$.
In general the following behaviour is found:
$\Psi_{0}\,\sim\,O(\Omega^{5})\;,\;\,\Psi_{1}\,\sim\,O(\Omega^{4})\;,\;\,\Psi_{2}\,\sim\,O(\Omega^{3})\;,\;\,\Psi_{3}\,\sim\,O(\Omega^{2})\;,\;\,\Psi_{4}\,\sim\,O(\Omega)\,.$
Since $\Omega=-r^{-1}$, the above relations along with table 2.1 implies the
following result known as the peeling theorem [27]:
###### Theorem 3
Let $(M,\boldsymbol{g})$ be an asymptotically flat space-time. Then if we
approach the null infinity, $r\rightarrow\infty$, along a null geodesic whose
affine parameter is $r$ and whose tangent vector is $\boldsymbol{l}$ then the
Weyl tensor has the following fall off behaviour:
$\boldsymbol{C}\,=\,\frac{\boldsymbol{C}_{N}}{r}\,+\,\frac{\boldsymbol{C}_{III}}{r^{2}}\,+\,\frac{\boldsymbol{C}_{II}}{r^{3}}\,+\,\frac{\boldsymbol{C}_{I}}{r^{4}}\,+\,O(r^{-5})\,.$
Where the tensors $\boldsymbol{C}_{N}$, $\boldsymbol{C}_{III}$,
$\boldsymbol{C}_{II}$, and $\boldsymbol{C}_{I}$ have the symmetries of a Weyl
tensor and are respectively of Petrov type $N$, $III$, $II$ and $I$ (or more
special). The vector field $\boldsymbol{l}$ is a repeated PND of the first
three terms of the above expansion and a PND of the tensor
$\boldsymbol{C}_{I}$ (see figure 3.3).
Figure 3.3: According to the peeling theorem, as we approach the null infinity
of an asymptotically flat space-time the Petrov type of the Weyl tensor
becomes increasingly special. The blue arrows represent the principal null
directions of the Weyl tensor, while the red axis represents the null
direction along which null infinity is approached.
The peeling theorem has been generalized to higher dimensions just quite
recently [71]. It was proved that the fall off behaviour of the Weyl tensor in
higher dimensions is both qualitatively and quantitatively different from the
4-dimensional case. Indeed, concerning asymptotic infinity the dimension 4 is
a very special one, as the definition of asymptotically flat in other
dimensions proved to be fairly tricky [72, 73]. The physical justification for
a non-trivial definition of asymptotic flatness in higher dimensions comes
from the fact that such definition must be stable under small perturbations,
it should be compatible with the existence of a generator for the Bondi energy
and it might allow the existence of gravitational radiation.
#### 3.5 Symmetries
Given the Petrov type of a space-time occasionally it is possible to say which
symmetries the manifold might have and, conversely, given the symmetries of a
space-time sometimes we can guess its Petrov classification. The intent of
this section is to present some theorems connecting the Petrov classification
with the existence of symmetry tensors. One of the first results on these
lines was obtained by Kinnersley in [24], where he explicitly found all type
$D$ vacuum solutions and, as a bonus, arrived at the following result:
###### Theorem 4
Every type $D$ vacuum space-time admits either 4 or 2 independent Killing
vector fields.
Another remarkable result about type $D$ solutions was then found by Walker
and Penrose in ref. [10], where it was proved that these space-times have a
hidden symmetry:
###### Theorem 5
Every type $D$ vacuum space-time with less than 4 independent Killing vectors
admits a non-trivial conformal Killing tensor (CKT) of order two. Furthermore,
if the metric is not a C-metric111This is an important class of type $D$
vacuum solutions representing a pair of Black Holes accelerating away from
each other due to structures represented by conical singularities. The
C-metric is a generalization of the Schwarzschild solution with one extra
parameter in addition to the mass, so that the Schwarzschild metric is a
particular member of this class. For a thorough analysis of these metrics see
[50]. then this CKT is, actually, a Killing tensor.
The second part of the above theorem can be found in [74, 49]. Later,
Collinson [8] and Stephani [75] investigated whether these Killing tensors can
be constructed out of Killing-Yano tensors (see section 1.4), arriving at the
following result:
###### Theorem 6
Every type $D$ vacuum space-time possessing a non-trivial Killing tensor of
order two, $K_{\mu\nu}$, also admits a Killing-Yano tensor $Y_{\mu\nu}$ such
that $K_{\mu\nu}=Y_{\mu}^{\phantom{\mu}\sigma}Y_{\sigma\nu}$.
As defined in section 3.3, a real bivector $B_{\mu\nu}$ is called null when it
can be written as $B_{\mu\nu}=l_{[\mu}e_{\nu]}$, where $\boldsymbol{l}$ is
null, $\boldsymbol{e}$ is space-like and $l^{\mu}\,e_{\mu}=0$. On the other
hand, if $B^{\prime}_{\mu\nu}$ is a real non-null bivector then it is always
possible to arrange a null tetrad frame such that
$B^{\prime}_{\mu\nu}=a\,l_{[\mu}n_{\nu]}+ib\,m_{[\mu}\overline{m}_{\nu]}$,
where $a$ and $b$ are real functions (this can be easily seen using spinors).
Using this along with the results of [76] we can state:
###### Theorem 7
A vacuum space-time admitting a null Killing-Yano tensor of order two,
$Y_{\mu\nu}=l_{[\mu}e_{\nu]}$, must be of Petrov type $N$ with
$\boldsymbol{l}$ being the repeated PND. On the other hand, a vacuum space-
time admitting a non-null Killing-Yano tensor of order two,
$Y^{\prime}_{\mu\nu}=a\,l_{[\mu}n_{\nu]}+ib\,m_{[\mu}\overline{m}_{\nu]}$,
must have type $D$ with $\boldsymbol{l}$ and $\boldsymbol{n}$ being repeated
PNDs.
Actually, this theorem remains valid if instead of vacuum we consider electro-
vacuum space-times [76]. For more theorems on the same line see [49] and
references therein.
Regarding higher-dimensional space-times, it is appropriate mentioning
references [77, 66] which, inspired by theorem 5, have suggested that a
suitable generalization of the Petrov type $D$ condition for manifolds of
dimension $d=2n+\epsilon$, with $\epsilon=0,1$, should be the existence of
$2^{n}$ integrable maximally isotropic distributions. For interesting results
concerning hidden symmetries and Killing-Yano tensors in higher-dimensional
black holes see the nice paper [78].
## Part II Original Research
### Chapter 4 Generalizing the Petrov Classification and the Goldberg-Sachs
Theorem to All Signatures
In the previous chapters it was defined the Petrov classification, an
algebraic classification for the Weyl tensor valid in 4-dimensional Lorentzian
manifolds that is related to very important theorems. In particular, such
classification proved to be helpful in the search of new exact solutions to
Einstein’s equation, a remarkable example being the Kerr metric [22]. The aim
of this chapter is to generalize the Petrov classification to 4-dimensional
spaces of arbitrary signature. The strategy adopted here is to work with
complexified spaces, interpreting the various signatures as different reality
conditions. This approach is based on the reference [33] and yields a unified
classification scheme to all signatures. Generalizations of the Petrov
classification were already known before the article [33]: In [79] the complex
case was treated using spinors, Euclidean manifolds were investigated in [80,
81], while the split signature was studied in [30, 82, 83, 84, 85]. But none
of these previous works attempted to provide a unified classification scheme
such that each signature is just a special case of the complex classification.
The Goldberg-Sachs theorem is the most important result on the Petrov
classification. Particularly, it enabled the complete integration of
Einstein’s vacuum equation for type $D$ space-times [24]. In ref. [60]
Plebański and Hacyan proved a beautiful generalization of this theorem valid
in complexified manifolds. They realised that a suitable complex
generalization of a shear-free null geodesic congruence is an integrable
distribution of isotropic planes. Here such generalized theorem will be used
to show that certain algebraic restrictions on the Weyl tensor imply the
existence of important geometric structures on 4-dimensional manifolds of any
signature, results that were presented on the article [35].
#### 4.1 Null Frames
Before proceeding it is important to establish the notation that will be
adopted throughout this chapter. In particular, let us see explicitly how one
can use a complexified space in order to obtain results on real manifolds of
arbitrary signature. We shall define a null frame on a 4-dimensional manifold
as a frame of vector fields
$\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2},\boldsymbol{E}_{3},\boldsymbol{E}_{4}\\}$
such that the only non-zero inner products are:
$\boldsymbol{g}(\boldsymbol{E}_{1},\boldsymbol{E}_{3})\,=\,1\quad\textrm{ and
}\quad\boldsymbol{g}(\boldsymbol{E}_{2},\boldsymbol{E}_{4})\,=\,-1\,.$ (4.1)
Particularly, note that all vector fields on this frame are null. Depending on
the signature of the manifold the vectors of a null frame obey to different
reality conditions, let us see this explicitly.
$\bullet$ Euclidean Signature, $\boldsymbol{s=4}$
In such a case, by definition, it is possible to introduce a real frame
$\\{\hat{\boldsymbol{e}}_{a}\\}$ such that
$\boldsymbol{g}(\hat{\boldsymbol{e}}_{a},\hat{\boldsymbol{e}}_{b})=\delta_{ab}$.
Thus it is straightforward to see that the following vectors form a null
frame:
$\boldsymbol{E}_{1}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}+i\hat{\boldsymbol{e}}_{3})\,;\;\boldsymbol{E}_{2}=\frac{i}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}+i\hat{\boldsymbol{e}}_{4})\,;\;\boldsymbol{E}_{3}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}-i\hat{\boldsymbol{e}}_{3})\,;\;\boldsymbol{E}_{4}=\frac{i}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}-i\hat{\boldsymbol{e}}_{4})\,.$
Note that the following reality conditions hold:
$\boldsymbol{E}_{3}\,=\,\overline{\boldsymbol{E}_{1}}\quad;\quad\boldsymbol{E}_{4}\,=\,-\overline{\boldsymbol{E}_{2}}\,.$
(4.2)
$\bullet$ Lorentzian Signature, $\boldsymbol{s=2}$
As shown on appendix B in this signature we can introduce a null tetrad
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$,
which is a frame such that the only non-zero inner products are
$l^{\mu}n_{\mu}=1$ and $m^{\mu}\overline{m}_{\mu}=-1$. Therefore, the
following vector fields form a null frame:
$\boldsymbol{E}_{1}\,=\,\boldsymbol{l}\;;\quad\boldsymbol{E}_{2}\,=\,\boldsymbol{m}\;;\quad\boldsymbol{E}_{3}\,=\,\boldsymbol{n}\;;\quad\boldsymbol{E}_{4}\,=\,\boldsymbol{\overline{m}}$
(4.3)
So a null frame is just a null tetrad reordered. Since, by definition, in a
null tetrad $\boldsymbol{l}$ and $\boldsymbol{n}$ are both real, it follows
that on Lorentzian case the reality conditions are:
$\boldsymbol{E}_{1}\,=\,\overline{\boldsymbol{E}_{1}}\quad;\quad\boldsymbol{E}_{3}\,=\,\overline{\boldsymbol{E}_{3}}\quad;\quad\boldsymbol{E}_{4}\,=\,\overline{\boldsymbol{E}_{2}}\,.$
(4.4)
$\bullet$ Split Signature, $\boldsymbol{s=0}$
In such signature there exists a real frame $\\{\hat{\boldsymbol{e}}_{a}\\}$
such that
$\boldsymbol{g}(\hat{\boldsymbol{e}}_{a},\hat{\boldsymbol{e}}_{b})=\operatorname{diag}(1,1,-1,-1)$.
Then the following vectors form a null frame:
$\boldsymbol{E}^{\prime}_{1}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}+\hat{\boldsymbol{e}}_{3})\,;\;\boldsymbol{E}^{\prime}_{2}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{4}+\hat{\boldsymbol{e}}_{2})\,;\;\boldsymbol{E}^{\prime}_{3}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}-\hat{\boldsymbol{e}}_{3})\,;\;\boldsymbol{E}^{\prime}_{4}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{4}-\hat{\boldsymbol{e}}_{2})\,.$
Note that all vectors on this frame are real:
$\boldsymbol{E}^{\prime}_{1}\,=\,\overline{\boldsymbol{E}^{\prime}_{1}}\quad;\quad\boldsymbol{E}^{\prime}_{2}\,=\,\overline{\boldsymbol{E}^{\prime}_{2}}\quad;\quad\boldsymbol{E}^{\prime}_{3}\,=\,\overline{\boldsymbol{E}^{\prime}_{3}}\quad;\quad\boldsymbol{E}^{\prime}_{4}\,=\,\overline{\boldsymbol{E}^{\prime}_{4}}\,.$
(4.5)
When the metric has split signature it is also possible to introduce a complex
null frame. Indeed, note that the vector fields
$\boldsymbol{E}_{1}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}+i\hat{\boldsymbol{e}}_{2})\,;\;\boldsymbol{E}_{2}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{3}+i\hat{\boldsymbol{e}}_{4})\,;\;\boldsymbol{E}_{3}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{1}-i\hat{\boldsymbol{e}}_{2})\,;\;\boldsymbol{E}_{4}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{3}-i\hat{\boldsymbol{e}}_{4})$
form a null frame. The reality conditions on this frame are
$\boldsymbol{E}_{3}=\overline{\boldsymbol{E}}_{1}$ and
$\boldsymbol{E}_{4}=\overline{\boldsymbol{E}}_{2}$.
Therefore, a wise path to obtain results valid in any signature is to assume
that the tangent bundle is complexified and when necessary use a suitable
reality condition to specify the signature. This can easily be understood as
follows: if we work over the complex field the signature is not fixed, because
a vector $\hat{\boldsymbol{e}}$ whose norm squared is $1$,
$\boldsymbol{g}(\hat{\boldsymbol{e}},\hat{\boldsymbol{e}})=1$, can be
multiplied by $i$ and yield a vector whose norm squared is $-1$, so that the
apparent signature can be changed.
Once fixed a null frame $\\{\boldsymbol{E}_{a}\\}$, one can define the dual
frame $\\{\boldsymbol{E}^{a}\\}$, which is a set of 1-forms such that
$\boldsymbol{E}^{a}(\boldsymbol{E}_{b})=\delta^{\,a}_{b}$ (see section 1.7).
By means of eq. (4.1) it is trivial to note that the components of such
1-forms are:
$E^{1\,\mu}\,=\,E_{3}^{\phantom{3}\mu}\,\,;\,\;E^{2\,\mu}\,=\,-E_{4}^{\phantom{4}\mu}\,\,;\,\;E^{3\,\mu}\,=\,E_{1}^{\phantom{1}\mu}\,\,;\,\;E^{4\,\mu}\,=\,-E_{2}^{\phantom{2}\mu}\,.$
(4.6)
The dual frame can be used to define the following 2-forms constituting a
basis for the space of bivectors:
$\displaystyle\boldsymbol{Z}^{1+}\,=\,\boldsymbol{E}^{4}\wedge\boldsymbol{E}^{3}\,\,;\;\,\boldsymbol{Z}^{2+}\,=\,\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{2}\,\,;\;\,\boldsymbol{Z}^{3+}\,=\,\frac{1}{\sqrt{2}}\left(\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{3}+\boldsymbol{E}^{4}\wedge\boldsymbol{E}^{2}\right)$
$\displaystyle\,\boldsymbol{Z}^{1-}\,=\,\boldsymbol{E}^{2}\wedge\boldsymbol{E}^{3}\,\,;\;\,\boldsymbol{Z}^{2-}\,=\,\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{4}\,\,;\;\,\boldsymbol{Z}^{3-}\,=\,\frac{1}{\sqrt{2}}\left(\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{3}+\boldsymbol{E}^{2}\wedge\boldsymbol{E}^{4}\right).$
By means of eq. (4.6) we see that the components of the 2-form
$\boldsymbol{Z}^{1+}$ are
$Z^{1+\,\mu\nu}=2E_{1}^{\phantom{1}[\mu}E_{2}^{\phantom{1}\nu]}$, which
sometimes is written as
$\boldsymbol{Z}^{1+}=\boldsymbol{E}_{1}\wedge\boldsymbol{E}_{2}$. Because of
this we say that $\boldsymbol{Z}^{1+}$ generates the family of planes spanned
by the vector fields $\boldsymbol{E}_{1}$ and $\boldsymbol{E}_{2}$. Note that
since
$\boldsymbol{g}(\boldsymbol{E}_{1},\boldsymbol{E}_{1})=\boldsymbol{g}(\boldsymbol{E}_{2},\boldsymbol{E}_{2})=\boldsymbol{g}(\boldsymbol{E}_{1},\boldsymbol{E}_{2})=0$,
all vectors tangent to these planes are null. This kind of plane is called
totally null or isotropic and $\boldsymbol{Z}^{1+}$ is then called a null
bivector. More about isotropic subspaces can be found in [86]. In the same
vein $\boldsymbol{Z}^{2+}$, $\boldsymbol{Z}^{1-}$ and $\boldsymbol{Z}^{2-}$
generate the isotropic planes spanned by
$\\{\boldsymbol{E}_{3},\boldsymbol{E}_{4}\\}$,
$\\{\boldsymbol{E}_{1},\boldsymbol{E}_{4}\\}$ and
$\\{\boldsymbol{E}_{2},\boldsymbol{E}_{3}\\}$ respectively. From now on a
bivector $\boldsymbol{Z}$ will be called a null bivector when it can be
written as $Z^{\mu\nu}=2l^{[\mu}k^{\nu]}$ with
$Span\\{\boldsymbol{l},\boldsymbol{k}\\}$ being a distribution of isotropic
planes111Note that in section 3.3 the definition of a null bivector was
broader than this, there a bivector
$\boldsymbol{B}=\boldsymbol{l}\wedge\boldsymbol{e}$ with $\boldsymbol{e}$
being space-like and orthogonal to the null vector $\boldsymbol{l}$ was also
called null. But if we are working with arbitrary signature it is more useful
to define a null bivector as a simple bivector that generates an isotropic
distribution..
Since the determinant of the matrix
$g_{ab}=\boldsymbol{g}(\boldsymbol{E}_{a},\boldsymbol{E}_{b})$ is $g=1$, the
components of the volume-form on the null frame $\\{\boldsymbol{E}_{a}\\}$ are
given by
$\epsilon_{abcd}\,=\,\varepsilon_{abcd}\,,\quad\textrm{where}\quad\varepsilon_{abcd}=\varepsilon_{[abcd]}\;\textrm{
and }\;\varepsilon_{1234}\equiv-1\,.$
Thus if $\boldsymbol{Z}$ is a bivector, $Z_{ab}=Z_{[ab]}$, then its Hodge dual
is given by:
$\star Z_{cd}\,=\,\frac{1}{2}\,Z^{ab}\,\varepsilon_{abcd}\,.$ (4.7)
With the aim of improving the notation, let us define $\mathcal{H}$ as an
operator that acts on the space of bivectors in some open set of the manifold,
$\mathcal{H}:\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)$, and
implements the Hodge dual map,
$\mathcal{H}(\boldsymbol{Z})\equiv\star\boldsymbol{Z}$. Then using equation
(4.7) it is simple matter to verify that $\mathcal{H}^{2}=\boldsymbol{1}$,
where $\boldsymbol{1}$ is the identity operator. Thus the eigenvalues of
$\mathcal{H}$ are $\pm 1$ and the bivector space at such neighborhood can be
split as the following direct sum222All results in this thesis are local, so
that it is always being assumed that we are in the neighborhood of some point.
Thus, formally, instead of $\Gamma(\wedge^{2}M)$ we should have written
$\Gamma(\wedge^{2}M)|_{N_{x}}$, which is the restriction of the space of
sections of the bivector bundle to some neighborhood $N_{x}$ of a point $x\in
M$. So we are choosing a particular local trivialization of the bivector
bundle.:
$\Gamma(\wedge^{2}M)\,=\,\Lambda^{2+}\,\oplus\,\Lambda^{2-}\,.$
Where $\Lambda^{2\pm}$ is spanned by the bivectors with eigenvalue $\pm 1$
with respect to $\mathcal{H}$. $\Lambda^{2+}$ is called the space of self-dual
bivectors, while $\Lambda^{2-}$ is the space of anti-self-dual 2-forms. It is
simple matter to prove that $\Lambda^{2+}$ is generated by
$\\{\boldsymbol{Z}^{i+}\\}$, while $\Lambda^{2-}$ is generated by
$\\{\boldsymbol{Z}^{i-}\\}$, with $i\in\\{1,2,3\\}$. For instance, let us
prove that $\boldsymbol{Z}^{1+}$ is self-dual:
$\star
Z^{1+}_{\phantom{1+}cd}\,=\,\frac{1}{2}\,Z^{1+}_{\phantom{1+}ab}\,\varepsilon^{ab}_{\phantom{ab}cd}\,=\,\varepsilon^{43}_{\phantom{43}cd}\,=\,\varepsilon_{12cd}\,=\,-\left(\delta^{\,3}_{c}\delta^{\,4}_{d}-\delta^{\,4}_{c}\delta^{\,3}_{d}\right)=Z^{1+}_{\phantom{1+}cd}\,.$
Particularly, note that every null bivector must be an eigenbivector of the
Hodge operator $\mathcal{H}$. It is worth remarking that what we call a self-
dual bivector will be an anti-self-dual bivector if we change the sign of the
volume-form. So the spaces $\Lambda^{2+}$ and $\Lambda^{2-}$ can be
interchanged by a simple change of sign on the volume-form
$\boldsymbol{\epsilon}$.
It is useful to introduce the following symmetric inner product on the space
of bivectors:
$\langle\boldsymbol{Z},\boldsymbol{B}\rangle\,\,\equiv\,\,Z_{\mu\nu}\,B^{\mu\nu}\,.$
It is simple matter to prove that the operator $\mathcal{H}$ is self-adjoint
with respect to this inner product,
$\langle\boldsymbol{Z},\mathcal{H}(\boldsymbol{B})\rangle=\langle\mathcal{H}(\boldsymbol{Z}),\boldsymbol{B}\rangle$.
In particular this implies that the inner product of a self-dual bivector and
an anti-self-dual bivector vanishes. Indeed, the only non-vanishing inner
products of the bivector basis introduced above are:
$\langle\boldsymbol{Z}^{1\pm},\boldsymbol{Z}^{2\pm}\rangle\,=\,2\quad\textrm{and}\quad\langle\boldsymbol{Z}^{3\pm},\boldsymbol{Z}^{3\pm}\rangle\,=\,-2\,.$
(4.8)
#### 4.2 Generalized Petrov Classification
Now let us define an algebraic classification for the Weyl tensor valid for
any signature and that naturally generalizes the Petrov classification. To
this end we shall define the Weyl operator at a point $x\in M$,
$\mathcal{C}:\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)$, by the
following action:
$\boldsymbol{Z}\,\longmapsto\,\boldsymbol{B}\,=\,\mathcal{C}(\boldsymbol{Z})\,,\,\textrm{
with }\,B_{\mu\nu}\,=\,Z^{\rho\sigma}\,C_{\rho\sigma\mu\nu}\,.$
Where $\boldsymbol{Z}$ and $\boldsymbol{B}$ are bivectors. Note that the
operator $\mathcal{C}$ is self-adjoint with respect to the inner product on
the space of bivectors,
$\langle\boldsymbol{Z},\mathcal{C}(\boldsymbol{B})\rangle=\langle\mathcal{C}(\boldsymbol{Z}),\boldsymbol{B}\rangle$.
Now let us prove that the Weyl operator has a fundamental property, it
commutes with the Hodge dual operator $\mathcal{H}$:
$\displaystyle[\mathcal{C}\,\mathcal{H}-\mathcal{H}\,\mathcal{C}](\boldsymbol{Z})=0\quad\forall\,\,\boldsymbol{Z}\;\;\Leftrightarrow\;\;C_{\phantom{\rho\sigma}\mu\nu}^{\rho\sigma}\,\epsilon_{\alpha\beta\rho\sigma}=\epsilon_{\rho\sigma\mu\nu}\,C^{\rho\sigma}_{\phantom{\rho\sigma}\alpha\beta}\;\Leftrightarrow\;$
$\displaystyle\epsilon^{\alpha\beta\kappa\gamma}C_{\phantom{\rho\sigma}\mu\nu}^{\rho\sigma}\,\epsilon_{\alpha\beta\rho\sigma}=\epsilon^{\alpha\beta\kappa\gamma}\epsilon_{\rho\sigma\mu\nu}\,C^{\rho\sigma}_{\phantom{\rho\sigma}\alpha\beta}\;\Leftrightarrow\;$
$\displaystyle(-1)^{s/2}\,2!\,2!\,C_{\phantom{\rho\sigma}\mu\nu}^{\rho\sigma}\,\delta^{\,[\kappa}_{\rho}\delta^{\,\gamma]}_{\sigma}=(-1)^{s/2}\,4!\,\delta^{\,[\alpha}_{\rho}\delta^{\,\beta}_{\sigma}\delta^{\,\kappa}_{\mu}\delta^{\,\gamma]}_{\nu}\,C^{\rho\sigma}_{\phantom{\rho\sigma}\alpha\beta}\;\Leftrightarrow\;$
$\displaystyle
4\,C_{\phantom{\rho\sigma}\mu\nu}^{\kappa\gamma}=4\,\delta^{\,[\alpha}_{\mu}\delta^{\,\beta]}_{\nu}\delta^{\,[\kappa}_{\rho}\delta^{\,\gamma]}_{\sigma}C^{\rho\sigma}_{\phantom{\rho\sigma}\alpha\beta}=4\,C_{\phantom{\mu\nu}\mu\nu}^{\kappa\gamma}\,.$
Where equations (1.15) and (2.1) were used. Thus we conclude that the
operators $\mathcal{C}$ and $\mathcal{H}$ commute. This implies that the
eigenspaces of $\mathcal{H}$ are preserved by the operator $\mathcal{C}$,
i.e., if $\boldsymbol{Z}^{\pm}\in\Lambda^{2\pm}$ then
$\mathcal{C}(\boldsymbol{Z}^{\pm})\in\Lambda^{2\pm}$. Thus the operator
$\mathcal{C}$ can be written as
$\mathcal{C}\,=\,\mathcal{C}^{+}\,\oplus\,\mathcal{C}^{-}\,,$
where $\mathcal{C}^{\pm}$ is the restriction of $\mathcal{C}$ to
$\Lambda^{2\pm}$. In other words, the operators $\mathcal{C}^{\pm}$ act on the
3-dimensional spaces generated by $\\{\boldsymbol{Z}^{i\pm}\\}$. When
$\mathcal{C}^{-}=0$ the Weyl tensor is said to be self-dual, while if
$\mathcal{C}^{+}=0$ it is anti-self-dual.
In 4 dimensions the Weyl tensor has 10 independent components, these can be
chosen to be the following scalars:
$\displaystyle\Psi^{+}_{0}\equiv C_{1212}\;;\;\Psi^{+}_{1}\equiv
C_{1312}\;;\;\Psi^{+}_{2}\equiv C_{1243}\;;\;\Psi^{+}_{3}\equiv C_{1343}\;;\;$
$\displaystyle\Psi^{+}_{4}\equiv C_{3434}$ $\displaystyle\Psi^{-}_{0}\equiv
C_{1414}\;;\;\Psi^{-}_{1}\equiv C_{1314}\;;\;\Psi^{-}_{2}\equiv
C_{1423}\;;\;\Psi^{-}_{3}\equiv C_{1323}\;;\;$
$\displaystyle\Psi^{-}_{4}\equiv C_{3232}\,.$ (4.9)
Where $C_{abcd}\equiv
C_{\mu\nu\rho\sigma}E_{a}^{\,\,\mu}E_{b}^{\,\,\nu}E_{c}^{\,\,\rho}E_{d}^{\,\,\sigma}$
are the components of the Weyl tensor on the null frame
$\\{\boldsymbol{E}_{a}\\}$. In order to see that these components of the Weyl
tensor are indeed independent of each other it is necessary to verify whether
the symmetries of the Weyl tensor impose any relation between them. After some
straightforward algebra it can be proved that the trace-less condition,
$C^{a}_{\phantom{a}bad}=0$, and the Bianchi identity, $C_{a[bcd]}=0$, are
equivalent to the following equations:
$\displaystyle C_{2123}$ $\displaystyle=$ $\displaystyle
C_{4143}=C_{1214}=C_{3234}=0\;;$ $\displaystyle C_{2124}$ $\displaystyle=$
$\displaystyle\Psi^{+}_{1}\;;\;C_{4142}=\Psi_{1}^{-}\;;\;C_{2324}=\Psi_{3}^{-}\;;\;C_{4342}=\Psi^{+}_{3}\;;$
$\displaystyle C_{2424}$ $\displaystyle=$ $\displaystyle
C_{1313}=\Psi^{+}_{2}+\Psi_{2}^{-}\;;\;C_{1324}=\Psi_{2}^{-}-\Psi^{+}_{2}.$
Which proves that the scalars defined on (4.2) can, indeed, represent the 10
degrees of freedom of the Weyl tensor. These scalars can also be conveniently
written as follows:
$\displaystyle\Psi^{\pm}_{0}=\frac{1}{4}\langle\boldsymbol{Z}^{1\pm},\mathcal{C}(\boldsymbol{Z}^{1\pm})\rangle\;;\;\Psi^{\pm}_{1}=\frac{-1}{4\sqrt{2}}\langle\boldsymbol{Z}^{1\pm},\mathcal{C}(\boldsymbol{Z}^{3\pm})\rangle$
$\displaystyle\Psi^{\pm}_{2}=\frac{1}{4}\langle\boldsymbol{Z}^{1\pm},\mathcal{C}(\boldsymbol{Z}^{2\pm})\rangle\,=\frac{1}{8}\langle\boldsymbol{Z}^{3\pm},\mathcal{C}(\boldsymbol{Z}^{3\pm})\rangle$
(4.10)
$\displaystyle\Psi^{\pm}_{3}=\frac{-1}{4\sqrt{2}}\langle\boldsymbol{Z}^{2\pm},\mathcal{C}(\boldsymbol{Z}^{3\pm})\rangle\;;\;\Psi^{\pm}_{4}=\frac{1}{4}\langle\boldsymbol{Z}^{2\pm},\mathcal{C}(\boldsymbol{Z}^{2\pm})\rangle\,.$
By means of equations (4.10) and (4.8) it can be easily proved that the matrix
representations of the operators $\mathcal{C}^{\pm}$ on the basis
$\\{\boldsymbol{Z}^{i\pm}\\}$ are given by:
$\mathcal{C}^{\pm}\,=\,2\left[\begin{array}[]{ccc}\Psi^{\pm}_{2}\vspace{0.15cm}&\Psi^{\pm}_{4}&-\sqrt{2}\Psi^{\pm}_{3}\\\
\vspace{0.15cm}\Psi^{\pm}_{0}&\Psi^{\pm}_{2}&-\sqrt{2}\Psi^{\pm}_{1}\\\
\sqrt{2}\Psi^{\pm}_{1}&\sqrt{2}\Psi^{\pm}_{3}&-2\Psi^{\pm}_{2}\\\
\end{array}\right].$ (4.11)
Since the operators $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ have vanishing
trace it follows that the possible algebraic types of these operators
according to the refined Segre classification are the ones listed on equation
(2.7). It is also worth noting that $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$
are independent of each other. So, for instance, we might say that the Weyl
tensor is type $(I,N)$ when $\mathcal{C}^{+}$ is type $I$ and
$\mathcal{C}^{-}$ is type $N$. Note that the type $(I,N)$ is intrinsically
equivalent to the type $(N,I)$, since the operators $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ are interchanged if we multiply the volume-form by $-1$. So
we conclude that on a complexified 4-dimensional manifold the Weyl tensor can
have 21 algebraic types [33]:
$\begin{array}[]{ccccccc}(O,O)&(O,D)&(O,N)&(O,III)&(O,II)&(O,I)&(D,D)\\\
(D,N)&(D,III)&(D,II)&(D,I)&(N,N)&(N,III)&(N,II)\\\
(N,I)&(III,III)&(III,II)&(III,I)&(II,II)&(II,I)&(I,I)\end{array}$ (4.12)
As proved in ref. [33], the same classification can be attained using the
boost weight approach. Up to now the metric was not assumed to be real, so
that the Weyl tensor is generally complex. But some of these types are
forbidden when the metric is real, as we shall see in what follows.
##### 4.2.1 Euclidean Signature
Let us suppose that $\boldsymbol{g}$ is a real metric with Euclidean
signature. Then the components $C_{\mu\nu\rho\sigma}$ of the Weyl tensor on a
real coordinate frame are real. By means of this fact along with equations
(4.2) and (4.2), one can easily prove that in this signature the Weyl scalars
obey the following reality conditions:
$\overline{\Psi^{\pm}_{0}}\,=\,\Psi^{\pm}_{4}\;\;;\;\;\;\overline{\Psi^{\pm}_{1}}\,=\,-\Psi^{\pm}_{3}\;\;;\;\;\;\overline{\Psi^{\pm}_{2}}\,=\,\Psi^{\pm}_{2}\,.$
(4.13)
This together with (4.11) implies that the matrix representation of the
operators $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ are Hermitian and
independent of each other. So these matrices can be diagonalized and,
therefore, the algebraic types $II$, $III$ and $N$ are forbidden. Thus if the
signature is Euclidean the Weyl tensor must have one of the following
algebraic types [33]:
$(O,O)\quad(O,D)\quad\,(O,I)\quad(D,D)\quad(D,I)\quad(I,I)\,.$ (4.14)
An equivalent classification was obtained in [80] using a mixture of null
tetrad and spinorial formalisms. The same classification was also found in
[81] by means of splitting the Weyl tensor as a sum of two 3-dimensional
tensors of rank two and using the group $SO(4,\mathbb{R})$ to find canonical
forms for such tensors.
##### 4.2.2 Lorentzian Signature
Now assume that the metric $\boldsymbol{g}$ is real and Lorentzian. Then the
Weyl tensor is real, so that equations (4.4) and (4.2) immediately imply the
following reality conditions:
$\overline{\Psi^{+}_{0}}\,=\,\Psi^{-}_{0}\;;\;\;\overline{\Psi^{+}_{1}}\,=\,\Psi^{-}_{1}\;;\;\;\overline{\Psi^{+}_{2}}\,=\,\Psi^{-}_{2}\;;\;\;\overline{\Psi^{+}_{3}}\,=\,\Psi^{-}_{3}\;;\;\;\overline{\Psi^{+}_{4}}\,=\,\Psi^{-}_{4}\,.$
Which along with equation (4.11) guarantees that the matrix representation of
$\mathcal{C}^{-}$ is the complex conjugate of the matrix representation of
$\mathcal{C}^{+}$. Therefore, in this signature $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ must have the same algebraic type. So from the 21 types
listed on eq. (4.12) just the following six types are allowed in the
Lorentzian case [33]:
$(O,O)\quad(D,D)\quad(N,N)\quad(III,III)\quad(II,II)\quad(I,I)\,.$ (4.15)
These types correspond respectively to the Petrov types $O$, $D$, $N$, $III$,
$II$ and $I$, retrieving the Petrov classification. In particular, note that
in this signature if $\mathcal{C}^{-}$ identically zero then $\mathcal{C}^{+}$
must also vanish, so that non-trivial self-dual Weyl tensors do not exist on
the Lorentzian case.
##### 4.2.3 Split Signature
Suppose that $\boldsymbol{g}$ is a real metric with split signature. In this
case it is possible to find a real null frame, as shown in (4.5). Thus the
Weyl scalars, defined on (4.2), are all real and generally independent of each
other. So the matrix representations of $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ are real and generally independent of each other. Therefore,
in this case there is no algebraic restriction on the matrices that represent
$\mathcal{C}^{\pm}$, which implies that all the 21 types of eq. (4.12) are
allowed [33]. A classification deeply related to this one was obtained in [84]
using spinorial calculus. Other, inequivalent, classifications for the Weyl
tensor in manifolds of split signature were defined in [30, 82, 83].
##### 4.2.4 Annihilating Weyl Scalars
In section 2.2 it was proved that when the signature is Lorentzian each Petrov
type can be characterized by the possibility of annihilating some of the Weyl
scalars. It turns out that the same thing happens on the generalized
classification presented in this chapter, as we shall prove.
Every transformation that maps a null frame into a null frame can be written
as a composition of the following three kinds of transformations:
(i) Lorentz Boosts
$\boldsymbol{E}_{1}\mapsto\lambda_{+}\lambda_{-}\,\boldsymbol{E}_{1}\;;\;\;\boldsymbol{E}_{2}\mapsto\lambda_{+}\lambda_{-}^{-1}\,\boldsymbol{E}_{2}\;;\;\;\boldsymbol{E}_{3}\mapsto\lambda_{+}^{-1}\lambda_{-}^{-1}\,\boldsymbol{E}_{3}\;;\;\;\boldsymbol{E}_{4}\mapsto\lambda_{+}^{-1}\lambda_{-}\,\boldsymbol{E}_{4}$
(ii) Null rotation around $\boldsymbol{E}_{1}$
$\boldsymbol{E}_{1}\mapsto\boldsymbol{E}_{1}\,;\;\boldsymbol{E}_{2}\mapsto\boldsymbol{E}_{2}+w_{-}\boldsymbol{E}_{1}\,;\;\boldsymbol{E}_{3}\mapsto\boldsymbol{E}_{3}+w_{+}\boldsymbol{E}_{2}+w_{-}\boldsymbol{E}_{4}+w_{+}w_{-}\boldsymbol{E}_{1}\,;\;\boldsymbol{E}_{4}\mapsto\boldsymbol{E}_{4}+w_{+}\boldsymbol{E}_{1}$
(iii) Null rotation around $\boldsymbol{E}_{3}$
$\boldsymbol{E}_{1}\mapsto\boldsymbol{E}_{1}+z_{-}\boldsymbol{E}_{2}+z_{+}\boldsymbol{E}_{4}+z_{+}z_{-}\boldsymbol{E}_{3}\,;\;\boldsymbol{E}_{2}\mapsto\boldsymbol{E}_{2}+z_{+}\boldsymbol{E}_{3}\,;\;\boldsymbol{E}_{3}\mapsto\boldsymbol{E}_{3}\,;\;\boldsymbol{E}_{4}\mapsto\boldsymbol{E}_{4}+z_{-}\boldsymbol{E}_{3}$
Where $\lambda_{\pm}$, $w_{\pm}$ and $z_{\pm}$ are complex numbers, the six
parameters of the group $SO(4;\mathbb{C})$. It is interesting to note that
under these transformations the Weyl scalars change as:
$\Psi_{A}^{+}\,\longmapsto\,F_{A}(\lambda_{+},w_{+},z_{+},\Psi_{B}^{+})\quad;\quad\Psi_{A}^{-}\,\longmapsto\,F_{A}(\lambda_{-},w_{-},z_{-},\Psi_{B}^{-})\,.$
So the parameters $\lambda_{-}$, $w_{-}$ and $z_{-}$ do not appear on the
transformation of the operator $\mathcal{C}^{+}$ while the transformation of
$\mathcal{C}^{-}$ does not depend on $\lambda_{+}$, $w_{+}$ and $z_{+}$.
Thanks to this property, the same argument used in section 2.2 in order to
show which Weyl scalars could be made to vanish by a suitable choice of null
tetrad remains valid here for both operators $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ individually. Table 4.1 summarizes this analysis. Thus, for
example, if the Weyl tensor is type $(I,II)$ then it is possible to choose a
null frame in which the Weyl scalars $\Psi_{0}^{+}$, $\Psi_{4}^{+}$,
$\Psi_{0}^{-}$ , $\Psi_{1}^{-}$ and $\Psi_{4}^{-}$ vanish simultaneously.
$\mathcal{C}^{\pm}$ type $I$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{4}^{\pm}$ | $\mathcal{C}^{\pm}$ type $II$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{1}^{\pm},\Psi_{4}^{\pm}$
---|---
$\mathcal{C}^{\pm}$ type $D$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{1}^{\pm},\Psi_{3}^{\pm},\Psi_{4}^{\pm}$ | $\mathcal{C}^{\pm}$ type $III$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{1}^{\pm},\Psi_{2}^{\pm},\Psi_{4}^{\pm}$
$\mathcal{C}^{\pm}$ type $N$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{1}^{\pm},\Psi_{2}^{\pm},\Psi_{3}^{\pm}$ | $\mathcal{C}^{\pm}$ type $O$ $\rightarrow$ $\Psi_{0}^{\pm},\Psi_{1}^{\pm},\Psi_{2}^{\pm},\Psi_{3}^{\pm},\Psi_{4}^{\pm}$
Table 4.1: Weyl scalars that can be made to vanish, by a suitable choice of
null frame, depending on the algebraic type of the operators
$\mathcal{C}^{\pm}$.
In this generalized classification the Weyl tensor shall be called
algebraically special when its type is different from $(I,I)$. In such a case
one can conveniently choose the signal of the volume-form so that
$\mathcal{C}^{+}$ is not type $I$. Therefore, table 4.1 implies that the Weyl
tensor is algebraically special if, and only if, it is possible to find a null
frame in which $\Psi_{0}^{+}=\Psi_{1}^{+}=0$. This along with eq. (4.11) yield
that the Weyl tensor is algebraically special if, and only if,
$\boldsymbol{Z}^{1+}$ is an eigenbivector of the Weyl operator. Since every
self-dual null bivector can be written as
$\boldsymbol{E}^{4}\wedge\boldsymbol{E}^{3}=\boldsymbol{Z}^{1+}$ on a suitable
null frame, we arrive at the following theorem [35]:
###### Theorem 8
The Weyl tensor of a 4-dimensional manifold is algebraically special if, and
only if, the Weyl operator admits a null eigenbivector.
#### 4.3 Generalized Goldberg-Sachs Theorem
In this section it will be presented a beautiful generalization of the
Goldberg-Sachs (GS) theorem valid in 4-dimensional vacuum333Throughout this
thesis the expressions vacuum manifold and Ricci-flat manifold will be
interchanged, they both mean a manifold with vanishing Ricci tensor. manifolds
of arbitrary signature, a result first proved by Plebański and Hacyan in [60].
To this end the notation introduced in section 1.7 will be used. In
particular, let us recall the following important equations:
$V^{\mu}\nabla_{\mu}\boldsymbol{E}^{a}\equiv-\boldsymbol{\omega}^{a}_{\phantom{a}b}(\boldsymbol{V})\,\boldsymbol{E}^{b}\;;\quad\omega_{ab}^{\phantom{ab}c}\equiv\boldsymbol{\omega}^{c}_{\phantom{c}b}(\boldsymbol{E}_{a})\;;\quad\nabla_{a}\boldsymbol{E}_{b}=\omega_{ab}^{\phantom{ab}c}\boldsymbol{E}_{c}\,.$
(4.16)
Where $\boldsymbol{\omega}^{a}_{\phantom{a}b}$ are the so-called connection
1-forms. Since for a null frame the matrix
$g_{ab}=\boldsymbol{g}(\boldsymbol{E}_{a},\boldsymbol{E}_{b})$ is constant it
follows that $\boldsymbol{\omega}_{ab}=-\boldsymbol{\omega}_{ba}$ and
$\omega_{abc}=-\omega_{acb}$, where $\boldsymbol{\omega}_{ab}\equiv
g_{ac}\boldsymbol{\omega}^{c}_{\phantom{c}b}$ and
$\omega_{abc}\equiv\omega_{ab}^{\phantom{ab}d}g_{dc}$. Using this notation,
the generalized Goldberg-Sachs theorem is given by [60]:
###### Theorem 9
Let $(M,\boldsymbol{g})$ be a 4-dimensional manifold with vanishing Ricci
tensor. If $\omega_{112}=\omega_{221}=0$ then $\Psi_{0}^{+}=\Psi_{1}^{+}=0$.
Conversely, if $\Psi_{0}^{+}=\Psi_{1}^{+}=0$ then it is possible to find a
null frame in which the scalars $\Psi_{0}^{+}$, $\Psi_{1}^{+}$, $\omega_{112}$
and $\omega_{221}$ all vanish.
Before proceeding, let us prove that this theorem is equivalent to the
Goldberg-Sachs theorem when the signature is Lorentzian. Indeed, using
equations (4.3) and (4.16) along with the definition of the shear parameter,
eq. (3.5), we find:
$\displaystyle\boldsymbol{l}^{\mu}\nabla_{\mu}\boldsymbol{l}=\nabla_{1}\boldsymbol{E}_{1}=\omega_{11}^{\phantom{11}a}\boldsymbol{E}_{a}=\omega_{113}\,\boldsymbol{l}-\omega_{114}\,\boldsymbol{m}-\omega_{112}\,\overline{\boldsymbol{m}}$
$\displaystyle\sigma=\boldsymbol{g}(m^{\mu}\,\nabla_{\mu}\boldsymbol{l},\boldsymbol{m})=\boldsymbol{g}(\nabla_{2}\boldsymbol{E}_{1},\boldsymbol{E}_{2})=-\omega_{21}^{\phantom{21}4}=\omega_{212}=-\omega_{221}\,.$
From which we conclude that the congruence generated by the null vector field
$\boldsymbol{l}=\boldsymbol{E}_{1}$ is geodesic and shear-free if, and only
if, the connection components $\omega_{114}$, $\omega_{112}$ and
$\omega_{221}$ all vanish. But equation (4.4) implies that on the Lorentzian
signature $\omega_{114}$ is the complex conjugate of $\omega_{112}$. Thus
$\boldsymbol{l}$ will be geodesic and shear-free if, and only if,
$\omega_{112}=\omega_{221}=0$, proving that theorem 9 reduces to the usual GS
theorem on the Lorentzian signature, see theorem 1.
The condition $\omega_{112}=\omega_{221}=0$ has a nice geometric
interpretation, it is equivalent to say that the complexified manifold can be
foliated by totally null leafs. Indeed, using eq. (4.16) we find that the Lie
bracket of $\boldsymbol{E}_{1}$ and $\boldsymbol{E}_{2}$ is:
$\displaystyle[\boldsymbol{E}_{1},\boldsymbol{E}_{2}]\,=\,$
$\displaystyle\nabla_{1}\boldsymbol{E}_{2}-\nabla_{2}\boldsymbol{E}_{1}=(\omega_{12}^{\phantom{12}a}-\omega_{21}^{\phantom{21}a})\boldsymbol{E}_{a}$
$\displaystyle\,=\,$
$\displaystyle(\omega_{123}-\omega_{213})\boldsymbol{E}_{1}-(\omega_{124}-\omega_{214})\boldsymbol{E}_{2}-\omega_{112}\boldsymbol{E}_{3}-\omega_{221}\boldsymbol{E}_{4}\,.$
(4.17)
Thus the condition $\omega_{112}=\omega_{221}=0$ is equivalent to say that the
Lie bracket $[\boldsymbol{E}_{1},\boldsymbol{E}_{2}]$ is a linear combination
of $\boldsymbol{E}_{1}$ and $\boldsymbol{E}_{2}$. Since
$[\boldsymbol{E}_{1},\boldsymbol{E}_{1}]$ and
$[\boldsymbol{E}_{2},\boldsymbol{E}_{2}]$ are trivially zero this, in turn, is
equivalent to the integrability of the distribution generated by
$\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2}\\}$, see section 1.8. Since such
vector fields are null and orthogonal to each other it follows that the
vectors tangent to this distribution are all null, this kind of distribution
is named isotropic. Therefore, theorem 9 guarantees that a vacuum manifold
admits an integrable distribution of isotropic planes if, and only if, the
Weyl tensor is algebraically special [60]. Since
$Z^{1+\,\mu\nu}=2E_{1}^{\,[\mu}E_{2}^{\,\nu]}$ we shall write
$\boldsymbol{Z}^{1+}=\boldsymbol{E}_{1}\boldsymbol{\wedge}\boldsymbol{E}_{2}$
and say that $\boldsymbol{Z}^{1+}$ generates the distribution of isotropic
planes spanned by the vector fields $\boldsymbol{E}_{1}$ and
$\boldsymbol{E}_{2}$. As noticed on the paragraph before theorem 8,
$\boldsymbol{Z}^{1+}$ is an eigenbivector of the Weyl operator if, and only
if, $\Psi_{0}^{+}=\Psi_{1}^{+}=0$, which lead us to the following result [35]:
###### Corollary 2
A distribution of isotropic planes in a Ricci-flat 4-dimensional manifold is
integrable if, and only if, the null bivector that generates such distribution
is an eigenbivector of the Weyl operator.
This fact is illustrated in figure 4.1. If the metric is real then whenever a
distribution is integrable the complex conjugate of such distribution will
also be integrable. Particularly, on the Lorentzian signature if $\Delta$ is
an integrable distribution of isotropic planes then $\overline{\Delta}$ will
also be integrable and $\Delta\cap\overline{\Delta}=Span\\{\boldsymbol{l}\\}$,
where $\boldsymbol{l}$ is a real null vector field generating a geodesic and
shear-free congruence, see figure 4.1.
Figure 4.1: In vacuum, the Weyl tensor admits a null eigenbivector if, and
only if, the isotropic distribution generated by such bivector is integrable,
as depicted on the left hand side of the picture. The vector fields
$\boldsymbol{E}_{1}$ and $\boldsymbol{E}_{2}$ are null and orthogonal to each
other, generating isotropic planes. On the right hand side of this figure we
have the Lorentzian case, where the intersection of a totally null plane and
its complex conjugate gives a real null direction $\boldsymbol{E}_{1}$. In
this signature if the distribution generated by
$\boldsymbol{E}_{1}\wedge\boldsymbol{E}_{2}$ is integrable so will be the
distribution generated by $\boldsymbol{E}_{1}\wedge\boldsymbol{E}_{4}$.
Moreover, $\boldsymbol{E}_{1}$ will be geodesic and shear-free.
One can also express such integrability result using the dual form of the
Frobenius theorem, seen in section 1.8. In this language the corollary 2 is
equivalent to the claim that given a null bivector $\boldsymbol{Z}$, it is
possible to find some scalar function $f\neq 0$ such that
$d(f\boldsymbol{Z})=0$ if, and only if, $\boldsymbol{Z}$ is an eigenbivector
of the Weyl operator. Let us state this as a corollary:
###### Corollary 3
In a Ricci-flat manifold, the Weyl scalars $\Psi_{0}^{+}$ and $\Psi_{1}^{+}$
vanish if, and only if, it is possible to find a scalar function $f\neq 0$
such that $d(f\boldsymbol{Z}^{1+})=0$ in a suitable null frame.
On the Lorentzian signature a real null vector field $\boldsymbol{l}$ is said
to be a principal null direction (PND) of the Weyl tensor when it is possible
to find a null tetrad
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
such that $\Psi_{0}\equiv\Psi_{0}^{+}$ vanishes, in general there exists 4
distinct PNDs. Moreover, this vector field is said to be a repeated PND when,
in addition to $\Psi_{0}$, the Weyl scalar $\Psi_{1}\equiv\Psi_{1}^{+}$ do
also vanish. On the general formalism presented in this chapter the concept of
privileged null directions might be substituted by privileged null bivectors
[35]. Looking at the definition of $\Psi_{0}^{+}$ on eq. (4.10) it is natural
to define a null bivector $\boldsymbol{Z}$ to be a principal null bivector
(PNB) when $\langle\boldsymbol{Z},\mathcal{C}(\boldsymbol{Z})\rangle=0$.
Furthermore, because of eq. (4.11) and theorem 8, $\boldsymbol{Z}$ shall be
called a repeated PNB if $\boldsymbol{Z}$ is null and
$\mathcal{C}(\boldsymbol{Z})\propto\boldsymbol{Z}$. In general the Weyl tensor
will admit 4 self-dual PNBs and 4 anti-self-dual PNBs, as can be verified
using the group $SO(4;\mathbb{C})$. In the Lorentzian case $\boldsymbol{l}$ is
a PND if, and only if, $\boldsymbol{Z}_{1}=\boldsymbol{l}\wedge\boldsymbol{m}$
is a self-dual PNB, which, in turn, is equivalent to say that
$\overline{\boldsymbol{Z}_{1}}=\boldsymbol{l}\wedge\overline{\boldsymbol{m}}$
is an anti-self-dual PNB.
As a last comment it is worth pointing out that the generalized GS theorem is
also valid in less restrictive situations than the Ricci-flat case. Indeed, on
the original article of Plebański and Hacyan [60] it was observed that such
theorem remains valid for Einstein manifolds, the ones such that the Ricci
tensor is proportional to the metric. Furthermore, in [83] it was worked out
the least restrictive version of the generalized GS theorem.
#### 4.4 Geometric Consequences of the Generalized
Goldberg-Sachs Theorem
The goal of this section is to use the generalized Goldberg-Sachs theorem in
order to prove that certain algebraic types of the Weyl tensor are
characterized by the existence of important geometric structures on the
manifold. Here it will be assumed that the Ricci tensor of the manifold is
identically zero. The results obtained in the present section are based on the
article [35]. Important attempts on the same line can also be found in [83,
87]. Before proceeding some definitions and tools of complex differential
geometry shall be introduced.
##### 4.4.1 Complex Manifolds
Let $(M,\boldsymbol{g})$ be an even-dimensional manifold, then an almost
complex structure on this manifold is an endomorphism of the tangent bundle,
$\mathcal{J}:TM\rightarrow TM$, whose square is minus the identity map,
$\mathcal{J}^{2}=-\mathbf{1}$. Note that the almost complex structure can be
seen as a tensor of rank two, $\mathcal{J}^{\mu}_{\phantom{\mu}\nu}$, defined
by the following relation:
$\mathcal{J}(\boldsymbol{V})\,=\,\boldsymbol{X}\quad\Longleftrightarrow\quad
X^{\mu}\,=\,\mathcal{J}^{\mu}_{\phantom{\mu}\nu}\,V^{\nu}\,.$
If $\boldsymbol{V}$ is some vector field then defining
$\boldsymbol{V}^{\pm}\equiv[\boldsymbol{V}\mp i\mathcal{J}(\boldsymbol{V})]$
we find that $\boldsymbol{V}^{\pm}$ is an eigenvector of $\mathcal{J}$ with
eigenvalue $\pm i$. Thus $\mathcal{J}$ splits the tangent bundle as follows:
$TM=TM^{+}\oplus TM^{-}\;,\;\quad TM^{\pm}\equiv\\{\boldsymbol{V}\in
TM\,|\,\mathcal{J}(\boldsymbol{V})=\pm i\,\boldsymbol{V}\\}\,.$
The almost complex structure is said to be integrable when the distributions
$TM^{+}$ and $TM^{-}$ are both integrable, in which case $\mathcal{J}$ is
called a complex structure. By means of $\mathcal{J}$ we can define a tensor
$\boldsymbol{N}$, called the Nijenhuis tensor, whose action on two vector
fields yields another vector field as follows:
$\boldsymbol{N}(\boldsymbol{V},\boldsymbol{X})=[\boldsymbol{V},\boldsymbol{X}]-[\mathcal{J}(\boldsymbol{V}),\mathcal{J}(\boldsymbol{X})]+\mathcal{J}\left([\mathcal{J}(\boldsymbol{V}),\boldsymbol{X}]\right)+\mathcal{J}\left([\boldsymbol{V},\mathcal{J}(\boldsymbol{X})]\right).$
It can be proved that $\mathcal{J}$ is integrable if, and only if,
$\boldsymbol{N}$ vanishes [55]. When the almost complex structure leaves the
inner products invariant,
$\boldsymbol{g}\left(\mathcal{J}(\boldsymbol{V}),\mathcal{J}(\boldsymbol{X})\right)=\boldsymbol{g}(\boldsymbol{V},\boldsymbol{X})$
for all vector fields $\boldsymbol{V}$ and $\boldsymbol{X}$, the metric is
said to be Hermitian with respect to $\mathcal{J}$. In this case one can
introduce a 2-form, called the Kähler form, defined by
$\Omega_{\mu\nu}=g_{\rho\nu}\mathcal{J}^{\rho}_{\phantom{\rho}\mu}$. Note that
if the metric is Hermitian with respect to $\mathcal{J}$ then the subbundles
$TM^{+}$ and $TM^{-}$ are both isotropic.
On the chapter 1 a manifold of dimension $n$ was defined to be a topological
set such that the neighborhood of each point can be smoothly mapped by a
coordinate system into a patch of $\mathbb{R}^{n}$. In addition, it must be
required that the transition functions between the coordinate systems of
overlapping neighborhoods are smooth. An $n$-dimensional complex manifold444Do
not confuse with a complexified manifold, which is just a regular manifold
with all its tensor bundles complexified. is, likewise, defined as a
topological set such that the neighborhood of each point can be smoothly
mapped by a coordinate system into a patch of $\mathbb{C}^{n}$ and such that
the transition functions between the coordinates systems of overlapping
neighborhoods are not only smooth but also analytic [55]. This last
requirement is more restrictive than it sounds. Indeed, a celebrated theorem
on differential geometry, the Newlander-Nirenberg theorem [88], states that a
manifold admits an integrable and real almost complex structure if, and only
if, it is a complex manifold. When a complex manifold is endowed with a metric
that is invariant by the action of the almost complex structure on the vector
fields the manifold is called Hermitian. In this case one can define a 2-form
$\boldsymbol{\Omega}$, called the Kähler form of the Hermitian manifold, as
defined in the preceding paragraph. If the exterior derivative of the Kähler
form vanishes, $d\boldsymbol{\Omega}=0$, the manifold is said to be a Kähler
manifold. If in addition the Ricci tensor vanishes, as assumed in this
chapter, the manifold is called a Calabi-Yau manifold555Actually, a Calabi-Yau
manifold is defined to be a Kähler manifold with vanishing first Chern class,
which is less restrictive than the Ricci-flat condition.. The Calabi-Yau
manifolds are of great relevance for string theory compactifications [15].
##### 4.4.2 General Results
Now let $(M,\boldsymbol{g})$ be a complexified 4-dimensional manifold of
arbitrary signature and $\\{\boldsymbol{E}_{a}\\}$ a null frame. Then we can
define the following almost complex structure [35]:
$\boldsymbol{J}\,\equiv\,i\,\left(\boldsymbol{E}_{1}\otimes\boldsymbol{E}^{1}+\boldsymbol{E}_{2}\otimes\boldsymbol{E}^{2}\right)-i\,\left(\boldsymbol{E}_{3}\otimes\boldsymbol{E}^{3}+\boldsymbol{E}_{4}\otimes\boldsymbol{E}^{4}\right)\,.$
(4.18)
Note that the metric $\boldsymbol{g}$ is Hermitian with respect to this almost
complex structure. For example,
$\boldsymbol{g}\left(\boldsymbol{J}(\boldsymbol{E}_{1}),\boldsymbol{J}(\boldsymbol{E}_{3})\right)\,=\,\boldsymbol{g}(i\boldsymbol{E}_{1},-i\boldsymbol{E}_{3})\,=\,\boldsymbol{g}(\boldsymbol{E}_{1},\boldsymbol{E}_{3})\,.$
It is also immediate to see that $\boldsymbol{E}_{1}$ and $\boldsymbol{E}_{2}$
are eigenvectors of $\boldsymbol{J}$ with eigenvalue $i$, while
$\boldsymbol{E}_{3}$ and $\boldsymbol{E}_{4}$ are eigenvectors with eigenvalue
$-i$. This means that $TM^{+}=Span\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2}\\}$
and $TM^{-}=Span\\{\boldsymbol{E}_{3},\boldsymbol{E}_{4}\\}$. So, using
equation (4.17), we conclude that $TM^{+}$ is integrable if, and only if,
$\omega_{112}=\omega_{221}=0$. Analogously, $TM^{-}$ is integrable if, and
only if, $\omega_{334}=\omega_{443}=0$. Therefore, we can state:
$\boldsymbol{J}\;\,\textrm{is
Integrable}\quad\Longleftrightarrow\quad\omega_{112}=\omega_{221}=\omega_{334}=\omega_{443}=0\,.$
(4.19)
But theorem 9 and equation (4.2) imply that the right hand side of (4.19)
holds if, and only if, the Weyl scalars $\Psi^{+}_{0}$, $\Psi^{+}_{1}$,
$\Psi^{+}_{3}$ and $\Psi^{+}_{4}$ vanish. Equation (4.11), in turn, guarantees
that the annihilation of these Weyl scalars is equivalent to say that
$\mathcal{C}^{+}$ is type $D$ or type $O$. So $\boldsymbol{J}$ is integrable
if, and only if, $\mathcal{C}^{+}$ is type $D$ or type $O$. In the same vein,
it can be proved that if $(M,\boldsymbol{g})$ admits an integrable almost
complex structure such that $\boldsymbol{g}$ is Hermitian with respect to it
then the Weyl tensor must be type $(D,\lozenge)$ or type $(O,\lozenge)$, where
$\lozenge$ represents an arbitrary Petrov type [35]. So the following theorem
holds:
###### Theorem 10
A Ricci-flat 4-dimensional manifold $(M,\boldsymbol{g})$ admits an integrable
almost complex structure with $\boldsymbol{g}$ being Hermitian with respect to
it if, and only if, the algebraic type of the Weyl tensor is $(D,\lozenge)$ or
$(O,\lozenge)$. Moreover, if such complex structure exists we can always
manage to find a null frame in which it takes the form shown on eq. (4.18).
The Kähler form is the 2-form $\boldsymbol{\Omega}$ such that
$\boldsymbol{X}\lrcorner\boldsymbol{V}\lrcorner\boldsymbol{\Omega}=\boldsymbol{g}(\boldsymbol{J}(\boldsymbol{V}),\boldsymbol{X})$
for all vector fields $\boldsymbol{V}$ and $\boldsymbol{X}$. It is simple
matter to prove that it is given by:
$\boldsymbol{\Omega}\,=\,i\left(\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{3}\,+\,\boldsymbol{E}^{4}\wedge\boldsymbol{E}^{2}\right)\,=\,i\,\sqrt{2}\,\boldsymbol{Z}^{3+}\,.$
(4.20)
We can calculate the exterior derivative of this 2-form by means of the first
Cartan’s structure equation,
$d\boldsymbol{E}^{a}+\boldsymbol{\omega}^{a}_{\phantom{a}b}\wedge\boldsymbol{E}^{b}=0$.
The bottom line is:
$d\boldsymbol{\Omega}\,=\,-2i\,\boldsymbol{\omega}_{12}\wedge\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{2}\,+\,2i\,\boldsymbol{\omega}_{34}\wedge\boldsymbol{E}^{3}\wedge\boldsymbol{E}^{4}\,.$
Since $\boldsymbol{\omega}_{ab}=\omega_{cba}\boldsymbol{E}^{c}$, it follows
that $d\boldsymbol{\Omega}=0$ if, and only if, the connection components
$\omega_{321}$, $\omega_{421}$, $\omega_{143}$ and $\omega_{243}$ all vanish.
This along with equation (4.19) yields:
$\boldsymbol{J}\;\,\textrm{is integrable
and}\;\,d\boldsymbol{\Omega}=0\quad\Longleftrightarrow\quad\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0\,.$
(4.21)
Furthermore, let us calculate the covariant derivative of the Kähler form.
Using the identity
$\nabla_{a}\boldsymbol{E}^{b}=\omega_{a\phantom{b}c}^{\phantom{a}b}\boldsymbol{E}^{c}$
and eq. (4.20) it is straightforward to prove that:
$\nabla_{a}\boldsymbol{\Omega}\,=\,-2i\,\omega_{a21}\,\boldsymbol{E}^{1}\wedge\boldsymbol{E}^{2}\,+\,2i\,\omega_{a43}\,\boldsymbol{E}^{3}\wedge\boldsymbol{E}^{4}\,.$
Thus $\boldsymbol{\Omega}$ is covariantly constant if, and only if,
$\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0$. This together with
(4.21) then imply the following useful equivalences:
$\boldsymbol{J}\;\,\textrm{Integrable,}\;\,d\boldsymbol{\Omega}=0\quad\Leftrightarrow\quad\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0\quad\Leftrightarrow\quad\nabla_{a}\boldsymbol{\Omega}\,=\,0\,.$
(4.22)
In order to make a connection between these results and the algebraic
classification of the Weyl tensor we need to use the second Cartan’s structure
equation, which in vacuum is:
$\frac{1}{2}\,C_{abcd}\,\boldsymbol{E}^{c}\wedge\boldsymbol{E}^{d}\,=\,d\boldsymbol{\omega}_{ab}+\boldsymbol{\omega}_{ac}\wedge\boldsymbol{\omega}^{c}_{\phantom{c}b}\,.$
Using the definition of the Weyl scalars, this equation can be proved to be
equivalent to the following ones:
$\begin{cases}\begin{array}[]{cl}\;\;\,d\boldsymbol{\omega}_{12}+\boldsymbol{\omega}_{12}\wedge(\boldsymbol{\omega}_{24}-\boldsymbol{\omega}_{13})&=\;\Psi_{2}^{+}\,\boldsymbol{Z}^{1+}+\Psi_{0}^{+}\,\boldsymbol{Z}^{2+}+\sqrt{2}\Psi_{1}^{+}\,\boldsymbol{Z}^{3+}\\\
-d\boldsymbol{\omega}_{34}+\boldsymbol{\omega}_{34}\wedge(\boldsymbol{\omega}_{24}-\boldsymbol{\omega}_{13})&=\;\Psi_{4}^{+}\,\boldsymbol{Z}^{1+}+\Psi_{2}^{+}\,\boldsymbol{Z}^{2+}+\sqrt{2}\Psi_{3}^{+}\,\boldsymbol{Z}^{3+}\\\
-\frac{1}{2}d(\boldsymbol{\omega}_{24}-\boldsymbol{\omega}_{13})+\boldsymbol{\omega}_{12}\wedge\boldsymbol{\omega}_{34}&=\;\Psi_{3}^{+}\,\boldsymbol{Z}^{1+}+\Psi_{1}^{+}\,\boldsymbol{Z}^{2+}+\sqrt{2}\Psi_{2}^{+}\,\boldsymbol{Z}^{3+}\end{array}\\\
\\\
\begin{array}[]{cl}\;\;\,d\boldsymbol{\omega}_{14}+\boldsymbol{\omega}_{14}\wedge(\boldsymbol{\omega}_{42}-\boldsymbol{\omega}_{13})&=\;\Psi_{2}^{-}\,\boldsymbol{Z}^{1-}+\Psi_{0}^{-}\,\boldsymbol{Z}^{2-}+\sqrt{2}\Psi_{1}^{-}\,\boldsymbol{Z}^{3-}\\\
-d\boldsymbol{\omega}_{32}+\boldsymbol{\omega}_{32}\wedge(\boldsymbol{\omega}_{42}-\boldsymbol{\omega}_{13})&=\;\Psi_{4}^{-}\,\boldsymbol{Z}^{1-}+\Psi_{2}^{-}\,\boldsymbol{Z}^{2-}+\sqrt{2}\Psi_{3}^{-}\,\boldsymbol{Z}^{3-}\\\
-\frac{1}{2}d(\boldsymbol{\omega}_{42}-\boldsymbol{\omega}_{13})+\boldsymbol{\omega}_{14}\wedge\boldsymbol{\omega}_{32}&=\;\Psi_{3}^{-}\,\boldsymbol{Z}^{1-}+\Psi_{1}^{-}\,\boldsymbol{Z}^{2-}+\sqrt{2}\Psi_{2}^{-}\,\boldsymbol{Z}^{3-}\end{array}\end{cases}$
These two sets of three equations are the self-dual and anti-self-dual parts
of the second structure equation respectively. The first important thing to
note is that if $\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0$ then
$\Psi_{A}^{+}=0$, so that $\mathcal{C}^{+}$ is type $O$. Conversely, if
$\mathcal{C}^{+}=0$ then we can find a null frame such that
$\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0$ and
$\boldsymbol{\omega}_{24}-\boldsymbol{\omega}_{13}=0$. Thus we can state:
$\mathcal{C}^{+}\,=\,0\quad\Longleftrightarrow\quad\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0\;\textrm{
in some null frame}\,.$ (4.23)
A manifold such that $\mathcal{C}^{+}$ vanishes is dubbed an anti-self-dual
manifold. Consider now the isotropic distribution
$Span\\{\boldsymbol{e}_{{}_{1,\lambda\kappa}},\,\boldsymbol{e}_{{}_{2,\lambda\kappa}}\\}$
with
$\boldsymbol{e}_{{}_{1,\lambda\kappa}}\,\equiv\,\lambda\boldsymbol{E}_{1}\,+\,\kappa\boldsymbol{E}_{4}\quad\textrm{and}\quad\boldsymbol{e}_{{}_{2,\lambda\kappa}}\,\equiv\,\lambda\boldsymbol{E}_{2}\,+\,\kappa\boldsymbol{E}_{3}\,,$
where $\lambda$ and $\kappa$ are constant scalars. Then this distribution will
be integrable if, and only if, the Lie bracket of
$\boldsymbol{e}_{{}_{1,\lambda\kappa}}$ and
$\boldsymbol{e}_{{}_{2,\lambda\kappa}}$ is of the form
$f\boldsymbol{e}_{{}_{1,\lambda\kappa}}+h\boldsymbol{e}_{{}_{2,\lambda\kappa}}$
for some functions $f$ and $h$. Working out such Lie bracket explicitly it is
straightforward to prove that this distribution will be integrable for all
$\lambda$ and $\kappa$ if, and only if, the following conditions hold:
$\displaystyle\omega_{112}\,=\,\omega_{221}\,=\,0\;\,;\;\;\,\omega_{312}$
$\displaystyle=\omega_{224}-\omega_{213}\;;\;\;\omega_{412}=\omega_{124}-\omega_{113}$
$\displaystyle\omega_{334}\,=\,\omega_{443}\,=\,0\;\,;\;\;\,\omega_{143}$
$\displaystyle=\omega_{424}-\omega_{413}\;;\;\;\omega_{243}=\omega_{324}-\omega_{313}\,.$
(4.24)
In particular, note that if
$\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=(\boldsymbol{\omega}_{24}-\boldsymbol{\omega}_{13})=0$
then this infinite family of distributions is integrable. Conversely, if
$Span\\{\boldsymbol{e}_{{}_{1,\lambda\kappa}},\,\boldsymbol{e}_{{}_{2,\lambda\kappa}}\\}$
is integrable for all $\lambda$ and $\kappa$ then equation (4.4.2) holds, so
that theorem 9 implies that $\Psi_{0}^{+}$, $\Psi_{1}^{+}$, $\Psi_{3}^{+}$ and
$\Psi_{4}^{+}$ all vanish. Then inserting this and eq. (4.4.2) on the self-
dual part of the second structure equation we find, after some algebra, that
$\Psi_{2}^{+}$ must also vanish, so that $\mathcal{C}^{+}=0$. Using this
result as well as equations (4.22) and (4.23) we arrive at the following
theorem:
###### Theorem 11
In a Ricci-flat 4-dimensional manifold the following conditions are
equivalent:
(1) The Weyl tensor is type $(O,\lozenge)$, so that $\mathcal{C}^{+}=0$
(2) There exists a null frame in which
$\boldsymbol{\omega}_{12}=\boldsymbol{\omega}_{34}=0$
(3) $\boldsymbol{J}$ is integrable and $d\boldsymbol{\Omega}=0$
(4) The Kähler form, $\boldsymbol{\Omega}$, is covariantly constant
(5) There exists some null frame in which the isotropic distributions
$Span\\{\lambda\boldsymbol{E}_{1}+\kappa\boldsymbol{E}_{4},\,\lambda\boldsymbol{E}_{2}+\kappa\boldsymbol{E}_{3}\\}$
are integrable for all $\lambda$ and $\kappa$ constants.
As shown in chapter 1, in general relativity the gravitational field is
represented by a metric $\boldsymbol{g}$ of a 4-dimensional manifold while the
electromagnetic field is represented by a 2-form $\boldsymbol{F}$ on this
manifold, with the field equations of this system in the absence of sources
being:
$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}=2G\left(F_{\mu\sigma}F_{\nu}^{\phantom{\nu}\sigma}-\frac{1}{4}\,g_{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\right)\;;\quad
d\boldsymbol{F}=0\;;\quad d(\star\boldsymbol{F})=0\,.$
Thus if $\boldsymbol{Z}$ is a 2-form such that its energy-momentum tensor
vanishes,
$4Z_{\mu\sigma}Z_{\nu}^{\phantom{\nu}\sigma}=g_{\mu\nu}Z^{\rho\sigma}Z_{\rho\sigma}$,
and $\star\boldsymbol{Z}\propto\boldsymbol{Z}$ then the above field equations
become just $R_{\mu\nu}=0$ and $d\boldsymbol{Z}=0$. Note that the 2-forms
$\boldsymbol{Z}^{1+}$, $\boldsymbol{Z}^{2+}$ and $\boldsymbol{\Omega}$ are all
self-dual and have vanishing energy-momentum tensor. Furthermore, by what was
seen in section 4.3, when the Weyl tensor of a Ricci-flat 4-dimensional
manifold is algebraically special we can find a null frame in which
$d(f\boldsymbol{Z}^{1+})=0$ for some function $f\neq 0$. In addition, if
$\mathcal{C}^{+}$ is type $D$ then $\boldsymbol{Z}^{2+}$ also generates an
integrable distribution and, therefore, we can find a function $h\neq 0$ such
that $d(h\boldsymbol{Z}^{2+})=0$. Moreover, theorem 11 guarantees that if
$\mathcal{C}^{+}=0$ then $d\boldsymbol{\Omega}=0$. Thus the following theorem
holds:
###### Theorem 12
If the Ricci tensor of a 4-dimensional manifold vanishes then depending on the
algebraic type of the Weyl tensor it is possible to find a null frame and non-
zero functions $f$ and $h$ such that the following 2-forms are solutions to
the Einstein-Maxwell equations without sources:
$\bullet$ $\mathcal{C}^{+}$ type $II$, $III$ or $N$:
$\boldsymbol{F}_{1}=f\boldsymbol{Z}^{1+}$
$\bullet$ $\mathcal{C}^{+}$ type $D$:
$\boldsymbol{F}_{1}=f\boldsymbol{Z}^{1+}$ and
$\boldsymbol{F}_{2}=h\boldsymbol{Z}^{2+}$
$\bullet$ $\mathcal{C}^{+}$ type $O$:
$\boldsymbol{F}_{1}=f\boldsymbol{Z}^{1+}$,
$\boldsymbol{F}_{2}=h\boldsymbol{Z}^{2+}$ and
$\boldsymbol{F}_{3}=\boldsymbol{\Omega}=i\sqrt{2}\boldsymbol{Z}^{3+}$ .
In the present subsection no assumption was made about the signature of the
manifold, nor even it was assumed that the metric is real. In the forthcoming
subsections the general results obtained here will be specialized to the case
of a real metric for each possible signature.
##### 4.4.3 Euclidean Signature
When the metric is real and Euclidean the vectors of a null frame obey the
reality conditions shown on eq. (4.2). Particularly, this implies that the
almost complex structure $\boldsymbol{J}$ and the Kähler form
$\boldsymbol{\Omega}$ are both real. In addition, for this signature just the
six algebraic types shown on equation (4.14) are allowed. Thus if the Weyl
tensor is not type $(I,I)$ then it must be type $(D,\lozenge)$ or
$(O,\lozenge)$, which according to theorem 10 is equivalent to say that
$\boldsymbol{J}$ is integrable on some null frame. Since $\boldsymbol{J}$ is
real, the Newlander-Nirenberg theorem guarantees that if $\boldsymbol{J}$ is
integrable then the manifold over the complex field is a complex manifold,
more precisely an Hermitian manifold. Therefore we can state the following
theorem [35, 83]:
###### Theorem 13
In a 4-dimensional Euclidean manifold with vanishing Ricci tensor, the Weyl
tensor is algebraically special if, and only if, the manifold over the complex
field is Hermitian.
Furthermore, if the type of the Weyl tensor is $(O,\lozenge)$ then theorem 11
guarantees that $\boldsymbol{\Omega}$ is covariantly constant,
$\nabla_{a}\boldsymbol{\Omega}=0$. In particular the Kähler form is closed,
$d\boldsymbol{\Omega}=0$, which implies that the manifold is a Calabi-Yau
manifold. So the following theorem holds:
###### Theorem 14
An Euclidean 4-dimensional Ricci-flat manifold over the complex field is a
Calabi-Yau manifold if, and only if, the Weyl tensor is either self-dual,
$\mathcal{C}^{-}=0$, or anti-self-dual, $\mathcal{C}^{+}=0$.
This result was first proved in [89] using spinorial language and later in
[35] using vectorial formalism.
##### 4.4.4 Lorentzian Signature
If the metric is real and Lorentzian a special phenomenon arises, the self-
dual and anti-self-dual parts of the Weyl tensor are complex conjugates of
each other, $\mathcal{C}^{+}=\overline{\mathcal{C}^{-}}$. In particular, if a
null bivector generates an integrable distribution of isotropic self-dual
planes then its complex conjugate generates an integrable distribution of
isotropic anti-self-dual planes. Using (4.4) we easily find that in this
signature $\boldsymbol{Z}^{i+}$ is the complex conjugate of
$\boldsymbol{Z}^{i-}$. Thus if $d\boldsymbol{Z}^{1+}=0$ then the bivector
$\boldsymbol{F}=\boldsymbol{Z}^{1+}+\boldsymbol{Z}^{1-}$ is real and
$d\boldsymbol{F}=d(\star\boldsymbol{F})=0$. Note also that $\boldsymbol{F}$
has the form $\boldsymbol{F}=\boldsymbol{l}\wedge\boldsymbol{e}$ with
$\boldsymbol{l}$ being a null vector field whereas $\boldsymbol{e}$ is space-
like and orthogonal to $\boldsymbol{l}$, so that $\boldsymbol{F}$ is a
bivector representing electromagnetic radiation, see section 3.3. Theorem 12
then guarantees that if the Weyl tensor is algebraically special then the
space-time admits a real solution for the Maxwell’s equations without
sources666$\boldsymbol{F}$ is not a solution for the Einstein-Maxwell
equations, since its energy-momentum tensor is different from zero. In other
words, $\boldsymbol{F}$ is just a test field. corresponding to electromagnetic
radiation. This is a classical result obtained by Robinson in [68], see
section 3.3. As a last comment note that theorem 11 is trivial on the
Lorentzian signature, since whenever $\mathcal{C}^{+}=0$ the whole Weyl tensor
must be identically zero, so that if the Ricci tensor is assumed to vanish
then space-time is flat.
##### 4.4.5 Split Signature
Now let us assume that the metric is real and has split signature. As
explicitly shown in section 4.1, in this case we have two kinds of null frame
[20]: (1) a real null frame $\\{\boldsymbol{E}^{\prime}_{a}\\}$, so that
$\boldsymbol{E}^{\prime}_{a}=\overline{\boldsymbol{E}^{\prime}_{a}}$; (2) a
complex null frame $\\{\boldsymbol{E}_{a}\\}$ such that
$\boldsymbol{E}_{3}=\overline{\boldsymbol{E}_{1}}$ and
$\boldsymbol{E}_{4}=\overline{\boldsymbol{E}_{2}}$. As shown on table 4.1, if
$\mathcal{C}^{+}$ is algebraically special then we can find a null frame in
which $\Psi_{0}^{+}=\Psi_{1}^{+}=0$, this null frame can then be either real
or complex. Let us work out these two cases separately.
Suppose that the frame in which the Weyl scalars $\Psi_{0}^{+}$ and
$\Psi_{1}^{+}$ vanish is real, then theorem 9 implies that the real isotropic
distribution generated by
$\\{\boldsymbol{E}^{\prime}_{1},\boldsymbol{E}^{\prime}_{2}\\}$ is integrable.
Moreover, if $\mathcal{C}^{+}$ is type $D$ then the real isotropic
distribution $\\{\boldsymbol{E}^{\prime}_{3},\boldsymbol{E}^{\prime}_{4}\\}$
will also be integrable, so that $\boldsymbol{J}$ is integrable. Since in this
case $\boldsymbol{J}$ and $\boldsymbol{\Omega}$ are complex it is useful to
define the real tensors $\boldsymbol{J}^{\prime}\equiv-i\boldsymbol{J}$ and
$\boldsymbol{\Omega}^{\prime}\equiv-i\boldsymbol{\Omega}$. Note that, seen as
an operator on the tangent bundle, $\boldsymbol{J}^{\prime}$ is such that
$\boldsymbol{J}^{\prime}\boldsymbol{J}^{\prime}=\boldsymbol{1}$ and
$\boldsymbol{g}\left(\boldsymbol{J}^{\prime}(\boldsymbol{V}),\boldsymbol{J}^{\prime}(\boldsymbol{X})\right)=-\boldsymbol{g}(\boldsymbol{V},\boldsymbol{X})$
for all vector fields $\boldsymbol{V}$ and $\boldsymbol{X}$. Hence the tensor
$\boldsymbol{J}^{\prime}$ is called a paracomplex structure, more details
about this kind of tensor in this context is available in [90].
Now let $\mathcal{C}^{+}$ be algebraically special and assume that the null
frame in which $\Psi_{0}^{+}=\Psi_{1}^{+}=0$ is not real. Then besides to the
isotropic distribution $\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2}\\}$, the
complex conjugate distribution $\\{\boldsymbol{E}_{3},\boldsymbol{E}_{4}\\}$
will also be integrable, so that the almost complex structure $\boldsymbol{J}$
is integrable. Note also that, since
$\boldsymbol{E}_{3}=\overline{\boldsymbol{E}_{1}}$ and
$\boldsymbol{E}_{4}=\overline{\boldsymbol{E}_{2}}$, the complex structure
$\boldsymbol{J}$ is real. Therefore, the Newlander-Nirenberg theorem
guarantees that the manifold over the complex field is an Hermitian manifold.
Moreover, theorem 11 implies that if $\mathcal{C}^{+}=0$ then
$\boldsymbol{\Omega}$ is covariantly constant. Particularly, in this case the
Kähler form is closed, $d\boldsymbol{\Omega}=0$, so that over the real field
the manifold is symplectic777A symplectic manifold is an even-dimensional
manifold endowed with a non-degenerate closed 2-form. In the present case
$\boldsymbol{\Omega}$ plays the role of a symplectic form., while over the
complex field it is a Calabi-Yau manifold. In general, the following theorem
can be stated [35]:
###### Theorem 15
Let $(M,\boldsymbol{g})$ be a Ricci-flat manifold of split signature. Then it
admits an integrable distribution of non-real isotropic planes if, and only
if, over the complex field such manifold is Hermitian. In addition, over the
complex field such manifold will be Calabi-Yau if, and only if,
$\mathcal{C}^{+}$(or $\mathcal{C}^{-}$) vanishes.
### Chapter 5 Six Dimensions Using Spinors
In the previous chapters it has been shown that the Petrov classification and
the Goldberg-Sachs (GS) theorem have played a prominent role in the
development of general relativity in 4 dimensions. With the increasing
interest on higher-dimensional manifolds, see section 1.9, it is quite natural
to try to develop an algebraic classification for the Weyl tensor valid in
dimensions greater than 4, as well as searching for a suitable generalization
of the GS theorem. As emphasized in chapter 2, there are several distinct but
equivalent paths to attain the Petrov classification, so one might be tempted
to arbitrarily choose one of these methods in order to define an algebraic
classification for the Weyl tensor in higher dimensions. However, it turns out
that such different approaches lead to inequivalent classifications when the
dimension is different from 4. Hence it is important to take a wise path.
Undoubtedly the most neat an elegant route toward Petrov classification in 4
dimensions is the spinorial approach. Therefore, in this chapter the spinors
will be used in order to define an algebraic classification for the Weyl
tensor valid in 6 dimensions. Furthermore, it will be shown that a
generalization of the GS theorem proved in [66, 67] can be nicely expressed by
means of the spinorial language. The material presented here is based on the
article [91]. After this paper the same issues were explored in [92] using
spinorial formalism in manifolds of arbitrary dimension. Some previous work on
spinors in six dimensions are available in [93], where the formalism has been
applied to quantum field theory. General aspects of spinors in even-
dimensional space-times were also used in [69].
Over the last decade there have been several attempts to provide suitable
higher-dimensional versions of the Petrov classification and GS theorem. In
[94] it was defined an algebraic classification for the Weyl tensor in 5
dimensions using spinors and some applications were made. An algebraic
classification for tensors in Lorentzian spaces of arbitrary dimension was
defined in [36], the so-called CMPP classification. Posterior work then
attempted, with partial success, to generalize the GS theorem using the CMPP
classification [58, 63, 64, 65]. Further, in [66, 67] it was put forward an
algebraic classification for the Weyl tensor based on maximally isotropic
structures. There it was also proved a higher-dimensional version of the
Goldberg-Sachs theorem stating that if the Weyl tensor obeys to certain
algebraic restrictions then the manifold admits an integrable maximally
isotropic distribution. Here it will be taken advantage of the spinorial
formalism in order to express such theorem in an elegant form. Finally, in
[70] it was defined a classification for the Weyl tensor valid in any
dimension that naturally generalizes the 4-dimensional bivector approach,
there it was also proved a generalization of the GS theorem.
#### 5.1 From Vectors to Spinors
In this section it will be shown how the low-rank tensors of a 6-dimensional
vector space are represented in the spinorial formalism. Particularly, the
isotropic subspaces will prove to be elegantly expressed in terms of spinors.
The reader is assumed to be familiar with the basics of spinorial formalism
and group representation theory, if this is not the case see appendices C and
D respectively.
Let us first start with the Euclidean vector space $\mathbb{R}^{6}$, later the
results of this case will be extrapolated to the space $\mathbb{C}^{6}$ in
order to obtain results valid in any signature. As explained on appendix C,
the universal covering group of $SO(n)$ is $SPin(\mathbb{R}^{n})$. More
precisely, the latter group is a double covering of the former,
$SPin(\mathbb{R}^{n})\sim SO(n)\otimes\mathbb{Z}_{2}$. In particular, it can
be proved that $SPin(\mathbb{R}^{6})\sim SU(4)$ [95]. Thus every tensor
transforming on a representation of $SO(6)$ can be said to be on a certain
representation of $SU(4)$, called the spinorial representation of this tensor.
In order to determine the spinorial equivalents for some $SO(6)$ tensors we
first need to study the irreducible representations of the group $SU(4)$.
Following the notation adopted on appendix D, the basic representations of
$SU(4)$ are:
$\textbf{4}:\;\;\chi^{A}\,\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\,U^{A}_{\phantom{A}B}\,\chi^{B}\;\;\;\;\;;\,\;\;\;\;\;\overline{\textbf{4}}:\;\;\gamma_{A}\,\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\,\overline{U}_{A}^{\phantom{A}B}\,\gamma_{B}\;.$ (5.1)
Where the indices $A,B,\ldots$ run from 1 to 4 and $U^{A}_{\phantom{A}B}$ is a
$4\times 4$ unitary matrix of unit determinant, with
$\overline{U}_{A}^{\phantom{A}B}$ being its complex conjugate. Since a unitary
matrix $U$ obeys to $(U^{-1})^{t}=\overline{U}$, it follows that the
representation $\overline{\boldsymbol{4}}$ is the inverse of the
representation 4, see eq. (D.2). In particular, this implies that
$\chi^{A}\gamma_{A}$ is invariant under the action of $SU(4)$. From now on we
shall call the objects transforming on the representation 4 the spinors of
positive chirality, while an object transforming on the representation
$\overline{\textbf{4}}$ is a spinor of negative chirality. Taking the complex
conjugate of eq. (5.1) we find that if $\chi^{A}$ is a spinor of positive
chirality then its complex conjugate, $\overline{\chi^{A}}$, will be a spinor
of negative chirality. Therefore we conclude that the complex conjugation
lowers the upper spinorial indices and raises the lower indices,
$\overline{\chi^{A}}=\overline{\chi}_{A}$ and
$\overline{\gamma_{A}}=\overline{\gamma}^{A}$. A list of the low-dimensional
irreducible representations of $SU(4)$ is shown on table 5.1. Since all
representations of this group can be constructed by means of the direct
products of the representation 4 and its inverse, $\overline{\textbf{4}}$, we
say that the fundamental representation of $SU(4)$ is 4.
$\boldsymbol{1}$ | $\boldsymbol{4}$ | $\overline{\boldsymbol{4}}$ | $\boldsymbol{6}$ | $\boldsymbol{4}\otimes\boldsymbol{4}$ | $=\,\boldsymbol{6}\oplus\boldsymbol{10}$
---|---|---|---|---|---
$\boldsymbol{10}$ | $\overline{\boldsymbol{10}}$ | $\boldsymbol{15}$ | $\boldsymbol{20}$ | $\boldsymbol{4}\otimes\overline{\boldsymbol{4}}$ | $=\,\boldsymbol{1}\oplus\boldsymbol{15}$
$\overline{\boldsymbol{20}}$ | $\boldsymbol{20^{\prime}}$ | $\boldsymbol{20^{\prime\prime}}$ | $\overline{\boldsymbol{20^{\prime\prime}}}$ | $\boldsymbol{6}\otimes\boldsymbol{6}$ | $=\,\boldsymbol{1}\oplus\boldsymbol{15}\oplus\boldsymbol{20^{\prime}}$
$\boldsymbol{35}$ | $\overline{\boldsymbol{35}}$ | $\boldsymbol{36}$ | $\overline{\boldsymbol{36}}$ | $\boldsymbol{10}\otimes\boldsymbol{6}$ | $=\,\boldsymbol{15}\oplus\boldsymbol{45}$
$\boldsymbol{45}$ | $\overline{\boldsymbol{45}}$ | $\boldsymbol{50}$ | $\boldsymbol{56}$ | $\boldsymbol{10}\otimes\overline{\boldsymbol{10}}$ | $=\,\boldsymbol{1}\oplus\boldsymbol{15}\oplus\boldsymbol{84}$
$\overline{\boldsymbol{56}}$ | $\boldsymbol{60}$ | $\overline{\boldsymbol{60}}$ | $\boldsymbol{64}$ | $\boldsymbol{15}\otimes\boldsymbol{6}$ | $=\,\boldsymbol{6}\oplus\boldsymbol{10}\oplus\overline{\boldsymbol{10}}\oplus\boldsymbol{64}$
$\boldsymbol{70}$ | $\overline{\boldsymbol{70}}$ | $\boldsymbol{84}$ | $\boldsymbol{84^{\prime}}$ | $\boldsymbol{20^{\prime}}\otimes\boldsymbol{6}$ | $=\,\boldsymbol{6}\oplus\boldsymbol{50}\oplus\boldsymbol{64}$
$\overline{\boldsymbol{84^{\prime}}}$ | $\boldsymbol{84^{\prime\prime}}$ | $\overline{\boldsymbol{84^{\prime\prime}}}$ | $\boldsymbol{105}$ | $\boldsymbol{15}\otimes\boldsymbol{15}$ | $=\boldsymbol{1}\oplus\boldsymbol{15}\oplus\boldsymbol{15}\oplus\boldsymbol{20^{\prime}}\oplus\boldsymbol{45}\oplus\overline{\boldsymbol{45}}\oplus\boldsymbol{84}$
Table 5.1: On the left hand side of this table we have a list of all
irreducible representations of the group $SU(4)$ with dimension less than
$120$. In this list the inequivalent representations of the same dimension are
distinguished by primes. Note that the representations $\boldsymbol{1}$,
$\boldsymbol{6}$, $\boldsymbol{15}$, $\boldsymbol{20^{\prime}}$,
$\boldsymbol{50}$, $\boldsymbol{64}$, $\boldsymbol{84}$ and $\boldsymbol{105}$
are real. Thus, for example, $\overline{\boldsymbol{15}}=\boldsymbol{15}$. On
the right hand side of this table we have the decomposition in irreducible
parts of some direct products of the irreducible representations [96].
Now let us see how the tensors of $SO(6)$ transform under $SU(4)$. A vector of
$\mathbb{R}^{6}$, $V^{\mu}$, has six degrees of freedom and, therefore, might
be on a non-trivial six-dimensional and real representation of $SU(4)$, which
according to table 5.1 is unique, $\boldsymbol{6}$. The same table says that
this representation can be obtained by decomposing the direct product
$\boldsymbol{4}\otimes\boldsymbol{4}$ in irreducible parts. Indeed, if
$D^{AB}$ is on the representation $\boldsymbol{4}\otimes\boldsymbol{4}$ then
we can split it in two irreducible parts (see appendix D):
$\underbrace{D^{AB}}_{\boldsymbol{4}\otimes\boldsymbol{4}}\;\longrightarrow\;\;\underbrace{D^{[AB]}}_{\boldsymbol{6}}\quad+\quad\underbrace{D^{(AB)}}_{\boldsymbol{10}}\,.$
(5.2)
Thus a vector $V^{\mu}$ transforms as an object of the form $V^{AB}=V^{[AB]}$.
Another representation of dimension 6 could be provided by $V_{AB}=V_{[AB]}$,
let us denote such representation by $\overline{\boldsymbol{6}}$. However, it
is not hard to verify this representation is, actually, equivalent to the
representation $\boldsymbol{6}$. Indeed, let
$\varepsilon_{ABCD}=\varepsilon_{[ABCD]}$ be the unique completely anti-
symmetric symbol such that $\varepsilon_{1234}=1$. Then its contraction with
four arbitrary spinors, $\zeta^{A},\chi^{A},\varphi^{A}$ and $\xi^{A}$, is
invariant under $SU(4)$:
$\varepsilon_{ABCD}\zeta^{A}\chi^{B}\varphi^{C}\xi^{D}\,\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\,\det(U)\,\varepsilon_{EFGH}\zeta^{E}\chi^{F}\varphi^{G}\xi^{H}=\varepsilon_{ABCD}\zeta^{A}\chi^{B}\varphi^{C}\xi^{D}.$
(5.3)
In the same fashion we can define the object
$\varepsilon^{ABCD}=\varepsilon^{[ABCD]}$ with $\varepsilon^{1234}=1$ and
verify that an analogous relation holds for spinors of negative chirality.
Thus if $V^{AB}$ is on the representation $\boldsymbol{6}$ then, in order for
the combination $V^{AB}\varepsilon_{ABCD}V^{CD}$ be invariant under $SU(4)$,
the object $\varepsilon_{ABCD}V^{CD}$ must be on the inverse representation,
$\overline{\boldsymbol{6}}$. So that we can define:
$V_{AB}\,\equiv\,\frac{1}{2}\varepsilon_{ABCD}V^{CD}\;\,;\,\;V^{AB}\,\equiv\,\frac{1}{2}\varepsilon^{ABCD}V_{CD}\,.$
(5.4)
Since the representation $\overline{\boldsymbol{6}}$ can be obtained from the
representation $\boldsymbol{6}$ by a simple algebraic operation not involving
complex conjugation it follows that these representations are actually
equivalent, $\boldsymbol{6}=\overline{\boldsymbol{6}}$. Because of this we
might say that this representation is real. Thus in six dimensions we can
raise or low a skew-symmetric pair of indices without changing the
representation.
A bivector $B_{\mu\nu}=-B_{\nu\mu}$ in 6 dimensions has 15 degrees of freedom
and, therefore, must be in a $15$-dimensional and real representation of
$SU(4)$. According to table 5.1 both criteria are satisfied by the
representation $\boldsymbol{15}$. The identity
$\boldsymbol{4}\otimes\overline{\boldsymbol{4}}=\boldsymbol{1}\oplus\boldsymbol{15}$
says that this representation is given by the objects of the form
$B^{A}_{\phantom{A}B}$ with vanishing trace, $B^{A}_{\phantom{A}A}=0$. The
reality of this representation can be understood by the fact that when we take
the complex conjugate of $B^{A}_{\phantom{A}B}$ we obtain another trace-less
object with one index up and one down.
If $S_{\mu\nu}=S_{(\mu\nu)}$ is a trace-less symmetric tensor on
$\mathbb{R}^{6}$ then it has $20$ independent components. Since it has two
indices, it follows that from the $SO(6)$ point of view this tensor is
obtained by the direct product of two vectorial representations. Therefore its
spinorial equivalent might be contained on the direct product
$\boldsymbol{6}\otimes\boldsymbol{6}$. Table 5.1 furnish that
$\boldsymbol{6}\otimes\boldsymbol{6}=\boldsymbol{1}\oplus\boldsymbol{15}\oplus\boldsymbol{20^{\prime}}$,
so that the spinorial equivalent of $S_{\mu\nu}$ might be on the
representation $\boldsymbol{20^{\prime}}$, which has the form
$S^{AB}_{\phantom{AB}CD}=S^{[AB]}_{\phantom{AB}[CD]}$ with vanishing trace,
$S^{AB}_{\phantom{AB}CB}=0$. Note that this representation is real.
In six dimensions a 3-vector $T_{\mu\nu\rho}=T_{[\mu\nu\rho]}$ has 20 degrees
of freedom and can be obtained by the anti-symmetrization of the direct
product of a bivector and a vector. Therefore its spinorial equivalent must be
contained on the direct product $\boldsymbol{15}\otimes\boldsymbol{6}$. By
means of table 5.1 we have
$\boldsymbol{15}\otimes\boldsymbol{6}=\boldsymbol{6}\oplus\boldsymbol{10}\oplus\overline{\boldsymbol{10}}\oplus\boldsymbol{64}$.
Thus we conclude that the 3-vectors are on the representation
$\boldsymbol{10}\oplus\overline{\boldsymbol{10}}$ of $SU(4)$. From the eq.
(5.2) we see that the representation $\boldsymbol{10}$ is given by
$T^{AB}=T^{(AB)}$. So in the spinorial language a 3-vector $T_{\mu\nu\rho}$ is
represented by a pair $(T^{AB},\tilde{T}_{AB})$ of symmetric objects. It is
possible to prove that if $\tilde{T}_{AB}=0$ then the 3-vector is self-dual,
$\star\boldsymbol{T}=\boldsymbol{T}$. Analogously, whenever $T^{AB}=0$ the
3-vector is anti-self-dual, $\star\boldsymbol{T}=-\boldsymbol{T}$.
The Weyl tensor $C_{\mu\nu\rho\sigma}$ is a trace-less object with the
symmetries $C_{\mu\nu\rho\sigma}=C_{[\mu\nu][\rho\sigma]}$ and
$C_{\mu[\nu\rho\sigma]}=0$. It can be proved that in 6 dimensions it has 84
independent components. From the first symmetry we see that this tensor can be
obtained by a linear combination of the direct product of bivectors, so that
its spinorial representation must be contained in
$\boldsymbol{15}\otimes\boldsymbol{15}$. Looking at the expansion of this
direct product on table 5.1 we see that $C_{\mu\nu\rho\sigma}$ must be on the
representation $\boldsymbol{84}$ of $SU(4)$. Because of the equation
$\boldsymbol{10}\otimes\overline{\boldsymbol{10}}=\boldsymbol{1}\oplus\boldsymbol{15}\oplus\boldsymbol{84}$
we conclude that an object in this representation have the form
$\Psi^{AB}_{\phantom{AB}CD}$ with
$\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)}$ and
$\Psi^{AB}_{\phantom{AB}CB}=0$. The results obtained so far are summarized on
table 5.2 [91].
$SO(6)$ Tensor | Spinorial Representation | Symmetries
---|---|---
$V^{\mu}$ | $\boldsymbol{6}\rightarrow$ $V^{AB}$ | $V^{AB}=-V^{BA}$
$B_{\mu\nu}$ | $\boldsymbol{15}\rightarrow$ $B^{A}_{\phantom{A}B}$ | $B^{A}_{\phantom{A}A}=0$
$S_{\mu\nu}$ | $\boldsymbol{20^{\prime}}\rightarrow$ $S^{AB}_{\phantom{AB}CD}$ | $S^{AB}_{\phantom{AB}CD}=S^{[AB]}_{\phantom{AB}[CD]},\,S^{AB}_{\phantom{AB}CB}=0$
$T_{\mu\nu\rho}$ | $\boldsymbol{10}\oplus\overline{\boldsymbol{10}}\rightarrow$ $(T^{AB},\tilde{T}_{AB})$ | $T^{AB}=T^{BA},\,\tilde{T}_{AB}=\tilde{T}_{BA}$
$C_{\mu\nu\rho\sigma}$ | $\boldsymbol{84}\rightarrow$ $\Psi^{AB}_{\phantom{AB}CD}$ | $\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)},\,\Psi^{AB}_{\phantom{AB}CB}=0$
Table 5.2: Spinorial equivalent of some low rank $SO(6;\mathbb{R})$ tensors.
$V^{\mu}$ is a vector, $S_{\mu\nu}=S_{(\mu\nu)}$ is a trace-less symmetric
tensor, $B_{\mu\nu}=B_{[\mu\nu]}$ is a bivector,
$T_{\mu\nu\rho}=T_{[\mu\nu\rho]}$ is a 3-vector and $C_{\mu\nu\rho\sigma}$ is
a tensor with the symmetries of a Weyl tensor. Note that all these
representations are real.
##### 5.1.1 A Null Frame
Let $V^{\mu}$ and $K^{\mu}$ be two vectors of $\mathbb{R}^{6}$, then the inner
product $\boldsymbol{g}(\boldsymbol{V},\boldsymbol{K})=V^{\mu}K_{\mu}$ is the
only scalar, up to a multiplicative factor, that is invariant under $SO(6)$
and is linear on both vectors. Denoting by $V^{AB}$ and $K^{AB}$ the spinorial
equivalents of these vectors then it follows from equation (5.3) that the
scalar $\varepsilon_{ABCD}V^{AB}K^{CD}$ is invariant under $SU(4)$ and, hence,
invariant under $SO(6)$. Since such scalar is also linear in $\boldsymbol{V}$
and $\boldsymbol{K}$ it follows that it must be a multiple of the inner
product $V^{\mu}K_{\mu}$. Because of equation (5.4) one conclude that this
multiplicative factor might be $2$:
$V^{\mu}\,K_{\mu}\,=\,\frac{1}{2}\,\varepsilon_{ABCD}V^{AB}K^{CD}\,=\,V^{AB}K_{AB}\,.$
(5.5)
Now let $\\{\chi_{1}^{\,A},\chi_{2}^{\,A},\chi_{3}^{\,A},\chi_{4}^{\,A}\\}$ be
a basis for the space of positive chirality spinors obeying to the following
normalization condition:
$\varepsilon_{ABCD}\,\chi_{1}^{\,A}\chi_{2}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}\,=\,1\,.$
(5.6)
Note, in particular, that the choice $\chi_{p}^{\,A}=\delta_{p}^{\,A}$ satisfy
this constraint. We can use the basis $\\{\chi_{p}^{\,A}\\}$ in order to
define a dual basis for the space of spinors with negative chirality:
$\displaystyle\gamma^{1}_{\,A}=\varepsilon_{ABCD}\,\chi_{2}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}\;\;;$
$\displaystyle\;\;\gamma^{2}_{\,A}=-\,\varepsilon_{ABCD}\,\chi_{1}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}$
$\displaystyle\gamma^{3}_{\,A}=\varepsilon_{ABCD}\,\chi_{1}^{\,B}\chi_{2}^{\,C}\chi_{4}^{\,D}\;\;;$
$\displaystyle\;\;\gamma^{4}_{\,A}=-\,\varepsilon_{ABCD}\,\chi_{1}^{\,B}\chi_{2}^{\,C}\chi_{3}^{\,D}$
It is simple matter to verify that the relation
$\chi_{p}^{\,A}\gamma^{q}_{\,A}=\delta^{\,q}_{p}$ holds. Then we can define
the following frame of vectors, objects on the representation
$\boldsymbol{6}$:
$\displaystyle
e_{1}^{\,AB}=\chi_{1}^{\,[A}\chi_{2}^{\,B]}\;;\;\;e_{2}^{\,AB}=\chi_{1}^{\,[A}\chi_{3}^{\,B]}\;;\;\;e_{3}^{\,AB}=\chi_{1}^{\,[A}\chi_{4}^{\,B]}$
$\displaystyle\theta^{1\,AB}=\chi_{3}^{\,[A}\chi_{4}^{\,B]}\;;\;\;\theta^{2\,AB}=\chi_{4}^{\,[A}\chi_{2}^{\,B]}\;;\;\;\theta^{3\,AB}=\chi_{2}^{\,[A}\chi_{3}^{\,B]}\,.$
(5.7)
By means of equation (5.4) one can lower these pairs of skew-symmetric indices
yielding:
$\displaystyle
e_{1\,AB}=\gamma^{3}_{\,[A}\gamma^{4}_{\,B]}\;;\;\;e_{2\,AB}=\gamma^{4}_{\,[A}\gamma^{2}_{\,B]}\;;\;\;e_{3\,AB}=\gamma^{2}_{\,[A}\gamma^{3}_{\,B]}$
$\displaystyle\theta^{1}_{\,AB}=\gamma^{1}_{\,[A}\gamma^{2}_{\,B]}\;;\;\;\theta^{2}_{\,AB}=\gamma^{1}_{\,[A}\gamma^{3}_{\,B]}\;;\;\;\theta^{3}_{\,AB}=\gamma^{1}_{\,[A}\gamma^{4}_{\,B]}\,.$
(5.8)
Thus using equations (5.5), (5.7) and (5.8) we easily find that the inner
products of the frame vectors are:
$e_{a^{\prime}}^{\,\,\mu}\,\,e_{b^{\prime}\,\mu}\,=\,\theta^{a^{\prime}\,\mu}\,\theta^{b^{\prime}}_{\,\,\mu}\,=\,0\quad;\quad
e_{a^{\prime}}^{\,\,\mu}\,\theta^{b^{\prime}}_{\,\,\mu}\,=\,\frac{1}{2}\,\delta^{\,\,b^{\prime}}_{a^{\prime}}\,.$
(5.9)
In particular, all vectors of the frame
$\\{\boldsymbol{e}_{a^{\prime}},\boldsymbol{\theta}^{b^{\prime}}\\}$ are
null111Throughout this chapter the following index conventions will be
adopted: $A,B,C,\ldots$ are the spinorial indices and pertain to
$\\{1,2,3,4\\}$; $\mu,\nu,\rho,\ldots$ are coordinate indices of
$\mathbb{R}^{6}$, pertaining to $\\{1,2,\ldots,6\\}$; $a,b,c,\ldots$ are
labels for a null frame of $\mathbb{C}^{6}$ and take the values
$\\{1,2,\ldots,6\\}$; $a^{\prime},b^{\prime},c^{\prime}$ pertain to
$\\{1,2,3\\}$; $p,q$ label a basis of Weyl spinors and pertain to
$\\{1,2,3,4\\}$; $r,s$ label a basis of (anti-)self-dual 3-vectors, running
from 1 to 10.. For later convenience we shall denote such frame by
$\\{\boldsymbol{e}_{a}\\}$ with $\boldsymbol{e}_{4}=\boldsymbol{\theta^{1}}$,
$\boldsymbol{e}_{5}=\boldsymbol{\theta^{2}}$ and
$\boldsymbol{e}_{6}=\boldsymbol{\theta^{3}}$, or shortly
$\boldsymbol{e}_{a^{\prime}+3}=\boldsymbol{\theta^{a^{\prime}}}$. From now on,
a frame of vectors $\\{\boldsymbol{e}_{a}\\}$ in a 6-dimensional space obeying
to eq. (5.9) will be called a null frame. Defining
$g_{ab}\equiv\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})$ we have
$g_{14}=g_{25}=g_{36}=\frac{1}{2}$ while the other components vanish. Using
equations (5.7) and (5.8) it is straightforward to prove the following
relation:
$e_{a}^{\,AB}\,e_{b\,CB}\,+\,e_{b}^{\,AB}\,e_{a\,CB}\,=\,\frac{1}{2}\,g_{ab}\,\delta_{C}^{\,A}\,.$
(5.10)
Equation (5.7) enables us to find explicitly the spinorial equivalent of any
vector in a vector space of 6 dimensions. More precisely, if
$\\{\boldsymbol{e}_{a}\\}$ is a null frame on this space and $\boldsymbol{V}$
is a vector then:
$\boldsymbol{V}\,=\,V^{a}\,\boldsymbol{e}_{a}\quad\Longleftrightarrow\quad
V^{AB}\,=\,V^{a}\,e_{a}^{\,AB}\,.$ (5.11)
Where $V^{a}$ are the components of the vector $\boldsymbol{V}$ on this null
frame and $e_{a}^{\,AB}$ are the objects defined on equation (5.7). Actually,
equation (5.11) teaches us how to convert any tensor to the spinorial
language. For example, if $\boldsymbol{F}$ is a tensor of rank 2 then its
spinorial image will be contained on the representation
$\boldsymbol{6}\otimes\boldsymbol{6}$ and can be written in this formalism as
$F^{AB\,CD}=F^{[AB]\,[CD]}$ defined by:
$F^{AB\,CD}\,=\,F^{ab}\,e_{a}^{\,AB}e_{b}^{\,CD}\quad\Longleftrightarrow\quad\boldsymbol{F}\,=\,F^{ab}\,\boldsymbol{e}_{a}\otimes\boldsymbol{e}_{b}\,.$
(5.12)
In particular, if $S_{\mu\nu}$ is a symmetric and trace-less tensor then its
spinorial equivalent can be written as:
$S^{AB}_{\phantom{AB}CD}\,=\,S^{ab}\,e_{a}^{\,AB}\,e_{b\,CD}\,.$
Note that using equation (5.10) one can easily see that
$S^{AB}_{\phantom{AB}CB}=0$, which agrees with table 5.2. In the same vein, if
$B_{ab}$ is a bivector then its spinorial equivalent is:
$B^{AB\,CD}\,=\,B^{ab}\,e_{a}^{\,AB}\,e_{b}^{\,CD}\,.$
However, this does not seem to agree with table 5.2, since there the bivector
is said to be represented by an object of the form $B^{A}_{\phantom{A}B}$ with
vanishing trace. But after some algebra it can be proved that the following
relation holds:
$\left\\{\begin{array}[]{cl}B^{AB\,CD}&=\,B^{[A}_{\phantom{A}E}\,\,\varepsilon^{B]ECD}-B^{[C}_{\phantom{C}E}\,\,\varepsilon^{D]EAB}\\\
B^{A}_{\phantom{A}B}&\equiv\,\frac{1}{4}\,B^{AC\,DE}\,\varepsilon_{CDEB}\,=\,\frac{1}{2}\,B^{AC}_{\phantom{AC}CB}\,.\end{array}\right.$
(5.13)
So all degrees of freedom of $B^{AB\,CD}$ are contained on the trace-less
object $B^{A}_{\phantom{A}B}$. That is the beauty of representation theory, by
means of it one can anticipate how the degrees of freedom of a tensor are
stored. Following the same reasoning, if $T_{abc}=T_{[abc]}$ is a 3-vector
then its spinorial equivalent will be of the form $T^{AB\,CD\,EF}$,
analogously to eq. (5.12). Nonetheless, according to table 5.2 the degrees of
freedom of this tensor must be contained on a pair $(T^{AB},\tilde{T}_{AB})$
such that $T^{AB}=T^{(AB)}$ and $\tilde{T}_{AB}=\tilde{T}_{(AB)}$. By lack of
any other option one can assure that $T^{AB}\propto
T^{AC\,BD}_{\phantom{AC\,BD}CD}$ and $\tilde{T}_{AB}\propto
T_{AC\,BD}^{\phantom{AC\,BD}CD}$. In order to agree with the notation of [91]
we might choose the proportionality constants to be $1/9$ and $-1/9$
respectively. So we can schematically write [91]:
$T_{abc}=T_{[abc]}\;\Leftrightarrow\;T^{AB\,CD\,EF}\;\Leftrightarrow\;(T^{AB},\tilde{T}_{AB})\equiv\frac{1}{9}(T^{AC\,BD}_{\phantom{AC\,BD}CD},-T_{AC\,BD}^{\phantom{AC\,BD}CD})\,.$
In a similar fashion, if $C_{abcd}$ is a tensor with the symmetries of a Weyl
tensor then table 5.2 says that its degrees of freedom are stored in an object
$\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)}$ with vanishing
trace. By lack of any other possibility this object must be a multiple of
$C^{AF\phantom{CF\,GD}BG}_{\phantom{AF}CF\,GD}$, so that one can write [91]:
$C_{abcd}\;\;\Leftrightarrow\;\;C^{AB\,CD\,EF\,GH}\;\;\Leftrightarrow\;\;\Psi^{AB}_{\phantom{AB}CD}\equiv\frac{1}{16}\,C^{AF\phantom{CF\,GD}BG}_{\phantom{AF}CF\,GD}\,.$
(5.14)
Let $\\{\boldsymbol{e}_{a}\\}$ be a null frame, then using equations (5.7),
(5.8) and (5.13) it is straightforward to find the spinorial equivalents of
the bivectors
$\boldsymbol{e}_{a}\wedge\boldsymbol{e}_{b}\equiv({\boldsymbol{e}_{a}\otimes\boldsymbol{e}_{b}}-\boldsymbol{e}_{b}\otimes\boldsymbol{e}_{a})$,
this is summarized on table 5.3. Analogously, the relation between the Weyl
tensor components on a null frame and the components of the object
$\Psi^{AB}_{\phantom{AB}CD}$ can be obtained, after a lot of algebra, by means
of equations (5.7), (5.8) and (5.14), the bottom line is shown on table 5.4.
$(e_{1}\wedge e_{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{1}^{\,A}\gamma^{4}_{\,B}$ | $(e_{1}\wedge e_{3})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{1}^{\,A}\gamma^{3}_{\,B}$ | $(e_{1}\wedge\theta^{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{2}^{\,A}\gamma^{3}_{\,B}$
---|---|---
$(e_{1}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{2}^{\,A}\gamma^{4}_{\,B}$ | $(e_{2}\wedge e_{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{1}^{\,A}\gamma^{2}_{\,B}$ | $(e_{2}\wedge\theta^{1})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\gamma^{2}_{\,B}$
$(e_{2}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\gamma^{4}_{\,B}$ | $(e_{3}\wedge\theta^{1})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{4}^{\,A}\gamma^{2}_{\,B}$ | $(e_{3}\wedge\theta^{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{4}^{\,A}\gamma^{3}_{\,B}$
$(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\gamma^{1}_{\,B}-\chi_{2}^{\,A}\gamma^{2}_{\,B}+\chi_{3}^{\,A}\gamma^{3}_{\,B}+\chi_{4}^{\,A}\gamma^{4}_{\,B}]$ | $(\theta^{1}\wedge\theta^{2})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{4}^{\,A}\gamma^{1}_{\,B}$
$(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\gamma^{1}_{\,B}+\chi_{2}^{\,A}\gamma^{2}_{\,B}-\chi_{3}^{\,A}\gamma^{3}_{\,B}+\chi_{4}^{\,A}\gamma^{4}_{\,B}]$ | $(\theta^{1}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\gamma^{1}_{\,B}$
$(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\gamma^{1}_{\,B}+\chi_{2}^{\,A}\gamma^{2}_{\,B}+\chi_{3}^{\,A}\gamma^{3}_{\,B}-\chi_{4}^{\,A}\gamma^{4}_{\,B}]$ | $(\theta^{2}\wedge\theta^{3})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{2}^{\,A}\gamma^{1}_{\,B}$
Table 5.3: The spinorial representation of a basis of bivectors [91].
$C_{1212}=4\Psi^{44}_{\phantom{44}11}$ | $C_{1213}=-4\Psi^{34}_{\phantom{34}11}$ | $C_{1215}=4\Psi^{34}_{\phantom{34}12}$ | $C_{1216}=4\Psi^{44}_{\phantom{44}12}$
---|---|---|---
$C_{1313}=4\Psi^{33}_{\phantom{33}11}$ | $C_{1315}=-4\Psi^{33}_{\phantom{33}12}$ | $C_{1316}=-4\Psi^{34}_{\phantom{34}12}$ | $C_{1515}=4\Psi^{33}_{\phantom{33}22}$
$C_{1516}=4\Psi^{34}_{\phantom{34}32}$ | $C_{1616}=4\Psi^{44}_{\phantom{44}22}$ | $C_{1645}=-4\Psi^{14}_{\phantom{14}24}$ | $C_{1646}=4\Psi^{14}_{\phantom{14}23}$
$C_{2323}=4\Psi^{22}_{\phantom{22}11}$ | $C_{2326}=4\Psi^{24}_{\phantom{24}13}$ | $C_{2335}=4\Psi^{23}_{\phantom{23}14}$ | $C_{2356}=-4\Psi^{12}_{\phantom{12}12}$
$C_{5656}=4\Psi^{11}_{\phantom{11}22}$ | $C_{2626}=4\Psi^{44}_{\phantom{44}33}$ | $C_{2635}=4\Psi^{34}_{\phantom{34}34}$ | $C_{2656}=-4\Psi^{14}_{\phantom{14}23}$
$C_{3535}=4\Psi^{33}_{\phantom{33}44}$ | $C_{3556}=-4\Psi^{13}_{\phantom{13}24}$ | $C_{1242}=-4\Psi^{24}_{\phantom{24}13}$ | $C_{1243}=-4\Psi^{24}_{\phantom{24}14}$
$C_{1245}=-4\Psi^{14}_{\phantom{14}14}$ | $C_{1246}=4\Psi^{14}_{\phantom{14}13}$ | $C_{1342}=4\Psi^{23}_{\phantom{23}13}$ | $C_{1343}=4\Psi^{23}_{\phantom{23}14}$
$C_{1345}=4\Psi^{13}_{\phantom{13}14}$ | $C_{1346}=-4\Psi^{13}_{\phantom{13}13}$ | $C_{1542}=-4\Psi^{23}_{\phantom{23}23}$ | $C_{1543}=-4\Psi^{23}_{\phantom{23}24}$
$C_{1545}=-4\Psi^{13}_{\phantom{13}24}$ | $C_{1546}=4\Psi^{13}_{\phantom{13}23}$ | $C_{1642}=-4\Psi^{24}_{\phantom{24}23}$ | $C_{1643}=-4\Psi^{24}_{\phantom{24}24}$
$C_{1223}=4\Psi^{24}_{\phantom{24}11}$ | $C_{1226}=4\Psi^{44}_{\phantom{44}13}$ | $C_{1235}=4\Psi^{34}_{\phantom{34}14}$ | $C_{1256}=-4\Psi^{14}_{\phantom{14}12}$
---|---|---|---
$C_{1323}=-4\Psi^{23}_{\phantom{23}11}$ | $C_{1326}=-4\Psi^{43}_{\phantom{43}13}$ | $C_{1335}=-4\Psi^{33}_{\phantom{33}14}$ | $C_{1356}=4\Psi^{13}_{\phantom{13}12}$
$C_{1523}=4\Psi^{32}_{\phantom{32}12}$ | $C_{1526}=4\Psi^{34}_{\phantom{34}23}$ | $C_{1535}=4\Psi^{33}_{\phantom{33}24}$ | $C_{1556}=-4\Psi^{13}_{\phantom{13}22}$
$C_{1623}=4\Psi^{24}_{\phantom{24}21}$ | $C_{1626}=4\Psi^{44}_{\phantom{44}23}$ | $C_{1635}=4\Psi^{34}_{\phantom{34}24}$ | $C_{1656}=-4\Psi^{14}_{\phantom{14}22}$
$C_{1414}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{22}_{\phantom{22}22}+2\Psi^{12}_{\phantom{12}12})$ | $C_{1425}=4(\Psi^{23}_{\phantom{23}23}-\Psi^{14}_{\phantom{14}14})$
---|---
$C_{2525}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{33}_{\phantom{33}33}+2\Psi^{13}_{\phantom{13}13})$ | $C_{1436}=4(\Psi^{24}_{\phantom{24}24}-\Psi^{13}_{\phantom{13}13})$
$C_{3636}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{44}_{\phantom{44}44}+2\Psi^{14}_{\phantom{14}14})$ | $C_{2536}=4(\Psi^{34}_{\phantom{34}34}-\Psi^{12}_{\phantom{12}12})$
$C_{1225}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{34}_{\phantom{34}31})$ | $C_{1236}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{44}_{\phantom{44}41})$ | $C_{1325}=4(\Psi^{23}_{\phantom{23}21}+\Psi^{43}_{\phantom{43}41})$
---|---|---
$C_{1336}=4(\Psi^{23}_{\phantom{23}21}+\Psi^{33}_{\phantom{33}31})$ | $C_{1525}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{33}_{\phantom{33}32})$ | $C_{1536}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{43}_{\phantom{43}42})$
$C_{1625}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{34}_{\phantom{34}32})$ | $C_{1636}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{44}_{\phantom{44}42})$ | $C_{1412}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{24}_{\phantom{24}21})$
$C_{1413}=4(\Psi^{33}_{\phantom{33}31}+\Psi^{43}_{\phantom{43}41})$ | $C_{1415}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{23}_{\phantom{23}22})$ | $C_{1416}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{24}_{\phantom{24}22})$
$C_{1423}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{22}_{\phantom{22}21})$ | $C_{1426}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{24}_{\phantom{24}23})$ | $C_{1435}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{23}_{\phantom{23}24})$
---|---|---
$C_{1456}=4(\Psi^{31}_{\phantom{31}32}+\Psi^{41}_{\phantom{41}42})$ | $C_{2523}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{23}_{\phantom{23}13})$ | $C_{3623}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{24}_{\phantom{24}14})$
$C_{2526}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{34}_{\phantom{34}33})$ | $C_{3626}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{44}_{\phantom{44}34})$ | $C_{2535}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{33}_{\phantom{33}34})$
$C_{3635}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{34}_{\phantom{34}44})$ | $C_{2556}=4(\Psi^{12}_{\phantom{12}22}+\Psi^{14}_{\phantom{14}24})$ | $C_{3656}=4(\Psi^{12}_{\phantom{12}22}+\Psi^{13}_{\phantom{13}23})$
Table 5.4: This table displays the relation between Weyl tensor’s components
in a null frame and its spinorial equivalents [91]. The missing components of
the Weyl tensor can be obtained by making the changes $1\leftrightarrow 4$,
$2\leftrightarrow 5$ and $3\leftrightarrow 6$ on the vectorial indices while
performing the transformation
$\Psi^{AB}_{\;\;\;\;CD}\mapsto\Psi^{CD}_{\;\;\;\;AB}$. Thus, for example, the
relation $C_{1212}=4\Psi^{44}_{\;\;\;11}$ implies
$C_{4545}=4\Psi^{11}_{\;\;\;44}$.
##### 5.1.2 Clifford Algebra in 6 Dimensions
The aim of this subsection is to provide a connection between the spinorial
calculus introduced so far and the abstract formalism presented on appendix C.
Let us denote the 4-dimensional vector space spanned by the spinors
$\\{\chi_{1}^{\,A},\chi_{2}^{\,A},\chi_{3}^{\,A},\chi_{4}^{\,A}\\}$ by $S^{+}$
and call it the space of positive chirality Weyl spinors. Analogously, the
4-dimensional space spanned by
$\\{\gamma^{1}_{\,A},\gamma^{2}_{\,A},\gamma^{3}_{\,A},\gamma^{4}_{\,A}\\}$
will be denoted by $S^{-}$ and called the space of Weyl spinors with negative
chirality. The vector space $S=S^{+}\oplus S^{-}$ is then named the space of
Dirac spinors, so that a Dirac spinor $\boldsymbol{\psi}\in S$ is generally
written as $\boldsymbol{\psi}=\psi^{A}+\tilde{\psi}_{A}$. Let us define the
inner product of two Dirac spinors by:
$(\boldsymbol{\psi},\boldsymbol{\phi})\,=\,\psi^{A}\,\tilde{\phi}_{A}-\phi^{A}\,\tilde{\psi}_{A}\,.$
(5.15)
Note that this inner product is skew-symmetric and vanishes if the two spinors
$\boldsymbol{\psi}$ and $\boldsymbol{\phi}$ have the same chirality, as said
on appendix C.
On the Clifford algebra formalism the vectors of $V=\mathbb{R}^{6}$ are linear
operators that act on the space of spinors. Therefore, to each vector
$\boldsymbol{e}_{a}$ it is associated a linear operator
$\check{\boldsymbol{e}}_{a}:S\rightarrow S$ acting on the space of Dirac
spinors. The action of this operator is defined by:
$\check{\boldsymbol{e}}_{a}(\boldsymbol{\psi})\,=\,\boldsymbol{\phi}\,\quad\Longleftrightarrow\quad\phi^{A}=2\,e_{a}^{\,AB}\,\tilde{\psi}_{B}\;\,\textrm{
and }\,\;\tilde{\phi}_{A}=-2\,e_{a\,AB}\,\psi^{B}\,.$ (5.16)
In order to verify that this action is correct note that using equations
(5.10) and (5.16) we arrive at the following important relation:
$\check{\boldsymbol{e}}_{a}\,\check{\boldsymbol{e}}_{b}\,+\,\check{\boldsymbol{e}}_{b}\,\check{\boldsymbol{e}}_{a}\,=\,2\,g_{ab}\,\boldsymbol{1}\,,$
where $\boldsymbol{1}$ is the identity operator on $S$. Such relation is the
analogous of equation C.1 on appendix C. Note also that the inner product
defined on (5.15) is such that
$(\check{\boldsymbol{e}}_{a}(\boldsymbol{\psi}),\boldsymbol{\phi})=(\boldsymbol{\psi},\check{\boldsymbol{e}}_{a}(\boldsymbol{\phi}))$,
which also agrees with appendix C222Although the symmetric inner product
$\langle\boldsymbol{\psi}|\boldsymbol{\phi}\rangle\equiv\psi^{A}\tilde{\phi}_{A}+\phi^{A}\tilde{\psi}_{A}$
is also invariant under $SPin(\mathbb{R}^{6})\sim SU(4)$, it does not obey to
the property
$\langle\check{\boldsymbol{e}}_{a}(\boldsymbol{\psi})|\boldsymbol{\phi}\rangle=\langle\boldsymbol{\psi}|\check{\boldsymbol{e}}_{a}(\boldsymbol{\phi})\rangle$.
Instead, the identity
$\langle\check{\boldsymbol{e}}_{a}(\boldsymbol{\psi})|\boldsymbol{\phi}\rangle=-\langle\boldsymbol{\psi}|\check{\boldsymbol{e}}_{a}(\boldsymbol{\phi})\rangle$
holds, so that this inner product is not invariant under the group
$Pin(\mathbb{R}^{6})$.. One can define the pseudo-scalar $\boldsymbol{I}$ to
be the linear operator on $S$ given by:
$\boldsymbol{I}\,\equiv\,2^{3}(\check{\boldsymbol{e}}_{1}\wedge\check{\boldsymbol{\theta}}^{1})(\check{\boldsymbol{e}}_{2}\wedge\check{\boldsymbol{\theta}}^{2})(\check{\boldsymbol{e}}_{3}\wedge\check{\boldsymbol{\theta}}^{3})\,\equiv\,(\check{\boldsymbol{e}}_{1}\check{\boldsymbol{\theta}}^{1}-\check{\boldsymbol{\theta}}^{1}\check{\boldsymbol{e}}_{1})(\check{\boldsymbol{e}}_{2}\check{\boldsymbol{\theta}}^{2}-\check{\boldsymbol{\theta}}^{2}\check{\boldsymbol{e}}_{2})(\check{\boldsymbol{e}}_{3}\check{\boldsymbol{\theta}}^{3}-\check{\boldsymbol{\theta}}^{3}\check{\boldsymbol{e}}_{3})\,.$
Using (5.16) along with equations (5.7) and (5.8) it is possible to prove that
$\boldsymbol{I}(\boldsymbol{\chi})=\boldsymbol{\chi}$ for every spinor
$\boldsymbol{\chi}\in S^{+}$ and
$\boldsymbol{I}(\boldsymbol{\gamma})=-\boldsymbol{\gamma}$ for all
$\boldsymbol{\gamma}\in S^{-}$. This justifies calling the spinors of
$S^{\pm}$ the spinors of positive and negative chirality.
##### 5.1.3 Isotropic Subspaces
Recall that a subspace of $N\subset\mathbb{C}\otimes\mathbb{R}^{6}$ is said to
be isotropic when every vector $n^{\mu}\in N$ has zero norm,
$n^{\mu}n_{\mu}=0$. In particular, a null vector $V^{\mu}$,
$V^{\mu}V_{\mu}=0$, is said to generate the 1-dimensional isotropic subspace
$N_{1}$ defined by $N_{1}=\\{\lambda V^{\mu}|\lambda\in\mathbb{C}\\}$. In the
same vein, a simple bivector
$\boldsymbol{B}=\boldsymbol{V}_{1}\wedge\boldsymbol{V}_{2}$ is said to
generate the subspace $N_{2}=Span\\{\boldsymbol{V}_{1},\boldsymbol{V}_{2}\\}$.
Moreover, this bivector $\boldsymbol{B}$ is said to be null when $N_{2}$ is an
isotropic subspace, which means that
$V_{1}^{\,\mu}V_{1\,\mu}=V_{2}^{\,\mu}V_{2\,\mu}=V_{1}^{\,\mu}V_{2\,\mu}=0$.
Analogously, a simple 3-vector
$\boldsymbol{T}=\boldsymbol{V}_{1}\wedge\boldsymbol{V}_{2}\wedge\boldsymbol{V}_{3}$
is said to generate the 3-dimensional subspace
$N_{3}=Span\\{\boldsymbol{V}_{1},\boldsymbol{V}_{2},\boldsymbol{V}_{3}\\}$.
Such 3-vector will then be called null whenever $N_{3}$ is an isotropic
subspace. In 6 dimensions the maximum dimension that an isotropic subspace can
have is 3, because of this the 3-dimensional isotropic subspaces are called
maximally isotropic subspaces. In this subsection it will be shown that the
null vectors, bivectors and 3-vectors are elegantly expressed in the spinorial
language.
Let $V^{AB}=\chi^{[A}\eta^{B]}$ be the spinorial image of the vector
$V^{\mu}$. Then by means of equation (5.5) it is immediate to verify that
$V^{\mu}$ is a null vector. Conversely, if $V^{\mu}$ is null it is always
possible to find two spinors $\chi^{A}$ and $\eta^{A}$ such that the spinorial
image of such vector is $V^{AB}=\chi^{[A}\eta^{B]}$ [91]. Indeed, this can be
grasped from the fact that if $\boldsymbol{V}$ is null then we can arrange a
null frame such that $\boldsymbol{V}=\boldsymbol{e}_{1}$, in which case
$V^{AB}=\chi_{1}^{\,[A}\chi_{2}^{\,B]}$.
In a similar fashion, $\boldsymbol{B}$ is a null bivector if, and only if, its
spinorial image is $B^{A}_{\phantom{A}B}=\chi^{A}\gamma_{B}$ for some spinors
$\chi^{A}$ and $\gamma_{A}$ such that $\chi^{A}\gamma_{A}=0$ [91]. In this
case isotropic subspace generated by $\boldsymbol{B}$ is the one spanned by
the vectors $V^{AB}=\chi^{[A}\eta^{B]}$ for all $\eta^{A}$ such that
$\eta^{A}\gamma_{A}=0$. For instance, if $\\{\boldsymbol{e}_{a}\\}$ is a null
frame then $\boldsymbol{B}=\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}$ is a
null bivector such that
$B^{A}_{\phantom{A}B}\propto\chi_{1}^{\,A}\gamma^{4}_{\,B}$, see table 5.3.
Finally, a 3-vector $\boldsymbol{T}$ is a null 3-vector if, and only if, its
spinorial image $(T^{AB},\tilde{T}_{AB})$ is either of the form
$(\chi^{A}\chi^{B},0)$ or $(0,\gamma_{A}\gamma_{B})$. In the former case the
isotropic subspace generated by $\boldsymbol{T}$ is
$N_{3}^{+}=\\{V^{AB}=\chi^{[A}\eta^{B]}\,|\,\eta^{A}\in S^{+}\\}$, while on
the latter case the isotropic subspace is
$N_{3}^{-}=\\{V_{AB}=\gamma_{[A}\zeta_{B]}\,|\,\zeta_{A}\in S^{-}\\}$. Using
equations (5.5) and (5.16) one can easily see that if $\boldsymbol{n}\in
N_{3}^{+}$ then $\check{\boldsymbol{n}}(\boldsymbol{\chi})=0$. In the jargon
introduced in appendix C this means that the spinor $\boldsymbol{\chi}$ is the
pure spinor associated with the maximally isotropic subspace $N_{3}^{+}$.
Analogously, one can prove that if $\boldsymbol{m}\in N_{3}^{-}$ then
$\check{\boldsymbol{m}}(\boldsymbol{\gamma})=0$, which means that the
$\boldsymbol{\gamma}$ is the pure spinor associated with the maximally
isotropic subspace $N_{3}^{-}$. The results of this subsection are summarized
on the table 5.5.
Null Vector | $V^{AB}=\chi^{[A}\eta^{B]}$ | $Span\\{\;\chi^{[A}\eta^{B]}\;\\}$
---|---|---
Null Bivector | $B^{A}_{\phantom{A}B}=\chi^{A}\gamma_{B}$, $\chi^{A}\gamma_{A}=0$ | $Span\\{\,\chi^{[A}\eta^{B]}\;|\;\eta^{A}\gamma_{A}=0\,\\}$
Null 3-vector $\left\\{\begin{array}[]{l}\;T^{AB}=\chi^{A}\chi^{B}\,,\;\tilde{T}_{AB}=0\\\ \;T^{AB}=0\,,\;\tilde{T}_{AB}=\gamma_{A}\gamma_{B}\\\ \end{array}\right.$ | $\begin{array}[]{l}Span\\{\,\chi^{[A}\eta^{B]}\;|\;\eta^{A}\in S^{+}\,\\}\\\ Span\\{\,\gamma_{[A}\zeta_{B]}\;|\;\zeta_{A}\in S^{-}\,\\}\\\ \end{array}$
Table 5.5: On the central column we have the spinorial form of a null
$p$-vector. The column on the right shows the isotropic subspaces generated by
the respective null $p$-vectors.
#### 5.2 Other Signatures
So far we dealt only with the Euclidean space $\mathbb{R}^{6}$, now it is time
to investigate the other signatures. In the previous chapter it was shown that
in four dimensions one can grasp the distinct signatures as different reality
conditions on the complexified space, see section 4.1. The same thing is valid
in any dimension. Particularly, in 6 dimensions if $\\{\boldsymbol{e}_{a}\\}$
is a null frame then we can have the following reality conditions according to
the signature [20]:
$\begin{cases}\mathbb{R}^{6}\;\textrm{(Euclidean)}\rightarrow\;\;\overline{\boldsymbol{e}_{1}}=\boldsymbol{\theta}^{1}\;,\;\overline{\boldsymbol{e}_{2}}=\boldsymbol{\theta}^{2}\;,\;\overline{\boldsymbol{e}_{3}}=\boldsymbol{\theta}^{3}\\\
\\\
\mathbb{R}^{5,1}\;\textrm{(Lorentzian)}\rightarrow\;\;\overline{\boldsymbol{e}_{1}}=\boldsymbol{e}_{1}\;,\;\overline{\boldsymbol{\theta}^{1}}=\boldsymbol{\theta}^{1}\;,\;\overline{\boldsymbol{e}_{2}}=\boldsymbol{\theta}^{2}\;,\;\overline{\boldsymbol{e}_{3}}=\boldsymbol{\theta}^{3}\\\
\\\
\mathbb{R}^{4,2}\rightarrow\;\;\begin{cases}\overline{\boldsymbol{e}_{1}}=\boldsymbol{e}_{1}\;,\;\overline{\boldsymbol{\theta}^{1}}=\boldsymbol{\theta}^{1}\;,\;\overline{\boldsymbol{e}_{2}}=\boldsymbol{e}_{2}\;,\;\overline{\boldsymbol{\theta}^{2}}=\boldsymbol{\theta}^{2}\;,\;\overline{\boldsymbol{e}_{3}}=\boldsymbol{\theta}^{3}\\\
\overline{\boldsymbol{e}_{1}}=-\boldsymbol{\theta}^{1}\;,\;\overline{\boldsymbol{e}_{2}}=\boldsymbol{\theta}^{2}\;,\;\overline{\boldsymbol{e}_{3}}=\boldsymbol{\theta}^{3}\end{cases}\\\
\\\ \mathbb{R}^{3,3}\;\textrm{(Split)}\rightarrow\;\;\begin{cases}\textrm{Real
Basis}\\\
\overline{\boldsymbol{e}_{1}}=\boldsymbol{e}_{1}\;,\;\overline{\boldsymbol{\theta}^{1}}=\boldsymbol{\theta}^{1}\;,\;\overline{\boldsymbol{e}_{2}}=-\boldsymbol{\theta}^{2}\;,\;\overline{\boldsymbol{e}_{3}}=\boldsymbol{\theta}^{3}\,.\end{cases}\end{cases}$
(5.17)
Therefore, in order to obtain results valid in any signature we just have to
work on the vector space $\mathbb{C}^{6}$ and then choose the desired reality
condition according to eq. (5.17). So instead of working with the group
$SPin(\mathbb{R}^{6})\sim SU(4)$ we shall deal with its complexification,
which is the group $SPin(\mathbb{C}^{6})\sim SL(4;\mathbb{C})$333In order to
see that the complexification of $SU(4)$ is $SL(4;\mathbb{C})$ remember that
on the Lie algebra formalism the elements of $SU(4)$ are of the form
$U=e^{i(a^{j}H_{j})}$, where $\\{H_{j}\\}$ is a basis of Hermitian trace-less
matrices and $a^{j}$ are real numbers. Then, complexify $SU(4)$ means allow
the scalars $a^{j}$ to assume complex values. This implies that elements of
the complexified group are of the form $S=e^{iM}$, with $M$ being the sum of a
trace-less Hermitian matrix and a trace-less anti-Hermitian matrix. Thus $M$
can be any trace-less matrix, so that $S$ is a general $4\times 4$ matrix with
unit determinant.. The group $SL(4;\mathbb{C})$ has four inequivalent
irreducible representations of dimension 4:
$\textbf{4}:\;\;\chi^{A}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}S^{A}_{\phantom{A}B}\,\chi^{B}\;\;\;\;;\;\;\;\;\widetilde{\textbf{4}}:\;\;\gamma_{A}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}S^{-1\,B}_{\phantom{-1\,B}A}\,\gamma_{B}\,$
$\overline{\textbf{4}}:\;\;\gamma_{\dot{A}}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}\overline{S}_{\dot{A}}^{\phantom{A}\dot{B}}\,\gamma_{\dot{B}}\;\;\;\;;\;\;\;\;\widetilde{\overline{\textbf{4}}}:\;\;\chi^{\dot{A}}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}\overline{S}^{-1\phantom{B}\dot{A}}_{\phantom{-1}\dot{B}}\,\chi^{\dot{B}}\,.$
(5.18)
Where $S^{A}_{\phantom{A}B}$ is a $4\times 4$ matrix of unit determinant,
$S^{-1\,A}_{\phantom{-1\,A}B}$ is its inverse and
$\overline{S}_{\dot{A}}^{\phantom{A}\dot{B}}$ its complex conjugate. From
equation (5.18) we see that if $\chi^{A}$ transforms on the representation 4
then its complex conjugate will be on the representation
$\overline{\textbf{4}}$, so that we can write
$\overline{\chi^{A}}=\overline{\chi}_{\dot{A}}$. Note that if $S$ is unitary
then $S^{-1}=\overline{S}^{\,t}$, which implies that in this case the
transformations $\widetilde{\textbf{4}}$ and $\overline{\textbf{4}}$ are
equivalent, as well as the transformations 4 and
$\widetilde{\overline{\textbf{4}}}$. This is the reason of why the group
$SU(4)$ has just two inequivalent irreducible representations of dimension 4.
Since $SPin(\mathbb{C}^{6})\sim SL(4;\mathbb{C})$ is a double cover for the
group $SO(6;\mathbb{C})$ it follows that every tensor transforming on a
representation of the latter group can be seen as an object transforming on
some representation of the former. Furthermore, since 4 is the fundamental
representation of $SL(4;\mathbb{C})$ then, as long as we do not take complex
conjugates, every tensor of $SO(6;\mathbb{C})$ can be said to be on a
composition of the representations 4 and $\widetilde{\textbf{4}}$. Thus,
almost all the results obtained for the Euclidean space $\mathbb{R}^{6}$ can
be carried for the complex space $\mathbb{C}^{6}$. In particular, except for
the table 5.1, all the above tables remain valid on the complex case. Note
also that, since $\det(S)=1$, equation (5.5) is still valid.
The differences between the Euclidean case and the other signatures shows up
only when the operation of complex conjugation is performed. As explained
before, on the Euclidean case the complex conjugation of an object on the
representation 4 turns out to be on the representation
$\overline{\textbf{4}}=\widetilde{\textbf{4}}$, while on the other signatures
the complex conjugate will be on the representation
$\overline{\textbf{4}}\neq\widetilde{\textbf{4}}$. Thus on the Euclidean case
one can easily verify whether a tensor is real using the spinorial language.
For example, in this signature a vector $V^{AB}$ is real when
$\overline{V^{AB}}\equiv\overline{V}_{AB}=V_{AB}$, while a bivector is real if
$\overline{B}_{A}^{\phantom{A}B}=B^{B}_{\phantom{B}A}$. In the other
signatures one cannot directly compare $V^{AB}$ to its complex conjugate,
since the latter is on the representation $\overline{\boldsymbol{4}}$ and the
equation $\overline{V}_{\dot{A}\dot{B}}=V_{AB}$ is non-sense. This kind of
comparison can be done only after introducing a charge conjugation operator,
which provides a map between the representations $\overline{\boldsymbol{4}}$
and $\widetilde{\textbf{4}}$, see appendix C. If $\boldsymbol{\psi}$ is a
Dirac spinor then its charge conjugate is the spinor $\boldsymbol{\psi}^{c}$
such that
$[\check{\boldsymbol{e}}_{a}(\boldsymbol{\psi})]^{c}=\check{\overline{\boldsymbol{e}}}_{a}(\boldsymbol{\psi}^{c})$.
For instance, one can use equations (5.7), (5.8), (5.16) and (5.17) to prove
that on the Euclidean and Lorentzian cases the charge conjugation can be
respectively given by444Note that the inner product introduced on (5.15) is
such that
$\overline{(\boldsymbol{\psi},\boldsymbol{\phi})}=(\boldsymbol{\psi}^{c},\boldsymbol{\phi}^{c})$.:
$\displaystyle\textbf{{Euclidean}}\left\\{\begin{array}[]{cccc}\boldsymbol{\chi}_{1}^{c}\,=\,\boldsymbol{\gamma}^{1}&\boldsymbol{\chi}_{2}^{c}\,=\,\boldsymbol{\gamma}^{2}&\boldsymbol{\chi}_{3}^{c}\,=\,\boldsymbol{\gamma}^{3}&\boldsymbol{\chi}_{4}^{c}\,=\,\boldsymbol{\gamma}^{4}\\\
\boldsymbol{\gamma}^{1\,c}\,=\,-\boldsymbol{\chi}_{1}&\boldsymbol{\gamma}^{2\,c}\,=\,-\boldsymbol{\chi}_{2}&\boldsymbol{\gamma}^{3\,c}\,=\,-\boldsymbol{\chi}_{3}&\boldsymbol{\gamma}^{4\,c}\,=\,-\boldsymbol{\chi}_{4}\end{array}\right.$
(5.21)
$\displaystyle\textbf{{Lorentzian}}\left\\{\begin{array}[]{cccc}\boldsymbol{\chi}_{1}^{c}\,=\,\boldsymbol{\chi}_{2}&\boldsymbol{\chi}_{2}^{c}\,=\,-\boldsymbol{\chi}_{1}&\boldsymbol{\chi}_{3}^{c}\,=\,-\boldsymbol{\chi}_{4}&\boldsymbol{\chi}_{4}^{c}\,=\,\boldsymbol{\chi}_{3}\\\
\boldsymbol{\gamma}^{1\,c}\,=\,\boldsymbol{\gamma}^{2\,c}&\boldsymbol{\gamma}^{2\,c}\,=\,-\boldsymbol{\gamma}^{1\,c}&\boldsymbol{\gamma}^{3\,c}\,=\,-\boldsymbol{\gamma}^{4\,c}&\boldsymbol{\gamma}^{4\,c}\,=\,\boldsymbol{\gamma}^{3\,c}.\end{array}\right.$
(5.24)
But, as far as the $SO(6;\mathbb{C})$ tensors are concerned, one can avoid
using the charge conjugation operation by making direct use of equation
(5.17), which sometimes is profitable.
#### 5.3 An Algebraic Classification for the Weyl Tensor
The intent of the present section is to use the spinorial formalism just
introduced in order to define a natural algebraic classification for the Weyl
tensor. The role played by the spinorial language here is to uncover relations
that are hard to guess using the vectorial formalism.
As a warming example let us work out an algebraic classification for bivectors
in 6 dimensions. Note that the spinorial form of a bivector,
$B^{A}_{\phantom{A}B}$, enables us to associate to each bivector
$\boldsymbol{B}$ the following map on the space of Dirac spinors [91]:
$\mathcal{B}:S\rightarrow
S\;\,,\quad\boldsymbol{\psi}=\psi^{A}+\tilde{\psi}_{A}\,\stackrel{{\scriptstyle\mathcal{B}}}{{\longmapsto}}\,\boldsymbol{\phi}=\underbrace{B^{A}_{\phantom{A}B}\,\psi^{B}}_{\phi^{A}}\,+\,\underbrace{\tilde{\psi}_{B}\,B^{B}_{\phantom{B}A}}_{\tilde{\phi}_{A}}\,.$
It is simple matter to verify that this operator is self-adjoint with respect
to the inner product defined on (5.15), meaning that
$(\mathcal{B}(\boldsymbol{\psi}_{1}),\boldsymbol{\psi}_{2})=(\boldsymbol{\psi}_{1},\mathcal{B}(\boldsymbol{\psi}_{2}))$.
Note also that it preserves the spaces $S^{+}$ and $S^{-}$. Indeed, plugging
$\tilde{\psi}_{A}=0$ in the above equation we get $\tilde{\phi}_{A}=0$.
Analogously, if $\psi^{A}$ vanishes then $\phi^{A}=0$. Hence we have
$\mathcal{B}=\mathcal{B}^{+}\oplus\mathcal{B}^{-}$, where $\mathcal{B}^{\pm}$
are the restrictions of the operator $\mathcal{B}$ to the spaces $S^{\pm}$. If
$\\{\boldsymbol{\chi}_{p}\\}$ is a basis for the space of Weyl spinors of
positive chirality, $S^{+}$, then one can define its dual basis
$\\{\boldsymbol{\gamma}^{p}\\}$ for the space $S^{-}$ as the basis such that
$(\boldsymbol{\chi}_{p},\boldsymbol{\gamma}^{q})=\delta^{\,q}_{p}$. The matrix
representations of the operators $\mathcal{B}^{\pm}$ on these bases are then
easily seen to be
$\mathcal{B}^{+}_{pq}=(\mathcal{B}(\boldsymbol{\chi}_{q}),\boldsymbol{\gamma}^{p})$
and
$\mathcal{B}^{-}_{pq}=(\boldsymbol{\chi}_{p},\mathcal{B}(\boldsymbol{\gamma}^{q}))$.
Thus using the fact that $\mathcal{B}$ is self-adjoint we find
$\mathcal{B}^{+}_{pq}=\mathcal{B}^{-}_{qp}$.
One can use the operator $\mathcal{B}$ to algebraically classify the bivectors
in six dimensions according to the Segre type of this operator, see appendix
A. But since $\mathcal{B}=\mathcal{B}^{+}\oplus\mathcal{B}^{-}$, then classify
$\mathcal{B}$ is equivalent to classify $\mathcal{B}^{\pm}$. Furthermore, once
the matrix representation of $\mathcal{B}^{-}$ is the transpose of the matrix
representation of $\mathcal{B}^{+}$ it follows that the algebraic types of the
operators $\mathcal{B}^{+}$ and $\mathcal{B}^{-}$ are the same. Thus we just
really need to classify $\mathcal{B}^{+}$. As an example note that if the
bivector is null, $B^{A}_{\phantom{A}B}=\chi^{A}\gamma_{B}$ with
$\chi^{A}\gamma_{A}=0$, then one can always arrange a basis such that
$\boldsymbol{\chi}_{1}=\boldsymbol{\chi}$ and
$\boldsymbol{\gamma}_{2}=\boldsymbol{\gamma}$. In this basis we have
$\mathcal{B}^{+}_{pq}\,=\,\operatorname{diag}(\left[\begin{array}[]{cc}0&1\\\
0&0\\\ \end{array}\right],0,0)\,.$
So that the refined Segre classification of $\mathcal{B}^{+}$ is $[\,|2,1,1]$.
The converse of this result is also true, leading us to the conclusion that a
bivector in six dimensions is null if, and only if, its algebraic type is
$[\,|2,1,1]$. Note that such algebraic classification for bivectors heavily
depends on the spinors and can hardly be attained using just the vectorial
formalism.
Now let us try to define an algebraic classification for the Weyl tensor.
According to table 5.2, in six dimensions a tensor with the symmetries of the
Weyl tensor is represented by an object of the form
$\Psi^{AB}_{\phantom{AB}CD}$ that is symmetric on both pairs of indices,
$\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)}$, and trace-less,
$\Psi^{AB}_{\phantom{AB}CB}=0$. Then, since the 3-vectors are represented by a
pair of symmetric tensors $(T^{AB},\tilde{T}_{AB})$, it follows that the Weyl
tensor can be seen as an operator
$\mathcal{C}:\Lambda^{3}\rightarrow\Lambda^{3}$, with $\Lambda^{3}$ denoting
the space of 3-vectors, whose action is [91]:
$\left(\,T^{AB},\tilde{T}_{AB}\,\right)\,\stackrel{{\scriptstyle\mathcal{C}}}{{\longmapsto}}\,\left(\,T^{\prime
AB},\tilde{T}^{\prime}_{AB}\,\right)=\left(\,\Psi^{AB}_{\phantom{AB}CD}T^{CD},\tilde{T}_{CD}\Psi^{CD}_{\phantom{CD}AB}\,\right)\,.$
(5.25)
Let us denote the space of self-dual 3-vectors, $\tilde{T}_{AB}=0$, by
$\Lambda^{3+}$ and the space of anti-self-dual 3-vectors, $T^{AB}=0$, by
$\Lambda^{3-}$. Then it is immediate to verify the spaces $\Lambda^{3\pm}$ are
preserved by the operator $\mathcal{C}$. Indeed, plugging $\tilde{T}_{AB}=0$
on equation (5.25) we find that $\tilde{T}^{\prime}_{AB}=0$. Analogously, if
$T^{AB}=0$ then $T^{\prime AB}=0$. So the operator $\mathcal{C}$ that acts on
the 20-dimensional space $\Lambda^{3}$ can be seen as the direct sum of two
operators acting on 10-dimensional spaces,
$\mathcal{C}=\mathcal{C}^{+}\oplus\mathcal{C}^{-}$. Where $\mathcal{C}^{\pm}$
are the restrictions of $\mathcal{C}$ to the spaces $\Lambda^{3\pm}$.
Thus one can classify the Weyl tensor according to the refined Segre types of
the operators $\mathcal{C}^{\pm}$. However, let us see that the algebraic
types of $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ always coincide, so that we
just need to classify the operator $\mathcal{C}^{+}$. To this end it is useful
to introduce the following basis for the space of 3-vectors:
$\begin{array}[]{llll}T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}&T_{2}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{2}^{\,B)}&T_{3}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{3}^{\,B)}&T_{4}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{4}^{\,B)}\\\
T_{5}^{\;AB}=\chi_{2}^{\,A}\chi_{2}^{\,B}&T_{6}^{\;AB}=\sqrt{2}\,\chi_{2}^{\,(A}\chi_{3}^{\,B)}&T_{7}^{\;AB}=\sqrt{2}\,\chi_{2}^{\,(A}\chi_{4}^{\,B)}&T_{8}^{\;AB}=\chi_{3}^{\,A}\chi_{3}^{\,B}\\\
T_{9}^{\;AB}=\sqrt{2}\,\chi_{3}^{\,(A}\chi_{4}^{\,B)}&T_{10}^{\;AB}=\chi_{4}^{\,A}\chi_{4}^{\,B}&\tilde{T}^{1}_{\;AB}=\gamma^{1}_{\,A}\gamma^{1}_{\,B}&\tilde{T}^{2}_{\;AB}=\sqrt{2}\gamma^{1}_{\,(A}\gamma^{2}_{\,B)}\\\
\tilde{T}^{3}_{\;AB}=\sqrt{2}\gamma^{1}_{\,(A}\gamma^{3}_{\,B)}&\tilde{T}^{4}_{\;AB}=\sqrt{2}\gamma^{1}_{\,(A}\gamma^{4}_{\,B)}&\tilde{T}^{5}_{\;AB}=\gamma^{2}_{\,A}\gamma^{2}_{\,B}&\tilde{T}^{6}_{\;AB}=\sqrt{2}\gamma^{2}_{\,(A}\gamma^{3}_{\,B)}\\\
\tilde{T}^{7}_{\;AB}=\sqrt{2}\gamma^{2}_{\,(A}\gamma^{4}_{\,B)}&\tilde{T}^{8}_{\;AB}=\gamma^{3}_{\,A}\gamma^{3}_{\,B}&\tilde{T}^{9}_{\;AB}=\sqrt{2}\gamma^{3}_{\,(A}\gamma^{4}_{\,B)}&\tilde{T}^{10}_{\;AB}=\gamma^{4}_{\,A}\gamma^{4}_{\,B}\end{array}$
Abstractly we shall denote by $\boldsymbol{T}_{r}$ the self-dual 3-vector
whose spinorial image is $(T_{r}^{\;AB},0)$ and by
$\tilde{\boldsymbol{T}}^{r}$ the anti-self-dual 3-vector
$(0,\tilde{T}^{r}_{\;AB})$. Then $\\{\boldsymbol{T}_{r}\\}$ provides a basis
for $\Lambda^{3+}$, while $\\{\tilde{\boldsymbol{T}}^{r}\\}$ provides a basis
for $\Lambda^{3-}$. It is simple matter to verify that the following
identities hold:
$T_{r}^{\;AB}\,\tilde{T}^{s}_{\;AB}\,=\,\delta^{\,s}_{r}\quad;\quad
T_{r}^{\;AB}\,\tilde{T}^{r}_{\;CD}\,=\,\delta^{\,(A}_{C}\delta^{\,B)}_{D}\,.$
Using the first relation above we find that the actions of the operators
$\mathcal{C}^{\pm}$ are given by
$\displaystyle\mathcal{C}^{+}(\boldsymbol{T}_{s})\,=\,\boldsymbol{T}_{r}\,\mathcal{C}^{+}_{rs}\quad\textrm{with}\quad\mathcal{C}^{+}_{rs}\,\equiv\,\tilde{T}^{r}_{\;AB}\,\Psi^{AB}_{\phantom{AB}CD}\,T_{s}^{\;CD}\,$
$\displaystyle\mathcal{C}^{-}(\tilde{\boldsymbol{T}}^{s})\,=\,\tilde{\boldsymbol{T}}^{r}\,\mathcal{C}^{-}_{rs}\quad\textrm{with}\quad\mathcal{C}^{-}_{rs}\,\equiv\,\tilde{T}^{s}_{\;AB}\,\Psi^{AB}_{\phantom{AB}CD}\,T_{r}^{\;CD}\,.$
Thus we have that $\mathcal{C}^{+}_{rs}=\mathcal{C}^{-}_{sr}$ and, therefore,
the algebraic types of $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ are always the
same. Note also that these operators are trace-less,
$\mathcal{C}^{+}_{rr}=\tilde{T}^{r}_{\;AB}\,\Psi^{AB}_{\phantom{AB}CD}\,T_{r}^{\;CD}=\Psi^{AB}_{\phantom{AB}AB}=0$.
Thus the algebraic classification for the Weyl tensor proposed here amounts to
compute the refined Segre type of the trace-less operator $\mathcal{C}^{+}$
[91]. As an example let suppose that the Weyl tensor has the form
$\Psi^{AB}_{\phantom{AB}CD}=f^{AB}h_{CD}$ with $f^{AB}h_{CB}=0$. Then one can
choose a basis $\\{\boldsymbol{\textsf{T}}_{r}\\}$ for $\Lambda^{3+}$ such
that $\textsf{T}_{1}^{\;AB}=f^{AB}$ and
$\textsf{T}_{r}^{\;AB}h_{AB}=\delta^{\,2}_{r}$. In this basis the matrix
representation of $\mathcal{C}^{+}$ is given by:
$\mathcal{C}^{+}_{rs}\,=\,\operatorname{diag}(\left[\begin{array}[]{cc}0&1\\\
0&0\\\ \end{array}\right],0,0,0,0,0,0,0,0)\,.$
The refined Segre classification of this matrix is $[\,|2,1,1,1,1,1,1,1,1]$.
Thus, in this example, we shall say that the algebraic classification of the
Weyl tensor is $[\,|2,1,1,1,1,1,1,1,1]$.
A special phenomenon occurs when the signature is Euclidean. In this case
equation (5.21) enables us to say that the 3-vectors $\boldsymbol{T}_{r}$ are
the complex conjugates of the 3-vectors $\tilde{\boldsymbol{T}}^{r}$.
Furthermore, if the Weyl tensor is real then
$\overline{\Psi^{AB}_{\phantom{AB}CD}}=\Psi^{CD}_{\phantom{CD}AB}$, so that we
have:
$\overline{\mathcal{C}^{+}_{rs}}\,\,=\,\,\overline{\tilde{T}^{r}_{\;AB}}\;\overline{\Psi^{AB}_{\phantom{AB}CD}}\;\overline{T_{s}^{\;CD}}\,\,=\,\,T_{r}^{\;AB}\,\Psi^{CD}_{\phantom{CD}AB}\,\tilde{T}^{s}_{\;CD}\,\,=\,\,\mathcal{C}^{+}_{sr}\,.$
Hence, when the signature is Euclidean and the Weyl tensor is real, the matrix
representation of $\mathcal{C}^{+}$ is Hermitian and, therefore, can be
diagonalized. This is an enormous constraint for the possible algebraic types
of the Weyl tensor, since one can anticipate that all Jordan blocks of
$\mathcal{C}^{+}$ will have dimension one.
In spite of the resemblances, it is worth noting that there is one important
difference between the bivector classification and the Weyl tensor
classification introduced in the present section. While on the former the
operator $\mathcal{B}$ acts on the space of spinors, which has no vectorial
corresponding, on the latter the operator $\mathcal{C}$ acts on the space of
3-vectors, which does have a vectorial equivalent. Thus the operator
$\mathcal{C}$ must admit an expression without the use of spinors. Indeed, it
can be proved that this operator is proportional to the following map:
$T_{\mu\nu\alpha}\,\,\longmapsto\,\,T^{\prime}_{\mu\nu\alpha}\,=\,C^{\rho\sigma}_{\phantom{\rho\sigma}[\mu\nu}\,T_{\alpha]\rho\sigma}\,.$
(5.26)
Then the operator $\mathcal{C}^{+}$ is proportional to the restriction of the
above map to the subspace of self-dual 3-vectors,
$\star\boldsymbol{T}=\boldsymbol{T}$.
As last comment it is worth mentioning that in 6 dimensions one can also
classify the Weyl tensor using the fact that this tensor provides an operator
on the space of bivectors, $B_{\mu\nu}\mapsto
C_{\mu\nu\rho\sigma}B^{\rho\sigma}$. Actually such classification can
obviously be done in any dimension, a fact that was exploited in [34] with the
aim of refining the CMPP classification. The advantage of the Weyl tensor
classification using 3-vectors, introduced in this section, is that it turns
out to be nicely related to some integrability properties, as will be shown in
what follows.
#### 5.4 Generalized Goldberg-Sachs
On reference [67] it was proved a beautiful partial generalization of the
Goldberg-Sachs (GS) theorem valid in manifolds of all dimensions greater than
4, as well as in any signature. The goal of the presented section is to prove
that in 6 dimensions such theorem can be elegantly expressed and acquires a
beautiful geometrical interpretation when the spinorial formalism is used.
Moreover, it will be shown that this theorem is nicely related to the
algebraic classification of the Weyl tensor introduced in the previous
section. In what follows the spinorial objects will be fields over a
6-dimensional manifold $(M,\boldsymbol{g})$, so that the vector spaces treated
so far are now the tangent spaces of this manifold555In order for the manifold
admit a spinor bundle its topology must be constrained, see [97] for example.
However, since from the physical point of view we are interested on local
phenomena this fact will be ignored..
Let be $N$ be a maximally isotropic distribution over a Ricci-flat666Actually
the theorem proved in [67] is more general and remains valid even if certain
components of the Ricci tensor are different from zero. Its original version
is expressed in a conformally invariant way in terms of the Cotton-York
tensor. But, for simplicity, from now on we shall assume the Ricci tensor to
vanish. manifold of dimension greater than four and arbitrary signature. Then
the theorem presented in [67] states that if the Weyl tensor is such that
$C_{\mu\nu\rho\sigma}V_{1}^{\,\mu}V_{2}^{\,\nu}V_{3}^{\,\rho}=0$ for all
vector fields $\boldsymbol{V}_{1}$, $\boldsymbol{V}_{2}$ and
$\boldsymbol{V}_{3}$ tangent to $N$ and is generic otherwise777The proof of
this theorem requires that some generality conditions are satisfied by the
Weyl tensor, so the imposition of “generic otherwise” is certainly sufficient,
but it is not clear at all what is the necessary requirement. For example, in
the section 3.4.2 of reference [66] some cases are shown in which the
generality assumption can be relaxed. Also, at section 5.3 of [64] it is said
that in five dimensions there exist many cases such that the generality
conditions can be neglected if the Ricci identities are used. As such, we will
ignore this requirement in the present discussion. then the maximally
isotropic distribution $N$ is locally integrable. Note that this theorem is a
partial generalization of the GS theorem to higher dimensions [60]. In six
dimensions given a maximally isotropic distribution $N$, one can always
arrange a null frame $\\{\boldsymbol{e}_{a}\\}$ such that
$N=Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$. Thus,
supposing that $(M,\boldsymbol{g})$ is Ricci-flat and that the Weyl tensor
obeys the generality conditions then:
$C_{a^{\prime}b^{\prime}c^{\prime}d}\,=\,0\;\quad\Longrightarrow\;\quad
Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}\;\;\textrm{
is Integrable.}$ (5.27)
Where in the above equation the indices $a^{\prime},b^{\prime},c^{\prime}$
pertain to $\\{1,2,3\\}$, while the index $d$ runs from 1 to 6. A careful look
at table 5.4 reveals that the algebraic condition on the left hand side of eq.
(5.27) has the following equivalent in the spinorial language:
$C_{a^{\prime}b^{\prime}c^{\prime}d}\,=\,0\;\;\Longleftrightarrow\;\;\begin{cases}\Psi^{AE}_{\phantom{AE}11}=0\\\
\Psi^{AB}_{\phantom{AB}1D}=0\end{cases}\;\;\forall\;\;A,B\neq 1\,.$ (5.28)
Actually, it is an immediate consequence of the identity
$\Psi^{AB}_{\phantom{AB}CB}=0$ that the first constraint on the right side of
eq. (5.28) is contained on the second constraint. Thus one can say that the
condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ is tantamount to
$\Psi^{AB}_{\phantom{AB}1D}=0$ for all $A,B\neq 1$. This last constraint, in
turn, can be reexpressed as:
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0\;\Leftrightarrow\;(\,\varepsilon_{AEFG}\,\varepsilon_{BHIJ}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\,\chi_{1}^{\,A}\,\chi_{1}^{\,B}\,\chi_{1}^{\,C}=0\;\Leftrightarrow\;\chi_{1}^{\,[E}\Psi^{A][B}_{\phantom{A][B}CD}\,\chi_{1}^{\,F]}\chi_{1}^{\,C}=0\,.$
But note that the spinor $\boldsymbol{\chi}_{1}$ is just the pure spinor
associated to the maximally isotropic distribution
$Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$, see
subsection 5.1.3 and appendix C. Thus the theorem of reference [67] can be
elegantly expressed in terms of spinors as follows:
###### Theorem 16
Let $(M,\boldsymbol{g})$ be a Ricci-flat 6-dimensional manifold whose Weyl
tensor obeys the constraint
$\chi^{[E}\Psi^{A][B}_{\phantom{A][B}CD}\,\chi^{F]}\chi^{C}=0$ for some spinor
$\boldsymbol{\chi}\in S^{+}$ and is generic otherwise (see [67]). Then the
maximally isotropic distribution associated to the pure spinor
$\boldsymbol{\chi}$ is integrable.
For completeness, let us remark that by means of table 5.4 one can also prove
that the following equivalences hold:
$\begin{array}[]{lll}C_{a^{\prime}b^{\prime}cd}\,=\,0\;\;\Leftrightarrow&(\,\varepsilon_{AEFG}\,\Psi^{GB}_{\phantom{GB}CD}\,)\chi_{1}^{\,A}\,\chi_{1}^{\,C}=0&\Leftrightarrow\;\;\chi_{1}^{\,[E}\Psi^{A]B}_{\phantom{A]B}CD}\chi_{1}^{\,C}=0\\\
C_{a^{\prime}bcd}\,=\,0\;\;\Leftrightarrow&(\,\varepsilon_{AEFG}\,\Psi^{GB}_{\phantom{GB}CD}\,)\chi_{1}^{\,A}\,=0&\Leftrightarrow\;\;\chi_{1}^{\,[E}\Psi^{A]B}_{\phantom{A]B}CD}=0\,.\\\
\end{array}$
In the previous paragraph we oriented the null frame in such a way that the
maximally isotropic distribution was spanned by
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$. This is a
self-dual distribution, meaning that the 3-vector
$\boldsymbol{T}=\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}$
is self-dual. But we could also have assumed that the distribution was
generated by
$\\{\boldsymbol{\theta}^{1},\boldsymbol{\theta}^{2},\boldsymbol{\theta}^{3}\\}$,
which is an anti-self-dual distribution. In such a case the associated pure
spinor is $\boldsymbol{\gamma}^{1}$, which has negative chirality. In this
circumstance the integrability condition of theorem 16 might be replaced by
$\gamma^{1}_{[E}\Psi^{AB}_{\phantom{AB}C][D}\gamma^{1}_{F]}\gamma^{1}_{A}=0$.
Now let us see that theorem 16 can be expressed in terms of the map
$\mathcal{C}$ defined in section 5.3. Indeed, using equations (5.25) and
(5.28) we immediately find:
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0\;\Rightarrow\;\Psi^{AB}_{\phantom{AB}11}=0\;\textrm{
if}\;A\neq
1\;\Rightarrow\;\Psi^{AB}_{\phantom{AB}11}\propto\chi_{1}^{\,A}\chi_{1}^{\,B}\;\Rightarrow\;\mathcal{C}^{+}(\boldsymbol{T}_{1})\propto\boldsymbol{T}_{1}\,.$
Where in the above equation the 3-vector $\boldsymbol{T}_{1}$ is the one whose
spinorial equivalent is $(\chi_{1}^{\,A}\chi_{1}^{\,B},0)$. In the vectorial
language this 3-vector is proportional to
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}$. Thus we
proved that if the integrability condition for a maximally isotropic
distribution is satisfied then the null 3-vector that generates it is an
eigen-3-vector of the operator $\mathcal{C}^{+}$. This is a partial
generalization of the corollary 2 of chapter 4. Furthermore, using the above
results we have:
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0\;\Leftrightarrow\;\Psi^{AB}_{\phantom{AB}1C}=0\;\textrm{
if}\;A,B\neq
1\;\Leftrightarrow\;\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\chi_{p}^{\,D}=\chi_{1}^{\,(A}\eta_{p}^{\,B)}\,.$
Where $\\{\eta_{p}^{\,B}\\}$ is some set of four spinors. The above equation
means that if the integrability condition
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ is satisfied then the subspace formed
by the 3-vectors of the form $(\chi_{1}^{\,(A}\eta^{B)},0)$ for all
$\boldsymbol{\eta}\in S^{+}$ is invariant by the action of $\mathcal{C}^{+}$.
Using the 3-vector basis introduced in section 5.3 this is the subspace
spanned by888On the vectorial formalism the referred subspace is the one
spanned by the 3-vectors
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}$,
$\boldsymbol{e}_{1}\wedge(\boldsymbol{e}_{2}\wedge\boldsymbol{\theta}^{2}+\boldsymbol{e}_{3}\wedge\boldsymbol{\theta}^{3})$,
$\boldsymbol{e}_{2}\wedge(\boldsymbol{e}_{1}\wedge\boldsymbol{\theta}^{1}+\boldsymbol{e}_{3}\wedge\boldsymbol{\theta}^{3})$
and
$\boldsymbol{e}_{3}\wedge(\boldsymbol{e}_{1}\wedge\boldsymbol{\theta}^{1}+\boldsymbol{e}_{2}\wedge\boldsymbol{\theta}^{2})$.
$\\{\boldsymbol{T}_{1},\boldsymbol{T}_{2},\boldsymbol{T}_{3},\boldsymbol{T}_{4}\\}$.
The results of this paragraph enables us to rephrase theorem 16 as follows:
###### Theorem 17
Let $(M,\boldsymbol{g})$ be a Ricci-flat 6-dimensional manifold whose Weyl
operator $\mathcal{C}^{+}$ keeps invariant the subspace spanned by the
3-vectors of the form $T^{AB}=\chi^{(A}\eta^{B)}$ for all $\eta^{A}\in S^{+}$,
with $\mathcal{C}^{+}$ being generic otherwise. Then the maximally isotropic
distribution associated to the pure spinor $\boldsymbol{\chi}$ is integrable
and the 3-vector $T^{AB}=\chi^{A}\chi^{B}$ is an eigen-3-vector of
$\mathcal{C}^{+}$.
##### 5.4.1 Lorentzian Signature
Now let us assume that $(M,\boldsymbol{g})$ is a manifold whose metric
$\boldsymbol{g}$ is real and has Lorentzian signature. If the Weyl tensor
satisfies the integrability condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$
then, by the previous results, we know that
$\mathcal{C}^{+}(\boldsymbol{T}_{1})\propto\boldsymbol{T}_{1}$. Furthermore,
the subspace $\mathcal{A}\equiv
Span\\{\boldsymbol{T}_{1},\boldsymbol{T}_{2},\boldsymbol{T}_{3},\boldsymbol{T}_{4}\\}$
is invariant under $\mathcal{C}^{+}$, where
$T_{1}^{\,AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}\quad\textrm{and}\quad\mathcal{A}=\\{\,T^{AB}=\chi_{1}^{\,(A}\eta^{B)}\,|\;\eta^{A}\in\,S^{+}\,\\}\,.$
Since the metric is assumed to be real it follows that the Weyl tensor is also
real, as well as the operator $\mathcal{C}^{+}$. Thus the complex conjugate of
these constraints are likewise valid, leading us to the conclusion that
$\mathcal{C}^{+}(\overline{\boldsymbol{T}_{1}})\propto\overline{\boldsymbol{T}_{1}}$
and that the subspace $\overline{\mathcal{A}}$ is also invariant by the action
of $\mathcal{C}^{+}$. By means of equation (5.24) we have that
$\overline{T_{1}^{\,AB}}=T_{5}^{\,AB}=\chi_{2}^{\,A}\chi_{2}^{\,B}\quad\textrm{and}\quad\overline{\mathcal{A}}=\\{\,T^{AB}=\chi_{2}^{\,(A}\eta^{B)}\,|\;\eta^{A}\in\,S^{+}\,\\}\,.$
Note that since the subspaces $\mathcal{A}$ and $\overline{\mathcal{A}}$ are
invariant under $\mathcal{C}^{+}$ so will be
$\mathcal{A}\cap\overline{\mathcal{A}}=Span\\{T^{AB}=\chi_{1}^{\,(A}\chi_{2}^{\,B)}\\}$.
From which we conclude that the 3-vector $\boldsymbol{T}_{2}$ is an
eigen-3-vector of the operator $\mathcal{C}^{+}$. These results along with
theorem 17 lead us to the following corollary [91]:
###### Corollary 4
Let $(M,\boldsymbol{g})$ be a Ricci-flat Lorentzian manifold, then the
integrability conditions for the maximally isotropic distribution generated by
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$ are:
(1) The 3-vectors $\boldsymbol{T}_{1}$, $\boldsymbol{T}_{2}$ and
$\boldsymbol{T}_{5}$ are eigen-3-vectors of the Weyl operator
$\mathcal{C}^{+}$
(2) The subspaces
$\mathcal{A}=Span\\{\boldsymbol{T}_{1},\boldsymbol{T}_{2},\boldsymbol{T}_{3},\boldsymbol{T}_{4}\\}$
and
$\overline{\mathcal{A}}=Span\\{\boldsymbol{T}_{2},\boldsymbol{T}_{5},\boldsymbol{T}_{6},\boldsymbol{T}_{7}\\}$
are invariant by the action of $\mathcal{C}^{+}$.
If the metric is real then whenever a distribution is integrable the complex
conjugate of this distribution will also be integrable, that is the
geometrical origin of the above corollary. Using eq. (5.17) we conclude that
the complex conjugate of the distribution
$Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$ is the
distribution spanned by
$\\{\boldsymbol{e}_{1},\boldsymbol{\theta}^{2},\boldsymbol{\theta}^{3}\\}$.
The pure spinor associated to the latter maximally isotropic distribution is
$\boldsymbol{\chi}_{2}$. Note that the intersection of the distributions
$Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$ and
$Span\\{\boldsymbol{e}_{1},\boldsymbol{\theta}^{2},\boldsymbol{\theta}^{3}\\}$
is the 1-dimensional distribution tangent to the real and null vector field
$\boldsymbol{e}_{1}$. Since the leafs of an integrable maximally isotropic
distribution are totally geodesic [77], it follows that if these two
distributions are integrable then the vector field $\boldsymbol{e}_{1}$ is
geodesic. But, differently from the 4-dimensional case, the congruence
generated by $\boldsymbol{e}_{1}$ generally is not shear-free. Finally, it is
easy to verify that if $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ then the vector
field $\boldsymbol{e}_{1}$ turns out to be a multiple Weyl aligned null
direction, meaning that the components $C_{1\alpha 1\beta}$,
$C_{1\alpha\beta\kappa}$ and $C_{141\alpha}$ vanish for all
$\alpha,\beta,\kappa\neq 1,4$.
#### 5.5 Example, Schwarzschild in 6 Dimensions
In this section it will be used the spinorial formalism in order to analyze
the 6-dimensional Schwarzschild space-time, the unique spherically symmetric
vacuum solution in 6 dimensions. In a suitable coordinate system the metric of
this manifold is given by:
$\textrm{ds}^{2}=-h^{2}\textrm{dt}^{2}+h^{-2}\textrm{dr}^{2}+r^{2}\left\\{\textrm{d}\phi_{1}^{2}+\sin^{2}\phi_{1}\left[\textrm{d}\phi_{2}^{2}+\sin^{2}\phi_{2}\,(\textrm{d}\phi_{3}^{2}+\sin^{2}\phi_{3}\,\textrm{d}\phi_{4}^{2})\right]\right\\}\,,$
where $h^{2}=(1-\alpha\,r^{-3})$. The Schwarzschild metric in higher
dimensions is sometimes also called the Tangherlini metric [98]. A convenient
null frame on this space-time is defined by:
$\displaystyle\boldsymbol{e}_{1}=\frac{1}{2}\left(h\partial_{r}+h^{-1}\partial_{t}\right)\;\;;\;\;\boldsymbol{e}_{2}=\frac{1}{2}\left(\frac{1}{r}\partial_{\phi_{1}}+\frac{i}{r\sin\phi_{1}}\partial_{\phi_{2}}\right)$
$\displaystyle\boldsymbol{e}_{3}=\frac{1}{2}\left(\frac{1}{r\sin\phi_{1}\sin\phi_{2}}\partial_{\phi_{3}}+\frac{i}{r\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}}\partial_{\phi_{4}}\right)\;\;;$
$\displaystyle\boldsymbol{e}_{4}=\frac{1}{2}\left(h\partial_{r}-h^{-1}\partial_{t}\right)\;\;;\;\;\boldsymbol{e}_{5}=\frac{1}{2}\left(\frac{1}{r}\partial_{\phi_{1}}-\frac{i}{r\sin\phi_{1}}\partial_{\phi_{2}}\right)$
$\displaystyle\boldsymbol{e}_{6}=\frac{1}{2}\left(\frac{1}{r\sin\phi_{1}\sin\phi_{2}}\partial_{\phi_{3}}-\frac{i}{r\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}}\partial_{\phi_{4}}\right)\,.$
Since this space-time is a vacuum solution its Ricci tensor vanishes, so that
the Riemann tensor is equal to the Weyl tensor. Up to the trivial symmetries,
$C_{abcd}=C_{[ab][cd]}=C_{cdab}$, the non-vanishing components of the Weyl
tensor are:
$\displaystyle
C_{1414}=-\frac{3\alpha}{2r^{5}}\;;\;\;C_{1245}=C_{1346}=C_{1542}=C_{1643}=-\frac{3\alpha}{8r^{5}}\;;$
$\displaystyle C_{2356}=C_{2552}=C_{2653}=C_{3636}=\frac{\alpha}{4r^{5}}\,.$
This reveals that such tensor is of type $D$ on the CMPP classification, with
$\boldsymbol{e}_{1}$ and $\boldsymbol{e}_{4}$ being multiple WANDs [36]. One
can then use table 5.4 to prove that the spinorial equivalent of this Weyl
tensor is:
$\displaystyle\Psi^{AB}_{\phantom{AB}CD}\,=\,-\frac{\alpha}{8r^{5}}[\chi_{1}^{\,A}\chi_{1}^{\,B}\gamma^{1}_{\,C}\gamma^{1}_{\,D}+\chi_{2}^{\,A}\chi_{2}^{\,B}\gamma^{2}_{\,C}\gamma^{2}_{\,D}+\chi_{3}^{\,A}\chi_{3}^{\,B}\gamma^{3}_{\,C}\gamma^{3}_{\,D}+\chi_{4}^{\,A}\chi_{4}^{\,B}\gamma^{4}_{\,C}\gamma^{4}_{\,D}]\;+$
$\displaystyle-2\frac{\alpha}{8r^{5}}[\chi_{1}^{\,(A}\chi_{2}^{\,B)}\gamma^{1}_{\,(C}\gamma^{2}_{\,D)}+\chi_{3}^{\,(A}\chi_{4}^{\,B)}\gamma^{3}_{\,(C}\gamma^{4}_{\,D)}]\;+$
(5.29)
$\displaystyle+3\frac{\alpha}{8r^{5}}[\chi_{1}^{\,(A}\chi_{3}^{\,B)}\gamma^{1}_{\,(C}\gamma^{3}_{\,D)}+\chi_{1}^{\,(A}\chi_{4}^{\,B)}\gamma^{1}_{\,(C}\gamma^{4}_{\,D)}+\chi_{2}^{\,(A}\chi_{3}^{\,B)}\gamma^{2}_{\,(C}\gamma^{3}_{\,D)}+\chi_{2}^{\,(A}\chi_{4}^{\,B)}\gamma^{2}_{\,(C}\gamma^{4}_{\,D)}]\,.$
It is then immediate to verify that the matrix representation of the operator
$\mathcal{C}^{+}$ on the basis $\\{\boldsymbol{T}_{r}\\}$, defined in section
5.3, is given by:
$\mathcal{C}^{+}_{rs}\,=\,-\frac{\alpha}{16r^{5}}\,\textrm{diag}(2,2,-3,-3,2,-3,-3,2,2,2)\,.$
Leading us to the conclusion that the algebraic type of the Weyl tensor of the
6-dimensional Schwarzschild space-time is $[(1,1,1,1,1,1),(1,1,1,1)|\,]$.
Using the expressions for the null frame $\\{\boldsymbol{e}_{a}\\}$ defined
above, it is straightforward to compute the following Lie brackets:
$\displaystyle[\boldsymbol{e}_{1},\boldsymbol{e}_{2}]=-\frac{h}{2r}\boldsymbol{e}_{2}\;\;;\;\;[\boldsymbol{e}_{1},\boldsymbol{e}_{3}]=-\frac{h}{2r}\boldsymbol{e}_{3}\;\;;\;\;[\boldsymbol{e}_{1},\boldsymbol{e}_{4}]=\frac{3\alpha}{4r^{4}}h^{-1}(\boldsymbol{e}_{1}-\boldsymbol{e}_{4})\;;$
$\displaystyle[\boldsymbol{e}_{2},\boldsymbol{e}_{3}]=-\frac{1}{2r}(\cot\phi_{1}+i\frac{\cot\phi_{2}}{\sin\phi_{1}})\boldsymbol{e}_{3}\;\;;\;\;[\boldsymbol{e}_{2},\boldsymbol{e}_{4}]=\frac{h}{2r}\boldsymbol{e}_{2}\;;$
$\displaystyle[\boldsymbol{e}_{2},\boldsymbol{e}_{5}]=\frac{\cot\phi_{1}}{2r}(\boldsymbol{e}_{2}-\boldsymbol{e}_{5})\;\;;\;\;[\boldsymbol{e}_{2},\boldsymbol{e}_{6}]=-\frac{1}{2r}(\cot\phi_{1}+i\frac{\cot\phi_{2}}{\sin\phi_{1}})\boldsymbol{e}_{6}\;;$
$\displaystyle[\boldsymbol{e}_{3},\boldsymbol{e}_{4}]=\frac{h}{2r}\boldsymbol{e}_{3}\;\;;\;\;[\boldsymbol{e}_{3},\boldsymbol{e}_{6}]=\frac{\cot\phi_{3}}{2r\sin\phi_{1}\sin\phi_{2}}(\boldsymbol{e}_{3}-\boldsymbol{e}_{6})\,.$
The missing commutators can be obtained by taking the complex conjugate of
these relations and using eq. (5.17). From these commutation relations one
conclude that the distributions spanned by
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$,
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{5},\boldsymbol{e}_{6}\\}$,
$\\{\boldsymbol{e}_{4},\boldsymbol{e}_{2},\boldsymbol{e}_{6}\\}$,
$\\{\boldsymbol{e}_{4},\boldsymbol{e}_{5},\boldsymbol{e}_{3}\\}$,
$\\{\boldsymbol{e}_{4},\boldsymbol{e}_{5},\boldsymbol{e}_{6}\\}$,
$\\{\boldsymbol{e}_{4},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$,
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{5},\boldsymbol{e}_{3}\\}$ and
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{6}\\}$ are all
integrable. Since the pure spinors associated to these maximally isotropic
distributions are respectively $\boldsymbol{\chi}_{1}$,
$\boldsymbol{\chi}_{2}$, $\boldsymbol{\chi}_{3}$, $\boldsymbol{\chi}_{4}$,
$\boldsymbol{\gamma}_{1}$, $\boldsymbol{\gamma}_{2}$,
$\boldsymbol{\gamma}_{3}$ and $\boldsymbol{\gamma}_{4}$, it is natural to
wonder whether such spinors obey the algebraic condition of theorem 16. Using
eq. (5.29) it is simple matter to verify that the integrability constraints
$\chi_{p}^{\,[E}\Psi^{A][B}_{\phantom{A][B}CD}\,\chi_{p}^{\,F]}\chi_{p}^{\,C}=0\quad\textrm{and}\quad\gamma^{p}_{[E}\Psi^{AB}_{\phantom{AB}C][D}\gamma^{p}_{F]}\gamma^{p}_{A}=0$
are, indeed, valid for all $p\in\\{1,2,3,4\\}$. In addition to these eight
distributions, there exist infinitely many independent maximally isotropic
integrable distributions on this manifold999The author thanks Marcello
Ortaggio for pointing out this fact. Comments in the same lines can also be
found in section 8.3 of [65], where it was argued that Robinson-Trautman
space-times with transverse spaces of constant curvature admit infinitely many
isotropic structures. See also the footnote in the section 5.2 of reference
[64].. Since the 4-sphere is conformally flat, it follows that one can manage
to find a coordinate system in which the metric of this space-time takes the
form
$\textrm{ds}^{2}=-h^{2}\,\textrm{dt}^{2}\,+\,h^{-2}\,\textrm{dr}^{2}\,+\,r^{2}g(y_{p})\,\left[dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+dy_{4}^{2}\right]\,.$
Defining $\boldsymbol{k}_{1}=a^{p}\partial_{y_{p}}$ and
$\boldsymbol{k}_{2}=b^{p}\partial_{y_{p}}$ with $a^{p}$ and $b^{p}$ being
complex constants such that
$\delta_{pq}a^{p}a^{q}=\delta_{pq}b^{p}b^{q}=\delta_{pq}a^{p}b^{q}=0$, then it
is immediate to verify that the maximally isotropic distributions
$\\{\boldsymbol{e}_{1},\boldsymbol{k}_{1},\boldsymbol{k}_{2}\\}$ and
$\\{\boldsymbol{e}_{4},\boldsymbol{k}_{1},\boldsymbol{k}_{2}\\}$ are
integrable for all $a^{p},b^{p}$ [91]. As a final comment it is worth
remarking that there exist some pure spinors that obey the integrability
condition while the associated maximally isotropic distributions are not
integrable, which is possible because the Weyl tensor of the Schwarzschild
space-time does not satisfy the generality condition assumed on ref. [67]. For
instance, although the pure spinor
$\boldsymbol{\eta}=\boldsymbol{\chi}_{1}+f\boldsymbol{\chi}_{2}$ obeys the
constraint $\eta^{[E}\Psi^{A][B}_{\phantom{A][B}CD}\,\eta^{F]}\eta^{C}=0$ for
all functions $f$, its associated distribution,
$Span\\{\boldsymbol{e}_{1},(\boldsymbol{e}_{2}+f\boldsymbol{e}_{6}),(\boldsymbol{e}_{3}-f\boldsymbol{e}_{5})\\}$,
is not integrable if $f\neq 0$.
### Chapter 6 Integrability and Weyl Tensor Classification in All Dimensions
Throughout this thesis it has been repeatedly advocated that, since the Petrov
classification and the Goldberg-Sachs (GS) theorem have played a prominent
role in the development of general relativity in 4 dimensions, it is worth
looking for higher-dimensional generalizations of these results. Hopefully
this could be helpful in the search of new exact solutions to Einstein’s
equation in higher dimensions, as it proved to be in 4 dimensions [22, 24]. It
is also worth mentioning that recently it was made a connection between
Navier-Stokes’ and Einstein’s equations [99] in which the algebraic
classification of the Weyl tensor plays an important role, which gives a
further motivation for a investigation on these subjects.
In the previous chapter it was taken advantage of the spinorial language in
order to define an algebraic classification for the Weyl tensor. Such
classification proved to be valuable because it is connected to a
generalization of the GS theorem in 6 dimensions. Given the success of the
spinorial formalism in 4 and 6 dimensions it seems reasonable trying to use
this language in higher-dimensional spaces. However, it is hard to deal with
spinors in arbitrary dimensions since some important details can heavily
depend on the specific dimension. Moreover, in dimensions greater than 6 not
all Weyl spinors are pure, which represents a further drawback. In spite of
these difficulties this path was adopted in [92].
The aim of the present chapter is to define an algebraic classification for
the Weyl tensor valid in arbitrary dimension and associate such classification
with integrability properties using the vectorial formalism. Here the Weyl
tensor will be used to define operators acting on the bundle of differential
forms, so that the refined Segre classification of these operators provides an
algebraic classification for the Weyl tensor. In this approach the Petrov
classification and the spinorial classification defined in chapter 5 emerge as
special cases. The material presented here is based in the article [70].
As in the previous chapters it will be assumed that the manifold is
complexified, so that the results can be carried to any signature by a
suitable choice of reality condition. For simplicity the metric is supposed to
be real, so that the Weyl tensor is real. All calculations here are local,
therefore global issues shall be neglected.
#### 6.1 Algebraic Classification for the Weyl Tensor
In what follows the reader is assumed to be familiar with the formalism of
differential forms, for a quick review see section 1.6 of chapter 1. Let
$(M,\boldsymbol{g})$ be an $n$-dimensional manifold of signature $s$. Since we
are interested on local results we can always assume that such manifold is
endowed with a volume-form $\epsilon_{\mu_{1}\ldots\mu_{n}}$. By means of this
tensor one can define the Hodge dual of a $p$-form as in equation (1.16). For
clearness on the notation we shall abstractly denote the Hodge dual map by
$\mathcal{H}_{p}$:
$\left\\{\begin{array}[]{ll}\mathcal{H}_{p}:\Gamma(\wedge^{p}M)\rightarrow\Gamma(\wedge^{p}M)\\\
\boldsymbol{F}\;\mapsto\;\mathcal{H}_{p}(\boldsymbol{F})\,=\,\star\boldsymbol{F}\,.\\\
\end{array}\right.$
Where $\Gamma(\wedge^{p}M)$ is the space of $p$-forms111Actually this operator
is defined just locally. So that, formally, its domain should be written as
$\Gamma(\wedge^{p}M)|_{N_{x}}$, where $N_{x}\subset M$ is the neighborhood of
some point $x\in M$.. Denote the identity operator on $\Gamma(\wedge^{p}M)$ by
$\boldsymbol{1}_{p}$. Then using the complete skew-symmetry of the volume-form
along with equation (1.15) it is immediate to see that the following identity
holds:
$\mathcal{H}_{n-p}\,\mathcal{H}_{p}\,=\,(-1)^{[(n-p)p+\frac{n-s}{2}]}\,\,\boldsymbol{1}_{p}\,.$
(6.1)
The Weyl tensor $C_{\mu\nu\rho\sigma}$ is the trace-less part of the Riemann
tensor and, therefore, has the following symmetries:
$C_{\mu\nu\rho\sigma}=C_{[\mu\nu][\rho\sigma]}=C_{\rho\sigma\mu\nu}\;\;;\;\;C_{\mu[\nu\rho\sigma]}=0\;\;;\;\;C^{\mu}_{\phantom{\mu}\nu\mu\sigma}=0\,.$
Inspired by equation (5.26) one can use this tensor to introduce an operator
$\mathcal{C}_{p}$ acting on the bundle of $p$-forms, with $p\geq 2$, whose
definition is [70]:
$\left\\{\begin{array}[]{ll}\mathcal{C}_{p}:\Gamma(\wedge^{p}M)\rightarrow\Gamma(\wedge^{p}M)\\\
\boldsymbol{F}\;\mapsto\;\mathcal{C}_{p}(\boldsymbol{F})\,=\,\frac{1}{p!}\,\left(C^{\rho\sigma}_{\phantom{\rho\sigma}\nu_{1}\nu_{2}}F_{\nu_{3}\ldots\nu_{p}\,\rho\sigma}\right)\,dx^{\nu_{1}}\wedge
dx^{\nu_{2}}\wedge\ldots\wedge dx^{\nu_{p}}\,.\end{array}\right.$ (6.2)
Note that for $p=2$ this operator reduces to the well-known bivector operator,
$B_{\mu\nu}\mapsto C_{\mu\nu\rho\sigma}B^{\rho\sigma}$, whose properties in
arbitrary dimension were explored in [34]. Furthermore, in 6 dimensions when
$p=3$ such operator is proportional to the Weyl operator defined in the
previous chapter using spinors, see eq. (5.26). Now let us prove that
$\mathcal{C}_{p}$ commutes with the Hodge dual map.
$\displaystyle\left[\mathcal{H}_{p}\right.$
$\displaystyle\left.\mathcal{C}_{p}(F)\right]^{\nu_{1}\ldots\nu_{n-p}}\;=\;\frac{1}{p!}\,\epsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{1}\ldots\nu_{n-p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,F_{\mu_{3}\ldots\mu_{p}\alpha\beta}\,$
$\displaystyle=\,\frac{1}{p!}\,\epsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{1}\ldots\nu_{n-p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\left[\mathcal{H}_{n-p}\mathcal{H}_{p}(F)\right]_{\mu_{3}\ldots\mu_{p}\alpha\beta}\,(-1)^{[(n-p)p+\frac{n-s}{2}]}$
$\displaystyle=\,\frac{(-1)^{[(n-p)p+\frac{n-s}{2}]}}{p!\,(n-p)!}\,\epsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{1}\ldots\nu_{n-p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\,\epsilon_{\sigma_{1}\ldots\sigma_{n-p}\mu_{3}\ldots\mu_{p}\alpha\beta}\left[\mathcal{H}_{p}(F)\right]^{\sigma_{1}\ldots\sigma_{n-p}}$
$\displaystyle=\,\frac{(p-2)!\,(n-p+2)!}{p!\,(n-p)!}\,\delta_{\alpha}^{\;[\mu_{1}}\delta_{\beta}^{\;\mu_{2}}\delta_{\sigma_{1}}^{\;\nu_{1}}\ldots\delta_{\sigma_{n-p}}^{\;\nu_{n-p}]}\,C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\left[\mathcal{H}_{p}(F)\right]^{\sigma_{1}\ldots\sigma_{n-p}}$
$\displaystyle=\,C^{\phantom{\mu_{1}\mu_{2}}[\nu_{1}\nu_{2}}_{\mu_{1}\mu_{2}}\,\left[\mathcal{H}_{p}(F)\right]^{\nu_{3}\ldots\nu_{n-p}]\mu_{1}\mu_{2}}\;=\;\left[\mathcal{C}_{n-p}\,\mathcal{H}_{p}(F)\right]^{\nu_{1}\ldots\nu_{n-p}}\,.$
Where equations (1.15) and (6.1) were used. This proves that the following
important relation holds:
$\mathcal{H}_{p}\,\mathcal{C}_{p}\;=\;\mathcal{C}_{n-p}\,\mathcal{H}_{p}\,.$
(6.3)
In particular, since the operator $\mathcal{H}_{p}$ is invertible, see eq.
(6.1), the above relation implies that
$\mathcal{C}_{n-p}=\mathcal{H}_{p}\mathcal{C}_{p}\mathcal{H}_{p}^{-1}$. So the
operators $\mathcal{C}_{n-p}$ and $\mathcal{C}_{p}$ are connected by a
similarity transformation. Recall that on equation (6.2) the operator
$\mathcal{C}_{p}$ was not defined for $p=0$ and $p=1$. However, we can use
equation (6.3) in order to define these operators in terms of
$\mathcal{C}_{n}$ and $\mathcal{C}_{n-1}$. For instance,
$\displaystyle\left[\,\mathcal{C}_{1}(F)\,\right]^{\mu}=$
$\displaystyle\left[\,\mathcal{H}_{1}^{-1}\,\mathcal{C}_{n-1}\,\mathcal{H}_{1}(F)\,\right]^{\mu}\propto\,\left[\,\mathcal{H}_{n-1}\,\mathcal{C}_{n-1}\,\mathcal{H}_{1}(F)\,\right]^{\mu}$
$\displaystyle\propto$
$\displaystyle\;\,\epsilon^{\nu_{1}\ldots\nu_{n-1}\mu}\,C^{\alpha\beta}_{\phantom{\alpha\beta}\nu_{1}\nu_{2}}\,\epsilon_{\sigma\nu_{3}\ldots\nu_{n-1}\alpha\beta}\,F^{\sigma}$
$\displaystyle\propto$
$\displaystyle\;\,\delta_{\alpha}^{\;[\nu_{1}}\delta_{\beta}^{\;\nu_{2}}\delta_{\sigma}^{\;\mu]}\,C^{\alpha\beta}_{\phantom{\alpha\beta}\nu_{1}\nu_{2}}\,F^{\sigma}=C^{[\nu_{1}\nu_{2}}_{\phantom{[\nu_{1}\nu_{2}}\nu_{1}\nu_{2}}\,F^{\mu]}=0\,.$
Where equation (1.15) and the trace-less property of the Weyl tensor were
used. In the same fashion one can prove that the operator $\mathcal{C}_{0}$ is
identically zero. Therefore, using these results along with eq. (6.3), we
conclude that in a manifold of dimension $n$ we have:
$\mathcal{C}_{0}\equiv 0\quad;\quad\mathcal{C}_{1}\equiv
0\quad;\quad\mathcal{C}_{n-1}=0\quad;\quad\mathcal{C}_{n}=0\,.$ (6.4)
The refined Segre types of the operators $\mathcal{C}_{p}$, for all possible
values of $p$, provide an algebraic classification for the Weyl tensor. But,
because of equation (6.4), we do not need to worry about the cases $p=0$,
$p=1$, $p=n-1$ and $p=n$. Moreover, since $\mathcal{C}_{p}$ and
$\mathcal{C}_{n-p}$ are connected by a similarity transformation they have the
same algebraic type according to the refined Segre classification. Therefore,
we just need to consider the values of $p$ between 2 and $n/2$. _So the
algebraic classification for the Weyl tensor established here amounts to
gathering the refined Segre types of the operators $\mathcal{C}_{p}$ for the
integer values of $p$ contained on the interval $2\leq p\leq n/2$_ [70].
##### 6.1.1 Inner Product of $p$-forms
It will prove to be valuable introducing the following symmetric inner product
on the space of $p$-forms:
$\langle\boldsymbol{F},\boldsymbol{K}\rangle\,\equiv\,F^{\nu_{1}\nu_{2}\ldots\nu_{p}}\,K_{\nu_{1}\nu_{2}\ldots\nu_{p}}\,.$
(6.5)
Where in the above equation $\boldsymbol{F}$ and $\boldsymbol{K}$ are
$p$-forms. Since the metric $\boldsymbol{g}$ is non-degenerate it follows that
the inner product $\langle\,,\rangle$ is also non-degenerate. Moreover, using
the Weyl tensor symmetry $C_{\mu\nu\rho\sigma}=C_{\rho\sigma\mu\nu}$ it is
trivial verifying that the operator $\mathcal{C}_{p}$ is self-adjoint with
respect to such inner product:
$\langle\boldsymbol{F},\mathcal{C}_{p}(\boldsymbol{K})\rangle\,=\,\langle\mathcal{C}_{p}(\boldsymbol{F}),\boldsymbol{K}\rangle\,.$
Now let $\\{\boldsymbol{F}_{r}\\}$ be some basis for the space of
$p$-forms222Actually, because of topological obstructions, generally we can
define such basis just locally. Therefore, we have
$\boldsymbol{F}_{r}\in\Gamma(\wedge^{p}M)|_{N_{x}}$. Where, formally,
$\Gamma(\wedge^{p}M)|_{N_{x}}$ is the restriction of the space of sections of
the $p$-form bundle to the neighborhood $N_{x}$ of some point $x\in M$.
Roughly speaking, $\Gamma(\wedge^{p}M)|_{N_{x}}$ is the space spanned by the
$p$-form fields in the neighborhood $N_{x}$., with333The indices $r,s,\ldots$
run from 1 to $\frac{n!}{p!(n-p)!}$.
$\langle\boldsymbol{F}_{r},\boldsymbol{F}_{s}\rangle=f_{rs}$. Since this inner
product is non-degenerate it follows that the matrix $f_{rs}$ is invertible,
let us denote its inverse by $f^{rs}$. Thus defining the $p$-forms
$\boldsymbol{F}^{r}\equiv f^{rs}\boldsymbol{F}_{s}$ we find that
$\langle\boldsymbol{F}_{r},\boldsymbol{F}^{s}\rangle=\delta_{r}^{\,s}$. So if
$\boldsymbol{F}$ is some $p$-form then its expansion on the basis
$\\{\boldsymbol{F}_{r}\\}$ is given by
$\boldsymbol{F}=\langle\boldsymbol{F}^{r},\boldsymbol{F}\rangle\,\boldsymbol{F}_{r}$.
Using index notation, the latter equation is tantamount to:
$\left(F^{r}\right)_{\nu_{1}\nu_{2}\ldots\nu_{p}}\,\left(F_{r}\right)^{\mu_{1}\mu_{2}\ldots\mu_{p}}\,=\,\delta_{\nu_{1}}^{\;[\mu_{1}}\delta_{\nu_{2}}^{\;\mu_{2}}\ldots\delta_{\nu_{p}}^{\;\mu_{p}]}\,.$
(6.6)
The action of the operator $\mathcal{C}_{p}$ on this basis is given by:
$\mathcal{C}_{p}(\boldsymbol{F}_{s})\,\equiv\,\boldsymbol{F}_{r}\,\mathcal{C}_{rs}\;,\;\;\textrm{where
}\;\mathcal{C}_{rs}\,=\,\langle\boldsymbol{F}^{r},\mathcal{C}_{p}(\boldsymbol{F}_{s})\rangle\,.$
Using this one can easily prove that the trace of $\mathcal{C}_{p}$ is zero.
Indeed, by means of (6.6) we have
$\displaystyle\textrm{tr}(\mathcal{C}_{p})$
$\displaystyle=\mathcal{C}_{rr}=\left(F^{r}\right)^{\mu_{1}\mu_{2}\ldots\mu_{p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\left(F_{r}\right)_{\mu_{3}\mu_{4}\ldots\mu_{p}\alpha\beta}$
$\displaystyle=C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\delta_{\alpha}^{\;[\mu_{1}}\delta_{\beta}^{\;\mu_{2}}\delta_{\mu_{3}}^{\;\mu_{3}}\ldots\delta_{\mu_{p}}^{\;\mu_{p}]}\propto
C^{\alpha\beta}_{\phantom{\alpha\beta}\alpha\beta}=0\,.$ (6.7)
The signature of the inner product $\langle\,,\rangle$ depends on the
signature of the metric $\boldsymbol{g}$. In particular, if the metric is
Euclidean then it is immediate to verify that the inner product defined in
(6.5) is positive-definite. Therefore, since the operator $\mathcal{C}_{p}$ is
real and self-dual with respect to $\langle\,,\rangle$, it follows that on the
Euclidean signature $\mathcal{C}_{p}$ can be diagonalized. More explicitly, if
the metric $\boldsymbol{g}$ is positive-definite then so will be
$\langle\,,\rangle$, which means that, locally (in a neighborhood $N_{x}$),
one can find a real basis $\\{\hat{\boldsymbol{F}}_{r}\\}$ for
$\Gamma(\wedge^{p}M)$ such that
$\langle\hat{\boldsymbol{F}}_{r},\hat{\boldsymbol{F}}_{s}\rangle=\delta_{rs}$.
The matrix representation of $\mathcal{C}_{p}$ in this basis is then real and
symmetric and, therefore, can be diagonalized. This represents a huge
limitation on the possible algebraic types that the operator $\mathcal{C}_{p}$
can have. Let us state this as a theorem [70]:
###### Theorem 18
When the signature of $\boldsymbol{g}$ is Euclidean the operator
$\mathcal{C}_{p}$ admits a trace-less diagonal matrix representation with real
eigenvalues. Particularly, this guarantees that on the refined Segre
classification of this operator all numbers inside the square bracket are
equal to 1.
##### 6.1.2 Even Dimensions
In this subsection it will be proved that a particularly interesting
simplification occurs when the dimension of the manifold is even. If the
dimension of $(M,\boldsymbol{g})$ is $n=2m$, with $m$ being an integer, then
equation (6.1) implies that
$\mathcal{H}_{m}\,\mathcal{H}_{m}\,=\,(-1)^{\frac{s}{2}}\,\,\boldsymbol{1}_{m}\;\Longrightarrow\;\mathcal{H}_{m}\,\mathcal{H}_{m}\,=\,\varrho^{2}\,\boldsymbol{1}_{m}\;\;\left\\{\begin{array}[]{ll}\varrho=1\;\textrm{
if $\frac{s}{2}$ is even}\\\ \varrho=i\;\textrm{ if $\frac{s}{2}$ is odd.}\\\
\end{array}\right.$
So locally the space of $m$-forms can be split into the direct sum of two
subspaces of the same dimension, the eigenspaces of $\mathcal{H}_{m}$:
$\Gamma(\wedge^{m}M)\,=\,\Lambda^{m+}\oplus\Lambda^{m-}\,,\quad\Lambda^{m\pm}=\\{\,\boldsymbol{F}\in\Gamma(\wedge^{m}M)\,|\,\,\mathcal{H}_{m}(\boldsymbol{F})=\pm\varrho\,\boldsymbol{F}\,\\}\,.$
An element of $\Lambda^{m+}$ is said to be a self-dual $m$-form, while an
element of $\Lambda^{m-}$ is called an anti-self-dual $m$-form. Note that
these spaces are interchanged when we multiply the volume-form by $-1$. The
subspaces $\Lambda^{m\pm}$ can equivalently be defined as follows:
$\Lambda^{m\pm}\,=\,\left\\{\;\left(\boldsymbol{F}\,\pm\,\frac{1}{\varrho}\,\mathcal{H}_{m}(\boldsymbol{F})\right)\;|\;\,\boldsymbol{F}\in\Gamma(\wedge^{m}M)\;\right\\}\;;\quad\left\\{\begin{array}[]{ll}\varrho=1\;\textrm{
if $\frac{s}{2}$ is even}\\\ \varrho=i\;\textrm{ if $\frac{s}{2}$ is odd.}\\\
\end{array}\right.$
From which we see that if $\frac{s}{2}$ is even then the spaces
$\Lambda^{m\pm}$ are real, while if $\frac{s}{2}$ is odd then the elements of
$\Lambda^{m\pm}$ must be complex. Furthermore, since the operator
$\mathcal{H}_{m}$ is real, if $\frac{s}{2}$ is odd then the complex conjugate
of a self-dual $m$-form is anti-self-dual. Note also that the operator
$\mathcal{H}_{m}$ can be self-adjoint or anti-self-adjoint with respect to the
inner product $\langle\,,\rangle$ depending on the dimension of the manifold:
$\displaystyle\langle\boldsymbol{F},\mathcal{H}_{m}(\boldsymbol{K})\rangle\,$
$\displaystyle=\,\frac{1}{m!}\,\epsilon_{\nu_{1}\ldots\nu_{m}\mu_{1}\ldots\mu_{m}}\,F^{\mu_{1}\ldots\mu_{m}}\,K^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=\,\frac{(-1)^{m^{2}}}{m!}\,\epsilon_{\mu_{1}\ldots\mu_{m}\nu_{1}\ldots\nu_{m}}\,F^{\mu_{1}\ldots\mu_{m}}\,K^{\nu_{1}\ldots\nu_{m}}\,=\,(-1)^{m}\,\langle\mathcal{H}_{m}(\boldsymbol{F}),\boldsymbol{K}\rangle$
Using the above equation one can easily see that if $m$ is even then the inner
product $\langle\boldsymbol{F}^{+},\boldsymbol{K}^{-}\rangle$ vanishes
whenever $\boldsymbol{F}^{+}\in\Lambda^{m+}$ and
$\boldsymbol{K}^{-}\in\Lambda^{m-}$. Analogously, if $m$ is odd then the inner
products $\langle\boldsymbol{F}^{+},\boldsymbol{K}^{+}\rangle$ and
$\langle\boldsymbol{F}^{-},\boldsymbol{K}^{-}\rangle$ vanish for all
$\boldsymbol{F}^{+},\boldsymbol{K}^{+}\in\Lambda^{m+}$ and
$\boldsymbol{F}^{-},\boldsymbol{K}^{-}\in\Lambda^{m-}$. These results are
summarized by the below theorem [70].
###### Theorem 19
Let $(M,\boldsymbol{g})$ be a manifold of signature $s$ and dimension $n=2m$,
with $m$ being an integer. Then the Hodge dual map splits the space of
$m$-forms into a direct sum of its eigenspaces,
$\Gamma(\wedge^{m}M)=\Lambda^{m+}\oplus\Lambda^{m-}$. When $s$ is a multiple
of 4 the spaces $\Lambda^{m+}$ and $\Lambda^{m-}$ are both real, otherwise
they must be complex conjugates of each other. Furthermore, if $m$ is even
then the spaces $\Lambda^{m+}$ and $\Lambda^{m-}$ are orthogonal to each
other, while if $m$ is odd both spaces $\Lambda^{m\pm}$ are isotropic with
respect to the inner product $\langle\,,\rangle$.
Now plugging $n=2m$ and $p=m$ on equation (6.3) yields that the operators
$\mathcal{C}_{m}$ and $\mathcal{H}_{m}$ commute. Thus the spaces
$\Lambda^{m+}$ and $\Lambda^{m-}$ are both preserved by the action of
$\mathcal{C}_{m}$. So, the latter operator can be written as the direct sum of
its restrictions to the spaces $\Lambda^{m\pm}$:
$\mathcal{C}_{m}\,=\,\mathcal{C}^{+}\oplus\mathcal{C}^{-}\,,\quad\mathcal{C}^{\pm}\,\equiv\,\frac{1}{2}\left(\mathcal{C}_{m}\,\pm\,\frac{1}{\varrho}\,\mathcal{C}_{m}\mathcal{H}_{m}\right)\,.$
(6.8)
Note that the action of $\mathcal{C}^{+}$ on an element of $\Lambda^{m-}$
gives zero, as well as the restriction of $\mathcal{C}^{-}$ to $\Lambda^{m+}$
is identically zero. Therefore, generally it is useful to assume that the
domains of the operators $\mathcal{C}^{\pm}$ are the spaces $\Lambda^{m\pm}$,
instead of the whole bundle of $m$-forms. It is worth remarking that eq. (6.8)
imposes huge restrictions on the possible algebraic types of the operator
$\mathcal{C}_{m}$.
A special phenomenon happens when $m$ is odd. In this case, because of theorem
19, one can always introduce a basis444Now the indices $r,s$ and $t$ run from
1 to $\frac{1}{2}\cdot\frac{(2m)!}{m!m!}$ . $\\{\boldsymbol{F}^{+}_{r}\\}$ for
$\Lambda^{m+}$ and a basis $\\{\boldsymbol{F}^{-}_{r}\\}$ for $\Lambda^{m-}$
such that
$\langle\boldsymbol{F}^{+}_{r},\boldsymbol{F}^{-}_{s}\rangle=\delta_{rs}$.
Indeed, since $\langle\,,\rangle$ is non-degenerate we just need to start with
a basis for $\Lambda^{m+}$ and a basis for $\Lambda^{m-}$ and then use the
Gram-Schmidt process in order to redefine the latter. Thus when $m$ is odd the
operators have the following matrix representations:
$\left.\begin{array}[]{ll}\mathcal{C}_{rs}^{+}\,=\,\langle\,\boldsymbol{F}^{-}_{r},\mathcal{C}^{+}(\boldsymbol{F}^{+}_{s})\,\rangle\,=\,\langle\,\boldsymbol{F}^{-}_{r},\mathcal{C}_{m}(\boldsymbol{F}^{+}_{s})\,\rangle\\\
\mathcal{C}_{rs}^{-}\,=\,\langle\,\boldsymbol{F}^{+}_{r},\mathcal{C}^{-}(\boldsymbol{F}^{-}_{s})\,\rangle\,=\,\langle\,\boldsymbol{F}^{+}_{r},\mathcal{C}_{m}(\boldsymbol{F}^{-}_{s})\,\rangle\\\
\end{array}\;\right\\}\;\Longrightarrow\;\mathcal{C}_{rs}^{+}\,=\,\mathcal{C}_{sr}^{-}\,.$
Where on the last step it was used the fact that $\mathcal{C}_{m}$ is self-
adjoint. Thus, when $m$ is odd the matrix representation of $\mathcal{C}^{+}$
is the transpose of the matrix representation of $\mathcal{C}^{-}$ and,
therefore, these operators have the same algebraic type. So if the dimension
$n$ is even but not a multiple of four, classify $\mathcal{C}_{\frac{n}{2}}$
is tantamount to classify $\mathcal{C}^{+}$.
In the same vein, if the signature $s$ is not a multiple of 4 then the spaces
$\Lambda^{m+}$ and $\Lambda^{m-}$ are connected by complex conjugation, see
theorem 19. Therefore, in this case the degrees of freedom of the operators
$\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ are connected by a reality condition.
More precisely, the operator $\mathcal{C}^{+}$ is the complex conjugate of
$\mathcal{C}^{-}$, which can be easily seen from equation (6.8) along with the
fact that the operators $\mathcal{C}_{m}$ and $\mathcal{H}_{m}$ are both real:
$\frac{s}{2}\;\textrm{ is
odd}\quad\Longrightarrow\quad\mathcal{C}^{\pm}\,=\,\frac{1}{2}\,\left(\,\mathcal{C}_{m}\,\mp\,i\,\mathcal{C}_{m}\,\mathcal{H}_{m}\,\right)\quad\Longrightarrow\quad\mathcal{C}^{+}\,=\,\overline{\mathcal{C}^{-}}\,.$
Thus, in such a case $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ have the same
refined Segre type. So that in order to classify $\mathcal{C}_{m}$ we just
need to compute the algebraic type of $\mathcal{C}^{+}$.
Since there is no scalar that can be constructed using just the Weyl tensor
and the volume-form linearly, it is reasonable to expect that both operators
$\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ have vanishing trace. Indeed, using
(6.8) along with the fact that $\mathcal{C}_{m}$ is trace-less, see eq. (6.7),
it follows that:
$\displaystyle\operatorname{tr}(\mathcal{C}^{\pm})\,$
$\displaystyle=\,\frac{\pm
1}{2\varrho}\operatorname{tr}(\mathcal{C}_{m}\mathcal{H}_{m})\,\propto\,(F^{r})^{\;\mu_{1}\ldots\mu_{m}}\,C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\epsilon^{\nu_{1}\ldots\nu_{m}}_{\phantom{\nu_{1}\ldots\nu_{m}}\mu_{3}\ldots\mu_{m}\alpha\beta}\,(F_{r})_{\;\nu_{1}\ldots\nu_{m}}$
$\displaystyle\propto\,C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\epsilon^{\nu_{1}\ldots\nu_{m}}_{\phantom{\nu_{1}\ldots\nu_{m}}\mu_{3}\ldots\mu_{m}\alpha\beta}\,\delta_{[\nu_{1}}^{\;\mu_{1}}\ldots\delta_{\nu_{m}]}^{\;\mu_{m}}\,=\,C_{\alpha\beta\mu_{1}\mu_{2}}\,\epsilon^{\alpha\beta\mu_{1}\ldots\mu_{m}}_{\phantom{\alpha\beta\mu_{1}\ldots\mu_{m}}\mu_{3}\ldots\mu_{m}}\,=\,0\,.$
Where on the last step it was used the Bianchi identity,
$C_{[\mu\nu\rho]\sigma}=0$. The previous results then lead us to the following
theorem [70].
###### Theorem 20
In a manifold of even dimension $n=2m$ the operator $\mathcal{C}_{m}$ is the
direct sum of its restrictions to the spaces $\Lambda^{m\pm}$,
$\mathcal{C}_{m}=\mathcal{C}^{+}\oplus\mathcal{C}^{-}$. The operators
$\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ have vanishing trace. Moreover, they
carry the same degrees of freedom both when $m$ is odd and when the signature
of the manifold is not a multiple of 4, more precisely the following relations
hold:
(1) $m$ is odd $\;\Rightarrow\;$ $\mathcal{C}^{+}$ is the adjoint of
$\mathcal{C}^{-}$,
$\langle\boldsymbol{F},\mathcal{C}^{+}(\boldsymbol{K})\rangle=\langle\mathcal{C}^{-}(\boldsymbol{F}),\boldsymbol{K}\rangle$
(2) $\frac{s}{2}$ is odd $\;\Rightarrow\;$ $\mathcal{C}^{+}$ is the complex
conjugate of $\mathcal{C}^{-}$, $\mathcal{C}^{+}=\overline{\mathcal{C}^{-}}$.
On the other hand, if $m$ and $\frac{s}{2}$ are both even then the operators
$\mathcal{C}^{+}$ and $\mathcal{C}^{-}$ generally carry different degrees of
freedom. In particular, on the latter case the reality condition relates
$\mathcal{C}^{+}$ with itself as well as $\mathcal{C}^{-}$ with itself, so
that both operators are real.
An immediate consequence of this theorem is that whenever $m$ or $\frac{s}{2}$
are odd the refined Segre type of the operators $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ coincide. Thus, is such cases in order to classify
$\mathcal{C}_{m}$ we just need to compute the refined Segre type of
$\mathcal{C}^{+}$.
Note that the chapters 4 and 5 provide explicit examples for the theorems
proved in the present chapter, let us perform few comparisons. In the previous
chapters it was proved that respectively in 4 and 6 dimensions the operators
$\mathcal{C}_{2}$ and $\mathcal{C}_{3}$ can be diagonalized when the signature
is Euclidean, which agrees with theorem 18. In 4 dimensions we proved that the
operator $\mathcal{C}^{+}$ is the complex conjugate $\mathcal{C}^{-}$ if the
signature is Lorentzian, which endorses theorem 20, since in this case
$\frac{s}{2}=1$. In 6 dimensions it was proved, using the spinorial formalism,
that in a suitable basis $\mathcal{C}^{-}$ is the transpose of
$\mathcal{C}^{+}$, since in such case $m=3$ this agrees with theorem 20.
Finally, recall that in chapter 4 it was shown that in a 4-dimensional
manifold of split signature the operators $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ are both real and independent of each other. Since on the
latter case $m=2$ and $\frac{s}{2}=0$ are both even, this again supports
theorem 20.
In 4 dimensions a manifold is said to be self-dual if $\mathcal{C}^{-}=0$ and
$\mathcal{C}^{+}\neq 0$, see chapter 4. Such manifolds have been widely
studied in the past [100, 89], in particular it has been shown that Einstein’s
vacuum equation on self-dual manifolds reduces to a single second-order
differential equation [100]. Now it is natural wondering whether the notion of
self-dual manifolds can be extended to higher dimensions. According to theorem
20 this is not possible neither if $\frac{n}{2}$ is odd nor if $\frac{s}{2}$
is odd, since in these cases the constraint $\mathcal{C}^{-}=0$ implies
$\mathcal{C}^{+}=0$. However, if the dimension and the signature are both
multiples of four then the self-dual manifolds could, in principle, be
defined. Nevertheless, it turns out that laborious calculations reveal that in
8 dimensions if $\mathcal{C}^{-}$ vanishes then $\mathcal{C}^{+}=0$,
irrespective of the signature being a multiple of four. Although the present
author has worked out only the 8-dimensional case, such result seems to
indicate that the self-dual manifolds cannot be defined if the dimension is
different from 4.
##### 6.1.3 An Elegant Notation
In this subsection it will be introduced an elegant and useful notation to
manage the operators $\mathcal{C}_{p}$. To this end the formalism presented in
section (1.7) will be extensively used. Let $\\{\boldsymbol{e}_{a}\\}$ be a
frame of vector fields on the manifold $(M,\boldsymbol{g})$, with
$\\{\boldsymbol{e}^{a}\\}$ being the dual frame of 1-forms such that
$\boldsymbol{e}^{a}(\boldsymbol{e}_{b})=\delta^{a}_{\phantom{a}b}$. Assuming
that the Ricci tensor vanishes, so that the Riemann tensor is equal to the
Weyl tensor, the curvature 2-form is then defined by
$\mathbb{C}^{a}_{\phantom{a}b}\,\equiv\,\frac{1}{2}\,C^{a}_{\phantom{a}bcd}\,\boldsymbol{e}^{c}\wedge\boldsymbol{e}^{d}\,.$
(6.9)
Now let $\boldsymbol{F}$ be a $p$-form, with $p\geq 2$, then we can associate
to it a set of $(p-2)$-forms defined by
$\mathbb{F}^{\phantom{a}b}_{a}\,\equiv\,\frac{2}{p!}\,F^{\phantom{a}b}_{a\phantom{ab}c_{1}c_{2}\ldots
c_{p-2}}\,\boldsymbol{e}^{c_{1}}\wedge\boldsymbol{e}^{c_{2}}\wedge\ldots\wedge\boldsymbol{e}^{c_{p-2}}\,.$
(6.10)
In particular, note that
$\boldsymbol{F}=\frac{1}{2}\mathbb{F}_{ab}\wedge\boldsymbol{e}^{a}\wedge\boldsymbol{e}^{b}$,
where $\mathbb{F}_{ab}\equiv\mathbb{F}^{\phantom{a}c}_{a}g_{cb}$. Then using
equations (6.9) and (6.10) we have
$\displaystyle\mathbb{C}^{a}_{\phantom{a}b}\wedge\mathbb{F}^{\phantom{a}b}_{a}\,$
$\displaystyle=\,\frac{1}{p!}\,C^{a}_{\phantom{a}bc_{1}c_{2}}\,F^{\phantom{a}b}_{a\phantom{b}c_{3}c_{4}\ldots
c_{p}}\,\boldsymbol{e}^{c_{1}}\wedge\boldsymbol{e}^{c_{2}}\wedge\boldsymbol{e}^{c_{3}}\wedge\ldots\wedge\boldsymbol{e}^{c_{p}}$
$\displaystyle=\,\frac{1}{p!}\,C^{ab}_{\phantom{ab}c_{1}c_{2}}\,F_{c_{3}c_{4}\ldots
c_{p}ab}\,\boldsymbol{e}^{c_{1}}\wedge\ldots\wedge\boldsymbol{e}^{c_{p}}\,=\,\mathcal{C}_{p}(\boldsymbol{F})\,\,\Rightarrow$
$\mathcal{C}_{p}(\boldsymbol{F})\,=\,\mathbb{C}^{a}_{\phantom{a}b}\wedge\mathbb{F}^{\phantom{a}b}_{a}\,.$
(6.11)
Now let us define the $(p-1)$-form
$\textbf{D}\mathbb{F}^{\phantom{a}b}_{a}\equiv
d\mathbb{F}^{\phantom{a}b}_{a}+\boldsymbol{\omega}^{b}_{\phantom{b}c}\wedge\mathbb{F}_{a}^{\phantom{a}c}-\boldsymbol{\omega}^{c}_{\phantom{c}a}\wedge\mathbb{F}_{c}^{\phantom{c}b}$,
where $\boldsymbol{\omega}^{a}_{\phantom{a}b}$ are the connection 1-forms
defined on eq. (1.17). Then taking the exterior derivative of the identity
$\boldsymbol{F}=\frac{1}{2}\mathbb{F}_{ab}\wedge\boldsymbol{e}^{a}\wedge\boldsymbol{e}^{b}$
and using the first Cartan structure equation we find that
$d\boldsymbol{F}=\frac{1}{2}\boldsymbol{e}^{a}\wedge\boldsymbol{e}^{b}\wedge\textbf{D}\mathbb{F}_{ab}$,
where $\textbf{D}\mathbb{F}_{ab}\equiv
g_{bc}\textbf{D}\mathbb{F}^{\phantom{a}c}_{a}$. When the Ricci tensor
vanishes, as assumed here, the second Cartan structure equation is
$\mathbb{C}^{a}_{\phantom{a}b}=d\boldsymbol{\omega}^{a}_{\phantom{a}b}+\boldsymbol{\omega}^{a}_{\phantom{a}c}\wedge\boldsymbol{\omega}^{c}_{\phantom{c}b}$.
Taking the exterior derivative of this relation we easily find that
$d\mathbb{C}^{a}_{\phantom{a}b}=\mathbb{C}^{a}_{\phantom{a}c}\wedge\boldsymbol{\omega}^{c}_{\phantom{c}b}-\boldsymbol{\omega}^{a}_{\phantom{a}c}\wedge\mathbb{C}^{c}_{\phantom{c}b}$.
Then using this result while computing the exterior derivative of equation
(6.11) lead us to the identity
$d\left[\mathcal{C}_{p}(\boldsymbol{F})\right]=\mathbb{C}^{ab}\wedge\textbf{D}\mathbb{F}_{ab}$.
The results of this paragraph are summarized by the following equations:
$\displaystyle
d\boldsymbol{F}\,=\,\frac{1}{2}\,\boldsymbol{e}^{a}\wedge\boldsymbol{e}^{b}\wedge\textbf{D}\mathbb{F}_{ab}\quad\quad;\quad\quad
d\left[\mathcal{C}_{p}(\boldsymbol{F})\right]\,=\,\mathbb{C}^{ab}\wedge\textbf{D}\mathbb{F}_{ab}$
(6.12)
$\displaystyle\textbf{D}\mathbb{F}_{ab}\,\equiv\,g_{bc}\left(d\,\mathbb{F}^{\phantom{a}c}_{a}+\boldsymbol{\omega}^{c}_{\phantom{c}d}\wedge\mathbb{F}_{a}^{\phantom{a}d}-\boldsymbol{\omega}^{d}_{\phantom{d}a}\wedge\mathbb{F}_{d}^{\phantom{d}c}\right)\,.$
As a simple application of this notation, suppose that $\boldsymbol{F}$ is a
$p$-form such that
$\textbf{D}\mathbb{F}_{a}^{\phantom{a}b}=\boldsymbol{\varphi}\wedge\mathbb{F}_{a}^{\phantom{a}b}$
for some 1-form $\boldsymbol{\varphi}$. Then equation (6.12) immediately
implies that:
$d\,\boldsymbol{F}\,=\,\boldsymbol{\varphi}\wedge\boldsymbol{F}\quad\textrm{and}\quad
d\left[\mathcal{C}_{p}(\boldsymbol{F})\right]\,=\,\boldsymbol{\varphi}\wedge\mathcal{C}_{p}(\boldsymbol{F})\,.$
(6.13)
This, in turn, implies that if $\boldsymbol{F}$ is a simple form then,
according to the Frobenius theorem, the vector distribution annihilated by
$\boldsymbol{F}$ is integrable, see section 1.8. Analogously, if
$\mathcal{C}_{p}(\boldsymbol{F})$ is a simple $p$-form then eq. (6.13)
guarantees that the vector distribution annihilated by
$\mathcal{C}_{p}(\boldsymbol{F})$ is integrable.
#### 6.2 Integrability of Maximally Isotropic Distributions
Let $\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2}\\}$ be a vector distribution
generating isotropic planes on a Ricci-flat 4-dimensional manifold, then the
celebrated Goldberg-Sachs theorem states that such distribution is integrable
if, and only if, the 2-form
$\boldsymbol{B}=\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}$ is such that
$\mathcal{C}_{2}(\boldsymbol{B})\propto\boldsymbol{B}$, see chapter 4. A
partial generalization of this theorem was proved in chapter 5 with the help
of a theorem of Taghavi-Chabert [67]. More precisely, it was shown that in 6
dimensions if the operator $\mathcal{C}_{3}$ obeys to certain algebraic
constraints then the manifold admits an integrable maximally isotropic
distribution. The aim of the present section is to generalize this result to
all even dimensions, i.e., express the integrability condition for a maximally
isotropic distribution in terms of algebraic constraints on the operator
$\mathcal{C}_{m}$. From now on in this chapter, we shall assume that the
manifold $(M,\boldsymbol{g})$ has dimension $n=2m$, with $m$ being an integer.
Before proceeding let us set few conventions and recall some important
definitions. Up to a multiplicative factor there exists a one-to-one relation
between vector field distributions and simple forms. More explicitly, if
$Span\\{\boldsymbol{V}_{1},\boldsymbol{V}_{2},\ldots,\boldsymbol{V}_{p}\\}$ is
a $p$-dimensional distribution of vector fields then any non-zero $p$-form
proportional to
$F^{\nu_{1}\ldots\nu_{p}}=p!\,V_{1}^{[\nu_{1}}V_{2}^{\nu_{2}}\ldots
V_{p}^{\nu_{p}]}$ is said to generate such distribution. In abstract notation
we shall right
$\boldsymbol{F}=\boldsymbol{V}_{1}\wedge\boldsymbol{V}_{2}\wedge\ldots\wedge\boldsymbol{V}_{p}$.
A distribution of vector fields is called _isotropic_ if every vector field
$\boldsymbol{V}$ tangent to such distribution has zero norm,
$\boldsymbol{g}(\boldsymbol{V},\boldsymbol{V})=0$. In particular all vector
fields tangent to an isotropic distribution are orthogonal to each other. A
simple form $\boldsymbol{F}$ is then said to be _null_ if its associated
distribution is isotropic. Following the convention adopted in the previous
chapter, a frame
$\\{\boldsymbol{e}_{a}\\}=\\{\boldsymbol{e}_{a^{\prime}},\boldsymbol{e}_{a^{\prime}+m}=\boldsymbol{\theta}^{a^{\prime}}\\}$
of vectors fields is called a _null frame_ whenever the inner products between
the frame vectors are:
$\boldsymbol{g}(\boldsymbol{e}_{a^{\prime}},\boldsymbol{e}_{b^{\prime}})\,=\,0\,=\,\boldsymbol{g}(\boldsymbol{\theta}^{a^{\prime}},\boldsymbol{\theta}^{b^{\prime}})\quad;\quad\boldsymbol{g}(\boldsymbol{e}_{a^{\prime}},\boldsymbol{\theta}^{b^{\prime}})=\frac{1}{2}\,\delta^{\,b^{\prime}}_{a^{\prime}}\,,$
where the indices $a,b,c,\ldots$ run from 1 to $2m$, while the indices
$a^{\prime},b^{\prime},c^{\prime},\ldots$ pertain to the set
$\\{1,2,\ldots,m\\}$. In $n=2m$ dimensions, the maximum dimension that an
isotropic distribution can have is $m$. Therefore, an $m$-dimensional
isotropic distribution is called _maximally isotropic_. In particular, note
that if $\\{\boldsymbol{e}_{a}\\}$ is a null frame then
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$
is a null $m$-form and its associated distribution is maximally isotropic.
As commented in section 5.4, in reference [67] it was proved a theorem that
partially generalizes the GS theorem to higher dimensions. Using the notation
adopted here, such theorem can be conveniently stated as follows: _If the Weyl
tensor of a Ricci-flat manifold is such that
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$, and is generic otherwise555See
footnote 7 of chapter 5 for comments on this generality condition., then the
maximally isotropic distribution $Span\\{\boldsymbol{e}_{a^{\prime}}\\}$ is
integrable._ The intent of the present section is to express the algebraic
condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ in terms of the operator
$\mathcal{C}_{m}$. With this aim it is of particular help to define the
subspaces $\mathcal{A}_{q}\subset\Gamma(\wedge^{m}M)$ as follows:
$\mathcal{A}_{q}\,\equiv\,\\{\,\boldsymbol{F}\in\Gamma(\wedge^{m}M)\;|\;\boldsymbol{e}_{a^{\prime}_{q}}\lrcorner\ldots\boldsymbol{e}_{a^{\prime}_{2}}\lrcorner\boldsymbol{e}_{a^{\prime}_{1}}\lrcorner\boldsymbol{F}\,=\,0\;\;\forall\;a^{\prime}_{1},\ldots,a^{\prime}_{p}\in(1,\ldots,m)\,\\}\,.$
(6.14)
Where $\boldsymbol{e}\lrcorner\boldsymbol{F}$ means the interior product of
the vector field $\boldsymbol{e}$ on the differential form $\boldsymbol{F}$
(see section 1.6). These subspaces can be equivalently defined by:
$\mathcal{A}_{q}\,=\,\textsf{A}_{1}\oplus\textsf{A}_{2}\oplus\cdots\oplus\textsf{A}_{q}\;;\quad\textsf{A}_{q}\,\equiv\,Span\\{\boldsymbol{\theta}^{a^{\prime}_{1}}\wedge\cdots\wedge\boldsymbol{\theta}^{a^{\prime}_{q-1}}\wedge\boldsymbol{e}_{a^{\prime}_{q}}\wedge\cdots\wedge\boldsymbol{e}_{a^{\prime}_{m}}\\}\,.$
Now let us use the notation of section 6.1.3 in order to express the
invariance of the subbundle $\mathcal{A}_{1}$ under the action of
$\mathcal{C}_{m}$ in terms of the Weyl tensor components. If $\boldsymbol{F}$
is an $m$-form pertaining to $\mathcal{A}_{1}$ then
$\boldsymbol{F}\propto\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$.
In particular it follows that $\mathbb{F}_{a^{\prime}b}=0$ and
$\boldsymbol{e}_{a^{\prime}}\lrcorner\mathbb{F}_{bc}=0$, so that eq. (6.11)
implies:
$\displaystyle\boldsymbol{e}_{c^{\prime}}\lrcorner\,\mathcal{C}_{m}(\boldsymbol{F})\,$
$\displaystyle=\,\left(\boldsymbol{e}_{c^{\prime}}\lrcorner\,\mathbb{C}_{ab}\right)\wedge\mathbb{F}^{ab}\,+\,\mathbb{C}_{ab}\wedge\left(\boldsymbol{e}_{c^{\prime}}\lrcorner\,\mathbb{F}^{ab}\right)\,=\,\left(\boldsymbol{e}_{c^{\prime}}\lrcorner\,\mathbb{C}_{ab}\right)\wedge\mathbb{F}^{ab}$
$\displaystyle=\,C_{abc^{\prime}d}\,\boldsymbol{e}^{d}\wedge\mathbb{F}^{ab}\,=\,C_{a^{\prime}b^{\prime}c^{\prime}d}\,\boldsymbol{e}^{d}\wedge\mathbb{F}^{a^{\prime}b^{\prime}}$
(6.15)
From this equation we easily see that if
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ then
$\boldsymbol{e}_{a^{\prime}}\lrcorner\,\mathcal{C}_{m}(\boldsymbol{F})=0$,
which means that $\mathcal{C}_{m}(\boldsymbol{F})$ pertain to
$\mathcal{A}_{1}$. Thus the integrability condition for the distribution
generated by
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$
implies that such $m$-vector is an eigen-$m$-vector of the operator
$\mathcal{C}_{m}$. On the other hand, equation (6.15) guarantees that if
$\mathcal{C}_{m}(\boldsymbol{F})\in\mathcal{A}_{1}$ then
$C_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}=0$ for all
$a^{\prime},b^{\prime},c^{\prime},d^{\prime}$ and
$C_{a^{\prime}b^{\prime}c^{\prime}}^{\phantom{a^{\prime}b^{\prime}c^{\prime}}d^{\prime}}=0$
if either $d^{\prime}=a^{\prime}$ or $d^{\prime}=b^{\prime}$. Particularly, in
4 dimensions these two constraints imply that the whole integrability
condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ is satisfied, while in
higher dimensions this is not true anymore. Similar manipulations lead to the
following interesting theorem [70]:
###### Theorem 21
The three statements below are equivalent:
(1) The Weyl tensor obeys the integrability condition
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$
(2) The subbundles $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are invariant under
the action of $\mathcal{C}_{m}$
(3) All subbundles $\mathcal{A}_{q}$, $q\in\\{1,2,\ldots,m\\}$, are invariant
by the action of $\mathcal{C}_{m}$.
This theorem along with the theorem of reference [67] immediately imply the
following corollary:
###### Corollary 5
In a Ricci-flat manifold of dimension $n=2m$, if the operator
$\mathcal{C}_{m}$ preserves the spaces $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$, with $\mathcal{C}_{m}$ being generic otherwise, then the
maximally isotropic distribution generated by
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$
is integrable.
In 4 dimensions these results recover part of the corollary 2 obtained in
chapter 4, while in 6 dimensions we retrieve theorem 17 of chapter 5. For the
details see [70].
Since the operator $\mathcal{C}_{m}$ preserves the spaces $\Lambda^{m\pm}$
then it follows that if $\mathcal{A}_{q}$ is an eigenspace of
$\mathcal{C}_{m}$ so will be the subbundles
$\mathcal{A}_{q}^{\pm}\equiv\mathcal{A}_{q}\cap\Lambda^{m\pm}$. In 4
dimensions we have that $\mathcal{A}_{1}^{-}=0$ and
$\mathcal{A}_{2}^{-}=\Lambda^{m-}$. Since these spaces are trivially preserved
by the action of $\mathcal{C}_{2}$ it follows that the invariance of the
subbundles $\mathcal{A}_{q}$ under $\mathcal{C}_{2}$ imposes no constraint
over $\mathcal{C}^{-}$. Differently, in higher dimensions, $m>2$, we have
$\dim(\mathcal{A}^{-}_{2})=\frac{1}{2}(m+m^{2})<\frac{1}{2}\frac{(2m)!}{m!\,m!}=\dim(\Lambda^{m-})$.
So, in these cases, if $\mathcal{A}_{2}$ is invariant by $\mathcal{C}_{m}$
then the operator $\mathcal{C}^{-}$ must admit a non-trivial eigenspace,
leading us to the following theorem:
###### Theorem 22
While in 4 dimensions the integrability condition for the self-dual planes
generated by $\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}$ imposes restrictions
only over $\mathcal{C}^{+}$, with $\mathcal{C}^{-}$ being arbitrary; in higher
dimensions the integrability condition for the self-dual maximally isotropic
distribution generated by
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$
constrains both operators, $\mathcal{C}^{+}$ and $\mathcal{C}^{-}$.
#### 6.3 Optical Scalars and Harmonic Forms
In this section the 4-dimensional concept of optical scalars introduced in
chapter 3 will be generalized to higher dimensional manifolds. Moreover, it
will be shown that the existence of certain harmonic forms imposes constraints
on these scalars. To this end, and in order to match the standard notation
[38], let us define a semi-null frame
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m}_{i}\\}$ to be a frame of
vector fields whose inner products are666The indices $i,j,k,\ldots$ run from 2
to $n-1$, where $n$ is the dimension of the manifold.:
$\boldsymbol{g}(\boldsymbol{l},\boldsymbol{l})=\boldsymbol{g}(\boldsymbol{n},\boldsymbol{n})=\boldsymbol{g}(\boldsymbol{l},\boldsymbol{m}_{i})=\boldsymbol{g}(\boldsymbol{n},\boldsymbol{m}_{i})=0\;;\quad\boldsymbol{g}(\boldsymbol{l},\boldsymbol{n})=1\;;\quad\boldsymbol{g}(\boldsymbol{m}_{i},\boldsymbol{m}_{j})=\delta_{ij}\,.$
Then the optical scalars associated to the null congruence generated by
$\boldsymbol{l}$ are defined by:
$M_{0}\,=\,l^{\nu}n^{\mu}\,\nabla_{\nu}l_{\mu}\
\;\;\;;\;\;M_{i}\,=\,l^{\nu}m_{i}^{\mu}\,\nabla_{\nu}l_{\mu}\
\;\;\;;\;\;M_{ij}\,=\,m_{j}^{\nu}m_{i}^{\mu}\,\nabla_{\nu}l_{\mu}\ \,.$
It is simple matter to prove that $\boldsymbol{l}$ is geodesic if, and only
if, $M_{i}=0$, the parametrization being affine when $M_{0}=0$. Furthermore,
the congruence generated by $\boldsymbol{l}$ is hyper-surface-orthogonal,
$l_{[\mu}\nabla_{\nu}l_{\rho]}=0$, if, and only if, $M_{i}$ and $M_{[ij]}$
both vanish. In the Lorentzian signature the vector fields of a semi-null
frame can be chosen to be real, so that in such a case the optical scalars are
real. The $(n-2)\times(n-2)$ matrix $M_{ij}$ is dubbed the optical matrix of
the null congruence generated by $\boldsymbol{l}$. Analogously to what was
done in chapter 3 it is useful to split this matrix as a sum of a symmetric
and trace-less matrix, a skew-symmetric matrix and a term proportional to the
identity:
$M_{ij}\,=\,\sigma_{ij}\,+\,A_{ij}\,+\,\theta\,\delta_{ij}\;;\quad\theta\equiv\frac{1}{n-2}\,\delta^{ij}M_{ij}\;;\;\;\sigma_{ij}\equiv
M_{(ij)}-\theta\,\delta_{ij}\;;\;\;A_{ij}\equiv M_{[ij]}\,.$
The scalar $\theta$ is called the expansion, $\sigma_{ij}$ is named the shear
matrix, while $A_{ij}$ is called the twist matrix. In particular, if
$\sigma_{ij}=0$ we shall say that the congruence is shear-free.
Before proceeding let us introduce some jargon. A $p$-form $\boldsymbol{K}$ is
called harmonic if it is closed, $d\boldsymbol{K}=0$, and co-closed,
$d(\star\boldsymbol{K})=0$. In terms of components this means that the
following differential equations hold:
$\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}\,=\,0\quad\textrm{and}\quad\nabla^{\alpha}\,K_{\alpha\mu_{2}\ldots\mu_{p}}\,=\,0\,.$
(6.16)
Note, in particular, that if $\boldsymbol{L}$ is a closed 1-form then, by the
Poincaré lemma [55], it follows that locally there exists some scalar function
$f$ such that $L_{\mu}=\nabla_{\mu}f$. Thus the 1-form $\boldsymbol{L}$ will
be harmonic if $\nabla^{\mu}\nabla_{\mu}f=0$, which is the well-known equation
satisfied by a harmonic function. In the CMPP classification [36] we say that
a $p$-form $\boldsymbol{K}$ is type $N$ with $\boldsymbol{l}$ being a multiple
aligned null direction if $\boldsymbol{K}$ admits the following expansion:
$K^{\mu_{1}\mu_{2}\ldots\mu_{p}}=p!\,f_{j_{2}j_{3}\ldots
j_{p}}\,l^{[\mu_{1}}m_{j_{2}}^{\mu_{2}}m_{j_{3}}^{\mu_{3}}\ldots
m_{j_{p}}^{\mu_{p}]}\,.$ (6.17)
Where $f_{j_{2}j_{3}\ldots j_{p}}=f_{[j_{2}j_{3}\ldots j_{p}]}$ are scalars
and it is being assumed a sum over the indices $j_{2},\ldots,j_{p}$. In what
follows it will be proved that if a manifold admits a harmonic form that is
type $N$ then the optical scalars of its multiple aligned null direction are
constrained.
Let $\boldsymbol{K}\neq 0$ be a harmonic $p$-form of type $N$ on the CMPP
classification with $\boldsymbol{l}$ being its multiple aligned null
direction, which means that the equations (6.16) and (6.17) hold. Since
$K^{\alpha\beta\mu_{3}\ldots\mu_{p}}l_{\beta}=0$ it follows that:
$\displaystyle
0=\nabla_{\alpha}\left(K^{\alpha\beta\mu_{3}\ldots\mu_{p}}\,l_{\beta}\right)=K^{\alpha\beta\mu_{3}\ldots\mu_{p}}\,\nabla_{\alpha}l_{\beta}=p!\,f_{j_{2}j_{3}\ldots
j_{p}}\,l^{[\alpha}m_{j_{2}}^{\beta}m_{j_{3}}^{\mu_{3}}\ldots
m_{j_{p}}^{\mu_{p}]}\,\nabla_{\alpha}l_{\beta}$
$\displaystyle=h_{1}\,f_{j_{2}j_{3}\ldots
j_{p}}\,m_{j_{2}}^{[\beta}m_{j_{3}}^{\mu_{3}}\ldots
m_{j_{p}}^{\mu_{p}]}\,l^{\alpha}\,\nabla_{\alpha}l_{\beta}\,+\,h_{2}\,l^{\beta}\,\nabla_{\alpha}l_{\beta}\,(\,\cdots\,)\,+$
$\displaystyle+\,h_{3}\,f_{j_{2}j_{3}\ldots
j_{p}}\,l^{[\mu_{3}}m_{j_{4}}^{\mu_{4}}\ldots
m_{j_{p}}^{\mu_{p}]}\,m_{j_{2}}^{\alpha}m_{j_{3}}^{\beta}\,\nabla_{\alpha}l_{\beta}$
$\displaystyle=\,h_{4}\,f_{j_{2}j_{3}\ldots j_{p}}\,m_{j_{3}}^{\mu_{3}}\ldots
m_{j_{p}}^{\mu_{p}}\,M_{j_{2}}\,+\,0\,+\,h_{5}\,f_{j_{2}j_{3}\ldots
j_{p}}l^{[\mu_{3}}m_{j_{4}}^{\mu_{4}}\ldots
m_{j_{p}}^{\mu_{p}]}\,M_{j_{2}j_{3}}\,.$
Where in the above equation the $h$’s are non-zero unimportant constants. We,
thus, arrive at the following constraints:
$M_{i}\,f_{ij_{3}\ldots j_{p}}\,=\,0\quad;\quad A_{ij}f_{ijk_{4}\ldots
k_{p}}\,=\,0\,.$ (6.18)
In a similar fashion, expanding the equation
$\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}l^{\alpha}m_{j_{1}}^{\phantom{j_{1}}\mu_{1}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}=0$ we arrive, after some careful algebra,
at the following relation:
$M_{[j_{1}}\,f_{j_{2}\ldots j_{p}]}\,=\,0\,.$
In particular, the contraction of this identity with $M_{j_{1}}$ along with
equation (6.18) lead us to the relation $M_{i}M_{i}=0$. Analogously, working
out the equality
$0=(\nabla^{\alpha}\,K_{\alpha\mu_{2}\ldots\mu_{p}})m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}$ it easily follows that:
$K_{\alpha\mu_{2}\ldots\mu_{p}}\nabla^{\alpha}(m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}})\,=\,(p-1)!\,\,l^{\alpha}\nabla_{\alpha}f_{j_{2}\ldots
j_{p}}\,+\,(p-1)!\,f_{j_{2}\ldots j_{p}}\nabla^{\alpha}l_{\alpha}\,.$ (6.19)
Now expanding the relation
$\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}l^{\alpha}n^{\mu_{1}}m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}=0$ and using the identity
$\nabla^{\alpha}l_{\alpha}=M_{0}+(n-2)\theta$ along with equation (6.19) it
follows that:
$2(p-1)\,f_{i[j_{3}\ldots
j_{p}}\,\sigma_{j_{2}]i}\;=\;(n-2p)\,\theta\,f_{j_{2}\ldots j_{p}}\,.$
These results are summarized by the following theorem [70]:
###### Theorem 23
If $K^{\mu_{1}\mu_{2}\ldots\mu_{p}}=p!\,f_{j_{2}\ldots
j_{p}}\,l^{[\mu_{1}}m_{j_{2}}^{\mu_{2}}\ldots m_{j_{p}}^{\mu_{p}]}$ is a non-
zero $p$-form such that $d\boldsymbol{K}=0$ and $d(\star\boldsymbol{K})=0$
then the following relations hold:
(1) $M_{i}\,f_{ij_{3}\ldots j_{p}}\,=\,0$
(2) $M_{[j_{1}}\,f_{j_{2}\ldots j_{p}]}\,=\,0$
(3) $2(p-1)\,f_{i[j_{3}\ldots
j_{p}}\,\sigma_{j_{2}]i}\;=\;(n-2p)\,\theta\,f_{j_{2}\ldots j_{p}}$
(4) $M_{i}M_{i}\,=\,0$
(5) $A_{ij}f_{ijk_{4}\ldots k_{p}}\,=\,0$.
On the Lorentzian signature it is possible to introduce a real semi-null
frame, so that the optical scalars are real in such frame. In this case the
equation $M_{i}M_{i}=0$ implies that $M_{i}=0$, which means that the real
vector field $\boldsymbol{l}$ is geodesic. The particular case $p=2$ of the
above theorem in Lorentzian manifolds was obtained before on ref. [58].
Similar results for arbitrary $p$ on the Lorentzian signature were also
obtained, by means of the so-called GHP formalism, in ref. [101], where the
identities (1), (2) and (3) can be explicitly found on the proof of the Lemma
3 of [101]777The author thanks Harvey S. Reall for pointing out this
reference..
#### 6.4 Generalizing Mariot-Robinson and Goldberg-Sachs Theorems
As explained in section 3.3, the Mariot-Robinson theorem guarantees that a
4-dimensional Lorentzian manifold admits a null bivector
$\boldsymbol{F}\propto\boldsymbol{l}\wedge\boldsymbol{m}$ obeying to the
source-free Maxwell’s equations, $d\boldsymbol{F}=0$ and
$d\star\boldsymbol{F}=0$, if, and only if, the real null vector field
$\boldsymbol{l}$ is geodesic and shear-free. But in 4 dimensions the proper
geometric generalization to arbitrary signature of a geodesic and shear-free
null congruence is the existence of an integrable distribution of isotropic
planes, see section 4.3. Then it follows that the Mariot-Robinson theorem
provides a connection between the existence of null solutions for Maxwell’s
equations and the existence of an integrable maximally isotropic distribution
in 4 dimensions. By means of the results presented in section 1.8 it is not so
hard to generalize this theorem to arbitrary even dimensions. Let
$\boldsymbol{F}=\boldsymbol{e}_{1}\wedge\ldots\wedge\boldsymbol{e}_{m}$ be a
null $m$-form on a $2m$-dimensional manifold, so that it generates the
maximally isotropic distribution $Span\\{\boldsymbol{e}_{a^{\prime}}\\}$. Note
that since $\boldsymbol{e}_{a^{\prime}}\lrcorner\boldsymbol{F}=0$, this
distribution coincides with the distribution annihilated by $\boldsymbol{F}$.
Now from the results of section 1.8 it follows that the latter distribution is
integrable if, and only if, there exists some function $h\neq 0$ such that
$d(h\boldsymbol{F})=0$. But a null $m$-form must always be self-dual or anti-
self-dual, $\star\boldsymbol{F}=\pm\varrho\boldsymbol{F}$ with $\varrho$ equal
to 1 or $i$, which can be grasped from the discussion below equation C.10 on
appendix C. Thus we conclude that if $d(h\boldsymbol{F})=0$ then
$d\star(h\boldsymbol{F})=\pm\varrho d(h\boldsymbol{F})=0$, leading us to the
following generalized version of the Mariot-Robinson theorem [69, 70]:
###### Theorem 24
In a $2m$-dimensional manifold a null $m$-form $\boldsymbol{F}^{\prime}$
generates an integrable maximally isotropic distribution if, and only if,
there exists some function $h\neq 0$ such that
$\boldsymbol{F}=h\boldsymbol{F}^{\prime}$ obeys the equations
$d\boldsymbol{F}=0$ and $d(\star\boldsymbol{F})=0$.
Now let
$\\{\boldsymbol{e}_{a}\\}=\\{\boldsymbol{e}_{a^{\prime}},\boldsymbol{\theta}^{b^{\prime}}\\}$
be a null frame on a $2m$-dimensional manifold. Then we can use it in order to
define the following semi-null frame:
$\boldsymbol{l}=\boldsymbol{e}_{1}\;,\;\;\boldsymbol{n}=2\boldsymbol{\theta}^{1}\;,\;\;\boldsymbol{m}_{j}=(\boldsymbol{e}_{j}+\boldsymbol{\theta}^{j})\;,\;\;\boldsymbol{m}_{j+m-1}=-i(\boldsymbol{e}_{j}-\boldsymbol{\theta}^{j})\;;\;\;j\,\in\,\\{2,3,\ldots,m\\}\,.$
In such a basis the null $m$-form
$\boldsymbol{F}=\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{m}$
can be written as follows888For example, in 6 dimensions, $m=3$, we have the
following expression:
$\widehat{f}_{j_{2}j_{3}}\,\equiv\,\frac{2!}{4}\,\left(\delta_{[j_{2}}^{2}+i\delta_{[j_{2}}^{4}\right)\left(\delta_{j_{3}]}^{3}+i\delta_{j_{3}]}^{5}\right)\,=\,\frac{1}{2}\left(\delta_{[j_{2}}^{2}\delta_{j_{3}]}^{3}\,+\,i\,\delta_{[j_{2}}^{2}\delta_{j_{3}]}^{5}\,+\,i\,\delta_{[j_{2}}^{4}\delta_{j_{3}]}^{3}\,-\,\delta_{[j_{2}}^{4}\delta_{j_{3}]}^{5}\right)$
.:
$\left\\{\begin{array}[]{l}F^{\mu_{1}\mu_{2}\ldots\mu_{m}}\,\equiv\,m!\,e_{1}^{\,[\mu_{1}}\ldots
e_{m}^{\,\mu_{m}]}\,=\,m!\,\widehat{f}_{j_{2}j_{3}\ldots
j_{m}}\,l^{[\mu_{1}}m_{j_{2}}^{\,\mu_{2}}\ldots m_{j_{m}}^{\,\mu_{m}]}\\\ \\\
\widehat{f}_{j_{2}j_{3}\ldots
j_{m}}\,\equiv\,\frac{(m-1)!}{2^{m-1}}\,\left(\delta_{[j_{2}}^{2}+i\delta_{[j_{2}}^{m+1}\right)\left(\delta_{j_{3}}^{3}+i\delta_{j_{3}}^{m+2}\right)\cdots\left(\delta_{j_{m}]}^{m}+i\delta_{j_{m}]}^{2m-1}\right)\\\
\end{array}\right.$ (6.20)
Thus the $m$-form $\boldsymbol{F}$ is type $N$ on the CMPP classification with
$\boldsymbol{l}=\boldsymbol{e}_{1}$ being a multiple aligned null direction.
It is worth noting that the definition
$\boldsymbol{l}\equiv\boldsymbol{e}_{1}$ was quite arbitrary, since we could
have chosen $\boldsymbol{l}$ to be any non-zero vector field tangent to the
distribution generated by the null form $\boldsymbol{F}$. A special phenomenon
happens when the signature is Lorentzian, in this case the real part of a
maximally isotropic distribution is always 1-dimensional [20]. Thus on the
Lorentzian case we shall choose $\boldsymbol{l}$ to be tangent to the unique
real null direction on the distribution generated by $\boldsymbol{F}$. Now the
successive combination of theorem 24, then equation (6.20) and finally theorem
23 immediately lead us to the following corollary:
###### Corollary 6
If $Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\ldots,\boldsymbol{e}_{m}\\}$
is an integrable maximally isotropic distribution on a manifold of dimension
$n=2m$ then the optical scalars of the null congruences generated by vector
fields tangent to such distribution are constrained as follows:
(1) $M_{i}\,\widehat{f}_{ij_{3}\ldots j_{p}}\,=\,0$
(2) $M_{[j_{1}}\,\widehat{f}_{j_{2}\ldots j_{m}]}\,=\,0$
(3) $\widehat{f}_{i[j_{3}\ldots j_{m}}\,\sigma_{j_{2}]i}\,=\,0$
(4) $M_{i}M_{i}\,=\,0$
(5) $A_{ij}\widehat{f}_{ijk_{4}\ldots k_{m}}\,=\,0$.
Particularly, on the Lorentzian signature if $\boldsymbol{l}$ is a real vector
field tangent to such distribution then the item (4) implies that
$\boldsymbol{l}$ is geodesic. It is worth mentioning that in appendix C of
ref. [65] the integrability of a maximally isotropic distribution is expressed
in terms of the Ricci rotation coefficients of a null frame. Note that in the
above corollary no condition is assumed over the Ricci tensor. A simple
application of this result on 6-dimensional manifolds has been worked out on
[70].
The original version of the Goldberg-Sachs theorem establish an equivalence
between algebraic restrictions on the Weyl operator $\mathcal{C}_{2}$ and the
existence of a null congruence whose optical scalars are constrained in Ricci-
flat 4-dimensional space-times, see theorem 1 in chapter 3. Now by a simple
merger of corollaries 5 and 6 one can state an analogous result valid in even-
dimensional manifolds of arbitrary signature [70]:
###### Theorem 25
In a Ricci-flat manifold of dimension $n=2m$ if the operator $\mathcal{C}_{m}$
preserves the spaces $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, with
$\mathcal{C}_{m}$ being generic otherwise, then the optical scalars of the
null congruences generated by vectors fields tangent to the maximally
isotropic distribution
$Span\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\ldots,\boldsymbol{e}_{m}\\}$
are constrained as follows:
(1) $M_{i}\,\widehat{f}_{ij_{3}\ldots j_{p}}\,=\,0$
(2) $M_{[j_{1}}\,\widehat{f}_{j_{2}\ldots j_{m}]}\,=\,0$
(3) $\widehat{f}_{i[j_{3}\ldots j_{m}}\,\sigma_{j_{2}]i}\,=\,0$
(4) $M_{i}M_{i}\,=\,0$
(5) $A_{ij}\widehat{f}_{ijk_{4}\ldots k_{m}}\,=\,0$.
Where the subbundles $\mathcal{A}_{q}$ were defined in (6.14), while the
object $\widehat{f}_{j_{2}j_{3}\ldots j_{p}}$ was defined in equation (6.20).
Again, in the particular case of the Lorentzian signature if $\boldsymbol{l}$
is a real vector field tangent to such distribution then the equation
$M_{i}M_{i}=0$ guarantees that $\boldsymbol{l}$ is geodesic.
Theorem 25 is a partial generalization of the Goldberg-Sachs theorem to even-
dimensional manifolds. Note, however, that while in 4 dimensions the GS
theorem is an equivalence relation, the theorem presented here goes just in
one direction, stating that algebraic restrictions on the Weyl tensor imply
the existence of constrained null congruences, but not the converse.
Furthermore, while in 4-dimensional manifolds of Lorentzian signature the item
(3) of theorem 25 implies that the null congruence is shear-free, in higher
dimensions this is not true anymore. Indeed, a simple count of degrees of
freedom reveals that the higher the dimension the more restrictive the shear-
free condition becomes. Indeed, in $n$ dimensions the object
$\nabla_{\mu}l_{\nu}$ has $D=n(n-1)$ non-trivial components, since
$l^{\nu}\nabla_{\mu}l_{\nu}$ is automatically zero. On the other hand, the
shear matrix $\sigma_{ij}$ has $S=\frac{1}{2}(n-1)(n-2)-1$ independent
components. Note that the rate $S/D$ becomes higher and higher as the
dimension increases, approaching the limit $S/D\rightarrow\frac{1}{2}$ as the
dimension goes to infinity. This gives a hint that in dimensions greater than
four the Goldberg-Sachs theorem cannot be trivially generalized stating that a
simple algebraic restriction on the Weyl tensor is equivalent to the existence
of a null congruence that is geodesic and shear-free, since the latter
condition is too strong.
### Chapter 7 Conclusion and Perspectives
As demonstrated in chapter 2, there exist several ways to approach the Petrov
classification in 4-dimensional Lorentzian manifolds. One of these methods is
the bivector line of attack, which treats the Weyl tensor as an operator,
$\mathcal{C}_{2}$, on the space of bivectors. Although this was the original
path taken for defining such classification, during the past decades it has
been overlooked in favor of other methods like the spinorial approach.
However, in this thesis it was proved that the bivector method can be quite
fruitful and full of geometric significance. Indeed, in chapter 4 this
approach was used in order to generalize the Petrov classification to
4-dimensional manifolds of arbitrary signature in a unified way. Furthermore,
it was proved that the null eigen-bivectors of $\mathcal{C}_{2}$ generate
integrable isotropic planes, providing a convenient way to state the Goldberg-
Sachs (GS) theorem. In particular, this form of interpreting the GS theorem
yielded connections between the algebraic type of the Weyl tensor and the
existence of geometric structures as symplectic forms and complex structures.
In chapter 6 it was shown that the bivector operator $\mathcal{C}_{2}$ is just
a single member of an infinite class of linear operators $\mathcal{C}_{p}$
sending $p$-forms into $p$-forms that can be constructed out of the Weyl
tensor in arbitrary dimension. It was proved that such operators have nice
properties as commuting with the Hodge dual map and being self-dual with
respect to a convenient inner product. Particularly, when the signature is
Euclidean these operators can be diagonalized, which makes the algebraic
classification rather simple in this case. Moreover, when the dimension is
even, $n=2m$, the operator $\mathcal{C}_{m}$ plays a prominent role, as it can
be nicely used to express the integrability condition of maximally isotropic
distributions. In chapter 6 it was also proved a generalized version of the
Goldberg-Sachs theorem, valid in even-dimensional spaces of arbitrary
signature, stating that certain algebraic constraints on the operator
$\mathcal{C}_{m}$ imply the existence of null congruences with restricted
optical scalars. These results teaches us that while in 4 dimensions the
bivectors are featured objects, in $n=2m$ dimensions this role is played by
the $m$-forms.
Since the most elegant approach to the Petrov classification and its
associated theorems uses spinors, it is natural to employ such language in
order to provide a higher-dimensional generalization of these results. This
was the route taken in chapter 5, where the spinorial formalism in 6
dimensions was developed ab initio. There it is shown how to represent the
$SO(6;\mathbb{C})$ tensors in terms of spinors, which reveals the possibility
of classifying the bivectors and the Weyl tensor in a simple way. In
particular, this Weyl tensor classification coincides with the one attained by
means of the operator $\mathcal{C}_{3}$. An important feature of spinors is
that they constitute the most suitable tool to describe isotropic subspaces,
as explicitly illustrated on subsection 5.1.3. Particularly, the maximally
isotropic distributions are represented by the so-called pure spinors. Because
of this property, the spinorial formalism was shown to provide a simple and
elegant form to express the integrability condition of a maximally isotropic
distribution.
The work presented in this thesis can be enhanced in multiple forms. For
example, the operators $\mathcal{C}_{p}$ and their relation with integrability
properties deserve further investigation. Since in $2m$ dimensions the
operator $\mathcal{C}_{m}$ is connected to the integrability of
$m$-dimensional isotropic distributions, a natural question to be posed is
whether the operators $\mathcal{C}_{p}$ are, likewise, associated to the
integrability of $p$-dimensional isotropic distributions irrespective of the
manifold dimension. Another interesting quest is trying to provide links
between the algebraic type of the Weyl tensor and the existence of hidden
symmetries on the manifold. A more ambitious project would be to study which
algebraic conditions might be imposed to the operator $\mathcal{C}_{m}$ in
order for Einstein’s vacuum equation to be analytically integrable, just as in
4 dimensions the type $D$ condition allows the complete integration of
Einstein’s equation. Concerning the 6-dimensional spinorial formalism
introduced here, certainly further progress can be accomplished as soon as a
connection is introduced on the spinor bundle. In particular, the generality
condition referred to on the footnote 7 of chapter 5 can, probably, be better
understood by means of the spinorial language. In addition, once such
connection is introduced the 6-dimensional twistors can be investigated.
The main goal behind the research shown on the present thesis was to give a
better understanding of general relativity in higher dimensions, particularly
to provide further tools to study geometrical properties of higher-dimensional
black holes. But, besides general relativity, this piece of work can,
hopefully, be applied to other branches of physics and mathematics. For
instance, higher-dimensional manifolds are of great relevance in string theory
and supergravity, so that the results obtained here could be useful. More
broadly, this work can be applied to physical systems whose degrees of freedom
form a differentiable manifold with dimension greater than 3. In particular,
by means of Caratheodory’s formalism, it follows that integrable distributions
are of interest to thermodynamics (see section 1.8), which suggests a possible
application for the results presented here. Finally, since spinors are
acquiring increasing significance in physics it follows that the 6-dimensional
spinorial language developed here can have multiple utility. For instance, in
order to retrieve our 4-dimensional space-time out of a 10-dimensional
manifold of string theory one generally need to compactify 6 dimensions, so
that 6-dimensional manifolds are of particular relevance.
### Appendix A Segre Classification and its Refinement
Segre classification is a well-known form to classify square matrices (or
linear operators) over the complex field. Essentially this classification
amounts to specify the eigenvalue structure of the matrix in a compact code.
In this appendix such classification will be explained and a refinement will
be presented.
It is a standard result of linear algebra that given a square matrix $M$ over
the complex field it is always possible to find a basis in which such matrix
acquires the so-called Jordan canonical form [102]. This means that it is
always possible to find an invertible matrix $B$ such that
$M^{\prime}=BMB^{-1}$ assumes the following block-diagonal form:
$M^{\prime}\,=\,\operatorname{diag}(J_{1},J_{2},\ldots,J_{q})\;,\textrm{ where
}\;J_{i}=\left[\begin{array}[]{ccccc}\lambda_{i}&1&0&\ldots&0\\\
0&\lambda_{i}&1&&\vdots\\\ 0&0&\ddots&&0\\\ \vdots&\vdots&&\lambda_{i}&1\\\
0&0&\ldots&0&\lambda_{i}\\\
\end{array}\right]\,,\;\lambda_{i}\in\mathbb{C}\,.$ (A.1)
Note that $J_{i}$ can also be the $1\times 1$ matrix $J_{i}=\lambda_{i}$. The
blocks $J_{i}$ are called the Jordan blocks of the matrix $M$. Each block
$J_{i}$ admits just one eigenvector and its eigenvalue is $\lambda_{i}$. Thus,
for example, if we manage to put the $5\times 5$ matrix $G$ on the Jordan
canonical form
$G^{\prime}\,=\,\left[\begin{array}[]{ccccc}2&1&0&0&0\\\ 0&2&0&0&0\\\
0&0&3&0&0\\\ 0&0&0&5&1\\\ 0&0&0&0&5\\\ \end{array}\right]\,,\textrm{ then
}\;J_{1}=\left[\begin{array}[]{cc}2&1\\\ 0&2\\\
\end{array}\right]\,,\;J_{2}=3\,\textrm{ and
}\;J_{3}=\left[\begin{array}[]{cc}5&1\\\ 0&5\\\ \end{array}\right].$
In particular this canonical form implies that the matrix $G$ admits just
three different eigenvectors (apart from a multiplicative scale). The
eigenvalues of these eigenvectors are $\lambda_{1}=2$, $\lambda_{2}=3$ and
$\lambda_{3}=5$.
The Jordan canonical form of a matrix is unique up to the ordering of the
Jordan blocks $J_{i}$. In particular, the dimensions of the Jordan Blocks are
invariant under the change of basis, which opens up the possibility of
introducing an invariant classification. The Segre classification of a matrix
amounts to _list the dimensions of all the Jordan blocks and bound together,
inside round brackets, the dimensions of the blocks with the same eigenvalue._
This classification can be refined if we separate the dimensions of the blocks
with eigenvalue zero putting them on the right of the dimensions of the other
blocks, using a vertical bar to separate [91]. As a pedagogical example, let
us work out the Segre type (ST) and the refined Segre type (RST) of the matrix
$F$:
$F\,=\,\left[\begin{array}[]{cccccc}\kappa&1&0&0&0&0\\\ 0&\kappa&1&0&0&0\\\
0&0&\kappa&0&0&0\\\ 0&0&0&\alpha&0&0\\\ 0&0&0&0&\beta&1\\\ 0&0&0&0&0&\beta\\\
\end{array}\right]\,.$ (A.2)
The types depend on the values of $\kappa$, $\alpha$ and $\beta$. Some of the
possibilities are:
$\displaystyle\kappa,\alpha,\beta\neq 0\,\textrm{ and all different}$
$\displaystyle\Rightarrow\;\textrm{ST: }[3,2,1]\;\;;\;\;\;\textrm{RST:
}[3,2,1|\,]$ $\displaystyle\alpha,\beta\neq 0=\kappa\,\textrm{ and
}\alpha\neq\beta$ $\displaystyle\Rightarrow\;\textrm{ST:
}[3,2,1]\;\;;\;\;\;\textrm{RST: }[2,1|3]$ $\displaystyle\alpha=\beta\neq
0\,,\;\kappa=0\,$ $\displaystyle\Rightarrow\;\textrm{ST:
}[3,(2,1)]\;;\;\;\textrm{RST: }[(2,1)|3]$
$\displaystyle\alpha=\beta=0\,,\;\kappa\neq 0\,$
$\displaystyle\Rightarrow\;\textrm{ST: }[3,(2,1)]\;;\;\;\textrm{RST:
}[3|2,1]\,.$
Note that the order of the numbers between the square bracket and the vertical
bar does not matter. As a final example it is displayed below all the possible
refined Segre types that a trace-less $3\times 3$ matrix can have. This result
will be used in chapter 2.
$(A):\quad\left[\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\\\
\end{array}\right]\longrightarrow\left\\{\begin{array}[]{cl}\lambda_{i}\neq
0\textrm{ and
}\lambda_{i}\neq\lambda_{j}\;\;\forall\;i,j&\rightarrow\,[1,1,1|\,]\\\
\lambda_{1}=0\textrm{ and
}\lambda_{i}\neq\lambda_{j}\;\;\forall\;i,j&\rightarrow\,[1,1|1]\\\
\lambda_{1}=\lambda_{2}\neq
0,\;\lambda_{3}=-2\lambda_{1}&\rightarrow\,[(1,1),1|\,]\\\
\lambda_{1}=\lambda_{2}=\lambda_{3}=0&\rightarrow\,[\,|1,1,1]\\\
\end{array}\right.$
$(B):\quad\left[\begin{array}[]{ccc}\lambda&1&0\\\ 0&\lambda&0\\\
0&0&-2\lambda\\\
\end{array}\right]\longrightarrow\left\\{\begin{array}[]{cl}\lambda\neq
0&\rightarrow\;[2,1|\,]\\\ \lambda=0&\rightarrow\;[\,|2,1]\\\
\end{array}\right.$
$(C):\quad\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 0&0&0\\\
\end{array}\right]\longrightarrow\;[\,|3]$
It is worth noting that the trace-less condition restricted enormously the
number of possible algebraic types. For instance, the types $[(1,1)|1]$,
$[2|1]$ and $[3|\,]$ are some examples of types that are incompatible with the
trace-less assumption.
### Appendix B Null Tetrad Frame
In 1962 E. T. Newman and R. Penrose introduced a tetrad frame formalism in
which all basis vectors are null [103], which can be accomplished only if
complex vectors are used. This was a novelty at the time and since then this
kind of basis has proved to be useful in many general relativity calculations.
According to [12] the reason that led Penrose to introduce a null basis was
his faith that the fundamental structures of general relativity are the light-
cones.
If $(M,\boldsymbol{g})$ is a 4-dimensional Lorentzian manifold then a null
tetrad frame is a set of four null vector fields
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
that span the tangent space at every point. The vector fields $\boldsymbol{l}$
and $\boldsymbol{n}$ are real, while $\boldsymbol{m}$ and
$\overline{\boldsymbol{m}}$ are complex and conjugates to each other. In a
null tetrad frame the only non-zero inner products are assumed to be:
$\boldsymbol{g}(\boldsymbol{l},\boldsymbol{n})\,=\,1\quad\textrm{ and
}\quad\boldsymbol{g}(\boldsymbol{m},\overline{\boldsymbol{m}})\,=\,-1\,.$
Therefore the metric can be written as follows:
$g_{\mu\nu}\,=\,2\,l_{(\mu}n_{\nu)}\,-\,2\,m_{(\mu}\overline{m}_{\nu)}\,.$
Which can be easily verified by contracting this metric with the basis
vectors. Given an orthonormal frame
$\\{\hat{\boldsymbol{e}}_{0},\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2},\hat{\boldsymbol{e}}_{3}\\}$,
with
$\boldsymbol{g}(\hat{\boldsymbol{e}}_{a},\hat{\boldsymbol{e}}_{b})=\eta_{ab}=\operatorname{diag}(1,-1,-1,-1)$,
then we can easily construct a null tetrad by defining:
$\boldsymbol{l}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}_{1})\;;\;\boldsymbol{n}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}-\hat{\boldsymbol{e}}_{1})\;;\;\boldsymbol{m}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}+i\hat{\boldsymbol{e}}_{3})\;;\;\overline{\boldsymbol{m}}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}-i\hat{\boldsymbol{e}}_{3})$
The null tetrads can be elegantly expressed in terms of spinors. Let
$\\{\boldsymbol{o},\boldsymbol{\iota}\\}$ be a spinor frame, i.e., spinors
such that $o_{{}_{A}}\iota^{A}=1$ (see section 2.5), then it can be easily
shown that the following vectors form a null tetrad:
$l^{\mu}\,\sim\,o^{A}\overline{o}^{\dot{A}}\;;\;\;n^{\mu}\,\sim\,\iota^{A}\overline{\iota}^{\dot{A}}\;;\;\;m^{\mu}\,\sim\,o^{A}\overline{\iota}^{\dot{A}}\;;\;\;\overline{m}^{\mu}\,\sim\,\iota^{A}\overline{o}^{\dot{A}}\,.$
(B.1)
### Appendix C Clifford Algebra and Spinors
The Clifford Algebra, also called geometric algebra, was created by the
English mathematician William Kingdon Clifford around 1880. His intent was to
unify Hamilton’s work on quaternions and Grassmann’s work about exterior
algebra. Since the first paper of Clifford on the subject has been published
in an obscure journal at the time, it went unnoticed until the beginning of
the XX century, when Élie Cartan discovered the spinors [105], objects related
to unknown representations of the $SO(n)$ group. Actually, it seems that R.
Brauer and H. Weyl have been the first ones to connect Cartan’s spinors with
the geometric algebra [104].
An algebra is, essentially, a vector space in which an associative
multiplication between the vectors is defined. Clifford algebra is a special
kind of algebra defined on vector spaces endowed with inner products. Let $V$
be an $n$-dimensional vector space endowed with the non-degenerate inner
product $<\,,>$, then the Clifford product of two vectors
$\boldsymbol{a},\boldsymbol{b}\in V$ is defined to be such that its symmetric
part gives the inner product:
$\boldsymbol{a}\boldsymbol{b}\,+\,\boldsymbol{b}\boldsymbol{a}\,=\,2<\boldsymbol{a},\boldsymbol{b}>\,.$
(C.1)
If
$\\{\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2},\ldots,\hat{\boldsymbol{e}}_{n}\\}$
is an orthonormal basis for $V$,
$<\hat{\boldsymbol{e}}_{i},\hat{\boldsymbol{e}}_{j}>=\pm\delta_{ij}$, then it
follows from (C.1) that
$\hat{\boldsymbol{e}}_{i}\hat{\boldsymbol{e}}_{j}=-\hat{\boldsymbol{e}}_{j}\hat{\boldsymbol{e}}_{i}$
if $i\neq j$. Analogously,
$\hat{\boldsymbol{e}}_{i}\hat{\boldsymbol{e}}_{j}\hat{\boldsymbol{e}}_{k}$ is
totally skew-symmetric if $i\neq j\neq k\neq i$. Thus we conclude that a
general element of $\mathcal{C}l(V)$, the Clifford algebra of $V$, can always
be put in the following form:
$\boldsymbol{\omega}\,=\,w+w^{i}\,\hat{\boldsymbol{e}}_{i}+w^{ij}\hat{\boldsymbol{e}}_{i}\hat{\boldsymbol{e}}_{j}+\ldots+w^{i_{1}\ldots
i_{n}}\hat{\boldsymbol{e}}_{i_{1}}\ldots\hat{\boldsymbol{e}}_{i_{n}}\,,$
where $w$ is a real (or complex) number and $w^{i_{1}\ldots i_{p}}$ are skew-
symmetric tensors with values on the real (or complex) field. Thus we conclude
that the exterior algebra of $V$, $\wedge V$, provides a basis for
$\mathcal{C}l(V)$. In other words, the vector space of the Clifford algebra
associated to $V$ is $\wedge V$. By what was just seen it is natural to define
the wedge product of vectors to be the totally anti-symmetric part of the
Clifford product:
$\boldsymbol{a}_{1}\wedge\boldsymbol{a}_{2}\wedge\ldots\wedge\boldsymbol{a}_{p}\,=\,\frac{1}{p!}\,\sum_{\sigma}\,(-1)^{\epsilon_{\sigma}}\,\boldsymbol{a}_{\sigma(1)}\boldsymbol{a}_{\sigma(2)}\ldots\boldsymbol{a}_{\sigma(p)}\,,$
(C.2)
where the sum runs over all permutations of $\\{1,2,\ldots,p\\}$ and
$\epsilon_{\sigma}$ is even or odd depending on the parity of the permutation
$\sigma$. In particular, note that
$\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\ldots\hat{\boldsymbol{e}}_{p}=\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}\wedge\ldots\wedge\hat{\boldsymbol{e}}_{p}$.
With this definition we find that given the vectors
$\boldsymbol{a},\boldsymbol{b}\in V$ then
$\boldsymbol{a}\boldsymbol{b}-\boldsymbol{b}\boldsymbol{a}=2\boldsymbol{a}\wedge\boldsymbol{b}$.
Using this and eq. (C.1) we arrive at the following formula for the Clifford
product of two vectors:
$\boldsymbol{a}\boldsymbol{b}\,=\,<\boldsymbol{a},\boldsymbol{b}>\,+\,\boldsymbol{a}\wedge\boldsymbol{b}\,.$
(C.3)
Using equations (C.2) and (C.3) it can be proved, for instance, that
$\boldsymbol{a}\boldsymbol{b}\boldsymbol{c}\,=\,<\boldsymbol{b},\boldsymbol{c}>\boldsymbol{a}\,+\,<\boldsymbol{a},\boldsymbol{b}>\boldsymbol{c}\,\,-\,<\boldsymbol{a},\boldsymbol{c}>\boldsymbol{b}\,+\,\boldsymbol{a}\wedge\boldsymbol{b}\wedge\boldsymbol{c}\,.$
(C.4)
A non-zero linear combination of the wedge product of $p$ vectors,
$\boldsymbol{a}_{1}\wedge\boldsymbol{a}_{2}\wedge\ldots\wedge\boldsymbol{a}_{p}$,
is called a $p$-vector or an element of order $p$. Since the Clifford product
of two elements of even order yields another even order element, it follows
that the set of all elements of $\mathcal{C}l(V)$ with even order forms a
subalgebra, denoted $\mathcal{C}l(V)^{+}$.
Example:
As a simple example let us work out the Clifford algebra of the vector space
$\mathbb{R}^{0,2}$. $\mathcal{C}l(\mathbb{R}^{0,2})$ is generated by
$\\{1,\,\hat{\boldsymbol{e}}_{1},\,\hat{\boldsymbol{e}}_{2},\,\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}\\}$,
where
$\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{1}=-1=\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{2}$
and
$\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}=\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}$.
Note also that
$(\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2})(\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2})=\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}=-\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{2}=-1\,.$
Thus defining $\boldsymbol{i}=\hat{\boldsymbol{e}}_{1}$,
$\boldsymbol{j}=\hat{\boldsymbol{e}}_{2}$ and
$\boldsymbol{k}=\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}$, we
find that
$\boldsymbol{i}^{2}=\boldsymbol{j}^{2}=\boldsymbol{k}^{2}=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$,
i.e., $\mathcal{C}l(\mathbb{R}^{0,2})$ is the quaternion algebra. In
particular note that it admits the following matrix representation:
$1\sim\left[\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right]\;;\,\boldsymbol{i}\sim\left[\begin{array}[]{cc}0&i\\\
i&0\\\
\end{array}\right]\;;\,\boldsymbol{j}\sim\left[\begin{array}[]{cc}0&-1\\\
1&0\\\
\end{array}\right]\;;\,\boldsymbol{k}\sim\left[\begin{array}[]{cc}i&0\\\
0&-i\\\ \end{array}\right]\,.$
$\Box$
An important element of $\mathcal{C}l(V)$ is the so-called pseudo-scalar,
$\boldsymbol{I}=\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\ldots\hat{\boldsymbol{e}}_{n}$.
If $s$ is the signature of the inner product, it is not difficult to prove
that the Clifford product of $\boldsymbol{I}$ with itself is given by
$\boldsymbol{I}^{2}\,=\,(-1)^{\frac{1}{2}[n(n-1)+(n-s)]}\,\,.$ (C.5)
Defining the reversion operation by
$(\boldsymbol{a}_{1}\boldsymbol{a}_{2}\ldots\boldsymbol{a}_{p})^{t}\equiv\boldsymbol{a}_{p}\ldots\boldsymbol{a}_{2}\boldsymbol{a}_{1}$
it follows that the Hodge dual of an element of $\wedge V$ can be easily
expressed in terms of the Clifford algebra, more precisely we have that
$\star\boldsymbol{\omega}=(-1)^{\frac{1}{2}[n(n-1)+(n-s)]}\,(\boldsymbol{I}\boldsymbol{\omega})^{t}\,.$
(C.6)
Now let us see the deep connection between geometric algebra and rotations.
Let $\boldsymbol{n}\in V$ be a normalized vector,
$\boldsymbol{n}^{2}=<\boldsymbol{n},\boldsymbol{n}>=\pm 1$, and
$\boldsymbol{a}\in V$ be an arbitrary vector. Then by means of (C.1) it easily
follows that:
$-\boldsymbol{n}\,\boldsymbol{a}\,\boldsymbol{n}^{-1}\,=\,-(-\boldsymbol{a}\boldsymbol{n}+2<\boldsymbol{n},\boldsymbol{a}>)\boldsymbol{n}^{-1}\,=\,\boldsymbol{a}-2<\boldsymbol{n},\boldsymbol{a}>\boldsymbol{n}^{-1}\,.$
(C.7)
Where $\boldsymbol{n}^{-1}=\pm\boldsymbol{n}$ when $\boldsymbol{n}^{2}=\pm 1$.
The combination
$\boldsymbol{a}-2<\boldsymbol{n},\boldsymbol{a}>\boldsymbol{n}^{-1}$ is the
exactly the reflection of the vector $\boldsymbol{a}$ with respect to the
plane orthogonal to $\boldsymbol{n}$. Indeed, if $\boldsymbol{a}$ is
orthogonal to $\boldsymbol{n}$ then it gives $\boldsymbol{a}$, while if
$\boldsymbol{a}$ is parallel to $\boldsymbol{n}$ such combination yields
$-\boldsymbol{a}$. It can be proved that in $n$ dimensions any rotation can be
decomposed as a product of at most $n$ reflections [105]. Thus is natural to
define the following groups contained on the Clifford algebra:
$\displaystyle Pin(V)\,$
$\displaystyle=\,\\{\boldsymbol{\varphi}\in\mathcal{C}l(V)\,|\,\boldsymbol{\varphi}=\boldsymbol{n}_{p}\ldots\boldsymbol{n}_{2}\boldsymbol{n}_{1},\,\boldsymbol{n}_{i}\in
V\textrm{ and }\boldsymbol{n}_{i}^{2}=\pm 1\\}$ $\displaystyle SPin(V)\,$
$\displaystyle=\,\\{\boldsymbol{\varphi}\in\mathcal{C}l(V)\,|\,\boldsymbol{\varphi}=\boldsymbol{n}_{2p}\ldots\boldsymbol{n}_{2}\boldsymbol{n}_{1},\,\boldsymbol{n}_{i}\in
V\textrm{ and }\boldsymbol{n}_{i}^{2}=\pm 1\\}$ (C.8)
Note that $SPin(V)=Pin(V)\cap\mathcal{C}l(V)^{+}$, i.e, $SPin(V)$ is the
subgroup of $Pin(V)$ formed by the elements of even order. It is simple matter
to verify that $Pin(V)$ and $SPin(V)$ are indeed groups under the Clifford
multiplication. Then, by what was seen above, we conclude that the elements of
these groups can be used to implement reflections and pure rotations on an
arbitrary vector $\boldsymbol{a}\in V$.
Rotation $\boldsymbol{+}$ Reflection
$\displaystyle:\,\;(-1)^{p}\,\boldsymbol{\varphi}\,\boldsymbol{a}\,\boldsymbol{\varphi}^{-1}\;,\;\boldsymbol{\varphi}\in
Pin(V)$ Pure Rotation
$\displaystyle:\,\;\boldsymbol{\varphi}\,\boldsymbol{a}\,\boldsymbol{\varphi}^{-1}\;,\;\boldsymbol{\varphi}\in
SPin(V)$
Indeed, these transformations are just a composition of the reflections seen
on eq. (C.7). In particular, it is immediate to verify that the norm of
$\boldsymbol{a}$ is preserved. Note that $\boldsymbol{\varphi}$ and
$-\boldsymbol{\varphi}$ accomplish the same transformation on a vector, which
results on the following important relations:
$O(V)=Pin(V)/\mathbb{Z}_{2}\quad;\quad SO(V)=SPin(V)/\mathbb{Z}_{2}\,.$
Moreover, it can be proved that $Pin(V)$ and $SPin(V)$ are the universal
covering groups of the orthogonal groups $O(V)$ and $SO(V)$ respectively. We
can also define the group $SPin_{+}(V)$ as being the subgroup of $SPin(V)$
formed by the elements $\varphi_{+}\in SPin(V)$ such that
$\varphi_{+}^{t}\varphi_{+}=1$. Note that the action of the groups $Pin(V)$
and $Spin(V)$ on $V$ yield elements on $V$, thus the vector space $V$ provides
a representation for these groups. But this representation is quadratic and
therefore it is not faithful, since $\varphi$ and $-\varphi$ are represented
by the same operation on $V$. In what follows we will see that the space of
spinors gives a linear and faithful representation for these groups, actually
for the whole Clifford algebra. But before proceeding let us see an explicit
example of how the rotations shows up on the geometric algebra formalism.
Example:
Let
$\\{\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2},\ldots,\hat{\boldsymbol{e}}_{n}\\}$
be an orthonormal basis for the Euclidian vector space $\mathbb{R}^{n}$,
$<\hat{\boldsymbol{e}}_{i},\hat{\boldsymbol{e}}_{j}>=\delta_{ij}$. Now
defining $\boldsymbol{n}_{1}=\hat{\boldsymbol{e}}_{1}$,
$\boldsymbol{n}_{2}=\cos\theta\,\hat{\boldsymbol{e}}_{1}+\sin\theta\,\hat{\boldsymbol{e}}_{2}$
and $\boldsymbol{\varphi}_{{}_{\theta}}=\boldsymbol{n}_{2}\boldsymbol{n}_{1}$,
it is simple matter to prove the following relations:
$\displaystyle\boldsymbol{\varphi}_{{}_{\theta}}\,\hat{\boldsymbol{e}}_{1}\,\boldsymbol{\varphi}_{{}_{\theta}}^{-1}\,$
$\displaystyle=\,\boldsymbol{n}_{2}\boldsymbol{n}_{1}\,\hat{\boldsymbol{e}}_{1}\,\boldsymbol{n}_{1}\boldsymbol{n}_{2}\,=\,\cos(2\theta)\,\hat{\boldsymbol{e}}_{1}+\sin(2\theta)\,\hat{\boldsymbol{e}}_{2}$
$\displaystyle\boldsymbol{\varphi}_{{}_{\theta}}\,\hat{\boldsymbol{e}}_{2}\,\boldsymbol{\varphi}_{{}_{\theta}}^{-1}\,$
$\displaystyle=\,\boldsymbol{n}_{2}\boldsymbol{n}_{1}\,\hat{\boldsymbol{e}}_{2}\,\boldsymbol{n}_{1}\boldsymbol{n}_{2}\,=\,-\sin(2\theta)\,\hat{\boldsymbol{e}}_{1}+\cos(2\theta)\,\hat{\boldsymbol{e}}_{2}$
$\displaystyle\boldsymbol{\varphi}_{{}_{\theta}}\,\hat{\boldsymbol{e}}_{j}\,\boldsymbol{\varphi}_{{}_{\theta}}^{-1}\,$
$\displaystyle=\,\boldsymbol{n}_{2}\boldsymbol{n}_{1}\,\hat{\boldsymbol{e}}_{j}\,\boldsymbol{n}_{1}\boldsymbol{n}_{2}\,=\,\hat{\boldsymbol{e}}_{j}\textrm{
if }j\geq 3$
Thus $\boldsymbol{\varphi}_{{}_{\theta}}\in SPin(\mathbb{R}^{n})$ accomplish a
rotation of $2\theta$ on the plane generated by
$\\{\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2}\\}$. As a final remark
note that
$\boldsymbol{\varphi}_{{}_{\theta}}=\boldsymbol{n}_{2}\boldsymbol{n}_{1}=(\cos\theta-\sin\theta\,\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2})$
can be formally represented by
$\boldsymbol{\varphi}_{{}_{\theta}}=e^{-\theta\,\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}}$,
as can be easily verified expanding the exponential in series. Thus, in
general, the element
$\boldsymbol{\varphi}=e^{-\theta\,\hat{\boldsymbol{e}}_{i}\wedge\hat{\boldsymbol{e}}_{j}}$
undertakes a rotation of $2\theta$ on the plane generated by
$\\{\hat{\boldsymbol{e}}_{i},\hat{\boldsymbol{e}}_{j}\\}$. $\Box$
Spinors can be roughly defined as the elements of a vector space on which the
less-dimensional faithful representation of the Clifford algebra acts. In
order to be more precise we shall define what a minimal left ideal is. In what
follows it will be assumed, for simplicity, that the dimension of $V$ is even,
$n=2r$ with $r\in\mathbb{N}$. We call $L\subset\mathcal{C}l(V)$ a left ideal
of the algebra $\mathcal{C}l(V)$ when $L$ is invariant under the action on the
left of the whole algebra:
$\textrm{$L$ is a left
ideal}\;\Leftrightarrow\;\boldsymbol{\omega}\,\boldsymbol{\zeta}\,=\,\boldsymbol{\zeta}^{\prime}\in
L\quad\forall\quad\boldsymbol{\zeta}\in L\,\textrm{ and
}\,\boldsymbol{\omega}\in\mathcal{C}l(V)\,.$
In particular, note that a left ideal is a subalgebra. A minimal left ideal is
a left ideal that as an algebra admits no proper left ideal, i.e, is a left
ideal that admits no left ideal other than itself and the zero element.
Note that a left ideal $L\subset\mathcal{C}l(V)$ provides a representation of
the Clifford algebra, sice $L$ is a vector space and, by definition, this
algebra maps $L$ into $L$. A minimal left ideal $S\subset\mathcal{C}l(V)$
furnish the less-dimensional faithful representation of $\mathcal{C}l(V)$, the
so-called spinorial representation of the Clifford algebra. Therefore the
elements of $S$ are called spinors. It can be proved that if $n=2r$ is the
dimension of the vector space $V$ then the dimension of the spinor space is
$2^{r}$ [106, 97]. Particularly, this implies that the algebra
$\mathcal{C}l(V)$ and the groups $Pin(V)$, $SPin(V)$, $O(V)$ and $SO(V)$ can
all be faithfully represented by $2^{r}\times 2^{r}$ matrices.
Although the pseudo-scalar $\boldsymbol{I}$ always commutes with the elements
of even order, when the dimension is even it does not commute with the
elements of odd order, so in this case the spinorial representation of
$\boldsymbol{I}$ is not a multiple of the identity. From equation (C.5) we see
that $\boldsymbol{I}^{2}=\varepsilon^{2}$, with $\varepsilon=1$ or
$\varepsilon=i$ depending on the dimension and on the signature. Thus when
acting on $S$ the pseudo-scalar $\boldsymbol{I}$ splits this space into a
direct sum of two subspaces of dimension $2^{r-1}$.
$S\,=\,S^{+}\oplus S^{-}\;;\quad S^{\pm}=\\{\boldsymbol{\psi}\in
S\,|\,\boldsymbol{I}\boldsymbol{\psi}=\pm\varepsilon\boldsymbol{\psi}\\}$
The elements of $S^{\pm}$ are called Weyl spinors (or semi-spinors) of
positive and negative chirality. Since $\boldsymbol{I}$ commutes with
$\mathcal{C}l(V)^{+}$ it follows that if $\boldsymbol{\psi}^{\pm}\in S^{\pm}$
and $\boldsymbol{\omega}_{+}\in\mathcal{C}l(V)^{+}$ then
$\boldsymbol{\omega}_{+}\boldsymbol{\psi}^{\pm}$ will also pertain to
$S^{\pm}$. This means that in even dimensions the spinorial representation of
$\mathcal{C}l(V)^{+}$ splits in two blocks of dimension $2^{r-1}\times
2^{r-1}$.
$\mathcal{C}l(V)^{+}\,\sim\,\left[\begin{array}[]{cc}R_{+}&0\\\ 0&R_{-}\\\
\end{array}\right]$
Where $R_{\pm}$ is the restriction of the spinorial representation of
$\mathcal{C}l(V)^{+}$ to $S^{\pm}$. The representations $R_{\pm}$ are
generally faithful and independent of each other. Since the group $SPin(V)$ is
formed just by elements of even order it then follows that it generally admits
representations of dimension $2^{r-1}$ and, consequently, the same is valid
for the group $SO(V)$. For instance, the following relations are valid [95]:
$SPin(\mathbb{R}^{2})\sim U(1)\quad;\;SPin(\mathbb{R}^{3,1})\sim
Sl(2,\mathbb{C})\quad;\;SPin(\mathbb{R}^{6})\sim SU(4)\,.$
In order to make clear the concepts introduced so far, let us work out a
simple example.
Example:
Let $\\{\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2}\\}$ be an
orthonormal basis for the space $V=\mathbb{R}^{2}$, so that
$\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{1}=\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{2}=1$
and
$\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}=\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}$.
In particular
$\\{1,\,\hat{\boldsymbol{e}}_{1},\,\hat{\boldsymbol{e}}_{2},\,\boldsymbol{I}=\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\\}$
forms a basis for $\mathcal{C}l(\mathbb{R}^{2})$. A general element of
$SPin(\mathbb{R}^{2})$ has the following form:
$\Phi=[\cos(\phi_{2})\hat{\boldsymbol{e}}_{1}+\sin(\phi_{2})\hat{\boldsymbol{e}}_{2}][\cos(\phi_{1})\hat{\boldsymbol{e}}_{1}+\sin(\phi_{1})\hat{\boldsymbol{e}}_{2}]=\cos\theta-\sin\theta\,\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}\,,$
where $\theta=\phi_{1}-\phi_{2}$. Hence the elements of $SPin(\mathbb{R}^{2})$
are labeled by a single real number $\theta\in[0,2\pi)$. Moreover, since
$\displaystyle\Phi_{\theta_{1}}\,\Phi_{\theta_{2}}\,$
$\displaystyle=\,(\cos\theta_{1}-\sin\theta_{1}\,\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2})(\cos\theta_{2}-\sin\theta_{2}\,\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2})$
$\displaystyle=\,\cos(\theta_{1}+\theta_{2})-\sin(\theta_{1}+\theta_{2})\,\hat{\boldsymbol{e}}_{1}\wedge\hat{\boldsymbol{e}}_{2}\,=\,\Phi_{(\theta_{1}+\theta_{2})}\,,$
it follows that $SPin(\mathbb{R}^{2})\sim U(1)$. The rotation implemented by
$\Phi_{\theta}$ is the following:
$\displaystyle\hat{\boldsymbol{e}}_{1}\,\rightarrow\,\hat{\boldsymbol{e}}^{\prime}_{1}=\Phi_{\theta}\,\hat{\boldsymbol{e}}_{1}\,\Phi_{\theta}^{-1}\,$
$\displaystyle=\,\cos(2\theta)\hat{\boldsymbol{e}}_{1}+\sin(2\theta)\hat{\boldsymbol{e}}_{2}$
$\displaystyle\hat{\boldsymbol{e}}_{2}\,\rightarrow\,\hat{\boldsymbol{e}}^{\prime}_{2}=\Phi_{\theta}\,\hat{\boldsymbol{e}}_{2}\,\Phi_{\theta}^{-1}\,$
$\displaystyle=\,-\sin(2\theta)\hat{\boldsymbol{e}}_{1}+\cos(2\theta)\hat{\boldsymbol{e}}_{2}\,.$
Now let us see that
$S=\\{\boldsymbol{\psi}\in\mathcal{C}l(\mathbb{R}^{2})\,|\,\boldsymbol{\psi}=\alpha(1+\hat{\boldsymbol{e}}_{1})+\beta\hat{\boldsymbol{e}}_{2}(1+\hat{\boldsymbol{e}}_{1})\;\forall\,\alpha,\beta\in\mathbb{C}\\}$
is a minimal left ideal of this Clifford algebra. Indeed, defining
$\boldsymbol{\psi}_{1}\equiv(1+\hat{\boldsymbol{e}}_{1})$ and
$\boldsymbol{\psi}_{2}\equiv\hat{\boldsymbol{e}}_{2}(1+\hat{\boldsymbol{e}}_{1})$
we easily prove that
$\hat{\boldsymbol{e}}_{1}\,(\alpha\boldsymbol{\psi}_{1}+\beta\boldsymbol{\psi}_{2})=\alpha\boldsymbol{\psi}_{1}-\beta\boldsymbol{\psi}_{2}\;\textrm{
and
}\;\hat{\boldsymbol{e}}_{2}\,(\alpha\boldsymbol{\psi}_{1}+\beta\boldsymbol{\psi}_{2})=\beta\boldsymbol{\psi}_{1}+\alpha\boldsymbol{\psi}_{2}\,,$
which implies that $S$ is invariant by the left action of
$\mathcal{C}l(\mathbb{R}^{2})$. It is also simple matter to verify that $S$
admits no proper left ideal, which implies that
$\\{\boldsymbol{\psi}_{1},\boldsymbol{\psi}_{2}\\}$ can be seen as a basis for
the spinor space. The spinors
$\boldsymbol{\psi}^{\pm}=\boldsymbol{\psi}_{1}\pm i\boldsymbol{\psi}_{2}$ are
Weyl spinors, since they obey the relation
$\boldsymbol{I}\boldsymbol{\psi}^{\pm}=\pm i\boldsymbol{\psi}^{\pm}$. The
action of the group $SPin(\mathbb{R}^{2})$ on the semi-spinors is the
following:
$\Phi_{\theta}\,\boldsymbol{\psi}^{+}\,=\,e^{-i\theta}\boldsymbol{\psi}^{+}\quad;\;\Phi_{\theta}\,\boldsymbol{\psi}^{-}\,=\,e^{i\theta}\boldsymbol{\psi}^{-}\,.$
Particularly, note that taking $\theta=\pi$ the vectors remain unchanged by
the action of the group $SPin(\mathbb{R}^{2})$ while the spinors change the
sign. This is an example of a well-known property of spinors, they are
multiplied by $-1$ when a rotation of $2\pi$ is executed on the space. $\Box$
Given a spinor $\boldsymbol{\psi}\in S$ we can associate to it a vector
subspace $N_{\boldsymbol{\psi}}\subset V$ called the null subspace of
$\boldsymbol{\psi}$ and defined by
$N_{\boldsymbol{\psi}}\,=\,\\{\boldsymbol{a}\in
V\,|\,\boldsymbol{a}\,\boldsymbol{\psi}\,=\,0\\}$. This vector subspace has
the property of being totally null (isotropic), i.e., all vectors of
$N_{\boldsymbol{\psi}}$ are orthogonal to each other. Indeed, assuming that
$\boldsymbol{\psi}\neq 0$ it follows that
$2<\boldsymbol{a},\boldsymbol{b}>\,\boldsymbol{\psi}\,=\,(\boldsymbol{a}\boldsymbol{b}+\boldsymbol{b}\boldsymbol{a})\,\boldsymbol{\psi}\,=\,0\quad\forall\;\boldsymbol{a},\boldsymbol{b}\in
N_{\boldsymbol{\psi}}\;\Rightarrow\;<\boldsymbol{a},\boldsymbol{b}>\,=\,0\,.$
In a vector space of complex dimension $n=2r$, the maximal dimension that an
isotropic subspace can have is $r$. Therefore a totally null subspace with
this dimension is dubbed maximally isotropic. When the subspace
$N_{\boldsymbol{\psi}}$ is maximally isotropic the spinor $\boldsymbol{\psi}$
is said to be a pure spinor. Apart from a multiplicative constant, the
association between pure spinors and maximally isotropic subspaces is one-to-
one. It is worth noting that in general the sum of two pure spinors is not a
pure spinor, indeed the purity condition is a quadratic constraint on the
spinor [106].
Now let us prove that every pure spinor must be a Weyl spinor. Let $V$ be a
complexified vector space and
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\ldots,\boldsymbol{e}_{r}\\}$ be the
basis of a maximally isotropic subspace $N_{\boldsymbol{\psi}}$, thus
$<\boldsymbol{e}_{a},\boldsymbol{e}_{b}>=0$. We can complete this basis with
$r$ other vectors $\\{\boldsymbol{\theta}^{a}\\}$ in order to form a basis for
the whole vector space $V$ such that
$<\boldsymbol{e}_{a},\boldsymbol{\theta}^{b}>=\frac{1}{2}\delta_{a}^{\,b}$ and
$<\boldsymbol{\theta}^{a},\boldsymbol{\theta}^{b}>=0$. Then we have that
$\boldsymbol{I}\,\propto\,(\boldsymbol{e}_{1}\wedge\boldsymbol{\theta}^{1})(\boldsymbol{e}_{2}\wedge\boldsymbol{\theta}^{2})\ldots(\boldsymbol{e}_{r}\wedge\boldsymbol{\theta}^{r})\,.$
(C.9)
By definition $\boldsymbol{e}_{a}\,\boldsymbol{\psi}=0$, therefore
$\displaystyle(\boldsymbol{e}_{a}\boldsymbol{\theta}^{b})\,\boldsymbol{\psi}\,=\,(\boldsymbol{e}_{a}\boldsymbol{\theta}^{b}+\boldsymbol{\theta}^{b}\boldsymbol{e}_{a})\,\boldsymbol{\psi}\,=\,2<\boldsymbol{e}_{a},\boldsymbol{\theta}^{b}>\boldsymbol{\psi}\,=\,\delta_{a}^{\,b}\,\boldsymbol{\psi}\;\Rightarrow$
$\displaystyle(\boldsymbol{e}_{a}\wedge\boldsymbol{\theta}^{b})\,\boldsymbol{\psi}=\frac{1}{2}(\boldsymbol{e}_{a}\boldsymbol{\theta}^{b}-\boldsymbol{\theta}^{b}\boldsymbol{e}_{a})\,\boldsymbol{\psi}=\frac{1}{2}(\boldsymbol{e}_{a}\boldsymbol{\theta}^{b})\,\boldsymbol{\psi}=\frac{1}{2}\delta_{a}^{\,b}\,\boldsymbol{\psi}\,.$
(C.10)
Then equations (C.9) and (C.10) imply that
$\boldsymbol{I}\,\boldsymbol{\psi}\propto\boldsymbol{\psi}$. This, in turn,
guarantees that the pure spinor $\boldsymbol{\psi}$ must be a Weyl spinor.
Conversely, if $n=2,4,6$ then all Weyl spinors are pure, but in higher
dimensions this is not true [106]. Using (C.6) it is also simple matter to
prove that the Hodge dual of the $r$-vector
$\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\ldots\wedge\boldsymbol{e}_{r}$
is a multiple of this $r$-vector.
The space of spinors, $S$, can be endowed with an operation called charge
conjugation, $c:S\rightarrow S$. This is an anti-linear operation whose
action, $\boldsymbol{\psi}\mapsto\boldsymbol{\psi}^{c}$, is such that the
following property holds:
$(\boldsymbol{\omega}\,\boldsymbol{\psi})^{c}\,=\,\overline{\boldsymbol{\omega}}\,\boldsymbol{\psi}^{c}\quad\forall\;\boldsymbol{\omega}\,\in\,\mathcal{C}l(V)\textrm{
and }\;\boldsymbol{\psi}\,\in\,S\,,$
where $\overline{\boldsymbol{\omega}}$ is the complex conjugate of
$\boldsymbol{\omega}$. The charge conjugation has different features depending
on the signature and on the dimension of the vector space, see [106] for
example. For instance, on the Minkowski space, $\mathbb{R}^{1,3}$, such
operation changes the chirality of a Weyl spinor and its square gives the
identity, while for $\mathbb{R}^{5,1}$ the spaces $S^{\pm}$ are invariant and
$(\boldsymbol{\psi}^{c})^{c}=-\boldsymbol{\psi}$.
Another important property of the spinor space is that it is always possible
to introduce a non-degenerate bilinear inner product, $(\,,):S\times
S\rightarrow\mathbb{C}$, that is invariant by the group $SPin_{+}(V)$. Indeed,
defining
$(\boldsymbol{\psi},\boldsymbol{\chi})=f(\boldsymbol{\psi}^{t}\boldsymbol{\chi})$
for some function $f:\mathcal{C}l(V)\rightarrow\mathbb{C}$ we find that
$(\boldsymbol{\omega}\boldsymbol{\psi},\boldsymbol{\chi})=(\boldsymbol{\psi},\boldsymbol{\omega}^{t}\boldsymbol{\chi})$.
Hence making a $SPin_{+}(V)$ transformation on the spinors,
$S\mapsto\boldsymbol{\varphi}_{+}S$, we find that
$(\boldsymbol{\psi},\boldsymbol{\chi})\mapsto(\boldsymbol{\varphi}_{+}\boldsymbol{\psi},\boldsymbol{\varphi}_{+}\boldsymbol{\chi})=(\boldsymbol{\psi},\boldsymbol{\varphi}_{+}^{t}\boldsymbol{\varphi}_{+}\boldsymbol{\chi})=(\boldsymbol{\psi},\boldsymbol{\chi})$,
since $\boldsymbol{\varphi}_{+}\in SPin_{+}(V)$. A particularly simple choice
for $f$ would be $f(\boldsymbol{\omega})=[\boldsymbol{\omega}]_{0}$, where
$[\boldsymbol{\omega}]_{0}$ is the scalar part (zero order term) of
$\boldsymbol{\omega}\in\mathcal{C}l(V)$. But in order for the inner product to
be non-degenerate we must judiciously choose the function $f$, as $f=[\,]_{0}$
may not obey to this criterium. The general formalism for the choice of an
adequate $f$ is very tricky and more details can be found in [97, 95].
The inner product $(\,,\,)$ can be symmetric or skew-symmetric depending on
the dimension of $V$. For example, in two dimensions it is symmetric, while in
four and six dimensions it is skew-symmetric [106]. Furthermore, in four
dimensions the inner product of two semi-spinors of _opposite_ chirality
vanishes, while in two and six dimensions the inner product of Weyl spinors of
the _same_ chirality vanish [106].
In Physics, the Clifford algebra and the spinor formalism is usually used in a
less abstract way, making use of the so-called Dirac matrices [107]. If the
metric of a $2r$-dimensional vector space $V$ is $g_{ab}$ then the Dirac
matrices, $\gamma_{a}$, are defined to be $2^{r}\times 2^{r}$ matrices such
that
$\\{\gamma_{a},\gamma_{b}\\}=(\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a})=2g_{ab}$.
The $2^{r}$-dimensional vector space on which these matrices act is called the
space of spinors. Using the tools presented in this appendix it is not hard to
guess the origin this practical approach. The matrices $\gamma_{a}$ are just a
matrix representation of the vectors $\\{\hat{\boldsymbol{e}}_{a}\\}$ of
$\mathcal{C}l(V)$ and the anti-commutation relation
$\\{\gamma_{a},\gamma_{b}\\}=2g_{ab}$ is the matrix realization of equation
(C.1). Since the Dirac matrices provide a faithful representation of minimal
dimension for the vectors of the Clifford algebra they must be $2^{r}\times
2^{r}$ matrices and the column vectors on which these matrices act should be
called spinors.
The material presented in this appendix is just a scratch on the rich field of
geometric algebra. There are many nice references on Clifford algebra and
spinors. The classical reference that presents the “modern” approach to the
subject is the book of C. Chevalley [108]. Introductory texts with
applications in Physics can be found in [42, 43], while geometric applications
and historical notes are available in [109]. More advanced and rigorous
treatments are found in [97, 95].
### Appendix D Group Representations
In this appendix it will be explained what is a representation of a group and
how to construct higher-dimensional representations out of a lower-dimensional
one. First let us recall some basic definitions on group theory. Let $G$ be a
set endowed with a product $g_{1}\cdot g_{2}=g_{3}$ such that $g_{3}\in G$ for
all $g_{1},g_{2}\in G$. Then $G$ is called a group when the following three
properties hold: (1) There exists an element $e\in G$, called the identity
element, such that $e\cdot g=g$ for all $g\in G$; (2) For every element $g\in
G$ there exists an element $g^{-1}\in G$, called the inverse of $g$, such that
$g\cdot g^{-1}=e$; (3) The product is associative, $g_{1}\cdot(g_{2}\cdot
g_{3})=(g_{1}\cdot g_{2})\cdot g_{3}$ for all $g_{1},g_{2},g_{3}\in G$. A map
$H:G\rightarrow G^{\prime}$ between two groups $G$ and $G^{\prime}$ is called
a homomorphism if $H(g_{1})\cdot H(g_{2})=H(g_{1}\cdot g_{2})$ for all
$g_{1},g_{2}\in G$.
Whenever a physical system has a symmetry the group theory can be used in
order to simplify the analysis. Although sometimes it is possible to move on
just using the abstract concept of a group, generally it is necessary to use a
down-to-earth approach, such as expressing the group elements by matrices. A
representation of a group $G$ on the vector space $V$ is a homomorphism
$L:G\rightarrow GL(V)$, where $GL(V)$ is the group formed by all invertible
linear operators acting on $V$. Since vector spaces are ubiquitous in physics
it follows that representation theory is a quite helpful tool in many branches
of this science. If $\dim(V)=n$ we say that $L$ is an $n$-dimensional
representation. Note that every group admits a trivial representation of
dimension $1$ given by $I:G\rightarrow GL(\mathbb{R})=\mathbb{R}^{*}$ with
$I(g)=1$ for all $g\in G$. Two representations $L_{1}$ and $L_{2}$ of the
group $G$ on the vector space $V$ are said to be equivalent when there exists
some $B\in GL(V)$ such that $L_{2}(g)=BL_{1}(g)B^{-1}$ for all $g\in G$.
Let us adopt the index notation and denote a vector of the $n$-dimensional
vector space $V$ by $v^{a}$, with $a\in\\{1,2,\ldots,n\\}$. Then a
representation of the group $G$ on this vector space is an association of a
matrix $L^{a}_{\phantom{a}b}(g)$ to every $g\in G$. Since this association is,
by definition, a homomorphism, the identity
$L^{a}_{\phantom{a}c}(g_{1})L^{c}_{\phantom{c}b}(g_{2})=L^{a}_{\phantom{a}b}(g_{1}\cdot
g_{2})$ must hold for all $g_{1},g_{2}\in G$. Once specified a representation
$L$ of the group $G$ on the vector space $V$, we then say that the action of a
group element $g$ on a vector $v^{a}$ amounts to the following transformation:
$v^{a}\,\stackrel{{\scriptstyle
g}}{{\longrightarrow}}\,L^{a}_{\phantom{a}b}(g)\,v^{b}\,.$ (D.1)
In abstract notation we can write $\boldsymbol{v}\rightarrow
L(g)\boldsymbol{v}$. Given such representation one can define another
representation $P:G\rightarrow GL(V)$ called the inverse representation and
defined by $\boldsymbol{v}\rightarrow P(g)\boldsymbol{v}$, with $P(g)$ being
the transpose of $L(g)$ inverse, $P(g)\equiv(L(g)^{-1})^{t}$. Let us verify
that this is, indeed, a representation:
$\displaystyle P(g_{1})P(g_{2})=$
$\displaystyle\,(L(g_{1})^{-1})^{t}(L(g_{2})^{-1})^{t}\,=\,\left(L(g_{2})^{-1}L(g_{1})^{-1}\right)^{t}$
$\displaystyle=$
$\displaystyle\,[\left(L(g_{1})L(g_{2})\right)^{-1}]^{t}\,=\,\left(L(g_{1}\cdot
g_{2})^{-1}\right)^{t}\,=\,P(g_{1}\cdot g_{2})\,.$
Note that generally the representations $L$ and $P$ are not equivalent. By
definition the representation $P$ acts on the same vector space of the
representation $L$, but it is useful to pretend that $P$ acts on a different
vector space $V^{\prime}$ that is isomorphic to $V$ and whose vectors are
denoted with an index down, $u_{a}\in V^{\prime}$. So the representation $P$
has the following action:
$u_{a}\,\stackrel{{\scriptstyle
g}}{{\longrightarrow}}\,P_{a}^{\phantom{a}b}(g)\,u_{b}\quad;\quad
P_{a}^{\phantom{a}b}(g)\equiv[L(g)^{-1}]^{b}_{\phantom{b}a}\,.$ (D.2)
On the jargon we say that $v^{a}$ is on the $L$ representation while $u_{a}$
is on the $P$ representation. Note that in this case the contraction
$v^{a}u_{a}$ is invariant by the action of the group $G$, which is equivalent
to say that $v^{a}u_{a}\in\mathbb{R}$ is on the trivial representation, $I$.
Suppose that the vector space $V$ has a proper subspace $K\subset V$ such that
$L(g)\boldsymbol{k}\in K$ for all $\boldsymbol{k}\in K$ and for all $g\in G$.
Then the restriction of $L(g)$ to this subspace provides a representation for
the group $G$ on the lower-dimensional vector space $K$. When this happens we
say that the representation $L$ is reducible, otherwise it is called
irreducible. The irreducible representations of a group are the building
blocks of a general representation, since every representation of $G$ can be
understood as a composition of some irreducible representations of this group.
For instance, it is well-known that the irreducible representations of the
rotation group on $\mathbb{R}^{3}$, $SO(3)$, are labeled by
$l\in\\{0,\,\frac{1}{2},\,1,\,\frac{3}{2},\,2,\cdots\\}$, the angular momentum
quantum number. The dimension of the representation dubbed $l$ is $(2l+1)$.
Here we shall label an irreducible representation of a group by its dimension
in bold face. Thus the representations $\boldsymbol{2}$ and $\boldsymbol{3}$
of $SO(3)$ mean the ones with $l=\frac{1}{2}$ and $l=1$ respectively.
Moreover, the trivial representation $I$ might be denoted by $\boldsymbol{1}$.
Given an irreducible representation $\boldsymbol{n}$ of a group $G$, generally
it is possible to generate other irreducible representations of $G$ by means
of the direct products of the representation $\boldsymbol{n}$ with itself. We
can understand this as follows, the representation $\boldsymbol{n}$ associates
to every $g\in G$ an $n\times n$ matrix $L(g)$. Then taking the direct product
$L(g)\otimes L(g)$ we obtain an $n^{2}\times n^{2}$ matrix for every $g$.
These matrices also provide a representation for the group $G$, but generally
this representation is not irreducible, since in general such $n^{2}\times
n^{2}$ matrices will admit proper invariant subspaces. Then looking for the
invariant subspaces of these matrices one can split the new representation
into its irreducible parts. For example, the direct product of the irreducible
representations $l^{\prime}$ and $l^{\prime\prime}$ of the group $SO(3)$ is
equal to the direct sum of all irreducible representations contained on the
interval $|l^{\prime\prime}-l^{\prime}|\leq
l\leq(l^{\prime}+l^{\prime\prime})$. This is usually written as [110]:
$l^{\prime}\otimes
l^{\prime\prime}\,=\,(l^{\prime}+l^{\prime\prime})\,\oplus\,(l^{\prime}+l^{\prime\prime}-1)\,\oplus\,(l^{\prime}+l^{\prime\prime}-2)\,\oplus\cdots\oplus\,|l^{\prime}-l^{\prime\prime}|\,.$
(D.3)
As an instructive example let us work out the direct product of some
irreducible representations of the group $SO(n)$. Let $R:SO(n)\rightarrow
GL(\mathbb{R}^{n})$ be the usual representation of this group that associates
to every element of $SO(n)$ an $n\times n$ orthogonal matrix $R$ with unit
determinant, $RR^{t}=\boldsymbol{1}$ and $\det(R)=1$. This irreducible
representation is denoted by $\boldsymbol{n}$ and its action on
$\mathbb{R}^{n}$ is given by:
$v^{a}\,\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\,R^{a}_{\phantom{a}b}\,v^{b}\,.$
We say that the tensor $T^{ab}$ is on the representation
$\boldsymbol{n}\otimes\boldsymbol{n}$ if its transformation under the group
$SO(n)$ is given by:
$T^{ab}\,\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\,R^{a}_{\phantom{a}c}\,R^{b}_{\phantom{b}d}\,T^{cd}\,.$
It is simple matter to verify that this representation is reducible. Indeed,
note that the subspace formed by the symmetric tensors $T^{ab}=T^{(ab)}$ is
invariant under the action of the representation
$\boldsymbol{n}\otimes\boldsymbol{n}$. Suppose that $S^{ab}$ is symmetric,
then
$\displaystyle S^{ab}\,\stackrel{{\scriptstyle R}}{{\longrightarrow}}\,$
$\displaystyle\;R^{a}_{\phantom{a}c}\,R^{b}_{\phantom{b}d}\,S^{cd}=R^{a}_{\phantom{a}c}\,R^{b}_{\phantom{b}d}\,\frac{1}{2}\,[S^{cd}+S^{dc}]$
$\displaystyle=\frac{1}{2}\,[R^{a}_{\phantom{a}c}\,R^{b}_{\phantom{b}d}+R^{a}_{\phantom{a}d}\,R^{b}_{\phantom{b}c}]\,S^{cd}=R^{(a}_{\phantom{a}c}\,R^{b)}_{\phantom{b}d}\,S^{cd}\,,$
which is also symmetric. In the same vein, the space of skew-symmetric tensors
$T^{ab}=T^{[ab]}$ is, likewise, invariant under the action of the
representation $\boldsymbol{n}\otimes\boldsymbol{n}$. Moreover, we can easily
convince ourselves that the restriction of the representation
$\boldsymbol{n}\otimes\boldsymbol{n}$ to the space of skew-symmetric tensors
is irreducible. Differently, the representation provided by the symmetric
tensors can be split in two irreducible representations. Indeed, note that the
symmetric tensors of the form $T^{ab}=\lambda\,\delta^{ab}$ are invariant by
$SO(n)$:
$\lambda\,\delta^{ab}\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\,\lambda\,R^{a}_{\phantom{a}c}\,R^{b}_{\phantom{b}d}\,\delta^{cd}=\lambda\,\delta^{ab}\,,$
where it was used that fact that $R$ is an orthogonal matrix. Note that the
inverse of the representation $\boldsymbol{n}$ for the group $SO(n)$ is the
representation $\boldsymbol{n}$ itself, which can be verified using equation
(D.2) and the identity $(R^{-1})^{t}=R$ valid for orthogonal matrices. Thus a
general tensor $T^{ab}$ on the representation
$\boldsymbol{n}\otimes\boldsymbol{n}$ of the group $SO(n)$ can be written as
the following sum of irreducible parts:
$T^{ab}\,=\,\left(T^{(ab)}-\lambda\,\delta^{ab}\right)\,+\,T^{[ab]}\,+\,\lambda\,\delta^{ab}\;\;;\quad\lambda\equiv\frac{1}{n}\,\delta_{cd}T^{cd}\,.$
These irreducible parts are respectively called the symmetric trace-less part,
the skew-symmetric part and the trace of the representation
$\boldsymbol{n}\otimes\boldsymbol{n}$. In terms of dimensions this is written
as:
$\boldsymbol{n}\otimes\boldsymbol{n}\,=\,\left[\boldsymbol{\frac{1}{2}n(n+1)-1}\right]\,\,\oplus\,\,\boldsymbol{\frac{1}{2}n(n-1)}\,\,\oplus\,\,\boldsymbol{1}\,.$
(D.4)
Where $[\frac{1}{2}n(n+1)-1]$ is the number of components of a symmetric
tensor with vanishing trace, $S^{ab}=S^{ba}$ and $\delta_{ab}S^{ab}=0$,
$\frac{1}{2}n(n-1)$ is the number of independent components of a skew-
symmetric tensor, $A^{ab}=-A^{ba}$, and $1$ represents the single degree of
freedom contained in $\lambda$, the trace of $T^{ab}$. Note that for $n=3$
this is consistent with the formula (D.3) valid for the group $SO(3)$:
$[\,l^{\prime}=1\,]\otimes[\,l^{\prime\prime}=1\,]\,\,\boldsymbol{=}\,\,[\,l=2\,]\,\,\oplus\,\,[\,l=1\,]\,\,\oplus\,\,[\,l=0\,]\,.$
Since the dimension of the irreducible representation labeled by $l$ is
$(2l+1)$, it follows that the above equation is equivalent to:
$\boldsymbol{3}\otimes\boldsymbol{3}\,=\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{1}\,,$
which agrees with equation (D.4) when $n=3$. As a last example let us look for
the irreducible parts of the representation
$\boldsymbol{n}\otimes\boldsymbol{n}\otimes\boldsymbol{n}$ of the group
$SO(n)$. An object in this representation is a tensor with three indices,
$N^{abc}$, transforming as follows:
$N^{abc}\,\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\,R^{a}_{\phantom{a}d}\,R^{b}_{\phantom{b}e}\,R^{c}_{\phantom{c}f}\,N^{def}\,.$
(D.5)
Let us try to separate the parts of this tensor that are invariant under this
transformation for a general orthogonal matrix $R^{a}_{\phantom{a}b}$. In what
follows we shall display the dimension of each representation below the
respective invariant terms, with the irreducible representations being denoted
by bold face. The first trivial separation of the tensor $N^{abc}$ in parts
that are invariant under the transformation (D.5) is given by:
$\underbrace{N^{abc}}_{n^{3}}\;\longrightarrow\;\;\underbrace{N^{a(bc)}}_{\frac{1}{2}n^{2}(n+1)}\quad,\quad\underbrace{N^{a[bc]}}_{\frac{1}{2}n^{2}(n-1)}\,.$
Then the first term on the right hand side of the above equation splits on the
following invariant parts:
$\underbrace{N^{a(bc)}}_{\frac{1}{2}n^{2}(n+1)}\;\longrightarrow\;\;\underbrace{\delta_{ab}N^{a(bc)}}_{\boldsymbol{n}}\quad,\quad\underbrace{\delta_{bc}N^{a(bc)}}_{\boldsymbol{n}}\quad,\quad\underbrace{\hat{N}^{a(bc)}}_{\frac{1}{2}n^{2}(n+1)-2n}\,.$
Where $\hat{N}^{a(bc)}$ is a tensor such that $\delta_{ab}\hat{N}^{a(bc)}=0$
and $\delta_{bc}\hat{N}^{a(bc)}=0$. This tensor, in turn, gives rise to the
following irreducible parts:
$\underbrace{\hat{N}^{a(bc)}}_{\frac{1}{2}n^{2}(n+1)-2n}\;\longrightarrow\;\;\underbrace{\hat{N}^{(abc)}}_{\boldsymbol{\frac{1}{3!}n(n+1)(n+2)-n}}\quad,\quad\underbrace{\tilde{N}^{a(bc)}}_{\boldsymbol{\frac{1}{3}n(n^{2}-4)}}\,.$
Where $\tilde{N}^{a(bc)}$ is a tensor obeying to the following constraints
$\tilde{N}^{(abc)}=0$, $\delta_{ab}\tilde{N}^{a(bc)}=0$ and
$\delta_{bc}\tilde{N}^{a(bc)}=0$. In the same vein, the tensor $N^{a[bc]}$
splits on the following invariant parts:
$\underbrace{N^{a[bc]}}_{\frac{1}{2}n^{2}(n-1)}\;\longrightarrow\;\;\underbrace{\delta_{ab}N^{a[bc]}}_{\boldsymbol{n}}\quad,\quad\underbrace{\hat{N}^{a[bc]}}_{\frac{1}{2}n^{2}(n-1)-n}\,.$
With $\hat{N}^{a[bc]}$ being a trace-less tensor,
$\delta_{ab}\hat{N}^{a[bc]}=0$. This tensor, in turn, lead to the following
irreducible parts:
$\underbrace{\hat{N}^{a[bc]}}_{\frac{1}{2}n^{2}(n-1)-n}\;\longrightarrow\;\;\underbrace{\hat{N}^{[abc]}}_{\boldsymbol{\frac{1}{3!}n(n-1)(n-2)}}\quad,\quad\underbrace{\tilde{N}^{a[bc]}}_{\boldsymbol{\frac{1}{3}n(n^{2}-4)}}\,.$
Where $\tilde{N}^{a[bc]}$ is a tensor such that $\tilde{N}^{[abc]}=0$ and
$\delta_{ab}\tilde{N}^{a[bc]}=0$. Therefore, the representation
$\boldsymbol{n}\otimes\boldsymbol{n}\otimes\boldsymbol{n}$ splits on the
following irreducible parts:
$\displaystyle\boldsymbol{n}\otimes\boldsymbol{n}\otimes\boldsymbol{n}\,=\,\,$
$\displaystyle\boldsymbol{n}\,\,\oplus\,\,\boldsymbol{n}\,\,\oplus\,\,\boldsymbol{n}\,\,\oplus\,\,\boldsymbol{\frac{1}{3}n(n^{2}-4)}\,\,\oplus\,\,\boldsymbol{\frac{1}{3}n(n^{2}-4)}$
$\displaystyle\oplus\,\,\boldsymbol{\frac{n(n-1)(n-2)}{6}}\,\,\oplus\,\,\boldsymbol{\left[\frac{n(n+1)(n+2)}{6}-n\right]}\,.$
(D.6)
In particular, for the group $SO(3)$ we have:
$\boldsymbol{3}\otimes\boldsymbol{3}\otimes\boldsymbol{3}\,=\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{1}\,\,\oplus\,\,\boldsymbol{7}\,.$
(D.7)
One can easily use equation (D.3) in order to verify that this result is
correct:
$\displaystyle\boldsymbol{3}\otimes\boldsymbol{3}\otimes\boldsymbol{3}\,=\,$
$\displaystyle\,\boldsymbol{3}\otimes\left[\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{1}\,\right]=\left[\,\boldsymbol{3}\otimes\boldsymbol{5}\,\right]\,\,\oplus\,\,\left[\,\boldsymbol{3}\otimes\boldsymbol{3}\,\right]\,\,\oplus\,\,\left[\,\boldsymbol{3}\otimes\boldsymbol{1}\,\right]$
$\displaystyle=\,$
$\displaystyle\left[\,\boldsymbol{7}\,\,\oplus\,\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{3}\,\right]\,\,\oplus\,\,\left[\,\boldsymbol{5}\,\,\oplus\,\,\boldsymbol{3}\,\,\oplus\,\,\boldsymbol{1}\,\right]\,\,\oplus\,\,\left[\,\boldsymbol{3}\,\right]\,,$
which agrees with equation (D.7). Equations (D.4) and (D.6) show us that
starting with the irreducible representation $\boldsymbol{n}$ of the group
$SO(n)$ we can take direct products in order to construct other irreducible
representations. In general this kind of procedure can be used for any group.
If a group $G$ admits an irreducible representation $\boldsymbol{f}$ such that
all irreducible representations of $G$ can be constructed using the direct
products of this representation, its inverse and its complex conjugate, then
$\boldsymbol{f}$ is called the fundamental representation of $G$. For
instance, the fundamental representation of the group $SO(3)$ is the one with
$l=\frac{1}{2}$, in which the rotations of $\mathbb{R}^{3}$ are represented by
$2\times 2$ unitary matrices of unit determinant (spinorial representation).
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### Index
* Bel-Debever criteria §2.4
* Boost weight §2.3, §4.2
* Calabi-Yau manifold §4.4.1, §4.4.3, §4.4.5
* Cartan structure equation §1.7
* Charge Conjugation Appendix C, §5.2
* Clifford algebra Appendix C, §2.6, §5.1.2
* CMPP classification §2.3, Chapter 5, §5.5, §6.3
* Complex manifold §4.4.1
* Dirac spinor §5.1.2
* Distribution §1.8, §3.2, §3.3, §4.3, §6.2
* Frobenius theorem §1.8
* Goldberg-Sachs theorem §3.2, §4.3, §5.4, §6.2, §6.4
* Graviton §1.1
* Harmonic form §6.3, Theorem 23
* Hidden symmetries §1.4, §3.5, §3.5
* Hodge dual Appendix C, §1.6, §2.1, §4.1, §6.1
* Interior product §1.6
* Irreducible representations Appendix D, §5.1
* Isotropic Appendix C, §3.3, §4.1, §4.3, §4.4.1, §4.4.2, §5.1.3, §6.2
* Killing vector §1.4, §2.8
* Killing-Yano tensor §1.4, §3.5
* Lie bracket §1.8
* Mariot-Robinson theorem §3.3, §6.4
* Maximally isotropic Appendix C, §3.5, Chapter 5, §5.1.3, §5.4, §6.2, Theorem 17, Theorem 22
* Null bivector §3.3, §3.3, footnote 1, §4.1, §5.1.3, §6.4, Corollary 2
* Null form §5.1.3, §5.1.3, §6.2, Theorem 24
* Null frame §4.1, §5.1.1, §6.2
* Optical scalars §3.1, §3.1, §6.3
* Peeling property §3.4, §3.4
* Principal null directions(PND) §2.2, §2.4, §2.7, §3.2, §4.3, Theorem 3
* Pseudo-scalar Appendix C, Appendix C, §2.6, §5.1.2
* Pure spinor Appendix C, §5.1.3, §5.5, Theorem 17
* Segre classification Appendix A, §2.1, §5.3, §5.5
* Self-dual bivector §2.1, §2.5, §4.1
* Self-dual form §5.1, §6.1.2
* Self-dual manifold §4.4.2, §6.1.2
* Shear §3.1, §3.1, §4.3, §6.3
* Signature §1.2, §4.1, §5.2
* Simple form §1.6
* SL(4;C) §5.2
* SO(3,1) §1.7, §2.2, §4.2.4
* Spinor Appendix C, Appendix C, Appendix C, §2.5, §5.1
* SU(4) §5.1, §5.1
* Volume-form §1.6, §4.1, §6.1
* Weyl spinor Appendix C, Appendix C, Appendix C, §5.1.2
* Weyl tensor §1.2, §2.1, §2.5, §4.2, §5.1, §6.1
List of Symbols
$\displaystyle\partial_{\mu}\quad\quad$ $\displaystyle\textrm{ Partial
derivative }\frac{\partial}{\partial x^{\mu}}:\small{\textsf{ Page
\ref{PartialD}.}}$ $\displaystyle C_{\mu\nu\rho\sigma}\quad\quad$ Weyl Tensor:
Page 1.2. $\displaystyle T_{[a_{1}a_{2}\ldots a_{p}]}\quad\quad$
$\displaystyle\textrm{ Skew-symmetric part of the tensor }T_{a_{1}a_{2}\ldots
a_{p}}:\small{\textsf{ Page \ref{Symmetrization}.}}$ $\displaystyle
T_{(a_{1}a_{2}\ldots a_{p})}\quad\quad$ $\displaystyle\textrm{ Symmetric part
of the tensor }T_{a_{1}a_{2}\ldots a_{p}}:\small{\textsf{ Page
\ref{Symmetrization}.}}$
$\displaystyle\epsilon_{\mu_{1}\mu_{2}\ldots\mu_{n}}\quad\quad$ Volume-form of
the $n$-dimensional manifold: Page 1.6.
$\displaystyle\boldsymbol{g}\quad\quad$ The metric of the manifold: Page 1.2.
$\displaystyle\lrcorner\;,\;\;\boldsymbol{V}\lrcorner\boldsymbol{F}\quad\quad$
Interior product: Page 1.6. $\displaystyle\star\boldsymbol{F}\quad\quad$ Hodge
dual of a differential form: Page 1.15.
$\displaystyle\boldsymbol{\omega}^{a}_{\phantom{a}b}\,,\;\omega_{ab}^{\phantom{ab}c}\,,\;\omega_{abc}\quad\quad$
Connection 1-form and its components: Page 1.7.
$\displaystyle\Psi_{0}\,,\;\Psi_{1}\,,\;\ldots,\Psi_{4}\quad\quad$ Weyl
scalars in 4 dimensions: Pages 2.1 and 4.2. $\displaystyle\sigma\quad\quad$
Shear of a null congruence: Pages 3.1 and 3.5.
$\displaystyle\Gamma(\wedge^{p}M)\quad\quad$ Space of local sections of the
$p$-form bundle: Pages 2 and 2. $\displaystyle\Lambda^{m+}\quad\quad$ Space of
self-dual $m$-forms in $2m$ dimensions: Page 6.1.2.
$\displaystyle\mathcal{H}_{p}\quad\quad$ Hodge dual operator on $p$-forms:
Page 6.1. $\displaystyle\mathcal{C}_{p}\quad\quad$ Weyl operator on $p$-forms:
Page 6.2. $\displaystyle\mathcal{C}^{\pm}\quad\quad$ Restriction of the Weyl
operator to $\Lambda^{m\pm}$: Page 6.8.
$\displaystyle\mathcal{A}_{q}\quad\quad$ Particular subbundle of
$\Gamma(\wedge^{m}M)$: Page 6.14. $\displaystyle
M_{i}\,,\;\sigma_{ij}\,,\;A_{ij}\,,\;\theta\quad\quad$ Optical scalars of a
null congruence: Page 6.3. $\displaystyle\Psi^{AB}_{\phantom{AB}CD}\quad\quad$
Spinorial representation of the Weyl tensor in 6D: Page 5.1.
$\displaystyle(T^{AB}\,,\,\tilde{T}_{AB})\quad\quad$ Spinorial representation
of a 3-vector in 6D: Page 5.1. $\displaystyle
Span\\{\boldsymbol{V}_{i}\\}\quad\quad$ Vector distribution generated by the
vector fields $\boldsymbol{V}_{i}$: Page 1.8.
|
arxiv-papers
| 2013-11-27T20:22:02 |
2024-09-04T02:49:54.408280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Batista",
"submitter": "Carlos A. Batista da S. Filho",
"url": "https://arxiv.org/abs/1311.7110"
}
|
1311.7165
|
###### Abstract
The purpose of this paper is to extend the embedding theorem of Sobolev spaces
involving general kernels and we provide a sharp critical exponent in these
embeddings. As an application, solutions for equations driven by a general
integro-differential operator, with homogeneous Dirichlet boundary conditions,
is established by using the Mountain Pass Theorem.
Sharp embedding of Sobolev spaces involving general kernels and its
application
Huyuan [email protected] Hichem [email protected]
1Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi
330022, PR China
and
2Department of Mathematics, College of Science, King Saud University P.O. Box
2455, Riyadh 11451, Saudi Arabia.
Key words: Sobolev space involving general kernel, Sobolev embedding, Integro-
differential operator, Mountain Pass Theorem
MSC2010: 35R09, 35J61, 46E35
## 1 Introduction
In the study of weak solutions for semilinear elliptic equations, the
embedding from corresponding Sobolev space to $L^{q}$ space plays a
fundamental role, especially the compact embedding. In a recent work, Di
Nazza, Palatucci and Valdinoci in [11] made a clear description for the
fractional Sobolev space $W^{s,p}(\Omega)$ and gave an elegant proof for the
embedding theorem from $W^{s,p}(\Omega)$ to $L^{q}(\Omega)$, which is
continuous when $q\in[1,\frac{Np}{N-sp}]$ and compact when
$q\in[1,\frac{Np}{N-sp})$, where $s\in(0,1)$, $sp<N$ and
$\Omega\subset\mathbb{R}^{N}$ is a bounded extension domain with $N\geq 2$.
Motivated by the above work, our purpose of this paper is to build a sharp
embedding theorem of Sobolev space involving general kernel $K$ and by using
this embedding theorem to search for weak solutions to problem
$\begin{array}[]{lll}\mathcal{L}_{K}u+f(x,u)=0&\rm{in}\quad\Omega,\\\\[5.69054pt]
\phantom{\mathcal{L}_{K}+u^{p}--}u=0&\rm{in}\quad\Omega^{c},\end{array}$ (1.1)
where $\Omega\subset\mathbb{R}^{N}$ is an open bounded $C^{2}$ domain with
$N\geq 2$ and the nonlocal operator $\mathcal{L}_{K}$ is defined by
$\mathcal{L}_{K}u(x)=\frac{1}{2}\int_{\mathbb{R}^{N}}[u(x+y)+u(x-y)-2u(x)]K(y)dy$
with the kernel $K:\mathbb{R}^{N}\setminus\\{0\\}\to(0,+\infty)$ satisfying
$\int_{\mathbb{R}^{N}}\min\\{|x|^{2},1\\}K(x)dx<+\infty$ (1.2)
and
$K(x)=K(-x),\qquad x\in\mathbb{R}^{N}\setminus\\{0\\}.$ (1.3)
Moreover, we assume that $K$ is decreasing monotone in the following sense
$K(x)\geq K(y)\qquad{\rm{if}}\ \ |x|\leq|y|.$ (1.4)
A typical example for $K$ is given by $K(x)=|x|^{-(N+2s)}$ with $s\in(0,1)$
and then $\mathcal{L}_{K}$ is the fractional Laplacian operator
$-(-\Delta)^{s}$.
During the last years, non-linear equations involving general integro-
differential operators, especially, fractional Laplacian, have been studied by
many authors. Caffarelli and Silvestre [4] studied the fractional Laplacian
through extension theory. Caffarelli and Silvestre [5, 6], Ros-Oton and Serra
[18] investigated regularity results for fractional elliptic equations. Sire
and Valdinoci in [21], Felmer and Wang in [13], Hajaiej [15, 16] and Felmer,
Quaas and Tan [12] obtained symmetry property of solutions for semilinear
equation involving the fractional Laplacin. More interests on fractional
elliptic equations see [7, 8, 9, 10, 14] and the references therein.
Recently, Servadei and Valdinoci in [20] obtained a solution of (1.1) via
Mountain Pass Theorem under the hypothesis that there exist $\lambda>0$ and
$s\in(0,1)$ such that
$K(x)\geq\lambda|x|^{-(N+2s)},\quad x\in\mathbb{R}^{N}\setminus\\{0\\}$
and nonlinear term $f$ is subcritical, that is,
$|f(x,t)|\leq a_{1}+a_{2}|t|^{q-1}\quad{\rm a.e.}\ x\in\Omega,\
t\in\mathbb{R}$
with $q\in(2,\frac{2N}{N-2s})$ and constants $a_{1},a_{2}>0$. We say that
$\frac{2N}{N-2s}$ is the critical exponent, denoted by $2^{*}(s)$.
In this paper, we are also interested in studying problem (1.1) with more
general kernels and our purpose is to find new criterion for critical
exponent, where we could deal with the following case
$\liminf_{|x|\to 0^{+}}K(x)|x|^{N}\in(0,\infty).$ (1.5)
To this end, we define
$s_{0}=\sup\\{s\geq 0\ |\ \lim_{r\to
0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=+\infty\\}.$ (1.6)
We remark that if $K$ satisfies (1.2) and is nonnegative, then the definition
in (1.6) is equivalent to
$s_{0}=\sup\\{s\geq 0\ |\ \lim_{r\to 0^{+}}r^{2s}\int_{B_{1}(0)\setminus
B_{r}(0)}K(x)dx=+\infty\\}$
By the fact that $\int_{B_{1}^{c}(0)}K(x)dx$ is bounded from (1.2).
Our basic setting is that $s_{0}>0.$ In section 2, we will prove that
$s_{0}\leq 1$ and exhibit an example in which the kernel $K$ satisfying (1.5)
makes $s_{0}\in(0,1)$. We note that the limit of
$r^{2s_{0}}\int_{B_{r}^{c}(0)}K(y)dy$, as $r\to 0$, could be in $[0,\infty]$
or even no exists. Denote
$l_{\infty}:=\liminf_{r\to 0^{+}}r^{2s_{0}}\int_{B_{r}^{c}(0)}K(y)dy,$ (1.7)
then it occurs one of the cases: Case 1: $l_{\infty}=0$ and Case 2:
$l_{\infty}\in(0,\infty]$,
Our first aim is to study the Sobolev space involving general kernel $K$.
Denote by $X$ the linear space of Lebesgue measurable functions from
$\mathbb{R}^{N}$ to $\mathbb{R}$ such that the restriction to $\Omega$ of any
function $g$ in $X$ belongs to $L^{2}(\Omega)$ and
$\int_{\mathbb{R}^{2N}\setminus\mathcal{O}}(g(x)-g(y))^{2}K(x-y)dxdy<+\infty,$
where $\mathcal{O}:=\Omega^{c}\times\Omega^{c}$. The space $X$ is endowed with
the norm as
$\|g\|_{X}=(\|g\|^{2}_{L^{2}(\Omega)}+\int_{\mathbb{R}^{2N}\setminus\mathcal{O}}(g(x)-g(y))^{2}K(x-y)dxdy)^{1/2}.$
(1.8)
Now we define the following Sobolev space
$X_{0}=\\{g\in X\ |\ g=0\ \ {\rm a.e.\ in}\ \Omega^{c}\\}$
equipped the norm (1.8). From (1.2), we stress that
$C^{2}_{0}(\Omega)\subseteq X_{0},$ see [20], and so $X$ and $X_{0}$ are
nonempty.
Now we are ready for an embedding theorem.
###### Theorem 1.1
Assume that $K$ satisfies (1.2), (1.4), (1.6) with $s_{0}\in(0,1]$,
$2^{*}(s_{0})=\frac{2N}{N-2s_{0}}$ and $l_{\infty}$ is defined by (1.7). Then
$(X_{0},\|\cdot\|_{X})$ is a Hilbert space and
$(i)$ if $l_{\infty}=0$, the embedding
$X_{0}\hookrightarrow L^{q}(\Omega)$ (1.9)
is continuous and compact for $q\in[1,2^{*}(s_{0}))$. Moreover, for
$q\in[1,2^{*}(s_{0}))$ there exists $C>0$ such that
$\|g\|_{L^{q}}\leq C\|g\|_{X},\quad\forall g\in X_{0};$ (1.10)
$(ii)$ if $l_{\infty}\in(0,\infty]$, the embedding (1.9) is continuous for
$q\in[1,2^{*}(s_{0})]$ and compact for $q\in[1,2^{*}(s_{0}))$, and the
embedding inequality (1.10) holds for for $q\in[1,2^{*}(s_{0})]$.
###### Example 1.1
Let
$K(x)=\frac{1}{|x|^{N+2s_{0}}}\left[(-\log|x|)_{+}+1\right]^{\sigma},\quad
x\in\mathbb{R}^{N}\setminus\\{0\\},$ (1.11)
where $\sigma\in\mathbb{R}$ and $(-\log|x|)_{+}=\max\\{-\log|x|,0\\}$. When
$s_{0}\in(0,1)$ $\sigma\in\mathbb{R}$ or $s_{0}=1$ $\sigma<-1$, the kernel $K$
defined by (1.11) satisfies (1.2) and (1.4).
We note that $l_{\infty}=0$ if $\sigma<0$, $l_{\infty}\in(0,\infty)$ if
$\sigma=0$ and $l_{\infty}=\infty$ if $\sigma>0$. In particular,
$s_{0}\in(0,1)$ and $\sigma=0$, the embedding (1.9) coincides the results in
[11]. Especially, when $s_{0}=1$ and $\sigma<-1$, $2^{*}(s_{0})=2^{*}$ the
critical exponent for $H^{1}_{0}(\Omega)\Subset L^{2^{*}}(\Omega)$.
Now we are able to make use of Theorem 1.1 to study the existence of weak
solutions of (1.1). Before stating the existence result we make precise the
definition of weak solution that we use in the article. We say that a function
$u\in X_{0}$ is a weak solution of (1.1) if
$\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}[u(x)-u(y)][\varphi(x)-\varphi(y)]K(x-y)dxdy=\int_{\Omega}f(x,u(x))\varphi(x)dx,$
(1.12)
for any $\varphi\in X_{0}$.
The existence result can be stated as follows.
###### Theorem 1.2
Assume that $f(x,u)=|u|^{p-2}u$, $K$ satisfies (1.2-1.4),
$2^{*}(s_{0})=\frac{2N}{N-2s_{0}}$, where $s_{0}\in(0,1]$ defined in (1.6).
Then problem (1.1) admits a nontrivial weak solution for
$p\in(2,2^{*}(s_{0}))$.
###### Remark 1.1
Take $K$ as example 1.1 with $s_{0}\in(0,1)$ and $\sigma\in\mathbb{R}$ or
$s_{0}=1$ and $\sigma<-1$, then problem (1.1) admits a weak solution for
$f(x,u)=|u|^{p-2}u$ with $p\in(2,2^{*}(s_{0}))$.
Take $K$ as example 2.1, problem (1.1) admits a weak solution for
$f(x,u)=|u|^{p-2}u$ with $p\in(2,2^{*}(s_{0}))$.
The paper is organized as follows. In Section 2, we analyze some basic
properties of the kernel $K$ and give an example showing that $s_{0}$ makes
sense. In Section 3, we study the Sobolev embedding theorem in our setting.
Finally, we prove the existence of weak solution to (1.1) in Section 4.
## 2 Discussion to the kernel $K$
This section is devoted to the properties of the kernel $K$.
###### Proposition 2.1
Assume that $s_{0}$ is defined by (1.6) and $K$ satisfies (1.2). Then $(i)$
for any $s<s_{0}$, we have
$\lim_{r\to 0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=+\infty;$
$(ii)$
$s_{0}\leq\inf\\{s\geq 0\ |\ \lim_{r\to
0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=0\\}\leq 1;$ (2.1)
$(iii)$ if there exists $s_{1}\leq s_{2}$ such that
$\liminf_{|x|\to 0^{+}}K(x)|x|^{N+2s_{1}}>0\quad{\rm and}\quad\limsup_{|x|\to
0^{+}}K(x)|x|^{N+2s_{2}}<\infty,$ (2.2)
then $s_{0}\in[s_{1},s_{2}]$.
Proof. _$(i)$_ By the definition of $s_{0}$, there at least are a sequence of
positive numbers $\\{s_{n}\\}$ such that
$s_{n}<s_{0},\quad\lim_{n\to\infty}s_{n}=s_{0},\quad\lim_{r\to
0^{+}}r^{2s_{n}}\int_{B_{r}(0)}K(y)dy=+\infty.$
Then for any $s<s_{0}$, there exists $s_{n}$ such that $s<s_{n}$ and then
$\lim_{r\to 0^{+}}r^{2s}\int_{B_{r}(0)}K(y)dy\geq\lim_{r\to
0^{+}}r^{2s_{n}}\int_{B_{r}(0)}K(y)dy=+\infty.$
_$(ii)$_ By (1.2) and $K$ being nonnegative, we have that for any $r\in(0,1)$,
$\displaystyle\infty$ $\displaystyle>$
$\displaystyle\int_{\mathbb{R}^{N}}\min\\{|x|^{2},1\\}K(x)dx$ $\displaystyle>$
$\displaystyle\int_{B_{1}(0)\setminus
B_{r}(0)}|x|^{2}K(x)dx+\int_{\mathbb{R}^{N}\setminus B_{1}(0)}K(x)dx$
$\displaystyle\geq$ $\displaystyle r^{2}\int_{\mathbb{R}^{N}\setminus
B_{r}(0)}K(x)dx.$
Then for any $s>1$, we have that
$\displaystyle
r^{2s}\int_{B_{r}^{c}(0)}K(x)dx=r^{2(s-1)}[r^{2}\int_{B_{r}^{c}(0)}K(x)dx]\to
0\quad{\rm as}\ r\to 0.$
Thus, $\inf\\{s\geq 0\ |\ \lim_{r\to
0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=0\\}\leq 1$.
We now prove the first inequality (2.1). We denote
$s_{00}=\inf\\{s\geq 0\ |\ \lim_{r\to
0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=0\\}.$
Since for any $s>s_{00}$, we have that
$\lim_{r\to 0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(x)dx=0.$
By the definition of $s_{0}$, we have $s_{0}\leq s$ and then by arbitrary of
$s>s_{00}$, we obtain that $s_{0}\leq s_{00}$.
_$(iii)$_ For any $s<s_{1}$ by (2.2), we have
$\displaystyle r^{2s}\int_{B_{r}^{c}(0)}K(x)dx$ $\displaystyle=$
$\displaystyle r^{2(s-s_{1})}[r^{2s_{1}}\int_{B_{r}^{c}(0)}K(x)dx]$
$\displaystyle\geq$ $\displaystyle
r^{2(s-s_{1})}\inf_{|x|\in(0,1)}(K(x)|x|^{N+2s_{1}})[r^{2s_{1}}\int_{r}^{1}\tau^{-2s_{1}-1}d\tau]$
$\displaystyle\geq$ $\displaystyle
r^{2(s-s_{1})}\int_{r}^{1}\tau^{-1}d\tau\inf_{|x|\in(0,1)}(K(x)|x|^{N+2s_{1}})$
$\displaystyle\to$ $\displaystyle\infty\quad{\rm as}\ r\to 0.$
By the definition of $s_{0}$, we have $s_{0}\geq s$ and then by arbitrary of
$s<s_{1}$, we obtain that $s_{0}\geq s_{1}$. Similarly to prove $s_{0}\leq
s_{2}$. $\Box$
###### Lemma 2.1
$(i)$ Assume that the kernel $K$ satisfies (1.4) and is continuous in
$\mathbb{R}^{N}\setminus\\{0\\}$, then $K$ is radially symmetric about the
origin.
$(ii)$ Assume that the kernel $K$ satisfies (1.2), (1.4) and (1.6) with
$s_{0}>0$. Then for any $s\in(0,s_{0})$, there exists a sequence $\\{r_{n}\\}$
of positive numbers which converges to 0 and
$\lim_{r_{n}\to 0^{+}}r_{n}^{N+2s}\inf_{|x|=r_{n}}K(x)=+\infty.$ (2.3)
Proof. $(i)$ By contradiction, we may assume that there exist
$x_{1},y_{1}\in\mathbb{R}^{N}\setminus\\{0\\}$ such that $|x_{1}|=|y_{1}|$ and
$K(x_{1})>K(y_{1})$. Since $K$ is continuous in
$\mathbb{R}^{N}\setminus\\{0\\}$, then there exists
$x_{2}\in\mathbb{R}^{N}\setminus\\{0\\}$ such that $|x_{2}|>|x_{1}|$ and
$K(x_{2})\geq K(x_{1})-\frac{K(x_{1})-K(y_{1})}{2}>K(y_{1}),$
which is impossible with the assumption (1.4).
$(ii)$ By Proposition 2.1 $(i)$, we have that for $s\in(0,s_{0})$ and
$\epsilon\in(0,s_{0}-s)$,
$\lim_{r\to 0^{+}}r^{2(s+\epsilon)}\int_{B_{r}^{c}(0)}K(x)dx=+\infty.$ (2.4)
Let $\tilde{K}(r)=\inf_{|x|=r}K(x)$, then by (1.4), we have
$\tilde{K}(r_{1})\leq\tilde{K}(r_{2})$ for $r_{1}\geq r_{2}$ and
$K(x)\leq\tilde{K}(r)$ for any $|x|>r$.
If (2.3) doesn’t hold, then there no exist any sequence $\\{r_{n}\\}$
converging to zero such that (2.3) holds, that is
$\limsup_{r\to 0^{+}}r^{N+2s}\tilde{K}(r)<+\infty.$
Together with $\tilde{K}$ is decreasing, then there exists $C>0$ such that
$\tilde{K}(r)\leq Cr^{-N-2s},\quad r\in(0,1).$
For any $x\in B_{1}(0)\setminus\\{0\\}$, we have
$K(x)\leq\tilde{K}(\frac{|x|}{2})$,
$\displaystyle r^{2(s+\epsilon)}\int_{B_{1}(0)\setminus B_{r}(0)}K(x)dx$
$\displaystyle\leq$ $\displaystyle r^{2(s+\epsilon)}\int_{B_{1}(0)\setminus
B_{r}(0)}\tilde{K}(\frac{|x|}{2})dx$ $\displaystyle\leq$ $\displaystyle
C2^{N+2s}r^{2(s+\epsilon)}\int_{r}^{1}\tau^{-1-2s}d\tau$ $\displaystyle\leq$
$\displaystyle Cr^{2\epsilon}.$
Together with (1.2), we have
$\lim_{r\to 0^{+}}r^{2(s+\epsilon)}\int_{B_{r}^{c}(0)}K(x)dx=0.$
which contradicts with (2.4). The proof is complete. $\Box$
To end this section, we construct an example of $K$ satisfying (1.5) for which
$s_{0}\in(0,1)$.
###### Example 2.1
Let
$K(x)=\left\\{\begin{array}[]{lll}a_{n}^{-N-2s},\ \
|x|\in[a_{n+1},a_{n}),\\\\[5.69054pt] |x|^{-N},\ \
|x|\in[a_{1},1),\\\\[5.69054pt] |x|^{-N-2s},\ \
|x|\in[1,+\infty).\end{array}\right.$ (2.5)
where $s\in(0,1)$, $a_{0}\in(0,1)$, $a_{n}=a_{0}^{b^{n}}$ with
$n\in\mathbb{N}$ and $b=\frac{N+2s}{N}$.
Then
$\liminf_{r\to 0^{+}}K(r)r^{N}=1\quad{\rm and}\quad s_{0}\in(0,s).$
Proof. We observe that $\lim_{n\to+\infty}a_{n}=0$ and
$\displaystyle
K(a_{n})a_{n}^{N}=a_{n-1}^{-N-2s}a_{n}^{N}=a_{0}^{-b^{n-1}(N+2s)}a_{n}^{N}=(a_{0}^{b^{n}})^{-N}a_{n}^{N}=1,$
then we have
$\liminf_{r\to 0^{+}}K(r)r^{N}=1.$
Combining Proposition 2.1 $(iii)$ and the fact of $\limsup_{r\to
0^{+}}K(r)r^{N+2s}\leq 1$, we have that
$s_{0}\in[0,s).$
Now we prove that $s_{0}>0$. For $r\in(0,a_{1})$, there exists
$n\in\mathbb{N}$ such that $a_{n+1}\leq r<a_{n}.$ If $n$ big enough, we have
$a_{n}\leq\frac{1}{2}a_{n-1}$. Combining with $b>1$, then
$\displaystyle\int_{B_{a_{1}(0)}\setminus B_{r}(0)}K(y)dy$ $\displaystyle=$
$\displaystyle|\omega_{N}|[(a_{n}-r)^{N}a_{n}^{-N-2s}+\sum_{k=2}^{n}(a_{k-1}-a_{k})^{N}a_{k-1}^{-N-2s}]$
$\displaystyle\geq$
$\displaystyle|\omega_{N}|\sum_{k=2}^{n}(a_{k-1}-a_{k})^{N}a_{k-1}^{-N-2s}$
$\displaystyle\geq$ $\displaystyle|\omega_{N}|2^{-N}a_{n-1}^{-2s},$
where $w_{N}$ is the unit sphere of $\mathbb{R}^{N}$. Choose
$\beta=b^{-2}s>0$, then we obtain that
$a_{n-1}^{-2s}\geq a_{n+1}^{-2\beta}.$
Therefore,
$\liminf_{r\to 0^{+}}r^{2\beta}\int_{B_{a_{1}(0)}\setminus B_{r}(0)}K(y)dy\geq
2^{-N}|\omega_{N}|.$
By Proposition 2.1 $(iii)$, we obtain that $s_{0}\geq\beta>0.$ $\Box$
## 3 Sobolev spaces
In this section, we will consider some embedding results inspired from [11].
First we introduce some basic spaces and some useful tools to prove embedding
theorems.
###### Lemma 3.1
Assume that $K$ satisfies (1.2), (1.4), (1.6) with $s_{0}>0$ and $l_{\infty}$
is defined by (1.7). Let $x\in\mathbb{R}^{N}$ and $E\subset\mathbb{R}^{N}$ be
a measurable set with $|E|\in(0,+\infty)$, then
$(i)$ if $l_{\infty}=0$, for any $s\in(0,s_{0})$, there exists $C>0$ such that
$\int_{E^{c}}K(x-y)dy\geq C|E|^{-\frac{2s}{N}};$ (3.1)
$(ii)$ if $l_{\infty}\in(0,\infty]$, there exists $C>0$ such that (3.1) holds
with $s\in(0,s_{0}]$.
Proof. We just need to prove that the conclusion of Lemma 3.1 holds for a
sequence $E_{n}$ with $|E_{n}|>0$ and $\lim_{n\to\infty}|E_{n}|=0$. Let
$\rho_{n}=(\frac{|E_{n}|}{\omega_{N}})^{1/N}$, then it follows that
$|E_{n}^{c}\cap B_{\rho_{n}}(x)|=|E_{n}\cap B_{\rho_{n}}^{c}(x)|$. Therefore,
by (1.4), we have that
$K(x-y)\geq\inf_{|z|=\rho_{n}}K(z),\quad y\in E_{n}^{c}\cap B_{\rho_{n}}(x),$
$K(x-y)\leq\inf_{|z|=\rho_{n}}K(z),\quad y\in
E_{n}\cap\bar{B}_{\rho_{n}}^{c}(x).$
Thus
$\displaystyle\int_{E_{n}^{c}}K(x-y)dy$ $\displaystyle=$
$\displaystyle\int_{E_{n}^{c}\cap
B^{c}_{\rho_{n}}(x)}K(x-y)dy+\int_{E_{n}^{c}\cap B_{\rho_{n}}(x)}K(x-y)dy$
(3.2) $\displaystyle\geq$ $\displaystyle\int_{E_{n}^{c}\cap
B^{c}_{\rho_{n}}(x)}K(x-y)dy+\inf_{|z|=\rho_{n}}K(z)|E_{n}^{c}\cap
B_{\rho_{n}}(x)|$ $\displaystyle\geq$ $\displaystyle\int_{E_{n}^{c}\cap
B^{c}_{\rho_{n}}(x)}K(x-y)dy+\inf_{|z|=\rho_{n}}K(z)|E_{n}\cap\bar{B}_{\rho_{n}}^{c}(x)|$
$\displaystyle=$ $\displaystyle\int_{B^{c}_{\rho_{n}}}K(x-y)dy.$
_$(i)$_ By Proposition 2.1 $(i)$ and $s_{0}>0$, we observe that for any
$s\in(0,s_{0})$
$\lim_{r\to 0^{+}}r^{2s}\int_{B_{r}^{c}(0)}K(y)dy=\infty.$ (3.3)
Then by (3.2), there exists $C>0$ such that
$\int_{E_{n}^{c}}K(x-y)dy\geq C|E_{n}|^{-\frac{2s}{N}}.$
_$(ii)$_ Since $l_{\infty}>0$, then there exists $\sigma\in(0,1)$ such that
for $r\in(0,1)$
$r^{2s_{0}}\int_{B_{r}^{c}(0)}K(y)dy\geq\sigma l_{\infty},$
which, together with (3.2), implies that
$\int_{E_{n}^{c}}K(x-y)dy\geq\sigma l_{\infty}|E_{n}|^{-\frac{2s_{0}}{N}}.$
For $s\in(0,s_{0})$, it is the same as the proof of $(i)$. $\Box$
###### Lemma 3.2
[11, Lemma 6.2] Assume that $s\in(0,1)$, $2s<N$ and $T>1$. Let
$n\in\mathbb{Z}$ and $\\{a_{k}\\}$ be a bounded, nonnegative, decreasing
sequence with $a_{k}=0$ for any $k\geq n$. Then,
$\displaystyle\sum_{k\in\mathbb{Z}}a_{k}^{1-\frac{2s}{N}}T^{k}\leq
C\sum_{k\in\mathbb{Z},a_{k}\not=0}a_{k+1}a_{k}^{-\frac{2s}{N}}T^{k},$
for a suitable constant $C=C(s,T,N)>0$, independent of $n$.
###### Lemma 3.3
Assume that $K$ satisfies (1.2), (1.4), (1.6) with $s_{0}\in(0,1)$ and
$l_{\infty}$ is defined by (1.7). Let $f\in L^{\infty}(\mathbb{R}^{N})$ be
compactly supported, then
$\int_{\mathbb{R}^{2N}}|f(x)-f(y)|^{2}K(x-y)dxdy\geq
C\sum_{k\in\mathbb{Z},a_{k}\not=0}a_{k+1}a_{k}^{-\frac{2s}{N}}2^{2k},$
where $a_{k}=|\\{|f|>2^{k}\\}|$, $k\in\mathbb{Z}$, $C=C(N,K)>0$ and the choice
of $s$ is the same as in Lemma 3.1.
Proof. The proof is similar to Lemma 6.3 in [11] just replaced the kernel by
$K$. For reader’s convenience, we give the detail below. Firstly, we assume
that $f$ is nonnegative. If not, we replace $f$ by $|f|$. Let
$A_{k}:=\\{f>2^{k}\\}$, $D_{k}:=A_{k}\setminus A_{k+1}$, $d_{k}:=|D_{k}|$ and
$S:=\sum_{j\in\mathbb{Z},a_{j-1}\not=0}2^{2j}a_{j-1}^{-\frac{2s}{N}}d_{j}.$
Then
$\\{(i,j)\in\mathbb{Z}^{2}\ s.t.\ a_{i-1}\not=0\ {\rm and}\
a_{j-1}^{-\frac{2s}{N}}d_{j}\not=0\\}\subset\\{(i,j)\in\mathbb{Z}^{2}\ s.t.\
a_{j-1}\not=0\\}.$ (3.4)
Then we have that
$\displaystyle\sum_{i\in\mathbb{Z},a_{i-1}\not=0}\sum_{j\in\mathbb{Z},j\geq
i+1}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{j}$ $\displaystyle=$
$\displaystyle\sum_{i\in\mathbb{Z},a_{i-1}\not=0}\sum_{j\in\mathbb{Z},j\geq
i+1,a_{i-1}^{s}d_{j}\not=0}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{i\in\mathbb{Z}}\sum_{j\in\mathbb{Z},j\geq
i+1,a_{i-1}\not=0}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{j}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z},a_{j-1}\not=0}\sum_{i\in\mathbb{Z},i\leq
j-1}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{j}$ $\displaystyle\leq$
$\displaystyle\sum_{j\in\mathbb{Z},a_{j-1}\not=0}\sum_{i\in\mathbb{Z},i\leq
j-1}2^{2i}a_{j-1}^{-\frac{2s}{N}}d_{j}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z},a_{j-1}\not=0}\sum_{k=0}^{+\infty}2^{2j-2}2^{-2k}a_{j-1}^{-\frac{2s_{0}}{N}}d_{j}$
$\displaystyle\leq$ $\displaystyle S.$
Fixed $i\in\mathbb{Z}$ and $x\in D_{i}$, for any $l\in\mathbb{Z}$ with $l\leq
i-2$ and any $y\in D_{l}$, we have that
$|f(x)-f(y)|\geq 2^{i-1}$
and therefore,
$\displaystyle\sum_{l\in\mathbb{Z},l\leq
i-2}\int_{D_{j}}|f(x)-f(y)|^{2}K(x-y)dy$ $\displaystyle\geq$ $\displaystyle
2^{2i-2}\sum_{l\in\mathbb{Z},l\leq i-2}\int_{D_{j}}K(x-y)dy$ $\displaystyle=$
$\displaystyle 2^{2i-2}\int_{A^{c}_{i-1}}K(x-y)dy.$
By Lemma 3.1, we have
$\sum_{l\in\mathbb{Z},l\leq i-2}\int_{D_{l}}|f(x)-f(y)|^{2}K(x-y)dy\geq
c_{0}2^{2i}a_{i-1}^{-\frac{2s}{N}},$
for some suitable $c_{0}>0$. As a consequence, for any $i\in\mathbb{Z}$,
$\sum_{l\in\mathbb{Z},l\leq i-2}\int_{D_{i}\times
D_{l}}|f(x)-f(y)|^{2}K(x-y)dxdy\geq c_{0}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{i}$
and then,
$\displaystyle\sum_{i\in\mathbb{Z},a_{i-1}\not=0}\sum_{l\in\mathbb{Z},l\leq
i-2}\int_{D_{i}\times D_{l}}|f(x)-f(y)|^{2}K(x-y)dxdy\geq c_{0}S.$
Thus, we obtain
$\displaystyle\sum_{i\in\mathbb{Z},a_{i-1}\not=0}\sum_{l\in\mathbb{Z},l\leq
i-2}\int_{D_{i}\times D_{l}}|f(x)-f(y)|^{2}K(x-y)dxdy$ $\displaystyle\geq
c_{0}[\sum_{i\in\mathbb{Z},a_{i-1}\not=0}2^{2i}a_{i-1}^{-\frac{2s}{N}}a_{i}-\sum_{i\mathbb{Z},a_{i-1}\not=0}\sum_{j\in\mathbb{Z},j\geq
i+1}2^{2i}a_{i-1}^{-\frac{2s}{N}}d_{j}]$ $\displaystyle\geq
c_{0}(2^{2i}a_{i-1}^{-\frac{2s}{N}}a_{i}-S).$
So, it follows that
$\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy$
$\displaystyle\geq$ $\displaystyle
2\sum_{i\in\mathbb{Z},a_{i-1}\not=0}\sum_{l\in\mathbb{Z},l\leq
i-2}\int_{D_{i}\times D_{l}}|f(x)-f(y)|^{2}K(x-y)dxdy$ $\displaystyle\geq$
$\displaystyle
c_{0}(\sum_{i\in\mathbb{Z},a_{i-1}\not=0}2^{2i}a_{i-1}^{-\frac{2s}{N}}a_{i}).$
$\Box$
###### Lemma 3.4
Assume that $q\in[1,+\infty)$, $f:\mathbb{R}^{N}\to\mathbb{R}$ is a measurable
function. For any $n\in\mathbb{N}$,
$f_{n}(x):=\max\\{\min\\{f(x),n\\},-n\\},\quad\forall x\in\mathbb{R}^{N}.$
Then
$\lim_{n\to+\infty}\|f_{n}\|_{L^{q}(\mathbb{R}^{N})}=\|f\|_{L^{q}(\mathbb{R}^{N})}.$
Proof. The details of the proof refers to [11, Lemma 6.4] or [2]. $\Box$
Now we can give the statement of embedding theorem as follows:
###### Theorem 3.1
Assume that $K$ satisfies (1.2), (1.4), (1.6) with $s_{0}>0$ and $l_{\infty}$
is defined by (1.7). Then
$(i)$ if $l_{\infty}=0$, then for $s\in(0,s_{0})$ there exists $C>0$ such that
for any $f\in X_{0}$, we have
$\|f\|_{L^{2^{*}(s)}(\Omega)}\leq
C(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy)^{\frac{1}{2}};$
(3.5)
$(ii)$ if $l_{\infty}\in(0,\infty]$, then (3.5) holds with $s=s_{0}$.
Proof. First we note that
$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy<+\infty.$
(3.6)
Without loss of generality, we can assume that $f\in
L^{\infty}(\mathbb{R}^{N})$. Indeed, let $f_{n}$ be defined as in Lemma 3.4,
then combining with Lemma 3.4 and (3.6), we make use of the Dominated
Convergence Theorem to imply
$\lim_{n\to\infty}\int_{\mathbb{R}^{2N}}|f_{n}(x)-f_{n}(y)|^{2}K(x-y)dxdy=\int_{\mathbb{R}^{2N}}|f(x)-f(y)|^{2}K(x-y)dxdy,$
which allows us to obtain estimate for function $f\in X_{0}$.
Take $s$, $a_{k}$ and $A_{k}$ defined as in Lemma 3.3, then we have that
$\displaystyle\|f\|^{2^{*}(s)}_{L^{2^{*}(s)}(\mathbb{R}^{N})}=\sum_{k\in\mathbb{Z}}\int_{A_{k}\setminus
A_{k+1}}|f(x)|^{2^{*}(s)}dx\leq\sum_{k\in\mathbb{Z}}2^{2^{*}(s)(k+1)}a_{k},$
that is,
$\|f\|^{2}_{L^{2^{*}(s)}(\mathbb{R}^{N})}\leq
4(\sum_{k\in\mathbb{Z}}2^{2^{*}(s)k}a_{k})^{2/2^{*}(s)}.$
Since $2<2^{*}(s)$, then
$\|f\|^{2}_{L^{2^{*}(s)}(\mathbb{R}^{N})}\leq
4\sum_{k\in\mathbb{Z}}2^{2k}a_{k}^{2/2^{*}(s)}.$
By Lemma 3.2 with $T=4$, it follows that
$\|f\|^{2}_{L^{2^{*}(s)}(\mathbb{R}^{N})}\leq
C\sum_{k\in\mathbb{Z}}2^{2k}a_{k+1}a_{k}^{-\frac{2s}{N}}.$
for a suitable constant $C$ depending on $N,K$.
Finally, it suffices to apply Lemma 3.4 to obtain that
$\|f\|_{L^{2^{*}(s)}(\mathbb{R}^{N})}\leq
C(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy)^{\frac{1}{2}},$
up to relabeling the constant $C$. Since $f\in X_{0}$, $f=0$ in $\Omega^{c}$,
then (3.5) holds. $\Box$
###### Corollary 3.1
The norm (1.8) in $X_{0}$ is equivalent to
$\|f\|_{X_{0}}:=(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy)^{\frac{1}{2}}.$
(3.7)
Proof. We only need to prove that there exists $C>0$ such that for any $f\in
X_{0}$,
$\|f\|_{X}\leq C\|f\|_{X_{0}}.$
It follows by Theorem 3.1 that
$\displaystyle\|f\|_{X}^{2}=\int_{\Omega}f^{2}(x)dx+\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy$
$\displaystyle\leq|\Omega|^{1-\frac{2}{2^{*}(s)}}(\int_{\Omega}|f|^{2^{*}(s)}(x)dx)^{\frac{2}{2^{*}(s)}}+\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy$
$\displaystyle\leq
C\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|f(x)-f(y)|^{2}K(x-y)dxdy.$
The proof is complete. $\Box$
###### Theorem 3.2
Assume that $K$ satisfies (1.2), (1.4), (1.6) with $s_{0}>0$ and $\mathcal{T}$
is a bounded subset of $X_{0}$.
Then $\mathcal{T}$ is pre-compact in $L^{q}(\Omega)$, $q\in[1,2^{*}(s_{0}))$.
Proof. We first prove that $\mathcal{T}$ is pre-compact in $L^{2}(\Omega)$. To
this end, we only show that $\mathcal{T}$ is totally bounded in
$L^{2}(\Omega)$. By Lemma 2.1$(ii)$, there exists $\\{r_{n}\\}$ positive and
convergent to 0 such that
$\lim_{n\to\infty}r_{n}^{N}K(r_{n})=+\infty.$
Let $\rho:\mathbb{R}_{+}\to\\{\frac{r_{n}}{2},n\in\mathbb{N}\\}$ such that,
denoting $\rho_{\epsilon}=\rho(\epsilon)$, for any $\epsilon>0$,
$\rho_{\epsilon}=r_{n}$ for some $n$ and
$\lim_{\epsilon\to 0^{+}}\rho_{\epsilon}=0.$
It is obvious that
$\lim_{\epsilon\to 0^{+}}(2\rho_{\epsilon})^{N}K(2\rho_{\epsilon})=+\infty.$
(3.8)
Let $\eta_{\epsilon}=\epsilon\rho_{\epsilon}^{\frac{N}{2}}$ and take a
collection of disjoints cubes $Q_{1},....,Q_{M}$ of side $\rho_{\epsilon}$
such that
$\Omega\subset\bigcup_{j=1}^{N}Q_{j}.$
For any $x\in\Omega$, there exists a unique integer $j(x)$ in $\\{1,...,M\\}$
such that $x\in Q_{j(x)}$. Let
$P(f)(x):=\frac{1}{|Q_{j(x)}|}\int_{Q_{j(x)}}f(y)dy,$
then $P$ is linear and $P(f)$ is constant in $Q_{j}$, which we denote by
$q_{j}(f)$. We define the linear operator $R$ by
$R(f)=\rho_{\epsilon}^{\frac{N}{2}}(q_{1}(f),...,q_{M}(f))\in\mathbb{R}^{M}$
and
$\|v\|_{2}:=(\sum^{M}_{j=1}|v_{j}|^{2})^{\frac{1}{2}},\quad
v\in\mathbb{R}^{M}.$
We observe that for any $f\in\mathcal{T}$,
$\displaystyle\|P(f)\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{M}\int_{Q_{j}}|P(f)(x)|^{2}dx=\rho_{\epsilon}^{N}\sum_{j=1}^{M}|q_{j}(f)|^{2}$
$\displaystyle=$ $\displaystyle\|R(f)\|_{2}^{2}=\int_{\Omega}|f(y)|^{2}dy$
$\displaystyle=$ $\displaystyle\|f\|_{L^{2}(\Omega)}^{2}\ \leq C_{0}^{2}.$
Therefore, there exist $b_{1},.....b_{I}\in\mathbb{R}^{M}$ such that
$R(\mathcal{T})\subset\bigcup_{i=1}^{I}B_{\eta_{\epsilon}}(b_{i}),$
where the balls $\\{B_{\eta_{\epsilon}}\\}$ are taken in $\mathbb{R}^{M}$. For
any $x\in\Omega$, we set
$\beta_{j}(x)=\rho_{\epsilon}^{-\frac{N}{2}}b_{i,j(x)},$
where $b_{i,j(x)}$ is the $j(x)$th coordinates of $b_{i}$. Noticing that
$\beta_{j}$ is constant on $Q_{j}$, i.e. for $x\in Q_{j}$, it follows that
$P(\beta_{i})(x)=\rho_{\epsilon}^{-\frac{N}{2}}b_{i,j}=\beta_{i}(x)$
and so $q_{j}(\beta_{i})=\rho_{\epsilon}^{-\frac{N}{2}}b_{i,j}$. Thus
$R(\beta_{i})=b_{i}$. Furthermore, for any $f\in\mathcal{T}$
$\displaystyle\|f-P(f)\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{M}\int_{Q_{j}}|f(x)-P(f)(x)|^{2}dx$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{M}\int_{Q_{j}}\frac{1}{|Q_{j}|^{2}}|\int_{Q_{j}}f(x)-f(y)dy|^{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{\rho_{\epsilon}^{2N}}\sum_{j=1}^{M}\int_{Q_{j}}[\int_{Q_{j}}|f(x)-f(y)|dy]^{2}dx$
and for any fixed $j\in\\{1,...,M\\}$, by Hölder inequality, we get
$\displaystyle\frac{1}{\rho_{\epsilon}^{2N}}[\int_{Q_{j}}|f(x)-f(y)|dy]^{2}\leq\frac{1}{\rho_{\epsilon}^{2N}}|Q_{j}|\int_{Q_{j}}|f(x)-f(y)|^{2}dy$
$\displaystyle\qquad\qquad\leq\frac{1}{\rho_{\epsilon}^{N}}\frac{1}{K(2\rho_{\epsilon})}\int_{Q_{j}}|f(x)-f(y)|^{2}K(x-y)dy$
$\displaystyle\qquad\qquad\qquad\qquad\leq\frac{1}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}\|f\|_{X}^{2},$
where $K(2\rho_{\epsilon})=\inf_{|x|=2\rho_{\epsilon}}K(x)$. Therefore,
$\displaystyle\|f-P(f)\|^{2}_{L^{2}(\Omega)}\leq\frac{1}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}\|f\|_{X}^{2}\sum_{j=1}^{M}|Q_{j}|\leq\frac{C}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}.$
(3.9)
Consequently, for any $f$, there exists $j\in\\{1,....M\\}$ such that $P(f)\in
B_{\eta_{\epsilon}}(b_{j})$ and then we derive that
$\displaystyle\|f-\beta_{j}\|_{L^{2}(\Omega)}$
$\displaystyle\quad\leq\|f-P(f)\|_{L^{2}(\Omega)}+\|P(f)-P(\beta_{j})\|_{L^{2}(\Omega)}+\|P(\beta_{j})-\beta_{j}\|_{L^{2}(\Omega)}$
$\displaystyle\quad\leq\frac{C}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}+\frac{\|R(f)-R(\beta_{j})\|_{L^{2}(\Omega)}}{\rho_{\epsilon}^{\frac{N}{2}}}$
$\displaystyle\quad\leq\frac{C}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}+\frac{\eta_{\epsilon}}{\rho_{\epsilon}^{\frac{N}{2}}},$
where by (3.8), $\frac{1}{(2\rho_{\epsilon})^{N}K(2\rho_{\epsilon})}\to 0$ as
$\epsilon\to 0$ and $\frac{\eta_{\epsilon}}{\rho_{\epsilon}^{N/2}}=\epsilon$.
As a consequence, $\mathcal{T}$ is pre-compact in $L^{2}(\Omega)$.
Now we are in the position to prove that $\mathcal{T}$ is pre-compact in
$L^{q}(\Omega)$ with $q\in[1,2^{*}(s_{0}))$. Since $L^{2}(\Omega)\subset
L^{q}(\Omega)$ with $q\in[1,2)$, then $\mathcal{T}$ is pre-compact in
$L^{q}(\Omega)$. For $q\in(2,2^{*}(s_{0}))$, there exists $s\in(0,s_{0})$ such
that $q<2^{*}(s)$, then using Hölder inequality with
$\theta=\frac{2(2^{*}(s)-q)}{q(2^{*}(s)-2)}$, we get that
$\displaystyle\|f-\beta_{j}\|_{L^{q}(\Omega)}$ $\displaystyle=$
$\displaystyle\left(\int_{\Omega}|f-\beta_{j}|^{\theta
q}|f-\beta_{j}|^{q(1-\theta)}dx\right)^{\frac{1}{q}}$ $\displaystyle\leq$
$\displaystyle\||f-\beta_{j}|\|^{\frac{\theta}{2}}_{L^{2}(\Omega)}\||f-\beta_{j}|\|^{\frac{1}{q}-\frac{\theta}{2}}_{L^{2^{*}(s)}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\left(\frac{C}{\rho_{\epsilon}^{N}K(2\rho_{\epsilon})}+\frac{\eta_{\epsilon}}{\rho_{\epsilon}^{\frac{N}{2}}}\right)^{\frac{\theta}{2}}\|f\|_{X}^{\frac{1}{q}-\frac{\theta}{2}},$
thus, $\mathcal{T}$ is pre-compact in $L^{q}(\Omega)$ with
$q\in(2,2^{*}(s_{0}))$. The proof ends. $\Box$
Proof of Theorem 1.1. For Theorem 1.1 part $(i)$, let $(f_{n})$ be a sequence
functions in $X_{0}$ such that
$\|f_{n}\|_{X}\leq C,\quad\forall n\in\mathbb{N}$
where $C>0$. By Theorem 3.1, Inequality (1.10) follows by (3.5). We obtain
that the sequence $(f_{n})$ is pre-compact in $L^{q}$ with
$q\in[1,2^{*}(s_{0}))$, then the compactness in Theorem 1.1 follows. $\Box$
## 4 Existence of weak solution to (1.1)
For the proof of Theorem 1.1, we observe that problem (1.1) has a variational
structure, indeed it is the Euler-Lagrange equation of the functional
$\mathcal{J}:X_{0}\to\mathbb{R}$ defined as follows
$\mathcal{J}(u)=\frac{1}{2}\|u\|_{X_{0}}^{2}-\frac{1}{p}\int_{\Omega}|u|^{p}dx.$
Note the functional $\mathcal{J}$ is Fréchet differentiable in $u\in X_{0}$
and for any $\varphi\in X_{0}$,
$\langle\mathcal{J}^{\prime}(u),\varphi\rangle=\int_{Q}\big{(}u(x)-u(y)\big{)}\big{(}\varphi(x)-\varphi(y)\big{)}K(x-y)dxdy-\int_{\Omega}|u|^{p-2}u(x)\varphi(x)dx.$
We will make use of Mountain Pass theorem to obtain the weak solution. In what
follows, we check the structure condition of Mountain Pass theorem. It is
obvious that $\mathcal{J}(0)=0$.
###### Proposition 4.1
Under the hypotheses of Theorem 1.2, there exist $\rho>0$ and $\beta>0$ such
that $\mathcal{J}(u)\geq\beta$, for any $u\in X_{0}$ with
$\|u\|_{X_{0}}=\rho$.
Proof. Let $u\in X_{0}$, then
$\displaystyle\mathcal{J}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\|u\|_{X_{0}}^{2}-\frac{1}{p}\int_{\Omega}|u(x)|^{p}\,dx$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\|u\|_{X_{0}}^{2}-C\|u\|_{X_{0}}^{p}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\|u\|_{X_{0}}^{2}(1-C\|u\|_{X_{0}}^{p-2}),$
where we used Theorem 1.1 and Corollary 3.1 for the inequality. We choose
$\sigma>0$ such that $1-C\sigma^{\frac{p-2}{2}}=\frac{1}{2}$, since $p>2$.
Then for $\|u\|_{X_{0}}^{2}=\sigma$, $1-C\|u\|_{X_{0}}^{p-2}=\frac{1}{2}$,
then we have
$\mathcal{J}(u)\geq\frac{1}{4}\sigma.$
The proof is complete. $\Box$
###### Proposition 4.2
Under the hypotheses of Theorem 1.2, there exists $e\in X_{0}$ such that
$\|e\|_{X_{0}}>\rho$ and $\mathcal{J}(e)\leq 0$, where $\rho$ is given in
Proposition 4.1.
Proof. We fix a function $u_{0}\in X_{0}$ with $\|u_{0}\|=1$ in $\Omega$.
Since the space of $\\{tu_{0}:t\in\mathbb{R}\\}$ is a subspace of $X_{0}$ with
dimension 1 and all the norms are equivalent, then
$\int_{\Omega}|u_{0}(x)|^{p}dx>0$. Then there exists $t_{0}>0$ such that for
$t\geq t_{0}$,
$\displaystyle\mathcal{J}(tu_{0})$ $\displaystyle=$
$\displaystyle\frac{t^{2}}{2}\|u_{0}\|_{X_{0}}^{2}-\frac{t^{p}}{p}\int_{\Omega}|u_{0}(x)|^{p}dx$
$\displaystyle\leq$ $\displaystyle C(t^{2}-t^{p})\leq 0.$
We choose $e=t_{0}u_{0}$. The proof is complete. $\Box$
We say that $\mathcal{J}$ has $P.S.$ condition, if for any sequence
$\\{u_{n}\\}$ in $X_{0}$ satisfying $\mathcal{J}(u_{n})\to c$ and
$\mathcal{J}^{\prime}(u_{n})\to 0$ as $n\to\infty$, there is a convergent
subsequence, where $c\in\mathbb{R}$.
###### Proposition 4.3
Under the hypotheses of Theorem 1.2, $\mathcal{J}$ has $P.S.$ condition in
$X_{0}$.
Proof. Let $\\{u_{n}\\}$ be a $P.S.$ sequence, then we need to show that there
are a subsequence $\\{u_{n_{k}}\\}$ and $u$ such that
$u_{n_{k}}\to u\quad{\rm in}\ \ L^{p}(\Omega)\quad{\rm as}\ k\to\infty.$
For some $C>0$, we have that
$\displaystyle
C\|u_{n}\|_{X_{0}}\geq\mathcal{J}^{\prime}(u_{n})u_{n}=\|u_{n}\|^{2}_{X_{0}}-\int_{\Omega}|u_{n}|^{p}dx$
(4.1)
and
$\displaystyle
c-1\leq\mathcal{J}(u_{n})=\frac{1}{2}\|u_{n}\|^{2}_{X_{0}}-\frac{1}{p}\int_{\Omega}|u_{n}|^{p}dx.$
(4.2)
Then $p\times$(4.2)-(4.1) implies that
$(\frac{p}{2}-1)\|u_{n}\|^{2}_{X_{0}}\leq c+C\|u_{n}\|_{X_{0}},$
then $u_{n}$ is uniformly bounded in $X_{0}$.
Thus, by Theorem 1.1 and Corollary 3.1, there exists a subsequence
$(u_{n_{k}})$ and $u$ such that
$u_{n_{k}}\rightharpoonup u,\quad{\rm in}\quad X_{0},$ $u_{n_{k}}\to
u,\quad{\rm a.e.\ in}\ \Omega\quad{\rm and\ \ in}\quad L^{p}(\Omega),$
when $k\to\infty$. Together with $\lim_{k\to\infty}\mathcal{J}(u_{n_{k}})=c$,
we have $\|u_{n_{k}}\|_{X_{0}}\to\|u\|_{X_{0}}$ as $k\to\infty$. Then we have
$u_{n_{k}}\to u$ in $X_{0}$ as $k\to\infty$. $\Box$
Proof of Theorem 1.2. By Proposition 4.1, Proposition 4.2 and Proposition 4.3,
we may use Mountain Pass Theorem (for instance, [22, Theorem 6.1]; see also
[1, 17]) to obtain that there exists a critical point $u\in X_{0}$ of
$\mathcal{J}$ at some value $c\geq\beta>0$. By $\beta>0$, we have $u$ is
nontrivial. Therefore, (1.1) admits a nonnegative weak solution. The proof is
complete. $\Box$
###### Remark 4.1
Suppose that $s_{0}\in(0,1)$ and $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a
Carathéodory function verifying the following hypothesis:
* $(f_{1})\ $
1. there exist $a_{1},a_{2}>0$ and $q\in(2,2^{*}(s_{0}))$ such that
$|f(x,t)|\leq a_{1}+a_{2}|t|^{q-1}\quad a.e.\ x\in\Omega,\ t\in\mathbb{R};$
* $(f_{2})\ $
1. $\lim_{t\to 0}\frac{f(x,t)}{|t|}=0\quad{\rm uniformly\ in}\ x\in\Omega;$
* $(f_{3})\ $
1. there exist $\mu>2$ and $r>0$ such that a.e. $x\in\Omega,t\in\mathbb{R},|t|\geq r$
$0<\mu F(x,t)\leq tf(x,t),$
where the function $F$ is the primitive of $f$ with respect to the variable
$t$, that is
$F(x,t)=\int_{0}^{t}f(x,\tau)d\tau.$
Then fractional elliptic problem (1.1) admits a nontrivial weak solution.
Proof. Using the technique in the proof of Theorem 1 in [20] and Theorem 1.1
part $(ii)$, we derive a nontrivial weak solution of (1.1) by Mountain Pass
Theorem. $\Box$
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|
arxiv-papers
| 2013-11-27T21:46:50 |
2024-09-04T02:49:54.456358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huyuan Chen and Hichem Hajaiej",
"submitter": "Huyuan Chen",
"url": "https://arxiv.org/abs/1311.7165"
}
|
1311.7235
|
Wp Wh
# Downscaling of global solar irradiation in R
F. Antonanzas-Torres111Corresponding author: [email protected]
Edmans Group
ETSII, University of La Rioja, Logroño, Spain F. J. Martínez-de-Pisón
Edmans Group
ETSII, University of La Rioja, Logroño, Spain J. Antonanzas
Edmans Group
ETSII, University of La Rioja, Logroño, Spain O. Perpinan
Electrical Engineering Department
ETSIDI, Universidad Politecnica de Madrid, Spain
###### Abstract
A methodology for downscaling solar irradiation from satellite-derived
databases is described using R software. Different packages such as raster,
parallel, solaR, gstat, sp and rasterVis are considered in this study for
improving solar resource estimation in areas with complex topography, in which
downscaling is a very useful tool for reducing inherent deviations in
satellite-derived irradiation databases, which lack of high global spatial
resolution. A topographical analysis of horizon blocking and sky-view is
developed with a digital elevation model to determine what fraction of hourly
solar irradiation reaches the Earth’s surface. Eventually, kriging with
external drift is applied for a better estimation of solar irradiation
throughout the region analyzed. This methodology has been implemented as an
example within the region of La Rioja in northern Spain, and the mean absolute
error found is a striking 25.5% lower than with the original database.
Keywords: Solar irradiation, R, raster, solaR, digital elevation model, shade
analysis, downscaling.
## 1 Introduction
During the last few years the development of photovoltaic energy has
flourished in developing countries with both multi-megawatt power plants and
micro installations. However, the scarcity of long-term, reliable solar
irradiation data from pyranometers in many of these countries makes it
necessary to estimate solar irradiation from other meteorological variables or
satellite photographs [Schulz et al., 2009]. In such cases, models need to be
validated via nearby pyranometer records, since they lack spatial
generalization. Thus, in some regions in which there are no pyranometers
nearby these models are ruled out as an option and irradiation data must be
obtained from satellite estimates. Although satellite-derived irradiation
databases such as NASA’s Surface meteorology and Solar Energy
(SSE)222http://maps.nrel.gov/SWERA, the National Renewable Energy Laboratory
(NREL)333http://www.nrel.gov/gis/solar.html, INPE444http://www.inpe.br,
SODA555http://www.soda-is.com/eng/index.html and the Climate Monitoring
Satellite Application Facility (CM SAF)666http://www.cmsaf.eu provide wide
spatial coverage, only NASA and some CM SAF climate data sets give global
coverage, albeit at a reduced spatial resolution (Table LABEL:tab:databases).
Table 1: Summary of solar irradiation databases Database | Product | Spatial coverage | Spatial resolution | Temporal coverage | Temporal resolution
---|---|---|---|---|---
CM SAF | SIS Climate Data Set (GHI) | Global | 0.25x0.25∘ | 1982-2009 | Daily means
CM SAF | SIS Climate Data Set (GHI) | 70S-70N, 70W-70E | 0.03x0.03∘ | 1983-2005 | Hourly means
CM SAF | SID Climate Data Set (BHI) | 70S-70N, 70W-70E | 0.03x0.03∘ | 1983-2005 | Hourly means
SODA | Helioclim 3 v2 and v3 (GHI) | 66S-66N,66W-66E | 5km | 2005 | 15 minutes
SODA | Helioclim 3 v2 and v3 (GHI) | 66S-66N,66W-66E | 5km | 2005 | 15 minutes
NREL | GHI Moderate resolution | Central and South America, Africa, India, East Asia | 40x40km | 1985-1991 | Monthly means of daily GHI
NASA | SSE | Global | 1x1∘ | 1983-2005 | Daily means
The spatial resolutions of satellite estimates are generally in the range of
kilometers: they tend to average solar irradiation and omit the impact of
topography within each cell. As a result, intra-cell variations can be very
significant in areas with local micro-climatic characteristics and in areas
with complex topography (which are often one and the same). In this case, the
irradiation data might not be accurate enough to enable a photovoltaic
installation to be designed. [Perez et al., 1994] analyze the spatial behavior
of solar irradiation and conclude that the break-even distance from satellite
estimates to pyranometers is in the order of 7 km and that variations are hard
to estimate for distances greater than 40 km. [Antonanzas-Torres et al., 2013]
reject ordinary kriging as a spatial interpolation method for solar
irradiation in Spain with stations more than 50 km apart in mountainous
regions, as a result of the high spatial variability in such areas. The NASA-
SSE and CM SAF SIS Climate Data Sets (GHI) provide global coverage with
resolutions of 1x1∘ and 0.25x0.25∘ (Table LABEL:tab:databases), which in most
latitudes implies a grosser resolution than the previously mentioned 40-50 km.
One of the alternatives for obtaining higher spatial resolution of solar
irradiation is the downscaling of satellite estimates. Irradiation downscaling
can be based on interpolation kriging techniques when pyranometer records are
available, with the implementation of continuous irradiation-related variables
such as elevation, sky-view-factor and other meteorological variables as
external drifts [Alsamamra et al., 2009; Batlles et al., 2008]. Downscaling is
generally based on digital elevation models (DEM) with satellite-derived
irradiation data to account for the effect of complex topography. It has
previously been applied in mountainous areas such as the Mont Blanc Massif
(France) [Corripio, 2003] and Sierra Nevada (Spain) [Bosch et al., 2010; Ruiz-
Arias et al., 2010] with image resolutions of 3.5x3.5 km. However, the NASA-
SSE and CM SAF SIS Climate Data Sets are based on much lower resolutions and
are the only irradiation datasets in numerous countries where there has been
recent interest in solar energy. In this paper, a downscaling methodology of
global solar irradiation is explained by means of R software and studied in
the region of La Rioja (a very mountainous region in northern Spain). Data
from the CM SAF with 0.03x0.03∘ resolution is considered and then downscaled
to a higher resolution (200x200 m). In a second step, _kriging with external
drift_ , also referred to as _universal kriging_ , is applied to interpolate
data from 6 on-ground pyranometers in the region, and this downscaled CM SAF
data is considered as an explanatory variable. Finally, a downscaled map of
annual global solar radiation throughout this region is obtained.
## 2 Data
The CM SAF was funded in 1992 as a joint venture of several European
meteorological institutes, with the collaboration of the European Organization
for the Exploitation of Meteorological Satellites (EUMETSAT) to retrieve,
archive and distribute climate data to be used for climate monitoring and
climate analysis [Posselt et al., 2012]. Two categories are provided:
operational products and climate data. Operational products are built on data
validated with on-ground stations and provided in near-to-present time and
climate data are long-term series for evaluating inter-annual variability.
This study is built on hourly surface incoming solar radiation and direct
irradiation climate data, denoted as SIS and SID by CM SAF respectively, for
the year 2005\. These climate data are derived from Meteosat first generation
satellites (Meteosat 2 to 7, 1982-2005) and validated using on-ground records
from the Baseline Surface Radiation Network (BSRN) as a reference. The target
accuracy of SIS and SID in hourly means is 15 $W/m^{2}$ [Posselt et al.,
2011], providing a maximum spatial resolution of 0.03x0.03∘. In the study, SIS
and SID data are selected with spatial resolution of 0.03x0.03∘. Data is
freely accessible via FTP through the CM SAF website. Hourly GHI records from
SOS
Rioja777http://www.larioja.org/npRioja/default/defaultpage.jsp?idtab=442821,
taken from 6 meteorological stations (shown in Figure 1 and Table
LABEL:tab:stations) in 2005 serve as complementary measurements for
downscaling within the region studied. These stations have First Class
pyranometers (according to ISO 9060) with uncertainty levels of 5% in daily
totals. These data are filtered from spurious, assuming when relevant the
average between the previous and following hourly measurements. The digital
elevation model (DEM) is also freely obtained from product MDT-200 by the
©Spanish Institute of Geography888http://www.ign.es with a spatial resolution
of 200x200 m.
Figure 1: Region analyzed and meteorological stations considered Table 2: Summary of the meteorological stations selected. # | Name | Net. | Lat.(º) | Long.(º) | Alt. | $GHI_{a}$
---|---|---|---|---|---|---
1 | Ezcaray | SOS | 42.33 | -3.00 | 1000 | 1479
2 | Logroño | SOS | 42.45 | -2.74 | 408 | 1504
3 | Moncalvillo | SOS | 42.32 | -2.61 | 1495 | 1329
4 | San Roman | SOS | 42.23 | -2.45 | 1094 | 1504
5 | Ventrosa | SOS | 42.17 | -2.84 | 1565 | 1277
6 | Yerga | SOS | 42.14 | -1.97 | 1235 | 1448
## 3 Method
This section describes the methodology proposed. Figure 2 displays the method
diagram using red ellipses and lines for data sources, blue ellipses and lines
for derived rasters (results), and black rectangles and lines for operations.
Figure 2: Methodology of downscaling: this figure uses red ellipses and lines
for data sources, blue ellipses and lines for derived rasters (results), and
black rectangles and lines for operations.
### 3.1 Irradiation decomposition
Initially, diffuse horizontal irradiation (_DHI_) is obtained from the
difference between global horizontal irradiation (_GHI_) and beam horizontal
irradiation (_BHI_) rasters, previously obtained from CM SAF. _DHI_ and _BHI_
are firstly disaggregated from the original gross resolution (0.03x0.03∘) into
the DEM resolution (200x200 m), leading to similar values remaining in
disaggregated pixels to the original gross resolution pixel. In a second step,
_DHI_ can be divided in two components: isotropic diffuse irradiation
($DHI_{iso}$), and anisotropic diffuse irradiation ($DHI_{ani}$) as per the
model by Hay & Mckay [Hay and Mckay, 1985] (Equation 1). This model is based
on the anisotropy index ($k_{1}$), defined as the ratio of the beam irradiance
($B(0)$) to the extra-terrestrial irradiance ($B_{0}(0)$), as shown in
Equation 2. High $k_{1}$ values are typical in clear sky atmospheres, while
low $k_{1}$ values are frequent in overcast atmospheres and those with a high
aerosol density.
$DHI=DHI_{iso}+DHI_{ani}$ (1) $k_{1}=\frac{B(0)}{B_{0}(0)}$ (2)
The $DHI_{iso}$ accounts for the incoming diffuse irradiation portion from an
isotropic sky, and is more significant on very cloudy days (Equation 3).
$DHI_{iso}=DHI\cdot{}(1-k_{1})$ (3)
$DHI_{ani}$, also denoted as circumsolar diffuse irradiation, considers the
incoming portion from the circumsolar disk and can be analyzed as beam
irradiation [Perpiñán-Lamigueiro, 2013] (Equation 4).
$DHI_{ani}=DHI\cdot{}k_{1}$ (4)
### 3.2 Sky view factor and horizon blocking
Topographical analysis is performed accounting for the visible sky sphere (sky
view) and horizon blocking. The $DHI_{iso}$ is directly dependent on the sky-
view factor (SVF), which computes the proportion of visible sky related to a
flat horizon. The sky-view factor is considered in earlier irradiation
assessments [Ruiz-Arias et al., 2010; Corripio, 2003]. It is calculated in
each DEM pixel by considering 72 vectors (separated by 5∘ each) and evaluating
the maximum horizon angle ($\theta_{hor}$) over 20 km in each vector (Equation
5). The $\theta_{hor}$ stands for the maximum angle between the altitude of a
location and the elevation of the group of points along each vector, related
to a horizontal plane on the location. Locations without horizon blocking have
SVFs close to 1, which means a whole visible semi-sphere of sky.
$SVF=1-\int_{0}^{2\pi}sin^{2}\theta_{hor}d\theta$ (5)
Eventually, the downscaled $DHI_{iso}$ ($DHI_{iso,down}$) is computed with
Equation 6.
$DHI_{iso,down}=DHI_{iso}\cdot{}SVF$ (6)
Horizon blocking is analyzed by evaluating the solar geometry in 15 minute
samples, particularly the solar elevation ($\gamma_{s}$) and the solar azimuth
($\psi_{s}$). Secondly, the mean hourly $\gamma_{s}$ and $\psi_{s}$ (from
those 15 minute rasters) are calculated and then disaggregated as explained
above for _DHI_ and _BHI_ rasters. The decision to solve the solar geometry
with low resolution rasters enables computation time to be reduced
significantly without penalizing the results. The $\theta_{hor}$ corresponding
to each $\psi_{s}$ is compared with the $\theta_{zs}$. As a consequence, if
the $\theta_{zs}$ is greater than the $\theta_{hor}$, then there is horizon
blocking on the surface analyzed and therefore, _BHI_ and $DHI_{ani}$ are
blocked. Finally, the sum of $DHI_{ani,down}$, $DHI_{iso,down}$ and
$BHI_{iso,down}$ constitutes the downscaled global horizontal irradiation
$GHI_{down}$.
### 3.3 Post-processing: kriging with external drift
The fact that this downscaling accounts for the irradiation loss due to
horizon blocking and the sky-view factor leads us to introduce a trend in
estimates (lowering them) compared to the original data (gross resolution
data). However, satellite-derived irradiation data implicitly considers shade,
as a consequence of the lower albedo recorded in these zones, although it is
later averaged over the pixel. $GHI_{down}$ can be considered as a useful bias
of the behavior of solar irradiation within the region studied. _Universal
kriging_ or _kriging with external drift_ (KED) includes information from
exhaustively-sampled explanatory variables in the interpolation. As a result,
$GHI_{down}$ is considered as the explanatory variable for interpolating
measured irradiation data from on-ground calibrated pyranometers, which is
denoted as _post-processing_. $GHI_{down}$ is correlated with the DEM as a
consequence of the major influence of horizon blocking with topography,
estimations can be derived by separating the deterministic
($\hat{m}(\mathbf{s}_{\theta})$) and stochastic components
($\hat{\epsilon}(\mathbf{s}_{\theta})$ (Equations 7 and 8).
$\hat{z}(\mathbf{\mathbf{s}}_{\theta})=\hat{m}(\mathbf{s}_{\theta})+\hat{\epsilon}(\mathbf{s}_{\theta})$
(7)
$\hat{z}(\mathbf{s}_{\theta})=\sum_{k=0}^{p}\hat{\beta}_{k}q_{k}(\mathbf{s}_{\theta})+\sum_{i=1}^{n}\lambda_{i}\epsilon(\mathbf{s}_{i})$
(8)
where $\hat{\beta}_{k}$ are the estimated coefficients of the deterministic
model, $q_{k}(\mathbf{s}_{\theta})$ are the auxiliary predictors obtained from
the fitted values of the explanatory variable at the new location,
$\lambda_{i}$ are the kriging weights determined by the spatial dependence
structure of the residual, and $\epsilon(\mathbf{s}_{i})$ are the residual at
location $\mathbf{s}_{i}$ [Antonanzas-Torres et al., 2013].
The semivariogram is a function defined as Equation 9 based on a constant
variance of $\epsilon$ and also on the assumption that spatial correlation of
$\epsilon$ depends on the distance amongst instances ($\mathbf{h}$) rather
than on their position [Pebesma, 2004].
$\gamma(\mathbf{h})=\frac{1}{2}\textrm{E}(\epsilon(\mathbf{s})-\epsilon(\mathbf{s}+\mathbf{h}))^{2}$
(9)
Given that the above sample variogram only collates estimates from observed
points, a fitting model of this variogram is generally considered to
extrapolate the spatial behavior of observed points to the area studied. In
the literature different variogram functions are commonly defined such as the
exponential, Gaussian or spherical models. Along these lines, different
parameters such as the sill, range and nugget of the model must be adjusted to
best fit the sample variogram [Hengl, 2009]. The nugget effect, generally
associated with intrinsic micro-variability and measurement error, models the
discontinuity of the variogram at the source. It must be highlighted that when
the nugget effect is recorded, kriging differs from a regular interpolation
and as a result estimates are different from measured values. The variogram
model of solar horizontal irradiation is evaluated in Spain, and the
conclusion reached is that a pure nugget fitting behaves best, which implies
no spatial auto-correlation on residuals [Antonanzas-Torres et al., 2013].
## 4 Implementation
The method proposed is applied in the region of La Rioja (northern Spain).
Figure 3 shows the corresponding annual global horizontal irradiation from CM
SAF with resolution 0.03x0.03∘.
Figure 3: Annual GHI of 2005 from CM SAF estimates (0.03x0.03∘) in La Rioja
### 4.1 Packages
The downscaling described in this paper has been implemented using the free
software environment R [R Development Core Team, 2013] and various contributed
packages:
* •
raster [Hijmans and van Etten, 2013] for spatial data manipulation and
analysis.
* •
solaR [Perpiñán-Lamigueiro, 2012] for solar geometry.
* •
gstat [Pebesma and Graeler, 2013] and sp [Pebesma et al., 2013] for
geostatistical analysis.
* •
parallel for multi-core parallelization.
* •
rasterVis [Perpiñán-Lamigueiro and Hijmans, 2013] for spatial data
visualization methods.
⬇
R> library(sp)
R> library(raster)
R> rasterOptions(todisk=FALSE)
R> rasterOptions(chunksize = 1e+06, maxmemory = 1e+07)
R> library(maptools)
R> library(gstat)
R> library(lattice)
R> library(latticeExtra)
R> library(rasterVis)
R> library(solaR)
R> library(parallel)
### 4.2 Data
Satellite data can be freely downloaded after registration from CM
SAF999www.cmsaf.eu by going to the data access area, selecting _web user
interface_ and _climate data sets_ and then choosing the hourly climate data
sets named _SIS_ (Global Horizontal Irradiation)) and _SID_ (Beam Horizontal
Irradiation) for 2005. Both rasters are projected to the UTM projection for
compatibility with the DEM.
⬇
R> projUTM <- CRS(’+proj=utm␣+zone=30’)
R> projLonLat <- CRS(’␣+proj=longlat␣+ellps=WGS84’)
R> listFich <- dir(pattern=’SIShm2005’)
R> stackSIS <- stack(listFich)
R> stackSIS <- projectRaster(stackSIS,crs=projUTM)
R> listFich <- dir(pattern=’SIDhm2005’)
R> stackSID <- stack(listFich)
R> stackSID <- projectRaster(stackSID, crs=projUTM)
We compute the annual global irradiation, which will be used as a reference
for subsequent steps.
⬇
R> SISa2005 <- calc(stackSIS, sum, na.rm=TRUE)
The Spanish Digital Elevation Model can be obtained after registration from
the ©Spanish Institute of Geography101010http://www.ign.es by going to the
_free download of digital geographic information for non-commercial use_ area,
and then cropping to the region analyzed (La Rioja). As stated above, this DEM
uses the UTM projection.
⬇
R> elevSpain <- raster(’elevSpain.grd’)
R> elev <- crop(elevSpain, extent(479600, 616200, 4639600, 4728400))
R> names(elev)<-’elev’
### 4.3 Sun geometry
The first step is to compute the sun angles (height and azimuth) and the
extraterrestrial solar irradiation for each cell of the CM SAF rasters. The
function calcSol from the solaR package calculates the daily and intradaily
sun geometry. By means of this function and overlay from the raster package,
three multilayer raster objects are generated with the sun geometry needed for
the next steps. For the sake of brevity we show only the procedure for
extraterrestrial solar irradiation. The sun geometry is calculated with the
resolution of CM SAF. First, it is defined a function to extract the hour for
aggregation, choose the annual irradiation raster as reference, and define a
raster with longitude and latitude coordinates.
⬇
R> hour <- function(tt)as.POSIXct(trunc(tt, ’hours’))
R> r <- SISa2005
R> latlon <- stack(init(r, v=’y’), init(r, v=’x’))
R> names(latlon) <- c(’lat’, ’lon’)
The extraterrestrial irradiation is calculated with 5-min samples. Each point
is a column of the data frame locs. Its columns are traversed with lapply, so
for each point of the raster object a time series of extraterrestrial solar
irradiation is computed. The result, B05min, is a RasterBrick object with a
layer for each element of the time index BTi, which is aggregated to an hourly
raster with zApply and transformed to the UTM projection.
⬇
R> BTi <- seq(as.POSIXct(’2005-01-01␣00:00:00’),
+ as.POSIXct(’2005-12-31␣23:55:00’), by=’5␣min’)
R> B05min <- overlay(latlon, fun=function(lat, lon){
+ locs <- as.data.frame(rbind(lat, lon))
+ b <- lapply(locs, function(p){
+
+ hh <- local2Solar(BTi, p[2])
+ sol <- calcSol(p[1], BTi=hh)
+ Bo0 <- as.data.frameI(sol)$Bo0
+ Bo0 })
+ res <- do.call(rbind, b)})
R> B05min <- setZ(B05min, BTi)
R> names(B05min) <- as.character(BTi)
R> B0h <- zApply(B05min, by=hour, fun=mean)
R> projectRaster(B0h,crs=projUTM)
### 4.4 Irradiation components
The CM SAF rasters must be transformed to the higher resolution of the DEM
(UTM 200x200 m). Because of the differences in pixel geometry between DEM
(square) and irradiation rasters (rectangle) the process is performed in two
steps.
The first step increases the spatial resolution of the irradiation rasters to
a similar and also larger pixel size than the DEM with disaggregated data,
where sf is the scale factor. The second step post-processes the previous step
by means of a bilinear interpolation which resamples the raster layer and
achieves the same DEM resolution (resample). This two-step disaggregation
prevents the loss of the original values of the gross resolution raster that
would be directly interpolated with the one-step disaggregation.
⬇
R> sf <- res(stackSID)/res(elev)
R> SIDd <- disaggregate(stackSID, sf)
R> SIDdr <- resample(SIDd, elev)
R> SISd <- disaggregate(stackSIS, sf)
R> SISdr <- resample(SISd, elev)
On the other hand, the diffuse irradiation is obtained from the global and
beam irradiation rasters. The two components of the diffuse irradiation,
isotropic and anisotropic, can be separated with the anisotropy index,
computed as the ratio between beam and extraterrestrial irradiation.
⬇
R> Difdr <- SISdr - SIDdr
R> B0hd <- disaggregate(B0h, sf)
R> B0hdr <- resample(B0hd, elev)
R> k1 <- SIDdr/B0hdr
R> Difiso <- (1-k1) * Difdr
R> Difani <- k1 * Difdr
### 4.5 Sky view factor and horizon blocking
#### 4.5.1 Horizon angle
The maximum horizon angle required for the horizon blocking analysis and to
derive the SVF is obtained with the next code. The alpha vector is visited
with mclapply (using parallel computing). For each direction angle (elements
of this vector) the maximum horizon angle is calculated for a set of points
across that direction from each of the locations defined in xyelev (derived
from the DEM raster and transformed in the matrix locs visited by rows).
⬇
R> xyelev <- stack(init(elev, v=’x’),
+ init(elev, v=’y’),
+ elev)
R> names(xyelev) <- c(’x’, ’y’,’elev’)
R> inc <- pi/36
R> alfa <- seq(-0.5*pi,(1.5*pi-inc), inc)
R> locs <- as.matrix(xyelev)
Separations between the source locations and points along each direction are
defined by resD, the maximum resolution of the DEM, d, maximum distance to
visit, and consequently in the vector seps.
⬇
R> resD <- max(res(elev))
R> d <- 20000
R> seps <- seq(resD, d, by=resD)
The elevation (z1) of each point in xyelev is converted into the horizon
angle: the largest of these angles is the horizon angle for that direction.
The result of each apply step is a matrix, which is used to fill in a
RasterLayer (r). The result of mclapply is a list, hor, of RasterLayer which
can be converted into a RasterStack with stack. Each layer of this RasterStack
corresponds to a different direction.
⬇
R> hor <- mclapply(alfa, function(ang){
+ h <- apply(locs, 1, function(p){
+ x1 <- p[1]+cos(ang)*seps
+ y1 <- p[2]+sin(ang)*seps
+ p1 <- cbind(x1,y1)
+ z1 <- elevSpain[cellFromXY(elevSpain,p1)]
+ hor <- r2d(atan2(z1-p[3], seps))
+ maxHor <- max(hor[which.max(hor)], 0)
+ })
+ r <- raster(elev)
+ r[] <- matrix(h, nrow=nrow(r), byrow=TRUE)
+ r}, mc.cores=8)
R> horizon <- stack(hor)
This operation is very time-consuming as it is necessary to work with high
resolution files. Computation time can be decreased by increasing the sampling
space (200 m) or the sectoral angle (5 ∘) or by reducing the maximum distance
(20 km).
#### 4.5.2 Horizon blocking
Horizon blocking is analyzed by evaluating the solar geometry in 15 minute
samples, particularly the solar elevation and azimuth angles from the original
irradiation raster. Secondly, the hourly averages are calculated,
disaggregated and post-processed as explained above for the irradiation
rasters. The decision to solve the solar geometry with low resolution rasters
enables a significant reduction to be obtained in computation time without
penalizing the results.
First, the azimuth raster is cut into different classes according to the alpha
vector (directions). The values of the horizon raster corresponding to each
angle class are extracted using stackSelect.
⬇
R> idxAngle <- cut(AzShr, breaks=r2d(alfa))
R> AngAlt <- stackSelect(horizon, idxAngle)
The number of layers of AngAlt is the same as idxAngle and can therefore be
used for comparison with the solar height angle, AlShr. If AngAlt is greater,
there is horizon blocking (dilogical=0).
⬇
R> dilogical <- ((AngAlt-AlShr) < 0)
With this binary raster, beam irradiation and diffuse anisotropic irradiation
can be corrected with horizon blocking.
⬇
R> Dirh <- SIDdr * dilogical
R> Difani <- Difani * dilogical
#### 4.5.3 Sky view factor
The sky-view factor can be easily computed from the horizon object with the
equation proposed above. This factor corrects the isotropic component of the
diffuse irradiation.
⬇
R> SVFRuizArias <- calc(horizon, function(x) sin(d2r(x))^2)
R> SVF <- 1 - mean(SVFRuizArias)
R> Difiso <- Difiso * SVF
Finally, the global irradiation is the sum of the three corrected components,
beam and anisotropic diffuse irradiation including horizon blocking, and
isotropic diffuse irradiation with the sky view factor.
⬇
R> GHIh <- Difanis + Difiso + Dirh
R> GHI2005a <- calc(GHIh, fun=sum)
### 4.6 Kriging with external drift
The downscaled irradiation rasters can be improved by using kriging with
external drift. Irradiation data from on-ground meteorological stations is
interpolated with the downscaled irradiation raster as the explanatory
variable. To define the variogram here we use the results previously published
in [Antonanzas-Torres et al., 2013].
⬇
R> load(’Stations.RData’)
R> UTM <- SpatialPointsDataFrame(Stations[,c(2,3)], Stations[,-c(2,3)],
+ proj4string=CRS(’+proj=utm␣+zone=30␣+ellps=WGS84’))
R> vgmCMSAF <- variogram(GHImed ~ GHIcmsaf, UTM)
R> fitvgmCMSAF <- fit.variogram(vgmCMSAF, vgm(model=’Nug’))
R> gModel <- gstat(NULL, id=’G0yKrig’,
+ formula= GHImed ~ GHIcmsaf,
+ locations=UTM, model=fitvgmCMSAF)
R> names(GHI2005a) <- ’GHIcmsaf’
R> G0yKrig <- interpolate(GHI2005a, gModel, xyOnly=FALSE)
### 4.7 Analysis of the results
Figure 3 shows the annual GHI as per CM SAF with the gross resolution analyzed
(0.03x0.03∘) and Figures 4 and 5 show the downscaled maps (200x200 m) without
and with the KED.
Figure 4: Annual GHI of 2005 downscaled without KED (0.03x0.03∘) in La Rioja
Figure 5: Annual GHI of 2005 downscaled with KED (0.03x0.03∘) in La Rioja
#### 4.7.1 Model performance
In order to evaluate the performance of the method proposed, relative
differences evaluated with station measurements are shown in Figure 6. As can
be deduced from this Figure, relative differences are smaller in _downscaling
with KED_ than in CM SAF or _downscaling without KED_ , at $\pm$ 15%. The mean
absolute error (MAE) and root mean square error (RMSE), described in Equations
10 and 11, are used as indicators of model performance.
$MAE=\frac{\sum_{i=1}^{n}{\left|{x_{est}-x_{meas}}\right|}}{n}$ (10)
$RMSE=\sqrt{\frac{\sum_{i=1}^{n}{(x_{est}-x_{meas})^{2}}}{n}}$ (11)
where _n_ is number of stations and $x_{est}$ and $x_{meas}$ the annual
estimated and measured irradiation, respectively.
Figure 6: Annual relative differences evaluated with station measurements.
Table 3 shows the MAE and RMSE obtained with CM SAF and with the methodology
proposed before and after the KED. The KED leads to a significant improvement
in estimates: the MAE is down by 25.5% and the RMSE by 27.4% compared to CM
SAF.
| CM SAF | without KED | with KED
---|---|---|---
MAE | 101.35 | 175.63 | 75.54
RMSE | 118.65 | 196.53 | 86.18
Table 3: Summary of errors obtained in $kWh/m^{2}$.
The higher MAE recorded in station locations in CM SAF and _downscaling
without KED_ is also explained in the irradiation maps shown in Figures 3 and
4. The $GHI_{annual}$ is lowered too far in certain regions of the area
studied with _downscaling without KED_ compared to $GHI_{down,ked}$, which is
also shown in Figure 6.
#### 4.7.2 Zonal variability
The intrapixel variability due to the downscaling procedure is indicative of
the importance of the topography as an attenuator of solar irrradiation. As a
result, this zonal variability is higher in pixels with complex topographies
and downscaling is more useful. Figure 7 shows the relative difference between
downscaling with KED and CM SAF. As might be deduced, CM SAF over-estimates
GHI in this region by between 11 and 22%. Figures 8 and 9 display the standard
deviations of the downscaled maps within each cell of the original CM SAF
raster (0.03x0.03∘). The zonal function from the raster library permits this
calculation, explaining the intrinsic variability of solar radiation within
gross resolution pixels. Consequently, in those pixels with higher standard
deviations there will be greater variability . Figure 9 shows how the KED
method smooths the deviation within pixels and also the range of solar
irradiation in the region (Figures 4 and 5).
Figure 7: Relative difference of $GHI_{KED}$ and $GHI_{CMSAF,down}$ related to
$GHI_{CMSAF,down}$ Figure 8: Difference of zonal standard deviations
($kWh/m^{2}$) between downscaling without KED and with KED. Figure 9: Density
plot of zonal standard deviations between CM SAF and downscaling.
## 5 Concluding comments
A methodology for downscaling solar irradiation is described and presented
using R software. This methodology is useful for increasing the accuracy and
spatial resolution of gross resolution satellite-estimates of solar
irradiation.
It has been proved that areas whose topography is complex show greater
differences with the original gross resolution data as a consequence of
horizon blocking and lower sky-view factors, so downscaling is highly
recommended in these areas.
_Kriging with external drift_ with the gstat package has proved very useful in
downscaling solar irradiation when on-ground registers are available and an
explanatory variable is provided.
This methodology is implemented as an example in the region of La Rioja in
northern Spain, and striking reductions of 25.5% and 27.4% in MAE and RMSE are
obtained compared to the original gross resolution database. The high
repeatability of this methodology and the reduction in errors obtained might
be also very useful in the downscaling of meteorological variables other than
solar irradiation.
## Software information
The source code is available at https://github.com/EDMANSolar/downscaling. The
results discussed in this paper were obtained in a R session with these
characteristics:
* •
R version 2.15.2 (2012-10-26), `x86_64-apple-darwin9.8.0`
* •
Locale: `es_ES.UTF-8/es_ES.UTF-8/es_ES.UTF-8/C/es_ES.UTF-8/es_ES.UTF-8`
* •
Base packages: base, datasets, graphics, grDevices, grid, methods, parallel,
stats,utils
* •
Other packages: foreign 0.8-51, gstat 1.0-16, hexbin 1.26.0, lattice 0.20-15,
latticeExtra 0.6-19, maptools 0.8-14, raster 2.1-16, rasterVis 0.20-01,
RColorBrewer 1.0-5, rgdal 0.8-01, solaR 0.33, sp 1.0-8, zoo 1.7-9
* •
Loaded via a namespace (and not attached): intervals 0.14.0, spacetime 1.0-4,
tools 2.15.2, xts 0.9-3
## Acknowledgements
We are indebted to the University of La Rioja (fellowship FPI2012) and the
Research Institute of La Rioja (IER) for funding parts of this research.
## References
* Alsamamra et al. [2009] Husain Alsamamra, Jose Antonio Ruiz-Arias, David Pozo-Vázquez, and Joaquin Tovar-Pescador. A comparative study of ordinary and residual kriging techniques for mapping global solar radiation over southern spain. _Agricultural and Forest Meteorology_ , 149(8):1343 – 1357, 2009.
* Antonanzas-Torres et al. [2013] Fernando Antonanzas-Torres, Federico Cañizares, and Oscar Perpiñán. Comparative assessment of global irradiation from a satellite estimate model (CM SAF) and on-ground measurements (SIAR): a spanish case study. _Renewable and Sustainable Energy Reviews_ , 21:248–261, 2013.
* Batlles et al. [2008] F.J. Batlles, J.L. Bosch, J. Tovar-Pescador, M. Martínez-Durbán, R. Ortega, and I. Miralles. Determination of atmospheric parameters to estimate global radiation in areas of complex topography: Generation of global irradiation map. _Energy Conversion and Management_ , 49(2):336 – 345, 2008.
* Bosch et al. [2010] J.L. Bosch, F.J. Batlles, L.F. Zarzalejo, and G. López. Solar resources estimation combining digital terrain models and satellite images techniques. _Renewable Energy_ , 35(12):2853 – 2861, 2010\.
* Corripio [2003] J. Corripio. Vectorial algebra algorithms for calculating terrain parameters from dems and solar radiation modelling in mountainous terrain. _International Journal of Geographical Information Science_ , 17:1–23, 2003.
* Hay and Mckay [1985] E. Hay and D.C. Mckay. Estimating solar irradiance on inclined surfaces: a review and assessment of methodologies. _International Journal of Solar Energy_ , 3:203–240, 1985\.
* Hengl [2009] T. Hengl. _A Practical Guide to Geostatistical Mapping_. 2009\. URL http://spatial-analyst.net/book/.
* Hijmans and van Etten [2013] Robert J. Hijmans and Jacob van Etten. _raster : Geographic Data Analysis and Modeling_, 2013. URL http://CRAN.R-project.org/package=raster. R package version 2.1-25.
* Pebesma [2004] E. J. Pebesma. Multivariable geostatistics in S: the gstat package. _Computers & Geosciences_, 30:683–691, 2004.
* Pebesma and Graeler [2013] Edzer Pebesma and Benedikt Graeler. _gstat : Spatial and Spatio-Temporal Geostatistical Modelling, Prediction and Simulation_, 2013. URL http://CRAN.R-project.org/package=gstat. R package version 1.0-16.
* Pebesma et al. [2013] Edzer Pebesma, Roger Bivand, Barry Rowlingson, and Virgilio Gomez-Rubio. _sp : Classes and Methods for Spatial Data_, 2013. URL http://CRAN.R-project.org/package=sp. R package version 1.0-9.
* Perez et al. [1994] Richard Perez, Robert Seals, Ronald Stewart, Antoine Zelenka, and Vicente Estrada-Cajigal. Using satellite-derived insolation data for the site/time specific simulation of solar energy systems. _Solar Energy_ , 53(6):491 – 495, 1994.
* Perpiñán-Lamigueiro [2012] Oscar Perpiñán-Lamigueiro. solaR: Solar radiation and photovoltaic systems with R. _Journal of Statistical Software_ , 50(9):1–32, 8 2012. ISSN 1548-7660. URL http://www.jstatsoft.org/v50/i09.
* Perpiñán-Lamigueiro [2013] Oscar Perpiñán-Lamigueiro. _Energía Solar Fotovoltaica_. 2013\. URL http://procomun.wordpress.com/documentos/libroesf/.
* Perpiñán-Lamigueiro and Hijmans [2013] Oscar Perpiñán-Lamigueiro and Robert J. Hijmans. _rasterVis : Visualization Methods for the Raster Package_, 2013. URL http://CRAN.R-project.org/package=rasterVis. R package version 0.20-07.
* Posselt et al. [2011] R. Posselt, R. Muller, J. Trentmann, and R. Stockli. Meteosat (mviri) solar surface irradiance and effective cloud albedo climate data sets. the cm saf validation report. Technical report, The EUMETSAT Network of Satellite Application Facilities, 2011.
* Posselt et al. [2012] R. Posselt, R.W. Mueller, R. Stöckli, and J. Trentmann. Remote sensing of solar surface radiation for climate monitoring — the cm-saf retrieval in international comparison. _Remote Sensing of Environment_ , 118(0):186 – 198, 2012.
* R Development Core Team [2013] R Development Core Team. _R: A Language and Environment for Statistical Computing_. R Foundation for Statistical Computing, Vienna, Austria, 2013. URL http://www.R-project.org. ISBN 3-900051-07-0.
* Ruiz-Arias et al. [2010] José A. Ruiz-Arias, Tomáš Cebecauer, Joaquín Tovar-Pescador, and Marcel Šúri. Spatial disaggregation of satellite-derived irradiance using a high-resolution digital elevation model. _Solar Energy_ , 84(9):1644 – 1657, 2010.
* Schulz et al. [2009] J. Schulz, P. Albert, H.-D. Behr, D. Caprion, H. Deneke, S. Dewitte, B. Dürr, P. Fuchs, A. Gratzki, P. Hechler, R. Hollmann, S. Johnston, K.-G. Karlsson, T. Manninen, R. Müller, M. Reuter, A. Riihelä, R. Roebeling, N. Selbach, A. Tetzlaff, W. Thomas, M. Werscheck, E. Wolters, and A. Zelenka. Operational climate monitoring from space: the eumetsat satellite application facility on climate monitoring (cm-saf). _Atmospheric Chemistry and Physics_ , 9(5):1687–1709, 2009.
|
arxiv-papers
| 2013-11-28T08:13:20 |
2024-09-04T02:49:54.466790
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Antonanzas-Torres and F.J. Mart\\'inez de Pis\\'on and J. Antonanzas\n and O. Perpi\\~n\\'an",
"submitter": "Oscar Perpinan",
"url": "https://arxiv.org/abs/1311.7235"
}
|
1311.7255
|
# Liouvillian integrability of polynomial differential systems
Xiang Zhang Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong
University, Shanghai 200240, People’s Republic of China. [email protected]
###### Abstract.
M.F. Singer [Liouvillian first integrals of differential equations, Trans.
Amer. Math. Soc. 333 (1992), 673–688] proved the equivalence between
Liouvillian integrability and Darboux integrability for two dimensional
polynomial differential systems. In this paper we will extend Singer’s result
to any finite dimensional polynomial differential systems. We prove that if an
$n$–dimensional polynomial differential system has $n-1$ functionally
independent Darboux Jacobian multiplier then it has $n-1$ functionally
independent Liouvillian first integrals. Conversely if the system is
Liouvillian integrable then it has a Darboux Jacobian multiplier.
###### Key words and phrases:
Liouville integrability; Darboux integrability; Jacobian multiplier; Galois
group.
To appear in Transactions of the American Mathematical Society
###### 2010 Mathematics Subject Classification:
34A34, 37C10, 34C14, 37G05.
## 1\. Background and statement of the main results
The theory of integrability for differential systems is classic and it is
useful in the study of dynamics of differential system. Integrability has
different definitions in different fields. Here we mainly concern the
algebraic aspects of integrability for polynomial differential systems, which
involves analysis, algebraic geometry, the field extension and so on. For
further information on this subject, we refer readers to Daboux [7, 8],
Jouanolou [12], Prelle and Singer [21], Singer [23], Schlomiuk [22], Llibre
[14], Dumortier and Llibre et al [9], Christopher et al [3, 6] and Llibre and
Zhang [16, 18, 19].
Darboux theory of integrability was established by Darboux [7, 8] in 1878 for
polynomial differential systems of degree $n$ by using the invariant algebraic
curves (resp. surfaces or hypersurfaces) in dimension 2 (resp. 3 or $n>3$).
Jouanolou [12] in 1979 extended the Darboux’s theory to construct rational
first integrals with the help of algebraic geometry. An elementary proof of
Jouanolou’s result was provided respectively by Christopher and Llibre [5] in
2000 for two dimensional differential systems and by Llibre and Zhang [17] in
2010 for any finite dimensional differential systems. On further extensions to
Darboux theory of integrability, Christopher, Llibre and Pereira [6] in 2007
took into account not only the number of invariant algebraic curves but also
their multiplicities for two dimensional differential systems. Llibre and
Zhang [16] further extended Christopher et al’s result in [6] to any finite
dimensional differential systems, where there are some deep characterizations
on the number of exponential factors and the multiplier of invariant algebraic
hypersurfaces. Darboux theory of integrability has important applications in
the center–focus problem, dynamical analysis and so on, see for instance [4,
14, 22, 24] and the reference therein.
Darboux theory of integrability has a nice extension to Weierstrass
integrability, see e.g. [10], where they used Weierstrass polynomials to
replace the usual polynomials. By definition the former include the latter as
a special one. In [1] Blázquez–Sanz and Pantazi provided a new approach to
study the Darboux integrability of polynomial differential systems of degree
$m$, where they replaced the dimension of $\mathbb{C}_{m-1}[x]$ which the
cofactors of Darboux polynomials and exponential factors are located in by the
rank of a matrix associated to these cofactors. Here $\mathbb{C}_{m-1}[x]$
denotes the linear space formed by polynomials in $x\in\mathbb{C}^{n}$ of
degree no more than $m-1$. Recently Darboux theory of integrability was also
successfully extended to nonautonomous differential systems which are
polynomial ones in space variables with coefficients the smooth functions of
the time, see e.g. [15, 11], where they extended the notion of invariant
algebraic hypersurfaces in the phase space to polynomial invariant
hypersurfaces in the extended space including the time.
Prelle and Singer [21] in 1983 proved that if a polynomial differential system
has an elementary first integral then it has a first integral of a very simple
form. As a corollary of their results, one gets that if a planar polynomial
differential system has an elementary first integral, then it has an
integrating factor of the form $f_{1}^{m_{1}}\ldots f_{p}^{m_{p}}$ with
$f_{i}\in\mathbb{C}[x,y]$ and $m_{i}\in\mathbb{Z}$. This shows the equivalence
between the existence of elementary first integrals and the Darboux
integrating factors for planar polynomial differential systems.
Singer [23] in 1992 proved that a planar polynomial differential system has a
Liouvillian first integral if and only if it has an integrating factor of the
form
$R(x,y)=\exp\left(\int U(x,y)dx+V(x,y)dy\right),$
where $U(x,y),V(x,y)$ are rational functions in $x,y$. For a simple proof to
Singer’s result, see [3] and [4, Theorem 3.2].
In this paper we will extend Singer’s result to any finite dimensional
polynomial differential systems.
Consider polynomial differential systems
(1.1) $\dot{x}=P(x),\qquad x\in\mathbb{C}^{n},$
where $P(x)=(P_{1}(x),\ldots,P_{n}(x))$ are vector–valued polynomial
functions. We call $m:=\max\\{\mbox{deg}P_{1},\ldots,\mbox{deg}P_{n}\\}$ the
degree of polynomial differential systems (1.1). In what follows we also use
$\mathcal{X}_{P}=P_{1}(x)\frac{\partial}{\partial
x_{1}}+\ldots+P_{n}(x)\frac{\partial}{\partial x_{n}},$
to represent the vector field associated to system (1.1). For simplifying
notations, in what follows we denote ${\partial}/{\partial x_{i}}$ by
$\partial_{i}$ and $(\partial_{1},\ldots,\partial_{n})$ by $\partial$.
Denote by $\mathbb{C}[x]$ the ring of polynomials in $x$ with coefficients in
$\mathbb{C}$. A polynomial $f(x)\in\mathbb{C}[x]$ is called a Darboux
polynomial of $\mathcal{X}_{P}$ if there exists a $k(x)\in\mathbb{C}[x]$ such
that
$\mathcal{X}_{P}(f)(x)=k(x)\,f(x),\qquad x\in\mathbb{C}^{n}.$
The polynomial $k$ is called cofactor of $f$. A function of the form
$\exp\left(\frac{g}{h}\right)f_{1}^{l_{1}}\ldots f_{r}^{l_{r}},\quad\mbox{
with }g,\,h,\,f_{i}\in\mathbb{C}[x],\,\,\,l_{i}\in\mathbb{C},\,\,i=1,\ldots,r$
is called a Darboux function. For a Darboux function, we always require that
its factors $f_{i}$ are irreducible and relatively different, and $g,h$ are
relative coprime.
A Darboux first integral of (1.1) is a Darboux function and it is a first
integral of (1.1). Note that a first integral is not necessary to be defined
in the full space but in a full Lebesgue measure subset of $\mathbb{C}^{n}$.
System (1.1) is Darboux integrable if it has $n-1$ functionally independent
Darboux first integrals.
A smooth function $J(x)$ is a Jacobian multiplier of system (1.1) if
$\partial_{1}(JP_{1})+\ldots+\partial_{n}(JP_{n})=0.$
A Darboux Jacobian multiplier of system (1.1) is a Jacobian multiplier of the
system and it is a Darboux function. For planar polynomial differential
systems, a Jacobian multiplier is usually called an integrating factor. If a
planar polynomial differential system has a Darboux integrating factor, it is
also called Darboux integrable.
For stating our results we recall the definition of Liouvillian functions.
A differential field $(K,\,\Delta)$ consists of the field $K$ and the set
$\Delta$ of commutative derivatives defined on $K$. In this paper all
mentioned fields have characteristic $0$.
A differential field extension of a differential field $(K,\,\Delta)$ is a
differential field $(L,\,\Delta^{\prime})$ with the properties that $K\subset
L$ and for $\forall\,\delta^{\prime}\in\Delta^{\prime}$ we have
$\left.\delta^{\prime}\right|_{K}\in\Delta$. Because of the relation between
the derivatives of differential field $(K,\,\Delta)$ and its field extension
$(L,\,\Delta^{\prime})$, we also use $\Delta$ to represent $\Delta^{\prime}$.
For simplifying notations we also use $L/K$ to denote the differential field
extension $(L,\Delta)$ of $(K,\Delta)$.
For a field extension $L/K$,
* •
$\alpha\in L$ is called
* –
an algebraic element of $K$, if there exists a polynomial with coefficients in
$K$ such that $F(\alpha)=0$.
* –
a transcendental element of $K$, if $\alpha$ is not an algebraic element over
$K$.
* •
If each element of $L$ is algebraic over $K$, we call $L/K$ an algebraic
extension of field $K$.
* •
$L$ can be considered as a vector space over $K$: the elements of $L$ are
treated as vectors, and elements of $K$ are treated as scalars, and the
summation of vectors is that of elements of field and the product of elements
of $L$ and $K$ is that of elements of field $L$.
* –
The dimension of this vector space is called degree of this differential field
extension, denoted by $[L:K]$.
* •
If $[L:K]\in\mathbb{N}$, we call $L/K$ finite field extension.
* •
Let $S\subset L$,
* –
$K(S)$ denotes the minimal subfield of $L$ including $K$ and $S$.
* –
If $S$ contains only one element, we call $K(S)$ the minimal field extension
of $K$.
A differential field extension $L/K$ is Liouvillian, if this differential
field extension can be written in the tower form
$K=K_{0}\subset K_{1}\subset\ldots\subset K_{r}=L,$
such that
* $(a)$
$K_{i+1}$ is a finite algebraic extension of $K_{i}$, or
* $(b)$
$K_{i+1}=K_{i}(t)$, where $t$ is a transcendental element of $K_{i}$
satisfying: for each $\delta\in\Delta$, $\dfrac{\delta t}{t}\in K_{i}$, or
* $(c)$
$K_{i+1}=K_{i}(t)$, where $t$ is a transcendental element of $K_{i}$
satisfying: for $\delta\in\Delta$, $\delta t\in K_{i}$.
A Liouvillian first integral of (1.1) is a Liouvillian function and is a first
integral of (1.1). System (1.1) is Liouvillian integrable if it has $n-1$
functionally independent Liouvillian first integrals.
Now we can state our main results. The first one characterizes the existence
of Liouvillian first integrals via Darboux Jacobian multipliers.
###### Theorem 1.1.
If polynomial differential system (1.1) has $n-1$ functionally independent
Darboux Jacobian multipliers, then they have $n-1$ functionally independent
Liouvillian first integrals.
The next one shows that Liouvillian integrability implies the existence of
Darboux Jacobian multipliers.
###### Theorem 1.2.
If system (1.1) is Liouvillian integrable, i.e. it has $n-1$ functionally
independent Liouvillian first integrals, then the system has a Darboux
Jacobian multiplier.
In the rest of this paper we will prove our main results.
## 2\. Proof of the main results
### 2.1. Proof of Theorem 1.1
Let $J_{1}(x),\,\ldots,\,J_{n-1}(x)$ be $n-1$ functionally independent Darboux
Jacobian multipliers of system (1.1). Then we have
(2.1) $\mathcal{X}_{P}(J_{l})=-J_{l}\,\mbox{div}P,\qquad l=1,\ldots,n-1,$
where $\mbox{div}P=\partial_{1}P_{1}(x)+\ldots+\partial_{n}P_{n}(x)$ is the
divergence of the vector fields $P(x)$. Recall that $\partial_{i}P_{i}$
denotes the partial derivative of the function $P_{i}$ with respect to
$x_{i}$.
From the definition of Darboux functions and some direct calculations we get
that
$\frac{\partial_{i}J_{l}}{J_{l}}\in\mathbb{C}(x),\qquad
l\in\\{1,\ldots,n-1\\},\,\,\,i\in\\{1,\ldots,n\\}.$
Recall that $\mathbb{C}(x)$ is the field of rational functions in $x$. So it
follows from the condition $(b)$ of Liouvillian extension of field that
$J_{l}$ for $l=1,\ldots,n-1$ are Liouvillian functions. Furthermore, some easy
calculations show that
$\frac{J_{l}}{J_{k}},\qquad\mbox{for }\quad 1\leq l\neq k\leq n-1,$
are non–trivial Liouvillian first integrals of the vector field
$\mathcal{X}_{P}$, i.e. $\frac{J_{l}}{J_{k}}$ is not a constant and
$\mathcal{X}_{P}\left(\frac{J_{l}}{J_{k}}\right)\equiv 0$.
We claim that
$\frac{J_{1}}{J_{n-1}},\,\ldots,\,\frac{J_{n-2}}{J_{n-1}},$
are functionally independent. Indeed, assume that
$c_{1}\partial\left(\frac{J_{1}}{J_{n-1}}\right)+\ldots+c_{n-2}\partial\left(\frac{J_{n-2}}{J_{n-1}}\right)=0.$
Since
$\partial\left(\frac{J_{l}}{J_{n-1}}\right)=\frac{J_{n-1}\,\partial
J_{l}-J_{l}\partial J_{n-1}}{J_{n-1}^{2}},$
we have
$\displaystyle c_{1}J_{n-1}\partial J_{1}+\ldots+c_{n-2}J_{n-1}\partial
J_{n-2}$
$\displaystyle\qquad\qquad\,\,-(c_{1}J_{1}+\ldots+c_{n-2}J_{n-2})\partial
J_{n-1}=0.$
So by the functional independence of $J_{1},\ldots,J_{n-1}$ we must have
$c_{1}J_{n-1}=\ldots=c_{n-2}J_{n-1}=c_{1}J_{1}+\ldots+c_{n-2}J_{n-2}=0,$
in a full Lebesgure measure subset of $\mathbb{C}^{n}$. Consequently
$c_{1}=\ldots=c_{n-2}=0,$
in a full Lebesgure measure subset of $\mathbb{C}^{n}$. This proves the claim.
Using the last claim, we assume without loss of generality that
$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\frac{J_{i}}{J_{n-1}},\qquad i=1,\ldots,n-2,$ $\displaystyle
y_{n-1}$ $\displaystyle=$ $\displaystyle x_{n-1},$ $\displaystyle y_{n}$
$\displaystyle=$ $\displaystyle x_{n},$
are invertible, at least in some full Lebesgue measure subset $\Omega$ of
$\mathbb{C}^{n}$. Denote by $y=G(x)$ this last transformation. Then under it
the differential system (1.1) is equivalent to
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle 0,\quad\qquad
i=1,\ldots,n-2,$ (2.2) $\displaystyle\dot{y}_{n-1}$ $\displaystyle=$
$\displaystyle P_{n-1}\circ G^{-1}(y),$ $\displaystyle\dot{y}_{n}$
$\displaystyle=$ $\displaystyle P_{n}\circ G^{-1}(y).$
Clearly system (2.1) has the first integrals $I_{i}(y)=y_{i}$,
$i=1,\ldots,n-2$. In addition, we can prove that system (2.1) has the Jacobian
multiplier
$M(y)=J_{n-1}\circ G^{-1}(y)D_{y}G^{-1}(y),$
where $D_{y}G^{-1}(y)$ denotes the Jacobian matrix of $G^{-1}$ with respect to
$y$. This shows that the two dimensional differential system
$\displaystyle\dot{y}_{n-1}$ $\displaystyle=$ $\displaystyle P_{n-1}\circ
G^{-1}(I_{1},\ldots,I_{n-2},y_{n-1},y_{n})=:g_{n-1}(y_{n-1},y_{n}),$
$\displaystyle\dot{y}_{n}$ $\displaystyle=$ $\displaystyle P_{n}\circ
G^{-1}(I_{1},\ldots,I_{n-2},y_{n-1},y_{n})=:g_{n}(y_{n-1},y_{n}),$
has the integrating factor
$V(y_{n-1},y_{n})=J_{n-1}\circ
G^{-1}(y)D_{y}G^{-1}(I_{1},\ldots,I_{n-2},y_{n-1},y_{n}),$
where we take $I_{1},\ldots,I_{n-2}$ as constants. Hence this last two
dimensional differential system has the first integral.
$I_{n-1}(y_{n-1},y_{n})=\int Vg_{n}dy_{n-1}-Vg_{n-1}dy_{n}.$
Obviously, $I_{n-1}$ is functionally independent of $I_{1},\dots,I_{n-2}$,
because the latter are independent of $y_{n-1}$ and $y_{n}$.
Next we prove that $I_{n-1}$ is a Liouvillian function. First we prove that
$G^{-1}(x)$ is a Liouvillian function. Indeed, by
$G^{-1}\circ G(x)=x,$
we have
$D_{y}G^{-1}(G(x))D_{x}G(x)=E.$
where $E$ is the $n$–dimensional unit matrix. Since $G$ is Liouvillian, and so
is $(D_{x}G(x))^{-1}$. Hence we have
$D_{y}G^{-1}(G(x))=(D_{x}G(x))^{-1},$
is a Liouvillian function. This shows that $G^{-1}(y)$ is a Liouvillian
function. Furthermore it follows from the above construction that
$g_{n-1},g_{n}$ and $V$ are Liouvillian functions. This proves that $I_{n-1}$
is a Liouvillian function.
Applying the transformation $y=G(x)$ to $I_{1}(y),\ldots,I_{n-1}(y)$, we get
$n-1$ functionally independent Liouvillian first integrals
$H_{1}(x):=I_{1}\circ G(x),\quad\ldots,\quad H_{n-1}(x)=I_{n-1}\circ G(x),$
of differential system (1.1). We complete the proof of the theorem.
### 2.2. Proof of Theorem 1.2
For proving Theorem 1.2 and readers’s convenience, we recall some notions.
Given a field $K$,
* •
A separating field of a polynomial $p(x)$ over $K$ is a minimal field
extension of $K$ such that $p(x)$ can be decomposed into product of linear
factors over this field extension, i.e. $p(x)=\prod(x-a_{i})$, $a_{i}\in L$,
$L/K$ is the minimal field extension such that this decomposition can happen.
* •
We say that an algebraic field extension $L/K$ of $K$ is normal, if $L$ is a
separating field of polynomials in $K[x]$.
* •
The normal closure of an algebraic field extension $L/K$ is a field extension
$\overline{L}$ of $L$ such that $\overline{L}/K$ is normal, and $\overline{L}$
is the minimal field extension satisfying this property.
* •
Field automorphism over field $K$ is a bijective map $\varphi:\,\,K\rightarrow
K$ which keeps the algebraic properties of $K$, i.e. $\varphi$ satisfies that
$\varphi(0_{K})=0_{K}$, $\varphi(1_{K})=1_{K}$,
$\varphi(a+b)=\varphi(a)+\varphi(b)$ and $\varphi(ab)=\varphi(a)\varphi(b)$.
* •
The set of all field automorphisms over field $K$ fixing elements of a
subfield $K^{\prime}\subset K$ forms a group under the composition of maps.
This group is called Galois group.
* •
The order of a group is the number of elements of a group $G$, denoted by
$|G|$.
Now we can prove Theorem 1.2. It will follows from the following lemmas.
###### Lemma 2.1.
If system (1.1) is Liouvillian integrable, then it has a Jacobian multiplier
of the form
$J=\exp\left(\int U_{1}dx_{1}+\ldots+U_{n}dx_{n}\right),$
with $U_{i}\in\mathbb{C}(x)$, $i=1,\ldots,n$, and
$\partial_{i}U_{j}=\partial_{j}U_{i},\qquad 1\leq j<i\leq n.$
###### Proof.
Assume that system (1.1) has the functionally independent Liouvillian first
integrals $H_{1},\ldots,H_{n-1}$, which are defined in a full Lebesgue measure
subset $\Omega$ of $\mathbb{C}^{n}$. By definition of first integrals we have
(2.3) $\mathcal{X}_{P}(H_{i})(x)\equiv 0,\quad x\in\Omega,\qquad
i=1,\ldots,n-1.$
From independence of $H_{1},\ldots,H_{n-1}$ we can assume without loss of
generality that
$\Gamma:=\det\left(\partial_{1}\mathcal{H},\cdots,\partial_{n-1}\mathcal{H}\right)\neq
0,\qquad x\in\Omega,$
with
$\mathcal{H}:=(H_{1},\ldots,H_{n-1})^{T},$
where $T$ denotes the transpose of a matrix, and
$\partial_{i}\mathcal{H}:=(\partial_{i}H_{1},\ldots,\partial_{i}H_{n-1})^{T},\qquad
i=1,\ldots,n.$
Set for $i=1,\ldots,n-1$
$\Gamma_{i}:=\det\left(\partial_{1}\mathcal{H},\cdots,\partial_{i-1}\mathcal{H},\partial_{n}\mathcal{H},\partial_{i+1}\mathcal{H},\cdots,\partial_{n-1}\mathcal{H}\right).$
Clearly $\Gamma$ and $\Gamma_{i}$, $i=1,\ldots,n$, are Liouvillian functions.
By the Cramer’s rule we get from (2.3) that
$P_{i}(x)=-\frac{\Gamma_{i}}{\Gamma}P_{n}(x),\quad i=1,\ldots,n-1.$
Hence we have
$\Gamma(P_{1}(x),\ldots,P_{n-1})=-(\Gamma_{1},\ldots,\Gamma_{n-1})P_{n}(x).$
Since $P_{1},\ldots,P_{n-1},P_{n}$ are relative coprime, and $\Gamma$ and
$\Gamma_{i}$ are Liouvillian functions, so there exists a Liouvillian function
$h(x)$ such that
$h(x)\Gamma=P_{n}(x).$
Consequently we have
$P_{i}(x)=-h(x)\Gamma_{i},\quad i=1,\ldots,n-1.$
Set
(2.4) $A_{i}:=\frac{\partial_{i}h}{h},\quad i=1,\ldots,n;\qquad
A:=(A_{1},\ldots,A_{n}).$
Then $A_{i}$ is Liouvillian for each $i\in\\{1,\ldots,n\\}$.
We claim that
(2.5) $\displaystyle\partial_{i}A_{j}$ $\displaystyle=$
$\displaystyle\partial_{j}A_{i},\qquad 1\leq j<i\leq n,$ (2.6)
$\displaystyle\langle A,P\rangle$ $\displaystyle:=$ $\displaystyle
A_{1}P_{1}+\ldots+A_{n}P_{n}=\mbox{div}P.$
The equality (2.5) can be proved by direct calculations via the fact
$\partial_{j}\partial_{i}h=\partial_{i}\partial_{j}h\qquad\mbox{ for all
}\,\,\,i,j\in\\{i,\ldots,n\\}.$
For proving (2.6), some computations show that
$\displaystyle\mbox{div}P$ $\displaystyle=$
$\displaystyle\partial_{1}(-h\Gamma_{1})+\ldots+\partial_{n-1}(-h\Gamma_{n-1})+\partial_{n}(h\Gamma)$
$\displaystyle=$ $\displaystyle
A_{1}P_{1}+\ldots+A_{n}P_{n}+h(\partial_{n}\Gamma-\partial_{1}\Gamma_{1}-\ldots-\partial_{n-1}\Gamma_{n-1}).$
Next we only need to prove that
$\partial_{n}\Gamma-\partial_{1}\Gamma_{1}-\ldots-\partial_{n-1}\Gamma_{n-1}=0.$
Since
$\displaystyle\partial_{n}\Gamma$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{n-1}\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\partial_{n}\partial_{i}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right),$
$\displaystyle\partial_{i}\Gamma_{i}$ $\displaystyle=$
$\displaystyle\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\partial_{i}\partial_{n}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right)$
$\displaystyle+\sum\limits_{j=1,j\neq
i}\limits^{n-1}\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{j-1}\mathcal{H},\partial_{i}\partial_{j}\mathcal{H},\partial_{j+1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\partial_{n}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right),$
we have
$\displaystyle\partial_{1}\Gamma_{1}+\ldots+\partial_{n-1}\Gamma_{n-1}-\partial_{n}\Gamma$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{n-1}\sum\limits_{j=1,j\neq
i}\limits^{n-1}\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{j-1}\mathcal{H},\partial_{i}\partial_{j}\mathcal{H},\partial_{j+1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.\partial_{n}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right)$
$\displaystyle=$ $\displaystyle 0,$
where in the last equality we have used the fact that
$\displaystyle\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{j-1}\mathcal{H},\partial_{i}\partial_{j}\mathcal{H},\partial_{j+1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\quad\left.\partial_{n}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right)$
$\displaystyle\qquad+\det\left(\partial_{1}\mathcal{H},\ldots,\partial_{j-1}\mathcal{H},\partial_{n}\mathcal{H},\partial_{j+1}\mathcal{H},\ldots,\partial_{i-1}\mathcal{H},\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\quad\left.\partial_{j}\partial_{i}\mathcal{H},\partial_{i+1}\mathcal{H},\ldots,\partial_{n-1}\mathcal{H}\right)=0.$
This proves the claim.
Set
$U_{i}=-A_{i},\qquad i=1,\ldots,n,$
with $A_{i}$ defined in (2.4). By (2.5) and the Stokes’s theorem (see e.g. [2,
p.3]), we get that
(2.7) $J=\exp\left(\int U_{1}dx_{1}+\ldots+U_{n}dx_{n}\right),$
is well defined in any connected subsets where $U_{1},\ldots,U_{n}$ are
defined. Furthermore we can check that $J$ in (2.7) is a Jacobian multiplier
of system (1.1) if and only if (2.6) holds. So, in what follows we only need
to prove that there exist rational functions $B_{i}$ for $i=1,\ldots,n$
instead of $A_{i}$ such that the equalities (2.5) and (2.6) hold.
According to the Liouvillian extension of field in the tower form, all the
$A_{i}$ belong to some tower for $i=1,\ldots,n$. We distinguish three
different cases according to the definition of the tower.
$(a)$ $A_{i}\in K_{l+1}$, $i=1,\ldots,n$, and $K_{l+1}$ is an algebraic
extension of the field $K_{l}$. We will prove that there exist $B_{i}\in
K_{l}$ instead of $A_{i}$ for $i=1,\ldots,n$ such that (2.5) and (2.6) hold.
Let $\overline{K}_{l+1}$ be the normal closure of $K_{l+1}$, and $\mathcal{G}$
be the Galois group formed by the automorphisms of $\overline{K}_{l+1}$ fixing
$K_{l}$. Then it follows from a result of Artin (see Lang [13, Theorem 1.1])
that $\mathcal{G}$ is of finite order, and denote by $N=|\mathcal{G}|$ the
order of the group. Note that $N\leq[K_{l+1}:K_{l}]$, the degree of the
algebraic extension of the field.
Since $P\in\mathbb{C}(x)^{n}$ and $\mathbb{C}(x)\subset K_{l}$, we get from
(2.5) and (2.6) that
(2.8)
$\displaystyle\begin{array}[]{rcl}\partial_{i}g(A_{j})&=&\partial_{j}g(A_{i}),\qquad\forall
g\,\in\mathcal{G},\\\
\left\langle\sum\limits_{g\in\mathcal{G}}g(A),P\right\rangle&=&\sum\limits_{g\in\mathcal{G}}g(A_{1})P_{1}+\ldots+\sum\limits_{g\in\mathcal{G}}g(A_{n})P_{n}=N\mbox{div}P,\end{array}$
where in the second equality we have used the fact that $g\in\mathcal{G}$
fixes $K_{l}$. Set
$B_{i}=\frac{1}{N}\sum\limits_{g\in\mathcal{G}}g(A_{i}),\qquad i=1,\ldots,n.$
Then $B_{i}\in K_{l}$ for $i=1,\ldots,n$, because all $B_{i}$ are fixed under
the action of all elements of the Galois group $\mathcal{G}$. Furthermore, we
get from (2.8) that (2.5) and (2.6) hold for $B_{i}$ instead of $A_{i}$ for
$i=1,\ldots,n$.
$(b)$ Assume $K_{l+1}=K_{l}(t)$ with $t$ a transcendental element over $K_{l}$
and $\partial_{i}t/t\in K_{l}$ for $i=1,\ldots,n$.
Since $A_{i}\in K_{l}(t)$ for $i=1,\ldots,n$, we can assume without loss of
generality that
$A_{i}=a_{i}(t)\in K_{l}(t),\qquad i=1,\ldots,n.$
Expanding $a_{j}(t)$ into Laurent series in $t$ gives
(2.9)
$a_{j}(t)=a_{0}^{(j)}+\sum\limits_{s\in\mathbb{Z}\setminus\\{0\\}}a_{s}^{(j)}t^{s},\qquad
j=1,\ldots,n,$
with $a_{s}^{(j)}\in K_{l}$ for $j=1,\dots,n$ and all $s$. Then we have
(2.10)
$\partial_{i}A_{j}=\partial_{i}a_{0}^{(j)}+\sum\limits_{s\in\mathbb{Z}\setminus\\{0\\}}\left(\partial_{i}a_{s}^{(j)}+sa_{s}^{(j)}p_{i}\right)t^{s},$
where $p_{i}\in K_{l}$ satisfying $\partial_{i}t/t=p_{i}\in K_{l}$ for
$i=1,\ldots,n$. Set
$B_{i}=a_{0}^{(i)},\qquad i=1,\ldots,n,$
we have $B_{i}\in K_{l}$. Substituting (2.9) and (2.10) into (2.5) and (2.6),
and equating the coefficients of $t^{0}$, we get that $B_{i}$ for
$i=1,\ldots,n$ satisfy (2.5) and (2.6) instead of $A_{i}$.
$(c)$ Assume that $K_{l+1}=K_{l}(t)$ with $t$ a transcendental element over
$K_{l}$ and $\partial_{i}t\in K_{l}$ for all $i\in\\{1,\ldots,n\\}$.
Similar to $(b)$ we set
$A_{j}=a_{j}(t)\in K_{l}(t),\qquad j=1,\ldots,n.$
Now the Laurent expansion in $t$ does not work, we choose the Laurent
expansion in $1/t$ of $a_{j}(t)$. Since $A_{i}\in K_{l}(t)$, we get from its
construction (2.4), i.e. $A_{i}=\partial_{i}h/h$, that the degree of numerator
in $t$ of $a_{j}(t)$ is less than or equal to the degree of its denominator.
Write $a_{j}(t)$ as
$\displaystyle a_{j}(t)$ $\displaystyle=$
$\displaystyle\frac{a_{j0}+a_{j1}t+\ldots+a_{jk}t^{k}}{b_{j0}+b_{j1}t+\ldots+b_{jl}t^{l}}$
$\displaystyle=$
$\displaystyle\frac{a_{j0}t^{-l}+a_{j1}t^{-(l-1)}+\ldots+a_{jk}t^{-(l-k)}}{b_{j0}t^{-l}+b_{j1}t^{-(l-1)}+\ldots+b_{jl}}.$
Since $l\geq k$, so the Laurent expansion in $t^{-1}$ of $a_{j}(t)$ has the
form
(2.11) $a_{j}(t)=\sum\limits_{s=-\infty}\limits^{0}a_{s}^{(j)}t^{s},\qquad
j=1,\ldots,n,$
with $a_{s}^{(j)}\in K_{l}$ for $j=1,\dots,n$ and all $s$. Direct calculation
shows that
(2.12)
$\partial_{i}A_{j}=\sum\limits_{s=-\infty}\limits^{0}\left(\partial_{i}a_{s-1}^{(j)}+sa_{s}^{(j)}q_{i}\right)t^{s-1}+\partial_{i}a_{0}^{(j)},$
where $q_{i}\in K_{l}$ satisfying $q_{i}=\partial_{i}t\in K_{l}$ for
$i=1,\ldots,n$.
Set
$B_{i}=a_{0}^{(i)},\qquad i=1,\ldots,n.$
Then $B_{i}\in K_{l}$. Using the expansions (2.11) and (2.12) we get from
(2.5) and (2.6) that
$\displaystyle\partial_{i}B_{j}$ $\displaystyle=$
$\displaystyle\partial_{j}B_{i},\qquad 1\leq j<i\leq n,$
$\displaystyle\mbox{div}P$ $\displaystyle=$ $\displaystyle
B_{1}P_{1}+\ldots+B_{n}P_{n}.$
Of course if all $B_{i}=0$, then $\mbox{div}P=0$, and so $J=1$ is a Jacobian
multiplier.
Summarizing the cases $(a)$, $(b)$ and $(c)$, and combining the definition of
Liouvillian functions in the tower form, by induction we get that there exist
$U_{1},\ldots,U_{n}\in K_{0}=\mathbb{C}(x)$ for which (2.5) and (2.6) hold
instead of $A_{1},\ldots,A_{n}$. We complete the proof of the lemma. ∎
The next result shows that the existence of Jacobian multipliers of the form
given in Lemma 2.1 implies the existence of Darboux Jacobian multipliers.
###### Lemma 2.2.
If polynomial differential system (1.1) has a Jacobian multiplier
$J=\exp\left(\int U_{1}dx_{1}+\ldots+U_{n}dx_{n}\right),$
with $U_{i}\in\mathbb{C}(x)$, and $\partial_{j}U_{i}=\partial_{i}U_{j}$ for
$1\leq i,j\leq n$, then it has a Darboux Jacobian multiplier
$\exp\left(\frac{g}{h}\right)\prod\limits_{i}f_{i}^{l_{i}},$
where $g,h,f_{i}\in\mathbb{C}[x,y]$, $l_{i}\in\mathbb{C}$.
###### Proof.
Since $U_{1},\ldots,U_{n}\in\mathbb{C}(x)$, we treat their numerators and
denominators as polynomials in $x_{1}$ with coefficients in
$\mathbb{C}[x_{2},\ldots,x_{n}]$. Let $K$ be the minimal normal algebraic
field extension of $\mathbb{C}(x_{2},\ldots,x_{n})$ such that it is the
separating field of the numerators and denominators of $U_{1},\ldots,U_{n}$.
By the properties on normal algebraic field extension, the rational functions
$U_{1},\ldots,U_{n}$ over $K$ can be expanded in
(2.13)
$U_{k}(x)=\sum\limits_{i=1}\limits^{r}\sum\limits_{j=1}\limits^{m}\frac{\alpha_{ij}^{(k)}}{(x_{1}-\beta_{i})^{j}}+\sum\limits_{i=0}\limits^{p}\xi_{i}^{(k)}x_{1}^{i},\qquad
k=1,\ldots,n,$
where $\alpha_{ij}^{(k)},\beta_{i},\xi_{i}^{(k)}\in K$, and parts of them can
be zero. Direct calculations show that for $l\in\\{2,\ldots,n\\}$
$\displaystyle\partial_{l}U_{1}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{r}\sum\limits_{j=1}\limits^{m}\left(\frac{\partial_{l}\alpha_{ij}^{(1)}}{(x_{1}-\beta_{i})^{j}}+\frac{j\alpha_{ij}^{(1)}\partial_{l}\beta_{i}}{(x_{1}-\beta_{i})^{j+1}}\right)+\sum\limits_{i=0}\limits^{p}\partial_{l}\xi_{i}^{(1)}x^{i},$
$\displaystyle\partial_{1}U_{l}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{r}\sum\limits_{j=1}\limits^{m}\frac{-j\alpha_{ij}^{(l)}}{(x_{1}-\beta_{i})^{j+1}}+\sum\limits_{i=0}\limits^{p}i\xi_{i}^{(l)}x_{1}^{i-1}.$
Using the assumption $\partial_{1}U_{l}=\partial_{l}U_{1}$, and comparing the
coefficients of $(x_{1}-\beta_{i})^{-j}$ and $x^{i}$, we get that for
$l\in\\{2,\ldots,n\\}$
(2.14)
$\partial_{l}\alpha_{i,j+1}^{(1)}+j\alpha_{ij}^{(1)}\partial_{l}\beta_{i}+j\alpha_{ij}^{(l)}=0,\qquad\partial_{l}\xi_{i}^{(1)}=(i+1)\xi_{i+1}^{(l)}.$
The first equality with $j=0$ of (2.14) shows that
$\alpha_{i1}^{(1)}\in\mathbb{C}$, because $\alpha_{ij}^{(k)}$ are functions in
$x_{2},\ldots,x_{n}$ as prescribed.
Set
$\displaystyle\Phi(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{r}\alpha_{i1}^{(1)}\log(x_{1}-\beta_{i})+\sum\limits_{i=1}\limits^{r}\sum\limits_{j=2}\limits^{m}\frac{-1}{j-1}\frac{\alpha_{ij}^{(1)}}{(x_{1}-\beta_{i})^{j-1}}$
$\displaystyle+\sum\limits_{i=1}\limits^{p}\frac{\xi_{i}^{(1)}}{i+1}x_{1}^{i+1}+\int\xi_{0}^{(2)}dx_{2}+\ldots+\xi_{0}^{(n)}dx_{n},$
where the integration represents any primitive function of
$\xi_{0}^{(2)}dx_{2}+\ldots+\xi_{0}^{(n)}dx_{n}$. Direct calculations show
that $\partial_{1}\Phi=U_{1}$. For $l>1$ since
$\displaystyle\partial_{l}\Phi(x)$ $\displaystyle=$
$\displaystyle-\sum\limits_{i}\frac{\alpha_{i1}^{(1)}\partial_{l}\beta_{i}}{x_{1}-\beta_{i}}+\sum\limits_{i,j}\frac{-1}{j-1}\left(\frac{\partial_{l}\alpha_{ij}^{(1)}}{(x_{1}-\beta_{i})^{j-1}}+\frac{(j-1)\alpha_{ij}^{(1)}\partial_{l}\beta_{i}}{(x_{1}-\beta_{i})^{j}}\right)$
$\displaystyle+\sum\limits_{i}\frac{\partial_{l}\xi_{i}^{(1)}}{i+1}x^{i+1}+\xi_{0}^{(l)}.$
Using the equalities (2.14) and by some calculations, we get that
$\partial_{l}\Phi=U_{l},\qquad l=2,\ldots,n.$
These show that
$\Phi(x)=\int U_{1}dx_{1}+\ldots+U_{n}dx_{n},$
with possible a constant difference.
Denote by $\mathcal{G}$ the group of automorphisms over $K$ which keep
$\mathbb{C}(y)$, where $y=(x_{2},\ldots,x_{n})$. Since $K$ is the minimal
normal algebraic field extension of $\mathbb{C}(y)$, we get from the
properties of field extensions that $\mathcal{G}$ is a finite group. Denote by
$N=|\mathcal{G}|$, the order of $\mathcal{G}$. Set
$\Psi=\frac{1}{N}\sum\limits_{\sigma\in\mathcal{G}}\sigma(\Phi).$
Since $\sigma\in\mathcal{G}$ is an automorphism over the algebraic field
extension $K$ of $\mathbb{C}(y)$, it follows that
$\displaystyle\sigma\left(\alpha_{i1}\log(x_{1}-\beta_{i})\right)$
$\displaystyle=$ $\displaystyle\alpha_{i1}\log(x_{1}-\sigma(\beta_{i})),$
$\displaystyle\sigma\left(\int\gamma_{0}^{(2)}dx_{2}+\ldots+\gamma_{0}^{(n)}dx_{n}\right)$
$\displaystyle=$
$\displaystyle\int\sigma(\gamma_{0}^{(2)})dx_{2}+\ldots+\sigma(\gamma_{0}^{(n)})dx_{n},$
where the second equality may have a constant difference.
Since $\sigma(\partial_{l}\Phi)=\sigma(U_{l})$, we have
(2.15) $\partial_{l}\sigma(\Phi)=\sigma(U_{l})=U_{l},\qquad l=1,\ldots,n,$
where in the second equality we have used the facts that the numerators and
denominators of $U_{l}$’$s$ can be written in the polynomials of $x_{1}$ with
coefficients in $\mathbb{C}(y)$ and $\sigma$ keeps $\mathbb{C}(y)$.
The equalities (2.15) show that
$\partial_{l}\Psi=U_{l},\qquad l=1,\ldots,n.$
Moreover we have
$\Psi(x)=\sum\limits_{i=1}\limits^{r_{0}}c_{i}\log R_{i}(x)+R(x)+\int
S_{2}(y)dx_{2}+\ldots+S_{n}(y)dx_{n},$
where $c_{i}\in\mathbb{C}$, $R_{i},\,R\in\mathbb{C}(x)$ and
$S_{i}\in\mathbb{C}(y)$. Recall that $y=(x_{2},\ldots,x_{n})$. By the
expansions of $U_{k}$’$s$ in (2.13) and $\partial_{l}U_{k}=\partial_{k}U_{l}$,
it follows that
$\partial_{l}\xi_{0}^{(k)}(y)=\partial_{k}\xi_{0}^{(l)}(y),\qquad 2\leq
k,l\leq n.$
So we have
$\sum\limits_{\sigma\in\mathcal{G}}\sigma(\xi_{0}^{(k)})\in\mathbb{C}(y),\qquad
k=2,\ldots,n,$
and for $2\leq k,l\leq n$
$\partial_{l}\left(\sum\limits_{\sigma\in\mathcal{G}}\sigma(\xi_{0}^{(k)})\right)=\sum\limits_{\sigma\in\mathcal{G}}\sigma(\partial_{l}\xi_{0}^{(k)})=\sum\limits_{\sigma\in\mathcal{G}}\sigma(\partial_{k}\xi_{0}^{(l)})=\partial_{k}\left(\sum\limits_{\sigma\in\mathcal{G}}\sigma(\xi_{0}^{(l)})\right).$
These show that
$\partial_{x_{i}}S_{j}(y)=\partial_{x_{j}}S_{i}(y),\quad 2\leq i,j\leq n.$
Now for the integration $\int S_{2}(y)dx_{2}+\ldots+S_{n}(y)dx_{n}$ with
$y=(x_{2},\ldots,x_{n})$ we are in the same conditions as those of integration
$\int U_{1}(y)dx_{1}+\ldots+U_{n}(y)dx_{n}$, so working in a similar way as
that in the above proof we get that there exists a function $\Psi_{1}(y)$ such
that
$\partial_{l}\Psi_{1}(y)=S_{l}(y),\qquad l=2,\ldots,n,$
and
$\Psi_{1}(y)=\sum\limits_{i=1}\limits^{r_{1}}d_{i}\log T_{i}(y)+T(y)+\int
W_{3}(z)dx_{3}+\ldots+W_{n}(z)dx_{n},$
where $d_{i}\in\mathbb{C}$, $T_{i},\,T\in\mathbb{C}(y)$, and
$W_{i}\in\mathbb{C}(z)$ with $z=(x_{3},\ldots,x_{n})$ satisfy
$\partial_{x_{i}}W_{j}(z)=\partial_{x_{j}}W_{i}(z),\qquad 3\leq i,\,j\leq n.$
By induction we can prove that
$\displaystyle\Psi(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}\limits^{r_{0}}c_{i}\log R_{i}(x)+R(x)$
$\displaystyle+\sum\limits_{j=2}^{n}\left(\sum\limits_{i=1}\limits^{r_{j}}c_{i}^{(j)}\log
R_{i}^{(j)}(x_{j},\ldots,x_{n})+R^{(j)}(x_{j},\ldots,x_{n})\right),$
where $c_{i},c_{i}^{(j)}\in\mathbb{C}$, and
$R_{i}^{(j)},R^{(j)}\in\mathbb{C}(x_{j},\ldots,x_{n})$. Recall that
$\partial_{l}\Psi(x)=U_{l}(x)$ for $l=1,\ldots,n$. Furthermore we have
$\displaystyle\exp\left(\Psi\right)$ $\displaystyle=$
$\displaystyle\exp\left(R(x)+\sum\limits_{j=2}^{n}R^{(j)}(x_{j},\ldots,x_{n})\right)$
$\displaystyle\times\prod\limits_{i=1}\limits^{r_{0}}\left(R_{i}(x)\right)^{c_{i}}\prod\limits_{j=2}^{n}\prod\limits_{i=1}^{r_{j}}\left(R_{i}^{(j)}(x_{j},\ldots,x_{n})\right)^{c_{i}^{(j)}}.$
Since $R,\,R^{(j)},\,R_{i},\,R_{i}^{(j)}\in\mathbb{C}(x)$, it follows that
$\exp(\Psi(x))$ is a Darboux function, and consequently is a Darboux Jacobian
multiplier. This proves Lemma 2.2. ∎
Summarizing Lemmas 2.1 and 2.2, we complete the proof of Theorem 1.2.
Acknowledgements. The author is partially supported by NNSF of China grant
11271252, RFDP of Higher Education of China grant 20110073110054, and
FP7-PEOPLE-2012-IRSES-316338 of Europe.
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* [23] M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673–688.
* [24] X. Zhang, Global structure of quaternion polynomial differential equations, Commun. Math. Phys. 303 (2011), 301–316.
|
arxiv-papers
| 2013-11-28T09:55:28 |
2024-09-04T02:49:54.476359
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiang Zhang",
"submitter": "Xiang Zhang",
"url": "https://arxiv.org/abs/1311.7255"
}
|
1311.7295
|
# Glasgow’s Stereo Image Database of Garments
Gerardo Aragon-Camarasa, Susanne B. Oehler, Yuan Liu,
Sun Li, Paul Cockshott and J. Paul Siebert Sir Alwyn Williams Building,
Lilybank Gardens, Glasgow, G12 8QQ, Scotland
###### Abstract
To provide insight into cloth perception and manipulation with an active
binocular robotic vision system, we compiled a database of 80 stereo-pair
colour images with corresponding horizontal and vertical disparity maps and
mask annotations, for 3D garment point cloud rendering has been created and
released. The stereo-image garment database is part of research conducted
under the EU-FP7 Clothes Perception and Manipulation (CloPeMa) project and
belongs to a wider database collection released through CloPeMa††thanks:
www.clopema.eu. This database is based on 16 different off-the-shelve
garments. Each garment has been imaged in five different pose configurations
on the project’s binocular robot head. A full copy of the database is made
available for scientific research only at
https://sites.google.com/site/ugstereodatabase/.
## 1 Introduction
The CloPeMa project is advancing the state of the art in clothes perception
and manipulation by delivering a novel robotic system that accomplishes
automatic sorting and folding of a laundry heap. To this end, CloPeMa is using
a prototype robot composed mainly of off-the-shelf components comprising an
active binocular vision robot head. This active binocular robot head, which is
inspired by the system developed in Aragon-Camarasa et al. [1], has been
designed by the Computer Vision and Graphics Group (CV&G) at the University of
Glasgow. This robotic head, as created for CloPeMa, is not only able to
provide high-resolution intensity images of the robot’s workspace, as required
for intensity based computer vision algorithms, but is capable of automatic
vergence and gaze control, hand eye calibration and 2.5D reconstruction of
areas-of-interest. Data captured by this robotic head can be used in a wide
variety of applications such garment spreading and flatteningSun et al. [6],
automatic visual inspection and exploration of cluttered scenesAragon-Camarasa
et al. [1], selection of better grasping points or more detailed feature
extraction and classification. In order to provide a first insight into the
type and quality of data produced by the binocular robot head in the CloPeMa
robot system, we have compiled and released a freely available database of
stereo-pair images of garments. The aim of this dataset is to serve as a
benchmark tool for algorithms for recognition, segmentation and various range
image properties of non-rigid objects. For instance, it will be used to
improve the Vector-Pascal Cockshott et al. [2] Glasgow parallel stereo matcher
and its GPU implementations. This dataset is the first high resolution stereo-
pair garment image dataset that is released for research purposes and
potentially allows for a variety of research applications. Therefore, the
Glasgow’s Stereo Image Database of Garments can be downloaded from:
https://sites.google.com/site/ugstereodatabase/.
This database comprises images of 16 different off-the-shelve garments
selected from the official CloPeMa cloth heap, defined in Molfino et al. [5].
The CloPeMa heap features a wide variety of textile materials with different
texture, colour and reflectance characteristics in order to give a realistic
sample of the real world clothing variety. For the released database, the
chosen garments where imaged in five possible pose configurations: _flat on
the table, folded in half, completely folded, randomly wrinkled and hanging
over the robot’s arm_. These configurations are an approximation of the most
representative pose configurations a robot may encounter while sorting and
folding clothes. Each of the selected five configurations were imaged under
software-control capture synchronisation. The database therefore yields a
total of 80 stereo-pairs of garment images. For completeness, the horizontal
and vertical disparities without mask of the Glasgow’s Stereo Image Database
of Garments can also be downloaded.
The 80 stereo-pairs in the database have all been processed using the Glasgow
stereo matcher Cyganek and Siebert [3], in order to compute the horizontal and
vertical disparities. A new version of the Glasgow stereo matcher has been
integrated in CloPeMa’s robot system as a ROS node within the CloPeMa robot
head package
collection111http://clopema.felk.cvut.cz/redmine/projects/clopema/wiki/Technical_Stuff.
Additionally, the data-set’s image pairs are accompanied by mask annotations
for the left as well as the right image. The camera calibration, which has
been computed using CloPeMa’s integrated OpenCV compatible robot head
calibration system, is also released as part of the database. This enables the
research community to use the database for 3D garment point cloud projection.
Specifically for this purpose, Matlab-based reconstruction software is also
distributed within this database.
It must be noted that the above algorithms and methods have been integrated as
part of a collection of ROS nodes distributed in the official CloPeMa package
collection. Specifically, the CloPeMa active robot head system software
includes ROS nodes for directing the robot’s gaze under program control,
automatic vergence, acquiring synchronised stereo-pair images, camera and
hand-eye calibration routines, stereo image processing algorithms (including a
GPU stereo matcher based on the Glasgow Stereo Matcher) , real-time SIFT
feature extraction and user interactive interfaces for gaze control and
calibration routines. The robot head ROS packages can be downloaded from:
http://clopema.felk.cvut.cz/redmine/projects/clopema/wiki/Packages_instalation_%28hydro%29_
## 2 Database Acquisition
The CloPeMa robotic test-bed is equipped with two Yaskawa robotic arms mounted
on a computer controllable tailor-made Yaskawa turn table, two RGB-D sensors
for wide vision mounted on the wrists of the robotic arms, two prototype
grippers designed by the University of Genoa Le et al. [4] and an active
binocular robot head for foveated vision designed by the University of
Glasgow. Figure 1 shows the University of Glasgow robotic infrastructure. The
database subject of this report has been captured using the active binocular
robot head. This robot head comprises two Nikon DSLR cameras (D5100) that are
capable of capturing images at 16 Mega Pixels at different zoom settings
(manually selected, 35mm used for this database). These are mounted on two pan
and tilt units (PTU-D46) with their corresponding controllers as depicted in
Figure 2. The cameras are separated by a pre-defined baseline for optimal
stereo capturing within the robot’s workplace. The baseline separation between
cameras is 30 centimetres.
Figure 1: CloPeMa test-bed at the University of Glasgow. Figure 2: CloPeMa
robot head.
Garments were placed on a planar surface at an average distance of 1.8 meters
from the binocular robot head. For each garment, five different garment pose
configurations were captured as showed in Figure 3. Figure 4 shows an example
of the 35-mm zoom setting. The cameras of the robotic head were converged at
the centre point in the left and right cameras prior capturing the stereo
images. For this purpose, the vergence algorithm reported in Aragon-Camarasa
et al. [1] was used and integrated as a ROS node as described in Section 1.
|
---|---
(a) Spread | (b) Half-way folded
|
(c) Folded | (d) Wrinkled
(e) Hanging
Figure 3: Garment states captured. Figure 4: Garment zoom setting at 35-mm. A
standard Nikon 8-55mm VR lens was used for capturing the database.
For each image, a manually segmented mask of the same image resolution has
been provided for annotation purposes. In the database creation, the mask was
applied as part of the vertical and horizontal disparities computation using
the Glasgow stereo matcher Cyganek and Siebert [3]. _Gimp 2.8_
222http://www.gimp.org/ was used to segment and annotate the stereo-pair
images.
The underlying objective of the stereo matching algorithm is to locate for
each pixel in one image of a stereo pair, the corresponding location on the
other image of the pair. The correspondence problem is solved by constructing
a displacement field (also termed parallax or disparity map) that maps points
in the left image to the corresponding location on the right image. These
displacement fields are expressed in terms of two disparity maps for storing
horizontal and vertical displacements mapping pixels in the left image to the
corresponding location in the right image. Computed disparities can then be
used to reconstruct highly detailed point clouds and/or range images. Range
image preview examples can be depicted in Figure 5. It should be noted that
point clouds and range images are not included in the database as the file
size of each stereo-pair sample is roughly in the order of 1GB; however,
source code to recover the 3D geometry from the disparity maps is included in
the database.
|
---|---
Figure 5: Examples of range images computed at different zoom settings.
## 3 Database File Description and Organisation
The database (https://sites.google.com/site/ugstereodatabase/) is firstly
divided according to the captured garments. These are organised and stored in
folders using a numeric index from 1 to 16. In each of these folders, garment
pose configurations are organised in folders which follow the the following
file format: XX_S; where XX denotes the garment class and the folder number
where the image is stored and S, the garment pose configurations. S can take
the following classification indices which correspond to how the garment was
captured:
* 1.
0 - Cloth is spread on the table (Figure 3(a)).
* 2.
1 - Cloth is half-way folded (Figure 3(b)).
* 3.
2 - Cloth is completely folded (Figure 3(c)).
* 4.
3 - Cloth is wrinkled (Figure 3(d)).
* 5.
4 - The robot is holding the cloth in the air and close to the table (Figure
3(e)).
Within the above folders, the following is stored (it can also be depicted in
Figure 6):
* 1.
Stereo-pair images (left and right camera images) are stored as 16Mpixel
colour TIFF image files (4928 x 3264 x 24 BPP).
* 2.
Annotated image masks for the stereo-pair are stored as black and white TIFF
files, i.e. (4928 x 3264 x 8 BPP).
* 3.
Horizontal (_dispMH_) and vertical (_dispMV_) disparity maps and a confidence
matching map (_dispMConfidence_) are stored as text files, in ASCII format, as
matrices of 4928 by 3264 floating point values. These maps are compressed as
7zip format.
* 4.
A JPEG compressed preview of the garment range image.
Figure 6: Example of the file organisation of the stereo database.
Camera calibration parameters are stored as XML files for each of the captured
garments. Calibration files are saved as _calL.xml_ and _calR.xml_ for the
left and right cameras, respectively, as showed in Figure 6. These XML files
can be easily read using OpenCV I/O XML functions. The companion source code
provides an example on how to load these calibration files. Calibration
parameters in each file include:
* 1.
Camera matrix, $K$, as a 3 by 3 matrix that stores the focal point and
principal point in pixels.
* 2.
Distortion coefficients, $D$, as a 1 by 4 vector. The Glasgow stereo matcher
and stereo reconstruction does not use this information; however, this
coefficients are included for completeness.
* 3.
Projection matrix, $P$, as a 3 by 4 matrix. This matrix is defined for the
left (Equation 1) and right (Equation 2) cameras as follows:
$P_{L}=K_{L}\left[\mathrm{I}|0\right]$ (1) $P_{R}=K_{R}[R|t]$ (2)
where $\mathrm{I}$ is a 3 by 3 identity matrix, $R$ and $t$, the rotation and
translation matrices that transforms the right camera reference frame into the
left camera reference frame. $P_{L}$ and $P_{R}$ are used to recover the 3D
structure of the captured scene.
* 4.
Fundamental matrix, $F$, as a 3 by 3 matrix that relates corresponding points
between the stereo-pair. The same numeric matrix is defined in both files.
## Acknowledgements
We would like to thank the European Community’s Seventh Framework Programme
(FP7/2007-2013) to support this research work under grant agreement no 288553,
CloPeMa.
## References
## References
* Aragon-Camarasa et al. [2010] Aragon-Camarasa, G., Fattah, H., Siebert, J. P., Mar. 2010. Towards a unified visual framework in a binocular active robot vision system. Robotics and Autonomous Systems 58 (3), 276–286.
* Cockshott et al. [2012] Cockshott, W., Oehler, S., Camarasa, G. A., Siebert, J., Xu, T., 2012. A parallel stereo vision algorithm. In: Many-Core Applications Research Community Symposium 2012.
URL http://eprints.gla.ac.uk/72079/
* Cyganek and Siebert [2011] Cyganek, B., Siebert, J. P., 2011. An introduction to 3D computer vision techniques and algorithms. Wiley.
* Le et al. [2013] Le, T.-H.-L., Jilich, M., Landini, A., Zoppi, M., Zlatanov, D., Molfino, R., 2013\. On the development of a specialized flexible gripper for garment handling. Journal of automation and control engineering 1 (3), 255–259.
* Molfino et al. [2012] Molfino, R., Zoppi, M., Jilich, M., Hong Loan, L. T., Cannata, G., Maiolino, P., Denei, S., Malassiotis, S., Triantafilou, D., Gorpas, D., Hlavac, V., Donner, M., Aragon-Camarasa, G., Siebert, J. P., 2012. D1.1 scenarios and detailed specification of m12 demonstration. Tech. rep., EU-FP7 Clothes Percpetion and Manipulation (CloPeMa) project under grant agreement no. 288553\.
* Sun et al. [2013] Sun, L., Aragon-Camarasa, G., Cockshott, P., Rogers, S., Siebert, J., August 2013\. A heuristic-based approach for flattening wrinkled clothes. In: Towards Autonomous Robotic Systems, TAROS 2013 (in press). LNCS Springer.
|
arxiv-papers
| 2013-11-28T12:09:28 |
2024-09-04T02:49:54.486035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gerardo Aragon-Camarasa, Susanne B. Oehler, Yuan Liu, Sun Li, Paul\n Cockshott and J. Paul Siebert",
"submitter": "Gerardo Aragon Camarasa",
"url": "https://arxiv.org/abs/1311.7295"
}
|
1311.7362
|
Ed Greening111The workshop was supported by the University of Manchester,
IPPP, STFC, and IOP
on behalf of the LHCb collaboration
The University of Oxford
> Studies of rare decays are an indirect probe of New Physics (NP). This
> document presents recent measurements of rare decays in the charm sector by
> the LHCb experiment. The analyses are performed with proton-proton collision
> data at $\sqrt{s}$ = 7 ${\mathrm{\,Te\kern-1.00006ptV\\!}}$ recorded in
> 2011.
> PRESENTED AT
>
>
>
>
> The 6th International Workshop on Charm Physics
> (CHARM 2013)
> Manchester, UK, 31 August – 4 September, 2013
## 1 Introduction
Flavour-changing neutral current (FCNC) processes are rare within the Standard
Model (SM) as they cannot occur at tree level. At loop level, they are
suppressed by the both the Glashow-Iliopoulos-Maiani (GIM) [1] and the
Cabibbo-Kobayashi-Maskawa (CKM) [2, 3] mechanisms but are nevertheless well
established in processes that involve $K$ and $B$ mesons. In contrast to the
$B$ meson system, where the near-unity value of $|V_{ub}|$ and very high mass
of the top quark in the loop weaken the suppression, the cancellation is
almost exact in $D$ meson decays leading to lower SM branching fractions
($\cal B$). This suppression provides a unique opportunity to probe the
effects of NP on the coupling of up-type quarks in electroweak processes. NP
models may introduce additional diagrams that a priori need not be suppressed
in the same manner as the SM contributions. Enhancement in the $\cal B$ of
such decays would therefore be a sign of NP. The large number of $D$ mesons
created at the LHC and LHCb’s excellent ability to discriminate between pions
and muons [4, 5] mean that the detector is in a outstanding position to
investigate rare charm decays.
## 2 $D^{0}\rightarrow\mu^{+}\mu^{-}$
The decay $D^{0}\rightarrow\mu^{+}\mu^{-}$ is very rare in the SM because of
additional helicity suppression. The short distance perturbative contribution
to the $\cal B$ is of the order of $10^{-18}$ while the long distance non-
perturbative contribution, dominated by the two-photon intermediate state, is
estimated to be of the order $10^{5}$ higher [6]. A search for the decay is
performed with 0.9 $\mathrm{\,fb^{-1}}$ of data [7]. By taking the $D^{0}$
from $D^{+*}\rightarrow D^{0}\pi^{+}$ decays, a two-dimensional fit is
performed in m($\mu^{+}\mu^{-}$) and $\Delta m$ ($\equiv
m(\pi^{+}\mu^{+}\mu^{-})-m(\mu^{+}\mu^{-})$). The measured $\cal B$ is
normalised with the decay $D^{0}\rightarrow\pi^{+}\pi^{-}$. Peaking
backgrounds from the misidentified hadronic decays
$D^{0}\rightarrow\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$ and the
misidentified and partially reconstructed semileptonic decay
$D^{0}\rightarrow\pi^{-}\mu^{+}\nu_{\mu}$ are also taken into account. The
observed number of events is consistent with the background expectations and
corresponds to an upper limit of $\cal B$($D^{0}\rightarrow\mu^{+}\mu^{-}$)
$<6.2\,(7.6)\times 10^{-9}$ at 90% (95%) CL. This represents an improvement of
more than a factor twenty with respect to previous measurements [8] but
remains several orders of magnitude larger than the SM prediction.
## 3 $D_{(s)}^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$
The decay $D_{(s)}^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ proceeds via short and
long distance contributions. The long-distance contributions are mediated by
intermediate resonances,
$D_{(s)}^{+}\rightarrow\pi^{+}(V\rightarrow\mu^{+}\mu^{-})$, where $V$ $=$
$\phi$, $\eta$, $\rho^{0}$ or $\omega$, whose large $\cal B$ mask any
deviation from the much smaller non-resonant SM prediction, caused by NP. A
search for the decay is performed with 1.0 $\mathrm{\,fb^{-1}}$ of data [9].
The data is binned in m($\mu^{+}\mu^{-}$) allowing the long and short distance
contributions to be separated. The binning definitions are shown in Table 1.
The contribution from the intermediate $\rho^{0}$ and $\omega$ resonances are
grouped together as it is non-trivial to separate them. The signal yields in
each bin are determined with a simultaneous fit to the
m($\pi^{+}$$\mu^{+}\mu^{-}$) distribution of the m($\mu^{+}\mu^{-}$) bins and
shown in Table 1. The parameters of the shapes defining the $D_{(s)}^{+}$
signals are determined simultaneously across all bins. Candidates from the
kinematically similar $D_{(s)}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decay form
an important peaking background. Data-driven methods are used to parameterise
their contributions. The observed data, away from resonant structures, is
compatible with the background-only hypothesis, and no enhancement is
observed. Upper limits in the low and high m($\mu^{+}\mu^{-}$) bins are
calculated by normalising with the $\phi$ resonances. The upper limits in the
low and high m($\mu^{+}\mu^{-}$) bins, assuming a phase space $\mu^{+}\mu^{-}$
distribution, are extrapolated across the entirety of m($\mu^{+}\mu^{-}$) by
taking into account the relative efficiencies in each bin. Upper limits on the
non-resonant signal component of $\cal
B$($D^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$) $<7.3\,(8.3)\times 10^{-8}$ and
$\cal B$($D_{s}^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$) $<4.1\,(4.8)\times
10^{-9}$ at 90% (95%) CL are set. These represent an improvement of a factor
50 with respect to the previous limits [10, 11], but $\cal
B$($D^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$) is still an order of magnitude
larger than the SM prediction.
Table 1: Signal yields for the $D_{(s)}^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ fits. Bin description | m($\mu^{+}\mu^{-}$) range [${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$] | $D$ yield | $D_{s}^{+}$ yield
---|---|---|---
low-m($\mu^{+}\mu^{-}$) | $\phantom{1}250-\phantom{1}525$ | $\phantom{00}-3\pm 11$ | $\phantom{-000}1\pm\phantom{0}6$
$\eta$ | $\phantom{1}525-\phantom{1}565$ | $\phantom{-00}29\pm\phantom{0}7$ | $\phantom{-00}22\pm\phantom{0}5$
$\rho^{0}/\omega$ | $\phantom{1}565-\phantom{1}850$ | $\phantom{-00}96\pm 15$ | $\phantom{-00}87\pm 12$
$\phi$ | $\phantom{1}850-1250$ | $\phantom{-}2745\pm 67$ | $\phantom{-}3855\pm 86$
high-m($\mu^{+}\mu^{-}$) | $1250-2000$ | $\phantom{-00}16\pm 16$ | $\phantom{0}-17\pm 16$
## 4 $D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$
The non-resonant component of the decay
$D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ has an expected SM $\cal B$ of
the order $10^{-9}$ [12]. The branching fraction for these decays is expected
to be dominated by long-distance contributions involving resonances, such as
$D^{0}\rightarrow\pi^{+}\pi^{-}(V\rightarrow\mu^{+}\mu^{-})$, where $V$ can be
any of the light mesons $\eta$, $\rho^{0}$, $\omega$ or $\phi$. The
corresponding branching fractions can reach O($10^{-6}$) [12]. A search for
the decay is performed with 1.0 $\mathrm{\,fb^{-1}}$ of data [13]. The
$D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ data are split into four
m($\mu^{+}\mu^{-}$) bins: two bins containing the $\rho/\omega$ and $\phi$
resonances and two signal bins. No $\eta$ bin is defined due to a lack of
events after the analysis’s offline selection. The bin definitions are shown
in Table 2. By taking the $D^{0}$ from $D^{+*}\rightarrow D^{0}\pi^{+}$
decays, a two-dimensional fit is performed in m($\mu^{+}\mu^{-}$) and $\Delta
m$ ($\equiv
m(\pi^{+}_{s}\pi^{+}\pi^{-}\mu^{+}\mu^{-})-m(\pi^{+}\pi^{-}\mu^{+}\mu^{-})$)
in each bin. $D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ forms an important
peaking background and data-driven methods are used to parameterise the
contribution of this misidentified decay in each bin. The $\Delta m$ and
m($\pi^{+}\pi^{-}\mu^{+}\mu^{-}$) fits can be seen in Fig. 1 and Fig. 2,
respectively. The fitted yields are provided in Table 2. The observed data,
away from resonant structures, in both the low and high m($\mu^{+}\mu^{-}$)
bins, is compatible with the background-only hypothesis, and no enhancement is
observed. Upper limits in the low and high m($\mu^{+}\mu^{-}$) bins are
calculated by normalising with $\cal
B$($D^{0}\rightarrow\pi^{+}\pi^{-}(\phi\rightarrow\mu^{+}\mu^{-})$). The
normalisation $\cal B$ is estimated with the results of the amplitude analysis
of the $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ decay performed at CLEO
[14] and the known value of $\cal B$($\phi$ $\rightarrow$
$\mu^{+}\mu^{-}$)$/$$\cal B$($\phi$ $\rightarrow$ $K^{+}K^{-}$) [15]. The
upper limits in the low and high m($\mu^{+}\mu^{-}$) bins, assuming a phase
space $\mu^{+}\mu^{-}$ distribution, are extrapolated across the entirety of
m($\mu^{+}\mu^{-}$) by taking into account the relative efficiencies in each
bin. An upper limit on the non-resonant signal component of $\cal
B$($D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$) $<5.5\,(6.7)\times 10^{-7}$
at 90% (95%) CL is set.
Table 2: Signal yields for the $D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ fits. Bin description | m($\mu^{+}\mu^{-}$) range [${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$] | $D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ yield
---|---|---
low-m($\mu^{+}\mu^{-}$) | $\phantom{1}250-\phantom{1}525$ | $\phantom{0}2\pm\phantom{0}2$
$\rho/\omega$ | $\phantom{1}565-\phantom{1}950$ | $23\pm\phantom{0}6$
$\phi$ | $\phantom{1}950-1100$ | $63\pm 10$
high-m($\mu^{+}\mu^{-}$) | $>1100$ | $\phantom{0}3\pm\phantom{0}2$
Figure 1: Distributions of $\Delta m$ for
$D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the (a)
low-m($\mu^{+}\mu^{-}$), (b) $\rho^{0}/\omega$, (c) $\phi$, and (d)
high-m($\mu^{+}\mu^{-}$) bins, with the $D^{0}$ invariant mass in the range
$1840-1888$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The data are shown as
points (black) and the fit result (dark blue line) is overlaid. The components
of the fit are also shown: the signal (black double-dashed double-dotted
line), the peaking background (green dashed line) and the non-peaking
background (red dashed-dotted line).
Figure 2: Distributions of m($\pi^{+}\pi^{-}\mu^{+}\mu^{-}$) for
$D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the (a)
low-m($\mu^{+}\mu^{-}$), (b) $\rho^{0}/\omega$, (c) $\phi$, and (d)
high-m($\mu^{+}\mu^{-}$) bins, with $\Delta m$ in the range $144.4-146.6$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The data are shown as points
(black) and the fit result (dark blue line) is overlaid. The components of the
fit are also shown: the signal (black double-dashed double-dotted line), the
peaking background (green dashed line) and the non-peaking background (red
dashed-dotted line).
## 5 Conclusion
Before the second long shutdown of the LHC in 2017, LHCb expects to record an
additional 5 $\mathrm{\,fb^{-1}}$ at $\sqrt{s}$ = 13
${\mathrm{\,Te\kern-1.00006ptV\\!}}$. This is in addition to the 1 and 2
$\mathrm{\,fb^{-1}}$ of data at $\sqrt{s}$ = 7 and 8
${\mathrm{\,Te\kern-1.00006ptV\\!}}$, respectively, that LHCb already has on
tape. Together with anticipated improvements in LHCb’s trigger system and
analysis strategies, the higher centre-of-mass energy increases heavy flavour
production cross-sections. In comparison to the analyses detailed in this
report, a factor of twenty increase in the number of observed decays can
optimistically be hoped for. A naive $\sqrt{20}$ scaling, would then give the
following limits: $\cal B$($D^{0}\rightarrow\mu^{+}\mu^{-}$) $=1\times
10^{-8}$, an order of magnitude above the indirect bound; $\cal
B$($D_{(s)}^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$) $=2\times 10^{-8}$, an order
of magnitude above the SM expectation; and $\cal
B$($D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$) $=2\times 10^{-7}$, two
orders of magnitude above the SM expectation. So although one would not expect
to observe a SM signal before the LHCb upgrade, the phase space available to
NP is set to be further probed.
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|
arxiv-papers
| 2013-11-28T16:32:40 |
2024-09-04T02:49:54.498747
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ed Greening",
"submitter": "Ed Greening",
"url": "https://arxiv.org/abs/1311.7362"
}
|
1311.7364
|
Antimo Palano
on behalf of the LHCb Collaboration
INFN and University of Bari, Italy
> A study of $D^{+}\pi^{-}$, $D^{0}\pi^{+}$ and $D^{*+}\pi^{-}$ final states
> is performed using $pp$ collision data, corresponding to an integrated
> luminosity of 1.0$\mbox{\,fb}^{-1}$, collected at a centre-of-mass energy of
> $7\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector. The
> $D_{1}(2420)^{0}$ resonance is observed in the $D^{*+}\pi^{-}$ final state
> and the $D^{*}_{2}(2460)$ resonance is observed in the $D^{+}\pi^{-}$,
> $D^{0}\pi^{+}$ and $D^{*+}\pi^{-}$ final states. For both resonances, their
> properties and spin-parity assignments are obtained. In addition, two
> natural parity and two unnatural parity resonances are observed in the mass
> region between 2500 and 2800 $\mathrm{\,Me\kern-1.00006ptV}$. Further
> structures in the region around 3000 $\mathrm{\,Me\kern-1.00006ptV}$ are
> observed in all the $D^{*+}\pi^{-}$, $D^{+}\pi^{-}$ and $D^{0}\pi^{+}$ final
> states. Using three- and four-body decays of $D$ mesons produced in
> semileptonic $b$-hadron decays, precision measurements of $D$ meson mass
> differences are made together with a measurement of the $D^{0}$ mass.
> PRESENTED AT
>
>
>
>
> The 6th International Workshop on Charm Physics
> (CHARM 2013)
> Manchester, UK, 31 August – 4 September, 2013
## 1 Introduction
Charm meson spectroscopy provides a powerful test of the quark model
predictions of the Standard Model. Many charm meson states, predicted in the
1980s [1], have not yet been observed experimentally. The $J^{P}$ states
having $P=(-1)^{J}$ and therefore $J^{P}=0^{+},1^{-},2^{+},...$ are called
natural parity states and are labelled as $D^{*}$, while unnatural parity
indicates the series $J^{P}=0^{-},1^{+},2^{-},...$. Apart from the ground
states ($D,D^{*}$), only two of the 1P states, $D_{1}(2420)$ and
$D^{*}_{2}(2460)$, are experimentally well established since they have
relatively narrow widths ($\sim$30$\mathrm{\,Me\kern-1.00006ptV}$). ***We work
in units where $c=1$. In contrast, the broad $L=1$ states, $D^{*}_{0}(2400)$
and ${D}^{\prime}_{1}(2430)$, have been established by the Belle and BaBar
experiments in exclusive $B$ decays [2, 3]. A search for excited charmed
mesons, labelled $D_{J}$, has been performed by BaBar [4]. They observe four
signals, labelled ${D}(2550)^{0}$, ${D^{*}}(2600)^{0}$, ${D}(2750)^{0}$ and
${D^{*}}(2760)^{0}$, and the isospin partners ${D^{*}}(2600)^{+}$ and
${D^{*}}(2760)^{+}$. This study [5] reports a search for $D_{J}$ mesons in a
data sample, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, of $pp$ collisions collected at a centre-of-mass energy
of $7\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector.
## 2 Event selection
The search for $D_{J}$ mesons is performed using the inclusive reactions
$pp\rightarrow D^{+}\pi^{-}X,\ pp\rightarrow D^{0}\pi^{+}X,\ pp\rightarrow
D^{*+}\pi^{-}X,$ (1)
where $X$ represents a system composed of any collection of charged and
neutral particles †††Throughout the paper use of charge-conjugate decay modes
is implied.. The charmed mesons in the final state are reconstructed in the
decay modes ${\mbox{$D^{+}$}\rightarrow K^{-}\pi^{+}\pi^{+}}$,
$D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{*+}\rightarrow D^{0}\pi^{+}$. Charged
tracks are required to have good track fit quality, momentum
$p>3\mathrm{\,Ge\kern-1.00006ptV}$ and $\mbox{$p_{\rm
T}$}>250\mathrm{\,Me\kern-1.00006ptV}$. These conditions are relaxed to lower
limits for the pion originating directly from the $D^{*+}$ decay. The cosine
of the angle between the momentum of the $D$ meson candidate and its
direction, defined by the positions of the primary vertex and the meson decay
vertex, is required to be larger than 0.99999. This ensures that the $D$ meson
candidates are produced at the primary vertex and reduces the contribution
from particles originating from $b$-hadron decays. The purity of the charmed
meson candidates is enhanced by requiring the decay products to be identified
by the RICH detectors. The reconstructed $D^{+}$, $D^{0}$ and $D^{*+}$
candidates are combined with all the right-sign charged pions in the event.
Each of the $D^{+}\pi^{-}$, the $D^{0}\pi^{+}$, and the $D^{*+}\pi^{-}$
candidates are fitted to a common vertex with $\chi^{2}/{\rm ndf}<8$, where
ndf is the number of degrees of freedom. In order to reduce combinatorial
background, the cosine of the angle between the momentum direction of the
charged pion in the $D^{(*)}\pi^{\pm}$ rest frame and the momentum direction
of the $D^{(*)}\pi^{\pm}$ system in the laboratory frame is required to be
greater than zero. It is also required that the $D^{(*)}$ and the $\pi^{\pm}$
point to the same primary vertex.
## 3 Mass spectra
The $D^{+}\pi^{-}$, $D^{0}\pi^{+}$ and $D^{*+}\pi^{-}$ mass spectra are shown
in Fig. 1. A further reduction of the combinatorial background is achieved by
performing an optimization of the signal significance and purity as a function
of $p_{\rm T}$ of the $D^{(*)}\pi^{\pm}$ system using the well known
$D_{1}(2420)$ and $D^{*}_{2}(2460)$ resonances. ‡‡‡We use the generic notation
$D$ to indicate both neutral and charged $D$ mesons. After the optimization
7.9$\times 10^{6}$, 7.5$\times 10^{6}$ and 2.1$\times 10^{6}$ $D^{+}\pi^{-}$,
$D^{0}\pi^{+}$ and $D^{*+}\pi^{-}$ candidates are obtained. We analyze, for
comparison and using the same selections, the wrong-sign $D^{+}\pi^{+}$,
$D^{0}\pi^{-}$ and $D^{*+}\pi^{+}$ combinations which are also shown in Fig.
1.
Figure 1: Invariant mass distribution for (a) $D^{+}\pi^{-}$, (b)
$D^{0}\pi^{+}$ and (c) $D^{*+}\pi^{-}$ candidates (points). The full line
histograms (in red) show the wrong-sign mass spectra for (a) $D^{+}\pi^{+}$,
(b) $D^{0}\pi^{-}$ and (c) $D^{*+}\pi^{+}$ normalized to the same yield at
high $D^{(*)}\pi$ masses.
The $D^{+}\pi^{-}$ mass spectrum, Fig. 1(a), shows a double peak structure
around 2300 $\mathrm{\,Me\kern-1.00006ptV}$ due to cross-feed from the decay
$D_{1}(2420)^{0}\ {\rm or}\
{D}^{*}_{2}(2460)^{0}\rightarrow\pi^{-}D^{*+}(\rightarrow
D^{+}\pi^{0}/\gamma)\ (32.3\%),$ (2)
where the $\pi^{0}/\gamma$ is not reconstructed; the last number, in
parentheses, indicates the branching fraction of $D^{*+}\rightarrow
D^{+}\pi^{0}/\gamma$ decays. We observe a strong ${D}^{*}_{2}(2460)^{0}$
signal and weak structures around 2600 and 2750
$\mathrm{\,Me\kern-1.00006ptV}$. The wrong-sign $D^{+}\pi^{+}$ mass spectrum
does not show any structure. The $D^{0}\pi^{+}$ mass spectrum, Fig. 1(b),
shows an enhanced double peak structure around 2300
$\mathrm{\,Me\kern-1.00006ptV}$ due to cross-feed from the decays
$D_{1}(2420)^{+}\ {\rm or}\
{D}^{*}_{2}(2460)^{+}\begin{array}[]{l}\rightarrow\pi^{+}D^{*0}\begin{array}[]{l}\\\
(\rightarrow D^{0}\pi^{0})\ (61.9\%)\\\ (\rightarrow D^{0}\gamma)\ (38.1\%)\
.\end{array}\end{array}$ (3)
The ${D}^{*}_{2}(2460)^{+}$ signal and weak structures around 2600 and 2750
$\mathrm{\,Me\kern-1.00006ptV}$ are observed. In comparison, the wrong-sign
$D^{0}\pi^{-}$ mass spectrum does show the presence of structures in the 2300
$\mathrm{\,Me\kern-1.00006ptV}$ mass region, similar to those observed in the
$D^{0}\pi^{+}$ mass spectrum. These structures are due to cross-feed from the
decay
$D_{1}(2420)^{0}\ {\rm or}\
{D}^{*}_{2}(2460)^{0}\rightarrow\pi^{-}D^{*+}(\rightarrow D^{0}\pi^{+})\
(67.7\%)\ .$ (4)
The $D^{*+}\pi^{-}$ mass spectrum, Fig. 1(c), is dominated by the presence of
the $D_{1}(2420)^{0}$ and ${D}^{*}_{2}(2460)^{0}$ signals. At higher mass,
complex broad structures are evident in the mass region between 2500 and 2800
$\mathrm{\,Me\kern-1.00006ptV}$.
## 4 Mass fit model
Using Monte Carlo simulations, We estimate resolutions which, in the mass
region between 2000 and 2900 $\mathrm{\,Me\kern-1.00006ptV}$, are similar for
the three mass spectra and range from 1.0 to 4.5
$\mathrm{\,Me\kern-1.00006ptV}$ as a function of the mass. Since the widths of
the resonances appearing in the three mass spectra are much larger than the
experimental resolutions, resolution effects are neglected. Binned $\chi^{2}$
fits to the three mass spectra are performed. The $D^{*}_{2}(2460)$ and
$D^{*}_{0}(2400)$ signal shapes in two-body decays are parameterized with a
relativistic Breit-Wigner that includes the mass-dependent factors for a
D-wave and S-wave decay, respectively. The radius entering in the Blatt-
Weisskopf [6] form factor is fixed to 4 $\mathrm{\,Ge\kern-1.00006ptV}^{-1}$.
Other resonances appearing in the mass spectra are described by Breit-Wigner
lineshapes. All Breit-Wigner expressions are multiplied by two-body phase
space. The cross-feed lineshapes from $D_{1}(2420)$ and $D^{*}_{2}(2460)$
appearing in the $D^{+}\pi^{-}$ and $D^{0}\pi^{+}$ mass spectra are described
by a Breit-Wigner function fitted to the data. The background $B(m)$ is
described by an empirical shape [4]
$\displaystyle B(m)=$ $\displaystyle P(m)e^{a_{1}m+a_{2}m^{2}}\ {\rm for}\
m<m_{0},$ $\displaystyle B(m)=$ $\displaystyle
P(m)e^{b_{0}+b_{1}m+b_{2}m^{2}}\ {\rm for}\ m>m_{0},$ (5)
where $P(m)$ is the two-body phase space and $m_{0}$ is a free parameter. The
two functions and their first derivatives are required to be continuous at
$m_{0}$ and therefore the background model has four free parameters.
Table 1: Definition of the categories selected by different ranges of $\cos\theta_{\rm H}$, and fraction of the total natural parity contribution. Category | Selection | natural parity fraction (%)
---|---|---
Enhanced unnatural parity sample | $|\cos\theta_{\rm H}|>0.75$ | 8.6
Natural parity sample | $|\cos\theta_{\rm H}|<0.5$ | 68.8
Unnatural parity sample | $|\cos\theta_{\rm H}|>0.5$ | 31.2
## 5 Fit to the $D^{*+}\pi^{-}$ mass spectrum
Figure 2: Fit to the $D^{*+}\pi^{-}$ mass spectrum, enhanced unnatural parity
sample. The dashed (blue) line shows the fitted background, the dotted lines
the $D_{1}(2420)^{0}$ (red) and ${D}^{*}_{2}(2460)^{0}$ (blue) contributions.
The inset displays the $D^{*+}\pi^{-}$ mass spectrum after subtracting the
fitted background. The full line curves (red) show the contributions from
$D_{J}(2580)^{0}$, $D_{J}(2740)^{0}$, and $D_{J}(3000)^{0}$. The dotted (blue)
lines display the $D^{*}_{J}(2650)^{0}$ and $D^{*}_{J}(2760)^{0}$
contributions. The top window shows the pull distribution where the horizontal
lines indicate $\pm 3\sigma$. The pull is defined as $(N_{\rm data}-N_{\rm
fit})/\sqrt{N_{\rm data}}$.
Due to the three-body decay and the availability of the helicity angle
information, the fit to the $D^{*+}\pi^{-}$ mass spectrum allows a spin
analysis of the produced resonances and a separation of the different spin-
parity components. We define the helicity angle $\theta_{\rm H}$ as the angle
between the $\pi^{-}$ and the $\pi^{+}$ from the $D^{*+}$ decay, in the rest
frame of the $D^{*+}\pi^{-}$ system. Full detector simulations are used to
measure the efficiency as a function of $\theta_{\rm H}$, which is found to be
uniform. It is expected that the angular distributions are proportional to
$\sin^{2}\mbox{$\theta_{\rm H}$}$ for natural parity resonances and
proportional to $1+h\cos^{2}\mbox{$\theta_{\rm H}$}$ for unnatural parity
resonances, where $h>0$ is a free parameter. The $D^{*}\pi$ decay of a
$J^{P}=0^{+}$ resonance is forbidden. Therefore candidates selected in
different ranges of $\cos\theta_{\rm H}$ can enhance or suppress the different
spin-parity contributions. We separate the $D^{*+}\pi^{-}$ data into three
different categories, summarized in Table 1. The data and fit for the
$D^{*+}\pi^{-}$ enhanced unnatural parity sample are shown in Fig. 2(a) and
the resulting fit parameters are summarized in Table 2. The mass spectrum is
dominated by the presence of the unnatural parity $D_{1}(2420)^{0}$ resonance.
The fitted natural parity ${D}^{*}_{2}(2460)^{0}$ contribution is consistent
with zero, as expected. To obtain a good fit to the mass spectrum, three
further resonances are needed. We label them $D_{J}(2580)^{0}$,
$D_{J}(2740)^{0}$, and $D_{J}(3000)^{0}$. The presence of these states in this
sample indicates unnatural parity assignments. The masses and widths of the
unnatural parity resonances are fixed in the fit to the natural parity sample.
The fit is shown in Fig. 2(b) and the obtained resonance parameters are
summarized in Table 2. The mass spectrum shows that the unnatural parity
resonance $D_{1}(2420)^{0}$ is suppressed with respect to that observed in the
enhanced unnatural parity sample. There is a strong contribution of the
natural parity ${D}^{*}_{2}(2460)^{0}$ resonance and contributions from the
$D_{J}(2580)^{0}$, $D_{J}(2740)^{0}$ and $D_{J}(3000)^{0}$ states. To obtain a
good fit, two additional resonances are needed, which we label
$D^{*}_{J}(2650)^{0}$ and $D^{*}_{J}(2760)^{0}$. Table 2 summarizes the
measured resonance parameters and yields. The significances are computed as
$\sqrt{\Delta\chi^{2}}$ where $\Delta\chi^{2}$ is the difference between the
$\chi^{2}$ values when a resonance is included or excluded from the fit while
all the other resonances parameters are allowed to vary. All the statistical
significances are well above 5$\sigma$.
Table 2: Resonance parameters, yields and statistical significances. The first uncertainty is statistical, the second systematic. Resonance | Final state | Mass (MeV) | Width (MeV) | Yields $\times 10^{3}$ | Significance
---|---|---|---|---|---
$D_{1}(2420)^{0}$ | $D^{*+}\pi^{-}$ | 2419.6 $\pm$ | 0.1 | $\pm$ 0.7 | 35.2 $\pm$ | 0.4 | $\pm$ 0.9 | 210.2 $\pm$ | 1.9 | $\pm$ 0.7 |
${D}^{*}_{2}(2460)^{0}$ | $D^{*+}\pi^{-}$ | 2460.4 $\pm$ | 0.4 | $\pm$ 1.2 | 43.2 $\pm$ | 1.2 | $\pm$ 3.0 | 81.9 $\pm$ | 1.2 | $\pm$ 0.9 |
$D^{*}_{J}(2650)^{0}$ | $D^{*+}\pi^{-}$ | 2649.2 $\pm$ | 3.5 | $\pm$ 3.5 | 140.2 $\pm$ | 17.1 | $\pm$ 18.6 | 50.7 $\pm$ | 2.2 | $\pm$ 2.3 | 24.5
$D^{*}_{J}(2760)^{0}$ | $D^{*+}\pi^{-}$ | 2761.1 $\pm$ | 5.1 | $\pm$ 6.5 | 74.4 $\pm$ | 3.4 | $\pm$ 37.0 | 14.4 $\pm$ | 1.7 | $\pm$ 1.7 | 10.2
$D_{J}(2580)^{0}$ | $D^{*+}\pi^{-}$ | 2579.5 $\pm$ | 3.4 | $\pm$ 5.5 | 177.5 $\pm$ | 17.8 | $\pm$ 46.0 | 60.3 $\pm$ | 3.1 | $\pm$ 3.4 | 18.8
$D_{J}(2740)^{0}$ | $D^{*+}\pi^{-}$ | 2737.0 $\pm$ | 3.5 | $\pm$11.2 | 73.2 $\pm$ | 13.4 | $\pm$ 25.0 | 7.7 $\pm$ | 1.1 | $\pm$ 1.2 | 7.2
$D_{J}(3000)^{0}$ | $D^{*+}\pi^{-}$ | 2971.8 $\pm$ | 8.7 | | 188.1 $\pm$ | 44.8 | | 9.5 $\pm$ | 1.1 | | 9.0
${D}^{*}_{2}(2460)^{0}$ | $D^{+}\pi^{-}$ | 2460.4 $\pm$ | 0.1 | $\pm$ 0.1 | 45.6 $\pm$ | 0.4 | $\pm$ 1.1 | 675.0 $\pm$ | 9.0 | $\pm$ 1.3 |
$D^{*}_{J}(2760)^{0}$ | $D^{+}\pi^{-}$ | 2760.1 $\pm$ | 1.1 | $\pm$ 3.7 | 74.4 $\pm$ | 3.4 | $\pm$19.1 | 55.8 $\pm$ | 1.3 | $\pm$ 10.0 | 17.3
$D^{*}_{J}(3000)^{0}$ | $D^{+}\pi^{-}$ | 3008.1 $\pm$ | 4.0 | | 110.5 $\pm$ | 11.5 | | 17.6 $\pm$ | 1.1 | | 21.2
${D}^{*}_{2}(2460)^{+}$ | $D^{0}\pi^{+}$ | 2463.1 $\pm$ | 0.2 | $\pm$ 0.6 | 48.6 $\pm$ | 1.3 | $\pm$ 1.9 | 341.6 $\pm$ | 22.0 | $\pm$ 2.0 |
$D^{*}_{J}(2760)^{+}$ | $D^{0}\pi^{+}$ | 2771.7 $\pm$ | 1.7 | $\pm$ 3.8 | 66.7 $\pm$ | 6.6 | $\pm$10.5 | 20.1 $\pm$ | 2.2 | $\pm$ 1.0 | 18.8
$D^{*}_{J}(3000)^{+}$ | $D^{0}\pi^{+}$ | 3008.1 | (fixed) | | 110.5 | (fixed) | | 7.6 $\pm$ | 1.2 | | 6.6
## 6 Spin-parity analysis of the $D^{*+}\pi^{-}$ system
In order to obtain information on the spin-parity assignment of the states
observed in the $D^{*+}\pi^{-}$ mass spectrum, the data are subdivided into
ten equally spaced bins in $\cos\theta_{\rm H}$. The ten mass spectra are then
fitted with the model described above with fixed resonance parameters to
obtain the yields as functions of $\cos\theta_{\rm H}$ for each resonance. The
resulting distributions for $D_{1}(2420)^{0}$ and ${D}^{*}_{2}(2460)^{0}$ are
shown in Fig. 3(a)-(b). A good description of the data is obtained in terms of
the expected angular distributions for $J^{P}=1^{+}$ and $J^{P}=2^{+}$
resonances.
Figure 3: Distributions of (a) $D_{1}(2420)^{0}$, (b)
${D}^{*}_{2}(2460)^{0}$, (c) $D^{*}_{J}(2650)^{0}$ and (d)
$D^{*}_{J}(2760)^{0}$ candidates as functions of the helicity angle
$\cos\theta_{\rm H}$. The distributions are fitted with natural parity (black
continuous), unnatural parity (red, dashed) and $J^{P}=0^{-}$ (blue, dotted)
functions.
Figure 3(c)-(d) shows the resulting distributions for the
$D^{*}_{J}(2650)^{0}$ and $D^{*}_{J}(2760)^{0}$ states. In this case we
compare the distributions with expectations from natural parity, unnatural
parity and $J^{P}=0^{-}$. In the case of unnatural parity, the $h$ parameter,
in $1+h\cos^{2}\theta_{\rm H}$, is constrained to be positive and therefore
the fit gives $h=0$. In both cases, the distributions are best fitted by the
natural parity hypothesis. Figure 4 shows the angular distributions for the
$D_{J}(2580)^{0}$, $D_{J}(2740)^{0}$ and $D_{J}(3000)^{0}$ states. The
distributions are fitted with natural parity and unnatural parity. The
$J^{P}=0^{-}$ hypothesis is also considered for $D_{J}(2580)^{0}$. In all
cases unnatural parity is preferred over a natural parity assignment.
Figure 4: Distributions of (a) $D_{J}(2580)^{0}$, (b) $D_{J}(2740)^{0}$ and
(c) $D_{J}(3000)^{0}$ candidates as functions of the helicity angle
$\cos\theta_{\rm H}$. The distributions are fitted with natural parity (black
continuous) and unnatural parity (red, dashed) functions. In (a) the
$J^{P}=0^{-}$ (blue, dotted) hypothesis is also tested.
## 7 Fit to the $D^{+}\pi^{-}$ and $D^{0}\pi^{+}$ mass spectra
The $D^{+}\pi^{-}$ and $D^{0}\pi^{+}$ mass spectra consist of natural parity
resonances. However these final states are affected by cross-feed from all the
resonances that decay to the $D^{*}\pi$ final state. Figures 1(a)-(b) show (in
the mass region around 2300 MeV) cross-feed contributions from $D_{1}(2420)$
and $D^{*}_{2}(2460)$ decays. However we also expect (in the mass region
between 2400 and 2600 MeV) the presence of structures originating from the
complex resonance structure present in the $D^{*}\pi$ mass spectrum in the
mass region between 2500 and 2800 $\mathrm{\,Me\kern-1.00006ptV}$. To obtain
an estimate of the lineshape and size of the cross-feed, we normalize the
$D^{*+}\pi^{-}$ mass spectrum to the $D^{+}\pi^{-}$ mass spectrum using the
sum of the $D_{1}(2420)^{0}$ and ${D}^{*}_{2}(2460)^{0}$ yields in the
$D^{*+}\pi^{-}$ mass spectrum and the sum of the cross-feed in the
$D^{+}\pi^{-}$ mass spectrum. To obtain the expected lineshape of the cross-
feed in the $D^{+}\pi^{-}$ final state, we perform a study based on a
generator level simulation. We generate $D^{*}_{J}(2650)^{0}$,
$D^{*}_{J}(2760)^{0}$, $D_{J}(2580)^{0}$ and $D_{J}(2740)^{0}$ decays
according to the chain described in Eq. (2). We then compute the resulting
$D^{+}\pi^{-}$ mass spectra and normalize each contribution to the measured
yields. The overall resulting structures are then properly scaled and
superimposed on the $D^{+}\pi^{-}$ mass spectrum shown in Fig. 5(a). A similar
method is used for the $D^{0}\pi^{+}$ final state and the resulting
contribution is superimposed on the $D^{0}\pi^{+}$ mass spectrum shown in Fig.
5(b). To obtain good quality fits we add broad structures around 3000
$\mathrm{\,Me\kern-1.00006ptV}$, which we label $D^{*}_{J}(3000)^{0}$ and
$D^{*}_{J}(3000)^{+}$.
Figure 5: (a) Fit to the $D^{+}\pi^{-}$ mass spectrum and (b) to the
$D^{0}\pi^{+}$ mass spectrum. The filled histogram (in red) shows the
estimated cross-feeds from the high mass $D^{*}\pi$ resonances.
The fits to the $D^{+}\pi^{-}$ and $D^{0}\pi^{+}$ mass spectra are shown in
Fig. 5(a) and Fig. 5(b), respectively. Several cross-checks are performed to
test the stability of the fits and their correct statistical behaviour. We
first repeat all the fits, including the spin-parity analysis, lowering the
$p_{\rm T}$ requirement from 7.5 to 7.0 GeV. We find that all the resonance
parameters vary within their statistical uncertainties and that the spin-
parity assignments are not affected by this selection. Then we perform fits
using random variations of the histogram contents and background parameters.
The various estimated systematic uncertainties are added in quadrature.
## 8 Precision measurement of $D$ meson mass differences
Using three- and four-body decays of $D$ mesons produced in semileptonic
$b$-hadron decays, precision measurements of $D$ meson mass differences are
made together with a measurement of the $D^{0}$ mass [8]. The selection uses
only well reconstructed charged particles that traverse the entire tracking
system. Further background suppression is achieved by exploiting the fact that
the products of heavy flavour decays have a large distance of closest approach
(‘impact parameter’) with respect to the $pp$ interaction vertex in which they
were produced. The impact parameter $\chi^{2}$ with respect to any primary
vertex is required to be larger than nine. Charged particles are combined to
form $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$, $D^{0}\rightarrow
K^{+}K^{-}K^{-}\pi^{+}$ and $D^{+}_{(s)}\rightarrow K^{+}K^{-}\pi^{+}$
candidates. To eliminate kinematic reflections due to misidentified pions, the
invariant mass of at least one kaon pair is required to be within $\pm
12~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal value of the
$\phi$ meson mass. Each candidate $D$ meson is combined with a well-identified
muon that is displaced from the $pp$ interaction vertex to form a $B$
candidate, requiring the muon and the $D$ candidate to originate from a common
point. The $D$ meson masses are determined by performing extended unbinned
maximum likelihood fits to the invariant mass distributions. In these fits the
background is modelled by an exponential function and the signal by the sum of
a Crystal Ball [9] and a Gaussian function. The Crystal Ball component
accounts for the presence of the QED radiative tail. The fits for the $D^{0}$
decay modes and the $K^{+}K^{-}\pi^{+}$ final state are shown in Fig. 6. The
resulting values of the $D^{+}$ and $D^{+}_{s}$ masses are in agreement with
the current world averages. These modes have relatively large $Q$-values and
consequently the systematic uncertainty due to the knowledge of the momentum
scale is at the level of $0.3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Hence, it is chosen not to quote these values as measurements. Similarly, the
systematic uncertainty due to the momentum scale for the $D^{0}\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$ mode is estimated to be
$0.2~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the measured mass in this
mode is not used in the $D^{0}$ mass determination.
Figure 6: Invariant mass distributions for the (a) $K^{+}K^{-}\pi^{+}\pi^{-}$
and (b) $K^{+}K^{-}K^{-}\pi^{+}$ final states. Invariant mass distribution for
the $K^{+}K^{-}\pi^{+}$ final state.
We obtain
$M(D^{0})$ $~{}=~{}$ | 1864.75 $\,\,\pm\,$ | 0.15 (stat) $\pm\,$ | 0.11 (syst) MeV/$c^{2}$ | ,
---|---|---|---|---
$M(D^{+})$ $-$ $M(D^{0})$ $~{}=~{}$ | 1114.76 $\,\,\pm\,$ | 0.12 (stat) $\pm\,$ | 0.07 (syst) MeV/$c^{2}$ | ,
$M(D^{+}_{s})$ $-$ $M(D^{+})$ $~{}=~{}$ | 98.68 $\,\,\pm\,$ | 0.03 (stat) $\pm\,$ | 0.04 (syst) MeV/$c^{2}$ | ,
where dominant systematic uncertainty is related to the knowledge of the
momentum scale. The measurements presented here, together with those given in
Ref. [7] for the $D^{+}$ and $D^{0}$ mass, and the mass differences
$M(D^{+})-M(D^{0})$, $M(D^{+}_{s})-M(D^{+})$ can be used to determine a more
precise value of the $D^{+}_{s}$ mass
$M(D^{+}_{s})=1968.19\pm 0.20\pm 0.14\pm
0.08{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
where the first uncertainty is the quadratic sum of the statistical and
uncorrelated systematic uncertainty, the second is due to the momentum scale
and the third due to the energy loss. This value is consistent with, but more
precise than, that obtained from the fit to open charm mass data,
$M(D^{+}_{s})=1968.49\pm 0.32~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [7].
ACKNOWLEDGEMENTS
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
## References
* [1] S. Godfrey and N. Isgur, Phys. Rev. D32 (1985) 189.
* [2] Belle collaboration, K. Abe et al., Phys. Rev. D69 (2004).
* [3] BaBar collaboration, B. Aubert et al., Phys. Rev. D79 (2009) 112004.
* [4] BaBar collaboration, P. del Amo Sanchez et al., Phys. Rev. D82 (2010) 111101.
* [5] LHCb collaboration, R. Aaij et al., JHEP 09 (2013) 145.
* [6] J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics, John Wiley & Sons, New York, 1952.
* [7] Particle Data Group, J. Beringer et al., Phys. Rev. D86 (2012) 010001.
* [8] LHCb collaboration, R. Aaij et al., JHEP 06 (2013) 065.
* [9] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986.
|
arxiv-papers
| 2013-11-28T16:35:21 |
2024-09-04T02:49:54.505019
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antimo Palano (for the LHCb Collaboration)",
"submitter": "Antimo Palano",
"url": "https://arxiv.org/abs/1311.7364"
}
|
1311.7507
|
# Maximal subfields of a division algebra
Mai Hoang Bien Mathematisch Instituut, Leiden Universiteit, Niels Bohrweg
1,2333 CA Leiden,The Netherlands. Dipartimento di Matematica, Università degli
Studi di Padova, Via Trieste 63, 35121 Padova, Italy.
[email protected]
###### Abstract.
Let $D$ be a division algebra over a field $F$. In this paper, we prove that
there exist $a,b,x,y\in D^{*}=D\backslash\\{0\\}$ such that $F(ab-ba)$ and
$F(xyx^{-1}y^{-1})$ are maximal subfields of $D$, which answers questions
posted in [5].
###### Key words and phrases:
Maximal subfield, division algebra, commutator, algebraic.
2010 Mathematics Subject Classification. 12F05, 12F10, 12E15, 16K20.
The author would like to thank his supervisor Prof. H.W. Lenstra for the
comments.
## 1\. Introduction
Let $F$ be a field. A ring $D$ is called a division algebra over $F$ if the
center $Z(D)=\\{\,a\in D\mid ab=ba,\forall b\in D\,\\}$ of $D$ is equal to
$F$, $D$ is a finite dimensional vector space over $F$ and $D$ has neither
proper left ideal nor proper right ideal. In other words, $D$ is a division
ring with the center $F$ and $\dim_{F}D<\infty$. In some books and papers, $D$
is also called centrally finite [4, Definition 14.1]. A central simple algebra
over $F$ is an algebra isomorphic to $M_{n}(D)$ for some positive integer $n$
and division algebra $D$ over $F$. For any central simple algebra $A$ over
$F$, $\sqrt{\dim_{F}A}$ is said to be degree of $A$.
For any division algebra $D$ over $F$, it is well known from Kothe’s Theorem
that there exists a maximal subfield $K$ of $D$ such that the extension of
fields $K/F$ is separable [4, Th. 15.12]. In [1, Theorem 7], authors proved
that for any separable extension of fields $K/F$ in $D$, there exists an
element $c\in[D,D]$, the group of additive commutators of $(D,+)$, such that
$K=F(c)$ unless $\operatorname{Char}(F)=[K:F]=2$ and $4$ does not divide the
degree of $D$. Hence, if $K$ is a maximal subfield of $D$ which is separable
over $F$ then there exists $c\in[D,D]$ such that $K=F(c)$. In particular,
there exists a maximal subfield of $D$ such that it is of the form $F(c)$ for
some element $c$ in $[D,D]$. We have a natural question: is it true that there
exists a commutator $ab-ba\in[D,D]$ such that $F(ab-ba)$ is a maximal subfield
of $D$ (see [5, Problem 28])? Almost similarly, if $K/F$ is a separable
extension of fields in $D$ then there exists an element $d\in
D^{\prime}=[D^{*},D^{*}]$, the group of multiplicative commutators of
$D^{*}=D\backslash\\{0\\}$, such that $K=F(d)$ (see [5, Theorem 2.26]). Again,
the author asked whether $F(xyx^{-1}y^{-1})$ is a maximal subfield of $D$ for
some $x,y\in D^{*}$ (see [5, Problem 29]).
The goal of this paper is to answer in the affirmative for both questions. The
main tools used in this paper are generalized rational identities over a
central simple algebra. Readers can find their definitions and notaions in
detail in [3] and [6].
## 2\. Results
Let $R$ be a ring. Recall that an element $a$ of $R$ is called algebraic of
degree $n$ over a subring $S$ of $R$ if there exists a polynomial $f(x)$ of
degree $n$ over $S$ such that $f(a)=0$ and there is no polynomial of degree
less than $n$ vanishing on $a$. In general, $f(x)$ is not necessary unique and
irreducible even if $S$ is a field. For example, the matrix
$A=\left({\begin{array}[]{*{20}{c}}1&0\\\ 0&2\end{array}}\right)\in M_{2}(F),$
where $F$ is a field, satisfies the polynomial $f(x)=(x-1)(x-2)$. Since
$A\notin F$, $2$ is the smallest degree of all the polynomials vanishing on
$A$.
Recall that a generalized rational expression over $R$ is an expression
contructed from $R$ and a set of noncommutative inderteminates using addition,
substraction, multiplication and division. A generalized rational expression
$f$ over $R$ is called a generalized rational identity if it vanishes on all
permissible substitutions from $R$. In this case, one says $R$ satisfies $f$.
We consider the following example which is important in this paper. Given a
positive integer $n$ and $n+1$ noncommutative indeterminates
$x,y_{1},\cdots,y_{n}$, put
$g_{n}(x,y_{1},y_{2},\cdots,y_{n})=\sum\limits_{\delta\in{S_{n+1}}}{\operatorname{sign}(\delta){x^{\delta(0)}}{y_{1}}{x^{\delta(1)}}{y_{2}}{x^{\delta(2)}}\ldots{y_{n}}{x^{\delta(n)}}},$
where $S_{n+1}$ is the symmetric group of $\\{\,0,1,\cdots,n\,\\}$ and
$\operatorname{sign}(\delta)$ is the sign of permutation $\delta$. This is a
generalized rational expression defined in [3] to connect an algebraic element
of degree $n$ and a polynomial of $n+1$ indeterminates.
###### Lemma 2.1.
Let $F$ be a field and $A$ be a central simple algebra over $F$. For any
element $a\in A$, the following conditions are equivalent.
1. (1)
The element $a$ is algebraic over $F$ of degree less than $n$.
2. (2)
$g_{n}(a,r_{1},r_{2},\cdots,r_{n})=0$ for any $r_{1},r_{2},\cdots,r_{n}\in A$.
Proof. This is a corollary of [3, Corollary 2.3.8].
In particular, a central simple algebra of degree $m$ satisfies the expression
$g_{m}$ since every central simple algebra of degree $m$ over a field $F$ can
be considered as a $F$-subalgebra of the ring $M_{m}(F)$ and elements of
$M_{m}(F)$ are algebraic of degree less than $m$ over $F$. In other words,
$g_{m}$ is a generalized rational indentity of any central simple algebra of
degree $m$.
For any central simple algebra $A$, denote ${\mathcal{G}}(A)$ the set of all
generalized rational identities of $A$. Then ${\mathcal{G}}(A)\neq\emptyset$
because $g_{m}\in{\mathcal{G}}(A)$. The following theorem gives us a relation
between the set of all generalized rational identities of a central simple
algebra and the ring of matrices over a field.
###### Theorem 2.2.
[2, Theorem 11] Let $F$ be an infinite field and $A$ be a central simple
algebra of degree $n$ over $F$. Assume that $L$ is an extension field of $F$.
Then ${\mathcal{G}}(A)={\mathcal{G}}(M_{n}(F))={\mathcal{G}}(M_{n}(L))$.
Now we are going to prove the main results of this paper. The following lemma
is basic.
###### Lemma 2.3.
Let $D$ be a division algebra of degree $n$ over a field $F$. Assume that $K$
is a subfield of $D$ containing $F$. Then $\dim_{F}K\leq n$. The quality holds
if and only if $K$ is a maximal sufield of $D$.
Proof. See [4, Corollary 15.6 and Proposition 15.7]
###### Lemma 2.4.
Let $F$ be an infinite field and $n\geq 2$ be an integer. There exist two
matrices $A,B\in M_{n}(F)$ such that the commutator $ABA^{-1}B^{-1}$ is an
algebraic element of degree $n$ over $F$.
Proof. Put
$A=\left({\begin{array}[]{*{20}{c}}0&0&\cdots&0&{{a_{1}}}\\\
1&0&\cdots&0&{{a_{2}}}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&1&0&{{a_{n-1}}}\\\ 0&0&0&1&0\end{array}}\right)$ and
$B=\left({\begin{array}[]{*{20}{c}}{{b_{1}}}&0&\cdots&0&0\\\
0&{{b_{2}}}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&{{b_{n-1}}}&0\\\ 0&0&0&0&{{b_{n}}}\end{array}}\right),$ where
$a_{i},b_{j}\neq 0$. One has
$ABA^{-1}B^{-1}=\left({\begin{array}[]{*{20}{c}}{{b_{n}}b_{1}^{-1}}&0&\cdots&0&0\\\
&{{b_{1}}b_{2}^{-1}}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
&*&*&{{b_{n-2}}b_{n-1}^{-1}}&0\\\
&*&*&*&{{b_{n-1}}b_{n}^{-1}}\end{array}}\right)$. If we choose
$b_{n}b_{1}^{-1},b_{1}b^{-1}_{2},\cdots,b_{n-1}b^{-1}$ all distinct (it is
possible since $F$ is infinite), then the characteristic polynomial of
$ABA^{-1}B^{-1}$ is a polynomial of smallest degree which vanishes on
$ABA^{-1}B^{-1}$. That is, $ABA^{-1}B^{-1}$ is an algebraic element of degree
$n$ over $F$.
The following theorem answers Problem 29 in [5, Page 83].
###### Theorem 2.5.
Let $D$ be a central division algebra over a field $F$. There exist $x,y\in
D^{*}$ such that $F(xyx^{-1}y^{-1})$ is a maximal subfield of $D$.
Proof. If $F$ is finite then $D$ is also finite, so that there is nothing to
prove. Suppose that $F$ is infinite and $D$ is of degree $n$ over $F$. By
Lemma 2.3, it suffices to show that there exist $x,y\in D^{*}$ such that
$\dim_{F}F(xyx^{-1}y^{-1})\geq n$. Indeed, put
$\ell=\max\\{\,\dim_{F}F(xyx^{-1}y^{-1})\mid x,y\in D^{*}\,\\}.$ Then from
Lemma 2.3,
$g_{\ell}(rsr^{-1}s^{-1},r_{1},r_{2},\cdots,r_{\ell})=0$
for any $r_{1},r_{2},\cdots,r_{\ell}\in D$ and $r,s\in D^{*}$. Hence,
$g_{\ell}(xyx^{-1}y^{-1},y_{1},y_{2},\cdots,y_{\ell})$ is a generalized
rational idenity of $D$, so that, by Lemma 2.2, $g_{\ell}(xy-
yx,y_{1},y_{2},\cdots,y_{\ell})$ is also a generalized rational idenity of
$M_{n}(F)$. Since $g_{\ell}(ABA^{-1}B^{-1},r_{1},r_{2},\cdots,r_{\ell})=0,$
for any $r_{i}\in M_{n}(F)$ and $A,B$ are chosen in Lemma 2.4. Therefore
$n\leq\ell$ because Lemma 2.1 and $AB-BA$ is an algebraic element of degree
$n$.
###### Lemma 2.6.
Let $F$ be an infinite field and $n>2$ be an integer. There exist two matrices
$A,B\in M_{n}(F)$ such that $AB-BA$ is an algebraic element of degree $n$ over
$F$.
Proof. Put
$A=\left({\begin{array}[]{*{20}{c}}0&0&\cdots&0&{{a_{1}}}\\\
1&0&\cdots&0&{{a_{2}}}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&1&0&{{a_{n-1}}}\\\ 0&0&0&1&0\end{array}}\right)$ and
$B=\left({\begin{array}[]{*{20}{c}}0&{{b_{1}}}&0&\cdots&0&0\\\
0&0&{{b_{2}}}&\cdots&0&0\\\ 0&0&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&0&{{b_{n-1}}}\\\
0&0&0&\cdots&0&0\end{array}}\right)$. One has $AB-
BA=\left({\begin{array}[]{*{20}{c}}{{b_{1}}}&*&\cdots&*&*\\\
0&{{b_{1}}-{b_{2}}}&\cdots&*&*\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&{{b_{n-2}}-{b_{n-1}}}&*\\\
0&0&\cdots&0&{{b_{n-1}}}\end{array}}\right)$. Since $F$ is infinite, we can
choose $b_{1},b_{2},\cdots,b_{n-1}\in F$ such that
$b_{1},b_{1}-b_{2},\cdots,b_{n-2}-b_{n-1},b_{n-1}$ all distinct. Hence, the
characteristic polynomial of $AB-BA$ is a polynomial of smallest degree
vanishing on $AB-BA$. Therefore, $AB-BA$ is an algebraic element of degree $n$
over $F$.
Almost similar to the proof of Theorem 2.5, we have the following theorem,
which answers Problem 28 in [5, Page 83].
###### Theorem 2.7.
Let $D$ be a central division algebra over a field $F$. There exist $x,y\in D$
such that $F(xy-yx)$ is a maximal subfield of $D$.
Proof. If $F$ is finite then $D$ is also finite, so that there is nothing to
prove. Suppose that $F$ is infinite and $D$ is of degree $n$. By Lemma 2.3, it
suffices to show that there exist $x,y\in D$ such that $\dim_{F}F(xy-yx)\geq
n$. Indeed, if $n=2$, by [4, Corollary 13.5], then there exist $x,y\in D$ such
that $xy-yx\notin F$, which implies $F(xy-yx)=2=n$. Assume that $n>2$. Then
put $\ell=\max\\{\,\dim_{F}F(xy-yx)\mid x,y\in D\,\\}.$ By Lemma 2.1,
$g_{\ell}(rs-sr,r_{1},r_{2},\cdots,r_{\ell})=0$
for any $r_{1},r_{2},\cdots,r_{\ell}\in D$ and $r,s\in D^{*}$. It follows
$g_{\ell}(xy-yx,y_{1},y_{2},\cdots,y_{\ell})$ is a generalized rational
idenity of $D$. From Lemma 2.2, $g_{\ell}(xy-yx,y_{1},y_{2},\cdots,y_{\ell})$
is also a generalized rational idenity of $M_{n}(F)$. But because there exist
$A,B\in M_{n}(F)$ such that $AB-BA$ is algebraic of degree $n$ (Lemma 2.4),
one has
$g_{\ell}(AB-BA,r_{1},r_{2},\cdots,r_{\ell})=0$
for any $r_{i}\in M_{n}(F)$. Therefore, by Lemma 2.1, $n\leq\ell$.
## References
* [1] S. Akbari, M. Arian-Nejad, M. L. Mehrabadi, On additive commutator groups in division rings, Results Math., 33 (1-2), 9–21, 1998.
* [2] S. A. Amitsur, Rational identities and applications to algebra and geometry, J. Algebra 3, 304–359, 1966.
* [3] K. I. Beidar, W. S. Martindale 3rd and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc., New York- Basel-Hong Kong, 1996.
* [4] T. Y. Lam, A first course in noncommutative rings, MGT 131, Springer, 1991.
* [5] M. Mahdavi-Hezavehi, Commutators in division rings revisited. Bull. Iranian Math. Soc, 26(3): 7–88, 2000.
* [6] L. H. Rowen, Polynomial identities in ring theory, Academic Press, Inc., New York, 1980.
|
arxiv-papers
| 2013-11-29T09:59:06 |
2024-09-04T02:49:54.514344
|
{
"license": "Public Domain",
"authors": "Mai Hoang Bien",
"submitter": "Bien Mai",
"url": "https://arxiv.org/abs/1311.7507"
}
|
1311.7518
|
# Power Penalty Due to First-order PMD in Optical OFDM/QAM and FBMC/OQAM
Transmission System
Jianping Wang, Ke Zhang, Xianyu Du, He Zhen, Jing Yan Department of
communication Engineering,30 Xueyuan Road, Haidian District, Beijing 100083 P.
R.China
###### Abstract
Polarization mode dispersion (PMD) is a challenge for high-data-rate optical-
communication systems. More researches are desirable for impairments that is
induced by PMD in high-speed optical orthogonal frequency division
multiplexing (OFDM) transmission system. In this paper, an approximately
analytical method for evaluating the power penalty due to first-order PMD in
optical OFDM with quadrature amplitude modulation (OFDM/QAM) and filter bank
based multi-carrier with offset quadrature amplitude modulation (FBMC/OQAM)
transmission system is presented. The simulation results show that, compared
with the single carrier with quadrature phase shift keying(SC-QPSK), both the
OFDM/QAM and the FBMC/OQAM can decrease the power penalty caused by PMD by
half. Furthermore, the FBMC/OQAM shows better power penalty immunity than the
OFDM/QAM under the influence of first order PMD.
###### keywords:
polarization mode dispersion , OFDM/QAM , FBMC/OQAM , power penalty
## 1 Introduction
Optical fiber system has become a hot spot because of its ultra high speed,
huge capacity and long haul transmission ability. Nowadays optical signal
amplification and fiber dispersion compensation techniques are increasingly
developed, and PMD has become the key limitation of transmission speed and
distance[1, 2, 3, 4]. PMD is a physical phenomenon caused by the birefringence
of the optical fiber. When transmitting optical signal, PMD will cause
different delays for different polarizations and the group delay difference
between the slow and the fast modes is called differential group delay(DGD).
When DGD is getting larger and can not be neglected compared with the signal
bit duration, it will cause the pulse broadening and Inter-symbol
Interference(ISI)[1], then pulse distortion and system penalties occur.
Different from the fiber chromatic dispersion(CD), PMD is a stochastic
quantity influenced by the external conditions such as temperature or fiber
vibration, et.al, which makes PMD particularly difficult to manage or
compensate. In a specific system, dissipation due to PMD is determined by the
theoretical structure model and physical factors (infrastructure,
environmental factor, et.al), and different systems have different modulation
techniques as well as PMD compensation techniques.
Many researches in recent years are concentrated on PMD tolerance which based
on different modulation modes and coding schemes[5, 6]. In [5], the article
proposed a 20Gb/s high-speed optical fiber transmission systems with Non
Return to Zero (NRZ) and Return to Zero (RZ) code and the differences of the
PMD-induced fiber channel is discussed by numerical simulation. In [6],
compared with on-off keying(OOK), it is demonstrated that differential phase
shift keying(DPSK) signal has large first-order PMD tolerate ability in a
40Gb/s optical fiber transmission system. Other researches are focused on the
PMD compensation techniques[7, 8, 9], and the compensation of system PMD is
demonstrated experimentally by electronical and optical compensators. Adaptive
optical PMD compensation, which based on feedback, courts a balance between
speed and accuracy. Recently, investigations pay more attention to the devised
electronic PMD compensation schemes, such as iterative decoding techniques,
maximum likelihood sequence estimation (MLSE), and transversal digital
filtering[10, 11, 12]. In [10], PMD equalizers based on constant modulus
algorithm (CMA) is presented in coherent optical polarization-division-
multiplexed (PDM) QPSK systems taking account of PMD effect.
Recently, OFDM has been recommended as an effective PMD-resilient modulation
format for high speed optical fiber transmission systems[13, 14, 15]. OFDM
employs multi-carrier transmission of orthogonal, and has lower data rate
subcarriers, therefore it simplifies PMD equalizer structure, and achieves
high spectral efficiency in frequency-selective channels via the fast Fourier
transform(FFT)[16]. In [15], the possibility of PMD compensation in fiber-
optic communication systems with direct detection using a simple channel
estimation technique and low-density parity-check (LDPC)-coded OFDM is
demonstrated. In [16], it has presented that, if it is not necessary for RF
guard bands, OFDM capacitates high speed transmission with PMD tolerance which
is at least twice greater than that of uncompensated OOK at a given bit rate,
while in systems with RF guard bands, a PMD tolerance trade-off proportional
to guard band size was shown, where in guard band and constellation sizes may
be viewed as design parameters. In [17], it has shown that, without requiring
any feedback, OFDM can mitigate pulse distortion caused by all-order PMD in
long-haul optic fiber communication systems. Like the OFDM, FBMC is another
multi-carrier technology with higher spectral efficiency and has been
perceived as an alternative to OFDM in recent years. While compared with OFDM,
FBMC has a larger PMD tolerance because of its large stop-band attenuation and
the frequency selective fading channel. Due to the advanced digital modulation
technique, the FBMC technique is quite fit for the high speed optical fiber
transmission systems. As a result, in optical FBMC communication system, more
research should be done on the impairment caused by the PMD.
In this paper, we focus on the system power penalty due to first-order PMD in
multi-carrier optical transmission system which use OFDM/QAM and FBMC/OQAM
modulation format respectively, by comparing it with single carrier QPSK
modulation, the theoretical model of power penalty in multi-carrier optical
system impacted by first-order PMD is proposed, and then numerical simulation
verification are given. The rest of the paper is organized as follow. In
Section 2, A brief introduction to the basic theory of OFDM/QAM and FBMC/OQAM
modulation has been given. And in Section 3, the PMD principles, as well as
the derivation of power penalty in optical OFDM/QAM and FBMC/OQAM is proposed.
In Section 4, the derivation results verified by the simulation results is
proposed and finally the conclusion is given in Section 5.
## 2 Optical OFDM/QAM and FBMC/OQAM System Model
OFDM/QAM and FBMC/OQAM are kinds of multi-carrier modulation technique that
can modulate and demodulate signals in frequency-domain by Inverse Fast
Fourier Transform/Fast Fourier Transform (IFFT/FFT). OFDM technique can reduce
the effects of dispersion and ISI efficiently and now is considered to be an
effective solution to high speed optical communication in the future. High
speed optical OFDM transmission will also be influenced by PMD effect as it is
in single carrier modulation. Optical OFDM transmission systems have better
anti-PMD effect ability compared with single carrier system because of the
OFDM principles that separate a high speed data stream into several orthogonal
low speed stream.
As we known that OFDM/QAM (i.e. CP-OFDM) has been widely used and considered
as the core technique solution for next generation wireless communication, and
FBMC/OQAM (i.e. OFDM/IOTA) is an alternative approach according to 3GPP
protocols. Compared with traditional OFDM/QAM based on cyclic prefix (CP),
FBMC/OQAM without CP can achieve greater spectral efficiency, furthermore,
FBMC/OQAM has better performance in wireless channel via choosing well time-
frequency localized pulse shaping prototype filters[18]. Applying OFDM
techniques into high-capacity and high-speed optical fiber communication
systems is a major research direction[19, 20], and it will achieve a high
flexibility and capacity in dynamic resource allocation and user access by
combining with new technologies like PON, et.al[21]. The basic principles of
these two OFDM techniques are introduced bellow:
### 2.1 OFDM/QAM System Model
High speed information bit stream with bite rate $R_{b}=1/T_{b}$ is modulated
in baseband using M-QAM modulation with symbol duration
$T_{s}=T_{b}\log_{2}M$, and then divided in to $N$ parallel symbol streams
which are filtered by a pulse shape function $g_{n,k}(t)$, the time-domain
OFDM/QAM signal can be written in the following analytic form[18]
$s_{\textrm{QAM}}(t)=\sum_{k=1}^{+\infty}\sum_{n=1}^{N}a_{n,k}g_{n,k}(t)$ (1)
where
$g_{n,k}(t)=e^{j2\pi nFt}g(t-kT)$ (2)
$F$ denotes the inter-carrier frequency spacing and $T$ is the OFDM symbol
duration. $a_{n,k}$ represents the QAM baseband modulation output data on the
$n$th subcarrier at time index $k$. In a OFDM/QAM system,
$F=1/NT_{s}=\nu_{0}$, $T=\tau_{0}$ and in order to satisfy the orthogonality,
$\tau_{0}\nu_{0}=1$, and the prototype function is defined as
$g(t)=\left\\{\begin{array}[]{lc}1/\sqrt{\tau_{0}},&0\leq t<\tau_{0}\\\
0,&\textrm{elsewhere}\end{array}\right.$ (3)
### 2.2 FBMC/OQAM System Model
Under the same initial conditions as OFDM/QAM, the time domain FBMC/OQAM
signal can be expressed as Eq.(4)[18, 22]
$s_{\textrm{OQAM}}(t)=\sum_{k=1}^{+\infty}\sum_{n=1}^{N}a_{n,k}g_{n,k}(t)$ (4)
where
$g_{n,k}(t)=e^{j2\pi n\nu_{0}t}g(t-k\tau_{0})\times
e^{j(n+k)\pi/2},\>\nu_{0}\tau_{0}=1/2$ (5)
$g(t)$ is the prototype pulse shaping function that can be different from
rectangular window, for example, Extended Gaussian Function (EGF) and
Isotropic Orthogonal Transform Algorithm (IOTA) Function, et.al. Unlike the
original OFDM/QAM, FBMC/OQAM employs a modified inner product by taking a real
component to maintain the orthogonality among the synthesis and analysis
basis, as show in bellow
$\mathcal{R}\left\\{g_{n,k}^{*}\times
g_{n^{\prime},k^{\prime}}\right\\}=\left\\{\begin{array}[]{lc}1,&(n,k)=(n^{\prime},k^{\prime})\\\
0,&(n,k)\neq(n^{\prime},k^{\prime})\end{array}\right.$ (6)
The purpose of pulse shaping in FBMC/OQAM is to find an efficient transmitter
and a corresponding receiver waveform for the current channel condition[23,
24], a well time-frequency localized waveform should satisfy
$\frac{\tau_{0}}{\Delta t}=\frac{\nu_{0}}{\Delta f}$ (7)
where $\Delta t$ and $\Delta f$ is the RMS delay spread and Doppler spread,
respectively.
## 3 Power Penalty duo to First-order PMD
Currently, OFDM is proposed to be a promising modulation technique for high-
speed optical transmission systems, owing to high tolerance to CD and PMD.
However, PMD still degrades the performance of the high-speed optical
transmisson systems, due to lacking of mature compensation techniques.
Subsequent bit error and system power penalty analysis seeks to assess in
order to evaluate the PMD tolerance of high speed fiber OFDM/QAM and FBMC/OQAM
transmission system.
In single mode fiber (SMF) transmission, optical signals are composed by two
orthogonally polarized $HE_{11}$ mode. If the SMF is ideal, the two polarized
mode have the same refractive index and transmitting speed, so there won’t be
any PMD as it shown in Fig. 1. However, in practical fiber, it’s impossible to
achieve identical refractive index, thus there will be a different delay
between the two polarized mode and causes the DGD, as shown in Fig. 1, this
phenomenon is called PMD. In long haul and high speed optical fiber
communication system, pulse broadening caused by PMD effect can lead to
serious ISI, which will degrade the system performance, and that is why PMD
has been considered as a key factor after CD and fiber attenuation.
Figure 1: Time-domain behavior of PMD in a short birefringent fiber. Figure 2:
System model and corresponding lowpass equivalent.
Fig. 2 illustrate the block diagram of basic optical fiber transmission system
and it’s frequency-domain equivalent, the input electronic signal after
electro-optical modulation (EOM) is set as $i_{in}(t)$, the envelope of the
input signal is $\tilde{x}(t)=\alpha i_{in}(t)$ and $\alpha$ is a
proportionality coefficient. The Jones vector of the resulting electronic
field at the fiber input is given by $\tilde{E}_{in}(t)$ as[16]
$\tilde{E}_{in}(t)=\tilde{E}_{in}(t)\hat{e}_{in}=R[\sqrt{\tilde{x}(t)}e^{j\omega_{0}t}]\hat{e}_{in}$
(8)
where $\hat{e}_{in}=[\hat{e}_{1},\hat{e}_{2}]^{T}$ is the polarization state
Jones vector, whose entries denote the two orthogonal Principle State of
Polarizations (PSPs) at the fiber input. Under the First-order PMD
approximation, PMD vector
$\vec{\Omega}=\vec{\Omega}_{0}=\Delta\tau\hat{e}_{1}$, $\hat{e}_{1}$ is the
unit vector in fast PSP. The Fourier transform of fiber output signal
$\hat{E}_{out}(t)$ is given by[16]
$\hat{E}_{out}(\omega)=\textbf{F}(\omega)\hat{E}_{in}(\omega)=\textbf{RU}(\omega)\textbf{R}^{-1}\hat{E}_{in}(\omega)$
(9)
where $\hat{E}_{in}(\omega)=\mathcal{F}(\tilde{E}_{in}(t))$,
$\textbf{F}(\omega)=\textbf{RU}(\omega)\textbf{R}^{-1}$ denotes the fiber
Jones matrix. R denotes the random, frequency-independent rotation matrix and
$\textbf{U}(\omega)$ denotes the time delay matrix caused by first-order PMD
which can be expressed as [25]:
$\textbf{U}(\omega)=\left[\begin{array}[]{cc}e^{j\omega\Delta\tau/2}&0\\\
0&e^{-j\omega\Delta\tau/2}\end{array}\right]$ (10)
where $\Delta\tau$ is the differential group delay (DGD). And the explicit
expression for R is given by[25]
$\textbf{R}=\left[\begin{array}[]{cc}r_{1}&-r_{2}^{*}\\\
r_{2}&r_{1}^{*}\end{array}\right]$ (11)
where
$\begin{split}r_{1}=&\cos{\theta}\cos{\phi}-j\sin{\theta}\sin{\phi}\\\
r_{2}=&\sin{\theta}\cos{\phi}+j\cos{\theta}\sin{\phi}\end{split}$ (12)
where $\theta$,$\phi$ are independent random variables,representing the fast
PSP azimuth and ellipticity angle respectively.
And then we can get[16]
$\hat{E}_{out}(t)=c_{1}E_{in}(t+\Delta\tau/2)\hat{e}_{1}+c_{2}E_{in}(t-\Delta\tau/2)\hat{e}_{2}$
(13)
where $c_{1},c_{2}$ depend on the rotation matrix R, and can be expressed by
$\begin{split}c_{1}=&|\cos{\varphi}|\\\ c_{2}=&|\sin{\varphi}|\end{split}$
(14)
where $2\varphi$ denotes the angle between the fast PSP and the signal
polarization state in Stokes space.After photoelectric detection (PD) and
post-detection filtering, the electronic signal[16]
$\begin{split}i_{out}(t)&=\rho|\hat{E}_{out}(t)|^{2}\\\
&=\rho\left[|c_{1}\sqrt{\tilde{x}(t+\Delta\tau/2)}|^{2}+|c_{2}\sqrt{\tilde{x}(t-\Delta\tau/2)}|^{2}\right]\\\
&=\rho\alpha\left[\gamma
i_{in}(t+\Delta\tau/2)+(1-\gamma)i_{in}(t-\Delta\tau/2)\right]\end{split}$
(15)
where $\rho$ is photoelectric detector sensitivity and $\gamma=|c_{1}|^{2}$ is
the PSP power splitting ratio.
In Eq.(15), it is obvious that first-order PMD causes pulse broadening and
results in adjacent pulses overlap, which will finally cause the Power Penalty
at the receiver. System Power Penalty due to PMD effect is defined as the
difference of receiver sensitivity (in dB) between two conditions, with or
without PMD effect. Considering first-order PMD approximation, the input
optical signal is only divided into two orthogonal polarized mode and causes
one DGD $\Delta\tau$, and leads to power penalty[26].
Under this assumption, we assume that the received signal $i_{out}(t)$ can be
determined by input signal $i_{in}(t)$ after fiber transmission via Eq.(15)
$i_{out}(t)=\gamma i_{in}(t+\Delta\tau/2)+(1-\gamma)i_{in}(t-\Delta\tau/2)$
(16)
The Root-Mean-Square (RMS) pulse width $\delta_{2}$ of the output signal is
given by[26]
$\delta_{2}^{2}=\frac{\int_{-\infty}^{+\infty}t^{2}i_{out}(t)dt}{\int_{-\infty}^{+\infty}i_{out}(t)dt}-[\frac{\int_{-\infty}^{+\infty}ti_{out}(t)dt}{\int_{-\infty}^{+\infty}i_{out}(t)dt}]^{2}$
(17)
Assuming that $i_{in}(t)$ is symmetrical about $t=0$, therefore
$i_{in}(t+\Delta\tau/2)$ and $i_{in}(t-\Delta\tau/2)$ are symmetrical about
$t=0$ too, so Eq.(17) can be expressed as
$\delta_{2}^{2}=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}+\Delta\tau^{2}\gamma(1-\gamma)$
(18)
When $\Delta\tau=0$ (without PMD), RMS pulse width $\delta_{1}$ of the input
signal is given by
$\begin{split}\delta_{1}^{2}&=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}-[\frac{\int_{-\infty}^{+\infty}ti_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}]^{2}\\\
&=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}\end{split}$
(19)
Furthermore, we can observe that[26]
$\delta_{2}^{2}=\delta_{1}^{2}+\Delta\tau^{2}\gamma(1-\gamma)$ (20)
Irrespective of the polarization dependent loss (PDL), power penalty due to
PMD effect $\epsilon$ can be represented as
$\epsilon(dB)=10lg\frac{\delta_{2}}{\delta_{1}}=5lg(1+\frac{\Delta\tau^{2}\gamma(1-\gamma)}{\delta_{1}^{2}})$
(21)
where $\Delta\tau$ is very small in a general way, so
$\epsilon(dB)\approx 5\frac{\Delta\tau^{2}\gamma(1-\gamma)}{\delta_{1}^{2}}$
(22)
For general system, the RMS pulse width $\delta_{1}$ of input signal is
proportional to bit interval $T_{b}$, power penalty $\epsilon$ can be
represented as[26]
$\epsilon=\frac{A}{T_{b}^{2}}\Delta\tau^{2}\gamma(1-\gamma)$ (23)
where $A$ is a coefficient concerned with pulse shape, modulation format and
receiver characters, et.al. $T_{b}$ is bit interval and $\Delta\tau$ is the
instant DGD. The PSP power splitting ratio $0<\gamma<1$. For a fixed DGD,
power penalty is maximized when $\gamma=0.5$, so it should be considered as a
requirement while designing a system to ensure performance. In the follow
simulations, $\gamma$ is fixed to be 0.5.
In original single carrier optical fiber communication systems, power penalty
due to PMD effect $\epsilon$ (in dB) can be seen in Eq.(23) , and when it
comes to the multi-carrier optical fiber transmission, power penalty due to
PMD can be deduced by the following content.
Assuming an OFDM/QAM system with symbol duration $\tau_{0}=T$, sub-carrier
number $N$ and inter-carrier frequency spacing $\nu_{0}=F=1/T$. $\Delta\tau$
represents the DGD caused by first-order PMD. The baseband modulation scheme
is SC-QPSK and the OFDM signal bandwidth $BW=N\times F$, bit rate
$R_{b}=N\times F\times\log_{2}M$ (for QPSK, 4QAM, $M=4$ and for 16QAM, $M=16$,
et.al) and bit duration $T_{b}=1/R_{b}$.
For FBMC/OQAM, inter-carrier frequency spacing $\nu_{0}=F=1/T$ and
$\tau_{0}=T/2$. Firstly, we can get the power penalty $\epsilon_{n}$ in $n$th
sub-carrier via its angular frequency $\omega_{n}$
$\omega_{n}=2\pi\nu_{0}n$ (24)
For OFDM/QAM and FBMC/OQAM, the cycle of the $n$th sub-carrier
$T_{n}=1/\nu_{0}n=T/n$ and put it into Eq.(23)
$\epsilon_{n}=\frac{A}{T_{n}^{2}}\Delta\tau^{2}\gamma(1-\gamma)=\frac{A\gamma(1-\gamma)n^{2}\Delta\tau^{2}}{T^{2}}$
(25)
Assuming that transmitting power of each sub-carrier is $P_{0}$ without PMD
can satisfy the requirement of receiver PMD while the $n$th sub-carrier
transmitting power is $P_{n}$ with first-order PMD to achieve the same
performance. According to the definition of power penalty we can define the
power penalty of the $n$th sub-carrier as
$\epsilon_{n}=10\log\frac{P_{n}}{P_{0}}$ (26)
then the total power penalty of the multi-carrier system is given by
$\epsilon=10\log\frac{\sum_{n=1}^{N}P_{n}}{NP_{0}}$ (27)
put Eq.(26) into Eq.(27)
$\epsilon=10\log\left(\frac{1}{N}\sum_{n=1}^{N}10^{\frac{\epsilon_{n}}{10}}\right)$
(28)
Equation Eq.(27) is the conclusion that the theoretical expression of power
penalty due to first-order PMD in OFDM/QAM and FBMC/OQAM system and in the
next section, numerical simulation will be given to prove its correctness.
## 4 Numerical Simulation
In this section,focused on studying the power penalty due to first-order PMD
in SC-QPSK, OFDM/QAM and FBMC/OQAM system respectively. Specially, in OFDM/QAM
and FBMC/OQAM simulation, we fixed inter-carrier frequency spacing
$\nu_{0}=100$MHz and PSP power splitting ratio $\gamma=0.5$, sub-carrier
number $N=64$ and $N=128$ respectively. The prototype pulse shaping function
in FBMC/OQAM is set to be Square Root Raised Cosine (SRRC) filter with the
length of $L=4N$. Obviously, in order to compare with the multi-carrier
condition on the PMD tolerance problem, we set the SC-QPSK modulation with the
same transmission bit rate as the OFDM/QAM and FBMC/OQAM. For example, SC-QPSK
modulation bit rate $R_{b}=2N\nu_{0}=25.6$Gb/s for $N=128$ and
$R_{b}=2N\nu_{0}=12.8$Gb/s for $N=64$. Other factors (like FEC et.al) remain
unchanged when comparing OFDM/QAM with FBMC/OQAM.
The Bit Error Rate (BER) vs Signal to Noise Ratio (SNR) simulation results of
the OFDM/QAM system with $N=128$ and inter-carrier frequency spacing
$\nu_{0}=100$MHz is shown in Fig. 3, DGD with 0, 0.2, 0.4, 0.8 and 1 times of
the bit duration $T_{b}$. caused by first-order PMD is separately simulated.
It’s clearly shown in the figure that system BER can reach $10^{-9}$ when
$E_{b}/N_{0}$ is about 6.8dB without PMD(DGD$=0$) and with the growth of DGD,
we have to increase the transmitter power to improve the channel SNR in order
to maintain the system BER performance in $10^{-9}$. When
DGD$=0.4\times/2N\nu_{0}=15.6ps$, thus 0.4 times of bit duration, the
$E_{b}/N_{0}$ is 7.7dB at the point that BER is $10^{-9}$, and the power
penalty due to PMD in this situation $\epsilon=7.8-7.1=0.7(dB)$. Comparing all
the simulation results, We can find that the power penalty due to PMD effect
is growing more faster when DGD is getting bigger, which means the signal
distortion is getting more worse.
Figure 3: Simulation BER versus $E_{b}/E_{o}$ results for several
$\Delta\tau/T_{b}$ values under OFDM/QAM. Figure 4: Simulational BER versus
$E_{b}/E_{o}$ results for several $\Delta\tau/T_{b}$ values under FBMC/OQAM.
Figure 5: Simulation BER versus $E_{b}/E_{o}$ results for several
$\Delta\tau/T_{b}$ values under SC-QPSk.
The BER performance of the FBMC/OQAM modulation scheme is show in Fig. 4. The
simulation condition is the same with that of the OFDM/QAM system Without the
influence of PMD effect, the system BER reaches $10^{-9}$ at
$E_{b}/N_{0}=$7.1dB, and when DGD is 0.4 times of bit duration ,$E_{b}/N_{0}$
should be 7.8dB to achieve the same BER performance. The power penalty
$\epsilon=7.8-7.1=0.7$dB at that moment. Same with the OFDM/QAM the power
penalty in the FBMC/OQAM caused by the first-order PMD is growing faster when
DGD is getting bigger and leads to more serious signal distortion.
The difference of PMD tolerance between single carrier and multi carrier
system can be seen from Fig. 5 which demonstrate the BER performance of a
25.6Gb/s SC-QPSK signal with different DGD due to first-order PMD . Same as
above, $E_{b}/N_{0}$ is 7.2dB and 10.1dB when DGD is 0 and $0.4T_{b}$
respectively, and derivatives the power penalty is about 2.9dB. By observing
Fig. 3, Fig. 4 and Fig. 5, we can find that both OFDM/QAM and FBMC/OQAM has
better anti-PMD ability than SC-QPSK.
Figure 6: Power penalty under SC-QPSK,OFDM/QAM and FBMC/OQAM for N=128. Figure
7: Power penalty under SC-QPSK,OFDM/QAM and FBMC/OQAM for N=64.
Fig. 6 presents the power penalty with different normalized mean DGD with the
SC-QPSK, OFDM/QAM and FBMC/OQAM modulation at the same transmission bit rate
of 25.6Gb/s, and the sub-carrier number $N=128$. Correctness can be verified
by the simulation results that the SC-QPSK power penalty matches well with the
theoretical curve which is given by Eq.(23). Theoretical curves of OFDM/QAM
and FBMC/OQAM power penalty due to first-order PMD which are given by our
derivation in Eq.(28) are also shown in this figure , and it’s clear to see
that the simulation results are consistent with theoretical values pretty
well. The coefficient $A$ is set as 68, 64 and 60 in SC-QPSK, OFDM/QAM and
FBMC/OQAM respectively as its value is dependent on multi-factors like
modulation schemes, pulse shaping techniques and receive modes, et.al, that
is, $A$ is a different value for different systems.
When power penalty is 1dB for SC-QPSK, OFDM/QAM and FBMC/OQAM systems,
$\Delta\tau/T_{b}$ is about 0.24, 0.41 and 0.45 in Fig. 6, and OFDM/QAM and
FBMC/OQAM have about twice more PMD tolerance than SC-QPSK from this view
point and show better anti-PMD abilities. FBMC/OQAM shows better performance
than OFDM/QAM at the same time because of its out of band attenuation. For
example, the power penalty of OFDM/QAM and FBMC/OQAM is 0.9dB and 0.7dB
respectively when $\Delta\tau/T_{b}=0.4$, thus the latter one is 0.2dB less
than the former. And when $\Delta\tau/T_{b}=0.6$, the gap between these two
schemes is increased to about 1dB. We can deduce a conclusion that FBMC/OQAM
will achieve better anti-PMD ability with the growth of PMD effect.
Fig. 7 presents the same situation with sub-carrier number $N=64$, and the
same conclusion can be deduced from this figure compared with Fig. 6.
comparing these two figure we can find that the difference between OFDM/QAM
and FBMC/OQAM is turning smaller with a lower number of sub carrier which give
us the conclusion that FBMC/OQAM shows better anti-PMD performance in power
penalty than OFDM/QAM with larger sub-carrier number $N$.
## 5 Conclusion
In this paper, we discussed the System Power Penalty due to first-order PMD
effect in multi-carrier optical communication system, especially two
modulation schemes OFDM/QAM and FBMC/OQAM. Theoretical derivation of the
multi-carrier condition were given by comparing with original single carrier
situation at the first, and then confirmed its validity via the numerical
simulation. Through the simulation results we can find that the power penalty
due to first-order PMD in OFDM/QAM and FBMC/OQAM systems is about a half
smaller than that of the single carrier SC-QPSK system at the same
transmitting bit rate, and with the growth of sub-carrier number and bit rate,
the latter one can achieve better PMD resistance ability than the the former.
## Acknowledgments
This research is supported by the Fundamental Research Funds for the Central
Universities (No.FRF-TP-09-015A), and also supported by the National Natural
Science Foundation of P.R.China (No.61272507).
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|
arxiv-papers
| 2013-11-29T10:56:10 |
2024-09-04T02:49:54.520455
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianping Wang and Ke Zhang and Xianyu Du and He Zhen and Jing Yan",
"submitter": "Ke Zhang",
"url": "https://arxiv.org/abs/1311.7518"
}
|
1311.7585
|
Patrick Spradlin
on behalf of the LHCb collaboration
School of Physics and Astronomy
University of Glasgow, Glasgow, UK
> The LHCb experiment has fully reconstructed close to $10^{9}$ charm hadron
> decays—by far the world’s largest sample. During the 2011-2012 running
> periods, the effective $pp$ beam crossing rate was 11-15${\rm\,MHz}$ while
> the rate at which events were written to permanent storage was
> 3-5${\rm\,kHz}$. Prompt charm candidates (produced at the primary
> interaction vertex) were selected using a combination of exclusive and
> inclusive high level (software) triggers in conjunction with low level
> hardware triggers. The efficiencies, background rates, and possible biases
> of the triggers as they were implemented will be discussed, along with plans
> for the running at 13$\mathrm{\,Te\kern-1.00006ptV}$ in 2015 and
> subsequently in the upgrade era.
> PRESENTED AT
>
>
>
>
> The 6th International Workshop on Charm Physics
> (CHARM 2013)
> Manchester, UK, 31 August – 4 September, 2013
## 1 Introduction
The LHCb experiment has rapidly become one of the foremost high-precision
flavor physics experiments, collecting the world’s largest samples of several
decay modes of $c$ and $b$-hadrons (e.g.[1, 2]). This success would have been
impossible without LHCb’s flexible and efficient trigger system. The task of
rapidly selecting which events will be stored permanently for subsequent
analysis and which will be discarded forever—triggering—presents a formidable
challenge in the high-energy hadronic collision environment of the Large
Hadron Collider (LHC). In 2012 the LHCb detector witnessed $pp$ collisions
with a center-of-mass energy of $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ at a
mean instantaneous luminosity of approximately $4\times
10^{32}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$. Given that the heavy flavor hadron
production cross-sections into the LHCb acceptance were measured to be
$\sigma_{b\overline{}b,\mathrm{acc}}=75.3\pm 14.1\rm\,\upmu b$ [3] and
$\sigma_{c\overline{}c,\mathrm{acc}}=1419\pm 134\rm\,\upmu b$ [4] for $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, the rate of heavy
flavor production into the LHCb acceptance exceeded 30${\rm\,kHz}$ for
$b$-hadrons and 600${\rm\,kHz}$ for $c$-hadrons. Because events are written to
permanent storage at just 3-5${\rm\,kHz}$, the trigger must be highly
selective even among events with a real heavy-flavor hadron. This article
discusses the structure and performance of the trigger components for
selecting events that contain open charm hadrons—the first and fundamental
building block for most precision charm measurements at LHCb. We also sketch
prospective improvements to the trigger that will extend our physics reach
when the LHC returns to operation after its first long shutdown period (LS1)
and in the era of the upgraded LHCb detector.
## 2 Detector
The LHCb detector [5] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. Charm hadron triggering uses information from
each of the detector subsystems. The detector includes a high-precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with large transverse momentum.
Different types of charged hadrons are distinguished by information from two
ring-imaging Cherenkov detectors [6]. Photon, electron, and hadron candidates
are identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [7].
## 3 Trigger overview
Figure 1: A diagrammatic overview of the trigger structure.
Although the global structure of the LHCb trigger system—a hardware trigger
system followed by a full detector readout and one or more layers of software
triggers—has remained unchanged since its initial design [8], the
implementation continues to evolve. The trigger system as it performed in 2011
is described in detail in Ref. [9], but the interval between 2011 and 2012 saw
the introduction of a major new feature, HLT deferral (Sec. 4). The steady
evolution of the trigger has led to and has been encouraged by an expansion of
LHCb’s physics program. Relative to the initial design, the 2012 LHCb trigger
processed twice the instantaneous luminosity of events ($4\times
10^{32}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$ vs. $2\times
10^{32}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$) with a much greater complexity (a
mean of $1.6$ visible $pp$ interactions per visible bunch crossing vs. $0.4$)
and recorded events to permanent storage at over twice the rate (5${\rm\,kHz}$
vs. 2${\rm\,kHz}$). As a consequence, LHCb is making an impact in areas far
outside its initial core physics program [10, 11], particularly in the realm
of charm physics. Though charm physics measurements were previously absent
from LHCb’s primary goals, approximately $40\%$ of the trigger output is now
dedicated to them. Figure 1 outlines the structure of the trigger system for
2012 data collection. The chain begins with a bunch crossing in which a bunch
of protons from each of the counter-rotating beams of the LHC meet at the LHCb
interaction point. The separation between successive potential sites for
bunches of protons in the beams of the LHC is $25{\rm\,ns}$, thus bunch
crossings may occur at a maximum rate of 40${\rm\,MHz}$ [12]. In much of 2012
the actual bunch crossing rate at the LHCb interaction point was
11-15${\rm\,MHz}$. The first layer of triggering happens in bespoke hardware.
Since the maximum rate at which the full detector response can be digitized
and read out is 1${\rm\,MHz}$, the purpose of this level-0 trigger system (L0)
is to select just 1${\rm\,MHz}$ of potentially interesting events from the
11-15${\rm\,MHz}$ of bunch crossings. L0 analyzes the response of selected
subdetectors to evaluate measures of event complexity and to identify
signatures of particles with large momentum components transverse to the $pp$
collision axis ($p_{\rm T}$). It contains a number of independent parallel
configurable channels that are tuned to balance the requirements of the
physics program and the readout constraint. If any one of the channels returns
a positive decision, the full detector response is digitized, read out, and
recorded to the temporary storage of the Event Filter Farm (EFF), a large farm
of multiprocessor computers, until the trigger processing is complete and a
final decision made on the fate of the event. Most of the events accepted by
L0 and transferred to the EFF are processed immediately by the subsequent and
final triggering layer, the High Level Trigger (HLT). For 20% of the events
the HLT processing is deferred until the interfill period (see Sec. 4). HLT is
implemented in software that runs on the EFF. Due to limitations of computing
resources available for permanent storage and data analysis, the rate at which
events are accepted for permanent storage is restricted to 5${\rm\,kHz}$.
Internally, HLT is segmented into two sequential stages of processing, HLT1
and HLT2. Each stage is composed of several independent parallel channels
(lines) that are sequences of event reconstruction algorithms and selection
criteria. Each line executes its sequence of elements either until the
decision of the line is known to be negative, e.g., by the failure of a
reconstruction element or selection criterion, or until the sequence is
complete and the event accepted by the line. The lines of HLT2 are executed
only for events that are accepted by at least one of the lines of HLT1. Events
accepted by at least one HLT2 line are preserved in permanent storage. The
lines of HLT1 are simple selections based on the properties of one or two
reconstructed tracks. The lines of HLT2 can be quite sophisticated,
incorporating complicated reconstruction elements and multivariate
discriminants, and are generally tailored to the requirements of a group of
physics analyses. The lines of HLT2 are generally better suited to the needs
of LHCb measurements than those of HLT1. However, they also require
substantial computing resources. The EFF has the computing power to execute
the lines of HLT2 on only a fraction of the L0-accepted events. Thus the two-
stage structure of HLT is a compromise, with HLT1 rapidly selecting a subset
of the L0-accepted events to be further analyzed by HLT2.
## 4 HLT deferral
The trigger system in 2012 featured a new facility for deferring HLT
processing for a fraction of the events accepted by L0. This represents a
significant improvement in the efficiency with which the EFF is used. Prior to
the implementation of HLT deferral, all events were processed immediately
after they were transferred to the EFF. In normal operation, the beams of the
LHC are dumped when their intensity decays below some threshold. New beams
with renewed intensity are then injected and accelerated to the target energy
before collisions resume. This interfill period in which no recordable
collisions occur can take a few hours during which the EFF would remain
largely idle. With the HLT deferral system, most events are processed
immediately, as before, but a configurable fraction of the incoming events are
cached in EFF storage instead of processed. During the interfill period, HLT
processes these cached events. The net result is a more efficient use of the
EFF that effectively increased the available computing power by approximately
20% in 2012.
## 5 Performance measures
We measure the performance of trigger lines in data with the method described
in Ref. [9]. The data sets for the measurements are collections of ‘offline’
candidate decays that have been reconstructed by LHCb’s analysis software from
the collected events. We require that at least one channel at each level of
the trigger accepted each event independently of the offline candidate in
order to mitigate biases due to the de facto triggering of the events. In
order to measure the efficiency with which these offline candidate decays
satisfy the criteria of a trigger line under investigation, we must compare
the underlying information from the detector that was used in reconstructing
the offline candidate to that used in the decision of the trigger line. This
is done by a direct comparison of the set of detector elements—the strips,
straws, cells, and pads of the sub-detectors—that contributed to each. As most
HLT1 and HLT2 lines are based on sets of reconstructed tracks, this is
effectively a comparison of the set of tracks constituting the offline
candidate decay and the set of tracks used by the line. We classify an offline
candidate as Triggered On Signal ($\mathrm{TOS}$) for a given trigger line if
the set of detector elements that was used in its reconstruction is sufficient
to satisfy the selection criteria of that line. An offline candidate is
classified as Triggered Independently of Signal ($\mathrm{TIS}$) for a given
trigger line if the set of detector elements that was used in its
reconstruction is disjoint with at least one of the combinations of elements
that led to a positive decision by that trigger line, that is, if the rest of
the event excluding the offline signal candidate was sufficient to satisfy the
criteria of that line. These are not mutually exclusive classifications. A
given offline candidate decay can be both $\mathrm{TOS}$ and $\mathrm{TIS}$
with respect to a given trigger line as there may be multiple sets of detector
elements whose response led to a positive decision for the line. The offline
candidate decays of the data sets for trigger performance are $\mathrm{TIS}$
with respect to at least one physics line at each level of the trigger. The
candidates of these data sets are largely unbiased by the trigger line under
investigation. A subset of these candidates will also be $\mathrm{TOS}$ with
respect to the target line. After determining the number of signal decays in
the set of $\mathrm{TIS}$ candidates ($N^{\mathrm{TIS}}$) and of its
$\mathrm{TOS}$ subset ($N^{\mathrm{TOS}\wedge\mathrm{TIS}}$), we define our
measure of the performance of a line as its $\mathrm{TOS}$ efficiency,
$\epsilon^{\mathrm{TOS}}=N^{\mathrm{TOS}\wedge\mathrm{TIS}}/N^{\mathrm{TIS}}$.
The $\mathrm{TOS}$ efficiency defined in this way should be considered a
relative measure of performance rather than an absolute efficiency. It is
sensitive to the criteria with which the set of offline candidate decays were
selected. Further, the $\mathrm{TIS}$ classification includes some bias due to
the pairwise production mechanisms of heavy hadrons. Despite these
limitations, $\epsilon^{\mathrm{TOS}}$ is an excellent measure of the relative
performance of a trigger line. In Sections 6 to 8 we will show
$\epsilon^{\mathrm{TOS}}$ for offline reconstructed decays of three charmed
hadrons to final states involving kaons and pions: $D^{0}\\!\rightarrow
K^{-}\pi^{+}$, $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$, and
$D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$. The
corresponding charge-conjugate decays are implied here and throughout the
remainder of this article. These modes were selected due to their large
abundance and in order to show the dependence of trigger efficiencies on the
multiplicity of the final state. Rare open charm hadron decays to final states
with two muons are expected to have a significantly better performance,
comparable to that of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays
(see Ref. [9]). However, their $\epsilon^{\mathrm{TOS}}$ performance cannot be
evaluated until sufficiently large samples are available. All of the plots and
performance estimates in the following sections are based on data collected by
LHCb in 2012 in $pp$ collisions at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$.
## 6 L0 performance
Figure 2: The efficiencies of L0Hadron for various reconstructed decay modes as functions of $p_{\rm T}$ of the signal $B$ and $D$ candidate based on $\sqrt{s}=8\mathrm{\,Te\kern-0.90005ptV}$ data collected in 2012. Table 1: Mean $\epsilon^{\mathrm{TOS}}$ efficiencies of L0Hadron for selected charm hadron decays. Decay mode | Mean $\epsilon^{\mathrm{TOS}}$
---|---
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ | $0.26894$ ± | $0.00069$
$D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ | $0.15766$ ± | $0.00016$
$D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ | $0.22045$ ± | $0.00043$
The L0 hardware trigger system is described more completely in Refs. [5, 9].
The decisions of the parallel channels are based on comparisons of a small
number of estimated quantities to specified configurable thresholds. The
primary physics quantities are the estimated transverse momenta for track
segments in the muon system and estimated transverse energy ($E_{\rm
T}$)***For a calorimeter cell centered at polar coordinates
$\vec{x}=(r,\theta,\phi)$ in the LHCb coordinate system in which the origin is
at the center of the $pp$ interaction envelope and the $z$-axis is the
laboratory-frame collision axis, a measured deposited energy of $E$
corresponds to $\mbox{$E_{\rm T}$}=E\sin{\theta}$. for clusters in the
calorimeter system. The overall activity in the scintillating-pad detector
enters many L0 channels as a proxy measure of event complexity. The primary
channel of interest for hadronic decays of charmed hadrons is the single-
cluster hadron line L0Hadron. It accepts events that have a scintillating-pad
detector activity below a certain threshold and that contain at least one
cluster in the hadron calorimeter that has a total transverse energy in all
calorimeters of $\mbox{$E_{\rm T}$}>3.5\mathrm{\,Ge\kern-1.00006ptV}$. In
2012, approximately 45% of the events accepted by L0 were accepted by
L0Hadron. Figure 2 shows $\epsilon^{\mathrm{TOS}}$ of L0Hadron as a function
of signal hadron $p_{\rm T}$ for $D^{0}\\!\rightarrow K^{-}\pi^{+}$,
$D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$, and
$D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ decays. It also
shows $\epsilon^{\mathrm{TOS}}$ of L0Hadron for two hadronic $B$ decay modes,
$B^{+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+})$ and $B^{0}\\!\rightarrow
K^{+}\pi^{-}$. The efficiencies of L0Hadron are strongly dependent on the
$p_{\rm T}$ of the signal hadron. Charm hadrons are predominantly produced in
the region of low efficiency [4], thus the mean efficiency for the set of
offline candidate decays is correspondingly low, as shown in Table 1. One of
the important ways in which the redesigned trigger for the upgraded LHCb
detector will benefit LHCb’s charm physics program by removing the limitations
of the L0 system (see Section 9.2).
## 7 HLT1 performance
HLT1, the initial stage of the HLT software trigger, is composed of parallel
independent lines—sequences of processing steps that include reconstruction
elements and selection criteria. The decisions of the L0 channels are
available to HLT1 lines, so the trigger history of an event can enter the
decision-making process of a line. Although the lines of HLT1 are independent,
most lines begin with a fast reconstruction of $pp$ primary interaction
vertexes (PVs) and charged particle tracks that is common to all lines that
use it. The details of this fast reconstruction are fully described in Ref.
[9]. Most HLT1 lines are simple selections based on the properties of one or
two of these reconstructed tracks. The single displaced-track line
Hlt1TrackAllL0, which is the primary HLT1 line of interest for charmed hadron
decays to hadronic final states, is of this type. Hlt1TrackAllL0 accepts
events that were accepted by any L0 channel and that have at least one track
that satisfies a number of track quality criteria (see Ref. [9]), that is
displaced from every reconstructed PV in the event (impact parameter with
respect to each PV $>0.1\rm\,mm$), and that has a relatively large estimated
$p_{\rm T}$ ($\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$).
Such tracks are typically produced by the decay products of $c$ and
$b$-hadrons and are excellent signatures of long-lived heavy hadrons.
LABEL:sub@fig:hlt1:etos:pt
(a)
LABEL:sub@fig:hlt1:etos:tau
(b)
Figure 3: The efficiency Hlt1TrackAllL0 for various reconstructed decay modes
as functions of LABEL:sub@fig:hlt1:etos:pt $p_{\rm T}$ and
LABEL:sub@fig:hlt1:etos:tau $\tau$ of the signal $B$ or $D$ candidate based on
$\sqrt{s}=8\mathrm{\,Te\kern-0.90005ptV}$ data collected in 2012. For the
decay mode $D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$,
$\tau$ is the measured decay time of the $D^{0}$ candidate.
We evaluate the performance of Hlt1TrackAllL0 relative to the output of L0
with a set of offline candidate decays that are from events accepted by L0 and
that are $\mathrm{TIS}$ with respect at least one of the HLT1 lines for
physics analyses. Figure 3 shows $\epsilon^{\mathrm{TOS}}$ of Hlt1TrackAllL0
as functions of $p_{\rm T}$ of the signal candidate and of measured decay
time, $\tau$, of the signal $D^{0}$ or $D^{+}$ candidate for
$D^{0}\\!\rightarrow K^{-}\pi^{+}$, $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$,
and $D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ decays. It
also shows $\epsilon^{\mathrm{TOS}}$ of Hlt1TrackAllL0 for two hadronic $B$
decay modes, $B^{+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+})$ and
$B^{0}\\!\rightarrow K^{+}\pi^{-}$. The mean efficiencies for the L0-accepted
HLT1-$\mathrm{TIS}$ offline candidate decays appear in Table 2.
Table 2: Mean $\epsilon^{\mathrm{TOS}}$ efficiencies of Hlt1TrackAllL0 relative to L0-accepted events for selected charm hadron decays. Decay mode | Mean $\epsilon^{\mathrm{TOS}}$
---|---
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ | $0.66853$ ± | $0.00054$
$D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ | $0.58580$ ± | $0.00014$
$D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ | $0.60802$ ± | $0.00038$
## 8 HLT2 performance
Like HLT1, HLT2 is composed of several independent parallel lines, each of
which is executed on each event accepted by at least one of the lines of HLT1.
The decisions of each of the L0 channels and HLT1 lines are available to HLT2
and can enter the decision making of a line. Also like HLT1, most of the lines
of HLT2 begin with a common reconstruction of PVs and charged particle tracks.
This reconstruction is more sophisticated, complete, and precise than that
used by HLT1 lines, but it also takes more computing power per event.
Reference [9] describes the HLT2 reconstruction for data collection in 2011.
Several improvements were made to the 2012 HLT2 reconstruction, chief among
them a reduction of the minimum $p_{\rm T}$ for reconstructed charged tracks
from $500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ to
$300{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The HLT deferral system provided
the additional computational power necessary for this more complete track
reconstruction.
### 8.1 Exclusive charm hadron lines
HLT2 lines are generally tailored to the needs of groups of analyses. Because
the precision and efficiency of HLT2’s track reconstruction approach those of
LHCb’s analysis software, HLT2 lines can use the same methods and selection
variables for fully reconstructing signal decays, with the exception of the
charged hadron identification. The algorithms for the charged hadron
identification require significant computational power and are executed only
for a small number of HLT2 lines on a relatively small number of events after
extensive filtering. Among the lines for charm hadron physics, only the lines
for $\mathchar 28931\relax_{c}^{+}$ decays used the charged hadron
identification. The mass distributions of Figure 4 demonstrate the purity with
which charm hadron decays are reconstructed by their HLT2 lines. We evaluate
the performance of these lines relative to the output of HLT1 with sets of
offline candidate decays that are from events that are TOS with respect to one
of the HLT1 lines for physics and that are $\mathrm{TIS}$ with respect at
least one of the HLT2 lines for physics. Figure 5 shows
$\epsilon^{\mathrm{TOS}}$ of the HLT2 lines as functions of $p_{\rm T}$ of the
signal candidate and of measured decay time, $\tau$, of the signal $D^{0}$ or
$D^{+}$ candidate. The mean efficiencies for the L0-accepted
HLT1-$\mathrm{TOS}$ HLT2-TIS offline candidate decays appear in Table 3.
LABEL:sub@fig:hlt2:mass2:Kpi
(a)
LABEL:sub@fig:hlt2:mass3:3h
(b)
LABEL:sub@fig:hlt2:mass3:4h
(c)
Figure 4: Mass distributions of reconstructed $D$ meson decay candidates in HLT2: LABEL:sub@fig:hlt2:mass2:Kpi $D^{0}\\!\rightarrow K^{-}\pi^{+}$ candidates reconstructed in the line Hlt2CharmHadD02HH_D02KPi, LABEL:sub@fig:hlt2:mass3:3h $D^{+}_{(s)}\\!\rightarrow h^{-}{h^{\prime}}^{+}{h^{\prime\prime}}^{+}$ candidates, where $h,h^{\prime},h^{\prime\prime}\in\left\\{K,\pi\right\\}$, reconstructed in the line Hlt2CharmHadD2HHH, and LABEL:sub@fig:hlt2:mass3:4h $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ candidates from the $D^{*+}\\!\rightarrow\pi^{+}D^{0}$ candidates reconstructed in the line Hlt2CharmHadD02HHHHDst_K3pi. Table 3: Mean $\epsilon^{\mathrm{TOS}}$ efficiencies of HLT2 lines relative to HLT1-TOS events for selected charm hadron decays. Decay mode | HLT2 line | Mean $\epsilon^{\mathrm{TOS}}$
---|---|---
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ | Hlt2CharmHadD02HH_D02KPi | $0.9069$ ± | $0.0015$
$D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ | Hlt2CharmHadD2HHH | $0.6588$ ± | $0.0005$
$D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ | Hlt2CharmHadD02HHHHDst_K3pi | $0.1989$ ± | $0.0004$
| Hlt2CharmHadD02HHXDst_hhX | $0.1712$ ± | $0.0005$
| Hlt2CharmHadD02HHHHDst_K3pi |
| or Hlt2CharmHadD02HHXDst_hhX | $0.2556$ ± | $0.0005$
LABEL:sub@fig:hlt2:etos:pt
(a)
LABEL:sub@fig:hlt2:etos:tau
(b)
Figure 5: The efficiency of various HLT2 lines for appropriate reconstructed
decay modes as functions of LABEL:sub@fig:hlt2:etos:pt $p_{\rm T}$ and
LABEL:sub@fig:hlt2:etos:tau $\tau$ of the signal $D$ candidate based on
$\sqrt{s}=8\mathrm{\,Te\kern-0.90005ptV}$ data collected in 2012. For the
decay mode $D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$,
$\tau$ is the measured decay time of the $D^{0}$ candidate.
### 8.2 Inclusive $D^{*+}$ line
Figure 6: Mass difference distribution of reconstructed candidates for the
HLT2 inclusive $D^{*+}$ line Hlt2CharmHadD02HHXDst_hhX.
Although highly successful, HLT2 lines for exclusive reconstruction of decay
modes are necessarily limited. Inclusive selections that do not depend on a
complete reconstruction of signal decays can allow for efficient selection of
a broader range of decay modes, including modes for which a full
reconstruction is impossible. The inclusive $D^{*+}$ HLT2 line is a first
example of inclusive triggering for charm hadrons. The inclusive $D^{*+}$
line,
Hlt2CharmHadD02HHXDst_hhX, selects decays of
$D^{*+}\\!\rightarrow\pi^{+}D^{0}$, where $D^{0}$ decays into at least two
charged final state particles. Partial $D^{0}$ decay candidates are
reconstructed as two-track vertexes that are significantly displaced from all
PVs. These two-track vertexes are combined with $\pi^{+}$ candidates to form
$D^{*+}$ candidates, and additional basic kinematic and reconstruction quality
criteria are applied to the system. For true $D^{*+}$ decays, the mass
difference between the reconstructed $D^{*+}$ and $D^{0}$ candidates peaks
strongly at the true value, even when the $D^{0}$ decays are not fully
reconstructed. Thus $D^{*+}$ decays can be successfully identified for a wide
array of $D^{0}$ decay modes. The method has also been applied to $\mathchar
28934\relax_{c}^{0(++)}\\!\rightarrow\mathchar 28931\relax_{c}^{+}\pi^{-(+)}$
decay modes with partially reconstructed $\mathchar 28931\relax_{c}^{+}$
decays in additional HLT2 lines. Figure 6 shows the prominent signal component
in the $D^{*+}$-$D^{0}$ candidate mass differences for the $D^{*+}$ candidates
selected by the inclusive $D^{*+}$ HLT2 line. Figure 5 includes a comparison
of the performance of the inclusive $D^{*+}$ line with that of the exclusive
line for $D^{*+}\\!\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+}\pi^{+}\pi^{-})$
decays. The inclusive line has a comparable efficiency and, furthermore,
selects a complementary set of decays as can be seen in the efficiencies of
Table 3. Approximately 33% of the signal decays selected by the inclusive line
were not selected by the exclusive line. Most of these are decays for which
one of the final state particles has $\mbox{$p_{\rm
T}$}<300{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, the lower limit for the track
reconstruction in the exclusive lines.
## 9 Future developments
### 9.1 Post-LS1 LHCb triggering
In 2015 LHCb will resume data collection after LHC’s LS1 at the greater $pp$
collision energy of $\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$. The L0
hardware trigger will be tuned to satisfy its $1{\rm\,MHz}$ output limit under
the new conditions, but its operation will remain unchanged. The HLT software
trigger will be substantially reorganized in order to improve the quality of
the event reconstruction in HLT2. The internal structures of HLT1 and HLT2
will remain largely unchanged, with the possible addition of lines to expand
LHCb’s physics program. However, an additional calibration step will be
inserted between HLT1 and HLT2. In 2010-2012, the calibration and the fine
alignment of detector elements that was used by the HLT2 reconstruction were
measured in an earlier data-taking period. Since the calibration and alignment
for analysis is always up-to-date, there may be small differences between the
measured parameters of identical candidates as reconstructed in HLT2 and as
reconstructed for analysis. This can be a source of irreducible systematic
uncertainty. By performing the calibration and alignment step before the
execution of HLT2, this source of uncertainty is reduced or eliminated. HLT1
will run immediately on all L0-accepted events. The events accepted by HLT1
will be cached on the storage of the EFF by a system similar to the HLT
deferral until an update of the a detector alignment and calibration is
complete. Then HLT2 will process the cached HLT1-accepted events and render
the final trigger decisions.
### 9.2 Triggering in an upgraded LHCb detector
Following the conclusion of LHC Run II, the LHCb experiment will be upgraded
for a higher rate of data collection [13, 14]. The upgraded experiment will
feature a substantially improved trigger. Inefficiency in the L0 trigger is
one of the main limitations of the current system for $b$ and $c$-hadron
decays to hadronic final states. This inefficiency is necessitated by
$1{\rm\,MHz}$ maximum readout rate for the detector electronics. The upgraded
LHCb detector will be capable of a full detector readout at $40{\rm\,MHz}$,
largely obviating the need for L0. L0 will be upgraded to a Low Level Trigger
that will function as a pass-through during normal operation. All trigger
decisions will be made by the more flexible and efficient HLT, which will
evolve to process the higher input rate. The rate at which events are accepted
by the trigger for permanent storage will increase from the current
$5{\rm\,kHz}$ to an estimated $20{\rm\,kHz}$. The combination of a more
efficient software trigger and the increased rate of data collection is
estimated to increase the annual yield of many charm decay modes by an order
of magnitude.
## 10 Summary
The current performance of the LHCb charm triggering, as documented in this
article, is the product of steady iterative improvement made with the goal of
expanding the scope and impact of LHCb’s physics program. Development of the
trigger system continues, with further important enhancements anticipated for
LHC Run II and for the subsequent upgrade of the LHCb experiment. The LHCb
trigger will continue to deliver world-class charm data sets for many years.
## References
* [1] LHCb collaboration, R. Aaij et al., Measurement of $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing parameters and search for $C\\!P$ violation using $D^{0}\rightarrow K^{+}\pi^{-}$ decays, arXiv:1309.6534, submitted to Phys. Rev. Lett.
* [2] LHCb collaboration, R. Aaij et al., Measurement of form factor independent observables in the decay $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$, arXiv:1308.1707, to appear in Phys. Rev. Lett.
* [3] LHCb collaboration, R. Aaij et al., Measurement of $\sigma(pp\rightarrow b\overline{}bX)$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ in the forward region, Phys. Lett. B694 (2010) 209, arXiv:1009.2731
* [4] LHCb collaboration, R. Aaij et al., Prompt charm production in $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Nucl. Phys. B871 (2013) 1, arXiv:1302.2864
* [5] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [6] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [7] A. A. Alves Jr. et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
* [8] LHCb collaboration, R. Antunes-Nobrega et al., LHCb trigger system : Technical design report, CERN-LHCC-2003-031 (2003), LHCb TDR 10
* [9] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
* [10] LHCb collaboration, B. Adeva et al., Roadmap for selected key measurements of LHCb, arXiv:0912.4179
* [11] LHCb collaboration, R. Aaij, _et al._ , and A. Bharucha et al., Implications of LHCb measurements and future prospects, Eur. Phys. J. C73 (2013) 2373, arXiv:1208.3355
* [12] L. Evans and P. Bryant, LHC Machine, JINST 3 (2008) S08001
* [13] LHCb collaboration, R. Aaij et al., Letter of Intent for the LHCb Upgrade, CERN-LHCC-2011-001, LHCC-I-018 (2011)
* [14] LHCb collaboration, I. Bediaga et al., Framework TDR for the LHCb Upgrade: Technical Design Report, CERN-LHCC-2012-007, LHCb-TDR-12 (2012)
|
arxiv-papers
| 2013-11-29T14:57:49 |
2024-09-04T02:49:54.529653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Patrick Spradlin (on behalf of the LHCb collaboration)",
"submitter": "Patrick Spradlin",
"url": "https://arxiv.org/abs/1311.7585"
}
|
1311.7636
|
# A simplified discharging proof of Grötzsch theorem
Zdeněk Dvořák Computer Science Institute, Charles University, Prague, Czech
Republic. E-mail: [email protected].
###### Abstract
In this note, we combine ideas of several previous proofs in order to obtain a
quite short proof of Grötzsch theorem.
Grötzsch [2] proved that every planar triangle-free graph is $3$-colorable,
using the discharging method. This proof was simplified by Thomassen [3] (who
also gave a principally different proof [4]). Dvořák et al. [1] give another
variation of the discharging proof. Both of the later arguments were developed
in order to obtain more general results (the Thomassen’s proof gives
extensions to girth $5$ graphs in the torus and the projective plane, while
the proof of Dvořák et al. aims at algorithmic applications), and thus their
presentation of the proof of Grötzsch theorem is not the simplest possible. In
this note, we provide a streamlined version of the proof, suitable for
teaching purposes.
We use the discharging method. Thus, we consider a hypothetical minimal
counterexample to Grötzsch theorem (or more precisely, its generalization
chosen so that we are able to deal with short separating cycles) and show that
it does not contain any of several “reducible” configurations. Then, we assign
charge to vertices and edges so that the total sum of charges is negative, and
redistribute the charge (under the assumption that no reducible configuration
appears in the graph) so that the final charge of each vertex and face is non-
negative. This gives a contradiction, showing that there exists no
counterexample to Grötzsch theorem.
A $3$-coloring $\varphi$ of a cycle $C$ of length at most $6$ is _valid_ if
either $|C|\leq 5$, or $|C|=6$ and there exist two opposite vertices $u,v\in
V(C)$ (i.e., both paths in $C$ between $u$ and $v$ have length three) such
that $\varphi(u)\neq\varphi(v)$. If $G$ is a plane triangle-free graph whose
outer face is bounded by an induced cycle $C$ of length at most $6$ and
$\varphi$ is a valid coloring of $C$, then we say that the pair $(G,\varphi)$
is _valid_. We define a partial ordering $<$ on valid pairs as follows. We
have $(G_{1},\varphi_{1})<(G_{2},\varphi_{2})$ if either
$|V(G_{1})|<|V(G_{2})|$, or $|V(G_{1})|=|V(G_{2})|$ and
$|E(G_{1})|>|E(G_{2})|$. A valid pair $(G,\varphi)$ is a _minimal
counterexample_ if $\varphi$ does not extend to a $3$-coloring of $G$, but for
every valid pair $(G^{\prime},\varphi^{\prime})<(G,\varphi)$, the coloring
$\varphi^{\prime}$ extends to a $3$-coloring of $G^{\prime}$.
Let us start with several basic reductions (eliminating short separating
cycles, $4$\- and $6$-faces), which are mostly standard. Usually, $6$-faces
are eliminated by collapsing similarly to $4$-faces, which is necessary in the
proofs that first eliminate the $4$-cycles and then maintain girth five; in
our setting, adding edges to transform them to $4$-faces is more convenient.
###### Lemma 1.
If $(G,\varphi)$ is a minimal counterexample, then $G$ is $2$-connected,
$\delta(G)\geq 2$, all vertices of degree two are incident with the outer
face, and every $(\leq\\!5)$-cycle in $G$ bounds a face.
###### Proof.
If $G$ contained a vertex $v$ of degree at most two not incident with the
outer face, then since $(G,\varphi)$ is a minimal counterexample, the coloring
$\varphi$ extends to a $3$-coloring of $G-v$. However, we can then color $v$
differently from its (at most two) neighbors, obtaining a $3$-coloring of $G$
extending $\varphi$. This is a contradiction, and thus $G$ contains no such
vertex. Note that all vertices of $G$ incident with the outer face have degree
at least two, since the outer face is bounded by a cycle.
Suppose that a $(\leq\\!5)$-cycle $K$ of $G$ does not bound a face. Since $G$
is triangle-free, the cycle $K$ is induced. Let $G_{1}$ be the subgraph of $G$
drawn outside (and including) $K$, and let $G_{2}$ be the subgraph of $G$
drawn inside (and including) $K$. We have $(G_{1},\varphi)<(G,\varphi)$, and
thus there exists a $3$-coloring $\psi_{1}$ of $G_{1}$ extending $\varphi$.
Furthermore, $(G_{2},\psi_{1}\restriction V(K))<(G,\varphi)$, and thus there
exists a $3$-coloring $\psi_{2}$ of $G_{2}$ that matches $\psi_{1}$ on $K$.
The union of $\psi_{1}$ and $\psi_{2}$ is a $3$-coloring of $G$ extending
$\varphi$, which is a contradiction. Hence, every $(\leq\\!5)$-cycle of $G$
bounds a face.
Suppose that $G$ is not $2$-connected, and thus there exist graphs $G_{1}$,
$G_{2}$ intersecting in at most one vertex such that $G=G_{1}\cup G_{2}$,
$C\subseteq G_{1}$ and $|V(G_{1})|,|V(G_{2})|\geq 4$. Observe that for
$i\in\\{1,2\\}$, there exists a vertex $v_{i}\in V(G_{i})$ incident with the
common face of $G_{1}$ and $G_{2}$ such that if $G_{1}$ and $G_{2}$ intersect,
then the distance between $v_{i}$ and the vertex in $G_{1}\cap G_{2}$ is at
least two. Then $G+v_{1}v_{2}$ is triangle-free and
$(G+v_{1}v_{2},\varphi)<(G,\varphi)$. However, this implies that there exists
a $3$-coloring of $G+v_{1}v_{2}$ extending $\varphi$, which also gives such a
$3$-coloring of $G$. This is a contradiction. ∎
###### Lemma 2.
If $(G,\varphi)$ is a minimal counterexample with the outer face bounded by a
cycle $C$, then $G$ contains no induced $6$-cycle other than $C$.
###### Proof.
Suppose that $G$ contains an induced $6$-cycle $K\neq C$. Let $G_{1}$ be the
subgraph of $G$ drawn outside (and including) $K$, and let $G_{2}$ be the
subgraph of $G$ drawn inside (and including) $K$. Since $K\neq C$ and $C$ is
an induced cycle, we have $V(K)\not\subseteq V(C)$. Let us label the vertices
of $K$ by $v_{1}$, $v_{2}$, …$v_{6}$ in order so that $v_{1}\not\in V(C)$ and
subject to that, the degree of $v_{1}$ in $G_{1}$ is as small as possible. Let
$G^{\prime}_{1}=G_{1}+v_{1}v_{4}$.
Note that $C$ is an induced cycle bounding the outer face of $G^{\prime}_{1}$.
If $G^{\prime}_{1}$ contains a triangle, then $G$ contains a $5$-cycle
$Q=v_{1}v_{2}v_{3}v_{4}x$ with $x\in V(G_{1})\setminus V(K)$, which bounds a
face by Lemma 1. Hence, the path $v_{1}v_{2}v_{3}$ is contained in boundaries
of two distinct faces ($K$ and $Q$) in $G_{1}$, and thus $v_{2}$ has degree
two in $G_{1}$. However, $v_{1}$ has at least three neighbors $v_{2}$, $v_{3}$
and $x$ in $G_{1}$, which contradicts the choice of the labels of the vertices
of $K$. Therefore, $G^{\prime}_{1}$ is triangle-free. Note also that either
$|V(G^{\prime}_{1})|<|V(G)|$ (if $K$ does not bound a face), or
$|V(G^{\prime}_{1})|=|V(G)|$ and $|E(G^{\prime}_{1})|>|E(G)|$ (if $K$ bounds a
face). Hence, $(G^{\prime}_{1},\varphi)<(G,\varphi)$, and thus there exists a
$3$-coloring $\psi_{1}$ of $G^{\prime}_{1}$ extending $\varphi$. Because of
the edge $v_{1}v_{4}$, $\psi_{1}\restriction V(K)$ is a valid coloring of $K$.
Since $K$ is an induced cycle, we have $V(C)\not\subseteq V(K)$, and thus
$|V(G_{2})|<|V(G)|$ and $(G_{2},\psi_{1}\restriction V(K))<(G,\varphi)$.
Therefore, there exists a $3$-coloring $\psi_{2}$ of $G_{2}$ that matches
$\psi_{1}$ on $K$. The union of $\psi_{1}$ and $\psi_{2}$ is a $3$-coloring of
$G$ extending $\varphi$, which is a contradiction. ∎
###### Lemma 3.
If $(G,\varphi)$ is a minimal counterexample with the outer face bounded by a
cycle $C$, then $G$ contains no $4$-cycle other than $C$.
###### Proof.
Suppose that $G$ contains a $4$-cycle $K\neq C$. By Lemma 1, $K$ bounds a
face. Let $v_{1}$, …, $v_{4}$ be the vertices of $K$ in order. Since $K\neq C$
and $C$ is an induced cycle, we can assume that $v_{3}\not\in V(C)$. Let
$G_{1}$ be the graph obtained from $G$ by identifying $v_{1}$ with $v_{3}$.
Note that each $3$-coloring of $G_{1}$ corresponds to a $3$-coloring of $G$,
and thus $\varphi$ does not extend to a $3$-coloring of $G_{1}$. Since
$|V(G_{1})|<|V(G)|$, it follows that the pair $(G_{1},\varphi)$ is not valid.
There are two possibilities: either $G_{1}$ contains a triangle or its outer
face is not an induced cycle.
If $G_{1}$ contains a triangle, then $G$ contains a $5$-cycle
$Q=v_{1}v_{2}v_{3}xy$. By Lemma 1, $Q$ bounds a face, hence the path
$v_{1}v_{2}v_{3}$ is contained in boundaries of two distinct faces ($K$ and
$Q$). It follows that $v_{2}$ has degree two, and by Lemma 1, $v_{2}$ is
incident with the outer face. However, this implies that $v_{3}$ is incident
with the outer face as well, contrary to its choice.
It remains to consider the case that the outer face of $G_{1}$ is not an
induced cycle. Since $G_{1}$ contains no triangle, it follows that the outer
face of $G_{1}$ has length $6$. Hence, $C=v_{1}w_{2}w_{3}w_{4}w_{5}w_{6}$ and
$v_{3}$ is adjacent to $w_{4}$. We choose the labels so that either
$v_{2}=w_{2}$ or $v_{2}$ is contained inside the $6$-cycle
$Q=v_{1}v_{4}v_{3}w_{4}w_{3}w_{2}$. By Lemma 2, $Q$ is not an induced cycle,
and since $C$ is an induced cycle and $G$ is triangle-free, we conclude that
$v_{3}w_{2}\in E(G)$. The symmetric argument for the $6$-cycle
$v_{1}v_{2}v_{3}w_{4}w_{5}w_{6}$ implies that $v_{3}w_{6}\in E(G)$. By Lemma
1, $w_{2}v_{1}w_{6}v_{3}$, $w_{2}v_{3}w_{4}w_{3}$ and $w_{6}v_{3}w_{4}w_{5}$
bound faces, hence $V(G)=V(C)\cup\\{v_{3}\\}$. Since $\varphi$ is a valid
coloring of $C$, two opposite vertices of $C$ have different colors; say
$\varphi(v_{1})\neq\varphi(w_{4})$. Then, we can properly color $v_{3}$ by
$\varphi(v_{1})$. This is a contradiction. ∎
###### Corollary 4.
If $(G,\varphi)$ is a minimal counterexample with the outer face bounded by a
cycle $C$, then $G$ contains no $6$-cycle other than $C$.
###### Proof.
No $6$-cycle in $G$ other than $C$ is induced by Lemma 2. However, a non-
induced $6$-cycle would imply the presence of at least two $4$-cycles,
contradicting Lemma 3. ∎
The following is the main reduction enabling us to eliminate $5$-faces
incident with too many vertices of degree three. Thomassen [3] uses a
different reduction in this case, which however is slightly more difficult to
argue about.
###### Lemma 5.
Let $(G,\varphi)$ be a minimal counterexample whose outer face is bounded by a
cycle $C$. Let $K=v_{1}v_{2}v_{3}v_{4}v_{5}$ be a cycle bounding a $5$-face in
$G$ such that $v_{1}$, $v_{2}$, $v_{3}$ and $v_{4}$ have degree three and do
not belong to $V(C)$. Then at least one of the neighbors of $v_{1}$, …,
$v_{4}$ outside $K$ belongs to $V(C)$.
###### Proof.
Let $x_{1}$, …, $x_{4}$ be the neighbors of $v_{1}$, …, $v_{4}$, respectively,
outside of $K$. Suppose that none of these vertices belongs to $V(C)$. Let
$G^{\prime}$ be the graph obtained from $G-\\{v_{1},v_{2},v_{3},v_{4}\\}$ by
adding the edge $x_{1}x_{4}$ and by identifying $x_{2}$ with $x_{3}$. Note
that $C$ is an induced cycle bounding the outer face of $G^{\prime}$.
If $G^{\prime}$ contained a triangle, then $G$ would contain a $6$-cycle
$x_{2}v_{2}v_{3}x_{4}ab$ or $x_{1}v_{1}v_{5}v_{4}x_{4}a$ (contrary to
Corollary 4) or a matching between $\\{x_{1},x_{4}\\}$ and $\\{x_{2},x_{3}\\}$
(contrary to either planarity or Lemma 3). Hence,
$(G^{\prime},\varphi)<(G,\varphi)$ is valid and there exists a $3$-coloring
$\psi$ of $G^{\prime}$ extending $\varphi$. Note that
$\psi(x_{1})\neq\psi(x_{4})$; hence, we can choose colors
$\psi(v_{1})\not\in\\{\psi(x_{1}),\psi(v_{5})\\}$ and
$\psi(v_{4})\not\in\\{\psi(x_{4}),\psi(v_{5})\\}$ so that
$\psi(v_{1})\neq\psi(v_{4})$. Since $\psi(x_{2})=\psi(x_{3})$, observe that we
can extend this coloring to $v_{2}$ and $v_{3}$. This gives a $3$-coloring of
$G$ extending $\varphi$, which is a contradiction. ∎
We can now proceed with the discharging phase of the proof.
###### Lemma 6.
If $(G,\varphi)$ is a valid pair, then $\varphi$ extends to a $3$-coloring of
$G$.
###### Proof.
Suppose that $\varphi$ does not extend to a $3$-coloring of $G$; choose a
valid pair $(G,\varphi)$ with this property that is minimal with respect to
$<$. Thus, $(G,\varphi)$ is a minimal counterexample. Clearly, $G$ has a
vertex not incident with its outer face. Let the _initial charge_ $c_{0}(v)$
of a vertex $v$ of $G$ be defined as $\deg(v)-4$ and the initial charge
$c_{0}(f)$ of a face $f$ of $G$ as $|f|-4$.
Let $C$ be the cycle bounding the outer face of $G$. A $5$-face $Q$ is _tied_
to a vertex $z\in V(C)$ if $z\not\in V(Q)$ and $z$ has a neighbor in
$V(Q)\setminus V(C)$ of degree three. Let us redistribute the charge as
follows: each face other than the outer one sends $1/3$ to each incident
vertex that either has degree two, or has degree three and does not belong to
$V(C)$. Each vertex of $C$ sends $1/3$ to each $5$-face tied to it. Let the
charge obtained by these rules be called _final_ and denoted by $c$.
First, let us argue that the final charge of each vertex $v\in V(G)\setminus
V(C)$ is non-negative: by Lemma 1, $v$ has degree at least three. If $v$ has
degree at least four, then $c(v)\geq c_{0}(v)=\deg(v)-4\geq 0$. If $v$ has
degree three, then it receives $1/3$ from each incident face, and
$c(v)=c_{0}(v)+1=0$.
Next, consider the charge of a face $f$ distinct from the outer one. By Lemma
3, we have $|f|\geq 5$. The face $f$ sends at most $1/3$ to each incident
vertex, and thus its final charge is $c(f)\geq c_{0}(f)-|f|/3=2|f|/3-4$.
Hence, $c(f)\geq 0$ unless $|f|=5$. Suppose that $|f|=5$ and let $k$ be the
number of vertices to that $f$ sends charge. We have
$c(f)=c_{0}(f)-k/3=1-k/3$. If $k\leq 3$, then $c(f)\geq 0$, and thus we can
assume that $k\geq 4$. If $f$ is incident with a vertex $v$ of degree two,
then note that $v\in V(C)$ by Lemma 1. Furthermore, since $G$ is $2$-connected
and $G\neq C$, we conclude that $f$ is incident with at least two vertices of
degree three belonging to $V(C)$, to which $f$ does not send charge. This
contradicts the assumption that $k\geq 4$. Hence, no vertex of degree two is
incident with $f$, and thus $k$ is the number of vertices of $V(f)\setminus
V(C)$ of degree three. By Lemma 5, $f$ is tied to at least $k-3$ vertices of
$C$, and thus $c(f)\geq c_{0}(f)-k/3+(k-3)/3=0$.
The final charge of the outer face is $|C|-4$. Consider a vertex $v\in V(C)$.
If $\deg(v)=2$, then $v$ receives $1/3$ from the incident non-outer face and
$c(v)=-5/3$. If $\deg(v)\geq 3$, then $v$ sends $1/3$ to at most $\deg(v)-2$
faces tied to it, and thus $c(v)\geq
c_{0}(v)-(\deg(v)-2)/3=2\deg(v)/3-10/3\geq-4/3$.
Note that since $G$ is $2$-connected and $G\neq C$, the outer face is incident
with at least two vertices of degree greater than two. Therefore, the sum of
the final charges is at least $(|C|-4)-5(|C|-2)/3-2\cdot 4/3=-10/3-2|C|/3>-8$,
since $|C|\leq 6$. On the other hand, the sum of final charges is equal to the
sum of the initial charges, which (if $G$ has $n$ vertices, $m$ edges and $s$
faces) is
$\displaystyle\sum_{v}c_{0}(v)+\sum_{f}c_{0}(f)$ $\displaystyle=$
$\displaystyle\sum_{v}(\deg(v)-4)+\sum_{f}(|f|-4)$ $\displaystyle=$
$\displaystyle(2m-4n)+(2m-4s)=4(m-n-s)$ $\displaystyle=$ $\displaystyle-8$
by Euler’s formula. This is a contradiction. ∎
The proof of Grötzsch theorem is now straightforward.
###### Theorem 7.
Every planar triangle-free graph is $3$-colorable.
###### Proof.
Suppose for a contradiction that $G$ is a planar triangle-free graph that is
not $3$-colorable, chosen with as few vertices as possible. Clearly, $G$ has
minimum degree at least three (as otherwise we can remove a vertex $v$ of
degree at most two, $3$-color the rest of the graph by the minimality of $G$,
and color $v$ differently from its neighbors). Hence, Euler’s formula implies
that every drawing of $G$ in the plane has a face of length at most $5$. Fix a
drawing of $G$ such that the outer face is bounded by a cycle $C$ of length at
most $5$. Since $G$ is triangle-free, the cycle $C$ is induced. Let $\varphi$
be an arbitrary $3$-coloring of $C$. By Lemma 6, $\varphi$ extends to a
$3$-coloring of $G$, which is a contradiction. ∎
## References
* [1] Z. Dvořák, K. Kawarabayashi, R. Thomas, 3-coloring triangle-free planar graphs in linear time, ACM Transactions on Algorithms 7 (2011), article no. 41.
* [2] H. Grötzsch, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 8 (1959), 109–120.
* [3] C. Thomassen, Grötzsch’s $3$-color theorem and its counterparts for the torus and the projective plane, J. Combin. Theory Ser. B 62 (1994), 268–279.
* [4] C. Thomassen, A short list color proof of Grötzsch’s theorem, J. Combin. Theory Ser. B 88 (2003), 189–192.
|
arxiv-papers
| 2013-11-29T17:10:51 |
2024-09-04T02:49:54.538813
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zden\\v{e}k Dvo\\v{r}\\'ak",
"submitter": "Zdenek Dvorak",
"url": "https://arxiv.org/abs/1311.7636"
}
|
1312.0098
|
# The 3-rainbow index of graph operations
TINGTING LIU Tianjin University Department of Mathematics 300072 Tianjin CHINA
[email protected] YUMEI HU111corresponding author Tianjin University Department
of Mathematics 300072 Tianjin CHINA [email protected]
Abstract: A tree $T$, in an edge-colored graph $G$, is called a rainbow tree
if no two edges of $T$ are assigned the same color. A $k$-rainbow coloring of
$G$ is an edge coloring of $G$ having the property that for every set $S$ of
$k$ vertices of $G$, there exists a rainbow tree $T$ in $G$ such that
$S\subseteq V(T)$. The minimum number of colors needed in a $k$-rainbow
coloring of $G$ is the $k$-rainbow index of $G$, denoted by $rx_{k}(G)$. Graph
operations, both binary and unary, are an interesting subject, which can be
used to understand structures of graphs. In this paper, we will study the
$3$-rainbow index with respect to three important graph product operations
(namely cartesian product, strong product, lexicographic product) and other
graph operations. In this direction, we firstly show if $G^{*}=G_{1}\Box
G_{2}\cdots\Box G_{k}$ ($k\geq 2$), where each $G_{i}$ is connected, then
$rx_{3}(G^{*})\leq\sum_{i=1}^{k}rx_{3}(G_{i})$. Moreover, we also present a
condition and show the above equality holds if every graph $G_{i}~{}(1\leq
i\leq k)$ meets the condition. As a corollary, we obtain an upper bound for
the 3-rainbow index of strong product. Secondly, we discuss the 3-rainbow
index of the lexicographic graph $G[H]$ for connected graphs $G$ and $H$. The
proofs are constructive and hence yield the sharp bound. Finally, we consider
the relationship between the 3-rainbow index of original graphs and other
simple graph operations : the join of $G$ and $H$, split a vertex of a graph
and subdivide an edge. Key–Words: $3$-rainbow index; cartesian product; strong
product; lexicographic product.
## 1 Introduction
All graphs considered in this paper are simple, connected and undirected. We
follow the terminology and notation of Bondy and Murty [7]. Let $G$ be a
nontrivial connected graph of order $n$ on which is defined an edge coloring,
where adjacent edges may be the same color. A path $P$ is a rainbow path if no
two edges of $P$ are colored the same. The graph $G$ is rainbow connected if
$G$ contains a $u$-$v$ rainbow path for every pair $u,v$ of distinct vertices
of $G$. If by coloring $c$ the graph $G$ is rainbow connected , the coloring
$c$ is called a rainbow coloring of $G$. The rainbow connection number $rc(G)$
of $G$, introduced by Chartrand et al. in [5], is the minimum number of colors
that results in a rainbow connected graph $G$.
Rainbow connection has an interesting application for the secure transfer of
classified information between agencies (cf. [2]). Although the information
needs to be protected since it is vital to national security, procedures must
be in place that permit access between appropriate parties. This two fold
issues can be addressed by assigning information transfer paths between
agencies which may have other agencies as intermediaries while requiring a
large enough number of passwords and firewalls that is prohibitive to
intruders, yet small enough to manage (that is, enough so that one or more
paths between every pair of agencies have no password repeated). An immediate
question arises: What is the minimum number of passwords or firewalls needed
that allows one or more secure paths between every two agencies so that the
passwords along each path are distinct? This situation can be modeled by a
graph and studied by the means of rainbow coloring.
Later, another generalization of rainbow connection number was introduced by
Chartrand et al.[4] in 2009. A tree $T$ is a rainbow tree if no two edges of
$T$ are colored the same. Let $k$ be a fixed integer with $2\leq k\leq n$. An
edge coloring of $G$ is called a $k$-rainbow coloring if for every set $S$ of
$k$ vertices of $G$, there exists a rainbow tree in $G$ containing the
vertices of $S$. The $k$-rainbow index $rx_{k}(G)$ of $G$ is the minimum
number of colors needed in a $k$-rainbow coloring of $G$. It is obvious that
$rc(G)=rx_{2}(G)$. A tree $T$ is called a concise tree if $T$ contains $S$ and
$T-v$ is not a tree containing $S$, where $v$ is any vertex of $T$. In the
paper, we suppose the tree containing $S$ be concise. Since if the given tree
$T$ is not concise, we can get a concise tree by deleting some vertices from
$T$.
As we know, the diameter is a natural lower bound of the rainbow connection
number. Similarly, we consider the Steiner diameter in this paper, which is a
nice generalization of the concept of diameter. The Steiner distance $d(S)$ of
a set $S$ of vertices in $G$ is the minimum size of a tree in $G$ containing
$S$. Such a tree is called a Steiner S-tree or simply a Steiner tree. The
$k$-Steiner diameter $sdiam_{k}(G)$ of $G$ is the maximum Steiner distance of
$S$ among all sets $S$ with $k$ vertices in $G$. The $k$-Steiner diameter
provides a lower bound for the $k$-rainbow index of $G$, i.e.,
$sdiam_{k}(G)\leq rx_{k}(G)$. It follows, for every nontrivial connected graph
$G$ of order $n$, that
$rx_{2}(G)\leq rx_{3}(G)\leq\cdots\leq rx_{k}(G).$
For general $k$, Chartrand et al. [4] determined the $k$-rainbow index of
trees and cycles. They obtained the following theorems.
###### Theorem 1.1
[4] Let $T$ be a tree of order $n\geq 3$. For each integer $k$ with $3\leq
k\leq n$,
$rx_{k}(T)=n-1.$
###### Theorem 1.2
[4] For integers $k$ and $n$ with $3\leq k\leq n$,
$rx_{k}(C_{n})=\left\\{\begin{array}[]{lll}n-2,&\mbox{ if~{} $k=3$ and $n\geq
4$;}\\\ n-1,&\mbox{ if~{} $k=n=3$ or $4\leq k\leq n$.}\\\ \end{array}\right.$
In the paper, we focus our attention on $rx_{3}(G)$. For 3-rainbow index of a
graph, Chartrand et al. [4] derive the exact value for the complete graphs.
###### Theorem 1.3
[4] For any integer $n\geq 3$,
$rx_{3}(K_{n})=\left\\{\begin{array}[]{lll}2,&\mbox{ if~{} $3\leq n\leq
5$;}\\\ 3,&\mbox{ if~{} $n\geq 6$;}\\\ \end{array}\right.$
Chakraborty et al. [11] showed that computing the rainbow connection number of
a graph is NP-hard. So it is also NP-hard to compute $k$-rainbow index of a
connected graph. For rainbow connection number $rc(G)$, people aim to give
nice upper bounds for this parameter, especially sharp upper bounds, according
to some parameters of the graph $G$ [9, 18, 19, 25].
Many researchers have paid more attention to rainbow connection number of some
graph products [10, 12, 16, 20, 21]. There is one way to bound the rainbow
connection number of a graph product by the rainbow connection number of the
operand graphs. Li and Sun [21] adopted the method to study rainbow connection
number with respect to Cartesian product and lexicographic product. They got
the following conclusions.
###### Theorem 1.4
[21] Let $G^{*}=G_{1}\Box G_{2}\cdots\Box G_{k}$ ($k\geq 2$), where each
$G_{i}$ is connected, then
$rc(G^{*})\leq\sum_{i=1}^{k}rc(G_{i})$
Moreover, if $rc(G_{i})=diam(G_{i})$ for each $G_{i}$, then the equality
holds.
###### Theorem 1.5
[21] If $G$ and $H$ are two graphs and $G$ is connected, then we have
1\. if $H$ is complete, then
$rc(G[H])\leq rc(G).$
In particular, if $diam(G)=rc(G)$, then $rc(G[H])=rc(G)$.
2\. if $H$ is not complete,then
$rc(G[H])\leq rc(G)+1.$
In particular, if $diam(G)=rc(G)$, then $diam$$(G[H])=2$ if $G$ is complete
and $rc(G)\leq diam(G)+1$ if $G$ is not complete.
In this paper, we study the $3$-rainbow index with respect to three important
graph product operations (namely cartesian product, lexicographic product and
strong product) and other operations of graphs. Moreover, we present the class
of graphs which obtain the upper bounds.
### 1.1 Preliminaries
We use $V(G)$, $E(G)$ for the set of vertices and edges of $G$, respectively.
For any subset $X$ of $V(G)$, let $G[X]$ be the subgraph induced by $X$, and
$E[X]$ the edge set of $G[X]$; Similarly, for any subset $E^{\prime}$ of
$E(G)$, let $G[E^{\prime}]$ be the subgraph induced by $E^{\prime}$. For any
two disjoint subsets $X$, $Y$ of $V(G)$, we use $G[X,Y]$ to denote the
bipartite graph with vertex set $X\cup Y$ and edge set $E[X,Y]=\\{uv\in
E(G)|u\in X,v\in Y\\}$. The distance between two vertices $u$ and $v$ in $G$
is the length of a shortest path between them and is denoted by $d_{G}(u,v)$.
The distance between a vertex $u$ and a path $P$ is the shortest distance
between $u$ and the vertices in $P$. Given a graph $G$, the eccentricity of a
vertex, $v\in V(G)$ is given by $ecc(v)=max\\{d_{G}(v,u):u\in V(G)\\}$. The
diameter of $G$ is defined as $diam(G)=max\\{ecc(v):v\in V(G)\\}$. The length
of a path is the number of edges in that path. The length of a tree $T$ is the
numbers of edges in that tree, denoted by $size(T)$. $G\setminus e$ denotes
the graph obtained by deleting an edge $e$ from the graph $G$ but leaving the
vertices and the remaining edges intact. $G-v$ denotes the graph obtained by
deleting the vertex $v$ together with all the edges incident with $v$ in $G$.
###### Definition 1
(The Cartesian Product) Given two graphs $G$ and $H$, the Cartesian product of
$G$ and $H$, denoted by $G\Box H$, is defined as follows: $V(G\Box
H)=V(G)\times V(H)$. Two distinct vertices $(g_{1},h_{1})$ and $(g_{2},h_{2})$
of $G\Box H$ are adjacent if and only if either $g_{1}=g_{2}$ and
$h_{1}h_{2}\in E(H)$ or $h_{1}=h_{2}$ and $g_{1}g_{2}\in E(G)$.
###### Definition 2
(The Lexicographic Product) The Lexicographic Product $G[H]$ of graphs $G$ and
$H$ has the vertex set $V(G[H])=V(G)\times V(H)$. Two vertices
$(g_{1},h_{1}),(g_{2},h_{2})$ are adjacent if $g_{1}g_{2}\in E(G)$, or if
$g_{1}=g_{2}$ and $h_{1}h_{2}\in E(H)$.
###### Definition 3
(The Strong Product) The Strong Product $G\boxtimes H$ of graphs $G$ and $H$
is the graph with $V(G\boxtimes H)=V(G)\times V(H)$. Two distinct vertices
$(g_{1},h_{1})$ and $(g_{2},h_{2})$ of $G\boxtimes H$ are adjacent whenever
$g_{1}=g_{2}$ and $h_{1}h_{2}\in E(H)$ or $h_{1}=h_{2}$ and $g_{1}g_{2}\in
E(G)$ or $g_{1}g_{2}\in E(G)$ and $h_{1}h_{2}\in E(H)$.
Clearly, the resultant graph is isomorphic to $G$ (respectively $H$) if
$H=K_{1}$ (respectively $G=K_{1}$). Therefore, we suppose $V(G)\geq 2$ and
$V(H)\geq 2$ when studying the 3-rainbow index of these three graph products.
###### Definition 4
(The union of graphs) The union of two graphs, by starting with a disjoint
union of two graphs $G$ and $H$ and adding edges joining every vertex of $G$
to every vertex of $H$, the resultant graph is the join of $G$ and $H$,
denoted by $G\vee H$.
###### Definition 5
(To split a vertex) To split a vertex $v$ of a graph $G$ is to replace $v$ by
two adjacent vertices $v_{1}$ and $v_{2}$, and to replace each edge incident
to $v$ by an edge incident to either $v_{1}$ or $v_{2}$ (but not both), the
other end of the edge remaining unchanged.
### 1.2 Some basic observations
It is easy to see that if the graph $H$ has a $3$-rainbow coloring with
$rx_{3}(H)$ colors, then the graph $G$, which is obtained from $H$ by adding
some edges to $H$, also has a $3$-rainbow coloring with $rx_{3}(H)$ colors
since the new edges of $G$ can be colored arbitrarily with the colors used in
$H$. So we have:
###### Observation 1
Let $G$ and $H$ be connected graphs and $H$ be a spanning subgraph of $G$.
Then $rx_{3}(G)\leq rx_{3}(H)$.
To verify a 3-rainbow index, we need to find a rainbow tree containing any set
of three vertices. So it is necessary to know the structure of concise trees.
Next we consider the structure of concise trees $T$ containing three vertices,
which will be very useful in the sequel.
###### Observation 2
Let $G$ be a connected graph and $S=\\{v_{1},v_{2},v_{3}\\}\subseteq V(G)$. If
$T$ is a concise tree containing $S$, then $T$ belongs to exactly one of Type
$I$ and Type $II$( see Figure 1).
Type $I$: $T$ is a path such that one vertex of $S$ as its origin, one of $S$
as its terminus, other vertex of $S$ as its internal vertex.
Type $II$: $T$ is a tree obtained from the star $S_{3}$ by replacing each edge
of $S_{3}$ with a path $P$.
Figure 1: Two types of concise trees, where
$\\{v_{i_{1}},v_{i_{2}},v_{i_{3}}\\}=\\{v_{1},v_{2},v_{3}\\}$, $v_{4}\in V(G)$
Proof: Firstly, we claim that the leaves of $T$ belong to $S$. Since if there
exists a leaf $v$ such that $v\notin S$, then we can get the more minimal tree
$T^{\prime}=T-v$ containing $S$, a contradiction. Thus the $T$ has at most
three leaves. If the $T$ has exactly two leaves, then it is easy to verify
that $T$ is a path. In this case, $T$ belongs to Type $I$. Otherwise there is
a $v_{1}v_{2}$-path $P$ in $T$ such that $v_{3}\notin P$. Since $T$ is
connected, there a path $P^{\prime}$ in $T$ connecting $v_{3}$ and $P$. Let
$v_{4}$ be the vertex of $P^{\prime}$ such that
$d_{T}(v_{3},v_{4})$=$d_{T}(v_{3},P)$. Then we get $T\supseteq P\cup
P^{\prime}$. On the other hand, we know, $P\cup P^{\prime}$ is a tree
containing $S$. Furthermore, since $T$ is a concise tree, $T=P\cup
P^{\prime}$, which belongs to Type $II$. $\sqcap\\!\\!\\!\\!\sqcup$
## 2 Cartesian product
In this section, we do some research on the relationship between the
$3$-rainbow index of the original graphs and that of the cartesian products.
Recall that the Cartesian product of $G$ and $H$, denoted by $G\Box H$, is
defined as follows: $V(G\Box H)=V(G)\times V(H)$. Two distinct vertices
$(g_{1},h_{1})$ and $(g_{2},h_{2})$ of $G\Box H$ are adjacent if and only if
either $g_{1}=g_{2}$ and $h_{1}h_{2}\in E(H)$ or $h_{1}=h_{2}$ and
$g_{1}g_{2}\in E(G)$. Let $V(G)=\\{g_{i}\\}_{i\in[s]}$,
$V(H)=\\{h_{j}\\}_{j\in[t]}$. Note that $H_{i}=G\Box
H[\\{(g_{i},h_{j})\\}_{j\in[t]}]\cong H,G_{j}=G\Box
H[\\{(g_{i},h_{j})\\}_{i\in[s]}]\cong G$. Any edge
$(g_{i},h_{j_{1}})(g_{i},h_{j_{2}})$ of $H_{i}$ corresponds to edge
$h_{j_{1}}h_{j_{2}}$ of $H$ and $(g_{i_{1}},h_{j})(g_{i_{2}},h_{j})$ of
$G_{j}$ corresponds to edge $g_{i_{1}}g_{i_{2}}$ of $G$. For the sake of our
results, we give some useful and fundamental conclusions about the Cartesian
product.
###### Lemma 2.1
[17] The Cartesian product of two graphs is connected if and only if these two
graphs are both connected.
###### Lemma 2.2
[17] The Cartesian product is associative.
###### Lemma 2.3
[17] Let $(g_{1},h_{1})$ and $(g_{2},h_{2})$ be arbitrary vertices of the
Cartesian product $G\Box H$. Then
$d_{G\Box
H}((g_{1},h_{1}),(g_{2},h_{2}))=d_{G}(g_{1},g_{2})+d_{H}(h_{1},h_{2}).$
With the aid of Observation 2 and above Lemmas, we derive the following lemma,
which is useful to show the sharpness of our main result.
###### Lemma 2.4
Let $G^{*}=G_{1}\Box G_{2}\cdots\Box G_{k}$ ($k\geq 2$), where each $G_{i}$ is
connected. Then
$Sdiam_{3}(G^{*})=\sum_{i=1}^{k}Sdiam_{3}(G_{i}).$
Proof: We first prove the conclusion holds for the case $k=2$. Let $G=G_{1}$,
$H=G_{2}$, $V(G)=\\{g_{i}\\}_{i\in[s]}$, $V(H)=\\{h_{j}\\}_{j\in[t]}$,
$V(G^{*})=\\{g_{i},h_{j}\\}_{i\in[s],j\in[t]}=\\{v_{i,j}\\}_{i\in[s],j\in[t]}$.
Let
$S=\\{(g_{1},h_{1}),(g_{2},h_{2}),(g_{3},h_{3})\\},S_{1}=\\{g_{1},g_{2},g_{3}\\},S_{2}=\\{h_{1},h_{2},h_{3}\\}$
be a set of any three vertices of $V(G^{*})$, $V(G)$, $V(H)$, respectively.
Suppose that $T$, $T_{1}$ and $T_{2}$ be Steiner trees containing $S$,
$S_{1}$, $S_{2}$, respectively. Next, we only need to show
$size(T)$=$size(T_{1})$+$size(T_{2})$.
On the one hand, by the definition of the Cartesian product of graphs, each
edge of $G^{*}$ is exactly one element of $\\{H_{i},G_{j}\\}$,
$i\in[s],j\in[t]$. Then we can regard $T$ as the union $G^{\prime}$ and
$H^{\prime}$, where $G^{\prime}$ is induced by all the edges of $G_{j}\cap T$,
$j\in[t]$, $H^{\prime}$ is induced by all the edges of $H_{i}\cap T$,
$i\in[s]$. Let $G^{\prime\prime}$ and $H^{\prime\prime}$ be the graphs induced
by the corresponding edges of all edges of $G_{j}\cap T$ and $H_{i}\cap
T$($i\in[s],j\in[t]$) in $G$ and $H$, respectively. Clearly,
$G^{\prime\prime}$ and $H^{\prime\prime}$ are connected and containing $S_{1}$
and $S_{2}$, respectively. Hence, we have,
$size(T)$=$size(G^{\prime})$+$size(H^{\prime})$=$size(G^{\prime\prime})$+$size(H^{\prime\prime})$
$\geq$ $size(T_{1})$+$size(T_{2})$.
On the other hand, we try to construct a tree $T^{\prime}$ containing $S$ with
$size(T^{\prime})=$ $size(T_{1})$+$size(T_{2})$. Notice that, for every
subgraph in $G$ (or $H$), we can find the corresponding subgraph in any copy
$G_{j}$ ( or $H_{i}$). If $T_{1}$ or $T_{2}$ belongs to Type $I$, without loss
of generality, say $T_{1}=P_{1}\cup P_{2}$, where $P_{1}$ is the path
connecting $g_{i_{1}}$ and $g_{i_{2}}$, $P_{2}$ is the path connecting
$g_{i_{2}}$ and $g_{i_{3}}$,
$\\{g_{i_{1}},g_{i_{2}},g_{i_{3}}\\}=\\{g_{1},g_{2},g_{3}\\}$. We can find a
tree $T^{\prime}=P_{1}^{\prime}\cup T_{2}^{\prime}\cup P_{2}^{\prime}$
containing $S$, where the path $P_{1}^{\prime}$ is the corresponding path of
$P_{1}$ in $G_{i_{1}}$ and the path $P_{2}^{\prime}$ is the corresponding path
of $P_{2}$ in $G_{i_{3}}$, the tree $T_{2}^{\prime}$ is the corresponding tree
of $T_{2}$ in $H_{i_{2}}$, (see Figure 2).
Figure 2 : $T_{1}$ belongs to Type $I$
If not, that is to say, $T_{1}$, $T_{2}$ belong to Type $II$, we suppose
$T_{1}=P_{1}\cup P_{2}\cup P_{3}$, where $P_{i}$ is the path connecting
$g_{4}$ and $g_{i}$ ($1\leq i\leq 3$), $g_{4}$ is other vertex of $G$ except
the vertices of $S_{1}$. Then the tree $T^{\prime}=P_{1}^{\prime}\cup
P_{2}^{\prime}\cup P_{3}^{\prime}\cup T_{2}^{\prime}$ containing $S$ can also
be found in $G\Box H$, where $P_{i}^{\prime}$ is the corresponding path of
$P_{i}$ in $G_{i}$ ($1\leq i\leq 3$), the $T_{2}^{\prime}$ is the
corresponding tree of $T_{2}$ in $H_{4}$ (see Figure $3$). Thus, $size(T)\leq$
$size(T^{\prime})$=$size(T_{1})$+$size(T_{2})$.
Figure 3 : $T_{1}$ and $T_{2}$ belong to Type $II$.
So we get $size(T)$=$size(T_{1})$+$size(T_{2})$. Hence, $Sdiam_{3}(G_{1}\Box
G_{2})$= $Sdiam_{3}(G_{1})$+$Sdiam_{3}(G_{2})$. By Lemma 2.2,
$Sdiam_{3}(G^{*})$= $Sdiam_{3}(G_{1}\Box G_{2}$ $\Box\cdots$ $\Box
G_{k-1})$+$Sdiam_{3}(G_{k})$=$\sum_{i=1}^{k}$ $Sdiam_{3}(G_{i})$.
$\sqcap\\!\\!\\!\\!\sqcup$
###### Theorem 2.1
Let $G^{*}=G_{1}\Box G_{2}\cdots\Box G_{k}$ ($k\geq 2$), where each $G_{i}$ is
connected, then
$rx_{3}(G^{*})\leq\sum_{i=1}^{k}rx_{3}(G_{i})$
Moreover, if $rx_{3}(G_{i})=Sdiam_{3}(G_{i})$ for each $G_{i}$, then the
equality holds.
Proof: We first show the conclusion holds for the case $k=2$. Let $G=G_{1}$,
$H=G_{2}$, $V(G)=\\{g_{i}\\}_{i\in[s]}$, $V(H)=\\{h_{j}\\}_{j\in[t]}$,
$V(G^{*})=\\{g_{i},h_{j}\\}_{i\in[s],j\in[t]}=\\{v_{i,j}\\}_{i\in[s],j\in[t]}$.
Since $G$ and $H$ are connected, $G^{*}$ is connected by Lemma 2.1. For
example, Figure $4$ shows the case for $G=P_{4}$ and $H=P_{3}$.
Figure 4 : An example in Theorem 2.1.
Since for an edge $v_{i_{1},j_{1}}v_{i_{2},j_{2}}\in G^{*}$, we have
$i_{1}=i_{2}$ or $j_{1}=j_{2}$; if the former, then
$v_{i_{1},j_{1}}v_{i_{1},j_{2}}\in H_{i_{1}}$, otherwise,
$v_{i_{1},j_{1}}v_{i_{2},j_{1}}\in G_{j_{1}}$. Hence, we only give a coloring
of each graph $G_{j}~{}(j\in[t])$ and $H_{i}~{}(i\in[s])$.
We give $G$ a $3$-rainbow coloring with $rx_{3}(G)$ colors (see Figure 4 in
which $G$ obtains a $3$-rainbow coloring with colors 1, 2, 3), and $H$ a
$3$-rainbow coloring with $rx_{3}(H)$ fresh colors (see Figure 4 in which $H$
obtains a 3-rainbow coloring with other two fresh colors, 4, 5). Then we color
edges of $G^{*}$ as follow: if the edge belongs to some $H_{i}$, then assign
the edge with the same color with its corresponding edge of $H$ (for example,
edge $v_{1,1}v_{1,2}$ belong to $H_{1}$ and corresponds to the edge
$h_{1}h_{2}$ in $H$, so it receives the color $4$), otherwise, the edge
belongs to some $G_{j}$, then assign the edge with the same color with its
corresponding edge of $G$. Now we will show that the given coloring is
$3$-rainbow coloring of $G^{*}$. It suffices to show that for every set $S$ of
three vertices of $G^{*}$, there is a rainbow tree containing $S$. Let
$S=\\{(g_{1},h_{1}),(g_{2},h_{2}),(g_{3},h_{3})\\}$. we distinguish three
cases:
Case 1 The vertices of $S$ lie in some $G_{j}$ (or $H_{i}$), where
$i,j\in\\{1,2,3\\}$
That is, $g_{1}=g_{2}=g_{3}$ or $h_{1}=h_{2}=h_{3}$, without loss of
generality, we say, $g_{1}=g_{2}=g_{3}$. Under the given coloring of $H$, we
can find a rainbow tree $T$ containing $h_{1},~{}h_{2},~{}h_{3}$ in $H$. By
the strategy of the above coloring, the corresponding tree $T^{\prime}$ of $T$
in $H_{1}$ is also rainbow and contains $S$.
Case 2 The vertices of $S$ lie in two different copies $G_{j}^{\prime}$
,$G_{j}^{\prime\prime}$ (or $H_{i}^{\prime}$, $H_{i}^{\prime\prime}$). where
$j^{\prime},~{}j^{\prime\prime}\in\\{1,~{}2,~{}3\\}$(or
$i^{\prime},i^{\prime\prime}\in\\{1~{},2,~{}3\\}$.
Without loss of generality, we assume $g_{1}=g_{2}\neq g_{3}$. Note that if a
coloring is $3$-rainbow coloring, then it is also rainbow coloring, that is,
there is a rainbow path connecting any two vertices of graphs. If $h_{1}\neq
h_{2}\neq h_{3}$ ($h_{1}=h_{3}\neq h_{2}$ or $h_{2}=h_{3}\neq h_{1}$), we can
find a rainbow tree $T_{1}$ in $H$ containing $h_{1},h_{2},h_{3}$
($h_{1},h_{2}$). By the strategy of coloring, we can find a rainbow tree
$T_{1}^{\prime}$ in $H_{1}$ containing $\\{v_{1,1},v_{2,2},v_{1,3},\\}$
($\\{v_{1,1},v_{2,2}\\}$). So we can find a rainbow path $P_{1}^{\prime}$ in
$G_{3}$ connecting $v_{1,3}$ ($v_{1,1}$ or $v_{2,2}$) and $v_{3,3}$. Thus
there is a rainbow tree $T=T_{1}^{\prime}\cup P_{1}^{\prime}$ in $G\Box H$
containing $S$.
Case 3 The vertices of $S$ lie in three different copies $G_{1}$, $G_{2}$,
$G_{3}$ and $H_{1}$, $H_{2}$, $H_{3}$.
Let $T_{1}$ be a rainbow tree containing $g_{1},g_{2},g_{3}$ and $T_{2}$ be a
rainbow tree containing $h_{1},h_{2},h_{3}$.
If $T_{1}$ or $T_{2}$ belongs to Type $I$, say $T_{1}$, let $T_{1}=P_{1}\cup
P_{2}$. Then the tree $T=P_{1}^{\prime}\cup T_{2}^{\prime}\cup P_{2}^{\prime}$
containing $S$ can be constructed by the way of Figure $2$. And by the
character of the given coloring, the tree $T$ is a rainbow tree.
If $T_{1}$ and $T_{2}$ belong to Type $II$, let $T_{1}=P_{1}\cup P_{2}\cup
P_{3}$. Then the tree $T=P_{1}^{\prime}\cup P_{2}^{\prime}\cup
P_{3}^{\prime}\cup T_{2}^{\prime}$ can also be obtained by the way of Figure
$3$. Furthermore, it is easy to see that the it is also a rainbow tree.
Since we use $rx_{3}(G)+rx_{3}(H)$ colors totally, we have $rx_{3}(G^{*})\leq
rx_{3}(G)+rx_{3}(H)$. From Lemma 2.4, if $rx_{3}(G)=Sdiam_{3}(G)$ and
$rx_{3}(H)=Sdiam_{3}(H)$, then
$Sdiam_{3}(G^{*})=Sdiam_{3}(G)+Sdiam_{3}(H)=rx_{3}(G)+rx_{3}(H)\geq
rx_{3}(G^{*})$. On the other hand, $Sdiam_{3}(G^{*})\leq rx_{3}(G^{*})$, so
the conclusion holds for $k=2$.
For general $k$, by the Lemma 2.2, $rx_{3}(G^{*})=rx_{3}(G_{1}\Box
G_{2}\Box\cdots\Box G_{k-1}\Box G_{k})\leq rx_{3}(G_{1}\Box G_{2}\Box$
$\cdots\Box G_{k-1})+rx_{3}(G_{k})\leq\sum_{i=1}^{k}rx_{3}(G_{i})$. Moreover,
if $rx_{3}(G)=Sdiam_{3}(G_{i})$ for each $G_{i}$, then $rx_{3}(G^{*})\geq
Sdiam_{3}(G^{*})=\sum_{i=1}^{k}Sdiam_{3}(G_{i})=\sum_{i=1}^{k}rx_{3}(G_{i})\geq
rx_{3}(G^{*})$. So if $rx_{3}(G_{i})=Sdiam_{3}(G_{i})$ for each $G_{i}$, then
the equality holds. $\sqcap\\!\\!\\!\\!\sqcup$
###### Corollary 2.1
Let $G=P_{n_{1}}\Box P_{n_{2}}\Box\cdots\Box P_{n_{k}}$, where $P_{n_{i}}$ is
a path with $n_{i}$ vertices ($1\leq i\leq k$). Then
$rx_{3}(G)=\sum_{i=1}^{k}n_{i}-k.$
Proof: For every path $P_{n_{i}}$, by Theorem 1.1, we have
$Sdiam_{3}(P_{n_{i}})=rx_{3}(P_{n_{i}})=n_{i}-1$. Thus, by the Theorem 2.1,
$rx_{3}(G)=\sum_{i=1}^{k}rx_{3}(P_{n_{i}})=\sum_{i=1}^{k}n_{i}-k$.
$\sqcap\\!\\!\\!\\!\sqcup$
Recall that the strong product $G\boxtimes H$ of graphs $G$ and $H$ has the
vertex set $V(G)\times V(H)$. Two vertices $(g_{1},h_{1})$ and $(g_{2},h_{2})$
are adjacent whenever $g_{1}=g_{2}$ and $h_{1}h_{2}\in E(H)$ or $h_{1}=h_{2}$
and $g_{1}g_{2}\in E(G)$ or $g_{1}g_{2}\in E(G)$ and $h_{1}h_{2}\in E(H)$. By
the definition, the graph $G\Box H$ is the spanning subgraph of the graph
$G\boxtimes H$ for any graphs $G$ and $H$. With the help of Observation 1,
then we have the following result.
###### Corollary 2.2
Let $\overline{G^{*}}$ =$G_{1}\boxtimes G_{2}\boxtimes\cdots\boxtimes G_{k}$,
$(k\geq 2)$, where each $G_{i}~{}(1\leq i\leq k)$ is connected. Then we have
$rx_{3}(\overline{G^{*}})\leq\sum_{i=1}^{k}rx_{3}(G_{i}).$
## 3 Lexicographic Product
Recall that the lexicographic product $G[H]$ of graphs $G$ and $H$ has the
vertex set $V(G[H])=V(G)\times V(H)$. Two vertices
$(g_{1},h_{1}),(g_{2},h_{2})$ are adjacent if $g_{1}g_{2}\in E(G)$, or if
$g_{1}=g_{2}$ and $h_{1}h_{2}\in E(H)$. By definition, $G[H]$ can be obtained
from $G$ by submitting a copy $H_{1}$ for every $g_{1}\in V(G)$ and by joining
all vertices of $H_{1}$ with all vertices of $H_{2}$ if $g_{1}g_{2}\in E(G)$.
In this section, we consider the relationship between 3-rainbow index of the
original graphs and their lexicographic product. Since the rainbow connection
and 3-rainbow index is only defined in connected graphs, it is nature to
assume the original graphs are connected. Note that if $V(G)=1$ (or $V(H)=1$),
then $G[H]$=$H$ (or $G$). So in the following discussion, we suppose $V(G)\geq
2$ and $V(H)\geq 2$. By definition, if $G$ and $H$ are complete, then $G[H]$
is also complete.
So for some special cases of $G$ and $H$, we have the following lemma.
###### Lemma 3.1
If $G,H\cong K_{2}$, then
$rx_{3}(G[H])=2.$
If $G$ and $H$ are complete with $V(G)\geq 3$ or $V(H)\geq 3$, then
$rx_{3}(G[H])=3.$
Proof: If $G,H\cong K_{2}$, then $G[H]$=$K_{4}$. Hence, we have
$rx_{3}(G[H])=2$ by Theorem 1.3. If $G$ and $H$ are complete with $V(G)\geq 3$
or $V(H)\geq 3$, then $G[H]=K_{n}$ ($n\geq 6$). We get immediately
$rx_{3}(G[H])=3$ from the Theorem 1.3. $\sqcap\\!\\!\\!\\!\sqcup$
For the remaining cases, we obtain the following theorem.
###### Theorem 3.1
Let $G$ and $H$ be two connected graphs with $V(G)\geq 2$, $V(H)\geq 2$, and
at least one of $G$, $H$ be not complete. Then
$rx_{3}(G[H])\leq rx_{3}(G)+rc(H).$
In particular, if $diam(G)=rx_{3}(G)$, and $H$ is complete, then the equality
holds.
Proof: Let $V(G)=\\{g_{i}\\}_{i\in[s]}$, $V(H)=\\{h_{j}\\}_{j\in[t]}$,
$V(G[H])=\\{g_{i},h_{j}\\}_{i\in[s],j\in[t]}=\\{v_{i,j}\\}_{i\in[s],j\in[t]}$.
Let $S=\\{(g_{1},h_{1}),(g_{2},h_{2}),(g_{3},h_{3})\\}$ be any three different
vertices of $G[H]$. We derive the theorem from two parts: 1\. $V(H)=2$ and $G$
is not complete; 2\. $V(H)\geq 3$ and $G$ or $H$ is not complete.
1\. If $V(H)=2$ and $G$ is not complete, we firstly give $G$ a $3$-rainbow
coloring with $rx_{3}(G)$ colors. Then we can give $G[H]$ a $rx_{3}(G)$+1-edge
coloring as follows: the edge belongs to some $G_{j}$, then assign the edge
with the same color with its corresponding edge in $G$. Otherwise, assign the
edge a fresh color.
If $h_{1}=h_{2}=h_{3}$, then we can find a rainbow tree $T^{\prime}$
containing $S$ , which is the corresponding tree of $T$ containing
$g_{1},~{}g_{2},~{}g_{3}$ in $G_{1}$. Otherwise the vertices of $S$ lie in two
different graphs $G_{1}$ and $G_{2}$. Without loss of generality, we suppose
$h_{1}=h_{3}\neq h_{2}$. In this case, $(g_{1},h_{1}),(g_{3},h_{3})\in G_{1}$,
$(g_{2},h_{2})\in G_{2}$. Then we can find the corresponding vertex
$(g_{2},h_{1})$ (or $(g_{1},h_{1})$ or $(g_{3},h_{3})$) of $(g_{2},h_{2})$ in
$H_{1}$ and a rainbow tree $T^{\prime}$ containing
$(g_{1},h_{1}),(g_{3},h_{3})$ and $(g_{2},h_{1})$ (or $\emptyset$). Clearly,
there is a rainbow tree $T=T^{\prime}\cup e$ containing $S$, where
$e=(g_{2},h_{2})(g_{2},h_{1})$ (or $(g_{1},h_{1})$ or $(g_{3},h_{3})$). Hence
the above coloring is $3$-rainbow coloring of $G[H]$. So $rx_{3}(G[H])\leq
rx_{3}(G)+1=rx_{3}(G)+rc(H)$.
2\. Let $c_{1}=\\{0,1,\cdots,rx_{3}(G)-1\\}$ be a $3$-rainbow coloring of $G$.
Let $c_{2}$ be a rainbow coloring of $H$ using $rc(H)$ fresh colors. For every
$h_{j}\in H$ color the copy $G_{j}$ the same as $G$. By the same way, there is
a rainbow tree containing any three vertices
$(g_{1},h_{i}),(g_{2},h_{i}),(g_{3},h_{i})\in V(G[H])$. Every edge of the form
$(g_{1},h_{1})(g_{2},h_{2})$ get color $k+1$ mod($rx_{3}(G))$, where
$g_{1}g_{2}\in E(G)$, $h_{1}\neq h_{2}$, and $c_{1}(g_{1}g_{2})=k$. Finally,
color edges from $H_{i}$ the same as $H$ such that any two vertices
$(g_{i},h_{j})(g_{i},h_{k})$ are connected by a rainbow path. The figure $5$
shows an example of the coloring.
Figure 5 : An example in Theorem 3.1. 2.
Now we show the above coloring is $3$-rainbow coloring of $G[H]$. We
distinguish the following three cases.
Case 1 $g_{1}=g_{2}=g_{3}$
Since $G$ is a connected graph, there exists an edge $g_{1}g_{4}\in
E(G),g_{4}\in V(G)$. Then we can find a rainbow path $P$ connecting
$(g_{2},h_{2})(g_{1},h_{1})$ in $H_{1}$, which uses the colors of $H$. By the
coloring of strategy, the tree $T=P\cup v_{1,1}v_{4,1}\cup v_{4,1}v_{3,3}$ is
a rainbow tree containing $S$.
Case 2 $g_{1}=g_{2}\neq g_{3}$ or $g_{1}=g_{3}\neq g_{2}$ or $g_{2}=g_{3}\neq
g_{1}$
Without loss of generality, we assume $g_{1}=g_{2}\neq g_{3}$.
Subcase 2.1 $h_{1}=h_{3}$ (or $h_{2}=h_{3}$)
Then $T=P_{1}\cup P_{2}$ is a rainbow tree containing $S$, where $P_{1}$ is a
rainbow path connecting $(g_{1},h_{1})$ and $(g_{2},h_{2})$ in $H_{1}$,
$P_{2}$ is a rainbow path connecting $(g_{1},h_{1})$ (or $(g_{2},h_{2})$) and
$(g_{3},h_{3})$ in $G_{3}$.
Subcases 2.2 $h_{1}\neq h_{2}\neq h_{3}$
As we know, there is a rainbow path $P_{1}$ connecting $g_{3}$ and $g_{1}$ in
$G$. The case that $P_{1}$=$g_{3}g_{1}$ is trivial, so we assume
$P_{1}$=$g_{3}g_{1}^{\prime}$$g_{2}^{\prime},$$\cdots,g_{k}^{\prime}g_{1}$,
$g_{i}^{\prime}\in V(G)$ $(1\leq i\leq k)$. We claim that
$P_{1}^{\prime}=(g_{3},h_{3})(g_{1}^{\prime},h_{2})(g_{2}^{\prime},h_{3})(g_{3}^{\prime},h_{2}),\cdots,(g_{k}^{\prime},u)(g_{1},h_{1})$
is a rainbow path connecting $(g_{3},h_{3})$ and $(g_{1},h_{1})$, where
$u=h_{3}$ if $k$ is even and $u=h_{2}$ otherwise. It is easy to see that the
path only use the edge of the form $(g_{i},h_{j})(g_{j},h_{l})$, where
$g_{i}g_{j}\in E(G)$, $h_{j}\neq h_{l}$. By the character of coloring, the
path is also a rainbow path and only uses the colors of $G$. Thus, there is a
rainbow tree $T=P^{\prime}\cup P_{2}$ containing $S$, where $P_{2}$ is a
rainbow path connecting $(g_{1},h_{1})$ and $(g_{2},h_{2})$ in $H_{1}$.
Case 3 $g_{1}\neq g_{2}\neq g_{3}$
Subcase 3.1 $h_{1}=h_{2}=h_{3}$
Then the $S$ lie in the copy $G_{1}$. So by the given coloring, we can claim
there is a rainbow tree $T$ containing $S$.
Subcase 3.2 $h_{1}=h_{2}\neq h_{3}$ or $h_{1}=h_{3}\neq h_{2},$ or
$h_{2}=h_{3}\neq h_{1}$
We suppose $h_{1}=h_{2}\neq h_{3}$. In this case, we first find the
corresponding vertex $(g_{3},h_{1})$ of $(g_{3},h_{3})$ in $G_{1}$. Then there
is a rainbow tree $T^{\prime}$ containing
$(g_{1},h_{1})(g_{2},h_{2})(g_{3},h_{1})$ in $G_{1}$ and a rainbow path $P$
connecting $(g_{3},h_{1})(g_{3},h_{3})$ in $H_{3}$. Thus, the rainbow tree
$T=T^{\prime}\cup P$ is our desire tree.
Subcase 3.3 $h_{1}\neq h_{2}\neq h_{3}$
Suppose $T_{1}$ be a rainbow tree containing $g_{1},g_{2},g_{3}$.
If $T_{1}$ or $T_{2}$ belongs to Type $I$, without loss of generality, we say
$T_{1}$. In order to describe graphs simply, we might suppose the leaves of
$T_{1}$ are $g_{1}$ and $g_{3}$, $T_{1}=P_{1}\cup P_{2}$, where $P_{1}$ is a
rainbow path connecting $g_{1}$ and $g_{2}$, $P_{2}$ is a rainbow path
connecting $g_{2}$ and $g_{3}$. If $P_{1}$ or $P_{2}$ is an edge, it is
trivial. So we suppose $P_{1}=g_{1}g_{1}^{\prime}g_{2}^{\prime}$ $\cdots
g_{k}^{\prime}$ $g_{2}$ and
$P_{2}=g_{2}g_{1}^{\prime\prime}g_{2}^{\prime\prime}$ $\cdots
g_{l}^{\prime\prime}g_{3}$. Thus we can construct a rainbow tree
$T_{1}^{\prime}=P_{1}^{\prime}\cup P_{2}^{\prime}$ containing $S$, where
$P_{1}^{\prime}=(g_{1},h_{1})$$(g_{1}^{\prime},h_{3})(g_{2}^{\prime},h_{1})$
$\cdots$ $(g_{k}^{\prime},u)(g_{2},h_{2})$,
$P_{2}^{\prime}=(g_{2},h_{2})(g_{1}^{\prime\prime},h_{1})(g_{2}^{\prime\prime},h_{2})$
$\cdots(g_{l}^{\prime\prime},v)(g_{3},h_{3})$, $u=h_{3}$, if $k$ is odd,
$u=h_{1}$ otherwise; $v=h_{1}$, if $l$ is odd; $v=h_{2}$ otherwise.
If $T_{1}$ and $T_{2}$ belong to Type $II$, suppose $T_{1}=P_{1}\cup P_{2}\cup
P_{3}$ and $T_{2}=Q_{1}\cup Q_{2}\cup Q_{3}$, where $P_{i},Q_{i}~{}(1\leq
i\leq 3)$ is a rainbow path connecting $g_{4}$ and $g_{i}$, $h_{4}$ and
$h_{i}$. If $P_{i}~{}(1\leq i\leq 3)$ is an edge, then it is trivial. Now we
suppose $P_{i}$ ($1\leq i\leq 3$) are not edges, then
$P_{1}$=$g_{4}l_{1}^{\prime}$$l_{2}^{\prime}\cdots l_{k}^{\prime}g_{1}$,
$P_{2}=g_{4}l_{1}^{\prime\prime}l_{2}^{\prime\prime}\cdots
l_{p}^{\prime\prime}g_{2}$, $P_{3}$=
$g_{4}l_{1}^{\prime\prime\prime}l_{2}^{\prime\prime\prime}\cdots
l_{q}^{\prime\prime\prime}g_{3}$. Similarly, the corresponding rainbow tree
$T_{1}^{\prime}=P_{1}^{\prime}\cup P_{2}^{\prime}\cup P_{3}^{\prime}$ can be
obtained containing $S$, where
$P_{1}^{\prime}=(g_{4},h_{4})(l_{1}^{\prime},h_{2})$
$(l_{2}^{\prime},h_{4})\cdots(l_{k}^{\prime},u_{1})$ $(g_{1},h_{1})$,
$P_{2}^{\prime}=(g_{4},h_{4})$$(l_{1}^{\prime\prime},h_{3})$
$(l_{2}^{\prime\prime},h_{4})\cdots(l_{p}^{\prime\prime},u_{2})$
$(g_{2},h_{2})$, $P_{3}=(g_{4},h_{4})(l_{1}^{\prime\prime\prime},h_{2})$
$(l_{2}^{\prime\prime\prime},h_{4})\cdots(l_{q}^{\prime\prime\prime},u_{3})(g_{3},h_{3})$,
$u_{1},u_{3}=h_{2},$ $u_{2}=h_{3}$ if $k,p,q$ is odd,
$u_{1},u_{2},u_{3}=h_{4}$, otherwise.
From the above discussion, we have, the given coloring is $3$-rainbow coloring
and we use $rx_{3}(G)+rc(H)$ colors totally. Thus, $rx_{3}(G[H])\leq
rx_{3}(G)+rc(H)$.
If $diam(G)=rx_{3}(G)$, and $H$ is complete, then $rx_{3}(G[H])\leq
rx_{3}(G)+rc(H)=diam(G)+1$. On the other hand, let $g,g^{\prime}\in V(G)$ such
that $d_{G}(g,g^{\prime})=diam(G)$. Let
$S=\\{(g^{\prime},h),(g,h)(g,h^{\prime})\\}$. By the Lemma 2.3,it is easy to
check that the tree containing $S$ has size at least $diam(G)+1$. So
$rx_{3}(G[H])\geq Sdiam_{3}(G[H])\geq diam(G)+1$. Thus,
$rx_{3}(G[H])=rx_{3}(G)+rc(H)$. $\sqcap\\!\\!\\!\\!\sqcup$
## 4 Other graph operations
We first consider the union of two graphs. Recall that the union of two
graphs, by starting with a disjoint union of two graphs $G$ and $H$ and adding
edges jointing every vertex of $G$ to every vertex of $H$, the resultant graph
is the join of $G$ and $H$, denoted by $G\vee H$. Note that if
$E(G)=\emptyset$ and $E(H)=\emptyset$, then the resultant graph is complete
bipartite graph. So we need some results about the 3-rainbow index of complete
bipartite graph. Li et al. got the following theorem for regular complete
bipartite graphs $K_{r,r}$.
###### Lemma 4.1
[8] For integer $r$ with $r\geq 3$, $rx_{3}(K_{r,r})=3$.
For complete bipartite graph, we obtained the following Lemmas.
###### Lemma 4.2
[14] For any complete bipartite graphs $K_{s,t}$ with $3\leq s\leq t$,
$rx_{3}(K_{s,t})\leq min\\{6,s+t-3\\}$, and the bound is tight.
In the proof of above Lemma 4.2, we showed the claim that for any $s\geq 3$,
$t\geq 2\times 6^{s}$, $rx_{3}(K_{s,t})=6$.
###### Lemma 4.3
[15] For any integer $t\geq 2$,
$rx_{3}(K_{2,t})=\left\\{\begin{array}[]{lll}2,&\mbox{ if ~{}~{}$t=2$;}\\\
3,&\mbox{ if ~{}~{}$t=3,4$;}\\\ 4,&\mbox{ if ~{}~{}$5\leq t\leq 8$;}\\\
5,&\mbox{ if ~{}~{}$9\leq t\leq 20$;}\\\ k,&\mbox{ if ~{}~{}$C_{k-1}^{2}+1\leq
t\leq C_{k}^{2}$,~{}($k\geq 6$).}\\\ \end{array}\right.$
Then, we derive the relationship between the $3$-rainbow index of the original
two graphs and that of their join graph. Note that if $G$ and $H$ are both
complete graphs, then $G\vee H$ is also the complete graph. By the Theorem
1.3, $rx_{3}(G\vee H)=3$ if $|V(G)|$+$|V(H)|$$\geq 6$; $rx_{3}(G\vee H)=2$ if
$|V(G)|$+$|V(H)|\leq 5$. So we consider the remaining cases in following
theorem.
###### Theorem 4.1
If $G$, $H$ are connected and at least one of $G$, $H$ are not complete, with
$|V(G)|=s$, $|V(H)|=t$, $s\leq t$, then we have
1\. if $s=1$, then
$rx_{3}(G\vee H)\leq rx_{3}(H)+1.$
2\. if $2=s\leq t$, then
$rx_{3}(G\vee H)\leq min\\{rc(H)+3,rx_{3}(K_{2,t})\\}.$
3\. if $3\leq s\leq t$, then
$rx_{3}(G\vee H)\leq min\\{c_{1}+1,rx_{3}(K_{s,t})\\}$
Where $c_{1}=max\\{rx_{3}(G),rx_{3}(H)\\}$.
In particular, if $s=t\geq 3$, then $rx_{3}(G\vee H)=rx_{3}(K_{s,t})=3$.
Proof: Let $G^{\prime}=G\vee H$, $V(G^{\prime})=V_{1}\cup V_{2}$ such that
$G^{\prime}[V_{1}]\cong G$, $G^{\prime}[V_{2}]\cong H$, where
$V_{1}=\\{v_{1},v_{2},\cdots,v_{s}\\}$,
$V_{2}=\\{u_{1},u_{2},\cdots,u_{t}\\}$.
1\. If $s=1$, then $G^{\prime}[V_{1}]$ is singleton vertex, we give an edge
coloring of $G^{\prime}$ as follows : we first give a 3-rainbow coloring of
$G^{\prime}[V_{2}]$ using $rx_{3}(H)$ colors. And for the other edges, that
is, elements of $E[V_{1},V_{2}]$, we use a fresh color. It is easy to show the
above coloring of $G^{\prime}$ is 3-rainbow coloring.
2\. If $2=s\leq t$, then $G^{\prime}[V_{1},V_{2}]\cong K_{2,t}$ is a spanning
subgraph of $G^{\prime}$. We have $rx_{3}(G^{\prime})\leq
rx_{3}(G^{\prime}[V_{1},V_{2}])=rx_{3}(K_{2,t})$. On the other hand, we give
an edge coloring of $G^{\prime}$ as follows: we first color the edges of the
subgraph $G^{\prime}[V_{2}]$ with $rc(H)$ colors such that it is rainbow
connected; we give the elements of $E[V_{1},V_{2}]$ incident with
$v_{i}$($i=1,2$) with color $rc(H)+i$ ($i=1,2$); for the element of
$G^{\prime}[V_{1}]$, we use a fresh color $rc(H)+3$. It is easy to show the
above coloring of $G^{\prime}$ is 3-rainbow coloring. Thus, we have
$rx_{3}(G\vee H)\leq min\\{rc(H)+3,rx_{3}(K_{2,t})\\}$.
3\. If $3\leq s\leq t$, by Lemma 4.2, we have $rx_{3}(G^{\prime})\leq
rx_{3}(G^{\prime}[V_{1},V_{2}])=rx_{3}(K_{s,t})$, similarly. On the other
hand, we color the edges of $G^{\prime}$ as follows: we first color the edges
of the subgraph $G^{\prime}[V_{i}]$ with $c_{1}$ colors such that it is
3-rainbow coloring of $G^{\prime}[V_{i}]$ ($i=1,2$). For the rest edges, that
is, elements of $E[V_{1},V_{2}]$, we use a fresh color $c_{1}+1$. It is easy
to verify that the coloring is a $3$-rainbow coloring. Thus, we get
$rx_{3}(G\vee H)\leq min\\{rx_{3}(K_{s,t}),c_{1}+1\\}$.
If $s=t\geq 3$, by Lemma 4.1, then $rx_{3}(G^{\prime})\leq rx_{3}(K_{s,s})=3$;
On the other hand, by Observation 1 and Theorem 1.3, $rx_{3}(G^{\prime})\geq
rx_{3}(K_{s+t})=3$, so the conclusion holds.
Note that $rx_{3}(K_{2,t})$ may be larger than $rc(H)+3$; for example, we
choose $H\cong K_{t}\setminus e~{}(t\geq 21)$. Then
$rx_{3}(K_{2,t})>5=rc(H)+3$ by Lemma 4.3. But $rx_{3}(K_{2,t})$ is not always
larger than $rc(H)+3$; for example, we choose $H\cong P_{t}~{}$, then
$rx_{3}(K_{2,t})<t+2=rc(H)+3$. Moreover, $rx_{3}(K_{s,t})~{}(3\leq s<t)$ may
be larger than $max\\{rx_{3}(G),rx_{3}(H)\\}+1$, since we suppose $G\cong
K_{s}\setminus e~{}(s\geq 3)$ and $H\cong K_{t}$, where $t\geq 2\times 6^{s}$.
Then $rx_{3}(K_{s,t})=6>max\\{rx_{3}(G),rx_{3}(H)\\}+1$. But $rx_{3}(K_{s,t})$
is not always larger than $max\\{rx_{3}(G),rx_{3}(H)\\}+1$. Similarly, for
example, $G,H\cong P_{s}$ ($s\geq 7$), we can get
$max\\{rx_{3}(G),rx_{3}(H)\\}+1=s>6\geq rx_{3}(K_{s,t})$. So the bounds we
give in the theorem are reasonable. $\sqcap\\!\\!\\!\\!\sqcup$
Recall that to split $v$ of a graph $G$ is to replace $v$ by two adjacent
vertices $v_{1}$ and $v_{2}$ by an edge incident to either $v_{1}$ or $v_{2}$
(but not both), the other end of the edge remaining unchanged. The Figure $6$
shows the operation of $G$. Let $N_{G}(v)$ be the neighbor sets of $v$. The
set is partitioned into two disjoint sets $N_{1}$ and $N_{2}$ such that
$N_{1}$ and $N_{2}$ are the neighbor sets of $v_{1}$ and $v_{2}$ in the
resultant graph, respectively.
Figure 6 : The operation for vertex spliting.
###### Theorem 4.2
If $G$ is a connected graph and $G^{\prime}$ is obtained from $G$ by splitting
a vertex $v$, then
$rx_{3}(G^{\prime})\leq rx_{3}(G)+1.$
Proof: We first give $G$ a $3$-rainbow coloring with $rx_{3}(G)$ colors, then
we give $G^{\prime}$ a $rx_{3}(G)$+1-edge coloring as follows: we give the
edge $e=v_{1}v_{2}$ a color $rx_{3}(G)$+1; for any edge $uv_{1}\in G^{\prime}$
with $uv_{1}\neq e$, let the color of $uv_{1}$ be the same as that of $uv$ in
$G$; for any edge $v_{2}w\in G^{\prime}$ with $v_{2}w\neq e$, let the color of
$v_{2}w$ be the same as that of $vw$ in $G$; color of the rest edges of
$G^{\prime}$ are the same as in $G$. Next, we will show the given coloring of
$G^{\prime}$ is a 3-rainbow coloring. It suffices to show that there is a
rainbow tree containing any three vertices of $G^{\prime}$. Let
$S=\\{x,y,z\\}$.
Case 1 Two vertices of $S$ belongs to $\\{v_{1},v_{2}\\}$, say $x=v_{1}$,
$y=v_{2}$.
By the above coloring, there a rainbow $v-z$ path $P:v=u_{1},\cdots,u_{t}=z$.
If $u_{2}\in N_{1}$, then $P^{\prime}:v_{1},u_{2},u_{3},\cdots,u_{t}=z$ is a
rainbow connecting $z$ and $x(v_{1})$. Thus, $T=P^{\prime}\cup e$ is the
rainbow tree containing $S$. If $u_{2}\in N_{2}$, it is similar to verify that
there is a rainbow tree containing $S$.
Case 2 Exactly one of $S$ belongs to $\\{v_{1},v_{2}\\}$, say $x=v_{1}$.
We know that, in graph $G$, there is a rainbow tree $T_{1}$ containing
$y,z,v$.
subcase 2.1 $d_{T_{1}}(v)=1$.
Then there is an edge $uv\in E(T_{1})$. If $u\in N_{1}$, the tree obtained
from $T_{1}$ by replacing $v$ with $v_{1}$ is rainbow and contains $S$. If
$u\in N_{2}$, the tree obtained from $T_{1}$ by replacing $v$ with
$v_{2},~{}v_{1}$ is a rainbow tree containing $S$.
subcase 2.2 $d_{T_{1}}(v)\neq 1$.
From the Observation 2, we claim $d_{T_{1}}(v)=2$. Let $u_{1}$ and $u_{2}$ be
the two neighbors of $v$ in $T_{1}$. If $u_{1}$ and $u_{2}$ belong to the
$N_{1}$, then let $T$ be obtained from $T_{1}$ by replacing $v$ with $v_{1}$.
If $u_{1}$ and $u_{2}$ belong to the $N_{2}$, then we can find a rainbow tree
$T=T_{2}\cup e$, where $T_{2}$ is obtained from $T_{1}$ by replacing $v$ with
$v_{2}$. If $u_{1}$ and $u_{2}$ belong to the different $N_{i}~{}(i=1,2)$,
then $T$ obtained from $T_{1}$ by replacing $v$ with subgraph $v_{1}v_{2}$ is
rainbow.
Case 3 None of vertices in $S$ belongs to $\\{v_{1},v_{2}\\}$.
We know that there is a rainbow $T_{3}$ containing $S$ in $G$. If $v$ does not
belong to $T_{3}$, then $T_{3}$ is also a rainbow tree containing $S$ in
$G^{\prime}$.
If $v$ belong to the tree $T_{3}$, by the Observation 2, then
$d_{T_{3}}(v)=2,~{}3$. Similar to the Subcase 2.2, we can find a rainbow tree
containing $S$.
So $G^{\prime}$ receives a 3-rainbow coloring. Since we use $rx_{3}(G)+1$
colors totally, then $rx_{3}(G^{\prime})\leq rx_{3}(G)+1$.
$\sqcap\\!\\!\\!\\!\sqcup$
A special case of vertex splitting occurs when exactly one link is assigned to
either $v_{1}$ or $v_{2}$. The resulting graph can be viewed as having been
obtained by subdividing an edge of the original graph, where to subdivide an
edge is to delete $e$, add a new vertex $x$, and join $x$ to the ends of $e$.
So by Theorem 4.2, we have
###### Corollary 4.1
If $G$ is a connected graph, and $G^{\prime}$ is obtained from $G$ by
subdividing an edge $e$, then
$rx_{3}(G^{\prime})\leq rx_{3}(G)+1.$
## 5 Conclusion
Rainbow connection number $rc(G)$ ($rx_{2}(G)$) comes from the communication
of information between agencies of government. $3$-rainbow index, $rx_{3}(G)$,
is a generalization of rainbow connection number. Chakraborty et al. have
proved that computing $rc(G)$($rx_{2}(G)$) is NP-hard. Hence, To get the exact
value for 3-rainbow index of general graph $G$ is also NP-hard. Thus,
researchers tend to get some better upper for 3-rainbow index of some classes
of graphs. Graph operations, both binary and unary, are interesting subjects,
which can be used to understand structures of graphs. In this paper, we will
study the $3$-rainbow index with respect to three important graph product
operations (namely cartesian product, strong product, lexicographic product)
and other graph operations. In this direction, we firstly show if
$G^{*}=G_{1}\Box G_{2}\cdots\Box G_{k}$ ($k\geq 2$), where each $G_{i}$ is
connected, then $rx_{3}(G^{*})\leq\sum_{i=1}^{k}rx_{3}(G_{i})$. Moreover, we
also present a condition and show the above equality holds if every graph
$G_{i}~{}(1\leq i\leq k)$ meets the condition. As a corollary, we obtain an
upper bound for the 3-rainbow index of strong product. Secondly, we discuss
the 3-rainbow index of the lexicographic graph $G[H]$ for connected graph $G$
and $H$. The proofs are constructive and hence yield the sharp bound. Finally,
we consider the relationship between the 3-rainbow index of original graphs
and other simple graph operations : the join of $G$ and $H$, split a vertex of
a graph and subdivide an edge and get the upper bounds.
Acknowledgements: The corresponding author, Yumei Hu, is supported by NSFC No.
11001196.
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|
arxiv-papers
| 2013-11-30T12:11:01 |
2024-09-04T02:49:54.552321
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tingting Liu and Yumei Hu",
"submitter": "Liu Tingting",
"url": "https://arxiv.org/abs/1312.0098"
}
|
1312.0135
|
# On annulus containing all the zeros of a polynomial
N. A. Rather and Suhail Gulzar Department of Mathematics
University of kashmir
Srinagar, Hazratbal 190006
India
###### Abstract.
In this paper, we obtain an annulus containing all the zeros of the polynomial
involving binomial coefficients and generalized Fibonacci numbers. Our result
generalize some of the recently obtained results in this direction.
###### Key words and phrases:
Polynomials; Location of zeros of polynomials.
###### 2010 Mathematics Subject Classification:
primary: 30C10, 30C15.
Department of Mathematics, University of Kashmir Hazratbal
Srinagar 190006, India
emails: [email protected], [email protected],
## 1\. Introduction and Statements
Gauss and Cauchy were the earliest contributors in the theory of the location
of zeros of a polynomial, since then this subject has been studied by many
people (for example, see [3, 4]). There is always a need for better and better
results in this subject because of its application in many areas, including
signal processing, communication theory and control theory.
A classical result due to Cauchy (see [3, p. 122]) on the distribution of
zeros of a polynomial may be stated as follows:
###### Theorem A.
If $P(z)=z^{n}+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots+a_{0}$ is a polynomial
with complex coefficients, then all zeros of $P(z)$ lie in the disk $|z|\leq
r$ where $r$ is the unique positive root of the real-coefficient polynomial
$Q(x)=x^{n}-|a_{n-1}|x^{n-1}-|a_{n-2}|x^{n-2}-\cdots-|a_{1}|x-|a_{0}|.$
Recently Díaz-Barrero [1] improved this estimate by identifying an annulus
containing all the zeros of a polynomial, where the inner and outer radii are
expressed in terms of binomial coefficients and Fibonacci numbers. In fact he
has proved the following result.
###### Theorem B.
Let $P(z)=\sum_{j=0}^{n}a_{j}z^{j}$ be a non-constant complex polynomial. Then
all its zeros lie in the annulus $C=\\{z\in\mathbb{C}:r_{1}\leq|z|\leq
r_{2}\\}$ where
$r_{1}=\frac{3}{2}\underset{1\leq k\leq
n}{\min}\left\\{\dfrac{2^{n}F_{k}\binom{n}{k}}{F_{4n}}\left|\frac{a_{0}}{a_{k}}\right|\right\\}^{\frac{1}{k}},\,\,\,r_{2}=\frac{2}{3}\underset{1\leq
k\leq
n}{\max}\left\\{\dfrac{F_{4n}}{2^{n}F_{k}\binom{n}{k}}\left|\frac{a_{n-k}}{a_{n}}\right|\right\\}^{\frac{1}{k}}.$
Here $F_{j}$ are Fibonacci’s numbers, that is, $F_{0}=0,$ $F_{1}=1$ and for
$j\geq 2,$ $F_{j}=F_{j-1}+F_{j-2}.$
More recently, M. Bidkham et. al. [2] considered $t$-Fibonacci numbers, namely
$F_{t,n}=tF_{t,n-1}+F_{t,n-2}$ for $n\geq 2$ with initial condition
$F_{t,0}=0,\,F_{t,1}=1$ where $t$ is any positive real number and obtained the
following generalization of Theorem B.
###### Theorem C.
Let $P(z)=\sum_{j=0}^{n}a_{j}z^{j}$ be a non-constant complex polynomial of
degree $n$ and
$\lambda_{k}=\dfrac{(t^{3}+2t)^{k}(t^{2}+1)^{n}F_{t,k}\binom{n}{k}}{(t^{2}+1)^{k}F_{t,4n}}$
for any real positive number $t.$ Then all the zeros of $P(z)$ lie in the
annulus $R=\\{z\in\mathbb{C}:s_{1}\leq|z|\leq s_{2}\\}$ where
$s_{1}=\underset{1\leq k\leq
n}{\min}\left\\{\lambda_{k}\left|\dfrac{a_{0}}{a_{k}}\right|\right\\}^{\frac{1}{k}},\quad
s_{2}=\underset{1\leq k\leq
n}{\max}\left\\{\dfrac{1}{\lambda_{k}}\left|\dfrac{a_{n-k}}{a_{n}}\right|\right\\}^{\frac{1}{k}}.$
In this paper, we determine in the complex plane an annulus containing all the
zeros of a polynomial involving binomial coefficients and generalized
Fibonacci numbers (see [5]) defined recursively by
$\displaystyle F_{0}^{(a,b,c)}$ $\displaystyle=0,\,\,\,F_{1}^{(a,b,c)}=1,$
(1.1) $\displaystyle F_{n}^{(a,b,c)}$
$\displaystyle=\begin{cases}a\,F_{n-1}^{(a,b,c)}+c\,F_{n-2}^{(a,b,c)}\quad\textnormal{if
n is even,}\\\ b\,F_{n-1}^{(a,b,c)}+c\,F_{n-2}^{(a,b,c)}\quad\textnormal{if n
is odd,}\end{cases}(n\geq 2)$
where $a,b,c$ are any three positive real numbers. Our result include Theorems
B, C as special cases. More precisely, we prove the following result.
###### Theorem 1.1.
Let $P(z)=\sum_{j=0}^{n}a_{j}z^{j}$ be a non-constant complex polynomial of
degree $n.$ Then all its zeros lie in the annulus
$C=\\{z\in\mathbb{C}:r_{1}\leq|z|\leq r_{2}\\}$ where
$r_{1}=\dfrac{uv+2w}{uvw+w^{2}}\,\underset{1\leq k\leq
n}{\min}\left\\{\dfrac{(uvw+w^{2})^{n}u^{\xi(k)}(uv)^{\lfloor\frac{k}{2}\rfloor}F_{k}^{(u,v,w)}\binom{n}{k}}{F_{4n}^{(u,v,w)}}\left|\dfrac{a_{0}}{a_{k}}\right|\right\\}^{\frac{1}{k}},$
$r_{2}=\dfrac{abc+c^{2}}{ab+2c}\,\underset{1\leq k\leq
n}{\max}\left\\{\dfrac{F_{4n}^{(a,b,c)}}{(abc+c^{2})^{n}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}F_{k}^{(a,b,c)}\binom{n}{k}}\left|\dfrac{a_{n-k}}{a_{n}}\right|\right\\}^{\frac{1}{k}},$
$a,b,c,u,v,w$ are any positive real numbers,
$\xi(k):=k-2\lfloor\frac{k}{2}\rfloor$ and $F_{m}^{(a,b,c)}$ is defined as in
(1).
###### Remark 1.2.
By taking $a,b,c$ and $u,v,w$ suitably, we shall obtain Theorems B, C. For
example, if we take $a=b=u=v=t$ and $c=w=1,$ we obtain Theorem C.
###### Example 1.3.
We consider the polynomial $P(z)=z^{3}+0.1z^{2}+0.3z+0.7,$ which is the only
example considered by Díaz-Barrero [1] and by using Theorem B, the annulus
containing all the zeros of $P(z)$ comes out to be $0.58<|z|<1.23$. We
improved the upper bound of this annulus by taking $a=1/2,$ $b=1$ and $c=3/8$
in Theorem 1.1 and obtained the disk, $|z|<1.185,$ which contains all the
zeros of polynomial $P(z).$ We can similarly improve the lower bound by
choosing $u,$ $v,$ $w$ suitably.
## 2\. Lemma
To prove the above theorem, we need the following lemma.
###### Lemma 2.1.
If $F_{k}^{(a,b,c)}$ is defined as in (1), then
(2.1)
$\displaystyle\sum\limits_{k=1}^{n}(ab+c)^{n-k}(ab+2c)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}c^{n-k}F_{k}^{(a,b,c)}\binom{n}{k}=F_{4n}^{(a,b,c)}$
where $\xi(k)=k-2\lfloor\frac{k}{2}\rfloor.$
###### Proof.
For $F_{k}^{(a,b,c)},$ we have [5]
$F_{k}^{(a,b,c)}=\dfrac{a^{1-\xi(k)}}{(ab)^{\lfloor\frac{k}{2}\rfloor}}\left(\dfrac{\alpha^{k}-\beta^{k}}{\alpha-\beta}\right)$
where $\alpha=\frac{ab+\sqrt{(ab)^{2}+4abc}}{2},$
$\beta=\frac{ab-\sqrt{(ab)^{2}+4abc}}{2}$ and
$\xi(k)=k-2\lfloor\frac{k}{2}\rfloor.$
Consider,
$\displaystyle\sum\limits_{k=1}^{n}$
$\displaystyle\binom{n}{k}(abc)^{n-k}\big{[}(ab)^{2}+abc\big{]}^{n-k}\big{[}(ab)^{3}+2(ab)^{2}c\big{]}^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}F_{k}^{(a,b,c)}$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{n}\binom{n}{k}(-1)^{n-k}(\alpha\beta)^{n-k}\Bigg{(}\sum\limits_{j=0}^{2}\alpha^{j}\beta^{2-j}\Bigg{)}^{n-k}\Bigg{(}\sum\limits_{j=0}^{3}\alpha^{j}\beta^{3-j}\Bigg{)}^{k}a\left(\dfrac{\alpha^{k}-\beta^{k}}{\alpha-\beta}\right)$
$\displaystyle=$
$\displaystyle\dfrac{a\alpha^{n}}{\alpha-\beta}\Bigg{\\{}\sum\limits_{k=1}^{n}\binom{n}{k}(-1)^{n-k}\Bigg{(}\sum\limits_{j=0}^{2}\alpha^{j}\beta^{3-j}\Bigg{)}^{n-k}\Bigg{(}\sum\limits_{j=0}^{3}\alpha^{j}\beta^{3-j}\Bigg{)}^{k}\Bigg{\\}}$
$\displaystyle-\dfrac{a\beta^{n}}{\alpha-\beta}\Bigg{\\{}\sum\limits_{k=1}^{n}\binom{n}{k}(-1)^{n-k}\Bigg{(}\sum\limits_{j=0}^{2}\alpha^{1+j}\beta^{2-j}\Bigg{)}^{n-k}\Bigg{(}\sum\limits_{j=0}^{3}\alpha^{j}\beta^{3-j}\Bigg{)}^{k}\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\dfrac{a\alpha^{n}}{\alpha-\beta}\Bigg{\\{}\sum\limits_{j=0}^{3}\alpha^{j}\beta^{3-j}-\sum\limits_{j=0}^{2}\alpha^{j}\beta^{3-j}\Bigg{\\}}^{n}-\dfrac{a\beta^{n}}{\alpha-\beta}\Bigg{\\{}\sum\limits_{j=0}^{3}\alpha^{j}\beta^{3-j}-\sum\limits_{j=0}^{2}\alpha^{1+j}\beta^{2-j}\Bigg{\\}}^{n}$
$\displaystyle=$ $\displaystyle
a\left(\dfrac{\alpha^{n}(\alpha^{3})^{n}-\beta^{n}(\beta^{3})^{n}}{\alpha-\beta}\right)=(ab)^{2n}F_{4n}^{(a,b,c)}.$
Equivalently, we have
$\displaystyle\sum\limits_{k=1}^{n}\binom{n}{k}(ab+c)^{n-k}(ab+2c)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}c^{n-k}F_{k}^{(a,b,c)}=F_{4n}^{(a,b,c)}.$
∎
## 3\. Proof of Theorem
###### Proof of Theorem 1.1.
We first show that all the zeros of $P(z)$ lie in
(3.1) $\displaystyle|z|\leq r_{2}=\underset{1\leq k\leq
n}{\max}\left\\{\dfrac{(ab+c)^{k}c^{k}F_{4n}^{(a,b,c)}}{(ab+c)^{n}(ab+2c)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}c^{n}F_{k}^{(a,b,c)}\binom{n}{k}}\left|\frac{a_{n-k}}{a_{n}}\right|\right\\}^{\frac{1}{k}}$
where $a,b,c$ are any three positive real numbers. From (3.1), it follows that
$\displaystyle\left|\frac{a_{n-k}}{a_{n}}\right|\leq
r_{2}^{k}\dfrac{(ab+c)^{n}(ab+2c)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}c^{n}F_{k}^{(a,b,c)}\binom{n}{k}}{(ab+c)^{k}c^{k}F_{4n}^{(a,b,c)}},\quad
k=1,2,3,\cdots,n$
or
(3.2)
$\displaystyle\sum\limits_{k=1}^{n}\left|\frac{a_{n-k}}{a_{n}}\right|\dfrac{1}{r_{2}^{k}}\leq\sum\limits_{k=1}^{n}\dfrac{(ab+c)^{n}(ab+2c)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}c^{n}F_{k}^{(a,b,c)}\binom{n}{k}}{(ab+c)^{k}c^{k}F_{4n}^{(a,b,c)}}.$
Now, for $|z|>r_{2},$ we have
$\displaystyle|P(z)|=$
$\displaystyle|a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0}|$
$\displaystyle\geq$
$\displaystyle|a_{n}||z|^{n}\left\\{1-\sum\limits_{k=1}^{n}\left|\frac{a_{n-k}}{a_{n}}\right|\dfrac{1}{|z|^{k}}\right\\}$
$\displaystyle>$
$\displaystyle|a_{n}||z|^{n}\left\\{1-\sum\limits_{k=1}^{n}\left|\frac{a_{n-k}}{a_{n}}\right|\dfrac{1}{r_{2}^{k}}\right\\}.$
Using (2.1) and (3.2), we have for $|z|>r_{2},$ $|P(z)|>0.$ Consequently all
the zeros of $P(z)$ lie in $|z|\leq r_{2}$ and this proves the second part of
theorem.
To prove the first part of the theorem, we will use second part. If $a_{0}=0,$
then $r_{1}=0$ and there is nothing to prove. Let $a_{0}\neq 0,$ consider the
polynomial
$Q(z)=z^{n}P(1/z)=a_{0}+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_{n}.$
By second part of the theorem for any three positive real numbers $u,v,w$, if
$Q(z)=0,$ then
$\displaystyle|z|\leq$ $\displaystyle\underset{1\leq k\leq
n}{\max}\left\\{\dfrac{(uv+w)^{k}w^{k}F_{4n}^{(u,v,w)}}{(uv+w)^{n}(uv+2w)^{k}u^{\xi(k)}(uv)^{\lfloor\frac{k}{2}\rfloor}w^{n}F_{k}^{(au,v,w)}\binom{n}{k}}\left|\frac{a_{k}}{a_{0}}\right|\right\\}^{1/k}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\underset{1\leq k\leq
n}{\min}\left\\{\dfrac{(uv+w)^{k}w^{k}F_{4n}^{(u,v,w)}}{(uv+w)^{n}(uv+2w)^{k}a^{\xi(k)}(ab)^{\lfloor\frac{k}{2}\rfloor}w^{n}F_{k}^{(u,v,w)}\binom{n}{k}}\left|\frac{a_{0}}{a_{k}}\right|\right\\}^{1/k}}$
$\displaystyle=$ $\displaystyle\frac{1}{r_{1}}.$
Now replacing $z$ by $1/z$ and observing that all the zeros of $P(z)$ lie in
$|z|\geq r_{1}=\underset{1\leq k\leq
n}{\min}\left\\{\dfrac{(uv+w)^{k}w^{k}F_{4n}^{(u,v,w)}}{(uv+w)^{n}(uv+2w)^{k}u^{\xi(k)}(uv)^{\lfloor\frac{k}{2}\rfloor}w^{n}F_{k}^{(u,v,w)}\binom{n}{k}}\left|\frac{a_{0}}{a_{k}}\right|\right\\}^{\frac{1}{k}}.$
This completes the proof of theorem 1.1. ∎
Acknowledgement
The second author is supported by Council of Scientific and Industrial
Research, New Delhi, under grant F.No. 09/251(0047)/2012-EMR-I.
## References
* [1] J. L. Díaz-Barrero, An annulus for the zeros of polynomials, J. Math. Anal. Appl., 273 (2002) 349-352.
* [2] M. Bidkham, E. Shashahani, An annulus for the zeros of polynomials, Appl. Math. Lett., 24 (2011) 122-125.
* [3] M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math. Soc. Providence R. I. 1966.
* [4] G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore, (1994).
* [5] Omer Yayenie, A note on generalized Fibonacci sequences, Appl. Math. Comput., 208 (2009) 180-185.
|
arxiv-papers
| 2013-11-30T18:11:08 |
2024-09-04T02:49:54.568822
|
{
"license": "Public Domain",
"authors": "N. A. Rather and Suhail Gulzar",
"submitter": "Suhail Gulzar Mattoo",
"url": "https://arxiv.org/abs/1312.0135"
}
|
1312.0169
|
###### Abstract
Using mobile phone records and information theory measures, our daily lives
have been recently shown to follow strict statistical regularities, and our
movement patterns are, to a large extent, predictable. Here, we apply entropy
and predictability measures to two datasets of the behavioral actions and the
mobility of a large number of players in the virtual universe of a massive
multiplayer online game. We find that movements in virtual human lives follow
the same high levels of predictability as offline mobility, where future
movements can, to some extent, be predicted well if the temporal correlations
of visited places are accounted for. Time series of behavioral actions show
similar high levels of predictability, even when temporal correlations are
neglected. Entropy conditional on specific behavioral actions reveals that in
terms of predictability, negative behavior has a wider variety than positive
actions. The actions that contain the information to best predict an
individual’s subsequent action are negative, such as attacks or enemy
markings, while the positive actions of friendship marking, trade and
communication contain the least amount of predictive information. These
observations show that predicting behavioral actions requires less information
than predicting the mobility patterns of humans for which the additional
knowledge of past visited locations is crucial and that the type and sign of a
social relation has an essential impact on the ability to determine future
behavior.
###### keywords:
human behavior; mobility; computational social science; online games; time-
series analysis; social dynamics
10.3390/e16010543 16 Received: 01 December 2013; in revised form: 16 December
2013 / Accepted: 30 December 2013 /
Published: 16 January 2014 Entropy and the Predictability of Online Life
Roberta Sinatra 1 and Michael Szell 2,* E-Mail: [email protected]; Tel.:
+1-617-324-4474; Fax: +1-617-258-8081.
## 1 Introduction
Capturing the regularities of our daily lives and the occasional deviations
from the steady diurnal patterns has traditionally eluded an all-encompassing
approach, due to tremendous efforts in monitoring detailed human activities
over long times and the bias in behavior caused by obtrusive methods of
observation Rosenthal (1991). However, the recent ability to address questions
in social science by using huge datasets that have emerged over the past
decades as a result of digitalization has opened previously unimaginable ways
of conducting research in the field Lazer et al. (2009).
On the one hand, these new datasets give a highly detailed protocol of our
ordinary lives, for example, in the form of mobile phone data, which enables a
deeper understanding of the regularities in our mobility patterns González et
al. (2008); Schneider et al. (2013), and how the regularities in human
behavior are reflected in the geographic regions that emerge from our
interactions Sobolevsky et al. (2013). On the other hand, from a previous
point of view, “extraordinary” new forms of human behavior can now be observed
online, where the full set of all actions performed in the system is typically
available for study, spanning an even deeper level of detail. Online social
networking services, such as Twitter or Facebook, or discussion forums allow
new insights into the rhythms of social actions and interactions, as expressed
online Golder and Macy (2011); Golder et al. (2007); Mitrović and Tadić
(2010); Tadić et al. (2013), and how these interactions relate to the
underlying offline events Szell et al. (2013). An even richer insight can be
gained into human-led lives that unfold _entirely_ in artificial online
environments, such as in persistent, massive multiplayer online games, where
human-controlled characters spend their whole virtual lives within an online
world interacting with other characters Bainbridge (2007). The playing of
online games is one of the most wide-spread forms of collective human behavior
in the world; the “massive multiplayer” aspect allows one to not only study
single individuals, but also collective behavioral phenomena that typically
emerge in complex social systems Ball (2003). Here, data can be available of
all actions, decisions and interactions between many thousands of individuals
over long time spans Szell and Thurner (2012), allowing understanding of the
structure and evolution of socio-economic networks Szell et al. (2010); Szell
and Thurner (2010); Klimek and Thurner (2013), mobility Szell et al. (2012) or
the emergence of good conduct Thurner et al. (2012) and elite structures
Corominas-Murtra et al. (2013) in large social systems.
Going hand in hand with the new availability of large-scale, longitudinal
behavioral datasets of various kinds, well-known methods from the mathematical
and physical sciences, especially statistical physics and information theory
Castellano et al. (2009); Sinatra et al. (2010); Gallotti et al. (2012), have
been extended and/or re-applied successfully in this context. In particular,
principal component analysis and the concept of “eigenbehavior” has been used
to quantify behavioral regularities and to predict future activities in the
daily lives of a group of 100 subjects Eagle and Pentland (2009). Similarly,
information theory measures provide an adequate quantification between uniform
distributions (maximal entropy) and maximally uneven distributions of states
(minimal entropy), which, in the case of human behavior, can inform us about
the extent of uniformity and, thus, predictability in our activity patterns.
The concept of entropy has been applied specifically to assess the
predictability of mobility patterns Song et al. (2010); Gallotti et al.
(2013), of economic behavior Krumme et al. (2010), the order of human-built
structures, such as urban street networks Gudmundsson and Mohajeri (2013), or
the complexity of online chatting behavior Takaguchi et al. (2011); Wang and
Huberman (2012); Tadić et al. (2013). Further, a theoretical framework for
non-extensive entropies has been recently developed that might be well
applicable tocomplex systems Hanel and Thurner (2011); Hanel et al. (2012).
## 2 Behavior and Mobility Data of Human Players in the Online World, Pardus
Here, our goal is to apply classical entropy measures to study the patterns of
various kinds of behavior in a single, closed socio-economic system, as
generated by thousands of users in the online game “Pardus” Szell and Thurner
(2012), to provide an insight into the regularity of life in online worlds
and, eventually, to draw possible conclusions on how humans lead their offline
lives.
### 2.1 The Online World, Pardus
The online world Pardus, www.pardus.at, is a browser-based, massive
multiplayer online game open to the public for over nine years. Over 400,000
users have registered to play so far. The game features three independent,
persistent game universes, which had a defined starting time, but no scheduled
end. There are no predefined goals in the game: many aspects of social life
within Pardus are self-organized, for example, the emergence of social groups
(alliances) and the politics between them. Players are engaged in a multitude
of social activities, i.e., chatting, cultivating friendships, building up
alliances, but also negative interactions, such as destructive attacks, and
economic activities, such as producing commodities in factories and selling
them to other players. We focus on the “Artemis” game universe, in which we
recorded player actions over the first 1,238 consecutive days of the
universe’s existence. Communication between any two players can take place
directly, by using a one-to-one, e-mail-like private messaging system. We
focus on one-to-one interactions between players only and discard indirect
interactions, such as, e.g., participation in chats or forums. There are
global interactions, i.e., interactions that can be performed independently of
the spatial position of players in the game universe, which are communication,
setting and removing friendship or enemy links, or placing a bounty on another
player. The actions of trade or attack, however, need players to meet in
space. All data used in this study are fully anonymized.
### 2.2 Mobility Time Series
For studying regularities in mobility patterns, we use the same dataset of
Pardus player movements that has been used in Szell et al. (2012). A universe
in Pardus can be represented as a network with 400 nodes, called sectors, and
1,160 links. Each sector is like a city, where players can have social
relations or entertain economic activities. Typically, sectors adjacent on the
universe map, as well as a few far-apart sectors, are interconnected by links
that allow players to move from sector to sector. At any point in time, each
sector is usually attended by a large number of players. The universe network
has a large diameter of 27, which means that, on average, players have to move
through a non-negligible number of sectors to traverse the universe. Due to a
limited pool of actions that players can spend on movement, traveling large
distances can take a player several days. Using this dataset, we previously
studied the statistical movement patterns of players and found that locations
are visited in a specific order, leading to strong long-term memory effects
Szell et al. (2012). In detail, we extract player mobility data from day 200
to day 1,200 of the universe’s existence. We discard the first 200 days,
because social networks between players of Pardus have shown aging effects in
the beginning of the universe Szell and Thurner (2010). To make sure we only
consider active players, we select all who exist in the game between the days
200 and 1,200, yielding 1,458 players active over a time-period of 1,000 days.
The sector IDs of these players, i.e., their positions on the universe
network’s nodes, are logged every day at 05:35 GMT.
### 2.3 Behavioral Action Time Series
For studying regularities in behavioral time series, we use the same dataset
of Pardus player actions that has been used in Thurner et al. (2012). Players
can express their sympathy (distrust) toward other players by establishing so-
called friendship (enmity) links. These links are only seen by the player
marking another as a friend (enemy) and the respective recipient of that link.
For more details on the game, see Szell and Thurner (2010). We consider eight
different actions every player can execute at any time. These are
communication (C),trade (T), setting a friendship link (F), removing an enemy
link (forgiving) (X), attack (A), placing a bounty on another player
(punishment) (B), removing a friendship link (D) and setting an enemy link
(E). While C, T, F and X can be associated with positive actions, A, B, D and
E are hostile or negative actions. We classify communication as positive,
because only a negligible part of communication takes place between enemies
Szell and Thurner (2010). Following a previous formalism Szell and Thurner
(2010), we say that positive actions have a positive sign, and negative
actions have a negative sign. The alphabet, $\mathcal{X}$, of all possible
dyadic actions happening in each player’s life therefore spans 16 letters:
eight possible performed actions (four negative, four positive) and eight
possible received actions (four negative, four positive). We denote received
actions with the suffix, $r$, e.g., $A_{r}$ for a received attack. Due to the
heterogeneous activity patterns of players, we operate in action-time rather
than in actual time; for example, indices of $t$ and $t-1$ denote that two
actions were subsequent, regardless of whether the actual time difference was
seconds or weeks Thurner et al. (2012). From all sequences of all 34,055
Artemis players who performed or received an action at least once within 1,238
days, we removed players with a life history of less than 1,000 actions,
leading to the set of the most active 1,758 players that are considered
throughout this work.
## 3 Entropy and Predictability Measures
To study the regularity and predictability of behavior from the discrete time
series, we use three entropy measures. Following Song et al. (2010), we call
the binary logarithm of the number of distinct states, $N_{i}$, of a player,
$i$, the _random entropy_ :
$S^{\mathrm{rand}}_{i}=\log_{2}N_{i}$ (1)
In the case of mobility, “states” refer to the 400 possible sectors in the
universe visitable at a given point in time by a player. The maximal possible
random entropy is $S^{\mathrm{rand}}=\log_{2}400\approx 8.6$, reached when all
sectors are visited at least once. In the case of behavioral actions, a state
can be one of the 16 possible action or received action types; here, the
maximum possible random entropy is $S^{\mathrm{rand}}=\log_{2}16=4$.
The Shannon entropy, $S^{\mathrm{unc}}_{i}$, of a player, $i$, is defined as:
$S^{\mathrm{unc}}_{i}=-\sum_{x\in\mathcal{X}_{i}}p_{i}(x)\log_{2}p_{i}(x)$ (2)
where $p_{i}(x)$ is the measured probability over the respective time span
that player $i$ has occupied a state, $x$, and $\mathcal{X}_{i}$ is the
ensemble of the $N_{i}$ distinct states. In this context, we call the Shannon
entropy the _temporal-uncorrelated entropy_ , because it captures the entropy
when the temporal order of states is ignored Song et al. (2010). The random
and temporal-uncorrelated entropies are equal,
$S^{\mathrm{rand}}_{i}=S^{\mathrm{unc}}_{i}$, if all of the $N_{i}$ distinct
states, $x$, were occupied with uniform probability $p_{i}(x)=1/N_{i}$ by the
player, $i$. For mobility, the occupation of a single sector over the whole
time span of 1,000 days would result in the smallest possible random and
temporal-uncorrelated entropy of $S^{\mathrm{rand}}=S^{\mathrm{unc}}=0$.
Finally, we make use of the conditional entropy $S^{\mathrm{cond}}_{i}$ of a
player, $i$, capturing the entropy conditional on temporal short-term
correlations over one previous state in the time series,
$S^{\mathrm{cond}}_{i}=-\sum_{x_{t}\in\mathcal{X}_{i}}\sum_{x_{t-1}\in\mathcal{X}_{i}}p_{i}(x_{t-1},x_{t})\log_{2}p_{i}(x_{t}|x_{t-1})$
(3)
with $p_{i}(x_{t-1},x_{t})$ being the probability of occurrence of the pair of
subsequent states, $x_{t-1}$ and $x_{t}$,
$p_{i}(x_{t}|x_{t-1})=p_{i}(x_{t-1},x_{t})/p(x_{t-1})$, the probability of the
state, $x_{t}$, at time $t$ given a preceding state, $x_{t-1}$. The
conditional and temporal-uncorrelated entropies are equal,
$S^{\mathrm{cond}}_{i}=S^{\mathrm{unc}}_{i}$, if there are no temporal
correlations.
It is easy to show that we have $S^{\mathrm{cond}}\leq S^{\mathrm{unc}}\leq
S^{\mathrm{rand}}$ for each user Cover and Thomas (2006). The differences in
these two inequalities quantify the effects of short-term temporal
correlations and the uniformity of the occupation distribution, respectively.
To assess the predictability of specific states or of classes of states, we
also define the conditional entropy for the set of states, $\mathcal{Z}$,
$S^{\mathrm{cond}}_{i}(\mathcal{Z})=-\sum_{x_{t}\in\mathcal{X}_{i}}\sum_{x_{t-1}\in\mathcal{X}_{i}\cap\mathcal{Z}}p_{i}(x_{t-1},x_{t})\log_{2}p_{i}(x_{t}|x_{t-1})$
(4)
which is the conditional entropy given that the previous state belonged to
$\mathcal{Z}$, where $\mathcal{Z}$ can be fixed as any subset of all the
possible states, $\mathcal{X}$. Notice that $S^{\mathrm{cond}}_{i}\equiv
S^{\mathrm{cond}}_{i}(\mathcal{Z})+S^{\mathrm{cond}}_{i}(\mathcal{X}_{i}\setminus\mathcal{Z})$.
Complementary to entropy measures of information content or _unpredictability_
are measures of _predictability_ that denote in a percent value how likely an
appropriate predictive algorithm could foresee an individual’s future behavior
Song et al. (2010). The predictability, $\Pi_{i}^{\bullet}$, of an individual
$i$ is bounded above by:
$S_{i}^{\bullet}=H(\Pi_{i}^{\bullet})+(1-\Pi_{i}^{\bullet})\log_{2}(N_{i}-1)$
(5)
with the binary entropy function:
$H(\Pi_{i}^{\bullet})=\Pi_{i}^{\bullet}\log_{2}(\Pi_{i}^{\bullet})-(1-\Pi_{i}^{\bullet})\log_{2}(1-\Pi_{i}^{\bullet})$
(6)
where $\bullet$ is a placeholder for any of the types, $\mathrm{rand}$,
$\mathrm{unc}$ or $\mathrm{cond}$. Unlike the measure of entropy, which is
well established, the application of this predictability measure to practical
problems is relatively recent. It is based on the idea that predictability is
related to the error probability in guessing the outcome of a discrete random
variable Feder and Merhav (1994). The upper bound given in Equation (5) comes
from Fano’s inequality Fano (1961); Cover and Thomas (2006). For a detailed
discussion on this bound and on possible lower bounds, see Feder and Merhav
(1994); Song et al. (2010).
For being able to study in more detail the effects of memory in the system
Sinatra et al. (2011); Chierichetti et al. (2012), we generalize the
conditional entropy:
$S^{\mathrm{cond},k}_{i}=-\sum_{x_{t}\in\mathcal{X}_{i}}\cdots\\!\sum_{x_{t-k}\in\mathcal{X}_{i}}p_{i}(x_{t-k},\ldots,x_{t})\log_{2}p_{i}(x_{t}|x_{t-k},\ldots,x_{t-1})$
(7)
where $k$ is an integer denoting the memory window. Note that
$S^{\mathrm{\mathrm{cond},1}}_{i}\equiv S^{\mathrm{cond}}_{i}$ and that we can
identify $S^{\mathrm{\mathrm{cond},0}}_{i}$ with $S^{\mathrm{unc}}_{i}$. It
follows from Fano’s inequality Fano (1961) that
$S^{\mathrm{\mathrm{cond},1}}_{i}\geq S^{\mathrm{\mathrm{cond},2}}_{i}\geq
S^{\mathrm{\mathrm{cond},3}}_{i}\geq\cdots$. The differences between
subsequent values in this chain inform us about the gain of predictability
when we increase the memory window one by one. If such a difference starts
becoming negligible from a particular level, $k$ to $k+1$, it means that the
system does not exhibit relevant memory effects beyond a window of $k$ steps.
If this level is at $k=0$, the events are uncorrelated; if at $k=1$, the
system is Markovian, otherwise, it is non-Markovian.
## 4 Results and Discussion
### 4.1 Predictability in Mobility
We applied all entropy and predictability measures to the mobility time
series, Figure 1a,b, respectively. Results show almost identical
predictability behavior for humans in our online world as for the mobility of
humans in geographic space Song et al. (2010); Gallotti et al. (2012). The
distributions for $S^{\mathrm{unc}}$ and $S^{\mathrm{rand}}$ are both
qualitatively and quantitatively matching, showing that also online, movements
of human avatars have the same highly predictable patterns when temporal
correlations are accounted for, but are mostly unpredictable when the order of
visitations is ignored. In particular, also here, $S^{\mathrm{rand}}$ peaks
around six, indicating that an individual who chooses her next location
randomly could be found, on average, in any of $2^{S^{\mathrm{rand}}}\approx
64$ locations, which is a substantial part of the 400 possible sectors. The
contrasting peak of $S^{\mathrm{cond}}$ below two shows that the actual
uncertainty of a typical player’s location is not 64, but rather, less than
$2^{2}=4$ sectors. The conditional entropy, $S^{\mathrm{cond}}$, is not
directly comparable to the actual entropy, $S$, in Song et al. (2010), but
shows the same tendency in that temporal correlations are substantial, even if
just having a memory of one. However, for re-creating the statistical features
of mobility thoroughly, longer memory is needed Szell et al. (2012). The peak
of $\Pi^{\mathrm{cond}}$ around $0.9$ means that only in around $10\%$ of
cases does a player choose her location in a manner that appears to be random,
but in $90\%$ of the cases, we can hope to predict her whereabouts with an
appropriate predictive algorithm. This high predictability stands in contrast
to the moderately predictive case given by $\Pi^{\mathrm{unc}}$ peaking around
$0.5$ and the highly unpredictive case of $\Pi^{\mathrm{rand}}$ peaking
narrowly and close to zero.
Figure 1: The distribution of (a) entropy and (b) the predictability measures
of the mobility of the Pardus players. Both are almost identical to the
mobility of humans in geographic space Song et al. (2010): Each considered
entropy measure improves predictability substantially, from considering the
uniformity of occupation to additionally short-term temporal correlations.
### 4.2 Predictability in Behavioral Actions
A similar picture to mobility arises for behavioral actions. Figure 2a,b,
respectively, report the entropy and predictability distributions of all 16
types of actions and received actions. Here, $S^{\mathrm{rand}}$ is peaked at
four, showing that most players are making full use of their behavioral
possibilities of $16=2^{4}$ action and received action types in the course of
their online lives. However, the sharp drop to the distribution of
$S^{\mathrm{unc}}$, which peaks around two, shows that, in practice, most of
these actions and received actions are focused on around $2^{2}=4$ action or
received action types only. The even narrower curve of $S^{\mathrm{cond}}$,
which peaks around $1.5$, with a corresponding peak of $\Pi^{\mathrm{cond}}$
at $0.8$, demonstrates that the conditional information allows us to predict
$80\%$ of actions. This is only slightly more than the $73\%$ prediction rate
peak from $\Pi^{\mathrm{unc}}$; however, $\Pi^{\mathrm{cond}}$ is distributed
more widely. In conclusion, the predictability gained from considering the
uniformity of occupation is much larger than the predictability gained from
also considering Markovian temporal correlations, as opposed to the case of
mobility where temporal correlations add substantial predictive value.
Figure 2: Distribution of (a) entropy and (b) predictability measures of the
behavioral actions of the Pardus players. As in the case of mobility,
behavioral actions are highly regular and predictable. However, the
predictability gained from considering the uniformity of occupation is much
larger than the predictability gained from also considering temporal
correlations.
One previous key observation on Pardus players is the fundamental structural
and dynamic difference between positive and negative action types and their
interaction networks Szell et al. (2010); Szell and Thurner (2012, 2010);
Thurner et al. (2012). To see if this difference is also apparent in the
extent of predictability, we plotted the distribution of the conditional
entropy of the players given that the previous action or received action was
positive/negative (Figure 3a), i.e., the set $\mathcal{Z}$ in Equation (4)
corresponds to $\mathcal{Z}=\\{C,T,F,X,C_{r},T_{r},F_{r},X_{r}\\}$ or to
$\mathcal{Z}=\\{A,B,D,E,A_{r},B_{r},D_{r},E_{r}\\}$, respectively. We aim to
understand whether the actions that follow positive actions are more
predictable than those that follow negative actions. If the distributions were
identical, the sign of an action would cause no difference in the
predictability of the subsequent action. In fact, although both distributions
peak around $0.55$, showing that there is a moderate amount of predictive
value gained from the information of an action’s sign, the positive
distribution is much more narrow than the negative one, implicating that there
is a much wider range of negative behavior in terms of predictability than
positive behavior. This result suggests that “good” people are much alike, but
“bad” persons behave badly in more various and, sometimes, more unpredictable
ways.
Figure 3: The distribution of the conditional entropy measures of the
behavioral actions of the Pardus players, given that the previous action
belonged to a certain category. (a) Entropy given that the previous action or
received action was positive/negative. The positive and negative distributions
have their maxima both around $0.55$, but the former is much more narrow than
the latter one, showing that there is a much wider range of negative behavior
in terms of predictability than positive behavior. (b) Entropy given that the
previous action was performed/received. Both distributions peak very close to
one, showing that the information of whether an action was performed or
received does, in general, not have a high predictive value. The peak for the
received actions is slightly closer to one than for the performed actions.
The conditional entropy for performed or received actions, i.e.,
$\mathcal{Z}=\\{C,T,F,X,A,B,D,E\\}$ or
$\mathcal{Z}=\\{C_{r},T_{r},F_{r},X_{r},A_{r},B_{r},D_{r},E_{r}\\}$ in
Equation (4), respectively, is peaked very narrowly and close to one for both
cases and slightly more so for received actions; Figure 3b. This observation
shows that the directionality of actions contains much less predictive
information than the sign of an action.
We can further refine the conditional entropy measure by considering single
actions as the condition, i.e., where $\mathcal{Z}$ in Equation (4) is a
singleton, to assess how much each action or received action type allows one
to predict the subsequent action that is about to happen in a player’s life.
The conditional entropy of trade peaks around $1.3$; the distribution of
communication is more wide, peaking around six bits; Figure 4a. Distributions
of received trades and communications are almost identical, only received
communication is slightly more right-skewed than performed actions of
communication. The reason why communication is associated with higher
unpredictability might have to do with the game’s action point system Szell
and Thurner (2010): every action, except the action of communication, costs an
amount of so-called action points for which every player has only a limited
pool. Therefore, players are not limited in their communication behavior, but
are so for trade, friendship markings, etc. The entropy distribution of
friendship marking, $F$, Figure 4b, peaks around one bit and is, therefore,
much less unpredictable. The entropy of enemy marking $E$ peaks even closer to
zero (Figure 4d); all of the actions related to enemy markings, $E$, $E_{r}$,
$X$ and $X_{r}$, show a bimodal distribution with an extra peak at zero, but
this is clearly not the case for friendship markings $F$ or $F_{r}$. This
bimodality could hint towards two different kinds of effects that arise from
enemy marking, where, for example, either the person who makes or removes the
marking immediately predictably sends a message to the recipient in a fraction
of cases, or in the remaining fraction, this does not happen. Finally, the
conditional entropy of received attacks, $A_{r}$, peaks around one, and
performed attacks, $A$, are more wide peaking at a smaller value; Figure 4c.
In all the distributions that deal with friendship or enemy markings, $F$,
$D$, $E$ and $X$, we observe a right-shift of peaks for received actions,
meaning that a player’s next action is more predictable given that a
friend/enemy event happened to her, as opposed to when she performed such an
action towards somebody else. For attacks, however, we see the opposite. It is
unclear what causes this phenomenon or how relevant it is: we can only
speculate that a received attack could have a possibly stronger emotional
impact on a player and, therefore, a more adverse effect on the predictability
on her next action, while this is vice versa for friendship/markings. Further,
it is interesting to note that the removal of a friendship link has a similar
pattern to the addition of an enmity link, suggesting that these two actions
might be closely related, since they have a similar impact on future behavior.
In general, however, the removal of a positive/negative tie cannot always be
put on the same level as the addition of a negative/positive tie, as the
reversed case of friendship addition and enemy removal shows.
Figure 4: The distribution of conditional entropy measures of the behavioral
actions of the Pardus players, given that the previous action was of a certain
type. (a) The distributions for performed and received communication events
($C$ and $C_{r}$) and for performed and received trade events ($T$ and
$T_{r}$). Communication peaks around six bits, trade around $1.3$ bits.
Performed and received actions do not show substantial deviations here. (b)
The distributions for performed and received friendship marking events ($F$
and $F_{r}$) and for performed and received friendship removals ($D$ and
$D_{r}$). The curves peak around one or lower. (c) The distributions for
performed and received attacks ($A$ and $A_{r}$). The former curve peaks below
one; the latter peaks around one and is narrower. (d) The distributions for
performed and received enemy marking events ($E$ and $E_{r}$) and for
performed and received enemy removals ($X$ and $X_{r}$). All the curves peak
once around $0.6$ and another time close to zero.
Finally, we are interested in assessing the memory dependence of the
behavioral actions in the system Chierichetti et al. (2012), i.e., the gain of
predictability from conditional entropies with longer time windows, using the
measures, $S^{\mathrm{cond},k}$, for increasing $k$. Unfortunately, in
practice, these rely on the empirical probabilities,
$p_{i}(x_{t-k},\ldots,x_{t})$, of all possible substrings
$x_{t-k},\ldots,x_{t}$ (see Equation (7)), which would lead to combinatorial
explosion with our alphabet size of 16. For example, $k=3$ would mean
$16^{3}=4,096$ possible substrings of a length of three, many of which do not
exist at all or are statistically not reliable to assess from a dataset of
1,758 players, each having performed up to a few thousand actions. Therefore,
in the following, we used the simplified alphabet of a size of two of negative
or positive actions, allowing feasible calculation of $S^{\mathrm{cond},k}$ up
to $k=5$. The distributions of these entropies are shown in Figure 5. The
distributions converge quickly, showing only a small difference between
$S^{\mathrm{cond},1}$ and $S^{\mathrm{cond},2}$ and almost no difference
between higher order distributions. We quantify these differences via the
Kullback–Leibler divergence between the distributions of the conditional
entropy of subsequent memory levels, $S^{\mathrm{cond},k-1}$ and
$S^{\mathrm{cond},k}$,
$D(k)=D(S^{\mathrm{cond},k}||S^{\mathrm{cond},k-1})=\sum_{j}S^{\mathrm{cond},k}(j)\log\frac{S^{\mathrm{cond},k}(j)}{S^{\mathrm{cond},k-1}(j)}$
(8)
which provides the information gain for going from a memory of a length of
$k-1$ to $k$ Cover and Thomas (2006); Sinatra et al. (2011). A divergence of
zero means that two distributions are identical. The first values from $D(2)$
to $D(5)$ read $0.0097$, $0.0020$, $0.0006$ and $0.0005$. For comparison, the
Kullback–Leibler divergence between $S^{\mathrm{unc}}$ and
$S^{\mathrm{cond}}$, $D(1)$, yields the much higher value of $0.38$, showing
that the system is, to a large part, Markovian and that the predictability
gained from higher-order correlations is negligible.
Figure 5: The convergence of the conditional entropy of the positive and
negative behavioral actions of Pardus players with an increasing memory
window. The difference between $S^{\mathrm{cond},1}$ and $S^{\mathrm{cond},2}$
is small, $D(2)=0.0097$, showing that the system is almost Markovian. For
higher memory windows, we have $D(3)=0.0020$, $D(4)=0.0006$ and $D(5)=0.0005$,
indicating almost identical distributions, which implies that there are
practically no long-term correlations in the signs of behavioral actions.
## 5 Conclusions
We applied three measures of entropy to two sets of time series of the
behavioral actions and the movements of a large number of players in a virtual
universe of a massive multiplayer online game. We found that movements in
virtual human lives follow identical levels of predictability as offline
mobility. This result reasserts previous observations on the similarities
between the online and offline movements of humans Szell et al. (2012) and is
especially striking considering that in online worlds, individuals are not
performing physical movements, but rather, navigate a virtual avatar.
Extending the approach to behavioral time series, also, here, we were able to
provide evidence for high predictability. However, in this case, we found that
due to weaker temporal correlations, there is hope to more easily predict
behavioral actions than the temporally correlated mobility patterns of humans
for which information about previously visited locations is required. Findings
using entropy measures conditional on positive and negative actions suggest
that “good” people are much alike, but “bad” persons behave badly in more
various and, sometimes, more unpredictable ways. Actions containing the
highest predictive information for an individual’s next behavior are negative,
such as attacks or enemy markings, while the positive actions of friendship
marking, trade and communication contain the least amount of predictive
information. However, we show that the system is, to a large part, Markovian
and almost devoid of any higher order correlations when taking into account
the sign of the action, showing that positive or negative behavior is not more
predictable when a longer history of previous actions is accounted for.
The distributions of entropies and predictability found here is strikingly
similar to distributions found for datasets of offline mobility Song et al.
(2010), economic transactions Krumme et al. (2010), online conversations and
online location check-ins Wang and Huberman (2012), therefore suggesting a
possible universality in the limitations of human behavior and its
independence of the concrete medium or context. However, contrary to our
result of little high-order correlations in behavior, a recent study has shown
that the behavior of browsing web pages is, to a large extent, non-Markovian
Chierichetti et al. (2012). Non-extensive entropies have been recently
developed that might be well applied for non-Markovian settings in complex
social systems Hanel and Thurner (2011); Hanel et al. (2012).
Our observations also provide additional evidence for the fundamental
differences in positive and negative behavior that were previously found on
dynamic Thurner et al. (2012) and structural Szell et al. (2010); Szell and
Thurner (2012, 2010) levels. Although previously large-scale evidence has
confirmed in online human behavior a number of known or hypothesized
behavioral phenomena of offline behavior, it is not immediately clear how
asymmetries between positive and negative behavior in our, to some extent,
artificial, online world can be translated to the offline world. Future
research should aim to analyze positive and negative relationships and
behaviors that happen in real-life societies and organizations Labianca and
Brass (2006), especially considering the multi-relational aspect of social
organization Szell et al. (2010); Kivelä et al. (2013). Fine-grained datasets
of socio-economic behavior, such as the one presented, offer the further
possibility of going beyond observations and measurements, to study the
mechanisms and origins of behavior in the view of collective phenomena Tadić
et al. (2013).
## 6 Notes added in proof
During the redaction of this paper, we were made aware of a relevant study
that applied the conditional entropy of signed messages to model growth of
entropy in emotionally charged online dialogues Sienkiewicz et al. (2013).
###### Acknowledgements.
Acknowledgments Roberta Sinatra is supported by the James S. McDonnell
Foundation. Michael Szell thanks the National Science Foundation, the
Singapore-Massachusetts Institute of Technology Alliance for Research and
Technology (SMART) program, the Center for Complex Engineering Systems (CCES)
at King Abdulaziz City for Science and Technology (KACST) and Massachusetts
Institute of Technology (MIT), Audi Volkswagen, Banco Bilbao Vizcaya
Argentaria (BBVA), The Coca Cola Company, Ericsson, Expo 2015, Ferrovial and
all the members of the MIT Senseable City Lab Consortium for supporting the
research. Both authors also thank the Santa Fe Institute for the opportunities
offered during the Complex Systems Summer School 2010, where some ideas for
this project originated. Conflicts of Interest Michael Szell is an associate
of the company, Bayer & Szell OG, which is developing and maintaining the
online game, Pardus, from which the data was collected.
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|
arxiv-papers
| 2013-12-01T01:34:09 |
2024-09-04T02:49:54.575875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roberta Sinatra, Michael Szell",
"submitter": "Michael Szell",
"url": "https://arxiv.org/abs/1312.0169"
}
|
1312.0181
|
# Mass Hierarchies with $m_{h}=125$ GeV from Natural SUSY
Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
Our study starts with a sequence of puzzles that include $(a)$ at which level
$\mu$ problem involving electroweak symmetry breaking can be solved; $(b)$ in
which paradigm masses of superpartners in the third family can be lighter than
in the first two families; $(c)$ whether it is possible to accommodate 125 GeV
Higgs boson simultaneously; and $(d)$ how natural such paradigm is. These
issues are considered in the context of two-site SUSY models. Both the MSSM
and NMSSM as low-energy effective theory below the scale of two-site gauge
symmetry breaking are investigated. We find that the fine tuning can be indeed
reduced in comparison with ordinary MSSM with $m_{h}=125$ GeV. In general, the
fine tuning parameter $\Delta$ is in the range of $20-400$.
12/2013
## 1 Introduction
Given a framework of new physics beyond standard model (SM), it faces a few
mass hierarchies. The fine tuning required to solve these hierarchies measures
how natural the framework is.
Among these mass hierarchies, we start with the quadratic divergence of SM
Higgs boson discovered at the LHC [1, 2]. In order to solve this problem,
frameworks such as technicolor and supersymmetry (SUSY) have been proposed
decades ago. In the context of SUSY, as we will explore in this paper, the
quadratic divergences between electroweak (EW) and ultraviolet (UV) energy
scale are canceled. In particular, this cancelation still holds without need
of the total spectrum of MSSM appearing at low energy scale. Therefore, the
masses of superpartners in the first-two families can be heavier than in the
third one. Naturalness implies that superpartners in the third family should
be not far away from the EW scale. These SUSY models are referred as Effective
SUSY in the early literature [3, 4] and Natural SUSY [5, 6, 7] recently. For
this type of models, typically we have $m_{\tilde{f}_{1,2}}\sim 10-20$ TeV in
first-two families and $m_{\tilde{f}_{3}}\sim 1$ TeV in the third family. It
is distinctive from viewpoint of phenomenology [8, 9, 10, 11, 12, 13].
On the realm of SUSY the electroweak symmetry breaking (EWSB) is more complex
than in SM. There exists a well-known little hierarchy between soft masses
$\mu$ and $B_{\mu}$ that involve the two Higgs doublets. Take the gauge
mediated (GM)111For a recent review on gauge mediation, see, e.g., [20] and
references therein. SUSY breaking for example. When an one-loop $\mu$ term of
order $\sim$ EW scale is generated, we usually obtain the same order of
$B_{\mu}$ term, i.e, $B_{\mu}\sim 16\pi^{2}\mu^{2}$. This spoils the
naturalness of EWSB. In order to evade this little hierarchy a few frameworks
such as addition of SM singlets [14, 15, 16] and conformal dynamics [17, 18]
have been proposed.
The last hierarchy we would like to address involves masses of SM flavors of
three generations. It is very appealing if a framework can provide a natural
explanation to this issue.
Our motivation for this study are followed by a sequence of puzzles:
* •
In which paradigm mass hierarchies mentioned above can be addressed ?
* •
In which paradigm masses of superpartners in the third family can be lighter
than in the first-two families ?
* •
Is there possible to accommodate 125 GeV Higgs boson simultaneously ?
* •
How natural the paradigm is ?
Recently it is pointed out that the mass hierarchies of SM flavor can be (at
least partially) addressed in SUSY quiver models [19]. We take the two-site
flavor model for illustration. The first-two families and the third one locate
at different sites, respectively. If one assumes that the SUSY breaking
effects are only communicated to site $G_{SM}^{(2)}$ under which the first two
families are charged-in terms of gauge interaction, and further to the other
site $G_{SM}^{(1)}$ under which the third family is charged-in terms of the
link fields, we can obtain the spectrum of Effective SUSY. Simultaneously,
mass hierarchy between the first-two and the third families of SM flavors can
be addressed. Fig. 1 shows the paradigm that provides Effective SUSY in two-
site model. The differences among two-site flavor model and the other two
scenarios are illustrated there also 222 For gaugino mediation we refer the
reader to Refs. [21, 22, 23, 24].. Therefore, it is possible to address all
mass hierarchies in Effective SUSY, once the little $\mu-B_{\mu}$ hierarchy is
accommodated.
Figure 1: Three mediation scenarios of SUSY breaking. In the two-site model,
either or both of two Higgs doublets locate at the first site.
For a candidate of viable model, it should provide Higgs boson of 125 GeV and
satisfy experimental limits such as flavor violating neutral currents (FCNC)
and electroweak precision tests (EWPT). Being consistent with FCNCs requires
the heavy bosons from broken gauge symmetries should be of order $\sim$ 10
TeV. This sets the scale of gauge symmetry breaking $G^{(1)}\times
G^{2}\rightarrow G_{SM}$ . Being consistent with EWPTs, the masses of
superpartners in first-two families are roughly of order $10-20$ TeV, which
sets the overall magnitude of soft mass $F/M\sim 10^{3}$ TeV. As well known
the fit to $m_{h}=125$ GeV requires significant modification to what the
minimal supersymmetric model (MSSM) exhibits. Because $m_{\tilde{t}}\sim 1$
TeV can not provide radioactive correction to $m_{h}$ large enough in the
context of Effective SUSY. Actually, this should not be realized in terms of
large radioactive correction, which otherwise implies that large fine tuning
exists. As a result the only sensible option is through modification to
$m_{h}$ at tree level. In the text, we will consider in detail two
possibilities-MSSM and NMSSM as low energy theory- either of which should give
rise to a significant correction to tree-level $m_{h}$.
The paper is organized as follows. In section 2, we discuss the case for which
the total Higgs sector is charged under $G_{SM}^{(1)}$ and singlets of
$G_{SM}^{(2)}$. This is referred as chiral Higgs sector. We divide this
section into MSSM in subsection 2.1 and NMSSM in subsection 2.2. In section 3,
we discuss that doublets $H_{u}$ and $H_{d}$ are charged under $G^{(1)}_{SM}$
and $G^{(2)}_{SM}$ respectively, which is referred as vector Higgs sector. We
briefly review and comment on such paradigm. Finally, we conclude in section
4.
## 2 Vector Higgs sector
Throughout this section, we use $Z$ boson mass to define fine tuning 333For a
comprehensive study about counting fine tuning, see the recent work [25] and
reference therein.,
$\displaystyle{}\left|\frac{\partial\ln m^{2}_{Z}}{\partial\ln
a_{i}}\right|\leq\Delta,$ (2.1)
where $a_{i}$ refer to soft breaking mass parameters that include
$\mu^{2},B_{\mu},m^{2}_{H_{u}},m^{2}_{H_{d}},m^{2}_{Q_{3}},m^{2}_{u_{3}}$,
$\dots$ . Note that $m^{2}_{Z}$ connects to some of soft mass parameters above
via condition of EWSB directly. For those indirect connections, the estimate
of their fine tunings should be extracted via chain derivative. To see how the
extensions of MSSM improve the fine tuning, one can compare it with that of
MSSM. Regardless of the possible fine tuning involving $\mu$ problem,
$\Delta\simeq 200$ in the MSSM with $m_{h}=125$ GeV [7].
In this section, we will explore both the MSSM and NMSSM in the context of
two-site model. The spectrum in both cases delivers light superpartners in the
third family. We mainly focus on the realizations of EWSB and $m_{h}=125$ GeV.
We also compare the fine tuning in these models with traditionary MSSM. As for
the configurations of two-site models described in this section, we refer the
reader to Ref. [19] for details.
### 2.1 MSSM from broken gauge symmetries
For the case of vector Higgs sector, the paradigm in this subsection is shown
in the left plot in fig.2. Gauge symmetries forbid the Higgs doublets coupling
to messengers directly. We introduce two additional singlets in comparison
with the minimal content of two-site model that is shown in fig.1. These
singlets are necessary in order to induce $\mu$ and $B_{\mu}$ terms. If one
assumes adding single singlet, the little hierarchy between $\mu$ and
$B_{\mu}$ can not be solved in this simple extension [14].
One of singlets $N$ is assumed to couple to the Higgs doublets, the link
fields and singlet $S$ simultaneously. The other singlet $S$ is assumed to
directly couple to messengers. The superpotential for these two singlets is
therefore of form
$\displaystyle{}W_{singlet}=N\left(\lambda_{1}H_{u}H_{d}+\frac{1}{2}\lambda_{2}S^{2}-\lambda_{3}\chi\tilde{\chi}\right)+\lambda_{S}S\Phi_{2}\tilde{\Phi}_{1}$
(2.2)
Messengers $\Phi_{i}$ couple to the SUSY breaking sector $X=M+\theta^{2}F$ as
in the minimal gauge mediation,
$\displaystyle{}W_{X}=X\left(\Phi_{1}\tilde{\Phi}_{1}+\Phi_{2}\tilde{\Phi}_{2}\right).$
(2.3)
For simplicity, we consider the case that $\Phi_{i},\tilde{\Phi}_{i}$ are
fundamental under $SU(5)\supset G^{(2)}_{SM}$. We also show the setting of
mass scales involved in the right plot of fig. 2. The rational for this
arrangement will be obvious.
Figure 2: Left: paradigm for vector Higgs. Here it is obvious that the singlet
$S$ plays the role similar to gaugino in the second site, which communicates
the SUSY breaking effects to Higgs doublets in the first site. This guarantees
$\mu^{2}$ and $B_{\mu}$ generated of same order of $m^{2}_{H_{u,d}}$. Right:
The arrangement of dynamical scales in the model.
Now we examine the soft breaking masses in Higgs sector. Below messenger
scale, one obtains one-loop renormalized wave function $Z_{S}$ for singlet $S$
after integrating messengers out,
$\displaystyle{}Z_{S}=1-\frac{5\lambda_{S}}{16\pi^{2}}\log\frac{XX^{{\dagger}}}{\Lambda^{2}},$
(2.4)
which gives rise to two-loop $m^{2}_{S}$ and one-loop $A_{S}$. The effective
superpotential and effective potential is given by,
$\displaystyle{}W_{eff}$ $\displaystyle=$ $\displaystyle
N\left(\lambda_{1}H_{u}H_{d}+\frac{1}{2}\lambda_{2}S^{2}-\lambda_{3}\chi\tilde{\chi}\right)$
$\displaystyle V_{soft}$ $\displaystyle=$ $\displaystyle m^{2}_{S}\mid
S\mid^{2}+\left(\lambda_{S}A_{S}NS^{2}+h.c\right)$ (2.5)
respectively. Between messenger scale $M$ and $m_{S}$, the gauge symmetries
$G^{(1)}_{SM}\times G^{(2)}_{SM}$ is spontaneously broken into its diagonal
subgroup $G_{SM}$ via the link fields with superpotential444In this paper, we
don’t investigate the details of dynamics in SUSY-breaking sector such as
superpotential (2.6). For microscopic construction in terms of confining UV
dynamics, see e.g.,[26].
$\displaystyle{}W_{link}=A\left(\chi\tilde{\chi}-f^{2}\right).$ (2.6)
with $f$ being the scale of gauge symmetry breaking and $A$ being a Lagrangian
multiplier field. Therefore, below scale $f$, we obtain a superpotential
instead of that in (2.1)
$\displaystyle W_{eff}$ $\displaystyle=$ $\displaystyle
N\left(\lambda_{1}H_{u}H_{d}+\frac{1}{2}\lambda_{2}S^{2}-M^{2}_{S}\right)$
(2.7)
with $M_{S}^{2}=\lambda_{3}f^{2}$. Together with $V_{soft}$ in (2.1) this
model indeed gives rise to one-loop $\mu$ and two-loop $B_{\mu}$, which are
shown in appendix A in terms of expansion in $m^{2}_{S}/M^{2}_{S}$.
The soft breaking mass $m_{N}$ of singlet $N$ is induced through singlet $S$
in (2.2), which is $m_{S}/M_{S}$-suppressed compared with $m_{S}$. Therefore,
the results presented in appendix A which are at the leading order of
$m^{2}_{S}/M^{2}_{S}$ are unaffected.
The addition of two singlets for addressing $\mu$ problem was firstly proposed
in Ref.[16]. The authors of [16] noted that one-loop $\mu$ and two-loop
$B_{\mu}$ terms were generated. If soft masses squared $m^{2}_{H_{u,d}}$ are
two-loop order as in minimal GM, EWSB can be indeed realized without much fine
tuning. However, masses squared $m^{2}_{H_{u,d}}$ are three-loop order instead
for two-site model discussed here. There is a key observation to resolve this
problem. Due to the individual contrubtion with different sign to two-loop
$B_{\mu}$ term (see appendix A), it can be numerically suppressed to be higher
than three-loop order. For example, by setting $\lambda_{1}/\lambda_{2}\sim
3\times 10^{-3}$ and $\lambda_{S}\simeq\sqrt{\frac{16}{5}}$ which are allowed
from consideration of naturalness, we obtain $\mu^{2}\sim\mid
m^{2}_{H_{u}}\mid\sim m^{2}_{H_{d}}$ all of which are at four-loop level
555The contributions to $m^{2}_{H_{u}}$ are composed of positive three-loop
and negative four-loop contribution, the absolute value of latter is larger
than the former. Also note that in this model the corrections to
$m^{2}_{H_{u,d}}$ due to Yukawa couplings in (2.2) are tiny in comparison with
those from gaugino mediation. These two properties keep the EWSB safe., while
the magnitude of $B_{\mu}$ term can be higher than three-loop order. These are
exactly conditions what EWSB requires (for large value of $\tan\beta$).
Now we consider the fit to 125 GeV Higgs boson discovered at the LHC. The
tree-level correction to $m_{h}$ due to D-terms of heavy $W^{\prime}$ and
$Z^{\prime}$ is proportional to soft breaking mass $m_{\chi}$ [28]. It is
absent in SUSY limit. So, we need large SUSY breaking effects, i.e.,
$\sqrt{F}/M\rightarrow 1$ . For large $\tan\beta$ limit
($\tan\beta=\left<H^{0}_{u}\right>/\left<H^{0}_{d}\right>$),
$\displaystyle{}m^{2}_{h}\simeq\left(1+\frac{g^{2}\delta+g^{\prime
2}\delta^{\prime}}{g^{2}+g^{\prime 2}}\right)m^{2}_{Z},$ (2.8)
where
$\displaystyle{}\delta=\frac{g_{(1)}^{2}}{g^{2}_{(2)}}\frac{2m_{\chi}^{2}}{M^{2}_{2}+2m_{\chi}^{2}},~{}~{}~{}~{}\delta^{\prime}=\frac{g_{(1)}^{{}^{\prime}2}}{g^{{}^{\prime}2}_{(2)}}\frac{2m_{\chi}^{2}}{M^{2}_{2}+2m_{\chi}^{2}}.$
(2.9)
Here $m_{\chi}$ and $M_{2}$ are masses of link fields and heavy gauge boson
from broken gauge symmetries, respectively, as shown in appendix A. SM gauge
couplings $g^{\prime}$, $g$ and $g_{3}$ are related to gauge couplings of
$G^{(1)}_{SM}$ and $G^{(1)}_{SM}$ as
$\frac{1}{g^{2}_{i}}=\frac{1}{(g^{(1)}_{i})^{2}}+\frac{1}{(g^{(2)}_{i})^{2}}$.
We define $\tan\beta_{1}=g^{\prime}_{(1)}/g^{\prime}_{(2)}$,
$\tan\beta_{2}=g_{(1)}/g_{(2)}$ and $\tan\beta_{3}=g_{(1)3}/g_{(2)3}$ for
later discussion.
In terms of (2.8) the fit to $m_{h}=125$ GeV suggests that $\delta<<1$ and
$\delta^{\prime}\simeq 4$ is the most natural choice666Other choices aren’t
viable. Solutions with $\delta\simeq 1$ leads to $\tan\beta_{2}=4\pi$, which
spoils the perturbativity of gauge theory. Solutions with $\delta\simeq 1$ and
$\delta^{\prime}\simeq 1$ deliver similar phenomenon. . This leads to
requirements on relative ratios of dynamical scales and choices of
$\tan\beta_{i}$ ,
$\displaystyle{}\frac{\sqrt{F}}{M}\rightarrow
1,~{}~{}~{}\frac{f}{M}\simeq\frac{g}{(4\pi)^{3/2}},~{}~{}\tan^{2}\beta_{1}\simeq
4\pi,~{}~{}\tan\beta_{2}\simeq 1,~{}~{}\tan\beta_{3}\simeq 0.94.$ (2.10)
The choice of $\tan\beta_{3}$ in (2.10) is unrelated to the fit to 125 GeV
Higgs. It is required in order to suppress $m^{2}_{S}$ in (A) by large
cancelation between the two individual contributions with opposite sign.
Otherwise, $m^{2}_{S}$ is too large to spoil the validity of expansion in
$m^{2}_{S}/M^{2}_{S}$. Furthermore, the ratio $F/M^{2}$ is close to its
critical value. This will provide deviation to soft mass parameter shown in
appendix A, whose magnitude depends on the value of this ratio [20]. For
example, $F/M^{2}\sim 0.95$ which is sufficiently large for promoting Higgs
mass can contribute about $10\%$ deviations to scalar and gaugino masses. In
this sense, the results in appendix A are approximately valid.
In summary, naturalness in two-site model we consider heavily relies on the
choices of three dimensionless parameters, i.e, $\lambda_{1}/\lambda_{2}$,
$\lambda_{S}$ and $\tan\beta_{3}$. The smallness of first parameter guarantees
that the value of $\mu$ is numerically correct, the second and last one leads
to large cancellation between individual contributions with opposite sign to
$B_{\mu}$ and $m^{2}_{S}$ respectively. Fortunately, the choices required to
achieve this naturalness show that these hidden Yukawa couplings and broken
gauge couplings are still on the realm of perturbative theory, which makes our
prediction on Higgs boson mass and phenomenology to be discussed below
reliable. The magnitude of Yukawa coupling $\lambda_{S}$ between singlet and
messenger pair is around unity, which indicates that strong dynamics as the UV
completion is probably favored.
Since the definition (2.1) used to measure fine tuning is insensitive to the
possible fine tuning involving soft mass parameters, the choices of above
three paramters at least keep two-site model technically natural. Let us
summarize the distinctive features from viewpoint of phenomenology.
* •
The fit to LHC data requires that the dynamical scales satisfy
$\frac{f}{M}\simeq\frac{1}{(4\pi)^{3/2}}$, with $M\simeq
0.5\times(4\pi)^{5/2}$ TeV.
* •
There exists viable choice of fundamental parameters. From (A), setting
$\lambda_{S}\simeq\sqrt{\frac{16}{5}}$ results in a tiny and positive
$B_{\mu}$ term. Setting $\lambda_{1}/\lambda_{2}\sim 3\times 10^{-3}$ provides
$\mu$ term of order $\mathcal{O}(100)$ GeV. Setting $\tan\beta_{3}\simeq 0.94$
allows large cancellation between the positive and negative contributions to
$m^{2}_{S}$, which results in suppression of the ratio $m^{2}_{S}/M^{2}_{S}$.
Fig.3 shows the sensitivity of conditions of EWSB to these three parameters.
Significant deviations from above choices will not induce EWSB. The
arrangement of dynamical scales results in,
$\displaystyle{}M_{i}$ $\displaystyle\sim$
$\displaystyle\mathcal{O}(4\pi)~{}TeV,$ $\displaystyle m_{\chi}$
$\displaystyle\sim$ $\displaystyle m_{\tilde{f}_{1,2}}\sim
m_{\lambda_{i}}\sim\mathcal{O}(\sqrt{4\pi})~{}TeV,$ $\displaystyle
m_{\tilde{f}_{3}}$ $\displaystyle\sim$ $\displaystyle
m_{S}\sim\mathcal{O}(1)~{}TeV,$ (2.11) $\displaystyle\mid\mu\mid$
$\displaystyle\sim$
$\displaystyle\sqrt{B_{\mu}}\sim\mathcal{O}(100-200)~{}GeV.$
Here the heavy gauge boson masses $M_{i}$ are $\sqrt{4\pi}$ enhanced in
compared with gaugino masses, so they are $4\pi$ enhanced in compared with the
soft scalar masses in the third family. As in minimal GM, the absence of
mixing between left- and right-hand soft scalar masses makes the model
consistent with the experimental limits from FCNCs. Heavy gauge bosons with
masses $\sim 10$ TeV in (• ‣ 2.1) don’t produce excess of FCNCs that can be
detected at present status [29].
Figure 3: Sensitivity of EWSB to parameters $\lambda_{1}/\lambda_{2}$ and
$\kappa$, $\kappa\equiv
2\lambda^{2}_{S}/(\frac{16g^{2}_{s}}{5\sin^{2}\beta_{3}})$. We choose
$M=(4\pi)^{5/2}$ TeV for illustration. This input parameter precisely
determines $\mu=175$ GeV in terms of one of conditions of EWSB. The contour of
$\mu=175$ GeV is projected into the plane of
$\kappa-(\lambda_{1}/\lambda_{2})$. The blue contour represents the other
condition of EWSB for different value of $\tan\beta$ respectively. It shows
less the value of $\tan\beta$ for more significant deviation of $\kappa$ to
unity. However, $\tan\beta<20$ conflicts with the 125 GeV Higgs boson mass.
Thus significant deviations from choices in the text will not induce EWSB.
* •
The fit to $m_{h}=125$ GeV suggests little hierarchy of order
$\mathcal{O}(\sqrt{4\pi})$ between soft scalar masses in the third and first-
two families. This is one of main results in our study. This phenomenon is far
from trivial from recent studies in the context of MSSM with $m_{h}=125$ GeV
777In the MSSM, either super heavy stop $\sim 10$ TeV for zero mixing or stop
mass $\sim 1$ TeV and $A_{t}\sim 2-3$ TeV for maximal mixing is needed to
accommodate 125 GeV Higgs boson. The first choice isn’t favored by
naturalness, while the latter one requires large $A_{t}$ term. In the scenario
of gauge mediation this can be only achieved either for directly coupling the
Higgs doublets to messengers or assuming high messenger scale. We refer the
reader to [30] and references therein for details.. Furthermore, the smallness
of ratios [19] $\epsilon_{l}=\frac{<\chi_{l}>}{M}\sim\frac{1}{(4\pi)^{3/2}}$
and $\epsilon_{h}=\frac{<\chi_{h}>}{M}\sim\frac{1}{(4\pi)^{3/2}}$ suggests
that SM fermion mass hierarchy with nearly two order of magnitude can be
explained in this context .
* •
Due to the soft mass squared $m^{2}_{H_{d}}$ relatively heavy to
$-m^{2}_{H_{u}}$, the model predicts the mass of heavy CP-even scalar
$m_{H}>300$ GeV, which is nearly degenerate with $m_{A}$ and $m_{H^{\pm}}$.
This spectrum is consistent with the present limit set by colliders. As for
the indirect experimental limits such as electroweak precision tests, this
kind of spectrum in Higgs sector doesn’t induce so significant deviation to SM
expectation that any firm conclusion can be made [37].
### 2.2 NMSSM from broken gauge symmetries
In comparison with the MSSM, the NMSSM 888For a review, see, e.g., [31]. has
been extensively studied to accommodate 125 GeV Higgs boson naturally [32, 33,
34, 35, 36, 37]. The rational for studying this model has been mentioned
above. There is additional contribution to Higgs boson mass at tree level, the
magnitude of which is controlled by the Yukawa coupling $\lambda$ in the NMSSM
superpotential,
$\displaystyle{}W_{NMSSM}=\lambda SH_{u}H_{d}+\frac{k}{3}S^{3}.$ (2.12)
The soft breaking masses in the potential read 999One may consider adding a
tree-level mass term $m_{S}$ for singlet $S$. The appropriate range for
$m_{S}$ is $\sim$ beneath 1 TeV. If $\left<S\right>$ is around EW scale, this
term can be used as a new input parameter. If $\left<S\right>$ is far above EW
scale, adding such a term is negative other than positive from viewpoint of
EWSB. ,
$\displaystyle{}V$ $\displaystyle=$ $\displaystyle\mid\lambda
H_{u}H_{d}-kS^{2}\mid^{2}+\lambda^{2}\mid S\mid^{2}(\mid H_{u}\mid^{2}+\mid
H_{d}\mid^{2})$ (2.13) $\displaystyle+$ $\displaystyle\frac{g^{2}+g^{\prime
2}}{8}\left(\mid H_{u}\mid^{2}-\mid H_{d}\mid^{2}\right)$ $\displaystyle+$
$\displaystyle(\lambda A_{\lambda}SH_{u}H_{d}-\frac{k}{3}A_{k}S^{3}+h.c)$
$\displaystyle+$ $\displaystyle m^{2}_{H_{u}}\mid
H_{u}\mid^{2}+m^{2}_{H_{d}}\mid H_{d}\mid^{2}+m^{2}_{S}\mid S\mid^{2}$
If singlet $S$ doesn’t couple to messengers directly, soft breaking term
$A_{\lambda}$ is at least two-loop effect, and $m_{S}$ typically appears near
EW scale. It actually recovers the case we have discussed in the previous
subsection. In this subsection, we discuss superpotential involving
messengers, which directly couple to $S$ as
$\displaystyle{}W=X\sum_{i=1}^{2}\left(\tilde{\Phi}_{i}\Phi_{i}\right)+\lambda_{S}S\Phi_{2}\tilde{\Phi}_{1}$
(2.14)
Here $\Phi_{i}$($\tilde{\Phi}_{i}$) belong to fundamental representation of
$SU(5)$. With addition of singlet $S$, the minimization conditions for the
potential (2.13) now are given by,
$\displaystyle{}\mu^{2}$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{H_{d}}-m^{2}_{H_{u}}\tan^{2}\beta}{\tan^{2}\beta-1}-\frac{m^{2}_{Z}}{2},$
$\displaystyle\sin 2\beta$ $\displaystyle=$
$\displaystyle\frac{2B_{\mu}}{m^{2}_{H_{d}}+m^{2}_{H_{u}}+2\mu^{2}},$ (2.15)
$\displaystyle 2\frac{k^{2}}{\lambda^{2}}\mu^{2}$ $\displaystyle-$
$\displaystyle\frac{k}{\lambda}A_{k}\mu+m^{2}_{S}=\lambda^{2}v^{2}\left[-1+\left(\frac{B_{\mu}}{\mu^{2}}+\frac{k}{\lambda}\right)\frac{\sin
2\beta}{2}+\frac{\lambda^{2}v^{2}\sin^{2}2\beta}{4\mu^{2}}\right].$
Figure 4: Left: paradigm for NMSSM. Here singlet $S$ communicates the SUSY
breaking effects to Higgs doublets in the first site. Right: The arrangement
of dynamical scales in the model.
In paradigm of fig.4 as we will explore, for the soft breaking terms in (2.2)
(we leave the explicit calculation on them in appendix B), contributions due
to Yukawa couplings (2.14) are generated at one-loop for $A_{\lambda}$ and
$A_{\kappa}$, two-loop for $m^{2}_{S}$, and two-loop for $m^{2}_{H_{u,d}}$. If
$\lambda_{S}$ and $\lambda$ of order SM gauge couplings, the corrections to
$m^{2}_{H_{u,d}}$ in (B) will dominate over the three-loop induced
contributions arising from gaugino mediation. Secondly, as noted in [16], the
effective $\mu$ and $B_{\mu}$ terms can be produced in terms of
$\mu=\lambda\left<S\right>$ and $B_{\mu}=\lambda F_{S}\sim\left<S\right>^{2}$
respectively. In other words, two-loop $B_{\mu}$ is automatically induced for
one-loop $\mu$ term. Roughly speaking, for Yukawa couplings
$\lambda,\lambda_{S}$ and $k$ all of order one, soft breaking terms (mass
squared) are two-loop for the Higgs sector, three-loop for the third family,
two-loop for the first two families, and two-loop for the gauginos. Therefore,
the superparters of third family can be light $\sim$ a few hundred GeV,
together with all the other soft breaking terms heavier than $\mathcal{O}(1)$
TeV.
We should also take the RG corrections into account for realistic EWSB. If we
consider low-scale messenger scale, the radioactive corrections to soft
breaking terms in (2.2) are logarithmic. In particular, the leading
corrections to $m^{2}_{H_{u,d}}$ are given by, respectively
$\displaystyle{}\delta m^{2}_{H_{d}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\lambda^{2}}{8\pi^{2}}m^{2}_{S}\log\left(\frac{M}{1~{}TeV}\right),$
$\displaystyle\delta m^{2}_{H_{u}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\lambda^{2}}{8\pi^{2}}m^{2}_{S}\log\left(\frac{M}{1~{}TeV}\right)-\frac{3y_{t}^{2}}{8\pi^{2}}\left(m^{2}_{Q_{3}}+m^{2}_{u_{3}}\right)\log\left(\frac{M}{1~{}TeV}\right).$
(2.16)
Now we consider the fit to $m_{h}=125$ GeV. For soft breaking mass parameters
being larger than EW scale, one can work in the limit $\left<S\right>>>v$
101010It is also of interest to consider the case
$\left<S\right>\sim\mathcal{O}(v)$. In this case the mixing effect in the mass
matrix for three CP-even Higgs boson is obvious. Analytic method used to
measure eigenvalues and fit 125 GeV Higgs mass is inappropriate anymore. We do
not discuss this case in this paper. . In this limit, the mass of lightest CP-
even scalar is approximately given by [16],
$\displaystyle{}m^{2}_{h}=M^{2}_{Z}\cos^{2}2\beta+\lambda^{2}v^{2}\left\\{\sin^{2}2\beta-\frac{\left[\frac{\lambda}{k}+\left(\frac{1}{6\omega}-1\right)\sin
2\beta\right]^{2}}{\sqrt{1-8z}}\right\\}.$ (2.17)
where $\omega\equiv(1+\sqrt{1-8z})/4$, $z=m^{2}_{S}/A^{2}_{k}$. Apparently
$z<1/8$ (or equivalently $\omega>1/4$) in order to insure that the vacuum is
deeper than the origin $\left<S\right>=0$. Eq (2.17) also indicates that large
$\lambda$ is favored in order to uplift its mass to 125 GeV.
Figure 5: Parameter space for EWSB in the plane of $u-z$ for $\lambda=0.8$,
$\tan\beta=2$ and $M\simeq 10^{5}$ TeV. Here solutions to constraints
$(1)$-$(2)$ in (2.2) correspond to the gray curve for $\lambda_{s}=0.3$, the
blue curve for $\lambda_{s}=0.35$, and the green curve for $\lambda_{s}=0.4$,
respectively. Three contours represent Higgs boson with $m_{h}=125\pm 2$ GeV
for different choices of $\lambda_{s}$. A representative point $(-2.0,0.3)$ in
plane of $z-u$ corresponds to
$\sin\beta_{1}=\sin\beta_{2}\simeq\sin\beta_{3}=0.7$ and $k\simeq 1.33$.
We show the parameter space numerically in fig.5 for $\lambda=0.8$,
$\tan\beta=2$ and $M=10^{5}$ TeV. Smaller value of $M$ suppresses RG
corrections in (2.2), which could spoil EWSB. In fig.5 the gray, blue and red
contours corresponds to $m_{h}=125\pm 2$ GeV with $\lambda_{s}=0.3,0.35,0.4$,
respectively. The numerical result shows that either $z>0.1$ or $u>1.0$ is
excluded 111111Two assumptions have been adopted. At first, RG runnings of
Yukawa couplings aren’t taken into account. We limit to the case with low
messenger scale $M$. Secondly, the stop induced loop correction to Higgs mass
is ignored. Because in our model, the stop mass is $m_{\tilde{t}}<$ 1 TeV..
The purple, blue and green curves satisfy the first two conditions in (2.2),
which corresponds to $\lambda_{s}=0.3,0.35,0.4$, respectively. Note that we
have used the results $\mu=(\lambda/k)A_{k}\omega$ and and
$B_{\mu}=(k/\lambda)\mu^{2}-A_{\lambda}\mu-\lambda^{2}v^{2}\sin 2\beta/2$ for
above analysis, which are determined in terms of the last constraint in (2.2).
In what follows we focus on the case for $\lambda_{s}=0.3$ (gray contour and
purple curve in fig.5). In ordinary weakly coupled NMSSM-without gauge
extension beyond SM gauge groups and -without taking the stop induced loop
correction into account, there is impossible to accommodate Higgs with
$m_{h}>122$ GeV (see, e.g., [7]). Our numerical results are consistent with
this well known claim.
Figure 6: Origin of bound on $\lambda_{s}$. The curves from bottom to top
correspond to value of $u=1.0,0.5,0.3,0$, respectively. The horizontal line
refers to the critical value where $G^{(2)}_{SM}$ becomes confining theory.
The value of $u$ must be upper bounded since too large and positive $u$ spoils
EWSB. The value of $u$ is also lower bounded due to limit on value of
$\lambda_{s}$, which is rather large for negative $u$. The red curve
corresponds to the value of $\sin\beta_{1,2}$ for $u=0.3$ as chosen in fig.5.
From fig. 5, all of $\lambda_{s}$, $\lambda$ and $k$ are bounded as result of
$m_{h}=125$ GeV. In particular, $\lambda$ and $k$ close to critical values
beyond perturbative field theory, which implies that there is probably a
confining gauge theory between the messenger and Plank scale 121212We refer
the reader to Ref. [38] for discussion about issue.. In order to show the
origin of bound on $\lambda_{s}$, we recall that two ratios $u$ and $z$ used
in fig.6 read from (B), respectively,
$\displaystyle{}u$ $\displaystyle=$
$\displaystyle\frac{1}{\left(15\lambda^{2}_{S}\right)^{2}}\left(\frac{3g^{4}}{4\sin^{4}\beta_{2}}+\frac{5}{12}\frac{g^{\prime
4}}{\sin^{4}\beta_{1}}-\frac{5}{4}\lambda^{2}\lambda^{2}_{S}\right),$
$\displaystyle z$ $\displaystyle=$
$\displaystyle\frac{1}{\left(15\lambda_{S}\right)^{2}}\left(\frac{35}{2}\lambda^{2}_{S}-10k^{2}-\frac{8g^{2}_{3}}{\sin^{2}\beta_{3}}-\frac{3g^{2}}{\sin^{2}\beta_{2}}-\frac{5}{3}\frac{g^{\prime
2}}{\sin^{2}\beta_{1}}\right).$ (2.18)
We show the lower bound as function of $u$ in fig. 6. The curves from bottom
to top in fig. 6 correspond to different value of $u$ respectively. Since the
value of $u$ is upper bounded due to EWSB, $\lambda_{s}$ is therefore lower
bounded.
The mass spectrum and phenomenological consequences in this model are as
follows.
* •
Unlike in the MSSM we consider in the previous subsection, the dynamical
scales satisfy $\frac{F}{M^{2}}<\frac{1}{4\pi}$, with $F/M\simeq 3.0\times
10^{2}$ TeV and $M\geq 10^{5}$ TeV.
* •
Correspondingly, we have
$\displaystyle{}m_{\tilde{f}_{3}}$ $\displaystyle\sim$
$\displaystyle\mathcal{O}(1)~{}TeV,$ $\displaystyle m_{\chi}$
$\displaystyle\sim$ $\displaystyle m_{\tilde{f}_{1,2}}\sim
m_{\lambda_{i}}\sim\mathcal{O}(3-4)~{}TeV,$ $\displaystyle\mid\mu\mid$
$\displaystyle\sim$ $\displaystyle\sqrt{B_{\mu}}\sim\mathcal{O}(2)~{}TeV.$
(2.19)
The heavy gauge boson masses $M_{i}$ can be heavier compared with the case for
MSSM. The masses for other two CP-even and three CP-odd Higgs bosons can be
determined in the limit $\left<S\right>>>v$. All of them are of order
$\sim\mu$. So they easily escape searches at colliders such as LHC with
$\sqrt{s}=8$ TeV.
* •
From (• ‣ 2.2) we find the most significant contribution to fine tuning comes
from the heavy higgsinos. Typically, we have $\Delta\simeq 400$ for
conservative value $\mu=2$ TeV and $M=10^{5}$ TeV. With smaller value of
$F/M$, the fine tuning can be slightly reduced. It depends on the lower bound
on masses of superpartners in the third family. In this sense, the main
resource for fine tuning might change in different paradigms. However, it is
impossible to reduce the fine tuning totally for SUSY models with $m_{h}=125$
GeV.
## 3 Chiral Higgs doublets
Unlike the configurations described in the previous section, one can move the
Higgs doublet $H_{d}$ from site one to site two. Gauge anomaly free requires
either introducing new charged matters into SM or moving one lepton doublet to
site two also. We refer the reader to [19] for the latter choice. $H_{u}$
($H_{d}$) is now charged under $G^{(1)}_{SM}$ ($G^{(2)}_{SM}$) but singlet of
$G^{(2)}_{SM}$ ($G^{(1)}_{SM}$). An consequence of this configuration is that
gauge invariance forbids singlet extension of type $W\sim SH_{u}H_{d}$ as we
have discussed in section 2. For completion, we briefly review and comment on
such paradigm in what follows.
We focus on the contents of MSSM as discussed in [19]. As link fields are
charged under both two gauge groups, it can provide such a superpotential
$W\sim\lambda_{\chi}\chi H_{u}H_{d}$. As a result of gauge symmetries
breaking, an effective $\mu$ term is induced, with $\mu=\lambda_{\chi}f$, the
magnitude of which is controlled by the Yukawa coupling constant
$\lambda_{\chi}$. As for the other soft breaking terms in the Higgs sector,
they are generated at two-loop level for $m^{2}_{H_{d}}$, three-loop level for
$m^{2}_{H_{u}}$ and vanishing $B_{\mu}$ at the input scale due to the fact
$F_{\chi}=0$. In particular, the four-loop, negative contribution to
$m^{2}_{H_{u}}$ guarantees that its sign is negative. The $B_{\mu}$ term at EW
scale is generated by short RG running, and its magnitude is rather small.
Thus, for $\lambda_{\chi}\sim 0.01$ and $f\sim 10$ TeV, we obtain tiny
$B_{\mu}$, $\mu^{2}\sim-m^{2}_{H_{u}}\sim(100~{}GeV)^{2}$ and
$m^{2}_{H_{d}}\sim$ a few TeV2 for the third-family scalar superpartners of
order $\sim 1$ TeV. A few consequences are predicted. First, we have
$\tan\beta>10^{4}$, which realizes EWSB naturally for soft breaking terms
above. Secondly, the Higgs mass can be uplifted to 125 GeV due to D-terms of
heavy $Z^{\prime}$ and $W^{\prime}$. At last, a generic property in this model
is that the bottom and tau masses are too light. Because they nearly decouple
from $H_{d}$.
The bottom and tau masses can be improved in some cases. An option deserves
our attention. Instead of being charged under $SU(5)$, messengers are divided
into singlets $\Phi$, doublets $\Phi^{D}_{i}$ charged under $SU(2)_{(2)}$ and
triplets $\Phi^{T}_{i}$ charged under $SU(3)_{(2)}$. If so, we can directly
couple doublet $H_{d}$ to the messengers via superpotential
$\displaystyle{}W\sim\lambda_{d}H_{d}\Phi^{D}_{i}\Phi.$ (3.1)
The $m^{2}_{H_{u}}$ is unchanged because it doesn’t couple to the messengers
as before. The Yukawa coupling in (3.1) gives rise to one-loop negative, and
$(F/M^{2})$-suppressed contribution to $m^{2}_{H_{d}}$, the magnitude of which
is controlled by the Yukawa coupling constant $\lambda_{d}$. With
$\lambda_{d}$ for which the one-loop negative and two-loop positive
contributions nearly cancel, we have naturally suppressed $m^{2}_{H_{d}}$.
Consequently, the value of $\tan\beta$ is suppressed to acceptable level.
## 4 Conclusions
A few hierarchies plague the new physics beyond SM in particle physics. Most
of them are tied to parameters involving Higgs boson. A paradigm proposed to
solve these hierarchies can be classified from the viewpoint of naturalness.
Unless there are other more fundamental principles, naturalness is still a
useful tool for guiding new physics. In this paper, we discuss the $\mu$
problem, the mass hierarchies between SM third and first-two families, and the
discrepancy between the experimental value for Higgs boson mass and its tree-
level bound in the MSSM.
We present paradigms in which these mass hierarchies can be naturally
explained, with fine tuning of $\Delta=20\sim 400$. The ingredients in our
paradigms such as mechanism of communicating SUSY breaking effects, the
mechanism of generating $\mu$ term aren’t new. However, it is subtle to put
these together and uncover a viable parameter space.
In this paper, we show paradigms for both MSSM and NMSSM as the low-energy
effective theory. We find that the main source of fine tuning might change in
various paradigms. However, in comparison with traditionary MSSM that provides
125 GeV Higgs boson mass (with the little hierarchy and mass hierarchies
between SM flavors are often ignored in the literature), they both do better
from the viewpoint of naturalness.
While uncovering the parameter space, a byproduct needs our attention. For the
two representative natural SUSY models we explore, the UV completion is
probably a strong dynamics. There are also a few interesting issues along this
line we have missed in this paper. In particular, the case for chiral Higgs
sector deserves detailed study. And it might be meaningful to address the mass
hierarchies among SM flavors of three generations.
## Acknowledgement
The author thanks Z. Sun for discussions, and M.-x. Luo for reading the
manuscript. This work is supported in part by the National Natural Science
Foundation of China with Grant No. 11247031.
## Appendix A Soft breaking terms in the MSSM
In terms of renormalized wave function $Z_{S}(X,X^{{\dagger}})$, the soft
masses involving singlet $S$ are given by,
$\displaystyle{}A_{S}$ $\displaystyle=$
$\displaystyle\frac{-5\lambda^{2}_{S}}{16\pi^{2}}\frac{F}{M^{2}}M,$
$\displaystyle m^{2}_{S}$ $\displaystyle\simeq$ $\displaystyle
35\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{\lambda^{2}_{S}}{\lambda_{2}}\frac{g^{2}_{3}}{\cos^{2}\beta_{3}}\left(\frac{F}{M^{2}}\right)^{2}M^{2}-\frac{5}{48\pi^{2}}\left(\frac{F^{2}}{M^{4}}\right)^{2}M^{2}.$
(A.1)
The second part of $m^{2}_{S}$ in (A) corresponds to negative and one-loop
$(F/M^{2})$-suppressed contribution [16]. With $F/M^{2}\rightarrow 1$ as
selected from the requirement of $m_{h}=125$ GeV, this contribution should be
taken into account. The positive and negative contributions to (A) cancel each
other so that it is valid to expand in order of $m^{2}_{S}/M^{2}_{S}$.
As for $\mu$ and $B_{\mu}$ terms, they are given by at leading order
$\displaystyle{}\mu$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{5\lambda_{1}\lambda^{2}_{S}}{\lambda_{2}}\frac{F}{M^{2}}M+\mathcal{O}\left(\frac{m^{2}_{S}}{M^{2}_{S}}\right),$
$\displaystyle B_{\mu}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{5\lambda_{1}\lambda^{2}_{S}}{\lambda_{2}}\left(\frac{16}{5}\frac{g^{2}_{3}}{\sin^{2}\beta_{3}}+\frac{2}{3}\frac{g^{\prime
2}}{\sin^{2}\beta_{1}}-2\lambda^{2}_{S}\right)\left(\frac{F}{M^{2}}\right)^{2}M^{2}+\mathcal{O}\left(\frac{m^{2}_{S}}{M^{2}_{S}}\right).$
Setting $\lambda_{S}\simeq\sqrt{\frac{16}{5}}$ results in a tiny and positive
$B_{\mu}$ term. This value is close to the region valid for perturbative
analysis. Setting $\lambda_{1}/\lambda_{2}\sim 10^{-3}$ results in $\mu$ term
of order $\mathcal{O}(100)$ GeV. The three masses of heavy gauge bosons from
broken gauge symmetries read,
$\displaystyle{}M^{2}_{i}=2(g^{2}_{(1)i}+g^{2}_{(2)i})f^{2}\simeq\frac{2}{(4\pi)^{3}}(g^{2}_{(1)i}+g^{2}_{(2)i})M^{2}$
(A.3)
The second expression is from (2.10). The mass of link fields $m^{2}_{\chi}$
are generated at two-loop level, similarly to (A.3),
$\displaystyle{}m^{2}_{\chi}\simeq
2n\sum_{a=1}^{3}C_{a}(\chi)\left(\frac{\alpha_{a}}{4\pi}\right)^{2}\left(\frac{F}{M^{2}}\right)^{2}M^{2}.$
(A.4)
Finally, following calculation of soft scalar masses of superpartners in [27],
we obtain our final results from arrangement of dynamical scales in (2.10),
$\displaystyle{}m^{2}_{\tilde{Q}_{3}}$ $\displaystyle=$
$\displaystyle\frac{4}{3}K_{3}+\frac{3}{4}K_{2}+\frac{1}{60}K_{1},$
$\displaystyle m^{2}_{\tilde{u}_{3}}$ $\displaystyle=$
$\displaystyle\frac{4}{3}K_{3}+\frac{4}{15}K_{1},$ $\displaystyle
m^{2}_{\tilde{d}_{3}}$ $\displaystyle=$
$\displaystyle\frac{4}{3}K_{3}+\frac{3}{4}K_{2}+\frac{1}{15}K_{1},$
$\displaystyle m^{2}_{\tilde{L}_{3}}$ $\displaystyle=$
$\displaystyle\frac{3}{4}K_{2}+\frac{1}{20}K_{1},$ (A.5) $\displaystyle
m^{2}_{\tilde{e}_{3}}$ $\displaystyle=$ $\displaystyle\frac{3}{5}K_{1},$
$\displaystyle m^{2}_{H_{u}}$ $\displaystyle=$ $\displaystyle
m^{2}_{H_{d}}=\frac{3}{4}K_{2}+\frac{3}{20}K_{1},$
where
$\displaystyle{}K_{i}$ $\displaystyle=$
$\displaystyle\alpha_{i}\left(m^{2}_{\lambda_{i}}\left[\log(\frac{M^{2}_{i}}{m^{2}_{\lambda_{i}}})-1+\frac{1}{2}\cot^{2}\beta_{i}\right]+\frac{1}{2}\tan^{2}\beta_{i}m^{2}_{\chi}\right)$
(A.6)
The negative four-loop correction to $m^{2}_{H_{u}}$ due to stop
$m_{\tilde{t}}$ loop is larger than the three-loop contribution [27], which
should be considered in realistic EWSB. As for the soft scalar masses of
superpartners in the third family as well as the gaugino mass
$m_{\lambda_{i}}$, they are the same as in minimal gauge mediation . Since
$\tan^{2}\beta_{1}$ enhancement only affects $K_{1}$ in (A.6), the little
hierarch between soft scalar masses in the third and first two families
doesn’t be violated. Therefore, the spectrum are similar to what Natural SUSY
suggests.
## Appendix B Soft breaking terms in the NMSSM
In our paradigm, integrating out the messengers with Yukawa couplings defined
in (2.14) contributes to the soft terms at the messenger scale $M$
$\displaystyle{}A_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{3}A_{k}=-\frac{5n\lambda^{2}_{S}}{16\pi^{2}}\frac{F}{M},$
$\displaystyle\delta m^{2}_{H_{u}}$ $\displaystyle=$ $\displaystyle\delta
m^{2}_{H_{d}}=\frac{n}{(16\pi^{2})^{2}}\left[\frac{3}{2}\left(\frac{g^{2}}{\sin^{2}\beta_{2}}\right)^{2}+\frac{5}{6}\left(\frac{g^{\prime
2}}{\sin^{2}\beta_{1}}\right)^{2}-\frac{5\lambda^{2}\lambda^{2}_{S}}{n}\right]\frac{F^{2}}{M^{2}},$
(B.1) $\displaystyle m^{2}_{S}$ $\displaystyle=$
$\displaystyle\frac{n}{(16\pi^{2})^{2}}\left[35\lambda^{4}_{S}-20\lambda_{S}^{2}k^{2}-16\lambda^{2}_{S}\left(\frac{g_{3}^{2}}{\sin^{2}\beta_{3}}\right)-6\lambda^{2}_{S}\left(\frac{g^{2}}{\sin^{2}\beta_{2}}\right)-\frac{10}{3}\lambda^{2}_{S}\left(\frac{g^{\prime
2}}{\sin^{2}\beta_{1}}\right)\right]\frac{F^{2}}{M^{2}}.$
Here $n$ being the number of messenger pairs; in our case $n=2$. There is also
one-loop, $F/M^{2}$-suppressed and negative contribution to $m^{2}_{S}$, which
is tiny for small ratio $F/M^{2}$. The effective $\mu$ and $B_{\mu}$ in (2.13)
read respectively,
$\displaystyle{}\mu$ $\displaystyle=$ $\displaystyle\lambda\left<S\right>,$
$\displaystyle B_{\mu}$ $\displaystyle=$
$\displaystyle\frac{k}{\lambda}\mu^{2}-A_{\lambda}\mu-\frac{\lambda^{2}v^{2}}{2}\sin
2\beta.$ (B.2)
Other soft breaking terms such as superpartner masses are the same as in
appendix A.
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|
arxiv-papers
| 2013-12-01T06:09:20 |
2024-09-04T02:49:54.585128
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1312.0181"
}
|
1312.0205
|
# Generalized BF state in quantum gravity
Shinji Yamashita111 Email: [email protected] , Satoshi Yajima, and Makoto
Fukuda
Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan
###### Abstract
The BF state is known as a simple wave function that satisfies three
constraints in canonical quantum gravity without a cosmological constant. It
is constructed from a product of the group delta functions. Applying the
chiral asymmetric extension, the BF state is generalized to the state for real
values of the Barbero–Immirzi parameter.
## 1 Introduction
In modern canonical quantum gravity, the connections with the Barbero–Immirzi
parameter $\beta$ play the role of fundamental variables [1, 2, 3]. Wave
functions are required to solve the three constraints, i.e., Gauss,
diffeomorphism, and Hamiltonian constraints.
The Chern–Simons (CS) state, which is also called the Kodama state, is known
as an exact solution of these three constraints with a cosmological constant
[4]. In this case, the configuration variable is a complex
$sl(2,{\mathbb{C}})$-valued connection, which takes a left- or right-handed
form, namely, $\beta=\pm{\rm i}$ for the Lorentzian case. However, loop
quantum gravity (LQG) proposes that the Barbero–Immirzi parameter takes real
values for several technical reasons. The real value of $\beta$ gives a real
$su(2)$-valued connection, but it makes the Hamiltonian constraint more
complicated.
The generalization of the CS state to real values of $\beta$ was achieved by
Randono [5, 6, 7]. The generalized states are parameterized by the Levi-Civita
curvature, and solve some difficulties of the ordinary CS state, e.g., the
normalizability and the reality conditions.
On the other hand, the wave function without a cosmological constant was found
by Miković [8, 9]. It is called the BF state here. This state is given as a
product of the group delta functions of the curvature, and constructed from
the left-handed complex connection as well as the ordinary CS state. In this
paper, the generalization of the BF state is considered as an analog of the
generalization of the CS state. We would like to emphasize that the process of
the generalization follows Refs. [5, 6, 7]. Specifically, it is carried out
via the chiral asymmetric extension.
In Sect. 2, we briefly review the BF state for $\beta={\rm i}$. Then, using
the chiral asymmetric model, the BF state is extended to the case of generic
purely imaginary values of $\beta$. In Sect. 3, the BF state is extended
further to the case of generic real values of $\beta$. This state is expressed
in terms of the real $su(2)$-valued connection and the Levi-Civita curvature.
Making use of the appropriate inner product, three constraints with real
values of $\beta$ are solved. In Sect. 4, we present the conclusions and
discuss the results.
## 2 BF state and chiral asymmetric extension
### 2.1 BF state
The three constraints of canonical quantum gravity without a cosmological
constant can be derived from the Holst action [10]
$S_{\rm H}=\frac{1}{4k}\int\left(\epsilon_{IJKL}\,e^{I}\wedge
e^{J}\wedge\Omega^{KL}-\frac{2}{\beta}\,e^{I}\wedge
e^{J}\wedge\Omega_{IJ}\right)\ ,$ (1)
where $k=8\pi G$, $e^{I}$ is the tetrad, and $\Omega={\rm
d}\omega+\omega\wedge\omega$ is the curvature of the spin connection
$\omega^{IJ}$. Capital Latin indices $I,J,\dots$ are used as Lorentz indices.
Performing the Legendre transformation, one obtains
$S_{\rm H}=\frac{1}{k\beta}\int{\rm
d}^{4}x\left(E_{i}^{a}\dot{A}^{(\beta)}{}_{a}^{i}+\lambda^{i}G_{i}+N^{b}V_{b}+NH\right)\
,$ (2)
where $\dot{A}^{(\beta)}{}_{a}^{i}={\cal L}_{t}A^{(\beta)}{}_{a}^{i}$ ,
$\lambda^{i},N^{a}$, and $N$ are Lagrange multipliers, and $G_{i},V_{b}$, and
$H$ are Gauss, diffeomorphism, and Hamiltonian constraints respectively.
Letters $i,j,\dots$ and $a,b,\dots$ denote 3D internal and spatial indices,
respectively. The configuration variable
$A^{(\beta)}{}_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}$ is constructed from the
Levi-Civita spin connection $\Gamma_{a}^{i}$ and the extrinsic curvature
$K_{a}^{i}$. Choosing the time gauge $e_{a}^{I}|_{I=0}=0$, the canonical
momentum variable can be written as $E_{i}^{a}=\det(e_{b}^{j})e_{i}^{a}$.
For $\beta={\rm i}$, the action (1) can be written only with the left-handed
variables:
$S_{\rm H}^{(+)}=\frac{{\rm i}}{k}\int\Sigma^{(+)IJ}\wedge\Omega_{IJ}^{(+)}\
,$ (3)
where
$\Sigma^{(+)IJ}=\frac{1}{2}\left(e^{I}\wedge e^{J}-\frac{{\rm
i}}{2}\epsilon^{IJ}{}_{KL}e^{K}\wedge e^{L}\right)\ ,$ (4)
and
$\Omega_{IJ}^{(+)}=\frac{1}{2}\left(\Omega_{IJ}-\frac{{\rm
i}}{2}\epsilon_{IJ}{}^{KL}\Omega_{KL}\right)\ .$ (5)
The sign $(+)$ explicitly denotes that the variable is left-handed, namely,
$\beta={\rm i}$. The three constraints are
$\displaystyle G_{i}^{(+)}$ $\displaystyle=$
$\displaystyle(D_{a}^{(+)}E^{(+)a})_{i}=\partial_{a}E^{(+)}{}_{i}^{a}+\epsilon_{ij}{}^{k}A^{(+)}{}_{a}^{j}E^{(+)}{}_{k}^{a}\
,$ (6) $\displaystyle V_{b}^{(+)}$ $\displaystyle=$ $\displaystyle
E^{(+)}{}_{i}^{a}F^{(+)}{}_{ab}^{i}\ ,$ (7) $\displaystyle H^{(+)}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{2\sqrt{|\det(E^{(+)})}|}\epsilon^{ijk}E^{(+)}{}_{i}^{a}E^{(+)}{}_{j}^{b}F^{(+)}_{ab\,k}\
,$ (8)
where
$E^{(+)}{}_{i}^{a}=\epsilon^{abc}\epsilon_{ijk}\Sigma^{(+)}{}_{bc}^{jk}$, and
$F^{(+)}{}_{ab}^{i}$ is the curvature of the connection
$A^{(+)}{}_{a}^{i}=\Gamma_{a}^{i}+{\rm i}K_{a}^{i}$. The wave function has to
satisfy the quantized constraints, which are formally written as
$\hat{G}^{(+)}_{i}\Psi=\hat{V}^{(+)}_{b}\Psi=\hat{H}^{(+)}\Psi=0\ .$ (9)
In Ref. [8], it is suggested that the product of the group delta functions
$\Psi_{\rm BF}(A^{(+)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left({\rm
e}^{F^{(+)}_{ab}(x)}\right)$ (10)
is a solution of the constraints. This state is originally derived from the
formal integral $\int{\cal DB}\,\exp[\,{\rm i}S_{\rm BF}\,]=\delta({\rm
e}^{F})$, where $S_{\rm BF}=\int_{\Sigma}{\rm Tr\,}(B\wedge F)$ is the $SU(2)$
BF action in 3D Euclidean space $\Sigma$. Thus let us call state (10) the BF
state. The group delta function has the following properties:
$\displaystyle\delta\left(g\,{\rm e}^{F^{(+)}_{ab}}g^{-1}\right)$
$\displaystyle=$ $\displaystyle\delta\left({\rm e}^{F^{(+)}_{ab}}\right)\ ,$
(11) $\displaystyle F^{(+)}_{ab}\delta\left({\rm e}^{F^{(+)}_{ab}}\right)$
$\displaystyle=$ $\displaystyle 0\ ,$ (12)
where $g$ is an element of the gauge group. Therefore the state $\Psi_{\rm
BF}(A^{(+)})$ is gauge invariant and $\hat{V}^{(+)}_{b}\Psi_{\rm
BF}(A^{(+)})=\hat{H}^{(+)}\Psi_{\rm BF}(A^{(+)})=0$. This state is proposed as
a tool to construct a flat vacuum state [9].
### 2.2 Chiral asymmetric extension
Following the strategy of Ref. [6], we first consider the chiral asymmetric
model with purely imaginary values of $\beta$. The left-handed action (3) is
extended to the chiral asymmetric one as follows:
$\displaystyle S$ $\displaystyle=$ $\displaystyle\alpha^{(+)}S_{\rm
H}^{(+)}+\alpha^{(-)}S_{\rm H}^{(-)}$ (13) $\displaystyle=$
$\displaystyle\frac{1}{4k}\int\biggl{[}\left(\alpha^{(+)}+\alpha^{(-)}\right)\epsilon_{IJKL}\,e^{I}\wedge
e^{J}\wedge\Omega^{KL}+2{\rm
i}\left(\alpha^{(+)}-\alpha^{(-)}\right)e^{I}\wedge
e^{J}\wedge\Omega_{IJ}\biggr{]}.$
Here $\alpha^{(+)}$ and $\alpha^{(-)}$ are mixing parameters of the left- and
right-handed components. The sign $(-)$ means right-handed, i.e., $\beta=-{\rm
i}$. To identify the action (13) with (1), the following identities are
obtained:
$\alpha^{(+)}+\alpha^{(-)}=1\ ,\hskip
14.22636pt\alpha^{(+)}-\alpha^{(-)}=\frac{{\rm i}}{\beta}\ .$ (14)
Note that, in the case of the left-handed action, these parameters take
$\alpha^{(+)}=1$ and $\alpha^{(-)}=0$. One can find that imaginary $\beta$
controls the degree of the chiral asymmetry. In this model, the Poisson
brackets of the canonical variables $(A^{(+)},E^{(+)})$ and
$(A^{(-)},E^{(-)})$ are
$\left\\{A^{(\pm)}{}_{a}^{i}(x),E^{(\pm)}{}_{j}^{b}(y)\right\\}=\pm\frac{{\rm
i}k}{\alpha^{(\pm)}}\,\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\ .$ (15)
Here $E^{(+)}{}_{i}^{a}=E^{(-)}{}_{i}^{a}=\det(e)e_{i}^{a}$ in the time gauge
$e_{a}^{0}=0$; nevertheless, these variables are treated independently of each
other. Each of the three constraints separates into left- and right-handed
components independently. Therefore, the extended wave function is given by
$\Psi(A^{(+)},A^{(-)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left({\rm
e}^{\alpha^{(+)}F_{ab}^{(+)}(x)}\right)\delta\left({\rm
e}^{-\alpha^{(-)}F_{ab}^{(-)}{(x)}}\right)\ .$ (16)
## 3 Generalized BF state
### 3.1 Real values of $\beta$
To consider the extended BF state for generic real values of $\beta$, new
configuration variables are introduced:
$\displaystyle A^{(-\frac{1}{\beta})}{}_{a}^{i}$ $\displaystyle=$
$\displaystyle\alpha^{(+)}A^{(+)}+\alpha^{(-)}A^{(-)}=\Gamma_{a}^{i}-\frac{1}{\beta}K_{a}^{i}\
,$ (17) $\displaystyle A^{(\beta)}{}_{a}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha^{(+)}-\alpha^{(-)}}\left(\alpha^{(+)}A^{(+)}-\alpha^{(-)}A^{(-)}\right)=\Gamma_{a}^{i}+\beta
K_{a}^{i}\ .$ (18)
The corresponding momentum variables are
$\displaystyle C_{i}^{a}$ $\displaystyle=$ $\displaystyle\frac{1}{2{\rm
i}}\left(E^{(+)}{}_{i}^{a}-E^{(-)}{}_{i}^{a}\right)=\epsilon^{abc}e_{bi}e_{c}^{0}\
,$ (19) $\displaystyle E_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(E^{(+)}{}_{i}^{a}+E^{(-)}{}_{i}^{a}\right)=\det(e)e_{i}^{a}\
.$ (20)
One can obtain the Poisson bracket relations as follows:
$\displaystyle\left\\{A^{(-\frac{1}{\beta})}{}_{a}^{i}(x),C_{j}^{b}(y)\right\\}$
$\displaystyle=$ $\displaystyle k\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\
,$ (21) $\displaystyle\left\\{A^{(\beta)}{}_{a}^{i}(x),E_{j}^{b}(y)\right\\}$
$\displaystyle=$ $\displaystyle
k\beta\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\ .$ (22)
To construct the generalized BF state, we attempt to define the extended BF
(EBF) action
$S_{\rm EBF}=\int{\rm Tr\,}\left[\alpha^{(+)}e^{(+)}\wedge
F^{(+)}-\alpha^{(-)}e^{(-)}\wedge F^{(-)}\right]\ ,$ (23)
where $e^{(\pm)}$ are triads playing the role of the $B$ field of the BF
action and are written as
$e^{(\pm)}{}_{a}^{i}=e_{a}^{i}=\frac{1}{2\sqrt{|\det(E)|}}\epsilon_{abc}\epsilon^{ijk}E_{j}^{b}E_{k}^{c}\
.$ (24)
The action $S_{\rm EBF}$ is expressed in terms of the variables
$A^{(-\frac{1}{\beta})}$ and $A^{(\beta)}$ as
$\displaystyle S_{\rm EBF}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}}{\beta}\int{\rm Tr\,}\left[e\wedge\left(F-\left(1+\beta^{2}\right)K\wedge
K\right)\right]$ (25) $\displaystyle=$ $\displaystyle\frac{{\rm
i}}{\beta}\int{\rm
Tr\,}\left[e\wedge\left(\left(1+\frac{1}{\beta^{2}}\right)R-\frac{1}{\beta^{2}}F-\beta{\rm
d}_{\Gamma}K\right)\right]\ ,$
where $F$ and $R$ are the curvatures of the connections $A^{(\beta)}$ and
$\Gamma$ respectively, and ${\rm d}_{\Gamma}K={\rm d}K+[\Gamma,K]$. The last
term vanishes for the torsion-free condition ${\rm d}_{\Gamma}e=0$.
One can propose an extended BF state defined in the following form:
$\displaystyle\Psi(A^{(\beta)},A^{(-\frac{1}{\beta})})$ $\displaystyle=$
$\displaystyle\int{\cal D}e\ \exp\left[\frac{{\rm
i}}{\alpha^{(+)}-\alpha^{(-)}}\ S_{\rm EBF}\right]$ (26) $\displaystyle=$
$\displaystyle\prod_{x\in\Sigma}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)R_{ab}(x)-\frac{1}{\beta^{2}}F_{ab}(x)\right]\right)\
.$
Note that when $\beta={\rm i}$, state (26) keeps the ordinary form (10). This
state has a problem. Due to the gauge fixing $e_{a}^{0}=0$, the wave function
should satisfy the additional constraint:
$\hat{C}_{i}^{a}\Psi=-{\rm i}k\frac{\delta}{\delta
A^{(-\frac{1}{\beta})}{}_{a}^{i}}\Psi=0\ .$ (27)
This equation implies that the wave function does not depend on the variable
$A^{(-\frac{1}{\beta})}$. However, the connection $\Gamma$ included in the
curvature $R$ is the explicit function of both $A^{(\beta)}$ and
$A^{(-\frac{1}{\beta})}$, namely,
$\Gamma_{a}^{i}=\frac{A^{(\beta)}{}_{a}^{i}+\beta^{2}A^{(-\frac{1}{\beta})}{}_{a}^{i}}{1+\beta^{2}}\
.$ (28)
To avoid this problem, we regard the connection $\Gamma$ as the explicit
variable of $E$, instead of $A^{(\beta)}$ and $A^{(-\frac{1}{\beta})}$. This
can be done via the torsion-free condition ${\rm d}_{\Gamma}e=0$. Taking this
modification into account, the extended BF state is redefined:
$\Psi_{R}(A^{(\beta)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)R_{ab}(x)-\frac{1}{\beta^{2}}F_{ab}(x)\right]\right)\
.$ (29)
Although the state $\Psi_{R}(A^{(\beta)})$ has the same form as (26), it is
the explicit function of $A^{(\beta)}$ only, and is parameterized by the Levi-
Civita curvature $R$. It is an analog of the fact that the ordinary wave
function $\Psi_{p}(x)=\exp[\,-{\rm i}\,(Et-{\bf p}\cdot{\bf x})\,]$ can be
regarded as the position function parameterized by the momentum.
### 3.2 Constraints and inner products
Here, we confirm whether state (29) satisfies the three constraints with real
values of $\beta$. The Gauss constraint requires the wave function to be
invariant under the $SU(2)$ gauge transformation. The state
$\Psi_{R}(A^{(\beta)})$ is gauge invariant because of the property of the
group delta function (11).
The simple inner product between two states can be supposed as
$\displaystyle\langle\,\Psi_{R^{\prime}}|\Psi_{R}\,\rangle$ $\displaystyle=$
$\displaystyle\int{\cal DA}\
\Psi_{R^{\prime}}^{\dagger}(A^{(\beta)})\Psi_{R}(A^{(\beta)})$ (30)
$\displaystyle=$
$\displaystyle\prod_{x}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)\left(R_{ab}-R^{\prime}_{ab}\right)\right]\right)$
$\displaystyle\equiv$ $\displaystyle\delta\left(R-R^{\prime}\right)\ .$
Here ${\cal DA}$ is the appropriate measure of the connection $A^{(\beta)}$
normalized such that $\int{\cal DA}=1$. This inner product is too sensitive.
When $R^{\prime}$ takes a different value from $R$, it vanishes, even if $R$
and $R^{\prime}$ are in the equivalence class of $SU(2)$ gauge and
diffeomorphism transformations. To make the inner product more convenient, the
following new inner product is introduced:
$\displaystyle(\,\Psi_{R^{\prime}}|\Psi_{R}\,)$ $\displaystyle=$
$\displaystyle\int{\cal D}\phi\ \langle\,\Psi_{R^{\prime}}|\ {\cal
U}(\phi)\,|\Psi_{R}\,\rangle$ (31) $\displaystyle=$ $\displaystyle\int{\cal
D}\phi\ \delta\left(R-\phi R^{\prime}\right)\ .$
Here ${\cal U}(\phi)$ is an operator of the gauge and diffeomorphism
transformations. The integral $\int{\cal D}\phi$ is over all of both
transformations. This construction of the inner product is an analogy of LQG
[11]. One can find that the dual state
$(\,\Psi_{R^{\prime}}|=\int{\cal D}\phi\ \langle\,\Psi_{R^{\prime}}|\ {\cal
U}(\phi)=\int{\cal D}\phi\ \langle\,\Psi_{\phi R^{\prime}}|$ (32)
is a solution of the Gauss and diffeomorphism constraints.
Finally, we consider the Hamiltonian constraint
$H=-\frac{\beta}{2\sqrt{|\det(E)|}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[F_{abk}-\left(1+\beta^{2}\right)\epsilon_{klm}K_{a}^{l}K_{b}^{m}\right]\
.$ (33)
Performing a similar calculation to (25), the smeared Hamiltonian constraint
is deformed as
$\displaystyle H(N)$ $\displaystyle=$ $\displaystyle\int{\rm d}^{3}x\ NH$ (34)
$\displaystyle=$ $\displaystyle-\int{\rm d}^{3}x\
\frac{N\beta}{2\sqrt{|\det(E)|}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[\left(1+\frac{1}{\beta^{2}}\right)R_{abk}-\frac{1}{\beta^{2}}F_{abk}\right]\
.$
Therefore, if the Levi-Civita curvature operator $\hat{R}$ can be well
defined, i.e., $\hat{R}\Psi_{R}(A^{(\beta)})=R\Psi_{R}(A^{(\beta)})$, then the
state $\Psi_{R}(A^{(\beta)})$ will satisfy
$\int{\rm d}^{3}x\
\chi^{abk}\left[\left(1+\frac{1}{\beta^{2}}\right)\hat{R}_{abk}-\frac{1}{\beta^{2}}\hat{F}_{abk}\right]\
\Psi_{R}(A^{(\beta)})=0\ ,$ (35)
where $\chi$ is a test function. According to Ref. [6], the Levi-Civita
curvature operator $\hat{R}$ is defined as follows:
$\int{\rm d}^{3}x\ \chi^{abk}\hat{R}_{abk}=\int{\cal D}\phi{\cal
D}R^{\prime}\left[\int{\rm d}^{3}x\ \chi^{abk}\left(\phi
R^{\prime}_{abk}\right)\right]|\Psi_{\phi
R^{\prime}}\,\rangle\langle\,\Psi_{\phi R^{\prime}}|\ ,$ (36)
where the integral $\int{\cal D}R^{\prime}$ is over the Levi-Civita curvature
$R^{\prime}$ modulo the equivalence class of the gauge and diffeomorphism
transformations. The action of this operator on the state $|\Psi_{R}\,\rangle$
becomes
$\displaystyle\int{\rm d}^{3}x\ \chi^{abk}\hat{R}_{abk}|\Psi_{R}\,\rangle$
$\displaystyle=$ $\displaystyle\int{\cal D}\phi{\cal D}R^{\prime}\
\delta(R-\phi R^{\prime})\left[\int{\rm d}^{3}x\ \chi^{abk}\left(\phi
R^{\prime}_{abk}\right)\right]|\Psi_{\phi R^{\prime}}\,\rangle$ (37)
$\displaystyle=$ $\displaystyle\int{\rm d}^{3}x\
\chi^{abk}R_{abk}|\Psi_{R}\,\rangle\ .$
With this operator, one obtains
$\hat{H}(N)\Psi_{R}(A^{(\beta)})=0\ .$ (38)
Consequently, the state $\Psi_{R}(A^{(\beta)})$ satisfies all three
constraints.
## 4 Conclusions and discussion
In this paper, we have constructed the generalized BF state for real values of
$\beta$. This has been done via the chiral asymmetric extension. The
generalized state is an explicit function of the connection $A^{(\beta)}$, and
is parameterized by the Levi-Civita curvature $R$ as well as in the
generalized CS state. It is gauge invariant and solves all constraints with
the appropriate inner product and the operator.
This state would be associated with the space such that
$(1+\beta^{2})R_{ab}-F_{ab}=0$. It contains a special case, i.e., a flat 3D
space $R=F=0$. More discussions are necessary to obtain further specific
interpretations. Problems with the connection with generic $\beta$ may arise,
because this connection is not a pull-back of a space-time connection [12].
It would be interesting to consider a loop representation of the state
$\Psi_{R}(A^{(\beta)})$:
$\displaystyle\Psi_{R}(\gamma)$ $\displaystyle=$
$\displaystyle\langle\,\gamma|\Psi_{R}\,\rangle=\int{\cal DA}\
\langle\,\gamma|A^{(\beta)}\,\rangle\langle\,A^{(\beta)}|\Psi_{R}\,\rangle$
(39) $\displaystyle\sim$ $\displaystyle\int{\cal DA}\
W(A^{(\beta)},\gamma)\prod_{x}\prod_{a,b}\delta\left(\exp\left[F_{ab}-(1+\beta^{2})R_{ab}\right]\right)\
.$
Here $W(A^{(\beta)},\gamma)$ is a spin network with a graph $\gamma$, which is
a generalized Wilson loop constructed from holonomy edges and invariant
tensors. The part
$\int{\cal DA}\
\prod_{x}\prod_{a,b}\delta\left(\exp\left[F_{ab}-(1+\beta^{2})R_{ab}\right]\right)$
(40)
looks like a generating functional of the spin foam model with a source term,
which is known as the Freidel–Krasnov (FK) model [13]. Let us consider a
discretized 3D space with a triangulation $\Delta$. The corresponding dual
cell $\Delta^{*}$ has vertices $v$, edges $l$, and faces $f$. In the FK model,
the discretized generating functional is given by
$\displaystyle Z_{\rm FK}[J]$ $\displaystyle=$ $\displaystyle\int{\cal DADB}\
\exp\left[{\rm i}\int{\rm Tr\,}(B\wedge F+B\wedge J)\right]$ (41)
$\displaystyle=$ $\displaystyle\int\prod_{l}{\rm d}g_{l}\
\prod_{f}\sum_{\Lambda_{f}\in{\rm Irrep}}\dim(\Lambda_{f})\,{\rm
Tr\,}\left[R^{(\Lambda_{f})}\left(g_{l_{1}}{\rm e}^{J_{v_{1}}}\cdots
g_{l_{n}}{\rm e}^{J_{v_{n}}}\right)\right],$
where $J$ is a source term for $B$, $R^{(\Lambda)}(g_{l})$ is a representation
of the group element $g_{l}=\exp[\int_{l}A]$ with a spin label $\Lambda$,
vertices $v_{1},\cdots,v_{n}$ and edges $l_{1},\cdots,l_{n}$ are associated
with the $n$-polygonal $\partial f$, and the sum is taken over all irreducible
representations. Thus the loop representation (39) is expressed as
$\displaystyle\Psi_{R}(\gamma)$ $\displaystyle=$
$\displaystyle\int\prod_{l}{\rm d}g_{l}\
W(A_{l}^{(\beta)},\gamma)\prod_{f}\sum_{\Lambda_{f}\in{\rm
Irrep}}\dim(\Lambda_{f})\,{\rm Tr\,}\left[R^{(\Lambda_{f})}\left(g_{l_{1}}{\rm
e}^{r_{v_{1}}}\cdots g_{l_{n}}{\rm e}^{r_{v_{n}}}\right)\right],$ (42)
where $r=-(1+\beta^{2})R$. The limit $R\to 0$ is consistent with the spin
network invariant $\Psi_{R=0}(\gamma)$ in Ref. [8].
## Acknowledgements
The authors would like to thank H. Taira, T. Oka, and K. Eguchi for helpful
discussions.
## References
* [1] F. Barbero, Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51, 5507 (1995), arXiv:gr-qc/9410014.
* [2] G. Immirzi, Real and complex connections for canonical gravity, Class. Quantum Grav. 14, L177 (1997), arXiv:gr-qc/9612030.
* [3] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: a status report, Class. Quantum Grav. 21, R53 (2004), arXiv:gr-qc/0404018.
* [4] H. Kodama, Holomorphic wave function of the universe, Phys. Rev. D 42, 2548 (1990).
* [5] A. Randono, A generalization of the Kodama state for arbitrary values of the Immirzi parameter, arXiv:gr-qc/0504010.
* [6] A. Randono, Generalizing the Kodama state I: Construction, arXiv:gr-qc/0611073.
* [7] A. Randono, Generalizing the Kodama state II: Properties and physical interpretation, arXiv:gr-qc/0611074.
* [8] A. Miković, Quantum gravity vacuum and invariants of embedded spin networks, Class. Quantum Grav. 20, 3483 (2003), arXiv:gr-qc/0301047.
* [9] A. Miković, Flat spacetime vacuum in loop quantum gravity, Class. Quantum Grav. 21, 3909 (2004), arXiv:gr-qc/0404021.
* [10] S. Holst, Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53, 5966 (1996), arXiv:gr-qc/9511026.
* [11] A. Perez, Introduction to loop quantum gravity and spin foams, arXiv:gr-qc/0409061.
* [12] J. Samuel, Is Barbero’s Hamiltonian formulation a gauge theory of Lorentzian gravity?, Class. Quantum Grav. 17, L141 (2000), arXiv:gr-qc/0005095.
* [13] L. Freidel and K. Krasnov, Spin foam models and the classical action principle, Adv. Theor. Math. Phys. 2, 1183 (1999), arXiv:hep-th/9807092.
|
arxiv-papers
| 2013-12-01T12:15:06 |
2024-09-04T02:49:54.595229
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shinji Yamashita, Satoshi Yajima and Makoto Fukuda",
"submitter": "Shinji Yamashita",
"url": "https://arxiv.org/abs/1312.0205"
}
|
1312.0218
|
# Inequalities for eigenvalues of the weighted Hodge Laplacian
Daguang Chen∗ and Yingying Zhang [email protected] Department of
Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China.
[email protected] Department of Mathematics, Lehigh University, Bethlehem, PA
USA 18015.
###### Abstract.
In this paper, we obtain ”universal” inequalities for eigenvalues of the
weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These
inequalities generalize the Yang-type and Levitin-Parnovski inequalities for
eigenvalues of the Laplacian and Laplacian. From the recursion formula of
Cheng and Yang [12], the Yang-type inequality for eigenvalues of the weighted
Hodge Laplacian are optimal in the sense of the order of eigenvalues.
###### Key words and phrases:
Eigenvalues, Weighted Hodge Laplacian, Universal inequalities, Self-shrinker
###### 2000 Mathematics Subject Classification:
35P15; 58J50; 58C40; 58A10
∗ This work of the first named author was partially supported by NSFC grant
No. 11101234.
## 1\. Introduction
Let $M^{m}$ be an $m$-dimensional complete Riemannian manifold and $\Omega$ be
a bounded domain in $M^{m}$. The Dirichlet eigenvalue problem of Laplacian is
given by
$\left\\{\begin{aligned} &\Delta u=-\lambda u,\qquad\text{in $\Omega$}\\\
&u=0,\qquad\qquad\text{on $\partial\Omega$}.\end{aligned}\right.$ (1.1)
It is well known that the spectrum of this problem is real and discrete:
$0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots\nearrow\infty,$
where each $\lambda_{i}$ has finite multiplicity which is repeated according
to its multiplicity.
The main developments were obtained by Payne, Pólya and Weinberger [32], Hile
and Protter [24] and Yang [36]. In 1956, Payne, Pólya and Weinberger [32]
proved that
$\lambda_{k+1}-\lambda_{k}\leq\frac{4}{mk}\sum_{i=1}^{k}\lambda_{i}.$ (1.2)
In 1980, Hile and Protter [24] improved (1.2) to
$\sum_{i=1}^{k}\frac{\lambda_{i}}{\lambda_{k+1}-\lambda_{i}}\geq\frac{mk}{4}.$
(1.3)
In 1991, Yang (see [36] and more recently [11]) obtained a very sharp
inequality
$\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})(\lambda_{k+1}-(1+\frac{4}{m})\lambda_{i})\leq
0.$ (1.4)
There has been much work dedicated to extending and strengthening the
classical inequalities of Payne-Pólya-Weinberger, Hile-Protter and Yang. When
$M^{m}$ is an $m$-dimensional compact manifold, there are similar results
about the eigenvalue estimates for the Laplacian (see, e.g.[31, 11, 28, 10,
17, 37]). For the compact Riemannian manifolds isometrically immersed in an
Euclidean space or a sphere, J. M. Lee [27] proved Hile-Protter type bounds
for eigenvalues for Hodge Laplacian on $p$-forms. In 2002, B. Colbois [15]
derived a Payne-Pólya-Weinberger type inequality for the rough Laplacian. In
[25], S. Ilias and O. Makhoul obtained inequalities for the eigenvalues of the
Hodge Laplacian.
In 1991, N. Anghel [1] obtained the analogous estimate of (1.2) for the Dirac
operator. In 2009, the Yang-type inequality (1.4) was extended to the
eigenvalues of Dirac operator by the first author in [8].
In [19], Harrell gave an abstract algebraic argument involving operators,
their commutators and traces, which generalize the original PPW arguments.
These algebraic ideas were developed in different contexts to produce many new
universal eigenvalues inequalities (see [5, 21, 20, 22, 23, 30]).
In present paper, making use of a theorem of Ashbaugh and Hermi [5], we obtain
the Yang-type inequality for higher order eigenvalues of the weighted Hodge
Laplacian for submanifolds in Euclidean space.
###### Theorem 1.1.
Let $x:(M^{m},g)\longrightarrow(\mathbb{R}^{n},{\rm can})$ be a compact self-
shrinker, $\Delta_{p,x}=\Delta_{H}+\frac{1}{2}\mathcal{L}_{\nabla|x|^{2}}$
(see below (2.9)) be the weighted Hodge Laplacian acting on $p$-forms over
$M^{m}$. Assume that $\Big{\\{}\lambda^{(p)}_{i}\Big{\\}}_{i=1}^{\infty}$ are
the eigenvalues of $\Delta_{p,x}$ and $\\{\varphi_{i}\\}_{i=1}^{\infty}$ is a
corresponding orthonormal basis of $p$-eigenforms. We have, for any
$p\in\left\\{0,1,\dots,m\right\\}$,
$\displaystyle
m\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)^{2}\leq$
$\displaystyle\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left(4\lambda^{(p)}_{i}+2m\right.$
(1.5)
$\displaystyle-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle+\left.4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)$
where $\\{e_{i}\\}_{i=1}^{m}$ is a local orthonormal basis of $TM^{m}$ with
respect to the induced metric $g$ and
$\mathfrak{Ric}=-\omega^{i}\wedge\imath(e_{j})R(e_{i},e_{j})$ (see also 2.8)
is the curvature operator acting on $p$-forms.
###### Remark 1.1.
When $p=0$, i.e., $\lambda_{i}:=\lambda^{(0)}_{i}$ are the eigenvalues of the
operator $\mathfrak{L}:=\Delta_{0,x}=\Delta+\langle x,\cdot\rangle$ acting on
scalar functions, we have
$\displaystyle m\sum_{i=1}^{k}\left(\lambda_{k+1}-\lambda_{i}\right)^{2}\leq$
$\displaystyle\sum_{i=1}^{k}\left(\lambda_{k+1}-\lambda_{i}\right)\left(4\lambda_{i}+2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)$
(1.6) $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}\left(\lambda_{k+1}-\lambda_{i}\right)\left(4\lambda_{i}+2m-\min_{M^{n}}|x|^{2}\right),$
which is Theorem 1.1 in [13]. Therefore, Theorem 1.1 generalizes eigenvalue
estimates from the operator $\mathfrak{L}$ to the weighted Hodge Laplacian
$\Delta_{p,x}$.
###### Remark 1.2.
If $\left|x\right|=c,(c>0)$, the manifold $M^{m}$ is a submanifold of sphere
$\mathbb{S}^{n-1}(\frac{1}{c})$ in Euclidean space $\mathbb{R}^{n}$.
Furthermore, the weighted Hodge Laplacian $\Delta_{p,x}$ is reduced to the
ordinary one.
For a compact self-shrinker (see (2.4)) in Euclidean space, we have
###### Corollary 1.1.
Let $x:(M^{m},g)\longrightarrow(\mathbb{R}^{n},{\rm can})$ be a compact self-
shrinker, $H,h$ be the second fundamental form and the mean curvature of the
immersion $x$, respectively. We have, $p\in\left\\{1,\dots,m\right\\}$,
$\displaystyle\sum_{i=1}^{k}(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i})^{2}$ (1.7)
$\displaystyle\leq$
$\displaystyle\frac{4}{m}\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left[\lambda^{(p)}_{i}+\frac{m}{2}+1\right.$
$\displaystyle\left.+\int_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right]$
$\displaystyle\leq$
$\displaystyle\frac{4}{m}\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left[\lambda^{(p)}_{i}+\frac{m}{2}+1\right.$
$\displaystyle\left.+\max_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)\right],$
where $\Phi(H,h)$ is a function depending on the second fundamental form $h$
and the mean curvature $H$ defined in (3.10).
From Theorem 1.1, we can obtain the spectral gaps of the consecutive
eigenvalues of the weighted Hodge Laplacian $\Delta_{p,x}$.
###### Corollary 1.2.
Under the same assumption in Corollary 1.1, we have
$\displaystyle\lambda^{(p)}_{k+1}-\lambda^{(p)}_{k}\leq$ $\displaystyle
2\left[\left(\frac{2}{m}\frac{1}{k}\sum_{i=1}^{k}\lambda^{(p)}_{i}+\frac{2}{m}+\frac{2}{m}\max_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)\right)^{2}\right.$
$\displaystyle-\left.\left(1+\frac{4}{m}\right)\frac{1}{k}\sum_{j=1}^{k}\left(\lambda^{(p)}_{j}-\frac{1}{k}\sum_{i=1}^{k}\lambda^{(p)}_{i}\right)^{2}\right]^{\frac{1}{2}}$
For the lower order eigenvalues of (1.1), in 1956, Payne, Pólya and Weinberger
[32] proved that for $\Omega\subset{\mathbb{R}}^{2}$,
$\lambda_{2}+\lambda_{3}\leq 6\lambda_{1},$
which was extended to domains $\Omega\subset{\mathbb{R}}^{m}$ in [35](or see
Section 3.2 of [2])
$\sum_{i=1}^{m}(\lambda_{i+1}-\lambda_{1})\leq 4\lambda_{1}.$
There are also a variety of extensions of results of this type, for examples,
see [7, 10, 8, 9, 11, 2, 34]. Recently, S. Ilias and O. Makhoul [26] obtained
the universal inequality for eigenvalues of the Hodge Laplacian.
In the second part of this paper, by using an algebraic identity deduced by
Levitin and Parnovski [30], we can obtain
###### Theorem 1.2.
Let $x:(M^{m},g)\longrightarrow(\mathbb{R}^{n},{\rm can})$ be a compact self-
shrinker and $\Delta_{p,x}$ be the weighted Hodge Laplacian defined acting on
$p$-forms over $M^{m}$. Assume that
$\Big{\\{}\lambda^{(p)}_{i}\Big{\\}}_{i=1}^{\infty}$ are the eigenvalues of
$\Delta_{p,x}$ and $\\{\varphi_{i}\\}_{i=1}^{\infty}$ is a corresponding
orthonormal basis of $p$-eigenforms. We have, for any
$p\in\left\\{0,1,\dots,m\right\\}$,
$\displaystyle\sum_{l=1}^{m}\left(\lambda^{(p)}_{i+l}-\lambda^{(p)}_{i}\right)\leq$
$\displaystyle
4\lambda^{(p)}_{i}+2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
(1.8)
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle+4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
###### Remark 1.3.
When $p=0$, i.e., $\lambda_{i}=\lambda^{(0)}_{i}$ is the eigenvalues of the
operator $\mathfrak{L}=\Delta_{0,x}=\Delta+\langle x,\cdot\rangle$ acting on
scalar functions, we have
$\displaystyle\sum_{l=1}^{m}(\lambda_{i+l}-\lambda_{i})\leq$ $\displaystyle
4\lambda_{i}+2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
(1.9) $\displaystyle\leq$ $\displaystyle 4\lambda_{i}+2m-\min_{M^{m}}|x|^{2}.$
Since $i$ is arbitrary, (1.9) is more general than Proposition 4.1 in [13].
###### Corollary 1.3.
Let $x:(M^{m},g)\longrightarrow(\mathbb{R}^{n},{\rm can})$ be a self-shrinker,
$H,h$ be the second fundamental form and the mean curvature of the immersion
$x$, respectively. Assume that
$\Big{\\{}\lambda^{(p)}_{i}\Big{\\}}_{i=1}^{\infty}$ are the eigenvalues of
$\Delta_{p,x}$ and $\\{\varphi_{i}\\}_{i=1}^{\infty}$ is a corresponding
orthonormal basis of $p$-eigenforms. We obtain, for
$p\in\left\\{1,\dots,m\right\\}$,
$\displaystyle\sum_{l=1}^{m}\left(\lambda^{(p)}_{i+l}-\lambda^{(p)}_{i}\right)\leq$
$\displaystyle 4\lambda^{(p)}_{i}+2m+4$ (1.10)
$\displaystyle+4\int_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\leq$ $\displaystyle
4\lambda^{(p)}_{i}+2m+4+\max_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right).$
Furthermore, from the recursion formula of Cheng and Yang [12], we can obtain
an upper bound for eigenvalue $\lambda^{(p)}_{k}$:
###### Corollary 1.4.
Let $M^{m}$ be an $m$-dimensional compact self-shrinker in $\mathbb{R}^{n}$.
Then, eigenvalues of the weighted Hodge Laplacian $\Delta_{p,x}$ 2.9 satisfy,
for any $k\geq 1$,
$\mu_{k+1}\leq C_{0}(m)k^{\frac{2}{m}}\mu_{1}$
where $C_{0}(m)\leq 1+\frac{4}{m}$ is a constant and
$\mu_{i}=\lambda^{(p)}_{i}+\frac{m}{2}+1+\max_{M^{m}}\Big{(}p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\Big{)}$.
This paper is organized as follows: In Section 2, we present some formulas for
submanifolds in Euclidean space, the definitions of the weighted Hodge
Laplacian. In Section 3, in order to prove main theorems, we derive several
lemmas for differential forms. In Section 4 and Section 5, we give the proofs
of Theorem 1.1 and 1.2.
### Acknowledgments
The authors wish to express their gratitude to Professors Huaidong Cao and
Xiaofeng Sun for their suggestions and useful discussions. This work of the
first named author was done while the author visited Department of
Mathematics, Lehigh University, USA. He also would like to thank the institute
for its hospitality.
## 2\. Preliminaries
### 2.1. Submanifold in Euclidean space and self-shrinker
Let $x:M^{m}\to\mathbb{R}^{n}$ be an $m$-dimensional submanifold of
$n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let
$\\{e_{1},\cdots,e_{m}\\}$ be a local orthonormal basis of $TM^{m}$ with
respect to the induced metric, and $\\{\omega^{1},\cdots,\omega^{m}\\}$ be
their dual 1-forms. Let $\\{e_{m+1},\cdots,e_{n}\\}$ be the local orthonormal
unit normal vector fields. In this paper we make the following conventions on
the range of indices:
$1\leq i,j,k\leq m;\qquad m+1\leq\alpha,\beta,\gamma\leq n.$
Then we have the following structure equations (see [8, 13])
$\displaystyle dx=\omega^{i}e_{i},\qquad\omega^{\alpha}=0,$ (2.1)
$\displaystyle
de_{i}=\omega^{j}_{i}e_{j}+\omega^{\alpha}_{i}e_{\alpha},\qquad\omega^{\alpha}_{i}=h^{\alpha}_{ij}\omega^{j},$
$\displaystyle
de_{\alpha}=\omega^{j}_{\alpha}e_{j}+\omega^{\beta}_{\alpha}e_{\beta},$
where $h^{\alpha}_{ij}$ denote the the components of the second fundamental
form of $M^{m}$. We denote by
$|h|^{2}=\sum\limits_{\alpha,i,j}(h^{\alpha}_{ij})^{2},$ the norm square of
the second fundamental form,
$H=\sum\limits_{\alpha}H^{\alpha}e_{\alpha}=\sum\limits_{\alpha}(\sum\limits_{i}h^{\alpha}_{ii})e_{\alpha}$
the mean curvature vector field over $M^{m}$.
One can deduce that, pointwise on $M^{m}$,
$\sum_{A=1}^{n}|\nabla x^{A}|^{2}=m,$ (2.2)
and
$\frac{1}{2}|x|^{2}_{,ij}=\frac{1}{2}(\sum_{A=1}^{n}(x^{A})^{2})_{,ij}=\langle
h^{\alpha}_{ij}e_{\alpha},x\rangle+\delta_{ij}.$ (2.3)
The submanifold $M^{m}$ is called a self-shrinker [16] if it satisfies the
quasilinear elliptic system:
$H=-x^{\perp},$ (2.4)
where $H$ denotes the mean curvature vector field of the immersion and $\perp$
is the projection onto the normal bundle of $M^{m}$.
### 2.2. Differential forms and the weighted Hodge Laplacian
Let $(M^{m},g)$ be an $m$-dimensional compact Riemannian manifold. For any two
$p$-forms $\varphi$ and $\psi$, we let $\varphi_{i_{1}\cdots
i_{p}}=\varphi(e_{i_{1}},\cdots,e_{i_{p}})$ and $\psi_{i_{1}\cdots
i_{p}}=\psi(e_{i_{1}},\cdots,e_{i_{p}})$ denote the components of $\varphi$
and $\psi$, with respect to a local orthonormal frame $\\{e_{i}\\}_{i=1}^{m}$.
Their pointwise inner product with respect to Riemannian metric $g$ is given
by
$\displaystyle\langle\varphi,\psi\rangle=$ $\displaystyle{\sum_{1\leq
i_{1}<\cdots<i_{p}\leq
m}}\varphi_{{i_{1}}\cdots{i_{p}}}\;\psi_{{i_{1}}\cdots{i_{p}}}$
$\displaystyle=$ $\displaystyle\frac{1}{p!}\sum_{1\leq i_{1},\dots,i_{p}\leq
m}\varphi_{{i_{1}}\cdots{i_{p}}}\;\psi_{{i_{1}}\cdots{i_{p}}}.$
We denote by $\Delta_{p}$ the Hodge Laplacian acting on $p$-forms
$\Delta_{p}=(d\,\delta+\delta d),$ (2.5)
where $d$ is the exterior derivative acting on $p$-forms and $\delta$ is the
adjoint of $d$ with respect to Riemannian measure $dvol$.
In [6, 33], the operator (2.5) is generalized to the weighted Hodge Laplacian
acting on differential forms. Let $f\in C^{\infty}(M^{m},\mathbb{R})$ be a
smooth function defined on $M^{m}$. When the Riemannian measure is changed
from being dvol to $e^{-f}{dvol}$, it is natural to define the weighted Hodge
Laplacian by
$\Delta_{p,f}=d\delta^{\prime}+\delta^{\prime}d$ (2.6)
where $\delta^{\prime}=e^{f}\delta e^{-f}$, which is the adjoint operator of
the exterior derivative $d$ with respect to Riemannian measure $e^{-f}{dvol}$.
For the weighted Hodge Laplacian, we have the following Bochner-Weitzenböck
type formula [33]
$\displaystyle\Delta_{p,f}=$ $\displaystyle\Delta_{p}+\mathcal{L}_{\nabla f}$
(2.7) $\displaystyle=$
$\displaystyle\nabla^{*}\nabla-\omega^{i}\wedge\imath(e_{j})R(e_{i},e_{j})+\mathcal{L}_{\nabla
f}$ $\displaystyle=$
$\displaystyle\nabla^{*}_{f}\nabla-\omega^{i}\wedge\imath(e_{j})R(e_{i},e_{j})-\nabla(\nabla
f)$ $\displaystyle=$
$\displaystyle\nabla^{*}_{f}\nabla+\mathfrak{Ric}-\nabla(\nabla f)$
where $\mathcal{L}$ is the Lie derivative, $\imath(X)$ for
$X\in\Gamma(TM^{m})$ is inner product acting on forms, $\nabla X$ acting on
from $\varphi$ is given by
$\nabla X\varphi={X^{j}}_{,l}\omega^{l}\wedge\imath(e_{j})\varphi$
and
$\mathfrak{Ric}=-\omega^{i}\wedge\imath(e_{j})R(e_{i},e_{j}).$ (2.8)
With respect to the measure $e^{-f}{dvol}$, the spectrum of $\Delta_{p,f}$
consists of a nondecreasing, unbounded sequence of eigenvalues with finite
multiplicities
${\rm
Spec}(\Delta_{p,f})=\\{0\leq\lambda^{(p)}_{1}\leq\lambda^{(p)}_{2}\leq\lambda^{(p)}_{3}\leq\cdots\leq\lambda^{(p)}_{k}\leq\cdots\\}.$
Let $x=(x^{1},\cdots,x^{n}):M^{m}\to\mathbb{R}^{n}$ be an $m$-dimensional
submanifold of $\mathbb{R}^{n}$. In this article, we will consider the
operator (2.6) over $M^{m}$, for $f=e^{\frac{1}{2}|x|^{2}}$,
$\Delta_{p,x}=d\delta^{\prime}+\delta^{\prime}d.$ (2.9)
For $\Delta_{p,x}$ acting on the scalar functions, the operator (2.9) [16]
111The Laplacian operator is different in [16] with a minus sign. is given by
$\mathfrak{L}=\Delta+\langle
x,\cdot\rangle=-e^{\frac{|x|^{2}}{2}}\mbox{div}\left(e^{\frac{-|x|^{2}}{2}}d\right)=\delta^{\prime}d=\Delta_{0,x}.$
(2.10)
where $\Delta$ is the positive operator. If $M^{m}$ is a self-shrinker, we
have
$\mathfrak{L}x^{A}=x^{A},\qquad A=1,\cdots,n.$ (2.11)
## 3\. Some lemmas
In order to prove our main theorems, we will derive some lemmas in this
section.
By the direct calculations, we have
###### Lemma 3.1.
For $f,u\in C^{\infty}(M,\mathbb{R})$ and
$\varphi\in\bigwedge^{p}(T^{*}M^{m})$, we have
$\mathcal{L}_{\nabla f}(u\varphi)=g(\nabla f,\nabla
u)\varphi+u\mathcal{L}_{\nabla f}\varphi.$ (3.1)
$[\Delta_{p,f},u]\varphi=[\Delta_{p},u]\varphi+[\mathcal{L}_{\nabla
f},u]\varphi$ (3.2) $\delta_{f}(u\varphi)=-\imath(\nabla
u)\varphi+u\delta_{f}\varphi$ (3.3)
where $[\Delta_{p,f},u]\varphi=\Delta_{p,f}(u\varphi)-u\Delta_{p,f}\varphi$.
###### Lemma 3.2.
Assuming that $T_{ij}$ is a symmetric 2-tensor, we have, for any $p$-form
$\varphi$,
$\displaystyle T_{ij}\omega^{i}\wedge\imath(e_{j})\varphi=$
$\displaystyle\frac{1}{p!}\sum_{i_{1},\cdots,i_{p}}(T\varphi)_{i_{1}\cdots
i_{p}}\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}}$ (3.4) $\displaystyle=$
$\displaystyle\frac{1}{(p-1)!}\sum_{i_{1},\cdots,i_{p}}\left(\sum_{j}T_{ji_{1}}\varphi_{ji_{2}\cdots
i_{p}}\right)\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}},$
and
$\left\langle\sum_{i,j=1}^{m}T_{ij}\omega^{i}\wedge\imath(e_{j})\varphi,\varphi\right\rangle=\frac{1}{(p-1)!}\sum_{j,i_{1},\cdots,i_{p}}T_{ji_{1}}\varphi_{ji_{2}\cdots
i_{p}}\varphi_{i_{1}\cdots i_{p}}$ (3.5)
where $(T\varphi)_{i_{1}\cdots
i_{p}}=\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{p}T_{ji_{k}}\varphi_{i_{1}\cdots
j\cdots i_{p}}$.
###### Proof.
Assuming that
$\varphi=\frac{1}{p!}\sum\limits_{i_{1},\cdots,i_{p}}\varphi_{i_{1}\cdots
i_{p}}\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}}$, then we get
$\displaystyle T_{ij}\omega^{i}\wedge\imath(e_{j})\varphi$ $\displaystyle=$
$\displaystyle\frac{1}{p!}\sum_{i,j=1}T_{ij}\omega^{i}\wedge\imath(e_{j})\left(\sum_{i_{1},\cdots,i_{p}}\varphi_{i_{1}\cdots
i_{p}}\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{p!}\sum_{i,j,i_{1},\cdots,i_{p}}\sum_{k=1}^{p}(-1)^{k-1}T_{ij}\delta_{j}^{i_{k}}\varphi_{i_{1}\cdots
i_{p}}\omega^{i}\wedge\omega^{i_{1}}\wedge\cdots\wedge\widehat{\omega^{i_{k}}}\wedge\cdots\wedge\omega^{i_{p}}$
$\displaystyle=$
$\displaystyle\frac{1}{p!}\sum_{i,i_{1},\cdots,\hat{i_{k}},\cdots
i_{p}}\sum_{k=1}^{p}(-1)^{k-1}T_{ij}\varphi_{i_{1}\cdots j\cdots
i_{p}}\omega^{i}\wedge\omega^{i_{1}}\wedge\cdots\wedge\widehat{\omega^{i_{k}}}\wedge\cdots\wedge\omega^{i_{p}}$
$\displaystyle=$
$\displaystyle\frac{1}{p!}\sum_{i_{1},\cdots,i_{p}}(T\varphi)_{i_{1}\cdots
i_{p}}\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}}$ $\displaystyle=$
$\displaystyle\frac{1}{(p-1)!}\sum_{i_{1},\cdots,i_{p}}T_{ji_{1}}\varphi_{ji_{2}\cdots
i_{p}}\omega^{i_{1}}\wedge\cdots\wedge\omega^{i_{p}}.$
Therefore, we obtain
$\displaystyle\langle\sum_{i,j=1}^{m}T_{ij}\omega^{i}\wedge\imath(e_{j})\varphi,\varphi\rangle=$
$\displaystyle\frac{1}{p!}\sum_{i_{1},\cdots,i_{p}}(T\varphi)_{i_{1}\cdots
i_{p}}\varphi_{i_{1}\cdots i_{p}}$ $\displaystyle=$
$\displaystyle\frac{1}{(p-1)!}\sum_{j,i_{1},\cdots,i_{p}}T_{ji_{1}}\varphi_{ji_{2}\cdots
i_{p}}\varphi_{i_{1}\cdots i_{p}}.$
∎
###### Lemma 3.3.
Under the same assumptions in Lemma 3.2, then we have
$|\sum_{i,j,i_{2},\cdots,i_{p}}T_{ij}\varphi_{ii_{2}\cdots
i_{p}}\varphi_{ji_{2}\cdots i_{p}}|\leq p!|T|\varphi|^{2}$ (3.6)
and
$\left\langle\sum_{i,j=1}^{m}T_{ij}\omega^{i}\wedge\imath(e_{j})\varphi,\varphi\right\rangle\leq
p|T|\varphi|^{2}$ (3.7)
where $|T|=\big{(}\sum\limits_{i,j}T_{ij}^{2}\big{)}^{\frac{1}{2}}$.
###### Proof.
$\displaystyle\left|\sum_{i,j,i_{2},\cdots,i_{p}}T_{ij}\varphi_{ii_{2}\cdots
i_{p}}\varphi_{ji_{2}\cdots i_{p}}\right|=$
$\displaystyle\left|\sum_{i_{2},\cdots,i_{p}}\sum_{j}(\sum_{i}T_{ij}\varphi_{ii_{2}\cdots
i_{p}})(\varphi_{ji_{2}\cdots i_{p}})\right|$ $\displaystyle\leq$
$\displaystyle\sum_{i_{2},\cdots,i_{p}}\left(\sum_{j}(\sum_{i}T_{ij}\varphi_{ii_{2}\cdots
i_{p}})^{2}\sum_{k}\varphi_{ki_{2}\cdots i_{p}}^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{i_{2},\cdots,i_{p}}\left(\sum_{j}\sum_{i}T_{ij}^{2}\sum_{l}\varphi_{li_{2}\cdots
i_{p}}^{2}\sum_{k}\varphi_{ki_{2}\cdots i_{p}}^{2}\right)^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left(\sum_{i,j}T_{ij}^{2}\right)^{\frac{1}{2}}\sum_{k,i_{2},\cdots,i_{p}}\varphi_{ki_{2}\cdots
i_{p}}^{2}$ $\displaystyle=$
$\displaystyle|T|\sum_{i,i_{2},\cdots,i_{p}}\varphi_{ii_{2}\cdots i_{p}}^{2}$
$\displaystyle=$ $\displaystyle p!|T|\varphi|^{2}.$
∎
###### Lemma 3.4.
Assume that $x:M^{m}\longrightarrow\mathbb{R}^{n}$ is a compact self-shrinker,
$H,h$ are the second fundamental form and the mean curvature of the immersion
$x$, respectively. We have, for any $p$-form $\varphi$, $p\in\\{1,\dots,m\\}$
$\displaystyle\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2}))\varphi,\varphi\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ (3.8) $\displaystyle\leq$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\int_{M^{m}}|H||h||\varphi|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\leq$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\max_{M^{m}}|H||h|.$
###### Proof.
From (2.3), and taking $T_{ij}=\langle h^{\alpha}_{ij}e_{\alpha},x\rangle$ in
(3.7), we obtain
$\displaystyle\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2}))\varphi,\varphi\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ $\displaystyle=$
$\displaystyle\int_{M^{m}}\langle\langle
h^{\alpha}_{ij}e_{\alpha},x\rangle\omega^{i}\wedge\imath(e_{j})\varphi,\varphi\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\leq$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\int_{M^{m}}\Big{(}\sum_{i,j}\langle
h^{\alpha}_{ij}e_{\alpha},x\rangle^{2}\Big{)}^{\frac{1}{2}}|\varphi|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle=$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\int_{M^{m}}\Big{(}\sum_{i,j}\big{(}H^{\alpha}h^{\alpha}_{ij}\big{)}^{2}\Big{)}^{\frac{1}{2}}|\varphi|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\leq$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\int_{M^{m}}|H||h||\varphi|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\leq$
$\displaystyle\int_{M^{m}}\left|\varphi\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+p\max_{M^{m}}|H||h|.$
∎
Combining Proposition 4.1 in [18] and Theorem 1.1 in [29], we obtain the
estimate of $\mathfrak{Ric}$ (2.8) acting on $p$-forms. (c.f. Theorem 3.2 of
[25])
###### Lemma 3.5.
$\langle\mathfrak{Ric}(\varphi),\varphi\rangle\geq\Phi(h,H)|\varphi|^{2},\qquad\varphi\in\textstyle{\bigwedge^{p}}(T^{*}M^{m}),$
(3.9)
where
$\displaystyle\Phi(h,H)=$
$\displaystyle\bigg{\\{}-p^{2}\bigg{[}\Big{(}\frac{m-5}{4}\Big{)}|H|^{2}+|h|^{2}-\frac{1}{4m^{2}}\Big{(}\sqrt{m-1}(m-2)|H|$
(3.10)
$\displaystyle-2\sqrt{m|h|^{2}-|H|^{2}}\,\Big{)}^{2}\bigg{]}-\frac{1}{2}\sqrt{p}(p-1)\Big{(}|H|^{2}+|h|^{2}\Big{)}\bigg{\\}}.$
## 4\. Inequalities for eigenvalues
In order to obtain the extrinsic bounds of higher order eigenvalues of the
weighted Hodge Laplacian, we firstly introduce the abstract formula derived by
Ashbaugh and Hermi [5].
Let $\mathfrak{H}$ be a complex Hilbert space with inner product $(,)$,
$\mathcal{A}:\mathcal{D}\subset\mathfrak{H}\longrightarrow\mathfrak{H}$ a
self-adjoint operator defined on a dense domain $\mathcal{D}$ that is bounded
below and has a discrete spectrum
$\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots.$
Let
$\\{\mathcal{B}_{k}:\mathcal{A}(\mathcal{D})\longrightarrow\mathfrak{H}\\}_{k=1}^{N}$
be a collection of symmetric operators leaving $\mathcal{D}$ invariant and
$\\{\varphi_{i},\lambda_{i}\\}_{i=1}^{\infty}$ be the spectral resolution of
$\mathcal{A}$. Moreover, $\\{\varphi_{i}\\}_{i=1}^{\infty}$ consisting of the
orthnormal basis w.r.t. inner product $(,)$ for $\mathfrak{H}$ is assumed.
Define the commutator $[\mathcal{A},\mathcal{B}]$ and the norm $\|\varphi\|$
by, respectively
$[\mathcal{A},\mathcal{B}]=\mathcal{A}\mathcal{B}-\mathcal{B}\mathcal{A},\qquad\|\varphi\|^{2}=(\varphi,\varphi).$
Based on commutator algebra and the Rayleigh-Ritz principle, M.S. Ashbaugh and
L. Hermi[5] obtained
###### Theorem 4.1.
The eigenvalues $\lambda_{i}$ of the operator $\mathcal{A}$ satisfy the Yang-
type inequality
$\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\rho_{i}\leq\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\Lambda_{i}$
(4.1)
where $\rho_{i},\Lambda$ are defined by, respectively,
$\displaystyle\rho_{i}$
$\displaystyle=\sum_{k=1}^{N}\langle{[\mathcal{A},\mathcal{B}_{k}]\varphi_{i}},{\mathcal{B}_{k}\varphi_{i}}\rangle$
$\displaystyle\Lambda_{i}$
$\displaystyle=\sum_{k=1}^{N}\|[\mathcal{A},\mathcal{B}_{k}]\varphi_{i}\|^{2}.$
Applying Theorem 4.1 to the weighted Hodge Laplacian $\Delta_{p,f}$, we have
###### Lemma 4.1.
Let $(M^{m},g)$ be an $m$-dimensional Riemannian manifold with Riemannian
measure $e^{-f}\mbox{dvol}$ and $u$ be a smooth function defined on $M^{m}$.
For the eigenvalues $\Big{\\{}\lambda^{(p)}_{i}\\}_{i=1}^{\infty}$ of the
weighted Hodge Laplacian $\Delta_{p,f}$ (2.6) acting on $p$-forms, we have,
$p\in\left\\{0,1,\dots,m\right\\}$,
$\displaystyle\sum_{i=1}^{k}(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i})^{2}\int_{M^{m}}\left|\nabla
u\right|^{2}\left|\varphi_{i}\right|^{2}e^{-f}\mbox{dvol}$ (4.2)
$\displaystyle\leq\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\int_{M^{m}}\Big{(}(\Delta_{0,f}u)^{2}|\varphi_{i}|^{2}+4|\nabla_{\nabla
u}\varphi_{i}|^{2}$
$\displaystyle-4\langle\Delta_{0,f}u\varphi_{i},\nabla_{\nabla
u}\varphi_{i}\rangle\Big{)}e^{-f}\mbox{dvol}$
where $\\{\varphi_{i}\\}_{i=1}^{\infty}$ is a corresponding orthonormal basis
of $p$-eigenforms, i.e.
$\int_{M^{m}}\langle{\varphi_{i}},{\varphi_{j}}\rangle
e^{-f}\mbox{dvol}=\delta_{ij}.$
###### Proof.
It is easy to check that $\mathcal{A}=\Delta_{p,f}$ and $\mathcal{B}=u\in
C^{\infty}(M^{m},\mathbb{R})$ satisfy the conditions in Theorem 4.1.
Therefore, by the estimate of (4.1), we have
$\displaystyle\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)^{2}$
$\displaystyle\int_{M}\langle[\Delta_{p,f},u]\varphi_{i},u\varphi_{i}\rangle
e^{-f}\mbox{dvol}$ (4.3)
$\displaystyle\leq\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\|[\Delta_{p,f},u]\varphi_{i}\|^{2},$
where
$\|[\Delta_{p,f},u]\varphi_{i}\|^{2}=\int_{M^{m}}\langle{[\Delta_{p,f},u]\varphi_{i}},{[\Delta_{p,f},u]\varphi_{i}}\rangle
e^{-f}\mbox{dvol}.$
By direct calculations, we have
$\displaystyle[\Delta_{p,f},u]\varphi_{i}$
$\displaystyle=[\Delta_{p}+\mathcal{L}_{\nabla f},u]\varphi_{i}$
$\displaystyle=[\Delta_{p},u]\varphi_{i}+[\mathcal{L}_{\nabla
f},u]\varphi_{i}.$ (4.4)
From (3.1), we obtain
$[\mathcal{L}_{\nabla f},u]\varphi_{i}=g(\nabla f,\nabla u)\varphi_{i}.$ (4.5)
By (2.7), we have
$\displaystyle[\Delta_{p},u]\varphi_{i}$
$\displaystyle=[\nabla^{*}\nabla,u]\varphi_{i}$ $\displaystyle=\Delta
u\varphi_{i}-2\nabla_{\nabla u}\varphi_{i}.$ (4.6)
Therefore, from (4.4) to (4.6) we get
$[\Delta_{p,f},u]\varphi_{i}=\Delta_{0,f}u\varphi_{i}-2\nabla_{\nabla
u}\varphi_{i}$ (4.7)
From (4.7), we have
$\displaystyle\int_{M^{m}}\langle[\Delta_{p,f},u]\varphi_{i},u\varphi_{i}\rangle
e^{-f}\mbox{dvol}$ $\displaystyle=$
$\displaystyle\int_{M^{m}}\left\langle\Delta_{0,f}u\varphi_{i}-2\nabla_{\nabla
u}\varphi_{i},u\varphi_{i}\right\rangle e^{-f}\mbox{dvol}.$
By integration by parts, we have
$\displaystyle 2\int_{M^{m}}\langle\nabla_{\nabla
u}\varphi_{i},u\varphi_{i}\rangle e^{-f}\mbox{dvol}$
$\displaystyle=\frac{1}{2}\int_{M^{m}}\langle{\nabla|\varphi_{i}|^{2}},{\nabla
u^{2}}\rangle e^{-f}\mbox{dvol}$
$\displaystyle=\int_{M^{m}}(u\Delta_{0,f}u-|\nabla
u|^{2})|\varphi_{i}|^{2}e^{-f}\mbox{dvol}.$
Finally, we obtain
$\int_{M^{m}}\langle[\Delta_{p,f},u]\varphi_{i},u\varphi_{i}\rangle
e^{-f}\mbox{dvol}=\int_{M^{m}}|\nabla
u|^{2}|\varphi_{i}|^{2}e^{-f}\mbox{dvol}.$ (4.8)
On the other hand, using (4.7), we get
$\displaystyle\|[\Delta_{p,f},u]\varphi_{i}\|^{2}=$
$\displaystyle\int_{M^{m}}\Big{(}(\Delta_{0,f}u)^{2}|\varphi_{i}|^{2}+4|\nabla_{\nabla
u}\varphi_{i}|^{2}$ (4.9)
$\displaystyle-4\langle\Delta_{0,f}u\varphi_{i},\nabla_{\nabla
u}\varphi_{i}\rangle\Big{)}e^{-f}\mbox{dvol}.$
Inserting (4.8) and (4.9) into (4.3), we obtain (4.2). ∎
###### Proof of Theorem 1.1.
Letting $f=\frac{1}{2}|x|^{2}$ and therefore $\Delta_{p,x}=\Delta_{p,f}$,
substituting $u=x^{A},A=1,\cdots,n$, the $p^{th}$ component of the isometric
immersion $x=(x^{1},\cdots,x^{n}):M^{m}\longrightarrow\mathbb{R}^{n}$ in
(4.2), and taking summation on $p$ from $1$ to $n$, we have
$\displaystyle\sum_{A=1}^{n}\sum_{i=1}^{k}(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i})^{2}\int_{M^{m}}\left|\nabla
x^{A}\right|^{2}\left|\varphi_{i}\right|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
(4.10)
$\displaystyle\leq\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\int_{M^{m}}\sum_{A=1}^{n}\Big{(}(\mathfrak{L}x^{A})^{2}|\varphi_{i}|^{2}+4|\nabla_{\nabla
x^{A}}\varphi_{i}|^{2}$
$\displaystyle-4\langle\mathfrak{L}x^{A}\varphi_{i},\nabla_{\nabla
x^{A}}\varphi_{i}\rangle\Big{)}e^{-\frac{|x|^{2}}{2}}\mbox{dvol},$
where $\mathfrak{L}$ is the weighted Hodge Laplaican acting on functions given
by (2.10). From (2.2), we obtain
$\displaystyle\int_{M^{m}}\sum_{p=1}^{n}|\nabla
x^{A}|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle=m\int_{M^{m}}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
(4.11) $\displaystyle=m.$
From (2.11) and (4.9), we get
$\displaystyle\sum_{A=1}^{n}\|[\Delta_{p,x},x^{A}]\varphi_{i}\|^{2}$ (4.12)
$\displaystyle=$
$\displaystyle\sum_{A=1}^{n}\int_{M^{m}}\bigg{(}(\mathfrak{L}x^{A})^{2}|\varphi_{i}|^{2}+4|\nabla_{\nabla
x^{A}}\varphi_{i}|^{2}-4\langle\mathfrak{L}x^{A}\varphi_{i},\nabla_{\nabla
x^{A}}\varphi_{i}\rangle\bigg{)}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle=$
$\displaystyle\sum_{A=1}^{n}\int_{M^{m}}\bigg{(}(x^{A})^{2}|\varphi_{i}|^{2}+4|\nabla_{\nabla
x^{A}}\varphi_{i}|^{2}-4\langle x^{A}\varphi_{i},\nabla_{\nabla
x^{A}}\varphi_{i}\rangle\bigg{)}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
Since $M^{m}$ is a compact self-shrinker, by integration by parts and (2.3),
we have
$4\sum_{A=1}^{n}\int_{M^{m}}\langle x^{A}\varphi_{i},\nabla_{\nabla
x^{A}}\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}=-\int_{M^{m}}2(m-|x|^{2})|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
Since $\displaystyle\sum_{A=1}^{n}|\nabla_{\nabla
x^{A}}\varphi_{i}|^{2}=|\nabla\varphi|^{2}$, we have
$\displaystyle\sum_{A=1}^{n}\|[\Delta_{p,x},x^{A}]\varphi_{i}\|^{2}=$
$\displaystyle
2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
(4.13)
$\displaystyle+4\int_{M^{m}}|\nabla\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
By integration by parts, from (2.7), (3.8) and (3.9) , we have
$\displaystyle\int_{M^{m}}|\nabla\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}=$
$\displaystyle\int_{M^{m}}\langle\nabla^{\prime*}\nabla\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ (4.14) $\displaystyle=$
$\displaystyle\int_{M^{m}}\langle(\Delta_{p,x}-\mathfrak{Ric}+\nabla(\nabla\frac{|x|^{2}}{2}))\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ $\displaystyle=$
$\displaystyle\lambda^{(p)}_{i}-\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle+\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol},$
where $\nabla^{\prime*}$ is the adjoint operator of $\nabla$ with respect to
the Riemannian measure $e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$. Therefore, we
obtain
$\displaystyle\sum_{A=1}^{n}\|[\Delta_{p,x},x^{A}]\varphi_{i}\|^{2}=$
$\displaystyle 4\lambda^{(p)}_{i}+2m$ (4.15)
$\displaystyle+4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
From (4.3) and (4.15), we get
$\displaystyle m\sum_{i=1}^{k}(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left(4\lambda^{(p)}_{i}+2m\right.$
$\displaystyle+4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\left.-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right),$
which completes the proof of Theorem 1.1. ∎
###### Proof of Corollary 1.1.
From (4.10), (3.8) and (3.9), we have
$\displaystyle m\sum_{i=1}^{k}(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left(4\lambda^{(p)}_{i}+2m+4\right.$
$\displaystyle+4p\int_{M^{m}}|H||h|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle\left.-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)$ $\displaystyle\leq$ $\displaystyle
4\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left[\lambda^{(p)}_{i}+\frac{m}{2}+1\right.$
$\displaystyle\left.+\int_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right]$
$\displaystyle\leq$ $\displaystyle
4\sum_{i=1}^{k}\left(\lambda^{(p)}_{k+1}-\lambda^{(p)}_{i}\right)\left[\lambda^{(p)}_{i}+\frac{m}{2}+1\right.$
$\displaystyle\left.+\max_{M^{m}}\left(p|H||h|-\Phi(H,h)-\frac{1}{4}|x|^{2}\right)\right].$
∎
## 5\. Generalization of the Levitin-Parnovski inequality
In this section, we will give the proof of Theorem 1.2 by similar argument in
[26]. Firstly, we recall the following algebraic identity obtained by Levitin
and Parnovski (see identity 2.2 of Theorem 2.2 in [30]).
###### Lemma 5.1.
Let $\mathcal{L}$ and $\mathcal{G}$ be two self-adjoint operators with domains
$D_{\mathcal{L}}$ and $D_{\mathcal{G}}$ contained in a same Hilbert space and
such that $G(D_{\mathcal{L}})\subseteq D_{\mathcal{L}}\subseteq
D_{\mathcal{G}}$. Let $\lambda_{j}$ and $u_{j}$ be the eigenvalues and
orthonormal eigenvectors of $\mathcal{L}$. Then, for each $j$,
$\sum_{k=1}^{\infty}\frac{|\langle[\mathcal{L},\mathcal{G}]u_{j},u_{k}\rangle|^{2}_{L^{2}}}{\lambda_{k}-\lambda_{j}}=\displaystyle{-\frac{1}{2}\langle[[\mathcal{L},\mathcal{G}],\mathcal{G}]u_{j},u_{j}\rangle}_{L^{2}}$
(5.1)
(The summation is over all $k$ and is correctly defined even when
$\lambda_{k}=\lambda_{j}$ because in this case
$\langle[\mathcal{L},\mathcal{G}]u_{j},u_{k}\rangle=0$).
###### Proof of Theorem 1.2.
By applying Lemma 5.1 with $\mathcal{L}=\Delta_{p,x}$ and $\mathcal{G}=x^{A}$,
where $x^{A}$ is one of the components of the isometric immersion $x$, we have
$\sum_{k=1}^{\infty}\frac{|\langle[\Delta_{p,x},x^{A}]u_{j},u_{k}\rangle|^{2}_{L^{2}}}{\lambda_{k}-\lambda_{j}}=\displaystyle{-\frac{1}{2}\langle[[\Delta_{p,x},x^{A}],x^{A}]u_{j},u_{j}\rangle}_{L^{2}}.$
(5.2)
From (4.7), we have
$\displaystyle[[\Delta_{p,x},x^{A}],x^{A}]\varphi_{i}=$
$\displaystyle[\Delta_{p,x},x^{A}](x^{A}\varphi_{i})-x^{A}([\Delta_{p,x},x^{A}]\varphi_{i})$
$\displaystyle=$
$\displaystyle\mathfrak{L}(x^{A})x^{A}\varphi_{i}-2\nabla_{\nabla
x^{A}}(x^{A}\varphi_{i})-x^{A}(\mathfrak{L}x^{A}\varphi_{i}-2\nabla_{\nabla
x^{A}}\varphi_{i})$ $\displaystyle=$ $\displaystyle-2\nabla_{\nabla
x^{A}}(x^{A}\varphi_{i})+2x^{A}\nabla_{\nabla x^{A}}\varphi_{i}$
$\displaystyle=$ $\displaystyle-2|\nabla x^{A}|^{2}\varphi_{i},$
hence
$-\frac{1}{2}\int_{M^{m}}\langle[[\Delta_{p,x},x^{A}],x^{A}]\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}=\int_{M^{m}}|\nabla
x^{A}|^{2}|\varphi_{j}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
From (5.2), we have
$\displaystyle\int_{M^{m}}|\nabla
x^{A}|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ (5.3)
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{\lambda^{(p)}_{k}-\lambda^{(p)}_{i}}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}.$
For a fixed $i$, from the Gram-Schmidt orthogonalization, we can find the
coordinate system $\\{x^{A}\\}_{A=1}^{n}$ in Euclidean space $\mathbb{R}^{n}$
such that the matrix
$\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{i+k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)_{1\leq k,\;A\leq n}$
is a real upper triangular matrix. That is,
$\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{i+k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}=0,\qquad 1\leq k<A\leq n.$ (5.4)
By (5.4), we can estimate the right hand side of (5.3) in the following
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{\lambda^{(p)}_{k}-\lambda^{(p)}_{i}}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{i-1}\frac{1}{\lambda^{(p)}_{k}-\lambda^{(p)}_{i}}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}$
$\displaystyle+\sum_{k=i+A}^{\infty}\frac{1}{\lambda^{(p)}_{k}-\lambda^{(p)}_{i}}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{k=i+A}^{\infty}\frac{1}{\lambda^{(p)}_{k}-\lambda^{(p)}_{i}}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{(p)}_{i+A}-\lambda^{(p)}_{i}}\sum_{k=1}^{\infty}\left(\int_{M^{m}}\langle[\Delta_{p,x},x^{A}]\varphi_{i},\varphi_{k}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{(p)}_{i+A}-\lambda^{(p)}_{i}}\int_{M^{m}}|[\Delta_{p,x},x^{A}]\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
where Parceval’s identity is used in the last equality.
Taking summation on $A$ from $1$ to $n$, from (5.3), (4.13), (4.14) and (2.7),
we have
$\displaystyle\sum_{A=1}^{n}(\lambda^{(p)}_{i+A}-\lambda^{(p)}_{i})\int_{M^{m}}|\nabla
x^{A}|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$ (5.5)
$\displaystyle\leq$
$\displaystyle\sum_{A=1}^{n}\int_{M^{m}}|[\Delta_{p,x},x^{A}]\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle=$ $\displaystyle
2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}+4\int_{M^{m}}|\nabla\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle=$ $\displaystyle
4\lambda^{(p)}_{i}+2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle+4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
Since $M^{n}$ is isometrically immersed in $\mathbb{R}^{n}$, it is easy to
check
$\sum_{A=1}^{n}\lambda^{(p)}_{i+A}|\nabla
x^{A}|^{2}\geq\sum_{l=1}^{m}\lambda^{(p)}_{i+l}.$ (5.6)
Therefore, we have
$\displaystyle\sum_{l=1}^{m}(\lambda^{(p)}_{i+l}-\lambda^{(p)}_{i})\leq$
$\displaystyle
4\lambda^{(p)}_{i}+2m-\int_{M^{m}}|x|^{2}|\varphi_{i}|^{2}e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle-4\int_{M^{m}}\langle\mathfrak{Ric}\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}$
$\displaystyle+4\int_{M^{m}}\langle\nabla(\nabla\frac{|x|^{2}}{2})\varphi_{i},\varphi_{i}\rangle
e^{-\frac{|x|^{2}}{2}}\mbox{dvol}.$
∎
###### Proof of Corollary 1.3.
The proof of Corollary 1.3 follows directly from (3.8) and (3.9). ∎
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|
arxiv-papers
| 2013-12-01T13:22:24 |
2024-09-04T02:49:54.603419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daguang Chen and Yingying Zhang",
"submitter": "Daguang Chen",
"url": "https://arxiv.org/abs/1312.0218"
}
|
1312.0310
|
# Meson exchange effects in elastic $ep$ scattering at loop level and the
electromagnetic form factors of the proton
Hong-Yu Chen1 , Hai-Qing Zhou1,2111E-mail: [email protected]
1Department of Physics, Southeast University, NanJing 211189, China
2State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics,
Chinese Academy of Sciences, Beijing 100190, P. R. China
###### Abstract
A new form of two-photon exchange(TPE) effect is studied to explain the
discrepancy between unpolarized and polarized experimental data in elastic
$ep$ scattering. The mechanism is based on a simple idea that apart from the
usual TPE effects from box and crossed-box diagrams, the mesons may also be
exchanged in elastic $ep$ scattering by two-photon coupling at loop level. The
detailed study shows such contributions to reduced unpolarized cross section
($\sigma_{un}$) and polarized observables ($P_{t},P_{l}$) at fixed $Q^{2}$ are
only dependent on proton’s electromagnetic form factors $G_{E,M}$ and a new
unknown universal parameter $g$. After combining this contribution with the
usual TPE contributions from box and crossed-box diagrams, the ratio
$\mu_{p}G_{E}/G_{M}$ extracted from the recent precise unpolarized and
polarized experimental data can be described consistently.
###### pacs:
13.40.Gp,25.30.Bf
## I Introduction
As the basic constituent of our world and most elemental bound states of
strong interaction, the proton plays an important role in the physics. Up to
now, our knowledge on the structure of proton has still been poor, for
example, how big is the protonproton-size , how large are the electromagnetic
form factors $G_{E,M}$ of the protonEx-polarized ; Ex-polarized-Meziane-2011 ;
Ex-Rosenbluth-1994 ; Ex-Rosenbluth-2006 . Since the first measurement of
$R=\mu_{p}G_{E}/G_{M}$ by the polarization transfer (PT) methodEx-polarized ,
it becomes a serious problem for theoretical physicists to explain the large
discrepancy of extracted $R$ between the PT method and Rosenbluth or
longitudinal-transverse (LT) methodEx-Rosenbluth-1994 ; Ex-Rosenbluth-2006 .
In the Born approximation, the elastic $ep$ scattering is described by one-
photon exchange (OPE) shown in Fig. 1(a). By this approximation, the reduced
unpolarized cross section is expressed as
$\displaystyle\sigma_{un,th}^{1\gamma}\equiv\left.{\frac{d\sigma^{(un)}}{d\Omega}}\right|_{lab}\frac{\varepsilon(1+\tau)}{\tau\sigma_{ns}}={G_{M}^{2}+\frac{\varepsilon}{\tau}G_{E}^{2}},$
(1)
and the polarized observables $P_{t},P_{l}$ are expressed as
$\displaystyle P_{t,th}^{1\gamma}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sigma_{un,th}^{1\gamma}}\sqrt{2\varepsilon(1-\varepsilon)/\tau}G_{M}G_{E},$
(2) $\displaystyle P_{l,th}^{1\gamma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{un,th}^{1\gamma}}\sqrt{(1+\varepsilon)(1-\varepsilon)}G_{M}^{2},$
$\displaystyle R_{PT,th}^{1\gamma}$ $\displaystyle\equiv$
$\displaystyle-\mu_{p}\sqrt{\frac{\tau(1+\epsilon)}{2\epsilon}}\frac{P_{t,th}^{1\gamma}}{P_{l,th}^{1\gamma}}=\mu_{p}\frac{G_{E}}{G_{M}},$
with
$\sigma_{ns}=\frac{\alpha^{2}cos^{2}(\theta_{e}/2)}{4E^{2}sin^{4}(\theta_{2}/2)}\frac{E^{\prime}}{E}$,
$\tau=Q^{2}/4M_{N}^{2},Q^{2}=-q^{2},q=p_{1}-p_{3},\epsilon=[1+2(1+\tau
tan^{2}\theta_{e}/2)]^{-1}$, $M_{N}$ the mass of proton, $\alpha$ the fine
structure constant, $\theta_{e}$ the scattering angle of electron, $E$ and
$E^{\prime}$ the energies of initial and final electrons in the laboratory
frame, respectively. The detail of the physical meaning of $P_{t,l}$ can be
seen in the literature, for example, Ex-polarized .
Experimentally, the LT method extracts $R$ from the $\epsilon$ dependence of
an experimental unpolarized cross section at fixed $Q^{2}$ by Eq.(1) and the
PT method extracts $R$ from the experimental ratio $P_{t}/P_{l}$ at fixed
$Q^{2}$ and $\epsilon$ by Eq.(2). In the following we name such extracted $R$s
as $R_{LT,Ex}^{1\gamma}$ and $R_{PT,Ex}^{1\gamma}$, respectively. The current
precise experimental measurementsEx-polarized ; Ex-Rosenbluth-2006 show that
$R_{LT,Ex}^{1\gamma}$ are much larger than $R_{PT,Ex}^{1\gamma}$ when
$Q^{2}>$2GeV2.
In the literature, two-photon exchange (TPE) effects are suggested to explain
such a discrepancy TPE-review . Many model dependent methods are studied to
estimate the TPE corrections such as the simple hadronic model TPE-hadronic-
model , GPDs method TPE-GPDs , dispersion relation method TPE-dispersion-
relation , pQCD TPE-pQCD , and SCET TPE-SCEF . These model dependent
calculations gave similar TPE corrections to $R_{LT,Ex}^{1\gamma}$, and it is
usually concluded that the discrepancy is able to be explained by TPE
corrections TPE-hadronic-model ; Arrinton2007 . But the recent polarized
experimental data Ex-polarized-Meziane-2011 show very different properties of
TPE corrections to $R_{PT,Ex}^{1\gamma}$ with that predicted by these
theoretical models. For example, the experimental data showed that the TPE
corrections to $R_{PT,Ex}^{1\gamma}$ are almost a constant at
$\epsilon=(0.152,0.635,0.785)$ when $Q^{2}=2.49$ GeV2 Ex-polarized-
Meziane-2011 , while the theoretical estimations of TPE corrections are large
and positive at small $\epsilon$ by the hadronic model and dispersion relation
method TPE-hadronic-model ; TPE-dispersion-relation , and are large and
negative at small $\epsilon$ by the GPDs method and pQCD method TPE-GPDs ;
TPE-pQCD . This situation shows that we are still far away from the accurate
understanding of experimental data in elastic $ep$ scattering. And a further
careful study of TPE corrections or similar effects are strongly called for.
In this work, we consider a new form of TPE effect in elastic $ep$ scattering.
The main idea is from the theoretical estimations of virtual Compton
scattering(VCS) and photoproduction of the vector meson. For these two
processes, the contributions from the $s$, $u$, and $t$-channels shown in
Figs. 1(b,c,d) are usually all included in the effective models meson-exchange
; Regge-meson-exchange . When considering the radiative corrections in elastic
$ep$ scattering, it is natural that the corresponding similar contributions
shown as Figs. 2(a,b,c) will give contributions, where only the permitted spin
0 and 2 mesons are included in the $t$ channel. Figures 2(a,b) are just the
usual box and crossed-box diagrams studied in TPE-hadronic-model , while the
contribution from Fig.2(c) is usually ignored in the literature. In Sec. II,
at first we rewrite the contribution from Fig.2(c) in a simple and general
form by the effective interactions, and then present the expressions for the
reduced unpolarized cross section and polarized observables after including
this contribution. In Sec. III, we present our numerical analysis on the
recent experimental data, the TPE corrections to the extracted $R$ by LT and
PT methods, and the TPE contributions to the ratio between unpolarized cross
sections of elastic $e^{+}p$ and $e^{-}p$ scattering.
Figure 1: (a)The Born diagram in elastic $ep$ scattering. (b,c,d) The
$s$,$u$,$t$ channels in photoproduction of vector meson, the similar diagrams
in VCS are not shown. Figure 2: TPE contributions in $ep$ scattering. (a) box
diagram; (b) crossed-box diagram; (c) meson-exchange diagram by two-photon
coupling; (d) effective direct meson-exchange diagram.
## II Basic Formula
The formal gauge invariant couplings of $M\gamma\gamma$ in Fig.2(c) can be
written down similarly with those in meson-exchange ; Regge-meson-exchange ,
while in the case of Fig. 2(c), the two virtual photons are in the loop and
their momentums are not limited by any conditions except their sum. This is
different with the usual VCS case where the coupling constants are taken as
constants or multiplied by some special form factors in a special kinematic
region. To avoid the uncertainty from the momentum dependent coupling
constants and describe the effect in a reliable and universal form, we rewrite
the contributions from Fig.2(c) in a general effective direct meson-exchange
form shown as Fig.2(d) where all the momentum dependence of $M\gamma\gamma$
couplings and their integrations are absorbed into the effective couplings
between electron and mesons, and the new effective couplings now are only
dependent on $Q^{2}$. The most general form of the effective interactions for
$0^{++},0^{-+},2^{++}$ mesons can be written as
$\displaystyle\Gamma_{See}$ $\displaystyle=$ $\displaystyle-
ig_{See},~{}~{}~{}~{}\Gamma_{Spp}=-ig_{Spp},$ (3) $\displaystyle\Gamma_{Pee}$
$\displaystyle=$ $\displaystyle
g_{Pee,1}\gamma_{5}-ig_{Pee,2}\gamma_{5}(p\\!\\!\\!/_{f}-p\\!\\!\\!/_{i}),$
$\displaystyle\Gamma_{Tee,\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{Tee,1}(p_{f}+p_{i})_{\mu}\gamma_{\nu}-ig_{Tee,2}g_{\mu\nu},$
$\displaystyle\Gamma_{Ppp}$ $\displaystyle=$ $\displaystyle
g_{Ppp,1}\gamma_{5}-ig_{Ppp,2}\gamma_{5}(p\\!\\!\\!/_{f}-p\\!\\!\\!/_{i}),$
$\displaystyle\Gamma_{Tpp,\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{Tpp,1}(p_{f}+p_{i})_{\mu}\gamma_{\nu}-ig_{Tpp,2}g_{\mu\nu},$
where $S,P,T$ refer to the scalar, pseudoscalar, and tensor meson,
$p_{i},p_{f}$ refer to the initial and final momentums of electron and proton,
and all the couplings $g_{i}$ are only functions of $Q^{2}$. The propagators
of exchanged mesons are taken as the Regge form Regge-meson-exchange
$\displaystyle S_{S,P}(q)$ $\displaystyle=$
$\displaystyle\mathcal{P}_{S,P}(q),$ (4) $\displaystyle
S^{\mu\nu;\rho\omega}_{T}(q)$ $\displaystyle=$
$\displaystyle\Pi^{\mu\nu;\rho\omega}(q)\mathcal{P}_{T}(q),$
where
$\Pi^{\mu\nu;\rho\omega}(q)=\frac{1}{2}(\eta^{\mu\rho}\eta^{\nu\omega}+\eta^{\mu\omega}\eta^{\nu\rho})-\frac{1}{3}\eta^{\mu\nu}\eta^{\rho\omega}$,
$\eta^{\mu\nu}=-g^{\mu\nu}+q^{\mu}q^{\nu}/m_{T}^{2}$ and
$\displaystyle\mathcal{P}_{X}$ $\displaystyle=$
$\displaystyle\frac{\pi\alpha^{\prime}_{X}}{\Gamma[\alpha_{X}(t)-J_{X}+1]\sin[\pi\alpha_{X}(t)]}\left(\frac{s}{s_{0}}\right)^{\overline{\alpha}_{X}},$
(5)
with $\overline{\alpha}_{X}=\alpha^{\prime}_{X}(t-m^{2}_{X})$,
$\alpha_{X}(t)=J_{X}+\alpha^{\prime}_{X}(t-m^{2}_{X})$. Here $\alpha_{X}$
denotes the Regge trajectory for the meson $X$ as a function of $t=-Q^{2}$
with the slope $\alpha^{\prime}_{X}$, $J_{X}$ and $m_{X}$ stand for the spin
and mass of the meson, respectively. The phase factors of the propagators are
taken as positive unity since they do not affect the results.
With Eqs. (3)-(5), the contribution from interference of Figs. 2(d) and 1(a)
can be calculated directly. After combining it with the Born contribution, the
reduced unpolarized cross section is expressed as
$\displaystyle\sigma_{un,th}^{1\gamma+2\gamma(M)}$ $\displaystyle=$
$\displaystyle\sigma_{un,th}^{1\gamma}+gf_{0}s^{\overline{\alpha}_{T}}(G_{M}(1+\varepsilon)\tau+2G_{E}\varepsilon),$
(6)
and the polarized observables $P_{t},P_{l}$ are expressed as
$\displaystyle P_{t,th}^{1\gamma+2\gamma(M)}$ $\displaystyle=$ $\displaystyle
P_{t,th}^{1\gamma}\frac{\sigma_{un,th}^{1\gamma}}{\sigma_{un,th}^{1\gamma+2\gamma(M)}}-\frac{gf_{1}s^{\overline{\alpha}_{T}}(G_{E}+2G_{M})}{\sigma_{un,th}^{1\gamma+2\gamma(M)}},$
$\displaystyle P_{l,th}^{1\gamma+2\gamma(M)}$ $\displaystyle=$ $\displaystyle
P_{l,th}^{1\gamma}\frac{\sigma_{un,th}^{1\gamma}}{\sigma_{un,th}^{1\gamma+2\gamma(M)}}+\frac{gf_{2}s^{\overline{\alpha}_{T}}G_{M}}{\sigma_{un,th}^{1\gamma+2\gamma(M)}},$
(7)
where
$f_{0}=\sqrt{\tau(1+\tau)(1+\varepsilon)/(1-\varepsilon)},f_{1}=\tau\sqrt{\varepsilon(1+\varepsilon)(1+\tau)/2},f_{2}=\tau^{3/2}\sqrt{(1+\tau)}(2\varepsilon+1),\sqrt{s}$
is the center of mass of the $ep$ system and $g$ is expressed as
$\displaystyle
g=\textrm{Re}[\frac{-4iM_{N}^{4}g_{Tee,1}g_{Tpp,1}\alpha^{\prime}_{T}}{\alpha\Gamma[\alpha_{T}(t)-J_{T}+1]\sin[\pi\alpha_{X}(t)]}\left(\frac{1}{s_{0}}\right)^{\overline{\alpha}_{T}}].$
The most important property of the above three corrections is that only the
$2^{++}$ meson-exchange gives contributions due to the zero mass of the
electron. This property lead to the interesting result that the three
corrections to $\sigma_{un,th}^{1\gamma},P_{t,l,th}^{1\gamma}$ are only
dependent on one new parameter $g$ which is a constant at fixed $Q^{2}$. This
makes it possible to extract $g$ by fitting the unpolarized experimental data
with Eq.(6) and then use such extracted parameters to predict the TPE
corrections to $P_{t,l,th}^{1\gamma}$. A nenefit of such extracting and
prediction is its universality since we have not assumed any special model
dependent calculation for the coupling. If the extracted $g$ is zero then it
naturally means the meson-exchange mechanism can be neglected and the
extracted $G_{E,M}$ naturally return to those extracted by Eq.(1), and if the
extracted $g$ is not zero, then it means the meson-exchange effect really
exists or there are some other similar notable physical effects beyond the OPE
and usual TPE corrections from Fig.2(a,b). The second important property of
the corrections is that they all vanish when $\epsilon\rightarrow 1$ due to
the factor $s^{\overline{\alpha}_{T}}$ which is expected by unitarity.
In the practical calculation, we take
$\overline{\alpha}_{T}=0.8(t-1.3^{2}$GeV2)Regge-meson-exchange and the
detailed analysis shows that the results are not sensitive to the slope of
$\overline{\alpha}_{T}$ in the region [0.7,0.9].
To estimate the TPE contributions from Fig.2(a,b), we use the simple hadronic
model and include $N$ and $\Delta$ as the intermediate states. For the TPE
contributions from $N$, we take the same parameters as TPE-hadronic-model .
For the TPE contribution from $\Delta$, we improve the choice of the coupling
parameters and form factors of $\Gamma_{\gamma N\Delta}$ used in TPE-hadronic-
model by taking $(g_{1},g_{2},g_{3})$=$(6.59,9.06,7.16)$ and
$\displaystyle F^{(1)}_{\Delta}$ $\displaystyle=$ $\displaystyle
F^{(2)}_{\Delta}=\left(\frac{\Lambda_{1}^{2}}{q^{2}-\Lambda_{1}^{2}}\right)^{2}\frac{-\Lambda_{3}^{2}}{q^{2}-\Lambda_{3}^{2}},$
(8) $\displaystyle F^{(3)}_{\Delta}$ $\displaystyle=$
$\displaystyle\left(\frac{\Lambda_{1}^{2}}{q^{2}-\Lambda_{1}^{2}}\right)^{2}\frac{-\Lambda_{3}^{2}}{q^{2}-\Lambda_{3}^{2}}\left[a\frac{-\Lambda_{2}^{2}}{q^{2}-\Lambda_{2}^{2}}+(1-a)\frac{-\Lambda_{4}^{2}}{q^{2}-\Lambda_{4}^{2}}\right],$
with $\Lambda_{1,2,3,4}=(0.84,2,\sqrt{2},0.2)$GeV and $a=-0.3$. Such coupling
parameters and form factors of $\gamma N\Delta$ are much closer to the
physical results SNYang-PR than those used in TPE-hadronic-model . With these
inputs, the contribution from the interference of Figs. 2(a,b) and1(a) can be
calculated directly as TPE-hadronic-model and the detailed analysis of these
two contributions can see zhouhq2014 .
## III Numerical results and discussion
To show the meson-exchange corrections to the extracted $R$ in the LT method,
at first we apply the usual TPE corrections from Figs .2(a,b) 111In this
paper, all the TPE correction from $N$ intermediate state refers to the one
that the soft part has been deducted as done in TPE-hadronic-model . to the
experimental data sets of unpolarized cross sections as done in Arrinton2007 ,
and then extract the corresponding $R$ from the TPE-corrected data using Eqs.
(1) and (6), respectively. We name such extracted $R$ as
$R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta)}$ and
$R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$, respectively. The results are
presented in Fig.3 where only the recent precise experimental data Ex-
Rosenbluth-2006 are taken and the error bar of experimental data is taken as
the weight in the fitting.
Figure 3: Extracted $R$ by the LT and PT methods. $R_{LT,Ex}^{1\gamma}$ refers
to the extracted $R$ by Eq.(1) from the experimental data without any TPE
corrections , $R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta)}$ and
$R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ refer to the extracted $R$ by Eqs.
(1) and (6) after applying the usual TPE corrections from Fig.2(a,b) to the
experimental data, respectively. The unpolarized experimental data are taken
from Ex-Rosenbluth-2006 and $R_{PT,Ex}^{1\gamma}$ are taken from Ex-polarized
. The error bar of experimental data is taken as the weight in the fitting.
The results in Fig.3 clearly show that when no TPE contributions are
considered, the extracted $R_{LT,Ex}^{1\gamma}$ Ex-Rosenbluth-2006 are
totally inconsistent with that by the PT method $R_{PT,Ex}^{1\gamma}$ Ex-
polarized . After considering the usual TPE contributions from Figs. 2(a,b),
the extracted $R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta)}$ are much closer to
$R_{PT,Ex}^{1\gamma}$, while an obvious discrepancy still exists for
$Q^{2}=3.2,4.1$ GeV2 cases. When the meson-exchange contribution is also
considered, the extracted $R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ are
naturally close to $R_{PT,Ex}^{1\gamma}$.
In the following, we will show that in the region where most of the PT
experiment is measured, $R_{PT,Ex}^{1\gamma}$ are close to
$R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ with
$R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ defined as the extracted $R$ by the
PT method after applying the TPE correction to the experimental PT data. The
combination of the above two properties means
$R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ are consistent with
$R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ and the larger discrepancy of $R$
between the PT and LT methods can be well understood.
| results with the error bar as weight in the fitting | results without weight
in the fitting
---|---|---
$Q^{2}$(GeV${}^{2})$ | $G_{M}$ | $R$ | $g$ | $G_{M}$ | $R$ | $g$
2.46 | 0.136 | 0.704 | -0.439 | 0.136 | 0.704 | -0.461
3.2 | 0.101 | 0.639 | -1.203 | 0.101 | 0.639 | -1.213
4.1 | 0.066 | 0.556 | -6.377 | 0.067 | 0.352 | -8.590
Table 1: Extracted parameters $G_{M},R,g$ by Eq.(6) after applying the usual
TPE corrections from Fig.2(a,b) to experimental dataEx-Rosenbluth-2006 .
We list the extracted $G_{M},R,g$ by the above method in Tab.1, where, for
comparison, the extracted results without any weight are also presented. The
comparison shows the extracted results are almost independent on the weight at
$Q^{2}=2.64,3.2$ GeV2, this means the experimental data sets are very precise
at these two $Q^{2}$. From Table 1, we can see that the absolute magnitude of
$g$ increases when $Q^{2}$ increases. At first glance, this property seems
very un-natural, while actually the coupling $g$ is always accompanied by a
factor $s^{\overline{\alpha}_{T}}$ which decreases very quickly when $Q^{2}$
increases since $s\geq M_{N}^{2}(1+\tau)(1+2\tau+2\sqrt{\tau(1+\tau)})$.
In the following discussion, we take the $G_{M},R,g$ in the left side of Table
1 as the physical quantities to calculate the polarized observables
$P_{t,l,th}^{1\gamma+2\gamma(N,\Delta,M)}$ and their ratio
$R_{PT,th}^{1\gamma,1\gamma+2\gamma(N,\Delta,M)}$ which is defined as
$-\mu_{p}\sqrt{\tau(1+\epsilon)/2\epsilon}P_{t,th}^{1\gamma+2\gamma(N,\Delta,M)}/P_{l,th}^{1\gamma+2\gamma(N,\Delta,M)}$,
where the indexes $1\gamma$ and $2\gamma(N,\Delta,M)$ refer to the results
without and with corresponding TPE contributions, respectively. To compare the
theoretical TPE corrections with the polarized experimental results directly,
we define
$\displaystyle\Delta P_{t,l,th}^{N,\Delta,M}$ $\displaystyle\equiv$
$\displaystyle P_{t,l,th}^{1\gamma+2\gamma(N,\Delta,M)}/P_{t,l,th}^{1\gamma},$
$\displaystyle\Delta R_{PT,th}^{N,\Delta,M}$ $\displaystyle\equiv$
$\displaystyle R_{PT,th}^{1\gamma+2\gamma(N,\Delta,M)}/R_{PT,th}^{1\gamma}.$
(9)
After all the TPE corrections are included, we expect the following properties
if the TPE corrections are the right ones:
$\displaystyle P_{t,l,th}^{1\gamma+2\gamma(N+\Delta+M)}$ $\displaystyle=$
$\displaystyle P_{t,l,Ex},$ $\displaystyle
R_{PT,th}^{1\gamma+2\gamma(N+\Delta+M)}$ $\displaystyle=$ $\displaystyle
R_{PT,Ex}^{1\gamma},$ $\displaystyle R_{PT,th}^{1\gamma}$ $\displaystyle=$
$\displaystyle R_{PT,Ex}^{1\gamma+1\gamma(N+\Delta+M)}=\mu_{p}G_{E}/G_{M},$
(10)
where $P_{t,l,Ex}$ refer to the measured $P_{t,l}$ by experiment. This results
in
$\displaystyle\Delta P_{l,th}^{N+\Delta+M}$ $\displaystyle=$ $\displaystyle
P_{l,th}^{1\gamma+2\gamma(N+\Delta+M)}/P_{l,th}^{1\gamma}$ $\displaystyle=$
$\displaystyle P_{l,Ex}/P_{l,Ex}^{Born},$ $\displaystyle\Delta
R_{PT,th}^{N+\Delta+M}$ $\displaystyle=$ $\displaystyle
R_{PT,th}^{1\gamma+2\gamma(N+\Delta+M)}/R_{PT,th}^{1\gamma}$ (11)
$\displaystyle=$ $\displaystyle
R_{PT,Ex}^{1\gamma}/R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$
$\displaystyle\approx$ $\displaystyle
R_{PT,Ex}^{1\gamma}/R_{PT,Ex}^{1\gamma}|_{\epsilon\approx 1},$
where the approximate equal is due to the unitarity that TPE corrections to
the extracted $R$ by the PT method are assumed to be zero at $\epsilon=1$, and
$P_{l,Ex}^{Born}$ is estimated in a corresponding experiment Ex-polarized-
Meziane-2011 . By these relations, we can compare our theoretical results with
the experimental data directly. The numerical results are presented in Figs. 4
and5.
Figure 4: Theoretical estimations of TPE corrections to $R_{PT}$. $\Delta
R_{PT,th}^{N,\Delta,M,N+\Delta+M}$ refer to the corresponding theoretical
estimations of TPE contributions from $N,\Delta$ intermediate states, meson-
exchange and their sum, respectively. The experimental results are taken from
Ex-polarized-Meziane-2011 and normalized at $\epsilon=0.785$. Figure 5:
Theoretical estimations of TPE corrections to $P_{l}$. $\Delta
P_{l,th}^{N,\Delta,M,N+\Delta+M}$ refer to the theoretical estimations of TPE
corrections from $N,\Delta$ intermediate states, meson-exchange and their sum,
respectively. The experimental results are taken from Ex-polarized-
Meziane-2011 and normalized at $\epsilon=0.152$.
For the $Q^{2}=2.64$ GeV2 case, Fig. 4(a) shows that at small $\epsilon$ the
corrections from the usual TPE contributions $\Delta R^{N,\Delta}_{PT,th}$ are
large and positive while the corrections from meson-exchange $\Delta
R^{M}_{PT,th}$ are large and negative, and they are canceled to some degree
which results in the small magnitude of the full TPE corrections $\Delta
R^{N+\Delta+M}_{PT,th}$. At large $\epsilon>0.7$ all three corrections are
small. For the $Q^{2}=3.2$ GeV2 case, the situation is similar and the full
TPE correction $\Delta R^{N+\Delta+M}_{PT,th}$ shown in Fig. 4(b) are also
small for almost all $\epsilon$. For the $Q^{2}=4.1$ GeV2 case, the comparable
experimental $R_{PT,Ex}^{1\gamma}$ at $Q^{2}=4.0$ GeV2 is measured at
$\epsilon=0.71$ Ex-polarized , and the corresponding $\Delta
R^{N+\Delta+M}_{PT,th}$ is as small as about 3% in this region. By Eq. (III),
the smallness of $\Delta R^{N+\Delta+M}_{PT,th}$ means $R_{PT,Ex}^{1\gamma}$
are close to $R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ in the region we
discussed, combining with the property that
$R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ are close to $R_{PT,Ex}^{1\gamma}$,
we get the above conclusion that $R_{LT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$ are
consistent with $R_{PT,Ex}^{1\gamma+2\gamma(N+\Delta+M)}$.
Figure 4(b) also shows the full TPE correction $\Delta R^{N+\Delta+M}_{PT,th}$
decreases when $Q^{2}$ decreases. The behaviors of $\Delta
R^{N+\Delta+M}_{PT,th}$ at $Q^{2}=2.64,3.2,4.1$ GeV2 strongly suggest it may
be close to 1 for almost all $\epsilon$ at $Q^{2}=2.49$ GeV2 and are
consistent with the recent experimental results of $\epsilon$ dependence of
$R_{PT,Ex}^{1\gamma}$ Ex-polarized-Meziane-2011 which can not be explained by
other model dependent calculations such as the simple hadronic model, pQCD,
and GDPs method.
Figure 5 shows that the behavior of $\Delta P_{l,th}^{N+\Delta+M}$ is much
closer to the experiment results than $\Delta P_{l,th}^{N+\Delta}$, while a
considerable discrepancy with experimental data still exists at large
$\epsilon$. Since the experimental error bars of $P_{l,Ex}$ are not small, it
is a little difficult to give a certain conclusion on such a discrepancy at
present and further more precise experiments will be a good and interesting
test.
Figure 6: The theoretical estimation of ratio $R_{e^{+}/e^{-}}$ at
$Q^{2}=2.64,3.2,4.1$ GeV2 after considering the full TPE corrections from
$N,\Delta$ intermediate states and meson-exchange, the experimental data is
taken from Rpm-VEPP .
Using the parameters listed in Table 1 and including the usual TPE corrections
from $N$ and $\Delta$ intermediate states, the ratio
$R_{e^{+}/e^{-}}\equiv\sigma_{un,e^{+}p\rightarrow
e^{+}p}/\sigma_{un,,e^{-}p\rightarrow e^{-}p}$ can also be calculated directly
and the corresponding numerical results are presented in Fig. 6. The numerical
results at $Q^{2}=2.64,3.2,4.1$ GeV2 show a similar magnitude and properties
with that predicted by Vanderhaeghen-2011-EPJA where both the unpolarized and
polarization data are used for fitting. Comparing with the smallness of
$R_{e^{+}/e^{-}}$ at $Q^{2}<2$ GeV2 Rpm-VEPP , the results suggest the
measurement of $R_{e^{+}e^{-}}$ at $Q^{2}=2.5$ GeV2 and small $\epsilon$ will
be a good test to the theoretical study of TPE effects.
To summarize, we suggest a new dynamical form of TPE effect in elastic $ep$
scattering and estimate its contributions to extracted $R^{\prime}s$ by the LT
and PT methods, $P_{l}$ and $R_{e^{+}/e^{-}}$ with one unknown universal
coupling parameter $g$ at fixed $Q^{2}$. We find after combining such
contributions with the usual TPE contributions from box and crossed-box
diagrams, the extracted $R^{\prime}s$ by the LT method from the recent precise
experimental data Ex-Rosenbluth-2006 are naturally close to those measured by
the PT method. And using the extracted $G_{M},R$ and $g$ by LT method, the
$\epsilon$ dependence of $R$ by the PT method at $Q^{2}=2.49$ GeV2 Ex-
polarized-Meziane-2011 can be described well, also our results for
$R_{e^{+}e^{-}}$ are similar with those predicted by Vanderhaeghen-2011-EPJA .
The full results suggest the meson-exchange mechanism may play an important
role in elastic $ep$ scattering and more precise experimental data at
$Q^{2}=2.5$ GeV2 will be a good test.
## IV Acknowledgments
H.-Q.Z thanks S.N. Yang for helpful discussions. This work is supported by the
National Sciences Foundations of China under Grant No. 11375044 and in part by
the Fundamental Research Funds for the Central Universities under Grant No.
2242014R30012.
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|
arxiv-papers
| 2013-12-02T02:57:04 |
2024-09-04T02:49:54.619676
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-Yu Chen, Hai-Qing Zhou",
"submitter": "Haiqing Zhou",
"url": "https://arxiv.org/abs/1312.0310"
}
|
1312.0431
|
# Effect of pairing correlations on nuclear low-energy structure: BCS and
general Bogoliubov transformation
J. Xiang School of Nuclear Science and Technology, Lanzhou University,
Lanzhou 730000, China School of Physical Science and Technology, Southwest
University, Chongqing 400715, China Z. P. Li School of Physical Science and
Technology, Southwest University, Chongqing 400715, China J. M. Yao School
of Physical Science and Technology, Southwest University, Chongqing 400715,
China Department of Physics, Tohoku University, Sendai 980-8578, Japan W. H.
Long School of Nuclear Science and Technology, Lanzhou University, Lanzhou
730000, China P. Ring Physik-Department der Technischen Universität München,
D-85748 Garching, Germany State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing 100871, China J.
Meng State Key Laboratory of Nuclear Physics and Technology, School of
Physics, Peking University, Beijing 100871, China School of Physics and
Nuclear Energy Engineering, Beihang University, Beijing 100191, China
Department of Physics, University of Stellenbosch, Stellenbosch, South Africa
###### Abstract
Low-lying nuclear states of Sm isotopes are studied in the framework of a
collective Hamiltonian based on covariant energy density functional theory.
Pairing correlation are treated by both BCS and Bogoliubov methods. It is
found that the pairing correlations deduced from relativistic Hartree-
Bogoliubov (RHB) calculations are generally stronger than those by
relativistic mean-field plus BCS (RMF+BCS) with same pairing force. By simply
renormalizing the pairing strength, the diagonal part of the pairing field is
changed in such a way that the essential effects of the off-diagonal parts of
the pairing field neglected in the RMF+BCS calculations can be recovered, and
consequently the low-energy structure is in a good agreement with the
predictions of the RHB model.
###### pacs:
21.60.Jz, 21.60.Ev, 21.10.Re, 21.10.Tg
The study of nuclear low-lying states is of great importance to unveil the
low-energy structure of atomic nuclei and turns out to be essential to
understand the evolution of shell structure and collectivity Meng98 ; Hagen12
; Kshetri06 , nuclear shape phase transitions Meng05 ; Casten06 ; Cejnar10 ,
shape coexistence Heyde11 , the onset of new shell gaps Ozawa2000 , the
erosion of traditional magic numbers Sorlin08 , etc. The understanding and the
quantitative description of low-lying states in nuclei necessitate an accurate
modeling of the underlying microscopic nucleonic dynamics.
Density functional theory (DFT) is a reliable platform for studying the
complicated nuclear excitation spectra and electromagnetic decay patterns
BHR.03 ; JacD.11 ; Vretenar05 ; Meng06 ; Meng2013FrontiersofPhysics55 . Since
the DFT scheme breaks essential symmetries of the system, this requires to
include the dynamical effects related to the restoration of broken symmetries,
as well as the fluctuations in the collective coordinates. In recent years
several accurate and efficient models and algorithms, based on microscopic
density functionals or effective interactions, have been developed that
perform the restoration of symmetries broken by the static nuclear mean field,
and take the quadrupole fluctuations into account NVR.06a ; NVR.06b ; BH.08 ;
Yao08 ; Yao.09 ; Yao.10 ; RE.10 . This level of implementation is also
referred as the multi-reference (MR)-DFT Lacroix09 . Compared with MR-DFT, the
model of a collective Hamiltonian with parameters determined in a microscopic
way from self-consistent mean-field calculations turns out to be a powerful
tool for the systematical studies of nuclear low-lying states PR.04 ; Nik.09 ;
Nik.11 , with much less numerical demanding. Even for the heavy nuclei full
triaxial calculations can be relatively easily carried out with a five-
dimension collective Hamiltonian Li.10 . It has achieved great success in
describing the low-lying states in a wide range of nuclei, from $A\sim 40$ to
superheavy nuclei including spherical, transitional, and deformed ones Nik.09
; Li.09 ; Li.09b ; Li.10 ; Li.11 ; Nik.11 ; Yao11-lambda ; Mei12 ; Del10 .
For open-shell nuclei, pairing correlations between nucleons have important
influence on low-energy nuclear structure Dean03 . In the relativistic scheme
they could be taken into account using the BCS ansatz GRT.90 or full
Bogoliubov transformation KR.91 ; Rin.96 . Compared with the simple BCS
method, the consideration of pairing correlations through the Bogoliubov
transformation is numerically demanding for heavy triaxial deformed nuclei. It
has been demonstrated that there is no essential difference between BCS and
Bogoliubov methods for the descriptions of the ground-state of stable nuclei
Ring80 . Girod et al. have compared the results obtained from Hartree-Fock-
Bogoliubov (HFB) and Hartree-Fock plus BCS (HF+BCS) calculations, including
the potential energy surfaces (PESs), pairing gaps, and pairing energies as
functions of the axial deformation Girod83 . It has been shown that the PESs
given by these two methods are very similar. Moreover, the pairing gaps and
energies from the HF+BCS calculations are slightly smaller than those from the
HFB calculation. In view of these facts, it is natural to test the validity of
the BCS ansatz in describing the the low-energy structure of nuclei, as
referred to the RHB method. Aiming at this point, the comparisons are
performed within the covariant density functional based 5DCH model,
specifically between the triaxial deformed RMF+BCS and RHB calculations. Due
to the emergence of an abrupt shape-phase-transition Li.09 , the even-even Sm
isotopes with $134\leqslant A\leqslant 154$ are taken as the candidates in
this study.
Practically nuclear excitations determined by quadrupole vibrational and
rotational degrees of freedom can be treated by introducing five collective
coordinates, i.e., the quadrupole deformations $(\beta,\gamma)$ and Euler
angles ($\Omega=\phi,\theta,\psi$) Pro.99 . The quantized 5DCH that describes
the nuclear excitations of quadrupole vibration, rotation and their couplings
can be written as,
$\hat{H}=\hat{T}_{\textnormal{vib}}+\hat{T}_{\textnormal{rot}}+V_{\textnormal{coll}}\;,$
(1)
where $V_{\textnormal{coll}}$ is the collective potential, and
$\hat{T}_{\textnormal{vib}}$ and $\hat{T}_{\textnormal{rot}}$ are respectively
the vibrational and rotational kinetic energies,
$\displaystyle{V}_{\text{coll}}=$ $\displaystyle
E_{\text{tot}}(\beta,\gamma)-\Delta V_{\text{vib}}(\beta,\gamma)-\Delta
V_{\text{rot}}(\beta,\gamma),$ (2) $\displaystyle\hat{T}_{\textnormal{vib}}=$
$\displaystyle-\frac{\hbar^{2}}{2\sqrt{wr}}\left\\{\frac{1}{\beta^{4}}\left[\frac{\partial}{\partial\beta}\sqrt{\frac{r}{w}}\beta^{4}B_{\gamma\gamma}\frac{\partial}{\partial\beta}\right.\right.$
$\displaystyle\left.\left.-\frac{\partial}{\partial\beta}\sqrt{\frac{r}{w}}\beta^{3}B_{\beta\gamma}\frac{\partial}{\partial\gamma}\right]+\frac{1}{\beta\sin{3\gamma}}\left[-\frac{\partial}{\partial\gamma}\right.\right.$
(3)
$\displaystyle\left.\left.\sqrt{\frac{r}{w}}\sin{3\gamma}B_{\beta\gamma}\frac{\partial}{\partial\beta}+\frac{1}{\beta}\frac{\partial}{\partial\gamma}\sqrt{\frac{r}{w}}\sin{3\gamma}B_{\beta\beta}\frac{\partial}{\partial\gamma}\right]\right\\},$
$\displaystyle\hat{T}_{\textnormal{{{rot}}}}=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{3}{\frac{\hat{J}^{2}_{k}}{\mathcal{I}_{k}}}.$
(4)
In eq. (2), $E_{\textnormal{tot}}(\beta,\gamma)$ is the binding energy
determined by the constraint mean-field calculations, and the terms $\Delta
V_{\textnormal{vib}}$ and $\Delta V_{\textnormal{rot}}$, calculated in the
cranking approximation Ring80 , are zero-point-energies (ZPE) of vibrational
and rotational motions, respectively. In eq. (4), $\hat{J}_{k}$ denotes the
components of the angular momentum in the body-fixed frame of the nucleus.
Moreover the mass parameters $B_{\beta\beta}$, $B_{\beta\gamma}$,
$B_{\gamma\gamma}$ in eq. (3), as well as the moments of inertia
$\mathcal{I}_{k}$ in eq. (4), depend on the quadrupole deformation variables
$\beta$ and $\gamma$,
$\displaystyle\mathcal{I}_{k}=$ $\displaystyle
4B_{k}\beta^{2}\sin^{2}(\gamma-2k\pi/3),$ $\displaystyle k=$ $\displaystyle
1,2,3,$ (5)
where $B_{k}$ represents inertia parameter. In eq. (3), the additional
quantities $r=B_{1}B_{2}B_{3}$ and
$w=B_{\beta\beta}B_{\gamma\gamma}-B_{\beta\gamma}^{2}$ define the volume
element of the collective space. The corresponding eigenvalue problem is
solved by expanding the eigenfunctions on a complete set of basis functions in
the collective space of the quadrupole deformations $(\beta,\gamma)$ and Euler
angles $(\Omega=\phi,\theta,\psi)$.
The dynamics of the 5DCH is governed by seven functions of the intrinsic
deformations $\beta$ and $\gamma$: the collective potential $V_{\rm coll}$,
three mass parameters $B_{\beta\beta}$, $B_{\beta\gamma}$, $B_{\gamma\gamma}$,
and three moments of inertia $\mathcal{I}_{k}$. These functions are determined
using the cranking approximation formula based on the intrinsic triaxially
deformed mean-field states. The diagonalization of the Hamiltonian (1) yields
the excitation energies and collective wave functions that are used to
calculate observables Nik.09 .
The fact that, the 5DCH model using the collective inertia parameters
calculated based on the cranking approximation can reproduce the structure of
the experimental low-lying spectra Nik.09 up to an overall renormalization
factor, demonstrates such approximation is fair enough for the present study.
As it has been shown in Ref. LNRVYM12 , this factor takes into account the
contributions of the time-odd fields. A microscopic calculation of this factor
would go far beyond the scope of the present investigation.
The intrinsic triaxially deformed mean-field states are the solutions of the
Dirac (RMF+BCS) or RHB equations. The point-coupling energy functional PC-PK1
Zhao10 and the separable pairing force TMR.09a are used in the particle-hole
and particle-particle channels, respectively. In solving the Dirac and RHB
equations, the Dirac spinors are expanded on the three-dimension harmonic
oscillator basis with 14 major shells KR.89 ; Peng08 . A quadratic constraint
on the mass quadrupole moments is carried out to obtain the triaxially
deformed mean-field states with $\beta\in[0.0,0.8]$ and
$\gamma\in[0^{\circ},60^{\circ}]$,and the step sizes $\Delta\beta=0.05$ and
$\Delta\gamma=6^{\circ}$. More details about the calculations can be found in
Refs. NRV.10 ; Xiang12 .
Figure 1: (Color online) Comparison between the RHB and RMF+BCS calculations
on the binding energy per nucleon $E/A$ [plot (a)], quadrupole deformation
$\beta$ [plot (b)], neutron [plot (c)] and proton [plot (d)] average pairing
gaps weighted by the occupation probabilities $v^{2}$ Bender00 for even-even
Sm isotopes.
Figure 1 displays the comparison between the RHB and RMF+BCS calculations for
the binding energy per nucleon $E/A$ [plot (a)], quadrupole deformation
$\beta$ [plot (b)], neutron [plot (c)] and proton [plot (d)] average pairing
gaps weighted by the occupation probabilities $v^{2}$ Bender00 of even-even
Sm isotopes with $134\leqslant A\leqslant 154$. The binding energies and
deformations found in the two calculations are close to each other. However,
the average neutron and proton pairing gaps provided by the RHB calculations
are generally larger than those by the RMF+BCS ones. This is consistent with
the observations in Ref. Girod83 , which indicates that the BCS ansatz gives
slightly weaker pairing correlations with same pairing force. The underlying
reason is well-known that the BCS ansatz corresponds to a special Bogoliubov
transformation, which only considers pairing correlation between two nucleons
in time-reversed conjugate states Ring80 , and the off-diagonal matrix
elements of the pairing field $\Delta$ are neglected in this approach.
Figure 2: (Color online) Neutron [plot (a)] and proton [plot (b)] average
pairing gaps obtained from RMF+BCS calculations as a function of the pairing
strength factor $R_{\tau}$, where the horizontal lines indicate the RHB
results with the original pairing force. In the right plots are shown the
ratios of the average pairing gaps between the calculations of RHB with the
original and RMF+BCS with 6% enhanced pairing force along the isotopic chain
of Sm for neutron [plot (c)] and proton [plot (d)].
In the following we have to consider that neglecting the off-diagonal matrix
elements of the pairing field leads i) to a reduced configuration mixing and
ii) as a consequence of self-consistency also to an overall reduction of the
pairing strength in the diagonal matrix elements of the pairing field.
Therefore it is interesting to address two points: i) whether the additional
configuration mixing induced by the off-diagonal matrix elements of the
pairing field is really essential and ii) whether the reduced strength of
pairing caused by neglecting the off-diagonal matrix elements in the RMF+BCS
approach can recovered simply by multiplying a strength factor $R_{\tau}$ to
the diagonal pairing, i.e. whether the enhanced pairing strength is also able
to reproduce the low-lying structure properties, e.g. the PESs, inertia
parameters, as well as the low-lying spectra. Taking 152Sm as the example,
Fig. 2 shows the neutron and proton average pairing gaps of the global minimum
calculated by RMF+BCS as the functions of the pairing strength factor
$R_{\tau}$, as referred to the horizontal lines denoting the RHB results with
original pairing force. It is shown that the average pairing gaps increase
almost linearly with respect to the pairing strength factor $R_{\tau}$ and
cross the RHB results at $R_{\tau}\sim 1.06$. Moreover, as shown in Fig. 2 (c)
and (d) the RMF+BCS calculations with 6% enhanced pairing strength provide
nearly identical average pairing gaps with the RHB results for the selected
even-even Sm isotopes, with a relative deviation less than 5%.
Figure 3: (Color online) Potential energy surfaces (a), neutron (b) and proton
(c) average pairing gaps, moments of inertia ${\cal I}_{x}$ (d), collective
masses $B_{\beta\beta}$ (e) and $B_{\gamma\gamma}$ (f) of 152Sm as functions
of the quadrupole deformation parameter $\beta$ calculated by RHB with the
original pairing force (solid lines), and by RMF+BCS with the original (dashed
lines) and the enhanced (by 6%) (dash-dotted lines) pairing force.
As the further clarification, Fig. 3 displays the PESs, neutron and proton
average pairing gaps, moments of inertia ${\cal I}_{x}$, collective masses
$B_{\beta\beta}$ and $B_{\gamma\gamma}$ for 152Sm as functions of the
quadrupole deformation parameter $\beta$, where the results are calculated by
RHB with the original, and by RMF+BCS with the original and the enhanced (by
6%) pairing strength. It is well demonstrated that for the selected Sm
isotopes the deviations on the low-lying structure properties described by
RMF+BCS and RHB models can be eliminated by simply enhancing the pairing force
about 6% in the BCS ansatz. Specifically, as the pairing strength increases,
the average pairing gaps become larger, which leads to lower spherical barrier
of PES Rutz99 and reduced inertia parameter Sobiczewski69 .
In Fig. 4 we also compare the theoretical low-lying spectra of 152Sm
calculated by RMF+BCS with the original and the enhanced (by 6%) pairing
strength, to the RHB results. As seen from the left two panels,when the
pairing strength is enhanced by 6%, the low-lying spectrum is extended, and
systematically the intraband $B(E2)$ transitions become weaker, and the
interband transitions are strengthened, finally leading to an identical
prediction as the full RHB calculations (right panel). Quantitatively, the
relative deviations between the RHB and RMF+BCS predications are reduced to
less than 4% for the intraband transitions, and the main interband transitions
agree with each other within $\sim 2$ W.u.. We have also checked the results
for the other Sm isotopes, and very similar spectra are predicted by RHB with
the original and RMF+BCS with enhanced (6%) pairing forces.
Figure 4: The low-lying spectra of 152Sm calculated from RMF+BCS with the
original [plot a] and the enhanced (by 6%) [plot b] pairing strength, and
compared with results from full RHB calculations [plot c].
The similarity on the low-lying structure can be understood by analyzing the
underlying shell structure predicted by the two mean-field calculations.
Taking 152Sm as an example, in Fig. 5 we plot the single-particle
configurations (energy and occupation probability) around the Fermi surface
corresponding to the mean-field states of the global minimum in the PESs
determined by the calculations of RHB with the original and RMF+BCS
calculations with both original and enhanced (by 6%) pairing strength. Notice
that the RHB results correspond to the canonical single-particle
configurations, which are determined from the diagonalization of the density
matrix Ring80 . Consistent with the agreement on the low-lying structure
properties, the RMF+BCS calculations with the enhanced pairing strength also
provide nearly identical single-particle configurations as the RHB ones.
Figure 5: (Color online) Single-particle energy levels (horizontal lines) and
occupation probabilities (length of horizontal lines) of 152Sm calculated by
RHB with the original and RMF+BCS with both original and enhanced (by 6%)
pairing strength, where $E_{F}$ denotes the Fermi levels.
In conclusion, we have taken Sm isotopes as examples to carry out a detailed
comparison between the 5DCH calculations based on the RMF+BCS and the RHB
approaches for the nuclear low-lying structure properties. It has been shown
that the pairing correlations resulting from the RHB method are generally
stronger than those from the RMF+BCS method with the same effective pairing
force. However, by simply increasing the pairing strength by a factor 1.06 in
the RMF+BCS calculations, the low-energy structure becomes very close to that
of the full RHB calculations with the original pairing force. We have also
carried out similar calculations in other regions of the nuclear chart and
found that the necessary renormalization factor stays roughly constant up to
heavy nuclei (1.06 in the Pu region) and increases slightly for light ones
(1.10 in the Mg region).
This work was supported in part by the Major State 973 Program 2013CB834400,
the NSFC under Grant Nos. 11335002, 11075066, 11175002, 11105110, and
11105111, the Research Fund for the Doctoral Program of Higher Education under
Grant No. 20110001110087, the Natural Science Foundation of Chongqing
cstc2011jjA0376, the Fundamental Research Funds for the Central Universities
(XDJK2010B007, XDJK2011B002, and lzujbky-2012-k07), the Program for New
Century Excellent Talents in University of China under Grant No. NCET-10-0466,
and the DFG cluster of excellence “Origin and Structure of the Universe”
(www.universe-cluster.de).
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|
arxiv-papers
| 2013-12-02T11:53:26 |
2024-09-04T02:49:54.629603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Xiang, Z.P. Li, J.M. Yao, W.H. Long, P. Ring, and J. Meng",
"submitter": "Xiang Jian",
"url": "https://arxiv.org/abs/1312.0431"
}
|
1312.0469
|
# Explicit Barenblatt Profiles for Fractional Porous Medium Equations
Yanghong Huang
###### Abstract.
Several one-parameter families of explicit self-similar solutions are
constructed for the porous medium equations with fractional operators. The
corresponding self-similar profiles, also called _Barenblatt profiles_ , have
the same forms as those of the classic porous medium equations. These new
exact solutions complement current theoretical analysis of the underlying
equations and are expected to provide insights for further quantitative
investigations.
Department of Mathematics, Imperial College London, London SW7 2AZ, United
Kingdom. Email: [email protected]
## 1\. Introduction
The realistic modelling of phenomena in nature and science is usually
described by nonlinear Partial Different Equations (PDEs). Comparing to their
simplified linear counterparts, these nonlinear PDEs in general has no
explicit representation of the solutions in terms of initial and/or boundary
conditions. Special explicit solutions, if available, are often associated to
certain symmetry groups of the underlying equation [5, 14], including the most
important ones, the scaling symmetry induced self-similar solutions.
Although self-similar solutions arise as exact solutions only with compatible
initial and boundary conditions, they possess a unique position in the general
theory of nonlinear partial differential equations. Take the Porous Medium
Equation (PME) $u_{t}=\Delta u^{m}$ in $\mathbb{R}^{N}$ for example. As
summarized in the monographs [18, 19], the self-similar solutions, also called
Barenblatt-Kompaneets-Pattle-Zel’dovich solutions, characterize the long time
asymptotic behaviours with nonnegative initial data; they indicate the
parameter regimes where the finite versus infinite speed of propagation of
information is expected; they also provide a guidance to more refined
questions like optimal regularity and optimal constants in various functional
identities and inequalities.
In this paper, we investigate the existence of certain explicit self-similar
solutions of the porous medium equations with fractional operators, i.e.,
(1a) $u_{t}+(-\Delta)^{s}u^{m}=0,$ and (1b)
$u_{t}=\nabla\cdot\big{(}u^{m-1}\nabla(-\Delta)^{-s}u\big{)}.$
The definition and related properties of the fractional Laplacian
$(-\Delta)^{s}$ and its inverse $(-\Delta)^{-s}$, together with the associated
function spaces, can be found in the monographs [13, 17] or the survey paper
[11]. When $s=2$ in (1a) or $s=0$ in (1b), the classical PME is recovered
(with different diffusion coefficients). The latter is also closely related to
another variant with fractional pressure
(2) $u_{t}=\nabla\cdot\big{(}u\nabla(-\Delta)^{-s}u^{m-1}\big{)}.$
In fact, (1b) coincides with (2), when $m=2$ in both cases.
Despite the equivalence of (1a), (1b) and (2) to the classical PME in some
ranges of $s$ and $m$, the three equations exhibit quite different qualitative
properties. The basic theory of (1a) is studied in [9] for $s=1/2$ and in [10]
for general $s\in(0,1)$, followed by more refined quantitative estimates [6,
21, 22]. In contrast, the notable feature of (2) is the finite speed of
propagation, studied for $m=2$ by Caffarelli and Vázquez [7, 8] and for
general $m>1$ by Biler, Imbert and Karch [3, 4]. The variant (1b) has been
studied only recently [16]; depending on $m$, the equation can have both
finite (for $1<m<2$) and infinite speed of propagation (for $m>2$).
One of the most important approaches to the study of qualitative and
quantitative properties of PDEs is to examine their self-similar solutions,
whenever they exist. The self-similar solutions are related to the scaling
symmetry groups of the PDEs, leading to transformed equations in scale-
invariant similarity variables. After the reduction using similarity
variables, the resulting equations for the self-similar profiles, called
_Barenblatt profiles_ below, still inherit some of the remaining scaling
symmetries. As summarized in [2], for self-similarity of the first kind, the
scaling exponents can be determined a priori and explicit Barenblatt profiles
can often be obtained. For self-similarity of the _second kind_ , also called
_anomalous scaling_ , Barenblatt profiles are in general not available,
because of the unknown anomalous exponents. Second kind self-similarity can be
demonstrated by the PME $u_{t}=\Delta u^{m}$ in the fast diffusion regime
$m<m_{c}=(N-2)_{+}/N$ where solutions are known to vanish in finite time.
Although no explicit Barenblatt profiles are expected in this case, the
remaining scaling symmetry of the reduced equation allows one to give a
detailed phase plane analysis to study the existence, uniqueness and
monotonicity of the profiles [15, 18].
Unfortunately, there is limited usage of the remaining scaling symmetry of
profile equations from the fractional porous medium equations (1), for both
first and second kind self-similarities, because the local characterization of
Lie symmetry [5, 14] is destroyed by the nonlocal operator. As a consequence,
explicit Barenblatt profiles are much more difficult to find. Surprisingly,
all Barenblatt profiles of (2) are obtained by Biler, Imbert and Karch [3, 4]
for any $s\in(0,1)$ and $m>1$, which are shown to be proportional to
$(R^{2}-|y|^{2})_{+}^{\frac{1-s}{m-1}}$ for some $R>0$. In this paper, we will
focus on the less known explicit profiles for (1a) and (1b), despite the
existence, uniqueness and many qualitative properties presented in [20] for
(1a). In contrast to the explicit two-parameter family (for any $s$ and $m$)
of profiles for (2), we can only find isolated one-parameter families (for
certain combinations of $s$ and $m$) of profiles for (1a) or (1b).
The special types of Barenblatt profiles sought here are proportional to
$(R^{2}+|y|^{2})^{-q}$ or $(R^{2}-|y|^{2})_{+}^{q}$, for some $R>0$ and $q>0$.
This is motivated from the Barenblatt profiles of the classic PME
$u_{t}=\Delta u^{m}$, which take the form of $(R^{2}+|y|^{2})^{-1/(1-m)}$ for
$m\in(\frac{N-2}{N},1)$ or $(R^{2}-|y|^{2})_{+}^{1/(m-1)}$ for $m>1$. The main
result is summarized as follows.
For (1a), three families of explicit self-similar solutions of the form
$(R^{2}+|y|^{2})^{-q}$ are found for $s\in(0,1)$:
1. (1)
when $m=\frac{N+2-2s}{N+2s}>m_{c}:=\frac{N-2s}{N}$,
(3a) $u(x,t)=\lambda
t^{-N\beta}\big{(}R^{2}+|xt^{-\beta}|^{2}\big{)}^{-s-\frac{N}{2}},\qquad\beta=\frac{1}{N(m-1)+2s};$
2. (2)
when $m=\frac{N-2s}{N+2s}<m_{c}$,
(3b)
$u(x,t)=\lambda(T-t)^{\frac{N+2s}{4s}}\big{(}R^{2}+|x|^{2}\big{)}^{-\frac{N}{2}-s};$
3. (3)
when $m=\frac{N-2s}{N+2s-2}$,
(3c) $u(x,t)=\lambda
t^{-\frac{N+2s-2}{2(1-s)}}\big{(}R^{2}+|xt^{-\frac{1}{2(1-s)}}|^{2}\big{)}^{-\frac{N}{2}-s+1}.$
For (1b), only one family of self-similar of explicit self-similar solutions
of the form $(R^{2}+|y|^{2})^{-q}$ is found for $s\in(0,1)$: when
$m=\frac{N+6s-2}{N+2s}$,
(4) $u(x,t)=\lambda
t^{-N\beta}\big{(}R^{2}+|xt^{-\beta}|^{2}\big{)}^{-\frac{N}{2}-s},\qquad\beta=\frac{1}{N(m-1)+2-2s}.$
To derive these Barenblatt profiles, we need some preliminary results related
to hypergeometric functions and their fractional Laplacians, given in Section
2. The mass conserving Barenblatt profiles (3a) for (1a) are constructed in
Section 3, followed by mass conserving Barenblatt profiles (4) for (1b) in
Section 4. The more complicated Barenblatt profiles (3b) and (3c) for (1a)
with second-kind self-similarity are derived in Section 5.
## 2\. Fractional Laplacians of the Barenblatt profiles and other identities
In the search of Barenblatt profiles of the form
$\Phi(y)=(R^{2}-|y|^{2})_{+}^{q}$ or $\Phi(y)=(R^{2}+|y|^{2})^{-q}$, the
explicit expressions for the fractional Laplacian of $\Phi(y)$ are derived
using Fourier transform. Certain special functions enter during various stages
of the derivation, and therefore their definitions with related properties are
introduced here. Most of the properties used here can be consulted from
standard textbooks on special functions [1].
Bessel-type special functions appear in the Fourier transform of $\Phi(y)$.
The _Bessel functions of the first kind_ $J_{\nu}(x)$ is the solution of the
Bessel differential equation
$x^{2}\frac{d^{2}z}{dx^{2}}+x\frac{dz}{dx}+(x^{2}-\nu^{2})z=0,$
that is finite at the origin for positive $\nu$. The _modified Bessel function
of the second kind_ $K_{\nu}(x)$ is the exponentially decaying solution of the
modified Bessel differential equation
$x^{2}\frac{d^{2}z}{dx^{2}}+x\frac{dz}{dx}-(x^{2}+\nu^{2})z=0.$
In fact, besides the definitions, the only property we use below is
$K_{-\nu}(x)=K_{\nu}(x)$.
The (Gauss) _hypergeometric function_ appears in the fractional Laplacian of
$\Phi(y)$, which is a solution of Euler’s hypergeometric differential equation
$x(1-x)\frac{d^{2}z}{dx^{2}}+\big{[}c-(a+b+1)x\big{]}\frac{dz}{dx}-abz=0,$
for any complex number $a,b$ and $c$. It is often represented more
conveniently as a power series
(5)
${}_{2}F_{1}(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}x^{n},\qquad|x|<1,$
where $(a)_{n}=\Gamma(a+n)\big{/}\Gamma(a)$ is the Pochhammer symbol and
$\Gamma(x)$ is the Euler Gamma function. If $c$ is a non-positive integer,
${}_{2}F_{1}(a,b;c;x)$ becomes a polynomial of degree $-c$ in $x$. From the
series expansion (5), it is obvious that
${}_{2}F_{1}(a,b;c;x)={}_{2}F_{1}(b,a;c;x)$ and
(6) $\frac{d}{dx}{}_{2}F_{1}(a,b;c;x)=\frac{ab}{c}\
{}_{2}F_{1}(a+1,b+1;c+1;x).$
These two simple properties still hold on the complex plane, by analytical
continuation.
The hypergeometric function is prevalent in mathematical physics because it
represents many other common yet important special functions and it emerges
also in many special integrals. In fact, the candidate Barenblatt profiles
$(R^{2}-|y|^{2})_{+}^{q}$ or $(R^{2}+|y|^{2})^{-q}$ are also special
hypergeometric functions, i.e.,
(7) $\displaystyle(R^{2}-|y|^{2})_{+}^{q}$
$\displaystyle=R^{2q}{}_{2}F_{1}(-q,c;c;|y|^{2}/R^{2}),$ (8)
$\displaystyle(R^{2}+|y|^{2})^{-q}$
$\displaystyle=R^{-2q}{}_{2}F_{1}(q,c;c;-|y|^{2}/R^{2}),$
for any complex number $c$. In this paper, we will always choose $c$ to be
$N/2$, half of the space dimension, to match the parameters in the Fourier
transforms of $\Phi(y)$.
For the explicit Barenblatt profiles of (2) found in [3, 4], the key formula
is the _Weber-Schafheitlin integral_ [23, p. 401-403]
$\int_{0}^{\infty}\eta^{-\rho}J_{\mu}(\eta a)J_{\nu}(\eta
b)d\eta\cr=\frac{b^{\nu}a^{\rho-\nu-1}\Gamma\left(\frac{\nu-\rho+\mu+1}{2}\right)}{2^{\rho}\Gamma(\nu+1)\Gamma\left(\frac{1+\mu+\rho-\nu}{2}\right)}{}_{2}F_{1}\left(\frac{\nu-\rho+\mu+1}{2},\frac{\nu-\rho-\mu+1}{2};\nu+1;\frac{b^{2}}{a^{2}}\right),$
with $\nu+\mu-\rho+1>0$, $\rho>-1$ and $0<b\leq a$. It enables the authors to
derive explicitly the (inverse) fractional Laplacian of
$(R^{2}-|y|^{2})_{+}^{q}$ for any $q>0$, $s\in(0,1)$, i.e.,
(9) $\displaystyle\quad(-\Delta)^{-s}\big{(}(R^{2}-|y|^{2})_{+}^{q}\big{)}$
(10) $\displaystyle=\begin{cases}C_{q,s,N}R^{2q+2s}\
{}_{2}F_{1}\left(\frac{N}{2}-s,-q-s;\frac{N}{2};|y|^{2}/R^{2}\right),\qquad&|y|\leq
R,\cr\tilde{C}_{q,s,N}R^{N+2q}|y|^{2s-N}{}_{2}F_{1}\left(\frac{N}{2}-s,1-s;\frac{N}{2}+q+1;R^{2}/|y|^{2}\right),&|y|\geq
R,\end{cases}$
with
$C_{q,s,N}=\frac{2^{-2s}\Gamma(q+1)\Gamma(N/2-s)}{\Gamma(N/2)\Gamma(q+s+1)},\quad\tilde{C}_{q,s,N}=\frac{2^{-2s}\Gamma(q+1)\Gamma(N/2-s)}{\Gamma(s)\Gamma(N/2+q+1)}.$
In this paper, we obtain explicit expressions for the fractional Laplacians of
$(R^{2}+|y|^{2})^{-q}$, using the closely related _modified Weber-Schafheitlin
integral_ [23, p. 410], which reads
(11) $\int_{0}^{\infty}\eta^{-\rho}K_{\mu}(\eta a)J_{\nu}(\eta
b)d\eta\cr=\frac{b^{\nu}a^{\rho-\nu-1}\Gamma\left(\frac{\nu-\rho+\mu+1}{2}\right)\Gamma\left(\frac{\nu-\rho-\mu+1}{2}\right)}{2^{\rho+1}\Gamma(\nu+1)}{}_{2}F_{1}\left(\frac{\nu-\rho+\mu+1}{2},\frac{\nu-\rho-\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$
with $|\mu|<\nu-\rho+1$ and $a>0$.
Already observed in [3, 4], these special Weber-Schafheitlin integrals are
connected to the fractional Laplacians of $(R^{2}-|y|^{2})_{+}^{q}$ or
$(R^{2}+|y|^{2})^{-q}$ by the fact that the Fourier transform of
${}_{2}F_{1}\big{(}a,b;\frac{N}{2};\pm|y|^{2}\big{)}$ are $J_{\nu}(|\xi|)$ or
$K_{\nu}(|\xi|)$, multiplied with a power of $|\xi|$. The Barenblatt profiles
$\Phi(y)=(R^{2}+|y|^{2})^{-q}$ we are interested in this paper can be written
as
$R^{-2q}{}_{2}F_{1}\big{(}q,\frac{N}{2};\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\big{)}$,
suggesting the choices of parameters $\nu=\frac{N}{2}-1$, $a=R$ and $b=|y|$ in
(11) while the rest two parameters $\mu$ and $\rho$ are chosen according to
other parameters like $q$ and $N$. Comparing the expressions of the inverse
Fourier transform of radial functions given by (36) in Appendix A, we conclude
the following Fourier transform pair
${}_{2}F_{1}\left(\frac{N}{4}+\frac{\mu-\rho}{2},\frac{N}{4}-\frac{\mu+\rho}{2};\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\right),\quad\frac{2^{\rho+1}(2\pi)^{\frac{N}{2}}\Gamma(\frac{N}{2})R^{\frac{N}{2}-\rho}}{\Gamma(\frac{N}{4}+\frac{\mu-\rho}{2})\Gamma(\frac{N}{4}-\frac{\mu+\rho}{2})}|\xi|^{-\rho-\frac{N}{2}}K_{\mu}(|\xi|R).$
This Fourier pair once again implies the following relation (with some
restrictions on the parameters $\rho$, $\mu$ and $s$) for the fractional
Laplacian of general hypergeometric functions
$(-\Delta)^{s}\left[{}_{2}F_{1}\left(\frac{N}{4}+\frac{\mu-\rho}{2},\frac{N}{4}-\frac{\mu+\rho}{2};\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\right)\right]\cr=2^{2s}R^{-2s}\frac{\Gamma(\frac{N}{4}+\frac{\mu-\rho}{2}+s)\Gamma(\frac{N}{4}-\frac{\mu+\rho}{2}+s)}{\Gamma(\frac{N}{4}+\frac{\mu-\rho}{2})\Gamma(\frac{N}{4}-\frac{\mu+\rho}{2})}{}_{2}F_{1}\left(\frac{N}{4}+\frac{\mu-\rho}{2}+s,\frac{N}{4}-\frac{\mu+\rho}{2}+s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\right).$
In particular, when $\rho=-q$ and $\mu=\frac{N}{2}-q$, we get
(12) $\displaystyle(-\Delta)^{s}(R^{2}+|y|^{2})^{-q}$
$\displaystyle=R^{-2q}(-\Delta)^{s}\left[{}_{2}F_{1}\Big{(}q,\frac{N}{2};\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\Big{)}\right]$
(13)
$\displaystyle=2^{2s}R^{-2s-2q}\frac{\Gamma(q+s)\Gamma(\frac{N}{2}+s)}{\Gamma(q)\Gamma(\frac{N}{2})}{}_{2}F_{1}\Big{(}q+s,\frac{N}{2}+s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\Big{)}.$
For the explicit Barenblatt profiles we find for (1a) and (1b) below, only two
simple cases of (2) are needed, which are collected here:
1. (i)
when $q=\frac{N}{2}+1-s$,
(14) $(-\Delta)^{s}(R^{2}+|y|^{2})^{-\frac{N}{2}-1+s}\\\
=2^{2s-1}NR^{-N-2}\frac{\Gamma(\frac{N}{2}+s)}{\Gamma(\frac{N}{2}+1-s)}\
{}_{2}F_{1}\left(\frac{N}{2}+1,\frac{N}{2}+s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\right);\qquad$
2. (ii)
when $q=\frac{N}{2}-s$,
(15)
$(-\Delta)^{s}(R^{2}+|y|^{2})^{-\frac{N}{2}+s}=2^{2s}R^{2s}\frac{\Gamma(\frac{N}{2}+s)}{\Gamma(\frac{N}{2}-s)}(R^{2}+|y|^{2})^{-\frac{N}{2}-s}.$
###### Remark 2.1.
Since there is no restriction on the sign of $s$, the inverse fractional
Laplacian $(-\Delta)^{-s}$ of above functions can be obtained by changing $s$
to $-s$.
In the next three sections, we search for Barenblatt profiles $\Phi(y)$ of the
form $\lambda(R^{2}+|y|^{2})^{-q}$ or $\lambda(R^{2}-|y|^{2})_{+}^{q}$, by
looking at the local power series expansion of the governing equation for
$\Phi(y)$ at the origin. Moreover, for mass conserving self-similar solutions
in the next two sections, the governing equation can be simplified to an
identity involving two Gauss hypergeometric functions. The corresponding
profiles are obtained using the following lemma, which is proved easily also
using a power series expansion at the origin.
###### Lemma 2.2.
If the non-constant hypergeometric functions ${}_{2}F_{1}(a_{1},b_{1};c;x)$
and ${}_{2}F_{1}(a_{2},b_{2};c;x)$ are identical for $|x|<1$, then either
$a_{1}=a_{2},b_{1}=b_{2}$ or $a_{1}=b_{2},b_{1}=a_{2}$.
## 3\. Mass conserving Barenblatt profiles for $u_{t}+(-\Delta)^{s}u^{m}=0$
If $u(x,t)$ is a solution of (1a), so is
$T_{\lambda}u(x,t)=\lambda^{N\beta}u(\lambda^{\beta}x,\lambda t)$ with
(16) $\beta=\frac{1}{N(m-1)+2s}.$
This implies self-similar solutions of the form $u(x,t)=t^{-N\beta}\Phi(y)$
with $y=xt^{-\beta}$, where the Barenblatt profile $\Phi$ satisfies the
equation
(17) $(-\Delta)^{s}\Phi^{m}=\beta\nabla\cdot(y\Phi).$
The basic existence, uniqueness and many properties of $\Phi(y)$ are already
established by Vázquez [20], without any explicit expressions of $\Phi(y)$
(except the linear case $m=1$ and $s=1/2$). Since the solutions (1a) become
positive instantaneously [10], we do not expect Barenblatt profiles of the
form $\Phi(y)=\lambda(R^{2}-|y|^{2})_{+}^{q}$ and hence concentrate on
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$ only. In fact, we have the following
theorem.
###### Theorem 3.1.
For every $s\in(0,1)$, equation (1a) admits a self-similar solution
$u(x,t)=t^{-N\beta}\Phi(xt^{-\beta})$ with the special profile
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$ ($q>0$) and $\beta=\frac{1}{N(m-1)+2s}$
only when $m=\frac{N+2-2s}{N+2s}$. The corresponding self-similar solution
$u(x,t)=\lambda
t^{-N\beta}\big{(}R^{2}+|xt^{-\beta}|^{2}\big{)}^{-s-\frac{N}{2}}$
is a classical solution on $(0,\infty)\times\mathbb{R}^{N}$ with $u(x,t)\to
M\delta(x)$ as $t\to 0$ for some $M>0$.
To derive the Barenblatt profile, replacing $q$ with $mq$ in (2),
$(-\Delta)^{s}\Phi(y)^{m}=\lambda^{m}2^{2s}R^{-2s-2mq}\frac{\Gamma(mq+s)\Gamma(\frac{N}{2}+s)}{\Gamma(mq)\Gamma(\frac{N}{2})}{}_{2}F_{1}\big{(}mq+s,\frac{N}{2}+s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\big{)}.$
On the other hand, a simple calculation yields
$\nabla\cdot\big{(}y\Phi(y)\big{)}=\lambda
NR^{-2q}{}_{2}F_{1}\big{(}q,\frac{N}{2}+1;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\big{)}.$
As a result, the governing equation (17) reduces to the identity
(18)
${}_{2}F_{1}\big{(}mq+s,\frac{N}{2}+s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\big{)}={}_{2}F_{1}\big{(}q,\frac{N}{2}+1;\frac{N}{2};-\frac{|y|^{2}}{R^{2}}\big{)}$
and the algebraic equation
(19)
$\lambda^{m}2^{2s}R^{-2s-2mq}\frac{\Gamma(mq+s)\Gamma(\frac{N}{2}+s)}{\Gamma(mq)\Gamma(\frac{N}{2})}=\beta\lambda
NR^{-2q}.$
Since $\frac{N}{2}+s\neq\frac{N}{2}+1$ in (18), Lemma 2.2 implies that
$mq+s=\frac{N}{2}+1,\qquad\frac{N}{2}+s=q,$
or
(20) $m=\frac{N+2-2s}{N+2s},\qquad q=\frac{N}{2}+s.$
Consequently, the algebraic identity (19) can be simplified as
(21)
$\lambda^{1-m}R^{2-2s}\beta=2^{2s-1}\frac{\Gamma(\frac{N}{2}+s)}{\Gamma(\frac{N}{2}+1-s)}.$
Together with the total mass condition
(22)
$M=\int_{\mathbb{R}^{N}}\Phi(y)dy=\lambda\pi^{\frac{N}{2}}R^{-2s}\frac{\Gamma(s)}{\Gamma(\frac{N}{2}+s)},$
the two free parameters $\lambda$ and $R$ are determined uniquely.
###### Remark 3.2.
The special case $m=1$ and $s=1/2$ is well-known, and the corresponding
Barenblatt profile is the Poisson kernel. The new solutions above can be
viewed as a continuous branch from the point $s=1/2$ to the whole interval
$s\in(0,1)$.
###### Remark 3.3.
These Barenblatt profiles are obtained for
$m=\frac{N+2-2s}{N+2s}>m_{c}:=\frac{(N-2s)_{+}}{N}$, and have the solutions
$u(\cdot,t)\in L^{1}(\mathbb{R}^{N})$ for any $t>0$. The general functional
framework of existence and uniqueness developed in [10] applies here.
Moreover, the optimal decay rate $O(|y|^{-N-2s})$ of general Barenblatt
profiles governed by (17) is proved in [6, 20] for $m>m_{1}:=\frac{N}{N+2s}$,
which is also verified in above special cases since
$m=\frac{N+2-2s}{N+2s}>m_{1}$.
## 4\. Mass conserving Barenblatt profiles
$u_{t}=\nabla\cdot(u^{m-1}\nabla(-\Delta)^{-s}u)$
Since solutions of (1b) could have either finite (for $m\geq 2$) or infinite
speed of propagation (for $1<m<2$) as shown in [16], Barenblatt profiles of
both forms $\lambda(R^{2}+|y|^{2})^{-q}$ and $\lambda(R^{2}-|y|^{2})_{+}^{q}$
are sought in this section.
If $u(x,t)$ is a solution of (1b), so is
$T_{\lambda}u(x,t)=\lambda^{N\beta}u(\lambda^{\beta}x,\lambda t)$ with
(23) $\beta=\frac{1}{N(m-1)+2-2s}.$
This implies self-similar solutions of the form $u(x,t)=t^{-N\beta}\Phi(y)$
with $y=xt^{-\beta}$, where the Barenblatt profile $\Phi$ satisfies
(24)
$\nabla\cdot\big{(}\Phi^{m-1}\nabla(-\Delta)^{-s}\Phi\big{)}+\beta\nabla\cdot\big{(}y\Phi\big{)}=0.$
Since the special case $m=2$ is already covered in [3, 4], we only consider
the case $m\neq 2$ below.
###### Theorem 4.1.
If $m\neq 2$, for every $s\in(0,1)$, equation (1b) admits a self-similar
solution $u(x,t)=t^{-N\beta}\Phi(xt^{-\beta})$ with the special profile
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$ ($q>0$) only when
$m=\frac{N+6s-2}{N+2s}$. The corresponding self-similar solution
$u(x,t)=\lambda
t^{-N\beta}\big{(}R^{2}+|xt^{-\beta}|^{2}\big{)}^{-s-\frac{N}{2}}$
is a classical solution on $(0,\infty)\times\mathbb{R}^{N}$ with $u(x,t)\to
M\delta(x)$ as $t\to 0$ for some $M>0$. Furthermore, equation (1b) does not
admit any self-similar solution $u(x,t)=t^{-N\beta}\Phi(xt^{-\beta})$ with the
special profile $\Phi(y)=\lambda(R^{2}-|y|^{2})_{+}^{q}$.
To facilitate the calculation, the governing equation (24) can be integrated
once and then simplified as
(25) $\nabla(-\Delta)^{-s}\Phi+\beta y\Phi^{2-m}=0,$
whenever $\Phi\neq 0$.
### 4.1. Barenblatt profiles of the form
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$
In this case we get by (2)
$(-\Delta)^{-s}\Phi(y)=\lambda
2^{-2s}R^{2s-2q}\frac{\Gamma(q-s)\Gamma(\frac{N}{2}-s)}{\Gamma(q)\Gamma(\frac{N}{2})}{}_{2}F_{1}\big{(}q-s,\frac{N}{2}-s;\frac{N}{2};-\frac{|y|^{2}}{R^{2}})$
and consequently $\nabla(-\Delta)^{-s}\Phi(y)$ becomes
$-\lambda
2^{1-2s}R^{2s-2q-2}\frac{\Gamma(q-s+1)\Gamma(\frac{N}{2}-s+1)}{\Gamma(q)\Gamma(\frac{N}{2}+1)}y{}_{2}F_{1}\big{(}q-s+1,\frac{N}{2}-s+1;\frac{N}{2}+1;-\frac{|y|^{2}}{R^{2}}).$
On the other hand, $\Phi^{2-m}$ can be written as
$\lambda^{2-m}(R^{2}+|y|^{2})^{-q(2-m)}=\lambda^{2-m}R^{-2q(2-m)}{}_{2}F_{1}\big{(}q(2-m),\frac{N}{2}+1;\frac{N}{2}+1;-\frac{|y|^{2}}{R^{2}}\big{)}.$
Therefore, the simplified governing equation (25) reduces to the identity
(26)
${}_{2}F_{1}\big{(}q-s+1,\frac{N}{2}-s+1;\frac{N}{2}+1;-\frac{|y|^{2}}{R^{2}})={}_{2}F_{1}\big{(}q(2-m),\frac{N}{2}+1;\frac{N}{2}+1;-\frac{|y|^{2}}{R^{2}}\big{)}$
and the algebraic equation
(27) $-\lambda
2^{1-2s}R^{2s-2q-2}\frac{\Gamma(q-s+1)\Gamma(\frac{N}{2}-s+1)}{\Gamma(q)\Gamma(\frac{N}{2}+1)}+\beta\lambda^{2-m}R^{-2q(2-m)}=0.$
Since $\frac{N}{2}-s+1\neq\frac{N}{2}+1$, (26) holds if and only if
$q-s+1=\frac{N}{2}+1,\qquad\frac{N}{2}-s+1=q(2-m)$
or
(28) $q=\frac{N}{2}+s,\qquad m=\frac{N+6s-2}{N+2s}.$
As a result, (27) can be simplified as
$\lambda^{1-m}R^{2s}\beta=2^{1-2s}\frac{\Gamma(\frac{N}{2}-s+1)}{\Gamma(\frac{N}{2}+s)},$
which determines $\lambda$ and $R$ uniquely, together with (22) for the total
mass.
### 4.2. Barenblatt profiles of the form
$\Phi(y)=\lambda(R^{2}-|y|^{2})_{+}^{q}$
In this case, using (9), $\nabla(-\Delta)^{-s}\Phi(y)$ for $|y|<R$ can be
written as
$-2\lambda\frac{C_{q,s,N}(N-2s)(q+s)}{N}R^{2q+2s-2}y{}_{2}F_{1}\left(\frac{N}{2}-s+1,-q-s+1;\frac{N}{2}+1;\frac{|y|^{2}}{R^{2}}\right).$
On the other hand,
$\beta y\Phi(y)^{2-m}=\beta y(R^{2}-|y|^{2})_{+}^{(2-m)q}=\beta
yR^{2(2-m)q}{}_{2}F_{1}\left(-(2-m)q,\frac{N}{2}+1;\frac{N}{2}+1;\frac{|y|^{2}}{R^{2}}\right).$
Therefore, the simplified governing equation (25) is satisfied only if
${}_{2}F_{1}\left(\frac{N}{2}-s+1,-q-s+1;\frac{N}{2}+1;\frac{|y|^{2}}{R^{2}}\right)={}_{2}F_{1}\left(-(2-m)q,\frac{N}{2}+1;\frac{N}{2}+1;\frac{|y|^{2}}{R^{2}}\right).$
Since $m\neq 2$, the hypergeometric function on the right hand side is non-
constant. By Lemma 2.2, we must have
$-q-s+1=\frac{N}{2}+1,\quad\frac{N}{2}-s+1=-(2-m)q.$
Since both $q$ and $s$ are positive, the first equation is invalid and there
is no Barenblatt profiles of these equations. Therefore, there is no non-
trivial Barenblatt profiles of the type $\lambda(R^{2}-|x|^{2})_{+}^{q}$ when
$m>2$, despite the existence of solutions propagating with finite speed in one
dimension [16].
## 5\. Second-kind Barenblatt profiles for $u_{t}+(-\Delta)^{s}u^{m}=0$
In the previous two sections, explicit self-similar solutions
$u(x,t)=t^{-\alpha}\Phi(xt^{-\beta})$ are sought with the _a priori_ condition
$\alpha=N\beta$, reflecting the mass conservation of these special solutions.
However, this condition may break down, leading to the concept of self-similar
solutions of the _second kind_ [2]. For (1a), these anomalous self-similar
solutions could appear in two situations. In the fast diffusion regime
$m<(N-2s)_{+}/N$, it is known that the mass escapes to infinity and the
solution becomes identically zero at some finite time $T$ [10]. Here the self-
similar solution, if it exists, takes the form
(29) $u(x,t)=(T-t)^{\alpha}\Phi\big{(}x(T-t)^{\beta}\big{)},$
with the restriction $\alpha>N\beta$. On the other hand, the solution may have
infinite mass, and hence it does not make any sense to require the solution to
”conserve” the total mass. Here the self-similar solution takes the form
(30) $u(x,t)=t^{-\alpha}\Phi\big{(}xt^{-\beta}\big{)},$
where $\Phi(y)$ decays slower than $|y|^{-N}$ as $|y|\to\infty$ and the
relation between $\alpha$ and $\beta$ cannot be determined _a priori_. Since
the Barenblatt profiles $\Phi$ for both (29) and (30) satisfy the same
equation
(31) $(-\Delta)^{s}\Phi^{m}-\alpha\Phi-\beta y\cdot\nabla\Phi=0,$
we treat them at the same time below. Notice that there is only one condition
on the scaling exponents $\alpha$ and $\beta$, i.e., $\alpha(m-1)+2s\beta=-1$
for (29) or $\alpha(m-1)+2s\beta=1$ for (30), which is not enough to determine
$\alpha$ and $\beta$ explicitly as in the previous two sections.
###### Theorem 5.1.
For every $s\in(0,1)$, equation (1a) admits two self-similar solutions of the
_second kind_ with profile $\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$:
1. (a)
when $m=\frac{N-2s}{N+2s}$, the self-similar solution
(32)
$u(x,t)=\lambda(T-t)^{\frac{N+2s}{4s}}\big{(}R^{2}+|x|^{2}\big{)}^{-\frac{N}{2}-s}$
is a classical solution on $[0,T)\times\mathbb{R}^{N}$ and vanishes at finite
time $T>0$.
2. (b)
when $m=\frac{N-2s}{N+2s-2}$, the self-similar solution
(33) $u(x,t)=\lambda
t^{-\frac{N+2s-2}{2(1-s)}}\big{(}R^{2}+|xt^{-\frac{1}{2(1-s)}}|^{2}\big{)}^{-\frac{N}{2}-s+1}$
is a classical solution on $(0,\infty)\times\mathbb{R}^{N}$ and has infinite
mass at any $t>0$.
Because $\alpha$ is different from $N\beta$ for the second kind self-
similarity, the three terms in the governing equation (31) can not be
simplified as an equation with two hypergeometric functions as in the previous
two sections. Instead, we proceed in two steps. In the first step, we focus on
the relation between the parameter $m$ and the exponent $q$ in the rescaled
profile $\Phi(y)=(1+|y|^{2})^{-q}$ by a local series expansion for $r=|y|$.
Using these explicit values of $m$ and $q$, we get the condition on $\lambda$
and $R$ in the general profile $\Phi(y)=\lambda(R^{2}+|y|^{2})^{-q}$. In fact,
the same two steps can be applied in Section 3, to find the relation (20) from
the identity (18) by power series expansions and the condition (21) between
$\lambda$ and $R$.
When the simple, rescaled profile $\Phi(y)=(1+|y|^{2})^{-q}$ is used, the
governing equation (31) should be rescaled too. The key observation is that,
because the last two terms $\alpha\Phi$ and $\beta y\cdot\nabla\Phi$ have the
same scaling factor, the relation between $m$ and $q$ can be computed from
$g(r)=0$, where $g(r)$ is a rescaled version of (21) in the radial variable
$r=|y|$, i.e.,
(34) $\displaystyle
g(r)={}_{2}F_{1}\left(mq+s,\frac{N}{2}+s;\frac{N}{2};-r^{2}\right)-(1+r^{2})^{-q}-\tilde{\beta}r\frac{d}{dr}(1+r^{2})^{-q},$
for some $\tilde{\beta}$. Here the coefficient of $(-\Delta)^{s}\Phi$ or
$\alpha\Phi$ is scaled to unit, such that $g(0)=0$. The scale invariance of
$y\cdot\nabla\Phi(y)/\Phi(y)$ implies that $\tilde{\beta}=\beta/\alpha$, which
should be different from $1/N$ for the second kind self-similar solutions we
are looking for here. This scaling technique enables us to get the relation
between $m$ and $q$, without worrying too much about the complicated constants
or prefactors, while the remaining parameters in the Barenblatt profiles are
then determined, using only the relatively simple identities (14) or (15).
Finally, we can find the conditions that $g(r)$ vanishes identically from a
power series expansion around the origin111A computer algebra system like
MAPLE or MATHEMATICA is recommended to perform these symbolic calculations..
that is $g(r)=g_{0}+g_{2}r^{2}+g_{4}r^{4}+\cdots$. Obviously $g_{0}$ vanishes.
From
$g_{2}={\frac{2\,\tilde{\beta}\,qN-Nmq-2\,mqs+Nq-Ns-2\,{s}^{2}}{N}}=0,$
we get
$m={\frac{2\,\tilde{\beta}\,qN+Nq-Ns-2\,{s}^{2}}{q\left(N+2\,s\right)}}.$
Solving $q$ from
$g_{4}={\frac{\left(2\,{N}^{2}{\tilde{\beta}}^{2}q+4\,N{\tilde{\beta}}^{2}qs+4\,N{\tilde{\beta}}^{2}q-{N}^{2}\tilde{\beta}-8\,\tilde{\beta}\,qs+4\,\tilde{\beta}\,{s}^{2}-2\,N\tilde{\beta}+Ns-4\,\tilde{\beta}\,s-2\,qs+2\,{s}^{2}\right)q}{\left(N+2\right)\left(N+2\,s\right)}}=0,$
to obtain (the other solution $q=0$ is irrelevant)
$q=\frac{1}{2}\,{\frac{\left(N+2\,s\right)\left(N\tilde{\beta}-2\,\tilde{\beta}\,s+2\,\tilde{\beta}-s\right)}{{N}^{2}{\tilde{\beta}}^{2}+2\,N{\tilde{\beta}}^{2}s+2\,N{\tilde{\beta}}^{2}-4\,\tilde{\beta}\,s-s}}.$
Using the explicit expressions of $m$ and $q$, the coefficient $g_{6}$ can be
simplified as
$\displaystyle\frac{1}{3}\frac{s(2\tilde{\beta}+1)(N+2s+2)(N+2s)(N\tilde{\beta}-2\tilde{\beta}s+2\tilde{\beta}-s)}{\left(N+4\right)\left({N}^{2}{\tilde{\beta}}^{2}+2\,N{\tilde{\beta}}^{2}s+2\,N{\tilde{\beta}}^{2}-4\,\tilde{\beta}\,s-s\right)^{3}}.$
Here all the non-zero factors in $g_{6}$ are isolated in the fractions,
especially $N\tilde{\beta}-2\tilde{\beta}s+2\tilde{\beta}-s$ (otherwise
$q=0$). We discuss the different cases for $g_{6}=0$ below, or all
$\tilde{\beta}$ such that
$\tilde{\beta}(N\tilde{\beta}-1)(N\tilde{\beta}-s)(N\tilde{\beta}+2\tilde{\beta}s-2\tilde{\beta}-1)=0.$
Case $\tilde{\beta}=0$.:
Then $m=\frac{N-2s}{N+2s}$, $q=\frac{N}{2}+s$ and
$\alpha=\pm\frac{1}{1-m}=\pm\frac{N+2s}{4s}$. We have to choose
$\alpha=\frac{N+2s}{4s}>0$, otherwise the corresponding self-similar solutions
are growing in time, leading to the self-similar solution
$u(x,t)=(T-t)^{\alpha}\Phi(x)$. The two constants $\lambda$ and $R$ in the
Barenblatt profile $\Phi(y)=\lambda(R^{2}+|y|^{2})^{-\frac{N}{2}-s}$ are
related by only one equation, the matching condition of coefficients from the
identity (15), i.e.,
$\lambda^{1-m}R^{2s}=\frac{\alpha}{2^{2s}}\frac{\Gamma(\frac{N}{2}-s)}{\Gamma(\frac{N}{2}+s)}.$
This gives the self-similar solution (32) in Theorem 5.1, where $\lambda$ and
$R$ can be determined uniquely by the initial mass
$M_{0}=\int_{\mathbb{R}^{N}}u(x,0)dx=\lambda\pi^{\frac{N}{2}}T^{\frac{N+2s}{4s}}\frac{\Gamma(s)}{\Gamma(\frac{N}{2}+s)}.$
Case $N\tilde{\beta}-1=0$.:
This implies that $\tilde{\beta}=1/N=\beta/\alpha$ and it reduces the
Barenblatt profiles considered in Section 3.
Case $N\tilde{\beta}-s=0$.:
Then $q=-1<0$, leading to unacceptable solutions growing at infinity.
Case $N\tilde{\beta}+2\tilde{\beta}s-2\tilde{\beta}-1=0$ or
$\tilde{\beta}=\frac{1}{N+2s-2}$.:
The exponents $m$ and $q$ are simplified as
$m=\frac{N-2s}{N+2s-2},\quad q=\frac{N}{2}+s-1.$
The corresponding Barenblatt profiles
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-\frac{N}{2}-s+1}$ have infinite mass, as
$q=\frac{N}{2}+s-1<\frac{N}{2}$. Since $m$ is strictly larger than
$m_{c}=(N-2s)_{+}/N$, the solutions do not vanish in finite, and we expect the
self-similar solutions (30) instead of (29). This implies
$(m-1)\alpha+2s\beta=1$. Together with
$\alpha=\beta/\tilde{\beta}=(N+2s-2)\beta$, we obtain
$\alpha=\frac{N+2s-2}{2(1-s)},\qquad\beta=\frac{1}{2(1-s)}.$
Finally, we find the relation $\lambda$ and $R$ in the profile
$\Phi(y)=\lambda(R^{2}+|y|^{2})^{-\frac{N}{2}-s+1}$. Since
$(-\Delta)^{s}\Phi(y)^{m}=\lambda^{m}(-\Delta)^{s}(R^{2}+|y|^{2})^{-\frac{N}{2}-s}=\lambda^{m}2^{2s}R^{2s}\frac{\Gamma(\frac{N}{2}+s)}{\Gamma(\frac{N}{2}-s)}(R^{2}+|y|^{2})^{-\frac{N}{2}-s},$
and
$\alpha\Phi(y)+\beta y\cdot\nabla\Phi(y)=\alpha\lambda
R^{2}(R^{2}+|y|^{2})^{-\frac{N}{2}-s},$
the equation (31) for the profile is satisfied if
$\lambda^{1-m}R^{2-2s}=\frac{2^{2s}}{\alpha}\frac{\Gamma(\frac{N}{2}+s)}{\Gamma(\frac{N}{2}-s)}.$
This gives the self-similar solution (33) in Theorem 5.1, and in general
$\lambda$ and $R$ can not be determined uniquely.
The second kind self-similar solutions already appear in the literature in
various contexts. The finite-time extinction of solution for
$m<m_{c}:=\frac{N-2s}{N}$ is already considered in [10], with estimates on the
extinction time using functional inequalities [6] or comparison in
Marcinkiewicz norm [21]. For $m=\frac{N-2s}{N+2s}$, the self-similar solutions
constructed above is believed to better characterize the fine details right
before the extinction and provides a more accurate estimate on the extinction
time for certain initial data.
Solutions of (1a) with infinite mass do not fit into the general theoretical
framework for $L^{1}$ initial data developed in [9, 10] and have to be treated
in weighted space [6]. Therefore, in contrast to those first kind self-similar
solutions starting with a Dirac delta initial condition, self-similar
solutions with singular initial data like $u(x,0)=|x|^{-N/p}$ for
$p>\max(1,N(1-m)/2s)$ are shown to be second kind, with conserved $L^{p}$ norm
instead of $L^{1}$ norm (the mass). The solution (33) obtained above provides
another explicit example of anomalous scaling for large data.
Finally, it should be noted another _anomalous_ self-similar solutions, so-
called Very Singular Solutions (VSS), also constructed from separation of
variables. when $0<m<m_{c}$, the solution
$u(x,t)=C(T-t)^{1/(1-m)}|x|^{-2s/(1-m)}$
is used to estimate the finite extinction time [21]; when
$\frac{N-2s}{N}:=m_{c}<m<N/(N+2)$, the solution
$u(x,t)=Ct^{1/(1-m)}|x|^{-2s/(1-m)}$
arises in the limit when the total mass of the first-kind Barenblatt profiles
goes to infinity [20]. In the limit $R\to 0$, (32) reduces to the former in
the case $m=\frac{N-2s}{N+2s}$. However, (33) does not reduce to the latter
because the range $m=\frac{N-2s}{N+2s-2}$ is not inside the interval
$\big{(}m_{c},N/(N+2)\big{)}$ of existence in general.
## 6\. Conclusion
In this paper, several one-parameter families of explicit self-similar
solutions are obtained for fractional porous medium equations (1a) and (1b).
The special forms of the Barenblatt profiles are motivated from the classic
PMEs, and are determined from the matching conditions of certain
hypergeometric functions or local power series expansions. These special scale
invariant solutions can complement the qualitative and quantitative studies of
the underlying equations with explicit examples, and provide immense intuition
for further investigation. In addition, these exact solutions can also be used
to test the accuracy and efficiency of numerical methods for equations with
fractional operators.
By our construction, the explicit Barenblatt profiles are exhausted in the
forms $\lambda(R^{2}+|y|^{2})^{-q}$ or $\lambda(R^{2}-|y|^{2})_{+}^{q}$ for
the cases we sought. In contrast to those of (2) obtained for all $m$ and $s$
in [3, 4], these explicit profiles for (1a) or (1b) exist only for certain
combinations of $m$ and $s$. The profiles for general $m$ and $s$, whose
existence may be relatively easy to prove as in [20], are expected to have
much more complicated expressions (if they exist). The complexity can be
observed from the explicit Barenblatt profiles of the fractional heat equation
$u_{t}+(-\Delta)^{s}u=0$ via Fourier transform. Therefore, it is interesting
to see whether there are any explicit candidate profiles for the more general
cases.
## Acknowledgements
This work is supported by Engineering and Physical Sciences Research Council
grant number EP/K008404/1. The author would like to thank the hospitality of
Professor Juan Luis Vázquez and Universidad Autónoma de Madrid where this work
was initiated. The author also appreciates the anonymous referees for comments
and suggestions to improve the paper.
## Appendix A Fourier transform of radial functions
The fractional Laplacian of Barenblatt profiles in this paper is evaluated by
Fourier transform and inverse Fourier transform. These transforms are defined
as
$\hat{u}(\xi)=\mathcal{F}[u](\xi)=\int_{\mathbb{R}^{N}}u(x)e^{-i\xi\cdot
x}dx,\qquad
u(x)=\mathcal{F}^{-1}[u](x)=(2\pi)^{-N}\int_{\mathbb{R}^{N}}\hat{u}(\xi)e^{i\xi\cdot
x}dx.$
In particular, we need a few facts about the transforms of radially symmetry
functions [12].
Using explicit expression for the integration of $e^{i\omega\cdot x}$ over the
unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^{N}$, i.e.,
$\int_{\mathbb{S}^{N-1}}e^{i\omega\cdot
x}d\omega=(2\pi)^{\frac{N}{2}}|x|^{1-\frac{N}{2}}J_{\frac{N}{2}-1}(|x|),$
the Fourier transform of a radial function $u(|x|)$ becomes
(35)
$\mathcal{F}[u](\xi)=(2\pi)^{\frac{N}{2}}|\xi|^{1-\frac{N}{2}}\int_{0}^{\infty}r^{\frac{N}{2}}J_{\frac{N}{2}-1}(r|\xi|)u(r)dr.$
Similarly the inverse Fourier transform of a radial function $\hat{u}(|\xi|)$
becomes
(36)
$\mathcal{F}^{-1}[\hat{u}](x)=(2\pi)^{-\frac{N}{2}}|x|^{1-\frac{N}{2}}\int_{0}^{\infty}\eta^{\frac{N}{2}}J_{\frac{N}{2}-1}(\eta|x|)\hat{u}(\eta)d\eta.$
## References
* [1] G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
* [2] G. I. Barenblatt. Scaling, self-similarity, and intermediate asymptotics, volume 14 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1996.
* [3] P. Biler, C. Imbert, and G. Karch. Barenblatt profiles for a nonlocal porous medium equation. Comptes Rendus Mathematique, 349(11):641–645, 2011.
* [4] P. Biler, C. Imbert, and G. Karch. Nonlocal porous medium equation: Barenblatt profiles and other weak solutions. 2013\. arXiv:1302.7219.
* [5] G. W. Bluman and S. C. Anco. Symmetry and integration methods for differential equations, volume 154 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002.
* [6] M. Bonforte and J. L. Vázquez. Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Advances in Mathematics, 250(0):242 – 284, 2014.
* [7] L. A. Caffarelli and J. L. Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst., 29(4):1393–1404, 2011.
* [8] L. A. Caffarelli and J. L. Vázquez. Nonlinear porous medium flow with fractional potential pressure. Archive for rational mechanics and analysis, 202(2):537–565, 2011\.
* [9] A. de Pablo, F. Quirós, A. Rodríguez, and J. L. Vázquez. A fractional porous medium equation. Adv. Math., 226(2):1378–1409, 2011.
* [10] A. de Pablo, F. Quirós, A. Rodríguez, and J. L. Vázquez. A general fractional porous medium equation. Comm. Pure Appl. Math., 65(9):1242–1284, 2012.
* [11] E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012.
* [12] L. Grafakos. Classical and modern Fourier analysis. Pearson Education, Inc., Upper Saddle River, NJ, 2004.
* [13] N. S. Landkof. Foundations of modern potential theory. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 180.
* [14] P. J. Olver. Applications of Lie groups to differential equations, volume 107 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
* [15] M. A. Peletier and H. F. Zhang. Self-similar solutions of a fast diffusion equation that do not conserve mass. Differential Integral Equations, 8(8):2045–2064, 1995.
* [16] D. Stan, F. del Teso, and J. L. Vázquez. Finite and infinite speed of propagation for porous medium equations with fractional pressure. Comptes Rendus Mathematique, 352(2):123–128, 2014.
* [17] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N. J., 1970.
* [18] J. L. Vázquez. Smoothing and decay estimates for nonlinear diffusion equations, volume 33 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006. Equations of porous medium type.
* [19] J. L. Vázquez. The porous medium equation. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007. Mathematical theory.
* [20] J. L. Vázquez. Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. To appear in Journal Europ. Math. Society, 2013. arXiv:1205.6332.
* [21] J. L. Vázquez and B. Volzone. Optimal estimates for fractional fast diffusion equations. 2013\. arXiv:1310.3218.
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|
arxiv-papers
| 2013-12-02T14:27:03 |
2024-09-04T02:49:54.638079
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanghong Huang",
"submitter": "Yanghong Huang",
"url": "https://arxiv.org/abs/1312.0469"
}
|
1312.0547
|
Current Address: ]Time and Frequency Division, NIST, Boulder CO, 80305
# Capture and isolation of highly-charged ions in a unitary Penning trap
Samuel M Brewer [ University of Maryland, College Park, Maryland 20742, USA
Nicholas D Guise University of Maryland, College Park, Maryland 20742, USA
National Institute of Standards and Technology, 100 Bureau Drive,
Gaithersburg, Maryland 20899-8422, USA Joseph N Tan National Institute of
Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899-8422,
USA
###### Abstract
We recently used a compact Penning trap to capture and isolate highly-charged
ions extracted from an electron beam ion trap (EBIT) at the National Institute
of Standards and Technology (NIST). Isolated charge states of highly-stripped
argon and neon ions with total charge $Q\geq 10$, extracted at energies of up
to $4\times 10^{3}\,Q$ eV, are captured in a trap with well depths of
$\,\approx(4\,{\rm to}\,12)\,Q$ eV. Here we discuss in detail the process to
optimize velocity-tuning, capture, and storage of highly-charged ions in a
unitary Penning trap designed to provide easy radial access for atomic or
laser beams in charge exchange or spectroscopic experiments, such as those of
interest for proposed studies of one-electron ions in Rydberg states or
optical transitions of metastable states in multiply-charged ions. Under near-
optimal conditions, ions captured and isolated in such rare-earth Penning
traps can be characterized by an initial energy distribution that is $\approx$
60 times narrower than typically found in an EBIT. This reduction in thermal
energy is obtained passively, without the application of any active cooling
scheme in the ion-capture trap.
## I Introduction
Highly-charged ions (HCI) are of interest in the study of atomic structure,
astrophysics, and plasma diagnostics for fusion science Beyer and Shevelko
(2003). The high nuclear charge, $Z$, tends to amplify relativistic effects in
atoms, such as fine and hyperfine structure splitting Gillaspy (2001). For
example, the fine structure energy splitting is proportional to
$(Z\alpha)^{4}$, where $\alpha\approx 1/137$ is the fine structure constant,
and hence can be so large for some high $Z$ ions that the transition frequency
is scaled up from the microwave to the visible domain of the electromagnetic
spectrum Jentschura _et al._ (2008) – a useful feature for observing
astrophysical objects.
Apart from natural sources, highly-charged ions have become more widely
accessible with the development of laboratory facilities like heavy-ion
storage rings Habs and et. al. (1989) and more compact devices like the
electron-cyclotron resonance (ECR) ion source Geller (1996) and the electron
beam ion trap/source (EBIT/EBIS) Levine _et al._ (1988); Donets (1998);
Motohashi _et al._ (2000); Xiao _et al._ (2012). These ion sources are
useful in various research areas, including: spectroscopy (moments, spectral
lines, etc.), ion-surface interactions Gillaspy _et al._ (2001), plasma
diagnostics for next-generation tokamak fusion reactors such as the
International Thermonuclear Experimental Reactor (ITER) Gillaspy _et al._
(2009), and tests of astrophysical models (see Gillaspy _et al._ (2011) and
references therein).
The isolation of single species, highly-charged ions at low energy in traps
can enable some interesting studies of atomic and nuclear phenomena Yang and
Church (1993). As a recent example, high precision studies of HCIs have been
proposed to realize atomic clocks based on Nd13+ and Sm15+ Dzuba _et al._
(2012) for laboratory investigations of the variation (temporal and spatial)
of $\alpha$. Another possibility is to test theory in Rydberg states of one-
electron ions with comb-based spectroscopy, which could led to a Rydberg
constant determination that is independent of the proton radius. Jentschura
_et al._ (2008)
A broad survey of trap types and ion sources developed to advance measurements
of atomic and nuclear properties can be found in the 2003 review article by
Kluge, et al. Kluge _et al._ (2003). A variety of useful techniques have been
developed for the study of trapped positrons Surko and Greaves (2004),
antiprotons Gabrielse _et al._ (1989) and antihydrogen (see Ref. Gabrielse
_et al._ (2012) and references therein) as well as highly-charged ions in
Penning traps Penning (1936) with meter-long electrode structures surrounded
by multi-tesla solenoid magnets Andjelkovic _et al._ (2010); Repp _et al._
(2012). In some of the earliest experiments, a cryogenic Penning trap (RETRAP)
with a high-field superconductive magnet Schneider _et al._ (1994) was
employed to capture ions extracted from an EBIT at the Lawrence Livermore
National Laboratory (LLNL). More recently, SMILETRAP II demonstrated capture
and cooling of Ar16+ in a Penning trap utilizing a room-temperature 1.1 T
solenoid magnet Hobein _et al._ (2011).
Solenoidal magnets can generate a strong magnetic field for ion confinement,
but they also impose geometrical constraints that hinder the access of laser
or atomic beams to be directed at the stored ions. In our effort to produce
and study one-electron ions in Rydberg states, we have designed unitary
Penning traps for isolating single-species charge states of highly-stripped
ions extracted from an EBIT at NIST Tan _et al._ (2012). The unitary
architecture is useful also for studying long-lived transitions, as will be
discussed in forthcoming publications. Initial demonstrations Tan _et al._
(2012, 2011); Guise _et al._ (2013) reported the use of unitary Penning traps
to isolate and store various HCIs. In this work, we discuss the dynamical
considerations and experimental manipulations that are essential for optimized
performance to maximize the number of stored ions as well as minimize the
energy distribution for precise measurements.
A brief description of the system configuration is provided in Sec. §II.
Numerical simulations were carried out to guide the design of the compact
Penning trap and additional beam-conditioning components, as discussed in Sec.
§III, with emphasis on the deceleration of fast ($\approx$ 40 keV) ions
approaching the region $\approx$ 3 cm in front of the trap. Section §IV.1
describes charge state selection and ion pulse optimization, emphasizing the
importance of (a) minimizing the time width of the extracted ion pulse, and
(b) matching the deceleration potential near the Penning trap to ion
extraction energy. Results from recent ion capture experiments are presented,
illustrating ion capture optimization (Sec. §IV.2) and residual energy
measurement (Sec. §IV.3). Finally, a discussion of the ion capture efficiency
is presented in Sec. §V.
## II Experimental Setup
The experimental set-up, illustrated in Figure 1, consists of the EBIT with
its ion extraction beamline, and the recently-installed ion-capture apparatus.
Since some parts of the set-up have been described in detail elsewhere Tan
_et al._ (2012); Guise _et al._ (2013), only a brief overview is given here.
Figure 1: (Color online) Schematic overview of the experimental set-up (NOTE:
Not to scale). The ion source is the EBIT at NIST with its existing ion
extraction beamline, which has an analyzing magnet (AMag) for charge state
selection. The experiment apparatus at the end of the beamline houses a
unitary Penning trap to capture selected ions, and detectors to count ions
ejected after storage. Labels with asterisk indicate mounting on retractable
translators. Broken lines represent the boundary of evacuated space; vacuum
pumps are not shown. Ion-trajectory path-length from the EBIT to the Penning
trap is $\approx$ 8 meters.
Highly-charged ions are produced in the EBIT, bound radially to the energetic
electron beam along the axis. Axially the ions are trapped in an electrostatic
well created by applying electric potentials $(V_{i})$ to three cylindrical
electrodes, called drift tubes–labelled by their location: upper (UDT), middle
(MDT), and lower (LDT), with $V_{\rm MDT}<V_{\rm UDT}<V_{\rm LDT}$. To extract
an ion bunch, the MDT can be quickly raised to a value $V_{\rm UDT}<V_{\rm
MDT}<V_{\rm LDT}$ thus ejecting HCIs into the beamline Tan _et al._ (2011);
Pikin _et al._ (1996); Ratliff _et al._ (1998).
Electrostatic ion optics in the beamline guide (EB1, Defl 1-3, EB2) and focus
(BEL 1-4) the extracted ions, transporting them over an 8-meter trajectory
from the EBIT to the unitary Penning trap. At various points, retractable
Faraday cups (FC1-2) can be inserted to monitor the ion beam. About half-way
along the beamline, an analyzing electromagnet (AMag) selects a specific
charge state to be captured in the Penning trap. The beamline vacuum space has
a base pressure of $2.7\times 10^{-7}$ Pa ($2.0\times 10^{-9}$ Torr).
Figure 2: (Color online) Cross-sectional view of the compact Penning trap used
to capture ions (foreground). The ring electrode has 4 equidistant holes–one
hole concentric with a vacuum window in the background; a small lens is inside
the top hole for observing fluorescence from stored ions. Two rare-earth
(NdFeB) magnets are embedded within the electrode assembly, one on each side
of the ring electrode. Ions enter from the right-hand-side along the trap
axis, slowing in the deceleration electrodes (DR1 and DR2) before entering the
trap via the 8.00 mm hole in FEC. Stored ions can be counted by ejection to a
TOF detector, focussed and guided by an einzel lens (EL 1, EL 2, EL 3) and
steering plates.
At the entrance of the ion-capture apparatus, specialized components are used
to optimize on-axis injection of HCIs into the unitary Penning trap; a set of
four steering plates (SP1), a one-magnet trap/einzel lens, and a retractable
Faraday cup (FCA) allow fine adjustments in alignment and ion pulse
conditioning Guise _et al._ (2013). After confinement in the Penning trap
(Fig. 2), stored ions are detected by ejection to one of the ion detectors. A
retractable micro-channel plate (MCP) with fast response is used for ion
counting and time-of-flight (TOF) or charge state analysis. If the fast TOF
detector is retracted, as discussed in Guise _et al._ (2013), a position-
sensitive MCP ion detector (PSD) can be used during beam alignment and
conditioning. The TOF detector is a “Chevron”, or V-stack type Colson _et
al._ (1973) with a disc head (8.0 mm active diameter), which is operated in
either proportional (charge amplifying) mode, or in a fully-saturated, event
counting mode. An event pulse has rise/fall time $\approx$ 350 ps with a gain
of $>10^{6}$ per incident charge.
Figure 2 shows a half-cut view of the unitary Penning trap used to capture
ions. The unitary architecture Tan _et al._ (2012) makes the ion trap
extremely compact, with an electrode assembly volume of less than 150 cm3. The
magnetic field for radial confinement of stored ions is generated by two rare-
earth magnets that are yoked by the soft-iron electrodes (FEC, RING, and BEC).
The front endcap (FEC) and the back endcap (BEC) are maintained at a higher
potential than the RING electrode to form an axial trapping well. The two
deceleration electrodes (DR 1 and DR 2) adjacent to the front endcap are
crucial for slowing ions before they enter the unitary Penning trap; their
conical inner surfaces are tailored to produce near-planar equipotential
surfaces. Application of static and time-varying electrical potentials is
controlled through a computer interface, the details of which are provided in
Sec. IV. A separate vacuum chamber houses the room-temperature Penning trap,
allowing control of the background gas composition and pressure; the base
pressure of this vacuum chamber is $1.0\times 10^{-7}$ Pa ($7.6\times
10^{-10}$ Torr).
## III Simulations
Numerical simulations have been carried out to investigate: (a) the optimal
electrode geometry of a unitary Penning trap designed to slow, capture, and
store ions extracted from an EBIT; (b) the operation settings, such as
voltages and switching times for controlling electrodes; and (c) the ideal
conditions of an incoming ion bunch. Ion capture simulations involve
computations of both the magnetic field in the trap as well as the
electrostatic potential generated by the trap electrodes and focusing
elements, generally under time-varying potentials. The details of the magnetic
field calculations, including comparisons with measured trap fields, are
presented in Tan _et al._ (2012). The measured magnetic field strength is
$\approx$ 310 mT in the trapping region and is in good agreement with the
calculated field. The electric field in the trap assembly is calculated using
a numerical Boundary Element Method (BEM), originally developed for computing
properties of electrostatic lenses Harting and Read (1976).
An example of the calculated electrostatic potential along the axis of the ion
trap is shown in Fig. 3. The “open” condition in preparation for ion capture
is shown in (a) and the “closed” condition following ion capture is shown in
(b). The applied voltages for each electrode and the critical EBIT parameters
are listed in Table I. The EBIT shield voltage and MDT high voltage pulse
levels are included in Fig. 3a for comparison. As shown in Fig. 3b, the axial
potential well near the trap center is well approximated by an analytic
quadrupole potential, which in cylindrical coordinates takes the form Brown
and Gabrielse (1986); Tan _et al._ (2012)
$V=\lambda V_{0}\frac{z^{2}-\rho^{2}/2}{2d^{2}}+V_{C}.$ (1)
The field coordinates $z$ and $\rho$ are defined from the center of the trap;
$V_{0}$ is the applied potential difference between the endcaps and the
central ring electrode, $V_{C}$ is the common-mode or float potential, and $d$
is a geometric factor
$d^{2}\equiv\frac{1}{2}(z_{0}^{2}+\rho_{0}^{2}/2).$ (2)
The coefficient $\lambda$ (often referred to as $C_{2}$) is of order unity.
The characteristic dimensions $r_{0}$ and $z_{0}$ are from the center of the
trap to the ring and endcap electrodes, respectively. For the Penning trap
presented here, $\rho_{o}=8.5$ mm, $z_{o}=4.736$ mm, and $\lambda=0.854$.
Figure 3: (Color online) Calculated electrostatic potential along the trap
axis with the electrode positions indicated at the top of the figure. The
“open” trap condition is shown in part (a), with the EBIT shield voltage and
the MDT pulse voltage indicated. The “closed” trap condition is shown in part
(b), magnified near the trap center at z = 0 mm, with BEM calculation in
black, and analytic quadrupole fit in red. The ion pulse enters the apparatus
from the right. The applied voltages are given in Table 1; the difference of
30 V between FEC $=$ BEC and the ring electrode corresponds to an on-axis well
depth of $11.64$ V. Penning Trap Parameters
---
Trap Electrode | Applied Potential (V)
DR1 | 1300.0
DR2 | 1600.0
FEC | (Low) 2610.0
| (High) 2956.8
Ring | 2926.8
BEC | (Low) 2460.0
| (High) 2956.8
EL1 | 500.0
EL2 | 1500.0
EL3 | 500.0
EBIT Parameters
e- beam Energy | 2.5 keV
e- beam Current | 14.4 mA
LDT | 500 V
MDT | Trap Dump = 400 V
UDT | 220 V
Ionization Time | 76.0 ms
Analyzing B-field | 66.22 mT
Table 1: Typical applied trap potentials and EBIT parameters used in producing
and capturing Ar13+ ions. The EBIT conditions have been chosen to both
maximize ion production and minimize the time width of ion pulse.
Special care was taken in designing the two deceleration electrodes, DR1 and
DR2, to generate nearly planar equipotential surfaces with resulting
$\nabla\Phi$ gradient that tends to remove axial kinetic energy from ions
entering the trap. In order to attain the lowest possible residual energy
after capture it is important to minimize momentum transfer to transverse
motions as the ions are injected into the Penning trap.
With the computed electric and magnetic fields Tan _et al._ (2012) and a
given set of initial conditions (the ion position and velocity), an ion
trajectory is calculated by integrating the equations of motion using an
adaptive step-size Runge-Kutta technique such as provided by a commercial
code, Charged Particle Optics Harting and Read (1976); dis . A triangle mesh
ratio limit (side/length) of 20 yields fractional precision of 10-4 for the
electric field and ray tracing computations.
In this work, only single particle trajectories are computed to model the
properties of the system. An improved model would require the inclusion of the
inter-ion coulomb interaction, and is not practical for computational
resources available in this work. To first approximation, single-particle
trajectories have been useful in finding the optimal conditions for successful
ion capture. To illustrate, trajectories calculated for a range of impact
parameter values, $a_{i}$ (perpendicular distance from trap axis at $z>70$ mm)
are presented in Fig. 4. Each trajectory starts with the same initial velocity
entirely parallel to the trap axis (the direction of propagation),
representing the zero-emittance Humphries (1990) beam condition. Iterating
such computation for various trap parameters, the potentials on the
deceleration electrodes DR1 and DR2, as well as the electrode geometry, have
been optimized to capture ions in trajectories with the smallest amplitudes of
resulting bound motions. Fig. 5 shows the maximum ion kinetic energy after
capture, calculated as a function of impact parameter, for Ar13+ ions
($Q\,=\,13$; Ar XIV in spectroscopic notation). The deceleration is most
effective on-axis, for which the initial ion kinetic energy is removed more
completely. As the impact parameter increases, the residual energy after
capture increases.
Figure 4: (Color online) Classical single-ion trajectories computed for a
family of impact parameter values, ai, ranging from 0.1 mm to 2.1 mm in 0.2 mm
steps. The ion trap center is located at z = 0 mm. For the same velocity
parallel to the axis, the amplitudes of bound ion motions increase with
increasing ai. The trajectory shown in red dotted line, magnified in the inset
(b), corresponds to an impact parameter of 0.5 mm. Figure 5: (Color online)
Maximum kinetic energy of captured Ar13+ ions, calculated as a function of the
impact parameter, ai. Ions enter the capture apparatus with velocity parallel
to the axis. The total kinetic energy is shown as a solid line (–), the
transverse kinetic energy is shown as a dotted line ($\cdots$), and the axial
kinetic energy is shown as a dashed line ($--$).
Single particle simulation has been particularly useful for finding the
capture time ($t_{capture})$ at which the Penning trap must be switched from
the open configuration to the closed configuration to capture and store ions.
A rough estimate is the mean transit time of the ion pulse from the EBIT to
the Penning trap. The front endcap (FEC), momentarily held below the ring
potential to admit ions into the trap, must be switched to close the trap
within a certain arrival time tolerance. If FEC is switched to close the trap
too early, before the extracted ions enter the trapping region, the ions will
scatter off and not be captured. On the other hand, if FEC is closed too late,
ions will have entered the trap, turned around, and exited the trapping
region–before they can be captured. For a given initial energy and trap well
configuration, there is a range of arrival times wherein the FEC electrode can
be switched to successfully confine the ions that have entered the trap; the
width of this allowed range for ion capture is labeled the capture time width
(CTW). The CTW can be estimated by computing ion trajectories to find bound
motions for a family of times at which FEC is switched to close the trap, in
10 ns time steps, assuming the same initial kinetic energy in each
calculation. For ions injected on-axis, the probability of ion capture is a
flat-top function of the time when FEC is switched to close the trap. The
width of this function is an estimate of the capture time width. For the case
of Ar XIV, CTW $\approx$ 80(20) ns is calculated for the optimal trapping
conditions given above in Table 1. For comparison, in a high-field Penning
trap with a long electrode stack, the ions are captured in a nearly-flat
bottom (square-well) potential and the CTW is well approximated by the round-
trip time, which can range from $\approx$ 300 ns Fei _et al._ (1987) to about
$1\,\mu$s Schneider _et al._ (1994). The CTW of a compact Penning trap tends
to be shorter due to its size. However, as illustrated in this work, the CTW
of a unitary Penning trap is sufficient to capture a broad range of ions.
## IV Experiments
### IV.1 Pulsed extraction of ions
The energy available for electron impact ionization in an EBIT is set by a
common-mode, float voltage applied to the drift tube assembly. In this work,
the float voltage is adjusted to give an electron beam energy ($E_{e-}$) in
the range from 2.0 keV to 4.0 keV with an electron beam current ($I_{e-}$) in
the range from 6 mA to 150 mA. The NIST EBIT ion-extraction beamline has been
optimized for high ion flux Pikin _et al._ (1996) in ion-surface bombardment
experiments Lake _et al._ (2011), wherein the EBIT is typically operated in a
continuous, high-current mode with $I_{e-}=150$ mA. For the ion capture
experiments discussed here, it would be ideal for the extracted ions to be
bunched tightly in both space and time. Therefore, the EBIT is operated in a
low-current, pulsed extraction mode. The electron beam energy and current are
chosen to optimize the production and capture of selected ions. As an example
we present the case of Ar13+ extracted at an electron beam energy of
E${}_{e-}=$ 2.50 keV and electron beam current I${}_{e-}=$ 14.4 mA.
To extract ions in pulses, a fast (rise time $\approx$ 50 ns) voltage pulse of
0 V to 400 V is applied to the MDT electrode in addition to the float voltage.
As indicated in Table 1, the UDT electrode is biased at a lower potential than
the LDT electrode. Consequently, the rapid rise in MDT voltage pushes all ions
in the EBIT into the beamline. As illustrated in Figure 1, ions leaving the
EBIT are transported via the ion optics in the horizontal beamline to an
analyzing magnet that filters to select a specific charge state.
Figure 6: Detection of extracted ion bunch: (a) using a Faraday cup (FC2)
before the analyzing magnet; and (b) using a fast TOF detector after selection
of one charge state (Ar XIV) which is propagated through the Penning trap. The
detected Ar ions were produced with an electron beam energy (Ee-) and current
(Ie-) of 2.50 keV and 14.4 mA, respectively.
Figure 6 (a) shows a typical Faraday cup signal generated by ions of various
charge states striking FC2 immediately in front of the analyzing magnet. The
analyzing magnetic field is tuned to single out a specific charge state to
pass through the magnet, with its trajectories bent into the vertical beamline
segment while all other charge states will hit the chamber wall. Illustrative
examples are provided in Guise _et al._ (2013). The selected charge state is
guided further into the ion capture apparatus. For beam diagnostics, the
extracted ion pulse passes through the grounded Penning trap and is detected
using a fast TOF detector. As shown in Fig 6 (b), the charge-state-selected
ion signal amplitude is $\approx$ 1.3 V and has a full width at half maximum
(FWHM) of $\approx$ 110 ns, corresponding to $\approx$ 1435 ions per
extraction pulse passing through the trap. By fine tuning the electrostatic
elements in the ion beamline, the EBIT settings, and the analyzing magnet
field, this TOF signal is optimized for maximum ion pulse amplitude and
minimum time width.
### IV.2 Slowing and capture
Capturing the extracted ion pulse involves two key aspects: (1) closing the
trap at the right time; and (2) tuning the float potential ($V_{C}$) of the
unitary Penning trap to match the EBIT extraction energy. The timing diagram
for ion extraction and capture is shown in Fig. 7. Details of the ion
detection scheme are discussed in Tan _et al._ (2012); Guise _et al._
(2013).
Figure 7: (Color online) Timing pulse diagram for controlling ion capture and
detection. TTL pulses triggering various switches/scopes are shown in the
upper section (blue); corresponding high voltage outputs are shown in the
lower section (red). Stored ions are ejected to a detector when BEC is low. A
schematic diagram for TOF detection is given in Tan _et al._ (2012) and an
abridged timing scheme is shown in Guise _et al._ (2013)
Experimentally, the “capture time,” the time at which the entrance endcap
electrode is switched to close the trap, is varied to maximize the number of
ions captured per pulse. A measurement of the optimal ion capture time is
shown in Fig. 8. Ions are captured and stored for 1 ms before being counted by
ejection to the TOF detector. In contrast to the ideal case presented in §III,
the observed ion capture time profile (Fig. 8 top) is mainly shaped by the
characteristics of the ion pulse extracted from the EBIT. The observed peak
gives the optimal capture time. In the case of Ar13+ ions, the optimal capture
time occurs at 17.43 $\mu$s after pulsed extraction from the EBIT with a
nominal energy of 2.50 keV.
Figure 8: (Color online) Observed ion capture time profile for Ar XIV. Ion
counts obtained by integrating TOF signals, as illustrated with 3 cases: (a)
capture time below optimal value; (b) capture time at the optimal value; and
(c) capture time above optimal value. The TOF signals associated with these 3
cases (in red) are shown in three inset plots and labelled (a-c)
correspondingly. Ions were stored for 1 ms; the data represent the average of
64 trials each. Error bars represent 1 standard deviation.
Another important consideration that affects the residual energy of captured
ions is the deceleration of the ion pulse as it approaches the Penning trap,
which is controlled largely by the common-mode, float voltage $V_{C}$ applied
to all electrodes in the Penning trap assembly. In the continuous extraction
mode, ions escape into the beamline with an energy of $E_{ion}\approx
QU_{e-beam}$, where $U_{e-beam}$ is the electron beam energy; in contrast, for
pulsed extraction mode, the fast switching of the MDT electrode gives ions an
additional $\approx 400\,Q$ eV of kinetic energy. The float voltage on the
unitary Penning trap is adjusted to match the incoming ion energy, thus fine-
tuning the amount of energy that is to be removed from the ion bunch in the
process of being slowed and captured.
The influence of energy matching is illustrated in Figures 9 and 10. The trap
float voltage $V_{C}$ is adjusted to obtain the optimal ion capture signal.
The number of ions following 1 ms storage is measured as a function of the
trap float voltage. There is a broad maximum between 2880 V and 2940 V.
However, the width of the TOF signal drops steadily over that same voltage
interval. The narrowing of the TOF width as a function of the float voltage
indicates that as $V_{C}$ is increased, the energy matching between the
Penning trap and the extraction energy of the incoming ion pulse is improving.
As $V_{C}$ is further increased, the number of captured ions begins to
decrease significantly, because more of the incoming ions lack the kinetic
energy to reach the trapping region.
Figure 9: (Color online) Optimization of the common-mode, float voltage
($V_{C}$) applied on the compact Penning trap. Figure (a) shows the number of
ions detected as a function of float voltage, following 1 ms of ion storage,
averaged over 64 pulses. Figure (b) shows the TOF width of the ejected ion
pulse. The applied trap well is $V_{o}=30$ V, and the capture time is
$t_{capture}=17.43\mu$s. Error bars represent 1 standard deviation. Figure 10:
(Color online) Optimized TOF signal from captured Ar XIV. Captured ions are
ejected after 1 ms of storage in the Penning trap. The narrow TOF signal in
red solid line is for optimized float voltage $V_{C}$ = 2927 V, highlighted in
Figure 9. For comparison, a double-peaked TOF signal corresponding to a
detuned float voltage is also shown in black dashed line. Optimal capture time
$\approx 17.43\mu$s is used (see Figure 8).
Dramatic broadening in the TOF signal for ions ejected from the Penning trap
can result from mistuning of the float voltage, as illustrated in Figure 10.
For a float voltage that is well below optimal value, the captured ions can
have energy significantly higher than the bottom of the potential well, and a
double peak structure in the TOF signal is observed. For float voltages near
the optimal value, the TOF signal is single peaked and narrower, with an
optimal FWHM $\approx$ 18.5 ns. It is important for the TOF signal to be
single peaked for proper interpretation of lower charge states generated after
long storage times Tan _et al._ (2012). Furthermore, as the float voltage
approaches the optimal value from below, the TOF signal becomes narrower (see
Figure 9 b) indicating that the captured ions have less residual energy.
### IV.3 Energy of captured ions
Experiments and model simulations, discussed in previous sections, have been
useful in developing a unitary Penning trap for capturing multi-charged ions.
Trap parameters were deliberately sought to favor computed ion trajectories
which lead to bound motions with small amplitudes. Furthermore, the control
settings of the ion source, electrostatic ion optics, and compact Penning trap
have been tuned in an attempt to maximize the number of ions captured, as well
as to minimize the width of the time-of-flight signal. Consequently, Fig. 9b
indicates that the residual energy in bound ion motions can be significantly
reduced.
To measure the energy distribution of captured ions, we used an over-the-
barrier technique that is well-established in high-magnetic-field, multi-well
Penning traps Gabrielse _et al._ (1989). In the standard method, ions
escaping from confinement are guided by strong magnetic field lines to an ion
counter if they have sufficient energy to surmount a controlled potential
barrier. The ion count is correlated with the instantaneous height of the
potential barrier to obtain the energy distribution.
The use of this method in a unitary Penning trap, on the other hand, requires
some modification because of several features: (1) the magnetic field (maximum
0.31 T at the center) drops rapidly, particularly as the ions enter the
endcap; (2) the reentrant endcaps make the well minimum very sensitive to
asymmetrically applied voltages; (3) the ions are guided mainly by
electrostatic ion optics to the detector. Hence, in order to minimize the
transport losses during the energy measurement, the ring electrode has been
used to control the barrier height. The ion cloud energy, 1 ms after capture,
has been measured by slowly ramping up the trap ring electrode voltage at a
specified rate. As the ring voltage rises, the axial potential well depth
decreases, allowing successively slower ions to escape over a known potential
barrier in transit to the detector. An ion energy distribution of Ar13+ ions
escaping from a unitary Penning trap is shown in Fig. 11.
Figure 11: Observed energy distribution of Ar XIV ions escaping the
confinement barrier along the trap axis as the ring electrode voltage is
ramped linearly to shallower well depths. The energy width at half-maximum is
5.5(5) eV. Measured after 1 ms of storage.
The TOF detector was operated in the ion-counting mode, with a fully-saturated
bias voltage of -1730 V. A fast multichannel scaler was used to count events,
triggered to begin acquisition simultaneously with the ramping of the ring
electrode voltage. Since the ring electrode voltage is ramped at a controlled
rate of $V_{r}(t)=$ 0.1V / $\mu$s $\times$ t, we can convert the arrival time
of ions at the TOF detector to the corresponding ring electrode voltage, and
hence to the barrier height. An ion escaping along the trap axis must have
energy exceeding $Q\,e\,\Delta V$ to surmount the barrier potential $\Delta
V=\Delta V_{0}-0.388V_{r}(t)$ where $\Delta V_{0}$ is the depth of the
electrical potential well (maximum $-$ minimum) on axis. For the case
considered (Table 1), $\Delta V_{0}=0.388\times 30V=11.64V$.
The energy distribution of Ar XIV ions escaping from the unitary Penning trap
along its axis has a FWHM energy width of 5.5 $\pm$ 0.5 eV. This energy
distribution is a factor of $\approx$ 60 narrower than expected inside an EBIT
Lapierre _et al._ (2005). The over-the-barrier method generally gives an
upper limit for the ion energy since the escaping ions tend to heat up from
release of the ion cloud space-charge potential energy Gabrielse _et al._
(1989). It is worth noting also that this is an estimate of the residual
energy distribution shortly after capture, before any active cooling scheme
has been implemented.
Generally, a narrower energy distribution is favorable for spectroscopy
because the Doppler broadening of spectral lines tend to have a Gaussian
distribution with a FWHM line-width that is related to system parameters by
$\Delta f_{\rm FWHM}=2f_{o}\sqrt{(2kT/Mc^{2}){\rm ln2}}$ where $f_{o}$ is the
transition frequency, $k$ is the Boltzmann constant, $T$ is the ion cloud
temperature, $M$ is the mass of the radiator, and $c$ is the speed of
light.Griem (1997) For example, the spectral lines emitted by an Ar13+ ion
cloud with temperature $kT\approx 5.5$ eV are expected to have a fractional
Doppler line-width of $\Delta f/f_{o}\approx 2\times 10^{-5}$.
## V Ion capture efficiency
The number of extracted ions captured in the Penning trap is determined in
part by the fixed parameters chosen for the trap and beam-tuning structures
(e.g., sizes of apertures); it is also affected by adjustments in operating
conditions made during experiments to optimize energy and ion pulse width.
Trade-offs are made in optimization, as illustrated in Fig.9. Assuming an
incoming ion beam with no initial transverse momentum and neglecting space-
charge effects, simulations show that ions arriving at a common time can be
captured with 100 % efficiency provided the beam radius is less than 2 mm. In
practice, the capture efficiency is observed to be roughly 60 % largely
because of the velocity spread in the extracted ion bunch. Some ways of
reducing the velocity spread to improve capture efficiency are described
above. In this section, we present measurements for estimating the number of
stored ions and capture efficiency.
We measure the following quantities to characterize ion number in the Penning
trap region: (a) $N_{FCA}$, the number of ions striking Faraday cup FCA after
passing through the one-magnet Einzel lens with 11.11 mm inner diameter; (b)
$N_{0V}$, the number of ions passing through the grounded trap and hitting the
TOF detector; and (c) $N_{HV}$, the number of ions hitting the TOF detector
after passing through the trap floated at high voltage $V_{C}$ but with the
endcaps biased at low settings (Table 1). Column 3 of Table 2 gives these
measurements for extracted bunches of Ar13+ ions. The number of ions
determined from the Faraday cup signal $N_{FCA}$ is the largest since the ion
bunch at FCA has not been partially clipped by the 8.00 mm diameter holes in
the FEC and BEC electrodes. The active diameters of the FCA and TOF detectors
are 9.525 mm and 8.00 mm, respectively.
| Ar13+ ion count
---|---
Detector (set-up) | symbol | Measured | Simulated
FCA (before trap) | $N_{FCA}$ | 5275 | 5275
TOF (grounded trap) | $N_{0V}$ | 1435 | 1655
TOF (HV-biased trap) | $N_{HV}$ | 687 | 718
Table 2: Measurement of the number of Ar13+ ions entering the trap region
under three conditions. $N_{FCA}$ is the number of ions measured on a Faraday
cup before the trap. $N_{0V}$ and $N_{HV}$ are the number of ions measured on
the TOF detector when the Penning trap is fully grounded and floated for
capture, respectively.
For comparison, we computed the ion transport for a Gaussian radial
distribution of trajectories entering the one-magnet Einzel lens, passing
through the trap, and terminating at the TOF detector. An initial ion velocity
of $42\,840$ m/s is assigned entirely along the trap axis. Previous
experimentGuise _et al._ (2013) has shown evidence to support a Gaussian
density profile in a tightly-focussed beam. The cross-sectional density is
modeled by a Gaussian function:
$\sigma(r)=\frac{N_{o}}{2\pi
R_{B}^{2}}\exp{\left({-\frac{r^{2}}{2R_{B}^{2}}}\right)}$ (3)
where $N_{o}$ is the total number of ions and $R_{B}$ is the one-sigma beam
radius; the number of ions within radius $r$ is given by the integral
$N=\int_{0}^{r}2\pi r\sigma(r)\,dr$. The simulation results for $N_{o}=5336$
ions and $R_{B}=2.0$ mm are in the last column of Table 2, and agree well with
measurements (column 3) for the grounded trap and for the floated trap.
For Ar13+, Figures 9 and 10 indicate that about 400 ions were detected when
the ion cloud in the Penning trap was ejected to the TOF detector. To
determine the capture efficiency for the Penning trap system, independent of
the ion source and beamline used for production and transport of HCIs, we use
the number of ions entering the Penning trap while at high voltage, $N_{HV}$,
as the normalization. The resulting efficiency is 57(16)% for the Ar13+ ion
capture experiment.
This result agrees with a crude estimate of 61(10)% for capture efficiency
obtained from the simulations of Section III. Here the efficiency is
calculated as the percentage of total ion signal that arrives at the TOF
detector within $\pm$ CTW/2 of the TOF peak; i.e. $t_{peak}\pm 40$ ns in Fig.
6b. For an on-axis beam, this is the maximum fraction of incoming ions that
can be located inside the trap region at one time.
## VI Summary
Highly-charged ions produced by electron impact ionization within an EBIT,
with electron beam energy of a few keV, have been slowed and captured in a
unitary Penning trap deployed on the existing ion-extraction beamline at NIST.
The Penning trap is made very compact (less than 150 cm3 in volume) by a
unitary architecture that embeds two rare-earth permanent magnets within the
electrode structure in order for the trapping apparatus to fit within space
constraints, and to provide easy radial access to the stored ions.
The procedure for capturing energetic ions in a unitary Penning trap is
presented here with experimental results for the isolation of Ar13+ ions, and
is elucidated with simulations of single ion trajectories. Measurements
confirm the importance of energy matching and precise timing of capture to
achieve the lowest energy distribution for the isolated ions. Simulations
provide some insight in designing the set of conical, electrostatic
decelerators near the entrance endcap of the ion trap to aid in maximizing ion
capture and minimizing residual energy. As a demonstration, Ar13+ ions
extracted from the EBIT with $\approx$ 38 keV kinetic energy have been
decelerated and captured with a residual energy spread of $\approx$ 5.5(5) eV,
measured by ejecting the isolated ions to a TOF detector 1 ms after capture.
Without applying any active cooling, this observed energy distribution is
$\approx 60$ times smaller than typically expected for ions inside an EBIT.
Colder ion clouds may be attainable by applying evaporative or sympathetic
cooling techniques. Recent theoretical studies propose various potential
applications for isolated highly-charged ions, including optical frequency
standards Derevianko _et al._ (2012); Dzuba _et al._ (2012), tests of
fundamental symmetries Berengut _et al._ (2010), and measurement of
fundamental constants Jentschura _et al._ (2008).
## VII Acknowledgments
Portions of this work were completed while Nicholas D. Guise held a National
Research Council Associateship Award at NIST. We thank Yuri Ralchenko and
Craig J. Sansonetti for reading this manuscript carefully and providing useful
comments.
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|
arxiv-papers
| 2013-12-02T18:43:43 |
2024-09-04T02:49:54.648135
|
{
"license": "Public Domain",
"authors": "Samuel M Brewer, Nicholas D Guise, Joseph N Tan",
"submitter": "Samuel Brewer",
"url": "https://arxiv.org/abs/1312.0547"
}
|
1312.0581
|
# First-passage time of Brownian motion with dry friction
Yaming Chen [email protected] School of Mathematical Sciences, Queen
Mary University of London, London E1 4NS, United Kingdom Wolfram Just
[email protected] School of Mathematical Sciences, Queen Mary University of
London, London E1 4NS, United Kingdom
(March 18, 2014)
###### Abstract
We provide an analytic solution to the first-passage time (FPT) problem of a
piecewise-smooth stochastic model, namely Brownian motion with dry friction,
using two different but closely related approaches which are based on
eigenfunction decompositions on the one hand and on the backward Kolmogorov
equation on the other. For the simple case containing only dry friction, a
phase transition phenomenon in the spectrum is found which relates to the
position of the exit point, and which affects the tail of the FPT
distribution. For the model containing as well a driving force and viscous
friction the impact of the corresponding stick-slip transition and of the
transition to ballistic exit is evaluated quantitatively. The proposed model
is one of the very few cases where FPT properties are accessible by analytical
means.
###### pacs:
02.50.–r, 05.40.–a, 46.55.+d, 46.65.+g
## I Introduction
The study of first-passage time (FPT) problems has a very long tradition with
its roots in the first half of the last century by the seminal study of
Kramers on chemical kinetics Kramers (1940) (see also Ref. Hänggi et al.
(1990) for an excellent review). While FPT problems originated in physical
chemistry concepts of this type have turned out to be relevant in diverse
disciplines, like mathematical finance Mannella (2004), biological modelling
Tuckwell et al. (2002), complex media Condamin et al. (2007), and others. In
an abstract setting the FPT is defined as the time when a stochastic process,
often governed by a stochastic differential equation (SDE), exits a given
region for the first time. Beyond the classical setup problems of this type
are relevant in different subjects. Renewed interest in FPT problems has been
triggered by studies to characterize large deviation properties, extreme
events, and nonequilibrium processes in many particle systems (see, e.g.,
Refs. Redner (2001); Bray et al. (2013)). FPT problems are normally nontrivial
to solve and a deeper analytical understanding of FPT properties, e.g., the
dependence on parameters of the system is often hampered by the lack of
analytically tractable model systems. There exists a vast literature about
this topic, whereby applications often require the application of numerical
tools. Various simple model systems can be handled by analytical means. Among
those are the pure diffusion process Majumdar (2005), the Brownian motion with
constant drift Kearney and Majumdar (2005), to some extent the Ornstein-
Uhlenbeck process Siegert (1951); Alili et al. (2006) and Bessel processes
Göing-Jaeschke and Yor (2003); DeBlassie and Smits (2007). It is one aim of
the present study to provide analytic insight into a FPT problem which has
some relevance for the phenomenological description of friction processes
often used in the engineering context.
Dynamical systems with discontinuities are frequently used for the
phenomenological modelling in engineering science. The impact of such
discontinuities on dynamical behavior has attracted recently considerable
attention from the general dynamical systems point of view (see, e.g., Ref.
Makarenkov and Lamb (2012)). While the general mathematical theory as well as
the theory of corresponding stochastic models is still incomplete, models of
such a type have been used successfully in the engineering context for
decades. The most prominent examples are dry friction processes, which
themselves are not fully understood from the microscopic point of view (see,
e.g., Ref Vanossi et al. (2013)). Here we want to go beyond the deterministic
dynamical systems setup and intend to study the interrelation between noise
and discontinuities, in particular, with regards to FPT problems. We aim at an
analytic investigation of a simple piecewise-smooth stochastic model. While
some exact results for the propagator of a few simple piecewise-constant or
piecewise-linear SDEs have been known (see for instance Refs. Karatzas and
Shreve (1984); Touchette et al. (2010, 2012); Simpson and Kuske (2012)), exact
results for the FPT problems of piecewise-smooth SDEs are to the best of our
knowledge not available in the literature.
To investigate the effect of discontinuities on a FPT problem we take as a
motivation Brownian motion with dry (also called solid or Coulomb) friction de
Gennes (2005); Hayakawa (2005). We consider for our analytic investigations a
paradigmatic model system, the phenomenological description of a particle
subjected to dry and viscous friction, noise, and a static driving force,
resulting in a piecewise-linear SDE
$\dot{v}(t)=-\mu\sigma(v(t))-\gamma v(t)+b+\sqrt{D}\xi(t).$ (1)
Here $\sigma(v)$, denoting the sign of $v$, represents the dry friction force
with coefficient $\mu>0$, $\gamma\geq 0$ denotes the viscous friction
coefficient, $b$ is a constant biased force and $D>0$ is the strength of the
Gaussian white noise $\xi(t)$ characterized by
$\langle\xi(t)\rangle=0,\qquad\langle\xi(t)\xi(t^{\prime})\rangle=2\delta(t-t^{\prime}).$
(2)
The notation $\langle\cdots\rangle$ stands for the average over all possible
realizations of the noise. Physically, this model describes the velocity of a
solid object of unit mass sliding over an inclined surface with dry and
viscous friction. Since the motion of two solid objects over each other is a
ubiquitous problem in nature, the dry friction model (1) is important to
understand the underlying dynamics of the motion. Mathematically, Eq. (1) is a
piecewise-linear SDE, which allows us to obtain analytic results. For
instance, expressions for the propagator can be derived analytically by using
spectral decomposition methods Touchette et al. (2010) or Laplace transforms
Touchette et al. (2012). In particular the propagator of the pure dry friction
case (also called Brownian motion with two-valued drift, i.e., Eq. (1) with
$\gamma=b=0$) is available in closed analytic form (see, e.g., Refs. Karatzas
and Shreve (1984, 1991); Touchette et al. (2010)). The weak-noise limit of the
model (1) has also been studied in detail by using a path integral approach
Baule et al. (2010, 2011); Chen et al. (2013). As a piecewise-smooth SDE Chen
et al. (2013); Simpson and Kuske (2012), Eq. (1) shows many interesting
features such as stick-slip transitions Baule et al. (2010, 2011) and a noise-
dependent decay of correlation functions Touchette et al. (2010). Some of
these features have also been shown experimentally in Refs. Chaudhury and
Mettu (2008); Goohpattader et al. (2009); Gnoli et al. (2013); Goohpattader
and Chaudhury (2010).
Hence, for such a paradigmatic model it is obvious to have a closer look at
the corresponding FPT problem, which is the purpose of this paper. In our
investigation we consider the exit from a semi-infinite escape interval
$(a,\infty)$. We can confine the analysis to negative exit points, i.e.,
$a<0$. Otherwise, for $a\geq 0$ the discontinuity at $v=0$ will not enter the
FPT problem and we are left with the well known FPT problem of Brownian motion
with constant drift ($\gamma=0$) Kearney and Majumdar (2005) or the Ornstein-
Uhlenbeck process ($\gamma\neq 0$) Wang and Uhlenbeck (1945), respectively.
We address the FPT problem for Eq. (1) by solving a corresponding Fokker-
Planck equation via a spectral decomposition method on the one hand, and by
solving a corresponding backward Kolmogorov equation on the other (see, e.g.,
Refs. Risken (1989); Gardiner (1990)). To keep the presentation self-contained
these two methods will be briefly revisited in Section II. In Section III, we
apply these methods to solve the seemly trivial case without viscous friction
($\gamma=0$) and without bias ($b=0$), i.e., the pure dry friction case. This
simple example already shows a phase transition phenomenon in the spectrum
which is related to the position of the exit point. Thereafter, in Section IV
the distribution of the FPT is derived for the model including viscous
friction and external force. Here the focus will be on the stick-slip
transition and a transition to ballistic exit. Results are summarized in
Section V.
## II Remarks on the FPT problem
The approach to FPT problems is well documented in the literature, and
suitable expositions can be found in standard textbooks, e.g., Ref. Risken
(1989). Here we just summarize the essential ideas not only for the
convenience of the reader but also to address the few technical issues related
to piecewise-smooth drifts. We will focus on the Langevin equation
$\dot{v}=-\Phi^{\prime}(v)+\xi(t),$ (3)
where the potential $\Phi(v)$ is smooth everywhere apart from $v=0$ and its
derivative may have a discontinuity. In particular we will compare and
contrast two different but closely related approaches based on eigenfunction
decompositions on the one hand and on the backward Kolmogorov equation on the
other.
### II.1 Spectral decomposition
If one considers the stochastic dynamics according to Eq. (3) on the interval
$(a,\infty)$ it is well known that the corresponding distribution of the FPT
for orbits starting at $v(0)=v_{0}\in(a,\infty)$ is given by (see Ref. Risken
(1989))
$f(T,v_{0})=-\frac{\partial}{\partial T}\int_{a}^{\infty}p(v,T|v_{0},0)dv,$
(4)
where the propagator $p(v,t|v_{0},0)$ satisfies the corresponding Fokker-
Planck equation
$\frac{\partial}{\partial t}p(v,t|v_{0},0)=\frac{\partial}{\partial
v}[\Phi^{\prime}(v)p(v,t|v_{0},0)]+\frac{\partial^{2}}{\partial
v^{2}}p(v,t|v_{0},0)$ (5)
with an initial condition
$p(v,0|v_{0},0)=\delta(v-v_{0}),$ (6)
an absorbing boundary condition at the left interval endpoint
$p(a,t|v_{0},0)=0,$ (7)
and a reflecting boundary, i.e., a vanishing probability current at infinity.
To get the solution $p(v,t|v_{0},0)$ we follow a spectral decomposition method
for piecewise-smooth systems used, e.g., in Ref. Touchette et al. (2010), and
first solve the associated eigenvalue problem of Eqs. (5)–(7)
$-\Lambda
u_{\Lambda}(v)=[\Phi^{\prime}(v)u_{\Lambda}(v)]^{\prime}+u_{\Lambda}^{\prime\prime}(v)$
(8)
with the (formal) boundary conditions
$u_{\Lambda}(a)=0,\qquad\left.[\Phi^{\prime}(v)u_{\Lambda}(v)+u^{\prime}_{\Lambda}(v)]\right|_{v\rightarrow\infty}=0.$
(9)
Since we are here concerned with the piecewise-smooth potential $\Phi(v)$, we
have to solve Eq. (8) on the two domains $v>0$ and $v<0$, respectively, and
have to apply suitable matching conditions, i.e.,
$u_{\Lambda}(0-)=u_{\Lambda}(0+)$ (10)
coming from the continuity of the eigenfunction and
$\Phi^{\prime}(0-)u_{\Lambda}(0-)+u^{\prime}_{\Lambda}(0-)=\Phi^{\prime}(0+)u_{\Lambda}(0+)+u^{\prime}_{\Lambda}(0+)$
(11)
from the continuity of the probability current in Eq. (5). As in the standard
case of Fokker-Planck equations with reflecting boundary conditions the
eigenfunctions of the Fokker-Planck operator and the eigenfunctions of the
formally adjoint problem are related to each other by an exponential factor
containing the potential $\Phi(v)$. Furthermore, both types of eigenfunctions
are mutually orthogonal sets and thus result in the orthogonality relations
$\displaystyle\int_{a}^{\infty}u_{\Lambda_{m}}(v)u_{\Lambda_{n}}(v)e^{\Phi(v)}dv=Z_{\Lambda_{n}}\delta_{mn},$
(12)
$\displaystyle\int_{a}^{\infty}u_{\Lambda}(v)u_{\Lambda^{\prime}}(v)e^{\Phi(v)}dv=Z_{\Lambda}\delta(\Lambda-\Lambda^{\prime}),$
(13)
depending on whether the eigenvalue is contained in the discrete or the
continuous part of the spectrum. These conditions implicitly take the
reflecting boundary at infinity into account. Furthermore, it is worth
mentioning that the reasoning for Fokker-Planck equations with reflecting
boundary conditions can be also applied to map the eigenvalue problem to a
formally Hermitian positive operator (see Refs. Risken (1989); Gardiner
(1990)). Thus, all eigenvalues are positive, in particular they are real.
Finally, the solution of Eq. (5) is given by (see, e.g., Ref. Horsthemke and
Lefever (1984) for an accessible account on the completeness of the spectrum)
$\displaystyle p(v,t|v_{0},0)$ $\displaystyle=$ $\displaystyle
e^{\Phi(v_{0})}\bigg{(}\sum_{n}u_{\Lambda_{n}}(v_{0})u_{\Lambda_{n}}(v)e^{-\Lambda_{n}t}/Z_{\Lambda_{n}}$
(14) $\displaystyle+\int u_{\Lambda}(v_{0})u_{\Lambda}(v)e^{-\Lambda
t}/Z_{\Lambda}d\Lambda\bigg{)},$
where the sum is taken over the discrete eigenvalues and the integral is taken
over the continuous part of the spectrum. The normalization factors
$Z_{\Lambda_{n}}$ and $Z_{\Lambda}$ are determined by Eqs. (12) and (13),
respectively.
### II.2 Backward Kolmogorov equation
The propagator $p(v,t|v_{0},0)$, which determines the FPT distribution (4),
obeys the backward Kolmogorov equation Gardiner (1990) with absorbing boundary
condition at $v_{0}=a$ and reflecting boundary condition at infinity. Hence
the FPT distribution obeys the backward Kolmogorov equation as well, i.e.,
$\frac{\partial}{\partial
T}f(T,v_{0})=-\Phi^{\prime}(v_{0})\frac{\partial}{\partial
v_{0}}f(T,v_{0})+\frac{\partial^{2}}{\partial v_{0}^{2}}f(T,v_{0})$ (15)
with initial condition
$f(0,v_{0})=0\quad\mbox{for }v_{0}>a.$ (16)
The two boundary conditions, i.e., Eq. (7) and vanishing probability current
at infinity, translate into
$f(T,v_{0}\rightarrow a)=\delta(T)$ (17)
at the left interval endpoint, and into
$\frac{\partial}{\partial v_{0}}f(T,v_{0}\rightarrow\infty)=0$ (18)
at infinity. If we use the Laplace transform
$\tilde{f}(s,v_{0})=\int_{0}^{\infty}f(T,v_{0})e^{-sT}dT,$ (19)
the partial differential equation (15) turns into the ordinary boundary value
problem
$\frac{\partial^{2}}{\partial
v_{0}^{2}}\tilde{f}(s,v_{0})-\Phi^{\prime}(v_{0})\frac{\partial}{\partial
v_{0}}\tilde{f}(s,v_{0})-s\tilde{f}(s,v_{0})=0,$ (20)
where Eq. (17) obviously results in
$\tilde{f}(s,v_{0}\rightarrow a)=1.$ (21)
As for the other boundary condition we observe that the Laplace transform (19)
converges uniformly in $v_{0}$ for $s$ being in the right half plane, as the
integral converges uniformly at $s=0$. Hence Eq. (18) yields
$\frac{\partial}{\partial
v_{0}}\tilde{f}(s,v_{0}\rightarrow\infty)=0\quad\mbox{for }\mbox{Re}(s)>0.$
(22)
Intuitively the two boundary conditions (21) and (22) take care of the fact
that on the one hand the FPT is $\delta$-distributed in the limit
$v_{0}\rightarrow a$ and that on the other hand the particle cannot exit the
given region $(a,\infty)$ at infinity. In addition, Eq. (20) should be solved
for $v_{0}>0$ and $v_{0}<0$ separately with matching conditions at $v_{0}=0$,
i.e.,
$\tilde{f}(s,0-)=\tilde{f}(s,0+),\qquad\frac{\partial}{\partial
v_{0}}\tilde{f}(s,0-)=\frac{\partial}{\partial v_{0}}\tilde{f}(s,0+),$ (23)
where the first condition follows from the solution $\tilde{f}(s,v_{0})$ being
continuous at $v_{0}=0$ and the second one is derived by integrating Eq. (20)
across $v_{0}=0$.
The approach via the backward Kolmogorov equation enables us to obtain the
Laplace transform of the FPT distribution in closed analytic form. Even though
it may not be possible to perform the inverse transform by analytical means to
compute $f(T,v_{0})$, by taking derivatives the moments of the FPT, $\langle
T^{n}\rangle$, are then easily evaluated as
$\langle T^{n}\rangle=(-1)^{n}\left.\frac{\partial^{n}}{\partial
s^{n}}\tilde{f}(s,v_{0})\right|_{s=0}\quad\mbox{for }n=1,2,3,\dots$ (24)
## III The inviscid case
Let us first consider the seemingly trivial case without viscous friction
($\gamma=0$) and without any external bias ($b=0$), i.e., a particle which is
only exposed to dry friction with a piecewise-constant drift term. We consider
this simplest case as it already shows, somehow counterintuitively, the main
phase transition behavior in the FPT distribution. As a by-product we can also
illustrate all the analytical tools in a very transparent setup.
If we consider Eq. (1) for $\gamma=b=0$ we can specialize to the choice
$\mu=D=1$ without loss of generality. Other nonvanishing values are covered by
the appropriate rescaling
$x=\mu v/D,\qquad\tau=\mu^{2}t/D.$ (25)
Hence, in this case Eq. (1) can be written in the form (3) with
$\Phi(v)=|v|.$ (26)
The corresponding eigenvalue problem (8) consists of a discrete eigenvalue for
$\Lambda<1/4$ and a continuous spectrum for $\Lambda>1/4$ (cf. also Ref. Wong
(1964)). The details of the derivation are summarized in Appendix A for the
convenience of the reader.
For $\Lambda<1/4$, the sole eigenfunction is given by [see Eqs. (59) and (61)]
$u_{\Lambda}(v)=\left\\{\begin{array}[]{ll}2\lambda
e^{-(\lambda+1/2)v}&\quad\mbox{for }v>0\\\ (2\lambda-1)e^{-(\lambda-1/2)v}&\\\
\qquad+e^{(\lambda+1/2)v}&\quad\mbox{for }a<v<0,\end{array}\right.$ (27)
where $\lambda=\sqrt{1/4-\Lambda}>0$. The discrete eigenvalue is determined by
the absorbing boundary condition (9), which results in
$e^{2a\lambda}=1-2\lambda\quad\mbox{for }\lambda>0.$ (28)
It is obvious that Eq. (28) has no real solution for $\lambda$ in the region
$[1/2,\infty)$. Hence we have $\Lambda>0$ and can confine ourselves to search
the solution of Eq. (28) for $\lambda$ in the region $(0,1/2)$. Since
$\exp(2a\lambda)$ is convex as a function of $\lambda$ and the right hand side
of Eq. (28) is a straight line, it is easy to verify [see Fig. 1(a)] that Eq.
(28) has no real solution in $(0,1/2)$ when $a\geq-1$ and admits a unique
solution, denoted by $\lambda_{0}$, when $a<-1$. The unique eigenvalue
$\Lambda_{0}=1/4-\lambda_{0}^{2}$ can be obtained numerically from Eq. (28),
being a monotonic function of the parameter $a$ [see Fig. 1(b)]. As an aside
we remark that the solution of Eq. (28) can be expressed in terms of the main
branch of the Lambert W function Corless et al. (1996) by
$\lambda_{0}=1/2-W[a\exp(a)]/(2a)$. The other quantities which enter the FPT
distribution are easily computed. For the normalization factor, Eqs. (12) and
(27) yield
$\displaystyle Z_{\Lambda_{0}}$ $\displaystyle=$
$\displaystyle\int_{a}^{\infty}u_{\Lambda_{0}}^{2}(v)e^{|v|}dv$ (29)
$\displaystyle=$
$\displaystyle\left[-e^{2a\lambda_{0}}+(1/2-\lambda_{0})^{2}e^{-2a\lambda_{0}}\right]/\lambda_{0}$
$\displaystyle-4\lambda_{0}+2(1+a).$
The integral of the eigenfunction which enters the distribution [see Eqs. (4)
and (14)] is evaluated as
$\int_{a}^{\infty}u_{\Lambda_{0}}(v)dv=2e^{(1/2-\lambda_{0})a}-e^{(1/2+\lambda_{0})a}/(1/2+\lambda_{0}).$
(30)
Figure 1: (Color online) (a) Graphical solution of Eq. (28) in terms of the
convex function $\exp(2a\lambda)$ and the straight line $1-2\lambda$. As
examples, $a=-0.5$ and $a=-2$ are used here to illustrate the shapes of the
function $\exp(2a\lambda)$ for the two phases $a>-1$ and $a<-1$, respectively.
(b) The discrete eigenvalue $\Lambda_{0}$ for $a<-1$. When $a=-1$, the
discrete eigenvalue merges with the continuous spectrum $\Lambda\geq 1/4$.
For $\Lambda>1/4$, the eigenfunction can be obtained explicitly as [see Eqs.
(59) and (63)]
$\displaystyle u_{\Lambda}(v)=\left\\{\begin{array}[]{lll}\sin(\kappa
a)\sin(\kappa v)e^{-v/2}&&\\\
\qquad+\kappa\sin[\kappa(v-a)]e^{-v/2}&&\mbox{for }v>0\\\
\kappa\sin[\kappa(v-a)]e^{v/2}&&\mbox{for }a<v<0,\end{array}\right.$ (34)
where $\kappa=\sqrt{\Lambda-1/4}>0$. Moreover, the normalization factor in Eq.
(13) is given by [see Eq. (65)]
$Z_{\Lambda}=\pi[\kappa^{2}+\kappa\sin(2a\kappa)+\sin^{2}(a\kappa)]/2,$ (35)
and the integral over the eigenfunction which enters Eq. (14) is evaluated as
$\int_{a}^{\infty}u_{\Lambda}(v)dv=\kappa^{2}e^{a/2}/(1/4+\kappa^{2}).$ (36)
Thus, the spectrum consists of a continuous part $\Lambda>1/4$ and an
additional discrete lowest eigenvalue $\Lambda_{0}$ for $a<-1$ which merges
with the continuous spectrum at $a=-1$ [see Fig. 1(b)]. Hence we expect
qualitative changes to appear at such a critical value.
By using Eqs. (4) and (14) we obtain the distribution of the FPT as follows
$\displaystyle f(T,v_{0})$ $\displaystyle=$
$\displaystyle\chi_{\\{a\leq-1\\}}\Lambda_{0}u_{\Lambda_{0}}(v_{0})e^{|v_{0}|-\Lambda_{0}T}\int_{a}^{\infty}u_{\Lambda_{0}}(v)dv/Z_{\Lambda_{0}}$
(37)
$\displaystyle+\frac{2}{\pi}e^{|v_{0}|-T/4+a/2}\int_{0}^{\infty}\kappa^{2}u_{\Lambda}(v_{0})e^{-\kappa^{2}T}/[\kappa^{2}+\kappa\sin(2a\kappa)+\sin^{2}(a\kappa)]d\kappa,$
where $\chi_{\\{a\leq-1\\}}$ denotes the indicator function of the set
$\\{a\leq-1\\}$, $u_{\Lambda_{0}}(v_{0})$ the eigenfunction of the discrete
eigenvalue (27), and $u_{\Lambda}(v_{0})$ the eigenfunction of the continuous
part of the spectrum (34). The normalizations $Z_{\Lambda_{0}}$ and
$\int_{a}^{\infty}u_{\Lambda_{0}}(v)dv$ are given in Eqs. (29) and (30),
respectively. In the trivial case $a=0$ the discontinuity does not enter the
FPT problem and the pure dry friction model is equivalent to that of the one-
dimensional Brownian motion with constant drift Kearney and Majumdar (2005).
In such a case, the first term in Eq. (37) does not contribute and the
integral can be evaluated in closed analytic form to yield
$\displaystyle\\!\\!\\!f(T,v_{0})$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}e^{v_{0}/2-T/4}\int_{0}^{\infty}\kappa\sin(\kappa
v_{0})e^{-\kappa^{2}T}d\kappa$ (38) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi}}\frac{v_{0}}{T^{3/2}}e^{-(v_{0}-T)^{2}/(4T)}\quad\mbox{for
}v_{0}>0,$
a result which is consistent with Refs. Kearney and Majumdar (2005); Majumdar
and Comtet (2002). Apart from this trivial case it seems to be difficult to
obtain a closed analytic expression from the representation (37).
Certainly the FPT distribution changes qualitatively at $a=-1$ when the
contribution in Eq. (37) coming from the discrete eigenvalue comes into play.
That can be demonstrated by focussing on the tail behavior of the distribution
which in itself is of interest when rare events are of interest. First of all
it is obvious that for $a<-1$ the first term in Eq. (37) determines the decay
which is plainly exponential $\exp(-\Lambda_{0}T)$. For $a\geq-1$, the first
term in Eq. (37) vanishes, as the coefficient of the characteristic function
vanishes at $a=-1$, and the tail is determined by evaluating the Laplace-type
integral in the second term. If we have a closer look at the kernel entering
the Laplace-type integral
$\rho(\kappa,a)=\kappa^{2}u_{\Lambda}(v_{0})/[\kappa^{2}+\kappa\sin(2a\kappa)+\sin^{2}(a\kappa)],$
(39)
it is evident that for $a>-1$ the properties
$\displaystyle\lim_{\kappa\rightarrow 0}\rho(\kappa,a)=0,$ (40)
$\displaystyle\lim_{\kappa\rightarrow 0}\partial_{\kappa}\rho(\kappa,a)=0,$
(41) $\displaystyle\lim_{\kappa\rightarrow
0}\partial_{\kappa}^{2}\rho(\kappa,a)\neq 0$ (42)
hold (see Fig. 2). Hence it is straightforward to evaluate the Laplace-type
integral to obtain a decay as $T^{-3/2}\exp(-T/4)$ for $a>-1$. For the
critical case $a=-1$ the situation differs as
$\lim_{\kappa\rightarrow
0}\rho(\kappa,-1)=\left\\{\begin{array}[]{lll}1&&\mbox{for }v_{0}>0\\\
1+v_{0}&&\mbox{for }-1<v_{0}<0\end{array}\right.$ (43)
holds. Here the Laplace method yields $T^{-1/2}\exp(-T/4)$ for $a=-1$. To
summarize, in the long time limit we have
$f(T,v_{0})\sim\left\\{\begin{array}[]{lll}e^{-\Lambda_{0}T}&&\mbox{for
}a<-1\\\ T^{-1/2}e^{-T/4}&&\mbox{for }a=-1\\\ T^{-3/2}e^{-T/4}&&\mbox{for
}a>-1.\end{array}\right.$ (44)
Figure 2: (Color online) The kernel $\rho(\kappa,a)$ [see Eq. (39)] appearing
in the spectral decomposition (37) for two different values of $v_{0}$ and
various values of the exit point $a$. Here $u_{\Lambda}(v_{0})$ is given by
Eq. (34).
To obtain closed analytic expressions for the FPT distributions we
alternatively can resort to the Laplace transform of the backward Kolmogorov
equation. In this pure dry friction case Eq. (20) reads [see Eq. (26)]
$\frac{\partial^{2}}{\partial
v_{0}^{2}}\tilde{f}(s,v_{0})-\sigma(v_{0})\frac{\partial}{\partial
v_{0}}\tilde{f}(s,v_{0})-s\tilde{f}(s,v_{0})=0,$ (45)
where the solution has to satisfy the boundary conditions (21) and (22) as
well as the matching condition (23) at $v_{0}=0$. It is in fact rather
straightforward to compute the solution of this linear second order problem
and we end up with
$\tilde{f}(s,v_{0})=\left\\{\begin{array}[]{lll}\exp\big{\\{}[\sqrt{1+4s}(a-v_{0})+a+v_{0}]/2\big{\\}}\sqrt{1+4s}/\theta(s,a)&&\mbox{for
}v_{0}>0\\\
\exp[(1+\sqrt{1+4s})(a-v_{0})/2]\theta(s,v_{0})/\theta(s,a)&&\mbox{for
}a<v_{0}<0,\end{array}\right.$ (46)
where we have introduced the abbreviation
$\theta(s,a)=\exp\left(a\sqrt{1+4s}\right)+\sqrt{1+4s}-1$ (47)
for the contribution appearing mainly in the denominator. Clearly Eq. (46) has
a branch cut at $s=-1/4$ which relates with the continuous spectrum found
previously. In addition, the condition $\theta(s,a)=0$, which is equivalent to
Eq. (28), determines a pole for $a<-1$. Hence, when $a<-1$ the simple pole
dominates the FPT distribution in the tail to yield an exponential decay
Whitehouse et al. (2013). Overall, the analytical structure of the Laplace
transform reflects the spectral properties mentioned previously.
The inverse Laplace transform of Eq. (46) does not seem to be available in
closed analytic form. As before, only the trivial case $a=0$ can be handled
with ease which then results in Eq. (38). For the other cases we resort to a
so-called Talbot method Talbot (1979); Abate and Valkó (2004); Abate and Whitt
(2006) to compute the FPT distribution in the time domain 111A Mathematica
implementation of this method is available at
http://libray.wolfram.com/infocenter/MathSource/5026/. Fig. 3 shows that the
expressions (37) and (46) give identical results, as expected. In addition,
evaluation of those expressions confirm as well the asymptotic decay given by
Eq. (44) (see Fig. 4).
Figure 3: (Color online) The FPT distribution of the pure dry friction case
[see Eq. (26)] for two values of initial velocity, $v_{0}=0.2$ and
$v_{0}=-0.2$, and different escape ranges. Lines correspond to a numerical
inversion of Eq. (46), and points to the evaluation of Eq. (37). Figure 4:
(Color online) Comparison of the FPT distribution obtained from Eq. (37)
(solid) with the asymptotic result (44) (dashed) for the initial velocity
$v_{0}=-0.2$ and different escape ranges. Data are plotted on a doubly
logarithmic scale to uncover the power law corrections to the leading
exponential behavior.
The closed form of the characteristic function (46) allows us to obtain easily
the moments of the FPT via Eq. (24). For the first moment, i.e., for the mean
first-passage time (MFPT) we have
$\langle T\rangle=\left\\{\begin{array}[]{lll}2e^{-a}+a+v_{0}-2&&\mbox{for
}v_{0}>0\\\ 2e^{-a}+a-v_{0}-2e^{-v_{0}}&&\mbox{for
}a<v_{0}<0.\end{array}\right.$ (48)
The first moment clearly displays a transition when the initial condition
changes sign (see also Fig. 5). For $v_{0}>0$ the MFPT depends linearly on the
initial velocity. No particular feature is visible at the transition at
$a=-1$, as a change in the tail behavior has no impact on the low order
moments of the distribution.
Figure 5: (Color online) The MFPT $\langle T\rangle$ for different escape
ranges. Lines correspond to the analytic result (48), and points to a
numerical evaluation of the first moment by using the spectral representation
(37).
## IV Biased Brownian motion with dry and viscous friction
In this section, we consider the full model (1) and set $\gamma=D=1$ without
loss of generality. Other cases can be covered by using the appropriate
rescaling
$x=\left(\gamma/D\right)^{1/2}v,\qquad\tau=\gamma t.$ (49)
Thus the model (1) can be written as Eq. (3) with
$\Phi(v)=(|v|+\mu)^{2}/2-bv.$ (50)
The corresponding eigenvalue problem (8) with potential (50) can be solved by
using parabolic cylinder functions Buchholz (1969), which are denoted by
$D_{\nu}(z)$. For the convenience of the reader we summarize the details of
the derivation in Appendix B.
The eigenvalues are discrete and determined by the characteristic equation
[see Eq. (78)]
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\bar{\theta}(\Lambda,a,\mu,b)$
(51) $\displaystyle=$
$\displaystyle\Gamma(1-\Lambda)\\{[D_{\Lambda}(\mu+b)D_{\Lambda-1}(\mu-b)$
$\displaystyle+D_{\Lambda}(\mu-b)D_{\Lambda-1}(\mu+b)]D_{\Lambda}(a-\mu-b)$
$\displaystyle-[D_{\Lambda}(-\mu-b)D_{\Lambda-1}(\mu-b)$ $\displaystyle-
D_{\Lambda}(\mu-b)D_{\Lambda-1}(-\mu-b)]D_{\Lambda}(-a+\mu+b)\\}$
$\displaystyle=$ $\displaystyle 0.$
For $\mu=0$, the model considered here reduces to the Ornstein-Uhlenbeck
process and the characteristic equation (51) simply reads
$D_{\Lambda}(a-b)=0,$ (52)
which agrees with the standard result of the Ornstein-Uhlenbeck process (see
for instance Ref. Siegert (1951)). It is furthermore unexpected that the
characteristic equation (52) coincides with those of the odd part of the
spectrum for a Fokker-Planck equation subjected to dry and viscous friction
only Touchette et al. (2010). While odd eigenfunctions vanish at the origin
and thus fulfil some kind of absorbing boundary condition it is not
intuitively obvious why the argument of the parabolic cylinder function is in
one case the absorbing boundary and in the other case the dry friction itself.
To link the current result with the previous section let us first consider the
special case without bias ($b=0$). Intuitively, we expect that if the dry
friction term dominates the viscous friction force then the particle will
behave like the one subjected to dry friction only. Hence the spectrum
obtained from Eq. (51) for large values of $\mu$ should resemble the spectrum
described in the previous section [see, e.g., Fig. 1(b)]. In particular it
means that a large gap should develop between the lowest eigenvalue and a
quasicontinuous part for small negative values of $a$. For comparison of the
models with and without viscous friction [see Eqs. (26) and (50)] we observe
that a rescaling of the velocity by $\mu$ and of time by $\mu^{2}$ transforms
the stochastic differential equation with dry and viscous friction to the
model with dry friction and a small viscous part of $O(1/\mu^{2})$ which
vanishes in the limit $\mu\rightarrow\infty$. Thus, to compare the eigenvalues
obtained from the characteristic equation (51) with the spectrum computed in
the previous section we rescale velocities by $\mu$ and eigenvalues by
$1/\mu^{2}$. Then, indeed numerical evaluation of Eq. (51) confirms what one
expects intuitively (see Fig. 6). The eigenvalues as a function of the exit
position $a$ develop a gap if $\mu$ is sufficiently large, even though the
transition is smoothened by the finite viscous friction. If dry and viscous
friction become comparable, i.e., if $\mu$ becomes too small such a feature is
going to disappear.
Figure 6: (Color online) The first five rescaled eigenvalues
$\Lambda_{n}/\mu^{2}$ for the model without bias ($b=0$) as a function of the
rescaled exit point $\mu a$ for two different values of $\mu$, according to
Eq. (51). The dashed line in (b) depicts the discrete branch of the model with
dry friction only [see Fig. 1(b)].
If we impose a force on the particle the finite bias will cause a stick-slip
transition at $|b|=\mu$ where the minimum of the potential (50), i.e., the
deterministic stationary state, changes from vanishing to finite velocity. The
characteristics of such a transition are reflected by the eigenvalue spectrum
as well (see Fig. 7). For small value of the bias, $|b|<\mu$, a case which we
will call for brevity the dry phase, a substantial spectral gap appears
between the lowest and the subleading eigenvalues. This gap shrinks when the
transition at $|b|=\mu$ is approached. The spectral gap corresponds to a fast
decay of velocity correlations in the system with small bias (see Ref.
Touchette et al. (2010)). If the bias is sufficiently negative, i.e.,
$b<-\mu$, a case which we will call the wet phase, the potential (50) develops
a quadratic minimum and the spectrum resembles that of the Ornstein-Uhlenbeck
process. As with regards to the exit time problem a second transition will
occur when on decreasing the force further the quadratic minimum of the
potential moves beyond the exit point at $b=-\mu+a$. Then the exit from the
region occurs in a purely ballistic way which decreases the exit time
considerably. Hence that transition is related with an increase of the lowest
eigenvalue (see Fig. 7). These two transitions are clearly visible if the
diffusion is sufficiently small, i.e., $\mu$ sufficiently large. But they
become obscured by noise for large diffusion, i.e., if $\mu$ becomes too
small. Finally, in the dry phase the spectrum shows avoided level crossings
for small bias, which are reminiscent of spectral properties in nonintegrable
dynamical systems.
Figure 7: (Color online) The first five eigenvalues as a function of the bias
$b$ for exit point at $a=-5$ and dry friction coefficient $\mu=5$, obtained
from Eq. (51). The stick-slip transition, i.e., the narrowing of the spectral
gap at $b=\pm\mu=\pm 5$ and the transition to a ballistic exit at
$b=-\mu+a=-10$ are clearly visible.
As we have access to the entire spectrum we can derive from Eqs. (4) and (14)
the FPT distribution
$f(T,v_{0})=e^{\Phi(v_{0})}\sum_{\Lambda}\Lambda u_{\Lambda}(v_{0})e^{-\Lambda
T}\int_{a}^{\infty}u_{\Lambda}(v)dv/Z_{\Lambda},$ (53)
where the sum is taken over all the discrete eigenvalues [see Eq. (51)],
$u_{\Lambda}(v_{0})$ refers to the eigenfunction given by Eq. (69), the
integral $\int_{a}^{\infty}u_{\Lambda}(v)dv$ is stated in Eq. (79) and the
normalization factor $Z_{\Lambda}$ is given by Eq. (82). It is thus
straightforward to evaluate the shape of the distribution function (see, e.g.,
Fig. 8). While it seems to be difficult to obtain a closed analytic expression
for this distribution we may pursue the approach used in the previous section
and focus on the Laplace transform. In fact, Eq. (20) tells us that [see Eq.
(50)]
$\frac{\partial^{2}}{\partial
v_{0}^{2}}\tilde{f}(s,v_{0})-(v_{0}+\mu\sigma(v_{0})-b)\frac{\partial}{\partial
v_{0}}\tilde{f}(s,v_{0})-s\tilde{f}(s,v_{0})=0,$ (54)
where the Laplace transform has to obey the boundary conditions (21) and (22)
as well as the matching condition (23). Solving Eq. (54) is rather
straightforward, as the boundary value problem for the Laplace transform is
the formally adjoint of the eigenvalue problem [see Eqs. (67) and (68)]. It is
well known and easy to confirm that the solution of the adjoint problem can be
written in terms of the analytic expression for the eigenfunction (see Ref.
Gardiner (1990)) if we multiply the eigenfunction with an exponential factor
$\exp[\Phi(v_{0})]$ containing the potential (50). Thus, the solution of Eq.
(54) can be written down directly as
$\tilde{f}(s,v_{0})=\frac{e^{(a-\mu-b)^{2}/4-\Phi(a)}}{\bar{\theta}(-s,a,\mu,b)}u_{-s}(v_{0})e^{\Phi(v_{0})}\quad\mbox{for
}v_{0}>a,$ (55)
where $u_{-s}(v_{0})$ refers to Eq. (69), and the additional normalization
factor containing the characteristic equation (51) is obtained by using the
boundary condition (21). Obviously the poles of the Laplace transform are
determined by the characteristic equation (51) and thus reflect the spectral
structure discussed previously. In addition, the smallest simple pole
determines the exponential tail of $f(T,v_{0})$.
As stated before, for $\mu=0$ the model investigated here corresponds to the
exit time problem of the Ornstein-Uhlenbeck process, which has been paid much
attention to in the past (see for instance Refs. Alili et al. (2006); Wang and
Uhlenbeck (1945); Siegert (1951); Darling and Siegert (1953); Blake and
Lindsey (1973); Leblanc and Scaillet (1998)). In this case Eq. (55) simplifies
considerably and reads [see Eqs. (51) and (69)]
$\tilde{f}(s,v_{0})=\frac{e^{(v_{0}-b)^{2}/4}D_{-s}(v_{0}-b)}{e^{(a-b)^{2}/4}D_{-s}(a-b)}\quad\mbox{for
}v_{0}>a,$ (56)
which is consistent with the standard result stated, for instance, in Ref.
Siegert (1951).
The analytic expressions Eqs. (53) or (55) now allow us to discuss the
dependence of the exit time problem on the initial velocity $v_{0}$. Both
expressions, if properly evaluated, give of course identical results (see Fig.
8). Here we are going to pay particular attention to the impact of the
discontinuity appearing at the origin. Depending on the sign of the initial
velocity the particle has to pass the discontinuity at $v=0$ before exiting at
$a<0$. Thus, a qualitative change of the FPT distribution is expected
depending on the sign of $v_{0}$. In fact, such a feature is already visible
from Eq. (55), as different analytical branches of the eigenfunction (69) come
into play if $v_{0}$ changes sign. The dependence on $v_{0}$ is still smooth
but not differentiable of higher order. The FPT distributions for small
positive and small negative values of $v_{0}$ look distinctively different, as
shown in Fig. 8. For $v_{0}>0$ the particle has to pass through $v=0$ before
exiting and thus sticks at the origin at least if the bias is small, causing
larger exit times. Thus, the distribution overall is shifted to the right,
compared to the case $v_{0}<0$.
Figure 8: (Color online) The distribution of the FPT for $\mu=1$, $a=-1$, two
values of initial velocity, $v_{0}=0.2$ (solid) and $v_{0}=-0.2$ (dashed), and
different values of the bias $b$. Lines correspond to a numerical inversion of
the Laplace transform (55), and points to the evaluation of Eq. (53) taking
the first twenty modes into account. A larger number of modes would be
required to reproduce the exact result for very small values of $t$.
The just mentioned phenomenon can be better illustrated by looking at the MFPT
which can be obtained in closed analytic form via Eqs. (24) and (55) even for
very small values of the diffusion, i.e., for large values of $\mu$. While the
analytic expression can be written down we just refer to the graphical
evaluation of the expressions (see Fig. 9). For small bias, $|b|<\mu$, i.e.,
in the dry phase there is a possibility that the particle sticks at the origin
which will impact on the MFPT. If the particle starts at $v_{0}<0$ it has less
chance to stick at the origin when $v_{0}$ becomes smaller, and the change of
the MFPT with regards to $v_{0}$ becomes fairly large. On the contrary, if we
choose a positive initial velocity $v_{0}>0$, the particle has always to pass
$v=0$ before exiting at $a<0$. Thus no huge variation of the MFPT with $v_{0}$
is detected. If we decrease the bias and enter the wet phase $b<-\mu$, the
particle does not stick any more and the just mentioned feature almost
disappears. This scenario is much more pronounced if we look at the first
derivative $\partial_{v_{0}}\langle T\rangle$ [see Fig. 9(b)]. Like the
distribution function itself the MFPT is continuously differentiable, but
loses analyticity due to the discontinuity at $v_{0}=0$. A kink can be seen
clearly at the origin for small bias $|b|<\mu$, which separates the two
different regimes of the MFPT for negative and positive initial velocities.
This feature is suppressed if we decrease the bias and finally enter the wet
phase with $b<-\mu$ where the kink almost disappears.
Figure 9: (Color online) (a) MFPT $\langle T\rangle$ as a function of the
initial value $v_{0}$ for $\mu=1$, exit condition $a=-1$, and different values
of the bias, covering the dry phase $|b|<\mu$ as well as the wet phase
$b<-\mu$. (b) First derivative of the MFPT with respect to the initial value
for the same data.
## V Conclusion
In this paper we have studied the FPT problem of Brownian motion with dry and
viscous friction. There has been renewed interest in such exit time problems
from two different points of view. On the one hand prediction and forecasting
of extreme events and the related large deviation theory are closely related
to exit time problems. On the other hand, the particular setup studied here is
a special example of a piecewise-smooth dynamical system. While such systems
are extensively used in engineering sciences only recently the attempt has
been made to put this subject in the systematic framework of dynamical
system’s theory.
As a case study we have considered here a simple piecewise-linear model which
can be largely solved by analytical means. In physicists terms we have
considered a particle subjected to dry and viscous friction, to noise, and to
an external force. This is one of the few models for which the FPT
distribution can be obtained analytically either by solving the Fokker-Planck
equation via a spectral decomposition method or by solving the backward
Kolmogorov equation in the Laplace space. While the first method gives more
insight into the underlying dynamical mechanisms through the additional
spectral information, the second is able to deliver closed analytic
expressions for the MFPT.
The simplest case, where only dry friction acts on the particle, already shows
one of the main features, a phase transition phenomenon in the spectrum which
is related to the position of the exit point. A unique discrete eigenvalue
links up with the continuous part of the spectrum at a critical size of the
exit region. Such a transition translates into different asymptotic properties
of the FPT distribution. The signature of this transition persists if the
viscous friction and the external bias are taken into account, even though the
transition is blurred by the finite diffusion. In this full model two new
features occur, i.e., a stick-slip transition and a transition to a ballistic
exit of the particle. All three transitions are clearly visible in the
discrete spectrum of the full model, especially at low diffusion, signalling
the different rates of asymptotic decay of the FPT distribution. As an aside,
the analysis of this model covers as special cases the Ornstein-Uhlenbeck
process on the one hand, and the previously discussed dry friction case on the
other.
The availability of analytical results for higher dimensional stochastic
models is rather limited, contrary to the one-variable case. Even the
computation of the stationary distribution is often a challenge if detailed
balance is violated, and dynamical quantities, like correlations or exit
probabilities are certainly out of reach. Having said that, models with more
than one degree of freedom are prevalent in applications and any progress on
the analytical side is certainly welcomed, even if simple model systems are
considered. In that sense the inclusion of inertia in the model discussed here
is a rewarding goal, which could lead to predictions that are experimentally
relevant and could trigger corresponding experimental investigations. Progress
in that direction seems possible even though the analysis may not be entirely
straightforward.
###### Acknowledgements.
Y.C. was supported by the China Scholarship Council and NUDT’s Innovation
Foundation (Grant No. B110205). W.J. gratefully acknowledges support from
EPSRC through Grant No. EP/H04812X/1 and DFG through SFB910. We would also
like to thank Hugo Touchette for the useful discussions on large deviation
theory of Brownian motion.
## Appendix A Eigenvalue problem for the inviscid case
Without viscous damping and driving Eq. (8) reads [see Eq. (26)]
$\displaystyle-\Lambda
u_{\Lambda}(v)=u_{\Lambda}^{\prime}(v)+u_{\Lambda}^{\prime\prime}(v)$
$\displaystyle\quad\mbox{for }v>0$ (57) $\displaystyle-\Lambda
u_{\Lambda}(v)=-u_{\Lambda}^{\prime}(v)+u_{\Lambda}^{\prime\prime}(v)$
$\displaystyle\quad\mbox{for }a<v<0.$ (58)
Let
$u_{\Lambda}(v)=e^{-|v|/2}\varphi_{\Lambda}(v),$ (59)
then Eqs. (57) and (58) can be written as
$\varphi^{\prime\prime}_{\Lambda}(v)=(1/4-\Lambda)\varphi_{\Lambda}(v)\quad\mbox{for
}v\neq 0.$ (60)
On the one hand, for $\Lambda<1/4$ let us introduce the positive variable
$\lambda=\sqrt{1/4-\Lambda}$. Then the solution of Eq. (60) which results in a
finite normalization factor [see Eq. (12)] is given by
$\varphi_{\Lambda}(v)=\left\\{\begin{array}[]{lll}A_{\lambda}e^{-\lambda
v}&&\mbox{for }v>0\\\ B_{\lambda}e^{\lambda v}+C_{\lambda}e^{-\lambda
v}&&\mbox{for }a<v<0.\end{array}\right.$ (61)
Choose $A_{\lambda}=2\lambda$ and use the matching conditions (10) and (11) to
determine the other two coefficients in Eq. (61) as
$\displaystyle B_{\lambda}=1,\qquad C_{\lambda}=2\lambda-1.$ (62)
The eigenvalue is now determined by the absorbing boundary condition (9),
i.e., $\varphi_{\Lambda}(a)=0$, which results in Eq. (28).
On the other hand, for $\Lambda>1/4$ the solution of Eq. (60) which vanishes
at $v=a$, i.e., which satisfies the absorbing boundary condition (9), is given
by
$\varphi_{\Lambda}(v)=\left\\{\begin{array}[]{lll}\bar{A}_{\kappa}\sin(\kappa
v)+\bar{B}_{\kappa}\cos(\kappa v)&&\mbox{for }v>0\\\
\bar{C}_{\kappa}\kappa\sin[\kappa(v-a)]&&\mbox{for }a<v<0,\end{array}\right.$
(63)
where we have introduced the abbreviation $\kappa=\sqrt{\Lambda-1/4}>0$.
Choose $\bar{C}_{\kappa}=\kappa$, then by using the matching conditions (10)
and (11), the two parameters $\bar{A}_{\kappa}$ and $\bar{B}_{\kappa}$ are
evaluated as
$\displaystyle\bar{A}_{\kappa}=\kappa\cos(a\kappa)+\sin(a\kappa),\qquad\bar{B}_{\kappa}=-\kappa\sin(a\kappa).$
(64)
Hence Eq. (34) follows from substituting Eq. (63) into Eq. (59). For the
normalization, Eqs. (59) and (63) result in
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{a}^{\infty}u_{\Lambda}(v)u_{\Lambda^{\prime}}(v)e^{|v|}dv$
(65) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}[\bar{A}_{\kappa}\sin(\kappa
v)+\bar{B}_{\kappa}\cos(\kappa
v)][\bar{A}_{\kappa^{\prime}}\sin(\kappa^{\prime}v)+\bar{B}_{\kappa^{\prime}}\cos(\kappa^{\prime}v)]dv+\int_{a}^{0}\kappa\kappa^{\prime}\sin[\kappa(v-a)]\sin[\kappa^{\prime}(v-a)]dv$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\bigg{\\{}\frac{1}{2}\big{(}\bar{A}_{\kappa}\bar{A}_{\kappa^{\prime}}+\bar{B}_{\kappa}\bar{B}_{\kappa^{\prime}}\big{)}\cos[(\kappa-\kappa^{\prime})v]+\frac{1}{2}\big{(}\bar{B}_{\kappa}\bar{B}_{\kappa^{\prime}}-\bar{A}_{\kappa}\bar{A}_{\kappa^{\prime}}\big{)}\cos[(\kappa+\kappa^{\prime})v]$
$\displaystyle+\frac{1}{2}\big{(}\bar{A}_{\kappa}\bar{B}_{\kappa^{\prime}}-\bar{A}_{\kappa^{\prime}}\bar{B}_{\kappa}\big{)}\sin[(\kappa-\kappa^{\prime})v]+\frac{1}{2}\big{(}\bar{A}_{\kappa}\bar{B}_{\kappa^{\prime}}+\bar{A}_{\kappa^{\prime}}\bar{B}_{\kappa}\big{)}\sin[(\kappa+\kappa^{\prime})v]\bigg{\\}}dv-\frac{\kappa\bar{A}_{\kappa}\bar{B}_{\kappa^{\prime}}-\kappa^{\prime}\bar{A}_{\kappa^{\prime}}\bar{B}_{\kappa}}{\kappa^{2}-{\kappa^{\prime}}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\pi}{2}(\bar{A}_{\kappa}^{2}+\bar{B}_{\kappa}^{2})\delta(\kappa-\kappa^{\prime}),$
which shows that the normalization factor $Z_{\Lambda}$ satisfies Eq. (35) if
we take Eq. (64) into account. To derive Eq. (65), we have used the standard
identities for the $\delta$– and the principal value distribution
$\int_{0}^{\infty}\cos(\kappa
v)dv=\pi\delta(\kappa),\qquad\int_{0}^{\infty}\sin(\kappa
v)dv=P\left(\frac{1}{\kappa}\right).$ (66)
## Appendix B Eigenvalue problem for the general case
For the model (1), the eigenvalue problem (8) reads
$\displaystyle-\Lambda
u_{\Lambda}(v)=[(v+\mu-b)u_{\Lambda}(v)]^{\prime}+u_{\Lambda}^{\prime\prime}(v)$
$\displaystyle\mbox{ for }v>0$ (67) $\displaystyle-\Lambda
u_{\Lambda}(v)=[(v-\mu-b)u_{\Lambda}(v)]^{\prime}+u_{\Lambda}^{\prime\prime}(v)$
$\displaystyle\mbox{ for }a<v<0,$ (68)
if we adopt the notation used for Eq. (50). These two equations are a special
case of the so-called Kummer’s equation, which can be solved in terms of
parabolic cylinder functions Touchette et al. (2010). The solution of Eqs.
(67) and (68) which vanishes at infinity is given by (see Refs. Touchette et
al. (2012); Buchholz (1969))
$u_{\Lambda}(v)=\left\\{\begin{array}[]{lll}A_{\Lambda}e^{-(v+\mu-b)^{2}/4}D_{\Lambda}(v+\mu-b)&&\mbox{for
}v>0\\\
B_{\Lambda}e^{-(v-\mu-b)^{2}/4}D_{\Lambda}(v-\mu-b)+C_{\Lambda}e^{-(v-\mu-b)^{2}/4}D_{\Lambda}(-v+\mu+b)&&\mbox{for
}a<v<0,\end{array}\right.$ (69)
where $D_{\Lambda}$ denotes the parabolic cylinder function. Here we have used
a fundamental system in terms of $D_{\nu}(z)$ and $D_{\nu}(-z)$ to write down
the solution. Such a fundamental system degenerates for $\nu$ being an
integer. Thus, our expressions may contain spurious singularities at integer
values of $\Lambda$ which have to be taken care of. The coefficients
$A_{\Lambda}$, $B_{\Lambda}$ and $C_{\Lambda}$ depend on the parameters $b$
and $\mu$ as well, but are independent of $v$.
Using Eq. (69) the matching conditions (10) and (11) result in a set of linear
homogeneous equations
$\displaystyle
B_{\Lambda}D_{\Lambda}(-\mu-b)+C_{\Lambda}D_{\Lambda}(\mu+b)=e^{\mu
b}A_{\Lambda}D_{\Lambda}(\mu-b),$ (70) $\displaystyle
B_{\Lambda}D_{1+\Lambda}(-\mu-b)-C_{\Lambda}D_{1+\Lambda}(\mu+b)=e^{\mu
b}A_{\Lambda}[D_{1+\Lambda}(\mu-b)-2\mu D_{\Lambda}(\mu-b)]$ (71)
when the property
$\frac{de^{-z^{2}/4}D_{\nu}(z)}{dz}=-e^{-z^{2}/4}D_{\nu+1}(z)$ (72)
of the parabolic cylinder function is employed. For $A_{\Lambda}$ we choose
$A_{\Lambda}=\sqrt{2\pi}e^{-\mu b}.$ (73)
Then, the other two coefficients in Eq. (69) follow as
$\displaystyle B_{\Lambda}$ $\displaystyle=$
$\displaystyle-\Lambda\Gamma(-\Lambda)[D_{\Lambda}(\mu+b)D_{\Lambda-1}(\mu-b)$
(74) $\displaystyle+D_{\Lambda}(\mu-b)D_{\Lambda-1}(\mu+b)],$ $\displaystyle
C_{\Lambda}$ $\displaystyle=$
$\displaystyle\Lambda\Gamma(-\Lambda)[D_{\Lambda}(-\mu-b)D_{\Lambda-1}(\mu-b)$
(75) $\displaystyle-D_{\Lambda}(\mu-b)D_{\Lambda-1}(-\mu-b)],$
where we have used the identities
$\displaystyle
D_{\nu}(z)D_{\nu+1}(-z)+D_{\nu}(-z)D_{\nu+1}(z)=\frac{\sqrt{2\pi}}{\Gamma(-\nu)},$
(76) $zD_{\nu}(z)-D_{\nu+1}(z)-\nu D_{\nu-1}(z)=0$ (77)
to simplify the above two expressions.
The characteristic equation simply follows from the boundary condition (9),
and is thus given by
$B_{\Lambda}D_{\Lambda}(a-\mu-b)+C_{\Lambda}D_{\Lambda}(-a+\mu+b)=0.$ (78)
Using the identities (76) and (77) we arrive at Eq. (51).
For the integral over the eigenfunction which enters the FPT distribution (53)
we obtain by using, e.g., the differential identity (72)
$\displaystyle\int_{a}^{\infty}u_{\Lambda}(v)dv$ $\displaystyle=$
$\displaystyle
A_{\Lambda}e^{-(\mu-b)^{2}/4}D_{\Lambda-1}(\mu-b)-e^{-(\mu+b)^{2}/4}[B_{\Lambda}D_{\Lambda-1}(-\mu-b)-C_{\Lambda}D_{\Lambda-1}(\mu+b)]$
(79)
$\displaystyle+e^{-(a-\mu-b)^{2}/4}[B_{\Lambda}D_{\Lambda-1}(a-\mu-b)-C_{\Lambda}D_{\Lambda-1}(-a+\mu+b)].$
Finally to compute the normalization let us consider the integral
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\Lambda-\Lambda^{\prime})\int_{a}^{\infty}e^{(v+\mu\sigma(v))^{2}/2-bv}u_{\Lambda}(v)u_{\Lambda^{\prime}}(v)dv$
(80) $\displaystyle=$ $\displaystyle e^{-\mu
b-b^{2}/2}(\Lambda-\Lambda^{\prime})\int_{a}^{0}[B_{\Lambda}D_{\Lambda}(v-\mu-b)+C_{\Lambda}D_{\Lambda}(-v+\mu+b)]$
$\displaystyle\times[B_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(v-\mu-b)+C_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(-v+\mu+b)]dv$
$\displaystyle+e^{\mu
b-b^{2}/2}(\Lambda-\Lambda^{\prime})A_{\Lambda}A_{\Lambda^{\prime}}\int_{0}^{\infty}D_{\Lambda}(v+\mu-b)D_{\Lambda^{\prime}}(v+\mu-b)dv$
$\displaystyle=$ $\displaystyle e^{-\mu
b-b^{2}/2}(\Lambda-\Lambda^{\prime})\int_{a-\mu-b}^{-\mu-b}[B_{\Lambda}D_{\Lambda}(v)+C_{\Lambda}D_{\Lambda}(-v)][B_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(v)+C_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(-v)]dv$
$\displaystyle+e^{\mu
b-b^{2}/2}(\Lambda-\Lambda^{\prime})A_{\Lambda}A_{\Lambda^{\prime}}\int_{\mu-b}^{\infty}D_{\Lambda}(v)D_{\Lambda^{\prime}}(v)dv$
$\displaystyle=$ $\displaystyle e^{-\mu
b-b^{2}/2}\big{\\{}-B_{\Lambda}D_{\Lambda+1}(a-\mu-b)[B_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(a-\mu-b)+C_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(\mu-a+b)]$
$\displaystyle+B_{\Lambda^{\prime}}D_{\Lambda^{\prime}+1}(a-\mu-b)[B_{\Lambda}D_{\Lambda}(a-\mu-b)+C_{\Lambda}D_{\Lambda}(\mu-a+b)]$
$\displaystyle+C_{\Lambda}D_{\Lambda+1}(\mu-a+b)[B_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(a-\mu-b)+C_{\Lambda^{\prime}}D_{\Lambda^{\prime}}(\mu-a+b)]$
$\displaystyle-
C_{\Lambda^{\prime}}D_{\Lambda^{\prime}+1}(\mu-a+b)[B_{\Lambda}D_{\Lambda}(a-\mu-b)+C_{\Lambda}D_{\Lambda}(\mu-a+b)]\big{\\}}.$
For the last computational step we have used the properties (72) and the
analogous identity
$\frac{de^{z^{2}/4}D_{\nu}(z)}{dz}=\nu e^{z^{2}/4}D_{\nu-1}(z).$ (81)
Indeed, if we choose for $\Lambda$ and $\Lambda^{\prime}$ two different
eigenvalues we obtain (bi-)orthogonality of the eigenfunctions if the
characteristic equation (78) is taken into account. Furthermore dividing Eq.
(80) on both sides by $\Lambda-\Lambda^{\prime}$ and taking the limit
$\Lambda^{\prime}\rightarrow\Lambda$ we end up with the normalization factor
$\displaystyle Z_{\Lambda}=e^{-\mu
b-b^{2}/2}\left[B_{\Lambda}D_{\Lambda+1}(a-\mu-b)-C_{\Lambda}D_{\Lambda+1}(\mu-a+b)\right]\partial_{\Lambda}\left[B_{\Lambda}D_{\Lambda}(a-\mu-b)+C_{\Lambda}D_{\Lambda}(\mu-a+b)\right].$
(82)
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|
arxiv-papers
| 2013-12-02T20:33:13 |
2024-09-04T02:49:54.659220
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yaming Chen and Wolfram Just",
"submitter": "Yaming Chen",
"url": "https://arxiv.org/abs/1312.0581"
}
|
1312.0661
|
# Relativistic MHD Simulations of Poynting Flux-Driven Jets
Xiaoyue Guan 11affiliation: Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM; [email protected] , Hui Li 11affiliation: Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM; [email protected] , and
Shengtai Li 11affiliation: Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM; [email protected]
###### Abstract
Relativistic, magnetized jets are observed to propagate to very large
distances in many Active Galactic Nuclei (AGN). We use 3D relativistic MHD
(RMHD) simulations to study the propagation of Poynting flux-driven jets in
AGN. These jets are assumed already being launched from the vicinity ($\sim
10^{3}$ gravitational radii) of supermassive black holes. Jet injections are
characterized by a model described in Li et al. (2006) and we follow the
propagation of these jets to $\sim$ parsec scales. We find that these current-
carrying jets are always collimated and mildly relativistic. When $\alpha$,
the ratio of toroidal-to-poloidal magnetic flux injection, is large the jet is
subject to non-axisymmetric current-driven instabilities (CDI) which lead to
substantial dissipation and reduced jet speed. However, even with the presence
of instabilities, the jet is not disrupted and will continue to propagate to
large distances. We suggest that the relatively weak impact by the instability
is due to the nature of the instability being convective and the fact that the
jet magnetic fields are rapidly evolving on Alfvénic timescale. We present the
detailed jet properties and show that far from the jet launching region, a
substantial amount of magnetic energy has been transformed into kinetic energy
and thermal energy, producing a jet magnetization number $\sigma<1$. In
addition, we have also studied the effects of a gas pressure supported “disk”
surrounding the injection region and qualitatively similar global jet
behaviors were observed. We stress that jet collimation, CDIs, and the
subsequent energy transitions are intrinsic features of current-carrying jets.
galaxies:active, galaxies:jets, methods:numerical, instabilities, black hole,
magnetic fields, relativistic MHD
## 1 Introduction
Relativistic jets, such as the famous kpc jet in M87, are observed in many
active galactic nuclei (AGN) systems through multi-wavelength observations.
AGN jets are collimated, magnetized, mildly relativistic ($\gamma\sim 10$),
and can travel to large distances (kpc or even Mpc scales). Peculiar spatial
structures such as knots are often observed in various locations along the
direction of jet propagation (e.g. Biretta et al. (1991)). Monitoring of jet
radiation has also revealed a range of jet time variabilities (minutes to
years), including recently observed TeV flares with a variability timescale of
minutes (e.g. Aharonian et al. (2007); Albert et al. (2007)), although the
mechanisms that are responsible for variabilities are under debate. There are
still many unresolved problems associated with relativistic jets, such as jet
composition ($\rm{e^{+}/e^{-}}$pairs vs. $\rm{e^{-}/p^{+}}$ plasma), jet
stability, particle acceleration/deceleration mechanisms, and jet emission
mechanism.
It is widely accepted that relativistic jets in AGN systems are powered
through some magnetic processes, and the most likely mechanism is the so-
called Blandford-Znajek process (Blandford & Znajek 1977, B-Z hereafter),
where the primary energy source is the spin of black hole but transferred via
magnetic fields. In recent years, development in numerical general
relativistic magnetohydrodynamics (GRMHD) and force-free electrodynamics
(FFEM) techniques (e.g. Komissarov (1999); McKinney & Gammie (2004); De
Villiers et al. (2003, 2005); McKinney (2005); Beckwith et al. (2008);
McKinney & Blandford (2009)) has enabled time-dependent studies of the
formation and evolution of relativistic jets, sometimes in connection with the
detailed accretion processes. Moreover, it has been shown numerically that the
B-Z mechanism is capable of powering a magnetically dominated jet with a
relativistic Lorentz factor up to $\gamma\sim 10$. In some accretion-type
simulations such as McKinney & Blandford (2009), although current-driven
instabilities (CDI) with a $m=1$ kink mode are observed, jet can get
collimated and propagate to $\sim 10^{3}GM/c^{2}$, where $GM/c^{2}$ is the
gravitational radii of the black hole, without being disrupted nor having much
dissipation. These first-principle simulations have the advantages of
exploring the important dynamics of accretion together with magnetized jet
formation. However, due to the extreme numerical requirements to resolve the
accretion disk dynamics, it is very difficult to examine how these jets will
evolve beyond several thousands of gravitational radii and over astronomically
significant timescales. Furthermore, observations of jets down to several
thousand gravitational radii of the black hole have been very difficult to
obtain, making comparisons between theory/simulations and observations
challenging.
Another class of jet models is focused more on the detailed properties of jets
in their propagation process after they are launched (Lery et al., 2000; Baty
& Keppens, 2002; Nakamura & Meier, 2004; O’Neill et al., 2005; Li et al.,
2006; Nakamura et al., 2006, 2007, 2008; Komissarov et al., 2007; Moll et al.,
2008; Mignone et al., 2010; Mizuno et al., 2009, 2011; O’Neill et al., 2012).
They typically adopt an MHD or relativistic MHD (RMHD) approach, utilizing
some boundary conditions to represent a jet injection, and following the jet
propagation. Simulations of these models can be either on relatively smaller
scales, which are focussed on the local properties of the flow, or on
relatively large scales ($\sim$ kpc), where the jet interacts with the
surrounding intergalactic medium. When a high-velocity, magnetized jet travels
through its environment, it could be subject to instabilities such as magnetic
Kelvin-Helmholtz instability due to the shear (e.g., see discussions in Baty &
Keppens (2002); Hardee (2007)), and/or current-driving instabilities when
there are strong toroidal fields and/or rotation (e.g., see discussions in
Mizuno et al. (2009); Narayan et al. (2009)). However, the long-term
consequences of these instabilities and how the properties of the localized
jet can be transformed into observed jet features are not clear. One
particular focus of this type of research is to identify the energy transition
mechanism (sometimes called the jet $\sigma$ problem; $\sigma$ is the jet
magnetization parameter; see Rees & Gunn (1974)) which transforms a
magnetically dominated jet deep in the gravitational potential of the black
hole to possibly kinetically dominated jet on larger scales (e.g., as
discussed in Lind et al. (1989) for FR II jets $\sigma\ll 1$). Begelman (1998)
has suggested that current-driven instabilities can be used to tackle the
energy transition problem, and numerical simulations by Mizuno et al. (2009),
O’Neill et al. (2012) have shown CDIs can indeed transform jet magnetic energy
into kinetic energy.
Here we present new simulations of magnetic flux-driven relativistic AGN jet
using RMHD code LA-COMPASS (Los Alamos COMPutational AStrophysics Suite).
Assuming that a Poynting-flux dominated jet can steadily propagate to $\sim
10^{3}$ gravitational radii as suggested by current generation of GRMHD black
hole accretion simulations, we adopt the approach of using an injection region
with a size $\sim 10^{3}$ gravitational radii and follow the jet evolution out
to tens/hundreds pc scales. The injected magnetic field has a geometry of
“closed” field lines that are confined in spatial extent, different from the
classic split monopole configuration which has an unconfined flow (see
discussions in Komissarov et al. (2009); Tchekhovskoy et al. (2009)). To our
knowledge, this is the first time that a RMHD jet can be followed to this
observation scale. This paper is also the first of a series of papers studying
relativistic jets properties.
The paper is organized as follows. In §2 we give a brief description of the
RMHD code and how the injection is implemented in our models. In §3 we present
a fiducial model where we analyze the properties of the simulated jets in
detail, including jet morphologies, energetics, and instabilities. We then
describe how these properties depend on model parameters such as the injected
field geometry, disk confinement, and resolution. A summary and discussions
are given in §4.
## 2 Numerical Methods and Model Set-up
### 2.1 RMHD Code
We use a 3D RMHD code based on evolving fluid equations using higher-order
Godunov-type finite-volume methods. The ideal MHD code is part of the code LA-
COMPASS , which was first developed at Los Alamos National Laboratory (Li &
Li, 2003) and has been used on a range of astrophysical MHD simulations,
including the jet collimation and stability problems.
The set of relativistic MHD equations can be written in the following
conservative form,
$\partial_{t}\mbox{\boldmath$U$}+\partial_{i}\mbox{\boldmath$F$}^{i}=\mbox{\boldmath$S$},$
(1)
where $i$ denotes a spatial index. First, a set of conserved variables
$\mbox{\boldmath$U$}=(D,M_{x},M_{y},M_{z},B_{x},B_{y},B_{z},E)^{\rm T}$ is
${\mbox{\boldmath$U$}}\equiv\left({\begin{matrix}\rho\gamma\\\ (\rho
h\gamma^{2}+\mbox{\boldmath$B$}^{2})v_{x}-(\mbox{\boldmath$v$}\cdot\mbox{\boldmath$B$})B_{x}\\\
(\rho
h\gamma^{2}+\mbox{\boldmath$B$}^{2})v_{y}-(\mbox{\boldmath$v$}\cdot\mbox{\boldmath$B$})B_{y}\\\
(\rho
h\gamma^{2}+\mbox{\boldmath$B$}^{2})v_{z}-(\mbox{\boldmath$v$}\cdot\mbox{\boldmath$B$})B_{z}\\\
B_{x}\\\ B_{y}\\\ B_{z}\\\ \rho
h\gamma^{2}-p+\frac{\mbox{\boldmath$B$}^{2}}{2}+\frac{\mbox{\boldmath$v$}^{2}\mbox{\boldmath$B$}^{2}}{2}-\frac{\mbox{\boldmath$v$}\cdot\mbox{\boldmath$B$}}{2}\end{matrix}}\right),$
(2)
where $v_{i}$ and ${\bf B}^{i}$ are the usual velocity and magnetic field
three-vector, and $\gamma$ is the Lorentz factor
$\gamma=(1-v^{2}/c^{2})^{-1/2}$.
Second, a set of fluxes $\mbox{\boldmath$F$}^{i}$, where the flux in the
x-direction, is given as
${\mbox{\boldmath$F$}^{x}}\equiv\left({\begin{matrix}Dv_{x}\\\
M_{x}v_{x}-\gamma^{-1}b_{x}B_{x}+p\\\ M_{y}v_{x}-\gamma^{-1}b_{y}B_{x}\\\
M_{z}v_{x}-\gamma^{-1}b_{z}B_{x}\\\ 0\\\ B_{y}v_{x}-B_{x}v_{y}\\\
B_{z}v_{x}-B_{x}y_{z}\\\ M_{x}\end{matrix}}\right),$ (3)
where $b_{i}$ are the usual magnetic field four-vector.
Third, a set of source is
$\mbox{\boldmath$S$}=(\dot{D},\dot{M_{x}},\dot{M_{y}},\dot{M_{z}},\dot{B_{x}},\dot{B_{y}},\dot{B_{z}},\dot{E})^{\rm
T},$ (4)
where $h=1+\Gamma p/[(\Gamma-1)\rho]$ is the specific enthalpy, and $\Gamma$
is the adiabatic index. To solve the approximate Riemann problem, we use the
HLL flux with parabolic piece wise reconstruction method by Colella & Woodward
(1984). Note for RMHD code, the set of primitive variables used for
interpolation are
${\mbox{\boldmath$P$}}\equiv(\rho,v^{i},B^{i},u)^{\rm T},$ (5)
and they are recovered from conservative variables from an iterative algorithm
where Newton-Raphson method is implemented.
Together with no-monopole constrain
$\partial_{i}{{\bf B}^{i}}=0,$ (6)
and a description of thermal dynamics the equation system is complete.
Numerically, we use a staggered mesh for magnetic fields, and use Constrained-
Transport (CT) method to evolve induction equations.
In the models we use an ideal gas equation of state (EOS),
$p=(\Gamma-1)u,$ (7)
where $u$ is the internal energy density. In this work we use $\Gamma=5/3$. We
have found that using a relativistic EOS with $\Gamma=4/3$ gives very similar
results for the jet properties studied in the work.
Because the code conserves total energy and there is no explicit cooling, all
the heat generated by the dissipation (both physical and numerical) in the jet
propagation process will be captured by the code (see detailed discussion in
§3). For the jet problem, in the total energy equation we have adopted the
common practice to exclude the rest mass energy from the total energy and the
corresponding energy flux. This is because in the vast region where total
energy is dominated by the rest mass energy, when we need to get the other
energetics, the subtraction of a large number from the other one may not be
accurate.
### 2.2 Our Model
The basic framework of our 3D simulations involves two key parts: First, the
initiation of the jet is through a (continued) injection process within a
small volume of size $r_{\rm inj}$. This is supposed to mimic the outcome of
accretion on the supermassive black hole plus the magnetized jet formation.
Second, the Lorentz force of the injected magnetic fields (and mass) will
cause the magnetic fields to expand into a pre-existing low density, low
pressure and unmagnetized background plasma with a size that is several
hundred times larger than $r_{\rm inj}$ in all directions. This is supposed to
mimic the propagation of relativistic jet through the interstellar medium near
the galaxy center on $\sim$ tens of pc scales.
With this approach, the critical questions we hope to address include: 1)
whether the jet will be collimated on scales much larger than $r_{\rm inj}$;
2) whether the jet will be stable; and 3) how efficient the energy conversion
processes inside the jet will be. Ultimately, these results could contribute
to, among other things, understanding both the observed jet structures on
those scales and physical conditions for multi-wavelength jet emissions.
### 2.3 Injection of Magnetic Field and Mass
In order to drive an injection, we have implemented source terms in the RMHD
equations at each time-step, similar to the method used in Li et al. (2006).
The injected magnetic flux has both a poloidal and toroidal component. In
cylindrical coordinates $(r,\phi,z)$ the poloidal flux function is
axisymmetric and has a form of
$\Phi(r,z)=B_{{\rm inj},0}r^{2}\exp(-\frac{r^{2}+z^{2}}{r^{2}_{{\rm
inj},B}}),$ (8)
which relates to the $\phi$ component of vector potential $A_{\phi}$ with
$\Phi(r,z)=rA_{\phi}$. From $\Phi(r,z)$ one can calculate the poloidal field
injection functions
$B_{{\rm inj},r}=-\frac{1}{r}\frac{\partial\Phi}{\partial z}=2B_{{\rm
inj},0}\frac{zr}{r^{2}_{{\rm inj},B}}\exp(-\frac{r^{2}+z^{2}}{r^{2}_{{\rm
inj},B}}),$ (9)
and
$B_{{\rm inj},z}=\frac{1}{r}\frac{\partial\Phi}{\partial r}=2B_{{\rm
inj},0}(1-\frac{r^{2}}{r^{2}_{{\rm
inj},B}})\exp(-\frac{r^{2}+z^{2}}{r^{2}_{{\rm inj},B}}),$ (10)
where $B_{{\rm inj},0}$ is a normalization constant for field strength and
$r_{{\rm inj},B}$ is the characteristic radius of magnetic flux injection.
This form of magnetic fields contains closed poloidal field lines, which
causes $B_{z}$ to change directions beyond $r_{{\rm inj},B}$ with no net
poloidal flux.
The toroidal field injection function is
$B_{{\rm inj},\phi}=\frac{\alpha\Phi}{r}=B_{{\rm inj},0}\alpha
r\exp(-\frac{r^{2}+z^{2}}{r^{2}_{{\rm inj},B}})~{}.$ (11)
Here $\alpha$ is a constant parameter and it has the unit of inverse length
scale. This parameter specifies the ratio of toroidal to poloidal flux
injection rate. As demonstrated in Li et al. (2006), the poloidal and toroidal
fluxes are roughly equal when $\alpha\sim 2.6$. In our simulations, we
typically use $\alpha>>1$. The assumption here is that the rotation of the
black hole at the base of jet launching location will wind up the poloidal
field through the B-Z effect and introduce a large toroidal component. The
injected magnetic fields are given as
$\dot{B}_{\rm inj}=\gamma_{b}{\mbox{\boldmath$B$}}_{\rm inj},$ (12)
where $\gamma_{b}$ is the characteristic rate of magnetic injection. In all
our numerical models $\gamma_{b}$ is set to a constant so that the magnetic
energy injection rate is roughly constant as well.111Constant injection of
magnetic fields over a region of $r_{{\rm inj},B}$ can be inherently acausal.
However, since our simulations extend in spatial scales $\gg r_{{\rm inj},B}$
and in temporal scales $\gg r_{{\rm inj},B}/c$, the causality concern is
somewhat limited.
Our numerical model also has mass injection in the injection region. There are
two motivations to consider mass flux injection: the first is that it is
possible that matter can enter the jet at its launching location, although the
details of the mass loading is unknown; the second motivation is to maintain a
certain density floor in the computational domain as the magnetic dominated
flow expansion tends to introduce extremely low density region. The rest mass
density injection function is
$\dot{\rho}_{\rm inj}=\gamma_{\rho}\rho_{0}\exp(-\frac{r^{2}+z^{2}}{r_{{\rm
inj},\rho}^{2}}),$ (13)
where $r_{{\rm inj},\rho}$ and $\gamma_{\rho}$ are the characteristic radius
and rate of mass injection.
Our numerical models also allows a jet velocity injection in the $z$
direction, and the $v_{z}$ injection function at the central region is
$v_{{\rm inj},z}=v_{{\rm inj},0}\frac{z}{r_{{\rm
inj},\rho}}\exp(-\frac{r^{2}+z^{2}}{r_{{\rm inj},\rho}^{2}}),$ (14)
where $v_{{\rm inj},0}$ is the characteristic velocity, which is often taken
as $0.5c$. It turns out that both the total injected mass and total injected
kinetic energy are small so they do not affect the overall jet dynamics.
Notice that for simplicity we have chosen not to include initial plasma
rotation in our injection scheme. Rotation is certainly a factor to consider
in jet models, and it has been argued to be important in stabilizing jet (e.g.
Tomimatsu et al. (2001); McKinney & Blandford (2009)). However, it is not
clear whether rotation will play a significant role on the scales our models
correspond to, therefore we do not include rotation in the initial conditions
and just focus on the limit when the rotation is small. The $\phi$ component
of the Lorentz force $({\mbox{\boldmath$J$}}\times{\mbox{\boldmath$B$}})_{{\rm
inj},\phi}$ resulted from the injected magnetic flux is zero, the evolution of
the total magnetic flux, however, could still introduce rotation to the gas.
From the models we indeed find that the rotation effect is small (see
discussion in §4.)
Numerically, we treat injection as a source step at the end of each time step.
For RMHD, the most straightforward way of injection is to add source terms
directly to the updated primary variables ${B^{i}}$ and $\rho$, and add an
injected momentum source to the updated $z$ momentum, as $v_{{\rm inj},z}$
only applied to the injected mass at each step. Our code is formulated to
conserve the total energy. Since the injection step will increase total energy
at each time step, we calculate the new total energy at the end of each
injection step.
For the injection scheme, again for simplicity, we have chosen $r_{{\rm
inj},\rho}=r_{{\rm inj},B}=r_{\rm inj}$, therefore both matter and fields
injection are confined within $r_{\rm inj}$. The form of magnetic field
injection functions guaranties the divergence free nature of the injection
field. We have also observed $\mbox{\boldmath$\nabla$}\cdot{\bf B}<10^{-8}$
throughout the simulation in all the computational domain. The mass injection
rate is set to be very small to satisfy the plasma thermal $\beta\ll 1$ and
plasma $\sigma=B^{2}/(4\pi\gamma^{2}\rho c^{2})\gg 1$.
We adopt a uniform Cartesian $(x,y,z)$ grid with a size of
$x=[-L_{x}/2,L_{x}/2],y=[-L_{y}/2,L_{y}/2],z=[-L_{z}/2,L_{z}/2]$. Outflow
boundary conditions are enforced on the primary variables. The initial grid is
filled with a uniform plasma background with a finite gas density $\rho_{0}$
and pressure $P_{0}$. The initial magnetic field structure has the same form
as the magnetic injection function Eqn(9-10) with a strength normalization
$B_{0}$. The injection region is located at the origin of the box with an
injection radius of $r_{{\rm inj}}$.
In all models we choose $\rho_{0}=1,P_{0}=10^{-5},r_{\rm inj}=1,c=1$. Other
units of physical quantities for normalization are listed in Table 1. To put
these numbers in an astrophysical context, assuming a background number
density of $10^{2}~{}{\rm cm}^{-3}$ and background temperature of $5~{}{\rm
keV}$, the code sound speed is $c_{s}=0.0041c$ which corresponds to a physical
sound speed of $8.93\times 10^{7}{\rm cms}^{-1}$. The code magnetic strength
$B_{0}=1$ corresponds to a physical magnetic field of $1.38{\rm G}$ and a
physical Alfvén speed of $v_{{\rm A},0}\sim 0.707c$. Note in all our models we
have initial $c_{s}\ll v_{{\rm A},0}<c$. For the code length scale, we choose
injection region size $r_{{\rm inj}}=1$, and if this corresponds to
$1000GM/c^{2}$, then for a supermassive black hole like M87($M_{\rm
BH}=3\times 10^{9}{\rm\,M_{\odot}}$), the injection region has a physical size
of $\sim 0.143$pc. Our computational domain usually has a size of
$10^{2}-10^{3}r_{\rm inj}$, and this corresponds to a physical domain size of
$14.3-143$ pc. In the code, $t=1$ then equals to the light crossing time scale
for the injection region, and it corresponds to a physical time scale of
$0.47$yr. We usually follow the jet propagation for a few hundreds to
thousands of years.
## 3 Results: Relativistic Jet Propagation
In this work we follow the propagation of relativistic magnetic-flux driven
jets from $\sim 10^{3}$ gravitational radii to tens of pc scales where they
are often observed. We are particularly interested in the jet morphology,
whether current-driven instability will occur along the way, and if it does,
how these instabilities will affect the jet properties. Here we first present
a fiducial model to give the detailed accounts of the jet propagation.
### 3.1 Fiducial Model
In our fiducial model we have $\alpha=10$ for magnetic injection. The
injection rate is $\gamma_{\rho}=\gamma_{b}=1$. The initial magnetic field
strength is $B_{0}=0.3$, the magnetic field injection coefficient is $B_{{\rm
inj},0}=0.2$. The jet velocity injection coefficient is $v_{{\rm inj},0}=0.5$.
The computational grid has a size of $L_{x}=L_{y}=150,L_{z}=400$ with a
resolution $N_{x}=N_{y}=300,N_{z}=800$. We run the simulation to $t_{f}=1500$.
#### 3.1.1 Jet Properties
Figures 1 and 2 show the overall morphology and evolution of the jet
propagation. Over scales that are much larger than $r_{\rm inj}$, we find that
the magnetic fields form an elongated structure that stays highly collimated,
with the central axis (along $z$) having roughly a cylindrical shape without
an obvious opening angle. While the central axis of the jet undergoes
instabilities, the overall collimation and propagation still remain (to as
long as we have simulated). The magnetic structure is enclosed by a
hydrodynamic structure that consists mostly of a strong shock that is
propagating into the background and sweeping up the material into a shell.
Figure 1 shows several snapshots of the z-component of current density $j_{z}$
at the $y=0$ plane. Because the injected fields possess a dipole-like poloidal
field structure plus a toroidal field proportional to the flux function, the
$j_{z}$ distribution has the overall structure that it contains an “outgoing”
(positive) current along the central axis and a “return” current mostly in a
thin shell encasing the structure. The location of the return current
separates the magnetized interior from the non-magnetized outer region. Before
$t\leq 225$, the jet appears to be propagating with little signature of
nonlinear instabilities, while around $t\sim 300$, significant nonlinear
instabilities first start to appear at the jet front, indicated by small
wiggles with characteristic length scale $\sim$ a couple of tens $r_{\rm
inj}$. At the late time many filamentary structures start to appear in jet
front as the jet propagates further, while the central high $j_{z}$ region
keeps almost the same vertical extent. It is interesting to note that the
return current has maintained a quite axisymmetric cocoon-like shape
throughout the duration of the run. At the late time, along the axis, the
$j_{z}$ distribution splits into two parts: further away from the injection,
the $j_{z}$ current density becomes highly unstable; whereas closer to the
injection region with $|l_{z}|\leq 50~{}r_{\rm inj}$, it stays quasi-stable
with relatively high peak current values (up to $j_{z,{\rm max}}=3.2$, not
shown in the figure), presumably due to the strong injection.
To illustrate the jet properties at the late time, in Figure 2, we plot the
snapshots of gas density $\rho$, gas pressure $P$, $z-$component of the gas
three-velocity $v_{z}$, $y-$component of magnetic field $B_{y}$, and
$z-$component of magnetic field $B_{z}$. In the density plot, there is a very
thin layer of gas at the shock front with a maximum density $\rho_{\rm
max}=4.8$, while inside this shell there is an extremely low density region
with a minimum density $\rho_{\rm min}=7.9\times 10^{-4}$. This is a result of
most of the uniform background gas being pushed away by the magnetic-dominated
jet as it expands into the environment. Note in the inner $|l_{z}|\leq 40$
region there’s a small amount of gas which follows where the strong current
is. This is because we inject a small amount of gas into the computational
domain. At the end of this simulation, we have injected a total of $M_{\rm
inj}=8688\rho_{0}r_{\rm inj}^{3}$, which for the parameters we specified at
the end of §2, corresponds to a total mass injection of $1.3\times
10^{35}~{}{\rm g}$ and a mass injection rate of
$0.09{\rm\,M_{\odot}}{\rm/yr}$. For the gas pressure, it is evident that the
shock front has a higher pressure in the $z$ direction than in the horizontal
direction, presumably due to the stronger expansion along the $z$ direction.
For the gas velocity, we get the maximum Lorentz factor of the plasma flow is
$\gamma_{\rm max}\sim 2.7$ and the maximum Lorentz factor generally increases
with time during the run. For $v_{z}$, we see that while around the $r=0$ axis
the gas is mostly moving outward, there’s also a returning component at larger
$r$ due to the magnetic field structure we have used in our model. For the
$B_{y}$ and $B_{z}$ plots, they show that, along the radial direction, the jet
has a magnetic dominated core with $B_{z}$ being dominant at $r=0$ but $B_{y}$
becomes dominant at large $r$. Along the $z$ direction, there is a magnetic
dominated region with $|l_{z}|\leq 50r_{\rm inj}$ that is followed by a more
smoothly decreasing region out to the vertical extent of the jet. Overall, a
magnetized central spine is always present.
To calculate the jet speed, we can follow the jet front and record its
location as a function of time. Figure 3 shows how the location of jet front
changes over time. Evidently, the jet front starts with an almost constant
speed $\sim 0.3c$, and then its propagation speed changes at around $t\sim
300$, and gradually slows to $\sim 0.1c$. There is no slowing-down at the late
time. Compared to the $j_{z}$ snapshots sequence in Figure 1, the turning
point at the jet propagation occurs at the time when the nonlinear modes start
to grow significantly.
#### 3.1.2 Energy Transition
As the current-driven jet propagates further away from the injection region,
instabilities grow and non-linear structures develop. These features also
affect jet energetics, which is a central problem in jet physics. In Figure 4
we plot the evolution of volume-integrated total magnetic energy $E_{\rm B}$,
total kinetic energy $E_{\rm K}$, total internal energy $E_{\rm U}$, and total
energy $E_{\rm tot}=E_{\rm B}+E_{\rm K}+E_{\rm U}$. Note that magnetic energy
density $e_{\rm B}$ includes all terms222In most of our models, the first term
dominates by being an order of magnitude larger than the other terms.
containing magnetic field, and it has a form of $e_{\rm
B}={\mbox{\boldmath$B$}}^{2}/2+[|{\mbox{\boldmath$v$}}|^{2}|{\mbox{\boldmath$B$}}|^{2}-({\mbox{\boldmath$v$}}\cdot{\mbox{\boldmath$B$}})^{2}]/2$.
For the kinetic energy density, we have excluded the rest mass energy,
therefore $e_{\rm K}=(\gamma-1)\gamma\rho$. The internal energy density is
$e_{\rm u}=p/(\Gamma-1)$. As a reference, we have also plotted the time and
volume integrated injected magnetic energy $E_{\rm B,inj}$, injected kinetic
energy $E_{\rm K,inj}$, and injected internal energy $E_{\rm U,inj}$. Note
that for the injected energy the meaningful diagnostic here is to calculate
the total injection up to a certain time $t$, $E_{\rm
inj}(t)=\int_{0}^{t}\int\dot{E}_{\rm inj}dvdt$. It is obvious that although
all energetics are increasing with the constant energy injection, after $t\sim
300$, $E_{\rm B}$ increases with a much shallower slope compared to the growth
of $E_{\rm K}$ and $E_{\rm U}$. Before $t\sim 300$, the magnetic energy is
larger than the kinetic energy but after that kinetic energy takes over.
We have also monitored the total energy conservation during the simulation. In
Figure 4, the dotted magenta line represents $E^{{}^{\prime}}_{\rm
tot,inj}=E_{\rm tot,inj}+E_{\rm tot,0}$, the total injected energy (magnetic +
kinetic + thermal) plus the initial background energy, whereas the solid
magenta line represents the sum of various energy components in the simulation
domain. At the beginning they are quite close to each other, but as the
simulation progresses, the difference between $E_{\rm tot}$ and
$E^{{}^{\prime}}_{\rm tot,inj}$ continues to increase. The difference between
these two total energies, however, is always much smaller than the other
energy components in the simulation. This energy discrepancy is dominated by
numerical errors and the origin of these errors in MHD simulations is
relatively well known. For our simulations we have used both dual-energy
formulation (evolving both internal energy and total energy equations) and
energy fix after the constrained-transport to preserve the positivity of the
thermal pressure. Both procedures break the total energy conservation in low
pressure region and introduce energy error by a small amount. In addition, we
find that these errors decrease gradually when we increase the numerical
resolution. The sudden change in $E_{\rm tot}$ after $t\sim 1000$ is because
the expansion has reached the computational domain boundaries and materials
are flowing out of the box.
Our numerical model therefore gives an example of transferring jet’s magnetic
energy into kinetic energy as jet propagates. The magnetization parameter
$\sigma$, which we have chosen here as the ratio of Poynting energy flux to
the kinetic energy flux333Other forms of $\sigma$ exist. Note that the factor
of $4\pi$ has been absorbed in our numerical representation of the magnetic
field., is $\sigma\equiv F_{\rm Poynting}/F_{\rm P}=B^{2}/4\pi\gamma^{2}\rho
c^{2}$. In Figure 5 we plot several snapshots of $\sigma$ at the $y=0$ plane.
As the jet propagates from its core region, the magnetically dominated region
has been kept to be a region with a nearly constant extent $|l_{z}|\leq
50~{}r_{\rm inj}$. At late time, as the instability causes the jet fields to
have more random and small structures, the jet can be seen in a more or less
kinetically dominated state. Therefore, our numerical model illustrates a jet
which contains a near-region with a $\sigma\gg 1$ and a far-region with a
$\sigma\ll 1$. The jet does not stop nor get destroyed after this transition
occurs. The energy transition is likely a result of current-driven
instabilities.
#### 3.1.3 Current-Driven Instabilities
In this section we give more details of the CDIs in the fiducial model. The
primary candidate for CDIs is the kink instability. According to Kruskal-
Shafranov criterion (Kadomtsev, 1966), a cylindrical MHD plasma with a
constant current density $j_{z}$ in a confined radius is unstable to kink
modes when $q=2\pi rB_{p}/(L_{z}B_{\phi})<q_{\rm crit}$, where $r$ is the
cylindrical radius, $B_{p}$ is the poloidal component of the magnetic field
which is parallel to the axis of the cylinder, $B_{\phi}$ is the toroidal
field, and $L_{z}$ is the plasma column length. For ideal MHD, $q_{\rm
crit}=1$, for RMHD, this number is a few (Narayan et al., 2009). This
instability criterion indicates that when the jet is dominated by $B_{\phi}$,
the jet will be unstable to the $m=1$ kink mode. This is indeed what we have
observed in our simulations. In Figure 6 we have plotted $q$ at different
times in the fiducial run, where we have chosen $L_{z}$ to be the height of
the jet at the time. We can see that most of the near-axis and
$|l_{z}|<50r_{\rm inj}$ region with large-current has $q<1$ throughout the
simulation. Note that the Kruskal-Shafranov criterion is derived from the
highly ideal situations and we should concentrate on the near-axis region
where the large current is confined. The growth of CDIs is responsible for the
slow-down of the jet front and facilitates the energy-transition process. For
the physical parameters in our model, this growth period is $\gtrsim 100$ yrs.
One way to quantify the growth of the nonaxisymmetric modes is to calculate
the power in the current using Fourier transform
$f(m,k)=\frac{\int_{r_{\rm min}}^{r_{\rm max}}\int_{0}^{2\pi}\int_{z_{\rm
min}}^{z_{\rm max}}|{\mbox{\boldmath$J$}}|e^{i(m\phi+kz)}rdrd\phi
dz}{\int_{r_{\rm min}}^{r_{\rm max}}\int_{0}^{2\pi}\int_{z_{\rm min}}^{z_{\rm
max}}rdrd\phi dz}$ (15)
where $|{\mbox{\boldmath$J$}}|$ is the amplitude of the current density and
the integration is over a cylindrical volume which encloses the current. In
our calculation, we have used $r_{\rm min}=0$, $r_{\rm max}=10r_{\rm inj}$,
$z_{\rm min}=0$, and $z_{\rm max}=200r_{\rm inj}$. $m$ is the azimuthal mode
number and $k=2\pi/\lambda$ is the vertical wavenumber where $\lambda$ is a
characteristic wavelength. The volume-averaged mode power in the current
amplitude $|J|$ is then
$P(m,k)=|f(m,k)|^{2}=\\{{\rm Re}[f(m,k)]\\}^{2}+\\{{\rm Im}[f(m,k)]\\}^{2},$
(16)
where ${\rm Re}[f(m,k)]$ and ${\rm Im}[f(m,k)]$ are the cosine and sine
Fourier transformations of $|{\mbox{\boldmath$J$}}|$, respectively.
In Figure 7 we plot the time evolution of $P(m,k)$ for the $m=0,1,2$
components for the fiducial run. For $k$, we have chosen $\lambda=20r_{\rm
inj}$ for the characteristic wavelength (we have examined other wavenumbers
and found they experience similar exponential growth). The $m=0$ component
dominates throughout the run, although at late times the power in the
nonaxisymmetric components has grown to be close to the power in the $m=0$
mode. The dominant nonaxisymmetric mode is the $m=1$ mode, and there is an
exponential growth period between $t\sim 300-500$. After $t\sim 500$, the
power in non-axisymmetric modes continues to grow, but at a rate which is much
slower. There is also substantial power in the $m=2$ mode. Note that the
background perturbations affect the onset time of significant growth: we have
found that in another simulation with $50\%$ random background density
perturbations, the onset time has changed significantly to about $t\sim 100$.
We have also observed magnetic Kelvin-Helmholtz instabilities due to the large
shear that exists at various regions in the jet. The characteristic “cat eye”
features can be observed at the jet front (e.g. see current near $z\sim
50r_{\rm inj}$ in $j_{z}$ slice at $t=450$ in Figure 1).
It is noteworthy that although instabilities occur in our models, the jet does
not get totally disrupted and continues to propagate with an almost constant
speed. This is partially due to the constant magnetic flux injection which
continually drives the jet. The fact that the power in $m>0$ modes remaining
smaller than the power in $m=0$ mode during the nonlinear stage is consistent
with the non-disruption of the jet. We will discuss the possible explanation
for stabilization in §4.
### 3.2 Effect of $\alpha$
The detailed properties of current-driven jets depend on the model parameters,
one of which is the $\alpha$ parameter that represents the ratio of toroidal
to poloidal fields. Effects of other parameters on the jet propagation will be
examined in future studies.
In this simulation we use a higher $\alpha=40$, which gives a stronger
toroidal field injection. In order to make comparison with the fiducial run,
we try to keep the same magnetic energy injection rate, we have used a smaller
magnetic field injection coefficient $B_{\rm inj,0}=0.054$. We found the jet
propagates faster using this injection field configuration. We therefore have
used a bigger vertical box extent of $L_{z}=800$ while keeping
$L_{x}=L_{z}=150$ in order to accommodate the jet for the same run duration
$t_{f}=1500$. We have also increased the grid size to $300\times 300\times
1600$ to keep the same resolution as that used in the fiducial run.
Figure 8 plots $y=0$ slices of the z component of current density $j_{z}$ at
different times. Notice that the vertical size is twice as that in the
fiducial run, then this jet definitely moves much faster than the fiducial
jet. Compared to the $\alpha=10$ run, the non-linear features appear at a much
later time, at a higher $z$ location, takes longer to grow, and the jet also
has a leaner shape. In the $\alpha=10$ run, the non-axisymmetric modes appear
to grow exponentially from $t\sim 300-500$, while here the instabilities do
not start significant growth after $t\sim 500$. The current is also more
concentrated toward the z-axis, most likely due to increased hoop pressure
resulted from the larger $B_{\phi}$ component.
Figure 9 shows snapshots of $y=0$ plane cut-through for $\rho,P,v_{z},B_{y}$
at late time $t=1350$. Despite the more elongated jet shape, all the plotted
quantities show qualitatively similar behaviors compared to the smaller
$\alpha$ run. The Lorentz factor continues to increase over time and the
highest Lorentz factor achieved in this run is about $\gamma\sim 2.4$. We
suspect this number will increase more as the jet has not developed much non-
linear features at the end of run. However, it is not clear what determines
the terminal $\gamma$ in our models, as it needs a much bigger computational
domain size as well as longer simulation run time.
Figure 10 illustrates the propagation of jet front for $\alpha=40$ case. The
slowing down of jet front does not occur until $t\sim 1200-1300$, much later
compared to the smaller alpha case. Although the injected magnetic energy rate
is the same, the jet propagates with a larger bulk velocity because the
dominant toroidal components, consistent with predictions by the magnetic
tower models (see discussion in the §4).
We have observed similar behavior for total energetics in this model as in the
$\alpha=10$ case, as shown in Figure 11. Similar to the fiducial run, the
total kinetic energy takes over the magnetic energy after the instabilities
grow, and both the kinetic energy and internal energy increase with the
continuous conversion of magnetic energy into these two energies. $E_{\rm
K}>E_{\rm B}$ occurs at a later time compared to the fiducial run, consistent
with the onset of non-linear features. At the end of the simulation, the total
$E_{\rm K}$ is quite similar to the $E_{\rm K}$ in the fiducial run, $E_{\rm
B}$ is $\sim 34\%$ larger than that in the fiducial run, and the total
internal energy is $\sim 27\%$ smaller than in the fiducial run. This smaller
dissipation is also consistent with the later onset of non-linear features.
The smaller energy transition can also be seen from the magnetization
parameter $\sigma$ images. Figure 12 shows $\sigma$ at $y=0$ slices at
different times for this run. It is clear that, when compared to the fiducial
run, the energy transition occurs mainly at a later time too, consistent with
the onset time for the significant non-linear interactions. This means for the
same amount of total magnetic energy injection, when $\alpha$ is larger, the
energy transition will occur further away from the jet launching location.
How about CDIs? Figure 13 plots the snapshots of value of $q$ for the kink
instability limits at $y=0$ slices. For a certain cylindrical current, when
$\alpha$ increases, the $q$ value decreases for the same cylindrical shape.
Therefore, the jet will still be unstable due to the kink instabilities, and
this is what we have observed here.
To see the detailed interplay between axisymmetric and non-axisymmetric modes,
we have calculated the power of first few modes in this model. Figure 14 shows
the growth of mode power of the amplitude of current for this run. Similar to
the lower $\alpha$ model, the dominant non-axisymmetric mode is the $m=1$ kink
mode. Throughout the simulation the axisymmetric $m=0$ mode dominates,
although the $m=1$ mode almost grows to a similar magnitude at the late time,
which introduces the non-linear behaviors. However, the growth rates of non-
axisymmetric modes are smaller compared to the smaller $\alpha$ case. This is
somewhat surprising as the larger $\alpha$ is expected to lead to a stronger
instability. One possible explanation is that, while the linear analysis for
the growth rate of kink instability is based on the ideal setup of a constant
cylindrical current with well-defined geometry and fixed boundaries, here we
are dealing with an evolving jet with continuous magnetic injection at the
center and the jet itself is fast propagating in the vertical direction and
expanding in the transverse directions. Therefore instability analysis from
ideal plasma physics derivation may not be applied directly to our evolving
system. Further discussions on this result are given in §4.
To understand the dependence of the CDI’s on-set on injection parameters, we
also make a run where the poloidal field injection rate is the same as the
fiducial run ($B_{\rm inj,0}=0.2$) while keeping $\alpha=40$ (hence a higher
total magnetic energy injection rate), we find that instabilities grow at a
rate that is more close to that in the fiducial run, and the jet front
propagation speed turn-over occurs earlier, at $t\sim 400$ (see the dashed
line in Figure 10). This indicates that the growth of CDIs and the onset of
nonlinear features in these propagating current jet systems are a complex
process probably depending more on the parameters for the magnetic field
injection profile (both magnitude and shape), and we will explore this more in
the future.
### 3.3 Effect of a Disk
Our simulations show that the magnetic structure expands both along the
$z-$axis and sideways. As the jet is a consequence of accretion, and in the
spatial scales we are considering, the accretion disk should surround and
extend into the injection region. In this section, we use a toy model to
investigate the effect of possible disk confinement and whether the
instabilities will still occur when there is a gas-pressure-supported disk at
the jet base. All the jet parameters are the same as in the fiducial run.
The reason to choose a gas pressure-supported disk instead of a rotation-
supported disk is mainly of numerical consideration. For a more physical
accretion disk with rotation, the simulation requires a much smaller time
step, in order to resolve the disk rotation. We therefore choose a gas
pressure supported disk which is initially in a hydrostatic equilibrium, and
this is numerically much easier than evolving a rotating disk. We are not
modeling the accretion process itself, but focusing on how the gas pressure
will confine the jet shape and whether the disk will affect the instabilities.
We have solved the effective gravitational potential $\Phi_{\rm eff}$ which is
able to hold a gas disk with a density distribution
$\rho(r,z)=\rho_{\rm
bkg}+\frac{\rho_{0}}{(1+r/r_{0})^{3/2}}\exp{(-\frac{z^{2}}{2H^{2}})},$ (17)
where the disk is centered at $x=y=z=0$, $r=(x^{2}+y^{2})^{1/2}$, $\rho_{0}$
is the characteristic disk midplane (defined as $z=0$) gas density, $r_{0}$ is
a characteristic disk radius, and $H$ is the disk scale height. When choosing
$\rho_{0}\gg\rho_{\rm bkg}$, the first term in the density equation can be
omitted. $\Phi_{\rm eff}(r,z)$ can be solved by considering the Euler equation
in the radial and vertical directions. Because there is no rotation and we
seek steady-state solutions, the equations are a set of partial differential
equations (PDE) of a simple form:
$\begin{matrix}\partial_{r}\Phi_{\rm
eff}(r,z)=-\frac{1}{\rho(r,z)}\partial_{r}p,\\\ \partial_{z}\Phi_{\rm
eff}(r,z)=-\frac{1}{\rho(r,z)}\partial_{z}p.\end{matrix}$ (18)
Assuming a simple, constant sound speed $c_{s0}$, the solution of the above
PDE can be obtained by integrating separately along $r$ and $z$ directions.
$\Phi_{\rm eff}(r,z)$ has a form
$\Phi_{\rm
eff}(r,z)=c_{s0}^{2}[\ln(1+(\frac{r}{r_{0}})^{3/2})+\frac{z^{2}}{2H^{2}}].$
(19)
For simplicity we have omitted the constant term. Including a non-trivial
$\rho_{\rm bkg}$ term in the disk density distribution makes solving
$\Phi_{\rm eff}(r,z)$ much more complex.
To set up this disk, we have chosen $\rho_{0}=100$ which is much greater than
the background density in the whole simulation box. We choose $r_{0}=10r_{\rm
inj}$, $H=r_{\rm inj}$, and the same sound speed used for the background gas.
The inner edge of the disk is set at $r_{\rm inj}$ and outer edge of the disk
extends to the edge of the box. The disk is thin in most of the regions except
in the inner few $r_{\rm inj}$. We have tested our effective gravitational
potential $\Phi_{\rm eff}(r,z)$ and the associated disk density distribution
$\rho(r,z)$. In the case of zero injection, our disk can indeed be held in a
hydrostatic equilibrium by the effective potential. After injecting the strong
magnetic flux into the center region, the disk cannot be retained in its
original equilibrium, and will be pushed outward by the strong magnetic
pressure. Again, our emphasis of this toy model is to test whether the
inclusion of a gaseous disk will change the properties of the propagating jet,
especially the path of the return current profile.
Figure 15 shows the current density slices at different times when including
this gas disk. Compared to the fiducial run, near the base of the jet ($z\leq
10r_{\rm inj}$), the jet expands less in the equatorial plane. The return
current is also much closer to the axis in this region (which changes the
magnetic field shape more paraboloidal). The overall shape of the jet
resembles more of an observed astrophysical jet in this situation, with an
opening angle at its base due to the disk confinement. Other quantities are
shown in Figure 16, which gives snapshots of $\rho,P,v_{z},B_{y}$ at the late
time. The disk component can be clearly seen in these snapshots. The magnetic
pressure is gradually pushing the disk outward due to the constant flux
injection, even at the late stage of the simulation: our disk never reaches a
static state in this model and this is due to the fact that we are not
simulating a real accretion event here. However, our simple toy model provides
a glimpse into what a more realistic disk-jet simulation would illustrate in
the future.
More importantly, on the larger vertical distance, the jet displays a very
similar morphology as in the fiducial run. The jet is well collimated, the CDI
grows and non-linear features have developed as jet propagates beyond a few
tens of $r_{\rm inj}$. In Figure 17 we have plotted the propagation of jet
front. It is obvious that jet front has already reached the vertical edge of
the box at $t=1000$. The jet front propagates with a high speed for a longer
duration ($t\sim 450$) than in the fiducial case. After this stage the jet
front propagation slows down but is still slightly faster than in the fiducial
case, most likely due to the extra ”pinch” effect at its base.
For instabilities, from instability criterion and mode power analysis we find
their general properties are quite similar to the fiducial run, although the
instability growth rate is slightly larger. This is not surprising because the
instabilities are driven by the injected current, and how they grow is a
reflection of the intrinsic property of the jet current at large distance,
rather that the environment confinement provided at its base.
For energy transition, Figure 19 shows $\sigma$ at different time. This
illustrates that, even with a disk, at distances far from the disk and
injection region, the instabilities introduce large dissipation and magnetic
energy is transformed into kinetic and thermal energies. We have also
calculated the evolution of energetics of the total box, as shown in Figure
18. We get quite similar results compared to the fiducial run: total kinetic
energy takes over the magnetic energy after the instabilities grow, and both
the total kinetic energy and the total internal energy increase as magnetic
energy is converted into these two energies over time.
We have made additional runs by changing the disk scale height $H$ to a
different value ($H=5r_{\rm inj}$ which sets up a thicker disk), similar
results were obtained.
### 3.4 Resolution Study
In order to illustrate the effects of resolution, we have re-run the fiducial
case with a higher resolution $N_{x}=N_{y}=450,N_{z}=1200$, while keeping all
other parameters unchanged. Figure 20 shows the $j_{z}$ current density slices
at the $y=0$ plane. Compared to the fiducial run, the non-linear features
appear earlier, already apparent at $t\sim 200$. At the late time, the jet has
a more pronounced “spine”, where large scale wiggles in this spine are visible
near both sides of the jet front. The return current also exhibits asymmetric
morphology, and extends slightly further away from the axis in the equatorial
plane. Recently, Mignone et al. (2010) have studied resolution effects in RMHD
simulations of jets. They also observed that as jet propagates further its
trajectory becomes more curved, moving from the central axis. This effect is
more pronounced in their higher resolution runs. We note our findings are
consistent with their results.
Comparing the jet front location at different times for both runs, we find
that the higher resolution jet propagates first with a similar speed compare
to that in the fiducial run. Its slowing-down point, however, occurs earlier
at $t\sim 150$ due to the early onset of the non-linear stage. After $t\sim
150$, the jet propagates again with the similar speed as in the fiducial run.
This explains why the jet front reaches a lower $z$ height compared to the
fiducial run at the late time.
Although the resolution does not affect much of the overall jet dynamics, it
certainly affects the instabilities. From the mode analysis we find that the
higher resolution simulation also gives an almost doubled growth rate for non-
axisymetric modes, which causes the current profile to become nonlinear at
$t\leq 200$.
Also similar to Mignone et al. (2010), we find more and stronger shocks in the
high resolution run. This introduces more dissipation and gives a larger total
thermal energy. As a result, we also notice that both the total kinetic energy
$E_{\rm K}$ and the total magnetic energy $E_{\rm B}$ are smaller in the
higher resolution run: for example, $E_{\rm B}$ is $\sim 11\%$ smaller than
that in the fiducial run and $E_{\rm K}$ is $\sim 7.6\%$ smaller at $t\sim
600$ when both models are at the non-linear stage. The magnetic-to-kinetic
energy transition still occurs in the higher resolution run. We have plotted
$\sigma$ parameter at $y=0$ plane at different times for this model, as shown
in Figure 21. We can see that at the “spine” region of the jet $\sigma$ is
smaller, indicating higher resolution leading to a more efficient energy
transition.
Lastly, we want to stress that although our higher resolution simulation has
displayed quantitatively similar behaviors as those in the fiducial run, such
as the development of CDIs and the energy transition, our numerical model of
RMHD jet has not shown signs of convergence. The convergence issue is
therefore out of the scope of this paper, and needs further investigation.
## 4 Summary and Discussion
We have carried out new RMHD simulations for Poynting-flux driven jets in AGN
systems. The computational domain is relatively large so that both the
injected magnetic fluxes and their subsequent evolution are contained well
within the simulation domain. The fluxes which are responsible for driving the
jet are injected at the center of the box, with an injection region size
$r_{\rm inj}$. The flux injection rate is continuous and is taken to be
constant. Our injected magnetic fields have an axisymmetric geometry with
close field lines, consisting of a poloidal field plus a dominant toroidal
field component. We follow the propagation of the jet to a few hundreds of
$r_{\rm inj}$ in three dimensions. We proposed to scale the injection region
$r_{\rm inj}$ to $\sim 10^{3}$ gravitational radii of a black hole, thus our
simulations could be relevant to observations of AGN jets on from sub-pc to
tens of pc scales.
We find these jets are well-collimated. They have a concentrated “spine” that
is roughly of the same size of the injection region inside which the majority
of the out-going current is flowing, along with a significant fraction of the
injected poloidal flux. Driven by the strong magnetic pressure gradient in the
$z-$direction, it eventually develops relativistic speeds. The magnetic
structure also expands transversely, though at a much reduced speed. This
sideway expansion is limited by the inertia of the swept-up background
material.
To understand better why the magnetic structure is highly collimated along the
central axis, we consider the force balance in the radial direction for the
fiducial model, at $t=900$, and vertical height $z=40r_{\rm inj}$, as shown in
Figure 22. We choose $z=40r_{\rm inj}$ because at this height the jet is still
quite axisymmetric, has propagated far enough in the vertical direction, and
non-linear features from instabilities are not severe. At this height, the
magnetically dominated part of the jet extends from $x=0$ to $\sim 10r_{\rm
inj}$, with the return current located at $x\sim 40r_{\rm inn}$. The outer
edge of the hydrodynamic shock is located at $x\sim 55r_{\rm inj}$. The left
panel of Figure 22 shows that inside $x\sim 10r_{\rm inj}$, magnetic pressure
$p_{\rm m}$ dominates over gas pressure $p$ $(\beta\ll 1)$ while both keep a
relative flat distribution along the radial direction; outside $x\sim 10r_{\rm
inj}$, magnetic pressure starts to drop quickly while gas pressure continues
to rise until $x\sim 15r_{\rm inj}$. We can compare this result to the
analysis of non-relativistic MHD simulation of Nakamura et al. (2006) (their
Figure 10). At large radial distances, $x\sim 55r_{\rm inj}$, since the plasma
pressure is much larger than the background pressure $\sim 10^{-4}$, the
radial expansion of the jet structure is limited by plasma inertial.
The right panel shows the various forces in the radial direction: near the
inner jet edge, in the $10r_{\rm inj}\leq x\leq 15r_{\rm inj}$ region, the
dominant force is the outward magnetic pressure gradient $F_{\rm
mp}=-\partial_{r}(B_{\phi}^{2}+B_{z}^{2})$, and there is also a smaller inward
magnetic tension force $F_{\rm t}=-B_{\phi}^{2}/r$. The sum of the two, the
total Lorentz force $F_{\rm\bf J\times B}$ is slightly larger than the inward
gas pressure gradient $F_{\rm p}=-\partial_{r}p$, although the magnitudes of
the two are comparable. Inside $x\sim 10r_{\rm inj}$, the largest force is the
inward magnetic tension force $F_{\rm t}$ provided by the strong toroidal
field, which gives a pinch effect. This effect is largely consistent with the
effects of magnet hoop stress in the “magnetic tower” models (Lynden-Bell,
1996, 2003; Li et al., 2001). There is a small rotation of gas that has also
been produced near the axis as seen by a non-trivial outward centrifugal force
$F_{\rm c}=\gamma\rho v_{\phi}^{2}/r$. Further out from the jet axis, all the
magnetic forces varnish and we can see a few hydrodynamic shock wave fronts.
It is also worth pointing out that although our jet is magnetically dominated
(see the magenta curve, sum of gas pressure gradient and Lorentz force $F_{\rm
total}$), it is not exactly force-free, as ${\bf J\times B}$ is not exactly
zero inside the jet (black dotted curve). Furthermore, the non-zero total
force also implies that the jet is not in a force balance.
The jets we have obtained in these simulations are mildly relativistic, with
the largest Lorentz number about $\gamma\sim 3$ (although the jet front is
slowed down by the shocks), while the small amount of injected mass has an
injection velocity of $v_{\rm inj}=0.5c$ initially. Acceleration is therefore
achieved through magnetic processes and we have observed $\gamma_{\rm max}$
increases with time with no signs of slowing-down. Due to the limit of the
computational resources, we have not yet been able to determine the terminal
speed of the jet in our models. However, it is plausible that a higher flux
injection rate and/or a higher $\alpha$ can lead to a higher speed. Another
issue is purely numerical: in RMHD/GRMHD simulations there is a small amount
of mass loading, and the choice of density floor probably affects strongly
$\gamma_{\rm max}$ (e.g. McKinney & Gammie (2004)). In our simulations we have
also injected a small amount of gas in the injection region (see discussion
below), which helps us to maintain the validity of the RMHD integration
scheme, especially in the injection region where the magnetic field is the
strongest.
These jets also display current-driven instabilities and undergo subsequent
strong dissipations. However, the jets are not disrupted and are able to
propagate to large distances in our simulations. The cylindrical jet current
is unstable most to the $m=1$ kink mode, which undergoes an initial period of
exponential growth. Depending on the model parameters, outside a few tens to
hundreds of $r_{\rm inj}$, the mode growth slows down and the non-linear
interaction among the modes leads to apparent non-linear features such as
filaments in the current and occasional large scale “wiggles” in the jet
spine. Large amounts of dissipation are also introduced outside this region.
As a consequence, as the jet propagates further away from its launching
location, much of magnetic energy has been transformed into jet kinetic energy
and heat, although the jet is still collimated and continues to propagate,
albeit at a slower speed. We notice that although the $m=1$ mode grows
exponentially, its power remains smaller than the power in the $m=0$ mode
throughout the simulations. This is consistent with the fact that the jet is
not disrupted even with CDI present. Such non-disruption behavior of jet is
consistent with the past RMHD simulations. These results also support the idea
some other mechanisms may be at work to suppress the non-linear impact of CDIs
(e.g., Narayan et al. (2009)).
We suggest that the ability of jets to avoid the complete disruption is due to
both the rapid jet propagation and the fast evolution of the associated
underlying magnetic structure, which we collectively term “dynamic
stabilization”. Away from the injection region, the Alfvén speed in the
magnetized region decreases from $\sim 0.9c$ near the central spine to $\sim
0.2-0.6c$ near the boundaries. The background flow (except that near the jet
front), however, still has a relativistic speed of $>0.9c$. It is therefore
possible that this fast background flow has modified the physical quantities
faster than the instability growth timescale. The same arguments can be
applied to the large $\alpha$ runs when the magnetic structure tends to evolve
even faster. In other words, the CDIs developed in our simulated jets are
quite convective, rather than being absolute instability. To the extent we can
simulate the jet propagation, it remains collimated and propagating at a
steady speed. It therefore remains to be seen how dynamical stabilization will
continue to help jets survive the instabilities and whether the environmental
factors may play some additional roles in determining the fate of relativistic
jets.
We have also shown that as these current-carrying jets propagate far from the
injection region, magnetic energy can be transformed into kinetic energy of
the jet and also generates heat. The magnetization parameter $\sigma$,
although much larger than one at the jet base, can become much smaller with
$\sigma\ll 1$ in the region where CDIs have grown to display non-linear
features. Note that in our model the smaller $\sigma$ is not a result of the
jet shocking on the external medium, but a consequence of development of CDIs
in a current-carrying jet. Although many non-linear features of CDIs appear in
our models, the model has not reached a saturated state: all the energetics in
the models still increase over time and it is not clear what the jet dynamics
will be on an even longer time scale. Future simulations of larger
computational domain with longer evolution time are needed to give a more
comprehensive picture of the $\sigma$ question. Recent local simulations of
CDIs by O’Neill et al. (2012) have also shown that development of CDIs are
able to convert magnetic energy into kinetic energy and thermal energy, and
they also have not found a saturated state. Nevertheless, all these
simulations are starting to show that CDIs are indeed able to tackle the
$\sigma$ problem.
In our high resolution run we have observed some large scale wiggles near the
jet fronts444These wiggles are also seen in model with a thicker disk, with
the same resolution with the fiducial run, although not shown in the paper. In
the future it would be interesting to see whether these models are able to
produce knots and spots along the jet axis, which are often observed in AGN
jets. All our models also produce a central current (“spine”) along the
vertical axis, and a cocoon-like return current which locates at a large
distance from the jet axis and encloses the central jet. In the high
resolution run this return current also exhibits non-axisymmetric features.
These return currents have also been produced in the past MHD simulations
(e.g. Ustyugova et al. (2006); Li et al. (2006); Nakamura et al. (2006, 2007,
2008)). It would be interesting to see whether these large scale return
currents are observable (e.g. Kronberg et al. (2011)). Time-dependent jet
properties produced in this work, when combined with radiative processes, can
also be used to compare with observational features of AGN jets, such as their
time variability555The time resolution of our simulation is on the order of
days.. This work marks our first effort toward producing AGN jet diagnostics
from a numerical RMHD model.
It would be useful to scale the model parameters for a supermassive black hole
system. As discussed in §2, for a $3\times 10^{9}{\rm\,M_{\odot}}$ black hole
as the one at the center of M87, we have a magnetic energy injection rate of
$5.2\times 10^{46}{\rm\,ergs^{-1}}$ (the Eddington luminosity is $L_{\rm
Edd}\sim 3.9\times 10^{47}{\rm\,ergs^{-1}}$). This current-carrying jet can
propagate from its injection region of size $r_{\rm inj}\sim 0.14{\rm\,pc}$ to
a distance of $\sim 28{\rm\,pc}$ in the fiducial model, and to a distance of
$\sim 56$ pc in the $\alpha=40$ model, without being disrupted. The features
of CDIs show up on the pc scales. The magnetic field has a strength that is on
the order of $10^{-3}{\rm G}$ in the jet axis and far from the core. The total
current is estimated to be $I\sim 10^{18}{\rm amp}$ in the fiducial model. For
the background gas, we have adopted a uniform background density of
$10^{2}{\rm cm}^{-3}$ and temperature of $5{\rm keV}$. We will explore the
effect of background profile in the future investigation. We have also
injected a small amount of gas in the injection region, and in the fiducial
model the mass injection rate is $\dot{M}_{\rm inj}\sim
0.09{\rm\,M_{\odot}}{\rm yr^{-1}}$, which is much smaller than the Eddington
accretion rate $\dot{M}_{\rm Edd}\sim 13{\rm\,M_{\odot}}{\rm yr^{-1}}$.
(Usually we need to inject more mass if the magnetic energy injection rate is
increased due to numerical reasons.) Lastly, for the resolution, in the
fiducial model the smallest length scale is $\Delta l\sim 0.01{\rm\,pc}$ and
the smallest time scale is $\Delta t\sim 20$ days. Note this time scale is
still long compared to the time scale on which the TeV flares operate.
Therefore, pushing to higher resolution deserves more efforts in future
studies.
We have investigated the fiducial model with two different resolutions, and
both exhibit qualitatively similar behaviors. However, the convergence is not
achieved: this is especially true for the instability and the shocks; effect
of resolution on energy transition is not clear yet. We will leave the even
higher resolution studies to the future work. We have also investigated a
model with a higher toroidal-to-poloidal injection ratio. The details of the
injection function definitely affect jet properties. In the future, we will
explore more model parameters including magnetic field geometries, injection
functions, and external environment profiles (e.g. power-law external pressure
profiles used in Komissarov et al. (2009)).
We have chosen an injection model that has closed poloidal field lines, which
causes $B_{z}$ change directions beyond $r_{\rm inj}$ with no net-flux.
Different field injection configuration exists. For example, past GRMHD black
hole accretion simulations have explored models with initial configurations
with open field lines/net flux (e.g. ”Magnetically Arrested Disc” models).
However, whether the disk has net-flux or not is a un-resolved question,
largely owing to our lack of knowledge of disk dynamo. Since these past
simulations have not typically produced the jet structure at large scales
where comparison with observations becomes more feasible, it is therefore of
interest to explore the case with zero net-flux. Furthermore, studies of
large-scale jets in the intra-cluster medium (hereafter ICM; Li et al. (2006);
Nakamura et al. (2006, 2007, 2008)) have argued that magnetic tower model
provides good fits to observations of jet s morphologies in the ICM. Future
work is therefore needed to explore different initial field configurations and
their consequences in jet stability and dissipation.
Lastly, we want to point out that recently there has been great progress in
the laboratory experiments to study current-driven instabilities in jets (e.g.
Hsu & Bellan (2005); Bergerson et al. (2006)). Although the physical
conditions in our AGN jet models differ greatly from the parameters in
laboratory jets (e.g. density, current etc.), it would be of great interest to
see whether laboratory plasma experiments can teach us the general principles
in understanding astrophysical jets.
The authors are grateful to Stirling Colgate, Brenda Dingus, Ken Fowler, John
Hawley, Philipp Kronberg, and Masanori Nakamura for discussions. We also thank
the anonymous referee for insightful suggestions. This work is supported by
the LDRD and Institutional Computing Programs at LANL and by DOE/Office of
Fusion Energy Science through CMSO.
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Table 1: Units of Physical Quantities For Normalization Physical Quantities | Description | Normalization Units | Typical Values
---|---|---|---
$r$ $[=(x^{2}+y^{2}+z^{2})^{1/2}]$ | Length | $r_{\rm inj}$ | 0.143pc
$v$ | Velocity field | c | $3.0\times 10^{10}{\rm\,cm\,s^{-1}}$
$t$ | Time | $r_{\rm inj}/c$ | 0.47yr
$\rho$ | Density | $\rho_{0}$ | $1.67\times 10^{-22}{\rm\,g\,cm^{-3}}$
$P$ | Pressure | $\rho_{0}c^{2}$ | $0.15{\rm dyncm}^{-2}$
$B$ | Magnetic field | $(4\pi\rho_{0}c^{2})^{1/2}$ | $1.38{\rm G}$
$E$ | Energy | $B^{2}r_{\rm inj}^{3}/(8\pi)$ | $6.52\times 10^{51}{\rm\,erg}$
$P$ | Power | $B^{2}r_{\rm inj}^{2}c/(8\pi)$ | $4.42\times 10^{44}{\rm\,ergs^{-1}}$
Figure 1: Snapshots of $j_{z}$ for the fiducial model showing the evolution of
jet propagation and expansion. These snapshots are taken at the $y=0$ plane of
a 3D simulation, at $t=225,450,675,900,1125,1350$, respectively. The spatial
scales are normalized by $r_{\rm inj}$. Magnetic energy and flux are injected
at the origin $x=y=z=0$ within $r=1$. The magnetic structure expands to form
both a collimated jet along the $z-$axis that carries a strong (positive)
outgoing current (indicated by the white-red-yellow color) and a “cocoon”
enclosing the jet structure with the (negative) return current (blue color).
The jet continues to propagate despite becoming unstable. With the
instability, both the outgoing and return current paths show complicated
structures, although the overall outgoing and return current patterns remain.
Figure 2: Snapshots of $\rho,P,v_{z},B_{y},B_{z}$ at a relatively late time
$t=1125$ and at the $y=0$ plane for the fiducial run. The expansion is
obviously much faster along the vertical direction than that in the transverse
direction. A strong hydrodynamic shock is formed all around the (mildly)
relativistically expanding outer boundary. The jet velocity is relativistic
along the $z-$axis with $\gamma\sim$ a few but slows down significantly near
the jet fronts. Magnetic fields fill up the volume enclosed by the swept-up
hydrodynamic shell. The poloidal field dominates along the $z-$ axis but
toroidal field dominates elsewhere.
Figure 3: The location of jet front along the $z-$ direction as a function of
time in the fiducial run. The time when the jet slows down ($t\sim 300$) is
consistent with the appearance of instabilities as shown in Fig. 1. Figure 4:
Evolution of different energy components of the fiducial run. Solid lines
denote volume integrated energy and the dotted lines denote time and volume
integrated injected energy. Black solid: $E_{\rm B}$; blue solid:$E_{\rm K}$;
red solid: $E_{\rm U}$; magenta solid: $E_{\rm tot}$. Black dotted: $E_{\rm
B,inj}$; blue dotted:$E_{\rm K,inj}+E_{\rm U,inj}$; magenta dotted: $E_{\rm
tot,inj}+E_{\rm tot,0}$. The flattening at $t\sim 1000$ is due to energy
flowing out of the computational domain. Even though the injected energy is
predominantly magnetic, it gets converted into kinetic and thermal energies.
So, the jet appears as having a large amount of kinetic and thermal energy.
Note that the plotted quantities are volume integrated. In localized regions
such as jet’s axis, magnetic energy can still be comparable to other energy
components.
Figure 5: Snapshots of $\sigma$ for the fiducial model. Similar to Fig. 1,
snapshots are taken from $t=225,450,675,900,1125,1350$, at $y=0$ plane. Large
$\sigma$ indicates magnetic energy domination.
Figure 6: Snapshots of $q$ for the fiducial model. These snapshots are taken
from $t=225,450,675,900,1125,1350$, at $y=0$ plane. $q<1$ denotes where the
current is unstable to the kink mode.
Figure 7: Evolution of various mode power in the current distribution for the
fiducial run. Solid lines: $m=0$; dotted lines: $m=1$; dashed lines: $m=2$.
The axisymmetric component remains dominant throughout the jet evolution. The
non-axisymmetric modes show exponential growth but relatively low saturation
level at the nonlinear stage.
Figure 8: Similar to Fig. 1 but with snapshots of $j_{z}$ for the $\alpha=40$
model. These snapshots are taken from $t=225,450,675,900,1125,1350$, at the
$y=0$ plane. The jet is much strongly collimated, presumably due to the
stronger $B_{\phi}$ injections.
|
---|---
|
Figure 9: Similar to Fig. 2 but with snapshots of $\rho,P,v_{z},B_{y}$ at late
time for the $\alpha=40$ run. These snapshots are taken from $t=1350$, at
$y=0$ plane. Figure 10: The location of jet front as a function of time in
the $\alpha=40$ runs. Jet slows down after the non-axisymmetric modes become
significant compared to the axisymmetric mode. Solid: $\alpha=40$ with the
same total magnetic energy injection rate as the fiducial run; dotted:
$\alpha=10$ fiducial run; dashed: $\alpha=40$ but with a larger magnetic
energy injection rate. Figure 11: Energetics of the $\alpha=40$ run. Color
scheme is the same as in Figure 4.
Figure 12: Snapshots of $\sigma$ for the $\alpha=40$ model. These snapshots
are taken from $t=225,450,675,900,1125,1350$, at $y=0$ plane.
Figure 13: Snapshots of $q$ for the $\alpha=40$ model. These snapshots are
taken from $t=225,450,675,900,1125,1350$, at $y=0$ plane. $q<1$ denotes when
the current is unstable to the kink mode.
Figure 14: Evolution of mode power in the current for the $\alpha=40$ run.
Solid lines: $m=0$; dotted lines: $m=1$; dashed lines: $m=2$.
Figure 15: Snapshots of $j_{z}$ for the disk model. These snapshots are taken
from $t=225,450,675,900,1125,1350$, at $y=0$ plane.
|
---|---
|
Figure 16: Snapshots of $\rho,P,v_{z},B_{y}$ at late time for the disk run.
These snapshots are taken from $t=900$, at $y=0$ plane. Figure 17: The
location of jet front as a function of time in the disk run (solid line). Jet
slows down after the nonlinear modes start to grow. Dash line: jet front
locations in the fiducial run. Figure 18: Energetics of the disk run. Color
scheme is the same as in Figure 4.
Figure 19: Snapshots of $\sigma$ for the disk model. These are taken from
$t=225,450,675,900,1125,1350$, at $y=0$ plane.
Figure 20: Snapshots of $j_{z}$ for the fiducial model with higher
resolutions. Snapshots are taken from $t=225,450,675,900,1125,1350$ at $y=0$
plane.
Figure 21: Snapshots of magnetization parameter $\sigma$ for the fiducial
model with higher resolutions. These snapshots are taken from
$t=225,450,675,900,1125,1350$, at $y=0$ plane.
.
Figure 22: Radial profiles of physical quantities along the x-axis in the
equatorial plane with $(y,z)=(0,40)$ at $t=900$ in the fiducial run. Left:
pressures in the radial direction. Black: magnetic pressure $p_{\rm m}$; red:
gas pressure $p$; magenta: total pressure $p+p_{\rm m}$. Right: forces in the
radial direction. Black solid: magnetic pressure gradient $F_{\rm mp}$; red:
gas pressure gradient $F_{\rm p}$ ; blue: centrifugal force $F_{\rm c}$;
green: magnetic tension force $F_{\rm t}$; black dotted: sum of magnetic
pressure gradient and tension force $F_{\rm\bf J\times B}=F_{\rm mp}+F_{\rm
t}$; magenta: total of magnetic forces and gas pressure gradient $F_{\rm
total}=F_{\rm\bf J\times B}+F_{\rm p}$.
|
arxiv-papers
| 2013-12-02T23:41:42 |
2024-09-04T02:49:54.672111
|
{
"license": "Public Domain",
"authors": "Xiaoyue Guan, Hui Li, Shengtai Li",
"submitter": "Xiaoyue Guan",
"url": "https://arxiv.org/abs/1312.0661"
}
|
1312.0679
|
aainstitutetext: Department of Physics, Institute of Theoretical Physics,
Beijing Normal University,
Beijing, 100875, Chinabbinstitutetext: Institute of Theoretical Physics,
Zhanjiang Normal University,
Zhanjiang, Guangdong, 524048, China
# Phase transitions, geometrothermodynamics and critical exponents of black
holes with conformal anomaly
Jie-Xiong Mo a,1 Wen-Biao Liu 111Corresponding author [email protected]
[email protected]
###### Abstract
Conformal anomaly is an important concept which has various applications in
quantum field theory in curved space-time, string theory, black hole physics
and cosmology. Probing its influences in phase transitions of black holes is
of great physical importance. In this paper, we achieve this goal by
investigating the phase transitions of black holes with conformal anomaly in
canonical ensemble from different perspectives. Some interesting and novel
phase transition phenomena has been discovered. Firstly, we discuss the
behavior of the specific heat and the inverse of the isothermal
compressibility. It is shown that there are striking differences in Hawking
temperature and phase structure between black holes with conformal anomaly and
those without it. In the case with conformal anomaly, there exists local
minimum temperature corresponding to the phase transition point; Phase
transitions take place not only from an unstable large black hole to a locally
stable medium black hole but also from an unstable medium black hole to a
locally stable small black hole. Secondly, we probe in details the dependence
of phase transitions on the choice of parameters. The results show that black
holes with conformal anomaly have much richer phase structure than those
without it. There would be two, only one or no phase transition points
depending on the parameters we have chosen. The corresponding parameter region
are derived both numerically and graphically. Thirdly, geometrothermodynamics
are built up to examine the phase structure we have discovered. It is shown
that Legendre invariant thermodynamic scalar curvature diverges exactly where
the specific heat diverges. Furthermore, critical behaviors are investigated
by calculating the relevant critical exponents. And we proved that these
critical exponents satisfy the thermodynamic scaling laws, leading to the
conclusion that critical exponents and the scaling laws do not change even
when we consider conformal anomaly.
## 1 Introduction
Black hole thermodynamics has long been one of exciting and challenging
research fields ever since the pioneer research made by Bekenstein and Hawking
Bekenstein -Hawking1 . A variety of thermodynamic quantities of black holes
has been studied. In 1983, Hawking and Page Hawking2 discovered that pure
thermal radiation in AdS space becomes unstable above certain temperature and
collapses to form black holes. This is the well-known Hawking-Page phase
transition which describes the phase transition between the Schwarzschild AdS
black hole and the thermal AdS space. This phenomenon can be interpreted in
the AdS/CFT correspondence Maldacena9999 as the confinement/deconfinement
phase transition of gauge field Witten9999 . Since then, phase transitions of
black holes have been investigated from different perspectives. For recent
papers, see Sahay -Wenbiao1 .
One of the elegant approach is the thermodynamic geometry method, which was
first introduced by Weinhold Weinhold and Ruppeiner Ruppeiner . Weinhold
proposed metric structure in the energy representation as
$g_{i,j}^{W}=\partial_{i}\partial_{j}M(U,N^{a})$ while Ruppeiner defined
metric structure as $g_{i,j}^{R}=-\partial_{i}\partial_{j}S(U,N^{a})$. These
metric structures are respectively the Hessian matrix of the internal energy
$U$ and the entropy $S$ with respect to the extensive thermodynamic variables
$N^{a}$. And Weinhold’s metrics were found to be conformally connected to
Ruppeiner’s metrics through the map $dS^{2}_{R}=\frac{dS^{2}_{W}}{T}$ Janyszek
. Ruppeiner’s metric has been applied to investigate various thermodynamics
systems for its profound physical meaning. For more details, see the nice
review Ruppeiner2 . Recently Quevedo et al. Quevedo2 presented a new
formalism called geometrothermodynamics, which allows us to derive Legendre
invariant metrics in the space of equilibrium states. Geometrothermodynamics
presents a unified geometry where the metric structure describes various types
of black hole thermodynamics Quevedo3 -Wenbiao2 .
Apart from the thermodynamic geometry, critical behavior also plays a crucial
role in the study of black hole phase transitions. The critical points of
phase transitions are characterized by the discontinuity of thermodynamic
quantities. So it is important to investigate the behavior in the neighborhood
of the critical point, especially the divergences of various thermodynamic
quantities. In classical thermodynamics, this goal is achieved by taking into
account a set of critical exponents, from which we can gain qualitative
insights into the critical behavior. These critical exponents are found to be
universal to a large extent (only depending on the dimensionality, symmetry
etc) and satisfy scaling laws, which can be attributed to scaling hypothesis.
Critical behavior of black holes accompanied with their critical exponents
have been investigated not only in asymptotically flat space time Davies
-Arcioni1 but also in the de Sitter and anti de Sitter space Muniain -Liu99 .
In this paper, we would like to focus our attention on the critical behavior
and geometrothermodynamics of static and spherically symmetric black holes
with conformal anomaly. As we know, conformal anomaly, an important concept
with a long history, has various applications in quantum field theory in
curved spaces, string theory, black hole physics and cosmology. So it is worth
probing its influences in phase transitions of black holes. Recently, Cai et
al. Cai9999 found a class of static and spherically symmetric black holes
with conformal anomaly, whose thermodynamic quantities were also investigated
in the same paper. It was found that there exists a logarithmic correction to
the well-known Bekenstein-Hawking area entropy. This discovery is quite
important in the sense that with this term one is able to compare black hole
entropy up to the sub-leading order, in the gravity side and in the
microscopic statistical interpretation side Cai9999 . Based on the metrics in
that paper, phase transitions of a spherically symmetric Schwarzschild black
hole have been investigated by taking into account the back reaction through
the conformal anomaly of matter fields recently Son9999 . It has been shown
that there exists an additional phase transition to the conventional Hawking-
Page phase transition. The entropy of these black holes has also been
investigated by using quantum tunneling approach Liran . Moreover, Ehrenfest
equation has been applied to investigate this class of black holes Chenghongbo
and it has been found that the phase transition is a second order one. Despite
of these achievements, there are still many issues left to be explored. Ref.
Son9999 mainly focus on the uncharged case . So it is natural to ask what
would happen to the charged black holes. Ref. Chenghongbo concentrated their
efforts on the Ehrenfest equation in the grand canonical ensemble. So it is
worthwhile to study the phase transition in canonical ensemble. The dependence
of the phase structure on the parameter deserves to be further investigated.
One may also wonder whether the thermodynamic geometry and scaling laws still
works to reveal the phase structure and critical behavior when conformal
anomaly is taken into consideration. Motivated by these, we would like to
investigate the phase transition, geometrothermodynamics and critical
exponents in canonical ensemble.
The organization of our paper is as follows. In Sec. 2, the thermodynamics of
black holes with conformal anomaly will be briefly reviewed. In Sec. 3, phase
transitions will be investigated in details in canonical ensemble and some
interesting and novel phase transition phenomena will be disclosed. In Sec. 4,
geometrothermodynamics will be established to examine the phase structure we
find in Sec. 3. In Sec. 5, critical exponents will be calculated and the
scaling laws will be examined. In the end, conclusions will be drawn in Sec.
6.
## 2 A brief review of thermodynamics
The static and spherically symmetric black hole solution with conformal
anomaly has been proposed as Cai9999
$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}(d\theta^{2}+sin^{2}\theta
d\varphi^{2}),$ (1)
where
$f(r)=1-\frac{r^{2}}{4\tilde{\alpha}}(1-\sqrt{1-\frac{16\tilde{\alpha}M}{r^{3}}+\frac{8\tilde{\alpha}Q^{2}}{r^{4}}}\,).$
(2)
The Newton constant $G$ has been set to one. Both $M$ and $Q$ are integration
constants. And the coefficient $\tilde{\alpha}$ is positive. The physical
meanings of $M$ and $Q$ were discussed in Ref. Cai9999 . $M$ is nothing but
the mass of the solution while $Q$ should be interpreted as $U(1)$ charge of
some conformal field theory.
When $M=Q=0$, the metric above reduces to
$ds^{2}=dt^{2}-dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}),$ (3)
implying that the vacuum limit is the Minkowski space-time.
In the large $r$ limit, Eq.(2) becomes
$f(r)\approx 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+O(r^{-2}),$ (4)
which behaves like the Reissner-Norström solution.
When $\tilde{\alpha}\rightarrow 0$, Eq.(2) reduces into
$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (5)
Eqs.(1) and (5) consist of the metric of Reissner-Norström black hole.
Solving the equation $f(r)=0$, we can get the radius of black hole horizon
$r_{+}$, with which the mass of the black hole can be expressed as
$M=\frac{r_{+}}{2}+\frac{Q^{2}}{2r_{+}}-\frac{\tilde{\alpha}}{r_{+}}.$ (6)
The Hawking temperature can be derived as
$T=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{r_{+}^{2}+2\tilde{\alpha}-Q^{2}}{4\pi
r_{+}(r_{+}^{2}-4\tilde{\alpha})}.$ (7)
The potential difference between the horizon and the infinity can be written
as
$\Phi=\frac{Q}{r_{+}}.$ (8)
The entropy was reviewed in Ref. Chenghongbo as
$S=\pi r_{+}^{2}-4\pi\tilde{\alpha}ln{r_{+}^{2}}.$ (9)
## 3 Novel phase transition phenomena
In this section, we would like to investigate the phase transition of black
holes with conformal anomaly in canonical ensemble where the charge of the
black hole is fixed.
The corresponding specific heat can be calculated as
$C_{Q}=T(\frac{\partial S}{\partial
T})_{Q}=\frac{2\pi(r_{+}^{2}-4\tilde{\alpha})^{2}(Q^{2}-2\tilde{\alpha}-r_{+}^{2})}{r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}.$
(10)
Apparently, $C_{Q}$ may diverge when
$r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})=0,$
(11)
which suggests a possible phase transition. However, the phase transition
point characterized by Eq.(11) is not apparent. To gain an intuitive
understanding, we plot Fig.1 using Eq.(10). To check whether the phase
transition point locates in the physical region, we also plot the Hawking
temperature using Eq.(7) in Fig.1. It is shown that the phase transition point
locates in the positive temperature region. From Fig.1 and Fig.1, we find that
there have been striking differences between the case $\tilde{\alpha}\neq 0$
and the case $\tilde{\alpha}=0$. In the case $Q=1,\tilde{\alpha}=0.1$, there
are two phase transition point while there is only one in the case
$\tilde{\alpha}=0$. The temperature in the case $\tilde{\alpha}=0$ increases
monotonically while there exists local minimum temperature in the case
$Q=1,\tilde{\alpha}=0.1$. Fig.1 can be divided into four phases. The first one
is thermodynamically stable ($C_{Q}>0$)with small radius. The second one is
unstable ($C_{Q}<0$)with meidium radius. The third one is thermodynamically
stable ($C_{Q}>0$)with medium radius while the fourth one is thermodynamically
unstable ($C_{Q}<0$)with large radius. So the phase transition take place not
only from an unstable large black hole to a locally stable medium black hole
but also from an unstable medium black hole to a locally stable small black
hole.
Figure 1: (a)$C_{Q}$ vs. $r_{+}$ for $Q=1,\tilde{\alpha}=0.1$ (b)$T$ vs.
$r_{+}$ for $Q=1,\tilde{\alpha}=0.1$
From Fig.1,we notice that the Hawking temperature has a local minimum value.
And the corresponding location can be derived through
$\frac{\partial T}{\partial
r_{+}}=-\frac{r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}{4\pi(r_{+}^{3}-4r_{+}\tilde{\alpha})^{2}}=0.$
(12)
It is quite interesting to note that the numerator of Eq.(12) is the same as
Eq.(11), which implies that the location which corresponds to the minimum
Hawking temperature also witnesses the existence of phase transition.
To probe the dependence of phase transition location on the choice of
parameter, we solve Eq.(11) and obtain the location of phase transition point
as
$r_{c}=\sqrt{\frac{3Q^{2}-10\tilde{\alpha}\pm\sqrt{132\tilde{\alpha}^{2}-76\tilde{\alpha}Q^{2}+9Q^{4}}}{2}}.$
(13)
With Eq.(13) at hand, we plot Fig.2 and Fig.2 which show the influence of
parameters $Q$ and $\tilde{\alpha}$ respectively.
Figure 2: (a)$r_{c}$ vs. $Q$ for $\tilde{\alpha}=0.1$ and $\tilde{\alpha}=0$
(b)$r_{c}$ vs. $\tilde{\alpha}$ for $Q=1$
It can be observed from Fig.2 and Fig.2 that black holes with conformal
anomaly have much richer phase structure than that without conformal anomaly.
When $\tilde{\alpha}=0$, the location of the phase transition $r_{c}$ is
proportional to the charge $Q$. However, the cases of black holes with
conformal anomaly are quite complicated. For $\tilde{\alpha}=0.1$, the curve
can be divided into three regions. Through numerical calculation, we find that
black holes have only one phase transition point when $Q\subset(0,0.4472)$.
When $0.4472<Q<0.7746$, there would be no phase transition at all. When
$Q>0.7746$, there exist two phase transition points, just as what we show in
Fig.1. And the distance between these two phase transition point becomes
larger with the increasing of $Q$. Fig.2 shows the case that the charge $Q$
has been fixed at one. We notice that there would be two phase transition
points when $0<\tilde{\alpha}<\frac{1}{6}$, which is consistent with Fig.1.
And the distance between these two phase transition point becomes narrower
with the increasing of $\tilde{\alpha}$. When
$\tilde{\alpha}\subset(\frac{1}{6},\frac{1}{2})$, there would be no phase
transition. When $\tilde{\alpha}>\frac{1}{2}$, there would be only one phase
transition point. To gain a three-dimensional understanding, we also include a
three dimensional figure of $C_{Q}$ in Fig.3 and Fig.3.
Apart from the specific heat, we would also like to investigate the behavior
of the inverse of the isothermal compressibility, which is defined as
Figure 3: (a)$C_{Q}$ vs. $Q$ and $r_{+}$ for $\tilde{\alpha}=0.1$ (b)$C_{Q}$
vs. $\tilde{\alpha}$ and $r_{+}$ for $Q=1$
$\kappa_{T}^{-1}=Q(\frac{\partial\Phi}{\partial Q})_{T}.$ (14)
Utilizing the thermodynamic identity relation
$(\frac{\partial\Phi}{\partial T})_{Q}(\frac{\partial T}{\partial
Q})_{\Phi}(\frac{\partial Q}{\partial\Phi})_{T}=-1,$ (15)
we obtain
$(\frac{\partial\Phi}{\partial Q})_{T}=-(\frac{\partial\Phi}{\partial
T})_{Q}(\frac{\partial T}{\partial Q})_{\Phi},$ (16)
where the second term on the right hand side can be calculated through
$(\frac{\partial T}{\partial Q})_{\Phi}=(\frac{\partial T}{\partial
r_{+}})_{Q}(\frac{\partial r_{+}}{\partial Q})_{\Phi}+(\frac{\partial
T}{\partial Q})_{r_{+}}.$ (17)
Utilizing Eqs.(7), (8), (14), (16), (17), we obtain the explicit form of
$\kappa_{T}^{-1}$ as
$\kappa_{T}^{-1}=\frac{Qr_{+}^{4}-Q^{3}r_{+}^{2}-4Q^{3}\tilde{\alpha}+10Qr_{+}^{2}\tilde{\alpha}-8Q\tilde{\alpha}^{2}}{r_{+}[r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]}.$
(18)
We show the behavior of $\kappa_{T}^{-1}$ in Fig.4. Comparing Fig.4 with
Fig.1, we find that the inverse of the isothermal compressibility
$\kappa_{T}^{-1}$ also diverges at the critical point.
Figure 4: The inverse of the isothermal compressibility $\kappa_{T}^{-1}$ vs.
$r_{+}$ for $Q=1,\tilde{\alpha}=0.1$
## 4 Geometrothermodynamics
According to geometrothermodynamics Quevedo2 , the $(2n+1)$-dimensional
thermodynamic phase space $\mathcal{T}$ can be coordinated by the set of
independent quantities {$\phi,E^{a},I^{a}$}, where $\phi$ corresponds to the
thermodynamic potential, and $E^{a},I^{a}$ are the extensive and intensive
thermodynamic variables respectively. The fundamental Gibbs 1- form defined on
$\mathcal{T}$ can then be written as $\Theta=d\phi-\delta_{ab}I^{a}dE^{b}$,
where $\delta_{ab}=diag(1,\cdots,1)$. Considering a non-degenerate Riemannian
metric $G$, a contact Riemannian manifold can be defined from the set
$(\mathcal{T},\Theta,G)$ if the condition $\Theta\wedge(\Theta)^{n}\neq 0$ is
satisfied. Utilizing a smooth map $\varphi:\varepsilon\rightarrow\mathcal{T}$,
i.e. $\varphi:(E^{a})\mapsto(\phi,E^{a},I^{a})$, a submanifold $\varepsilon$
called the space of thermodynamic equilibrium states can be induced.
Furthermore, a thermodynamic metric $g$ can be induced in the equilibrium
manifold $\varepsilon$ by the smooth map $\varphi$.
As proposed by Quevedo, the non-degenerate metric $G$ and the thermodynamic
metric $g$ can be written as follows Quevedo7
$G=(d\phi-\delta_{ab}I^{a}dE^{b})^{2}+(\delta_{ab}E^{a}I^{b})(\eta_{cd}dE^{c}dI^{d}),$
(19) $g=\varphi^{*}(G)=(E^{c}\frac{\partial\phi}{\partial
E^{c}})(\eta_{ab}\delta^{bc}\frac{\partial^{2}\phi}{\partial E^{c}\partial
E^{d}}dE^{a}dE^{d}),$ (20)
where $\eta_{ab}=diag(-1,\cdots,1)$.
To construct geometrothermodynamics of black holes with conformal anomaly in
canonical ensemble, we choose $M$ to be the thermodynamic potential and $S,Q$
to be the extensive variables. Then the corresponding thermodynamic phase
space is a 5-dimensional one coordinated by the set of independent
coordinates{$M,S,Q,T,\Phi$}. The fundamental Gibbs 1- form defined on
$\mathcal{T}$ can then be written as
$\Theta=dM-TdS-\Phi dQ.$ (21)
The non-degenerate metric $G$ from Eq.(19) can be changed into
$G=(dM-TdS-\Phi dQ)^{2}+(TS+\Phi Q)(-dSdT+dQd\Phi).$ (22)
Introducing the map
$\varphi:\\{S,Q\\}\mapsto\\{M(S,Q),S,Q,\frac{\partial M}{\partial
S},\frac{\partial M}{\partial Q}\\},$ (23)
the space of thermodynamic equilibrium states can be induced. According to
Eq.(19), the thermodynamic metric $g$ can be written as follows
$g=(S\frac{\partial M}{\partial S}+Q\frac{\partial M}{\partial
Q})(-\frac{\partial^{2}M}{\partial S^{2}}dS^{2}+\frac{\partial^{2}M}{\partial
Q^{2}}dQ^{2}).$ (24)
Utilizing Eqs.(6) and (9), we can easily calculate the relevant quantities in
Eq.(24) as
$\displaystyle\frac{\partial M}{\partial S}$
$\displaystyle=\frac{r_{+}^{2}+2\tilde{\alpha}-Q^{2}}{4\pi
r_{+}(r_{+}^{2}-4\tilde{\alpha})},$ (25) $\displaystyle\frac{\partial
M}{\partial Q}$ $\displaystyle=\frac{Q}{r_{+}},$ (26)
$\displaystyle\frac{\partial^{2}M}{\partial S^{2}}$
$\displaystyle=\frac{8\tilde{\alpha}^{2}-r_{+}^{4}-10r_{+}^{2}\tilde{\alpha}-Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}{8\pi^{2}r_{+}(r_{+}^{2}-4\tilde{\alpha})^{3}},$
(27) $\displaystyle\frac{\partial^{2}M}{\partial Q^{2}}$
$\displaystyle=\frac{1}{r_{+}}.$ (28)
Comparing Eqs.(25),(26) with Eqs.(7),(8), we find
$\frac{\partial M}{\partial S}=T,\quad\frac{\partial M}{\partial Q}=\Phi,$
(29)
which proves the validity of the first law of black hole thermodynamics
$dM=TdS+\Phi dQ$. Substituting Eqs.(25)-(28) into Eq.(24), we can calculate
the component of the thermodynamic metric $g$ as
$\displaystyle g_{SS}=$
$\displaystyle\frac{1}{32\pi^{2}r_{+}^{2}(r_{+}^{2}-4\tilde{\alpha})^{4}}\times[r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]$
$\displaystyle\times[r_{+}^{4}+2r_{+}^{2}\tilde{\alpha}+Q^{2}(3r_{+}^{2}-16\tilde{\alpha})+8\tilde{\alpha}(Q^{2}-r_{+}^{2}-2\tilde{\alpha})ln{r_{+}}],$
(30) $\displaystyle g_{QQ}=$
$\displaystyle\frac{r_{+}^{4}-16Q^{2}\tilde{\alpha}+2r_{+}^{2}\tilde{\alpha}+3Q^{2}r_{+}^{2}+(8Q^{2}\tilde{\alpha}-8r_{+}^{2}\tilde{\alpha}-16\tilde{\alpha}^{2})ln{r_{+}}}{4r_{+}^{2}(r_{+}^{2}-4\tilde{\alpha})}.$
(31)
Utilizing Eqs.(30) and (31), we can obtain the Legendre invariant scalar
curvature as
$\mathfrak{R}_{Q}=\frac{A(x_{+},Q)}{B(x_{+},Q)},$ (32)
where
$\displaystyle B(x_{+},Q)=$
$\displaystyle[r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]^{2}$
$\displaystyle\times[r_{+}^{4}+Q^{2}(3r_{+}^{2}-16\tilde{\alpha})+2r_{+}^{2}\tilde{\alpha}+8\tilde{\alpha}(Q^{2}-r_{+}^{2}-2\tilde{\alpha})ln{r_{+}}]^{3}.$
(33)
Figure 5: Thermodynamic scalar curvature $R_{Q}$ vs. $r_{+}$ for
$Q=1,\tilde{\alpha}=0.1$
The numerator of the Legendre invariant scalar curvature is too lengthy to be
displayed here. From Eq.(33), we find that the Legendre invariant scalar
curvature shares the same factor
$r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})$
with the specific heat $C_{Q}$ in its denominator , which implies that it
would diverge when
$r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})=0$.
That is the exact condition that the phase transition point satisfies. To get
an intuitive sense on this issue, we plot Fig.5 showing the behavior of
thermodynamic scalar curvature $\mathfrak{R}_{Q}$. From Fig.5, we find that
thermodynamic scalar curvature $\mathfrak{R}_{Q}$ diverges at three locations.
Comparing Fig.5 with Fig.1, we find that the second diverging point which
corresponds to negative Hawking temperature does not have physical meaning.
Furthermore, the first and the third diverging points coincide exactly with
the phase transition point, which can be witnessed by comparing Fig.5 with
Fig.1. So we can safely draw the conclusion that the Legendre invariant metric
constructed in geometrothermodynamics correctly produces the behavior of the
thermodynamic interaction and phase transition structure of black holes with
conformal anomaly.
## 5 Critical exponents and scaling laws
In order to have a better understanding of the phase transition of black holes
with conformal anomaly, we would like to investigate their critical behavior
near the critical point by considering a set of critical exponents in this
section.
Before embarking on calculating critical exponents, we would like to reexpress
physical quantities near the critical point as
$\displaystyle r_{+}$ $\displaystyle=r_{c}(1+\Delta),$ (34) $\displaystyle
T(r_{+})$ $\displaystyle=T_{c}(1+\varepsilon),$ (35) $\displaystyle Q(r_{+})$
$\displaystyle=Q_{c}(1+\eta),$ (36)
where $|\Delta|\ll 1,|\varepsilon|\ll 1,|\eta|\ll 1$. Note that the footnote
”c” in this section denotes the value of the physical quantity (or the
expression) at the critical point. For example, $T_{c}$ corresponds to the
temperature at the critical point.
Critical exponent $\alpha$ is defined through
$C_{Q}\sim|T-T_{c}|^{-\alpha}.$ (37)
To obtain $T-T_{c}$, we would like to carry out Taylor expansion as below
$T(r_{+})=T_{c}+[(\frac{\partial T}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})+\frac{1}{2}[(\frac{\partial^{2}T}{\partial
r_{+}^{2}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})^{2}+higher\,order\,terms,$
(38)
from which we obtain
$\Delta=\frac{1}{r_{c}}\sqrt{\frac{2\varepsilon T_{c}}{D}},$ (39)
where
$D=[(\frac{\partial^{2}T}{\partial
r_{+}^{2}})_{Q=Q_{c}}]_{r_{+}=r_{c}}=\frac{r_{c}^{6}+24r_{c}^{4}\tilde{\alpha}-24r_{c}^{2}\tilde{\alpha}^{2}+32\tilde{\alpha}^{3}-2Q_{c}^{2}(3r_{c}^{4}-6r_{c}^{2}\tilde{\alpha}+8\tilde{\alpha}^{2})}{2\pi(r_{c}^{3}-4r_{c}\tilde{\alpha})^{3}}.$
(40)
In the above derivation, we have considered the fact that $C_{Q}$ diverges at
the critical point, making the second term at the right hand side of Eq.(38)
vanish. Substituting Eq.(34) into Eq.(10) and keeping only the linear terms in
its denominator, we obtain
$C_{Q}\simeq\frac{2\pi(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{\Delta(4r_{c}^{4}+20r_{c}^{2}\tilde{\alpha}-6Q_{c}^{2}r_{c}^{2})},$
(41)
which can be transformed via Eq.(39) into
$C_{Q}\simeq\frac{\pi\sqrt{2D}(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{(4r_{c}^{3}+20r_{c}\tilde{\alpha}-6Q_{c}^{2}r_{c})(T-T_{c})^{1/2}},$
(42)
Comparing Eq.(42) with Eq.(37), we can obtain $\alpha=1/2$.
Critical exponent $\beta$ is defined through the following relation when $Q$
is fixed,
$\Phi(r_{+})-\Phi(r_{c})\sim|T-T_{c}|^{\beta}.$ (43)
The above definition motivates us to carry out the Taylor expansion as
$\Phi(r_{+})=\Phi_{c}+[(\frac{\partial\Phi}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms,$ (44)
Utilizing Eq.(8), (44) and neglecting higher order terms of Eq.(43), we get
$\Phi(r_{+})-\Phi_{c}=[(\frac{\partial\Phi}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}\sqrt{\frac{2}{D}}(T-T_{c})^{1/2}=-\frac{Q_{c}}{r_{c}^{2}}\sqrt{\frac{2}{D}}(T-T_{c})^{1/2}.$
(45)
Comparing Eq.(43) with Eq.(45), we can obtain $\beta=1/2$.
Critical exponent $\gamma$ is defined through the following relation
$\kappa_{T}^{-1}\sim|T-T_{c}|^{-\gamma}.$ (46)
Substituting Eq.(34) and (39) into Eq.(18) and keeping only the linear term of
$\Delta$, we obtain
$\kappa_{T}^{-1}=\frac{\sqrt{D}(Q_{c}r_{c}^{4}-Q_{c}^{3}r_{c}^{2}-4Q_{c}^{3}\tilde{\alpha}+10Q_{c}r_{c}^{2}\tilde{\alpha}-8Q_{c}\tilde{\alpha}^{2})}{\sqrt{2}[5r_{c}^{4}-8\tilde{\alpha}^{2}+30r_{c}^{2}\tilde{\alpha}+Q_{c}^{2}(4\tilde{\alpha}-9r_{c}^{2})](T-T_{c})^{\frac{1}{2}}}.$
(47)
From Eq.(46) and (47), we find that $\gamma=1/2$
Critical exponent $\delta$ is defined for the fixed temperature $T_{c}$
through
$\Phi(r_{+})-\Phi(r_{c})\sim|Q-Q_{c}|^{1/\delta}.$ (48)
To obtain $Q-Q_{c}$, we would like to carry out Taylor expansion as
$Q(r_{+})=Q_{c}+[(\frac{\partial Q}{\partial
r_{+}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})+\frac{1}{2}[(\frac{\partial^{2}Q}{\partial
r_{+}^{2}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})^{2}+higher\,order\,terms,$ (49)
Utilizing the thermodynamic identity again, we get
$[(\frac{\partial Q}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}=-[(\frac{\partial
T}{\partial r_{+}})_{Q}]_{r_{+}=r_{c}}[(\frac{\partial Q}{\partial
T})_{r_{+}}]_{r_{+}=r_{c}}=0.$ (50)
In the above derivation, we have taken into account the fact that $C_{Q}$
diverges at the critical point, making the first term at the right hand side
of Eq.(38) vanish. Substituting Eq.(34) and Eq.(36) into Eq.(49)and neglecting
the high order terms, we obtain
$\Delta=\sqrt{\frac{2Q_{c}\eta}{Er_{c}^{2}}},$ (51)
where
$E=[(\frac{\partial^{2}Q}{\partial
r_{+}^{2}})_{T}]_{r_{+}=r_{c}}=\frac{22r_{c}^{4}\tilde{\alpha}+32\tilde{\alpha}^{3}+16\tilde{\alpha}^{2}(r_{c}^{2}-Q_{c}^{2})-r_{c}^{4}(3Q_{c}^{2}+r_{c}^{2})}{2Q_{c}(r_{c}^{3}-4r_{c}\tilde{\alpha})^{2}}.$
(52)
Taylor expanding $\Phi$ near the critical point, we get
$\Phi(r_{+})=\Phi_{c}+[(\frac{\partial\Phi}{\partial
r_{+}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms,$ (53)
where the coefficient of the second term on the right hand side can be derived
as follows
$[(\frac{\partial\Phi}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}=[(\frac{\partial
Q}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}[(\frac{\partial\Phi}{\partial
Q})_{r_{+}}]_{r_{+}=r_{c}}+[(\frac{\partial\Phi}{\partial
r_{+}})_{Q}]_{r_{+}=r_{c}}=-\frac{Q_{c}}{r_{c}^{2}}.$ (54)
Utilizing Eq.(51), (53), (54),we get
$\Phi(r_{+})-\Phi_{c}\simeq-\frac{Q_{c}}{r_{c}^{2}}\sqrt{\frac{2(Q-Q_{c})}{E}},$
(55)
from which we can draw the conclusion that $\delta=2$.
Critical exponent $\varphi$ is defined through
$C_{Q}\sim|Q-Q_{c}|^{-\varphi}.$ (56)
Substituting Eq.(51) into Eq.(41), we obtain
$C_{Q}\simeq\frac{\pi
r_{c}\sqrt{2E}(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{\sqrt{Q-Q_{c}}(4r_{c}^{4}+20r_{c}^{2}\tilde{\alpha}-6Q_{c}^{2}r_{c}^{2})},$
(57)
Comparing Eq.(57) and (56), we find that $\varphi=1/2$.
Critical exponent $\psi$ is defined through
$S(r_{+})-S_{c}\sim|Q-Q_{c}|^{\psi}.$ (58)
Taylor expanding $S$ near the critical point, we obtain
$S(r_{+})=S_{c}+[(\frac{\partial S}{\partial
r_{+}})_{Q}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms.$ (59)
Utilizing Eq.(9), (34), (51) and (59), we get
$S(r_{+})-S_{c}\simeq(2\pi
r_{c}-\frac{8\pi\tilde{\alpha}}{r_{c}})\sqrt{\frac{2(Q-Q_{c})}{E}},$ (60)
from which we obtain $\psi=1/2$.
Till now, we have finished the calculations of six critical exponents. They
are also equal to $1/2$ except $\delta=2$. Our results are in accordance with
those in classical thermodynamics. And it can be easily proved that critical
exponents we obtain in our paper satisfy the following thermodynamic scaling
laws
$\displaystyle\alpha+2\beta+\gamma=2,\,\alpha+\beta(\delta+1)=2,\,(2-\alpha)(\delta\psi-1)+1=(1-\alpha)\delta,$
$\displaystyle\gamma(\delta+1)=(2-\alpha)(\delta-1),\,\gamma=\beta(\delta-1),\,\varphi+2\psi-\delta^{-1}=1.$
(61)
## 6 Conclusions
The phase transition of black holes with conformal anomaly has been
investigated in canonical ensemble. Firstly, we calculate the relevant
thermodynamic quantities and discuss the behavior of the specific heat at
constant charge. We find that there have been striking differences between
black holes with conformal anomaly and those without conformal anomaly. In the
case $Q=1,\tilde{\alpha}=0.1$, there are two phase transition point while
there is only one in the case $\tilde{\alpha}=0$. The temperature in the case
$\tilde{\alpha}=0$ increases monotonically while there exists local minimum
temperature in the case $Q=1,\tilde{\alpha}=0.1$. This local minimum
temperature corresponds to the phase transition point. We also find that the
phase transitions of black holes with conformal anomaly take place not only
from an unstable large black hole to a locally stable medium black hole but
also from an unstable medium black hole to a locally stable small black hole.
We also study the behavior of the inverse of the isothermal compressibility
$\kappa_{T}^{-1}$ and find that $\kappa_{T}^{-1}$ also diverges at the
critical point.
Secondly, we probe the dependence of phase transitions on the choice of
parameters. The results show that black holes with conformal anomaly have much
richer phase structure than that without conformal anomaly. When
$\tilde{\alpha}=0$, the location of the phase transition $r_{c}$ is
proportional to the charge $Q$. By contrast, the case of black holes with
conformal anomaly is more complicated. For $\tilde{\alpha}=0.1$, the curve can
be divided into three regions. Through numerical calculation, we find that
black holes has only one phase transition point when $Q\subset(0,0.4472)$.
When $0.4472<Q<0.7746$, there would be no phase transition at all. When
$Q>0.7746$, there exist two phase transition points. And the distance between
these two phase transition points becomes larger with the increasing of $Q$.
In the case that the charge $Q$ has been fixed at one, we notice that there
would be two phase transition point when $0<\tilde{\alpha}<\frac{1}{6}$. And
the distance between these two phase transition points becomes narrower with
the increasing of $\tilde{\alpha}$. When
$\tilde{\alpha}\subset(\frac{1}{6},\frac{1}{2})$, there would be no phase
transition. When $\tilde{\alpha}>\frac{1}{2}$, there would be only one phase
transition point.
Thirdly, we build up geometrothermodynamics in canonical ensembles. We choose
$M$ to be the thermodynamic potential and build up both thermodynamic phase
space and the space of thermodynamic equilibrium states. We also calculate the
Legendre invariant thermodynamic scalar curvature and depict its behavior
graphically. It is shown that Legendre invariant thermodynamic scalar
curvature diverges exactly where the specific heat diverges. Based on this, we
can safely conclude that the Legendre invariant metrics constructed in
geometrothermodynamics can correctly produce the behavior of the thermodynamic
interaction and phase transition structure even when conformal anomaly is
taken into account.
Furthermore, we calculate the relevant critical exponents. They are also equal
to $1/2$ except $\delta=2$. Our results are in accordance with those of other
black holes. And it has been proved that critical exponents we obtain in our
paper satisfy the thermodynamic scaling laws. We conclude that the critical
exponents and the scaling laws do not change even when we consider conformal
anomaly. This may be attributed to the mean field theory.
###### Acknowledgements.
This research is supported by the National Natural Science Foundation of China
(Grant Nos.11235003, 11175019, 11178007). It is also supported by ”Thousand
Hundred Ten” project of Guangdong Province and Natural Science Foundation of
Zhanjiang Normal University under Grant No. QL1104.
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|
arxiv-papers
| 2013-12-03T01:30:21 |
2024-09-04T02:49:54.685116
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie-Xiong Mo, Wen-Biao Liu",
"submitter": "Wen-Biao Liu",
"url": "https://arxiv.org/abs/1312.0679"
}
|
1312.0714
|
# On sufficient conditions for expressibility of constants in the 4-valued
extension of the propositional provability logic $GL$
Andrei RUSU
Information Society Development Institute
Academy of Sciences of Moldova
[email protected]
( )
###### Abstract
In the present paper we consider the simplest non-classical extension $GL4$ of
the well-known propositional provability logic $GL$ together with the notion
of expressibility of formulas in a logic proposed by A. V. Kuznetsov.
Conditions for expressibility of constants in $GL4$ are found out, which were
first announced in a author’s paper in 1996.
_In memory of Professor Mefodie Rață_
## 1 Introduction
The criteria of completeness with respect to expressibility is well-known in
the case of boolean functions [1, 2]. A. V. Kuznetsov [3, 4] has specified the
notion of expressibility to the case of formulas in logical calculi, using the
rule of replacement by its equivalent in the given logic. Professor Mefodie
Rață has obtained the criterion of completeness relativ to expressibility in
propositional intuitionistic logic and its extensions [5, 6].
We consider the simplest non-classical 4-valued extension of the propositional
provability logic of Gödel-Löb $GL$ [7] and found out the sufficient
conditions for expressibility of constant formulas of this logic.
## 2 Definitions and notations
Propositional provability logic $GL$. The formulas of the propositional
provability calculus of $GL$ are built from the symbols of propositional
variables $p,q,r,\dots$ (may be also indexed), by means of the symbols of
logical connectives $\&,\vee,\supset,\neg$ and $\Delta$ (represent the unary
modal operation of provability by Gödel), and parentheses. For example, the
expressions $(p\&\neg p)$, $(p\supset p)$, $(\Delta(p\&\neg p))$ and
$(\neg(\Delta(p\&\neg p)))$ are formulas in the calculus of $GL$, representing
the constant formulas denoted in the following by $0,1,\sigma,\rho$, and we
denote the formulas $(p\&\Delta p)$ and $((p\supset q)\&(q\supset p))$ as
$\square p$ (box $p$) and $(p\sim q)$ (equivalence of $p$ and $q$). External
parentheses are usually omitted. The calculus of the $GL$ is determined by the
axioms of the classical calculus of propositions, three $\Delta$-axioms
$\Delta(p\supset q)\supset(\Delta p\supset\Delta q),\ \Delta(\Delta p\supset
p)\supset\Delta p,\ \Delta p\supset\Delta\Delta p$
and the next three rules of inference: 1) the rule of substitution, 2) the
modus ponens rule, and 3) the rule of necessitation which allows to pass from
formula $A$ to formula $\Delta A$.
In the present paper we consider the extension of $GL$, denoted by $GL4$,
which can be obtained from $GL$ considering an additional axiom:
$\Delta\Delta 0\&(\Delta(\Delta p\supset q)\vee(\Delta(\Delta q\supset p)).$
Magari’s algebras. A Magari’s algebra [8] (also referred to as diagonalizable
algebra) $\mathfrak{D}$ is a boolean algebra
$\mathfrak{B}=(B;\with,\vee,\supset,\neg,\mathbb{0},\mathbb{1})$ with an
additional operator $\Delta$ satisfying the following identities:
$\displaystyle\Delta(x\supset y)\supset(\Delta x\supset\Delta y)$
$\displaystyle=\mathbb{1},$ $\displaystyle\Delta x\supset\Delta\Delta x$
$\displaystyle=\mathbb{1},$ $\displaystyle\Delta(\Delta x\supset x)$
$\displaystyle=\Delta x,$ $\displaystyle\Delta\mathbb{1}$
$\displaystyle=\mathbb{1},$
where $\mathbb{1}$ is the unit of $\mathfrak{B}$.
Interpreting logical connectives of a formula $F$ by corresponding operations
on a Magari’s algebra $\mathfrak{D}$ we can evaluate any formula of $GL$ on
any algebra $\mathfrak{D}$. If for any evaluation of variables of $F$ by
elements of $\mathfrak{D}$ the resulting value of the formula $F$ on
$\mathfrak{D}$ is $\mathbb{1}$ they say $F$ _is valid on $\mathfrak{D}$_. The
set of all valid formulas on the given Magari’s algebra $\mathfrak{D}$ is an
extension of $GL$ [10].
We consider the 4 valued Magari’s algebra
$\mathfrak{B}_{2}=(\\{\mathbb{0},\rho,\sigma,\mathbb{1}\\};\with,\vee,\supset,\neg,\Delta)$,
its boolean operations $\with,\vee,\supset,\neg$ are defined as usual, and the
operation $\Delta$ is defined as:
$\Delta\mathbb{0}=\Delta\rho=\sigma,\
\Delta\sigma=\Delta\mathbb{1}=\mathbb{1}.$
Expressibility of formulas [9]. Suppose in the logic $L$ we can define the
equivalence of two formulas. The formula $F$ is said to be (explicitly)
expressible via a system of formulas $\Sigma$ in the logic $L$ if $F$ can be
obtained from variables and formulas of $\Sigma$ using two rules: a) the rule
of weak substitution, which allows to pass from two formulas, say $A$ and $B$
to the result of substitution of one of them in another in place of any
variable $p$ of the formula $\frac{A,B}{A[p/B]}$ (where we denote by $A[p/B]$
the thought substitution); b) if we already get formulas $A$ and we know $A$
is equivalent in $L$ to $B$, then we have also formula $B$.
Relations on algebras. They say the formula $F(p_{1},\dots,p_{n})$ preserves
on the Magari’s algebra $\mathfrak{D}$ the relation $R(x_{1},\dots,x_{m})$ if
for any elements $\alpha_{11},\dots,\alpha_{mn}$ of $\mathfrak{D}$ the
relations
$R(\alpha_{11},\dots,\alpha_{m1}),\dots,(\alpha_{1n},\dots,\alpha_{mn})$
implies
$R(F(\alpha_{11},\dots,\alpha_{1n}),\dots,F(\alpha_{m1},\dots,\alpha_{mn}))$
The relation $R(x_{1},\dots,x_{m})$ on a finite algebra $\mathfrak{D}$ can be
substituted by a corresponding matrix $\beta_{ik}$ $(i=1,\dots,m,\
k=1,\dots,l)$ of all elements of $\mathfrak{D}$ such that the statement
$R(\beta_{1k},\dots,\beta_{mk})$ holds. In this case we speak about preserving
of a matrix instead of preserving of a relation on $\mathfrak{D}$.
## 3 Preliminary results
Representatin of 4-valued operations by formulas. Next theorem gives necessary
and sufficient conditions for a 4-valued operation on the set
$\\{\mathbb{0},\rho,\sigma,\mathbb{1}\\}$ to be expressible via a formula of
the propositional provability calculus.
###### Theorem 1.
A function $f$ of the general 4-valued logic can be expressed by a formula of
the calculus of the logic $L\mathfrak{B}_{2}$ if and only if it conserves the
relation $\Delta x=\Delta y$ on the algebra $\mathfrak{B}_{2}$.
###### Proof.
Necessity. It can be easily verified the formulas $p\&q$, $p\vee q$, $p\supset
q$, $\neg p$ şi $\Delta p$ conserve the relation $\Delta x=\Delta y$ on the
algebra $\mathfrak{B}_{2}$. Since any formula $F$ is directly expressible by
them, and, so, the formula $F$ must also preserve the same relation on
$\mathfrak{B}_{2}$.
Sufficiency. Let us to note that to any element of the algebra
$\mathfrak{B}_{2}$ corresponds a constant of the logic $L\mathfrak{B}_{2}$,
so, in the sequel we denote the elements of the algebra $\mathfrak{B}_{2}$ and
the constants of the logic $L\mathfrak{B}_{2}$ by the same symbols. Suppose
the operation $f(p_{1},\dots,p_{n})$ conserves the relation $\Delta x=\Delta
y$ on the algebra $\mathfrak{B}_{2}$. We will show in the following how to
design the formula $F(p_{1},\dots,p_{n}$), which represent the operation $f$
on the algebra $\mathfrak{B}_{2}$.
Examine an arbitrary fixed set $\alpha=(\alpha_{1},\dots,\alpha_{n})$ of
elements of $\mathfrak{B}_{2}$. Let $f(\alpha_{1},\dots,\alpha_{n})=\delta$
and consider the formula $(\&_{i=1}^{n}\square(p_{i}\sim\alpha_{i}))\&\delta$
denoted by $C^{\alpha}(p_{1},\dots,p_{n})$. It can be verified that
$C^{\alpha}$ satisfies the following conditions:
$C^{\alpha}(p_{1},\dots,p_{n})=\begin{cases}\delta,&\text{if
$p_{i}=\alpha_{i}$, $i=1,\dots,n$}\\\ \square 0\&\sigma,&\text{if $\forall
i:\Delta p_{i}=\Delta\alpha_{i}$, \c{s}i $\exists i:p_{i}\not=\alpha_{i},$}\\\
0,&\text{if $\exists j:\Delta p_{j}\not=\Delta\alpha_{j}$}\end{cases}.$
Denote with $\Gamma$ the set of all ordered sets of 4 elements from the set
$\\{\mathbb{0},\rho,\sigma,\mathbb{1}\\}$. Consider the formula
$F(p_{1},\dots,p_{n})=\bigvee_{\gamma\in\Gamma}C^{\gamma}(p_{1},\dots,p_{n})$
(1)
Let us show the formula $F$ is the thought for one. To prove this it is
sufficient to convince ourselves that $F[\alpha_{1},\dots,\alpha_{n}]$ $=$
$f(\alpha_{1}$, $\dots$, $\alpha_{n})$ since the set of elements $\alpha$ is
taken arbitrarily. The relation (1) can be rewritten as:
$\begin{split}F(p_{1},\dots,p_{n})=\bigvee_{\gamma\in\Gamma,\gamma=\alpha}&C^{\gamma}(p_{1},\dots,p_{n})\vee\\\
\bigvee_{\gamma\in\Gamma,\exists
i:\Delta\gamma_{i}\not=\Delta\alpha_{i}}&C^{\gamma}(p_{1},\dots,p_{n})\vee\\\
\bigvee_{\alpha\not=\gamma,\Delta\gamma_{i}=\Delta\alpha_{i}}&C^{\gamma}(p_{1},\dots,p_{n}).\end{split}$
(2)
The last relation (2) implies, taking into consideration the properties of the
formula $C^{\alpha}$, the following equality:
$\displaystyle F[\alpha_{1},\dots,\alpha_{n}]=$ $\displaystyle
C^{\alpha}(\alpha_{1},\dots,\alpha_{n})\vee$
$\displaystyle\bigvee_{\gamma\in\Gamma,\exists
i:\Delta\gamma_{i}\not=\Delta\alpha_{i}}C^{\gamma}(\alpha_{1},\dots,\alpha_{n})\vee$
$\displaystyle\bigvee_{\alpha\not=\gamma,\Delta\gamma_{i}=\Delta\alpha_{i}}C^{\gamma}(\alpha_{1},\dots,\alpha_{n})=$
$\displaystyle\delta\vee
0\vee(\square\delta\&\sigma)=\delta=f(\alpha_{1},\dots,\alpha_{n}).$
Hence, for an arbitrary set of elements $\alpha\in\Gamma$ we have
$F[\alpha_{1},\dots,\alpha_{n}]=f(\alpha_{1},\dots,\alpha_{n}).$
So, the formula $F$ realizes the operation $f$ on the algebra
$\mathfrak{B}_{2}$.
The theorem 1 is proved. ∎
The next statement is a consequence of the above theorem.
###### Proposition 1.
There are 64 unary formulas in the calculus of the logic $L\mathfrak{B}_{2}$
which are not equivalent each other in $L\mathfrak{B}_{2}$ and realize the
corresponding unary operations of the algebra $\mathfrak{B}_{2}$.
Table 1: Unary operations of $\mathfrak{B}_{2}$ $p$ | $I_{1j}$ | $I_{2j}$ | $I_{3j}$ | $I_{4j}$ | $I_{5j}$ | $I_{6j}$ | $I_{7j}$ | $I_{8j}$
---|---|---|---|---|---|---|---|---
$0$ | $0$ | $0$ | $\rho$ | $\rho$ | $\sigma$ | $\sigma$ | $1$ | $1$
$\rho$ | $0$ | $\rho$ | $0$ | $\rho$ | $\sigma$ | $1$ | $\sigma$ | $1$
$p$ | $I_{i1}$ | $I_{i2}$ | $I_{i3}$ | $I_{i4}$ | $I_{i5}$ | $I_{i6}$ | $I_{i7}$ | $I_{i8}$
$\sigma$ | $0$ | $0$ | $\rho$ | $\rho$ | $\sigma$ | $\sigma$ | $1$ | $1$
$1$ | $0$ | $\rho$ | $0$ | $\rho$ | $\sigma$ | $1$ | $\sigma$ | $1$
In order to describe the derived unary operations of the algebra
$\mathfrak{B}_{2}$ we use the table 1, where $I_{ij}(p)$ $(i=1,\dots,8;\
j=1,\dots,8)$ denotes the unary operation which for $p=0$ and $p=\rho$ takes
values from the $i$-th column, and for $p=\sigma$ and $p=1$ it takes values
from the $j$-th column.
For example, $I_{11}=0$, $I_{16}=p$, $I_{73}=\neg p$, $I_{58}=\Delta p$,
$I_{88}=1$.
## 4 Main result
Consider the following relations on $\mathfrak{B}_{2}$ (read symbols "$==$" as
"defined by"):
1) $R_{1}(x)==(\Delta x=\sigma)$;
2) $R_{2}(x)==(\Delta x=1)$;
3) $R_{3}(x)==I_{15}(x)=x)$;
4) $R_{4}(x)==I_{18}(x)=x)$;
5) $R_{5}(x)==I_{45}(x)=x)$;
6) $R_{6}(x)==I_{48}(x)=x)$;
7) $R_{7}(x)==I_{25}(x)=x)$;
8) $R_{8}(x)==I_{28}(x)=x)$;
9) $R_{9}(x)==I_{16}(x)=x)$;
10) $R_{10}(x)==I_{46}(x)=x)$;
11) $R_{11}(x,y)==(I_{37}(x)=y)$;
12) $R_{12}(x,y)==(\Delta x\not=\Delta y)$;
We denote by $\mathfrak{M}_{i}$ the corresponding matrix to the relation
$R_{i}$ on the algebra $\mathfrak{B}_{2}$ and denote with $\Pi_{i}$ the class
of all formulas, which preserves the relation $R_{i}$ on the algebra
$\mathfrak{B}_{2}$, i.e. the class of all formulas, which conserves the matrix
$\mathfrak{M}_{i}$ on $\mathfrak{B}_{2}$ for any $i=1,\dots,12$.
The table LABEL:tabel:21 presents the list of all classes
$\Pi_{1},\dots,\Pi_{12}$ and their corresponding matrix.
Table 2: The class of formulas and the corresponding matrix The class | Defining matirx
---|---
$\Pi_{1}$ | $\left(0\rho\right)$
$\Pi_{2}$ | $\left(\sigma 1\right)$
$\Pi_{3}$ | $\left(0\sigma\right)$
$\Pi_{4}$ | $\left(01\right)$
$\Pi_{5}$ | $\left(\rho\sigma\right)$
$\Pi_{6}$ | $\left(\rho 1\right)$
$\Pi_{7}$ | $\left(0\rho\sigma\right)$
$\Pi_{8}$ | $\left(0\rho 1\right)$
$\Pi_{9}$ | $\left(0\sigma 1\right)$
$\Pi_{10}$ | $\left(\rho\sigma 1\right)$
$\Pi_{11}$ | $\left({{\displaystyle 0}\atop{\displaystyle\rho}}{{\displaystyle\rho}\atop{\displaystyle 0}}{{\displaystyle\sigma}\atop{\displaystyle 1}}{{\displaystyle 1}\atop{\displaystyle\sigma}}\right)$
$\Pi_{12}$ | $\left({{\displaystyle 0}\atop{\displaystyle\sigma}}{{\displaystyle 0}\atop{\displaystyle 1}}{{\displaystyle\rho}\atop{\displaystyle\sigma}}{{\displaystyle\rho}\atop{\displaystyle 1}}{{\displaystyle\sigma}\atop{\displaystyle 0}}{{\displaystyle\sigma}\atop{\displaystyle\rho}}{{\displaystyle 1}\atop{\displaystyle 0}}{{\displaystyle 1}\atop{\displaystyle\rho}}\right)$
###### Theorem 2.
Suppose the formulas $F_{1},\dots,F_{12}$ do not preserve the corresponding
relations $R_{1},\dots,R_{12}$ on the Magari’s algebra $\mathfrak{B}_{2}$. The
constants $\mathbb{0},\rho,\sigma,\mathbb{1}$ are expressible in the logic
$L\mathfrak{B}_{2}$ via formulas $F_{1},\dots,F_{12}$.
The proof of the theorem follows from the next 5 lemmas.
###### Lemma 1.
The formula $A(p)$, where
$A[0]\in\\{\sigma,1\\}$ (3)
is expressible $L\mathfrak{B}_{2}$ via formula $F_{1}$.
###### Proof.
Really, the formula $F_{1}$ does not conserve the relation $R_{1}$ on the
algebra $\mathfrak{B}_{2}$. Then there exists an ordered set of elements
$\left<\alpha_{1},\dots,\alpha_{n}\right>$ from $\mathfrak{B}_{2}$ such that
$\displaystyle\alpha_{i}\in\left\\{0,\rho\right\\}\quad(i=1,\dots,n)$ (4)
$\displaystyle F_{1}[\alpha_{1},\dots,\alpha_{n}]\in\left\\{\sigma,1\right\\}$
(5)
Since $F_{1}$ conserves the predicate $\Delta x=\Delta y$ on the algebra
$\mathfrak{B}_{2}{}$, in view of relations (4) şi (5) we also have that
$F_{1}[0,\dots,0]\in\left\\{\sigma,1\right\\}$ (6)
Let $A(p)=F_{1}[p_{1}/p,\dots,p_{n}/p]$. In virtue of (6) we obtain
$A[0]\in\left\\{\sigma,1\right\\}$. ∎
###### Lemma 2.
The formula $B(p)$, where
$B[1]\in\\{0,\rho\\}$ (7)
is expressible in $L\mathfrak{B}_{2}$ via $F_{2}$.
###### Proof.
The validity of lemma 2 follows from the fact that its formulation is
dualistic to the formulation of lemma 1 with respect to $\neg p$, where
formula $B$ is considered in place of the corresponding formula $A$. ∎
###### Lemma 3.
Let the formulas $A$ and $B$ satisfy the relations (3), (7) and
$B[\sigma]=B[1].$ (8)
Then at one of the constants $0$ or $\rho$ is expressible via formulas $A,B$
and $F_{12}$ in the logic $L\mathfrak{B}_{2}{}$.
###### Proof.
Let $B$ satisfies the relations (7) and (8). Then two cases are possible for
the formula $B$: 1) $B[0]\in\left\\{0,\rho\right\\}$; 2)
$B[0]\in\left\\{\sigma,1\right\\}$. Let us observe in the first case the
formula $B[A[B(p)]]$ is equivalent to one of the constants $0$ or $\rho$.
Consider case 2), i.e. $B[0]\in\left\\{\sigma,1\right\\}$. Consider formula
$F_{12}$, which does not preserve $R_{12}$ on $\mathfrak{B}_{2}{}$. Then there
are exist two ordered sets of elements
$\left<\alpha_{1},\dots,\alpha_{n}\right>$ and
$\left<\beta_{1},\dots,\beta_{n}\right>$ from the algebra $\mathfrak{B}_{2}{}$
such that
$\displaystyle\Delta\alpha_{i}\not=\Delta\beta_{i}\quad(i=1,\dots,n)$ (9)
$\displaystyle\Delta F_{12}[\alpha_{1},\dots,\alpha_{n}]=\Delta
F_{12}[\beta_{1},\dots,\beta_{n}]$ (10)
We build the formula $D(p_{1},\dots,p_{8})=F_{12}[D_{1},\dots,D_{n}]$, where
for every $i=1,\dots,n$
$D_{i}(p_{1},\dots,p_{8})=p_{1},\mbox{if }\alpha_{i}=0,\beta_{i}=\sigma,$
$D_{i}(p_{1},\dots,p_{8})=p_{2},\mbox{if }\alpha_{i}=0,\beta_{i}=1,$
$D_{i}(p_{1},\dots,p_{8})=p_{3},\mbox{if }\alpha_{i}=\rho,\beta_{i}=\sigma,$
$D_{i}(p_{1},\dots,p_{8})=p_{4},\mbox{if }\alpha_{i}=\rho,\beta_{i}=1,$
$D_{i}(p_{1},\dots,p_{8})=p_{5},\mbox{if }\alpha_{i}=\sigma,\beta_{i}=0,$
$D_{i}(p_{1},\dots,p_{8})=p_{6},\mbox{if }\alpha_{i}=\sigma,\beta_{i}=\rho,$
$D_{i}(p_{1},\dots,p_{8})=p_{7},\mbox{if }\alpha_{i}=1,\beta_{i}=0,$
$D_{i}(p_{1},\dots,p_{8})=p_{8},\mbox{if }\alpha_{i}=1,\beta_{i}=\rho$
(by the power of relation (9) other cases are impossible). It is clear that
$D_{i}[0$, $0$, $\rho$, $\rho$, $\sigma$, $\sigma$, $1$, $1]=\alpha_{i}$ and
$D_{i}[\sigma,1,\sigma,1,0,\rho,0,\rho]=\beta_{i}$. Then, taking into account
the design of the formula $D$, the relation (LABEL:eq:2-9) and the last
equalities, we obtain
$\left(\begin{array}[]{c}D[0,0,\rho,\rho,\sigma,\sigma,1,1]\\\
D[\sigma,1,\sigma,1,0,\rho,0,\rho]\end{array}\right)\subseteq\left(\begin{array}[]{c}00\rho\rho\sigma\sigma
11\\\ 0\rho 0\rho\sigma 1\sigma 1\end{array}\right)$ (11)
Consider now the formula $D^{*}(p,q)=D[p,p,p,p,q,q,q,q]$. By (11) and the fact
that $D$ conserves on the algebra $\mathfrak{B}_{2}{}$ the predicate $\Delta
x=\Delta y$, we obtain
$\left(\begin{array}[]{c}D^{*}[0,1]\\\
D^{*}[1,0]\end{array}\right)\subseteq\left(\begin{array}[]{c}00\rho\rho\sigma\sigma
11\\\ 0\rho 0\rho\sigma 1\sigma 1\end{array}\right)$ (12)
Let us examine the formula $D^{\prime}(p,q)$, defined by the scheme
$D^{\prime}(p,q)=\left\\{\begin{array}[]{ll}D^{*}(p,q),&\mbox{if
}D^{*}[0,1]\in\left\\{0,\rho\right\\},\\\ B[D^{*}(p,q)],&\mbox{if
}D^{*}[0,1]\in\left\\{\sigma,1\right\\}.\end{array}\right.$
By power of the relation (8) and taking into consideration (12), the formula
$D^{\prime}$ satisfies the inclusion
$\left\\{D^{\prime}[0,1],D^{\prime}[1,0]\right\\}\subseteq\left\\{0,\rho\right\\}$.
Therefore, in the second case, taking into consideration (7), the relation
$B[D^{\prime}[p,B(p)]]]\in\left\\{\sigma,1\right\\}$ holds. Hence, on the
basis of the conditions (7) and (8), the formula $B[B[D^{\prime}[p$, $B(p)]]]$
is equivalent to one of the constants $0$ or $\rho$. ∎
###### Lemma 4.
Let formulas $A$ and $B$ satisfy the relations (3), (7) and
$B[\sigma]\not=B[1].$ (13)
Then at least one of the constants $0$ or $\rho$ is expressible via formulas
$A$, $B$, $F_{3},F_{7},F_{11},F_{12}$ in the logic $L\mathfrak{B}_{2}{}$.
###### Proof.
The relation (7) and the fact that $B$ conserves the predicate
$\Delta{x}=\Delta{y}$ on the algebra $\mathfrak{B}_{2}{}$ implies that there
are two possible situations: 1) $B[1]=\rho$, and 2) $B[1]=0$.
Let us consider the first case. On the basis of the relation (13) we have that
$B[\sigma]=0$. We consider the formula $F_{3}$. Since it does not conserve
$R_{3}$ on $\mathfrak{B}_{2}{}$there exists an ordered set
$\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ of elements of
$\mathfrak{B}_{2}{}$ such that
$\displaystyle\alpha_{i}\in\left\\{0,\sigma\right\\},\quad i=1,\dots,n$ (14)
$\displaystyle F_{3}[\alpha_{1},\dots,\alpha_{n}]\in\left\\{\rho,1\right\\}.$
(15)
We design the formula $E(p)=F_{3}[{E}_{1},\dots,{E}_{n}]$, where for every
$i={1},\dots,{n}$
$E_{i}(p)=\left\\{\begin{array}[]{ll}B(p),&\mbox{if }\alpha_{i}=0,\\\
p,&\mbox{if }\alpha_{i}=\sigma\end{array}\right.$
(in accordance to (14) other cases are impossible for the elements
$\alpha_{i}$). The formula $E$ is direct expressible via formulas $F_{3}$ and
$B$. Obviously, $E_{i}[\sigma]=\alpha_{i}$ and the view of relation (15) we
have $E[\sigma]=F_{3}[E_{1}[\sigma],\dots$,
$E_{n}[\sigma]]=F_{3}[\alpha_{1},\dots,\alpha_{n}]\in\left\\{\rho,1\right\\}$.
Consider the formula $E^{*}(p)$, defined by the scheme
$E^{*}(p)=\left\\{\begin{array}[]{ll}E(p),&\mbox{if }E[\sigma]=\rho,\\\
B[E(p)],&\mbox{if }E[\sigma]=1.\end{array}\right.$
The formula $E^{*}(p)$ is directly expressible via formulas $B$ and $E$ and
satisfies the condition
$E^{*}[\sigma]=\rho.$ (16)
Two sub-cases are possible: 1.1) $E^{*}[1]=\rho$ and 1.2) $E^{*}[1]=0$. In the
sub-case 1.1) the formula $E^{*}(p)$ satisfies analogous conditions to (7) for
the formula $B(p)$ from lemma 3 and then the proof will follow the
corresponding proof of the lemma 3, thus one of two constants $0$ or $\rho$ is
obtained.
Consider now the sub-case 1.2) when $E^{*}[1]=0$. Consider formula $F_{7}$.
Since $F_{7}$ does not conserve the relation $R_{7}$ on $\mathfrak{B}_{2}{}$,
then there exists an ordered set of elements
$\left<{\beta}_{1},\dots,{\beta}_{n}\right>$ from $\mathfrak{B}_{2}{}$ such
that
$\displaystyle\beta_{i}\in\left\\{0,\rho,\sigma\right\\},\quad i=1,\dots,n$
(17) $\displaystyle F_{7}[{\beta}_{1},\dots,{\beta}_{n}]=1$ (18)
Take the formula $H(p)=F_{7}[{H}_{1},\dots,{H}_{n}]$, where for every
$i=1,\dots,n$
$H_{i}(p)=B(p),\mbox{if }\beta_{i}=0,$ $H_{i}(p)=E^{*}(p),\mbox{if
}\beta_{i}=\rho,$ $H_{i}(p)=p,\mbox{if }\beta_{i}=\sigma.$
(obviously, other cases are missed for the elements $\beta_{i}$). The formula
$H$ is directly expressible via $F_{7},B$ and $E^{*}$. It is clear
$H_{i}[\sigma]=\beta_{i}$ and in agreement with relation (18) we have
$H[\sigma]=1.$ (19)
If $H[1]=1$ then the formula $B[H(p)]$ satisfies analogous conditions to
conditions (7) and (8) for the formula $B$ from lemma 3. That is why we can
obtain one of the constants $0$ or $\rho$ in the case when $H[1]=1$ in the
same way as in lemma 3.
Let $H[1]=\sigma$. Use the formula $F_{11}$. It follows from its properties
that there exist two ordered sets of elements
$({\gamma}_{1},\dots,{\gamma}_{n})$ and $({\delta}_{1},\dots,{\delta}_{n})$
from $\mathfrak{B}_{2}{}$, such that the next relation holds
$I_{37}[\gamma_{i}]=\delta_{i},\quad i=1,\dots,n$ (20)
Taking also into consideration the theorem 1 we have
$F_{11}[{\gamma}_{1},\dots,{\gamma}_{n}]=F_{11}[{\delta}_{1},\dots,{\delta}_{n}].$
(21)
Design the formula $J(p)=F_{11}[{J}_{1}(p),\dots,{J}_{n}(p)]$, where for any
$i=1,\dots,n$ we get
$J_{i}(p)=\left\\{\begin{array}[]{ll}B(p),&\mbox{if
}\gamma_{i}=0,\delta_{i}=\rho,\\\ E^{*}(p),&\mbox{if
}\gamma_{i}=\rho,\delta_{i}=0,\\\ p,&\mbox{if
}\gamma_{i}=\sigma,\delta_{i}=1,\\\ B(p),&\mbox{if
}\gamma_{i}=1,\delta_{i}=\sigma\end{array}\right.$
(by properties of the relation (20) the elements $\gamma_{i}$ and $\delta_{i}$
do not take other values). $J(p)$ is directly expressible via $B$, $E^{*}$,
$H$ and $F_{11}$. Let us notice that $J_{i}[\sigma]=\gamma_{i}$,
$J_{i}[1]=\delta_{i}$ şi, hence, by relation (21), we obtain $J[\sigma]=J[1]$.
So, the formula $J^{*}(p)$, defined by the scheme
$J^{*}(p)=\left\\{\begin{array}[]{ll}J(p),&\mbox{if
}J[1]\in\left\\{0,\rho\right\\},\\\ B[J(p)],&\mbox{if
}J[1]\in\left\\{\sigma,1\right\\},\end{array}\right.$
satisfies the relations
$J[\sigma]=J[1],\;J[1]\in\left\\{0,\rho\right\\}$ (22)
Let us notice that conditions (22) are analogous to conditions (7) and (8)
from lemma 3. Hence, we can obtain in a similar way one of the constants $0$
or $\rho$. So, the proof of the lemma (4) in the case 1) is finished.
Let us consider the second case, when $B[1]=0$. Examine the formula $F_{4}$.
Since it does not conserve the relation $R_{4}$ on $\mathfrak{B}_{2}{}$, then
there is an ordered set of elements
$({\varepsilon}_{1},\dots,{\varepsilon}_{n})$ on $\mathfrak{B}_{2}{}$ such
that
$\displaystyle\varepsilon_{i}\in\left\\{0,1\right\\},\quad i=1,\dots,n$ (23)
$\displaystyle
F_{4}[{\varepsilon}_{1},\dots,{\varepsilon}_{n}]\in\left\\{\rho,\sigma\right\\}$
(24)
Design the formula $S(p)=F_{4}[{S}_{1},\dots,{S}_{n}]$, where for any
$i=1,\dots,n$ we have
$S_{i}(p)=\left\\{\begin{array}[]{ll}B(p),&\mbox{if }\varepsilon_{i}=0,\\\
p,&\mbox{if }\varepsilon_{i}=1.\end{array}\right.$
(by properties of (23) we do not have other cases). The formula $S$ is
directly expressible via $F_{4}$ şi $B$. Obviously $S_{i}[1]=\varepsilon_{i}$
and in agreement with relation (24) we have
$S[1]=F_{4}[{S}_{1}[1],\dots,{S}_{n}[1]]=F_{4}[{\varepsilon}_{1},\dots,{\varepsilon}_{n}]\in\left\\{\rho,\sigma\right\\}.$
Consider the formula $S^{*}(p)$ defined by the scheme
$S^{*}(p)=\left\\{\begin{array}[]{ll}S(p),&\mbox{if }S[1]=\rho,\\\
B[S(p)],&\mbox{if }S[1]=\sigma.\end{array}\right.$
The formula $S^{*}(p)$ is directly expressible via $B$ and $S$ and verifies
the condition $S^{*}[1]=\rho$. taking into consideration the theorem 1 we also
have the relation $S^{*}[\sigma]\in\left\\{0,\rho\right\\}$. If
$S^{*}[\sigma]=\rho$, then we obtain one of the constants $0$ or $\rho$ as in
the case 1). It remains to consider the case when $S^{*}[\sigma]=0$. But in
this case we are already under conditions of the first case, which was
successfully considered already. ∎
###### Lemma 5.
All constants $0,\rho,\sigma,1$ are expressible in the logic
$L\mathfrak{B}_{2}{}$ via formulas $F_{i},\;i=3,\dots,10$, via any unary
formulas $A$ and $B$, which verify the corresponding conditions (3) and (7),
and via any constant $0$ or $\rho$.
###### Proof.
Let us convince ourselves that one of the following systems of formulas (25)
is expressible via one of the constants $0$ or $\rho$ and the formula $A$:
$\left\\{0,\sigma\right\\},\;\left\\{0,1\right\\},\;\left\\{\rho,\sigma\right\\},\;\left\\{\rho,1\right\\}.$
(25)
Let us consider we have the constant $0$. By properties of (3) we have
$A[0]\in\left\\{\sigma,1\right\\}$, which means we have at least one of the
first two systems of (25). Suppose we have the constant $\rho$. Then by
theorem 1 we have $A[\rho]\in\left\\{\sigma,1\right\\}$, and by the similar
reasons as in the case of the constant $0$ we can conclude analogously we have
at least one the the last two systems of (25).
We wil show in the following that via every system of formulas of the list
(25) and via corresponding formulas $F_{3}$, $F_{4}$, $F_{5}$, $F_{6}$ is
expressible one of the following systems of constants
$\left\\{0,\rho,\sigma\right\\},\;\left\\{0,\rho,1\right\\},\;\left\\{0,\sigma,1\right\\},\;\left\\{\rho,\sigma,1\right\\}.$
(26)
Let us consider the system of formulas $\left\\{0,\sigma\right\\}$. Examine
the formula $F_{3}$. Obviously via $F_{3}$ and constants $0$ and $\sigma$ is
expressible some formula $F^{*}_{3}(p,q)$, which satisfies the condition
$F^{*}_{3}[0,\sigma]\in\left\\{\rho,1\right\\}$. Hence, we obtain one of the
systems of formulas $\left\\{0,\rho,\sigma\right\\}$ or
$\left\\{0,\sigma,1\right\\}$. In a similar way we obtain:
* •
the system $\left\\{0,\rho,1\right\\}$ or the system
$\left\\{0,\sigma,1\right\\}$ via $\left\\{0,1\right\\}$ and $F_{4}$;
* •
the system $\left\\{0,\rho,\sigma\right\\}$ or the system
$\left\\{\rho,\sigma,1\right\\}$ via $\left\\{\rho,\sigma\right\\}$ and
$F_{5}$;
* •
the system $\left\\{0,\rho,1\right\\}$ or the system
$\left\\{\rho,\sigma,1\right\\}$ via $\left\\{\rho,1\right\\}$ and $F_{6}$.
In a similar manner we obtain that all constants of the system
$\left\\{0,\rho,\sigma,1\right\\}$ are expressible in $L\mathfrak{B}_{2}{}$
via every system of formulas of (26) and corresponding formulas
$F_{7},F_{8},F_{9},F_{10}$. ∎
## 5 Conclusions
Theorem 2 provide us only sufficient conditions for expressibility of
constants of the propositional provability logic $L\mathfrak{B}_{2}$. We can
consider a slice of extensions of $GL$ [12], which also has an additional
axiom $\Delta\Delta p$ and examine the conditions of expressibility of
constants in these logics too. Note the logic $L\mathfrak{B}_{2}$ is an
element of this slice of extensions. Also we can examine other types of
expressibility of formulas: implicit expressibility, parametric
expressibility, existential expressibility, etc.
## References
* [1] Post E. L. _Introduction to a general theory of elementary propositions_ , Amer. J. Math., 1921, v. 43, p. 163 - 185.
* [2] Post E. L. _Two-valued iterative systems of mathematical logic_. Princeton, 1941.
* [3] Кузнецов А. В., _Аналоги штриха Шеффера в конструктивной логике_ , Доклады АН СССР, 1965, 160, (2), 274–277
* [4] Кузнецов А. В., _О функциональной выразимости в суперинтуиционистских логиках_ , Математические исследования, 1971, 6, (4), 75–122
* [5] Раца М. Ф., _Критерий функциональной полноты в интуиционистской логике высказываний_ , Доклады АН СССР, 1971, 201, (4), 794–797
* [6] Раца М. Ф., _О функциональной полноте в интуиционистской логике высказываний_ , Проблемы кибернетики, 1982, 39, 107–150
* [7] Solovay R. M., _Provability interpretations of modal logic_ , Israel J. Math., 1975, 25, p. 287 - 304.
* [8] Magari R., _The diagonalizable algebras (the algebraization of the theories which express Theor.: II)_ , Boll. Unione Mat. Ital. , 12 (1975) (suppl. fasc 3) pp. 117–125.
* [9] Kuznetsov A. V., _On detecting non-deducibility and non-expressibility_ in: Locical deduction, Nauka, Moscow (1979), 5–33 (in russian)
* [10] Maksimova, L.L. _Continuum of normal extensions of the modal logic of provability with the interpolation property_ // Sib. Math. J. 30, No.6, 935-944 (1989)
* [11] Кузнецов А.В., Муравицкий А.Ю., _Доказуемость как модальность_ , в кн. Актуальные проблемы логики и методологии науки, Киев: Наукова думка, 1980, с. 193–230.
* [12] Blok W.J. Pretabular varieties of modal algebras // Studia Logica, 1980, v. 39, no. 2-3, p. 101 - 124.
|
arxiv-papers
| 2013-12-03T06:54:59 |
2024-09-04T02:49:54.694779
|
{
"license": "Public Domain",
"authors": "Andrei Rusu",
"submitter": "Andrei Rusu",
"url": "https://arxiv.org/abs/1312.0714"
}
|
1312.0752
|
# Tropical Grassmannian
and Tropical Linear Varieties from phylogenetic trees
Aritra Sen and Ambedkar Dukkipati [email protected]
[email protected] Dept. of Computer Science & Automation
Indian Institute of Science, Bangalore - 560012
###### Abstract.
In this paper we study tropicalization of Grassmannian and linear varieties.
In particular, we study the tropical linear spaces corresponding to the
phylogenetic trees. We prove that corresponding to each subtree of the
phylogenetic tree there is a point on the tropical grassmannian. We deduce a
necessary and sufficient condition for it to be on the facet of the tropical
linear space.
## 1\. Introduction
Tropical algebraic geometry is a new area that studies objects from algebraic
geometry using tools of combinatorics. The key process in tropical geometry is
that of tropicalization, where an algebraic variety is degenerated to a
polyhedral complex. The resulting polyhedral complex encodes information about
the original algebraic variety that can now be studied using the tools of
combinatorics (Maclagan & Sturmfels, 2009). One of the main achievements of
this field was due to the works of Mikhalkin (2003), where it was shown that
Gromov-Witten invariants of a curve in plane can be calculated by counting
lattice paths in polygons. This approach led to combinatorial proofs of many
identities in enumerative geometry (Gathmann & Markwig, 2008).
The tropical grassmanian is obtained from the tropicalization of the
grassmanian. It is known that the tropicalization of $\operatorname{Gr}(n,2)$
is a polyhedral complex, in which each point corresponds to a phylogenetic
tree. Just like the classical grassmanian that parametrizes the linear
varieties, the tropical grassmanian parametrizes the tropical linear varieties
(Speyer & Sturmfels, 2004). It has been shown, using the representation theory
of $SL_{n}(\mathbb{C})$, that the image of the tropicalization of
$\operatorname{Gr}(n,2)$ under the generalized dissimilarity map is contained
in the tropicalization of $\operatorname{Gr}(n,r)$ (Manon, 2011). Here, we
study the tropical linear varieties that correspond to these images. We show
that for each sub-tree of a tree there is a point on the tropical linear
space. We then prove a necessary and sufficient condition for it to be on the
facet of the tropical linear space.
## 2\. Grassmannian
Let $V$ be an $n$-dimensional vector space over the field $\mathbb{K}$ i.e.,
$V\cong\mathbb{K}^{n}$, then the Grassmannian $\operatorname{Gr}(n,r)$ is the
set of all $r$-dimensional subspaces of $V$. $\operatorname{Gr}(n,1)$ is the
set of all one-dimensional subspaces of V. If $k=\mathbb{R}$ or $\mathbb{C}$,
this is nothing but the projective space $\mathbb{P}(\mathbb{R})$ or
$\mathbb{P}(\mathbb{C})$.
Let $a_{1},\ldots,a_{r}$ be $r$ linear independent vectors in
$\mathbb{K}^{n}$, therefore they span a $r$-dimensional subspace. Let
$M_{r\times n}$ be the matrix with row vectors $a_{1},\ldots,a_{r}$. Since,
$a_{1},\ldots,a_{r}$ are linearly independent the rank of $M_{r\times n}$ is
$r$. So, to each $r$-rank $r\times n$ matrix one can associate an
$r$-dimensional subspace of $\mathbb{K}^{n}$. But this mapping is not one-one
as there can be more than one $r-$rank $r\times n$ matrix that can give rise
to the same subspace.
Let $\sigma$ be an $r$-element subset of $[n]={1,2,\ldots,n}$. Let
$M_{\sigma}$ denote the $r\times r$ submatrix of $M_{r\times n}$ such that
column indices coming from $\sigma$. Now consider the list
$m=(\mathbf{det}(M_{\sigma}):\sigma\subset[n])$. Let $N_{r\times n}$ be any
other $r\times n$ matrix . Then $N_{r\times n}$ and $M_{r\times n}$ have the
same row span (therefore represent the same $r$-dimensional subspace of
$\mathbb{K}^{n}$ if and only if the list
$m=(\mathbf{det}(M_{\sigma}):\sigma\subset[n])$ and
$n=(\mathbf{det}(N_{\sigma}):\sigma\subset[n])$ are multiple of each other.
###### Theorem 2.1.
(Miller & Sturmfels, 2005) Two $r\times n$ matrices $N_{r\times n}$ and
$M_{r\times n}$ have the same row space if and only if there exists
$a\in\mathbb{K}^{*}$ such that for all r-element subset $\sigma\subset[n]$
$\mathbf{det}(M_{\sigma})=a\mathbf{det}(N_{\sigma})\;.$
From this we can say that each $r$-dimensional subspace of ${\mathbb{K}}^{n}$
corresponds to a $n\choose r$ vector upto a constant multiple. Therefore each
$r$-dimensional subspace of $\mathbb{K}^{n}$ corresponds to a point in
$\mathbb{P}^{{n\choose r}-1}$. Hence, $\operatorname{Gr}(n,r)$ can be thought
of as a subset of $\mathbb{P}^{{n\choose r}-1}$.
Consider the map
$f:\operatorname{Gr}(V,r)\rightarrow\mathbb{P}\\{\bigwedge_{i=1}^{i=m}V\\}$,
where $\bigwedge$ represents a wedge (or exterior) product. Let $w_{1},\ldots
w_{r}$ be the basis of a $r$-dimensional vector subspace $W$ of $V$, then
$f(W)=w_{1}\ldots\wedge w_{r}$. Now, suppose $w^{\prime}_{1},\ldots
w^{\prime}_{r}$ is a basis for $W$. Consider the column vector $W_{c}$
consisting of $w_{1},\ldots w_{r}$ as its elements and the column vector
$W^{\prime}_{c}$ consisting of $w^{\prime}_{1},\ldots,w^{\prime}_{r}$. Then
there exists an invertible matrix $A$, such that $W^{\prime}=AW$. Using the
Leibnitz formula for determinants, we can see that $w_{1}\wedge\ldots\wedge
w_{r}=\det(A)w_{1}\wedge\ldots\wedge w^{\prime}_{r}$. Therefore the map $f$ is
well-defined.
Now, a vector lies in the image of $f$, if and only if $u\in\bigwedge^{r}V$
can be written in the form of $u=w_{1}\wedge\ldots\wedge w_{r}$. If
$e_{1},\ldots,e_{n}$ is a basis of V, then $e_{I}=e_{i_{1}}\wedge\ldots\wedge
e_{i_{r}}$ where $\\{i_{1},\ldots i_{r}\\}\in{[n]\choose r}$ forms a basis of
$\bigwedge^{r}V$. If $x\in\bigwedge^{r}V$ then $x=\Sigma_{I\in{[n]\choose
r}}a_{I}e_{I}$ and $a_{I}$ are the homogeneous coordinates of $x$. Now
consider the map $m_{u}(v)=v\wedge u$. So, $u$ lies in the image of $f$ if and
only the kernel of $m_{u}$ is r-dimensional.The homogeneous coordinates of $u$
in $\mathbb{P}{\bigwedge^{r}V}$ are the entries of the matrix of $m_{u}$ and
since it nullity $r$ every $(n-r+1)\times(n-r+1)$ submatrix of $m_{u}$ will
have zero determinant. Therefore, $\operatorname{Gr}(n,r)$ is a projective
variety in $\mathbb{P}^{{n\choose r}-1}$ and $\operatorname{Gr}(n,r)$ is the
zero set of a homogeneous ideal in $\mathbb{K}[X_{1},\ldots,X_{n\choose r}]$.
This homogeneous ideal is called the Plucker ideal.
## 3\. Dissimilarity maps and Tree
A map D: ${[n]\choose 2}\rightarrow\mathbb{R}$ is called a dissimilarity map.
Let $T$ be a weighted tree with $n$ nodes labeled by the set
$[n]=\\{1,2,3,\ldots,n\\}$. Every tree induces a dissimilarity map such that
$D({i,j})$ is the length of the path between the leaves $i$ and $j$. A natural
question one can pose is given a dissimilarity map when does it come from a
tree. The answer is given by the tree metric theorem .
###### Theorem 3.1.
(Buneman, 1974) Let $D$ be a dissimilarity map. The map $D$ comes from a tree
if and only if the the four-point condition holds i.e. for all $i,j,k$ and
$l\in[n]$ (not necessarily distinct) then the maximum of three number is
achieved at least twice $D({i,j})+D({k,l}),D({i,k})+D({j,l})$ and
$D({i,l})+D({j,k})$. The tree realizing $w$ is distinct.
Now we look at a further generalization of the dissimilarity map. Let
$D^{\prime}$ be a map from $[n]\choose r$ to $\mathbb{R}$. Let
$i_{1},\ldots,i_{m}\in[n]$ be distinct. Consider the dissimilarity map
$D^{\prime}$, such that $D^{\prime}({i_{1},\ldots,i_{r}})$ equals the weight
of the smallest tree containing the leaf nodes ${i_{1},\ldots,i_{r}}$. The
following theorem tells us how can we calculate $D^{\prime}$ from $D$.
###### Theorem 3.2.
Let $\phi^{m}:\mathbb{R}^{[n]\choose 2}\rightarrow\mathbb{R}^{[n]\choose r}$
such that $D\rightarrow D^{\prime}$
$D^{\prime}(\\{i_{1},\ldots,i_{m}\\})=\mathrm{min}\frac{1}{2}(D({i_{1},i_{\sigma(1)}})+D({i_{\sigma(1)},i_{\sigma^{2}(1)}})+\ldots+D({i_{\sigma^{m-1}(1)},i_{\sigma^{m}(1)}}))\;,$
where $\sigma$ is a cyclic permutation.
## 4\. Tropical Algebraic Geometry
Let $K$ represent the field of puiseux series over $\mathbb{C}$ i.e.,
$K=\mathbb{C}\\{\\{t\\}\\}=\bigcup\limits_{n\geq 1}\mathbb{C}((t^{1/n}))\;.$
Let $\operatorname{val}:K\rightarrow\mathbb{R}$ represent the valuation map
which takes a series to its lowest exponent. Let $f\in K[X_{1},\ldots,X_{n}]$
and $f=\sum\limits_{a\in\mathbb{N}}c_{a}X^{a},c_{a}\in K$.
The tropicalization of the polynomial $f$ is defined as
$\operatorname{trop}(f)=\operatorname{min}(\operatorname{val}(c_{a})+X.a).$
###### Definition 4.1.
Let $f\in K[X_{1},\ldots X_{n}]$. The tropical hypersurface
$\operatorname{trop}(V(f))$ is the set
$\\{w\in\mathbb{R}^{n}:\text{the minimum in $\operatorname{trop}(f)$ is
achieved at least twice}\\}\;.$
###### Definition 4.2.
Let $I$ be an ideal of $K[X_{1}\ldots X_{n}]$ and $X=V(I)$ be variety of $I$.
The tropicalization of X is defined as
$\operatorname{trop}(X)=\bigcap\limits_{f\in
I}\operatorname{trop}(V(f))\subset\mathbb{R}^{n}\;.$
Now we present the various characterization of the set
$\operatorname{trop}(V(f))$. The following theorem is also called the
fundamental theorem of tropical geometry.
###### Theorem 4.3.
Let $I$ be ideal of in $K[X_{1},\ldots X_{n}]$ and $X=V(I)$ be the variety
defined by I. Then the following sets coincide
1\. The tropical variety $trop(X)$, and
2\. the closure in $\mathbb{R}^{n}$ (euclidean topology) of the set
$\operatorname{Val}(X)$
$\operatorname{Val}(X)=\\{(\operatorname{val}(u_{1}),\ldots,\operatorname{val}(u_{n})):(u_{1},\ldots,u_{n})\in
X\\}\;.$
So, the tropicalization of a variety is the image of the variety under the
valuation map.
## 5\. Tropicalization of Grassmannian
###### Definition 5.1.
For any two sequences $1\leq i_{1}<i_{2}<\ldots<i_{k-1}\leq n$ and $1\leq
j_{1}\leq j_{2}<\ldots<j_{n}$, the following relation is called Plucker
relation
$\sum_{a=1}^{k+1}(-1)^{a}p_{i_{1},i_{2},\ldots
i_{k-1},j_{a}}p_{j_{1},j_{2},,\widehat{j_{a}}\ldots j_{k+1},}$
Here $\widehat{j_{a}}$ means that it is omitted.
Let $I_{k,n}$ denote the homogeneous ideal generated by all the plucker
relations. We have already stated that $\operatorname{Gr}(k,n)$ is a
projective variety in $\mathbb{P}^{{n\choose k}-1}$. $\operatorname{Gr}(k,n)$
is the zero set of the plucker ideal, i.e.
$\operatorname{Gr}(k,n)=V(I_{k,n})$. So, the tropical Grassmannian is the
$\operatorname{trop}(V(I_{k,n}))$ and is denoted by $\mathcal{G}_{k.n}$.
### 5.1. $\mathcal{G}_{k,n}$ and the space of phylogenetic trees
When $k$=2, the plucker ideal $I_{2,n}$ is generated by three term plucker
relations, $p_{i,j}p_{k,\ell}-p_{i,k}p_{j,l}+p_{i,l}p_{j,k}$, i.e.,
$I_{2,n}=({p_{i,j}p_{k,l}-p_{i,k}p_{j,l}+p_{i,l}p_{j,k}:i,j,k,l\in[n]})\;.$
Therefore,
$\mathcal{G}_{k.n}=\operatorname{Trop}(I_{2,n})=\bigcap\operatorname{trop}(V(p_{i,j}p{k,l}-p_{i,k}p_{j,l}+p_{i,l}p_{j,k}))$.
But $\operatorname{trop}(V(p_{i,j}p{k,l}-p_{i,k}p_{j,l}+p_{i,l}p_{j,k}))$ is
the set of all points where the minimum of $p_{i,j}+p{k,l},p_{i,k}+p_{j,l}$
and ${p_{i},l}+p_{j,k}$ is achieved twice, that is exactly the four-point
condition of the tree metric theorem mentioned above. So, we get the following
result
###### Theorem 5.2.
$\mathcal{G}_{2.n}=T_{n}$=space of all trees (phylogenetic trees).
### 5.2. Tropical Linear spaces
The Grassmannian is the simplest example of modulli space as each point of the
Grassmannian corresponds to a linear variety. In a similar way we can think of
the tropical Grassmannian as parametrizing the tropical linear spaces. Each
point of the tropical Grassmannian corresponds to a tropical linear space. In
this section, we look at tropical linear spaces which are in the image of the
$\mathcal{G}_{2,k}$ under the generalized dissimilarity map.
###### Theorem 5.3.
(Manon, 2011) $\phi^{k}(\mathcal{G}_{2.n})\subset\mathcal{G}_{n,k}$
Let $v\in\mathcal{G}_{2,k}$. Consider the point
$\phi^{k}(v)\in\mathcal{G}_{n,k}$. Let $TL_{v}$ denote the tropical linear
space associated to $v$.
###### Theorem 5.4.
Let $T$ be the tree realizing $v$ and $(v_{1},\ldots,v_{n})$ be the distance
of the leaf nodes from the root of $T$. Then the point $(v_{1},\ldots,v_{n})$
lies in the tropical linear space $TL_{v}$.
###### Proof.
We use theorem 2.2 to deduce the above result. Let $\sigma$ be the permutation
for which
$\displaystyle D^{r}(\\{i_{1},\ldots
i_{r}\\})=\frac{1}{2}((D(i_{1},\sigma(i_{1}))+D(\sigma(i_{1}),\sigma^{2}(i_{1}))$
$\displaystyle+D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots+D(\sigma^{r-1}(i_{1}),\sigma^{r}(i_{r}))$
Now $x_{i_{k}}+x_{i_{k}^{\prime}}\geq D(i_{k},i_{k}^{\prime})$, since
$D(i_{k},i_{k}^{\prime})$ is the length of the shortest path between $i_{k}$
and $i_{k}^{\prime}$.Therefore,
$\displaystyle(($ $\displaystyle
x_{1}+x_{\sigma(1)})+(x_{\sigma(i_{1})}+x_{\sigma(i_{2})})+(x_{\sigma(i_{2})}+x_{\sigma{i_{3}}}+\ldots+x_{\sigma(i_{r-1}}+x_{\sigma(i_{r})}\geq$
$\displaystyle(D(i_{1},\sigma(i_{1}))+D(\sigma(i_{1}),\sigma^{2}(i_{1}))+D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots+$
$\displaystyle D(\sigma^{r-1}(i_{r-1}),\sigma^{r}(i_{r}))$
From which we get
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}\geq\frac{1}{2}(D(i_{1},\sigma(i_{1}))+$
$\displaystyle D(\sigma(i_{1}),\sigma^{2}(i_{1}))+$ $\displaystyle
D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots+D(\sigma^{r-1}(i_{r-1}),\sigma^{r}(i_{r})))$
Now using theorem 2.2 We get $x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}\geq
D({i_{1},\ldots i_{r}})$. ∎
The above statement is actually a special case of a more general theorem. Let
$x$ be any internal node in our tree, let $x_{TL}\in\mathbb{R}^{n}$ represent
the $(w(l_{1},x),\ldots w(l_{n},x))$.
###### Theorem 5.5.
Every internal node of $T$ corresponds to a distinct point in the $TL(T)$.
###### Proof.
We show that each of the $x_{TL}$ belong $TL(T)$. Let $x_{TL}=(x_{1},\ldots
x_{n})$.
We proceed as above, let $\sigma$ be the permutation for which
$\displaystyle D^{r}(\\{i_{1},\ldots,i_{r}\\})=$
$\displaystyle\frac{1}{2}((D(i_{1},\sigma(i_{1}))+D(\sigma(i_{1}),\sigma^{2}(i_{1}))$
$\displaystyle+D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots+D(\sigma^{r-1}(i_{1}),\sigma^{r}(i_{r})).$
Now $x_{i_{k}}+x_{i_{k}^{\prime}}\geq D(i_{k},i_{k}^{\prime})$, since
$D(i_{k},i_{k}^{\prime})$ is the length of the shortest path between $i_{k}$
and $i_{k}^{\prime}$ and $w(x,i_{k})+w(x,i_{k}^{\prime})\geq
D(i_{k},i_{k}^{\prime})$. Therefore,
$\displaystyle((x_{1}+x_{\sigma(1)})+(x_{\sigma(i_{1})}+x_{\sigma(i_{2})})+(x_{\sigma(i_{2})}+x_{\sigma{i_{3}}}+\ldots+x_{\sigma(i_{r-1}}+x_{\sigma(i_{r})}\geq$
$\displaystyle(D(i_{1},\sigma(i_{1}))+D(\sigma(i_{1}),\sigma^{2}(i_{1}))+D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots$
$\displaystyle+D(\sigma^{r-1}(i_{r-1}),\sigma^{r}(i_{r})).$
From which we get
$\displaystyle x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}\geq$
$\displaystyle\frac{1}{2}(D(i_{1},\sigma(i_{1}))+D(\sigma(i_{1}),\sigma^{2}(i_{1}))$
$\displaystyle+D(\sigma^{2}(i_{1}),\sigma^{3}(i_{1}))+\ldots+D(\sigma^{r-1}(i_{r-1}),\sigma^{r}(i_{r})).$
Now using theorem 2.2 We have $x_{i_{1}}+x_{i_{2}}\ldots+x_{i_{r}}\geq
D^{r}({i_{1},\ldots,i_{r}})$.
Now, we prove that $x_{TL}$ and $x^{\prime}_{TL}$ are distinct if $x$ and
$x^{\prime}$ are distinct nodes. To see this, first note that the smallest
subtree of $T$ containing the leaf nodes of $T$=
$\\{l_{1},l_{2},\ldots,l_{n}\\}$ is $T$ itself, because if we remove any
vertex from $T$, then both the connected components of the tree after deletion
contain leaf nodes. Therefore, there exists a leaf node $l$ such that the
shortest path from $l$ to $x$ must pass through $x^{\prime}$ ,so $x_{\ell}$
must be greater than $x^{\prime}_{\ell}$ and we immediately get the result. ∎
Now we extend this result from the nodes of $T$ to sub-trees of $T$. Let
$T^{\prime}$ be the sub-tree of $T$ consisting only internal nodes. Consider
$x_{T}^{\prime}\in\mathbb{R}^{n}$ and
$x_{T}^{\prime}=(w(1,T^{\prime})+cT^{\prime},w(2,T^{\prime})+cT^{\prime},w(3,T^{\prime})+cT^{\prime},\ldots,w(r,T^{\prime})+cT^{\prime})$.
###### Theorem 5.6.
For every $T^{\prime}$ in $T$, $x_{T}^{\prime}$ lies in $TL(T)$
###### Proof.
Consider the leaf nodes $i_{1},\ldots i_{r}$. Suppose $d=0$ be the shortest
distance between the smallest tree containing $i_{1},\ldots i_{r}$ and $T$.
Let the shortest path between $i_{k}$ and $T^{\prime}$ be
$(i_{k},\ldots,d_{k})$. Since, $d=0$, $d_{k}$ lies in the shortest tree
containing $i_{1},\ldots i_{r}$. Also, no other vertex of $T^{\prime}$ lies in
the path $(i_{k}\ldots,d_{k})$ other than $d_{k}$, otherwise it will
contradict the minimality criteria. Now, let $v$ be a node contained both in
tree $T^{\prime}$ and the smallest tree containing $i_{1},\ldots i_{r}$. Now
$\displaystyle
w(i_{1},v_{1})+w(i_{2},v_{2})+\ldots+w(i_{r},v_{r})+w(v_{1},v)+$
$\displaystyle w(v_{2},v)+\ldots+w(v_{r},v)$ $\displaystyle\geq
D^{r}(\\{i_{1},\ldots,i_{r}\\}).$
So, we get
$\displaystyle w(i_{1},v_{1})+w(i_{2},v_{2})+\ldots+w(i_{r},v_{r})$
$\displaystyle\geq
D^{r}(\\{i_{1},\ldots,i_{r}\\})-\\{w(v_{1},v)+w(v_{2},v)+\ldots+w(v_{r},v)\\}.$
Now,adding $T^{\prime}$ on both side, we get
$\displaystyle w(i_{1},v_{1})+w(i_{2},v_{2})+\ldots+w(i_{r},v_{r})+T^{\prime}$
$\displaystyle\geq D^{r}(\\{i_{1},\ldots,i_{r}\\})-\\{w(i_{2},v_{2})+\ldots$
$\displaystyle+w(i_{r},v_{3})+w(v_{1},v)+w(v_{2},v)+\ldots+w(v_{r},v)\\}+T^{\prime}.$
Since, $v,v_{1},\ldots v_{r}$ belong to $T^{\prime}$,
$T^{\prime}-\\{w(i_{2},v_{2})+\ldots+w(i_{r},v_{3})+w(v_{1},v)+w(v_{2},v)+\ldots+w(v_{r},v)\\}$
is positive.
Now, let $x_{T}^{\prime}=(x_{1},\ldots,x_{n})$. We get
$\displaystyle x_{i_{1}}+x_{i_{2}}+\ldots
x_{i_{r}}=w(i_{1},s)+w(i_{2},s)+w(i_{3},s)+\ldots w(i_{r},s)+rd+T^{\prime}$
$\displaystyle\geq D^{r}(\\{i_{1},\ldots i_{r}\\})+rd+T^{\prime}\geq
D^{r}(\\{i_{1},\ldots i_{r}\\}).$
Now let us assume $d>0$ be the shortest distance between the smallest tree
containing $i_{1},\ldots i_{r}$ and $T^{\prime}$. Now let the shortest path
from the smallest tree containing $i_{1},\ldots,i_{r}$ and $T^{\prime}$ be
$s,v_{1},v_{2},\ldots,t$. Then shortest the path from $i_{k}$ to $T^{\prime}$
is $i_{k},\ldots,s,\ldots,t$, because if the path is something different
$i_{k},\ldots,s^{\prime},\ldots,t^{\prime}$, then either
$i_{k},\ldots,s,\ldots,s^{\prime}$ will form a cycle or
$i_{k},\ldots,d,\ldots,d^{\prime}$ will form a cycle. Now, let
$x_{T}^{\prime}=(x_{1},\ldots+x_{n})$. $w(i_{r},T)=w(i_{r},s)+d$. Therefore,
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}=w(i_{1},s)+w(i_{2},s)+w(i_{3},s)+\ldots+w(i_{r},s)+rd+T^{\prime}$
$\displaystyle\geq D^{r}(\\{i_{1},\ldots i_{r}\\})+rd+T^{\prime}\geq
D^{r}(\\{i_{1},\ldots i_{r}\\}).$
∎
Now, we study the points which lie on facets of the $TL(T)$. We deduce a
necessary and sufficient condition on $T^{\prime}$ for $x_{T^{\prime}}$ to be
on the facet of $TL(T)$.
###### Theorem 5.7.
A necessary condition for $x_{T}^{\prime}$ to lie on the facet of $TL(T)$ is
that there exists $\\{i_{1},\ldots,i_{r}\\}\in{[n]\choose r}$ such that
smallest tree containing $\\{i_{1},\ldots,i_{r}\\}$ also contains
$T^{\prime}$.
###### Proof.
We prove it by contradiction. Suppose that $T$ is not contained in the
smallest tree containing $\\{i_{1},\ldots i_{r}\\}$ for any
$\\{i_{1},\ldots,i_{r}\\}\in{[n]\choose r}$. Now, there are two cases the
distance.
Suppose the distance between $T$ and smallest tree containing $\\{i_{1},\ldots
i_{r}\\},d>0$. Now let the shortest path from the smallest tree containing
$i_{1},\ldots,i_{r}$ and $T^{\prime}$ be $s,v_{1},v_{2}\ldots,t$. As in the
proof above we get
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}=w(i_{1},s)+w(i_{2},s)+w(i_{3},s)+\ldots+w(i_{r},s)+rd+T^{\prime}$
$\displaystyle\geq D^{r}(\\{i_{1},\ldots i_{r}\\})+rd+T^{\prime}\geq
D^{r}(\\{i_{1},\ldots i_{r}\\}).$
Now in this both $rd$ and $T^{\prime}$ are non-zero positive integers. From
which we get
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}=w(i_{1},s)+w(i_{2},s)+w(i_{3},s)+\ldots+w(i_{r},s)+rd+T^{\prime}$
$\displaystyle>D^{r}(\\{i_{1},\ldots,i_{r}\\}).$
Hence we get the result.
Now, suppose $d=0$. In that case we get
$\displaystyle
w(i_{1},v_{1})+w(i_{2},v_{2})+\ldots+w(i_{r},v_{r})+T^{\prime}\geq
D^{r}(\\{i_{1},\ldots i_{r}\\})-\\{w(i_{2},v_{2})$
$\displaystyle+\ldots+w(i_{r},v_{3})+w(v_{1},v)+w(v_{2},v)+\ldots+w(v_{r},v)\\}+T^{\prime}.$
Now, since we $T^{\prime}$ is not contained in the smallest tree containing
$\\{i_{1},\ldots,i_{r}\\}$,
$T^{\prime}-\\{w(i_{2},v_{2})+\ldots+w(i_{r},v_{3})+w(v_{1},v)+w(v_{2},v)+\ldots+w(v_{r},v)\\}$
is strictly greater than zero. So,
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}>D^{r}(\\{i_{1},\ldots,i_{r}\\}).$
∎
Let $\mathrm{in}(v,T^{\prime})$ denote the set of all vertices appearing in
the shortest path from $v$ and $T$ except the beginning and the end vertices.
Now we get our necessary and sufficient condition for $x_{T}^{\prime}$ to lie
on the facet.
###### Theorem 5.8.
$x_{T}^{\prime}$ lie on the facet of $TL(T)$ iff there exists a
$S=\\{i_{1},\ldots i_{r}\\}\in{[n]\choose r}$ such that $T^{\prime}$ is
contained in the smallest tree containing ${i_{1},\ldots i_{r}}$ and
$\bigcap_{k\in T}\mathrm{in}(v,T^{\prime})=\phi$.
###### Proof.
We have $x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}\geq D^{r}(\\{i_{1},\ldots
i_{r}\\})$. Now, $T^{\prime}\;\cap\;\mathrm{in}(i_{k},T^{\prime})=\phi$ for
all $k\in{1,2,\ldots,r}$ because otherwise it will contradict the minimality
of the path from $i_{k}$ to $T^{\prime}$. Now since $\bigcap_{k\in
T}in(v,T^{\prime})=\phi$, every vertex of appears at most once in
$x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}$ which implies
$\displaystyle x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}\leq D^{r}(\\{i_{1},\ldots
i_{r}\\}).$
Therefore, we get
$\displaystyle
x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{r}}=D^{r}(\\{i_{1},\ldots,i_{r}\\}).$
and hence the result. ∎
## 6\. Conclusion
We have shown here how the tropical linear spaces corresponding to a
phylogenetic tree encodes various information about the tree.
## References
* Buneman (1974) Buneman, P. (1974). A note on the metric properties of trees. _Journal of Combinatorial Theory, Series B_ 17(1), 48–50.
* Gathmann & Markwig (2008) Gathmann, A. & Markwig, H. (2008). Kontsevich’s formula and the wdvv equations in tropical geometry. _Advances in Mathematics_ 217(2), 537–560.
* Maclagan & Sturmfels (2009) Maclagan, D. & Sturmfels, B. (2009). Introduction to tropical geometry. _Book in preparation_ 34.
* Manon (2011) Manon, C. (2011). Dissimilarity maps on trees and the representation theory of sl m (ℂ). _Journal of Algebraic Combinatorics_ 33(2), 199–213.
* Mikhalkin (2003) Mikhalkin, G. (2003). Counting curves via lattice paths in polygons. _Comptes Rendus Mathematique_ 336(8), 629–634.
* Miller & Sturmfels (2005) Miller, E. & Sturmfels, B. (2005). _Combinatorial commutative algebra_ , vol. 227. Springer.
* Speyer & Sturmfels (2004) Speyer, D. & Sturmfels, B. (2004). The tropical grassmannian. _Advances in Geometry_ 4(3), 389–411.
|
arxiv-papers
| 2013-12-03T09:55:44 |
2024-09-04T02:49:54.702967
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ambedkar Dukkipati, Aritra Sen",
"submitter": "Aritra Sen",
"url": "https://arxiv.org/abs/1312.0752"
}
|
1312.0910
|
# MPWide: a light-weight library for efficient message passing over wide area
networks
Derek Groen1, Steven Rieder2,3,4, Simon Portegies Zwart2
1 Centre for Computational Science, University College London, London, United
Kingdom
2 Leiden Observatory, Leiden University, Leiden, The Netherlands
3 System and Network Engineering research group, University of Amsterdam,
Amsterdam, the Netherlands
4 Kapteyn Instituut, Rijksuniversiteit Groningen, Groningen, the Netherlands
E-mail: [email protected]
###### Abstract
We present MPWide, a light weight communication library which allows efficient
message passing over a distributed network. MPWide has been designed to
connect application running on distributed (super)computing resources, and to
maximize the communication performance on wide area networks for those without
administrative privileges. It can be used to provide message-passing between
application, move files, and make very fast connections in client-server
environments. MPWide has already been applied to enable distributed
cosmological simulations across up to four supercomputers on two continents,
and to couple two different bloodflow simulations to form a multiscale
simulation.
Keywords: communication library, distributed computing, message passing, TCP,
model coupling, communication performance, data transfer, co-allocation
## 1 Overview
### 1.1 Introduction
Modern scientific software is often complex, and consists of a range of hand-
picked components which are combined to address a pressing scientific or
engineering challenge. Traditionally, these components are combined locally to
form a single framework, or used one after another at various locations to
form a scientific workflow, like in AMUSE 111AMUSE - http://www.amusecode.org
[17, 16]. However, these two approaches are not universally applicable, as
some scientifically important functionalities require the use of components
which run concurrently, but which cannot be placed on the same computational
resource. Here we present MPWide, a library specifically developed to
facilitate wide area communications for these distributed applications.
The main use of MPWide is to flexibly manage and configure wide area
connections between concurrently running applications, and to facilitate high-
performance message passing over these connections. These functionalities are
provided to application users and developers, as MPWide can be installed and
used without administrative privileges on the (super)computing resources. We
initially reported on MPWide in 2010 [9], but have since extended the library
considerably, making it more configurable and usable for a wider range of
applications and users. Here we describe MPWide, its implementation and
architecture, requirements and reuse potential.
MPWide was originally developed as a supporting communication library in the
CosmoGrid project [15]. Within this project we constructed and executed large
cosmological N-body simulations across a heterogeneous global network of
supercomputers. The complexity of the underlying supercomputing and network
architectures, as well as the high communication performance required for the
CosmoGrid project, required us to develop a library that was both highly
configurable and trivial to install, regardless of the underlying
(super)computing platform.
There are a number of tools which have similarities to MPWide. ZeroMQ [1], is
a socket library which supports a wide range of platforms. However, compared
with MPWide it does have a heavier dependency footprint. Among other things it
depends on uuid-dev, a package that requires administrative privileges to
install. In addition, there are several performance optimization parameters
which can be tweaked with MPWide but not with ZeroMQ. Additionally, the
NetIBIS [3] and the PadicoTM [5] tools provide functionalities similar to
MPWide, though NetIBIS is written in Java, which is not widely supported on
the compute nodes of supercomputers, and PadicoTM requires the presence of a
central rendez-vous server. For fast file transfers, alternatives include
GridFTP and various closed-source file transfer software solutions. There are
also dedicated tools for running MPI applications across clusters [11, 14, 2]
and for coupling applications to form a multiscale simulation (e.g., MUSCLE
[4] and the Jungle Computing System [6]).
### 1.2 Summary of research using MPWide
MPWide has been applied to support several research and technical projects so
far. In this section we summarize these projects, the purpose for which MPWide
has been used in these projects, and the performance that we obtained using
MPWide.
#### 1.2.1 The CosmoGrid project
MPWide has been used extensively in the CosmoGrid project, for which it was
originally developed. In this project we required a library that enabled fast
message passing between supercomputers and which was trivial to install on
PCs, clusters, little Endian Cray-XT4 supercomputers and big Endian IBM Power6
supercomputers. In addition, we needed MPWide to deliver solid communication
performance over light paths and dedicated 10Gbps networks, even when these
networks were not optimally configured by administrators.
In CosmoGrid we ran large cosmological simulations, and at times in parallel
across multiple supercomputers, to investigate key properties of small dark
matter haloes [13]. We used the GreeM cosmological N-body code [12], which in
turn relied on MPWide to facilitate the fast message-passing over wide area
networks.
Our initial production simulation was run distributed, using a supercomputer
at SurfSARA in Amsterdam, and one at the National Astronomical Observatory of
Japan in Tokyo [15]. The supercomputers were interconnected by a lightpath
with 10 Gigabit/s bandwidth capacity. Our main simulation consisted of
$2048^{3}$ particles, and required about 10% of its runtime to exchange data
over the wide area network.
We subsequently extended the GreeM code, and used MPWide to run cosmological
simulations in parallel across up to 4 supercomputers [8]. We also performed a
distributed simulation across 3 supercomputers, which consisted of $2048^{3}$
particles and used 2048 cores in total [10]. These machines were located in
Espoo (Finland), Edinburgh (Scotland) and Amsterdam (the Netherlands). The run
used MPWide version 1.0 and lasted for about 8 hours in total. We present some
performance results of this run in Fig. 1, and also provide the performance of
the simulation using one supercomputer as a reference. The distributed
simulation is only 9% slower than the one executing on a single site, even
though simulation data is exchanged over a baseline of more than 1500
kilometres at every time step. A snapshot of our distributed simulation, which
also features dynamic load balancing, can be found in Fig. 2. The results from
the CosmoGrid project have been used for the analysis of dark matter haloes
[13] as well as for the study of star clusters in a cosmological dark matter
environment [19, 18].
Figure 1: Comparison of the wallclock time required per simulation step
between a run using 2048 cores on one supercomputer (given by the teal line),
and a nearly identical run using 2048 cores distributed over three
supercomputers (given by the red line). The two peaks in the performance of
the single site run were caused by the writing of 160GB snapshots during those
iterations. The run over three sites used MPWide to pass data between
supercomputers. The communication overhead of the run over three sites is
given by the black line. See Groen et al. [10] for a detailed discussion on
these performance measurements. Figure 2: Snapshot of the cosmological
simulation discussed in Fig. 1, taken at redshift $z$ = 0 (present day). The
contents have been colored to match the particles residing on supercomputers
in Espoo (green, left), Edinburgh (blue, center) and Amsterdam (red, right)
respectively [10].
#### 1.2.2 Distributed multiscale modelling of bloodflow
We have also used MPWide to couple a three-dimensional cerebral bloodflow
simulation code to a one-dimensional discontinuous Galerkin solver for
bloodflow in the rest of the human body [7]. Here, we used the 1D model to
provide more realistic flow boundary conditions to the 3D model, and relied on
MPWide to rapidly deliver updates in the boundary conditions between the two
codes. We ran the combined multiscale application on a distributed
infrastructure, using 2048 cores on the HECToR supercomputer to model the
cerebral bloodflow and a local desktop at University College London to model
the bloodflow in the rest of the human body. The two resources are connected
by regular internet, and messages require 11 ms to traverse the network back
and forth between the desktop and the supercomputer. We provide an overview of
the technical layout of the codes and the communication processes in Fig. 3
The communications between these codes are particularly frequent, as the codes
exchanged data every 0.6 seconds. However, due to latency hiding techniques we
achieve to run our distributed simulations with neglishible coupling overhead
(6 ms per coupling exchange, which constituted 1.2% of the total runtime). A
full description of this run is provided by Groen et al. [7].
Figure 3: Overview of the codes and communication processes in the distributed
multiscale bloodflow simulation. Here the 1D pyNS code uses MPWide to connect
to an MPWide data forwarding process on the front-end node of the HECToR
supercomputer. The 3D HemeLB code, which is executed on the compute nodes of
the HECToR machine also connects to this data forwarding process. The
forwarding process allows us to construct this simulation, even when the
incoming ports of HECToR are blocked, and when the nodes where HemeLB will run
are not known in advance. Once the connections are established, the
simulations startd and boundary data is exchanged between the codes at
runtime.
#### 1.2.3 Other research and technical projects
We have used MPWide for several other purposes. First, MPWide is part of the
MAPPER software infrastructure [20], and is integrated in the MUSCLE2 coupling
environment 222MUSCLE2 - http://www.qoscosgrid.org/trac/muscle. Within
MUSCLE2, MPWide is used to improve the wide area communication performance in
coupled distributed multiscale simulation [4]. Additionally, we applied the
mpw-cp file transferring tool to test the network performance between the
campuses of University College London and Yale University. In these throughput
performance tests we were able to exchange 256 MB of data at a rate of $\sim$8
MB/s using scp, a rate of $\sim$40 MB/s using MPWide, and a rate of $\sim$48
MB/s using a commercial, closed-source file transfer tool named Aspera.
We have conducted a number of basic performance tests over regular internet,
comparing the performance of MPWide with that of ZeroMQ 333ZeroMQ -
http://www.zeromq.org, MUSCLE 1 and regular scp. During each test we exchanged
64MB of data (in memory in the case of MPWide, MUSCLE and ZeroMQ, and from
file in the case of scp), measuring the time to completion at least 20 times
in each direction. We then took the average value of these communications in
each direction. In these tests we used ZeroMQ with the default autotuned
settings.
Endpoint 1 | Endpoint 2 | Name of tool | average throughput in each direction
---|---|---|---
| | | MB $s^{-1}$
London, UK | Poznan, PL | scp | 11/16
London, UK | Poznan, PL | MPWide | 70/70
London, UK | Poznan, PL | ZeroMQ | 30/110
Poznan, PL | Gdansk, PL | scp | 13/21
Poznan, PL | Gdansk, PL | MPWide | 115/115
Poznan, PL | Gdansk, PL | ZeroMQ | 64/-
Poznan, PL | Amsterdam, NL | scp | 32/9.1
Poznan, PL | Amsterdam, NL | MPWide | 55/55
Poznan, PL | Amsterdam, NL | MUSCLE 1 | 18/18
Table 1: Summary of the throughput performance tests using MPWide and several
other tools to exchange data between resources in the United Kingdom (UK), the
Netherlands (NL) and Poland (PL) using regular internet. Tests over individual
connections were performed in quick succession to mitigate potential bias due
to background load on the internet backbone. A full report on these tests can
be found at http://www.mapper-project.eu, Deliverable 4.2 version 0.7.
### 1.3 Implementation and architecture
We present a basic overview of the MPWide architecture in Fig. 4. MPWide has
been implemented with a strong emphasis on minimalism, relying on a small and
flexible codebase which is used for a range of functionalities.
#### 1.3.1 Core MPWide library
The core MPWide functionalities are provided by the MPWide C++ API, the
communication codebase, and the Socket class. Together, these classes comprise
about 2000 lines of C++ code. The Socket class is used to manage and use
individual tcp connections, while the role of the communication codebase is to
provide the MPWide API functionalities in C++, using the Socket class. We
provide a short listing of functions in the C++ API in Table 2. More complete
information can be found in the MPWide manual, which resides in the /doc
subdirectory of the source code tree.
MPWide relies on a number of data structures, which are used to make it easier
to manage the customized connections between endpoints. The most
straightforward way to construct a connection in MPWide is to create a
communication path. Each path consists of 1 or more tcp streams, each of which
is used to facilitate actual communications over that path. Using a single tcp
stream is sufficient to enable a connection, but in many wide area networks,
MPWide will deliver much better performance when multiple streams are used.
MPWide supports the presence of multiple paths, and the creation and deletion
of paths at runtime. In addition, any messages can be passed from one path to
another using MPW_Cycle(), or MPW_Relay() for sustained dedicated data
forwarding processes (See Tab. 2).
MPWide comes with a number of parameters which allow users to optimize the
performance of individual paths. Aside from varying the number of streams,
users can modify the size of data sent and received per low-level
communication call (the chunk size), the tcp window size, and limit the
throughput for individual streams by adjusting the communication pacing rate.
The number of streams will always need to provided by the user when creating a
path, but users can choose to have the other parameters automatically tuned by
enabling the MPWide autotuner. The autotuner, which is enabled by default, is
useful for obtaining fairly good performance with minimal effort, but the best
performance is obtained by testing different parameters by hand. When choosing
the number of tcp streams to use in a path, we recommend using a single stream
for connections between local programs, and at least 32 streams when
connecting programs over long-distance networks. We have found that MPWide can
communicating efficiently over as many as 256 tcp streams in a single path.
Figure 4: Overview of MPWide functionalities and their links to underlying components. Functionalities available to the user are given by black arrows, links of these functionalities to the corresponding MPWide API by red lines, and internal codebase dependencies by dark blue lines. function name | summary description
---|---
MPW_Barrier() | Synchronize between two ends of the network.
MPW_CreatePath() | Create and open a path consisting of 1+ tcp streams.
MPW_Cycle() | Send buffer over one set of channels, receive from other.
MPW_DCycle() | As Cycle(), but with buffers of unknown size using caching.
MPW_DestroyPath() | Close and destroy a path consisting of 1+ tcp streams.
MPW_DNSResolve() | Obtain an IP address locally, given a hostname.
MPW_DSendRecv() | Send/receive buffers of unknown size using caching.
MPW_Init() | Initialize MPWide.
MPW_Finalize() | Close connections and delete MPWide buffers.
MPW_Recv() | Receive a single buffer (merging the incoming data).
MPW_Relay() | Forward all traffic between two channels.
MPW_Send() | Send a single buffer (splitted evenly over the channels).
MPW_SendRecv() | Send/receive a single buffer.
MPW_ISendRecv() | Send and/or receive data in a non-blocking mode.
MPW_Has_NBE_Finished() | Check if a particular non-blocking call has completed.
MPW_Wait() | Wait until a particular non-blocking call has completed.
MPW_setAutoTuning() | Enable or disable autotuning (default: enabled)
MPW_setChunkSize() | Change the size of data sent and received per low-level tcp send command.
MPW_setPacingRate() | Adjust the software-based communication pacing rate.
MPW_setWin() | Adjust the TCP window size within the constraints of the site configuration.
Table 2: List of available functions in the MPWide API.
#### 1.3.2 Python extensions
We have constructed a Python interface, allowing MPWide to be used through
Python 444Python - http://www.python.org. We construct the interface using
Cython 555Cython - http://www.cython.org, so as a result a recent version of
Cython is recommended to allow a smooth translation. The interface works
similar to the C++ interface, but supports only a subset of the MPWide
features. It also includes a Python test script. We also implemented an
interface using SWIG, but recommend Cython over SWIG as it is more portable.
#### 1.3.3 Forwarder
It is not uncommon for supercomputing infrastructures to deny direct
connections from the outside world to compute nodes. In privately owned
infrastructures, administrators commonly modify firewall rules to facilitate
direct data forwarding from outside to the compute nodes. The Forwarder is a
small program that mimicks this behavior, but is started and run by the user,
without the need for administrative privileges. Because the Forwarder operates
on a higher level in the network architecture, it is generally slightly less
efficient than conventional firewall-based forwarding. An extensive example of
using multiple Forwarder instances in complex networks of supercomputers can
be found in Groen et al. [8]
#### 1.3.4 mpw-cp
mpw-cp is a command-line file transfer tool which relies on SSH. Its
functionality is basic, as it essentially uses SSH to start a file transfer
process remotely, and then links that process to a locally executed one. mpw-
cp works largely similar to scp, but provides superior performance in many
cases, allowing users to tune their connections (e.g., by using multiple
streams) using command-line arguments.
#### 1.3.5 DataGather
The DataGather is a small program that allows users to keep two directories
synchronized on remote machines in real-time. It synchronizes in one direction
only, and it has been used to ensure that the data generated by a distributed
simulation is collected on a single computational resource. The DataGather can
be used concurrently with other MPWide-based tools, allowing users to
synchronize data while the simulation takes place.
#### 1.3.6 Constraints in the implementation and architecture
MPWide has a number of constraints on its use due to the choices we made
during design and implementation. First, MPWide has been developed to use the
tcp protocol, and is not able to establish or facilitate messages using other
transfer protocols (e.g. UDP). Second, compared to most MPI implementations,
MPWide has a limited performance benefit (and sometimes even a performance
disadvantage) on local network communications. This is because vendor MPI
implementations tend to contain architecture-specific optimizations which are
not in MPWide.
Third, MPWide does not support explicit data types in its message passing, and
treats all data as an array of characters. We made this simplification,
because data types vary between different architectures and programming
environments. Incorporating the management of these in MPWide would result in
a vast increase of the code base, as well as a permanent support requirement
to update the type conversions in MPWide, whenever a new platform emerges. We
recommend that users perform this serialization task in their applications,
with manual code for simple data types, and relying on a high-quality
serialization libraries for more complex data types.
### 1.4 Quality Control
Due to the small size of the codebase and the development team, MPWide has a
rather simplistic quality control regime. Prior to each public release, the
various functionalities of MPWide are tested manually for stability and
performance. Several test scripts (those which do not involve the use of
external codes) are available as part of the MPWide source distribution,
allowing users to test the individual functionalities of MPWide without
writing any new code of their own. These include:
* •
MPWUnitTests - A set of basic unit tests, can be run without any additional
arguments.
* •
MPWTestConcurrent - A set of basic functional tests, can be run without any
additional arguments.
* •
MPWTest - A benchmark suite which requires to be started manually on both end
points.
More details on how to use these tests can be found in the manual, which is
supplied with MPWide.
## 2 Availability
### 2.1 Operating system
MPWide is suitable for most Unix environments. It can be installed and used
as-is on various supercomputer platforms and Linux distributions. We have also
been able to install and use this version of MPWide successfully on Mac OS X.
### 2.2 Programming language
MPWide requires a C++ compiler with support for pthreads and UNIX sockets.
### 2.3 Additional system requirements
MPWide has no inherent hardware requirements.
### 2.4 Dependencies
MPWide itself has no major dependencies. The mpw-cp functionality relies on
SSH and the Python interface has been tested with Python 2.6 and 2.7. The
Python interface has been created using SWIG, which is required to generate a
new interface for different types of Python, or for non 64-bit and/or non-
Linux platforms.
### 2.5 List of contributors
* •
Derek Groen, has written most of MPWide and is the main contributor to this
writeup.
* •
Steven Rieder, assisted in testing MPWide, provided advice during development,
and contributed to the writeup.
* •
Simon Portegies Zwart, provided supervision and support in the MPWide
development, and contributed to the writeup.
* •
Joris Borgdorff, provided advice on the recent enhancements of MPWide, and
made several recent contributions to the codebase.
* •
Cees de Laat, provided advice during development and helped arrange the
initial Amsterdam-Tokyo lightpath for testing and production.
* •
Paola Grosso, provided advice during development and in the initial writeup of
MPWide.
* •
Tomoaki Ishiyama, contributed in the testing of MPWide and implemented the
first MPWide-enabled application (the GreeM N-body code).
* •
Hans Blom, provided advice during development and conducted preliminary tests
to compare a TCP-based with a UDP-based approach.
* •
Kei Hiraki, provided advice during development and infrastructural support
during the initial wide area testing of MPWide.
* •
Keigo Nitadori, provided advice during development.
* •
Junichiro Makino, for providing advice during development.
* •
Stephen L.W. McMillan, provided advice during development.
* •
Mary Inaba, provided infrastructural support during the initial wide area
testing of MPWide.
* •
Peter Coveney, provided support on the recent enhancements of MPWide.
### 2.6 Software location
We have made MPWide available on GitHub at: https://github.com/djgroen/MPWide.
### 2.7 Code Archive
Name: MPWide version 1.8.1 Persistent identifier:
http://dx.doi.org/10.6084/m9.figshare.866803 Licence: MPWide has been released
under the Lesser GNU Public License version 3.0. Publisher: Derek Groen Date
published: 3rd of December 2013
### 2.8 Code Repository
Name: MPWide Identifier: https://github.com/djgroen/MPWide Licence: MPWide has
been released under the Lesser GNU Public License version 3.0. Publisher:
Derek Groen (account name: djgroen) Date published: 15th of October 2013.
### 2.9 Language
GitHub uses the git repository system. The full MPWide distribution contains
code written primarily in C++, but also contains fragments written in C and
Python.
The code has been commented and documented solely in English.
## 3 Reuse potential
MPWide has been designed with a strong emphasis on reusability. It has a small
codebase, with minimal dependencies and does not make use of the more obscure
C++ features. As a result, users will find that MPWide is trivial to set up in
most Unix-based environments. MPWide does not receive any official funding for
its sustainability, but the main developer (Derek Groen) is able to respond to
any queries and provide basic assistance in adapting MPWide for new
applications.
### 3.1 Reuse of MPWide
MPWide can be reused for a range of different purposes, which all share one
commonality: the combination of light-weight software with low latency and
high throughput communication performance.
MPWide can be reused to parallelize an application across supercomputers and
to couple different applications running on different machines to form a
distributed multiscale simulation. A major advantage of using MPWide over
regular TCP is the more easy-to-use API (users do not have to cope with
creating arrays of sockets, or learn low-level TCP calls such as listen() and
accept()), and built-in optimizations that deliver superior performance over
long-distance networks.
In addition, users can apply MPWide to facilitate high speed file transfers
over wide area networks (using mpw-cp or the DataGather). MPWide provides
superior performance to existing open-source solutions on many long-distance
networks (see e.g., section 1.2.3). MPWide could also be reused to stream
visualization data from an application to a visualization facility over long-
distances, especially in the case when dedicated light paths are not
available.
Users can also use MPWide to link a Python program directly to a C or C++
program, providing a fast and light-weight connection between different
programming languages. However, the task of converting between data types is
left to the user (MPWide works with character buffers on the C++ side, and
strings on the Python side).
### 3.2 Support mechanisms for MPWide
MPWide is not part of any officially funded project, and as such does not
receive sustained official funding. However, there are two mechanisms for
unofficial support. When users or developers run into problems we encourage
them to either raise an issue on the GitHub page or, if urgent, to contact the
main developer (Derek Groen, [email protected]) directly.
### 3.3 Possibilities of contributing to MPWide
MPWide is largely intended as stand-alone and a very light-weight
communication library, which is easy to maintain and support. To make this
possible, we aim to retain a very small codebase, a limited set of features,
and a minimal number of dependencies in the main distribution.
As such, we are fairly strict in accepting new features and contributions to
the code on the central GitHub repository. We primarily aim to improve the
performance and reliability of MPWide, and tend to accept new contributions to
the main repository only when these contributions boost these aspects of the
library, and come with a limited code and dependency footprint.
However, developers and users alike are free to branch MPWide into a separate
repository, or to incorporate MPWide into higher level tools and services, as
allowed by the LGPL 3.0 license. We strongly recommend integrating MPWide as a
library module directly into higher level services, which then rely on the
MPWide API for any required functionalities. MPWide has a very small code
footprint, and we aim to minimize any changes in the API between versions,
allowing these high-level services to easily swap their existing MPWide module
for a future updated version of the library. We have already used this
approach in codes such as SUSHI, HemeLB and MUSCLE 2.
## Funding statement
This research is supported by the Netherlands organization for Scientific
research (NWO) grants #614.061.608 (AMUSE), #614.061.009 (LGM), #639.073.803,
#643.000.803 and #643.200.503, the European Commission grant for the
QosCosGrid project (grant number: FP6-2005-IST-5 033883), the Qatar National
Research Fund (QNRF grant code NPRP 5-792-2-328) and the MAPPER project (grant
number: RI-261507), SURFNet with the GigaPort project, NAOJ, the International
Information Science Foundation (IISF), the Netherlands Advanced School for
Astronomy (NOVA), the Leids Kerkhoven-Bosscha fonds (LKBF) and the Stichting
Nationale Computerfaciliteiten (NCF). SR acknowledges support by the John
Templeton Foundation, grant nr. FP05136-O. We thank the organizers of the
Lorentz Center workshop on Multiscale Modelling and Computing 2013 for their
support. We also thank the DEISA Consortium (www.deisa.eu), co-funded through
the EU FP6 project RI-031513 and the FP7 project RI-222919, for support within
the DEISA Extreme Computing Initiative (GBBP project).
## References
* [1] ZeroMQ - www.zeromq.org, 2013.
* [2] E. Agullo, C. Coti, T. Herault, J. Langou, S. Peyronnet, A. Rezmerita, F. Cappello, and J. Dongarra. QCG-OMPI: MPI applications on grids. Future Generation Computer Systems, 27(4):357 – 369, 2011.
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* [4] J. Borgdorff, M. Mamonski, B. Bosak, D. Groen, M. Ben Belgacem, K. Kurowki, and A. G. Hoekstra. Multiscale computing with the multiscale modeling library and runtime environment. In accepted by the International Conference for Computational Science, 2013.
* [5] A. Denis, C. Perez, and T. Priol. PadicoTM: An open integration framework for communication middleware and runtimes. Future Generation Computer Systems, 19(4):575–585, May 2003.
* [6] N. Drost, J. Maassen, M. van Meersbergen, H. Bal, I. Pelupessy, S. Portegies Zwart, M. Kliphuis, H. Dijkstra, and F. Seinstra. High-performance distributed multi-model / multi-kernel simulations: A case-study in jungle computing. In Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops & PhD Forum, IPDPSW ’12, pages 150–162, Washington, DC, USA, 2012. IEEE Computer Society.
* [7] D. Groen, J. Borgdorff, C. Bona-Casas, J. Hetherington, R. W. Nash, S. J. Zasada, I. Saverchenko, M. Mamonski, K. Kurowski, M. O. Bernabeu, A. G. Hoekstra, and P. V. Coveney. Flexible composition and execution of high performance, high fidelity multiscale biomedical simulations. Interface Focus, 3(2), April 6, 2013.
* [8] D. Groen, S. Portegies Zwart, T. Ishiyama, and J. Makino. High Performance Gravitational N-body simulations on a Planet-wide Distributed Supercomputer. Computational Science and Discovery, 4(015001), January 2011.
* [9] D. Groen, S. Rieder, P. Grosso, C. de Laat, and P. Portegies Zwart. A light-weight communication library for distributed computing. Computational Science and Discovery, 3(015002), August 2010.
* [10] D. Groen, S. Rieder, and S. Portegies Zwart. High performance cosmological simulations on a grid of supercomputers. In Proceedings of INFOCOMP 2011. Thinkmind.org, September 2011.
* [11] R. W. Hockney. The communication challenge for mpp: Intel paragon and meiko cs-2. Parallel Computing, 20(3):389 – 398, 1994.
* [12] T. Ishiyama, T. Fukushige, and J. Makino. GreeM: Massively Parallel TreePM Code for Large Cosmological N -body Simulations. Publications of the Astronomical Society of Japan, 61:1319–1330, December 2009.
* [13] T. Ishiyama, S. Rieder, J. Makino, S. Portegies Zwart, D. Groen, K. Nitadori, C. de Laat, S. McMillan, K. Hiraki, and S. Harfst. The cosmogrid simulation: Statistical properties of small dark matter halos. The Astrophysical Journal, 767(2):146, 2013.
* [14] S. Manos, M. Mazzeo, O. Kenway, P. V. Coveney, N. T. Karonis, and B. R. Toonen. Distributed mpi cross-site run performance using mpig. In HPDC, pages 229–230, 2008.
* [15] S. Portegies Zwart, T. Ishiyama, D. Groen, K. Nitadori, J. Makino, C. de Laat, S. McMillan, K. Hiraki, S. Harfst, and P. Grosso. Simulating the universe on an intercontinental grid. Computer, 43:63–70, 2010.
* [16] S. Portegies Zwart, S. McMillan, S. Harfst, D. Groen, M. Fujii, B. Ó Nualláin, E. Glebbeek, D. Heggie, J. Lombardi, P. Hut, V. Angelou, S. Banerjee, H. Belkus, T. Fragos, J. Fregeau, E. Gaburov, R. Izzard, M. Juric, S. Justham, A. Sottoriva, P. Teuben, J. van Bever, O. Yaron, and M. Zemp. A multiphysics and multiscale software environment for modeling astrophysical systems. New Astronomy, 14(4):369 – 378, 2009.
* [17] S. Portegies Zwart, S. L. W. McMillan, E. van Elteren, I. Pelupessy, and N. de Vries. Multi-physics simulations using a hierarchical interchangeable software interface. Computer Physics Communications, 183:456–468, March 2013.
* [18] S. Rieder, T. Ishiyama, P. Langelaan, J. Makino, S. L. W. McMillan, and S. Portegies Zwart. Evolution of star clusters in a cosmological tidal field. ArXiv e-prints, September 2013.
* [19] S. Rieder, R. van de Weygaert, M. Cautun, B. Beygu, and S. Portegies Zwart. Assembly of filamentary void galaxy configurations. MNRAS, 435:222–241, October 2013.
* [20] S. J. Zasada, M. Mamonski, D. Groen, J. Borgdorff, I. Saverchenko, T. Piontek, K. Kurowski, and P. V. Coveney. Distributed infrastructure for multiscale computing. In Distributed Simulation and Real Time Applications (DS-RT), 2012 IEEE/ACM 16th International Symposium on, pages 65 –74, oct. 2012.
|
arxiv-papers
| 2013-12-03T19:17:57 |
2024-09-04T02:49:54.728834
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Derek Groen, Steven Rieder and Simon Portegies Zwart",
"submitter": "Derek Groen",
"url": "https://arxiv.org/abs/1312.0910"
}
|
1312.0916
|
# The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic
Quantum Field Theory
Tadafumi Ohsaku
(today)
In der Welt habt ihr Angst; aber seid getrost, ich habe die Welt überwunden. (
Johannes, Kapitel 16 )
It always seems impossible until it is done. ( Nelson Mandela )
This Paper is Dedicated for Our Brave Fighters and Super-Heroes for the
Fundamental Human Rights Around the World.
Abstract: The anomalous Nambu-Goldstone ( NG ) theorem which is found as a
violation of counting law of the number of NG bosons of the standard ( normal
) NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic
theories is studied in detail, with emphasis on its mathematical aspect from
Lie algebras, geometry to number theory. The basis of counting law of NG
bosons in the anomalous NG theorem is examined by Lie algebras ( local ) and
Lie groups ( global ). A quasi-Heisenberg algebra is found generically in
various symmetry breaking schema of the anomalous NG theorem, and it indicates
that it causes a violation/modification of the Heisenberg uncertainty relation
in an NG sector which can be experimentally confirmed. This fact implies that
we might need a framework ”beyond” quantum mechanical apparatus to describe
quantum fluctuations in the phenomena of the anomalous NG theorem which might
affect formations of orderings in quantum critical phenomena. The formalism of
effective potential is presented for understanding the mechanism of anomalous
NG theorem with the aid of our result of Lie algebras. After an investigation
on a bosonic kaon condensation model with a finite chemical potential as an
explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on
the anomalous NG theorem is given for our general discussion. Not only the
condition of the counting law of true NG bosons, but also the mechanism to
generate a mass of massive NG boson is also found by our examination on the
kaon condensation model. Furthermore, the generation of a massive mode in the
NG sector is understood by the quantum uncertainty relation of the Heisenberg
algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in
the effective potential of the kaon condensation model. Hence the relation
between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode
coupling, and the mechanism of mass generation in an NG sector is established.
Finally, some relations between the Riemann hypothesis and the anomalous NG
theorem are presented.
KEYWORDS: Normal and anomalous Nambu-Goldstone theorem, Lie algebras and Lie
groups, Heisenberg algebra, the Heisenberg uncertainty relation, quantum
fluctuation and phase transition, differential geometry, number theory, the
Riemann hypothesis, spin systems, QCD.
## 1 Introduction
This paper belongs to the recent efforts to intend to make a final answer on
the controversial ( or, not well-organized ) issue on the counting law of the
number of Nambu-Goldstone ( NG ) bosons/modes in a nonrelativistic and/or
Lorentz-symmetry-broken system. We mainly concentrate on the cases of so-
called continuous internal symmetries expressed by Lie groups, commute with
the spacetime 4-translations and the Lorentz 4-rotations. We restrict
ourselves on the cases of four-dimensional spacetime: Two- and three-
dimensional cases are outside of our discussion due to the Mermin-Wagner-
Coleman theorem [11,48,52]. The Nambu-Goldstone theorem ( established around
1960-1962 [26,27,60,61] ) states that a zero-mass/gap particle ( Nambu-
Goldstone boson ) naturally arises from a theory associated with a
spontaneously-broken-symmetry generator, with one-to-one correspondences
between broken generators and the NG bosons. Here we call this standard
situation as the ”normal” NG theorem. It is a famous fact that, in an
anomalous case, the number of NG bosons does not coincide with the number of
broken generators of ${\rm Lie}(G)$ in a case of nonrelativistic field theory
( $G$ is a Lie group which gives a symmetry of the system, Lie$(G)$ denotes
its corresponding Lie algebra ). Nielsen and Chadha found this phenomena in
1976 [64]. After those discoveries in fundamental physics, almost four
decades, there is no precise solution or any explanation of reasoning of the
Nielsen-Chadha anomaly. Quite recently ( 2002-2013 ), this controversy has
been started to be resolved by several works in theoretical physics.
In this section, we summarize those recent results of their works appeared in
literature. To clarify our discussions and perspectives, one should employ a
classification: Lorentz = symmetric, spontaneously broken, explicitly broken;
Lie group = symmetric, spontaneously broken, explicitly broken. Thus totally
$3\times 3=9$ cases we have in general. In those cases, we consider here some
examples where the Lorentz symmetry is explicitly broken, while a Lie group
symmetry is spontaneously/explicitly broken. We will show that these two
examples are understood by a single framework. In principle, our anomalous NG
theorem is fall into the category of ”spontaneous” cases without an explicit
symmetry breaking. Other cases remain for our further study in future. In
fact, recent several works given in literature also consider the cases that
the Lorentz symmetry is explicitly broken or a nonrelativistic case, while a
Lie group of internal symmetry is spontaneously broken.
First, we discuss the main result of Refs.[8,34,81,82,83,84] which is
summarized into the following inequality for the number of NG bosons given by
Watanabe and Brauner: They gave it without any derivation [81]:
$\displaystyle n_{NG}$ $\displaystyle\geq$ $\displaystyle
n_{BS}-\frac{1}{2}{\rm rank}\langle 0|[Q^{A},Q^{B}]|0\rangle.$ (1)
Here, $n_{NG}$ gives the number of NG bosons they would be observed ( when a
norm is positive and physical ), $n_{BS}$ is the number of broken generators
of ${\rm Lie}(G)\simeq T_{e}G$ ( $G$; a Lie group which describes a symmetry
of system we consider, $e$; the origin ), $Q^{A}$ and $Q^{B}$ (
$A,B=1,\cdots,n_{BS}$ ) imply the conserved charges of broken generators, and
$|0\rangle$ is the vacuum of a theory. It should be mentioned that the rank is
defined for a matrix with matrix entries of indices $AB$. While, we know the
very simple and universal law that the number of broken generators $n_{BS}$
equals the sum of the number of ”true” NG bosons and the number of ”massive”
NG bosons:
$\displaystyle n_{true-NG}$ $\displaystyle=$ $\displaystyle n_{BS}-n_{massive-
NG}.$ (2)
It is clear for us that $\frac{1}{2}{\rm rank}\langle[Q^{A},Q^{B}]\rangle$
must give the number of massive modes ( the rank of Hessian of effective
action expanded by NG modes around a VEV, see the paper of S. Weinberg in our
references [86] ): The matrix is given in a quadratic form of broken
generators. Thus, our issue is now understood how to find a law in which the
broken generators contain massive modes. Let us compare the anomalous behavior
with a case of explicit symmetry breaking. The paper of Weinberg [86]
discusses a chiral Lagrangian with an explicit symmetry breaking mass term
proportional to
$\displaystyle{\rm tr}(e^{iQ}\Phi^{-iQ})$ $\displaystyle=$ $\displaystyle{\rm
tr}(g\Phi g^{-1})={\rm tr}{\rm Ad}(G(\Phi)),$ (3)
( $\Phi$; some bosonic fields ) and gives a formula of square of mass
parameters of NG bosons by the second-order derivative of the mass term. It is
given by the bracket like the following form, namely, a term given by twice
actions of adjoints:
$\displaystyle[[\Phi,Q^{A}],Q^{B}]$ (4)
and if we regard this as a matrix of indices $(A,B)$, then its dimension is
exactly equal with $[Q^{A},Q^{B}]$ appeared in the formula for counting the
number of NG bosons. Therefore, there is a similarity between an anomalous and
an explicit symmetry breaking at the Lie algebra level beside the factor 1/2.
Later, we will see how this factor 1/2 arises in our anomalous NG theorem. If
a theory has $n_{ex}$ explicit symmetry breaking mass parameters, and if
$\displaystyle n_{ex}\geq\frac{1}{2}{\rm rank}\langle[Q^{A},Q^{B}]\rangle,$
(5)
then
$\displaystyle n_{NG}=n_{BS}-n_{ex}$ (6)
may holds, since an explicit symmetry breaking parameter enforces that an NG
boson always has a finite mass.
While, after the work of Watanabe and Brauner, Hidaka [34] derived the
equation ( replaces $\geq$ to $=$ in the above inequality (1) ) via the
generalized Langevin formalism: His formalism is essentially the same with the
method of effective action. Thus, later we utilize the effective action
formalism, both Lorentz-violating relativistic and nonrelativistic cases.
Note that in a Poincaré invariant theory, a charge $Q$ is Lorentz scalar when
it is conserved, $[P^{\mu},Q]=0$. This is due to the Coleman-Mandula no-go
theorem [12]. Schaefer et al. pointed out in their paper [77] that if $Q$ is
given from the zeroth-component of conserved vector current of an internal
symmetry, then it cannot have a nonvanishing VEV for a Lorentz-symmetric
vacuum. They argue that we need $\langle Q\rangle\neq 0$ for realizing an
anomalous behavior of NG theorem. We need a careful discussion on it. In fact,
we now consider a theory of Lorentz-violating system, hence we cannot restrict
$Q$ as a Lorentz scalar. Moreover, we should distinguish the cases of Wigner
phase and Nambu-Goldstone ( NG ) phase. In a Wigner phase, $Q|0\rangle=0$ is
immediately concluded since $Q$ is a conserved quantity, while $Q|0\rangle\neq
0$ in an NG phase, and it does not look like a conserved quantity: This is the
essential part of the NG theorem. The symmetry of a Lagrangian ( theory ) and
its vacuum do not coincide with each other in an NG phase. Due to the unitary
inequivalence, one can not definitely say about what $e^{iQ}|0\rangle$ gives.
For example, the vector $Q|0\rangle$ cannot be normalized in an NG phase. It
might be possible to say that $|0\rangle$ or $Q|0\rangle$ are not $G$-modules
in the naive sense. While, if a theory spontaneously breaks its vacuum
symmetry, and if the Lorentz symmetry is broken under a certain manner, then
we lost the basis of the statement of $\langle 0|Q|0\rangle=0$ even though it
will be defined as an integral of three-dimensional total volume/space: The
physical situation of those VEVs may be determined self-consistently. We
emphasize the fact that this discussion is valid for a quantum theory but not
for classical systems, since we take a VEV of a quantum operator. We will also
give some insights on the case where a broken charge is a Lorentz-violating
generic tensor. Hence we have another classification: Vacuum = Lorentz
symmetric/Lorentz violated, Nöther charge = Lorentz symmetric/Lorentz
violated.
According to our short observation of several previous results in literature,
we classify NG bosons into, (i) true NG bosons as exactly massless particles,
(ii) massive NG bosons in our anomalous NG theorem, (iii) pseudo NG bosons
which have finite masses due to explicit symmetry breaking parameters in the
Lagrangian of the beginning. Several examples of (iii) have been studied, for
example, in Refs. [16,70]. Our terminology presented here is not the same with
the famous classification of NG bosons given by S. Weinberg for Lorentz-
invariant relativistic cases [86]: Our present discussion should be understood
as a generalization/extension of it.
Our several classifications are summarized into the table given in the next
page:
Theory (Lagrangian) | LS, explicitly LV
---|---
Vacuum | LS, spontaneously LV, explicitly LV
Lie group | symmetric, SB, AB, EB
Discrete ( C, P, T ) | symmetric, SB, EB
Nöther charge | LS, LV
NG boson | true, anomalously massive, pseudo
Here, several abbreviations mean: LS = Lorentz symmetric, LV = Lorentz
violated, SB = spontaneously broken, AB = anomalously broken, EB = explicitly
broken. ”Discrete” indicates a discrete symmetry, typically as a charge
conjugation ( C ), a parity ( P ), and a time-reversal ( T ).
This paper is organized as follows: In sec. 2, several typical symmetry
breaking schema will be studied from their Lie algebra/group aspects, and will
find several characteristic features of them in our anomalous NG theorem,
which never takes place in the standard NG case. In sec. 3, an effective
potential formalism will be investigate to understand the mechanism of our
anomalous NG theorem by employing our Lie-algebra results. In sec. 4, a kaon
condensation model with a finite chemical potential will be examined to obtain
our viewpoint on a generic Lagrangian of NG bosons which gives the phenomenon
of anomalous NG theorem. Then we will construct a generic Lagrangian which
cause the anomalous NG theorem, in sec. 5. Some relations between our
anomalous NG theorem and number theory, especially the Riemann zeta function,
will be presented in sec. 6. Finally, the concluding remarks will be given in
sec. 7.
## 2 Lie Algebras, Lie Groups, and Symmetry Breaking Schema
In this section, our anomalous NG theorem is examined by Lie algebras ( give
some local characters of NG sectors ) and Lie groups ( contain informations on
global aspects/structures of NG manifolds ), with employing several examples.
First, we would like to pay attention on the following fact before obtaining a
general discussion of symmetry breakings. In a breaking scheme of a symmetry,
it is not always the case that a Lie group $G$ is broken to a Lie subgroup $H$
to give a coset $G/H$: Thus, an examination on cosets as results of symmetry
breakings is not enough for studying the ( global ) nature of (
normal/explicit+dynamical/anomalous ) NG theorem. For example, let us consider
some examples of $SO(3)$ or $SU(2)$. ( You can compare with the case of
$U(2)$, or the electroweak symmetry breaking of the Standard Model! ) The
Hamiltonian of spin systems of ferro- and antiferromagnets may belong to
$SO(3)$ and sometimes $SU(2)$ [87] ( they are locally isomorphic, ${\rm
Lie}SO(3)\simeq{\rm Lie}SU(2)\simeq{\rm Lie}USP(2)$, and thus one has a
freedom to choose one of them at least at the Lie algebra level ). The isospin
space also be described by $SU(2)$. In a ferromagnetic case, quite a lot of
works consider broken generators as $s_{1}$ and $s_{2}$ of $SU(2)$ while
$s_{3}$ remains ”unbroken.” ( $s_{a}$, $a=1,2,3$ $\in{\rm Lie}(SU(2))$, by
using the representation of Pauli matrices. ) This breaking scheme is
schematically denoted as $SU(2)\to U(1)$, but it does not give a coset: This
breaking scheme does not have a coset ( quotient ) topology, since the set of
$g_{3}=e^{i\theta\sigma_{3}}$ does not form a closed normal subgroup. In this
case, two massless NG bosons may be expected but we find only one due to the
Nielsen-Chadha anomaly.
Let $G$ be a Lie group which gives the symmetry of a system, and its Lie
algebra as ${\bf g}={\rm Lie}(G)$. Let $S^{\alpha}$ ( $\alpha=1,\cdots,{\rm
dim}(G)-n_{SB}$ ) denote the generators ( a set of bases of ${\rm Lie}(G)$ )
correspond to remaining symmetries, and let $X^{\beta}$ (
$\beta=1,\cdots,n_{SB}$ ) imply the broken generators. From the orthogonality
condition of the bases of ${\bf g}={\rm Lie}(G)$, the Lie brackets
$[S^{\alpha},X^{\beta}]$ always belong to the linear space of broken
generators. While, any commutator of broken generators will be given by a
linear combination of all generators,
$\displaystyle[X^{\beta},X^{\gamma}]=\sum c^{\alpha}S^{\alpha}+\sum
c^{\delta}X^{\delta}.$ (7)
Hence, if the corresponding charges $Q^{S^{\alpha}}$ of $S^{\alpha}$ are
conserved and simultaneously they are Lorentz symmetric, and if the vacuum of
the theory is also Lorentz symmetric, then $Q^{S^{\alpha}}|0\rangle=0$ is
concluded immediately. On the contrary, $Q^{X^{\delta}}|0\rangle\neq 0$ ( for
$\forall\delta$ ) even if they are Lorentz scalar. If the breaking scheme
$G\to H$ ( $G$, $H$; Lie groups ) forms a coset $G/H$ and if it is a symmetric
space, then any Lie bracket of broken generators belongs to ${\rm Lie}(H)$
[20,32,43]:
$\displaystyle[X^{\alpha},X^{\beta}]\subset{\bf h}={\rm Lie}(H),\quad
S^{\alpha}\in{\bf h},\quad X^{\beta}\in{\bf m},\quad{\rm Lie}(G)={\bf g}={\bf
h}+{\bf m}.$ (8)
In that case, the VEV of any $[X^{\alpha},X^{\beta}]$ always vanishes in the
case of Lorentz symmetric conserved charges belong to ${\bf h}$. ( Therefore,
if the relation of Watanabe and Brauer is correct and is valid also in a
symmetric space, then the number of massive modes is given as a function of
VEVs of symmetric generators. ) It is interesting for us to consider several
models defined over Riemannian ( global ) symmetric spaces of the Cartan
classification [32]. Later, we will discuss how a local nature of anomalous NG
theorem is extended to a global structure in a symmetric space.
The equation of Lie algebra $[X^{\beta},X^{\gamma}]=\sum
c^{\alpha}S^{\alpha}+\sum c^{\delta}X^{\delta}$ means that the left hand side
of commutator is expanded by the linear form of the right hand side: Namely,
this formula counts the dimension of the linear space which the commutator
belongs to, and especially after taking a VEV of both side of this equation,
it gives a subspace which the VEV of commutator is described, such like a two-
dimensional surface in a three-dimensional space. In this sense, the VEV of
this equation is ”algebro-geometric.” After employing a method of
compactification suitable for a breaking scheme, a deformation theory and
moduli space for such an algebraic variety [30,31] could be introduced ( but
not always ). Especially in the symmetric space mentioned above, ${\rm
dim}[X^{\alpha},X^{\beta}]={\rm dim}{\rm Lie}(H)$. If the matrix $\langle
0|[X^{\alpha},X^{\beta}]|0\rangle$ is obtained from the second-order
derivative of an effective potential expanded by NG bosons associated with
broken generators, the matrix might contain a nonvanishing part, embedded in
the total part of the second-order derivative, which gives the finite mass
eigenvalues for the NG bosons. Namely, the dimension of the matrix of
nonvanishing part is the dimension of the linear space of massive NG bosons. (
In fact, the proof given in the paper of Watanabe and Murayama, Ref. [82], can
be interpreted as a calculation of basis set of the linear space which the
mass matrix of NG bosons belongs. ) Any type of non-vanishing VEV of $\sum
c^{\alpha}S^{\alpha}+\sum c^{\delta}X^{\delta}$ defines which pair
$[X^{\alpha},X^{\beta}]$ forms a non-vanishing matrix element. You should
notice that the rank of a matrix implies the dimension of a linearly-
independent components of a matrix. It should be mentioned that the dimension
of mass matrix is obtained after taking a VEV of the vacuum $|0\rangle$, and
thus at the moment one can say nothing about the mass matrix when one takes a
displacement from the vacuum $|0\rangle$ ( a mass matrix is given by
displacements of displacements ). It also should be examined how these
conditions of counting the dimension of massive modes in a mass matrix defined
in the linear space of Lie algebra has the validity, in a Higgs-type bosonic
model, an effective action of composite model like the Nambu$-$Jona-Lasinio (
NJL ) model or QCD, or in the case of Coleman-Weinberg mechanism [13]. In this
paper, we mainly consider a Goldstone-Higgs-type bosonic Lagrangian/theory to
investigate our anomalous NG theorem. Since a displacement caused by a Lie
group action to an order parameter ( where it is a composite or an elementary
field ) is given by an adjoint action of a Lie group, thus, our result should
be valid also in an NJL-type composite model. This is, of course, also the
case in a Coleman-Weinberg mechanism of symmetry breaking. Later, we examine a
Goldstone-Higgs type bosonic model of kaon condensation, while an examination
of NJL or QCD demands us further investigation as another paper, due to the
fact that those theories demand us some heavy calculations.
A lot of discussions use an $SU(2)$ model with two broken generators
$\sigma_{x}$, $\sigma_{y}$ while $\sigma_{z}$ is symmetric (
$(\sigma_{x},\sigma_{y},\sigma_{z})\in{\rm Lie}(SU(2))$ ). In this case,
$Q^{z}\propto\sigma_{z}$ can be regarded as an unbroken charge even though
this breaking scheme does not give a coset,
$\displaystyle[Q^{x},Q^{y}]=iQ^{z}\to{\rm symmetric}$ (9)
and thus the commutator always vanishes for a Lorentz-symmetric vacuum if
$Q^{z}$ is Lorentz-scalar and a conserved quantity. Note that only a symmetric
generator ( namely $Q^{z}$ ) appears in the right hand side in this expansion,
even though the breaking scheme does not give a coset ( quotient ) and of
course not a symmetric space: This breaking scheme is special from several
points. Moreover, if the following relations of VEVs holds,
$\displaystyle\langle[Q^{x},Q^{y}]\rangle=i\langle Q^{z}\rangle\neq 0,$ (10)
$\displaystyle\langle[Q^{x},Q^{z}]\rangle=-i\langle Q^{y}\rangle=0,$ (11)
$\displaystyle\langle[Q^{y},Q^{z}]\rangle=i\langle Q^{x}\rangle=0,$ (12)
then, they are isomorphic with the three-dimensional Heisenberg algebra,
$\displaystyle[x,y]=z,\qquad[x,z]=0,\qquad[y,z]=0.$ (13)
Thus, the insight of Nambu given in Ref. [62] which states that $Q^{x}$ and
$Q^{y}$ form a canonical conjugate pair in the case of ferromagnet $\langle
Q^{z}\rangle\neq 0$ is mathematically natural. Since a vacuum of the theory
must be chosen to evaluate VEVs of these brackets, the theory of $SU(2)$ is
expanded at the origin of the Lie group manifold by the NG bosonic
coordinates. Thus, the transformation from the Lie algebra to the Heisenberg
algebra is achieved at the origin of the Lie group and the corresponding
Heisenberg group. Physically, such a Heisenberg-algebra relation directly
concludes the Heisenberg uncertainty principle in the ”dynamical degrees of
freedom”, hence two NG-bosonic coordinates generated by the conserved charges
over a group manifold may acquire a quantum uncertainty. A comment on the
appearance of a symplectic vector space or group might be possible as some
literature already have done ( Refs. [81-84] ), though the notion and
structure of (quasi-)Heisenberg algebras/groups are better to emphasize the
quantum nature, since a Poisson bracket of classical mechanics also satisfies
a symplectic structure. In fact, a Heisenberg algebra is a central extension
of an algebra of symplectic vector space, and an automorphism of Heisenberg
algebra is given by a group of symplectic type. Such a symplectic vector space
of course defines a symplectic structure such like $\omega=\sum dp_{i}\wedge
dq_{i}$ up to an isomorphism, which is given by a so-called Lagrangian
subspace/submanifold. It should be emphasized that the transform from a Lie
algebra to a Heisenberg algebra under the prescription given above is not
achieved by some kind of perturbation or an analytic expansion ( such as a
deformation quantization of Poisson manifold [45] ) but by a functor, a
functorial manner provided by quantum field theory. It is a known fact that
both a three-dimensional Heisenberg group and $SU(2)$ can be embedded into
$SU(2,1)$ [46]. Thus, both the three-dimensional Heisenberg algebra and ${\rm
Lie}(SU(2))$ can be derived from the same Lie group, and it is interesting for
us to know how those ”submanifolds” are related with each other inside a
larger Lie group, and how an automorphism of ${\rm Lie}(SU(2))$ are related
with that of the Heisenberg algebra, vice versa. ( The groups of automorphisms
of $G$ and ${\rm Lie}(G)$ are isomorphic in general. Thus, the transform from
${\rm Lie}(SU(2))$ to the three-dimensional Heisenberg algebra may give a
global correspondence via their automorphism groups. ) Let us investigate this
problem by ourselves. ${\rm Lie}(SU(2,1))$ is an eight-dimensional algebra,
while ${\rm Lie}(SU(2))\simeq{\rm Lie}(SU(1,1))\simeq{\rm
Lie}(Sp(2))\simeq{\rm Lie}(SL(2,{\bf R}))\simeq{\rm Lie}(SL(2,{\bf C}))$ and
the three-dimensional Heisenberg algebra define three-dimensional linear
spaces. Thus, the three-dimensional spaces of ${\rm Lie}(SU(2))$ and the
Heisenberg algebra are embedded in the eight-dimensional space of ${\rm
Lie}(SU(2,1))$, and the anomalous NG theorem gives a mapping ( possibly a
bijection ) between two spaces. $SU(2)$ defines a sphere $S^{2}$, namely a
curve or a compact Riemann surface, and thus the corresponding three-
dimensional Heisenberg group should also define a curve or a Riemann surface.
Hence, the correspondence ( functor ) between ${\rm Lie}(SU(2))$ and the
three-dimensional Heisenberg algebra cause a correspondence between two curves
or Riemann surfaces with some globalization of those Lie algebras, especially
via exponential mappings. The NG bosons of the breaking scheme $SU(2)\to U(1)$
define a subset of $S^{2}$ ( a set of circles ). This implies that the NG
bosons of this breaking scheme gives a subspace of a Riemann surface, and thus
the three-dimensional Heisenberg algebra also gives a subspace of a Riemann
surface as the Heisenberg group manifold.
Let us examine the case of $SU(3)$ ( its Lie algebra is isomorphic with
Lie$SL(3,{\bf C})$ ). The definition of the Gell-Mann matrix representation of
${\rm Lie}(SU(3))$ is
$\displaystyle[Q^{A},Q^{B}]=if^{ABC}Q^{C},\quad Q^{C}\in{\rm
Lie}(SU(3)),\quad(A,B,C=1,2,\cdots,8)$ $\displaystyle f^{123}=1,\quad
f^{147}=f^{165}=f^{246}=f^{257}=f^{345}=f^{376}=\frac{1}{2},\quad
f^{458}=f^{678}=\frac{\sqrt{3}}{2}.$ (14)
For example, in the case of $\langle Q^{3}\rangle\neq 0$, $\langle
Q^{8}\rangle\neq 0$ with all of other generators have vanishing VEVs, the set
of following VEVs gives a ”quasi” Heisenberg algebra:
$\displaystyle\langle[Q^{1},Q^{2}]\rangle=i\langle
Q^{3}\rangle,\quad\langle[Q^{1},Q^{3}]\rangle=\langle[Q^{2},Q^{3}]\rangle=0,$
$\displaystyle\langle[Q^{4},Q^{5}]\rangle=\frac{i}{2}\langle
Q^{3}\rangle+\frac{i\sqrt{3}}{2}\langle Q^{8}\rangle,$
$\displaystyle\langle[Q^{4},Q^{3}]\rangle=\langle[Q^{5},Q^{3}]\rangle=\langle[Q^{4},Q^{8}]\rangle=\langle[Q^{5},Q^{8}]\rangle=0,$
$\displaystyle\langle[Q^{6},Q^{7}]\rangle=-\frac{i}{2}\langle
Q^{3}\rangle+\frac{i\sqrt{3}}{2}\langle Q^{8}\rangle,$
$\displaystyle\langle[Q^{6},Q^{3}]\rangle=\langle[Q^{7},Q^{3}]\rangle=\langle[Q^{6},Q^{8}]\rangle=\langle[Q^{7},Q^{8}]\rangle=0.$
(15)
The VEVs of all other brackets vanish and ”commute.” Strictly speaking, the
set of VEVs in this case does not give a Heisenberg algebra in the sense of
its definition given below, and we need to remove $Q^{3}$ or $Q^{8}$ from the
algebra to set $\langle Q^{3}\rangle=0$ or by $\langle Q^{8}\rangle=0$. While
we observe that a pairwise decoupling takes place. All of
$(Q^{1},Q^{2},Q^{4},Q^{5},Q^{6},Q^{7})$ are broken at the case $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle\neq 0$, or at the case $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle=0$ ( those cases give the scheme
$SU(3)\to U(1)\otimes U(1)$ ), while $(Q^{4},Q^{5},Q^{6},Q^{7})$ are broken
and $(Q^{1},Q^{2},Q^{3},Q^{8})$ remain unbroken ( this case gives the breaking
scheme $SU(3)\to SU(2)\otimes U(1)$ ) in the case $\langle Q^{3}\rangle=0$ and
$\langle Q^{8}\rangle\neq 0$. However, if we change the representation of
Lie$SU(3)$ from the Gell-Mann-type to others such as canonical basis, then we
find all generators except the Cartan subalgebra will be broken when we give a
finite VEV for one of elements of Cartan subalgebra of Lie$SU(3)$. Since a
physical phenomenon which depends on our choice of representation of a Lie
algebra never takes place in the nature, we conclude that the choice $\langle
Q^{3}\rangle=0$ and $\langle Q^{8}\rangle\neq 0$ is a special case of the
Gell-Mann representation, never occurs in the nature. Moreover, when we seek a
Heisenberg algebra in a Goldstone-type bosonic model of $SU(3)$, we need to
choose the form of VEV to make these charges broken when $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle=0$ ( the case $SU(3)\to
SU(2)\otimes U(1)$ mentioned above ). In such a case, the bosonic field $\Phi$
belongs to ${\bf 3}$-representation, and $\Phi^{\dagger}Q^{3}\Phi\neq 0$ and
$\Phi^{\dagger}Q^{8}\Phi=0$ gives an additional condition to the three
components of $\Phi$. Namely, we have to perform a variation of a subspace of
complex three ( real six ) dimensional space: This is not natural. Thus, we
conclude that the diagonal breaking of $SU(3)$ always gives not a Heisenberg
but a quasi-Heisenberg algebra. This fact indicates that the Heisenberg
uncertainty relation might be modified in quantum mechanical description of
fluctuations ( namely, NG bosons ) of the NG sector of the diagonal breaking
of $SU(3)$, such that,
$\displaystyle\Delta\chi^{1}\Delta\chi^{2}\geq
C^{a},\quad\Delta\chi^{4}\Delta\chi^{5}\geq C^{b},\cdots.$ (16)
Hence we speculate the deviation from the Heisenberg algebra given by a set of
VEVs of the Cartan subalgebra measures which pair of NG modes is more
”classical” and which pair of NG modes has a quantum fluctuation stronger than
others. A deviation from the Heisenberg-type uncertainty relation might affect
on quantum fluctuation in quantum phase transition.
The definition of Heisenberg algebra is
$\displaystyle[p_{i},q_{j}]=\delta_{ij}z,\quad[p_{i},z]=[q_{j},z]=0,$ (17)
where $(p_{1},\cdots,p_{n},q_{1},\cdots,q_{n},z)$ gives the generator of the
algebra. Thus, the number of generators must be odd in the Heisenberg algebra.
Here, $z$ is a central element of the Heisenberg algebra. Hence the finite
VEVs of elements of Cartan subalgebra give the center of the ( quasi )
Heisenberg algebra we have obtained, namely a central extension. A quasi-
Heisenberg algebra is defined to be
$\displaystyle[p_{i},q_{j}]=\delta_{ij}\sum_{\alpha}z_{\alpha},\quad[p_{i},z_{\alpha}]=[q_{j},z_{\alpha}]=0.$
(18)
Moreover, the expansion
$\displaystyle\langle[X^{\beta},X^{\gamma}]\rangle$ $\displaystyle=$
$\displaystyle\sum c^{\alpha}\langle S^{\alpha}\rangle+\sum c^{\delta}\langle
X^{\delta}\rangle$ (19)
must be pairwise decoupled to obtain a Heisenberg algebra: This is in general
not the case. Hence, to obtain a Heisenberg algebra via VEVs of Lie brackets
of generators of a Lie algebra, a subset of generators of odd number must give
a subalgebra. In this sense, an algebra generically obtained from the VEVs of
conserved charges should be called as a deformed/quasi Heisenberg algebra. The
uncertainty relation in a quasi-Heisenberg algebra should be investigated in
detail, since it might give a violation or a modification of the ordinary
Heisenberg uncertainty relation in a spontaneous symmetry breaking system, and
it would be confirmed experimentally from physical behaviors of NG sectors in
a condensed matter or a nucleus, especially in their quantum critical
phenomena [10].
Let us examine how a quasi-Heisenberg algebra arises by using the general
theory of Cartan decomposition, Cartan matrix, and canonical basis in the case
of semisimple Lie algebra Lie$(G)$ [20,32,43]. In this case, via the root
system ( so-called Cartan-Weyl basis ),
$\displaystyle{\bf g}$ $\displaystyle=$ $\displaystyle{\bf
h}\oplus\bigoplus_{\lambda\in R}{\bf g}_{\lambda},$ (20) $\displaystyle{\bf
g}_{\lambda}$ $\displaystyle=$ $\displaystyle\bigl{\\{}a\in{\bf
g}:[h_{j},a]=\lambda(h_{j})a,\quad\forall h_{j}\in{\bf h}\bigr{\\}},$ (21)
( $\lambda(h_{j})$: Cartan matrix, $R$: roots ), the Lie algebra is
generically defined by
$\displaystyle{\bf g}={\bf h}\oplus{\bf e}\oplus{\bf f},\quad
h_{i},h_{j}\in{\bf h},\quad e_{i},e_{j}\in{\bf e},\quad f_{j}\in{\bf f},$ (22)
$\displaystyle[h_{i},h_{j}]=0,$ (23)
$\displaystyle[e_{i},f_{j}]=\delta_{ij}h_{i},$ (24)
$\displaystyle[h_{i},e_{j}]=a_{ij}e_{j},$ (25)
$\displaystyle[h_{i},f_{j}]=-a_{ij}f_{j}.$ (26)
Here, $a_{ij}$ denote the Cartan matrix, and ${\bf h}$ is the Cartan
subalgebra. Thus, if $\langle e_{i}\rangle=\langle f_{j}\rangle=0$ ( $\forall
i,j$ ) while some of the bases of Cartan subalgebra take finite VEVs, $\langle
h_{j}\rangle\neq 0$, then a pairwise decoupling takes place and a quasi-
Heisenberg algebra is embedded in the total algebra. A Heisenberg pair is
given by the algebra basis of a positive and a negative roots. Thus, we obtain
the following theorem:
Theorem: Let $G$ be a semisimple Lie group and let Lie$(G)$ be its semisimple
Lie algebra. Let us assume the case where a theory only has VEVs of generators
toward the directions of Cartan subalgebra. Then the Cartan subgroup remains
unbroken, and the generators of Lie algebra will be pairwisely decoupled by
taking their VEVs, they form a quasi-Heisenberg algebra. This type of
decoupling never takes place in a Lorentz-invariant system due to the
vanishing condition $\langle Q\rangle=0$ of a conserved charge.
In such a situation of this theorem, a Lagrangian of NG bosons may be pairwise
decomposed inside the linear space of NG bosons at least in the quadratic part
of the Lagrangian of NG boson fields ( we discuss how and when such a
decomposition takes place in the NG sector of a theory in sec. 5 ). In the
case of breaking scheme $G\to$ Cartan subgroup, where all group elements
except the Cartan subgroup are broken, and all generators of Cartan subalgebra
take non-vanishing VEVs, then one can count the number of pairs ( namely, the
number of mode-mode couplings of NG bosons ) which give a quasi-Heisenberg
algebra:
$\displaystyle n_{pair}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm rank}{\rm
Lie}(G)\Bigr{]}.$ (27)
Note that the rank of Lie$(G)$ coincides with the dimension of Cartan
subalgebra. If $G\to H$ gives a symmetric space $G/H$, then
$\displaystyle{\rm rank}{\rm Lie}(G)$ $\displaystyle=$ $\displaystyle{\rm
dim}{\rm Lie}(H)={\rm dim}[X^{\alpha},X^{\beta}]$ (28)
holds. Thus,
$\displaystyle n^{symmetric-space}_{pair}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm dim}{\rm
Lie}(H)\Bigr{]}$ (29) $\displaystyle=$ $\displaystyle\frac{1}{2}\Bigl{[}{\rm
dim}{\rm Lie}(G)-{\rm dim}[X^{\alpha},X^{\beta}]\Bigr{]}.$
Moreover, if the Lagrangian of any pair of NG bosons is given in the form as
only one mode of a pair is massive by the mixing of modes inside the pair (
this situation will be given by an NG boson Lagrangian in sec. 5 ), then the
number of massive NG bosons equals the number of pairs, and
$\displaystyle n_{BS}$ $\displaystyle=$ $\displaystyle
n_{NG}+\frac{n_{pair}}{2}=n_{NG}+\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm
rank}{\rm Lie}(G)\Bigr{]}.$ (30)
Since $[X^{\alpha},X^{\beta}]=\sum c^{\gamma}S^{\gamma}+\sum
c^{\delta}X^{\delta}$,
$\displaystyle{\rm dim}\langle[X^{\alpha},X^{\beta}]\rangle={\rm
dim}\Bigl{(}\sum c^{\gamma}\langle S^{\gamma}\rangle\Bigr{)}={\rm rank}{\rm
Lie}(G)$ (31)
holds in the diagonal breaking case. Hence we get the following equation for a
diagonal breaking:
$\displaystyle n_{BS}$ $\displaystyle=$ $\displaystyle
n_{NG}+\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm
dim}\langle[X^{\alpha},X^{\beta}]\rangle\Bigr{]}.$ (32)
Here, ${\rm dim}{\rm Lie}(G)-{\rm dim}\langle[X^{\alpha},X^{\beta}]\rangle$
gives the number of Heisenberg pairs in the diagonal breaking scheme. Since
$n_{BS}=$dimLie$(G)-$dimLie$(H)$, we get
$\displaystyle n_{NG}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Bigl{[}{\rm
dim}{\rm Lie}(G)-{\rm rank}{\rm Lie}(G)\Bigr{]}.$ (33)
More general case is examined by utilizing (20)-(26). When $\langle
h\rangle\neq 0$, $\langle e\rangle=\langle f\rangle=0$ ( a diagonal breaking
), then $e$ and $f$ are broken since $[\Phi,e]\neq 0$, $[\Phi,f]\neq 0$,
$\Phi\in h$. If $\langle h\rangle\neq 0$, $\langle e\rangle\neq 0$, $\langle
f\rangle=0$, then $h$, $e$, $f$ are broken since $[\Phi,h]\neq 0$,
$[\Phi,e]\neq 0$, $[\Phi,f]\neq 0$, $\Phi\in h\oplus e$. In the latter case,
the algebra of VEVs is not a (quasi) Heisenberg-type. An investigation on the
case of all generators are broken, $G\to$ nothing, becomes complicated to give
a general theory.
Since a Heisenberg group and its Lie algebra are realized on a symplectic
vector space, one can introduce a Darboux basis of a symplectic vector space,
corresponds to the canonical coordinates, to express the Heisenberg algebra.
Then the Heisenberg algebra acquires a geometric implication. Moreover, one
can introduce an operator algebra analysis of the anomalous NG theorem via the
Stone-von Neumann theorem [75]. Therefore, a unification of algebra, analysis,
and geometry takes place in our anomalous NG theorem. The Heisenberg algebra
$(p,q,z)$ obtained from our $SU(2)$ model is a special example of,
$\displaystyle X\stackrel{{\scriptstyle
p}}{{\rightarrow}}Y\stackrel{{\scriptstyle q}}{{\rightarrow}}Z,\quad
X\stackrel{{\scriptstyle q}}{{\rightarrow}}Y^{\prime}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Z^{\prime},\quad Z\neq Z^{\prime}.$ (34)
Here, $X,Y,Z,Y^{\prime},Z^{\prime}$ implies some mathematical sets, and we
regard the canonical pair $(p,q)$ is given by certain types of morphisms. The
center $z$ given by the VEV $\langle Q^{z}\rangle$ in the $SU(2)\to U(1)$ case
measures how $Z$ and $Z^{\prime}$ are different. This kind of noncommutativity
appears in algebras of monodromy, holonomy, etc. This is the geometric nature
of the Heisenberg algebra as the essence of quantum mechanics. From the aspect
of Heisenberg groups, our Heisenberg algebra coming from the anomalous NG
theorem of an $SU(2)$ model can be related with the three-dimensional compact
Iwasawa manifold obtained from $\Gamma\backslash G/H$ of the three-dimensional
Heisenberg group $G$, where
$\displaystyle G$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&a&b\\\ 0&1&c\\\
0&0&1\end{array}\right),\quad a,b,c\in{\bf R},$ (38) $\displaystyle H$
$\displaystyle=$ $\displaystyle\\{e\\},$ (39) $\displaystyle\Gamma$
$\displaystyle=$ $\displaystyle G\cap GL(3,{\bf Z}).$ (40)
It is a known fact that a complex structure is found in a three-dimensional
Iwasawa manifold [40]. From the Kodaira-Spencer theory of deformation of
complex structure of a complex manifold [44], we know the obstruction of
deformation of a complex structure is determined by the cohomology group of
the manifold, namely it is one of global aspects/structures of the NG manifold
of our anomalous NG theorem.
An interesting fact is that a three-dimensional Heisenberg algebra $[p,q]=z$,
$[p,z]=[q,z]=0$ is constructed by the differential operators such as
$p=\partial_{1}-\frac{x^{2}}{2}\partial_{3}$,
$q=\partial_{2}+\frac{x^{1}}{2}\partial_{3}$, $z=\partial_{3}$: In that case,
$(p,q,z)$ forms an orthogonal frame of an appropriate manifold. It is
interesting for us to compare this fact with the representation of
differential operator expression of the $sl_{2}$-triple. It is a well-known
fact that a Heisenberg algebra can be expressed by a Weyl algebra,
$\displaystyle[x_{i},\partial_{j}]=-\delta_{ij},\quad[x_{i},x_{j}]=[\partial_{i},\partial_{j}]=0,$
(41)
and the Weyl-algebra expression of our quasi-Heisenberg algebra gives us a
further implication of mathematical and geometric nature of our Lorentz-
violating NG boson Lagrangian ( see sec. 5 ). A Weyl algebra is a simple
Nötherian integral domain, and it has a global dimension $n$. It should be
mentioned that any term higher than the second-order of the expansion of an
adjoint action expressed by an exponential mapping, a similarity
transformation $g^{-1}Qg$ ( $Q$; a conserved charge ), or in a Killing form (
kinetic term ) of the Lagrangian or the effective potential, are fall into the
fundamental relation of quasi-Heisenberg algebra (18) after taking their VEVs,
and thus those expansions are effectively ”terminated” at the quasi-Heisenberg
algebra ( such a truncation can take place, of course, in a quantum theory ),
and thus, only the subset of (quasi-)Weyl algebra of universal enveloping
algebra appears in a theory: This case is coming from the fact that our
theoretical framework is suitable in the vicinity of the ground state of the
system determined by a choice of the form of VEVs and we consider a Heisenberg
algebra, not a Heisenberg group. The Weyl algebra itself is isomorphic with
the Moyal algebra of deformation quantization, thus the global character of
our theory of anomalous NG theorem will acquire a connection with the Moyal-
Weyl deformation quantization. Moreover, a Weyl algebra defines several
differential operators which directly connects with theory of $D$-modules [4].
A Weyl algebra of an infinite order, possibly isomorphic with a deformation
quantization, will be entered into our anomalous NG theorem when we consider
the corresponding Heisenberg group: Naively, the Weyl algebra is terminated at
the order of the elementary relations ( Lie brackets, (38) ) of the
corresponding quasi-Heisenberg algebra, as we have stated, while a linear
transformation of a basis of representation space of the quasi-Heisenberg
algebra caused by an operation of Heisenberg group ( an adjoint action to the
Heisenberg algebra ) gives the Weyl algebra which can acquire its higher-order
products and derivatives of algebras expanded by the set
$(x_{j},\partial_{j})$. To make this matter consistent, we need to obtain the
notion of quasi-Heisenberg group. It should be investigated that a Moyal-Weyl
type deformation quantization for the quasi-Heisenberg/quasi-Weyl algebra
starting from a Poisson manifold defined by a Poisson structure of broken
generators, which might give some results confirmed by experiments. From our
observation of $SU(3)$, we can propose the following modified Moyal-Weyl
product:
$\displaystyle f*g$ $\displaystyle=$ $\displaystyle
f\exp\Bigg{(}(\sum_{j}\langle
h_{j}\rangle)\sum_{A,B}\frac{\overleftarrow{\partial}}{\partial\chi^{A}}\frac{\overrightarrow{\partial}}{\partial\chi^{B}}-\frac{\overleftarrow{\partial}}{\partial\chi^{B}}\frac{\overrightarrow{\partial}}{\partial\chi^{A}}\Bigg{)}g$
(42) $\displaystyle=$ $\displaystyle fg+\sum_{j}\langle
h_{j}\rangle\\{f,g\\}_{PB}+\cdots,$ $\displaystyle\\{f,g\\}_{PB}$
$\displaystyle=$ $\displaystyle\frac{\partial
f}{\partial\chi^{A}}\frac{\partial g}{\partial\chi^{B}}-\frac{\partial
f}{\partial\chi^{B}}\frac{\partial g}{\partial\chi^{A}}$ (43)
This modified Moyal-Weyl product provides an example of a multi-
Planck-”constant” model which has been proposed by the author in Ref. [70].
The set of VEVs of the Cartan subalgebra generators gives several Planck
”constants” which depend on a position of the Lie group manifold or a local
geometry, mainly curvatures of $V_{eff}$. ( An example of deformation
quantization of Heisenberg manifolds, see [73]. ) It should be mentioned that
our approach to the anomalous NG theorem heavily depends on Lie algebras of
conserved charges and they are the special cases of current algebras, for
example, $[j^{A}(x_{0},{\bf x}),j^{B}(x_{0},{\bf
y})]=if^{ABC}\delta^{(3)}({\bf x}-{\bf y})j^{C}(x)$. Hence, our result may be
reformulated by those current algebras and their globalizations ( it may be
called as ”current groups”, and they might introduce new mathematics in our NG
theorem, especially from the context of integral geometry and operator
algebras ). In Ref [46], a quasiconformal mapping of a Heisenberg group is
studied. While, a quasiconformal mapping is obtained in a Moyal-Weyl
deformation quatization [67]. It is interesting for us to unify these
approaches.
In more generic case, a Lie algebra is defined as a direct sum of finite
number of Lie subalgebras:
$\displaystyle{\bf g}$ $\displaystyle=$ $\displaystyle{\bf g}_{1}+\cdots+{\bf
g}_{l},$ (44)
and each of them has a root space decomposition. Thus the theorem given above
can be stated differently:
Theorem: Any diagonal breaking which remains the Cartan subalgebra unbroken
converts the Lie algebra into a direct sum of a finite number of quasi-
Heisenberg algebras and Abelian algebras, via taking the VEVs of algebra
generators. Turn to the group theory, such a breaking scheme gives a direct
product of quasi-Heisenberg groups and Abelian Lie groups via the anomalous NG
theorem in quantum field theory. The quasi-Heisenberg group and Abelianized
group act on the effective action/potential of the theory and its low energy
effective theory.
Since this paper considers internal symmetries mainly for our anomalous NG
theorem, we choose a semisimple Lie group such as $SU(n)$, $SO(n)$ and $Sp(n)$
as the main subject, though we can examine more generic cases of $SL(n,F)$ and
$GL(n,F)$ ( $F$: a number field of characteristic zero ). Note that any Lie
group has an associated Lie algebra. Moreover, especially ${\rm Lie}(SL(n,F))$
and ${\rm Lie}(GL(n,F))$ are finite dimensional, we have a functor which gives
corresponding simply-connected Lie groups. Hence our analysis presented here
is valid to those Lie algebras. The Ado-Iwasawa theorem states that a finite-
dimensional Lie algebra defined over a field $F$ has a faithful finite-
dimensional representation. A general consideration of some cases of ${\rm
Lie}(SL(n,F))$ and ${\rm Lie}(GL(n,F))$ contains theories of $SU(N)$, $SO(N)$,
…, as their special examples. Especially, $GL(2,{\bf R})$ is disconnected into
two parts according to the signature $\pm 1$ of determinant, thus a problem of
covering on the space of NG bosons, a problem of the global nature of the NG
theorem similar to the case of gauge orbits, gives an interesting subject
which has a strong connection with number theory. Though, for examining the
mechanism to generate massive NG bosons, group elements in the vicinity of
identity are important since we examine small fluctuations of bosonic fields
from a stationary point. It is a known fact that if a Lie group is simply
connected, its global structure is determined by the corresponding Lie
algebra. The global structure of the effective action/potential of a theory
will be known by both its group theoretical nature and quantum field theory.
In a global aspect, $Spin(N)$ is the double-covering group of $SO(N)$, and
thus, if $SO(N)$ gauge theory has a unique vacuum in its $SO(N)$ fundamental
domain, the corresponding $Spin(N)$-gauge theory must have two exactly
degenerate vacua, though they cannot be distinguished by the Lie algebra (
namely, a locally defined quantity ) in general.
Our anomalous NG theorem has not only a locally characteristic aspect ( quasi-
Heisenberg algebra, massive NG bosons, Weyl algebra ) but also some
interesting global nature, which reflect some number theoretical aspects such
as a fundamental group of covering group or a Galois group. Such a global
aspect of our anomalous NG theorem directly reflects to symmetry between
stationary points and geometry of stationary points, given by the effective
potential $V_{eff}$. For our understanding of the global nature of an NG
manifold/variety ( probably some class of quotients of breaking schema, if it
has a coset topology, may have fixed points and singularities ) of the
anomalous NG theorem, we need further investigation. Some exotic mathematical
nature of NG manifolds provides us a new subject of study on submanifolds in (
differential ) geometry. The global character of an NG manifold, namely its
compactness or the fundamental group, is understood by neglecting the local
details of effective potential, and then convert our problem to a problem of
Lie groups and homogeneous spaces. Sometimes $G\to H$ gives a
Riemannian/Hermitian symmetric space, while it might be possible to generate a
pseudo-Riemannian space as a result of symmetry breaking ( this is not
familiar from a context of physics ). Topological/global nature of pseudo-
Riemannian spaces is still not yet understood enough in modern mathematics
[43]. Since there is a common understanding that a fundamental group is a
Galois group [72], our problem continues to the region of number theory. From
this aspect of the global character of geometry of symmetry breaking,
especially the Clifford-Klein form $\Gamma\backslash G/H$ is important since
it has some nice properties, and the case where $\Gamma$ is proper
discontinuous and free is interesting for us. If a breaking scheme $G\to H$
gives a Hermitian symmetric space $G/H$, then it is a known fact that the
$G/H$ has a uniform lattice $\Gamma$ ( i.e., $\Gamma\backslash G/H$ is compact
).
A more complicated situation will arise when we consider a breaking scheme
under some generators of a Lie algebra are already broken by an explicit
symmetry breaking parameter, namely so-called ”explicit+spontaneous” symmetry
breakings [70]. ( We give an example ( the kaon condensation model ) of it (
anomalous+explicit+spontaneous symmetry breaking ) later in this paper. ) In
such situations, for example, a breaking scheme such as
$\displaystyle SU(N)\to({\bf Z}/N{\bf Z})^{\times}\simeq Gal({\bf
Q}(\zeta_{N})/{\bf Q})$ (45)
can take place. Here $\zeta_{N}$ is the $N$-th root of unity, and $({\bf
Z}/N{\bf Z})^{\times}$ gives the center of $SU(N)$. ( Note that $\det[{\rm
diag}(\underbrace{\zeta_{N},\cdots,\zeta_{N}}_{N})]=1$. ) Namely, it gives the
following central extension:
$\displaystyle 1\to({\bf Z}/N{\bf Z})^{\times}\to SU(N)\to PSU(N)\to 1.$ (46)
( $PSU(N)$; projective special unitary group. ) $Gal({\bf Q}(\zeta_{N})/{\bf
Q})$ is the Galois group of the cyclotomic extension. This exact sequence is
useful to consider a breaking scheme which gives a Grassmannian
$SU(N)/SU(N-M)SU(M)$ and then successively ${\bf Z}/(N-M){\bf Z}\times{\bf
Z}/N{\bf Z}$. This breaking scheme may have a quite interesting mathematical
implication in a quantum group by utilizing the quasi-Heisenberg and quasi-
Weyl algebra representations: The discrete Heisenberg group ( all of the
matrix elements of representation of a discrete Heisenberg group are integers
) is given by the algebraic relations of generators such that $xy=zyx$,
$[x,z]=[y,z]=0$. The author speculate this is the first time to find a quantum
algebra in the NG theorem. Needless to say, the relation $xy=zyx$ is
consistent with the canonical Heisenberg algebra $[x,y]=1$ by setting
$xy=z/(z-1)$ and $yx=1/(z-1)$ ( they recover the canonical commutation
relation ). Moreover, Bost and Connes construct a theory of dynamical system
which gives a Galois group $Gal({\bf Q}(\zeta_{N})/{\bf Q})$ associated with a
spontaneous symmetry breaking, and the partition function below the critical
temperature is the Riemann zeta function [7]. It is emphasized that their work
has a strong connection with the Riemann hypothesis. In fact, our theory of NG
theorem contains some parts of algebraic aspects of their work naturally. For
example, a global nature of our anomalous NG theorem can give $Gal({\bf
Q}(\zeta_{N})/{\bf Q})$. Hence our theory of NG theorem might provide an
approach toward the solution of the Riemann hypothesis: This point will be
discussed later in this paper. We need a systematic investigation on the
relation between several symmetry-breaking schema and Galois representations,
with a perspective of (non)commutative class field theory, i.e., the Langlands
conjecture [22,23,24,29,58,80]. From similar perspective, the following short
exact sequences are also interesting:
$\displaystyle 1\to{\bf Z}/2{\bf Z}\to{\rm Spin}(N)\to SO(N)\to 1,$ (47)
$\displaystyle 1\to{\bf Z}/2{\bf Z}\to{\rm Spin}^{\bf C}(N)\to SO(N)\otimes
U(1)\to 1.$ (48)
In fact, ${\rm Spin}(N)$ is a double covering group of $SO(N)$, and ${\rm
Spin}^{\bf C}(N)$ is its complexification. Also, ${\rm Spin}(N,{\bf R})$ is
the group in the theory of Clifford algebra. ${\bf Z}/2{\bf Z}$ is a Galois
group. The upper exact sequence is frequently used for an explanation of a
Stiefel-Whitney class which judges whether a manifold is orientable. A central
extension of Lie group by a discrete group corresponds to the covering space,
directly related with the fundamental group. Moreover, a central extension of
Lie group induces a central extension of Lie algebra ( but its inverse is not
true in general ). The Lie’s third theorem states that a simply connected Lie
group exists for a given finite dimensional Lie algebra. We make a brief
comment on a central extension of Lie algebra [63]. For example,
$\displaystyle 0\to{\bf R}\to\widetilde{{\rm Lie}(G)}\to{\rm Lie}(G)\to 0.$
(49)
Here, $\widetilde{{\rm Lie}(G)}$ is the central extension of ${\rm Lie}(G)$ by
${\bf R}$. Some literature given as our references discuss possible roles of
central extensions to Lie brackets which might affect on the anomalous
behavior of NG theorem. It is a well-known fact that there is no nontrivial
central extension if ${\rm Lie}(G)$ is semisimple. A central extension may
have a role when we consider a symmetry of a Kac-Moody group or a Heisenberg
group. The following isomorphism is useful for us:
$\pi_{1}(G/H)\simeq\pi_{1}(G)$ where $G$ is a connected Lie group, and $H$ is
a simply connected closed subgroup of $G$. Thus, the nature of covering space
which is implied by a central extension of a Lie group conserves under a
breaking scheme $G\to H$, and it is enough for us to consider a
fundamental/Galois group of covering space of $G$. Those covering groups and
Galois groups describe symmetries of stationary points of NG sectors. For
example, the set of stationary points inside the fundamental domain of $G/H$
acquires the symmetry of $\pi_{1}(G/H)$.
From our examination, there are functors of algebra cohomologies associated
with a breaking scheme of anomalous NG theorem, such that,
$\displaystyle{\rm Lie\,algebra\,cohomology}\to{\rm
Heisenberg\,algebra\,cohomology}\to{\rm Galois\,cohomology}.$ (50)
This is a remarkable fact since, for example, a Lie algebra cohomology
describes the topological nature of underlying Lie group. If a symmetry of a
set of stationary points in an NG sector is a Galois type, the set gives a
Galois representation controlled by a Galois cohomolgy. These cohomology,
especially a Galois cohomology may have an overlap with an étale cohomology
since a Galois cohomology is a special case of étale cohomology which implies
an underlying algebraic variety. This fact may help us to understand the
underlying mechanism of the relations of those cohomologies and algebras. The
relationship between a Heisenberg algebra and a Galois group is a
characteristic aspect of our anomalous NG theorem, while other relations may
be contained also in the normal NG theorem. A Galois group appears in various
geometric examples but of particular interest here is geometric expressions of
class field theory, several Galois representations, and étale fundamental
groups by our anomalous NG theorem. It may be noteworthy to mention that the
Abelianized part of the total Lie algebra reflects the flatness of the
effective action/potential of the theory, while the quasi-Heisenberg relation
lifts partly the degeneracy of the vacua of the theory along with some NG-
bosonic coordinates. Since an apparent discrete symmetry between stationary
points takes place in a massive NG-bosonic coordinate/space, a Galois
representation will be found in the space of a quasi-Heisenberg relation.
Now, we list some breaking schema interesting for us from the context of this
paper:
$\displaystyle SU(4)\to SU(2)\otimes SU(2)\simeq SO(4)\to SU(2)_{diag}\to
U(1),$ (51) $\displaystyle SU(5)\to SU(3)\otimes SU(2)\otimes U(1)\to
U(1)\otimes U(1)\otimes U(1)\otimes U(1),$ (52) $\displaystyle SU(6)\to
SU(3)_{L}\otimes SU(3)_{R}\to SU(3)_{V}\to U(1)\otimes U(1),$ (53)
$\displaystyle SO(10)\to SU(4)\otimes SU(2)\otimes SU(2),$ (54) $\displaystyle
E_{6}\to SO(10)\otimes U(1)\to SU(5),$ (55) $\displaystyle E_{8}-{\rm
ferromagnet},\quad({\rm experimentally\,observed\,spin\,system}),$ (56)
$\displaystyle G_{2}\to SO(4),$ (57) $\displaystyle SU(N)\to
SU(N-M),\quad({\rm Stiefel\,manifold}),$ (58) $\displaystyle SO(N)\to
SO(N-M),\quad({\rm Stiefel\,manifold}),$ (59) $\displaystyle SU(N)\to
SU(N-M)\otimes SU(M),\quad({\rm symmetric\,space,\,Grassmann}),$ (60)
$\displaystyle SO(N)\to SO(N-M)\otimes SO(M),\quad({\rm
symmetric\,space,\,Grassmann}),$ (61) $\displaystyle Spin(6)=SU(4)\to
something,$ (62) $\displaystyle Spin(4,2)=SU(2,2)\to something.$ (63)
In those examples, if a symmetry breaking takes place under a breaking scheme
in which some elements of the Cartan subgroup of the total group remains
unbroken, then it is trivial that a ( quasi ) Heisenberg algebra arises. For
example, the breaking scheme $SU(N)\to SU(N-M)\otimes SU(M)$ will take place
by an order parameter of the form
diag$(\underbrace{a,\cdots,a}_{N-M},\underbrace{b,\cdots,b}_{M})$ which should
be proportional to a linear combination of VEVs of the Cartan subalgebra of
$SU(N)$. A large part of breaking schema listed above are fall into this class
of symmetry breakings. The breaking scheme $SU(N)_{L}\otimes SU(N)_{R}\to
SU(N)_{V}$ is famous in a chiral symmetry breaking of left-right symmetric
$N$-flavor model. In this case, the Lie algebra one considers is
$\displaystyle[\theta^{a}T^{a}\otimes
1+\varphi^{b}T^{b}\otimes\sigma^{3},\Phi],$
$\displaystyle\Phi\propto\sigma^{1},\quad\theta^{a}T^{a}\otimes 1\in{\rm
Lie}(SU(N)_{V}),$
$\displaystyle(\theta^{a}+\varphi^{a})T^{a}\otimes\frac{1+\sigma^{3}}{2}\in{\rm
Lie}(SU(N)_{L}),$
$\displaystyle(\theta^{a}-\varphi^{a})T^{a}\otimes\frac{1-\sigma^{3}}{2}\in{\rm
Lie}(SU(N)_{R}).$ (64)
Here, $\theta^{a}T^{a}\otimes 1$ remains symmetric while
$\varphi^{b}T^{b}\otimes\sigma^{3}$ is broken. Hence, the VEV takes its value
toward $\sigma^{1}$ direction ( as you know, $\gamma^{0}$ is frequently used,
while $\sigma^{3}\to\gamma_{5}$ ), and this case does not belong to the class
of diagonal breaking we have studied in this paper.
$E_{8}$ might have several exotic breaking schema due to its Dynkin diagram
and Cartan martrix while our observation of anomalous NG theorem in a generic
case should valid to it. An example of spin system of $E_{8}$ symmetry has
been observed experimentally quite recently [6,9,88]. Zamolodchikov seems to
use his theory ( affine Toda field theory of $E_{8}$ ) to give a mass spectrum
of mesons, two quark ( kink ) bound states and thus his theory is constructed
in a (1+1)-dimensional model, though some part of the mechanism of generating
an $E_{8}$ spectrum is independent from the dimensionality of a system,
determined by the Lie algebra Lie$(E_{8})$. Hence an $E_{8}$ spin system has
an importance from its own right, beyond its dimensionality. ( A breaking
scheme of $E_{8}$ is also interesting for us from the context of the Kazhdan-
Lusztig-Vogan polynomials for $E_{8}$ [38,39,51]. )
### 2.1 Riemannian and Hermitian Symmetric Spaces
If a breaking scheme $G\to H$ gives a symmetric space [32], several geometric
properties will be introduces to our NG theorem more concretely. Especially, a
local nature ( Lie algebra ) and a global structure ( Lie group ) is bridged
more clearly. First, we summarize the basic well-known fact of a symmetric
space. Let $M$ be a symmetric space. A Lie group $G$ acts transitively on $M$.
In addition, an involution $s$ is defined for any local point of $M$, as an
automorphism of $M$, and $s$ acts on any group element of $G$ as an adjoint
$sgs^{-1}$. This involution is an automorphism of $G$ itself, and of course it
acts on ${\rm Lie}(G)$ as an automorphism. Then ${\bf g}={\rm Lie}(G)$ is
decomposed into ${\bf h}+{\bf m}$ by their eigenvalues of operations of $s$ (
${\bf h}\to+1$, the Cartan subalgebra, and ${\bf m}\to-1$ ). Hence, by the
number of odd elements ${\bf m}$, the relations $[{\bf h},{\bf h}]\subset{\bf
h}$, $[{\bf h},{\bf m}]\subset{\bf m}$, $[{\bf m},{\bf m}]\subset{\bf h}$ are
immediately obtained ( this is a kind of grading of the algebra by the set of
odd elements ${\bf m}$ ). ${\bf m}$, an orthogonal complement space of ${\bf
h}$, is isomorphic with $TM$, and a curve $t\to e^{it{\bf m}}\cdot o$ ( $o$: a
point of $M$, $t\in{\bf R}$ ) is geodesic. As stated above, the tangent bundle
of a Riemannian manifold has $O(n)$ as the structure group. If a spontaneous
symmetry breaking $G\to K$ gives a symmetric space, $M=G/K=e^{i{\bf m}}$, and
the Cartan subalgebra remains unbroken, then the NG manifold is expanded only
by the basis of broken generators ${\bf m}$ which is isomorphic with
$TM=T(G/K)$ which may have the structure group $O(n)$, and the NG bosons
$\\{\chi^{A}\\}$ as the local coordinate system expressed by $\Phi\to
e^{i\chi^{A}m^{A}}\Phi$ are all geodesic, whether the normal or anomalous
cases of NG theorem. The VEVs $\langle{\bf h}\rangle\neq 0$ are always normal
( vertical ) with the tangent space given by the NG boson space ${\bf m}$. A
quasi-Heisenberg algebra is globally defined, inside the linear space $TM$.
Since an NG boson gives a geodesic over the manifold $G$, a Jacobi field is
associated along with the geodesic curve. The curvature tensor is given by
$R(X,Y)Z=[[X,Y],Z]\subset{\bf m}$ ( here, $X,Y,Z\in{\bf m}$ and we have used
the physics convention of $g=e^{i{\bf m}}$, where $i=\sqrt{-1}$ ) at the
origin. If an order parameter $\Phi$ belongs to ${\bf h}$, then the Riemann
curvature $R$ appears at the fourth order displacement of a Lagrangian or an
effective potential caused by broken generators. Here, a symmetric space is
defined by a Lie algebra, thus it does not depend on details of the manifold.
For example, $SU(N)\to SU(N-M)\otimes SU(M)$ is a symmetric space, thus broken
generators form the algebra $[X^{a},X^{b}]\subset S^{c}$, and if $\langle
S^{c}\rangle\neq 0$, a ( quasi ) Heisenberg algebra arises. In this case,
Lie$SU(N)$ is projected to the quasi-Heisenberg algebra globally via our
anomalous NG theorem. Especially interesting for us is the fact that a unitary
group $U(N)$ is a Hermitian manifold ( keeps a Hermitian structure of a
quadratic form ), while a Heisenberg group is possibly be described by a
Riemannian manifold. Hence, our anomalous NG theorem may bridge between a
complex ( Kähler ) structure and a quantized symplectic ( quasi-Heisenberg )
structure in a symmetric space.
### 2.2 A Heisenberg Group as the Symmetry of the Beginning
If a Heisenberg group ( sometimes used in a flavor dynamics, a flavor-symmetry
breaking ) is an internal symmetry from the beginning of a theory, and if one
considers its spontaneous symmetry breaking, a situation similar with the
cases of semisimple classical Lie groups takes place, since a set of VEVs of a
Heisenberg algebra can again give a Heisenberg algebra. For example,
$\Xi=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\quad{\rm or}\quad\left(\begin{array}[]{ccc}0&1&0\\\
0&0&0\\\ 0&0&0\end{array}\right)\quad{\rm
or}\quad\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 0&0&0\end{array}\right),$
(65)
with an action of a Heisenberg group $G$ ( see, (35) ) from the left side
gives $G\Xi=\Xi$, namely $\Xi$ is $G$-singlet, can be utilized to make an
invariant theory. This form of $\Xi$ may be attractive for an attempt to
generate a flavor degree of freedom: For example, the following
$\widetilde{\Xi}$ which breaks a Heisenberg-group symmetry can generate a
flavor hierarchy by an action of $G$:
$\widetilde{\Xi}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\epsilon&0\\\
0&0&\epsilon^{2}\end{array}\right),\quad|\epsilon|\ll 1.$ (66)
Therefore, one can consider a symmetry breaking which generates a flavor
hierarchy via our anomalous NG theorem:
SU(2) ( SU(N) ) $\to$ Heisenberg group $\to$ something.
An important issue is coming from the fact that the Killing form of a
nilpotent Lie algebra is identically zero, while the Killing form of a
solvable Lie group gives a different result from it. A quotient of Heisenberg
group gives a solvmanifold [3]. A fermion model of solvable Lie group symmetry
with a dynamical symmetry breaking which generates a sigma model might be
possible. In a bosonic field theory of a compact Lie group, the Killing form
is negative definite, while in the case of indefinite signature of Killing
form, a model Lagrangian should be regarded as an ”analytic continuation” from
a physical model.
### 2.3 Kac-Moody Algebras, Generalized Kac-Moody Algebras, and Affine Lie
Algebras
It should be noticed that a Lie group which derives a Nöther charge can be
treated ( at least, formally in a certain sense ) as an internal symmetry of a
model Lagrangian. We assume a Kac-Moody Lie algebra [25,37] has its
corresponding Lie group via an exponential mapping, surjectively, even though
this assumption is sometimes violated, and it is a non-trivial issue to define
a Haar measure for a Kac-Moody group. This issue affects to define a path-
integral measure over a Kac-Moody group. From our perspective of this paper,
the most interesting fact is that, after taking VEVs of brackets of a Kac-
Moody algebra ( now we have an infinite number of conserved charges ),
especially an affine Lie algebra, we obtain a direct sum of quasi-Heisenberg
algebras of polynomial growths ( would be called as an infinite-dimensional
quasi-Heisenberg algebra ) and Abelianized subalgebras. The method to
construct a Kac-Moody algebra ( a Cartan subalgebra, a root system, a Cartan
matrix, etc. ) is parallel with the general theory of finite-dimensional
simple Lie algebras [37]: Hence, our discussion given above can
straightforwardly be applied to several cases of Kac-Moody algebras. Our
theory can be regarded as a higher-dimensional version of a ( affine ) Toda
field theory, in which a scalar field of the theory takes its value on the
Cartan subalgebra of Kac-Moody algebra. After a diagonal breaking scheme takes
place, such a higher-dimensional version of affine Toda field theory-like
model acquires a quasi-Heisenberg algebra. A case beyond a diagonal breaking
causes more complicated result. An affine Lie algebra can be interpreted as a
special form of trivial fiber bundle, ${\bf g}\otimes{\bf C}[t,t^{-1}]$. Hence
a $G$-bundle and a Maurer-Cartan form of Cartan geometry can, at least
formally, be considered.
Another different perspective is coming from an ${\cal N}=2$ superconformal
algebra, especially the so-called coset construction. It is constructed by the
affine Kac-Moody algebra of $SU(2)$ at level $l$,
$\displaystyle[h_{m},h_{n}]=2ml\delta_{n+m,0},\quad[e_{m},f_{n}]=h_{m+n}+ml\delta_{m+n,0},$
$\displaystyle[h_{m},e_{n}]=2e_{m+n},\quad[H_{m},f_{n}]=-2f_{m+n},$ (67)
with an associated set of complex Grassmann variables. Thus, if a generic
Lagrangian of an NG sector ( for example, see (104) ) is pairwisely decomposed
via taking VEVs, as the direct sum of VEVs of algebra of Lie$(SL(2))$-triple
$(h,e,f)$, then the algebra inside the Lagrangian is a finite and special
version of the $SU(2)$ affine Lie algebra: The algebra arised in the generic
NG-boson Lagrangian can be embedded into the superconformal algebra. This type
of discussion is useful for us to consider several relations between our
generic NG-boson Lagrangian and other theoretical models. It is a known fact
that a class of infinite-dimensional simple linearly compact Lie superalgebras
contain the Standard Model gauge group $SU(3)\otimes SU(2)\otimes U(1)$ as the
algebra of level zero. Our argument presented here has some similarity with
such a situation. The important issue is to know how geometries of these Lie
groups are related with each other. It might be possible to obtain affine Lie
groups starting from Lie algebras of Riemann/Hermitian symmetric spaces.
### 2.4 Graded Lie Algebras and Lie Superalgebras
Lie superalgebras and Lie supergroups have quite interesting characters, and
they have importances in their own right [36,89], while they also acquire
attention from some particle phenomenological point of view. An extension of
our anomalous NG theorem to supersymmetric theory is an interesting subject
for us to complete our theorem. This will be done in another paper by the
author ( in preparation ), and here we will see some perspectives especially
from mathematics. For example, we can consider the following diagram:
Lie superalgebra $\to$ Heisenberg superalgebra ( bosonic/fermionic ) $\to$
Galois supergroup.
The notion of Galois supergroup is not strange, if we consider a non-trivial
central extension to give the supergroup. Via an effective potential and an
order parameter, notions of supermodules, superschemes would be introduced in
our theory of NG theorem. For our context of this paper, the following diagram
is considered:
Lie superalgebras $\to$ Lie supergroups $\to$ supergroup-schemes $\to$
superschemes $\to$ super-étale cohomology $\to$ super-Galois representations,
and,
super-sheaves and supermodules $\to$ superschemes $\to$ perverse super-sheaves
$\to$ super-intersection cohomology $\to$ stratified super-Morse theory,
and,
supergroups $\to$ Maurer-Cartan superforms $\to$ Cartan supergeometry
## 3 The Effective Potential Formalism
Let us examine the effective action $\Gamma_{eff}$ and effective potential
$V_{eff}$ of a general situation. ( See the book of Kugo [49]. ) Let $\Phi$ be
a matrix order parameter. $\Phi$ can be regarded as a left $G$-module [20,30]:
$g(\Phi_{1}+\Phi_{2})=g\Phi_{1}+g\Phi_{2}$, $g\in G$,
$\Phi_{1},\Phi_{2}\in\Phi$. $\Phi$ is assumed as $G$-equivariant. Both $G$ and
$\Phi$ are defined over the same field $F$, usually ${\bf C}$ or ${\bf R}$,
and thus $G$ and $\Phi$ acquire the same topology with $F$. Since $G$ acts on
$\Phi$ continuously, $\Phi$ is a topological $G$-module. A group (co)homology
$H_{n}(G,\Phi)$ and $H^{n}(G,\Phi)$ can be considered by modules generated by
$g\Phi_{1}-\Phi_{1}$ ( difference ), $\Phi_{1}\in\Phi$ and $g\in G$, under the
systematic manner. There are several nontrivial issues in our situation due to
the nature of quantum field theory ( $\Phi$ is a quantum field, not exactly
the same with ${\bf R}$, ${\bf C}$, or ${\bf Z}$ ), as a physical system. If
we regard the effective potential $V_{eff}$ as a scheme, or when $V_{eff}$
defines an algebraic variety, then the (co)homology groups $H_{n}(X,{\cal
O}_{X})$ and $H^{n}(X,{\cal O}_{X})$ ( ${\cal O}_{X}$; a sheaf ) can also be
considered [30,31]. ( Such a cohomology group is introduced anywhere we meet a
sheaf in our theory. The étale cohomology is a cohomology theory of sheaves in
the étale topology [53]. It may be possible to apply the method of étale
cohomology to study a topological nature of our NG theorem. ) If $V_{eff}$ is
included in a line bundle, the Borel-Weil theory can be applied [43].
Let us consider a linear displacement of a field $\Phi$ caused by a conserved
charge $Q^{A}$:
$\displaystyle\delta_{A}\Phi$ $\displaystyle=$
$\displaystyle[Q^{A},\Phi]=\theta^{A}T^{A}\Phi,\quad(A=1,\cdots,N),\quad
T^{A}\in{\rm Lie}(G).$ (68)
A typical example is the chiral $\gamma_{5}$ transformation:
$[Q^{5},\bar{\psi}i\gamma_{5}\psi]=-2\bar{\psi}\psi$, and
$\langle\bar{\psi}\psi\rangle\neq 0$ gives an order parameter with a fixed
phase of the chiral rotation. By takings its VEV, we get
$\displaystyle\langle 0|[Q^{A},\Phi]|0\rangle$ $\displaystyle=$
$\displaystyle\langle 0|\theta^{A}T^{A}\Phi|0\rangle.$ (69)
With taking into account $Q^{A}|0\rangle=0$ ( symmetric ) and
$Q^{A}|0\rangle\neq 0$ ( broken ), usually we conclude that $\langle
0|\theta^{A}T^{A}\Phi|0\rangle=0$ ( symmetric ) and $\langle
0|\theta^{A}T^{A}\Phi|0\rangle\neq 0$ ( broken ). However, in the case of
$SU(2)\to U(1)$ of a ferromagnet, $\langle 0|Q^{z}|0\rangle=\int d^{3}{\bf
x}\langle 0|j^{z}_{0}(x_{0},{\bf x})|0\rangle\neq 0$ may take place even
though $Q^{z}$ is unbroken. Next, we give a formal expansion of
$V_{eff}[\Phi]$ by the set of vectors of ${\rm Lie}(G)$ around the VEV:
$\displaystyle V_{eff}[\Phi]$ $\displaystyle=$ $\displaystyle
V_{eff}[v]+\Bigl{(}\frac{\partial
V_{eff}}{\partial\Phi}\Bigr{)}_{\Phi=v}(\theta^{A}T^{A}\Phi)+\frac{1}{2!}\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}(\theta^{A}T^{A}\Phi)^{2}+\cdots.$
(70)
Here $v$ implies a certain type of VEV of $\Phi$. The effective potential
$V_{eff}$ belongs to a germ of a sheaf of smooth function ${\cal O}_{D}$ (
$D$: a domain ). Thus, after taking a VEV of this expansion, the stationary
condition gives the following criterion ( Eqs. (16)-(18) of the paper of
Goldstone, Salam, and Weinberg [27] ):
$\displaystyle\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}\langle
0|[Q^{A},\Phi]|0\rangle=\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}\langle
0|\theta^{A}T^{A}\Phi|0\rangle=0.$ (71)
This equation is coming from the Nöther theorem of a conserved current
combining with the stationary a condition of $V_{eff}$. Since the zeroth
cohomology group of a Lie algebra is defined by the set of invariants (
annihilated ) under the algebra operation on a module [20],
$\displaystyle H^{0}({\rm Lie}(G),M)$ $\displaystyle=$ $\displaystyle M^{{\rm
Lie}(G)}=\bigl{\\{}m\in M|gm=0,\,\forall g\in{\rm Lie}(G)\bigr{\\}}.$ (72)
Thus the matrix $\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$ is interpreted
as an ”effective” invariant module, the zeroth cohomology of the Lie algebra,
via taking VEVs in the quantum theory.
$\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$ is regarded to take its value
in the sheaf of germs of continuous functions. ( For example, when $M={\bf
R}$, any connected compact semisimple Lie group $G$ has $H^{0}({\rm
Lie}(G),{\bf R})={\bf R}$, $H^{1}({\rm Lie}(G),{\bf R})=H^{2}({\rm
Lie}(G),{\bf R})=0$. ) In the usual case, the equation (68) has exactly the
same dimension with the number of broken generators and closed in the linear
space, while in the case such as a ferromagnet, $\langle 0|Q^{z}|0\rangle\neq
0$. Note that this VEV will be rewritten after a change of basis set of Lie
algebra, an algebra homomorphism, i.e., $\langle 0|Q^{z}|0\rangle=\langle
0|Q^{x^{\prime}}+Q^{y^{\prime}}+Q^{z^{\prime}}|0\rangle\neq 0$, and thus the
final result obtained from any calculation must not depend on the choice of
Lie algebra representation, $\langle Q^{z}\rangle\neq 0$. This fact means that
a rotation of the frame $(x,y,z)$ must not affect on the physical content of
this equation, of course. The important point is that this equation (68)
cannot be written down only by the set of broken generators in the
ferromagnetic case: This case apparently breaks the condition of proof of the
ordinary NG theorem, and one cannot conclude the existence of a zero-mass
bosonic particle. In other words, the NG boson subspace interacts with the
”symmetric” subspace in a breaking scheme caused by a quantum effect: The
author argues that this is a kind of quantum geometry. Thus, the mass matrix
of NG bosons,
$\displaystyle\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}$
$\displaystyle=$ $\displaystyle\Delta^{-1}_{F}(p=0)$ (73)
( $\Delta_{F}(p)$; a matrix Feynman propagator of NG bosons ) ”should” have a
nonzero value. This is just the mechanism of famous Nielsen-Chadha anomaly in
the NG theorem [1,8,34,55,77,81,82,83,84]. Simultaneously, it is also clear
from our discussion, a Lorentz-invariant system with a breaking scheme $G/H$
which gives a symmetric space never has a ”spontaneous violation” of the
ordinary/normal NG theorem. ( The Nielsen-Chadha anomaly never takes place. )
Hence, the ordinary NG theorem is protected by the Coleman-Mandula theorem of
$S$-matrix. Not only the mass matrix of NG bosons, but the dispersion
relations themselves should be derived from
$\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$. It should be mentioned that
there might be a similar situation in a relativistic model with a Lorentz-
violating parameter: For example, the following VEV in an $SU(2)$ isospin
space could be considered,
$\displaystyle\langle\bar{\psi}\tau_{3}\psi\rangle\neq 0,$ (74)
in a NJL type model. In this VEV, $\tau_{3}$ is symmetric while
$(\tau_{1},\tau_{2})$ is broken. It might give a similar situation with the
ferromagnet with $\langle Q^{3}_{isospin}\rangle\neq 0$.
Let us discuss further on $V_{eff}$. From the general theory of effective
action, the displacement (65) derives the following equation, from a second-
order derivative of the effective potential $V_{eff}$ after taking a VEV:
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle(V^{\prime\prime}_{eff})_{1}\theta^{1}T^{1}\Phi+\cdots+(V^{\prime\prime}_{eff})_{N}\theta^{N}T^{N}\Phi\rangle$
(75) $\displaystyle=$
$\displaystyle\langle(V^{\prime\prime}_{eff})_{1}\theta^{1}T^{1}v+\cdots+(V^{\prime\prime}_{eff})_{N}\theta^{N}T^{N}v\rangle,$
$\displaystyle V^{\prime\prime}_{eff}$ $\displaystyle=$
$\displaystyle\frac{\delta^{2}V_{eff}[\Phi]}{\delta\Phi^{2}},$ (76)
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
c_{1}T^{1}+\cdots+c_{N}T^{N},\quad\\{c_{j}\\}\in{\bf C},\,(j=1,\cdots,N).$
(77)
Here $v\in M_{n}({\bf C})$ ( matrix ) indicates a VEV. $\Phi$ is a $G$-module
expanded by the basis of ${\rm Lie}(G)$. Some normalization condition for
$\Phi$ is set aside for a while. Since $T^{A}\Phi$ is cased by taking
adjoints, the above equation implicitly contains the root space of the Lie
algebra ( i.e., from $[{\bf h},{\bf g}]=\lambda({\bf h}){\bf g}$,
$\lambda({\bf h})$; root, ${\bf h}$; Cartan subalgebra, ${\bf g}\in{\rm
Lie}(G)$ ), and the corresponding Weyl group acts implicitly. In a case of
Riemannian symmetric space, the adjoints, the Killing form, and the Jacobi
field are obtained from its root system ( see the book of Helgason [32] ).
Then they are related with a harmonic mapping and a harmonic analysis. Later,
we will mention that an NG boson gives a geodesic in a case of Riemannian
symmetric space. Note that the ${\rm Lie}(G)$ itself can be regarded as a
$G$-module, by satisfying the axiom of a $G$-module with a certain type of
group operations $G\times{\rm Lie}(G)\to{\rm Lie}(G)$. This formula (72)
includes the ”off-diagonal” contributions of the second-order derivative of
$V_{eff}$ in the space of Lie algebra generators which may cause mode-mode
couplings of NG bosons: Later, we will observe that a mode-mode coupling
between bosonic fields modifies dispersion relations of NG bosons and as a
consequence, an NG boson acquires a finite mass. Needless to say, the bases
$T^{A}\in{\rm Lie}(G)$ are always linearly independent. For example, in a case
of $SU(2)\to U(1)$,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle(V^{\prime\prime}_{eff})_{1}\langle[Q^{1},\Phi]\rangle+(V^{\prime\prime}_{eff})_{2}\langle[Q^{2},\Phi]\rangle+(V^{\prime\prime}_{eff})_{3}\langle[Q^{3},\Phi]\rangle,$
(78) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
c_{1}\sigma^{1}+c_{2}\sigma^{2}+c_{3}\sigma^{3}.$ (79)
Here, $\Phi\in{\bf 2}$-representation. We write the Lie brackets in the above
equation as follows:
$\displaystyle\langle[Q^{1},c_{2}\sigma^{2}]\rangle=(c_{2})_{1}\sigma_{3},\quad\langle[Q^{1},c_{3}\sigma^{3}]\rangle=(c_{3})_{1}\sigma_{2},$
$\displaystyle\langle[Q^{2},c_{1}\sigma^{1}]\rangle=(c_{1})_{2}\sigma_{3},\quad\langle[Q^{2},c_{3}\sigma^{3}]\rangle=(c_{3})_{2}\sigma_{1},$
$\displaystyle\langle[Q^{3},c_{1}\sigma^{1}]\rangle=(c_{1})_{3}\sigma_{2},\quad\langle[Q^{3},c_{2}\sigma^{2}]\rangle=(c_{2})_{3}\sigma_{1}.$
(80)
Then we get
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bigl{\\{}(V^{\prime\prime}_{eff})_{1}(c_{2})_{1}+(V^{\prime\prime}_{eff})_{2}(c_{1})_{2}\bigr{\\}}\sigma_{3}$
(81)
$\displaystyle+\bigl{\\{}(V^{\prime\prime}_{eff})_{2}(c_{3})_{2}+(V^{\prime\prime}_{eff})_{3}(c_{2})_{3}\bigr{\\}}\sigma_{1}$
$\displaystyle+\bigl{\\{}(V^{\prime\prime}_{eff})_{1}(c_{3})_{1}+(V^{\prime\prime}_{eff})_{3}(c_{1})_{3}\bigr{\\}}\sigma_{2}$
Due to the linear independence of $\sigma_{1,2,3}$, all of the coefficients
vanish. If $\langle[Q^{1},\Phi]\rangle\neq 0$ and/or
$\langle[Q^{2},\Phi]\rangle\neq 0$, both of them have contributions to rotate
$\Phi$ to the third direction proportional to $\sigma^{3}$, and there is a
freedom to take $(V^{\prime\prime}_{eff})_{1,2,3}$ finite in those vanishing
coefficients ( the case of our anomalous NG theorem of a ferromagnet ), while
if $\langle[Q^{1},\Phi]\rangle=\langle[Q^{2},\Phi]\rangle=0$, then
$(V^{\prime\prime}_{eff})_{1,2,3}$ will vanish independently ( the case of
ordinary NG theorem ). More general case is understood by the algebra we have
discussed in the previous section: At least in the case of quasi-Heisenberg
algebra, a mode-mode coupling takes place in $V_{eff}$ which becomes apparent
from its second-order derivatives. Hence, we arrive at the following theorem:
Theorem: Any type of mode-mode coupling between NG bosons modifies their
dispersion relations and mass spectra, gives an anomalous behavior of the NG
theorem.
We also yield another simple but important result:
Theorem: Any spontaneous symmetry breaking of isolated $U(1)$ Abelian Lie
group cannot give an anomalous behavior of the NG theorem, due to the lack of
mode-mode coupling.
In the theory of itinerant (anti)ferromagnetism of Moriya [56,57], he pointed
out that a mode-mode coupling of magnetic ( i.e., spin ) fluctuations is
important which can be experimentally observed by the method of magnetic
resonance. Our anomalous NG theorem of a ferromagnet may have a physical
implication in the Moriya theory. If the mass of massive NG boson is
controlled as a function of a strength external field or temperature, the
energy split of two NG bosons might be made small relative to the
characteristic energy scale. In the vicinity of the critical region, those
bosons acquire special importance. In the theory of Moriya, the method of
self-consistent renormalization theory is employed, which takes into account
diagrammatically higher-order interactions between electrons. It is
interesting from our context that how such higher-order interactions affect
the mass spectra of NG bosons, simultaneously to the value of $T_{c}$ and
correlation lengths.
In a composite particle model, a Schwinger-Dyson equation determines an order
parameter which is non-local, for example a two-point function $\langle
T\phi(x)\phi(y)\rangle\neq 0$ [15]. In that case, the gap equation is derived
by ( in a translation-invariant case )
$\displaystyle\frac{\delta V_{eff}(G(p))}{\delta G(p)}$ $\displaystyle=$
$\displaystyle 0.$ (82)
Here, $G(p)$ is a propagator. Now, the mass matrix of NG bosons may be
determined by the examination of the following equation:
$\displaystyle\sum^{N}_{A=1}\Bigl{(}\frac{\delta^{2}V_{eff}(G(p))}{\delta
G(p)^{2}}\Bigr{)}_{A}T^{A}G(p)$ $\displaystyle=$ $\displaystyle 0.$ (83)
Here, $T^{A}G(p)$ imply VEVs, for example,
$\langle\bar{\psi}T^{A}\psi\rangle$. Now, the Lie group $G$ acts on the
propagator $G(p)$ nontrivially, though the algebra we consider here is an
adjoint type, and thus this equation is essentially the same with (68) as an
equation of Lie algebra. Hence we obtain a mathematically similar equation
with (72). We conclude that the essential part of the basis of our anomalous
NG theorem is the same in case of both composite and elementary fields.
## 4 The Kaon Condensation Model
Let us start from the following $SU(2)$ Higgs-Kibble-type model Lagrangian:
$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}{\rm
tr}F^{a}_{\mu\nu}F^{a\mu\nu}+\tilde{D}^{\dagger}_{\nu}\Phi^{\dagger}\tilde{D}^{\nu}\Phi+m^{2}_{0}\Phi^{\dagger}\Phi-\frac{\lambda}{2}(\Phi^{\dagger}\Phi)^{2}+\epsilon
m^{2}_{ex}\Phi^{\dagger}\sigma^{3}\Phi,$ $\displaystyle\tilde{D}_{\nu}$
$\displaystyle=$
$\displaystyle\partial_{\nu}-i\mu\delta_{0\nu}-\frac{i}{2}g\sigma^{a}A^{a}_{\nu},$
(84)
Here, we introduce $SU(2)$-gauge fields $A^{a}_{\nu}$ ( $a=1,2,3$ ), and the
mass term $\epsilon m^{2}_{ex}\Phi^{\dagger}\sigma^{3}\Phi$ with a relatively
small parameter $\epsilon$ explicitly breaks the symmetry. Since this explicit
symmetry breaking parameter breaks the symmetries of $\sigma^{1}$ and
$\sigma^{2}$, both of them acquire finite masses. The complex bosonic field is
defined as $\Phi\equiv(\phi_{1},\phi_{2})^{T}$ as a ${\bf 2}$-representation,
and $\mu$ is a Lorentz-symmetry violating chemical potential. Let us mention
the fact that the anomalous behavior of NG bosons cannot be understandable by
the hypercharge model of $U(2)$ used in the paper of Schaefer et al [77]: Due
to the special nature of $SU(2)$, this model has an additional symmetry, so-
called ”custodial symmetry” of $SU(2)$. Hence the breaking scheme of this
model is $SU(2)\otimes SU(2)\simeq SO(4)\to SU(2)_{diag}$. This breaking
scheme is essentially the same with the Lie algebra of $SU(N)_{L}\otimes
SU(N)_{R}\to SU(N)_{V}$ discussed at (61). Namely,
$[\theta^{a}\sigma^{a}\otimes
1+\varphi^{b}\sigma^{b}\otimes\sigma^{3},\sum^{3}_{j=1}a_{j}\sigma^{j}]$ will
be examined. We consider the broken and symmetric generators carefully
according to this breaking scheme. First, let us consider the case where all
of the gauge fields are dropped from this model. We assume the model chooses
$\Phi_{0}=\langle\Phi\rangle=(0,v)^{T}/\sqrt{2}$ as one of its vacua: Then we
yield the following relation for the VEV $v$:
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial
V^{(tree)}_{eff}}{\partial v},$ (85) $\displaystyle V^{(tree)}_{eff}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})v^{2}+\frac{\lambda}{8}v^{4},$ (86) $\displaystyle v$
$\displaystyle=$ $\displaystyle\pm\sqrt{\frac{2(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})}{\lambda}}.$ (87)
Now a small displacement around the vacuum solution we have obtained, consists
with the amplitude mode and three NG modes, is described by the following ’t
Hooft parametrization:
$\Phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\chi_{2}+i\chi_{1}\\\
v+\psi-i\chi_{3}\end{array}\right)$ (88)
Then we get
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}\psi+\tilde{\partial}^{\dagger}_{\nu}\chi_{a}\tilde{\partial}_{\nu}\chi_{a}+\mu^{2}v^{2}$
(89)
$\displaystyle\quad+i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{1}-\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{2}\Bigr{)}+i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{3}\tilde{\partial}_{\nu}v-\tilde{\partial}^{\dagger}_{\nu}v\tilde{\partial}_{\nu}\chi_{3}\Bigr{)}$
$\displaystyle\quad-i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}\chi_{3}-\tilde{\partial}^{\dagger}_{\nu}\chi_{3}\tilde{\partial}_{\nu}\psi\Bigr{)}+\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}v+\tilde{\partial}^{\dagger}_{\nu}v\tilde{\partial}_{\nu}\psi\Bigg{]}$
$\displaystyle\quad+\frac{m^{2}_{0}}{2}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}+(v+\psi)^{2}\bigr{)}$
$\displaystyle\quad-\frac{\lambda}{8}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}+(v+\psi)^{2}\bigr{)}^{2}$
$\displaystyle\quad+\frac{\epsilon
m^{2}_{ex}}{2}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}-\chi^{2}_{3}-(v+\psi)^{2}\bigr{)},$
$\displaystyle\tilde{\partial}_{\nu}$ $\displaystyle=$
$\displaystyle\partial_{\nu}-i\mu\delta_{0\nu}.$ (90)
After using the expression of the VEV $v$ ( Eq.(84) ) of $\Phi$, we get the
quadratic part of the Lagrangian in the following form:
$\displaystyle{\cal L}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}\partial_{\nu}\psi\partial^{\nu}\psi+\partial_{\nu}\chi_{a}\partial^{\nu}\chi_{a}\bigr{)}$
(91)
$\displaystyle+\mu\bigl{(}\chi_{1}\partial_{0}\chi_{2}-\chi_{2}\partial_{0}\chi_{1}+\chi_{3}\partial_{0}\psi-\psi\partial_{0}\chi_{3}\bigr{)}$
$\displaystyle+\epsilon
m^{2}_{ex}(\chi^{2}_{1}+\chi^{2}_{2})-(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})\psi^{2}.$
$\epsilon>0$ must be excluded to avoid tachyonic modes in
$(\chi_{1},\chi_{2})$, and this condition holds for the dispersion relations
after diagonalization of ${\cal L}^{(2)}$ ( see, (92) ) Note that the total
Lagrangian ${\cal L}$ is a class function ${\cal L}(\Phi)={\cal L}(g\Phi
g^{-1})$, $g\in G$, while both the effective potential
$V_{eff}(\Phi=\Phi_{0})$ and ${\cal L}^{(2)}(\delta\Phi=\Phi-\Phi_{0})$ are
not class functions, even if the explicit symmetry breaking parameter
vanishes. This fact is crucially important for us to understand the physics of
our anomalous NG theorem. In the case of normal NG theorem,
$V_{eff}(\Phi=\Phi_{0})$ is ”effectively” a constant under the action of $g\in
G$ since it does not contain any coordinate of the NG manifold explicitly.
While, in our anomalous NG theorem, $V_{eff}(\Phi=\Phi_{0})$ will acquire an
energy=mass by the action of $g\in G$, and the energy which corresponds to the
mass of an NG boson is provided from ${\cal L}^{(2)}(\delta\Phi)$. Therefore,
the effective potential of the theory is certainly periodically modulated
under the action of $g\in G$ if $G$ is compact. We will see this is the case
in our discussion given below. By using the path-integral formalism $\int{\cal
D}\Phi{\cal D}\Phi^{\dagger}e^{i\int{\cal L}(\Phi,\Phi^{\dagger})}$, we
immediately recognize that this phenomenon of anomalous NG theorem is a pure
quantum effect, since the Lagrangian of the beginning, ${\cal
L}(\Phi,\Phi^{\dagger})$, is also a class function of any group operation
$g\in G$ when any explicit symmetry breaking parameter vanishes. Namely, it is
impossible to understand it by a tree-level, and we need at least the one-loop
level to obtain the finite curvature along with a massive NG-bosonic
coordinate. This is a remarkable fact since the treatment of the symmetry
breaking of the traditional bosonic Goldstone model ( and also the Standard
Model Higgs sector ) is understandable at the tree level, while a fermion
composite model such as an NJL-type model has to evaluate at least a one-loop
effective potential. To make the Lagrangian in a Hermitian matrix form
explicitly, we perform partial integrations in time-derivatives and rearrange
the Lagrangian:
${\cal
L}^{(2)}=\frac{1}{2}\tilde{\Phi}\left(\begin{array}[]{cccc}k^{2}+\epsilon
m^{2}_{ex}&ik_{0}\mu&0&0\\\ -ik_{0}\mu&k^{2}+\epsilon m^{2}_{ex}&0&0\\\
0&0&k^{2}&ik_{0}\mu\\\
0&0&-ik_{0}\mu&k^{2}-M^{2}\end{array}\right)\tilde{\Phi},$ (92)
where,
$\displaystyle\tilde{\Phi}$ $\displaystyle=$
$\displaystyle(\chi_{1},\chi_{2},\chi_{3},\psi)^{T},$ (93) $\displaystyle
M^{2}$ $\displaystyle=$ $\displaystyle\mu^{2}+m^{2}_{0}-\epsilon m^{2}_{ex}.$
(94)
Then we get
$\displaystyle E^{\chi_{1},\chi_{2}}_{\pm}$ $\displaystyle=$
$\displaystyle\sqrt{{\bf k}^{2}-\epsilon
m^{2}_{ex}+\frac{\mu^{2}}{2}\pm\frac{\mu}{2}\sqrt{\mu^{2}+4{\bf
k}^{2}-4\epsilon m^{2}_{ex}}},$ (95) $\displaystyle E^{\chi_{3},\psi}_{\pm}$
$\displaystyle=$ $\displaystyle\sqrt{{\bf
k}^{2}+\frac{M^{2}+\mu^{2}}{2}\pm\frac{1}{2}\sqrt{4\mu^{2}{\bf
k}^{2}+(M^{2}+\mu^{2})^{2}}}.$ (96)
Now it is clear for us from these dispersion relations at the limit ${\bf
k}\to 0$ and $\epsilon\to 0$, $E^{\chi_{1},\chi_{2}}_{+}$ and
$E^{\chi_{3},\psi}_{+}$ are massive while $E^{\chi_{1},\chi_{2}}_{-}$ and
$E^{\chi_{3},\psi}_{-}$ are massless.
Therefore, we find that the masses of NG bosons in the $SU(2)$-Higgs-Kibble-
type model are coming from the mode-mode coupling of a pair of broken
generators. This fact provides us a confirmation on our general discussion
given in the previous section. Hence, $\langle[Q^{1},Q^{2}]\rangle\neq 0$ and
other commutators must have vanishing VEVs in this case. This is achieved by
the form of the vacuum $\Phi_{0}=(0,v)/\sqrt{2}$. It must be distinguished
that the commutators $[Q^{A},Q^{B}]$ are given by conserved charges and now
$\langle Q^{1}\rangle=\langle Q^{2}\rangle=0$, $\langle Q^{3}\rangle\neq 0$,
while an order parameter can take the form $\Phi\sim
a_{1}\sigma^{1}+a_{2}\sigma^{2}+a_{3}\sigma^{3}$, $a_{1}\neq 0$, $a_{2}\neq
0$, $a_{3}\neq 0$. A Heisenberg algebra is obtained, and a pairwise decoupling
takes place. In this example of kaon condensation, the amplitude mode $\psi$
is defined toward the direction of VEV, and it couples with $\chi_{3}$. This
amplitude mode might be expressed more generally such as
$\psi\to\exp\psi((1-\sigma^{3})/2)\Phi_{0}$, though this is not an element of
the Lie group $SU(2)$, rather a projection operator, and then the manner of
mode-mode coupling between $\psi$ and $\chi_{3}$ is not given in the same
manner of commutator $[Q^{1},Q^{2}]$, different from the mode-mode coupling of
$\chi_{1}$ and $\chi_{2}$. It should be noticed that
$\chi_{1}=\Re\delta\phi_{1}$, $\chi_{2}=\Im\delta\phi_{1}$,
$\psi=\Re\delta\phi_{2}$, $\chi_{3}=\Im\delta\phi_{2}$, where
$(\delta\phi_{1},\delta\phi_{2})$ are fluctuations in the vicinity of the
stationary point. Thus, the mode-mode couplings relevant for our anomalous NG
theorem take place between the real and imaginary parts of $\phi_{1}$ and
$\phi_{2}$ separately. This fact in our kaon condensation model is quite
interesting toward a classification of several possible types of mode-mode
couplings in NG bosons. In the above example, the mode-mode couplings of the
NG bosons and the amplitude are described over a two independent discs of
Gaussian plane. It may be possible to generate a mode-mode coupling which
gives a finite mass to an NG boson via a radiative correction: This can be
regarded as a kind of Coleman-Weinberg mechanism in an NG boson mass matrix.
This can be understandable if there is an interaction between two NG bosons (
such as an electromagnetic interaction ): Especially, a Rayleigh-Schrödinger
or a quasi-degenerate perturbation theory can apply to the case where the
spectrum of an NG sector is split by small but finite masses.
The periodicity of the NG sector inside the Lagrangian can be understood as
follows. Since the massive NG bosons arises from the pair
$(\chi^{1},\chi^{2})$, we prepare
$g=e^{i(\chi_{1}\sigma^{1}+\chi_{2}\sigma^{2})}=\left(\begin{array}[]{cc}\cos|\chi|&i\frac{\chi_{-}}{|\chi|}\sin|\chi|\\\
i\frac{\chi_{+}}{|\chi|}\sin|\chi|&\cos|\chi|\end{array}\right),$ (97)
where,
$\displaystyle|\chi|$ $\displaystyle=$
$\displaystyle\sqrt{\chi^{2}_{1}+\chi^{2}_{2}},\quad\chi_{\pm}=\chi_{1}\pm
i\chi_{2}.$ (98)
Then we evaluate
$\displaystyle\tilde{\partial}_{\nu}g\Phi_{0}$ $\displaystyle=$ $\displaystyle
i\Bigl{[}(\tilde{\partial}_{\nu}\chi_{1})\sigma^{1}+(\tilde{\partial}_{\nu}\chi_{2})\sigma^{2}\Bigr{]}g\Phi_{0},$
(99)
and take the following inner product, we get
$\displaystyle\tilde{\partial}^{\dagger}_{\nu}(\Phi_{0}g^{-1})\cdot\tilde{\partial}_{\nu}(g\Phi_{0})$
$\displaystyle\quad=v^{2}\Bigl{[}\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{1}+\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{2}+i\bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{2}-\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{1}\bigr{)}\cos
2|\chi|\Bigr{]}.$ (100)
Therefore, the chemical potential $\mu$ acquires a periodic modulation
proportional to the trigonometric function such that $\sim\mu^{2}\cos
2|\chi|$, and thus the mass of the dispersion $E^{\chi_{1},\chi_{2}}_{+}$
becomes periodic as a function of $|\chi|$. Namely, the periodic modulation of
the effective potential is kinematically generated in our anomalous NG theorem
by the mode-mode-coupling caused by a finite chemical potential: This is a
quite remarkable result, since our result explains both the mechanism of
generation of a finite mass of an NG bosons, and a kinematically generated
periodicity of the effective potential beyond the tree-level defined over the
$SU(2)$ group manifold. It should be mentioned that this expression of the
kinetic part of the Lagrangian is obtained by choosing a specific form of the
VEV of $\Phi$, i.e., $\langle\Phi\rangle=\Phi_{0}=(0,v)^{T}/\sqrt{2}$, and
thus this expression is a ”function” of the form of VEV, the special form of
local coordinates $|\chi|$, and the chemical potential $\mu$. Hence, if we
choose another type of VEV to $\Phi$, then in fact we will obtain another
expression different from (97): From this sense, both $V^{(tree)}_{eff}$ and
${\cal L}^{(2)}$ are not class functions of $SU(2)$ ( caution: the kinetic
term given above contains all of the orders of fluctuations
$(\chi_{1},\chi_{2})$, not only ${\cal L}^{(2)}$ ). The form $|\chi|$ implies
that the theory is isotropic toward the directions $\chi_{1}$ and $\chi_{2}$ (
axial symmetric ), similar to the case of a ferromagnet. One should notice
that the Lagrangian of (88) or (89) is defined locally, at a specific point
over the $SU(2)$ Lie group manifold. To see the periodicity, we need a group
element which is defined globally, as we have used above. Now we obtain the
effective potential of the model as $V_{eff}\sim
V^{(tree)}_{eff}+f(v)\mu^{2}\cos 2|\chi|$ by using the result of
$E^{\chi_{1},\chi_{2}}_{+}$ ( $f(v)$: a scalar function of the VEV $v$ ),
which shows a periodicity toward the direction of ”amplitude”
$|\chi|=\sqrt{\chi^{2}_{1}+\chi^{2}_{2}}$, while it is flat to the direction
of the phase ( precession mode ) of $\chi_{1}+i\chi_{2}$ ( the amplitude and
the phase defines an infinite number of $S^{1}$ circles of a Gaussian plane ),
as we have stated above. This fact is parallel with the case of ferromagnet,
where the massless NG mode ( spin wave ) is the precession described by a
linear combination of the two modes $(\sigma_{1},\sigma_{2})$. Absolutely
interesting fact we have found here is that this global structure of the
effective potential ( periodic toward the radial direction $|\chi|$ while
exactly flat along with the phase variable ) is coming from the uncertainty
relation arises from the Heisenberg algebra obtained from $SU(2)$: There is a
strong uncertainly toward the phase direction, while the motion toward the
radial direction is well localized and the ”position” is determined by the set
of periodic stationary points: The set of stationary points gives a Galois
symmetry. In the vicinity of a stationary point ( valley ), a representation
point has a small fluctuation toward the radial direction while it strongly
fluctuate along with the phase coordinate. Namely,
Theorem: The uncertainty relation of the Heisenberg algebra obtained from the
Lie algebra of $SU(2)$ realizes in the global structure of the effective
potential of a theory. One degree of freedom is almost fixed/determined while
another degree of freedom of the Heisenberg pair of shows a strong
uncertainty.
We argue that a quasi-Heisenberg algebra generically obtained in various
symmetry breaking schema of our anomalous NG theorem will determine the
structure of effective potential according to satisfy the uncertainty
relations. Therefore, a mass generation in an NG sector in our anomalous NG
theorem reflects the uncertainty principle! Furthermore, an explicit symmetry
breaking mass parameter such as the prescription of explicitly+dynamical
symmetry breakings also acts to fix a phase degree of freedom, and thus, a
mass spectrum any meson ( mesonic state ) in a non-Abelian Lie group symmetry
is generically a result of uncertainty relation.
The kaon condensation model we consider here has a lot of similar aspects with
physics of a ferromagnet. Needless to say, the time-reversal symmetry is
broken in a ferromagnet, and a precession of magnetization reflects this time-
reversal symmetry breaking. The system of kaon condensation we discuss here
might have an effect of time-reversal symmetry breaking, caused by the NG
bosons, at its low-energy excited state.
We make a brief comment on the Higgs ( Anderson-Higgs-Brout-Englert-Guralnik-
Hagen-Kibble ) phenomenon in the Lagrangian (81) [2,21,28,34]. If we put a
$U(1)$ gauge field to the Lagrangian, it also gives a Higgs phenomenon with
the field redefinition $U_{0}=\mu+A_{0}-\partial_{0}\theta$,
$U_{i}=A_{i}-\partial_{i}\theta$ ( $\theta$: a $U(1)$ phase ) and thus they
give a massive Proca theory. While if we put a set of $SU(2)$ gauge fields as
(81), the chemical potential $\mu$ plays no role and the usual Higgs
phenomenon is observed.
## 5 The Model Lagrangian Approach
In all of the above classification of types of NG bosons given in the end of
the introduction of this paper, the dispersion relations and mass spectrum of
NG bosons should be obtained from an analysis of effective action $V_{eff}$.
In general, an effective action and/or a potential are expanded by bosonic
fields, thus we restrict ourselves to the case of bosonic ( bosonized )
theory. Since the NG boson Lagrangian will be obtained from $V_{eff}$, we
construct a Lagrangian to make our problem more tractable. Let ${\cal
L}(\Phi)$ be a Lagrangian, and let us take a small displacement of bosonic
field as $\Phi=\Phi_{0}+\delta\Phi$. $\Phi$ is assumed to belong to a
representation of $G$. $\delta\Phi$ contains the NG bosons and the amplitude
mode. In the case of $SU(N)$, one frequently use a fundamental representation,
${\bf N}\ni\Phi,\Phi_{0},\delta\Phi$. Let us consider the case of symmetry
given by a Lie group $G$ with ${\rm dim}{\rm Lie}(G)=N$. Then ${\rm
dim}(\delta\Phi)=N+1$, where the additional one degree of freedom is the
amplitude mode of $\Phi$. Then the Lagrangian is expanded into the following
form:
$\displaystyle{\cal L}(\Phi_{0}+\delta\Phi)$ $\displaystyle=$
$\displaystyle{\cal L}(\Phi_{0})+\frac{\partial{\cal
L}(\Phi_{0})}{\partial(\delta\Phi)}\delta\Phi+\frac{1}{2!}\frac{\partial^{2}{\cal
L}(\Phi_{0})}{\partial(\delta\Phi)^{2}}(\delta\Phi)^{2}+\cdots.$ (101)
The first-order derivative vanishes in the effective action, and the second-
order derivative gives the mass matrix and dispersion relations of NG bosons.
Namely,
$\displaystyle\frac{\partial^{2}{\cal L}(\Phi_{0})}{\partial(\delta\Phi)^{2}}$
$\displaystyle=$ $\displaystyle\Delta^{-1}_{F}(p_{\nu}).$ (102)
This equation corresponds to (70). The algebraic roots of ${\rm
det}\Delta^{-1}_{F}(p_{\nu})=0$ gives the dispersion relations of NG bosons.
One can also consider the case where the order parameter is a vector/tensor
$\Phi_{\mu\nu\cdots\rho}$, explicitly breaks the Lorentz symmetry. In that
case, we can consider the following formal expansion:
$\displaystyle{\cal L}(\Phi_{\mu\nu\cdots\rho})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{n=0}\frac{1}{n!}\frac{\partial^{n}{\cal
L}}{\partial(\delta\Phi_{\mu\nu\cdots\rho})^{n}}(\delta\Phi_{\mu\nu\cdots\rho})^{n}.$
(103)
Here, we do not consider contractions of the Lorentz indices
$(\mu,\nu,\cdots,\rho)$. A vectorial order parameter is frequently found in
superconductivity or 3He superfluidity [50,65,66,78]. The second-order
derivative term as the quadratic part of quantum fluctuations,
$\displaystyle{\cal L}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial^{2}{\cal
L}}{\partial(\delta\Phi_{\mu\nu\cdots\rho})^{2}}(\delta\Phi_{\mu\nu\cdots\rho})^{2}$
(104)
gives the mass matrix and dispersion relations of NG bosons. While, due to the
Coleman-Mandula theorem, a Poincaré-invariant theory can have only a conserved
charge of scalar type. We currently consider a Lorentz-violating system of
(non)relativistic theory, and thus we assume a theory can have vector/tensor-
charges $Q^{A}_{\mu\nu\cdots\rho}$ of internal symmetries. Hence, formally,
$\displaystyle\delta^{A}_{\mu\nu\cdots\rho}\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}$
$\displaystyle=$
$\displaystyle[Q^{A}_{\mu\nu\cdots\rho},\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
(105)
will be considered. Hence, we find that the Lie brackets
$[Q^{A}_{\mu\nu\cdots\rho},\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
define how ${\cal L}^{(2)}$ is given in terms of the NG bosons, similar to the
case of $V_{eff}$ we have discussed in the previous section. Namely, we will
consider the Lie brackets of internal symmetries with Lorentz indices. Thus, a
quasi-Heisenberg algebra should be obtained from
$[Q^{A}_{\mu\nu\cdots\rho},Q^{B}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
in a diagonal symmetry breaking scheme. If this algebra causes a mode-mode
coupling in quantum fluctuations of NG bosons, then our anomalous NG theorem
takes place. Hence, we argue it is enough for us to consider a scalar field to
study the mechanism of our anomalous NG theorem.
Now, we can systematically construct a generic Lagrangian which may show the
phenomenon of anomalous NG theorem. We know from our observation on the model
of kaon condensation, the relevant part of the Lagrangian to give the
anomalous behavior of NG bosons is its kinetic term. For example, in the case
of two-component real bosonic field:
${\cal
L}=\frac{1}{2}(\phi_{1},\phi_{2})\left(\begin{array}[]{cc}\partial_{\nu}\partial^{\nu}&a_{\nu}\partial^{\nu}\\\
a^{*}_{\nu}\partial^{\nu}&\partial_{\nu}\partial^{\nu}\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\right).$ (106)
The off-diagonal part describes a mode-mode coupling: From our examination on
the Lagrangian of NG sector of the kaon condensation model, it is now obvious
fact that the explicit symmetry breaking mass parameter enters into the
diagonal part of the Lagrangian matrix, while the mode-mode coupling matrix
elements take of course several off-diagonal elements. We do not consider
kinetic terms of derivatives higher than the second-order, since they may
cause a tachyonic mode or a negative norm state. Therefore, this mechanism of
anomalous NG theorem must break the Lorentz symmetry.
With respect to the general theory of nonlinear sigma models, the following
Lagrangian is examined:
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta\Phi{\cal M}\delta\Phi$ (110) $\displaystyle=$
$\displaystyle\frac{1}{2}g_{ab}\partial_{\nu}(\Psi)^{a}\partial^{\nu}(\Psi)^{b}-\frac{1}{2}\Psi
M^{2}\Psi+\Psi\left(\begin{array}[]{ccc}0&&\sum\tilde{a}_{\nu}\partial^{\nu}\\\
&\ddots&\\\
\sum\tilde{a}^{\dagger}_{\nu}\partial^{\nu}&&0\end{array}\right)\Psi,$
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(\chi_{1},\cdots,\chi_{N},\phi)^{T},\quad(a=1,2,\cdots,N).$
(111)
Here, the real matrix $g_{ab}$, which defines a Riemannian metric
$\hat{g}=g_{ab}dx^{a}\otimes dx^{b}$ ( $dx^{a},dx^{b}\in{\bf R}$), must
satisfy a condition given by the Lie group $G$. Usually, it is taken to be a
unit matrix, i.e., which defines a Euclidean space, a typical example of a
simply connected Riemannian symmetric space, with its dimension equal to the
number of broken generators. It is known fact that, in a small displacement
$\Psi\to\Psi+\delta\Psi$, $\delta\Psi$ must be a Killing vector defined over
the target space of a nonlinear sigma model. Here, we have put explicit
symmetry breaking mass parameters $M^{2}$ ( a Hermitian matrix ).
$\tilde{a}_{\nu}$ are matrices, and they can contain both explicit symmetry
breaking parameters ( like a chemical potential ) or spontaneously generated
parameters from its underlying theory. Therefore, the number of roots of
$\displaystyle{\rm det}{\cal M}(p_{\nu}\neq 0)$ $\displaystyle=$
$\displaystyle 0$ (112)
gives the number of NG bosons plus 1. From the mathematical structure of
${\cal M}$, we find that the matrix can be expressed by a Borel subalgebra (
given by the direct sum of Cartan subalgebra ${\bf h}$ and the positive-weight
part of ${\bf g}$ ) of ${\rm Lie}GL(N+1)$ or ${\rm Lie}O(N+1)$ [43]. Probably,
we can employ a flag variety to study a representation of such an expression
of Borel subalgebra, Borel subgroup $B$, the Borel-Weil theorem, and a Bruhat
decomposition of $G=\cup_{\omega\in W}B\omega B\ni g$ as a disjoint union (
$G$; a connected reductive algebraic group, $W$; a Weyl group ): The mode-mode
coupling part of the matrix ${\cal M}$ is decomposed into a strictly upper
triangular matrix and a strictly lower triangular matrix, and they can be
expressed by a Borel subalgbera. ( Note: Any strictly triangular matrix is
nilpotent, and a set of strictly upper/lower triangular matrices forms a
nilpotent Lie algebra. Borel subgroup = invertible triangular matrices = all
diagonal entries must be nonzero. Borel subalgebra = not necessarily
invertible. ) Of particular interest is a homogeneous space $G/P=BWB/P$ ( $P$;
a parabolic subgroup ) which gives a parabolic geometry. Since ${\cal
L}\in{\rm Lie}O(N+1)$ and the local coordinate system ( NG bosons ) are all
real, one can consider an adjoint group action $g^{-1}{\cal L}g$ ( $g\in
O(N+1)$ ). The equation ${\rm det}{\cal M}(p_{\nu}\neq 0)=0$ defines a
hypersurface in such a Borel subgroup representation space. The Borel-Weil
theorem states that the global section $\Gamma(G/B,L_{\lambda})$ (
$L_{\lambda}$ is a $G$-equivariant holomorphic line bundle over $G/B$,
$\lambda$ denotes a dominant integral highest weight ) gives an irreducible
representation of $G$. Hence, the mechanism of generating massive NG bosons
reflects a breaking of $O(N+1)$ ( more precisely, $O(N+1,{\bf R})$ ) symmetry
as a result of a breaking scheme and a Lorentz-violating parameter, and the
breaking structure of $O(N+1)$ reflects the dispersion relations and numbers
of massive/massless NG bosons. We will see later that this $O(N+1)$ structure
gives a symmetry ( in fact, degeneracy ) of the mass spectrum of NG bosons: In
fact, as we will see at (116) of the decomposition into several Heisenberg
pairs, the number and structure of Heisenberg pairs embedded in ${\cal L}$ is
determined by the $O(N+1)$ structure. When we observe our Lagrangian from the
context of chiral perturbation theory, the expansion of a chiral perturbation
may be given in a form of a series of symplectic matrices ( a series of the
form proportional to $\sum_{n}c_{n}(1+c_{A}T^{A})^{n}$, $T^{A}\in{\rm Lie}(G)$
).
Our result and discussion given here depends on our observation of the kaon
condensation model with the general theory of effective action/potential, and
thus we cannot say definitely whether there is another mechanism ( different
from the mode-mode coupling mechanism ) which gives a similar
phenomenon/result of anomalous NG theorem at a Lagrangian level. Our result
given here is derived from a relativistic model, though it can be applied to
the cases of (anti)ferromagnets, since their low energy excitations are
described by the class of $O(3)$ ( and mathematically generally, $O(N)$ )
nonlinear sigma models. In that case, according to our examination of the
Heisenberg algebra coming from $SU(2)$, mode-mode coupling terms similar to
the case of relativistic model with a finite chemical potential may be
introduced in the sigma model of a nonrelativistic case. Therefore, if such
type of nonlinear sigma models can be derived from Ginzburg-Landau-Goldstone-
Higgs-Kibble-type theories, then similar phenomenon may be observed. In fact,
the Lagrangian of ferromagnet given in the paper of Watanabe and Murayama [82]
takes almost a special example of our generic Lagrangian (104). Namely,
Theorem: The mechanism of anomalous NG theorem in a nonrelativistic case is
the same with that of a Lorentz-violating relativistic case. The counting law
of the number of NG bosons of a nonrelativistic case is the same with that of
a Lorentz-violating relativistic case.
It is interesting for us to study a phase of magnon condensation by our
theoretical framework presented here, since a condition of Bose-Einstein
condensation is determined by a mass parameter and a chemical potential ( a
magnon energy in a real substance is typically $\mu$eV ).
Our generic Lagrangian given above can be discussed more systematically and
mathematically/geometrically [42]. With respect to the procedure to obtain the
quadratic part in terms of NG bosons from the Lagrangian of kaon condensation,
a kinetic part is expressed as follows:
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle g\Phi_{0},\qquad
g=e^{iQ^{A}\chi^{A}}\in G,$ (113)
$\displaystyle\tilde{\partial}^{\dagger}_{\nu}\Phi^{\dagger}\tilde{\partial}^{\nu}\Phi$
$\displaystyle=$
$\displaystyle-\Phi^{\dagger}_{0}(g^{-1}\tilde{\partial}_{\nu}g)(g^{-1}\tilde{\partial}^{\nu}g)\Phi_{0},$
(114) $\displaystyle=$
$\displaystyle-\Phi^{\dagger}_{0}\Bigl{(}g^{-1}(\partial_{\nu}\partial^{\nu}-2i\mu\partial_{0}-\mu^{2})g\Bigr{)}\Phi_{0}.$
Here, we have chosen the local coordinate system $\\{\chi^{A}\\}$ as the first
kind. The term of $2i\mu\partial_{0}$ is coming from the Leibniz rule of
derivation. In general, the kinetic term is given by
$g_{\alpha\bar{\beta}}\tilde{\partial}^{\dagger}_{\nu}(\Phi^{\dagger})^{\beta}\tilde{\partial}^{\nu}(\Phi)^{\alpha}$,
and $\tilde{g}=g_{\alpha\bar{\beta}}dz^{\alpha}\otimes d\bar{z}^{\beta}$ (
$dz^{\alpha}\in{\bf C}$, $d\bar{z}^{\beta}\in\bar{\bf C}$ ) gives a Hermitian
metric. This Lagrangian does not have the fluctuation of amplitude mode of
$\Phi$, which can couple with an NG boson as we have observed in the kaon
condensation model. In fact, the amplitude mode cannot be expressed by a
Maurer-Cartan form. It should be noticed that,
$\displaystyle g^{-1}\tilde{\partial}_{0}g=g^{-1}\partial_{0}g-i\mu.$ (115)
Note that this equation shows a ”deformation” of ( or, a deviation from ) the
Maurer-Cartan 1-form
$\displaystyle\omega(T^{A},\theta^{A})=g^{-1}\partial_{\nu}g=i\sum
T^{A}\otimes\theta^{A},\quad\theta^{A}=\partial_{\nu}\chi^{A},$ (116)
as a geometric object ( $\\{T^{A}\\}$ give the Maurer-Cartan frame,
$\\{\theta^{A}\\}$ are the Maurer-Cartan coframe ), or a deviation from the
Killing form given in terms of $g^{-1}\partial_{\nu}g$, namely ${\rm
tr}(g^{-1}\partial_{\nu}g,g^{-1}\partial^{\nu}g)$ in the Lagrangian. It must
be noticed that $g^{-1}\partial_{0}g\in{\rm Lie}(G)\simeq T_{e}G$ ( $e$: the
origin ), while the part $-i\mu$ is independent from a geometric structure of
${\rm Lie}(G)$. A Maurer-Cartan form is an adjoint orbit, defines essentially
a homogeneous space. Moreover, a Maurer-Cartan form defines a local section: A
vacuum state of the quantum field theory is given by the specific local
section of $V_{eff}$ in the sense of fiber bundle, and a structural group (
namely, a Lie group ) gives a transformation between two local sections. The
effective potential $V_{eff}$ is an example of so-called representation
function: $V_{eff}$ is a continuous function defined over a topological space
$X$, and has a continuous group action of a group element of $G$. Thus, the
group orbit is defined by the pair $(V_{eff},g\in G)$, which is a subset of
the space of continuous functions over $X$. In other words, $V_{eff}$ belongs
to a set of continuous sections of a $G$-bundle $E$. Namely,
$V_{eff}\in\Gamma(E)_{G}\subset\Gamma(E)$, where $\Gamma(E)$ is the space of
all continuous sections, $\Gamma(E)_{G}$ is its submodule. $\Gamma(E)_{G}$ is
dense in $\Gamma(E)$ when $G$ is compact. This fact is important to certify a
variational calculus of $V_{eff}$ to obtain a stationary point. Note that the
Killing form is now subjected to the Euler-Lagrange variation principle, and
of course defines a phase factor of path integral. In the usual case, a
connection $\xi$ is introduced by the form $g^{-1}(d+\xi)g$, and the part
$g^{-1}dg$ is the Maurer-Cartan form, namely the chemical potential is
included in our theory under the manner of a connection. Note that in our
case, $\chi^{A}$ have the additional dependence on spacetime coordinates
coming from the local ”wave” nature of quantum field theory which is not
contained in the traditional Lie theory. From our results, we know the mode-
mode couplings of NG bosons are given as VEVs of the space of Lie algebra of
the 1-form, such as $g^{-1}\mu(\partial_{0}\chi^{A})Q^{A}g$, namely a
displacement at the origin caused by a Lie group in the derivative of NG
boson, and the explicit symmetry breaking parameter $\mu$ acts on them as a
linear scaling factor. Moreover, any deformation theory of mathematics of
manifolds defined over a field of characteristic zero is described by Maurer-
Cartan elements of a differential graded Lie algebra ( for example, the theory
of deformation quantization of Kontsevich [45] ). In our case, since our
anomalous NG theorem quite often gives a ”quasi” Heisenberg algebra over a
Poisson manifold, from locally to globally, thus, we need a special study on a
deformation theory and a deformation quantization which can achieve a quasi-
Heisenberg algebra/group. In addition, our $\omega$ satisfies the Maurer-
Cartan structural equation, sometimes called as the deformation equation,
$\displaystyle
d\omega(T^{A},\theta^{A})+\frac{1}{2}[\omega(T^{A},\theta^{A}),\omega(T^{A},\theta^{A})]=0,\quad{\rm
or},$ (117) $\displaystyle
d\theta^{A}+i\sum_{B,C}f^{BCA}\theta^{B}\wedge\theta^{C}=0.$ (118)
The Maurer-Cartan equation is the vanishing condition of the curvature 2-form,
namely the vanishing condition of the curvature of Cartan connection
$\Omega=d\omega+\frac{1}{2}[\omega,\omega]=0$. ( Understood as Maurer-Cartan
form $\subset$ Cartan connection. ) The Maurer-Cartan equation always holds
for any $\omega$. This definition of curvature is independent from the notion
of ”curvature” we use in the second-order derivative of the effective
potential $V_{eff}$: In fact, $V_{eff}$ is a function of local coordinates,
while a transition function between two local coordinate systems over
$V_{eff}$ is determined by the geometric structure of $V_{eff}$ itself, thus
the transition function cannot be defined group theoretically a priori (
mostly, $GL(n,{\bf R})$ ). The symmetry in the vicinity of a point defined
over $V_{eff}$ can be found from the curvature matrix as the second order
derivative of $V_{eff}$ since the mass matrix of NG sector reflects the
symmetry at a point. If the mass spectrum of NG sector has a degeneracy, then
the symmetry at the point becomes higher. Now the Cartan connection is an
affine connection of frame bundle ( i.e., a tangent bundle ) of the base
manifold, and also be interpreted as a special example of connection of
principal bundle. Due to the Cartan-Ambrose-Hicks theorem, a manifold is
locally Riemannian symmetric if and only if its curvature is constant, and for
any simply connected, complete locally symmetric space is Riemannian
symmetric. The expressions of a Lagrangian given in terms of $\omega=g^{-1}dg$
are familiar in theory of nonlinear realization [49]: The reason is needless
to say. For a case of a Clifford-Klein form $\Gamma\backslash G/H$ of a
symmetric space, the following parametrization will be introduced:
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\gamma
gh\Phi_{0},\quad\gamma\in\Gamma,\,g\in G,\,h\in H.$ (119)
Hence, a lattice element $\gamma$ is contained in the Maurer-Cartan form
$(\gamma gh)^{-1}d(\gamma gh)$. Here, as we have mentioned above, a lattice
$\gamma$ gives a symmetry of a set of stationary points of $V_{eff}$ over
$G/H$, and thus $\gamma$ implies an equivalence of physics of $(gh)^{-1}d(gh)$
and $(\gamma gh)^{-1}d(\gamma gh)$ in an NG boson Lagrangian ( a kind of
replication of a theory takes place ). Due to involutions and $\gamma$, the
global structure of the NG manifold in a symmetric space is well-
characterized. The part of mode-mode couplings in our Lagrangian is coming
from the cross terms of a product of similarity transformations in the above
expression (108). In fact, these expressions have the advantage to consider
some geometric nature of the anomalous NG theorem, since the manner of how a
Lorentz-symmetry-violating parameter couples with an internal symmetry of Lie
group will be explained by the words of geometry. Note that $\partial_{\nu}$
are generators of subalgebra of Poincaré algebra. It should be mentioned that,
when $\Phi$ is an order-parameter of Dirac mass type which spontaneously
breaks a chiral $\gamma_{5}$ symmetry ( coming from an underlying fermion
system ), a transform $\Phi\to e^{2i\gamma_{5}\theta_{5}}\Phi$ can be
considered in the Lagrangian, though it does not give a mode-mode coupling
between NG bosonic fields inside the Lagrangian. Hence, at least in our
definition of bosonic Lagrangian, the structure of mode-mode couplings of NG
bosons we have revealed in this paper is closed inside the internal symmetry
of a Lie group. ( Or, can say an internal symmetry can couple with a Poincaré
generator $\partial_{\nu}$ but cannot couple with the $\gamma_{5}$-rotation. )
On the other hand, there are couplings between the chiral
$\gamma_{5}$-symmetry and NG bosons in an explicit+dynamical symmetry breaking
in an NJL-type $SU(N)$ model with $N>2$ ( see [70] ). In the latter case, the
$\gamma_{5}$-chiral symmetry and the flavor symmetry are coupled with each
other through the chiral projection operator
$P_{\pm}=\frac{1\pm\gamma_{5}}{2}$. In Ref. [70], the author discussed that a
chiral mass can be handled as a Riemann surface. Since a
$\gamma_{5}$-transformation is vertical with the internal symmetry of $SU(2)$,
$\Phi$ gives a direct sum of Riemann surfaces. ( In the case of $SU(N)$ with
the case when $\Phi$ takes a fundamental representation ${\bf N}$ or
$\overline{\bf N}$, a direct sum of $N$ Riemann surfaces will be given. ) In
such a case, since some NG bosons become massive and the effective potential
$V_{eff}$ is periodic with respect to the directions of massive modes in our
anomalous NG theorem, some Galois-group symmetries of both $\gamma_{5}$ and
massive modes arise simultaneously in a theory. In other words, the theory
gives a Galois representation of those degrees of freedom ( $\gamma_{5}$,
massive modes ) simultaneously.
Toward the understanding of counting law of massive NG bosons, we will examine
the kinetic term of bosonic Lagrangian given above, especially by using the
words of Cartan geometry. The relevant part of mode-mode couplings of bosonic
fields is obtained from the Lie-algebra expansion of the Maurer-Cartan 1-form
$g^{-1}\partial_{0}g$, namely,
$\displaystyle\Phi^{\dagger}_{0}\Bigl{\\{}e^{-i\sum_{A}Q^{A}\chi^{A}}2i\mu\sum_{B}Q^{B}(\partial_{0}\chi^{B})e^{i\sum_{A}Q^{A}\chi^{A}}\Bigr{\\}}\Phi_{0}$
(120)
Here we have used the fact that $\Phi_{0}$ is a constant ( VEV ), has no
dependence on spacetime coordinates. This term can arise only by the presence
of the Lorentz-symmetry-violating parameter $\mu$ in the Lagrangian. After
expanding the exponential mappings and picking up the quadratic terms of
bosonic fields, we get
$\displaystyle(114)$ $\displaystyle=$
$\displaystyle-2\mu\sum_{A}\sum_{B}\chi^{A}(\partial_{0}\chi^{B})\Phi^{\dagger}_{0}[Q^{B},Q^{A}]\Phi_{0}$
(121) $\displaystyle=$ $\displaystyle
2\mu{\sum\sum}_{A>B}\Bigl{\\{}\chi^{A}(\partial_{0}\chi^{B})-\chi^{B}(\partial_{0}\chi^{A})\Bigr{\\}}\Phi^{\dagger}_{0}[Q^{A},Q^{B}]\Phi_{0}.$
This is a sum of off-diagonal part ( upper triangular part without the
diagonal elements ) of matrix elements with indices $A,B$. The number of non-
vanishing terms of this sum, namely the part
$\sum_{A>B}\Phi^{\dagger}_{0}[Q^{A},Q^{B}]\Phi_{0}=i\sum_{A>B}f^{ABC}\Phi^{\dagger}_{0}Q^{C}\Phi_{0}$,
counts the number of pairs of two modes they are coupled with each other.
Maximally the number of terms is $N(N-1)/2$ when $N={\rm dim}{\rm Lie}(G)$,
and thus the number coincides with the number of generators of $SO(N)$: Hence
the linear space of mode-mode coupling matrix can be expanded by the basis set
of Lie$(SO(N))$. Moreover, each term of this sum has a correspondence with the
Lie algebra valued linear equations given in our discussion by using the
second-order derivatives of the effective potential $V_{eff}$, namely Eqs.
(68), (72) and (75). The symplectic structure discussed in Ref. [81] is
obvious in our expression (115) due to the anti-symmetric nature of structural
constants $f^{ABC}$. Thus, our Lagrangian defines a symplectic manifold with
an appropriately defined symplectic structure. A linear transformation in the
matrix space of our Lagrangian gives an isomorphism of the symplectic
structure, possiblly continuously. Moreover, if a symplectic transformation of
our Lagrangian is given over a symplectic manifold, then it is expressed as a
symplectic Lie group. For example, in the case of $SU(2)\to U(1)$ of a
ferromagnet with broken generators $(Q^{1},Q^{2})$ and a symmetric generator
$Q^{3}$, only $if^{123}\Phi^{\dagger}_{0}Q^{3}\Phi_{0}$ remains to give a
finite VEV. If we choose the representation that gives $Q^{3}$ in the form of
diagonal matrix such as the Pauli matrix $\sigma^{3}$, then this term remains
when the VEV of $\Phi$ takes the form $\Phi_{0}=(v_{1},v_{2})$ ( $v_{1}\neq
v_{2}$ ). In this case, the number of pair of bosonic fields they are coupled
is 1. When the VEVs of $[Q^{A},Q^{B}]$ are pairwise decoupled, then the set of
VEVs gives a set of Heisenberg algebras, and then the mode-mode couplings in
the space of NG bosons $(\chi_{1},\cdots,\chi_{N})$ are also decoupled into
subspaces pairwisely, and the problem of the matrix is reduced into the direct
sum of $2\times 2$ matrices ( i.e., block-diagonalized ) such that
$\displaystyle\left(\begin{array}[]{cc}k^{2}&2ic_{1}\mu k_{0}\\\ -2ic_{1}\mu
k_{0}&k^{2}\end{array}\right)\oplus\cdots\oplus\left(\begin{array}[]{cc}k^{2}&2ic_{l}\mu
k_{0}\\\ -2ic_{l}\mu k_{0}&k^{2}\end{array}\right).$ (126)
In such a case, our discussion on dispersion relations of NG bosons is reduced
very much. The decomposition to $2\times 2$ matrices can also be interpreted
as the result that a Lie algebra is constructed by the fundamental unit
$sl_{2}$-triple, apparent from a Cartan decomposition. This is the origin of
the counting law of Watanabe, Brauner, and Hidaka: It is obvious from our
analysis, rank$\langle[X^{A},X^{B}]\rangle$ counts the number of pairs of
mode-mode coupling, and thus it can estimate the dimension of a matrix of non-
vanishing mode-mode coupling elements, though the pairwise decoupling must
take place to conclude definitely that the number of massive NG bosons is the
half of the rank. The cases of
$\displaystyle\langle h_{i}\rangle\neq 0,\quad\langle e_{j}\rangle\neq
0,\quad\langle f_{k}\rangle=0,$ (127)
and
$\displaystyle\langle h_{i}\rangle\neq 0,\quad\langle e_{j}\rangle\neq
0,\quad\langle f_{k}\rangle\neq 0,$ (128)
give more complicated situation for the counting law.
The adjoint orbit $O(X)={\rm Ad}(G)X$ ( $X\in{\rm Lie}(G)$ ) defines a
submanifold of ${\rm Lie}(G)$. A typical example is the Maurer-Cartan form
$g^{-1}dg$, and the curvature 2-form will be defines as a function of the
adjoint orbit. Needless to say, an NG manifold consists of adjoint orbits:
$\displaystyle O(\Phi)$ $\displaystyle=$ $\displaystyle g^{-1}\Phi g\simeq
e^{i\chi^{A}X^{A}}\Phi,\quad\Phi\in{\rm Lie}(G).$ (129)
After choosing the specific form of $\Phi\in{\rm Lie}(G)$ ( for example, when
$\Phi$ takes its value in the Cartan subalgebra of ${\rm Lie}(G)$ ),
$O(\Phi)=g^{-1}\Phi g$ gives a homogeneous space $G/G(\Phi)$, where $G(\Phi)$
is the stabilizer $\\{g\in G|{\rm Ad}(g)\Phi=\Phi\\}$. If $G$ is compact, an
adjoint orbit is called as an elliptic orbit.
Especially the case of $SU(2)\to U(1)$ gives a more explicit geometric
interpretation. Let us consider a situation where a one-dimensional curve is
defined over a two-dimensional oriented surface, and the surface is embedded
into a Euclidean space ${\bf R}^{3}$. Let ${\bf T}$ be the unit tangent vector
of the curve, let ${\bf N}$ be the unit normal vector of the surface, and
${\bf S=N\times T}$ is the tangent normal. The ${\bf T}$, ${\bf N}$ and ${\bf
S}$ gives the Darboux frame, an orthogonal frame. Let $C_{normal}$ be the
normal curvature, $C_{geodesic}$ be the geodesic curvature, and let
$T_{geodesic}$ be the geodesic torsion. It is a known fact that those
quantities are given by the following linear transformation ( the Frenet-
Serret equation ):
$\frac{d}{ds}\left(\begin{array}[]{c}{\bf T}\\\ {\bf S}\\\ {\bf
N}\end{array}\right)=\left(\begin{array}[]{ccc}0&C_{geodesic}&C_{normal}\\\
-C_{geodesic}&0&T_{geodesic}\\\
-C_{normal}&-T_{geodesic}&0\end{array}\right)\left(\begin{array}[]{c}{\bf
T}\\\ {\bf S}\\\ {\bf N}\end{array}\right).$ (130)
Apparently, the matrix of this linear transformation is a group element of
$SO(3)$. Hence the off-diagonal part of our Lagrangian of the anomalous NG
theorem in the $SU(2)$ case has this geometric implication, a curve on a
surface in ${\bf R}^{3}$. An important difference is that ${\bf S}$ is defined
by ${\bf T}$ and ${\bf N}$, while $(\chi_{1},\chi_{2},\chi_{3})$ are linearly
independent with each other: Namely, the projective case
$\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}=1$ corresponds to the Frenet-Serret
equation. A matrix element of the off-diagonal part corresponds to a curvature
or a torsion, and the local coordinates of $SU(2)$ ( i.e., the NG bosons )
gives an orthonormal frame. In other words, $[Q^{A},Q^{B}]$ define the local
geometry of the NG manifold. Those mathematical structure may be hidden in the
back ground of physics of a ferromagnet. Hence, in a case of pairwise
decoupling, only one of $C_{geodesic}$, $C_{normal}$, and $T_{geodesic}$
remains finite, which indicates that the three dimensional space
$(\chi_{1}.\chi_{2},\chi_{3})$ is decomposed into a one- and a two-dimensional
spaces, and the mixing of them along with the curve only takes place in the
two-dimensional subspace, and the one-dimensional subspace is inert. Our
interpretation on the Lagrangian of NG sector in $SU(2)$ is quite natural and
not surprising one, since the cross terms of the kinetic part
$(g^{-1}(d+\xi)g)(g^{-1}(d+\xi)g)$ contains the off-diagonal elements of the
Lagrangian matrix, and the chemical potential $\mu$ acts like a connection
$\xi$ inside the Lagrangian. More explicitly, the NG bosons
$(\chi_{1},\chi_{2},\chi_{3})$ form an orthogonal local coordinate system of
the $SU(2)$ group manifold, and the ”curvatures” and ”torsions” reflect the
geometric effect on the local coordinates $(\chi_{1},\chi_{2},\chi_{3})$
displaced by the Lagrangian. Namely, they measure how the curve generated by a
collective motion of NG bosons are distorted. This mathematical/physical
interpretation of our Lagrangian is more clarified if the Lagrangian formalism
is converted into a Hamiltonian of equations of motion. Hence, the dynamical
equation of NG bosons itself keeps this geometric nature. This interpretation
of the geometric implication of our NG-bosonic Lagrangian can apparently be
applied to more general case, for example, $SU(N)$. Namely the off-diagonal
matrix elements proportional to chemical potential $\mu$, which give the mode-
mode couplings between NG bosons and cause massive spectra of them, act as
curvatures/torsions to the local coordinate system ( i.e., the NG bosons ) of
$SU(N)$ Lie group manifold. In other words, the chemical potential $\mu$ gives
a measure of how much the group manifold ( more precisely, the NG manifold as
the submanifold of $SU(N)$ ) have the finite curvatures/torsions. Therefore,
the kinetic part ${\cal L}_{K}$ can be rewritten symbolically as
$\displaystyle{\cal L}_{K}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
tr}\Phi\bigl{(}(g^{-1}dg)^{2}+\mu\Omega_{a}T^{a}\bigr{)}\Phi,$ (131)
where, $\\{T_{a}\\}$ denote the Lie algebra of orthogonal group, $\Omega_{a}$
indicate curvature 2-forms. The NG bosons give a Darboux frame in general, an
example of moving frame, which is closely related with the Maurer-Cartan form.
It should be noticed that an NG bosons as a Darboux frame is always holds, no
matter the case of normal or anomalous NG theorem. While a curvature matrix
vanishes in a normal case, and it takes finite matrix elements in an anomalous
case. The Cartan’s method of moving frames is applied to study the local
structure of a homogeneous space $G/H$: Hence now we find something about the
local nature of an NG manifold of the anomalous NG theorem. If we employ a
normalization condition of the vector $(\chi_{1},\cdots,\chi_{N})$ ( namely a
unit vector $\chi^{2}_{1}+\cdots+\chi^{2}_{N}=1$ ), with taking a special
orthogonal group for the algebra $T^{a}$ of the above Lagrangian, then it
contains an isometry group, may be expressed as a Killing vector field. It is
a known fact that Killing vector fields form a Lie algebra. Moreover, a set of
Killing vector fields is related with a curvature tensor.
The part of mode-mode coupling terms for NG bosons caused by the chemical
potential $\mu$ in our Lagrangian, (104) or (108), is not pairwisely decoupled
in general, and thus it does not show a Heisenberg algebra apparently. At
least, via VEVs $\Phi^{\dagger}[Q^{A},Q^{B}]\Phi$, a subspace of the
$N$-dimensional space must be decomposed such as
$(1,2)\oplus(3,4)\oplus\cdots\oplus(M-1,M)$, ( $M<N$ must be satisfied ) to
show a Heisenberg algebra. Thus, we cannot conclude the VEVs of Eq. (115) are
always Heisenberg-type, depend on cases and breaking schema. While, due to the
Hermitian nature of Eq. (115) and any non-vanishing off-diagonal matrix
element in the momentum space takes pure-imaginary, and the number of
independent matrix elements are $N(N-1)/2$, the matrix, namely the mode-mode
coupling part of our Lagrangian can be expressed by the generators $L_{j}$ of
$SO(N)$ ( angular momenta of ${\rm Lie}(SO(N))$, $j=1,\cdots,N(N-1)/2$ ):
$\displaystyle(\tilde{\partial}_{\nu}\Phi)^{\dagger}(\tilde{\partial}_{\nu}\Phi)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\Psi\Bigg{\\{}{\rm
diag}(\partial^{2}_{\nu},\cdots,\partial^{2}_{\nu},\partial^{2}_{\nu}-M^{2})+i\sum^{N(N-1)/2}_{j=1}c_{j}L_{j}\Bigg{\\}}\Psi+\cdots,$
(132)
( $c_{j}$; coefficients ). These angular momenta $L_{j}$ in an $N$-dimensional
real Euclidean space ( locally ${\bf R}^{N}$ ) describe rotations of
coordinates, namely the NG bosons, on the group manifold $G$. Since a quantum
mechanical mixing of NG bosons takes place in our theory of anomalous NG
theorem, it is quite interesting that those NG bosons may give a multiplet
structure in their energy spectrum! Hence it might be possible to introduce a
weight space ${\cal V}=\oplus_{\lambda}{\cal V}_{\lambda}$, starting from the
highest weight. In a case of ferromagnet, the highest weight state of
Lie$(SU(2))$ is the eigenstate of the Heisenberg Hamiltonian, and
$S_{\pm}=S_{1}\pm iS_{1}$ provides the ladder operators. A similar situation
can take place in a Lie$(SU(N))$ model under our anomalous NG theorem.
Since the chemical potential $\mu$ takes a similar form with a zeroth
component of gauge field $A_{0}(x)$, we speculate a similar situation takes
place when the NG sector couples with gauge fields. Let us give a general
theory for understanding this situation. Let us write a generic differential
operator $\cal{D}_{\nu}$, which gives a covariant derivative in the sense of
gauge theory as its special case, and make a similarity transformation:
$\displaystyle{\cal D}_{\nu}(\\{Q^{A}\\})$ $\displaystyle=$
$\displaystyle\partial_{\nu}(\\{Q^{A}\\})+\delta{\cal D}_{\nu}(\\{Q^{A}\\}),$
(133) $\displaystyle\partial_{\nu}(\\{Q^{A}\\})$ $\displaystyle=$
$\displaystyle g^{-1}\partial_{\nu}g,$ (134) $\displaystyle\delta{\cal
D}_{\nu}(\\{Q^{A}\\})$ $\displaystyle=$ $\displaystyle g^{-1}{\cal B}g,$ (135)
$\displaystyle g$ $\displaystyle=$ $\displaystyle e^{iQ^{A}\chi_{A}}\in G,$
(136) $\displaystyle{\cal B}$ $\displaystyle=$ $\displaystyle{\cal
B}^{0}\hat{1}+{\cal
B}^{\alpha}\tau^{\alpha}=B+B_{\nu}+B_{\nu\mu}+B_{\nu\mu\rho}+\cdots.$ (137)
Here, the part of $\delta{\cal D}_{\nu}(\\{Q^{A}\\})$ denote the
”displacement” from the Maurer-Cartan 1-form caused by some explicit symmetry
breaking parameters or gauge fields. The Lie algebra $\tau^{\alpha}$ in which
the gauge fields $B_{\nu}$ take their values are in principle different from (
no relation with ) the broken generator $Q^{A}$. The matrix ${\cal B}$ is
considered as a set of Lorentz-symmetry-violating parameters with various
tensors. We do not consider any gravitational effect but it can be
incorporated. Then the kinetic term is assumed to take the following
expression defined over a bosonic field $\Phi$, and one can expand it:
$\displaystyle({\cal D}_{\nu}(\\{Q^{A}\\})\Phi)^{\dagger}({\cal
D}^{\nu}(\\{Q^{A}\\})\Phi)$
$\displaystyle\quad=\Phi^{\dagger}\Bigg{\\{}-\partial_{\nu}(\\{Q^{A}\\})\partial^{\nu}(\\{Q^{A}\\})-\partial_{\nu}(\\{Q^{A}\\})\delta{\cal
D}_{\nu}(\\{Q^{A}\\})$ $\displaystyle\qquad+\delta{\cal
D}_{\nu}(\\{Q^{A}\\})^{\dagger}\partial_{\nu}(\\{Q^{A}\\})+\delta{\cal
D}_{\nu}(\\{Q^{A}\\})^{\dagger}\delta{\cal
D}_{\nu}(\\{Q^{A}\\})\Bigg{\\}}\Phi.$ (138)
The mode-mode couplings between bosonic fields including the NG bosons are
coming from
$\displaystyle\Phi^{\dagger}\Bigl{[}-\bigl{(}\delta{\cal
D}_{\nu}(\\{Q^{A}\\})-\delta{\cal
D}_{\nu}(\\{Q^{A}\\})^{\dagger}\bigr{)}\partial_{\nu}(\\{Q^{A}\\})-\bigl{\\{}\partial_{\nu}(\\{Q^{A}\\})\delta{\cal
D}_{\nu}(\\{Q^{A}\\})\bigr{\\}}\Bigr{]}\Phi.$ (139)
If we restrict ${\cal B}$ as vector components ( connection ), then the
expression of the kinetic term is reduced into the following form by using the
Maurer-Cartan 1-form:
$\displaystyle{\cal L}_{K}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi\Bigl{(}(g^{-1}(d+A)g)(g^{-1}(d+A)g)\Bigr{)}\Phi$
(140) $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi\Bigl{(}g^{-1}(\partial\cdot\partial+\partial\cdot
A+A\cdot\partial+A^{2})g\Bigr{)}\Phi.$
Since the group element of broken symmetry $g$ has no relation with the gauge
field $A$, and thus $A$ can be taken as a scalar matrix proportional to a unit
matrix ( namely, an electromagnetic field ) for our present purpose. Then $A$
gives a similar effect with $\mu$ inside the Lagrangian: This is a remarkable
result since this can be experimentally confirmed, and it might be possible to
confirm our anomalous NG theorem by mesons of QCD under an electromagnetic
field ( external, static, constant, modulated ). One should notice that ${\cal
L}^{(2)}$ is manifestly gauge invariant. For example, if $A=(A_{0}={\rm
const}.,0,0,0)$, then it gives the same situation with a finite $\mu$.
Therefore, the chemical potential $\mu$ plays a similar role with an external
magnetic field in a spin system: Again, we have met with a phenomenological
similarity between our anomalous NG theorem and an explicit symmetry breaking.
Since NG bosons can have finite masses in our anomalous NG theorem, they have
finite ranges for their propagations similar to the Yukawa pion. ( The
correlation length is finite to the direction of a massless mode, while
diverges toward a massive mode. ) Therefore, their corresponding orderings in
a matter may be of short ranges: A long-range ordering observed by an
experiment realizes via a remaining massless NG boson. Since the range of
propagation of a massive NG boson is shorter than a massless one, the
interaction between amplitude-mode particles mediated by a massive NG bosons
also has a finite range. It indicates that there is an anisotropy in an
ordered state in a matter. In the case of a ferromagnet, an off-diagonal
element causes a mixing between $\chi^{1}$ and $\chi^{2}$ modes with the same
weight, and then it results a massive and a massless modes. Thus, there is no
anisotropy toward the $x$ and $y$ directions, the axial symmetry around the
$z$-axis is kept.
Let us consider an $SU(4)$ spin-orbital model which has been studied in
condensed matter physics [41], with assuming a ”ferromagnetic” ordering takes
place. In addition, the diagonal breaking, namely, all Lie algebra generators
except the Cartan subalgebra are broken is assumed. Thus, a conserved charge
$Q^{A}$ which belongs to the Cartan subalgebra taking a nonvanishing VEV. Now
rank(Lie$(SU(4))$) is 3, while dim(Lie$(SU(4))$) is 15, then the number of
Heisenberg-algebra-like pair is (15-3)/2=6. Furthermore, if the low-energy
Lagrangian of this system takes the similar form with the kaon condensation
model with the pairwise decomposition like (89) or (116), then the
ferromagnetic state has 6 massless and 6 massive NG bosons: Each mass
eigenstate belongs to a sextet, would be called as NG-boson multiplets. If
each Lagrangian of 6 pairs are the same form, then an $SO(6)\otimes SO(6)$
symmetry arises in the spectrum of the theory. It should be noticed that the
number of generators of this group is 15+15=30, larger than that of $SU(4)$.
The reason is that the Ginzburg-Landau-Goldstone-Higgs-Kibble-type $SU(4)$
Lagrangian is given in terms of a complex scalar field, while the NG bosons
are given by the real and imaginary part of the complex field. At the massless
limit, the mass spectrum has $SO(12)$ symmetry ( now the dimension of
Lie$(SO(12))$ is 66 ). Since a quasi-Heisenberg relation will be obtained in
this case, we predict a modification of the Heisenberg uncertainty relation
may be observed in the NG sector of this $SU(4)$ spin-orbital model. Hence,
Theorem: A diagonal breaking of $SU(N)$ under the situation of anomalous NG
theorem discussed here gives the $SO((N^{2}-N)/2)\otimes SO((N^{2}-N)/2)$
symmetry ( massive and massless ) in the mass spectrum of NG bosons. At the
massless limit, the spectrum of NG bosons has the $SO(N^{2}-N)$ symmetry.
Thus, phenomenologically, the anomalous NG theorem is understood as a symmetry
breaking $SO(N^{2}-N)\to SO((N^{2}-N)/2)\otimes SO((N^{2}-N)/2)$. The $SO(M)$
symmetry of the NG manifold means the isotropy of an effective potential
$V_{eff}$ at a point of the NG manifold where a Lie algebra is defined and
examined ( local ). The $SO(M)$ itself can be used globally as a structural
group of vector bundle constructed by NG boson fields.
It is noteworthy to mention that the author observed an $SO(2)$ symmetry
arises clearly in the case of pseudo-NG bosons of the flavor symmetry breaking
of $SU(2)\to U(1)$ in an explicit+dynamical symmetry breaking of an NJL-type
four fermion model [70]. The $SO(M)$ symmetry arised from an $SU(N)$ model is
already discussed by in Ref. [70]. The appearance of orthogonal Lie group
symmetry in an NG-boson sector is not yet examined enough in spite of its
importance.
The importance of orthogonal group symmetry in a mass spectrum of NG bosons in
our anomalous NG theorem is that they are coming from the eigenvalues of
geometric curvature matrix. In our case of the anomalous NG theorem, the
effective potential depends on the local coordinates ( i.e., the NG bosons )
of a Lie group, and thus the potential has a nonvanishing curvature in
general: The curvature reflects the dependence of $V_{eff}$ on those
coordinates. The global behaviors of those curvatures will have the
correspondence with the Lie group manifold and the NG manifold as its
subspace, though, several local properties of them must be distinguished. If
the mass spectrum of the NG sector has a symmetry ( degeneracy ) such as an
orthogonal group discussed above, then the curvature matrix of the effective
potential has a symmetry. In general, an NG sector has a degeneracy in the
mass spectrum, whether the situation is normal or anomalous.
By using a four-dimensional Heisenberg-type spin model, some similarity
between anomalous and explicit symmetry breakings would be understood. A
Heisenberg-like spin model ${\cal H}_{spin}$ is obtained via a Killing form:
$\displaystyle{\cal H}_{spin}\sim{\cal J}{\rm tr}(S^{a},S^{a})\sim{\rm
tr}({\rm Ad}(G),{\rm Ad}(G))\simeq{\rm tr}(g^{-1}dg,g^{-1}dg),$ (141)
( where, $S^{a}\in{\rm Lie}(G)$, and ${\cal J}$ implies an isotropic coupling
constant with respect to the indices of the Lie algebra ). Then, with
including an external field or a small perturbation parallel with a
ferromagnetic mean field ( which is taken to the third direction in the
following form ) which acts as a term of an explicit symmetry breaking
parameter, the Hamiltonian is
$\displaystyle\widetilde{\cal H}$ $\displaystyle\sim$ $\displaystyle{\cal
J}{\rm tr}(g^{-1}dg,g^{-1}dg)+{\cal H}_{ex+mf},$ (142) $\displaystyle{\cal
H}_{ex+mf}$ $\displaystyle\propto$ $\displaystyle S^{3}\sim(g^{-1}dg)_{3}+{\rm
h.c.}$ (143)
( h.c. means the Hermitian conjugate ). ${\cal H}_{ex+mf}$ contains both the
contributions of an external field and a mean ( ”molecular” ) field. After
taking a derivative expansion of spacetime coordinates, we yield a nonlinear
sigma model of ferromagnet defined over $G$: This observation is similar with
the case of explicit symmetry breaking. From the form of ${\cal H}_{ex+mf}$,
it is apparent for us that ${\cal H}_{ex+mf}$ gives a term which may have a
similar role of the chemical potential discussed by our model Lagrangian in
this section. It is interesting for us to consider the case when $G$ is an
exceptional Lie group, from the perspective of geometric property of our
anomalous NG theorem.
Usually, the Langevin equation formalism is utilized as the canonical approach
to study an irreversible process and a dynamical critical phenomenon. The
Langevin equation ( a stochastic differential equation [71,74] ) is given as a
first-order differential equation, which has its theoretical back ground in
theory of nonrelativistic Brownian motions. The diffusion equation which will
be obtained at the long-wave-length/hydrodynamic limit of a Langevin equation,
is also a nonrelativistic equation, of course: We never have met with a
relativistic diffusion equation which might belongs to the world of
hydrodynamics. ( Recently, an attempt toward a theory of relativistic Brownian
motions and a relativistic Langevin equation has been published [19].
Mathematically, we need a framework of relativistic stochastic differential
equation and relativistic Ito diffusion. ) Thus, currently there is an
essential difficulty to adopt our anomalous NG theorem to those
nonrelativistic theoretical frameworks. In other words, a difference between
relativistic and nonrelativistic cases of our anomalous NG theorem might be
found in some problems of dynamics.
In summary, a violation of Lorentz symmetry by a certain mechanism (
explicitly or spontaneously ) in a Lagrangian causes a modification of its
low-energy effective theory which describes NG bosons, then the subset of NG
bosons acquires masses. Probably, a certain type of deformation of a sigma
model Lagrangian can generically give a massive mode. Our Lagrangian can also
be generalized to supersymmetric nonlinear sigma models of several types: In
such a model, a massive NG fermion might appear simultaneously with a massive
NG boson. A supersymmetric theory frequently used in particle phenomenology
has a usual Lie group/algebra, thus it may be the case that we will consider a
usual Lie group/algebra ( not Lie supergroup/superalgebra ) to investigate an
anomalous behavior of NG theorem. For examples of supersymmetric field theory
with finite chemical potentials, see [68,69]: Our anomalous NG theorem can be
extended to SUSY cases via the results of these references. To find and
establish the counting law for SUSY cases is an important subject for particle
phenomenology.
### 5.1 Poincaré, Conformal, Super-Poincaré, Superconformal Groups and Some
Lie Groups in the Anomalous NG Theorem
Until now, we examine the anomalous NG theorem by the following logic: (1) The
Lorentz symmetry is broken in a theory, (2) then a special coupling between
elements of a Lie algebra of internal symmetry is caused via the Lagrangian,
(3) then a violation of the normal NG theorem takes place. We try to extend
this logic to (1’) Poincaré, conformal, super-Poincaré, or superconformal
symmetries are broken in a theory, (2’) then some couplings between Lie
algebras of an internal symmetry are caused inside the Lagrangian of a theory,
(3’) then a violation of the normal NG theorem takes place.
At least a formal discussion is quite easy. Let us consider the largest case,
a superconformal group, its superconformal algebra, and a ( semisimple ) Lie
group of internal symmetry of a theory. Then let us assume a Lagrangian in
which some generators of the superconformal algebra are broken
spontaneously/explicitly by a VEV or an explicit symmetry breaking parameter.
Then the Lagrangian is assumed to have a mode-mode coupling term of the Lie
algebra of internal symmetry via the explicit symmetry breaking parameter.
From this logic, it is clear for us that a Lie bracket which will be examined
for studying our anomalous NG theorem belongs to the Lie algebra of internal
symmetry. Let ${\cal L}$ a be Lagrangian, and let ${\cal Q}^{A}$ (
$A=1,\cdots,S$ ) be the Nöther charges of internal symmetries associated with
Lie groups, and $j^{A}$ the corresponding conserved Nöther currents. Then we
add $j^{A}$ to ${\cal L}$ as a Legendre transform:
$\displaystyle{\cal
L}(\Phi,\Phi^{\dagger})-\sum^{S}_{A=1}\mu^{A}j^{A}(\Phi,\Phi^{\dagger}).$
(144)
The multipliers $\mu^{A}$ are explicit symmetry breaking parameters ( of a
superconformal group ), conjugates of conserved currents $j^{A}$. Then we
obtain the NG boson Lagrangian of the quadratic part:
$\displaystyle{\cal L}^{(2)}\sim\frac{1}{2}{\rm
tr}\Phi_{0}\Bigg{\\{}(g^{-1}dg)^{2}-\sum^{S}_{A=1}\mu^{A}\langle\frac{\delta^{2}j^{A}}{\delta\Phi^{2}}\rangle\Bigg{\\}}\Phi_{0}+\cdots.$
(145)
The second term inside the curly bracket may cause model-mode couplings
between NG bosons. $\langle\cdots\rangle$ indicates a VEV. Since an analysis
on the algebraic structure of mode-mode couplings of NG bosons is examined by
VEVs of Lie algebra of an internal symmetry, certainly a quasi-Heisenberg
algebra arises also in a (super)conformal/Poincaré-violating case.
## 6 The Riemann Hypothesis and the Nambu-Goldstone Theorem: Toward the
Solution
Here, we discuss an interesting aspect of mathematical implication of the
Nambu-Goldstone theorem to the Riemann hypothesis, the Bost-Connes model [7],
and class field theory [54]. In fact, when we consider an explicit+dynamical
symmetry breaking [70], the mathematical structure of the NG theorem acquires
a viewpoint closely related with the mechanism of the phenomena of the Riemann
hypothesis. This fact implies us a natural solution=proof on the Riemann
hypothesis, which has been unsolved 154 years, might be found along with the
direction of the mathematical structure of the NG theorem. ( In this paper, we
do not discuss a possible way toward the solution of the Riemann hypothesis,
which remains for our future efforts. ) Since an explicit+dynamical symmetry
breaking and our anomalous NG theorem share some similarities, we consider
here this problem.
In the paper of Connes and Marcolli [14], they discussed that the physical
implication of some algebra of the Bost-Connes model is understood by a phase
factor ( which takes a similar form to a coherent state representation ) which
takes its form as the $N$-th root of unity. Besides the ordinary Bost-Connes
model, the ”generated” cyclotomic field associated with a spontaneous symmetry
breaking should take place in quantum field theory, i.e., a system of an
infinite number of dynamical degrees of freedom. A quantum field theory is
usually defined over ${\bf R}^{n}$ or ${\bf C}^{n}$ with some quantum numbers
associated with symmetries of the theory, while a cyclotomic field is a Galois
extension of ${\bf Q}$. This fact implies that a model which generates a
Galois group ”effectively” changes a number field via a certain mechanism or a
functor, associated with a change of topology and cardinality of a number
field as a base space of the system. This phenomenon is quite often observed
also in the NG theorem, both its generalization [70] and our anomalous NG
theorem: This is the starting point of our discussion toward the mechanism of
the phenomena of the Riemann hypothesis. In our NG theorem, we can consider a
coset $({\bf Z}/N{\bf Z})\backslash G$, and the bosonic field is given by
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\zeta_{N}g\Phi_{0},\quad\zeta_{N}=e^{2\pi i/N},\,g\in G.$ (146)
Then the Lagrangian will be constructed by the formalism of nonlinear
realization [49]. In this case, the Galois group symmetry, a cyclotomic
extension, is introduced implicitly in the kinetic part ( the Killing form )
of the Lagrangian, while the potential/mass term may contain the Galois
symmetry explicitly. The $\zeta_{N}$ as a phase factor of a wavefunction will
vanish inside the Maurer-Cartan form and the Killing form ( since of course
$\zeta_{N}$ does not have a spacetime dependence ). Therefore, when
$\zeta_{N}$ is introduced to a theory explicitly, the theory acquires
something beyond the framework of Cartan geometry constructed by the 1-form
$g^{-1}dg$. It is noteworthy to mention that the quantity $\zeta_{N}=e^{2\pi
i/N}$ takes its value in a unit circle of ${\bf C}$, while ${\bf Z}/N{\bf Z}$
is arised as a symmetry defined by the quantity. This quite simple fact
indicates us that the current issue certainly occupy its place in the
mechanism of spontaneous symmetry breaking. Moreover, the symmetry ${\bf
Z}/N{\bf Z}$ is the symmetry of several vacua given by a theory ( as argued in
the ordinary Bost-Connes model ), while it cannot be an NG bosonic mode if we
keep ourselves inside the ordinary NG theorem: The statement of
ordinary/normal NG theorem translated by the words of effective potential is
that a spontaneous symmetry breaking gives a flat direction toward a local
coordinate of broken generator of a Lie group. In other words, all points on
the NG manifold are equivalent. While, such a discrete symmetry can be
obtained via the generalized NG theorem [70], or our anomalous NG theorem with
a breaking of equivalence between points of an NG manifold. Of course, we can
introduce $({\bf Z}/N{\bf Z})\backslash G$ as the structural group of a fiber
bundle ( for example, defined by a Higgs field ) of a theory.
Let $M$ be a homogeneous space. Then a study on the $k$-rational points in $M$
is the problem of Galois cohomology ( $k$: a field ). In general, a cohomology
of group studies a set of fixed points under group actions. Let $A$ be an
Abelian module, and assume $Gal(K/k)$ acts on $A$. Then the Galois cohomology
group is
$\displaystyle H^{n}(Gal(K/k),A),\quad n\geq 0.$ (147)
It is defined by the complex $(C^{n},d)$, where $C^{n}$ consists with all maps
$Gal(K/k)^{n}\to A$, and $d$ is the coboundary operator. If $A$ is a non-
Abelian case, only the zero-dimensional $H^{0}$ and one-dimensional $H^{1}$
cohomology can be defined. In that case, $H^{0}(Gal(K/k),A)=A^{Gal(K/k)}$ is
the set of fixed points under the action of $Gal(K/k)$ in A: Thus, the
invariant set $A^{Gal(K/k)}$ gives a representation of the Galois symmetry.
For example, in an explicit+dynamical symmetry breaking of a $U(1)$ group, an
embedding $A^{Gal({\bf Q}(\zeta_{N})/{\bf Q})}\to A^{U(1)}$ takes place by the
set of stationary points obtained from $V_{eff}$ [70]. This type of symmetry
will be discussed later, in our discussion on the relation between the NG
theorem and the Bost-Connes model, and the Riemann hypothesis. In fact, the
generalized NG theorem of explicit+dynamical symmetry breaking given by the
author in Ref [70] has some examples where the ground state of a quantum field
theory spontaneously acquires a Galois symmetry. In such a case, a set of
discrete vacua arises from the symmetry breaking, and they are in fact the
invariant subset $A^{Gal(K/k)}$ embedded in $A$. Here, $A$ is the NG manifold
expanded by the broken local coordinates coming from a subspace of the group
manifold. Hence, the set of discrete vacua of the generalized NG theorem gives
a Galois representation. Similar situation is realized in our anomalous NG
theorem, since it gives a massive NG mode, and the effective potential is
lifted along with the local coordinate ( i.e., the NG boson ) then the
effective potential must be periodic in the direction of local coordinate if
the Lie group is compact: This is a dynamical mechanism for generating a
Galois representation in quantum field theory. If we take a Maurer-Cartan form
from the group element $\zeta_{N}g$, then $\zeta_{N}$ will be canceled inside
the Maurer-Cartan form. Thus, a curvature 2-form derived from the Maurer-
Cartan form, a characteristic class [59] evaluated from the 2-form, and also
the kinetic part of the Lagrangian, i.e., a Killing form, cannot contain any
information of the Galois symmetry. This fact implies that it is difficult to
express a Galois symmetry by the modern differential-geometric setting. While
the mass ( potential energy ) term of a Lagrangian can explicitly give a
Galois symmetry such that
$\displaystyle V(\zeta_{N},g)$ $\displaystyle\propto$
$\displaystyle\zeta_{N}g\Phi+\Phi^{\dagger}g^{\dagger}\zeta^{*}_{N}.$ (148)
Let us show another perspective on the relation between the Bost-Connes model
and our anomalous NG theorem. The Hamiltonian $H_{BS}$ and the partition
function $Z_{BS}$ of the Bost-Connes-type model are defined as follows:
$\displaystyle H_{BS}$ $\displaystyle=$ $\displaystyle\ln N,$ (149)
$\displaystyle Z_{BS}$ $\displaystyle=$ $\displaystyle{\rm Tr}e^{-\beta
H_{BS}},$ (150)
( $\beta$; inverse temperature ). $N$ is the number operator of one-flavor
bosonic field, and thus $H_{BS}$ is defined by a Heisenberg algebra.
Especially, $N$ can be regarded as ( a part of ) the second-order Casimir
element ( Laplacian, the center of the universal enveloping algebra ${\cal
U}({\rm Lie}(G))$ ) of the Heisenberg algebra. It is known fact from the
result of Beilinson and Bernstein that there is a categorical correspondence
between the category of coherent $D$-modules and the category of finitely
generated ${\cal U}({\rm Lie}(G))$-modules with a certain condition given by
the center of ${\cal U}({\rm Lie}(G))$ [5]. Thus, the representation problem
of ${\rm Lie}(G)$ in our case discussed here can be translated to the problem
of $D$-modules. Let us consider, for example, our anomalous NG theorem of
$SU(2)\to U(1)$ of a ferromagnet. Let $a$ be an annihilation operator of the
Bost-Connes mode, and let $a^{\dagger}$ be its Hermitian conjugate. Then,
needless to say, we have the Heisenberg algebra $(a,a^{\dagger},c)$,
$[a,a^{\dagger}]=c$, $[a,c]=[a^{\dagger},c]=0$. By comparing this algebra with
the Lie$(SU(2))$ algebra, we find/set the correspondence $a\leftrightarrow
S_{1}$, $a^{\dagger}\leftrightarrow S_{2}$, $c\leftrightarrow S_{3}$ from the
context of our anomalous NG theorem, generating a Heisenberg algebra from the
Lie$(SU(2))$ algebra. Therefore we find
$\displaystyle
N\sim\frac{1}{2}(a^{\dagger}a+aa^{\dagger})\simeq\frac{1}{2}(S_{1}S_{2}+S_{2}S_{1}).$
(151)
Namely, $N$ is expressed in somewhat similar form of an $XY$-spin model (
$H_{XY}=\sum S_{x}(i)S_{x}(i\pm 1)+S_{y}(i)S_{y}(i\pm 1)$ ). The
algebraic/operator structure of $N$ given as a quadratic form of bosonic
operators might be interpreted as a non-interacting bosonic system, though we
can say $(a,a^{\dagger})$ are given from a Hartree-Fock-Bogoliubov mean field
theory. The crucial point is that the Hamiltonian is diagonalizable against a
Fock space, and the notion of occupation number is well-defined. From the
context of our anomalous NG theorem, this form gives a mode-mode coupling of
broken generators $(S_{1},S_{2})$ in the breaking scheme $SU(2)\to U(1)$ of a
ferromagnet. We can generalize our statement. Let ${\bf g}={\rm Lie}(G)$, and
decompose it as ${\bf g}={\bf h}+{\bf m}={\bf h}\oplus_{\alpha\in R}{\bf
g}_{\alpha}={\bf h}\oplus{\bf e}\oplus{\bf f}$. Then, at least in a case of
diagonal breaking scheme, we have the correspondence of Heisenberg and Lie
algebras as follows:
$\displaystyle N$ $\displaystyle\sim$ $\displaystyle{\cal C}({\rm Lie}(G)),$
(152) $\displaystyle{\cal C}({\rm Lie}(G))$ $\displaystyle\sim$
$\displaystyle\sum(e_{i}f_{i}+f_{i}e_{i})$ (153) $\displaystyle=$
$\displaystyle\sum(g_{\alpha}\otimes g_{-\alpha}+g_{-\alpha}\otimes
g_{\alpha})\in{\rm tr}({\bf m}\otimes{\bf m})$ $\displaystyle\sim$
$\displaystyle{\rm tr}\Bigl{[}(g^{-1}dg)_{\bf m}\otimes(g^{-1}dg)_{\bf
m}\Bigr{]}.$
A Weyl group implicitly acts on the Casimir element, which will also be
reflected to enforce a specification/restriction of the algebraic form of our
interpretation of the Bost-Connes model. One should notice that the part
$\sum(e_{i}f_{i}+f_{i}e_{i})$ is just the Casimir element of the universal
enveloping algebra of Lie$(G)$. Thus, we can write
$\displaystyle\zeta(\beta)$ $\displaystyle=$ $\displaystyle{\rm Tr}e^{-\beta
H}={\rm Tr}e^{-\beta\ln{\cal C}({\rm Lie}(G))}={\rm Tr}({\cal C}({\rm
Lie}(G)))^{-\beta}.$ (154)
Here, $\zeta$ is the Riemann zeta function. Especially in the case where a
symmetric space is spontaneously generated, we can write
$\displaystyle H^{G/H}_{BS}$ $\displaystyle=$ $\displaystyle\ln\bigl{[}{\rm
tr}({\bf m}\otimes{\bf m})\bigr{]}\simeq\ln\bigl{[}{\rm tr}(T_{e}(G/H)\otimes
T_{e}(G/H))\bigr{]},$ (155) $\displaystyle\zeta^{G/H}(\beta)$ $\displaystyle=$
$\displaystyle Z={\rm Tr}\bigl{[}{\rm tr}(T_{e}(G/H)\otimes
T_{e}(G/H))\bigr{]}^{-\beta}.$ (156)
This might be understood as a generalization of Riemann zeta function. Namely,
it is given by a trace of direct product of adjoint orbits or tangent spaces.
Therefore, our interpretation/generalization of the Bost-Conne-like model
resembles with the notion of dynamical zeta function [76], and also a chiral
perturbation theory. From our discussion given here, we can say a Riemann zeta
function is a function of a sum of the number of quantum states caused by
mode-mode couplings in our anomalous NG theorem. How a Galois symmetry of
cyclotomic extension will be found? In the case $SU(2)\to U(1)$ of a
ferromagnet of our anomalous NG theorem, the ground state of $V_{eff}$ of the
system is defined over a two-dimensional local coordinate system, where one is
”massless” and $V_{eff}$ is flat along with this direction, while another
direction is ”massive” and has a finite curvature, and $V_{eff}$ shows a
periodicity along with the massive direction since $SU(2)$ is compact. Then
the set of discrete vacua gives a Galois symmetry, $Gal({\bf
Q}(\zeta_{N})/{\bf Q})$. Therefore, a Galois symmetry arises from the symmetry
of several vacua of the theory, while our Bost-Connes-type model is evaluated
as a kind of ”invariant” or a ”character” of the theory in our case: This
point is different from the ordinary Bost-Connes model, in which the Riemann
zeta function arises as the partition function of the model itself, and the
cyclotomic Galois symmetry is the symmetry of the vacuum states of the model.
Since the effective potential $V_{eff}$ of the case $SU(2)\to U(1)$ of a
ferromagnet gives a set of discrete vacua, it defines a lattice of the
generated Heisenberg algebra, ${\bf Z}\otimes X^{a}$ ( $X^{a}$: the basis of
Lie algebra ). Our discussion is summarized as the following diagram:
Normal/generalized/anomalous NG theorem in quantum field theory $\to$ NG boson
Lagrangian/Hamiltonian $\to$ Heisenberg algebra, residual symmetry between
several vacua ( periodicity ) $\to$ Bost-Connes-type model, Galois symmetry
$\to$ the Riemann zeta function.
Our formalism of the Bost-Connes-type Hamiltonian by a Lie algebra can be
extended to a case of Kac-Moody algebra. ( Someone might recall the Shintani-
Witten zeta function from our result given above, but it is quite different. )
It can be stated that the Boltzmann factor $e^{-\beta H}$ is a kind of
exponential mapping of the Heisenberg algebra: Namely, the trace ( sum ) of
the exponential mappings of the universal enveloping algebra of the Heisenberg
algebra with an appropriate Hilbert space gives the Riemann zeta function.
Thus, the Boltzmann factor $e^{-\beta H}$ is a kind of globalization (
analytic continuation ) of a Lie algebra ( a tangent space at the origin )
from a geometric point of view. This simple observation is remarkable, since
such a globalization can be achieved only by a non-compact Lie group, to
acquire the continuation of the whole part of Gaussian plane ${\bf C}$ from
the perspective of the Riemann hypothesis. ( The set of zeroes of $\zeta$ is
non-compact. ) Our result is summarized by the following diagram:
Lie algebra, or a central extension of symplectic algebra $\to$ quasi-
Heisenberg algebra $\to$ Casimir element of the universal enveloping algebra
$\to$ continuation to the whole part of the Gaussian plane via the trace of
exponentiation of the logarithmic function of the Casimir element $\to$ the
Riemann zeta function, the Riemann hypothesis.
We will give the following theorem:
Theorem: A dynamical/spontaneous generation of a Heisenberg algebra of our
anomalous NG theorem of quantum field theory gives a Riemann zeta function via
the prescription of the Bost-Connes model. A bosonic Fock space is associated
with the Riemann zeta function automatically.
We would like to give some comments here. The famous Deligne-Lusztig theory
[17] is defined for a finite reductive group under applying a Frobenius
endomorphism, and thus it is not exactly the same with an $p$-adic analog of
local coordinates of a Lie algebra/group sometimes obtained in our
generalized/anomalous NG theorem. For example, for $SL(n,{\bf K})$ ( ${\bf
K}=\overline{\bf F}_{p}$ ), a Frobenius endomorphism $F:x_{ij}\to x^{q}_{ij}$
( $x_{ij}$: matrix elements, $q=p^{a}$, $a\in{\bf N}$ ) is applied and then
yield the finite reductive group $SL(n,{\bf F}_{q})$. While, in our case, we
will consider, for example, ${\bf F}_{q}\otimes{\bf g}$ or ${\bf
F}_{q}\otimes({\bf h}\oplus_{\alpha\in R}{\bf g}_{\alpha})$ ( ${\bf g}\in{\rm
Lie}(G)$ ), namely, a so-called Lie algebra lattice. From this aspect, the
geometry of a set of discrete stationary points is closer to arithmetic
geometry.
As we have discussed in the previous section, Lie$(SU(2))$ and the
corresponding Heisenberg algebra define curves. Due to $SU(2)\simeq SO(3)$,
the Casimir element of Lie$(SU(2))$ corresponds to that of Lie$(SO(3))$, i.e.,
$L^{2}=L^{2}_{x}+L^{2}_{y}+L^{2}_{z}$ as the magnitude of three-dimensional
angular momentum. Then we recognize that the Riemann zeta function of $SU(2)$
is expressed by $L^{2}$.
The general theory of Galois representation is as follows: Let $G$ be a
profinite group ( a typical example is a Galois group ), let $R$ be a locally
compact topological ring, and let $M$ be a finitely generated $R$-module. Then
one considers the following continuous homomorphism [22,29,33,80],
$\displaystyle\rho:G\to{\rm Aut}_{R}(M).$ (157)
This morphism is called as a linear representation of $G$. In our case, the
set of stationary points as fixed points of a Galois group gives an example of
$M$. Moreover, if the rank of $M$ is $n$ over $R$ ( $n=2$ in the case of
elliptic curve ), then
$\displaystyle\rho:G\to{\rm Aut}_{R}(M)\simeq GL(n,R)=\\{g\in
M_{n}(R)|\det(g)\in R^{\times}\\}$ (158)
is obtained. From this aspect, an automorphism of the set of stationary points
of the space of the NG bosonic coordinates gives a possibility toward a Galois
representation theory. It is possible to choose $U(n)$, $O(n)$ or $Sp(n)$ as
$GL(n,R)$ by adopting an appropriate condition ( algebraic structure ) in $M$.
All of the notions of decomposition group, inertia group, Frobenius morphism,
unramified/ramified, …, consider corresponding invariant sets under their
group actions [22,29,33,80]. For example, the cyclotomic extension of $U(1)$
case has an isomorphism with a finite field ${\bf F}_{q}$. Thus, those tools
of Galois representations and Galois cohomology ( hence, class field theory )
will be introduced into the framework of our NG theorem. This can be
understood by the fact that a Galois theory studies a symmetry of a number
field. In practice in number theory, usually one has to introduce a geometric
object such as elliptic curves or Abelian varieties, and an examination of a
geometric object by the method of étale cohomology gives a concrete example of
a Galois representation, especially an $l$-adic representation ( so-called
$p\neq l$ case ) [22,29,33,53,80]. In our NG theorem, a set of stationary
points ( vacua ) corresponds to a geometric object in the above prescription:
A set of stationary points give a cyclotomic extension, and it is an example
of Abelian extension due to the Kronecker-Weber theorem [54,85], then we yield
the $n=1$ case of (148).
Our perspective is summarized in the following diagram:
normal/generalized/anomalous NG theorem $\to$ class field theory, adele, idele
$\to$ Langrands correspondence.
## 7 Concluding Remarks
In conclusion, we have studied the mechanism, the counting law of the number
of true NG bosons, geometric and number theoretical aspects, of the anomalous
NG theorem. We have established the counting law of true NG bosons of the
diagonal breaking scheme of the anomalous NG theorem from several approaches,
while a more general case remains as an open question: Probably, from our
several observations in this paper, it seems not easy to give a general
formula/law of generic breaking schema in the anomalous NG theorem. Namely, it
seems the case that there is no universal counting law which is always valid
to any type of symmetry breaking scheme of the anomalous NG theorem. While,
our several results of formalisms, geometry of Lie algebras and Lie groups in
the anomalous NG theorem, Lagrangian and the effective potential show their
universality.
We have presented a generic Lagrangian which has a Lorentz-violating
parameter, which gives our anomalous NG theorem. Kostelecky et al. study on
Lorentz and CPT violations intensively, as a fundamental physics, especially
from the context of neutrino phenomenology [18,47]. It is interesting for us
to find some applications of our result in theory of Lorentz/CPT violations.
In a Poincaré invariant theory, Lorentz and CPT symmetries are deeply related
with each other. Thus, our anomalous NG theorem would be restricted by CPT
symmetries to apply it to several examples.
We have another interesting issue we will consider in the next step. In
several well-known substances ( metals ), some quantum fluctuations they may
be described as NG modes still survive temperature regions over $T_{c}$. Our
anomalous NG theorem could be applied to such situations with giving new
aspects to understand a mechanism of ordering in substances. To describe such
physically/experimentally observed situations, we can utilize several
mathematical and physical methods such as the Maurer-Cartan form and Cartan
geometry, the Stone-von Neumann theorem and Heisenberg
manifolds/groups/algebras, submanifold geometry and topology ( since we have
found the fact that there is an interaction between a submanifold and its
complement ), quantum uncertainties, quantum fluctuations and quantum critical
phenomena in quantum phase transitions.
Now we have arrived at the stage to modify/improve the traditional statement
of the NG theorem in nonrelativistic/Lorentz-violated systems, given usually
in several literatures, such as the paper of F. Strocchi [79]. The first
modification is to the usual statement that it argues the one-to-one
correspondence between broken generators and the NG bosons with vanishing
masses. In our case, the space of broken generators is ”reduced”, i.e.,
projected into a space of smaller dimensions. From our result given in this
paper, the notion of symmetry breaking is formulated as the following formal
statement ( see the theorem given below ). Usually, one employs the formalism
of axiomatic field theory in literature, namely, (a) the definition of local
quantum field theory, (b) the definition of spontaneous symmetry breaking ,
(c) the nonrelativistic NG theorem, (d) the relativistic NG theorem, in which
some restrictions is applied to the nonrelativistic formalism, especially due
to the definition of conserved charge: In a nonrelativistic case, a three-
dimensional support is used to define the integration domain of a Nöther
current, while a four-dimensional support will be prepared in a relativistic
case. Beside those delicate questions, in our formal statement, we do not need
to distinguish relativistic and nonrelativistic cases seriously since the
essential mechanism of anomalous NG theorem is the same between them.
The definition of spontaneously broken symmetry is improved to be:
Theorem: Let $\beta$ be an internal symmetry described as a Lie group
automorphism of an algebra $\cal{A}$, which commutes with any spacetime
translations. After a symmetry breaking takes place, $\beta$ is fall into a
representation $A\in\pi(\cal{A})$ and $\langle\beta{A}\rangle\neq\langle
A\rangle$ modulo a lattice of Lie group, and $\langle\beta{A}\rangle=\langle
A\rangle$ when $\beta$ coincides with a lattice.
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|
arxiv-papers
| 2013-12-01T05:35:34 |
2024-09-04T02:49:54.740662
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tadafumi Ohsaku",
"submitter": "Tadafumi Ohsaku",
"url": "https://arxiv.org/abs/1312.0916"
}
|
1312.0923
|
# Stanley depth on five generated, squarefree, monomial ideals
Dorin Popescu [email protected] Dorin Popescu, Simion Stoilow Institute
of Mathematics of Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest
014700, Romania
###### Abstract.
Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra
over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$
is either generated by four squarefree monomials of degrees $d$ and others of
degrees $\geq d+1$, or by five special monomials of degrees $d$. If the
Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq
d+1$ too.
Key words : Monomial Ideals, Depth, Stanley depth.
2010 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.
## Introduction
Let $K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ be the polynomial $K$-algebra
in $n$ variables. Let $I\supsetneq J$ be two squarefree monomial ideals of $S$
and suppose that $I$ is generated by squarefree monomials of degrees $\geq d$
for some positive integer $d$. After a multigraded isomorphism we may assume
either that $J=0$, or $J$ is generated in degrees $\geq d+1$.
Let $P_{I\setminus J}$ be the poset of all squarefree monomials of $I\setminus
J$ with the order given by the divisibility. Let $P$ be a partition of
$P_{I\setminus J}$ in intervals $[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$,
let us say $P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint.
Define $\operatorname{sdepth}P=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and the Stanley depth of $I/J$ given by
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{P}\operatorname{sdepth}P$,
where $P$ runs in the set of all partitions of $P_{I\setminus J}$ (see [3],
[19]). Stanley’s Conjecture says that
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$.
In spite of so many papers on this subject (see [3], [10], [17], [1], [4],
[18], [11], [7], [2], [12], [16]) Stanley’s Conjecture remains open after more
than thirty years. Meanwhile, new concepts as for example the Hilbert depth
(see [1], [20], [5]) proved to be helpful in this area (see for instance [18,
Theorem 2.4]). Using a Theorem of Uliczka [20] it was shown in [8] that for
$n=6$ the Hilbert depth of $S\oplus m$ is strictly bigger than the Hilbert
depth of $m$, where $m$ is the maximal graded ideal of $S$. Thus for $n=6$ one
could also expect $\operatorname{sdepth}_{S}(S\oplus
m)>\operatorname{sdepth}_{S}m$, that is a negative answer for a Herzog’s
question. This was stated later by Ichim and Zarojanu [6].
Suppose that $I\subset S$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degrees $d$, and a set $E$ of squarefree monomials of
degree $\geq d+1$. By [3, Proposition 3.1] (see [12, Lemma 1.1]) we have
$\operatorname{depth}_{S}I/J\geq d$. Thus if $\operatorname{sdepth}_{S}I/J=d$
then Stanley’s Conjecture says that $\operatorname{depth}_{S}I/J=d$. This is
exactly what [12, Theorem 4.3]) states. Next step in studying Stanley’s
Conjecture is to prove the following weaker one.
###### Conjecture 1.
Suppose that $I\subset S$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degrees $d$, and a set $E$ of squarefree monomials of
degree $\geq d+1$. If $\operatorname{sdepth}_{S}I/J=d+1$ then
$\operatorname{depth}_{S}I/J\leq d+1$.
This conjecture is studied in [14], [15], [16] either when $r=1$, or when
$E=\emptyset$ and $r\leq 3$. Recently, these results were improved in the next
theorem.
###### Theorem 1.
(A. Popescu, D.Popescu [9, Theorem 0.6]) Let $C$ be the set of the squarefree
monomials of degree $d+2$ of $I\setminus J$. Conjecture 1 holds in each of the
following two cases:
1. (1)
$r\leq 3$,
2. (2)
$r=4$, $E=\emptyset$ and there exists $c\in C$ such that
$\operatorname{supp}c\not\subset\cup_{i\in[4]}\operatorname{supp}f_{i}$.
The purpose of this paper is to extend the above theorem in the following
form.
###### Theorem 2.
Let $B$ be the set of the squarefree monomials of degree $d+1$ of $I\setminus
J$. Conjecture 1 holds in each of the following two cases:
1. (1)
$r\leq 4$,
2. (2)
$r=5$, and there exists $t\not\in\cup_{i\in[5]}\operatorname{supp}f_{i}$,
$t\in[n]$ such that $(B\setminus E)\cap(x_{t})\not=\emptyset$ and
$E\subset(x_{t})$.
The above theorem follows from Theorems 3, 4 (the case $r=4$, $E=\emptyset$ is
given already in Proposition 2). It is worth to mention that the idea of the
proof of Proposition 2, and Theorem 1 started already in the proof of [16,
Lemma 4.1] when $r=1$. Here path is a more general notion, the reason being to
suit better the exposition. However, the case $r=4$, $E\not=\emptyset$ is more
complicated (see Remark 8) and we have to study separately the special case
when $f_{i}\in(v)$, $i\in[4]$ for some monomial $v$ of degree $d-1$ (see the
proof of Theorem 3).
What can be done next? We believe that Conjecture 1 holds, but the proofs will
become harder with increasing $r$. Perhaps for each $r\geq 5$ the proof could
be done in more or less a common form but leaving some ”pathological” cases
which should be done separately. Thus to get a proof of Conjecture 1 seems to
be a difficult aim.
We owe thanks to a Referee, who noticed some mistakes in a previous version of
this paper, especially in the proof of Lemma 3.
## 1\. Depth and Stanley depth
Suppose that $I$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degrees $d$ for some $d\in{\mathbb{N}}$ and a set of
squarefree monomials $E$ of degree $\geq d+1$. Let $B$ (resp. $C$) be the set
of the squarefree monomials of degrees $d+1$ (resp. $d+2$) of $I\setminus J$.
Set $s=|B|$, $q=|C|$. Let $w_{ij}$ be the least common multiple of $f_{i}$ and
$f_{j}$ and set $W$ to be the set of all $w_{ij}$. Let $C_{3}$ be the set of
all $c\in C\cap(f_{1},\ldots,f_{r})$ having all divisors from $B\setminus E$
in $W$. In particular each monomial of $C_{3}$ is the least common multiple of
three of the $f_{i}$. The converse is not true as shown by [9, Example 1.6].
Let $C_{2}$ be the set of all $c\in C$, which are the least common multiple of
two $f_{i}$, that is $C_{2}=C\cap W$. Then $C_{23}=C_{2}\cup C_{3}$ is the set
of all $c\in C$, which are the least common multiple of two or three $f_{i}$.
We may have $C_{2}\cap C_{3}\not=\emptyset$ as shows the following example.
###### Example 1.
Let $n\geq 4$, $f_{i}=x_{i}x_{i+1}$, $i\in[3]$, $f_{4}=x_{1}x_{4}$ and
$I=(f_{1},\ldots,f_{4})$, $J=0$. Note that $m=x_{1}x_{2}x_{3}x_{4}$ is a least
common multiple of every three monomials $f_{j}$ and the divisors of $m$ with
degree $3$ are $w_{12},w_{23},w_{34},w_{14}$. Thus $m\in C_{3}$. But $m\in
C_{2}$ because $m=w_{13}=w_{24}$.
We start with a lemma, which slightly extends [9, Theorem 2.1].
###### Lemma 1.
Suppose that there exists $t\in[n]$,
$t\not\in\cup_{i\in[r]}\operatorname{supp}f_{i}$ such that $(B\setminus
E)\cap(x_{t})\not=\emptyset$ and $E\subset(x_{t})$. If Conjecture 1 holds for
$r^{\prime}<r$ and $\operatorname{sdepth}_{S}I/J=d+1$, then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
We follow the proof of [9, Theorem 2.1]. Apply induction on $|E|$, the case
$|E|=0$ being done in the quoted theorem. We may suppose that $E$ contains
only monomials of degrees $d+1$ by [14, Lemma 1.6]. Since Conjecture 1 holds
for $r^{\prime}<r$ we see that $C\not\subset(f_{2},\ldots,f_{r},E)$ implies
$\operatorname{depth}_{S}I/J\leq d+1$ by [16, Lemma 1.1]. If Conjecture 1
holds for $r$ and $E\setminus\\{a\\}$ with some $a\in E$ then
$C\not\subset(f_{1},\ldots,f_{r},E\setminus\\{a\\})$ implies again
$\operatorname{depth}_{S}I/J\leq d+1$ by the quoted lemma. Thus using the
induction hypothesis on $|E|$ we may assume that
$C\subset(W)\cup((E)\cap(f_{1},\ldots,f_{r}))\cup(\cup_{a,a^{\prime}\in
E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$. Let $I_{t}=I\cap(x_{t})$,
$J_{t}=J\cap(x_{t})$, $B_{t}=(B\setminus
E)\cap(x_{t})=\\{x_{t}f_{1},\ldots,x_{t}f_{e}\\}$, for some $1\leq e\leq r$.
If $\operatorname{sdepth}_{S}I_{t}/J_{t}\leq d+1$ then
$\operatorname{depth}_{S}I_{t}/J_{t}\leq d+1$ by [12, Theorem 4.3] because
$I_{t}$ is generated only by monomials of degree $d+1$. Thus
$\operatorname{depth}_{S}I/J\leq\operatorname{depth}_{S}I_{t}/J_{t}\leq d+1$
by [9, Lemma 1.1].
Suppose that $\operatorname{sdepth}_{S}I_{t}/J_{t}\geq d+2$. Then there exists
a partition on $I_{t}/J_{t}$ with sdepth $d+2$ having some disjoint intervals
$[x_{t}f_{i},c_{i}]$, $i\in[e]$ and $[a,c_{a}]$, $a\in E$. We may assume that
$c_{i},c_{a}$ have degrees $d+2$. We have either $c_{i}\in(W)$, or
$c_{i}\in((E)\cap(f_{1},\ldots,f_{r}))\setminus(W)$. In the first case
$c_{i}=x_{t}w_{ik_{i}}$ for some $1\leq k_{i}\leq r$, $k_{i}\not=i$. Note that
$x_{t}f_{k_{i}}\in B$ and so $k_{i}\leq e$. We consider the intervals
$[f_{i},c_{i}]$. These intervals contain $x_{t}f_{i}$ and possible a
$w_{ik_{i}}$. If $w_{ik_{i}}=w_{jk_{j}}$ for $i\not=j$ then we get
$c_{i}=c_{j}$ which is false. Thus these intervals are disjoint.
Let $I^{\prime}$ be the ideal generated by $f_{j}$ for $e<j\leq r$ and
$B\setminus(E\cup(\cup_{i=1}^{e}[f_{i},c_{i}]))$. Set
$J^{\prime}=I^{\prime}\cap J$. Note that $I^{\prime}\not=I$ because $e\geq 1$
. As we showed already $c_{i}\not\in I^{\prime}$ for any $i\in[e]$. Also
$c_{a}\not\in I^{\prime}$ because otherwise $c_{a}=x_{t}x_{k}f_{j}$ for some
$e<j\leq r$ and we get $x_{t}f_{j}\in B$, which is false. In the following
exact sequence
$0\to I^{\prime}/J^{\prime}\to I/J\to I/(J+I^{\prime})\to 0$
the last term has a partition of sdepth $d+2$ given by the intervals
$[f_{i},c_{i}]$ for $1\leq i\leq e$ and $[a,c_{a}]$ for $a\in E$. It follows
that $I^{\prime}\not=J^{\prime}$ because $\operatorname{sdepth}_{S}I/J=d+1$.
Then $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ using [17, Lemma
2.2] and so $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ by
Conjecture 1 applied for $r-e<r$. But the last term of the above sequence has
depth $>d$ because $x_{t}$ does not annihilate $f_{i}$ for $i\in[e]$. With the
Depth Lemma we get $\operatorname{depth}_{S}I/J\leq d+1$.
Next we give a variant of the above lemma.
###### Lemma 2.
Suppose that $r>2$, $E=\emptyset$, $C\subset(W)$ and there exists $t\in[n]$,
$t\not\in\cup_{i\in[r]}\operatorname{supp}f_{i}$ such that $x_{t}w_{ij}\in C$
for some $1\leq i<j\leq r$. If Conjecture 1 holds for $r^{\prime}\leq r-2$ and
$\operatorname{sdepth}_{S}I/J=d+1$, then $\operatorname{depth}_{S}I/J\leq
d+1$.
###### Proof.
We follow the proof of the above lemma, skipping the first part since we have
already $C\subset(W)$. Note that in our case $x_{t}f_{i},x_{t}f_{j}\in B$ and
so $e\geq 2$. Thus $I^{\prime}$ is generated by at most $(r-2)$ monomials of
degrees $d$ and some others of degrees $\geq d+1$. Therefore, Conjecture 1
holds for $I^{\prime}/J^{\prime}$ and so the above proof works in our case.
For $r\leq 3$ the following lemma is part from the proof of [9, Lemma 3.2] but
not in an explicit way. Here we try to formalize better the arguments in order
to apply them when $r=4$.
###### Lemma 3.
Suppose that $r\leq 4$ and for each $i\in[r]$ there exists $c_{i}\in
C\cap(f_{i})$ such that the intervals $[f_{i},c_{i}]$, $i\in[r]$ are disjoint.
Then $\operatorname{depth}_{S}I/J\geq d+1$.
###### Proof.
The proof consists of an induction part dealing with the case
$C\not\subset(W)$ followed by a case analysis covering the case $C\subset(W)$.
Case 1, $C\not\subset(W)$
Suppose that there exists $c\in C\setminus(W)$, let us say
$c\in(f_{1})\setminus(f_{2},\ldots,f_{r})$. Then $[f_{1},c]$ is disjoint with
respect to $[f_{i},c_{i}]$, $1<i\leq r$ and we may change $c_{1}$ by $c$, that
is we may suppose that $c_{1}\in(f_{1})\setminus(f_{2},\ldots,f_{r})$. Let
$B\cap[f_{1},c_{1}]=\\{b,b^{\prime}\\}$ and
$L=(f_{2},\ldots,f_{r},B\setminus\\{b,b^{\prime},E\\})$. In the following
exact sequence
$0\to L/(J\cap L)\to I/J\to I/(J,L)\to 0$
the first term has depth $\geq d+1$ by induction hypothesis and the last term
is isomorphic with $(f_{1})/((J,L)\cap(f_{1}))$ and has depth $\geq d+1$
because $b\not\in(J,L)$. Thus $\operatorname{depth}_{S}I/J\geq d+1$ by the
Depth Lemma.
Case 2, $r=2$
In this case, note that one from $c_{1},c_{2}$ is not in $(W)=(w_{12})$, that
is we are in the above case. Indeed, if $c_{1}\in(W)$ then either
$c_{1}=w_{12}$ and so $c_{2}$ cannot be in $(W)$, or $c_{1}=x_{j}w_{12}$ and
then $w_{12}\in[f_{1},c_{1}]$ cannot divide $c_{2}$ since the intervals are
disjoint.
From now on assume that $r>2$.
Case 3, $c_{1}\in(w_{12})$, $f_{i}\not|c_{1}$ for $i>2$ and
$c_{i}\not\in(w_{12})$ for $1<i\leq r$.
First suppose that $w_{12}\in B$. We have $c_{1}=x_{j}w_{12}$ for some $j$ and
we see that $b=f_{1}x_{j}\not\in(f_{2},\ldots,f_{r})$. Set
$T=(f_{2},\ldots,f_{r},B\setminus\\{b,E\\})$. In the following exact sequences
$0\to T/(J\cap T)\to I/J\to I/(J,T)\to 0$ $0\to(w_{12})/(J\cap(w_{12}))\to
T/(J\cap T)\to T/((J,w_{12})\cap T)\to 0$
the last terms have depth $\geq d+1$ since $b\not\in(J,T)$ and using the
induction hypothesis in the second situation. As the first term of the second
sequence has depth $\geq d+1$ we get $\operatorname{depth}_{S}T/(J\cap T)\geq
d+1$ and so $\operatorname{depth}_{S}I/J\geq d+1$ using the Depth Lemma in
both exact sequences.
If $w_{12}\in C$ then both monomials $b,b^{\prime}$ from $B\cap[f_{1},c_{1}]$
are not in $(f_{2},\ldots,f_{r})$ and the above proof goes with $b^{\prime}$
instead $w_{12}$.
Case 4, $r=3$.
By Case 1 we may suppose that $C\subset(W)$. Then $w_{12},w_{13},w_{23}$ are
different because otherwise only one $c_{i}$ can be in $(W)$. We may suppose
that $c_{1}\in(w_{12})$, $c_{2}\in(w_{23})$, $c_{3}\in(w_{13})$, because each
$c_{i}$ is a multiple of one $w_{ij}$ which can be present just in one
interval since these are disjoint. If $f_{3}|c_{1}$ then $w_{13}$ is present
in both intervals $[f_{1},c_{1}]$, $[f_{3},c_{3}]$. If let us say $w_{12}\in
C$, then $c_{2},c_{3}\not\in(w_{12})$ because $c_{3}\not=c_{1}\not=c_{2}$.
Thus we are in Case 3.
If $w_{12}\in B$ and $c_{2},c_{3}\not\in(w_{12})$ then we are in Case 3.
Otherwise, we may suppose that either $c_{2}\in(w_{12})$, or
$c_{3}\in(w_{12})$. In the first case, we have $w_{12}$ in both intervals
$[f_{1},c_{1}]$, $[f_{2},c_{2}]$, which is false. In the second case, we have
also $w_{23}$ present in both intervals $[f_{2},c_{2}]$, $[f_{3},c_{3}]$,
again false.
Case 5, $r=4$, $c_{1}\in(w_{12})$, $w_{12}\in B$, $f_{i}\not|c_{1}$ for
$2<i\leq 4$, $c_{3}\in(w_{12})$.
It follows that $c_{3}\in(w_{23})$. Thus $c_{2}\not\in(w_{23})$, that is
$f_{3}\not|c_{2}$, because otherwise the intervals $[f_{2},c_{2}]$,
$[f_{3},c_{3}]$ will contain $w_{23}$, which is false. If $c_{2}\in(w_{12})$
then the intervals $[f_{1},c_{1}]$, $[f_{2},c_{2}]$ will contain $w_{12}$. It
follows that $c_{2}\in(w_{24})$. Note that $c_{4}\not\in(w_{24})$ because
otherwise $w_{24}$ belongs to $[f_{2},c_{2}]\cap[f_{4},c_{4}]$. If
$c_{3}\not\in(w_{24})$ then we are in Case 3 with $w_{24}$ instead $w_{12}$
and $c_{2}$ instead $c_{1}$.
Remains to see the case when
$c_{3}\in(f_{1})\cap(f_{2})\cap(f_{3})\cap(f_{4})$. Then $c_{4}\not\in(f_{3})$
because otherwise $w_{34}$ is in $[f_{3},c_{3}]\cap[f_{4},c_{4}]$. In the
exact sequence
$0\to(f_{3})/(J\cap(f_{3}))\to I/J\to I/(J,f_{3})\to 0$
the last term has depth $\geq d+1$ by induction hypothesis. The first term has
depth $\geq d+1$ since for example $w_{23}\not\in J$. By the Depth Lemma we
get $\operatorname{depth}_{S}I/J\geq d+1$.
Case 6, $r=4$, the general case.
Since $|W|\leq 6$ there exist an interval, let us say $[f_{1},c_{1}]$,
containing just one $w_{ij}$, let us say $w_{12}$. Thus no $f_{i}$, $2<i\leq
4$ divides $c_{1}$. If $w_{12}\in C$ then no $c_{i}$, $i>1$ belongs to
$(w_{12})$ because otherwise $c_{i}=c_{1}$. If $w_{12}\in B$ and one
$c_{i}\in(w_{12})$, $i>1$ then we must have $i=2$ because otherwise we are in
Case 5. But if $c_{2}\in(w_{12})$ then $w_{12}$ is present in both intervals
$[f_{1},c_{1}]$, $[f_{2},c_{2}]$, which is false. Thus $c_{i}\not\in(w_{12})$
for all $1<i\leq 4$, that is Case 3.
###### Remark 1.
When $r>4$ the statement of the above lemma is not valid anymore, as shows the
following example.
###### Example 2.
Let $n=5$, $d=1$, $I=(x_{1},\ldots,x_{5})$,
$J=(x_{1}x_{3}x_{4},x_{1}x_{2}x_{4},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5},x_{2}x_{4}x_{5}).$
Set $c_{1}=x_{1}x_{2}x_{3}$, $c_{2}=x_{2}x_{3}x_{4}$, $c_{3}=x_{3}x_{4}x_{5}$,
$c_{4}=x_{1}x_{4}x_{5}$, $c_{5}=x_{1}x_{2}x_{5}$. We have
$C=\\{c_{1},\ldots,c_{5}\\}$ and $B=W$. Thus $s=2r$ and
$\operatorname{sdepth}_{S}I/J=3$ because we have a partition on $I/J$ given by
the intervals $[x_{i},c_{i}]$, $i\in[5]$. But $\operatorname{depth}_{S}I/J=1$
because of the following exact sequence
$0\to I/J\to S/J\to S/I\to 0$
where the last term has depth $0$ and the middle $\geq 2$.
The proposition below is an extension of [9, Lemma 3.2], its proof is given in
the next section.
###### Proposition 1.
Suppose that the following conditions hold:
1. (1)
$r=4$, $8\leq s\leq q+4$,
2. (2)
$C\subset(\cup_{i,j\in[4],i\not=j}(f_{i})\cap(f_{j}))\cup((E)\cap(f_{1},\ldots,f_{4}))\cup(\cup_{a,a^{\prime}\in
E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$,
3. (3)
there exists $b\in(B\cap(f_{1}))\setminus(f_{2},f_{3},f_{4})$ such that
$\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ for
$I_{b}=(f_{2},\ldots,f_{r},B\setminus\\{b\\})$, $J_{b}=J\cap I_{b}$,
4. (4)
the least common multiple $\omega_{1}$ of $f_{2},f_{3},f_{4}$ is not in
$(C_{3}\setminus W)\cap(E)$ (see Example 1).
Then either $\operatorname{sdepth}_{S}I/J\geq d+2$, or there exists a nonzero
ideal $I^{\prime}\subsetneq I$ generated by a subset of
$\\{f_{1},\ldots,f_{4}\\}\cup B$ such that
$\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$ and either
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap
I^{\prime}$ or $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ .
###### Proposition 2.
Conjecture 1 holds for $r=4$ when the least common multiples $\omega_{i}$ of
$f_{1},\ldots,f_{i-1},f_{i+1},\ldots,f_{4}$, $i\in[4]$ are not in
$(C_{3}\setminus W)\cap(E)$. In particular, Conjecture 1 holds when $r=4$ and
$E=\emptyset$.
###### Proof.
By Theorems [13, Theorem 1.3], [18, Theorem 2.4] (more precisely the
particular forms given in [9, Theorems 0.3, 0.4]) we may suppose that
$8=2r\leq s\leq q+4$ and we may assume that $E$ contains only monomials of
degrees $d+1$ by [14, Lemma 1.6]. We may assume that there exists $b\in
B\cap(f_{1},\ldots,f_{4})$ which is not in $W$ because otherwise
$B\cap(f_{1},\ldots,f_{4})\subset B\cap W$ and therefore
$|B\cap(f_{1},\ldots,f_{4})|\leq|B\cap W|\leq 6$. By [18, Theorem 2.4] this
implies the depth $\leq d+1$ of the first term of the exact sequence
$0\to(f_{1},\ldots,f_{r})/(J\cap(f_{1},\ldots,f_{r}))\to
I/J\to(E)/((J,f_{1},\ldots,f_{r})\cap(E))\to 0$
and then the middle has depth $\leq d+1$ too using the Depth Lemma.
Renumbering $f_{i}$ we may suppose that there exists
$b\in(f_{1})\setminus(f_{2},\ldots,f_{4})$. As in the proof of [9, Theorem
1.7] we may suppose that the first term of the exact sequence
$0\to I_{b}/J_{b}\to I/J\to I/(J,I_{b})\to 0$
has sdepth $\geq d+2$. Otherwise it has depth $\leq d+1$ by Theorem 1. Note
that the last term is isomorphic with $(f_{1})/((f_{1})\cap(J,I_{b}))$ and it
has depth $\geq d+1$ because $b\not\in(J,I_{b})$. Then the middle term of the
above exact sequence has depth $\leq d+1$ by the Depth Lemma.
Thus we may assume that the condition (3) of Proposition 1 holds. Also we may
apply [16, Lemma 1.1] and see that the condition (2) of Proposition 1 holds.
Applying Proposition 1 we get either $\operatorname{sdepth}_{S}I/J\geq d+2$
contradicting our assumption, or there exists a nonzero ideal
$I^{\prime}\subsetneq I$ generated by a subset $G$ of $B$, or by $G$ and a
subset of $\\{f_{1},\ldots,f_{4}\\}$ such that
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap
I^{\prime}$ and $\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$. In the
last case we see that $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$
by Theorem 1, or by induction on $s$, and so $\operatorname{depth}_{S}I/J\leq
d+1$ applying in the following exact sequence
$0\to I^{\prime}/J^{\prime}\to I/J\to I/(J,I^{\prime})\to 0$
the Depth Lemma.
## 2\. Proof of Proposition 1
Since $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ by (3), there exists a
partition $P_{b}$ on $I_{b}/J_{b}$ with sdepth $d+2$. We may choose $P_{b}$
such that each interval starting with a squarefree monomial of degree $d$,
$d+1$ ends with a monomial of $C$. In $P_{b}$ we have three disjoint intervals
$[f_{2},c^{\prime}_{2}]$, $[f_{3},c^{\prime}_{3}]$, $[f_{4},c^{\prime}_{4}]$.
Suppose that $B\cap[f_{i},c^{\prime}_{i}]=\\{u_{i},u^{\prime}_{i}\\}$,
$1<i\leq 4$. For all $b^{\prime}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ we have an
interval $[b^{\prime},c_{b^{\prime}}]$. We define
$h:B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}\to C$ by
$b^{\prime}\longmapsto c_{b^{\prime}}$. Then $h$ is an injection and
$|\operatorname{Im}h|=s-7\leq q-3$.
We follow the proofs of [9, Lemmas 3.1, 3.2]. A sequence $a_{1},\ldots,a_{k}$
is called a path from $a_{1}$ to $a_{k}$ if the following statements hold:
(i) $a_{l}\in
B\setminus\\{b,u_{2},u_{2}^{\prime},\ldots,u_{4},u_{4}^{\prime}\\}$,
$l\in[k]$,
(ii) $a_{l}\not=a_{j}$ for $1\leq l<j\leq k$,
(iii) $a_{l+1}|h(a_{l})$ for all $1\leq l<k$.
This path is weak if
$h(a_{j})\in(b,u_{2},u_{2}^{\prime},\ldots,u_{4},u^{\prime}_{4})$ for some
$j\in[k]$. It is bad if $h(a_{j})\in(b)$ for some $j\in[k]$ and it is maximal
if all divisors from $B$ of $h(a_{k})$ are in
$\\{b,u_{2},u_{2}^{\prime},\ldots,u_{4},u^{\prime}_{4},a_{1},\ldots,a_{k}\\}$.
We say that the above path starts with $a_{1}$. Note that here the notion of
path is more general than the notion of path used in [16] and [9].
By hypothesis $s\geq 8$ and there exists $a_{1}\in
B\setminus\\{b,u_{2},u_{2}^{\prime},\ldots,u_{4},u^{\prime}_{4}\\}$. We
construct below, as an example, a path with $k>1$. By recurrence choose if
possible $a_{p+1}$ to be a divisor from
$B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u_{4}^{\prime},a_{1},\ldots,a_{p}\\}$
of $m_{p}=h(a_{p})$, $p\geq 1$. This construction ends at step $p=e$ if all
divisors from $B$ of $m_{e}$ are in
$\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u_{4}^{\prime},a_{1},\ldots,a_{e}\\}$.
This is a maximal path. If one
$m_{p}\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u_{4}^{\prime})$ then the
constructed path is weak. If one $m_{p}\in(b)$ then this path is bad.
We start the proof with some helpful lemmas.
###### Lemma 4.
$P_{b}$ could be changed in order to have the following properties:
1. (1)
For all $1<i<j\leq 4$ with $u_{i},u_{j}\not\in W$ and $w_{ij}\in
B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ it holds
that $h(w_{ij})\not\in(u_{i})\cap(u_{j})$,
2. (2)
For each $1\leq i<j\leq 4$ with $u_{j}\in W$, $u^{\prime}_{j}\not\in W$ and
$w_{ij}\in B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
it holds that $h(w_{ij})\not\in(u_{j})$ and if $h(w_{ij})\in(u^{\prime}_{j})$
then $i>1$,
3. (3)
For each $1\leq i<j\leq 4$ with $u_{j},u^{\prime}_{j}\not\in W$ and $w_{ij}\in
B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ it holds
that $h(w_{ij})\not\in(u_{j},u^{\prime}_{j})$.
###### Proof.
Suppose that $w_{ij}\in
B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and
$h(w_{ij})\in(u_{i})$ for some $2\leq i\leq 4$ and $j\in[4]$, $j\not=i$. We
have $h(w_{ij})=x_{l}w_{ij}$ for some $l\not\in\operatorname{supp}w_{ij}$ and
it follows that $u_{i}=x_{l}f_{i}$. Changing in $P_{b}$ the intervals
$[f_{i},c^{\prime}_{i}]$, $[w_{ij},h(w_{ij})]$ with $[f_{i},h(w_{ij})]$,
$[u^{\prime}_{i},c^{\prime}_{i}]$ we may assume that the new
$u^{\prime}_{i}=w_{ij}$. We will apply this procedure several times eventually
obtaining a partition $P_{b}$ with the above properties. In case (1) we change
in this way $u^{\prime}_{i}$ by $w_{ij}$. Note that the number of elements
among $\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ which are from
$B\cap W$ is either preserved or increases by one. Applying this procedure
several time we get (1) fulfilled.
In case (3) the above procedure preserves among
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ the former elements
which were from $B\cap W$ and includes a new one $w_{ij}$. After several steps
we get fulfilled (3).
For case (2) if $u_{j}\in W$, $u^{\prime}_{j}\not\in W$ and
$h(w_{ij})\in(u_{j})$ we change as above $u^{\prime}_{j}$ by $w_{ij}$. Note
that the number of elements among
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ which are from $B\cap
W$ increases by one. If $h(w_{ij})\in(u^{\prime}_{j})$ then we may change in
this way $u_{j}$ by $w_{ij}$. We do this only if $i=1$. Note that the number
of elements among $\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
which are from $B\cap W$ is preserved. Our procedure does not affect those
$c^{\prime}_{i}$ with $u_{i},u^{\prime}_{i}\in W$ and does not affect the
property (1). After several such procedures we get also (2) fulfilled.
From now on we suppose that $P_{b}$ has the properties mentioned in the above
lemma. Moreover, we fix $a_{1}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and let
$a_{1},\ldots,a_{p}$ be a path which is not bad. For an $a^{\prime}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ set
$T_{a^{\prime}}=\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path}\ \
a^{\prime}_{1}=a^{\prime},\ldots,a^{\prime}_{e}\ \ \mbox{ not\ \ bad\ \ with}\
\ a^{\prime}_{e}=b^{\prime}\\},$
$U_{a^{\prime}}=h(T_{a^{\prime}})$, $G_{a^{\prime}}=B\setminus
T_{a^{\prime}}$. If $a^{\prime}=a_{1}$ we write simply $T_{1}$ instead
$T_{a_{1}}$ and similarly $U_{1}$, $G_{1}$.
###### Remark 2.
Any divisor from $B$ of a monomial of $U_{1}$ is in
$T_{1}\cup\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
###### Lemma 5.
If no weak path and no bad path starts with $a_{1}$ then the conclusion of
Proposition 1 holds.
###### Proof.
Assume that
$[r]\setminus\\{j\in[r]:U_{1}\cap(f_{j})\not=\emptyset\\}=\\{k_{1},\ldots,k_{\nu}\\}$
for some $1\leq k_{1}<\ldots<k_{\nu}\leq 4$, $0\leq\nu\leq 4$. Set
$k=(k_{1},\ldots,k_{\nu})$,
$I^{\prime}_{k}=(f_{k_{1}},\ldots,f_{k_{\nu}},G_{1})$,
$J^{\prime}_{k}=I^{\prime}_{k}\cap J$, and $I^{\prime}_{0}=(G_{1})$,
$J^{\prime}_{0}=I^{\prime}_{0}\cap J$ for $\nu=0$. Note that all divisors from
$B$ of a monomial $c\in U_{1}$ belong to $T_{1}$, and $I^{\prime}_{0}\not=0$
because $b\in I^{\prime}_{0}$. Consider the following exact sequence
$0\to I^{\prime}_{k}/J^{\prime}_{k}\to I/J\to I/(J,I^{\prime}_{k})\to 0.$
If $U_{1}\cap(f_{1},\ldots,f_{4})=\emptyset$ then the last term of the above
exact sequence given for $k=(1,\ldots,4)$ has depth $\geq d+1$ and sdepth
$\geq d+2$ because $P_{b}$ can be restricted to
$(T_{1})\setminus(J,I^{\prime}_{k})$ since $h(b)\notin I^{\prime}_{k}$ , for
all $b\in T_{1}$ (see Remark 2). If the first term has sdepth $\geq d+2$ then
by [17, Lemma 2.2] the middle term has sdepth $\geq d+2$. Otherwise, take
$I^{\prime}=I^{\prime}_{k}$.
If $U_{1}\cap(f_{1},f_{2},f_{3})=\emptyset$, but there exists $b_{4}\in
T_{1}\cap(f_{4})$, then set $k=(1,2,3)$. In the following exact sequence
$0\to I^{\prime}_{k}/J^{\prime}_{k}\to I/J\to I/(J,I^{\prime}_{k})\to 0$
the last term has sdepth $\geq d+2$ since $h(b^{\prime})\notin I^{\prime}_{k}$
for all $b^{\prime}\in T_{1}$ and we may substitute the interval
$[b_{4},h(b_{4})]$ from the restriction of $P_{b}$ by $[f_{4},h(b_{4})]$, the
second monomial from $[f_{4},h(b_{4})]\cap B$ being also in $T_{1}$. As above
we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or
$\operatorname{sdepth}_{S}I^{\prime}_{k}/J^{\prime}_{k}\leq d+1$,
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k})\geq d+1$.
Suppose that $U_{1}\cap(f_{j})\not=\emptyset$ if and only if $\nu<j\leq 4$,
for some $0\leq\nu\leq 4$ and set $k=(1,\ldots,\nu)$. We omit the subcases
$0<\nu<3$, since they go as in [9, Lemma 3.2], and consider only the worst
subcase $\nu=0$. Let $b_{j}\in T_{1}\cap(f_{j})$, $j\in[4]$ and set
$c_{j}=h(b_{j})$. For $1\leq l<j\leq 4$ we claim that we may choose
$b_{l}\not=b_{j}$ and such that one from $c_{l},c_{j}$ is not in $(w_{lj})$.
Indeed, if $w_{lj}\not\in B$ and $c_{l},c_{j}\in(w_{lj})$ then necessarily
$c_{l}=c_{j}$ and it follows $b_{l}=b_{j}=w_{lj}$, which is false. Suppose
that $w_{lj}\in B$ and $c_{j}=x_{p}w_{lj}$. Then choose $b_{l}=x_{p}f_{l}\in
T_{1}$. If $c_{l}=h(b_{l})\in(w_{lj})$ then we get $c_{l}=c_{j}$ and so
$b_{l}=b_{j}=w_{lj}$ which is impossible.
We show that we may choose $b_{j}\in T_{1}\cap(f_{j})$, $j\in[4]$ such that
the intervals $[f_{j},c_{j}]$, $j\in[4]$ are disjoint. Let $C_{2}$, $C_{3}$ be
as in the beginning of the previous section. Set $C^{\prime}_{2}=U_{1}\cap
C_{2}$, $C^{\prime}_{3}=U_{1}\cap C_{3}$, $C^{\prime}_{23}=C^{\prime}_{2}\cup
C^{\prime}_{3}$. Let ${\tilde{c}}\in C^{\prime}_{2}$, let us say $\tilde{c}$
is the least common multiple of $f_{1},f_{2}$. Then $\tilde{c}$ has as
divisors two multiples $g_{1},g_{2}$ of $f_{1}$ and two multiples of $f_{2}$.
If ${\hat{c}}\in C^{\prime}_{2}$ is also a multiple of $g_{1}$, let us say
$\hat{c}$ is the least common multiple of $f_{1},f_{3}$ then $g_{2}$ does not
divide $\hat{c}$ and the least common multiple of $f_{2},f_{3}$ is not in $C$.
Thus the divisors from $B\setminus E$ of $\tilde{c}$, $\hat{c}$ are at least
$7$. Since the divisors from $B\setminus E$ of $\tilde{c}$, $\hat{c}$ are in
$T_{1}\setminus E$ we see in this way that $|T_{1}\setminus
E|\geq|C^{\prime}_{2}|+3$. If $|C^{\prime}_{2}|\not=0$ then
$|C^{\prime}_{3}|\leq 1$ and so $|T_{1}\setminus E|\geq|C^{\prime}_{23}|+2$.
Assume that $|C^{\prime}_{2}|=0$. Then $|C^{\prime}_{3}|\leq 4$. Let
${\tilde{c}}\in C^{\prime}_{3}$ be the least common multiple of
$f_{1},f_{2},f_{3}$ then $w_{12},w_{23},w_{13}$ are the only divisors from
$T_{1}\setminus E$ of $\tilde{c}$ (this could be not true when
$|C^{\prime}_{2}|\not=0$ as shows Example 1). If ${\hat{c}}\in C^{\prime}_{3}$
is the least common multiple of $f_{1},f_{2},f_{4}$ we have also
$w_{14},w_{24}$ in $T_{1}\setminus E$. Similarly, if $|C^{\prime}_{3}|\geq 3$
we get also $w_{34}\in T_{1}\setminus E$. Thus $|T_{1}\setminus
E|\geq|C^{\prime}_{3}|+2=|C^{\prime}_{23}|+2$ also when $|C^{\prime}_{2}|=0$.
Then there exist two different $b_{j}\in T_{1}\cap(f_{j})$ such that
$c_{j}=h(b_{j})\not\in C^{\prime}_{23}$ for let us say $j=1,2$ and so each of
the intervals $[f_{j},c_{j}]$, $j=1,2$ has at most one monomial from
$T_{1}\cap W$. Suppose the worst subcase when $[f_{1},c_{1}]$ contains
$w_{12}\in B$, and $[f_{2},c_{2}]$ contains $w_{2j}\in B$ for some $j\not=2$.
First assume that $j\geq 3$, let us say $j=3$. Then choose as above $b_{3}\in
T_{1}\cap(f_{3})$, $b_{4}\in T_{1}\cap(f_{4})$ such that
$c_{3}\not\in(w_{23})$, $c_{4}\not\in(w_{34})$. Then $[f_{3},c_{3}]$ has from
$T_{1}\cap W$ at most $w_{13},w_{34}$ and $[f_{4},c_{4}]$ has from $T_{1}\cap
W$ at most $w_{14},w_{24}$. Thus the corresponding intervals are disjoint.
Otherwise, $j=1$ and we have $c_{j}=x_{p_{j}}w_{12}$, $j\in[2]$, for some
$p_{j}\not\in\operatorname{supp}w_{12}$, $p_{1}\not=p_{2}$. Take
$b^{\prime}_{1}=x_{p_{2}}f_{1}$, $b^{\prime}_{2}=x_{p_{1}}f_{2}$ and
$v_{1}=h(b^{\prime}_{1})$, $v_{2}=h(b^{\prime}_{2})$. Then $v_{1},v_{2}$ are
not in $C^{\prime}_{3}$ because otherwise $b^{\prime}_{1}$, respectively
$b^{\prime}_{2}$ is in $W$, which is false. Note that $v_{2}\not\in(w_{12})$,
because otherwise $v_{2}=x_{p_{1}}w_{12}=c_{1}$ which is false since
$b_{1}\not=b^{\prime}_{2}$. Similarly $v_{1}\not\in(w_{12})$. If let us say
$v_{2}\not\in C^{\prime}_{2}$ then we may take $b_{2}=b^{\prime}_{2}$ and we
see that for the new $c_{2}$ (namely $v_{2}$) the interval $[f_{2},c_{2}]$
contains at most a monomial from $W$, which we assume to be $w_{23}$ and we
proceed as above. If $v_{1},v_{2}\in C^{\prime}_{2}$, we may assume that
$v_{1}=w_{13}\in C$ and either $v_{2}=w_{23}\in C$, or $v_{2}=w_{24}\in C$. In
the first case we choose $b_{3},b_{4}$ such that $c_{3}\not\in(w_{34})$,
$c_{4}\not\in(w_{24})$ and we see that $[f_{3},c_{3}]$ has no monomial from
$W$. Indeed, if $c_{3}\in(w_{23})$ (the case $c_{3}\in(w_{13})$ is similar)
then $c_{3}=v_{2}$, which is false since then
$h(b^{\prime}_{2})=v_{2}=c_{3}=h(b_{3})$ and so
$b^{\prime}_{2}=b_{3}\in(w_{23})$, $h$ being injective. Also $[f_{4},c_{4}]$
has at most $w_{14},w_{34}$. Thus taking $b_{i}=b^{\prime}_{i}$, $c_{i}=v_{i}$
for $i\in[2]$ we have again the intervals $[f_{j},c_{j}]$, $j\in[4]$ disjoint.
Similarly in the second case choose $b_{3},b_{4}$ such that
$c_{3}\not\in(w_{23})$, $c_{4}\not\in(w_{34})$ and we see that $[f_{3},c_{3}]$
have at most $w_{34}$ and $[f_{4},c_{4}]$ have at most $w_{14}$, which is
enough, because as above $c_{3}\not=w_{13}$ and $c_{4}\not=w_{24}$.
Next we replace the intervals $[b_{j},c_{j}]$, $1\leq j\leq 4$ from the
restriction of $P_{b}$ to $(T_{1})\setminus(J,I^{\prime}_{0})$ with
$[f_{j},c_{j}]$, the second monomial from $[f_{j},c_{j}]\cap B$ being also in
$T_{1}$. Note that $I/(J,I^{\prime}_{0})$ has depth $\geq d+1$ by Lemma 3.
Thus, as above we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or
$\operatorname{sdepth}_{S}I^{\prime}_{0}/J^{\prime}_{0}\leq d+1$,
$\operatorname{depth}_{S}I/(J,I^{\prime}_{0})\geq d+1$.
###### Lemma 6.
Let $a_{1},\ldots,a_{e_{1}}$ be a bad path, $m_{j}=h(a_{j})$, $j\in[e_{1}]$
and $m_{e_{1}}=bx_{i}$. Suppose that
$m_{e_{1}}\not\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$. Then one
of the following statements holds:
1. (1)
$\operatorname{sdepth}_{S}I/J\geq d+2$,
2. (2)
there exists
$a_{e_{1}+1}\in(B\cap(f_{1}))\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
dividing $m_{e_{1}}$ such that every path $a_{e_{1}+1},\ldots,a_{e_{2}}$
satisfies
$\\{a_{1},\ldots,a_{e_{1}}\\}\cap\\{a_{e_{1}+1},\ldots,a_{e_{2}}\\}=\emptyset.$
###### Proof.
If $a_{e_{1}}=f_{1}x_{i}$ then changing in $P_{b}$ the interval
$[a_{e_{1}},m_{e_{1}}]$ by $[f_{1},m_{e_{1}}]$ we get a partition on $I/J$
with sdepth $d+2$. If $f_{1}x_{i}\in\\{a_{1},\ldots,a_{e_{1}-1}\\}$, let us
say $f_{1}x_{i}=a_{v}$, $1\leq v<e_{1}$ then we may replace in $P_{b}$ the
intervals $[a_{k},m_{k}],v\leq k\leq e_{1}$ with the intervals
$[a_{v},m_{e_{1}}],[a_{k+1},m_{k}],v\leq k\leq e_{1}-1$. Now we see that we
have in $P_{b}$ the interval $[a_{v},m_{v}]$ (the new $m_{v}$ is the old
$m_{e_{1}}$) and switching it with the interval $[f_{1},m_{v}]$ we get a
partition with sdepth $\geq d+2$ for $I/J$. Thus we may assume that
$f_{1}x_{i}\notin\\{a_{1},...,a_{e_{1}}\\}$. Note that $e_{1}$ could be also
$1$ as in Example 3 when we take $a_{1}=x_{5}x_{6}$, in this case we take
$f_{1}x_{i}=x_{1}x_{5}$ and $\\{x_{1}x_{5},x_{2}x_{5}\\}$ is a maximal path
which is weak but not bad.
By hypothesis
$m_{e_{1}}\not\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$ and so
$f_{1}x_{i}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Then set $a_{e_{1}+1}=f_{1}x_{i}$ and let $a_{e_{1}+1},\ldots,a_{e_{2}}$ be a
path starting with $a_{e_{1}+1}$ and set $m_{p}=h(a_{p}),p>e_{1}$. If
$a_{p}=a_{v}$ for $v\leq e_{1}$, $p>e_{1}$ then change in $P_{b}$ the
intervals $[a_{k},m_{k}],v\leq k\leq p-1$ with the intervals
$[a_{v},m_{p-1}],[a_{k+1},m_{k}],v\leq k\leq p-2$. We have in the new $P_{b}$
an interval $[f_{1}x_{i},m_{e_{1}}]$ and switching it to $[f_{1},m_{e_{1}}]$
we get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may suppose that
$a_{p+1}\not\in\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4},a_{1},\ldots,a_{p}\\}$
and so (2) holds.
###### Example 3.
Let $n=7$, $r=4$, $d=1$, $f_{i}=x_{i}$ for $i\in[4]$,
$E=\\{x_{5}x_{6},x_{5}x_{7}\\}$, $I=(x_{1},\ldots,x_{4},E)$ and
$J=(x_{1}x_{7},x_{2}x_{7},x_{3}x_{7},x_{4}x_{7},x_{1}x_{2}x_{4},x_{1}x_{2}x_{6},x_{1}x_{3}x_{4},x_{1}x_{3}x_{6},x_{2}x_{3}x_{4},x_{2}x_{4}x_{5},$
$x_{2}x_{5}x_{6},x_{3}x_{5}x_{6},x_{4}x_{5}x_{6}).$
Then $B=$
$\\{x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},x_{1}x_{6},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{2}x_{6},x_{3}x_{4},x_{3}x_{5},x_{3}x_{6},x_{4}x_{5},x_{4}x_{6}\\}\cup
E$
and
$C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{5},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},x_{1}x_{5}x_{6},x_{2}x_{3}x_{5},x_{2}x_{3}x_{6},x_{2}x_{4}x_{6},$
$x_{3}x_{4}x_{5},x_{3}x_{4}x_{6},x_{5}x_{6}x_{7}\\}.$
We have $q=12$ and $s=q+r=16$. Take $b=x_{1}x_{6}$ and
$I_{b}=(x_{2},x_{3},x_{4},B\setminus\\{b\\},E)$, $J_{b}=I_{b}\cap J$. There
exists a partition $P_{b}$ with sdepth $3$ on $I_{b}/J_{b}$ given by the
intervals $[x_{2},x_{1}x_{2}x_{3}]$, $[x_{3},x_{1}x_{3}x_{5}]$,
$[x_{4},x_{1}x_{4}x_{6}]$, $[x_{1}x_{5},x_{1}x_{2}x_{5}]$,
$[x_{2}x_{4},x_{2}x_{4}x_{6}]$, $[x_{2}x_{5},x_{2}x_{3}x_{5}]$,
$[x_{2}x_{6},x_{2}x_{3}x_{6}]$, $[x_{3}x_{4},x_{3}x_{4}x_{5}]$,
$[x_{3}x_{6},x_{3}x_{4}x_{6}]$,
$[x_{4}x_{5},x_{1}x_{4}x_{5}]$, $[x_{5}x_{6},x_{1}x_{5}x_{6}]$,
$[x_{5}x_{7},x_{5}x_{6}x_{7}]$. We have $c^{\prime}_{2}=x_{1}x_{2}x_{3}$,
$c^{\prime}_{3}=x_{1}x_{3}x_{5}$, $c^{\prime}_{4}=x_{1}x_{4}x_{6}$ and
$u_{2}=x_{2}x_{3}$, $u^{\prime}_{2}=x_{1}x_{2}$, $u_{3}=x_{3}x_{5}$,
$u^{\prime}_{3}=x_{1}x_{3}$, $u_{4}=x_{1}x_{4}$, $u^{\prime}_{4}=x_{4}x_{6}$.
Take $a_{1}=x_{2}x_{4}$, $m_{1}=x_{2}x_{4}x_{6}$. This is a weak path but not
bad. It can be extended to a maximal one
$x_{2}x_{4},x_{2}x_{6},x_{3}x_{6},x_{3}x_{4},x_{4}x_{5},x_{1}x_{5},x_{2}x_{5}$
which is not bad. Bad paths are for example $\\{x_{5}x_{6}\\}$,
$\\{x_{5}x_{7},x_{5}x_{6}\\}$,
$\\{x_{5}x_{7},x_{5}x_{6},x_{1}x_{5},x_{2}x_{5}\\}$, the last one being
maximal. Replacing in $P_{b}$ the intervals $[x_{4},x_{1}x_{4}x_{6}]$,
$[x_{2}x_{4},x_{2}x_{4}x_{6}]$ with $[x_{4},x_{2}x_{4}x_{6}]$,
$[x_{1},x_{1}x_{4}x_{6}]$ we get a partition on $I/J$ with sdepth $3$.
###### Lemma 7.
Let $a_{1},\ldots,a_{e_{1}}$ be a bad path, $m_{j}=h(a_{j})$, $j\in[e_{1}]$
and $m_{e_{1}}=bx_{i}$. Suppose that $a_{e_{1}}\in E$ and
$m_{e_{1}}\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$. Then one of
the following statements holds:
1. (1)
there exists $a_{e_{1}+1}\in
B\setminus(\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}\cup E)$
dividing $m_{e_{1}}$ such that every path $a_{e_{1}+1},\ldots,a_{e_{2}}$
satisfies
$\\{a_{1},\ldots,a_{e_{1}}\\}\cap\\{a_{e_{1}+1},\ldots,a_{e_{2}}\\}=\emptyset,$
2. (2)
there exist $j$, $2\leq j\leq 4$ and a new partition $P_{b}$ of $I_{b}/J_{b}$
for which $T_{1}$ is preserved such that $a_{e_{1}}\in(f_{j})$ and
$m_{e_{1}}\in(u_{j},u^{\prime}_{j})$.
###### Proof.
Assume that $m_{e_{1}}=x_{i}b$ for some $i$ and let us say
$m_{e_{1}}\in(u^{\prime}_{2})$. Then $f_{1}x_{i}=u^{\prime}_{2}=w_{12}$ and so
there exists another divisor $\tilde{a}$ of $m_{e_{1}}$ from $B\cap(f_{2})$
different of $w_{12}$. If ${\tilde{a}}\in[f_{2},c^{\prime}_{2}]$ then we get
$m_{e_{1}}=c^{\prime}_{2}$, which is false. If $\tilde{a}$ is not in
$\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ then set
$a_{e_{1}+1}={\tilde{a}}$. If let us say $\tilde{a}=u_{3}$ then
${\tilde{a}}=w_{23}$ and so $m_{e_{1}}$ is the least common multiple of
$f_{1},f_{2},f_{3}$. Clearly, $m_{e_{1}}\not\in C_{3}$ because otherwise $b\in
W$, which is false. Then $m_{e_{1}}=w_{13}\in C$ and we may find, let us say
another divisor $\hat{a}$ of $m_{e_{1}}$ from $B\cap(f_{3})$ which is not
$u^{\prime}_{3}$ because $m_{e_{1}}\not=c^{\prime}_{3}$. If $\hat{a}$ is in
$\\{u_{4},u^{\prime}_{4}\\}$ then we may find an $a^{\prime}$ in
$B\cap(f_{4})$ which is not in $\\{u_{4},u^{\prime}_{4}\\}$ because
$m_{e_{1}}\not=c^{\prime}_{4}$. Thus in general we may find an
$a^{\prime\prime}$ in $B\cap(f_{j})$ for some $2\leq j\leq 4$ which is not in
$\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and
$m_{e_{1}}\in(u_{j},u^{\prime}_{j})$. Set $a_{e_{1}+1}=a^{\prime\prime}$. Let
$a_{e_{1}+1},\ldots,a_{e_{2}}$ be a path. If we are not in the case (1) then
$a_{p}=a_{v}$ for $v\leq e_{1}$, $p>e_{1}$ and change in $P_{b}$ the intervals
$[a_{k},m_{k}],v\leq k\leq p-1$ with the intervals
$[a_{v},m_{p-1}],[a_{k+1},m_{k}],v\leq k\leq p-2$. Note that the new
$a_{e_{1}}$ is the old $a_{e_{1}+1}\in(f_{j})$, that is the case (2).
###### Lemma 8.
Suppose that $\operatorname{sdepth}_{S}I/J\leq d+1$. Then there exists a
partition $P_{b}$ of $I_{b}/J_{b}$ such that for any $a_{1}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and any
bad path $a_{1},\ldots,a_{e_{1}}$ , $m_{j}=h(a_{j})$, $j\in[e_{1}]$ with
$m_{e_{1}}=bx_{i}$ the following statements holds:
1. (1)
$m_{e_{1}}\not\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$,
2. (2)
there exists $a_{e_{1}+1}\in
B\setminus(\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}\cup E)$
dividing $m_{e_{1}}$ such that every path $a_{e_{1}+1},\ldots,a_{e_{2}}$
satisfies
$\\{a_{1},\ldots,a_{e_{1}}\\}\cap\\{a_{e_{1}+1},\ldots,a_{e_{2}}\\}=\emptyset.$
###### Proof.
If for any $a_{1}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ there
exist no bad path starting with $a_{1}$ there exists nothing to show. If for
any such $a_{1}$ for each bad path $a_{1},\ldots,a_{e_{1}}$, $m_{j}=h(a_{j})$,
$j\in[e_{1}]$ with $m_{e_{1}}\in(b)$ it holds
$m_{e_{1}}\not\in(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$ then then
to get (2) apply Lemma 6. Now suppose that there exists $a_{1}$ and a bad path
$a_{1},\ldots,a_{e_{1}}$, $m_{j}=h(a_{j})$, $j\in[e_{1}]$ with let us say
$m_{e_{1}}\in(b)\cap(u_{2})$. If we are not in case (2) then by Lemma 7 we may
change $P_{b}$ such that $T_{1}$ is preserved, $a_{e_{1}}\in(f_{j})$ and
$m_{e_{1}}\in(u_{j},u^{\prime}_{j})$ for some $2\leq j\leq 4$. Assume that
$j=2$ and so $m_{e_{1}}\in(w_{12})$, let us say $u^{\prime}_{2}=w_{12}$.
Replacing in $P_{b}$ the intervals $[f_{2},c^{\prime}_{2}]$,
$[a_{e_{1}},m_{e_{1}}]$ with $[f_{2},m_{e_{1}}]$, $[u_{2},c^{\prime}_{2}]$ the
new $c^{\prime}_{2}$ is the least common multiple of $b$ and $f_{2}$. Thus
there exists no path $a_{1},\ldots,a_{e_{1}}$ with
$h(a_{e_{1}})\in(b)\cap(u_{2},u^{\prime}_{2})$ because
$h(a_{e_{1}})\not=c^{\prime}_{2}$. Applying this procedure several time we see
that there exists no path $a_{1},\ldots,a_{e_{1}}$ with
$h(a_{e_{1}})\in(b)\cap(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$.
Then we may apply Lemma 6 as above.
###### Example 4.
Let $n=5$, $I=(x_{1},\ldots,x_{4})$,
$J=(x_{2}x_{3}x_{4},x_{2}x_{3}x_{5},x_{2}x_{4}x_{5},x_{3}x_{4}x_{5})$. So
$C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5}\\},$
$B=\\{x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{3}x_{4},x_{3}x_{5},x_{4}x_{5}\\}.$
Then $q=6$, $s=10=q+r$. Set $b=x_{1}x_{5}$, $a_{1}=x_{2}x_{5}$,
$a_{2}=x_{3}x_{5}$, $a_{4}=x_{4}x_{5}$, $m_{1}=x_{1}x_{2}x_{5}$,
$m_{2}=x_{1}x_{3}x_{5}$, $m_{3}=x_{1}x_{4}x_{5}$,
$c^{\prime}_{2}=x_{1}x_{2}x_{3}$, $c^{\prime}_{3}=x_{1}x_{3}x_{4}$,
$c^{\prime}_{4}=x_{1}x_{2}x_{4}$. We have on $I_{b}/J_{b}$ the partition
$P_{b}$ given by the intervals $[x_{i},c^{\prime}_{i}]$, $2\leq i\leq 4$ and
$[a_{j},m_{j}]$, $j\in[3]$. Clearly, $P_{b}$ has sdepth $3$ and
$m_{i}=bx_{i}$, $2\leq i\leq 4$. Using the above lemma we change in $P_{b}$
the intervals $[a_{i-1},m_{i-1}]$, $[x_{i},c^{\prime}_{i}]$ with
$[f_{i},m_{i-1}]$, $[x_{i}x_{5},c^{\prime}_{i}]$ for $2\leq i\leq 4$. Now we
see that all $m$ from the new $U_{1}$ are not in
$(b)\cap(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})$.
We have $\operatorname{sdepth}_{S}I/J\leq 2$. If
$\operatorname{sdepth}_{S}I/J=3$ then there exists an interval $[x_{1},c]$
with $c\in\\{m_{1},m_{2},m_{3}\\}$. If $c=m_{i}$ for some $2\leq i\leq 4$ then
for any interval $[x_{i},c^{\prime}]$ it holds
$[x_{1},c]\cap[x_{i},c^{\prime}]=\\{x_{1}x_{i}\\}$, which is impossible. Also
we have $\operatorname{depth}_{S}I/J\leq 2$ by Lemma 12.
###### Remark 3.
Suppose that $\operatorname{sdepth}_{S}I/J\leq d+1$. We change $P_{b}$ as in
Lemma 8. Moreover assume that there exists a bad path
$a_{e_{1}+1},\ldots,a_{e_{2}}$. Using the same lemma we find $a_{e_{2}+1}$
such that for each path $a_{e_{2}+1},\ldots,a_{e_{3}}$ one has
$\\{a_{e_{1}+1},\ldots,a_{e_{2}}\\}\cap\\{a_{e_{i_{2}}+1},\ldots,a_{e_{3}}\\}=\emptyset.$
The same argument gives also
$\\{a_{1},\ldots,a_{e_{1}}\\}\cap\\{a_{e_{i_{2}}+1},\ldots,a_{e_{3}}\\}=\emptyset.$
Thus we may find some disjoint sets of elements
$\\{a_{e_{j}+1},\ldots,a_{e_{j+1}}\\}$, $j\geq 0$, where $e_{0}=0$. It follows
that after some steps we arrive in the case when for some $l$ there exist no
bad path starting with $a_{l+1}$.
###### Lemma 9.
Suppose that $\operatorname{sdepth}_{S}I/J\leq d+1$ and ${\tilde{P}}_{b}$ is a
partition of $I_{b}/J_{b}$ given by Lemma 8. Assume that no bad path starts
with $a_{1}$, $U_{1}\cap(u_{2})\not=\emptyset$ and there exists a divisor
$\tilde{a}$ in
$(B\cap(f_{2}))\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
of a monomial $m\in U_{1}\cap(u_{2})$. Then there exist a partition $P_{b}$
and a (possible bad) path $a_{1},\ldots,a_{p}$ such that
$T_{a_{p}}\cap\\{a_{1},\ldots,a_{p-1}\\}=\emptyset$, $u_{2}$ and
$c^{\prime}_{i}$, $i=3,4$ are not changed in $P_{b}$, no bad path starts with
$a_{p}$ and one of the following statements holds:
1. (1)
$U_{a_{p}}\cap(u_{2})=\emptyset$,
2. (2)
$U_{a_{p}}\cap(u_{2})\not=\emptyset$ and there exists $b_{2}\in
T_{a_{p}}\cap(f_{2})$ with $h(b_{2})\in(u_{2})$,
3. (3)
$U_{a_{p}}\cap(u_{2})\not=\emptyset$ and every monomial of
$U_{a_{p}}\cap(u_{2})$ has all its divisors from $B\cap(f_{2})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Moreover, if also $U_{1}\cap(u^{\prime}_{2})\not=\emptyset$, then we may
choose $P_{b}$ and the path $a_{1},\ldots,a_{p}$ such that either
$U_{a_{p}}\cap(u^{\prime}_{2})=\emptyset$ when there exists a bad path
starting with a divisor from
$B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ of
$c^{\prime}_{2}$, or otherwise $u^{\prime}_{2}\in T_{a_{p}}$ and
$c^{\prime}_{2}=h(u^{\prime}_{2})$.
###### Proof.
Let $a_{1},\ldots,a_{e}$ be a weak path, $m_{j}=h(a_{j})$, $j\in[e]$ such that
$m_{e}=m$. If $a_{e}={\tilde{a}}$ then take $b_{2}=a_{e}$. If
$a_{e}\not={\tilde{a}}$ but there exists $1\leq v<e$ such that
$a_{v}={\tilde{a}}$. Then we may replace in $P_{b}$ the intervals
$[a_{p},m_{p}],v\leq p\leq e$ with the intervals
$[a_{v},m_{e}],[a_{p+1},m_{p}],v\leq p<e$. The old $m_{e}$ becomes the new
$m_{v}$, that is we reduce to the above case when $v=e$.
Now assume that there exist no such $v$ but there exists a path
$a_{e+1}={\tilde{a}},\ldots,a_{l}$ such that
$m_{l}=h(a_{l})\in(a_{v^{\prime}})$ for some $v^{\prime}\in[e]$. Then we
replace in $P_{b}$ the intervals $[a_{j},m_{j}],v^{\prime}\leq j\leq l$ with
the intervals $[a_{v^{\prime}},m_{l}]$, $[a_{j+1},m_{j}]$, $v^{\prime}\leq
j<l$. The new $m_{e+1}$ is the old $m_{e}$ but the new $a_{e+1}$ is the old
$a_{e+1}$ and we may proceed as above.
Finally, suppose that no path starting with $a_{e+1}$ contains an element from
$\\{a_{1},\ldots,a_{e}\\}$. Taking $p=e+1$ we see that $m\not\in
U_{a_{p}}\cap(u_{2})$. If there exists another monomial $m^{\prime}$ like $m$
then we repeat this procedure and after a while we may get (2), or (3).
Remains to see what happens when we have also
$U_{a_{p}}\cap(u^{\prime}_{2})\not=\emptyset$. Assume that there exist no bad
path starting with a divisor of $c^{\prime}_{2}$ from
$B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$. Then
changing in $P_{b}$ the intervals $[b_{2},h(b_{2}]$, $[f_{2},c^{\prime}_{2}]$
with $[f_{2},h(b_{2})]$, $[u^{\prime}_{2},c^{\prime}_{2}]$ we see that there
exists a path $a_{1},\ldots,a_{k}$, which is not bad, such that the old
$u^{\prime}_{2}=a_{k}$. We may complete $T_{a_{p}}$ such that $a_{k}\in
T_{a_{p}}$ and all divisors from $B$ of $c^{\prime}_{2}$ which are not in
$\\{u_{2},b_{2},u_{3},u_{3}^{\prime},u_{4},u^{\prime}_{4}\\}$ belong to
$T_{a_{p}}$. For this aim we complete $T_{a_{p}}$ with the elements connected
by a path with $u^{\prime}_{2}$ (see Example 5).
Next suppose that there exists a bad path $a_{k}=u^{\prime}_{2},\ldots,a_{l}$
with $h(a_{l})\in(b)$. We may assume that ${\tilde{P}}_{b}$ is given by Lemma
8 and so there exist no multiple of $b$ in
$U_{1}\cap(u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3},u_{4},u^{\prime}_{4})$.
Note that $u^{\prime\prime}_{2}=b_{2}$ the new $u^{\prime}_{2}$ considered
above has no multiple in $U_{1}\cap(b)$ because $b_{2}\in U_{1}$. By Lemma 6
there exists $a_{l+1}\in
B\setminus\\{b,u_{2},u^{\prime\prime}_{2},u_{3},u^{\prime}_{3},u_{4},u^{\prime}_{4}\\}$
dividing $h(a_{l})$ such that every path $a_{l+1},\ldots,a_{l_{1}}$ satisfies
$\\{a_{1},\ldots,a_{l}\\}\cap\\{a_{l+1},\ldots,a_{l_{1}}\\}=\emptyset.$ Using
Remark 3 if necessary we have
$T_{a_{p^{\prime}}}\cap\\{a_{1},\ldots,a_{p^{\prime}-1}\\}=\emptyset$ for some
$p^{\prime}>l$, and the above situation will not appear, that is the old
$u^{\prime}_{2}$ will not divide anymore a monomial from
$U_{a_{p^{\prime}}}\cap(u_{2},u^{\prime\prime}_{2},u_{3},u^{\prime}_{3},u_{4},u^{\prime}_{4})$.
It is also possible that $u_{2}$ will not divide a monomial from
$U_{a_{p^{\prime}}}$.
The following bad example is similar to [9, Example 3.3].
###### Example 5.
Let $n=7$, $r=4$, $d=1$, $f_{i}=x_{i}$ for $i\in[4]$,
$E=\\{x_{5}x_{6},x_{5}x_{7}\\}$, $I=(x_{1},\ldots,x_{4},E)$ and
$J=(x_{1}x_{7},x_{2}x_{4},x_{2}x_{6},x_{2}x_{7},x_{3}x_{6},x_{3}x_{7},x_{4}x_{6},x_{4}x_{7},x_{3}x_{4}x_{5}).$
Then
$B=\\{x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},x_{1}x_{6},x_{2}x_{3},x_{2}x_{5},x_{3}x_{4},x_{3}x_{5},x_{4}x_{5}\\}\cup
E$ and
$C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{5},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5},x_{1}x_{5}x_{6},x_{2}x_{3}x_{5},x_{5}x_{6}x_{7}\\}.$
We have $q=8$ and $s=q+r=12$. Take $b=x_{1}x_{6}$ and
$I_{b}=(x_{2},x_{3},x_{4},B\setminus\\{b\\},E)$, $J_{b}=I_{b}\cap J$. There
exists a partition $P_{b}$ with sdepth $3$ on $I_{b}/J_{b}$ given by the
intervals $[x_{2},x_{1}x_{2}x_{3}]$, $[x_{3},x_{1}x_{3}x_{4}]$,
$[x_{4},x_{1}x_{4}x_{5}]$, $[x_{1}x_{5},x_{1}x_{3}x_{5}]$,
$[x_{2}x_{5},x_{1}x_{2}x_{5}]$, $[x_{3}x_{5},x_{2}x_{3}x_{5}]$,
$[x_{5}x_{6},x_{1}x_{5}x_{6}]$, $[x_{5}x_{7},x_{5}x_{6}x_{7}]$. We have
$c^{\prime}_{2}=x_{1}x_{2}x_{3}$, $c^{\prime}_{3}=x_{1}x_{3}x_{4}$,
$c^{\prime}_{4}=x_{1}x_{4}x_{5}$ and $u_{2}=x_{1}x_{2}$,
$u^{\prime}_{2}=x_{2}x_{3}$, $u_{3}=x_{3}x_{4}$, $u^{\prime}_{3}=x_{1}x_{3}$,
$u_{4}=x_{1}x_{4}$, $u^{\prime}_{4}=x_{4}x_{5}$. Take $a_{1}=x_{1}x_{5}$,
$a_{2}=x_{3}x_{5}$, $a_{3}=x_{2}x_{5}$. This gives a maximal weak path but not
bad and defines $T_{1}=\\{x_{1}x_{5},x_{3}x_{5},x_{2}x_{5}\\}$,
$U_{1}=\\{x_{1}x_{3}x_{5},x_{2}x_{3}x_{5},x_{1}x_{2}x_{5}\\}$.
As in the above lemma we may change in $P_{b}$ the intervals
$[x_{2},x_{1}x_{2}x_{3}]$, $[x_{2}x_{5},x_{1}x_{2}x_{5}]$ with
$[x_{2},x_{1}x_{2}x_{5}]$, $[x_{2}x_{3},x_{1}x_{2}x_{3}]$. Note that the old
$u^{\prime}_{2}$ is not anymore in $[f_{2},c^{\prime}_{2}]$ and divides
$x_{2}x_{3}x_{5}\in U_{1}$. Moreover, we have the path
$\\{a_{1},x_{1}x_{5},x_{3}x_{5},x_{2}x_{3}\\}$ and so we must take
$T^{\prime}_{1}=(T_{1}\cup\\{x_{2}x_{3}\\})\setminus\\{x_{2}x_{5}\\}$,
$U^{\prime}_{1}=(U_{1}\cup\\{x_{1}x_{2}x_{3}\\})\setminus\\{x_{1}x_{2}x_{5}\\}$
as it is hinted in the above proof. The new $u_{2},u^{\prime}_{2}$ are all
divisors of $x_{1}x_{2}x_{5}$ \- the new $c^{\prime}_{2}$, which are not in
$T^{\prime}_{1}$. However, this change of $P_{b}$ was not necessary because
the new $u_{2},u^{\prime}_{2},u^{\prime}_{3}$ are all divisors from $B$ of the
old $c^{\prime}_{2}$ (see Remark 7 and Example 6). The same thing is true for
$c^{\prime}_{3}$ and $c^{\prime}_{4}$ has all divisors from $B$ among
$\\{a_{1},u_{4},u^{\prime}_{4}\\}$.
###### Remark 4.
Suppose that in Lemma 9 the partition ${\tilde{P}}_{b}$ satisfies also the
property (1) mentioned in Lemma 4. If ${\tilde{a}}=w_{2i}$ for some $i=3,4$
then $m\not\in(u_{i},u^{\prime}_{i})$. In particular
$b_{2}\not=w_{23},w_{24}$.
###### Lemma 10.
Assume that $U_{a_{p}}\cap(u_{2})\not=\emptyset$ and a monomial $m$ of
$U_{a_{p}}\cap(u_{2})$ has all its divisors from $B\cap(f_{2})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$. Then one of the
following statements holds:
1. (1)
$m$ has a divisor
${\tilde{a}}_{i}\in(B\cap(f_{i}))\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
for some $i=3,4$,
2. (2)
$m\in C_{3}\setminus W$ and it is the least common multiple of
$f_{2},f_{3},f_{4}$.
###### Proof.
There exists a divisor ${\hat{a}}\not\in\\{u_{2},u^{\prime}_{2}\\}$ of $m$
from $B\cap(f_{2})$, otherwise $m=c^{\prime}_{2}$. By our assumption we have
let us say ${\hat{a}}=u_{3}=w_{23}$. Then there exists a divisor
$a^{\prime}\not=u_{3}$ from $B\cap(f_{3})$. If
$a^{\prime}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ then
we are in (1). Otherwise, $a^{\prime}=u_{4}=w_{34}$. If $m\in W$ then
$m=w_{24}\in C_{2}$ and there exists a divisor of $m$ from
$(B\cap(f_{4}))\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$,
that is (1) holds. Thus we may suppose that $m\not\in W$ and all its divisors
from $B\setminus E$ are $w_{23},w_{34},w_{24}$, that is $m$ is in (2).
###### Remark 5.
Assume that in the above lemma $m$ has the form given in Example 1. Then
$m\not\in\\{c^{\prime}_{2},c^{\prime}_{3},c^{\prime}_{4}\\}$ and so
necessarily $w_{12},w_{13},w_{14}$ are divisors of $m$ from
$B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$, that is
$m$ is in case (1).
###### Lemma 11.
Suppose that $\operatorname{sdepth}_{S}I/J\leq d+1$ and ${\tilde{P}}_{b}$ is a
partition of $I_{b}/J_{b}$ given by Lemma 8. Assume that ${\tilde{P}}_{b}$
satisfies also the properties mentioned in Lemma 4 and no bad path starts with
$a_{1}$. Then there exist a partition $P_{b}$ which satisfies the properties
mentioned in Lemma 4 and a (possible bad) path $a_{1},\ldots,a_{p}$ such that
$T_{a_{p}}\cap\\{a_{1},\ldots,a_{p-1}\\}=\emptyset$, no bad path starts with
$a_{p}$, and for every $i=2,3,4$ such that there exists a divisor
${\tilde{a}}_{i}$ in
$(B\cap(f_{i}))\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
of a monomial from $U_{1}\cap(u_{i})$, one of the following statements holds:
1. (1)
$U_{a_{p}}\cap(u_{i})=\emptyset$,
2. (2)
$U_{a_{p}}\cap(u_{i})\not=\emptyset$ and there exists $b_{i}\in
T_{a_{p}}\cap(f_{i})$ with $h(b_{i})\in(u_{i})$,
3. (3)
$U_{a_{p}}\cap(u_{i})\not=\emptyset$ and every monomial of
$U_{a_{p}}\cap(u_{i})$ has all its divisors from $B\cap(f_{i})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Moreover, these possible $b_{i}$ are different and if for some $i=2,3,4$ it
holds also $U_{1}\cap(u^{\prime}_{i})\not=\emptyset$, then we may choose
$P_{b}$ and the path $a_{1},\ldots,a_{p}$ such that either
$U_{a_{p}}\cap(u^{\prime}_{i})=\emptyset$ when there exists a bad path
starting with a divisor from
$B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ of
$c^{\prime}_{i}$, or otherwise $u^{\prime}_{i}\in T_{a_{p}}$ and
$h(u^{\prime}_{i})$ is the old $c^{\prime}_{i}$.
###### Proof.
Suppose that there exists a divisor ${\tilde{a}}_{2}$ in
$(B\cap(f_{2}))\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
of a monomial from $U_{1}\cap(u_{2})$ with respect of ${\tilde{P}}_{b}$. Using
Lemma 9 we find a partition $P_{b}$ and a (possible bad) path
$a_{1},\ldots,a_{p_{1}}$ such that
$T_{a_{p_{1}}}\cap\\{a_{1},\ldots,a_{p_{1}-1}\\}=\emptyset$, no bad path
starts with $a_{p_{1}}$ and one of the following statements holds:
$j_{2})$ $U_{a_{p_{1}}}\cap(u_{2})=\emptyset$,
$j^{\prime}_{2})$ $U_{a_{p_{1}}}\cap(u_{2})\not=\emptyset$ and there exists
$b_{2}\in T_{a_{p_{1}}}\cap(f_{2})$ with $h(b_{2})\in(u_{2})$,
$j^{\prime\prime}_{2})$ $U_{a_{p_{1}}}\cap(u_{2})\not=\emptyset$ and every
monomial of $U_{a_{p_{1}}}\cap(u_{2})$ has all its divisors from
$B\cap(f_{2})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Moreover, if also $U_{1}\cap(u^{\prime}_{2})\not=\emptyset$, then we may
choose $P_{b}$ and the path $a_{1},\ldots,a_{p_{1}}$ such that either
$U_{a_{p_{1}}}\cap(u^{\prime}_{2})=\emptyset$ when there exists a bad path
starting with a divisor from
$B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ of
$c^{\prime}_{2}$, or otherwise $u^{\prime}_{2}\in T_{a_{p_{1}}}$ and
$c^{\prime}_{2}=h(u^{\prime}_{2})$. After a small change we may suppose that
$P_{b}$ satisfies the properties of Lemma 4 and so $b_{2}\not=w_{23},w_{24}$.
If $U_{a_{p_{1}}}\cap(u_{3},u_{4})=\emptyset$ then we are done. Now assume
that there exists a divisor ${\tilde{a}}_{3}$ in
$B\cap(f_{3})\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
of a monomial $m\in U_{a_{p_{1}}}\cap(u_{3})$, let us say $m=m_{e}$ for some
path $a_{p_{1}},\ldots,a_{e}$. If $a_{e}={\tilde{a}}_{3}$, or
$a_{e}\not={\tilde{a}}_{3}$ but there exists a path
$a_{e+1}={\tilde{a}}_{3},\ldots,a_{k}$ with $a_{k}=a_{v}$ for some $v\leq e$
then we change $P_{b}$ as in the proof of Lemma 9 to replace $c^{\prime}_{3}$
by $m$. Clearly, $c^{\prime}_{2},c^{\prime}_{3}$ satisfy (2) for $i=2,3$.
Otherwise, if $a_{e}\not={\tilde{a}}_{3}$ but there exists no path
$a_{e+1}={\tilde{a}}_{3},\ldots,a_{k}$ with $a_{k}=a_{v}$ for some $v\leq e$,
apply again the quoted lemma with $c^{\prime}_{3}$. We get a (possible bad)
path $a_{p_{1}},\ldots,a_{p_{2}}$ with $p_{2}>p_{1}$ such that
$T_{a_{p_{2}}}\cap\\{a_{1},\ldots,a_{p_{2}-1}\\}=\emptyset$, no bad path
starts with $a_{p_{2}}$ and one of the following statements holds:
$j_{3})$ $U_{a_{p_{2}}}\cap(u_{3})=\emptyset$,
$j_{3}^{\prime})$ $U_{a_{p_{2}}}\cap(u_{3})\not=\emptyset$ and there exists
$b_{3}\in T_{a_{p_{2}}}\cap(f_{3})$ with $h(b_{3})\in(u_{3})$,
$j_{3}^{\prime\prime})$ $U_{a_{p_{2}}}\cap(u_{3})\not=\emptyset$ and every
monomial $m\in U_{a_{p_{2}}}\cap(u_{3})$ has all its divisors from
$B\cap(f_{3})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
If we also have $U_{1}\cap(u^{\prime}_{3})\not=\emptyset$ then it holds a
similar statement as in case $i=2$. Note that $b_{2}\not=b_{3}$ since
$b_{2}\not=w_{23}$ by Remark 4 and so $h(b_{2})\not=h(b_{3})$. Very likely
meanwhile the corresponding statements of $j_{2})$, $j_{2}^{\prime})$,
$j_{2}^{\prime\prime})$ do not hold anymore because we could have
$b_{2}\not\in T_{a_{p_{2}}}$. If there exists another ${\tilde{a}}_{2}$ we
apply again Lemma 9 with $c^{\prime}_{2}$ obtaining a new partition $P_{b}$
and a path $a_{p_{2}},\ldots,a_{p_{3}}$ for which this situation is repaired.
If now $c^{\prime}_{3}$ does not satisfy (2) then the procedure could continue
with $c^{\prime}_{3}$ and so on. However, after a while we must get a path
$a_{1},\ldots,a_{p_{23}}$ such that
$T_{a_{p_{23}}}\cap\\{a_{1},\ldots,a_{p_{23}-1}\\}=\emptyset$, no bad path
starts with $a_{p_{23}}$ and for every $i=2,3$ one of the following statements
holds:
$j_{23})$ $U_{a_{p_{23}}}\cap(u_{i})=\emptyset$,
$j_{23}^{\prime})$ $U_{a_{p_{23}}}\cap(u_{i})\not=\emptyset$ there exist
$b_{i}\in T_{a_{p_{23}}}\cap(f_{i})$ with $h(b_{i})\in(u_{i})$,
$j_{23}^{\prime\prime})$ $U_{a_{p_{23}}}\cap(u_{i})\not=\emptyset$ and every
monomial $m\in U_{a_{p_{23}}}\cap(u_{i})$ has all its divisors from
$B\cap(f_{i})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
We end the proof applying the same procedure with $c^{\prime}_{4}$ together
with $c^{\prime}_{2}$, $c^{\prime}_{3}$ and if necessary Lemma 4.
###### Remark 6.
Using the properties (2), (3) mentioned in Lemma 4 we may have $b_{i}=w_{1i}$,
for some $2\leq i\leq 4$ only if $u_{i},u^{\prime}_{i}\in W$. Thus, let us say
$b_{2}=w_{12}$ only if $\\{u_{2},u^{\prime}_{2}\\}=\\{w_{23},w_{24}\\}$. Then
$\\{u_{i},u^{\prime}_{i}\\}\not\subset W$ for $i=3,4$ and so
$b_{3}\not=w_{13}$, $b_{4}\not=w_{14}$, in case $b_{3},b_{4}$ are given by
Lemma 11. Therefore at most one from $b_{i}$ could be $w_{1i}$.
The idea of the proof of Proposition 1 fails in a special case hinted by
Example 4. This case is solved directly by the following lemma.
###### Lemma 12.
Suppose that $b=x_{j}f_{1}$ and $(B\setminus E)\subset
W\cup\\{x_{j}f_{1},x_{j}f_{2},x_{j}f_{3},x_{j}f_{4}\\}$ for some
$j\not\in\operatorname{supp}f_{1}$. Then $\operatorname{depth}_{S}I/J\leq
d+1$.
###### Proof.
If $|B\setminus E|<2r=8$ then
$\operatorname{depth}_{S}I^{\prime\prime}/J^{\prime\prime}\leq 2$ by [18,
Theorem 2.4]. Assume that $|B\setminus E|\geq 8$. Our hypothesis gives $|B\cap
W|\geq 4$. First assume that $5\leq|B\cap W|\leq 6$ and we get that let us say
$f_{i}=vx_{i}$, $1\leq i\leq 4$ for some monomial $v$ of degree $d-1$ (see the
proof of [16, Lemma 3.2]). Then
$\operatorname{depth}_{S}I/J=\operatorname{deg}v+\operatorname{depth}_{S^{\prime}}((I:v)\cap
S^{\prime})/((J:v)\cap S^{\prime}),$
$S^{\prime}=K[\\{x_{i}:i\in([n]\setminus\operatorname{supp}v)\\}]$ and it is
enough to show the case $v=1$, that is $d=1$.
We may assume that $f_{i}=x_{i}$, $i\in[4]$ and $j=5$ since $b\not\in W$. It
follows that $(B\setminus E)\subset
W\cup\\{b,x_{2}x_{5},x_{3}x_{5},x_{4}x_{5}\\}$. Set
$I^{\prime\prime}=(x_{1},\ldots,x_{4})$, $J^{\prime\prime}=J\cap
I^{\prime\prime}$. Note that
$J\supset(x_{1},\ldots,x_{5})(x_{6},\ldots,x_{n})$ and so
$\operatorname{depth}_{S}I^{\prime\prime}/J^{\prime\prime}=\operatorname{depth}_{S^{\prime\prime}}(I^{\prime\prime}\cap
S^{\prime\prime})/(J^{\prime\prime}\cap S^{\prime\prime})$ for
$S^{\prime\prime}=K[x_{1},\ldots,x_{5}]$.
Then $J^{\prime\prime}\cap S^{\prime\prime}$ is generated by at most two
monomials and so
$\operatorname{depth}_{S^{\prime\prime}}S^{\prime\prime}/(J^{\prime\prime}\cap
S^{\prime\prime})\geq 3$. Since
$\operatorname{depth}_{S^{\prime\prime}}S^{\prime\prime}/(I^{\prime\prime}\cap
S^{\prime\prime})=1$ it follows that
$\operatorname{depth}_{S}I^{\prime\prime}/J^{\prime\prime}=\operatorname{depth}_{S^{\prime\prime}}(I^{\prime\prime}\cap
S^{\prime\prime})/(J^{\prime\prime}\cap S^{\prime\prime})=2$. Therefore
$\operatorname{depth}_{S}I/J\leq 2$ either when $E=\emptyset$ or by the Depth
Lemma since $I/(J,I^{\prime\prime})$ is generated by monomials of $E$ which
have degrees $2$.
Now assume that $|B\cap W|=4$, let us say $B\cap
W=\\{w_{14},w_{23},w_{24},w_{34}\\}$. Then we may suppose that
$f_{i}=vx_{i}x_{6}$, $2\leq i\leq 4$ and $f_{1}=vx_{1}x_{4}$ for some monomial
$v$ of degree $d-2$. As above we may assume that $v=1$ and $n=6$. If $j=6$
then $b=w_{14}$ which is impossible. If let us say $j=2$ then $(B\setminus
E)\subset W\cup\\{b,x_{2}x_{3}x_{6},x_{2}x_{4}x_{6}\\}$ and so $|B\setminus
E|<8$, which is false.
Thus $j\not\in\\{1,\ldots,4,6\\}$ and we may assume that $j=5$. It follows
that
$J\subset(x_{1}x_{2}x_{6},x_{1}x_{3}x_{6},x_{1}x_{2}x_{4},x_{1}x_{3}x_{4})$,
the inclusion being strict only if $|B\setminus E|<8$ which is not the case.
Thus $J=(x_{1}x_{2}x_{6},x_{1}x_{3}x_{6},x_{1}x_{2}x_{4},x_{1}x_{3}x_{4})$ and
a computation with SINGULAR shows that $\operatorname{depth}_{S}I/J=3$ in this
case.
Next we put together the above lemmas to get the proof of Proposition 1.
Assume that $\operatorname{sdepth}_{S}I/J\leq d+1$. We may suppose always that
$P_{b}$ satisfies the properties mentioned in Lemma 4. Applying Lemma 8 and
Remark 3 and changing $a_{1}$ if necessary we may suppose that no bad path
starts from $a_{1}$. By Lemma 11 changing $a_{1}$ by $a_{p}$ we may suppose
that for every $i=2,3,4$ one of the following statements holds
1) $U_{1}\cap(u_{i})=\emptyset$,
2) $U_{1}\cap(u_{i})\not=\emptyset$ and there exists $b_{i}\in
T_{1}\cap(f_{i})$ with $h(b_{i})\in(u_{i})$,
3) $U_{1}\cap(u_{i})\not=\emptyset$ and every monomial of $U_{1}\cap(u_{i})$
has all its divisors from $B\cap(f_{i})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Mainly we study case 3) the other two cases are easier as we will see later.
Suppose that $U_{1}\cap(u_{2})\not=\emptyset$ and every monomial of
$U_{1}\cap(u_{2})$ has all its divisors from $B\cap(f_{2})$ contained in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$. Let $m\in
U_{1}\cap(u_{2})$, let us say $m=h(a_{e})$ for some path $a_{1},\ldots,a_{e}$.
be as in case 3). We may suppose that $U_{1}\cap(u^{\prime}_{2})=\emptyset$
because otherwise we may assume as in Lemma 9 that all divisors of
$c^{\prime}_{2}$ are in the enlarged $T^{\prime}_{1}$ of $T_{1}$ and so
$c^{\prime}_{2}$ is preserved. As in the proof of Lemma 10 one of the
following statements holds:
$1^{\prime})$ $U_{1}\cap(u_{2})=\\{m\\}$, $m\in(u_{2})\cap(u_{3})$,
$u_{3}=w_{23}$, $m\not\in(u_{4},u^{\prime}_{4})$ and there exists
${\tilde{a}}_{3}\in T_{1}\cap(f_{3})$ dividing $m$ with
${\tilde{a}}_{3}=a_{e}$,
$2^{\prime})$ $U_{1}\cap(u_{2})=\\{m\\}$, $m\in(u_{2})\cap(u_{3})$,
$u_{3}=w_{23}$, $m\not\in(u_{4},u^{\prime}_{4})$ and there exists
${\tilde{a}}_{3}\in T_{1}\cap(f_{3})$ dividing $m$ with
${\tilde{a}}_{3}\not=a_{e}$,
$3^{\prime})$ $U_{1}\cap(u_{2})=\\{m\\}$, $m\in(u_{2})\cap(u_{4})$,
$u_{4}=w_{24}$, $m\not\in(u_{3},u^{\prime}_{3})$ and there exists
${\tilde{a}}_{4}\in T_{1}\cap(f_{4})$ dividing $m$ with
${\tilde{a}}_{4}=a_{e}$,
$4^{\prime})$ $U_{1}\cap(u_{2})=\\{m\\}$, $m\in(u_{2})\cap(u_{4})$,
$u_{4}=w_{24}$, $m\not\in(u_{3},u^{\prime}_{3})$ and there exists
${\tilde{a}}_{4}\in T_{1}\cap(f_{4})$ dividing $m$ with
${\tilde{a}}_{4}\not=a_{e}$,
$5^{\prime})$ $m=w_{24}\in(u_{2})\cap(u_{3})\cap(u_{4})$, $u_{3}=w_{23}$,
$u_{4}=w_{34}$ and there exists ${\tilde{a}}_{4}\in T_{1}\cap(f_{4})$ dividing
$m$ with $h({\tilde{a}}_{4})=m$,
$6^{\prime})$ $m=w_{24}\in(u_{2})\cap(u_{3})\cap(u_{4})$, $u_{3}=w_{23}$,
$u_{4}=w_{34}$ and there exists ${\tilde{a}}_{4}\in T_{1}\cap(f_{4})$ dividing
$m$ with $h({\tilde{a}}_{4})\not=m$,
$7^{\prime})$ $m=\omega_{1}\in C_{3}$, $u_{2}=w_{24}$, $u_{3}=w_{23}$.
In subcase $1^{\prime})$ change in $P_{b}$ the intervals
$[f_{3},c^{\prime}_{3}]$, $[{\tilde{a}}_{3},m]$ with $[f_{3},m]$,
$[u^{\prime}_{3},c^{\prime}_{3}]$. The new
$T_{1}^{\prime\prime}=T_{1}\setminus\\{{\tilde{a}}_{3}\\}$ corresponds to
$U_{1}^{\prime\prime}=U_{1}\setminus\\{m\\}$ which has empty intersection with
$(u_{2})$ by our assumption. If $T_{1}^{\prime\prime}$ is not empty then we
may go on with $T_{1}^{\prime\prime}$ instead $T_{1}$, the advantage being
that now we have no problem with $u_{2}$. If $T_{1}^{\prime\prime}=\emptyset$
then $e=1$ and the path $a_{1}$ is maximal. Since
$m\not\in(u_{4},u^{\prime}_{4})$ we must have $u_{2}=x_{k}f_{2}$ for some $k$
(we can also have $w_{12}=x_{k}f_{2}$) and so $m=x_{k}w_{23}$,
${\tilde{a}}_{3}=x_{k}f_{3}$. If $E\not=\emptyset$ then we may change $a_{1}$
by a monomial of $E$. Assume that $E=\emptyset$. If
$c^{\prime}_{3}=x_{t}w_{23}$ for some $t$ then $x_{t}f_{2}\in B$ since it
divides $c^{\prime}_{3}$. If $t=k$ then $m=c^{\prime}_{3}$. Thus $t\not=k$,
$x_{t}f_{2}\not\in\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
and we may change $a_{1}$ by $x_{t}f_{2}$ and the new $T_{1}^{\prime\prime}$
will be not empty. If $c^{\prime}_{3}\in C_{2}$ we may find also a divisor
$b^{\prime}\in
B\setminus\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ dividing
$c^{\prime}_{3}$ and changing $a_{1}$ by $b^{\prime}$ we will get the new
$T_{1}^{\prime\prime}$ not empty. Remains to assume that $c^{\prime}_{3}\in
C_{3}$. Then $u^{\prime}_{3}=w_{34}$ and $b^{\prime\prime}=w_{24}$ is either
in $\\{u_{2}^{\prime},u_{4},u_{4}^{\prime}\\}$, or we may change $a_{1}$ by
$b^{\prime\prime}$ as above. Suppose that $u^{\prime}_{2}=w_{24}$. Then
$x_{k}f_{4}\in B$. If
$x_{k}f_{4}\not\in\\{b,u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ we
may change $a_{1}$ by $x_{k}f_{4}$. Otherwise, let us say $u_{4}=x_{k}f_{4}$
and $c^{\prime}_{4}=x_{k}w_{14}$. We get $x_{k}f_{1}\in
B\setminus\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and if
$b\not=x_{k}f_{1}$ then we may change as above $a_{1}$ by $x_{k}f_{1}$. If
$b=x_{k}f_{1}$ then note that
$B\supset\\{w_{23},w_{24}.w_{34},w_{14},b,x_{k}f_{2},x_{k}f_{3},x_{k}f_{4}\\}$.
If there exists a monomial $b^{\prime}\in
B\setminus(W\cup\\{b,x_{k}f_{2},x_{k}f_{3},x_{k}f_{4}\\})$ then change $a_{1}$
by $b^{\prime}$. Otherwise $B\subset
W\cup\\{b,x_{k}f_{2},x_{k}f_{3},x_{k}f_{4}\\}$ and we apply Lemma 12.
Therefore in this subcase changing $P_{b}$ ($u_{3}$ is preserved and the new
$u^{\prime}_{3}$ is $b_{3}$) and passing from $T_{1}$ to
$T_{1}^{\prime\prime}$ there exist no problem with $u_{2}$. As in Lemma 9 we
may suppose that only one from $U_{1}^{\prime\prime}\cap(u_{3})$,
$U_{1}^{\prime\prime}\cap(u^{\prime}_{3})$ is nonempty because otherwise we
preserve the new $c^{\prime}_{3}$, that is $m$. If let us say
$U_{1}^{\prime\prime}\cap(u_{3})=\\{m^{\prime}\\}$, and all divisors of
$m^{\prime}$ from $B\cap(f_{3})$ are contained in
$\\{u_{3},u^{\prime}_{3},u_{4},u^{\prime}_{4}\\}$ then
$m^{\prime}\in(u_{3})\cap(u_{4})$, $u_{4}=w_{34}$ and there exists
${\tilde{a}}_{4}\in T_{1}^{\prime\prime}\cap(f_{4})$ dividing $m^{\prime}$. If
$h({\tilde{a}}_{4})=m^{\prime}$ then as above change in $P_{b}$ the intervals
$[f_{4},c^{\prime}_{4}]$, $[{\tilde{a}}_{4},m^{\prime}]$ with
$[f_{4},m^{\prime}]$, $[u_{4}^{\prime},c^{\prime}_{4}]$. Clearly
${\tilde{T}}_{1}=T_{1}^{\prime\prime}\setminus\\{{\tilde{a}}_{4}\\}$ has empty
intersection with $(u_{3})$ and similarly to above we may suppose that
${\tilde{T}}_{1}\not=\emptyset$. In this way we arrive to the situation when
we will not meet case 3) for $2\leq i\leq 4$.
In subcase $2^{\prime})$ we have $a_{e}\in E$ and $a_{e+1}={\tilde{a}}_{3}\in
T_{1}$. Take $T_{a_{e+1}}$ instead $T_{1}$. If $a_{e}$ will not appear anymore
in $T_{a_{e+1}}$ then $U_{a_{e+1}}\cap(u_{2})=\emptyset$ and the problem is
solved. Otherwise, if $a_{v}=a_{e}$ for some $v>e+1$ then change in $P_{b}$
the intervals $[a_{i},h(a_{i})]$, $e\leq i\leq v$ with $[a_{i+1},h(a_{i})]$,
$e\leq i<v$, $[a_{e},m_{v}]$ we see that the new $a_{e}$ is the old $a_{e+1}$,
that is we reduced to the subcase $1^{\prime})$. Subcases $3^{\prime})$,
$4^{\prime})$ are similar to $1^{\prime})$, $2^{\prime})$.
Change in subcase $5^{\prime})$ (as in subcase $1^{\prime})$) the intervals
$[f_{4},c^{\prime}_{4}]$, $[{\tilde{a}}_{4},m]$ of $P_{b}$ with $[f_{4},m]$,
$[u^{\prime}_{4},c^{\prime}_{4}]$. The new
$T_{1}^{\prime\prime}=T_{1}\setminus\\{{\tilde{a}}_{4}\\}$ corresponds to
$U_{1}^{\prime\prime}=U_{1}\setminus\\{m\\}$ which has empty intersection with
$(u_{2})$ by our assumption. The proof continues as in $1^{\prime})$.
Similarly, $6^{\prime})$ goes as $2^{\prime})$.
In subcase $7^{\prime})$ if $\omega_{1}\in W$ (see Example 1) then it has $4$
divisors from $B\setminus E$ and so one of them is not in
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and we may proceed as
in subcases $5^{\prime})$, $6^{\prime})$. So we may assume that
$\omega_{1}\not\in W$. Then either $u_{4}=w_{34}$ and then $a_{e}\in E$ which
is false by our assumption, or $w_{34}\in T_{1}$. Set $a_{e+1}=w_{34}$. We
proceed as in $2^{\prime})$ taking $T_{a_{e+1}}$ if $a_{e}\not\in T_{a_{e+1}}$
or otherwise changing $P_{b}$ we reduce to the situation when $h(a_{e+1})=m$.
Then change in $P_{b}$ the intervals $[f_{4},c^{\prime}_{4}]$, $[a_{e+1},m]$
with $[f_{4},m]$, $[u^{\prime}_{4},c^{\prime}_{4}]$ and as usual the new
$U_{1}^{\prime\prime}=U_{1}\setminus\\{m\\}$ has empty intersection with
$(u_{2})$.
Thus we may assume that for all $2\leq i\leq 4$ we are in cases 1), 2). When
we are in case 2) there exists $b_{i}\in T_{1}\cap(f_{i})$ with
$h(b_{i})\in(u_{i})$ and we may consider the intervals
$[f_{i},c^{\prime}_{i}]$, which are disjoint since $b_{i}$ are different by
Lemma 11. Moreover, they contain at most one monomial from
$w_{12},w_{13},w_{14}$ by Remark 6, which is useful next. Remains to study
those $i$ with $U_{1}\cap(f_{i})\not=\emptyset$ but
$U_{1}\cap(u_{i},u^{\prime}_{i})=\emptyset$. If
$U_{1}\cap(u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4})=\emptyset$ then
we apply Lemma 5. Suppose that $U_{1}\cap(f_{2})\not=\emptyset$ and
$U_{1}\cap(u_{2},u^{\prime}_{2})=\emptyset$ but we found already $b_{3}$ and
possible $b_{4}$ as in 2). If $h(b_{3})\not\in(f_{2})$ then choosing
$b^{\prime}\in B\cap(f_{2})$ we see that the intervals
$[f_{2},h(b^{\prime})]$, $[f_{3},h(b_{3})]$ are disjoint. A similar result
holds if there exists $b_{4}$ and $h(b_{4})\not\in(f_{2})$.
Assume that $h(b_{3})\in(f_{2})$. Then we may suppose that $u_{3}=w_{23}$ and
$h(b_{3})=x_{k}w_{23}$ for some $k\in[n]\setminus\operatorname{supp}w_{23}$.
We claim that
$b^{\prime\prime}=x_{k}f_{2}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
It is clear that
$b^{\prime\prime}\not\in\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$. If
$b^{\prime\prime}\in\\{u_{4},u^{\prime}_{4}\\}$ then
$b^{\prime\prime}=w_{24}=u_{4}$, let us say. Thus $h(b_{3})\in(u_{3},u_{4})$
but $h(b_{3})\not\in(u_{2},u^{\prime}_{2})$. This means that the monomial
$h(b_{3})\in U_{1}\cap(u_{4})$ is in the situation 3) (similarly to
$1^{\prime})$) which is not possible as we assumed. This shows our claim.
Therefore, $b^{\prime\prime}\in T_{1}\cap(f_{2})$ because it divides
$h(b_{3})$. If $h(b^{\prime\prime})\in(f_{3})$ then
$h(b^{\prime\prime})=kw_{23}=h(b_{3})$ which is impossible. If
$h(b^{\prime\prime})\in(f_{4})$ then $h(b^{\prime\prime})=x_{t}w_{24}$ for
some $t$. As we saw above $b^{\prime\prime}\not=w_{24}$ and so $t=k$. If
$b_{4}$ is not done by 2) then it is enough to note that the intervals
$[f_{2},h(b^{\prime\prime})]$, $[f_{3},h(b_{3})]$ are disjoint. Assume that
$b_{4}$ is given already from 2) and $u_{4}=w_{24}$. Then
${\tilde{b}}=x_{k}f_{4}\not=u^{\prime}_{4}$ because otherwise
$h(b^{\prime\prime})=h(b_{4})$. We see that
${\tilde{b}}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$ and
so $\tilde{b}$ is in $T_{1}\cap(f_{4})$. But $h({\tilde{b}})\not\in(u_{4})$
because it is different of $h(b_{4})$. Then the intervals
$[f_{2},h(b^{\prime\prime})]$, $[b_{3},h(b_{3})]$, $[f_{4},h({\tilde{b}})]$
are disjoint. As in Lemma 5 we find if necessary an interval $[f_{1},c]$
disjoint of the rest.
Suppose as in Lemma 5 that
$[r]\setminus\\{j\in[r]:U_{1}\cap(f_{j})\not=\emptyset\\}=\\{k_{1},\ldots,k_{\nu}\\}$
for some $1\leq k_{1}<\ldots<k_{\nu}\leq 4$, $0\leq\nu\leq 4$. Set
$I^{\prime}=(f_{k_{1}},\ldots,f_{k_{\nu}},G_{1})$, $J^{\prime}=I^{\prime}\cap
J$, With the help of the above disjoint intervals, $P_{b}$ induces on
$I/(I^{\prime},J)$ a partition $P^{\prime}_{b}$ with sdepth $d+2$. It follows
that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ using [17, Lemma
2.2]. By Lemma 3 we get $\operatorname{depth}_{S}I/(J,I^{\prime})\leq d+1$ and
we are done. $\hfill\ \square$
###### Remark 7.
Note that in $P^{\prime}_{b}$, all divisors from $B$ of the new
$c^{\prime}_{i}$ are in
$T_{1}\cup\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$. If one old
$c^{\prime}_{i}$ has already this property then we may keep it.
###### Remark 8.
If $\omega_{1}\in(C_{3}\setminus W)\cap(E)$ then we may have indeed a problem.
For example, if $u_{2}=w_{24}$, $u_{3}=w_{23}$, $u_{4}=w_{34}$,
$\omega_{1}=h(a_{1})$ for some $a_{1}\in E$ but $\omega_{1}\not\in
h(E\setminus\\{a_{1}\\})$ then the path $a_{1}$ is maximal,
$T_{1}=\\{a_{1}\\}$ and our theory fails to solve this case if we cannot
change $P_{b}$ in order to have
$\\{u_{2},u_{3},u_{4}\\}\not=\\{w_{24},w_{23},w_{34}\\}$.
###### Example 6.
We continue Example 5. If we take as in the above proof
$I^{\prime}=(b,x_{5}x_{6},x_{5}x_{7})$ and $J^{\prime}=I^{\prime}\cap J$ we
have the disjoint intervals $[x_{i},c^{\prime}_{i}]$, $2\leq i\leq 4$ and to
conclude that $h$ induces a partition on $I/(I^{\prime},J)$, which has sdepth
$3$ we need an interval $[x_{1},c^{\prime}_{1}]$ disjoint of the other ones.
But this is hard because there are too many $w_{1i}$ among
$\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$. We must change one
$c^{\prime}_{i}$ with one $m\in(U_{1}\cap(x_{i}))\setminus(x_{1})$. The only
possibility is to take $m_{2}=x_{2}x_{3}x_{5}$. Since
$m\in(u^{\prime}_{2})\setminus(u_{3},u_{3}^{\prime},u_{4},u^{\prime}_{4})$ we
may change somehow $c^{\prime}_{2}$ with $m$. This is not easy since
$m_{2}=h(a_{2})$, $a_{2}=x_{3}x_{5}\not\in(x_{2})$. As in Lemma 9 note that
$a_{1}|m_{3}=h(a_{3})$ and replacing in $P_{b}$ the intervals $[a_{i},m_{i}]$,
$i\in[3]$, $m_{1}=h(a_{1})$ with the intervals $[a_{1},m_{3}]$,
$[a_{2},m_{1}]$, $[a_{3},m_{2}]$ we see that $x_{2}x_{5}$ \- the new $a_{2}$,
belongs to $(x_{2})$. Thus we may change in $P_{b}$ the intervals
$[x_{2},c^{\prime}_{2}]$, $[x_{2}x_{5},m_{2}]$ with $[x_{2},m_{2}]$,
$[u_{2},c^{\prime}_{2}]$. The new $T_{1}$ is
$T^{\prime}_{1}=(T_{1}\cup\\{x_{1}x_{2}\\})\setminus\\{x_{2}x_{5}\\}$. Note
that all divisors from $B\cap(x_{2})$ of the new $c^{\prime}_{2}$ which are
different from the new $u_{2},u^{\prime}_{2}$ are contained in the new
$T_{1}$. As above $[x_{i},c^{\prime}_{i}]$ are disjoint intervals and changing
in $P_{b}$ the intervals $[x_{1}x_{2},x_{1}x_{2}x_{3}]$,
$[x_{1}x_{5},x_{1}x_{2}x_{5}]$ with $[x_{1},x_{1}x_{2}x_{5}]$ we get a
partition with sdepth $3$ on $I/(I^{\prime},J)$.
## 3\. Main results
We start with an elementary lemma closed to Lemma 12.
###### Lemma 13.
Let $r$ be arbitrarily chosen, $r^{\prime}\leq r$,
$t\in[n]\setminus\cup_{i=1}^{r^{\prime}}\operatorname{supp}f_{i}$ and
$I^{\prime}=(f_{1},\ldots,f_{r^{\prime}})$, $J^{\prime}=J\cap I^{\prime}$.
Suppose that all $w_{ij}$, $1\leq i<j\leq r^{\prime}$ are in $B$ and
different. Then the following statements hold
1. (1)
there exists a monomial $v$ of degree $d-1$ such that $f_{i}\in(v)$ for all
$i\in[r^{\prime}]$,
2. (2)
if $x_{k}(f_{1},\ldots,f_{r^{\prime})}\subset J$ for all
$k\in[n]\setminus(\\{t\\}\cup(\cup_{i=1}^{r^{\prime}}\operatorname{supp}f_{i}))$
then $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$.
###### Proof.
As in the proof of [16, Lemma 3.2] we may suppose that $f_{i}=vx_{i}$ for
$i\in[r]$ and some monomial $v$ of degree $d-1$, that is (1) holds. It follows
that
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=d-1+\operatorname{depth}_{S^{\prime\prime}}(x_{1},\ldots,x_{r^{\prime}})S^{\prime\prime}=d+1$
where $S^{\prime\prime}=K[x_{1},\ldots,x_{r^{\prime}},x_{t}]$.
###### Theorem 3.
Conjecture 1 holds for $r\leq 4$, the case $r\leq 3$ being given in Theorem 1.
###### Proof.
Suppose that $\operatorname{sdepth}_{S}I/J=d+1$ and $E\not=\emptyset$, the
case $E=\emptyset$ is given in Proposition 2. The proofs of Proposition 1 and
Proposition 2 show that we get $\operatorname{depth}_{S}I/J\leq d+1$, that is
Conjecture 1 holds, when we may choose $b_{i}\in(B\cap(f_{i}))\setminus W$
such that $\omega_{i}\not\in(C_{3}\setminus W)\cap(E)$. Suppose that we choose
$b_{1}\in(B\cap(f_{1}))\setminus W$ but $\omega_{1}\in(C_{3}\setminus
W)\cap(E)$. In the last part of the proof of Proposition 1 (see $7^{\prime})$
and also Remark 8) a problem appears when $m=\omega_{1}\in T_{1}$ and let us
say $u_{2}=w_{24}$, $u_{3}=w_{23}$, $u_{4}=w_{34}$. As in the proof of [PZ@,
Lemma 3.2] we may assume that $f_{i}=vx_{i}$ for $2\leq i\leq 4$ and some
monomial $v$ of degree $d-1$. If let us say $x_{t}f_{2}\in B$ for some
$t\not\in\cup_{i=2}^{4}\operatorname{supp}f_{i}$ then either $tf_{2}=w_{12}$,
or $tf_{2}\not\in W$. In the first case we may suppose, as in the proof of
Lemma 12, that one of the following statements hold:
1) $f_{i}=vx_{i}$, $i\in[4]$ for some monomial $v$ of degree $d-1$,
2) $f_{i}=px_{i}x_{5}$, $2\leq i\leq 4$, $f_{1}=px_{1}x_{2}$ for some monomial
$p$ of degree $d-2$.
In both cases we see that if $B\cap(f_{2},f_{3},f_{4})\subset W$ then we have
$x_{k}(f_{2},\ldots f_{4})\subset J$ for all
$k\in[n]\setminus(\\{1\\}\cup(\cup_{i=2}^{4}\operatorname{supp}f_{i}))$. By
Lemma 13 we get $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for
$I^{\prime}=(f_{2},f_{3},f_{4})$, $J^{\prime}=J\cap I^{\prime}$ which gives
$\operatorname{depth}_{S}I/J\leq d+1$ since
$\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$, $b$ being not in
$(J,I^{\prime})$. Thus $B\cap(f_{2},f_{3},f_{4})\not\subset W$ and we may
choose, let us say $b_{2}\in(B\cap(f_{2}))\setminus W$ and again we may get
$\operatorname{depth}_{S}I/J\leq d+1$ if $\omega_{2}\not\in(C_{3}\setminus
W)\cap(E)$.
Thus we may assume that $\omega_{1},\omega_{2}\in(C_{3}\setminus W)\cap(E)$.
In particular $B\cap W$ consists in at least $5$ different monomials and so we
may suppose that 1) above holds and $u^{\prime}_{2}=vx_{2}x_{k_{2}}$,
$u^{\prime}_{3}=vx_{3}x_{k_{3}}$, $u^{\prime}_{4}=vx_{4}x_{k_{4}}$ for some
$k_{i}\in([n]\setminus(\\{2,3,4\\}\cup\operatorname{supp}v)$. If
$k_{2}=k_{3}=k_{4}=1$ then $c^{\prime}_{2}=\omega_{3}$,
$c^{\prime}_{3}=\omega_{4}$, $c^{\prime}_{4}=\omega_{2}$, that is all
$\omega_{i}$ are in $C_{3}\setminus W$. If let us say $k_{3}>4$ then
$b^{\prime\prime}=x_{k_{3}}f_{3}\not\in W$ and we are ready if
$\omega_{3}\not\in(C_{3}\setminus W)\cap(E)$. Thus we may assume that
$\omega_{3}\in(C_{3}\setminus W)\cap(E)$. Consequently in all cases we may
assume that $3$ from $\omega_{i}$ are in $C_{3}\setminus W$. In particular
$|B\cap W|=6$. If $B\cap(f_{i})\subset W$ for some $i=3,4$ then $(J:f_{i})$ is
generated by $x_{j}$ with
$j\not\in(\\{1,\ldots,4\\}\cup\operatorname{supp}v)$. It follows that in the
exact sequence
$0\to(f_{i})/J\cap(f_{i})\to I/J\to I/(J,f_{i})\to 0$
the first term has depth $\operatorname{deg}v+4=d+3$ and sdepth $\geq d+2$. By
[17, Lemma 2.2] we get $\operatorname{sdepth}_{S}I/(J,f_{i})\leq d+1$ and so
the last term in the above sequence has depth $\leq d+1$ by Theorem 1. Using
the Depth Lemma we get $\operatorname{depth}_{S}I/J\leq d+1$ too.
Therefore, we may find $b_{i}\in(B\cap(f_{i}))\setminus W$, $i=3,4$ and as
above we may suppose that $\omega_{i}\in(C_{3}\setminus W)\cap(E)$, let us say
$\omega_{i}\in({\tilde{a}}_{i})$ for some ${\tilde{a}}_{i}\in E$. We consider
three cases depending on $k_{i}$.
Case 1, when $k_{i}=1$ and $k_{j}>4$ for some $i,j=2,3,4$, $i\not=j$.
Assume that $k_{2}=1$, that is $c^{\prime}_{2}=\omega_{3}$ and $k_{4}>1$. Then
$a_{1}=vx_{1}x_{4}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$
is a divisor of $c^{\prime}_{2}$. Start the usual proof with $a_{1}$ and if
$\omega_{1}\not\in U_{1}$ then we get $\operatorname{depth}_{S}I/J\leq d+1$.
Suppose that there exists a (possible bad) path $a_{1},\ldots,a_{e}$,
$m_{i}=h(a_{i})$ such that $m_{e}=\omega_{1}$. Changing in $P_{b}$ the
intervals $[a_{i},m_{i}]$, $i\in[e]$, $[f_{2},c^{\prime}_{2}]$,
$[f_{3},c^{\prime}_{3}]$ with $[a_{i+1},m_{i}]$, $i\in[e-1]$,
$[f_{1},c^{\prime}_{2}]$, $[f_{2},m_{e}]$, $[u^{\prime}_{3},c^{\prime}_{3}]$
we see that the new ${\tilde{c}}^{\prime}_{i}$, $i=1,2,4$ contain two from
$\omega_{i}$. Choose a new $a_{1}$ and start to build $U_{1}$. This time any
monomial from $U_{1}$ has at least one divisor from $B\setminus E$ which is
not in $\cup_{j=1,2,4}[f_{j},{\tilde{c}}^{\prime}_{j}]$ so the usual proof
goes.
Case 2, $k_{2},k_{3},k_{4}>4$.
Then
$a_{1}=vx_{1}x_{4}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
Let $m_{1}=h(a_{1})=a_{1}x_{k}$ for some $k$. If $k=k_{4}$ then changing in
$P_{b}$ the intervals $[f_{4},c^{\prime}_{4}]$, $[a_{1},m_{1}]$ with
$[f_{4},m_{1}]$, $[u_{4},c^{\prime}_{4}]$ we see that $u_{4}=w_{34}$ does not
divide the new $c^{\prime}_{4}$ and so we have no problem with $\omega_{1}$.
Suppose that $k\not=k_{4}$ and $k>4$ then
$a_{2}=vx_{4}x_{k}\not\in\\{u_{2},u^{\prime}_{2},\ldots,u_{4},u^{\prime}_{4}\\}$.
If there exists no path $a_{2},\ldots,a_{e}$, $m_{i}=h(a_{i})$ with
$m_{e}=\omega_{1}$ then we proceed as usual. Otherwise, let
$a_{2},\ldots,a_{e}$, $m_{i}=h(a_{i})$ be a (possible bad) path with
$m_{e}=\omega_{1}$. Changing in $P_{b}$ the intervals $[a_{i},m_{i}]$,
$i\in[e]$, $[f_{3},c^{\prime}_{3}]$, $[f_{4},c^{\prime}_{4}]$ with
$[a_{i+2},m_{i+1}]$, $i\in[e-2]$, $[f_{3},m_{e}]$, $[f_{4},m_{1}]$,
$[u^{\prime}_{3},c^{\prime}_{3}]$, $[u^{\prime}_{4},c^{\prime}_{4}]$ we see
that any monomial from $C$ has at least one divisor from $B\setminus E$ which
is not in $\cup_{j=2,3,4}[f_{j},{\tilde{c}}^{\prime}_{j}]$ so the usual proof
goes, where ${\tilde{c}}^{\prime}_{j}$ denotes the new $c^{\prime}_{j}$ for
$j=3,4$ and ${\tilde{c}}^{\prime}_{2}=c^{\prime}_{2}$.
Remains to study the case when $k\not=k_{4}$ and $k=2$ or $k=3$. Assume that
$k=2$, that is $m_{1}=\omega_{3}$. Similarly we may assume that
$a_{2}=vx_{1}x_{2}$, $m_{2}=h(a_{2})=a_{2}x_{3}=\omega_{4}$ and
$a_{3}=vx_{1}x_{3}$, $m_{3}=h(a_{3})=a_{3}x_{4}=\omega_{2}$. If there exists
no path $a_{3},\ldots,a_{e}$, $m_{i}=h(a_{i})$ with $m_{e}=\omega_{1}$ then we
proceed as usual. Otherwise, let $a_{3},\ldots,a_{e}$, $m_{i}=h(a_{i})$ be a
(possible bad) path with $m_{e}=\omega_{1}$. Changing in $P_{b}$ the intervals
$[a_{i},m_{i}]$, $i\in[e]$, $[f_{j},c^{\prime}_{j}]$, $j=2,3,4$ with
$[a_{i+3},m_{i+2}]$, $i\in[e-3]$, $[f_{1},m_{1}]$, $[f_{3},m_{2}]$,
$[f_{4},\omega_{1}]$, $[u^{\prime}_{2},c^{\prime}_{2}]$,
$[u^{\prime}_{3},c^{\prime}_{3}]$, $[u^{\prime}_{4},c^{\prime}_{4}]$ we arrive
in a case similar to the next one.
Case 3, $k_{2}=k_{3}=k_{4}=1$.
Thus $c^{\prime}_{2}=\omega_{3}\in(a_{1})$ for $a_{1}={\tilde{a}}_{3}$. If
there exists a path $a_{1},\ldots,a_{e}$, $m_{i}=h(a_{i})$ with
$m_{e}=\omega_{1}$ then changing in $P_{b}$ the intervals $[a_{i},m_{i}]$,
$i\in[e]$, $[f_{2},c^{\prime}_{2}]$, $[f_{3},c^{\prime}_{3}]$ with
$[a_{i+1},m_{i}]$, $i\in[e-1]$, $[a_{1},c^{\prime}_{2}]$,
$[f_{1},c^{\prime}_{3}]$, $[f_{2},\omega_{1}]$ we get the new
${\tilde{c}}^{\prime}_{1}=\omega_{4}$, ${\tilde{c}}^{\prime}_{2}=\omega_{1}$
and ${\tilde{c}}^{\prime}_{4}=c^{\prime}_{4}=\omega_{2}$. Thus we may change
the three $c^{\prime}_{i}$ to be any three monomials from $\omega_{j}$.
Assume that the above path is bad, let us say $m_{p}\in(b)$ for $p<e$ and as
in Lemma 8 we may suppose that $a_{p+1}\not\in E$,
$T_{a_{p+1}}\cap\\{a_{1},\ldots,a_{p}\\}=\emptyset$ and there exists no bad
path starting with $a_{p+1}$. Changing $P_{b}$ as above we see that the new
${\tilde{c}}_{i}^{\prime}$ are $\omega_{1},\omega_{2},\omega_{4}$ and the
$\omega_{3}\not\in U^{\prime}_{a_{p+1}}$, where $U^{\prime}_{a_{p+1}}$
corresponds to $T^{\prime}_{a_{p+1}}=T_{a_{p+1}}\setminus\\{a_{p+1}\\}$. Set
$b^{\prime}=a_{p+1}$. In fact changing in the new $P_{b}$ the intervals
$[b^{\prime},m_{p}]$ with $[b,m_{p}]$ we get a partition $P_{b^{\prime}}$ on
$I_{b^{\prime}}/J_{b^{\prime}}$, where $I_{b^{\prime}}J_{b^{\prime}}$ are
defined as usually but we could have $b^{\prime}\in W$. There exists no bad
path in $P_{b^{\prime}}$ because otherwise this induces one in $P_{b}$. We may
proceed as before since all monomials from $U^{\prime}_{b^{\prime}}$ has at
least one divisor from $B\setminus E$ which is not in
$\cup_{j=1,2,4}[f_{j},{\tilde{c}}^{\prime}_{j}]$. Similarly, we do for any
$a_{1}\in E$ dividing one from $c^{\prime}_{2},c^{\prime}_{3},c^{\prime}_{4}$
and remains to assume that there exists no bad path starting with a divisor
from $E$ of any $c^{\prime}_{i}$, $i=2,3,4$.
Now suppose that $a_{1}=b_{3}$ and consider $T_{1},U_{1}$ as usual and we may
suppose that we are still in Case 3 but with $(\tilde{c}^{\prime}_{j})$,
$j=1,3,4$. If there exists no bad path starting with $a_{1}$ and
$m_{1}=h(a_{1})\in(W)$, let us say $m_{1}\in(w_{13})$ then changing in $P_{b}$
the intervals $[a_{1},m_{1}]$, $[f_{1},\tilde{c}^{\prime}_{1}]$ with
$[f_{1},m_{1}]$, $[\tilde{u}_{1},\tilde{c}^{\prime}_{1}]$,
$\tilde{u}_{1}=w_{12}$ we arrive in a case similar to Case 1. If
$m_{1}\not\in(W)$ then assume that in $P_{b}$ there exist the intervals
$[f_{1},\omega_{2}]$, $[f_{2},\omega_{4}]$, $[f_{4},\omega_{1}]$. Then
$[f_{3},m_{1}]$ is disjoint of these intervals. Enlarge $T_{1}$ to
$\tilde{T}_{1}$ adding all monomials from $B$ connected by a path which is not
bad, with the divisors from $E$ of $(\omega_{j})$, $j=1,2,4$. Thus taking
$I^{\prime}=(B\setminus(\tilde{T}_{1}\cup W))$, $J^{\prime}=J\cap I^{\prime}$
we get $\operatorname{sdepth}_{S}I/(J,I^{\prime})\geq d+2$ which is enough as
usual.
If there exists a bad path $a_{1},\ldots,a_{e}$, $m_{i}=h(a_{i})$,
$m_{e}=\omega_{1}$, $m_{p}\in(b)$, $p<e$ then as above we may assume that
$a_{p+1}\not\in E$, $T_{a_{p+1}}\cap\\{a_{1},\ldots,a_{p}\\}=\emptyset$ and
there exists no bad path starting with $a_{p+1}$. Moreover, we may choose
$a_{p+2}\not\in E$ when $e>p+1$ because $m_{p+1}\not=\omega_{1}$. Taking as
above $b^{\prime}=a_{p+1}$ and the partition $P_{b^{\prime}}$ given on
$I_{b^{\prime}}/J_{b^{\prime}}$ we see that
$T_{a_{p+2}}\cap(f_{1},\ldots,f_{4})\not=\emptyset$ and we reduce to the above
situation with $T_{a_{p+2}}$ instead $T_{1}$. If $p\geq e-1$ then
$\omega_{1}\not\in U_{a_{p+2}}$ and so there exists no problem.
###### Theorem 4.
Conjecture 1 holds for $r=5$ if there exists $t\in[n]$ such that
$t\not\in\cup_{i\in[5]}\operatorname{supp}f_{i}$, $(B\setminus
E)\cap(x_{t})\not=\emptyset$ and $E\subset(x_{t})$.
###### Proof.
Apply Lemma 1, since Conjecture 1 holds for $r\leq 4$ by Theorem 3.
###### Example 7.
Let $n=8$, $E=\\{x_{6}x_{7},x_{7}x_{8}\\}$,
$I=(x_{1},x_{2},x_{3},x_{4},x_{5},E),$
$J=(x_{1}x_{6},x_{1}x_{8},x_{2}x_{8},x_{3}x_{6},x_{3}x_{8},x_{4}x_{6},x_{4}x_{7},x_{4}x_{8},x_{5}x_{6},x_{5}x_{7},x_{5}x_{8})$.
We see that we have $B=$
$\\{x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},x_{1}x_{7},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{2}x_{6},x_{2}x_{7},x_{3}x_{4},x_{3}x_{5},x_{3}x_{7},x_{4}x_{5}\\}\cup\\{E\\},$
$C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{2}x_{7},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{1}x_{3}x_{7},x_{1}x_{4}x_{5},x_{2}x_{3}x_{4},x_{2}x_{3}x_{5},$
$x_{2}x_{3}x_{7},x_{2}x_{4}x_{5},x_{2}x_{6}x_{7},x_{3}x_{4}x_{5},x_{6}x_{7}x_{8}\\}$
and so $r=5$, $q=15$, $s=16\leq q+r$. We have
$\operatorname{sdepth}_{S}I/J=2$, because otherwise the monomial $x_{2}x_{6}$
could enter either in $[x_{2},x_{2}x_{6}x_{7}]$, or in
$[x_{2}x_{6},x_{2}x_{6}x_{7}]$ and in both cases remain the monomials of $E$
to enter in an interval ending with $x_{6}x_{7}x_{8}$, which is impossible.
Then $\operatorname{depth}_{S}I/J\leq 2$ by the above theorem since
$E\subset(x_{7})$ and for instance $x_{1}x_{7}\in(B\setminus E)\cap(x_{7})$.
## References
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* [2] M. Cimpoeas, The Stanley conjecture on monomial almost complete intersection ideals, Bull. Math. Soc. Sci. Math. Roumanie, 55(103) (2012), 35-39.
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* [4] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc., 140 (2012), 493-504.
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in ”Bridging Algebra, Geometry, and Topology”, Editors Denis Ibadula, Willem
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* [16] D. Popescu, A. Zarojanu, Three generated, squarefree, monomial ideals, to appear in Bull. Math. Soc. Sci. Math. Roumanie, 58(106) (2015), no 3, arXiv:AC/1307.8292v6.
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|
arxiv-papers
| 2013-12-03T20:36:39 |
2024-09-04T02:49:54.763300
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1312.0923"
}
|
1312.1094
|
theoremrubric theorem corollaryrubric corollary lemmarubric lemma
propositionrubric proposition definitionrubric definition remarkrubric remark
# Interaction Graphs: Exponentials
Thomas Seiller111Institut des Hautes Études Scientifiques (IHÉS) Le Bois-
Marie, 35 Route de Chartres, 91440 Bures-sur-Yvette, France222This work was
partially supported by the ANR project ANR-10-BLAN-0213 LOGOI
[email protected]
# Interaction Graphs: Exponentials
Thomas Seiller111Institut des Hautes Études Scientifiques (IHÉS) Le Bois-
Marie, 35 Route de Chartres, 91440 Bures-sur-Yvette, France222This work was
partially supported by the ANR project ANR-10-BLAN-0213 LOGOI
[email protected]
###### Abstract
This paper is the fourth of a series [Sei12a, Sei14a, Sei14c] exposing a
systematic combinatorial approach to Girard’s Geometry of Interaction program
[Gir89b]. This program aims at obtaining particular realizability models for
linear logic that accounts for the dynamics of cut-elimination. This fourth
paper tackles the complex issue of defining exponential connectives in this
framework. In order to succeed in this, we use the notion of _graphings_ , a
generalization of graphs which was defined in earlier work [Sei14c]. We
explain how we can use this framework to define a GoI for Elementary Linear
Logic (ELL) with second-order quantification, a sub-system of linear logic
that captures the class of elementary time computable functions.
###### Contents
1. 1 Introduction
2. 2 Interaction Graphs
3. 3 Thick Graphs and Contraction
4. 4 Construction of an Exponential Connective on the Real Line
5. 5 Soundness for Behaviors
6. 6 Contraction and Soundness for Polarized Conducts
7. 7 Conclusion and Perspectives
## 1 Introduction
### 1.1 Geometry of Interaction
A Geometry of Interaction (GoI) model, i.e. a construction that fulfills the
GoI research program [Gir89b], is in a first approximation a representation of
linear logic proofs that accounts for the dynamics of cut-elimination. A proof
is no longer a morphism from $A$ to $B$ — a function from $A$ into $B$ — but
an operator acting on the space $A\oplus B$. As a consequence, the modus
ponens is no longer represented by composition. The operation representing
cut-elimination, i.e. the obtention of a cut-free proof of $B$ from a cut-free
proof of $A$ and a cut-free proof of $A\multimap B$, consists in constructing
the solution to an equation called the _feedback equation_ (illustrated in
Figure 2). A GoI model hence represents both the proofs and the dynamics of
their normalization. Contrarily to denotational semantics, a proof $\pi$ and
its normalized form $\pi^{\prime}$ are not represented by the same object.
However, they remain related since the normalization procedure has a
semantical counterpart — the execution formula $\textnormal{Ex}(\cdot)$ —
which satisfies $\textnormal{Ex}(\pi)=\pi^{\prime}$. This essential difference
between denotational semantics and GoI is illustrated in Figure 1.
$\pi$$\lVert\pi\rVert$$\rho$$\lVert\rho\rVert$$\lVert\cdot\rVert$$\lVert\cdot\rVert$cutelimination
(a) Denotational Semantics
$\pi$$\lVert\pi\rVert$$\rho$$\lVert\rho\rVert$$\lVert\cdot\rVert$$\lVert\cdot\rVert$cutelimination$\textnormal{Ex}(\cdot)$
(b) Geometry of Interaction
Figure 1: Denotational Semantics vs Geometry of Interaction
$P\in\mathcal{L}(\mathbb{H\oplus K})$ represents 333Here, $\mathbb{H}$ and
$\mathbb{K}$ are separable infinite-dimensional Hilbert spaces, and
$\mathcal{L}(\star)$ denotes the set of operators acting on the Hilbert space
$\star$: bounded (or, equivalently, continuous) linear maps from $\star$ to
$\star$. a program/proof of implication $A\in\mathcal{L}(\mathbb{H})$
represents an argument. $R\in\mathcal{L}(\mathbb{K})$ represents the result of
the computation if:
$R(\xi)=\xi^{\prime}\Leftrightarrow\exists\eta,\eta^{\prime}\in\mathbb{H},\left\\{\begin{array}[]{lcl}P(\eta\oplus\xi)&=&\eta^{\prime}\oplus\xi^{\prime}\\\
A(\eta^{\prime})&=&\eta\end{array}\right.$
(a) Formal statement
$\mathbb{H}$$\xi$$\mathbb{K}$$A$$P$$A(\eta)$$\mathbb{H}$$\xi^{\prime}$$\mathbb{K}$$\eta$
(b) Illustration of the equation
Figure 2: The Feedback Equation
The GoI program has a second aim: define by realizability techniques a
reconstruction of logical operations from the dynamical model just exposed.
The objects of study in a GoI construction are a generalization of the notion
of proof — paraproofs, in the same sense the proof structure where a
generalization of the notion of sequent calculus proof. This is reminiscent of
game semantics where not all strategies are interpretations of programs, or
Krivine’s classical realizability [Kri01, Kri09] where terms containing
continuation constants are distinguished from “proof-like terms”. This point
of view allows a reconstruction of logic as a description of how paraproofs
interact. It is therefore a sort of ”discursive syntax” where paraproofs are
opposed one to another, where they argue together in a way reminiscent of game
semantics, each one trying to prove the other wrong. This argument terminates
when one of them gives up. The discussion itself corresponds to the execution
formula, which describes the solution to the feedback equation and generalizes
the cut-elimination procedure to this generalized notion of proofs. Two
paraproofs are then said _orthogonal_ — denoted by the symbol $\simperp$ —
when this arguement (takes place and) terminates. A notion of formula is then
drawn from this notion of orthogonality: a formula is a set of paraproofs $A$
equal to its bi-orthogonal closure $A^{\simbot\simbot}$ or, equivalently, a
set of paraproofs $A=B^{\simbot}$ which is the orthogonal to a given set of
paraproofs $B$.
Drawing some intuitions from the Curry-Howard correspondence, one may propose
an alternative reading to this construction in terms of programs. Since proofs
correspond to well-behaved programs, paraproofs are a generalization of those,
representing somehow _badly-behaved programs_. If the orthogonality relation
represents negation from a logical point of view, it represents a notion of
_testing_ from a computer science point of view. The notion of formula defined
from it corresponds to a notion of type, defined interactively from how
(para)programs behave. This point of view is still natural when thinking about
programs: a program is of type $\mathbf{nat}\rightarrow\mathbf{nat}$ because
it produces a natural number when given a natural number as an argument. On
the logical side, this change may be more radical: a proof is a proof of the
formula $\text{Nat}\Rightarrow\text{Nat}$ because it produces a proof of Nat
each time it is cut (applied) to a proof of Nat.
Once the notion of type/formula defined, one can reconstruct the connectives:
from a ”low-level” — between paraproofs — definition, one obtains a ”high-
level” definition — between types. For instance, the connective $\otimes$ is
first defined between any two paraproofs $\mathfrak{a,b}$, and this definition
is then extended to types by defining $A\otimes
B=\\{\mathfrak{a}\otimes\mathfrak{b}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathfrak{a}\in A,\mathfrak{b}\in B\\}^{\simbot\simbot}$. As a consequence,
the connectives are not defined in an _ad hoc_ way, but their definition is a
consequence of their computational meaning: the connectives are defined on
proofs/programs and their definition at the level of types is just the
reflection of the interaction between the execution — the dynamics of proofs —
and the low-level definition on paraproofs. Logic thus arises as generated by
computation, by the normalization of proofs: types/formulas are not there to
tame the programs/proofs but only to describe their behavior. This is
reminiscent of realizability in the sense that a type is defined as the set of
its (para-)proofs. Of course, the fact that we consider a generalized notion
of proofs from the beginning has an effect on the construction: contrarily to
usual realizability models (except from classical realizability in the sense
of Krivine [Kri01, Kri09]), the types $A$ and $A^{\simbot}$ (the negation of
$A$) are in general both non-empty. This is balanced by the fact that one can
define a notion of successful paraproofs, which corresponds in a way to the
notion of winning strategy in game semantics. This notion on paraproofs then
yields a high-level definition: a formula/type is _true_ when it contains a
successful paraproof.
### 1.2 Interaction Graphs and Graphings
Interaction Graphs were first introduced [Sei12a] to define a combinatorial
approach to Girard’s geometry of interaction in the hyperfinite factor
[Gir11]. The main idea was that the execution formula — the counterpart of the
cut-elimination procedure — can be computed as the set of alternating paths
between graphs, and that the measurement of interaction defined by Girard
using the Fuglede-Kadison determinant [FK52] can be computed as a measurement
of a set of cycles.
The setting was then extended to deal with additive connectives [Sei14a],
showing by the way that the constructions were a combinatorial approach not
only to Girard’s hyperfinite GoI construction but also to all the earlier
constructions [Gir87, Gir89a, Gir88, Gir95a]. This result could be obtained by
unveiling a single geometrical property, which we called the _trefoil
property_ , upon which all the constructions of geometry of interaction
introduced by Girard are built.
In a third paper, we explored a wide generalization of the graph
framework444This generalization, or more precisely a fragment of it, already
appeared in the author’s PhD thesis [Sei12b].. We introduced the notion of
_graphing_ which we now informally describe. If $(X,\mathcal{B},\mu)$ is a
measured space and $\mathfrak{m}$ is a monoid of measurable maps555For
technical reasons, we in fact consider monoids of Borel-preserving non-
singular maps [Sei14c]. $X\rightarrow X$ (the internal law is composition),
then a graphing in $\mathfrak{m}$ is a countable family of restrictions of
elements of $\mathfrak{m}$ to measurable subsets. These restrictions of
elements of $\mathfrak{m}$ are regarded as edges of a graph _realized_ as
measurable (partial) maps. We showed that the notions of paths and cycles in a
graphing could be defined. As a consequence, one can define the _execution_ as
the set of alternating paths between graphings, mimicking the corresponding
operation of graphs. On the other hand, a more complex argument shows that one
can define appropriate measures of cycles in order to insure that the trefoil
property holds. As a consequence, we obtained whole hierarchies of models of
multiplicative-additive linear logic in this way. The purpose of this paper is
to exhibit a family of such models in which one can interpret Elementary
Linear Logic [Gir95b, DJ03] with second-order quantification.
### 1.3 Outline of the paper
In Section 2, we recall some important definitions and properties on directed
weighted graphs. This allows us to introduce important notations that will be
used later on. We then recall some definitions and properties about the
additive construction [Sei14a]. These properties are essential to the
understanding of the construction of the multiplicative-additive fragment of
linear logic in the setting of interaction graphs.
In Section 3, we define and study the notion of _thick graphs_ , and show how
it can be used to interpret the contraction $\mathbf{\oc A\multimap\oc
A\otimes\oc A}$ for some specific formulas $\mathbf{A}$. This motivates the
definition of a _perennisation_ $\Omega$ from which one can define an
exponential $\mathbf{A}\mapsto\mathbf{\oc_{\Omega}A}$. We also explain why it
is necessary to work with a generalization of graphs, namely graphings, in
order to define perennisations that are suitably expressive.
In Section 4, we give a definition of an exponential connective defined from a
suitable notion of perennisation. We show for this a result which allows us to
encode any bijection over the natural numbers as a measure-preserving map over
the unit interval of the real line. This result is then used to encode some
combinatorics as measure-preserving maps and show that functorial promotion
can be implemented for the exponential we defined.
We then prove a soundness result for a variant (in Section 5) of Elementary
Linear Logic (ELL). This result, though interesting, is not ideal since we
restrict to proofs that are in some sense ”intuitionnistic”. Indeed, for
technical reasons explained later on, the introduction of exponentials cannot
be performed without being associated to a tensor product. Since the
interpretation of elementary time functions in ELL relies heavily on those
proofs that are not intuitionnistic in this sense666This fact was pointed out
to the author by Damiano Mazza..
Consequently, we introduce (in Section 6) a notion of polarities which
generalize the notion of _perennial/co-perennial_ formulas defined before. The
discussion on polarities leads to a refinement of the sequent calculus
considered in the previous section which does not suffer from the drawbacks
explained above. We then prove a soundness result for this calculus.
## 2 Interaction Graphs
### 2.1 Basic Definitions
Departing from the realm of infinite-dimensional vector spaces and linear maps
between them, we proposed in previous work [Sei12a, Sei14a] a graph-
theoretical construction of GoI models. We give here a brief overview of the
main definitions and results. The graphs we consider are directed and
weighted, where the weights are taken in a _weight monoid_ $(\Omega,\cdot)$.
###### Definition .
A _directed weighted graph_ is a tuple $G$, where $V^{G}$ is the set of
vertices, $E^{G}$ is the set of edges, $s^{G}$ and $t^{G}$ are two functions
from $E^{G}$ to $V^{G}$, the _source_ and _target_ functions, and $\omega^{G}$
is a function $E^{G}\rightarrow\Omega$.
The construction is centered around the notion of alternating paths. Given two
graphs $F$ and $G$, an alternating path is a path $e_{1}\dots e_{n}$ such that
$e_{i}\in E^{F}$ if and only if $e_{i+1}\in E^{G}$. The set of alternating
paths will be used to define the interpretation of cut-elimination in the
framework, i.e. the graph $F\mathop{\mathopen{:}\mathclose{:}}G$ — the
_execution of $F$ and $G$_ — is defined as the graph of alternating paths
between $F$ and $G$ whose source and target are in the symmetric difference
$V^{F}\Delta V^{G}$. The weight of a path is naturally defined as the product
of the weights of the edges it contains.
###### Definition .
Let $F,G$ be directed weighted graphs. The set of alternating paths between
$F$ and $G$ is the set of paths $e_{0},e_{1},\dots,e_{n}$ such that $e_{i}\in
E^{G}\Rightarrow e_{i+1}\in E^{F}$ and $e_{i}\in E^{F}\Rightarrow e_{i+1}\in
E^{G}$. We write $\text{{Path}}(F,G)$ the set of such paths, and
$\text{{Path}}(F,G)_{V}$ the subset of $\text{{Path}}(F,G)$ containing the
paths whose source and target lie in $V$.
The execution $F\mathop{\mathopen{:}\mathclose{:}}G$ of $F$ and $G$ is then
defined by:
$\displaystyle V^{F\mathop{\mathopen{:}\mathclose{:}}G}$ $\displaystyle=$
$\displaystyle V^{F}\Delta V^{G}$ $\displaystyle
E^{F\mathop{\mathopen{:}\mathclose{:}}G}$ $\displaystyle=$
$\displaystyle\text{{Path}}(F,G)_{V^{F\mathop{\mathopen{:}\mathclose{:}}G}}$
where the source and target maps are naturally defined, and the weight of a
path is the product of the weights of the edges it is composed of.
As it is usual in mathematics, this notion of paths cannot be considered
without the associated notion of cycle: an _alternating cycle_ between two
graphs $F$ and $G$ is a cycle which is an alternating path $e_{1}e_{2}\dots
e_{n}$ such that $e_{1}\in V^{F}$ if and only if $e_{n}\in V^{G}$. For
technical reasons, we actually consider the related notion of $1$-circuit.
###### Definition .
A _$1$ -circuit_ is an alternating cycle $\pi=e_{1}\dots e_{n}$ which is not a
proper power of a smaller cycle. In mathematical terms, there do not exists a
cycle $\rho$ and an integer $k$ such that $\pi=\rho^{k}$, where the power
represents iterated concatenation.
We denote by $\mathcal{C}(F,G)$ the set of $1$-circuits in the following. We
show that these notions of paths and cycles satisfy a property we call the
_trefoil property_ which will turn out to be fundamental. The trefoil property
states that there exists weight-preserving bijections:
$\mathcal{C}(F\mathop{\mathopen{:}\mathclose{:}}G,H)\cup\mathcal{C}(F,G)\cong\mathcal{C}(G\mathop{\mathopen{:}\mathclose{:}}H,F)\cup\mathcal{C}(G,H)\cong\mathcal{C}(H\mathop{\mathopen{:}\mathclose{:}}F,G)\cup\mathcal{C}(H,F)$
We showed, based only on the trefoil property, how one can define the
multiplicative and additive connectives of Linear Logic, obtaining a model
fulfilling the GoI research program. This construction is moreover
parametrized by a map from the set $\Omega$ to $\mathbb{R}_{\geqslant
0}\cup\\{\infty\\}$, and therefore yields not only one but a whole family of
models. This parameter is introduced to define the notion of orthogonality in
our setting, a notion that account for linear negation. Indeed, given a map
$m$ and two graphs $F,G$ we define
$\mathopen{\llbracket}F,G\mathclose{\rrbracket}_{m}$ as the sum
$\sum_{\pi\in\mathcal{C}(F,G)}m(\omega(\pi))$, where $\omega(\pi)$ is the
weight of the cycle $\pi$. The orthogonality is then constructed from this
measurement.
We moreover showed how, from any of these constructions, one can obtain a
$\ast$-autonomous category $\mathfrak{Graph}_{MLL}{}$ with
$\parr\not\cong\otimes$ and $1\not\cong\bot$, i.e. a non-degenerate
denotational semantics for Multiplicative Linear Logic (MLL). However, as in
all the versions of GoI dealing with additive connectives, our construction of
additives does not define a categorical product. We solve this issue by
introducing a notion of _observational equivalence_ within the model. We are
then able to define a categorical product from our additive connectives when
considering classes of observationally equivalent objects, thus obtaining a
denotational semantics for Multiplicative Additive Linear Logic (MALL).
### 2.2 Models of MALL in a Nutshell
We recall the basic definitions of projects, and behaviors, which will be
respectively used to interpret proofs and formulas, as well as the definition
of connectives.
* •
a _project_ of carrier $V^{A}$ is a triple $\mathfrak{a}=(a,V^{A},A)$, where
$a$ is a real number, $A=\sum_{i\in I^{A}}\alpha^{A}_{i}A_{i}$ is a finite
formal (real-)weighted sum of graphings of carrier included in $V^{A}$;
* •
two projects $\mathfrak{a,b}$ are _orthogonal_ when:
$\mathopen{\ll}\mathfrak{a},\mathfrak{b}\mathclose{\gg}_{m}=a(\sum_{i\in
I^{A}}\alpha^{B}_{i})+b(\sum_{i\in I^{B}}\alpha^{B}_{i})+\sum_{i\in
I^{A}}\sum_{j\in
I^{B}}\alpha_{i}^{A}\alpha^{B}_{j}\mathopen{\llbracket}A_{i},B_{j}\mathclose{\rrbracket}_{m}\neq
0,\infty$
* •
the _execution_ of two projects $\mathfrak{a,b}$ is defined as:
$\mathfrak{a\mathop{\mathopen{:}\mathclose{:}}b}=(\mathopen{\ll}\mathfrak{a},\mathfrak{b}\mathclose{\gg}_{m},V^{A}\Delta
V^{B},\sum_{i\in I^{A}}\sum_{j\in
I^{B}}\alpha^{A}_{i}\alpha^{B}_{j}A_{i}\mathop{\mathopen{:}\mathclose{:}}B_{j})$
* •
if $\mathfrak{a}$ is a project and $V$ is a measurable set such that
$V^{A}\subset V$, we define the extension $\mathfrak{a}_{\uparrow V}$ as the
project $(a,V,A)$;
* •
a _conduct_ $\mathbf{A}$ of carrier $V^{A}$ is a set of projects of carrier
$V^{A}$ which is equal to its bi-orthogonal $\mathbf{A}^{\simbot\simbot}$;
* •
a _behavior_ $\mathbf{A}$ of carrier $V^{A}$ is a conduct such that for all
$\lambda\in\mathbf{R}$,
$\begin{array}[]{rcl}\mathfrak{a}\in\mathbf{A}&\Rightarrow&\mathfrak{a+\lambda
0}\in\mathbf{A}\\\
\mathfrak{b}\in\mathbf{A}^{\simbot}&\Rightarrow&\mathfrak{b+\lambda
0}\in\mathbf{A}^{\simbot}\end{array}$
* •
we define, for every measurable set the _empty_ behavior of carrier $V$ as the
empty set $\mathbf{0}_{V}$, and the _full behavior_ of carrier $V$ as its
orthogonal $\mathbf{T}_{V}=\\{\mathfrak{a}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathfrak{a}\text{ of support }V\\}$;
* •
if $\mathbf{A,B}$ are two behaviors of disjoint carriers, we define:
$\displaystyle\mathbf{A\otimes B}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{a\mathop{\mathopen{:}\mathclose{:}}b}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A},\mathfrak{b}\in\mathbf{B}\\}^{\simbot\simbot}$
$\displaystyle\mathbf{A\multimap B}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{f}\leavevmode\nobreak\ |\leavevmode\nobreak\
\forall\mathfrak{a}\in\mathbf{A},\mathfrak{f\mathop{\mathopen{:}\mathclose{:}}a}\in\mathbf{B}\\}$
$\displaystyle\mathbf{A\oplus B}$ $\displaystyle=$
$\displaystyle(\\{\mathfrak{a}_{\uparrow V^{A}\cup V^{B}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A}\\}^{\simbot\simbot}\cup\\{\mathfrak{b}_{\uparrow
V^{A}\cup V^{B}}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathfrak{b}\in\mathbf{B}\\}^{\simbot\simbot})^{\simbot\simbot}$
$\displaystyle\mathbf{A\with B}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{a}_{\uparrow V^{A}\cup V^{B}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A^{\simbot}}\\}^{\simbot}\cap\\{\mathfrak{b}_{\uparrow
V^{A}\cup V^{B}}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathfrak{b}\in\mathbf{B}^{\simbot}\\}^{\simbot}$
* •
two elements $\mathfrak{a,b}$ of a conduct $\mathbf{A}$ are _observationally
equivalent_ when:
$\forall\mathfrak{c}\in\mathbf{A}^{\simbot},\leavevmode\nobreak\
\mathopen{\ll}\mathfrak{a},\mathfrak{c}\mathclose{\gg}_{m}=\mathopen{\ll}\mathfrak{b},\mathfrak{c}\mathclose{\gg}_{m}$
One important point in this work is the fact that all results rely on a single
geometric property, namely the previously introduced _trefoil property_ which
describes how the sets of $1$-circuits evolve during an execution. This
property insures on its own the four following facts:
* •
we obtain a $\ast$-autonomous category $\mathfrak{Graph}_{MLL}{}$ whose
objects are conducts and morphisms are projects;
* •
the observational equivalence is a congruence on this category;
* •
the quotiented category $\mathfrak{Cond}{\leavevmode\nobreak\ }$ inherits the
$\ast$-autonomous structure;
* •
the quotiented category $\mathfrak{Cond}{\leavevmode\nobreak\ }$ has a full
subcategory $\mathfrak{Behav}{\leavevmode\nobreak\ }$ with products whose
objects are behaviors.
This can be summarized in the following two theorems.
###### Theorem .
For any map $m:\Omega\rightarrow\mathbf{R}\cup\\{\infty\\}$, the categories
$\mathfrak{Cond}{\leavevmode\nobreak\ }$and $\mathfrak{Graph}_{MLL}{}$ are
non-degenerate categorical models of Multiplicative Linear Logic with
multiplicative units.
###### Theorem .
For any map $m:\Omega\rightarrow\mathbf{R}\cup\\{\infty\\}$, the full
subcategory $\mathfrak{Behav}{\leavevmode\nobreak\ }$ of
$\mathfrak{Cond}{\leavevmode\nobreak\ }$ is a non-degenerate categorical model
of Multiplicative-Additive Linear Logic with additive units.
The categorical model we obtain has two layers (see Figure 3). The first layer
consists in a non-degenerate (i.e. $\otimes\neq\parr$ and
$\mathbf{1}\neq\mathbf{\bot}$) $\ast$-autonomous category
$\mathfrak{Cond}{\leavevmode\nobreak\ }$, hence a denotational model for MLL
with units. The second layer is the full subcategory
$\mathfrak{Behav}{\leavevmode\nobreak\ }$which does not contain the
multiplicative units but is a non-degenerate model (i.e. $\otimes\neq\parr$,
$\oplus\neq\with$ and $\mathbf{0}\neq\mathbf{\top}$) of MALL with additive
units that does not satisfy the mix and weakening rules.
$\mathfrak{Cond}{\leavevmode\nobreak\ }$
---
($\ast$-autonomous)
$\mathfrak{Behav}{\leavevmode\nobreak\ }$
---
(closed under $\otimes,\multimap,\with,\oplus,(\cdot)^{\simbot}$)
NO weakening, NO mix
$\bullet_{\bot}$$\bullet_{\mathbf{1}}$$\bullet_{\mathbf{T}}$$\bullet_{\mathbf{0}}$
Figure 3: The categorical models
We here recall some technical results obtained in our paper on additives
[Sei14a] and that will be useful in the following.
###### Proposition .
If $A$ is a non-empty set of projects of same carrier $V^{A}$ such that
$(a,A)\in A$ implies $a=0$, then $\mathfrak{b}\in A^{\simbot}$ implies
$\mathfrak{b}+\lambda\mathfrak{0}_{V^{A}}\in A^{\simbot}$ for all
$\lambda\in\mathbb{R}$.
###### Proposition .
If $A$ is a non-empty set of projects of carrier $V$ such that
$\mathfrak{a}\in A\Rightarrow\mathfrak{a+\lambda 0}_{V}\in A$, then any
project in $A^{\simbot}$ is wager-free, i.e. if $(a,A)\in A^{\simbot}$ then
$a=0$.
###### Lemma (Homothety).
Conducts are closed under homothety: for all $\mathfrak{a}\in\mathbf{A}$ and
all $\lambda\in\mathbf{R}$ with $\lambda\neq 0$,
$\lambda\mathfrak{a}\in\mathbf{A}$.
###### Proposition .
We denote by $\mathbf{A\odot B}$ the set
$\\{\mathfrak{a}\otimes\mathfrak{b}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A},\mathfrak{b}\in\mathbf{B}\\}$. Let $E,F$ be non-
empty sets of projects of respective carriers $V,W$ with $V\cap W=\emptyset$.
Then
$(E\odot F)^{\simbot\simbot}=(E^{\simbot\simbot}\odot
F^{\simbot\simbot})^{\simbot\simbot}$
###### Proposition .
Let $\mathbf{A,B}$ be conducts. Then:
$(\\{\mathfrak{a}\otimes\mathfrak{0}_{\mathnormal{B}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A}\\}\cup\\{\mathfrak{0}_{\mathnormal{A}}\otimes\mathfrak{b}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{b}\in\mathbf{B}\\})^{\simbot\simbot}=\mathbf{A\oplus B}$
###### Proposition (Distributivity).
For any behaviors $\mathbf{A,B,C}$, and delocations $\phi,\psi,\theta,\rho$ of
$\mathbf{A},\mathbf{A},\mathbf{B},\mathbf{C}$ respectively, there is a project
$\mathfrak{distr}$ in the behavior
$\mathbf{((\phi(A)\\!\multimap\\!\theta(B))\\!\with\\!(\psi(A)\\!\multimap\\!\rho(C)))\\!\multimap\\!(A\\!\multimap\\!(B\\!\with\\!C))}$
### 2.3 Graphings
In subsequent work [Sei14c], we introduced a generalization of graphs to which
the previously described results can extended. This generalization allows,
among other things, for the definition of second order quantification. The
main purpose of this generalization is that a vertex can always be cut in an
arbitrary (finite) number of sub-vertices, with the idea that these sub-
vertices are smaller (hence vertices have a _size_) and form a partition of
the initial vertex (where two sub-vertices have the same size). These notions
could be introduced and dealt with combinatorially, but we chose to use
measure-theoretic notions in order to ease the intuitions and some proofs. In
fact, a _graphing_ — the notion which is introduced as a generalization of the
notion of graph — can be though of and used as a graph. Another important
feature of the construction is the fact that it depends on a _microcosm_ — a
monoid of non-singular transformations — which somehow describes that
computational principles allowed in the model.
###### Definition .
Let $(X,\mathcal{B},\lambda)$ be a measured space. We denote by
$\mathcal{M}(X)$ the set of Borel-preserving non-singular transformations777A
non-singular transformation $f:X\rightarrow X$ is a measurable map which
preserves the sets of null measure, i.e. $\lambda(f(A))=0$ if and only if
$\lambda(A)=0$. A Borel-preserving map is a map such that the images of Borel
sets are Borel sets. $X\rightarrow X$. A _microcosm_ of the measured space $X$
is a subset $\mathfrak{m}$ of $\mathcal{M}(X)$ which is closed under
composition and contains the identity.
As in the graph construction described above, we will consider a notion of
graphing depending on a _weight-monoid_ $\Omega$, i.e. a monoid
$(\Omega,\cdot,1)$ which contains the possible weights of the edges.
###### Definition (Graphings).
Let $\mathfrak{m}$ be a microcosm of a measured space
$(X,\mathcal{B},\lambda)$ and $V^{F}$ a measurable subset of $X$. A _$\Omega$
-weighted graphing in $\mathfrak{m}$_ of carrier $V^{F}$ is a countable family
$F=\\{(\omega_{e}^{F},\phi_{e}^{F}:S_{e}^{F}\rightarrow T_{e}^{F}\\}_{e\in
E^{F}}$, where, for all $e\in E^{F}$ (the set of _edges_):
* •
$\omega_{e}^{F}$ is an element of $\Omega$, the _weight_ of the edge $e$;
* •
$S_{e}^{F}\subset V^{F}$ is a measurable set, the _source_ of the edge $e$;
* •
$T_{e}^{F}=\phi_{e}^{F}(S_{e}^{F})\subset V^{F}$ is a measurable set, the
_target_ of the edge $e$;
* •
$\phi_{e}^{F}$ is the restriction of an element of $\mathfrak{m}$ to
$S_{e}^{F}$, the _realization_ of the edge $e$.
It is natural, as we are working with measure-theoretic notions, to identify
two graphings that differ only on a set of null measure. This leads to the
definition of an equivalence relation between graphings: that of _almost
everywhere equality_. Moreover, since we want vertices to be _decomposable_
into any finite number of parts, we want to identify a graphing $G$ with the
graphing $G^{\prime}$ obtained by replacing an edge $e\in E^{F}$ by a finite
family of edges $e_{i}\in G^{\prime}$ ($i=1,\dots,n$) subject to the
conditions:
* •
the family $\\{S^{G^{\prime}}_{e_{i}}\\}_{i=1}^{n}$ (resp.
$\\{T^{G^{\prime}}_{e_{i}}\\}_{i=1}^{n}$) is a partition of $S_{e}^{G}$ (resp.
$T_{e}^{G}$);
* •
for all $i=1,\dots,n$, $\phi_{e_{i}}^{G^{\prime}}$ is the restriction of
$\phi_{e}^{G}$ on $S^{G^{\prime}}_{e_{i}}$.
Such a graphing $G^{\prime}$ is an example of a _refinement of $G$_, and one
can easily generalize the previous conditions to define a general notion of
refinement of graphings. Figure 4 gives the most simple example of refinement.
To be a bit more precise, we define, in order to ease the proofs, a notion of
refinement _up to almost everywhere equality_. We then define a second
equivalence relation on graphings by saying that two graphings are equivalent
if and only if they have a common refinement (up to almost everywhere
equality).
$[0,2]$$[3,5]$$x\mapsto 5-x$$[0,1]$$[1,2]$$[3,4]$$[4,5]$$x\mapsto
5-x$$x\mapsto 5-x$ Figure 4: A graphing and one of its refinements
The objects under study are thus equivalence classes of graphings modulo this
equivalence relation. Most of the technical results on graphings contained in
our previous paper [Sei14c] aim at showing that these objects can actually be
manipulated as graphs: one can define paths and cycles and these notions are
coherent with the quotient by the equivalence relation just mentioned. Indeed,
the notions of paths and cycles in a graphings are quite natural, and from two
graphings $F,G$ in a microcosm $\mathfrak{m}$ one can define its execution
$F\mathop{\mathopen{:}\mathclose{:}}G$ which is again a graphing in
$\mathfrak{m}$888As a consequence, a microcosm is a closed world for the
execution of programs.. A more involved argument then shows that the trefoil
property holds for a family of measurements
$\mathopen{\llbracket}\cdot,\cdot\mathclose{\rrbracket}_{m}$, where
$m:\Omega\rightarrow\mathbf{R}_{\geqslant 0}\cup\\{\infty\\}$ is any
measurable map. These results are obtained as a generalization of
constructions considered in the author’s thesis999In the cited work, the
results were stated in the particular case of the microcosm of measure-
preserving maps on the real line..
###### Theorem .
Let $\Omega$ be a monoid, $\mathfrak{m}$ a microcosm and
$m:\Omega\rightarrow\mathbf{R}_{\geqslant 0}\cup\\{\infty\\}$ be a measurable
map. The set of $\Omega$-weighted graphings in $\mathfrak{m}$ yields a model,
denoted by $\mathbb{M}[\Omega,\mathfrak{m}]_{m}$, of multiplicative-additive
linear logic whose orthogonality relation depends on $m$.
## 3 Thick Graphs and Contraction
In this section, we will define the notion of _thick graphs_ , and extend the
addictive construction defined in our earlier paper [Sei14a] to that setting.
The introduction of these objects will be motivated in Section 3.3, where we
will explain how thick graphs allows for the interpretation of the contraction
rule. This contraction rule being satisfied only for a certain kind of
conducts — interpretations of formulas, this will justify the definition of
the exponentials.
### 3.1 Thick Graphs
###### Definition .
Let $S^{G}$ and $D^{G}$ be finite sets. A directed weighted _thick graph_ $G$
of carrier $S^{G}$ and _dialect_ $D^{G}$ is a directed weighted graph over the
set of vertices $S^{G}\times D^{G}$.
We will call _slices_ the set of vertices $S^{G}\times\\{d\\}$ for $d\in
D^{G}$.
Figure 5 shows two examples of thick graphs. Thick graphs will be represented
following a graphical convention very close to the one we used for sliced
graphs:
* •
Graphs are once again represented with colored edges and delimited by hashed
lines;
* •
Elements of the carrier $S^{G}$ are represented on a horizontal scale, while
elements of the dialect $D^{G}$ are represented on a vertical scale;
* •
Inside a given graph, slices are separated by a _dotted_ line.
$1_{1}$$2_{1}$$1_{2}$$2_{2}$$2_{1}$$3_{1}$$2_{2}$$3_{2}$slice $2$slice
$1$Gslice $2$slice $1$H Figure 5: Two thick graphs $G$ and $H$, both with
dialect $\\{1,2\\}$
###### Remark .
If $G=\sum_{i\in I^{G}}\alpha^{G}_{i}G_{i}$ is a sliced graph such that
$\forall i\in I^{G},\alpha^{G}_{i}=1$, then $G$ can be identified with a thick
graph of dialect $I^{G}$. Indeed, one can define the thick graph $\\{G\\}$ by:
$\displaystyle V^{\\{G\\}}$ $\displaystyle=$ $\displaystyle V^{G}\times I^{G}$
$\displaystyle E^{\\{G\\}}$ $\displaystyle=$ $\displaystyle\uplus_{i\in
I^{G}}E^{G_{i}}$ $\displaystyle s^{\\{G\\}}$ $\displaystyle=$ $\displaystyle
e\in E^{G_{i}}\mapsto(s^{G_{i}}(e),i)$ $\displaystyle t^{\\{G\\}}$
$\displaystyle=$ $\displaystyle e\in E^{G_{i}}\mapsto(t^{G_{i}}(e),i)$
$\displaystyle\omega^{\\{G\\}}$ $\displaystyle=$ $\displaystyle e\in
E^{G_{i}}\mapsto\omega^{G_{i}}(e)$
###### Definition (Variants).
Let $G$ be a thick graph and $\phi:D^{G}\rightarrow E$ a bijection. One
defines $G^{\phi}$ as the graph:
$\displaystyle V^{G^{\phi}}$ $\displaystyle=$ $\displaystyle S^{G}\times E$
$\displaystyle E^{G^{\phi}}$ $\displaystyle=$ $\displaystyle E^{G}$
$\displaystyle s^{G^{\phi}}$ $\displaystyle=$
$\displaystyle(Id_{V^{G}}\times\phi)\circ s^{G}$ $\displaystyle t^{G^{\phi}}$
$\displaystyle=$ $\displaystyle(Id_{V^{G}}\times\phi)\circ t^{G}$
$\displaystyle\omega^{G^{\phi}}$ $\displaystyle=$ $\displaystyle\omega^{G}$
If $G$ and $H$ are two thick graphs such that $H=G^{\phi}$ for a bijection
$\phi$, then $H$ is called a _variant_ of $G$. The relation defined by $G\sim
H$ if and only if $G$ is a variant of $H$ can easily be checked to be an
equivalence relation.
###### Definition (Dialectal Interaction).
Let $G$ and $H$ be thick graphs.
1. 1.
We denote by $G^{\dagger_{D^{H}}}$ the thick graph of dialect $D^{G}\times
D^{H}$ defined as $\\{\sum_{i\in D^{H}}G\\}$;
2. 2.
We denote by $H^{\ddagger_{D^{G}}}$ the thick graph of dialect $D^{G}\times
D^{H}$ defined as $\\{\sum_{i\in D^{G}}H\\}^{\tau}$ where $\tau$ is the
natural bijection $D^{H}\times D^{G}\rightarrow D^{G}\times
D^{H},(a,b)\mapsto(b,a)$.
$1_{1,1}$$2_{1,1}$$1_{2,1}$$2_{2,1}$$1_{1,2}$$2_{1,2}$$1_{2,2}$$2_{2,2}$slices
$(\cdot,1)$slices
$(\cdot,2)$$2_{1,1}$$3_{1,1}$$2_{1,2}$$3_{1,2}$$2_{2,1}$$3_{2,1}$$2_{2,2}$$3_{2,2}$slices
$(1,\cdot)$slices $(2,\cdot)$ Figure 6: The graphs $G^{\dagger_{D^{H}}}$ and
$H^{\ddagger_{D^{G}}}$
We can then define the plugging $F\square G$ of two thick graphs as the
plugging of the graphs $F^{\dagger_{D^{G}}}$ and $G^{\ddagger_{D^{F}}}$.
Figure 7 shows the result of the plugging of $G$ and $H$, the thick graphs
represented in Figure 5.
$1_{1,1}$$2_{1,1}$$1_{2,1}$$2_{2,1}$$1_{1,2}$$2_{1,2}$$1_{2,2}$$2_{2,2}$$3_{1,1}$$3_{1,2}$$3_{2,1}$$3_{2,2}$
Figure 7: Plugging of the thick graphs $G$ and $H$
One can then define the execution $G\mathop{\mathopen{:}\cdot\mathclose{:}}H$
of two thick graphs $G$ and $H$ as the execution of the graphs
$G^{\dagger_{D^{H}}}$ and $H^{\ddagger_{D^{G}}}$. Figure 8 shows the set of
alternating paths in the plugging of the thick graphs $G$ and $H$ introduced
in Figure 5. Figure 9 and Figure 10 represent the result of the execution of
these two thick graphs, the first is three-dimensional representation which
can help make the connection with the set of alternating paths in Figure 8,
while the second is a two-dimensional representation of the same graph. In a
natural way, the measurement of the interaction between two thick graphs $G,H$
is defined as
$\mathopen{\llbracket}G^{\dagger_{D^{H}}},H^{\ddagger_{D^{G}}}\mathclose{\rrbracket}_{m}$.
###### Definition .
The execution $F\mathop{\mathopen{:}\cdot\mathclose{:}}G$ of two thick graphs
$F,G$ is the thick graph of carrier $S^{F}\Delta S^{G}$ and dialect
$D^{F}\times D^{G}$ defined as
$F^{\dagger_{D^{G}}}\mathop{\mathopen{:}\mathclose{:}}G^{\ddagger_{D^{F}}}$.
$1_{1,1}$$1_{2,1}$$1_{1,2}$$1_{2,2}$$3_{1,1}$$3_{1,2}$$3_{2,1}$$3_{2,2}$
Figure 8: Alternating paths in the plugging of thick graphs $G$ and $H$
$1_{1,1}$$1_{2,1}$$1_{1,2}$$1_{2,2}$$3_{1,1}$$3_{1,2}$$3_{2,1}$$3_{2,2}$slice
$(1,1)$slice $(2,1)$slice $(1,2)$slice $(2,2)$ Figure 9: Result of the
execution of the thick graphs $G$ and $H$
$1_{1,1}$$1_{2,1}$$1_{1,2}$$1_{2,2}$$3_{1,1}$$3_{1,2}$$3_{2,1}$$3_{2,2}$slice
$(2,1)$slice $(1,1)$slice $(2,2)$slice
$(1,2)$$G\mathop{\mathopen{:}\cdot\mathclose{:}}H$ Figure 10: The thick graph
$G\mathop{\mathopen{:}\cdot\mathclose{:}}H$ represented in two dimensions.
###### Remark .
Since we only modified the graphs before plugging them together, we can make
the following remark. Let $F,G,H$ be thick graphs. Then the thick graph
$F\mathop{\mathopen{:}\cdot\mathclose{:}}(G\mathop{\mathopen{:}\cdot\mathclose{:}}H)$
is defined as
$F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}(G^{\dagger_{D^{H}}}\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{G}}})^{\ddagger_{D^{F}}}=F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}((G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}}\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}})$
If one supposes that $S^{F}\cap S^{G}\cap S^{H}=\emptyset$, it is clear that
$(S^{F}\times D)\cap(S^{G}\times D)\cap(S^{H}\times D)=\emptyset$. We can
deduce from the associativity of execution that
$F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}((G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}}\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}})=(F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}((G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}}))\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}}$
But:
$(F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}((G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}})\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}}=((F^{\dagger_{D^{G}}}\mathop{\mathopen{:}\mathclose{:}}G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}})\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}}$
The latter is by definition the thick graph
$(F\mathop{\mathopen{:}\cdot\mathclose{:}}G)\mathop{\mathopen{:}\cdot\mathclose{:}}H$.
This shows that the associativity of $\mathop{\mathopen{:}\cdot\mathclose{:}}$
on thick graphs is a simple consequence of the associativity of
$\mathop{\mathopen{:}\mathclose{:}}$ on simple graphs.
###### Proposition (Associativity).
Let $F,G,H$ be thick graphs such that $S^{F}\cap S^{G}\cap S^{H}=\emptyset$.
Then:
$F\mathop{\mathopen{:}\cdot\mathclose{:}}(G\mathop{\mathopen{:}\cdot\mathclose{:}}H)=(F\mathop{\mathopen{:}\cdot\mathclose{:}}G)\mathop{\mathopen{:}\cdot\mathclose{:}}H$
###### Definition .
Let $F$ and $G$ be two thick graphs. We define $\text{{Cy}}^{e}(F,G)$ as the
set of circuits in $F^{\dagger_{D^{G}}}\square G^{\ddagger_{D^{F}}}$.
We also define, being given a dialect $D^{H}$,
* •
the set $\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}$ of circuits in the graph
$(F^{\dagger_{D^{G}}})^{\dagger_{D^{H}}}\square(G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}}$
* •
the set $\text{{Cy}}^{e}(F,G)^{\ddagger_{D^{H}}}$ of circuits in the graph
$(F^{\ddagger_{D^{G}}})^{\dagger_{D^{H}}}\square(G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}}$
###### Proposition .
Let $F,G,H$ be thick graphs and $\phi:D^{H}\rightarrow D$ a bijection. Then:
$\displaystyle\text{{Cy}}^{e}(F,H)$ $\displaystyle\cong$
$\displaystyle\text{{Cy}}^{e}(F,H^{\phi})$
$\displaystyle\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}$ $\displaystyle\cong$
$\displaystyle\text{{Cy}}^{e}(F,G)^{\dagger_{\phi(D^{H})}}$
$\displaystyle\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}$ $\displaystyle\cong$
$\displaystyle\text{{Cy}}^{e}(F,G)^{\ddagger_{D^{H}}}$
As in Section 3.1, one considers the three thick graphs $F,G,H$. We are
interested in the circuits in
$\text{{Cy}}^{e}(F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H)\cup(\text{{Cy}}^{e}(G,H)^{\ddagger{D^{F}}})$.
By definition, these are the circuits in one of the following graphs:
$F^{\dagger_{D^{G}\times
D^{H}}}\square((G^{\dagger_{D^{H}}}\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{G}}})^{\ddagger_{D^{F}}})=F^{\dagger_{D^{G}\times
D^{H}}}\square((G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}}\mathop{\mathopen{:}\mathclose{:}}H^{\ddagger_{D^{F}\times
D^{G}}})$ $(G^{\dagger_{D^{H}}}\square
H^{\ddagger_{D^{G}}})^{\ddagger{D^{F}}}=(G^{\dagger_{D^{H}}})^{\ddagger_{D^{F}}}\square
H^{\ddagger_{D^{F}\times D^{G}}}$
We can now use the trefoil property to deduce that these sets of circuits are
in bijection with the set of circuits in the following graphs:
$(F^{\dagger_{D^{G}\times
D^{H}}}\mathop{\mathopen{:}\mathclose{:}}(G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}})\square
H^{\ddagger_{D^{F}\times
D^{G}}}=(F^{\dagger_{D^{G}}}\mathop{\mathopen{:}\mathclose{:}}G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}}\square
H^{\ddagger_{D^{F}\times D^{G}}})$ $(F^{\dagger_{D^{G}\times
D^{H}}}\square(G^{\ddagger_{D^{F}}})^{\dagger_{D^{H}}}=(F^{\dagger_{D^{G}}}\square
G^{\ddagger_{D^{F}}})^{\dagger{D^{H}}}$
This shows that the trefoil property holds for thick graphs.
###### Proposition (Geometric Trefoil Property for Thick Graphs).
If $F$, $G$, $H$ are thick graphs such that $S^{F}\cap S^{G}\cap
S^{H}=\emptyset$, then:
$\text{{Cy}}^{e}(F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H)\cup\text{{Cy}}^{e}(G,H)^{\dagger_{D^{F}}}\cong\text{{Cy}}^{e}(F\mathop{\mathopen{:}\cdot\mathclose{:}}G,H)\cup\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}$
###### Corollary (Geometric Adjunction for Thick Graphs).
If $F$, $G$, $H$ are thick graphs such that $S^{G}\cap S^{H}=\emptyset$, we
have:
$\text{{Cy}}^{e}(F,G\cup
H)\cong\text{{Cy}}^{e}(F\mathop{\mathopen{:}\cdot\mathclose{:}}G,H)\cup\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}$
###### Definition .
Being given a circuit quantifying map $m$, one can define a measurement of the
interaction between thick graphs. For every couple of thick graphs $F,G$, it
is defined as:
$\mathopen{\llbracket}F,G\mathclose{\rrbracket}_{m}=\sum_{\pi\in\text{{Cy}}^{e}(F,G)}\frac{1}{\text{Card}(D^{F}\times
D^{G})}m(\omega(\pi))$
###### Proposition (Numerical Trefoil Property for Thick Graphs).
Let $F,G,H$ be thick graphs such that $S^{F}\cap S^{G}\cap S^{H}=\emptyset$.
Then:
$\mathopen{\llbracket}F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}=\mathopen{\llbracket}H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}F,G\mathclose{\rrbracket}_{m}$
###### Proof.
The proof is a simple calculation using the geometric trefoil property for
thick graphs (Section 3.1). We denote by $n^{F}$ (resp. $n^{G}$, $n^{H}$) the
cardinality of the dialect $D^{F}$ (resp. $D^{G}$, $D^{H}$).
$\displaystyle\mathopen{\llbracket}F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H)}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))+\sum_{\pi\in\text{{Cy}}^{e}(G,H)}\frac{1}{n^{G}n^{H}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H)}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))+\sum_{\pi\in\text{{Cy}}^{e}(G,H)^{\dagger_{D^{F}}}}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H)\cup\text{{Cy}}^{e}(G,H)^{\dagger_{D^{F}}}}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G)\cup\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G)}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))+\sum_{\pi\in\text{{Cy}}^{e}(F,G)^{\dagger_{D^{H}}}}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\sum_{\pi\in\text{{Cy}}^{e}(H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G)}\frac{1}{n^{F}n^{G}n^{H}}m(\omega(\pi))+\sum_{\pi\in\text{{Cy}}^{e}(F,G)}\frac{1}{n^{F}n^{G}}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\mathopen{\llbracket}H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}F,G\mathclose{\rrbracket}_{m}$
∎
###### Corollary (Numerical Adjunction for Thick Graphs).
Let $F,G,H$ be thick graphs such that $S^{G}\cap S^{H}=\emptyset$. Then:
$\mathopen{\llbracket}F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H\mathclose{\rrbracket}_{m}=\mathopen{\llbracket}H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}F,G\mathclose{\rrbracket}_{m}$
###### Remark .
About the Hidden Convention of the Numerical Measure
The measurement of interaction we defined hides a convention: each slice of a
thick graph $F$ is considered as having a ”weight” equal to $1/n^{F}$, so that
the total weight of the set of all slices have weight $1$. This convention
corresponds to the choice of working with a _normalized trace_ (such that
$tr(1)=1$) on the idiom in Girard’s hyperfinite geometry of interaction. It
would have been possible to consider another convention which would impose
that each slice have a weight equal to $1$ (this would correspond to working
with the usual trace on matrices in Girard’s hyperfinite geometry of
interaction). In this case, the measurement of the interaction between two
thick graphs $F,G$ is defined as:
$\mathopen{\llparenthesis}F,G\mathclose{\rrparenthesis}=\sum_{\pi\in\text{{Cy}}^{e}(F,G)}m(\omega(\pi))$
The numerical trefoil property is then stated differently: for all thick
graphs $F$, $G$, and $H$ such that $S^{F}\cap S^{G}\cap S^{H}=\emptyset$, we
have:
$\mathopen{\llparenthesis}F,G\mathop{\mathopen{:}\cdot\mathclose{:}}H\mathclose{\rrparenthesis}+n^{F}\mathopen{\llparenthesis}G,H\mathclose{\rrparenthesis}=\mathopen{\llparenthesis}H,F\mathop{\mathopen{:}\cdot\mathclose{:}}G\mathclose{\rrparenthesis}+n^{H}\mathopen{\llparenthesis}F,G\mathclose{\rrparenthesis}$
We stress the apparition of the terms $n^{F}$ and $n^{H}$ in this equality:
their apparition corresponds exactly to the apparition of the terms
$\textbf{1}_{F}$ and $\textbf{1}_{H}$ in the equality stated for the trefoil
property for sliced graphs.
### 3.2 Sliced Thick Graphs
One can of course apply the additive construction presented in our previous
paper [Sei14a] in the case of thick graphs. A _sliced thick graph_ $G$ of
carrier $S^{G}$ s a finite family $\sum_{i\in I^{G}}\alpha^{G}_{i}G_{i}$
where, for all $i\in I^{G}$, $G_{i}$ is a thick graph such that
$S^{G_{i}}=S^{G}$, and $\alpha^{G}_{i}\in\mathbf{R}$. We define the _dialect_
of $G$ to be the set $\uplus_{i\in I^{G}}D^{G_{i}}$. We will abusively call a
_slice_ a couple $(i,d)$ where $i\in I^{G}$ and $d\in D_{G_{i}}$; we will say
a graph $G$ is a _one-sliced graph_ when $I^{G}=\\{i\\}$ and
$D_{G_{i}}=\\{d\\}$ are both one-element sets.
One can extend the execution and the measurement of the interaction by
applying the thick graphs constructions slice by slice:
$\displaystyle(\sum_{i\in
I^{F}}\alpha^{F}_{i}F_{i})\mathop{\mathopen{:}\mathclose{:}}(\sum_{i\in
I^{G}}\alpha^{G}_{i}G_{i})$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{(i,j)\in I^{F}\times
I^{G}}\\!\\!\\!\\!\alpha^{F}_{i}\alpha^{G}_{j}F_{i}\mathop{\mathopen{:}\cdot\mathclose{:}}G_{j}$
$\displaystyle\mathopen{\llbracket}\sum_{i\in
I^{F}}\alpha^{F}_{i}F_{i},\sum_{i\in
I^{G}}\alpha^{G}_{i}G_{i}\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{(i,j)\in I^{F}\times
I^{G}}\\!\\!\\!\\!\alpha^{F}_{i}\alpha^{G}_{j}\mathopen{\llbracket}F_{i},G_{j}\mathclose{\rrbracket}_{m}$
Figure 11 shows two examples of sliced thick graphs. The graphical convention
we will follow for representing sliced and thick graphs corresponds to the
graphical convention for sliced graphs, apart from the fact that the graphs
contained in the slices are replaced by thick graphs. Thus, two slices are
separated by a dashed line, two elements in the dialect of a thick graph (i.e.
the graph contained in a slice) are separated by a dotted line.
$1_{1}$$1_{2}$$1_{3}$$2_{1}$$2_{2}$$2_{3}$$3_{1}$$3_{2}$$3_{3}$$\frac{1}{2}F$$3G$$1_{a}$$2_{a}$$1_{b}$$2_{b}$$F_{a}$$F_{b}$
Figure 11: Examples of sliced thick graphs: $\frac{1}{2}F+3G$ and
$F_{a}+F_{b}$
One should however notice that some sliced thick graphs (for instance the
graph $F_{a}+F_{b}$ represented in red in Figure 11) can be considered both as
a thick graph — hence a sliced thick graph with a single slice — or as a
sliced graph with two slices — hence a sliced thick graph with two slices.
Indeed, consider the graphs:
$\begin{array}[]{rcl|rcl|rcl}&F_{a}&&&F_{b}&&&F_{c}\\\ \hline\cr\hline\cr
V^{F_{a}}&=&\\{1,2\\}&V^{F_{b}}&=&\\{1,2\\}&V^{F_{c}}&=&\\{1,2\\}\times\\{a,b\\}\\\
E^{F_{a}}&=&\\{f,g\\}&E^{F_{b}}&=&\\{f,g\\}&E^{F_{c}}&=&\\{f_{a},f_{b},g_{a},g_{b}\\}\\\
s^{F_{a}}&=&\left\\{\begin{array}[]{l}f\mapsto 1\\\ g\mapsto
2\end{array}\right.&s^{F_{b}}&=&\left\\{\begin{array}[]{l}f\mapsto 1\\\
g\mapsto
1\end{array}\right.&s^{F_{c}}&=&\left\\{\begin{array}[]{l}f_{i}\mapsto
s^{F_{i}}(f)\\\ g_{i}\mapsto s^{F_{i}}(g)\end{array}\right.\\\
t^{F_{a}}&=&\left\\{\begin{array}[]{l}f\mapsto 2\\\ g\mapsto
2\end{array}\right.&t^{F_{b}}&=&\left\\{\begin{array}[]{l}f\mapsto 2\\\
g\mapsto
1\end{array}\right.&t^{F_{c}}&=&\left\\{\begin{array}[]{l}f_{i}\mapsto
t^{F_{i}}(f)\\\ g_{i}\mapsto t^{F_{i}}(g)\end{array}\right.\\\
\omega^{F_{a}}&\equiv&1&\omega^{F_{b}}&\equiv&1&\omega^{F_{c}}&\equiv&1\end{array}$
One can then define the the two sliced thick graphs $G_{1}=F_{c}$ and
$G_{2}=\frac{1}{2}F_{a}+\frac{1}{2}F_{b}$. These two graphs are represented in
Figure 12. They are similar in a very precise sense: one can show that if $H$
is any sliced thick graph, and $m$ is any circuit-quantifying map, then
$\mathopen{\llbracket}G_{1},H\mathclose{\rrbracket}_{m}=\mathopen{\llbracket}G_{2},H\mathclose{\rrbracket}_{m}$.
We say they are _universally equivalent_. Notice that this explains in a very
formal way the remark about the convention on the measurement of interaction
Section 3.1.
$1_{a}$$2_{a}$$1_{b}$$2_{b}$$F_{c}$$1_{a}$$2_{a}$$1_{b}$$2_{b}$$\frac{1}{2}F_{a}$$\frac{1}{2}F_{b}$
Figure 12: Les graphes $G_{1}$ et $G_{2}$
###### Definition (Universally equivalent graphs).
Let $F,G$ be two graphs. We say that $F$ and $G$ are _universally equivalent_
(for the measurement
$\mathopen{\llbracket}\cdot,\cdot\mathclose{\rrbracket}_{m}$) — which will be
denoted by $F\simeq_{u}G$ — if for all graph $H$:
$\mathopen{\llbracket}F,H\mathclose{\rrbracket}_{m}=\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
The next proposition states that if $F^{\prime}$ is obtained from a graph $F$
by a renaming of edges, then $F\simeq_{u}F^{\prime}$.
###### Proposition .
Let $F,F^{\prime}$ be two graphs such that $V^{F}=V^{F^{\prime}}$, and $\phi$
a bijection $E^{F}\rightarrow E^{F^{\prime}}$ such that:
$\displaystyle s^{G}\circ\phi=s^{F},\leavevmode\nobreak\ \leavevmode\nobreak\
t^{G}\circ\phi=t^{F},\leavevmode\nobreak\ \leavevmode\nobreak\
\omega^{G}\circ\phi=\omega^{F}$
Then $F\simeq_{u}F^{\prime}$.
###### Proof.
Indeed, the bijection $\phi$ induces, from the hypotheses in the source and
target functions, a bijection between the sets of cycles $\text{{Cy}}(F,H)$
and $\text{{Cy}}(G,H)$. The condition on the weight map then insures us that
this bijection is $\omega$-invariant, from which we deduce the proposition. ∎
###### Proposition .
Let $F,G$ be sliced graphs. If there exists a bijection $\phi:I^{F}\rightarrow
I^{G}$ such that $F_{i}=G_{\phi(i)}$ and
$\alpha^{F}_{i}=\alpha^{G}_{\phi(i)}$, then $F\simeq_{u}G$.
###### Proof.
By definition:
$\displaystyle\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{(i,j)\in I^{G}\times
I^{H}}\alpha^{G}_{i}\alpha^{H}_{j}\mathopen{\llbracket}G_{i},H_{j}\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{(i,j)\in I^{F}\times
I^{H}}\alpha^{G}_{\phi(i)}\alpha^{H}_{j}\mathopen{\llbracket}G_{\phi(i)},H_{j}\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{(i,j)\in I^{F}\times
I^{G}}\alpha^{F}_{i}\alpha^{G}_{j}\mathopen{\llbracket}F_{i},G_{j}\mathclose{\rrbracket}_{m}$
Thus $F$ and $G$ are universally equivalent. ∎
###### Proposition .
Let $F,G$ be thick graphs. If there exists a bijection $\phi:D^{F}\rightarrow
D^{G}$ such that $G=F^{\phi}$, then $F\simeq_{u}G$.
###### Proof.
Let $F,G$ be thick graphs such that $G=F^{\phi}$ for a bijection
$\phi:D^{G}\rightarrow D^{F}$, and $H$ an arbitrary thick graph. Then the
bijection $\phi\times\text{Id}:D^{G}\times D^{H}\rightarrow D^{F}\times D^{H}$
satisfies that
$G^{\dagger_{D^{H}}}=(F^{\dagger_{D^{H}}})^{\phi\times\text{Id}}$. One can
notice that the set of alternating circuits in $F^{\dagger}\square
H^{\ddagger}$ is the same as the set of alternating circuits in
$(F^{\dagger})^{\phi\times\text{Id}}\square(H^{\dagger})^{\phi\times\text{Id}}=G^{\dagger}\square
H^{\ddagger}$. Thus:
$\displaystyle\mathopen{\llbracket}F,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{\pi\in\text{{Cy}}(F,H)}m(\omega(\pi))$
$\displaystyle=$ $\displaystyle\sum_{\pi\in\text{{Cy}}(G,H)}m(\omega(\pi))$
$\displaystyle=$
$\displaystyle\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
And finally $F$ and $G$ are universally equivalent. ∎
###### Proposition .
Let $F=\sum_{i\in I^{F}}\alpha_{i}^{F}F_{i}$ be a sliced thick graph, and let
us define, for all $i\in I^{F}$, $n^{F_{i}}=\text{Card}(D^{F_{i}})$ and
$n^{F}=\sum_{i\in I^{F}}n^{F_{i}}$. Suppose that there exists a scalar
$\alpha$ such that for all $i\in I^{F}$,
$\alpha_{i}^{F}=\alpha\frac{n^{F_{i}}}{n^{F}}$. We then define the sliced
thick graph with a single slice $\alpha G$ of dialect $\uplus
D^{F_{i}}=\cup_{i\in I^{F}}D^{F_{i}}\times\\{i\\}$ and carrier $V^{F}$ by:
$\displaystyle V^{G}$ $\displaystyle=$ $\displaystyle V^{F}\times\uplus
D^{F_{i}}$ $\displaystyle E^{G}$ $\displaystyle=$ $\displaystyle\uplus
E^{F_{i}}=\cup_{i\in I^{F}}E^{F_{i}}\times\\{i\\}$ $\displaystyle s^{G}$
$\displaystyle=$ $\displaystyle(e,i)\mapsto(s^{F_{i}}(e),i)$ $\displaystyle
t^{G}$ $\displaystyle=$ $\displaystyle(e,i)\mapsto(t^{F_{i}}(e),i)$
$\displaystyle\omega^{G}$ $\displaystyle=$
$\displaystyle(e,i)\mapsto\omega^{F_{i}}(e)$
$\displaystyle\left((e,i)\coh^{G}(f,j)\right.$ $\displaystyle\Leftrightarrow$
$\displaystyle\left.(i\neq j)\vee(i=j\wedge e\coh^{F_{i}}f)\right)$
Then $F$ and $G$ are universally equivalent.
###### Proof.
Let $H$ be a sliced thick graph. Then:
$\displaystyle\mathopen{\llbracket}F,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{i\in I^{H}}\sum_{j\in
I^{F}}\alpha^{H}_{i}\alpha^{F}_{j}\mathopen{\llbracket}F_{i},H_{j}\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{i\in I^{H}}\sum_{j\in
I^{F}}\alpha^{H}_{i}\alpha\frac{n^{F_{i}}}{n^{F}}\mathopen{\llbracket}F_{i},H_{j}\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{i\in I^{H}}\sum_{j\in
I^{F}}\alpha^{H}_{i}\alpha\frac{n^{F_{i}}}{n^{F}}\frac{1}{n^{F_{i}}n^{H_{j}}}\sum_{\pi\in\text{{Cy}}(F_{i},H_{j})}m(\omega(\pi))$
$\displaystyle=$ $\displaystyle\sum_{i\in I^{H}}\sum_{j\in
I^{F}}\alpha^{H}_{i}\alpha\frac{1}{n^{F}n^{H_{j}}}\sum_{\pi\in\text{{Cy}}(F_{i},H_{j})}m(\omega(\pi))$
$\displaystyle=$ $\displaystyle\sum_{i\in
I^{H}}\alpha^{H}_{i}\alpha\frac{1}{n^{F}n^{H_{j}}}\sum_{j\in
I^{F}}\sum_{\pi\in\text{{Cy}}(F_{i},H_{j})}m(\omega(\pi))$
But one can notice that $\cup_{j\in
I^{F}}\text{{Cy}}(F_{i},H_{j})=\text{{Cy}}(G,H)$. We thus get:
$\displaystyle\mathopen{\llbracket}F,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle\sum_{i\in
I^{H}}\alpha^{H}_{i}\alpha\frac{1}{n^{F}n^{H_{j}}}\sum_{j\in
I^{F}}\sum_{\pi\in\text{{Cy}}(F_{i},H_{j})}m(\omega(\pi))$ $\displaystyle=$
$\displaystyle\sum_{i\in
I^{H}}\alpha^{H}_{i}\alpha\frac{1}{n^{F}n^{H_{j}}}\sum_{\pi\in\text{{Cy}}(F,H_{j})}m(\omega(\pi))$
$\displaystyle=$ $\displaystyle\mathopen{\llbracket}\alpha
G,H\mathclose{\rrbracket}_{m}$
Finally, we showed that $F$ and $\alpha G$ are universally equivalent. ∎
One of the consequences of Section 3.2, Section 3.2, and Section 3.2 is that
two graphs $F,G$ such that $G$ is obtained from $F$ by a renaming of the sets
$E^{F},I^{F},D^{F}$ are universally equivalent. We will therefore work from
now on with graphs modulo renaming of these sets.
### 3.3 Thick Graphs and Contraction
In this section, we will explain how the introduction of thick graphs allow
the definition of contraction by using the fact that edges can go from a slice
to another (contrarily to sliced graphs). In the following, we will be working
with sliced thick graphs. The way contraction is dealt with by using slice-
changing edges is quite simple, and the graph which will implement this
transformation is essentially the same as the graph implementing additive
contraction (i.e. the graph implementing distributivity — Section 2.2 —
restricted to the location of contexts) modified with a change of slices.
The graph we obtain is then the superimposition of two $\mathfrak{Fax}$, but
where one of them goes from one slice to the other.
$1_{1}$$2_{1}$$3_{1}$$4_{1}$$5_{1}$$6_{1}$$7_{1}$$8_{1}$$9_{1}$$1_{2}$$2_{2}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$$4_{2}$$5_{2}$$6_{2}$
Figure 13: The graph of a contraction project
###### Definition (Contraction).
Let $\phi:V^{A}\rightarrow W_{1}$ and $\psi:V^{A}\rightarrow W_{2}$ be two
bijections with $V^{A}\cap W_{1}=V^{A}\cap W_{2}=W_{1}\cap W_{2}=\emptyset$.
We define the project
$\mathfrak{Ctr}^{\phi}_{\psi}=(0,\text{{Ctr}}^{\phi}_{\psi})$, where the graph
$\text{{Ctr}}^{\phi}_{\psi}$ is defined by:
$\displaystyle V^{\text{{Ctr}}^{\phi}_{\psi}}$ $\displaystyle=$ $\displaystyle
V^{A}\cup W_{1}\cup W_{2}$ $\displaystyle D^{\text{{Ctr}}^{\phi}_{\psi}}$
$\displaystyle=$ $\displaystyle\\{1,2\\}$ $\displaystyle
E^{\text{{Ctr}}^{\phi}_{\psi}}$ $\displaystyle=$ $\displaystyle
V^{A}\times\\{1,2\\}\times\\{i,o\\}$ $\displaystyle
s^{\text{{Ctr}}^{\phi}_{\psi}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rcl}(v,1,o)&\mapsto&(\phi(v),1)\\\
(v,1,i)&\mapsto&(v,1)\\\ (v,2,o)&\mapsto&(\psi(v),1)\\\
(v,2,i)&\mapsto&(v,2)\end{array}\right.$ $\displaystyle
t^{\text{{Ctr}}^{\phi}_{\psi}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rcl}(v,1,o)&\mapsto&(v,1)\\\
(v,1,i)&\mapsto&(\phi(v),1)\\\ (v,2,o)&\mapsto&(v,2)\\\
(v,2,i)&\mapsto&(\psi(v),1)\end{array}\right.$
$\displaystyle\omega^{\text{{Ctr}}^{\phi}_{\psi}}$ $\displaystyle\equiv$
$\displaystyle 1$
Figure 13 illustrates the graph of the project $\mathfrak{Ctr}^{\psi}_{\phi}$,
where the functions are defined by
$\phi:\\{1,2,3\\}\rightarrow\\{4,5,6\\},x\mapsto x+3$ and
$\psi:\\{1,2,3\\}\rightarrow\\{7,8,9\\},x\mapsto 10-x$.
###### Proposition .
Let $\mathfrak{a}=(0,A)$ be a project in a behavior $\mathbf{A}$, such that
$D^{A}\cong\\{1\\}$. Let $\phi,\psi$ be two delocations $V^{A}\rightarrow
W_{1}$, $V^{A}\rightarrow W_{2}$ of disjoint codomains. Then
$\mathfrak{Ctr}^{\psi}_{\phi}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{a}\in\mathbf{\phi(A)\otimes\psi(A)}$.
###### Proof.
We will denote by Ctr the graph $\text{Ctr}_{\phi}^{\psi}$ to simplify the
notations. We first compute
$A\mathop{\mathopen{:}\cdot\mathclose{:}}\text{Ctr}$. We get
$A^{\ddagger_{\\{1,2\\}}}=(V^{A}\times\\{1,2\\},E^{A}\times\\{1,2\\},s^{A}\times
Id_{\\{1,2\\}},t^{A}\times Id_{\\{1,2\\}},\omega^{A}\circ\pi)$ where $\pi$ is
the projection: $E^{A}\times\\{1,2\\}\rightarrow E^{A},(x,i)\mapsto x$.
Moreover the graph $\text{Ctr}^{\dagger_{D^{A}}}$ is a variant of the graph
Ctr since $D^{A}\cong\\{1\\}$. Here is what the plugging of
$\text{Ctr}^{\dagger_{D^{A}}}$ with $A^{\ddagger_{\\{1,2\\}}}$ looks like:
$V^{A}\times\\{2\\}$$W_{1}\times\\{2\\}$$W_{2}\times\\{2\\}$$V^{A}\times\\{1\\}$$W_{1}\times\\{1\\}$$w_{2}\times\\{1\\}$$\phi$$\psi$
The result of the execution is therefore a two-sliced graph, i.e. a graph of
dialect $D^{A}\times\\{1,2\\}\cong\\{1,2\\}$, and which contains the graph
$\phi(A)\cup\psi(A)$ in the slice numbered $1$ and contains the empty graph in
the slice numbered $2$.
We deduce from this that
$\mathfrak{Ctr}^{\psi}_{\phi}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{a}$
is universally equivalent (Section 3.2) to the project
$\frac{1}{2}\mathfrak{\phi(a)\otimes\psi(a)}+\frac{1}{2}\mathfrak{0}$ from
Section 3.2. Since
$\mathfrak{\phi(a)\otimes\psi(a)}\in\mathbf{\phi(A)\otimes\psi(A)}$, then the
project $\frac{1}{2}(\mathfrak{\phi(a)\otimes\psi(a)})$ is an element in
$\mathbf{\phi(A)\otimes\psi(A)}$ by the homothety Lemma (Section 2.2).
Moreover, $\mathbf{A}$ is a behavior, hence $\mathbf{\phi(A)\otimes\psi(A)}$
is also a behavior and we can deduce that
$\frac{1}{2}\mathfrak{\phi(a)\otimes\psi(a)}+\frac{1}{2}\mathfrak{0}$ is an
element in $\mathbf{\phi(A)\otimes\psi(A)}$. ∎
Figure 15, Figure 16 and Figure 17 illustrate the plugging and execution of a
contraction with two graphs: the first — $A$ — having a single slice, and the
other — $B$ — having two slices (the graphs are shown in Figure 14). One can
see that the hypothesis $D^{A}\equiv\\{1\\}$ used in the preceding proposition
is necessary, and that slice-changing edges allow to implement contraction of
graphs with a single slice.
$1_{1}$$2_{1}$$3_{1}$$4_{1}$$5_{1}$$6_{1}$$1_{2}$$2_{2}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$
(a) The graph of the project $\mathfrak{Ctr}^{\phi}_{\psi}$
$1$$2$$A$$1_{1}$$1_{2}$$2_{1}$$2_{2}$$B$ (b) The graphs $A$ and $B$ of the
projects $\mathfrak{a}$ and $\mathfrak{b}$
Figure 14: The graphs of the projects $\mathfrak{Ctr}^{\phi}_{\psi}$,
$\mathfrak{a}$ and $\mathfrak{b}$.
$1_{1}$$2_{1}$$3_{1}$$4_{1}$$5_{1}$$6_{1}$$1_{2}$$2_{2}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$$A$$A$
(a) Plugging of $\text{{Ctr}}^{\phi}_{\psi}$ and $A$
$1_{1,1}$$2_{1,1}$$3_{1,1}$$4_{1,1}$$5_{1,1}$$6_{1,1}$$1_{2,1}$$2_{2,1}$$3_{2,1}$$4_{2,1}$$5_{2,1}$$6_{2,1}$$1_{1,2}$$2_{1,2}$$3_{1,2}$$4_{1,2}$$5_{1,2}$$6_{1,2}$$1_{2,2}$$2_{2,2}$$3_{2,2}$$4_{2,2}$$5_{2,2}$$6_{2,2}$
(b) Plugging of $\text{{Ctr}}^{\phi}_{\psi}$ and $B$
Figure 15: Plugging of $\text{{Ctr}}^{\phi}_{\psi}$ with the two graphs $A$
and $B$
$3_{1}$$4_{1}$$5_{1}$$6_{1}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$ (a) Result of the
execution of $\text{{Ctr}}^{\phi}_{\psi}$ and $A$
$3_{1}$$4_{1}$$5_{1}$$6_{1}$ (b) The graph of
$\mathfrak{\phi(a)\otimes\psi(a)}$
Figure 16: Graphs of the projects
$\mathfrak{Ctr}^{\phi}_{\psi}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{a}$
and $\mathfrak{\phi(a)\otimes\psi(a)}$
$3_{1}$$4_{1}$$5_{1}$$6_{1}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$$3_{3}$$4_{3}$$5_{3}$$6_{3}$$3_{4}$$4_{4}$$5_{4}$$6_{4}$
(a) Result of the execution of $\text{{Ctr}}^{\phi}_{\psi}$ and $B$
$3_{1}$$4_{1}$$5_{1}$$6_{1}$$3_{2}$$4_{2}$$5_{2}$$6_{2}$$3_{3}$$4_{3}$$5_{3}$$6_{3}$$3_{4}$$4_{4}$$5_{4}$$6_{4}$
(b) Graph of the project $\mathfrak{\phi(b)\otimes\psi(b)}$
Figure 17: Graphs of the projects
$\mathfrak{Ctr}^{\phi}_{\psi}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{b}$
and $\mathfrak{\phi(b)\otimes\psi(b)}$
We will use the following direct corollary of Section 2.2.
###### Proposition .
If $E$ is a non-empty set of project sharing the same carrier $V^{E}$,
$\mathbf{F}$ is a conduct and $\mathfrak{f}$ satisfies that
$\forall\mathfrak{e}\in E$,
$\mathfrak{f\mathop{\mathopen{:}\mathclose{:}}e}\in\mathbf{F}$, then
$\mathfrak{f}\in E^{\simbot\simbot}\multimap\mathbf{F}$.
This proposition insures us that if $\mathbf{A}$ is a conduct such that there
exists a set $E$ of one-sliced projects with $\mathbf{A}=E^{\simbot\simbot}$,
then the contraction project $\mathfrak{Ctr}^{\psi}_{\phi}$ belongs to the
conduct $\mathbf{A\multimap\phi(A)\otimes\psi(A)}$.
We find here a geometrical explanation to the introduction of exponential
connectives. Indeed, in order to use a contraction, we must be sure we are
working with one-sliced graphs. We will therefore define, for all behavior
$\mathbf{A}$, a conduct $\mathbf{\oc A}$ generated by a set of one-sliced
graphs.
One should notice that a conduct $\mathbf{\oc A}$ generated by a set of one-
sliced projects cannot be a behavior: the projects $(a,\emptyset)$ necessarily
belong to the orthogonal of $\mathbf{\oc A}$. We will therefore introduce
_perennial conducts_ as those conducts generated by a set of wager-free one-
sliced projects. Dually, we introduce the _co-perennial conducts_ as the
conducts that are the orthogonal of a perennial conduct.
But first, we will need a way to associate a wager-free one-sliced project to
any wager-free project. In order to do so, we will introduce the notion of
_thick graphing_.
## 4 Construction of an Exponential Connective on the Real Line
We now consider the microcosm $\mathfrak{mi}$ of measure-inflating maps101010A
_measure-inflating map_ on the real line with Lebesgue measure $\lambda$ is a
non-singular Borel-preserving transformation
$\phi:\mathbf{R}\rightarrow\mathbf{R}$ such that there exists a positive real
number $\mu_{\phi}$ with $\lambda(\phi^{-1}(A))=\mu_{\phi}\lambda(A)$. In
other terms, $\phi$ _transports the measure_ $\lambda$ onto
$\mu_{\phi}\lambda$. on the real line endowed with Lebesgue measure, we fix
$\Omega=]0,1]$ endowed with the usual multiplication and we chose any
measurable map $m:\Omega\rightarrow\mathbf{R}_{\geqslant 0}\cup\\{\infty\\}$
such that $m(1)=\infty$. We showed in a previous work how this framework can
interpret multiplicative-additive linear logic with second order
quantification111111We actually showed how one can interpret second-order
multiplicative-additive linear logic in the model
$\mathbb{M}[\Omega,\mathfrak{aff}]_{m}$ where
$\mathfrak{aff}\subsetneq\mathfrak{mi}$ is the microcosm of affine
transformations on the real line. The result is however valid for any super-
microcosm $\mathfrak{n}\supset\mathfrak{aff}$, hence for $\mathfrak{mi}$,
since a graphing in $\mathfrak{aff}$ can be considered as a graphing in
$\mathfrak{n}$ in a way that is coherent with execution, orthogonality, sums,
etc. [Sei14c]. We now show how to interpret elementary linear logic
exponential connectives in the model $\mathbb{M}[\Omega,\mathfrak{mi}]_{m}$
(defined in Section 2.3).
### 4.1 Sliced Thick Graphings
The sliced graphings are obtained from graphings in the same way we defined
sliced thick graphs from directed weighted graphs: we consider formal weighted
sums $F=\sum_{i\in I^{F}}\alpha^{F}_{i}F_{i}$ where the $F_{i}$ are graphings
of carrier $V^{F_{i}}$. We define the _carrier of $F$_ as the measurable set
$\cup_{i\in I^{F}}V^{F_{i}}$. The various constructions are then extended as
explained above:
$\displaystyle(\sum_{i\in
I^{F}}\alpha^{F}_{i}F_{i})\mathop{\mathopen{:}\mathclose{:}}(\sum_{i\in
I^{G}}\alpha^{G}_{i}G_{i})$ $\displaystyle=$ $\displaystyle\sum_{(i,j)\in
I^{F}\times
I^{G}}\alpha_{i}^{F}\alpha^{G}_{j}F_{i}\mathop{\mathopen{:}\mathclose{:}}G_{j}$
$\displaystyle\mathopen{\llbracket}\sum_{i\in
I^{F}}\alpha^{F}_{i}F_{i},\sum_{i\in
I^{G}}\alpha^{G}_{i}G_{i}\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\sum_{(i,j)\in I^{F}\times
I^{G}}\alpha_{i}^{F}\alpha^{G}_{j}\mathopen{\llbracket}F_{i}\mathop{\mathopen{:}\mathclose{:}}G_{j}\mathclose{\rrbracket}_{m}$
The trefoil property and the adjunction are then easily obtained through the
same computations as in the proofs of Section 3.1 and Section 3.1.
We will now consider the most general notion of _thick graphing_ one can
define. As it was the case in the setting of graphs, a thick graphing is a
graphing whose carrier has the form $V\times D$. The main difference between
graphings and thick graphings really comes from the way two such objects
interact.
###### Definition .
Let $(X,\mathcal{B},\lambda)$ be a measured space and $(D,\mathcal{D},\mu)$ a
probability space (a measured space such that $\mu(D)=1$). A thick graphing of
carrier $V\in\mathcal{B}$ and dialect $D$ is a graphing on $X\times D$ of
carrier $V\times D$.
###### Definition (Dialectal Interaction).
Let $(X,\mathcal{B},\lambda)$ be a measured space and $(D,\mathcal{D},\mu)$,
$(E,\mathcal{E},\nu)$ two probability spaces. Let $F,G$ be thick graphings of
respective carriers $V^{F},V^{G}\in\mathcal{B}$ and respective dialects $D,E$.
We define the graphings $F^{\dagger_{E}}$ and $G^{\ddagger_{D}}$ as the
graphings of respective carriers $V^{F},V^{G}$ and dialects $E\times F$:
$\displaystyle F^{\dagger_{E}}$ $\displaystyle=$
$\displaystyle\\{(\omega^{F}_{e},\phi^{F}_{e}\times\text{Id}_{E}:S_{e}^{F}\times
D\times E\rightarrow T_{e}^{F}\times D\times E)\\}_{e\in E^{F}}$
$\displaystyle G^{\ddagger_{D}}$ $\displaystyle=$
$\displaystyle\\{(\omega^{G}_{e},\text{Id}_{X}\times(\tau\circ(\phi^{G}_{e}\times\text{Id}_{D})\circ\tau^{-1}):S_{e}^{G}\times
D\times E\rightarrow T_{e}^{G}\times D\times E)\\}_{e\in E^{G}}$
where $\tau$ is the natural symmetry: $E\times D\rightarrow D\times E$.
###### Definition (Plugging).
The plugging $F\mathop{\mathopen{:}\mathclose{:}}G$ of two thick graphings of
respective dialects $D^{F},D^{G}$ is defined as
$F^{\dagger_{D^{G}}}\tilde{\square}G^{\ddagger_{D^{F}}}$.
###### Definition (Execution).
Let $F,G$ be two thick graphings of respective dialects $D^{F},D^{G}$. Their
execution is equal to
$F^{\dagger_{D^{G}}}\mathop{\mathopen{:}\text{\scriptsize{m}}\mathclose{:}}{G}^{\ddagger_{D^{F}}}$.
###### Definition (Measurement).
Let $F,G$ be two thick graphings of respective dialects $D^{F},D^{G}$, and $q$
a circuit-quantifying map. The corresponding measurement of the interaction
between $F$ and $G$ is equal to
$\mathopen{\llbracket}F^{\dagger_{D^{G}}},G^{\ddagger_{D^{F}}}\mathclose{\rrbracket}_{m}$.
As in the setting of graphs, one can show that all the fundamental properties
are preserved when we generalize from graphings to thick graphings.
###### Proposition .
Let $F,G,H$ be thick graphings such that $V^{F}\cap V^{G}\cap V^{H}$ is of
null measure. Then:
$\displaystyle
F\mathop{\mathopen{:}\mathclose{:}}(G\mathop{\mathopen{:}\mathclose{:}}H)$
$\displaystyle=$
$\displaystyle(F\mathop{\mathopen{:}\mathclose{:}}G)\mathop{\mathopen{:}\mathclose{:}}H$
$\displaystyle\mathopen{\llbracket}F,G\mathop{\mathopen{:}\mathclose{:}}H\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$
$\displaystyle\mathopen{\llbracket}G,H\mathop{\mathopen{:}\mathclose{:}}F\mathclose{\rrbracket}_{m}+\mathopen{\llbracket}H,F\mathclose{\rrbracket}_{m}$
In a similar way, the extension from thick graphings to sliced thick graphings
should now be quite clear. One extends all operations by ”linearity” to formal
weighted sums of thick graphings, and one obtains, when $F,G,H$ are sliced
thick graphings such that $V^{F}\cap V^{G}\cap V^{H}$ is of null measure:
$\displaystyle
F\mathop{\mathopen{:}\mathclose{:}}(G\mathop{\mathopen{:}\mathclose{:}}H)$
$\displaystyle=$
$\displaystyle(F\mathop{\mathopen{:}\mathclose{:}}G)\mathop{\mathopen{:}\mathclose{:}}H$
$\displaystyle\mathopen{\llbracket}F,G\mathop{\mathopen{:}\mathclose{:}}H\mathclose{\rrbracket}_{m}+\textbf{1}_{F}\mathopen{\llbracket}G,H\mathclose{\rrbracket}_{m}$
$\displaystyle=$
$\displaystyle\mathopen{\llbracket}G,H\mathop{\mathopen{:}\mathclose{:}}F\mathclose{\rrbracket}_{m}+\textbf{1}_{G}\mathopen{\llbracket}H,F\mathclose{\rrbracket}_{m}$
### 4.2 Perennial and Co-perennial conducts
Since we are working with sliced thick graphings, we can follow the
constructions of multiplicative and additive connectives as they are studied
in the author’s second paper on interaction graphs [Sei14a] and which were
quickly recalled in Section 2.2.
###### Definition (Projects).
A _project_ is a couple $\mathfrak{a}=(a,A)$ together with a support $V^{A}$
where:
* •
$a\in\mathbf{R}\cup\\{\infty\\}$ is called the wager;
* •
$A$ is a sliced and thick weighted graphing of carrier $V^{A}$, of dialect
$D^{A}$ a discrete probability space, and index $I^{A}$ a finite set.
###### Remark .
We made here the choice to stay close to the hyperfinite geometry of
interaction defined by Girard [Gir11]. This is why we restrict to discrete
probability spaces as dialects, a restriction that corresponds to the
restriction to finite von Neumann algebras of type I as idioms in Girard’s
setting. However, the results of the preceding section about execution and
measurement, and the definition of the family of circuit-quantifying maps do
not rely on this hypothesis. It should therefore be possible to consider a
more general set of project where the dialects may eventually be continuous.
It may turn out that this generalization could be used to define more
expressive exponential connectives than the one defined in this paper, such as
the usual exponentials of linear logic (recall that the exponentials defined
here are the exponentials of Elementary Linear Logic).
As we explained at the end of Section 3, we will need to consider a particular
kind of conducts which are the kind of conducts obtained from the application
of the exponential modality to a conduct and which are unfortunately not
behaviors. We now study these types of conducts.
###### Definition (Perennialisation).
A Perennialisation is a function that associates a one-sliced weighted
graphing to any sliced and thick weighted graphing.
###### Definition (Exponentials).
Let $\mathbf{A}$ be a conduct, and $\Omega$ a perennialisation. We define the
$\mathbf{\oc_{\Omega}A}$ as the bi-orthogonal closure of the following set of
projects:
$\sharp_{\Omega}\mathbf{A}=\\{\oc\mathfrak{a}=(0,\Omega(A))\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathfrak{a}=(0,A)\in\mathbf{A},I^{A}\cong\\{1\\}\\}$
The dual connective is of course defined as
$\mathbf{\wn_{\Omega}A}=\mathbf{(\sharp_{\Omega}A^{\simbot})^{\simbot}}$.
###### Definition .
A conduct $\mathbf{A}$ is a _perennial conduct_ when there exists a set $A$ of
projects such that:
1. 1.
$\mathbf{A}=A^{\simbot\simbot}$;
2. 2.
for all $\mathfrak{a}=(a,A)\in A$, $a=0$ and $A$ is a one-sliced graphing.
A _co-perennial_ conduct is a conduct $\mathbf{B}=\mathbf{A}^{\simbot}$ where
$\mathbf{A}$ is a perennial conduct.
###### Proposition .
A co-perennial conduct $\mathbf{B}$ satisfies the _inflation property_ : for
all $\lambda\in\mathbf{R}$,
$\mathfrak{b}\in\mathbf{B}\Rightarrow\mathfrak{b+\lambda b}\in\mathbf{B}$.
###### Proof.
The conduct $\mathbf{A}=\mathbf{B}^{\simbot}$ being perennial, there exists a
set $A$ of one-sliced wager-free projects such that
$\mathbf{A}=A^{\simbot\simbot}$. If $A$ is non-empty, the result is a direct
consequence of Section 2.2. If $A$ is empty, then
$\mathbf{B}=\mathbf{A}^{\simbot}=A^{\simbot}$ is the full behavior
$\mathbf{T}_{V^{B}}$ which obviously satisfies the inflation property. ∎
###### Proposition .
A co-perennial conduct is non-empty.
###### Proof.
Suppose that $\mathbf{A}^{\simbot}$ is a co-perennial conduct of carrier
$V^{A}$. Then there exists a set $A$ of one-sliced wager-free projects such
that $\mathbf{A}=\mathbf{A}^{\simbot\simbot}$. If $A$ is empty, then
$A^{\simbot}=\mathbf{A}^{\simbot}$ is the behavior $\mathbf{T}_{V^{A}}$. If
$\mathbf{A}$ is non-empty, then one can easily check that for all real number
$\lambda\neq 0$, the project
$\mathfrak{Dai}_{\lambda}=(\lambda,(V^{A},\emptyset))$ is an element of
$A^{\simbot}=\mathbf{A}^{\simbot}$. ∎
###### Corollary .
Let $\mathbf{A}$ be a perennial conduct. Then
$\mathfrak{a}=(a,A)\in\mathbf{A}\Rightarrow a=0$.
###### Proof.
Since $\mathbf{A}^{\simbot}$ is co-perennial, it is a non-empty set of
projects with the same carrier which satisfies the inflation property. The
result is then obtained by applying Section 2.2. ∎
###### Proposition .
If $\mathbf{A}$ is a co-perennial conduct, then for all $a\neq 0$, the project
$\mathfrak{Dai}_{a}=(a,(V^{A},\emptyset))$ is an element of $\mathbf{A}$.
###### Proof.
We write $B$ the set of one-sliced wager-free projects such that
$B^{\simbot}=\mathbf{A}$. Then for all element $\mathfrak{b}\in\mathbf{B}$, we
have that $\textbf{1}_{B}=1$, from which we conclude that
$\mathopen{\ll}\mathfrak{b},\mathfrak{Dai_{\text{$a$}}}\mathclose{\gg}_{m}=a\textbf{1}_{B}=a$.
Thus $\mathfrak{Dai}_{a}\in B^{\simbot}=\mathbf{A}$ for all $a\neq 0$. ∎
###### Proposition .
The tensor product of perennial conducts is a perennial conduct.
###### Proof.
Let $\mathbf{A,B}$ be perennial conducts. Then there exists two sets of one-
sliced wager-free projects $E,F$ such that $\mathbf{A}=E^{\simbot\simbot}$ and
$\mathbf{B}=F^{\simbot\simbot}$. Using Section 2.2, we know that
$\mathbf{A\otimes B}=(E\odot F)^{\simbot\simbot}$. But, by definition, $E\odot
F$ contains only projects of the form $\mathfrak{e}\otimes\mathfrak{f}$, where
$\mathfrak{e,f}$ are one-sliced and wager-free. Thus $E\odot F$ contains only
one-sliced wager-free projects and $\mathbf{A\otimes B}$ is therefore a
perennial conduct. ∎
###### Proposition .
If $\mathbf{A,B}$ are perennial conducts, then $\mathbf{A\oplus B}$ is a
perennial conduct.
###### Proof.
This is a consequence of Section 2.2. ∎
###### Proposition .
If $\mathbf{A}$ is a perennial conduct and $\mathbf{B}$ is a co-perennial
conduct, then $\mathbf{A\multimap B}$ is a co-perennial conduct.
###### Proof.
We recall that $\mathbf{A\multimap
B}=(\mathbf{A}\otimes\mathbf{B}^{\simbot})^{\simbot}$. Since $\mathbf{A}$ and
$\mathbf{B}^{\simbot}$ are perennial conducts,
$\mathbf{A}\otimes\mathbf{B}^{\simbot}$ is a perennial conduct, and therefore
$\mathbf{A\multimap B}$ is a co-perennial conduct. In particular,
$\mathbf{A\multimap B}$ is non-empty and satisfies the inflation property. ∎
###### Proposition .
If $\mathbf{A}$ is a perennial conduct and $\mathbf{B}$ is a behavior, then
$\mathbf{A\otimes B}$ is a behavior.
###### Proof.
If $\mathbf{A}=\mathbf{0}_{V^{A}}$ with $\mathbf{B}=\mathbf{0}_{V^{B}}$, then
$\mathbf{A\otimes B}=\mathbf{0}_{V^{A}\cup V^{B}}$ which is a behavior.
Let $A$ be the set of one-sliced wager-free projects such that
$\mathbf{A}=A^{\simbot\simbot}$. We have that $\mathbf{A\otimes
B}=(A\odot\mathbf{B})^{\simbot\simbot}$ by Section 2.2. If
$\mathbf{B}\neq\mathbf{0}_{V^{B}}$ and $A\neq 0$, then $A\odot\mathbf{B}$ is
non-empty and contains only wager-free projects. Thus $\mathbf{(A\otimes
B)^{\simbot}}$ satisfies the inflation property by Section 2.2.
Now suppose there exists $\mathfrak{f}=(f,F)\in\mathbf{(A\otimes
B)^{\simbot}}$ such that $f\neq 0$. Then for all $\mathfrak{a}\in A$ and
$\mathfrak{b}\in\mathbf{B}$, $\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
b}\mathclose{\gg}_{m}\neq 0,\infty$. But, since $\mathfrak{a}$ is wager-free
and $\textbf{1}_{A}=1$, $\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
b}\mathclose{\gg}_{m}=f\textbf{1}_{B}+b\textbf{1}_{F}+\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}$. We can then define
$\mu=(-\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}-b\textbf{1}_{F})/f-\textbf{1}_{B}$. Since
$\mathbf{B}$ is a behavior, $\mathfrak{b+\mu 0}\in\mathbf{B}$. However:
$\displaystyle\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes(b+\mu
0)}\mathclose{\gg}_{m}$ $\displaystyle=$ $\displaystyle
f(\textbf{1}_{B}+\mu)+b\textbf{1}_{F}+\mathopen{\llbracket}F,A\cup(B+\mu
0)\mathclose{\rrbracket}_{m}$ $\displaystyle=$ $\displaystyle
f(\textbf{1}_{B}+\mu)+b\textbf{1}_{F}+\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}$ $\displaystyle=$ $\displaystyle
f(\textbf{1}_{B}+(-\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}-b\textbf{1}_{F})/f-\textbf{1}_{B})+b\textbf{1}_{F}+\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle-\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}-b\textbf{1}_{F}+b\textbf{1}_{F}+\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}$ $\displaystyle=$ $\displaystyle 0$
But this is a contradiction. Therefore the elements in $\mathbf{(A\otimes
B)^{\simbot}}$ are wager-free.
If $\mathbf{(A\otimes B)^{\simbot}}$ is non-empty, it is a non-empty conduct
containing only wager-free projects and satisfying the inflation property: it
is therefore a (proper) behavior.
The only case left to treat is when $\mathbf{(A\otimes B)^{\simbot}}$ is
empty, but then $\mathbf{A\otimes B}=\mathbf{T}_{V^{A}\cup V^{B}}$ is clearly
a behavior. ∎
###### Corollary .
If $\mathbf{A}$ is perennial and $\mathbf{B}$ is a behavior, then
$\mathbf{A\multimap B}$ is a behavior.
###### Proof.
We recall that $\mathbf{A\multimap B}=(\mathbf{A\otimes
B^{\simbot}})^{\simbot}$. Using the preceding proposition, the conduct
$\mathbf{A\otimes B^{\simbot}}$ is a behavior since $\mathbf{A}$ is a
perennial conduct and $\mathbf{B}^{\simbot}$ is a behavior. Thus
$\mathbf{A\multimap B}$ is a behavior since it is the orthogonal of a
behavior. ∎
###### Corollary .
If $\mathbf{A}$ is a behavior and $\mathbf{B}$ is a co-perennial conduct, then
$\mathbf{A\multimap B}$ is a behavior.
###### Proof.
One just has to write $\mathbf{A\multimap B}=\mathbf{(A\otimes
B^{\simbot})^{\simbot}}$. Since $\mathbf{A\otimes B^{\simbot}}$ is the tensor
product of a behavior with a perennial conduct, it is a behavior. The result
then follows from the fact that the orthogonal of a behavior is a behavior. ∎
###### Proposition .
The weakening (on the left) of perennial conducts holds.
###### Proof.
Let $\mathbf{A,B}$ be conducts, and $\mathbf{N}$ be a perennial conduct. Chose
$\mathfrak{f}\in\mathbf{A\multimap B}$. We will show that
$\mathfrak{f}\otimes\mathfrak{0}_{V^{N}}$ is a project in $\mathbf{A\otimes
N\multimap B}$. For this, we pick $\mathfrak{a}\in\mathbf{A}$ and
$\mathfrak{n}\in\mathbf{N}$ (recall that $\mathfrak{n}$ is necessarily wager-
free). Then for all $\mathfrak{b^{\prime}}\in\mathbf{B^{\simbot}}$,
$\displaystyle\mathopen{\ll}\mathfrak{(f\otimes
0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f\otimes 0},\mathfrak{(a\otimes
n)\otimes b}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f\otimes 0},\mathfrak{(a\otimes
b^{\prime})\otimes n}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{F}(\textbf{1}_{A}\textbf{1}_{B^{\prime}}n+\textbf{1}_{N}\textbf{1}_{A}b^{\prime}+\textbf{1}_{N}\textbf{1}_{B^{\prime}}a)+\textbf{1}_{N}\textbf{1}_{A}\textbf{1}_{B^{\prime}}f+\mathopen{\llbracket}F\cup
0,A\cup B^{\prime}\cup N\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{F}(\textbf{1}_{N}\textbf{1}_{A}b^{\prime}+\textbf{1}_{N}\textbf{1}_{B^{\prime}}a)+\textbf{1}_{N}\textbf{1}_{A}\textbf{1}_{B^{\prime}}f+\mathopen{\llbracket}F\cup
0,A\cup B^{\prime}\cup N\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{N}(\textbf{1}_{F}(\textbf{1}_{A}b^{\prime}+\textbf{1}_{B^{\prime}}a)+\textbf{1}_{A}\textbf{1}_{B^{\prime}}f)+\textbf{1}_{N}\mathopen{\llbracket}F,A\cup
B^{\prime}\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{N}\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
b^{\prime}}\mathclose{\gg}_{m}$
Since $\textbf{1}_{N}\neq 0$, $\mathopen{\ll}\mathfrak{(f\otimes
0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}\neq 0,\infty$ if and only if
$\mathopen{\ll}\mathfrak{f\mathop{\mathopen{:}\mathclose{:}}a},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}\neq
0,\infty$. Thus for all $\mathfrak{a\otimes n}\in\mathbf{A\odot N}$,
$\mathfrak{(f\otimes 0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)}\in\mathbf{B}$. This shows that $\mathfrak{f\otimes 0}\in\mathbf{A\otimes
N\multimap B}$ by Section 3.3. ∎
### 4.3 A Construction of Exponentials
We will begin by showing a technical result that will allow us to define
measure preserving transformations from bijections of the set of integers.
This result will be used to show that functorial promotion can be implemented
for our exponential modality.
###### Definition .
Let $\phi:\mathbf{N}\rightarrow\mathbf{N}$ be a bijection and $b$ an integer
$\geqslant 2$. Then $\phi$ induces a transformation
$T^{b}_{\phi}:[0,1]\rightarrow[0,1]$ defined by $\sum_{i\geqslant
0}a_{k}2^{-k}\mapsto\sum_{i\geqslant 0}a_{\phi^{-1}(k)}2^{-k}$.
###### Remark .
Suppose that $\sum_{i\geqslant 0}a_{i}b^{-i}$ and $\sum_{i\geqslant
0}a^{\prime}_{i}b^{-i}$ are two distinct representations of a real number $r$.
Let us fix $i_{0}$ to be the smallest integer such that $a_{i_{0}}\neq
a^{\prime}_{i_{0}}$. We first notice that the absolute value of the difference
between these digits has to be equal to $1$:
$\mathopen{\lvert}a_{i_{0}}-a^{\prime}_{i_{0}}\mathclose{\rvert}=1$. Indeed,
if this was not the case, i.e. if
$\mathopen{\lvert}a_{i_{0}}-a^{\prime}_{i_{0}}\mathclose{\rvert}\geqslant 2$,
the distance between $\sum_{i\geqslant 0}a_{i}b^{-i}$ and $\sum_{i\geqslant
0}a^{\prime}_{i}b^{-i}$ would be greater than $b^{-i_{0}}$, which contradicts
the fact that both sums are equal to $r$. Let us now suppose, without loss of
generality, that $a_{i_{0}}=a^{\prime}_{i_{0}}+1$. Then $a_{j}=0$ for all
$j>i_{0}$ since if there existed an integer $j>i_{0}$ such that $a_{j}\neq 0$,
the distance between the sums $\sum_{i\geqslant 0}a_{i}b^{-i}$ and
$\sum_{i\geqslant 0}a^{\prime}_{i}b^{-i}$ would be greater than $b^{-j}$,
which would again be a contradiction. Moreover, $a^{\prime}_{j}=b-1$ for all
$j>i_{0}$: if this was not the case, one could show in a similar way that the
difference between the two sums would be strictly greater than $0$. We can
deduce from this that only the reals with a finite representation in base $b$
have two distinct representations.
Since the set of such elements is of null measure, the transformation
$T_{\phi}$ associated to a bijection $\phi$ of $\mathbf{N}$ is well defined as
we can define $T_{\phi}$ only on the set of reals that have a unique
representation. We can however chose to deal with this in another way:
choosing between the two possible representations, by excluding for instance
the representations as sequences that are almost everywhere equal to zero.
Then $T_{\phi}$ is defined at all points and bijective. We chose in the
following to follow this second approach since it will allow to prove more
easily that $T_{\phi}$ is measure-preserving. However, this choice is not
relevant for the rest of the construction since both transformations are
almost everywhere equal.
###### Lemma .
Let $T$ be a transformation of $[0,1]$ such that for all interval $[a,b]$,
$\lambda(T([a,b]))=\lambda([a,b])$. Then $T$ is measure-preserving on $[0,1]$.
###### Proof.
A classical result of measure theory states that if $T$ is a transformation of
a measured space $(X,\mathcal{B},\lambda)$, that $\mathcal{B}$ is generated by
$\mathcal{A}$, and that for all $A\in\mathcal{A}$, $\lambda(T(A))=\lambda(A)$,
then $T$ preserves the measure $\lambda$ on $X$. Applying this result with
$X=[0,1]$, and $\mathcal{A}$ as the set of intervals $[a,b]\subset[0,1]$, we
obtain the result. ∎
###### Lemma .
Let $T$ be a bijective transformation of $[0,1]$ that preserves the measure on
all interval $I$ of the shape
$[\sum_{k=1}^{p}a_{k}b^{-k},\sum_{k=1}^{p}a_{k}b^{-k}+b^{-p}]$. Then $T$ is
measure-preserving on $[0,1]$.
###### Proof.
Chose $[a,b]\subset[0,1]$. One can write $[a,b]$ as a union
$\cup_{i=0}^{\infty}[a_{i},a_{i+1}]$, where for all $i\geqslant 0$,
$a_{i+1}=a_{i}+b^{-k_{i}}$. We then obtain, using the hypotheses of the
statement and the $\sigma$-additivity of the measure $\lambda$:
$\displaystyle\lambda(T([a,b]))$ $\displaystyle=$
$\displaystyle\lambda(T(\cup_{i=0}^{\infty}[a_{i},a_{i+1}[))$ $\displaystyle=$
$\displaystyle\lambda(\cup_{i=0}^{\infty}T([a_{i},a_{i+1}[))$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\lambda(T([a_{i},a_{i+1}[))$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\lambda([a_{i},a_{i+1}[)$ $\displaystyle=$
$\displaystyle\lambda(\cup_{i=0}^{\infty}[a_{i},a_{i+1}[)$ $\displaystyle=$
$\displaystyle\lambda([a,b])$
We now conclude by using the preceding lemma. ∎
###### Theorem .
Let $\phi:\mathbf{N}\rightarrow\mathbf{N}$ be a bijection and $b\geqslant 2$
an integer. Then the transformation $T^{b}_{\phi}$ is measure-preserving.
###### Proof.
We recall first that the transformation $T^{b}_{\phi}$ is indeed bijective
(see Section 4.3).
By using the preceding lemma, it suffices to show that $T^{b}_{\phi}$
preserves the measure on intervals of the shape $I=[a,a+b^{-k}]$ with
$a=\sum_{i=0}^{k}a_{i}b^{-i}$. Let us define
$N=\max\\{\phi(i)\leavevmode\nobreak\ |\leavevmode\nobreak\ 0\leqslant
i\leqslant k\\}$. We then write $[0,1]$ as the union of intervals
$A_{i}=[i\times b^{-N},(i+1)\times b^{-N}]$ where $0\leqslant i\leqslant
b^{N}-1$.
Then the image if $I$ by $T^{b}_{\phi}$ is equal to the union of the $A_{i}$
for $i\times b^{-N}=\sum_{i=0}^{N}x_{i}b^{-i}$, where $x_{\phi(j)}=a_{j}$ for
all $0\leqslant j\leqslant k$. The number of such $A_{j}$ is equal to the
number of sequences $\\{0,\dots,b-1\\}$ of length $N-k$, i.e. $b^{N-k}$. Since
each $A_{j}$ has a measure equal to $b^{-N}$, the image of $I$ by
$T^{b}_{\phi}$ is of measure $b^{-N}b^{N-k}=b^{-k}$, which is equal to the
measure of $I$ since $\lambda(I)=b^{-k}$. ∎
###### Remark .
The preceding theorem can be easily generalized to bijections
$\mathbf{N}+\dots+\mathbf{N}\rightarrow\mathbf{N}$ (the domain being the
disjoint union of $k$ copies of $\mathbf{N}$, $k\in\mathbf{N}$) which induce
measure-preserving bijections from $[0,1]^{k}$ onto $[0,1]$. The particular
case $\mathbf{N}+\mathbf{N}\rightarrow\mathbf{N}$, $(n,i)\mapsto 2n+i$ defines
the well-known measure-preserving bijection between the unit square and the
interval $[0,1]$:
$(\sum_{i\geqslant 0}a_{i}2^{-i},\sum_{i\geqslant
0}b_{i}2^{-i})\mapsto\sum_{i\geqslant 0}a_{2i}2^{-2i}+b_{2i+1}2^{-2i-1}$
Let us now define the bijection:
$\psi:\mathbf{N}+\mathbf{N}+\mathbf{N}\rightarrow\mathbf{N},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (x,i)\mapsto 3x+i$
We also define the injections $\iota_{i}$ ($i=0,1,2$):
$\iota_{i}:\mathbf{N}\rightarrow\mathbf{N}+\mathbf{N}+\mathbf{N},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\mapsto(x,i)$
We will denote by $\psi_{i}$ the composite
$\psi\circ\iota_{i}:\mathbf{N}\rightarrow\mathbf{N}$.
###### Definition .
Let $A\subset\mathbf{N}+\mathbf{N}+\mathbf{N}$ be a finite set. We write $A$
as $A_{0}+A_{1}+A_{2}$, and define, for $i=0,1,2$, $n_{i}$ to be the
cardinality of $A_{i}$ if $A_{i}\neq$ and $n_{i}=1$ otherwise. We then define
a partition of $[0,1]$, denoted by
$\mathcal{P}_{A}=\\{P_{A}^{i_{1},i_{2},i_{3}}\leavevmode\nobreak\
|\leavevmode\nobreak\ \forall k\in\\{0,1,2\\},0\leqslant i_{k}\leqslant
n_{i}-1\\}$, by:
$P_{A}^{i_{1},i_{2},i_{3}}=\\{\sum_{j\geqslant
1}a_{j}2^{-j}\leavevmode\nobreak\ |\leavevmode\nobreak\ \forall
k\in\\{0,1,2\\},\frac{i_{k}}{n_{k}}\leqslant\sum_{j\geqslant
1}a_{\psi_{k}(j)}2^{-j}\leqslant\frac{i_{k}+1}{n_{k}}\\}$
When $A_{k}$ is empty or of cardinality $1$, we will not write the
corresponding $i_{k}$ in the triple $(i_{1},i_{2},i_{3})$ since it is
necessarily equal to $0$.
###### Proposition .
Let us keep the notations of the preceding proposition and let
$X=P_{A}^{i_{1},i_{2},i_{3}}$ and $Y=P_{A}^{j_{1},j_{2},j_{3}}$ be two
elements of the partition $\mathcal{P}_{A}$. For all $x=\sum_{l\geqslant
1}a_{l}2^{-l}$, we define
$T_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}(x)=\sum_{l\geqslant 1}b_{l}2^{-l}$
where the sequence $(b_{i})$ is defined by:
$\forall k\in\\{0,1,2\\},\sum_{l\geqslant
1}b_{\psi_{k}(l)}2^{-l}=\sum_{l\geqslant 1}a_{\psi_{k}(l)}2^{-l}+j_{k}-i_{k}$
Then $T_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}:X\rightarrow Y$ is a measure-
preserving bijection.
###### Proof.
For $k=0,1,2$, we will denote by $(m^{k}_{j})$ the sequence such that
$j_{k}-i_{k}=\sum_{l\geqslant 1}m^{k}_{l}2^{-l}$. We can define the real
number $t=\sum_{l\geqslant 1}\sum_{k=0,1,2}m^{k}_{l}2^{-3j+k}$. It is then
sufficient to check that $T_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}(x)=x+t$.
Since $T_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}$ is a translation
translation, it is a measure-preserving bijection. ∎
###### Definition .
Let $A\subset\mathbf{N}$ be a finite set endowed with the normalized — i.e.
such that $A$ has measure $1$ — counting measure, and
$X\subset\mathcal{B}(\mathbf{R}\times A)$ be a measurable set. We define the
measurable subset
$\mathopen{\ulcorner}A\mathclose{\urcorner}\in\mathbf{R}\times[0,1]$:
$\mathopen{\ulcorner}A\mathclose{\urcorner}=\\{(x,y)\leavevmode\nobreak\
|\leavevmode\nobreak\ \exists z\in A,(x,z)\in X,y\in P_{A}^{z}\\}$
We will write $\mathcal{P}^{-1}_{A}:[0,1]\rightarrow A$ the map that
associates to each $x$ the element $z\in A$ such that $x\in P_{A}^{z}$.
###### Proposition .
Let $D^{A}\subset\mathbf{N}$ be a finite set endowed with the normalized
counting measure $\mu$ (i.e. such that $\mu(A)=1$),
$S,T\in\mathcal{B}(\mathbf{R}\times D^{A})$ be measurable sets, and
$\phi:S\rightarrow T$ a measure-preserving transformation. We define
$\mathopen{\ulcorner}\phi\mathclose{\urcorner}:\mathopen{\ulcorner}S\mathclose{\urcorner}\rightarrow\mathopen{\ulcorner}T\mathclose{\urcorner}$
by:
$\mathopen{\ulcorner}\phi\mathclose{\urcorner}:(x,y)\mapsto(x^{\prime},y^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\phi(x,\mathcal{P}^{-1}_{A}(y))=(x^{\prime},z),\leavevmode\nobreak\
\leavevmode\nobreak\ y^{\prime}=T_{\mathcal{P}^{-1}_{A}(y)}^{z}$
Then $\mathopen{\ulcorner}\phi\mathclose{\urcorner}$ is a measure-preserving
bijection.
###### Proof.
For all $(a,b)\in D^{A}$ we define the set
$S_{a,b}=X\cap\mathbf{R}\times\\{a\\}\cap\phi^{-1}(Y\cap\mathbf{R}\times\\{b\\})$.
The family $(S_{a,b})_{a,b\in D^{A}}$ is a partition of $S$, and the family
$(\mathopen{\ulcorner}S_{a,b}\mathclose{\urcorner})_{a,b\in D^{A}}$ is a
partition of $\oc A$. The restriction of
$\mathopen{\ulcorner}\phi\mathclose{\urcorner}$ to
$\mathopen{\ulcorner}S_{a,b}\mathclose{\urcorner}$ can then be defined as the
composite $T_{a}\circ\phi_{1}$ with:
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle(\pi_{1}\circ\phi)\times\text{Id}$ $\displaystyle T_{a}$
$\displaystyle=$ $\displaystyle\text{Id}\times T_{a}^{b}$
Since the product (resp. the composition) of measure preserving bijections is
a measure preserving bijection, the restriction of
$\mathopen{\ulcorner}\phi\mathclose{\urcorner}$ to $X_{a}$ is a measure
preserving bijection. Moreover, it is clear that the image of
$\mathopen{\ulcorner}S\mathclose{\urcorner}$ by
$\mathopen{\ulcorner}\phi\mathclose{\urcorner}$ is equal to
$\mathopen{\ulcorner}T\mathclose{\urcorner}$ and we have finished the proof. ∎
###### Definition .
Let $A$ be a thick graphing, i.e. of support $V^{A}\subset\mathbf{R}\times
D^{A}$ measurable, where $D^{A}$ is a finite subset of $\mathbf{N}$ endowed
with the normalized counting measure. We define the graphing:
$\mathopen{\ulcorner}A\mathclose{\urcorner}=\\{(\omega_{e}^{A},\mathopen{\ulcorner}\phi_{e}^{A}\mathclose{\urcorner}:\mathopen{\ulcorner}S_{e}^{A}\mathclose{\urcorner}\rightarrow\mathopen{\ulcorner}T_{e}^{A}\mathclose{\urcorner}\\}_{e\in
E^{A}}$
###### Definition .
Let $A$ be a thick graphing of dialect $D^{A}$, and
$\Omega:\mathbf{R}\times[0,1]\rightarrow\mathbf{R}$ an isomorphism of measured
spaces. We define the graphing $\oc_{\Omega}A$ by:
$\oc_{\Omega}A=\\{(\omega_{e}^{A},\Omega\circ\mathopen{\ulcorner}\phi_{e}^{A}\mathclose{\urcorner}\circ\Omega^{-1}:\Omega(\mathopen{\ulcorner}S_{e}^{A}\mathclose{\urcorner})\rightarrow\Omega(\mathopen{\ulcorner}T_{e}^{A}\mathclose{\urcorner})\\}_{e\in
E^{A}}$
###### Definition .
A project $\mathfrak{a}$ is _balanced_ if $\mathfrak{a}=(0,A)$ where $A$ is a
thick graphing, i.e. $I^{A}$ is a one-element set, for instance
$I^{A}=\\{1\\}$, and $\alpha^{A}_{1}=1$.
###### Definition .
Let $\mathfrak{a}$ be a balanced project. We define
$\oc_{\Omega}\mathfrak{a}=(0,\oc_{\Omega}A)$. If $\mathbf{A}$ is a conduct, we
define:
$\oc_{\Omega}\mathbf{A}=\\{\oc_{\Omega}\mathfrak{a}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathfrak{a}=(0,A)\in\mathbf{A},\text{ $\mathfrak{a}$
balanced}\\}^{\simbot\simbot}$
We will now show that it is possible to implement the functorial promotion. In
order to do this, we define the bijections
$\tau,\theta:\mathbf{N}+\mathbf{N}+\mathbf{N}\rightarrow\mathbf{N}+\mathbf{N}+\mathbf{N}$:
$\displaystyle\tau$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{rcl}(x,0)&\mapsto&(x,1)\\\
(x,1)&\mapsto&(x,0)\\\ (x,2)&\mapsto&(x,2)\end{array}\right.$
$\displaystyle\theta$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{rcl}(x,0)&\mapsto&(2x,0)\\\
(x,1)&\mapsto&(2x+1,1)\\\ (2x,2)&\mapsto&(x,1)\\\
(2x+1,2)&\mapsto&(x,2)\end{array}\right.$
These bijections induce bijections of $\mathbf{N}$ onto $\mathbf{N}$ through
$\psi:(x,i)\mapsto 3x+i$. We will abusively denote by
$T_{\tau}=T_{\psi\circ\tau\circ\psi^{-1}}$ and
$T_{\theta}=T_{\psi\circ\theta\circ\psi^{-1}}$ the induced measure-preserving
transformations $[0,1]\rightarrow[0,1]$.
Pick $\mathfrak{a}\in\mathbf{\sharp\phi(A)}$ and
$\mathfrak{f}\in\sharp(\mathbf{A\multimap B})$, where $\phi$ is a delocation.
By definition,
$\mathfrak{a}=(0,\Omega(\mathopen{\ulcorner}A\mathclose{\urcorner}))$ and
$\mathfrak{f}=(0,\Omega(\mathopen{\ulcorner}F\mathclose{\urcorner}))$ where
$A,F$ are graphings of respective dialects $D^{A},D^{F}$. We define the
graphing $T=\\{(1,\Omega(\text{Id}\times T_{\tau})),(1,(\Omega(\text{Id}\times
T_{\tau}))^{-1})\\}$ of carrier $V^{\phi(A)}\cup V^{A}$, and denote by
$t,t^{\ast}$ the two edges in $E^{T}$. We fix $(x,y)$ an element of $V^{B}$
and we will try to understand the action of the path
$f_{0}ta_{0}t^{\ast}f_{1}\dots ta_{k-1}t^{\ast}f_{k}$.
We fix the partition $\mathcal{P}_{D^{F}+D^{A}}$ of $[0,1]$, and denote by
$(i,j)$ the integers such that $y\in\mathcal{P}_{D^{F}+D^{A}}^{i,j}$. By
definition of $\mathopen{\ulcorner}F\mathclose{\urcorner}$, the map
$\mathopen{\ulcorner}\phi^{F}_{f_{0}}\mathclose{\urcorner}$ sends this element
to $(x_{1},y_{1})$ which is an element of
$\mathcal{P}_{D^{F}+D^{A}}^{i_{1},j_{1}}$ with $j_{1}=j$. Then, the function
$\phi_{t}$ sends this element on $(x_{2},y_{2})$, where $x_{2}=x_{1}$ and
$y_{2}$ is an element of $\mathcal{P}_{D^{F}+D^{A}}^{j_{1},i_{1}}$. The
function $\mathopen{\ulcorner}\phi_{a_{0}}^{A}\mathclose{\urcorner}$ then
produces an element $(x_{3},y_{3})$ with $y_{3}$ in
$\mathcal{P}_{D^{F}+D^{A}}^{j_{2},i_{2}}$ and $i_{2}=i_{1}$. The element
produced by $\phi_{t^{\ast}}=\phi_{t}^{-1}$ is then $(x_{4},y_{4})$ where
$y_{4}$ is an element of $\mathcal{P}_{D^{F}+D^{A}}^{i_{2},j_{2}}$. One can
therefore see how the graphing $T$ simulates the dialectal interaction. The
following proposition will show how one can use $T$ to implement functorial
promotion.
In order to implement functorial promotion, we will make use of the three
bijections we just defined. Though it may seem a complicated, the underlying
idea is quite simple. We will be working with three disjoint copies of
$\mathbf{N}$, let us say $\mathbf{N}_{i}$ ($i=0,1,2$). When applying
promotion, we will encode the information contained in the dialect on the
first copy $\mathbf{N}_{0}$ (let us stress here that promotion is defined
through a non-surjective map, something that will be essential in the
following). Suppose now that we have two graphs obtained from two promotions:
all the information they contain is located in their first copy
$\mathbf{N}_{0}$. To simulate dialectal information, we need to make these two
sets disjoint: this is where the second copy $\mathbf{N}_{1}$ will be used.
Hence, we apply to one of these promoted graphs the bijection $\tau$ (in
practice we will of course use $\tau$ through the induced transformation
$T_{\tau}$) which exchanges $\mathbf{N}_{0}$ and $\mathbf{N}_{1}$. The
information coming from the dialects of the two graphs are now disjoint. We
then compute the execution of the two graphs to obtain a graph whose
information coming from the dialect is encoded on the two copies
$\mathbf{N}_{0}$ and $\mathbf{N}_{1}$! In order to be able to see this
obtained graph as a graph obtained from a promotion, we need now to move this
information so that it is encoded on the first copy $\mathbf{N}_{0}$ only.
This is where we use the third copy $\mathbf{N}_{2}$: we use the bijection
$\theta$ (once again, we use in practice the induced transformation
$T_{\theta}$) in order to contract the two copies $\mathbf{N}_{0}$ and
$\mathbf{N}_{1}$ on the first copy $\mathbf{N}_{0}$, while deploying the third
copy $\mathbf{N}_{2}$ onto the two copies $\mathbf{N}_{1}$ and
$\mathbf{N}_{2}$.
###### Proposition .
One can implement functorial promotion: for all delocations $\phi,\psi$ and
conducts $\mathbf{A,B}$ such that $\mathbf{\phi(A),A,B,\psi(B)}$ have pairwise
disjoint carriers, there exists a project $\mathfrak{prom}$ in the conduct
$\mathbf{\oc\phi(A)}\otimes\oc(\mathbf{A\multimap
B})\multimap\mathbf{\oc\psi(B)}$
###### Proof.
Let $\mathfrak{f}\in\mathbf{A\multimap B}$ be a balanced project, $\phi,\psi$
two delocations of $\mathbf{A}$ and $\mathbf{B}$ respectively. We define the
graphings $T=\\{(1,\Omega(\text{Id}\times
T_{\tau})),(1,(\Omega(\text{Id}\times T_{\tau}))^{-1})\\}$ of carrier
$V^{\phi(A)}\cup V^{A}$ and $P=\\{(1,\Omega(\text{Id}\times
T_{\theta})),(1,(\Omega(\text{Id}\times T_{\theta}))^{-1})\\}$ of carrier
$V^{B}\cup V^{\psi(B)}$. We define $\mathfrak{t}=(0,T)$ and
$\mathfrak{p}=(0,P)$, and the project:
$\mathfrak{prom}=(0,T\cup P)=\mathfrak{t}\otimes\mathfrak{p}$
We will now show that $\mathfrak{prom}$ is an element in
$\mathbf{(\oc\phi(A)\otimes\oc(A\multimap B))\multimap\oc\psi(B)}$.
We can suppose, up to choosing refinements of $A$ and $F$, that for all $e\in
E^{A}\cup E^{F}$, $(S_{e})_{2}$ and $(T_{e})_{2}$ are one-elements
sets121212The sets $S_{e}$ and $T_{e}$ being subsets of a product, w write
$(S_{e})_{2}$ (resp. $(T_{e})_{2}$) the result of their projection on the
second component..
Pick $\mathfrak{a}\in\mathbf{\sharp\phi(A)}$ and
$\mathfrak{f}\in\sharp(\mathbf{A\multimap B})$. Then, by definition
$\mathfrak{a}=(0,\Omega(\mathopen{\ulcorner}A\mathclose{\urcorner}))$ and
$\mathfrak{f}=(0,\Omega(\mathopen{\ulcorner}F\mathclose{\urcorner}))$ where
$A,F$ are graphings of dialects $D^{A},D^{F}$. We get that $\mathfrak{a\otimes
f}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{prom}=((\mathfrak{a}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{t})\mathop{\mathopen{:}\mathclose{:}}\mathfrak{f})\mathop{\mathopen{:}\mathclose{:}}\mathfrak{p}$
from the associativity and commutativity of
$\mathop{\mathopen{:}\mathclose{:}}$ (recall that $\mathfrak{a\otimes
f}=\mathfrak{a\mathop{\mathopen{:}\mathclose{:}}f}$).
We show that
$\mathopen{\ulcorner}A\mathclose{\urcorner}\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}T\mathclose{\urcorner}$
is the graphings composed of the $\oc^{\tau}\phi_{a}$ for $a\in E^{A}$, where
$\oc^{\tau}\phi_{a}$ is defined by:
$\oc^{\tau}\phi_{a}:(x,y)\mapsto(x^{\prime},y^{\prime}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\phi_{a}(x,\mathcal{P}_{\\{0\\}+D^{A}}^{-1}(y))=(x^{\prime},z),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
y^{\prime}=T_{\mathcal{P}_{\\{0\\}+D^{A}}^{-1}(y)}^{(z,1)}(y)$
This is almost straightforward. An element in
$\mathopen{\ulcorner}A\mathclose{\urcorner}\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}T\mathclose{\urcorner}$
is a path of the form $tat^{\ast}$. It is therefore the function
$\phi_{t}\circ\mathopen{\ulcorner}\phi_{a}\mathclose{\urcorner}\circ\phi_{t}^{-1}$.
By definition,
$\mathopen{\ulcorner}\phi_{a}\mathclose{\urcorner}:(x,y)\mapsto(x^{\prime},y^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
n=\mathcal{P}_{A}^{-1}(y),\leavevmode\nobreak\ \leavevmode\nobreak\
\phi_{a}(x,n)=(x^{\prime},k),\leavevmode\nobreak\ y^{\prime}=T_{n}^{k}(y)$
But $\phi_{t}:\text{Id}\times T_{\tau}$ and $T_{\tau}$ is a bijection from
$\mathcal{P}_{A}(y)$ to $\mathcal{P}_{\\{0\\}+A}(1,y)$.
We now describe the graphing
$G=(\mathopen{\ulcorner}A\mathclose{\urcorner}\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}T\mathclose{\urcorner})\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}F\mathclose{\urcorner}$.
It is composed of the paths of the shape
$\rho=f_{0}(ta_{0}t^{\ast})f_{1}(ta_{1}t^{\ast})f_{2}\dots
f_{n-1}(ta_{n-1}t^{\ast})f_{n}$. The associated function is therefore:
$\phi_{\rho}=\mathopen{\ulcorner}\phi_{f_{0}}\mathclose{\urcorner}(\oc^{\tau}\phi_{a_{0}})\mathopen{\ulcorner}\phi_{f_{1}}\mathclose{\urcorner}\dots\mathopen{\ulcorner}\phi_{f_{n-1}}\mathclose{\urcorner}(\oc^{\tau}\phi_{a_{n-1}})\mathopen{\ulcorner}\phi_{f_{n}}\mathclose{\urcorner}$
Let $\pi=f_{0}a_{0}f_{1}\dots f_{n-1}a_{n-1}f_{n}$ be the corresponding path
in $F\mathop{\mathopen{:}\mathclose{:}}A$. The function $\phi_{\pi}$ has, by
definition, as domain and codomain measurable subsets of $\mathbf{R}\times
D^{F}\times D^{A}$. We define, for such a function, the function
$\mathop{\rotatebox[origin={c}]{180.0}{$\oc$}}\phi_{\pi}$ by:
$\displaystyle\mathop{\rotatebox[origin={c}]{180.0}{$\oc$}}\phi_{\pi}:(x,y)\mapsto(x^{\prime},y^{\prime})$
$\displaystyle(n,m)=\mathcal{P}_{D^{F}+D^{A}}^{-1}(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \phi_{\pi}(x,n,m)=(x^{\prime},k,l),\leavevmode\nobreak\
\leavevmode\nobreak\ y^{\prime}=T_{(n,m)}^{(k,l)}(y)$
One can then check that
$\mathop{\rotatebox[origin={c}]{180.0}{$\oc$}}\phi_{\pi}=\phi_{\rho}$.
Finally,
$G\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}P\mathclose{\urcorner}$
is the graphing composed of paths that have the shape $p\rho p^{\ast}$ where
$\rho$ is a path in $G$. But $\phi_{p}=\text{Id}\times T_{\theta}$ applies a
bijection, for all couple $(k,l)\in D^{F}\times D^{A}$, from the set
$\mathcal{P}_{D^{F}+D^{A}}^{k,l}$ to the set
$\mathcal{P}_{\theta(D^{F}+D^{A})}^{\theta(k,l)}$ where:
$\theta(D^{F}+D^{A})=\\{\theta(f,a)\leavevmode\nobreak\ |\leavevmode\nobreak\
f\in D^{F},a\in D^{A}\\}$
We deduce that:
$\displaystyle\phi_{p\rho p^{\ast}}:(x,y)\mapsto(x^{\prime},y^{\prime})$
$\displaystyle
n=\theta(k,l)=\mathcal{P}_{\theta(D^{F}+D^{A})}^{-1}(y)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\phi_{\pi}(x,k,l)=(x^{\prime},k^{\prime},l^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
y^{\prime}=T_{n}^{\theta(k^{\prime},l^{\prime})}(y)$
Modulo the bijection $\mu:D^{F}\times
D^{A}\rightarrow\theta(D^{F}+D^{A})\subset\mathbf{N}$, we get that
$G\mathop{\mathopen{:}\mathclose{:}}\mathopen{\ulcorner}P\mathclose{\urcorner}$
is the delocation (along $\psi$) of the graphing
$\oc(F\mathop{\mathopen{:}\mathclose{:}}A)$.
Therefore, for all $\mathfrak{a},\mathfrak{f}$ in $\mathbf{\sharp
A},\mathbf{\sharp(A\multimap B)}$ respectively there exists a project
$\mathfrak{b}$ in $\mathbf{\sharp\psi(B)}$ such that
$\mathfrak{prom}\mathop{\mathopen{:}\mathclose{:}}(\mathfrak{a}\otimes\mathfrak{f})=\mathfrak{b}$.
We showed that for all $\mathfrak{g}\in\mathbf{\sharp{A}\odot\sharp(A\multimap
B)}$, one has
$\mathfrak{prom}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{g}\in\mathbf{\oc
B}$, and thus $\mathfrak{prom}$ is an element in $(\sharp
A\odot\sharp(A\multimap B))^{\simbot\simbot}\multimap\mathbf{B}$ by Section
3.3. But $(\sharp A\odot\sharp(A\multimap B))^{\simbot\simbot}=\mathbf{\oc
A\otimes\oc(A\multimap B)}$ by Section 2.2. ∎
$\oc\phi(A)$$\phi(V^{A})\times[0,1]$$\oc F$$\oc V^{A}$$\oc V^{B}$$(V^{A}\cup
V^{B})\times[0,1]$$\oc\psi(B)$$\psi(V^{B})\times[0,1]$$\Omega$$\Omega$$\Omega$$\text{Id}\times
T_{\tau}$$\text{Id}\times T_{\theta}$ (a) Global Picture
$\phi(V^{A})\times D^{A}$$\phi(V^{A})\times
D^{A}$$\phi(V^{A})\times[0,1]$$\phi(V^{A})\times[0,1]$$\mathcal{P}_{D^{A}}$$\mathcal{P}_{\\{0\\}+D^{A}}$$\text{Id}\times
T_{\tau}$$\text{Id}\times\tau$ (b) Action of $T_{\tau}$
$\psi(V^{B})\times D^{F}\times
D^{A}$$\psi(V^{B})\times\theta(D^{F}+D^{A})$$\phi(V^{A})\times[0,1]$$\phi(V^{A})\times[0,1]$$\mathcal{P}_{D^{F}+D^{A}}$$\mathcal{P}_{\theta(D^{F}+D^{A})}$$\text{Id}\times
T_{\theta}$$\text{Id}\times\theta$ (c) Action of $T_{\theta}$
Figure 18: Functorial Promotion
In the setting of its hyperfinite geometry of interaction [Gir11], Girard
shows how one can obtain the exponentials isomorphism as an equality between
the conducts $\oc(\mathbf{A\with B})$ and $\mathbf{\oc A\otimes\oc B}$. Things
are however quite different here. Indeed, if the introduction of behaviors in
place of Girard’s negative/positive conducts is very interesting when one is
interested in the additive connectives, this leads to a (small) complication
when dealing with exponentials. The first thing to notice is that the proof of
the implication $\oc A\otimes\oc B\multimap\oc(A\with B)$ in a sequent
calculus with functorial promotion and without dereliction and digging rules
cannot be written if the weakening rule is restrained to the formulas of the
form $\wn A$:
ax $\vdash A,A^{\simbot}$ weak $\vdash A,B^{\simbot},A^{\simbot}$ ax $\vdash
B,B^{\simbot}$ weak $\vdash B,B^{\simbot},A^{\simbot}$ $\with$ $\vdash
B^{\simbot},A^{\simbot},A\with B$ $\oc$ $\vdash\wn B^{\simbot},\wn
A^{\simbot},\oc(A\with B)$ $\vdash\oc A\otimes\oc B\multimap\oc(A\with B)$
In Girard’s setting, weakening is available for all positive conducts (the
conducts on which one can apply the $\wn$ modality), something which is
coherent with the fact that the inclusion $\mathbf{\oc A\otimes\oc
B\subset\oc(A\with B)}$ is satisfied. In our setting, however, weakening is
never available for behaviors and we think the latter inclusion is therefore
not satisfied. This question stays however open.
Concerning the converse inclusion, it does not seem clear at first that it is
satisfied in our setting either. This issue comes from the contraction rule.
Indeed, since the latter does not seem to be satisfied in full generality (see
Section 5.1), one could think the inclusion $\mathbf{\oc(A\with B)\subset\oc
A\otimes\oc B}$ is not satisfied either. We will show however in Section 6,
through the introduction of alternative ”additive connectives”, that it does
hold (a result that will not be used until the last section).
###### Proposition .
The conduct $\mathbf{1}$ is a perennial conduct, equal to $\oc\mathbf{T}$.
###### Proof.
By definition, $\mathbf{1}=\\{(0,\emptyset)\\}^{\simbot\simbot}$ is a
perennial conduct. Moreover, the balanced projects in $\mathbf{T}$ are the
projects of the shape $\mathfrak{t}_{D}=(0,\emptyset)$ with dialects
$D\subset\mathbf{N}$. Each of these satisfy
$\oc\mathfrak{t}_{D}=(0,\emptyset)$. Thus
$\sharp\mathbf{T}=\\{(0,\emptyset)\\}$ and $\mathbf{\oc T}=\mathbf{1}$. ∎
###### Corollary .
The conduct $\mathbf{\bot}$ is a co-perennial conduct, equal to
$\wn\mathbf{0}$.
###### Proof.
This is straightforward:
$\mathbf{\bot}=\mathbf{1}^{\simbot}=\mathbf{(\oc T)^{\simbot}}=\mathbf{(\sharp
T)^{\simbot\simbot\simbot}}=\mathbf{(\sharp T)^{\simbot}}=\mathbf{(\sharp
0^{\simbot})^{\simbot}}=\mathbf{\wn 0}$
∎
## 5 Soundness for Behaviors
### 5.1 Sequent Calculus
To deal with the three kinds of conducts we are working with (behaviors,
perennial and co-perennial conducts), we introduce three types of formulas.
###### Definition .
We define three types of formulas, (B)ehaviors, (N)egative — perennial, and
(P)ositive — co-perennial, inductively defined by the following grammar:
$\displaystyle B$ $\displaystyle:=$
$\displaystyle\mathbf{T}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathbf{0}\leavevmode\nobreak\ |\leavevmode\nobreak\ X\leavevmode\nobreak\
|\leavevmode\nobreak\ X^{\simbot}\leavevmode\nobreak\ |\leavevmode\nobreak\
B\otimes B\leavevmode\nobreak\ |\leavevmode\nobreak\ B\parr
B\leavevmode\nobreak\ |\leavevmode\nobreak\ B\oplus B\leavevmode\nobreak\
|\leavevmode\nobreak\ B\with B\leavevmode\nobreak\ |\leavevmode\nobreak\
\forall X\leavevmode\nobreak\ B\leavevmode\nobreak\ |\leavevmode\nobreak\
\exists X\leavevmode\nobreak\ B\leavevmode\nobreak\ |\leavevmode\nobreak\
N\otimes B\leavevmode\nobreak\ |\leavevmode\nobreak\ P\parr B$ $\displaystyle
N$ $\displaystyle:=$ $\displaystyle\mathbf{1}\leavevmode\nobreak\
|\leavevmode\nobreak\ P^{\simbot}\leavevmode\nobreak\ |\leavevmode\nobreak\
\oc B\leavevmode\nobreak\ |\leavevmode\nobreak\ N\otimes N\leavevmode\nobreak\
|\leavevmode\nobreak\ N\oplus N$ $\displaystyle P$ $\displaystyle:=$
$\displaystyle\mathbf{\bot}\leavevmode\nobreak\ |\leavevmode\nobreak\
N^{\simbot}\leavevmode\nobreak\ |\leavevmode\nobreak\ \wn
B\leavevmode\nobreak\ |\leavevmode\nobreak\ P\parr P\leavevmode\nobreak\
|\leavevmode\nobreak\ P\with P$
We will denote by $\tt FV\rm(\Gamma)$ the set of free variables in $\Gamma$,
where $\Gamma$ is a sequence of formulas (of any type).
###### Definition .
We define _pre-sequents_ $\Delta\Vvdash\Gamma;\Theta$ where $\Delta,\Theta$
contain negative (perennial) formulas, $\Theta$ containing at most one
formula, and $\Gamma$ contains only behaviors.
Section 3.3 supposes that we are working with behaviors, and cannot be used to
interpret contraction in full generality. It is however possible to show in a
similar way that contraction can be interpreted when the sequent contains at
least one behavior (this is the next proposition). This restriction of the
context is necessary: without behaviors in the sequent one cannot interpret
the contraction since the inflation property is essential for showing that
$(1/2)\phi(\oc\mathfrak{a})\otimes\psi(\oc\mathfrak{a})+(1/2)\mathfrak{0}$ is
an element of $\mathbf{\phi(\oc A)\otimes\psi(\oc A)}$.
###### Proposition .
Let $\mathbf{A}$ be a conduct and $\phi,\psi$ be disjoint delocations of $\oc
V^{A}$. Let $\mathbf{C}$ be a behavior and $\theta$ a delocation disjoint from
$\phi$ and $\psi$. Then the project $\mathfrak{Ctr}_{\phi\cup\theta}^{\psi}$
is an element of the behavior:
$\mathbf{(\oc A\otimes C)\multimap(\phi(\oc A)\otimes\psi(\oc
A)\otimes\theta(C))}$
###### Proof.
The proof follows the proof of Section 3.3. We show in a similar manner that
the project
$\mathfrak{Ctr}_{\phi\cup\theta}^{\psi}\mathop{\mathopen{:}\mathclose{:}}(\mathfrak{a\otimes
c})$ is universally equivalent to:
$\frac{1}{2}\phi(\oc\mathfrak{a})\otimes\psi(\oc\mathfrak{a})\otimes\theta(\mathbf{C})+\frac{1}{2}\mathfrak{0}$
Since $\mathbf{\oc A}$ is a perennial conduct and $\mathbf{C}$ is a behavior,
$\mathbf{(\phi(\oc A)\otimes\psi(\oc A)\otimes\theta(C))}$ is a behavior. Thus
$\mathfrak{Ctr}_{\phi\cup\theta}^{\psi}\mathop{\mathopen{:}\mathclose{:}}(\mathfrak{a\otimes
c})$ is an element in $\mathbf{(\phi(\oc A)\otimes\psi(\oc
A)\otimes\theta(C))}$. Finally we showed that the project
$\mathfrak{Ctr}_{\phi\cup\theta}^{\psi}$ is an element of $\mathbf{(\oc
A\otimes C)\multimap(\phi(\oc A)\otimes\psi(\oc A)\otimes\theta(C))}$, and
that the latter is a behavior. ∎
In a similar way, the proof of distributivity relies on the property that
$\mathbf{A+B}\subset\mathbf{A\with B}$ which is satisfied for behaviors but
not in general. It is therefore necessary to restrict to pre-sequents that
contain at least one behavior in order to interpret the $\with$ rule. Indeed,
we can think of a pre-sequent $\Delta\Vvdash\Gamma;\Theta$ as the
conduct131313This will actually be the exact definition of its interpretation.
$\left(\bigparr_{N\in\Delta}N^{\simbot}\right)\parr\left(\bigparr_{B\in\Gamma}B\right)\parr\left(\bigparr_{N\in\Theta}N\right)$
Such a conduct is a behavior when the set $\Gamma$ is non-empty and the set
$\Theta$ is empty, but it is neither a perennial conduct nor a co-perennial
conduct when $\Gamma=\emptyset$. We will therefore restrict to pre-sequents
such that $\Gamma\neq\emptyset$ and $\Theta=\emptyset$.
###### Definition (Sequents).
A sequent $\Delta\vdash\Gamma;$ is a pre-sequent $\Delta\Vvdash\Gamma;\Theta$
such that $\Gamma$ is non-empty and $\Theta$ is empty.
###### Definition (The Sequent Calculus
$\textnormal{ELL}_{\textnormal{comp}}$).
A proof in the sequent calculus $\textnormal{ELL}_{\textnormal{comp}}$ is a
derivation tree constructed from the derivation rules shown in Figure 19 page
19.
ax $\vdash C^{\simbot},C;$ $\Delta_{1}\vdash\Gamma_{1},C;$
$\Delta_{2}\vdash\Gamma_{2},C^{\simbot};$ cut
$\Delta_{1},\Delta_{2}\vdash\Gamma_{1},\Gamma_{2};$
(a) Identity Group
$\Delta_{1}\vdash\Gamma_{1},C_{1};$ $\Delta_{2}\vdash\Gamma_{2},C_{2};$
$\otimes$ $\Delta_{1},\Delta_{2}\vdash\Gamma_{1},\Gamma_{2},C_{1}\otimes
C_{2};$ $\Delta\vdash\Gamma,C_{1},C_{2};$ $\parr$
$\Delta\vdash\Gamma,C_{1}\parr C_{2};$ $\Delta,N_{1},N_{2}\vdash\Gamma;$
$\otimes^{pol}_{g}$ $\Delta,N_{1}\otimes N_{2}\vdash\Gamma;$
$\Delta,P^{\simbot}\vdash\Gamma,C;$ $\parr^{mix}$ $\Delta\vdash\Gamma,P\parr
C;$ $\Delta\vdash\Gamma,C;$ $\mathbf{1}_{d}$
$\Delta\vdash\Gamma,C\otimes\mathbf{1};$ $\Delta\vdash\Gamma;$
$\mathbf{1}_{g}$ $\Delta,\mathbf{1}\vdash\Gamma;$
(b) Multiplicative Group
$\Delta\vdash\Gamma,C_{i};$ $\oplus_{i}$ $\Delta\vdash\Gamma,C_{1}\oplus
C_{2};$ $\Delta\vdash\Gamma,C_{1};$ $\Delta\vdash\Gamma,C_{2};$ $\with$
$\Delta\vdash\Gamma,C_{1}\with C_{2};$ $\top$ $\Delta\vdash\Gamma,\top;$ No
rules for $0$.
(c) Additive Group
$\Delta_{1}\vdash\Gamma_{1},C_{1};$ $\Delta_{2}\vdash\Gamma_{2},C_{2};$ $\oc$
$\oc\Delta_{1},\Delta_{2},\oc\Gamma_{1}^{\simbot}\vdash\Gamma_{2},C_{1}\otimes\oc
C_{2};$ $\Delta,\oc A,\oc A\vdash\Gamma;$ ctr ($\Gamma\neq\emptyset$)
$\Delta,\oc A\vdash\Gamma;$ $\Delta\vdash\Gamma;$ weak $\Delta,N\vdash\Gamma;$
(d) Exponential Group
$\vdash\Gamma,C;$ $X\not\in\tt FV\rm(\Gamma)$ $\forall$ $\vdash\Gamma,\forall
X\leavevmode\nobreak\ C;$ $\vdash\Gamma,C[A/X];$ $\exists$
$\vdash\Gamma,\exists X\leavevmode\nobreak\ C;$
(e) Quantifier Group
Figure 19: Rules for the sequent calculus
$\textnormal{ELL}_{\textnormal{comp}}$
### 5.2 Truth
The notion of success is the natural generalization of the corresponding
notion on graphs [Sei12a, Sei14a]. The graphing of a successful project will
therefore be a disjoint union of ”transpositions”. Such a graphing can be
represented as a graph with a set of vertices that could be infinite, but
since we are working with equivalence classes of graphings one can always find
a simpler representation: a graphing with exactly two edges.
###### Definition .
A project $\mathfrak{a}=(a,A)$ is _successful_ when it is balanced, $a=0$ and
$A$ is a disjoint union of transpositions:
* •
for all $e\in E^{A}$, $\omega^{A}_{e}=1$;
* •
for all $e\in E^{A}$, $\exists e^{\ast}\in E^{A}$ such that
$\phi^{A}_{e^{\ast}}=(\phi_{e}^{A})^{-1}$ — in particular
$S_{e}^{A}=T_{e^{\ast}}^{A}$ and $T_{e}^{A}=S_{e^{\ast}}^{A}$;
* •
for all $e,f\in E^{A}$ with $f\not\in\\{e,e^{\ast}\\}$, $S^{A}_{e}\cap
S^{A}_{f}$ and $T^{A}_{e}\cap T^{A}_{f}$ are of null measure;
A conduct $\mathbf{A}$ is _true_ when it contains a successful project.
The following results were shown in our previous paper [Sei14c]. They ensure
that the given definition of truth is coherent.
###### Proposition (Consistency).
The conducts $\mathbf{A}$ and $\mathbf{A}^{\simbot}$ cannot be simultaneously
true.
###### Proof.
We suppose that $\mathfrak{a}=(0,A)$ and $\mathfrak{b}=(0,B)$ are successful
project in the conducts $\mathbf{A}$ and $\mathbf{A}^{\simbot}$ respectively.
Then:
$\mathopen{\ll}\mathfrak{a},\mathfrak{b}\mathclose{\gg}_{m}=\mathopen{\llbracket}A,B\mathclose{\rrbracket}_{m}$
If there exists a cycle whose support is of strictly positive measure between
$A$ and $B$, then $\mathopen{\llbracket}A,B\mathclose{\rrbracket}_{m}=\infty$.
Otherwise, $\mathopen{\llbracket}A,B\mathclose{\rrbracket}_{m}=0$. In both
cases we obtained a contradiction since $\mathfrak{a}$ and $\mathfrak{b}$
cannot be orthogonal. ∎
###### Proposition (Compositionnality).
If $\mathbf{A}$ and $\mathbf{A\multimap B}$ are true, then $\mathbf{B}$ is
true.
###### Proof.
Let $\mathfrak{a}\in\mathbf{A}$ and $\mathfrak{f}\in\mathbf{A\multimap B}$ be
successful projects. Then:
* •
If $\mathopen{\ll}\mathfrak{a},\mathfrak{f}\mathclose{\gg}_{m}=\infty$, the
conduct $\mathbf{B}$ is equal to $\mathbf{T}_{V^{B}}$, which is a true conduct
since it contains $(0,\emptyset)$;
* •
Otherwise $\mathopen{\ll}\mathfrak{a},\mathfrak{f}\mathclose{\gg}_{m}=0$ (this
is shown in the same manner as in the preceding proof) and it is sufficient to
show that $F\mathop{\mathopen{:}\mathclose{:}}A$ is a disjoint union of
transpositions. But this is straightforward: to each path there corresponds an
opposite path and the weights of the paths are all equal to $1$, the
conditions on the source and target sets $S_{\pi}$ and $T_{\pi}$ are then
easily checked.
Finally, if $\mathbf{A}$ and $\mathbf{A\multimap B}$ are true, then
$\mathbf{B}$ is true. ∎
### 5.3 Interpretation of proofs
To prove soundness, we will follow the proof technique used in our previous
papers [Sei12a, Sei14a, Sei14c]. We will first define a localized sequent
calculus and show a result of full soundness for it. The soundness result for
the non-localized calculus is then obtained by noticing that one can always
_localize_ a derivation. We will consider here that the variables are defined
with the carrier equal to an interval in $\mathbf{R}$ of the form $[i,i+1[$.
###### Definition .
We fix a set $\mathcal{V}=\\{X_{i}(j)\\}_{i,j\in\mathbf{N}\times\mathbf{Z}}$
of _localized variables_. For $i\in\mathbf{N}$, the set
$X_{i}=\\{X_{i}(j)\\}_{j\in\mathbf{Z}}$ will be called the _variable name
$X_{i}$_, and an element of $X_{i}$ will be called a _variable of name
$X_{i}$_.
For $i,j\in\mathbf{N}\times\mathbf{Z}$ we define the _location_ $\sharp
X_{i}(j)$ of the variable $X_{i}(j)$ as the set
$\\{x\in\mathbf{R}\leavevmode\nobreak\ |\leavevmode\nobreak\
2^{i}(2j+1)\leqslant m<2^{i}(2j+1)+1\\}$
###### Definition (Formulas of $\textnormal{locELL}_{\textnormal{comp}}$).
We inductively define the formulas of _localized polarized elementary linear
logic_ $\textnormal{locELL}_{\textnormal{comp}}$ as well as their _locations_
as follows:
* •
Behaviors:
* –
A variable $X_{i}(j)$ of name $X_{i}$ is a behavior whose location is defined
as $\sharp X_{i}(j)$;
* –
If $X_{i}(j)$ is a variable of name $X_{i}$, then $(X_{i}(j))^{\simbot}$ is a
behavior whose location is $\sharp X_{i}(j)$.
* –
The constants $\mathbf{T}_{\sharp\Gamma}$ are behaviors whose location is
defined as $\sharp\Gamma$;
* –
The constants $\mathbf{0}_{\sharp\Gamma}$ are behaviors whose location is
defined as $\sharp\Gamma$.
* –
If $A,B$ are behaviors with respective locations $X,Y$ such that $X\cap
Y=\emptyset$, then $A\otimes B$ (resp. $A\parr B$, resp. $A\with B$, resp.
$A\oplus B$) is a behavior whose location is $X\cup Y$;
* –
If $X_{i}$ is a variable name, and $A(X_{i})$ is a behavior of location
$\sharp A$, then $\forall X_{i}\leavevmode\nobreak\ A(X_{i})$ and $\exists
X_{i}\leavevmode\nobreak\ A(X_{i})$ are behaviors of location $\sharp A$.
* –
If $A$ is a perennial conduct with location $X$ and $B$ is a behavior whose
location is $Y$ such that $X\cap Y=\emptyset$, then $A\otimes B$ is a behavior
with location $X\cup Y$;
* –
If $A$ is a co-perennial conduct whose location is $X$ and $B$ is a behavior
with location $Y$ such that $X\cap Y=\emptyset$, then $A\parr B$ is a behavior
and its location is $X\cup Y$;
* •
Perennial conducts:
* –
The constant $\mathbf{1}$ is a perennial conduct and its location is
$\emptyset$;
* –
If $A$ is a behavior or a perennial conduct and its location is $X$, then $\oc
A$ is a perennial conduct and its location is $\Omega(X\times[0,1])$;
* –
If $A,B$ are perennial conducts with respective locations $X,Y$ such that
$X\cap Y=\emptyset$, then $A\otimes B$ (resp. $A\oplus B$) is a perennial
conduct whose location is $X\cup Y$;
* •
Co-perennial conducts:
* –
The constant $\mathbf{\bot}$ is a co-perennial conduct;
* –
If $A$ is a behavior or a co-perennial conduct and its location is $X$, then
$\wn A$ is a co-perennial conduct whose location is $\Omega(X\times[0,1])$;
* –
If $A,B$ are co-perennial conducts with respective locations $X,Y$ such that
$X\cap Y=\emptyset$, then $A\parr B$ (resp. $A\with B$) is a co-perennial
conduct whose location is $X\cup Y$;
If $A$ is a formula, we will denote by $\sharp A$ the location of $A$. A
sequent $\Delta\vdash\Gamma;$ of $\textnormal{locELL}_{\textnormal{comp}}$
must satisfy the following conditions:
* •
the formulas of $\Gamma\cup\Delta$ have pairwise disjoint locations;
* •
the formulas of $\Delta$ are all perennial conducts;
* •
$\Gamma$ is non-empty and contains only behaviors.
###### Definition (Interpretations).
An _interpretation basis_ is a function $\Phi$ which associates to each
variable name $X_{i}$ a behavior of carrier $[0,1[$.
###### Definition (Interpretation of
$\textnormal{locELL}_{\textnormal{comp}}$ formulas).
Let $\Phi$ be an interpretation basis. We define the interpretation
$I_{\Phi}(F)$ along $\Phi$ of a formula $F$ inductively:
* •
If $F=X_{i}(j)$, then $I_{\Phi}(F)$ is the delocation (i.e. a behavior) of
$\Phi(X_{i})$ defined by the function $x\mapsto 2^{i}(2j+1)+x$;
* •
If $F=(X_{i}(j))^{\simbot}$, we define the behavior
$I_{\Phi}(F)=(I_{\Phi}(X_{i}(j)))^{\simbot}$;
* •
If $F=\mathbf{T}_{\sharp\Gamma}$ (resp. $F=\mathbf{0}_{\sharp\Gamma}$), we
define $I_{\Phi}(F)$ as the behavior $\mathbf{T}_{\sharp\Gamma}$ (resp.
$\mathbf{0}_{\sharp\Gamma}$);
* •
If $F=\mathbf{1}$ (resp. $F=\mathbf{\bot}$), we define $I_{\Phi}(F)$ as the
behavior $\mathbf{1}$ (resp. $\mathbf{\bot}$);
* •
If $F=A\otimes B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\otimes
I_{\Phi}(B)$;
* •
If $F=A\parr B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\parr
I_{\Phi}(B)$;
* •
If $F=A\oplus B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\oplus
I_{\Phi}(B)$;
* •
If $F=A\with B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\with
I_{\Phi}(B)$;
* •
If $F=\forall X_{i}A(X_{i})$, we define the conduct
$I_{\Phi}(F)=\mathbf{\forall X_{i}}I_{\Phi}(A(X_{i}))$;
* •
If $F=\exists X_{i}A(X_{i})$, we define the conduct
$I_{\Phi}(F)=\mathbf{\exists X_{i}}I_{\Phi}(A(X_{i}))$.
* •
If $F=\oc A$ (resp. $\wn A$), we define the conduct $I_{\Phi}(F)=\oc
I_{\Phi}(A)$ (resp. $\wn I_{\Phi}(A)$).
Moreover, a sequent $\Delta\vdash\Gamma;$ will be interpreted as the $\parr$
of formulas in $\Gamma$ and negations of formulas in $\Delta$, which will be
written $\bigparr\Delta^{\simbot}\parr\bigparr\Gamma$. This formulas can also
be written in the equivalent form $\bigotimes\Delta\multimap(\bigparr\Gamma)$.
###### Definition (Interpretation of
$\textnormal{locELL}_{\textnormal{comp}}$ proofs).
Let $\Phi$ be an interpretation basis. We define the interpretation
$I_{\Phi}(\pi)$ — a project — of a proof $\pi$ inductively:
* •
if $\pi$ is a single axiom rule introducing the sequent
$\vdash(X_{i}(j))^{\simbot},X_{i}(j^{\prime})$, we define $I_{\Phi}(\pi)$ as
the project $\mathfrak{Fax}$ defined by the translation $x\mapsto
2^{i}(2j^{\prime}-2j)+x$;
* •
if $\pi$ is composed of a single rule $\mathbf{T}_{\sharp\Gamma}$, we define
$I_{\Phi}(\pi)=\mathfrak{0}_{\sharp\Gamma}$;
* •
if $\pi$ is obtained from $\pi^{\prime}$ by using a $\parr$ rule, a
$\parr^{mix}$ rule, a $\otimes_{g}^{pol}$ rule, or a $\mathbf{1}$ rule, then
$I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})$;
* •
if $\pi$ is obtained from $\pi_{1}$ and $\pi_{2}$ by performing a $\otimes$
rule, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi_{1})\otimes
I_{\Phi}(\pi^{\prime})$;
* •
if $\pi$ is obtained from $\pi^{\prime}$ using a weak rule or a $\oplus_{i}$
rule introducing a formula of location $V$, we define
$I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})\otimes\mathfrak{0}_{V}$;
* •
if $\pi$ of conclusion $\vdash\Gamma,A_{0}\with A_{1}$ is obtained from
$\pi_{0}$ and $\pi_{1}$ using a $\with$ rule, we define the interpretation of
$\pi$ in the same way it was defined in our previous paper [Sei14a];
* •
If $\pi$ is obtained from a $\forall$ rule applied to a derivation
$\pi^{\prime}$, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})$;
* •
If $\pi$ is obtained from a $\exists$ rule applied to a derivation
$\pi^{\prime}$ replacing the formula $\mathbf{A}$ by the variable name
$X_{i}$, we define
$I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})\mathop{\mathopen{:}\mathclose{:}}(\bigotimes[e^{-1}(j)\leftrightarrow
X_{i}(j)])$, using the notations of our previous paper [Sei14c];
* •
if $\pi$ is obtained from $\pi_{1}$ and $\pi_{2}$ through the use of a
promotion rule $\oc$, we think of this rule as the following ”derivation of
pre-sequents”:
$\vdots^{\pi_{1}}$ $\Delta_{1}\vdash\Gamma_{1},C_{1};$ $\vdots^{\pi_{2}}$
$\Delta_{2}\vdash\Gamma_{2},C_{2};$ $\oc$
$\oc\Delta_{2},\oc\Gamma_{2}^{\simbot}\Vvdash;\oc C_{2}$ $\otimes^{mix}$
$\oc\Delta_{2},\Delta_{1},\oc\Gamma_{2}^{\simbot}\vdash\Gamma_{1},C_{1}\otimes\oc
C_{2};$
As a consequence, we first define a delocation of $\oc I_{\Phi}(\pi)$ to which
we apply the implementation of the functorial promotion. Indeed, the
interpretation of
$\bigparr\Delta^{\simbot}\parr\bigparr\Gamma$
can be written as a sequence of implications. The exponential of a well-chosen
delocation is then represented as:
$\mathbf{\oc(\phi_{1}(A_{1})\multimap(\phi_{2}(A_{2})\multimap\dots(\phi_{n}(A_{n})\multimap\phi_{n+1}(A_{n+1}))\dots))}$
Applying $n$ instances of the project implementing the functorial promotion to
the interpretation of $\pi$, we obtain a project $\mathfrak{p}$ in:
$\mathbf{\oc(\phi_{1}(A_{1}))\multimap\oc(\phi_{2}(A_{2}))\multimap\dots\oc(\phi_{n}(A_{n}))\multimap\oc(\phi_{n+1}(A_{n+1}))}$
which is the same conduct as the one interpreting the conclusion of the
promotion ”rule” in the ”derivation of pre-sequents” we have shown. Now we are
left with taking the tensor product of the interpretation of $\pi_{2}$ with
the project $\mathfrak{p}$ to obtain the interpretation of the $\oc$ rule;
* •
if $\pi$ is obtained from $\pi$ using a contraction rule $ctr$, we write the
conduct interpreting the premise of the rule as $\mathbf{(\oc A\otimes\oc
A)\multimap D}$. We then define a delocation of the latter in order to obtain
$\mathbf{(\phi(\oc A)\otimes\psi(\oc A))\multimap D}$, and take its execution
with $\mathfrak{ctr}$ in $\mathbf{\oc A\multimap(\oc A\otimes\oc A)}$;
* •
if $\pi$ is obtained from $\pi_{1}$ and $\pi_{2}$ by applying a cut rule or a
$\text{cut}^{pol}$ rule, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi_{1})\pitchfork
I_{\Phi}(\pi_{2})$.
###### Theorem ($\textnormal{locELL}_{\textnormal{comp}}$ soundness).
Let $\Phi$ be an interpretation basis. Let $\pi$ be a derivation in
$\textnormal{locELL}_{\textnormal{comp}}$ of conclusion $\Delta\vdash\Gamma;$.
Then $I_{\Phi}(\pi)$ is a successful project in
$I_{\Phi}(\Delta\vdash\Gamma;)$.
###### Proof.
The proof is a simple consequence of of the proposition and theorems proved
before hand. Indeed, the case of the rules of multiplicative additive linear
logic was already treated in our previous papers [Sei12a, Sei14a]. The only
rules we are left with are the rules dealing with exponential connectives and
the rules about the multiplicative units. But the implementation of the
functorial promotion (Section 4.3) uses a successful project do not put any
restriction on the type of conducts we are working with, and the contraction
project (Section 3.3 and Section 5.1) is successful. Concerning the
multiplicative units, the rules that introduce them do not change the
interpretations. ∎
As it was remarked in our previous papers, one can chose an enumeration of the
occurrences of variables in order to ”localize” any formula $A$ and any proof
$\pi$ of $\textnormal{ELL}_{\textnormal{comp}}$: we then obtain formulas
$A^{e}$ and proofs $\pi^{e}$ of $\textnormal{locELL}_{\textnormal{comp}}$. The
following theorem is therefore a direct consequence of the preceding one.
###### Theorem (Full $\textnormal{ELL}_{\textnormal{comp}}$ Soundness).
Let $\Phi$ be an interpretation basis, $\pi$ an
$\textnormal{ELL}_{\textnormal{comp}}$ proof of conclusion
$\Delta\vdash\Gamma;$ and $e$ an enumeration of the occurrences of variables
in the axioms in $\pi$. Then $I_{\Phi}(\pi^{e})$ is a successful project in
$I_{\Phi}(\Delta^{e}\vdash\Gamma^{e};)$.
## 6 Contraction and Soundness for Polarized Conducts
### 6.1 Definitions and Properties
In this section, we consider a variation on the definition of additive
connectives, which is constructed from the definition of the formal sum
$\mathfrak{a+b}$ of projects. Let us first try to explain the difference
between the usual additives $\with$ and $\oplus$ considered until now and the
new additives $\tilde{\with}$ and $\tilde{\oplus}$ defined in this section.
The conduct $\mathbf{A\with B}$ contains all the tests that are necessary for
the set $\\{\mathfrak{a^{\prime}}\otimes\mathfrak{0}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a^{\prime}}\in\mathbf{A}^{\simbot}\\}\cup\\{\mathfrak{b^{\prime}}\otimes\mathfrak{0}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathfrak{b^{\prime}}\in\mathbf{B}^{\simbot}\\}$ to
generate the conduct $\mathbf{A\oplus B}$, something for which the set
$\mathfrak{a+b}$ is not sufficient. For the variant of additives considered in
this section, it is the contrary that happens: the conduct
$\mathbf{A\tilde{\with}B}$ is generated by the projects of the form
$\mathfrak{a+b}$, but it is therefore necessary to add to the conduct
$\mathbf{A\tilde{\oplus}B}$ all the needed tests.
###### Definition .
Let $\mathbf{A,B}$ be conducts of disjoint carriers. We define
$\mathbf{A\tilde{\with}B}=\mathbf{(A+B)^{\simbot\simbot}}$. Dually, we define
$\mathbf{A\tilde{\oplus}B}=\mathbf{(A^{\simbot}\tilde{\with}B^{\simbot})^{\simbot}}$.
These connectives will be useful for showing that the inclusion
$\oc\mathbf{(A\tilde{\with}B)}\subset\mathbf{\oc A\otimes\oc B}$ holds when
$\mathbf{A,B}$ are behaviors. We will first dwell on some properties of these
connectives before showing this inclusion. Notice that if one of the two
conducts $\mathbf{A,B}$ is empty, then $\mathbf{A\tilde{\with}B}$ is empty.
Therefore, the behavior $\mathbf{0}_{\emptyset}$ is a kind of absorbing
element for $\tilde{\with}$. But the latter connective also has a neutral
element, namely the neutral element $\mathbf{1}$ of the tensor product! Notice
that the fact that $\tilde{\with}$ and $\otimes$ share the same unit appeared
in Girard’s construction141414Our construction [Sei14a] differs slightly from
Girard’s, which explains why our additives don’t share the same unit as the
multiplicatives. of geometry of interaction in the hyperfinite factor [Gir11].
Notice that at the level of denotational semantics, this connective is almost
the same as the usual $\with$ (apart from units). The differences between them
are erased in the quotient operation.
###### Proposition .
Distributivity for $\tilde{\with}$ and $\tilde{\oplus}$ is satisfied for
behaviors.
###### Proof.
Using the same project than in the proof of Section 2.2, the proof consists in
a simple computation. ∎
###### Proposition .
Let $\mathbf{A,B}$ be behaviors. Then
$\\{\mathfrak{a}\otimes\mathfrak{0}_{V^{B}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A}\\}\cup\\{\mathfrak{b}\otimes\mathfrak{0}_{V^{A}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{b}\in\mathbf{B}\\}\subset\mathbf{A\tilde{\oplus}B}$
###### Proof.
We will show only one of the inclusions, the other one can be obtained by
symmetry. Chose $\mathfrak{f+g}\in\mathbf{A^{\simbot}+B^{\simbot}}$ and
$\mathfrak{a}\in\mathbf{A}$. Then:
$\displaystyle\mathopen{\ll}\mathfrak{f+g},\mathfrak{a\otimes
0}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
0}\mathclose{\gg}_{m}+\mathopen{\ll}\mathfrak{g},\mathfrak{a\otimes
0}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f},\mathfrak{a}\mathclose{\gg}_{m}$
Using the fact that $\mathfrak{g}$ and $\mathfrak{a}$ have null wagers. ∎
Recall (this notion is defined and studied in our second paper [Sei14a]) that
a behavior $\mathbf{A}$ is _proper_ if both $\mathbf{A}$ and its orthogonal
$\mathbf{A^{\simbot}}$ are non-empty. Proper behavior can be characterized as
those conducts $\mathbf{A}$ such that:
* •
$(a,A)\in\mathbf{A}$ implies that $a=0$;
* •
for all $\mathfrak{a}\in\mathbf{A}$ and $\lambda\in\mathbf{R}$, the project
$\mathfrak{a}+\lambda\mathfrak{0}\in\mathbf{A}$;
* •
$\mathbf{A}$ is non-empty.
###### Proposition .
Let $\mathbf{A,B}$ be proper behaviors. Then every element in
$\mathbf{A\tilde{\oplus}B}$ is observationally equivalent to an element in
$\\{\mathfrak{a}\otimes\mathfrak{0}_{V^{B}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A}\\}\cup\\{\mathfrak{b}\otimes\mathfrak{0}_{V^{A}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathfrak{b}\in\mathbf{B}\\}\subset\mathbf{A\tilde{\oplus}B}$.
###### Proof.
Let $\mathfrak{c}\in\mathbf{A\tilde{\oplus}B}$. Since
$\mathbf{(A^{\simbot}+B^{\simbot})^{\simbot}}=\mathbf{A\tilde{\oplus}B}$, we
know that $\mathfrak{c}\simperp\mathfrak{a+b}$ for all
$\mathfrak{a+b}\in\mathbf{A^{\simbot}+B^{\simbot}}$. By the homothety lemma
(Section 2.2), we obtain, for all $\lambda,\mu$ non-zero real numbers $0$:
$\displaystyle\mathopen{\ll}\mathfrak{c},\mathfrak{\lambda a+\mu
b}\mathclose{\gg}_{m}=\lambda\mathopen{\ll}\mathfrak{c},\mathfrak{a}\mathclose{\gg}_{m}+\mu\mathopen{\ll}\mathfrak{c},\mathfrak{b}\mathclose{\gg}_{m}\neq
0,\infty$
We deduce that one expression among
$\mathopen{\ll}\mathfrak{c},\mathfrak{a}\mathclose{\gg}_{m}$ and
$\mathopen{\ll}\mathfrak{c},\mathfrak{b}\mathclose{\gg}_{m}$ is equal to $0$.
Suppose, without loss of generality, that it is
$\mathopen{\ll}\mathfrak{c},\mathfrak{a}\mathclose{\gg}_{m}$. Then
$\mathopen{\ll}\mathfrak{c},\mathfrak{a^{\prime}}\mathclose{\gg}_{m}=0$ for
all $\mathfrak{a^{\prime}}\in\mathbf{A^{\simbot}}$. Thus
$\mathopen{\ll}\mathfrak{b},\mathfrak{c}\mathclose{\gg}_{m}\neq 0,\infty$ for
all $\mathfrak{b}\in\mathbf{B^{\simbot}}$. But
$\mathopen{\ll}\mathfrak{b\otimes
0},\mathfrak{c}\mathclose{\gg}_{m}=\mathopen{\ll}\mathfrak{b},\mathfrak{c\mathop{\mathopen{:}\mathclose{:}}0}\mathclose{\gg}_{m}$.
We finally have that
$\mathfrak{c\mathop{\mathopen{:}\mathclose{:}}0}\in\mathbf{B^{\simbot}}$ and
$\mathfrak{c\mathop{\mathopen{:}\mathclose{:}}0}\cong_{\mathbf{A\tilde{\oplus}B}}\mathfrak{c}$.
∎
###### Proposition .
Let $\mathbf{A,B}$ be proper behaviors. Then $\mathbf{A\tilde{\with}B}$ is a
proper behavior.
###### Proof.
By definition, $\mathbf{A\tilde{\with}B}=\mathbf{(A+B)}^{\simbot\simbot}$. But
$\mathbf{A,B}$ are non empty contain only one-sliced wager-free projects. Thus
$\mathbf{A+B}$ is non empty and contains only one-sliced wager-free projects.
Thus $\mathbf{(A+B)^{\simbot}}$ satisfies the inflation property. Moreover, if
$\mathfrak{a}+\mathfrak{b}\in\mathbf{A+B}$, we have that
$\mathfrak{a+b+\lambda 0}=\mathfrak{(a+\lambda 0)+b}$. Since $\mathbf{A}$ has
the inflation property, $\mathbf{A+B}$ has the inflation property. Thus
$\mathbf{(A+B)^{\simbot}}$ contains only wager-free projects. Moreover,
$\mathbf{(A+B)^{\simbot}}=\mathbf{A^{\simbot}\tilde{\oplus}B^{\simbot}}$ and
it is therefore non-empty by the preceding proposition (because
$\mathbf{A}^{\simbot},\mathbf{B^{\simbot}}$ are non empty). Then
$\mathbf{(A+B)^{\simbot}}$ is a proper behavior, which allows us to conclude.
∎
###### Proposition .
Let $\mathbf{A,B}$ be behaviors. Then
$\oc\mathbf{(A\tilde{\with}B)}\subset\mathbf{\oc A\otimes\oc B}$.
###### Proof.
If one of the behaviors among $\mathbf{A,B}$ is empty,
$\mathbf{\oc(A\tilde{\with}B)}=\mathbf{0}=\mathbf{\oc A\otimes\oc B}$. We will
now suppose that $\mathbf{A,B}$ are both non empty.
Chose $\mathfrak{f}=(0,F)$ a one-sliced wager-free project. We have that
$\mathfrak{f}^{\prime}=n_{F}/(n_{F}+n_{G})\mathfrak{f}\in\mathbf{A}$ if and
only if $\mathfrak{f}\in\mathbf{A}$ from the homothety lemma (Section 2.2).
Moreover, since $\mathbf{A}$ is a behavior,
$\mathfrak{f^{\prime}}\in\mathbf{A}$ is equivalent151515The implication
$\mathfrak{a}\in\mathbf{A}\Rightarrow\mathfrak{a+\lambda 0}\in\mathbf{A}$
comes from the definition of behaviors, its reciprocal is shown by noticing
that $\mathfrak{a+\lambda 0-\lambda 0}$ is equivalent to $\mathfrak{a}$. to
$\mathfrak{f^{\prime\prime}}=\mathfrak{f^{\prime}}+\sum_{i\leqslant
n_{G}}(1/(n_{F}+n_{G}))\mathfrak{0}\in\mathbf{A}$. Since the weighted thick
and sliced graphing
$\frac{n_{F}}{n_{F}+n_{G}}F+\sum_{i=1}^{n_{G}}\frac{1}{n_{F}+n_{G}}\emptyset$
is universally equivalent to (Section 3.2) a one-sliced weighted thick and
sliced graphing $F^{\prime}$, we obtain finally that the project
$(0,F^{\prime})$ is an element of $\mathbf{A}$ if and only if
$\mathfrak{f}\in\mathbf{A}$. We define in a similar way, being given a project
$\mathfrak{g}$, a weighted graphing with a single slice $G^{\prime}$ such that
$(0,G^{\prime})\in\mathbf{B}$ if and only if $\mathfrak{g}\in\mathbf{B}$.
We are now left to show that
$\oc(0,F^{\prime})\otimes\oc(0,G^{\prime})=\oc(\mathfrak{f+g})$. By
definition, the graphing of $\oc(0,F^{\prime})\otimes\oc(0,G^{\prime})$ is
equal to $\oc_{\Omega}F^{\prime}\uplus\oc_{\Omega}G^{\prime}$. By definition
again, the graphing of $\oc(\mathfrak{f+g})$ is equal to $\oc_{\Omega}(F\uplus
G)=\oc_{\Omega}F^{\iota_{1}}\uplus\oc_{\Omega}G^{\iota_{2}}$, where
$\iota_{1}$ (resp. $\iota_{2}$) denotes the injection of $D^{F}$ (resp.
$D^{G}$) into $D^{F}\uplus D^{G}$. We now are left to notice that
$\oc_{\Omega}F^{\iota_{1}}=\oc_{\Omega}F^{\prime}$ since $F^{\iota_{1}}$ and
$F^{\prime}$ are variants one of the other. Similarly,
$\oc_{\Omega}G^{\iota_{2}}=\oc_{\Omega}G^{\prime}$. Finally, we have that
$\sharp\mathbf{(A+B)}\subset\mathbf{\sharp A\odot\sharp B}$ which is enough to
conclude. ∎
###### Lemma .
Let $\mathbf{A}$ be a conduct, and $\phi,\psi$ disjoint delocations. There
exists a successful project in the conduct
$\mathbf{A\multimap\phi(A)\tilde{\with}\psi(A)}$
###### Proof.
We define
$\mathfrak{c}=\mathfrak{Fax}_{\phi}\otimes\mathfrak{0}_{\psi(V^{A})}+\mathfrak{Fax}_{\psi}\otimes\mathfrak{0}_{\phi(V^{A})}$.
Then for all $\mathfrak{a}\in\mathbf{A}$:
$\mathfrak{c}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{a}=\phi(\mathfrak{a})\otimes\mathfrak{0}_{\psi(V^{A})}+\psi(\mathfrak{a})\otimes\mathfrak{0}_{\phi(V^{A})}$
Thus $\mathfrak{c}\in\mathbf{A\multimap\phi(A)\tilde{\with}\psi(A)}$.
Moreover, $\mathfrak{c}$ is obviously successful. ∎
###### Proposition .
Let $\mathbf{A}$ be a behavior, and $\phi,\psi$ be disjoint delocations. There
exists a successful project in the conduct
$\mathbf{\wn\phi(A)\parr\wn\psi(A)\multimap\wn A}$
###### Proof.
If $\mathfrak{f}\in\mathbf{\wn\phi(A)\parr\wn\psi(A)}$, then we have
$\mathfrak{f}\in\wn\mathbf{(\phi(A)\tilde{\oplus}\psi(A))}$ by Section 6.1.
Moreover, we have a successful project $\mathfrak{c}$ in
$\mathbf{A^{\simbot}}\multimap\mathbf{\phi(A^{\simbot})\tilde{\with}\psi(A^{\simbot})}$
using the preceding lemma. Using the successful project implementing
functorial promotion we obtain a successful project
$\mathfrak{c^{\prime}}\in\mathbf{\oc
A^{\simbot}\multimap\oc(\phi(A^{\simbot})\tilde{\with}\psi(A^{\simbot}))}$.
Thus $\mathfrak{c^{\prime}}$ is a successful project in
$\wn\mathbf{\phi(A)\tilde{\oplus}\psi(A)}\multimap\mathbf{\wn A}$. Finally, we
obtain, by composition, that
$\mathfrak{f}\mathop{\mathopen{:}\mathclose{:}}\mathfrak{c^{\prime}}$ is a
successful project in $\mathbf{\wn A}$. ∎
###### Corollary .
Let $\mathbf{A,B}$ be behaviors, and $\phi,\psi$ be respective delocations of
$\mathbf{A}$ and $\mathbf{B}$. There exists a successful project in the
conduct
$\mathbf{\oc(A\with B)\multimap\oc\phi(A)\otimes\oc\psi(B)}$
###### Proof.
It is obtained as the interpretation of the following derivation (well formed
in the sequent calculus we define later on):
ax $\Vvdash A,A^{\simbot};$ $\oplus_{d,2}$ $\Vvdash A^{\simbot}\oplus
B^{\simbot},A$ $\oc$ $\oc(A\with B)\Vvdash;\oc A$ ax $\Vvdash B,B^{\simbot};$
$\oplus_{d,1}$ $\Vvdash A^{\simbot}\oplus B^{\simbot},B$ $\oc$ $\oc(A\with
B)\Vvdash;\oc B$ $\otimes^{pol}$ $\oc(A\with B),\oc(A\with B)\Vvdash;\oc
A\otimes\oc B$ ctr $\oc(A\with B)\Vvdash;\oc A\otimes\oc B$
The fact that it is successful is a consequence of the soundness theorem
(Section 6.3). ∎
### 6.2 Polarized conducts
The notions of perennial and co-perennial conducts are not completely
satisfactory. In particular, we are not able to show that an implication
$\mathbf{A\multimap B}$ is either perennial or co-perennial when $\mathbf{A}$
is a perennial conduct (resp. co-perennial) and $\mathbf{B}$ is a co-perennial
conduct (resp. perennial). This is an important issue when one considers the
sequent calculus: the promotion rule has to be associated with a rule
involving behaviors in order to in the setting of behaviors (using Section
4.2). Indeed, a sequent $\vdash\wn\Gamma,\oc A$ would be interpreted by a
conduct which is neither perennial nor co-perennial in general. The sequents
considered are for this reason restricted to pre-sequent containing behaviors.
We will define now the notions of negative and positive conducts. The idea is
to relax the notion of perennial conduct in order to obtain a notion _negative
conduct_. The main interest of this approach is that positive/negative
conducts will share the important properties of perennial/co-perennial
conducts while interacting in a better way with connectives. In particular, we
will be able to interpret the usual functorial promotion (not associated to a
$\otimes$ rule), and we will be able to use the contraction rule without all
the restrictions we had in the previous section.
###### Definition (Polarized Conducts).
A positive conduct $\mathbf{P}$ is a conduct satisfying the inflation property
and containing all daemons:
* •
$\mathfrak{p}\in\mathbf{P}\Rightarrow\mathfrak{p+\lambda 0}\in\mathbf{P}$;
* •
$\forall\lambda\in\mathbf{R}-\\{0\\},\leavevmode\nobreak\
\mathfrak{Dai}_{\lambda}=(\lambda,(V^{P},\emptyset))\in\mathbf{P}$.
A conduct $\mathbf{N}$ is negative when its orthogonal $\mathbf{N}^{\simbot}$
is a positive conduct.
###### Proposition .
A perennial conduct is negative. A co-perennial conduct is positive.
###### Proof.
We already showed that the perennial conducts satisfy the inflation property
(Section 4.2) and contain daemons (Section 4.2). ∎
###### Proposition .
A conduct $\mathbf{A}$ is negative if and only if:
* •
$\mathbf{A}$ contains only wager-free projects;
* •
$\mathfrak{a}\in\mathbf{A}\Rightarrow\textbf{1}_{A}\neq 0$.
###### Proof.
If $\mathbf{A}^{\simbot}$ is a positive conduct, then it is non-empty and
satisfies the inflation property, thus $\mathbf{A}$ contains only wager-free
projects by Section 2.2. As a consequence, if $\mathfrak{a}\in\mathbf{A}$, we
have that
$\mathopen{\ll}\mathfrak{a},\mathfrak{Dai_{\text{$\lambda$}}}\mathclose{\gg}_{m}=\lambda\textbf{1}_{A}$
thus the condition
$\mathopen{\ll}\mathfrak{a},\mathfrak{Dai}\mathclose{\gg}_{m}\neq 0$ implies
that $\textbf{1}_{A}\neq 0$.
Conversely, if $\mathbf{A}$ satisfies that stated properties, we distinguish
two cases. If $\mathbf{A}$ is empty, then is it clear that
$\mathbf{A}^{\simbot}$ is a positive conduct. Otherwise, $\mathbf{A}$ is a
non-empty conduct containing only wager-free projects, thus
$\mathbf{A}^{\simbot}$ satisfies the inflation property (Section 2.2).
Moreover,
$\mathopen{\ll}\mathfrak{a},\mathfrak{Dai}\mathclose{\gg}_{m}=\textbf{1}_{A}\lambda\neq
0$ as a consequence of the second condition and therefore
$\mathfrak{Dai}\in\mathbf{A}^{\simbot}$. Finally, $\mathbf{A}^{\simbot}$ is a
positive conduct, which implies that $\mathbf{A}$ is a negative conduct. ∎
The polarized conducts do not interact very well with the connectives
$\tilde{\with}$ and $\tilde{\oplus}$. Indeed, if $\mathbf{A,B}$ are negative
conducts, the conduct $\mathbf{A\tilde{\with}B}$ is generated by a set of
wager-free projects, but it does not satisfy the second property needed to be
a negative conduct. Similarly, if $\mathbf{A,B}$ are positive conducts, then
$\mathbf{A\tilde{\with}B}$ will obviously have the inflation property, but it
will contain the project $\mathfrak{Dai}_{0}$ (which implies that any element
$\mathfrak{c}$ in its orthogonal is such that $\textbf{1}_{C}=0$). We are also
not able to characterize in any way the conduct $\mathbf{A\tilde{\with}B}$
when $\mathbf{A}$ is a positive conduct and $\mathbf{B}$ is a negative
conduct, except that it is has the inflation property. However, the notions of
positive and negative conducts interacts in a nice way with the connectives
$\otimes,\with,\parr,\oplus$.
###### Proposition .
The tensor product of negative conducts is a negative conduct. The $\with$ of
negative conducts is a negative conduct. The $\oplus$ of negative conducts is
a negative conduct.
###### Proof.
We know that $\mathbf{A\otimes B}=\emptyset$ if one of the two conducts
$\mathbf{A}$ and $\mathbf{B}$ is empty, which leaves us to treat the non-empty
case. In this case, $\mathbf{A\otimes
B}=(\mathbf{A}\odot\mathbf{B})^{\simbot\simbot}$ is the bi-orthogonal of a
non-empty set of wager-free projects. Thus $(\mathbf{A\otimes B})^{\simbot}$
satisfies the inflation property. Moreover $\mathopen{\ll}\mathfrak{a\otimes
b},\mathfrak{Dai}\mathclose{\gg}_{m}=\textbf{1}_{B}\textbf{1}_{A}\lambda$
which is different from zero since $\textbf{1}_{A},\textbf{1}_{B}$ both are
different from zero. Thus $\mathfrak{Dai}\in(\mathbf{A\otimes B})^{\simbot}$,
which shows that $\mathbf{A\otimes B}$ is a negative conduct since
$\mathbf{(A\otimes B)^{\simbot}}$ is a positive conduct.
The set $\mathbf{A}^{\simbot}{\uparrow_{{B}}}$ contains all daemons
$\mathfrak{Dai}_{\lambda}\otimes\mathfrak{0}=\mathfrak{Dai}_{\lambda}$, and
$\mathfrak{Dai}\in\mathbf{A}^{\simbot}$. It has the inflation property since
$\mathfrak{(b+\lambda 0)\otimes
0}=\mathfrak{b}\otimes\mathfrak{0}+\lambda\mathfrak{0}$. Thus
$((\mathbf{A}^{\simbot}){\uparrow_{{B}}})^{\simbot}$ is a negative conduct.
Similarly, $((\mathbf{B}^{\simbot}){\uparrow_{{B}}})^{\simbot}$ is a negative
conduct, and their intersection is a negative conduct since the properties
defining negative conducts are are preserved by intersection. As a
consequence, $\mathbf{A\with B}$ is a negative conduct.
In the case of $\oplus$, we will use the fact that $\mathbf{A\oplus
B}=(\mathbf{A}{\uparrow_{{B}}}\cup\mathbf{B}{\uparrow_{{A}}})^{\simbot}$. If
$\mathfrak{a}\in\mathbf{A}$, $\mathfrak{a\otimes 0}=\mathfrak{b}$ has a null
wager and $\textbf{1}_{B}=\textbf{1}_{A}\neq 0$. If $\mathbf{A}$ is empty,
$(\mathbf{A}{\uparrow_{{B}}})^{\simbot}$ is a positive conduct. If
$\mathbf{A}$ is non-empty, then Section 2.2 allows us to state that
$(\mathbf{A}{\uparrow_{{B}}})^{\simbot}$ has the inflation property. Moreover,
the fact that all elements in $\mathfrak{a\otimes 0}=\mathfrak{b}$ satisfy
$\textbf{1}_{B}\neq 0$ implies that
$\mathfrak{Dai}_{\lambda}\in(\mathbf{A}{\uparrow_{{B}}})^{\simbot}$ for all
$\lambda\neq 0$. Therefore, $(\mathbf{A}{\uparrow_{{B}}})^{\simbot}$ is a
positive conduct. As a consequence, $\mathbf{A}{\uparrow_{{B}}}$ is a negative
conduct. We show in a similar way that $\mathbf{B}{\uparrow_{{A}}}$ is a
negative conduct. We can deduce from this that
$\mathbf{A}{\uparrow_{{B}}}\cup\mathbf{B}{\uparrow_{{A}}}$ contains only
projects $\mathfrak{c}$ with zero wager and such that $\textbf{1}_{C}\neq 0$.
Finally, we showed that $\mathbf{A\oplus B}$ is a negative conduct. ∎
###### Corollary .
The $\parr$ of positive conducts is a positive conduct, the $\with$ of
positive conducts is a positive conduct, and the $\oplus$ of positive conducts
is a positive conduct.
###### Proposition .
Let $\mathbf{A}$ be a positive conduct and $\mathbf{B}$ be a negative conduct.
Then $\mathbf{A\otimes B}$ is a positive conduct.
###### Proof.
Pick $\mathfrak{f}\in\mathbf{(A\otimes B)^{\simbot}}=\mathbf{B\multimap
A^{\simbot}}$. Then for all $\mathfrak{b}\in\mathbf{B}$,
$\mathfrak{f\mathop{\mathopen{:}\mathclose{:}}b}=(\textbf{1}_{B}f+\textbf{1}_{F}b,F\mathop{\mathopen{:}\mathclose{:}}B)$
is an element of $\mathbf{A}^{\simbot}$. Since $\mathbf{A}^{\simbot}$ is a
negative conduct, we have that $\textbf{1}_{F}\textbf{1}_{B}\neq 0$ and
$\textbf{1}_{B}f+\textbf{1}_{F}b=0$. Thus $\textbf{1}_{F}\neq 0$. Moreover,
$\mathbf{B}$ is a negative conduct, therefore $\textbf{1}_{B}\neq 0$ and
$b=0$. The condition $\textbf{1}_{B}f+\textbf{1}_{F}b=0$ then becomes
$\textbf{1}_{B}f=0$, i.e. $f=0$.
Thus $\mathbf{(A\otimes B)^{\simbot}}$ is a negative conduct, which implies
that $\mathbf{A\otimes B}$ is a positive conduct. ∎
###### Corollary .
If $\mathbf{A}$ is a positive conduct and $\mathbf{B}$ is a positive conduct,
$\mathbf{A\multimap B}=\mathbf{(A\otimes B^{\simbot})^{\simbot}}$ is a
positive conduct.
###### Corollary .
If $\mathbf{A,B}$ are negative conducts, then $\mathbf{A\multimap B}$ is a
negative conduct.
###### Proof.
We know that $\mathbf{A\multimap B}=\mathbf{(A\otimes
B^{\simbot})^{\simbot}}$. We also just showed that $\mathbf{A\otimes
B^{\simbot}}$ is a positive conduct, thus $\mathbf{A\multimap B}$ is a
negative conduct. ∎
###### Proposition .
The tensor product of a negative conduct and a behavior is a behavior.
###### Proof.
Let $\mathbf{A}$ be a negative conduct and $\mathbf{B}$ be a behavior. If
either $\mathbf{A}$ or $\mathbf{B}$ is empty (or both), $\mathbf{(A\otimes
B)^{\simbot}}$ equals $\mathbf{T}_{V^{A}\cup V^{B}}$ and we are done. We now
suppose that $\mathbf{A}$ and $\mathbf{B}$ are both non empty.
Since $\mathbf{A,B}$ contain only wager-free projects, the set
$\\{\mathfrak{a\otimes b}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathfrak{a}\in\mathbf{A},\mathfrak{b}\in\mathbf{B}\\}$ contains only wager-
free projects. Thus $\mathbf{(A\otimes B)^{\simbot}}$ has the inflation
property: this is a consequence of Section 2.2. Suppose now that there exists
$\mathfrak{f}\in\mathbf{(A\otimes B)^{\simbot}}$ such that $f\neq 0$. Chose
$\mathfrak{a}\in\mathbf{A}$ and $\mathfrak{b}\in\mathbf{B}$. Then
$\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
b}\mathclose{\gg}_{m}=f\textbf{1}_{B}\textbf{1}_{A}+\mathopen{\llbracket}F,A\mathop{\mathopen{:}\mathclose{:}}B\mathclose{\rrbracket}_{m}$.
Since $\textbf{1}_{A}\neq 0$, we can define $\mu=-\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}/(\textbf{1}_{A}f)$, and $\mathfrak{b+\mu
0}\in\mathbf{B}$ since $\mathbf{B}$ has the inflation property. We then have:
$\displaystyle\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes(b+\mu
0)}\mathclose{\gg}_{m}$ $\displaystyle=$ $\displaystyle
f\textbf{1}_{A}(\textbf{1}_{B}+\mu)+\mathopen{\llbracket}F,A\mathop{\mathopen{:}\mathclose{:}}(B+\mu
0)\mathclose{\rrbracket}_{m}$ $\displaystyle=$ $\displaystyle
f\textbf{1}_{A}\frac{-\mathopen{\llbracket}F,A\cup
B\mathclose{\rrbracket}_{m}}{\textbf{1}_{A}f}+\mathopen{\llbracket}F,A\mathop{\mathopen{:}\mathclose{:}}B\mathclose{\rrbracket}_{m}$
$\displaystyle=$ $\displaystyle 0$
This is a contradiction, since $\mathfrak{f}\in\mathbf{(A\otimes
B)^{\simbot}}$. Thus $f=0$.
Finally, we have shown that $\mathbf{(A\otimes B)^{\simbot}}$ has the
inflation property and contains only wager-free projects. ∎
###### Corollary .
If $\mathbf{A}$ is a negative conduct and $\mathbf{B}$ is a behavior,
$\mathbf{A\multimap B}$ is a behavior.
$\otimes$ | N | P
---|---|---
N | N | P
P | P | ?
(a) Tenseur
$\parr$ | N | P
---|---|---
N | ? | N
P | N | P
(b) Parr
$\with$ | N | P
---|---|---
N | N | ?
P | ? | P
(c) Avec(1)
$\oplus$ | N | P
---|---|---
N | N | ?
P | ? | P
(d) Plus(1)
Figure 20: Connectives and Polarization
###### Proposition .
The weakening (on the left) of negative conducts holds.
###### Proof.
Let $\mathbf{A,B}$ be conducts, $\mathbf{N}$ be a negative conduct, and pick
$\mathfrak{f}\in\mathbf{A\multimap B}$. We will show that
$\mathfrak{f}\otimes\mathfrak{0}_{V^{N}}$ is an element of $\mathbf{A\otimes
N\multimap B}$. For this, we pick $\mathfrak{a}\in\mathbf{A}$ and
$\mathfrak{n}\in\mathbf{N}$. Then for all
$\mathfrak{b^{\prime}}\in\mathbf{B^{\simbot}}$,
$\displaystyle\mathopen{\ll}\mathfrak{(f\otimes
0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f\otimes 0},\mathfrak{(a\otimes
n)\otimes b}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\mathopen{\ll}\mathfrak{f\otimes 0},\mathfrak{(a\otimes
b^{\prime})\otimes n}\mathclose{\gg}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{F}(\textbf{1}_{A}\textbf{1}_{B^{\prime}}n+\textbf{1}_{N}\textbf{1}_{A}b^{\prime}+\textbf{1}_{N}\textbf{1}_{B^{\prime}}a)+\textbf{1}_{N}\textbf{1}_{A}\textbf{1}_{B^{\prime}}f+\mathopen{\llbracket}F\cup
0,A\cup B^{\prime}\cup N\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{F}(\textbf{1}_{N}\textbf{1}_{A}b^{\prime}+\textbf{1}_{N}\textbf{1}_{B^{\prime}}a)+\textbf{1}_{N}\textbf{1}_{A}\textbf{1}_{B^{\prime}}f+\mathopen{\llbracket}F\cup
0,A\cup B^{\prime}\cup N\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{N}(\textbf{1}_{F}(\textbf{1}_{A}b^{\prime}+\textbf{1}_{B^{\prime}}a)+\textbf{1}_{A}\textbf{1}_{B^{\prime}}f)+\textbf{1}_{N}\mathopen{\llbracket}F,A\cup
B^{\prime}\mathclose{\rrbracket}_{m}$ $\displaystyle=$
$\displaystyle\textbf{1}_{N}\mathopen{\ll}\mathfrak{f},\mathfrak{a\otimes
b^{\prime}}\mathclose{\gg}_{m}$
Since $\textbf{1}_{N}\neq 0$, $\mathopen{\ll}\mathfrak{(f\otimes
0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}\neq 0,\infty$ if and only if
$\mathopen{\ll}\mathfrak{f\mathop{\mathopen{:}\mathclose{:}}a},\mathfrak{b^{\prime}}\mathclose{\gg}_{m}\neq
0,\infty$. Therefore, for all $\mathfrak{a\otimes n}\in\mathbf{A\odot N}$,
$\mathfrak{(f\otimes 0)\mathop{\mathopen{:}\mathclose{:}}(a\otimes
n)}\in\mathbf{B}$. This shows that $\mathfrak{f\otimes 0}$ is an element of
$\mathbf{A\otimes N\multimap B}$ by Section 3.3. ∎
### 6.3 Sequent Calculus and Soundness
We now describe a sequent calculus which is much closer to the usual sequent
calculus for Elementary Linear Logic. We introduce once again three types of
formulas: (B)ehaviors, (P)ositive, (N)egative. The sequents we will be working
with will be the equivalent to the notion of pre-sequent introduced earlier.
###### Definition .
We once again define three types of formulas — (B)ehavior, (P)ositive,
(N)egative — by the following grammar:
$\displaystyle B$ $\displaystyle:=$ $\displaystyle X\leavevmode\nobreak\
|\leavevmode\nobreak\ X^{\simbot}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathbf{0}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathbf{T}\leavevmode\nobreak\ |\leavevmode\nobreak\ B\otimes
B\leavevmode\nobreak\ |\leavevmode\nobreak\ B\parr B\leavevmode\nobreak\
|\leavevmode\nobreak\ B\oplus B\leavevmode\nobreak\ |\leavevmode\nobreak\
B\with B\leavevmode\nobreak\ |\leavevmode\nobreak\ \forall
X\leavevmode\nobreak\ B\leavevmode\nobreak\ |\leavevmode\nobreak\ \exists
X\leavevmode\nobreak\ B\leavevmode\nobreak\ |\leavevmode\nobreak\ B\otimes
N\leavevmode\nobreak\ |\leavevmode\nobreak\ B\parr P$ $\displaystyle N$
$\displaystyle:=$ $\displaystyle\mathbf{1}\leavevmode\nobreak\
|\leavevmode\nobreak\ \oc B\leavevmode\nobreak\ |\leavevmode\nobreak\ \oc
N\leavevmode\nobreak\ |\leavevmode\nobreak\ N\otimes N\leavevmode\nobreak\
|\leavevmode\nobreak\ N\with N\leavevmode\nobreak\ |\leavevmode\nobreak\
N\oplus N\leavevmode\nobreak\ |\leavevmode\nobreak\ N\parr P$ $\displaystyle
P$ $\displaystyle:=$ $\displaystyle\mathbf{\bot}\leavevmode\nobreak\
|\leavevmode\nobreak\ \wn B\leavevmode\nobreak\ |\leavevmode\nobreak\ \wn
P\leavevmode\nobreak\ |\leavevmode\nobreak\ P\parr P\leavevmode\nobreak\
|\leavevmode\nobreak\ P\with P\leavevmode\nobreak\ |\leavevmode\nobreak\
P\oplus P\leavevmode\nobreak\ |\leavevmode\nobreak\ N\otimes P$
###### Definition .
A sequent $\Delta\Vvdash\Gamma;\Theta$ is such that $\Delta,\Theta$ contain
only negative formulas, $\Theta$ containing at most one formula and $\Gamma$
containing only behaviors.
###### Definition (The System $\textnormal{ELL}_{\textnormal{pol}}$).
A proof in the system $\textnormal{ELL}_{\textnormal{pol}}$ is a derivation
tree constructed from the derivation rules shown in Figure 21.
ax $\Vvdash B^{\simbot},B;$ $\Delta_{1}\Vvdash\Gamma_{1};N$
$\Delta_{2},N\Vvdash\Gamma_{2};\Theta$ cutpol
$\Delta_{1},\Delta_{2}\Vvdash\Gamma_{1},\Gamma_{2};\Theta$
$\Delta_{1}\Vvdash\Gamma_{1},B;\Theta$
$\Delta_{2}\Vvdash\Gamma_{2},B^{\simbot};$ cut
$\Delta_{1},\Delta_{2}\Vvdash\Gamma_{1},\Gamma_{2};\Theta$
(a) Identity Group
$\Delta_{1}\Vvdash\Gamma_{1},B_{1};\Theta$
$\Delta_{2}\Vvdash\Gamma_{2},B_{2};$ $\otimes$
$\Delta_{1},\Delta_{2}\Vvdash\Gamma_{1},\Gamma_{2},B_{1}\otimes B_{2};\Theta$
$\Delta\Vvdash\Gamma,B_{1},B_{2};\Theta$ $\parr$
$\Delta\Vvdash\Gamma,B_{1}\parr B_{2};\Theta$
$\Delta,N_{1},N_{2}\Vvdash\Gamma;\Theta$ $\otimes^{pol}_{g}$
$\Delta,N_{1}\otimes N_{2}\Vvdash\Gamma;\Theta$
$\Delta_{1}\Vvdash\Gamma_{1};N_{1}$ $\Delta_{2}\Vvdash\Gamma_{2};N_{2}$
$\otimes^{pol}_{d}$
$\Delta_{1},\Delta_{2}\Vvdash\Gamma_{1},\Gamma_{2};N_{1}\otimes N_{2}$
$\Delta,P_{1}^{\simbot}\Vvdash\Gamma;N_{2}$ $\parr^{pol}_{d}$
$\Delta\Vvdash\Gamma;P_{1}\parr N_{2}$
$\Delta_{1}\Vvdash\Gamma_{1};P^{\simbot}_{1}$
$\Delta_{2},N_{2}\Vvdash\Gamma_{2};\Theta$ $\parr^{pol}_{g}$
$\Delta_{1},\Delta_{2},P_{1}\parr N_{2}\Vvdash\Gamma_{1},\Gamma_{2};\Theta$
$\Delta,P^{\simbot}\Vvdash\Gamma,B;\Theta$ $\parr^{mix}$
$\Delta\Vvdash\Gamma,P\parr B;\Theta$ $\Delta_{1}\Vvdash\Gamma_{1};N$
$\Delta_{2}\Vvdash\Gamma_{2},B;\Theta$ $\otimes^{mix}$
$\Delta_{1},\Delta_{2}\Vvdash\Gamma_{1},\Gamma_{2},N\otimes B;\Theta$
$\mathbf{1}_{d}$ $\Vvdash;\mathbf{1}$ $\Delta\Vvdash\Gamma;\Theta$
$\mathbf{1}_{g}$ $\Delta,\mathbf{1}\Vvdash\Gamma;\Theta$
(b) Multiplicative Group
$\Delta\Vvdash\Gamma,B_{i};\Theta$ $\oplus_{i}$
$\Delta\Vvdash\Gamma,B_{1}\oplus B_{2};\Theta$
$\Delta\Vvdash\Gamma,B_{1};\Theta$ $\Delta\Vvdash\Gamma,B_{2};\Theta$ $\with$
$\Delta\Vvdash\Gamma,B_{1}\with B_{2};\Theta$ $\top$
$\Delta\Vvdash\Gamma,\top;\Theta$ No rules for $0$.
(c) Additive Group
$\Delta\Vvdash\Gamma;N$ $\oc^{pol}$ $\oc\Delta,\oc\Gamma^{\simbot}\Vvdash;\oc
N$ $\Delta\Vvdash\Gamma,B;$ $\oc$ $\oc\Delta,\oc\Gamma^{\simbot}\Vvdash;\oc B$
$\Delta,\oc B,\oc B\Vvdash\Gamma;\Theta$ ctr (B Behavior) $\Delta,\oc
B\Vvdash\Gamma;\Theta$ $\Delta\Vvdash\Gamma;\Theta$ weak
$\Delta,N\Vvdash\Gamma;\Theta$
(d) Exponential Group
$\Delta\Vvdash\Gamma,C;\Theta$ $X\not\in\tt FV\rm(\Gamma,\Delta,\Theta)$
$\forall$ $\Delta\Vvdash\Gamma,\forall X\leavevmode\nobreak\ C;\Theta$
$\Delta\Vvdash\Gamma,C[A/X];\Theta$ $\exists$ $\Delta\Vvdash\Gamma,\exists
X\leavevmode\nobreak\ C;\Theta$
(e) Quantifier Group
Figure 21: Sequent Calculus $\textnormal{ELL}_{\textnormal{pol}}$
###### Remark .
Even though one can consider the conduct $\mathbf{A\with B}$ when
$\mathbf{A,B}$ are negative conducts, no rule of the sequent calculus
$\textnormal{ELL}_{\textnormal{pol}}$ allows one to construct such a formula.
The reason for that is simple: since in this case the set $\mathbf{A+B}$ is
not necessarily included in the conduct $\mathbf{A\with B}$, one cannot
interpret the rule in general (since distributivity does not necessarily
holds). The latter can be interpreted when the context contains at least one
behavior, but imposing such a condition on the rule could lead to difficulties
when considering the cut-elimination procedure (in case of commutations). We
therefore whose to work with a system in which one introduces additive
connectives only between behaviors. Notice however that a formula built with
an additive connective between negative sub-formulas can still be introduced
by a weakening rule.
The following proposition is obtained easily by standard proof techniques.
###### Proposition .
The system $\textnormal{ELL}_{\textnormal{pol}}$ possesses a cut-elimination
procedure.
We now define the interpretation of the formulas and proofs of the localized
sequent calculus in the model $\mathbb{M}[\Omega,\mathfrak{mi}]_{m}$.
###### Definition .
We fix $\mathcal{V}=\\{X_{i}(j)\\}_{i,j\in\mathbf{N}\times\mathbf{Z}}$ a set
of _localized variables_. For $i\in\mathbf{N}$, the set
$X_{i}=\\{X_{i}(j)\\}_{j\in\mathbf{Z}}$ will be referred to as _the name of
the variable $X_{i}$_, and an element of $X_{i}$ will be referred to as a
_variable of name $X_{i}$_.
For $i,j\in\mathbf{N}\times\mathbf{Z}$ we define the _location_ $\sharp
X_{i}(j)$ of the variable $X_{i}(j)$ as the set
$\\{x\in\mathbf{R}\leavevmode\nobreak\ |\leavevmode\nobreak\
2^{i}(2j+1)\leqslant m<2^{i}(2j+1)+1\\}$
###### Definition (Formulas of $\textnormal{locELL}_{\textnormal{pol}}$).
We inductively define the formulas of $\textnormal{locELL}_{\textnormal{pol}}$
together with their _locations_ as follows:
* •
Behaviors:
* –
A variable $X_{i}(j)$ of name $X_{i}$ is a behavior whose location is defined
as $\sharp X_{i}(j)$;
* –
If $X_{i}(j)$ is a variable of name $X_{i}$, then $(X_{i}(j))^{\simbot}$ is a
behavior of location $\sharp X_{i}(j)$.
* –
The constants $\mathbf{T}_{\sharp\Gamma}$ are behaviors of location
$\sharp\Gamma$;
* –
The constants $\mathbf{0}_{\sharp\Gamma}$ are behaviors of location
$\sharp\Gamma$.
* –
If $A,B$ are behaviors of respective locations $X,Y$ such that $X\cap
Y=\emptyset$, then $A\otimes B$ (resp. $A\parr B$, resp. $A\with B$, resp.
$A\oplus B$) is a behavior of location $X\cup Y$;
* –
If $X_{i}$ is a variable name, and $A(X_{i})$ is a behavior of location
$\sharp A$, then $\forall X_{i}\leavevmode\nobreak\ A(X_{i})$ and $\exists
X_{i}\leavevmode\nobreak\ A(X_{i})$ are behaviors of location $\sharp A$.
* –
If $A$ is a negative conduct of location $X$ and $B$ is a behavior of location
$Y$ such that $X\cap Y=\emptyset$, then $A\otimes B$is a behavior of location
$X\cup Y$;
* –
If $A$ is a positive conduct of location $X$ and $B$ is a behavior of location
$Y$ such that $X\cap Y=\emptyset$, then $A\parr B$ is a behavior of location
$X\cup Y$;
* •
Negative Conducts:
* –
The constant $\mathbf{1}$ is a negative conduct;
* –
If $A$ is a behavior or a negative conduct of location $X$, then $\oc A$ is a
negative conduct of location $\Omega(X\times[0,1])$;
* –
If $A,B$ are negative conducts of locations $X,Y$ such that $X\cap
Y=\emptyset$, then $A\otimes B$ (resp. $A\oplus B$, resp. $A\with B$) is a
negative conduct of location $X\cup Y$;
* –
If $A$ is a negative conduct of location $X$ and $B$ is a positive conduct of
location $Y$, $A\parr B$ is a negative conduct of location $X\cup Y$.
* •
Positive Conducts:
* –
The constant $\mathbf{\bot}$ is a positive conduct;
* –
If $A$ is a behavior or a positive conduct of location $X$, then $\wn A$ is a
positive conduct of location $\Omega(X\times[0,1])$;
* –
If $A,B$ are positive conducts of locations $X,Y$ such that $X\cap
Y=\emptyset$, then $A\parr B$ (resp. $A\with B$, resp. $A\oplus B$) is a
positive conduct of location $X\cup Y$;
* –
If $A$ is a negative conduct of location $X$ and $B$ is a positive conduct of
location $Y$, $A\otimes B$ is a positive conduct of location $X\cup Y$.
If $A$ is a formula, we will denote by $\sharp A$ its location. We also define
sequents $\Delta\Vvdash\Gamma;\Theta$ of
$\textnormal{locELL}_{\textnormal{pol}}$ when:
* •
formulas in $\Gamma\cup\Delta\cup\Theta$ have pairwise disjoint locations;
* •
formulas in $\Delta$ and $\Theta$ are negative conducts;
* •
there is at most one formula in $\Theta$;
* •
$\Gamma$ contains only behaviors.
###### Definition (Interpretations).
We define an _interpretation basis_ as a function $\Phi$ which maps every
variable name $X_{i}$ to a behavior of carrier $[0,1[$.
###### Definition (Interpretation of
$\textnormal{locELL}_{\textnormal{pol}}$ formulas).
Let $\Phi$ be an interpretation basis. We define the interpretation
$I_{\Phi}(F)$ along $\Phi$ of a formula $F$ inductively:
* •
If $F=X_{i}(j)$, then $I_{\Phi}(F)$ is the delocation (i.e. a behavior) of
$\Phi(X_{i})$ along the function $x\mapsto 2^{i}(2j+1)+x$;
* •
If $F=(X_{i}(j))^{\simbot}$, we define the behavior
$I_{\Phi}(F)=(I_{\Phi}(X_{i}(j)))^{\simbot}$;
* •
If $F=\mathbf{T}_{\sharp\Gamma}$ (resp. $F=\mathbf{0}_{\sharp\Gamma}$), we
define $I_{\Phi}(F)$ as the behavior $\mathbf{T}_{\sharp\Gamma}$ (resp.
$\mathbf{0}_{\sharp\Gamma}$);
* •
If $F=\mathbf{1}$ (resp. $F=\mathbf{\bot}$), we define $I_{\Phi}(F)$ as the
behavior $\mathbf{1}$ (resp. $\mathbf{\bot}$);
* •
If $F=A\otimes B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\otimes
I_{\Phi}(B)$;
* •
If $F=A\parr B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\parr
I_{\Phi}(B)$;
* •
If $F=A\oplus B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\oplus
I_{\Phi}(B)$;
* •
If $F=A\with B$, we define the conduct $I_{\Phi}(F)=I_{\Phi}(A)\with
I_{\Phi}(B)$;
* •
If $F=\forall X_{i}A(X_{i})$, we define the conduct
$I_{\Phi}(F)=\mathbf{\forall X_{i}}I_{\Phi}(A(X_{i}))$;
* •
If $F=\exists X_{i}A(X_{i})$, we define the conduct
$I_{\Phi}(F)=\mathbf{\exists X_{i}}I_{\Phi}(A(X_{i}))$.
* •
If $F=\oc A$ (resp. $\wn A$), we define the conduct $I_{\Phi}(F)=\oc
I_{\Phi}(A)$ (resp. $\wn I_{\Phi}(A)$).
Moreover a sequent $\Delta\vdash\Gamma;\Theta$ will be interpreted as the
$\parr$ of the formulas in $\Gamma$ and $\Theta$ and the negations of formulas
in $\Delta$, which we will write
$\bigparr\Delta^{\simbot}\parr\bigparr\Gamma\parr\bigparr\Theta$. We will also
represent this formula by the equivalent formula
$\bigotimes\Delta\multimap(\bigparr\Gamma\parr\bigparr\Theta)$.
###### Definition (Interpretation of
$\textnormal{locELL}_{\textnormal{pol}}$ proofs).
Let $\Phi$ be an interpretation basis. We define the interpretation
$I_{\Phi}(\pi)$ — a project — of a proof $\pi$ inductively:
* •
if $\pi$ consists in an axiom rule introducing
$\vdash(X_{i}(j))^{\simbot},X_{i}(j^{\prime})$, we define $I_{\Phi}(\pi)$ as
the project $\mathfrak{Fax}$ defined by the translation $x\mapsto
2^{i}(2j^{\prime}-2j)+x$;
* •
if $\pi$ consists solely in a $\mathbf{T}_{\sharp\Gamma}$ rule, we define
$I_{\Phi}(\pi)=\mathfrak{0}_{\sharp\Gamma}$;
* •
if $\pi$ consists solely in a $\mathbf{1}_{d}$ rule, we define
$I_{\Phi}(\pi)=\mathfrak{0}_{\emptyset}$;
* •
if $\pi$ is obtained from $\pi^{\prime}$ by a $\parr$ rule, a
$\otimes_{g}^{pol}$ rule, a $\parr_{d}^{pol}$ rule, a $\parr^{mix}$ rule, or a
$\mathbf{1}_{g}$ rule, then $I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})$;
* •
if $\pi$ is obtained from $\pi_{1}$ and $\pi_{2}$ by applying a $\otimes$
rule, a $\otimes^{pol}_{d}$ rule, a $\parr_{g}^{pol}$ rule or a
$\otimes^{mix}$ rule, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi_{1})\otimes
I_{\Phi}(\pi^{\prime})$;
* •
if $\pi$ is obtained from $\pi^{\prime}$ by a weak rule or a $\oplus_{i}$ rule
introducing a formula of location $V$, we define
$I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})\otimes\mathfrak{0}_{V}$;
* •
if $\pi$ of conclusion $\vdash\Gamma,A_{0}\with A_{1}$ is obtained from
$\pi_{0}$ and $\pi_{1}$ by applying a $\with$ rule, we define the
interpretation of $\pi$ as it was done in our earlier paper [Sei14a]: ;
* •
If $\pi$ is obtained from a $\forall$ rule applied to a derivation
$\pi^{\prime}$, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})$;
* •
If $\pi$ is obtained from a $\exists$ rule applied to a derivation
$\pi^{\prime}$ replacing the formula $\mathbf{A}$ by the variable name
$X_{i}$, we define
$I_{\Phi}(\pi)=I_{\Phi}(\pi^{\prime})\mathop{\mathopen{:}\mathclose{:}}(\bigotimes[e^{-1}(j)\leftrightarrow
X_{i}(j)])$, using the notations of our previous paper [Sei14c] for the
_measure-inflating faxes_ $[e^{-1}(j)\leftrightarrow X_{i}(j)]$ where $e$ is
an enumeration of the occurrences of $\mathbf{A}$ in $\pi^{\prime}$;
* •
if $\pi$ is obtained from $\pi^{\prime}$ by applying a promotion rule $\oc$ or
$\oc^{pol}$, we apply the implementation of the functorial promotion rule to
the project $\oc I_{\Phi}(\pi^{\prime})$ $n-1$ times, where $n$ is the number
of formulas in the sequent;
* •
if $\pi$ is obtained from $\pi$ by applying a contraction rule $ctr$, we
define the interpretation of $\pi$ as the execution between the interpretation
of $\pi^{\prime}$ and the project implementing contraction described in
Section 5.1;
* •
if $\pi$ is obtained from $\pi_{1}$ and $\pi_{2}$ by applying a $cut$ rule or
a $cut^{pol}$ rule, we define $I_{\Phi}(\pi)=I_{\Phi}(\pi_{1})\pitchfork
I_{\Phi}(\pi_{2})$.
Once again, one can chose an enumeration $e$ of the occurrences of variables
in order to ”localize” any formula $A$ and any proof $\pi$ of
$\textnormal{ELL}_{\textnormal{pol}}$: and define formulas $A^{e}$ and proofs
$\pi^{e}$ of $\textnormal{locELL}_{\textnormal{pol}}$. One easily shows a
soundness result for the localized calculus
$\textnormal{locELL}_{\textnormal{pol}}$ which implies the following result.
###### Theorem .
Let $\Phi$ be an interpretation basis, $\pi$ a proof of
$\textnormal{ELL}_{\textnormal{pol}}$ of conclusion
$\Delta\Vvdash\Gamma;\Theta$, and$e$ an enumeration of the occurrences of
variables in the axioms of $\pi$. Then $I_{\Phi}(\pi^{e})$ is a successful
project in $I_{\Phi}(\Delta^{e}\vdash\Gamma^{e};\Theta^{e})$.
## 7 Conclusion and Perspectives
In this paper, we extended the setting of Interaction Graphs in order to deal
with all connectives of linear logic. We showed how one can obtain a soundness
result for two versions of Elementary Linear Logic. The first system, which is
conceived so that the interpretation of sequents are behaviors, seems to lack
expressivity and it may appear that elementary functions cannot be typed in
this system. The second system, however, is very closed to usual ELL sequent
calculus, and, even though one should prove it, the proofs of type
$\oc\tt{nat}\multimap\tt{nat}$ to itself seem to correspond to elementary
functions from natural numbers to natural numbers, as it is the case with
traditional Elementary Linear Logic [DJ03].
Though the generalization from graphs to graphings may seem a big effort, we
believe the resulting framework to be extremely interesting. We should stress
that with little work on the definition of exponentials, one should be able to
show that interpretations of proofs can be described by finite means. Indeed,
the only operation that seems to turn an interval into an infinite number of
intervals is the promotion rule. One should however be able to show that, up
to a suitable delocation, the promotion of a project defined on a finite
number of rational intervals is defined on a finite number of rational
intervals.
Another interesting perspective would consist in considering continuous
dialects in addition to discrete ones. All the definitions and properties of
thick and sliced graphings obviously hold in this setting and one can obtain
all the results described in this paper, although no finite description of
projects could be expected in this case. The question of wether we would gain
some expressivity by extending the framework in this way is still open. We
believe that it may be a way to obtain more expressive exponentials, such as
the usual exponentials of linear logic.
More generally, now that this framework has been defined and that we have
shown its interest by providing a construction for _elementary exponentials_ ,
we believe the definition and study of other exponential connectives may be a
work of great interest. First, these new exponentials would co-exist with each
other, making it possible to study their interactions. Secondly, even if the
definition of exponentials for full linear logic may be a complicated task,
the definition of low-complexity exponentials may be of great interest.
Finally, we explained in our previous paper how the systematic construction of
models of linear logic based on graphings [Sei14c] give rise to a hierarchy of
models mirroring subtle distinctions concerning computational principles. In
particular, it gives rise to a hierarchy of models characterizing complexity
classes [Sei14b] by adapting results obtained using operator theory [AS12,
AS13]. The present work will lead to characterizations of larger complexity
classes such as Ptime or Exptime predicates and/or functions, following the
work of Baillot [Bai11].
## References
* [AS12] Clément Aubert and Thomas Seiller. Characterizing co-nl by a group action. CoRR, abs/1209.3422, 2012.
* [AS13] Clément Aubert and Thomas Seiller. Logarithmic space and permutations. CoRR, abs/1301.3189, 2013.
* [Bai11] Patrick Baillot. Elementary linear logic revisited for polynomial time and an exponential time hierarchy. In Hongseok Yang, editor, APLAS, volume 7078 of Lecture Notes in Computer Science, pages 337–352. Springer, 2011.
* [DJ03] Vincent Danos and Jean-Baptiste Joinet. Linear logic & elementary time. Information and Computation, 183(1):123–137, 2003.
* [FK52] Bent Fuglede and Richard V. Kadison. Determinant theory in finite factors. Annals of Mathematics, 56(2), 1952.
* [Gir87] Jean-Yves Girard. Multiplicatives. In Lolli, editor, Logic and Computer Science : New Trends and Applications, pages 11–34, Torino, 1987. Università di Torino. Rendiconti del seminario matematico dell’università e politecnico di Torino, special issue 1987.
* [Gir88] Jean-Yves Girard. Geometry of interaction II: Deadlock-free algorithms. In Proc. of COLOG’ 1988, LNCS 417, pages 76–93. Springer, 1988\.
* [Gir89a] Jean-Yves Girard. Geometry of interaction I: Interpretation of system F. In In Proc. Logic Colloquium 88, 1989.
* [Gir89b] Jean-Yves Girard. Towards a geometry of interaction. In Proceedings of the AMS Conference on Categories, Logic and Computer Science, 1989.
* [Gir95a] Jean-Yves Girard. Geometry of interaction III: Accommodating the additives. In Advances in Linear Logic, number 222 in Lecture Notes Series, pages 329–389. Cambridge University Press, 1995.
* [Gir95b] Jean-Yves Girard. Light linear logic. In Selected Papers from the International Workshop on Logical and Computational Complexity, LCC ’94, pages 145–176, London, UK, UK, 1995. Springer-Verlag.
* [Gir11] Jean-Yves Girard. Geometry of interaction V: Logic in the hyperfinite factor. Theoretical Computer Science, 412:1860–1883, 2011.
* [Kri01] Jean-Louis Krivine. Typed lambda-calculus in classical zermelo-fraenkel set theory. Archive for Mathematical Logic, 40(3):189–205, 2001.
* [Kri09] Jean-Louis Krivine. Realizability in classical logic. Panoramas et synthèses, 27:197–229, 2009.
* [Sei12a] Thomas Seiller. Interaction graphs: Multiplicatives. Annals of Pure and Applied Logic, 163:1808–1837, December 2012\.
* [Sei12b] Thomas Seiller. Logique dans le facteur hyperfini : géometrie de l’interaction et complexité. PhD thesis, Université de la Méditerranée, 2012.
* [Sei14a] Thomas Seiller. Interaction graphs: Additives. Accepted for publication in Annals of Pure and Applied Logic, 2014\.
* [Sei14b] Thomas Seiller. Interaction graphs and complexity. Extended Abstract, 2014.
* [Sei14c] Thomas Seiller. Interaction graphs: Graphings. CoRR, abs/1405.6331, 2014.
|
arxiv-papers
| 2013-12-04T10:13:24 |
2024-09-04T02:49:54.787409
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Thomas Seiller",
"submitter": "Christoph Rauch",
"url": "https://arxiv.org/abs/1312.1094"
}
|
1312.1116
|
# Coupling of (ultra-) relativistic atomic nuclei with photons
M. Apostol1,a and M. Ganciu2,b
###### Abstract
The coupling of photons with (ultra-) relativistic atomic nuclei is presented
in two particular circumstances: very high electromagnetic fields and very
short photon pulses. We consider a typical situation where the (bare) nuclei
(fully stripped of electrons) are accelerated to energies $\simeq 1TeV$ per
nucleon (according to the state of the art at LHC, for instance) and photon
sources like petawatt lasers $\simeq 1eV$-radiation (envisaged by ELI-NP
project, for instance), or free-electron laser $\simeq 10keV$-radiation, or
synchrotron sources, etc. In these circumstances the nuclear scale energy can
be attained, with very high field intensities. In particular, we analyze the
nuclear transitions induced by the radiation, including both one- and two-
photon proceses, as well as the polarization-driven transitions which may lead
to giant dipole resonances. The nuclear (electrical) polarization concept is
introduced. It is shown that the perturbation theory for photo-nuclear
reactions is applicable, although the field intensity is high, since the
corresponding interaction energy is low and the interaction time (pulse
duration) is short. It is also shown that the description of the giant nuclear
dipole resonance requires the dynamics of the nuclear electrical polarization
degrees of freedom.
PACS: 52.38.-r; 41.75.Jv; 52.27.Ny; 24.30.Cz; 25.20.-x; 25.75.Ag; 25.30.Rw
_Key words:__relativistic heavy ions; high-intensity laser radiation; photo-
nuclear reactions; giant nuclear dipole resonance_
__
_a_ Electronic mail: [email protected]
bElectronic mail: [email protected]
1Institute of Atomic Physics, Institute for Physics and Nuclear Engineering,
Magurele-Bucharest 077125, MG-6, POBox MG-35, Romania,2National Institute for
Lasers, Plasma and Radiation Physics, Magurele-Bucharest 077125, POBox MG-36,
Romania
## 1 Introduction. Accelerated ions
It is well known that the nuclear photoreactions occurr in the $keV-
MeV$-energy range. In particular, the characteristic energy of the giant
dipole resonance (which implies oscillations of protons with respect to
neutrons) is $10-20MeV$.1-4 In order to get this energy scale typical
mechanisms are used, like Compton backscattering (for instance a laser-
electron system), or electron bremsstrahlung (usually with the same nucleus
acting both as converter and target), etc.5-18 High intensity laser pulses can
be used for accelerating electrons in compact laser-plasma configurations.3,17
High-power and short-pulsed lasers are pursued nowadays for increasing the
intensity of the electromagnetic field.19 Photon-ion or photon-photon mediated
ion-ion interactions are also well known in the so-called peripheral
reactions.20,21 Vacuum polarization effects have also been discussed recently
in high-energy photon-proton collisions,22 or light-by-light scattering in
multi-photon Compton effect.23-25 We describe here a high-energy and high-
field intensity coupling of the atomic nucleus to photons from various sources
(_e.g._ , optical laser, free electron laser, synchrotron radiation) by using
(ultra-) relativistic atomic nuclei.
We consider (ultra-) relativistically acelerated ions moving with velocity $v$
along the $x$-axis. We envisage acceleration energies of the order
$\varepsilon=1TeV$ per nucleon (according to the state of the art at LHC, for
instance).26 At these energies the ion is fully stripped of its electrons, so
we have a bare atomic nucleus. We assume that a beam of photons of frequency
$\omega_{0}$ is propagating counterwise (from a laser source, or a free
electron laser, or a synchrotron source, etc), such that the photons suffer a
head-on collision with the nucleus. The moving nucleus will "see" a photon
frequency
$\omega=\omega_{0}\sqrt{\frac{1+\beta}{1-\beta}}\,\,,\,\,\beta=v/c$ (1)
in its rest frame, according to the Doppler effect. For (ultra-) relativistic
nuclei ($\beta\simeq 1$) this frequency may acquire high values. For instance,
we have
$\beta\simeq
1-\frac{\varepsilon_{0}^{2}}{2\varepsilon^{2}}\,\,,\,\,\omega\simeq
2\omega_{0}\frac{\varepsilon}{\varepsilon_{0}}\,\,\,,$ (2)
where $\varepsilon_{0}\simeq 1GeV$ is the nucleon rest energy; for
$\varepsilon=1TeV$ we get a photon frequency $\omega\simeq 2\times
10^{3}\omega_{0}$
($\gamma=(1-\beta^{2})^{-1/2}\simeq\varepsilon/\varepsilon_{0}=10^{3}$). We
can see that for a $1eV$-laser we get $2keV$-photons in the rest frame of the
accelerated nucleus; for a $10keV$-free electron laser we get $20MeV$-photons,
etc. The effect is tunable by varying the energy of the accelerated ions. This
idea has been discussed in relation to hydrogen-like accelerated heavy ions,
which may scatter resonantly $X$\- or gamma-rays photons.27 Similarly, a
frequency up-shift was discussed for photons reflected by a relativistically
flying plasma mirror generated by the laser-driven plasma wakefield,28 or
photons in the rest frame of an ultra-relativistic electron beam.24,29
For a typical laser radiation (see, for instance, ELI-NP project,30) we take a
photon energy $\hbar\omega_{0}=1eV$ (wavelength $\lambda\simeq 1\mu m$), an
energy $\mathcal{E}=50J$ and a pulse duration $\tau=50fs$. The pulse length is
$l=15\mu m$ (cca $15$ wavelengths), the power is $P=10^{15}w$ ($1$ pettawatt).
For a $d^{2}=(15\mu m)^{2}$-pulse cross-sectional area the intensity is
$I=P/d^{2}=4\times 10^{20}w/cm^{2}$. The electric field is $E\simeq
10^{9}statvolt/cm$ ($1statvolt/cm=3\times 10^{4}V/m$) and the magnetic field
is $H=10^{9}Gs$ ($1Ts=10^{4}Gs$). These are very high fields (higher than
atomic fields). The (ultra-) relativistic ion will see a shortened pulse of
length $l^{{}^{\prime}}=\sqrt{1-\beta^{2}}l$, with a shortened duration
$\tau^{{}^{\prime}}=\sqrt{1-\beta^{2}}\tau$ and an energy
$\mathcal{E}^{{}^{\prime}}=\mathcal{E}\sqrt{(1+\beta/(1-\beta)}$ (the number
of photons $N_{ph}\simeq 10^{20}$ is invariant). It follows that the power and
intensity are increased by the factor $(1-\beta)^{-1}$ ($\simeq 2\gamma^{2}$)
and the fields are increased by the factor $(1-\beta)^{-1/2}$; for instance,
$E^{{}^{\prime}}=E/\sqrt{1-\beta}=\sqrt{2}(\varepsilon/\varepsilon_{0})E\simeq
10^{12}statvolt/cm$; this figure is two orders of magnitude below Schwinger
limit.
A higher enhancement can be obtained by taking into account the aberration of
light, even from a collimated laser.31-33 Indeed, for a cross-sectional beam
area $D^{2}=(0.5mm)^{2}$ we get an intensity $I=P/D^{2}=4\times
10^{17}w/cm^{2}$ and an electric field $E\simeq 5\times 10^{7}statvolt/cm$
(all the other parameters being the same). In the rest frame of the ion the
power increases by a factor $(1-\beta)^{-1}$, as before, but the cross-
sectional area $D^{{}^{\prime}2}$ of the beam, decreases by a factor
$(1-\beta)/(1+\beta)$ ($\simeq 1/4\gamma^{2}$), as a consequence of the
"forward beaming" (aberration of light);28 we have
$D^{{}^{\prime}2}=D^{2}(1-\beta)/(1+\beta)$, which leads to an enhancement
factor $(1+\beta)/(1-\beta)^{2}$ for intensity and a factor
$(1+\beta)^{1/2}/(1-\beta)$ ($\simeq 2\sqrt{2}\gamma^{2}$) for field. We get,
for instance, $I^{{}^{\prime}}\simeq 3\times 10^{24}w/cm^{2}$ and an electric
field $E^{{}^{\prime}}\simeq 2\sqrt{2}\gamma^{2}E\simeq 10^{14}statvolt/cm$.
Similarly, we can take as typical parameters for a free electron laser the
photon energy $\hbar\omega_{0}=10keV$, the pulse duration $\tau=50fs$ and a
much lower energy $\mathcal{E}=5\times 10^{-5}J$ (power $P=10Gw$); the fields
may decrease by $3$ orders of magnitude, but still they are very high
$(10^{9}-10^{11}statvolt/cm$) in the rest frame of the accelerated ion.
Under these circumstances, the photons can attain energies sufficiently high
for photonuclear reactions, or giant dipole resonances, with additional
features arising from the electron-positron pair creation, vacuum
polarization, etc; indeed, above $\simeq 1MeV$ the pair creation in the
Coulomb field of the atomic nucleus becomes important. Vacuum polarization
effects at very high intensity fields and high field frequency are still
insufficiently explored. Beside, all these happen in two particular
cirumstances: very short times and very high electromagnetic fields. We
discuss here the effect of these particular circumstances on typical phenomena
related to photon-nucleus interaction.
## 2 Nuclear transitions
Let us cosider an ensemble of interacting particles, some of them with
electric charge, like protons and neutrons in the atomic nucleus, subjected to
an external radiation field. We envisage quantum processes driven by field
energy quantum of the order $\hbar\Omega=10MeV$, as discussed above. First, we
note that the motion of the particles at this energy is non-relativistic,
since the particle rest energy $\simeq 1GeV$ is much higher than the energy
quantum (we can check that the acceleration $qE/m$ is much smaller than the
"relativistic acceleration" $c\Omega$, where $q$ and $m$ is the particle
charge and, respectively, mass and $E$ denotes he electric field).
Consequently, we start with the classical lagrangian
$L=mv^{2}/2-V+q\mathbf{vA}/c-q\Phi$ of a particle with mass $m$ and charge
$q$, moving in the potential $V$ and subjected to the action of an
electromagnetic field with potentials $\Phi$ and $\mathbf{A}$; $\mathbf{v}$ is
the particle velocity. We get immediately the momentum
$\mathbf{p}=m\mathbf{v}+q\mathbf{A}/c$ and the hamiltonian
$H=\frac{1}{2m}p^{2}+V-\frac{q}{mc}\mathbf{pA}+\frac{q^{2}}{2mc^{2}}A^{2}+q\Phi\,\,.$
(3)
Usually, the particle hamiltonian $p^{2}/2m+V$ is separated and quantized ($V$
may be viewed as the mean-field potential of the nucleus), and the remaining
terms are treated as a perturbation. In the first order of the perturbation
theory we limit ourselves to the external radiation field, which is considered
sufficiently weak. Consequently, we put $\mathbf{A}=\mathbf{A}_{0}$ and
$\Phi=0$ in equation (3) and take approximately $\mathbf{p}\simeq
m\mathbf{v}$. We get the well known interaction hamiltonian
$H_{1}=-\frac{q}{c}\mathbf{v}\mathbf{A}_{0}=-\frac{1}{c}\mathbf{JA}_{0}\,\,\,,$
(4)
where $\mathbf{J}=q\mathbf{v}$ is the current; in the non-relativistic limit
we include also the spin currents in $\mathbf{J}$. If we leave aside the spin
currents, the interaction hamiltonian given by equation (4) can also be
written as $q\mathbf{r}(d\mathbf{A}_{0}/dt)/c$. Usually, the field does not
depend on position over the spatial extension of the ensemble of particles.
Indeed, in the present case the wavelength of the quantum $\hbar\Omega=10MeV$
is $\lambda\simeq 10^{-12}cm$, which is larger than the nucleus dimension
$\simeq 10^{-13}cm$; therefore we may neglect the spatial variation of the
field and write the interaction hamiltonian as
$H_{1}=\frac{q}{c}\mathbf{r}\frac{d\mathbf{A}_{0}}{dt}=\frac{q}{c}\mathbf{r}\frac{\partial\mathbf{A}_{0}}{\partial
t}=-q\mathbf{r}\mathbf{E}_{0}=-\mathbf{d}\mathbf{E}_{0}\,\,\,,$ (5)
where $\mathbf{d}=q\mathbf{r}$ is the dipole moment. This is the well-known
dipole approximation. For an ensemble of $N$ particles we write the
interaction hamiltonian given by equation (4) as
$H_{1}=-\frac{1}{c}\sum_{i}\mathbf{J}_{i}\mathbf{A}_{0}$ (6)
(within the dipole approximation) and its matrix elements between two states
$a$ and $b$ are given by
$\begin{array}[]{c}H_{1}(a,b)=-\frac{1}{c}\mathbf{J}(a,b)\mathbf{A}_{0}=\\\
\\\ =-\frac{1}{c}[\sum_{i}\int
d\mathbf{r}_{1}...d\mathbf{r}_{i}...d\mathbf{r}_{N}\psi_{a}^{*}(\mathbf{r}_{1}..\mathbf{r}_{i}..\mathbf{r}_{N})\mathbf{J}_{i}\psi_{b}(\mathbf{r}_{1}..\mathbf{r}_{i}..\mathbf{r}_{N})]\mathbf{A}_{0}\,\,\,,\end{array}$
(7)
where $\psi_{a,b}$ are the wavefunctions of the two states $a$ and $b$; the
notation $\mathbf{r}_{i}$ in equation (7) includes also the spin variable. As
it is well known, the transition amplitude is given by
$c_{ab}=-\frac{i}{\hbar}\int dtH_{1}(a,b)e^{i\omega_{ab}t}\,\,\,,$ (8)
where $\omega_{ab}=(E_{a}-E_{b})/\hbar$ is the frequency associated to the
transition between the two states $a$ and $b$ with energies $E_{a}$ and,
respectively, $E_{b}.$ We take
$\mathbf{A}_{0}(t)=\mathbf{A}_{0}e^{-i\Omega t}+\mathbf{A}_{0}^{*}e^{i\Omega
t}$ (9)
(with $\Omega>0$) and note that the pulse duration
$\tau^{{}^{\prime}}=\sqrt{1-\beta^{2}}\tau\simeq 5\times 10^{-17}s$ is much
longer than the transition time $1/\Omega\simeq 10^{-22}s$; we can extend the
integration in equation (8) to infinity and get
$c_{ab}=\frac{2\pi i}{\hbar
c}\mathbf{J}(a,b)\mathbf{A}_{0}\delta(\omega_{ab}-\Omega)\,\,;$ (10)
making use of $\delta(\omega=0)=t/2\pi$, we get the number of transitions per
unit time
$P_{ab}=\left|c_{ab}\right|^{2}/t=2\pi\left|\frac{\mathbf{J}(a,b)\mathbf{A}_{0}}{\hbar
c}\right|^{2}\delta(\omega_{ab}-\Omega)\,\,.$ (11)
This is a standard calculation. Usually, the field and the wavefunctions of
the atomic nuclei are decomposed in electric and magnetic multiplets, and the
selection rules of conservation of the parity and the angular momentum are
made explicit (see, for instance,34). It relates to the absorption (emission)
of one photon.
It is worth estimating the number of transitions per unit time as given by
equation (11). First, we may approximate $J(a,b)$ by $qv$. For an energy
$\hbar\Omega=10MeV$ and a rest energy $1GeV$ we have $v/c=10^{-1}$. Next, from
$\mathbf{E}_{0}=(-1/c)\partial\mathbf{A}_{0}/\partial t$ we deduce
$A_{0}\simeq 10^{-3}statvolt$ (for $E_{0}=10^{9}statvolt/cm$ and
$\Omega=10^{22}s^{-1}$); it follows that the particle energy in this field is
$qA_{0}\simeq 1eV$ (which is a very small energy). We get from equation (11)
$P_{ab}\simeq(10^{28}/\Delta\Omega)s^{-1}$, where $\Delta\Omega\simeq
1/\tau^{{}^{\prime}}\simeq 10^{16}s^{-1}$ is the uncertainty in the pulse
frequency, such that the number of transitions per unit time is $P_{ab}\simeq
10^{12}s^{-1}$(much smaller than $\Omega=10^{22}s^{-1}$). We can see that,
under these circumstances, the first-order calculations of the perturbation
theory are justified.
For higher fields we should include the second-order terms in the interaction
hamiltonian given by equation (3); this second-order interaction hamiltonian
reads
$H_{2}=-\frac{q^{2}}{2mc^{2}}\mathbf{A}_{0}^{2}\,\,.$ (12)
We can see that within the dipole approximation this interaction does not
contribute to the transition amplitude, since the field does not depend on
position and the wavefunctions are orthogonal. For field wavelengths shorter
than the dimension of the ensemble of particles (_i.e.,_ beyond the dipole
approximation) we write
$\mathbf{A}_{0}(\mathbf{r},t)=\mathbf{A}_{0}e^{-i\Omega
t+i\mathbf{kr}}+\mathbf{A}_{0}^{*}e^{i\Omega t-i\mathbf{kr}}\,\,\,,$ (13)
where $\mathbf{k}=\Omega/c$ is the wavevector, and get
$H_{2}(a,b)=-\frac{q^{2}}{2mc^{2}}\left[A_{0}^{2}(a,b)e^{-2i\Omega
t}+A_{0}^{*2}(b,a)e^{2i\Omega t}\right]\,\,\,,$ (14)
where
$A_{0}^{2}(a,b)=[\sum_{i}\int
d\mathbf{r}_{1}...d\mathbf{r}_{i}...d\mathbf{r}_{N}\psi_{a}^{*}(\mathbf{r}_{1}..\mathbf{r}_{i}..\mathbf{r}_{N})e^{2i\mathbf{k}\mathbf{r}_{i}}\psi_{b}(\mathbf{r}_{1}..\mathbf{r}_{i}..\mathbf{r}_{N})]A_{0}^{2}\,\,.$
(15)
This interaction gives rise to two-photon pocesses, with the transition
amplitude
$c_{ab}=\frac{2\pi
i}{\hbar}\frac{q^{2}}{2mc^{2}}A_{0}^{2}(a,b)\delta(\omega_{ab}-2\Omega)\,\,.$
(16)
Comparing the transition amplitudes produced by the interaction hamiltonians
$H_{1}$ (equation (10)) and $H_{2}$ (equation (16)) we may get an approximate
criterion: $qA_{0}/mc^{2}$ (two-photons) compared with $v/c$ (one photon).
Since $v/c\simeq 10^{-1}$ (as estimated above), we should have
$qA_{0}>10^{-1}\times 1GeV=100MeV$ in order to get a relevant contribution
from two-photon processes. As estimated above, $qA_{0}\simeq 1eV$, so we can
see that the second-order interaction hamiltonian and the two-photon processes
bring a very small contribution to the transition amplitudes.
## 3 Giant dipole resonance
There is another process of excitation of the ensemble of particles described
by the hamiltonian given by equation (3). Indeed, let us write the interaction
hamiltonian
$H_{int}=-\frac{q}{mc}\mathbf{pA}+\frac{q^{2}}{2mc^{2}}A^{2}+q\Phi\,\,\,,$
(17)
or
$H_{int}=-\frac{q}{c}\mathbf{vA}-\frac{q^{2}}{2mc^{2}}A^{2}+q\Phi\,\,.$ (18)
Under the action of the electromagnetic field the mobile charges (_e.g.,_
protons in atomic nucleus) acquire a displacement $\mathbf{u}$, which, in
general, is a function $\mathbf{u}(\mathbf{r},t)$ of position and time. This
is a collective motion associated with the particle-density degrees of
freedom; in the limit of long wavelengths (_i.e._ for $\mathbf{u}$ independent
of position) it is the motion of the center of mass of the charges. Therefore,
an additional velocity $\dot{\mathbf{u}}$ should be included in equation (18).
It is easy to see that this $\mathbf{u}$-motion implies a variation
$\rho_{p}=-nqdiv\mathbf{u}$ of the (volume) charge density and a current
density $\mathbf{j}_{p}=nq\dot{\mathbf{u}}$, where $n$ is the concentration of
mobile charges. Obviously, these are polarization charge and current densities
(the suffix $p$ comes from "polarization"). The charge and current densities
$\rho_{p}$ and $\mathbf{j}_{p}$ give rise to an internal, polarization
electromagnetic field, with the potentials $\mathbf{A}_{p}$ and $\Phi_{p}$
(related through the Lorenz gauge
$div\mathbf{A}_{p}+(1/c)\partial\Phi_{p}/\partial t=0$), which should be added
to the potential of the external field in equation (18). Indeed, the
retardation time $t_{r}=a/c\simeq 10^{-23}s$, where $a\simeq 10^{-13}cm$ is
the dimension of the atomic nucleus, is shorter than the excitation time
$\Omega^{-1}=10^{-22}s$, so the atomic nucleus gets polarized. In particular
the scalar potential $\Phi$ in equation (18) is the polarization scalar
potential $\Phi_{p}$. We get
$H_{int}=H_{1}-\frac{1}{c}\mathbf{J}\mathbf{A}_{p}-\frac{q}{c}\dot{\mathbf{u}}(\mathbf{A}_{0}+\mathbf{A}_{p})-\frac{q^{2}}{2mc^{2}}(\mathbf{A}_{0}+\mathbf{A}_{p})^{2}+q\Phi_{p}\,\,\,,$
(19)
where $H_{1}$ is given by equation (4). Within the dipole approximation we may
take $\mathbf{u}$ independent of position, except for the surface of the
particle ensemble, where the density falls abruptly to zero. A similar
behaviour extends to the vector and scalar polarization potentials (inside the
ensemble); in addition, through the Lorenz gauge, the scalar potential
$\Phi_{p}$ can be taken independent of time within this approximation. The
surface effects can be neglected as regards the scalar product of two
orthogonal wavefunctions. All these simplifications amount to neglecting all
the terms in equation (19) except the first two; therefore, we are left with
$H_{int}\simeq
H_{1}+H_{1p}\,\,,\,\,H_{1p}=-\frac{1}{c}\mathbf{J}\mathbf{A}_{p}\,\,;$ (20)
in order to get $\mathbf{A}_{p}$ we need a dynamics for the displacement field
$\mathbf{u}$.
We can construct a dynamics for the displacement field $\mathbf{u}$ by
assuming that it is subjected to internal forces of elastic type,
characterized by frequency $\omega_{c}$; the (non-relativistic) equation of
motion is given by
$m\ddot{\mathbf{u}}=q(\mathbf{E}_{0}+\mathbf{E}_{p})-m\omega_{c}^{2}\mathbf{u}\,\,\,,$
(21)
where $\mathbf{E}_{0}=-(1/c)\partial\mathbf{A}_{0}/\partial t$ is the external
electric field and $\mathbf{E}_{p}$ is the polarization electric field. Within
the dipole approximation, Gauss’s equation
$div\mathbf{E}_{p}=4\pi\rho_{p}=-4\pi nqdiv\mathbf{u}$ gives
$\mathbf{E}_{p}=-4\pi nq\mathbf{u}$ for matter of infinite extension
(polarization $\mathbf{P}=nq\mathbf{u}$). For polarizable bodies of finite
size there appears a (de-) polarizing factor $f$ within the same dipole
approximation, as a consequence of surface charges (for instance, $f=1/3$ for
a sphere). Therefore, we can write equation (21) as
$\ddot{\mathbf{u}}+(\omega_{c}^{2}+f\omega_{p}^{2})\mathbf{u}=\frac{q}{m}\mathbf{E}_{0}\,\,\,,$
(22)
where $\omega_{p}=\sqrt{4\pi nq^{2}/m}$ is the plasma frequency. For nucleons
we can estimate $\hbar\omega_{p}\simeq Z^{1/2}MeV$, where $Z$ is the atomic
number. An estimation for the characteristic frequency $\omega_{c}$ can be
obtained from $m\omega_{c}^{2}d^{2}/2=\mathcal{E}_{c}(d/a)$, where $d$ is the
displacement amplitude, $a$ is the dimension of the nucleus and
$\mathcal{E}_{c}$ ($\simeq 7-8MeV$) is the mean cohesion energy per nucleon;
the maximum value of $d$ is the mean inter-particle separation distance
$d=a/A^{1/3}$, where $A$ is the mass number. We get $\hbar\omega_{c}\simeq
10A^{1/6}MeV$. It is convenient to introduce the frequency
$\Omega_{0}=(\omega_{c}^{2}+f\omega_{p}^{2})^{1/2}$, which, as we can see from
the preceding estimations, is of the order of $10MeV$, and write the equation
of motion (22) as
$\ddot{\mathbf{u}}+\Omega_{0}^{2}\mathbf{u}=\frac{q}{m}\mathbf{E}_{0}\,\,.$
(23)
This is the equation of motion of a linear harmonic oscillator under the
action of an external force $q\mathbf{E}_{0}$. Making use of equation (9), we
get the external field
$\mathbf{E}_{0}=\frac{i\Omega}{c}\mathbf{A}_{0}e^{-i\Omega
t}-\frac{i\Omega}{c}\mathbf{A}_{0}^{*}e^{i\Omega t}\,\,;$ (24)
for frequency $\Omega$ approaching the oscillator frequency $\Omega_{0}$ the
motion described by equation (23) is a classical motion, and we get
$\mathbf{u}=-\frac{iq\Omega}{mc}\cdot\frac{1}{\Omega^{2}-\Omega_{0}^{2}}\left(\mathbf{A}_{0}e^{-i\Omega
t}-\mathbf{A}_{0}^{*}e^{i\Omega t}\right)\,\,.$ (25)
According to the discussion made above, the polarization field is
$\mathbf{E}_{p}=-4\pi
fnq\mathbf{u}=\frac{if\omega_{p}^{2}\Omega}{c}\cdot\frac{1}{\Omega^{2}-\Omega_{0}^{2}}\left(\mathbf{A}_{0}e^{-i\Omega
t}-\mathbf{A}_{0}^{*}e^{i\Omega t}\right)$ (26)
and the corresponding vector potential is
$\mathbf{A}_{p}=\frac{f\omega_{p}^{2}}{\Omega^{2}-\Omega_{0}^{2}}\left(\mathbf{A}_{0}e^{-i\Omega
t}+\mathbf{A}_{0}^{*}e^{i\Omega t}\right)\,\,.$ (27)
A damping factor $\Gamma$ can be included in equation (23),
$\ddot{\mathbf{u}}+\Omega_{0}^{2}\mathbf{u}+\Gamma\dot{\mathbf{u}}=\frac{q}{m}\mathbf{E}_{0}\,\,\,,$
(28)
and we can write the solution as
$\mathbf{u}=-\frac{q}{m}\mathbf{E}_{0}\frac{1}{\Omega^{2}-\Omega_{0}^{2}+i\Omega\Gamma}e^{-i\Omega
t}+c.c.\,\,;$ (29)
the polarization reads
$\mathbf{P}=nqf\mathbf{u}=-\frac{f\omega_{p}^{2}}{4\pi}\frac{1}{\Omega^{2}-\Omega_{0}^{2}+i\Omega\Gamma}\mathbf{E}_{0}e^{-i\Omega
t}+c.c.\,\,\,,$ (30)
so that we can define the polarizability
$\alpha=-\frac{f\omega_{p}^{2}}{4\pi}\frac{1}{\Omega^{2}-\Omega_{0}^{2}+i\Omega\Gamma}\,\,.$
(31)
Therefore, the vector potential $\mathbf{A}_{p}$ given by equation (27) can be
written as
$\mathbf{A}_{p}=-4\pi\left(\alpha\mathbf{A}_{0}e^{-i\Omega
t}+\alpha^{*}\mathbf{A}_{0}^{*}e^{i\Omega t}\right)\,\,.$ (32)
Now, we can estimate the transition amplitude between two states $a$ and $b$,
making use of the interaction hamiltonian $H_{1p}$ given by equation (20). We
get the amplitude
$c_{ab}=-\frac{8\pi^{2}i}{\hbar
c}\alpha\mathbf{J}(a,b)\mathbf{A}_{0}\delta(\omega_{ab}-\Omega)$ (33)
and the number of transitions per unit time
$P_{ab}=32\pi^{3}\left|\frac{\mathbf{J}(a,b)\mathbf{A}_{0}}{\hbar
c}\right|^{2}\left|\alpha\right|^{2}\delta(\omega_{ab}-\Omega)\,\,.$ (34)
Comparing this result with equation (11) we can see that, apart from a
numerical factor, the rate of polarization-driven transitions are modified by
the factor
$\left|\alpha\right|^{2}=\left(\frac{f\omega_{p}^{2}}{4\pi}\right)^{2}\frac{1}{(\Omega^{2}-\Omega_{0}^{2})^{2}+\Omega^{2}\gamma^{2}}\,\,.$
(35)
This is a typical resonance factor, which indicates that the polarization of
the particle ensemble is important for $\Omega\simeq\Omega_{0}$ (at
resonance), where the ensemble can be disrupted. Obviously, this is a giant
dipole resonance.35,36 For $\Omega$ far away from the resonance frequency
$\Omega_{p}$ the polarization is practically irrelevant, and it may be
neglected in comparison with the transitions brought about by the interaction
hamiltonian $H_{1}$ (equation (11)). It is worth noting that we can define an
electric susceptibility $\chi$ and a dielectric function $\varepsilon$ for the
polarizable ensemble of particles, by combining equations (4), (20) and (32).
We get
$H_{1}+H_{1p}=-\frac{1}{c}\mathbf{J}\left[(1-4\pi\alpha)\mathbf{A}_{0}e^{-i\Omega
t}+c.c.\right]=-\frac{1}{c}\mathbf{J}\left[\frac{1}{\varepsilon}\mathbf{A}_{0}e^{-i\Omega
t}+c.c\right]\,\,\,,$ (36)
since $1-4\pi\alpha=(1+4\pi\chi)^{-1}=1/\varepsilon$, as expected (according
to their definitions, we have
$\mathbf{P}=\alpha\mathbf{E}_{0}=\chi(\mathbf{E}_{0}-4\pi\mathbf{P})$, where
$\mathbf{P}$ is the polarization,_i.e._ the dipole moment per unit volume).
Therefore, the total interaction hamiltonian is proportional to
$1/\varepsilon=(\Omega^{2}-\omega_{c}^{2})/(\Omega^{2}-\Omega_{0}^{2})$, and
we note that, beside the $\Omega_{0}$-pole, it has a zero for
$\Omega=\omega_{c}$, where the transitions are absent.
A similar description holds for ions (or neutral atoms) in an external
electromagnetic field. Perhaps the most interesting case is a neutral, heavy
atom, for which we can estimate the plasma energy $\hbar\omega_{p}\simeq
10Z^{1/2}eV$. For the cohesion energy per electron we can use the Thomas-Fermi
estimation $16Z^{7/3}/ZeV=16Z^{4/4}eV$, which leads to $\hbar\omega_{c}\simeq
13Z^{5/6}eV$. We can see that the typical scale energy where we may expect to
occur a giant dipole resonance is $\hbar\Omega_{0}\simeq 1keV$. However, the
motion of the electrons under the action of a high-intensity electromagnetic
field is relativistic (see, for instance,37).
## 4 Discussion and conclusions
The direct photon-nucleus coupling processes described here are hampered by
electron-positron pairs creation in the Coulomb field of the nucleus. For
photons of energy $\hbar\Omega=10MeV$ we may consider the (ultra-)
relativistic limit of the pair creation cross-section. As it is well
known,38,39 in this case the cross-section is derived within the Born
approximation, the pair partners are generated mainly in the forward
direction, they have not very different energies from one another and the
recoil momentum (energy) trasmitted to the nucleus is small. For bare nuclei
(absence of screening) the total cross-section of pair production is given by
$\sigma_{pair}=\frac{Z^{2}r_{0}^{2}}{137}\left(\frac{28}{9}\ln\frac{2\hbar\Omega}{mc^{2}}-\frac{218}{27}\right)\simeq
10^{-28}Z^{2}cm^{2}\,\,\,,$ (37)
where $r_{0}=e^{2}/mc^{2}$ is the classical electron radius, $-e$ is the
electron charge and $m$ is the electron mass. We can get an order of magnitude
estimation of the efficiency of the processes described here by comparing this
cross-section with the nuclear cross-section $a^{2}$$\simeq 10^{-26}cm^{2}$.
We can see that $\sigma_{pair}/a^{2}\simeq 10^{-2}Z^{2}$, which may go as high
as $10^{2}$ for heavy nuclei.
In conclusion, we may say that in the rest frame of (ultra-) relativistically
accelerated heavy ions (atomic nuclei) the electromagnetic radiation field
produced by high-power optical or free electron lasers may acquire high
intensity and high energy, suitable for photonuclear reactions. In particular,
the excitation of dipole giant resonance may be achieved. Nuclear transitions
are analyzed here under such particular circumstances, including both one- and
two-photon processes. It is shown that the perturbation theory is applicable,
although the field intensity is high, since the interaction energy is low (as
a consequence of the high frequency) and the interaction time (pulse duration
is short). It is also shown that the giant nuclear dipole resonance is driven
by the nuclear (electrical) polarization degrees of freedom, whose dynamics
may lead to disruption of the atomic nucleus when resonance conditions are
met. The concept of nuclear (electrical) polarization is introduced, as well
as the concept of nuclear electrical polarizability and dielectric function.
ACKNOWLEDGMENTS
The authors are indebted to the members of the Seminar of the Laboratory of
Theoretical Physics at Magurele-Bucharest for useful discussions. The
collaborative atmosphere of the Institute for Physics and Nuclear Engineering
and the Institute for Lasers, Plasma and Radiation at Magurele-Bucharest is
also gratefully acknowledged. This work has been supported by UEFISCDI Grants
CORE Program #09370102/2009 and PN-II-ID-PCE-2011-3-0958 of the Romanian
Governmental Agency of Scientfic Research.
Both authors contributed equally to this work.
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|
arxiv-papers
| 2013-12-04T11:44:08 |
2024-09-04T02:49:54.809802
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Apostol and M. Ganciu",
"submitter": "Marian Apostol",
"url": "https://arxiv.org/abs/1312.1116"
}
|
1312.1182
|
Re-examination of globally flat space-time
Michael R. Feldman1,∗
1 Michael R. Feldman Private researcher, New York, NY, United States of
America
$\ast$ E-mail: [email protected]
## Abstract
In the following, we offer a novel approach to modeling the observed effects
currently attributed to the theoretical concepts of ‘dark energy’, ‘dark
matter’, and ‘dark flow’. Instead of assuming the existence of these
theoretical concepts, we take an alternative route and choose to redefine what
we consider to be inertial motion as well as what constitutes an inertial
frame of reference in flat space-time. We adopt none of the features of our
current cosmological models except for the requirement that special and
general relativity be local approximations within our revised definition of
inertial systems. Implicit in our ideas is the assumption that at “large
enough” scales one can treat objects within these inertial systems as point-
particles having an insignificant effect on the curvature of space-time. We
then proceed under the assumption that time and space are fundamentally
intertwined such that time- and spatial-translational invariance are not
inherent symmetries of flat space-time (i.e. observable clock rates depend
upon both relative velocity and spatial position within these inertial
systems) and take the geodesics of this theory in the radial Rindler chart as
the proper characterization of inertial motion. With this commitment, we are
able to model solely with inertial motion the observed effects expected to be
the result of ‘dark energy’, ‘dark matter’, and ‘dark flow’. In addition, we
examine the potential observable implications of our theory in a gravitational
system located within a confined region of an inertial reference frame,
subsequently interpreting the Pioneer anomaly as support for our redefinition
of inertial motion. As well, we extend our analysis into quantum mechanics by
quantizing for a real scalar field and find a possible explanation for the
asymmetry between matter and antimatter within the framework of these
redefined inertial systems.
## Introduction
The purpose of this paper is to present the foundational groundwork for a new
metric theory of flat space-time which takes into account the observed effects
currently expected to be the result of ‘dark energy’[1], ‘dark matter’[2], and
‘dark flow’ [3] without resorting to these theoretical concepts that we have
yet to observe in the laboratory. We emphasize above the fact that we are
working in flat space-time as this paper is not concerned with reformulating
gravity. Meaning, we assume gravity is the consequence of local curvature in
space-time resulting from the energy-momentum content associated with an
object as formulated by Einstein in his theory of general relativity. However,
for our discussion, we operate under the assumption that at “large enough”
scales we may treat massive objects in our proposed inertial reference frames
as point-particles having an insignificant effect on the curvature of space-
time for the purpose of examining the motion of said objects within the
context of these larger scales. Consequently, we assume that space-time is
essentially flat at these scales, and therefore, the energy density throughout
our inertial systems is taken to be approximately zero. Thus, we assume that
the deviation away from flat space-time inertial paths due to curvature in
space-time is insignificant in our analysis. Furthermore, it is assumed that
the observed motion of these large-scale objects about central points (e.g.
stars orbiting the center of a galaxy, galaxies orbiting the center of a
group/cluster, groups/clusters orbiting the center of a supercluster, etc.) is
not due to the presence of gravitational sources at these centers but is
instead a manifestation of the way in which objects move when no net external
forces are acting upon them. In other words, the following work is concerned
with reformulating our understanding of inertial motion. Furthermore, we focus
on reformulating the global properties of an inertial reference frame while
disregarding the potential local effects that objects moving within this
global inertial system may have on the curvature of space-time. To begin with
our reformulation, we explicitly state for the reader the assumptions of flat
space-time as given by Einstein’s special relativity [4]:
1\. An object will travel in a straight line at a constant speed when no net
external forces are acting upon this object (inertial motion adopted from
Newton; see section titled “Definitions” in [5]).
2\. An observer undergoing inertial motion has the freedom to describe events
by “carrying rulers” in any three arbitrarily chosen spatial directions
(perpendicular to one another) and calibrating clocks according to Einstein’s
prescription for synchronization (an inertial frame of reference). As well,
inertial reference frames moving with uniform (constant velocity) rectilinear
motion relative to one another are treated equally (i.e. there are no
preferred inertial frames of reference in flat space-time).
3\. The speed of light remains constant in all of these observer dependent
inertial frames.
While operating under these assumptions in addition to those of general
relativity [6], our cosmological models (e.g. $\Lambda$CDM [7]) then require a
‘Big-Bang’ event[8][9][10], ‘inflation’ [11], ‘dark energy’, ‘dark matter’,
and ‘dark flow’ as explanations for observed phenomena on cosmological scales
given our assumed understanding of inertial motion and inertial reference
frames as stated above. In contrast, our claims in this paper are that in
order to reproduce the observed behavior attributed to the theoretical
concepts of ‘dark energy’, ‘dark matter’ and ‘dark flow’, it is not necessary
to assume that these supplements must exist. Instead, it is possible to
reproduce this behavior by simply incorporating it into a revised
understanding of inertial motion and inertial reference frames in empty flat
space-time, thereby no longer assuming the three pillars of theoretical
physics as listed above and no longer requiring the occurrence of a ‘Big-
Bang’, ‘inflation’, and expansion of space. While seemingly rash at first
glance, we claim that in what we term as our “Theory of Inertial Centers”, as
laid out in the following work, one can reproduce with inertial motion in our
redefined inertial reference frames the following observed features:
1\. Accelerated redshifts [12] and the Hubble relation [13].
2\. Plateauing orbital velocity curves at large distances from a central point
about which objects orbit [14].
3\. Consistent velocity “flow” of objects toward a central point [3] [15].
4\. An orientation associated with a particular frame of reference [16] (i.e.
we do not take the cosmological principle to be a valid assumption as can be
seen from experimental evidence such as [17]).
In our theory of flat space-time, inertial motion remains defined to be the
motion of an object when it is subjected to no net external forces. In
addition, an inertial reference frame is defined to be a system within which
objects move along inertial trajectories when no net external forces are
acting upon them. We then make the following assumptions and requirements in
our theory:
1\. Inertial motion is not characterized by an object moving in a straight
line at a constant speed. Instead, inertial motion is characterized by
geodesics about “inertial center points” in the radial Rindler chart as
examined in the following discussion (the radial Rindler chart has been
mentioned in other contexts such as [18] and [19]). Note that implicit in this
assumption is the idea that time and space are fundamentally intertwined such
that time-translational invariance and spatial-translational invariance are
not inherent symmetries of flat space-time. Mathematically, this notion
reduces to incorporating both time and spatial distance into the invariant
interval associated with our metric. Meaning, the physically observable
elapsed time as measured by a clock carried along a given curve, denoted as
“proper time” $\tau$, is not our affine parameter and thus is not invariant.
Therefore, observable clock rates depend upon both spatial position in a
particular inertial frame as well as in which inertial frame the observer is
observing. Our affine parameter $\chi$ in the theory of inertial centers is
then taken as a function of proper time in a particular inertial frame to be
$\chi=\sqrt{\Lambda}\cdot\int r(\tau)d\tau$
where $r=r(\tau)$ represents the physical distance to the inertial center
about which the observer moves at a particular observable clock time $\tau$ in
the inertial system and $\sqrt{\Lambda}$ is taken to be the Hubble
constant[13][20]. In addition, these inertial center points define the centers
of our inertial reference frames.
2\. An observer does not have the freedom to describe an inertial reference
frame in whichever way he/she chooses as in special relativity. We, as
observers, are forced to adopt the orientation of the inertial reference frame
that nature provides for us at the particular scale in which we are describing
phenomena. As well, the inertial motion of an object must be thought of
relative to the inertial center point about which said object orbits
(throughout this paper, we will use the term “orbit” to refer to the inertial
motion of an object about an inertial center point).
3\. The speed of light is not constant throughout these inertial reference
frames.
4\. Locally within a confined region of each of these newly defined inertial
reference frames, our theory reduces to and abides by the axioms of special
relativity and general relativity.
Our analysis is organized in the following manner. First, we explore the
limiting behavior of our equations of motion with the radial Rindler chart in
flat space-time. Out of this, we come upon the ability to model the observed
features as listed above. Second, we determine the limit in which our theory
reduces to special relativity, while also proposing the form of our invariant
interval in terms of both time and distance to an inertial center. We have
stated the form of our affine parameter earlier in this introduction as a
preface to the logic used in this proposition. Third, we examine the potential
observable effects of this theory within our solar system and interpret the
Pioneer anomaly [21] as support for our ideas. Fourth, we extend our analysis
by quantizing our theory for a real massive scalar field. Within the context
of this extension, we find a potential explanation for the asymmetry between
matter and antimatter in our observable universe through the possibility of a
parallel region to each inertial system embodied mathematically by the “other”
radial Rindler wedge. We conclude by proposing future work including
addressing the source of the cosmic microwave background [22] in this theory,
attempting to explain other astrometric anomalies within our solar system
besides Pioneer [23], and extending our quantum mechanical analysis to complex
fields with spin.
## Discussion
### Geodesic paths
Adopting the signature $(-,+,+,+)$ and employing abstract index notation
throughout our analysis (see Chapter 2.4 of [24]), we work in the following
metric:
$-d\chi^{2}=-{\Lambda}r^{2}dt^{2}+dr^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)d{\Omega}^{2}$
(1)
where $d{\Omega}^{2}=d{\theta}^{2}+d{\phi}^{2}\sin^{2}{\theta}$;
$0\leq\theta\leq\pi$, $0\leq\phi<2\pi$, $-\infty<t<\infty$, $0<r^{2}<\infty$,
and $\Lambda$ is a positive constant. In a subsequent section, we’ll deduce
that $\Lambda$ must be the square of the Hubble constant. $d\chi^{2}$ denotes
the invariant interval associated with this metric where $d\chi^{2}\neq
c^{2}d\tau^{2}$ assuming $\tau$ denotes proper time, defined as the physically
observable elapsed time between two events as measured by a clock passing
through both events carried along a particular curve, and $c$ denotes the
constant associated with the speed of light in special relativity. Therefore,
in contrast with special relativity, our proper time interval is not assumed
to be invariant, and the speed of light in flat space-time is not assumed to
be constant. However, in subsequent sections, we shall show how special
relativity can be treated as a local approximation to our theory of inertial
centers. As in special and general relativity, massless particles travel along
null geodesics. Thus, with this radial Rindler chart as the description of our
inertial frame of reference and our redefinition of the invariant interval
associated with the metric, we implicitly assume that time and space are
fundamentally intertwined such that time-translational invariance and spatial-
translational invariance are not inherent symmetries of flat space-time. In
other words, one cannot progress coordinate time $t$ forward (i.e. replace
$t\rightarrow t+t_{0}$ where $t_{0}$ is a constant) without considering the
effect of this action on space and vice versa. As well, this concept requires
that we incorporate into the invariant interval associated with our metric
both distance to inertial centers as well as proper time. Later in our
analysis, we will express $d\chi^{2}$ for this theory of inertial centers in
terms of the proper time interval in a particular inertial frame.
For the affine connection terms, Ricci tensor elements, curvature scalar and
square Riemann tensor, we refer to Appendix A. From these calculations it is
clear that this space-time geometry is indeed flat. Taking the Rindler
transformation equations, $cT=r\sinh(\sqrt{\Lambda}t)$,
$R=r\cosh(\sqrt{\Lambda}t)$, we find our metric equation becomes
$-d\chi^{2}=-c^{2}dT^{2}+dR^{2}+R^{2}d\Omega^{2},\indent\forall
R,c^{2}T^{2}<R^{2}$
where $c=$ speed of light in the local Minkowski reference frame [25]. If one
operates under the assumptions of special relativity, $d\chi^{2}$ would in
fact equal $c^{2}d\tau^{2}$, and then the metric in (1) can be used to model
uniformly radially accelerated motion with respect to Minkowski space-time
confined to either of the Rindler wedges: left wedge for $|T|<-R/c$ and right
wedge for $|T|<R/c$ [26]. For the rest of our analysis, however, we no longer
assume that special relativity is valid throughout globally flat space-time
(again, $d\chi^{2}\neq c^{2}d\tau^{2}$) and instead examine the geodesic
motion of point-particles in this radial Rindler coordinate system with time
and radial distance from our inertial center point corresponding to the
coordinate labels $t$ and $r$, respectively. Additionally, as $d\chi^{2}\neq
c^{2}d\tau^{2}$, we do not assume that the reference frame itself is radially
accelerating. Instead, we are re-examining inertial motion under the
guidelines presented in our introduction keeping in mind that the form of our
invariant interval is different from that of special and general relativity.
And since our affine parameter is different from that of special and general
relativity, the geodesics of our theory will also be different. Consequently,
our employment of the radial Rindler chart in the following analysis is our
way of establishing that this coordinate system is the “natural” one for
describing an inertial system in the theory of inertial centers (i.e.
coordinate time in the radial Rindler chart progresses at the same rate as the
physical clock of a stationary observer in the inertial system). Thus, in the
following work, we abandon the idea that Minkowski coordinates can cover all
of an inertial system in flat space-time. Furthermore, we propose that the
radial Rindler chart should be our “natural” coordinate system for describing
an inertial frame of reference in the theory of inertial centers.
Referring to Appendix B, we find for the equations of motion of a particle
within a particular inertial system ($U^{a}{\nabla}_{a}U^{b}=0$, where our
‘proper velocity’ in component form is $U^{\mu}=dx^{\mu}/d\sigma$):
$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\frac{2}{r}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(2) $\displaystyle 0=\frac{d^{2}r}{d\sigma^{2}}+\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(3) $\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\frac{d\theta}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}\bigg{]}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
(4) $\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$
(5)
And our norm for the ‘four-velocity’ is given by
$-k=g_{ab}U^{a}U^{b}=-\Lambda
r^{2}\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}+\bigg{(}\frac{dr}{d\sigma}\bigg{)}^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(6)
where
$k=\left\\{\begin{array}[]{l l}0&\quad\textrm{massless particle}\\\
1&\quad\textrm{massive particle}\\\ \end{array}\right.$
and $\sigma=\chi$ for massive particles. Notice that our ‘four-velocity’
$U^{a}$ in this theory is dimensionless for spatial components and has units
of [time]/[distance] for our time component since $\chi$ (and therefore
$\sigma$) has units of [distance]. Multiplying each term in our radial
equation of motion by $r$ and plugging in (6),
$0=r\frac{d^{2}r}{d\sigma^{2}}+\bigg{(}\frac{dr}{d\sigma}\bigg{)}^{2}+k$ (7)
But to remain at a constant radial distance away from our inertial center:
$d^{2}r/d\sigma^{2}$, $dr/d\sigma=0$. Therefore, only massless particles can
have circular orbits.
Possible geodesic paths obey the relation $U^{a}U_{a}\leq 0$ from (6), and
solving for $d\theta/dt$ and $d\phi/dt$, we find that
$\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\Lambda
r^{2}-\bigg{(}\frac{dr}{dt}\bigg{)}^{2}\bigg{]}\geq\bigg{(}\frac{d\theta}{dt}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{dt}\bigg{)}^{2}$
Examining our $\theta$ equation of motion (4), we see that a particle remains
at a constant value of $\theta$ for non-zero angular velocity in $\phi$ if and
only if $d\theta/d\sigma=0$ and $\theta=0$, $\pi/2$, $\pi$. Consequently, the
angular velocity of a particle traveling in the equatorial plane
($\theta=\pi/2$) of this inertial reference frame is bound by the range:
$-\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}\leq\frac{d\phi}{dt}\bigg{|}_{\theta=\pi/2}\leq+\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}$
(8)
Then, for a photon traveling in a circular orbit in the equatorial plane, we
find
$\frac{d\phi}{dt}\bigg{|}_{k=0,\theta=\pi/2,{\rm
circular}}=\pm\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}$ (9)
Later, we’ll see that a massless particle can have circular orbits only for
$\theta=\pi/2$ (orbits with $\phi=\phi_{0}$ cannot be circular).
For massive particles nearly at rest with respect to the center of this
inertial system (i.e. spatial ‘velocity’ terms are much smaller than our
‘velocity’ term in time so that these spatial terms can be taken as nearly
zero), these four equations of motion (2), (3), (4), and (5) reduce to two:
$0=\frac{d^{2}t}{d\chi^{2}}\indent{\rm and}\indent
0=\frac{d^{2}r}{d\chi^{2}}+\Lambda r\bigg{(}\frac{dt}{d\chi}\bigg{)}^{2}$
And solving for the radial acceleration, we find that
$\frac{d^{2}r}{dt^{2}}=-\Lambda r$ (10)
In this limit, the inertial motion of our point-particle is described by a
spatial acceleration in $r$ pulling inward toward the center of this
particular reference frame scaled by the square of the time-scale constant.
Thus, slowly moving objects at large radial distances experience a large
radial acceleration pulling inward toward the center of the inertial system
about which the objects orbit.
Then, let us examine the case where the motion of particles far from an
inertial center (large $r$) is dominated by angular velocities with
approximately circular radial motion ($r\approx{\rm constant}$). Our equations
of motion reduce to
$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle 0=\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\theta}{d\sigma}\frac{dt}{d\sigma}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
$\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$
Plugging $d^{2}t/d\sigma^{2}$ and $dt/d\sigma$ into our expressions for
$d^{2}\phi/d\sigma^{2}$ and $d^{2}\theta/d\sigma^{2}$:
$\displaystyle
0=\frac{d^{2}\phi}{dt^{2}}+\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\phi}{dt}+2\cot{\theta}\frac{d\theta}{dt}\frac{d\phi}{dt}$
$\displaystyle
0=\frac{d^{2}\theta}{dt^{2}}+\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\theta}{dt}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{dt}\bigg{)}^{2}$
Now, if we assume $d\phi/dt\gg d\theta/dt$ and integrate:
$\frac{d\phi}{dt}=\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}$
where $\phi_{0}\rightarrow\pm\sqrt{\Lambda}$ for light taking a circular orbit
in the equatorial plane and thus $|\phi_{0}|\leq\sqrt{\Lambda}$ for
$\theta=\pi/2$. Plugging in for $d\phi/dt$, we have
$\frac{d^{2}\theta}{dt^{2}}=\sin{\theta}\cos{\theta}\bigg{(}\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}\bigg{)}^{2}$
(11)
where $\phi_{0}$ is a constant. In the large $d\phi/d\sigma$ limit:
$\displaystyle\frac{d^{2}\theta}{dt^{2}}>0\indent\textrm{for}\indent
0<\theta<\frac{\pi}{2}$ (12)
$\displaystyle\frac{d^{2}\theta}{dt^{2}}=0\indent\textrm{for}\indent\theta=\frac{\pi}{2}$
(13)
$\displaystyle\frac{d^{2}\theta}{dt^{2}}<0\indent\textrm{for}\indent\frac{\pi}{2}<\theta<\pi$
(14)
As long as the particle is not located at either of the poles ($\theta\neq 0$,
$\pi$), we see a sinusoidal spatial angular acceleration that decreases with
$t$ and moves the object toward $\theta=\pi/2$. One can then picture spiral
galaxy formation resulting from objects orbiting an inertial center with large
angular velocity in $\phi$.
If we refer back to our expression for $d\phi/dt$, we find for the orbital
velocity ($v=r\cdot d\phi/dt$) of a particle in this limit:
$v=\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}r$ (15)
And for $\sqrt{\Lambda}t\approx 0$, our particle’s speed is linearly
proportional to its radial distance away from the inertial center about which
it orbits. In this limit at large $r$, the relationship between orbital
velocity and radial distance mimics the relationship between orbital velocity
and radial distance found in our observed galaxy rotation curves[14] for
comparably small values of $\sqrt{\Lambda}$ and therefore $\phi_{0}$. However,
the analysis above will apply to the classical (in the sense that we are not
taking into account quantum mechanics) inertial motion of an object in any
particular inertial system (e.g. galaxies, groups, clusters, etc.). Later in
our analysis, we’ll provide an experimental scale for the time-scale constant
$\sqrt{\Lambda}$ by analyzing the inherent redshift that occurs in these
inertial frames (i.e. we’ll take $\sqrt{\Lambda}$ to be the Hubble constant).
Since $|\phi_{0}|\leq\sqrt{\Lambda}$, this value will also give us an upper
limit for the slope of our orbital velocity curves at large $r$. Thus, we
claim that the linear relationship found in (15) models the experimental
relationship found from our observed orbital velocity curves for objects far
from the center of the galaxy within which they orbit. We base this claim off
of the idea that the plateauing nature of our experimental curves would be
interpreted in our model to be the result of the small scale of
$\sqrt{\Lambda}$ relative to galactic distance and orbital velocity scales.
### Conservation laws
Since this metric is just a coordinate transformation away from Minkowski, we
expect to find ten linearly independent Killing vector fields as vector fields
are geometric objects independent of our coordinate parametrization. One could
obtain these using the radial Rindler transformation equations, but we find it
helpful to explicitly derive them. We refer to Appendix C for more detail as
well as a full list of all Killing vector fields given in the radial Rindler
chart. Rewriting here for reference the three we will be using:
$\displaystyle\rho^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}r}\cosh(\sqrt{\Lambda}t),-\sinh(\sqrt{\Lambda}t),0,0\rangle$
(16)
$\displaystyle\Theta^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}}\cos{\theta},0,-\sin{\theta}\tanh(\sqrt{\Lambda}t),0\rangle$
(17) $\displaystyle\psi^{\mu}\rightarrow\langle 0,0,0,1\rangle$ (18)
Applying Noether’s theorem ($U^{a}\xi_{a}={\rm constant}$),
$\displaystyle
E=\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\sinh(\sqrt{\Lambda}t)\frac{dr}{d\sigma}$
(19)
$\displaystyle\Omega=\sqrt{\Lambda}r^{2}\cos{\theta}\frac{dt}{d\sigma}+r^{2}\sin{\theta}\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)\frac{d\theta}{d\sigma}$
(20) $\displaystyle
L=r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}\frac{d\phi}{d\sigma}$ (21)
Plugging into (6) and solving for $dt/d\sigma$, we find that
$\frac{dt}{d\sigma}=\frac{1}{\sqrt{\Lambda}}\bigg{[}\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta}\pm\sqrt{}|_{dt/d\sigma}\bigg{]}$
(22)
where
$\displaystyle\sqrt{}|_{dt/d\sigma}=\bigg{\\{}(\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\bigg{(}\frac{E}{r}\bigg{)}^{2}+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{r}\bigg{)}^{2}\bigg{)}$
$\displaystyle+\bigg{(}\frac{\Omega}{r^{2}}\bigg{)}^{2}+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{r^{2}}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(23)
requiring
$(\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta})^{2}\geq\sin^{2}{\theta}\bigg{[}\bigg{(}\frac{E}{r}\bigg{)}^{2}+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{r}\bigg{)}^{2}\bigg{]}+\bigg{(}\frac{\Omega}{r^{2}}\bigg{)}^{2}+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{r^{2}}\bigg{)}^{2}$
Notice, for massive particles moving radially in the equatorial plane ($k=1$,
$\theta=\pi/2$, $\Omega=0$, and $L=0$), this constraint reduces to:
$E^{2}\geq 1$
which is just our analogue of the statement in special relativity that the
energy of an object must be greater than or equal to its rest mass [27] since
in special relativity one would assume this constant $E$ would equal the
energy of the particle divided by its rest mass (i.e. in special relativity,
$E$ would be equal to $\tilde{E}/mc^{2}$ where $\tilde{E}$ is the energy of
the particle). Using (19), (20), (21), and (22), we find
$\displaystyle\frac{dr}{dt}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{Er}\cos{\theta}\pm\sqrt{}|_{dr/dt}}-\cosh(\sqrt{\Lambda}t)\bigg{]}$
(24)
$\displaystyle\frac{d\theta}{dt}=\frac{\sqrt{\Lambda}}{\sin{\theta}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\cos{\theta}\pm\sqrt{}|_{d\theta/dt}}-\cos{\theta}\bigg{]}$
(25)
$\displaystyle\frac{d\phi}{dt}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\Omega\cos{\theta}\pm\sqrt{}|_{d\phi/dt}}\bigg{]}$
(26)
where
$\displaystyle\sqrt{}|_{dr/dt}=\bigg{\\{}(\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{Er}\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}1+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{E}\bigg{)}^{2}\bigg{)}+\bigg{(}\frac{\Omega}{Er}\bigg{)}^{2}$
$\displaystyle+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{Er}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(27)
$\displaystyle\sqrt{}|_{d\theta/dt}=\bigg{\\{}(\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\bigg{(}\frac{Er}{\Omega}\bigg{)}^{2}+k\bigg{(}\frac{r\sinh(\sqrt{\Lambda}t)}{\Omega}\bigg{)}^{2}\bigg{)}$
$\displaystyle+1+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{\Omega}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(28)
$\displaystyle\sqrt{}|_{d\phi/dt}=\bigg{\\{}(Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\Omega\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}(Er)^{2}+k(r\sinh(\sqrt{\Lambda}t))^{2}\bigg{)}$
$\displaystyle+{\Omega}^{2}+(L\tanh(\sqrt{\Lambda}t))^{2}\bigg{]}\bigg{\\}}^{1/2}$
(29)
For light traveling radially in the equatorial plane, $\Omega$, $L=0$ and (24)
reduces to
$\frac{dr}{dt}\bigg{|}_{k=0,\Omega=0,L=0}=\pm\sqrt{\Lambda}r$ (30)
giving us
$r(t)|_{k=0,\Omega=0,L=0}=r_{0}\textrm{exp}({\pm\sqrt{\Lambda}t})$ (31)
where $r_{0}$ is a constant signifying the radial position of the photon at
$t=0$. One could have arrived at this expression for the general case of light
traveling radially even outside of the equatorial plane simply from (6) for
null geodesics. Let us now pose the question of whether or not it is possible
for light to travel from the $r>0$ region of our inertial system to $r=0$
which we regard as our inertial center point. Integrating $dr/dt$ from $0$ to
$\Delta t$,
$\Delta r=\pm\sqrt{\Lambda}r_{0}\int^{\Delta
t}_{0}{\textrm{exp}(\pm\sqrt{\Lambda}t)dt}=r_{0}[\textrm{exp}(\pm\sqrt{\Lambda}\Delta
t)-1]$
Solving for $\Delta t$,
$\Delta t=\ln{\bigg{(}\frac{\Delta r+r_{0}}{r_{0}}\bigg{)}^{\pm
1/\sqrt{\Lambda}}}$ (32)
For a photon traveling radially inward, the sign of the root is negative, and
it reaches $r=0$ in
$\lim_{r_{{\rm final}}\rightarrow 0}\Delta
t=\ln{\bigg{(}\frac{(0-r_{0})+r_{0}}{r_{0}}\bigg{)}^{-1/\sqrt{\Lambda}}}=\ln{\bigg{(}\frac{r_{0}}{-r_{0}+r_{0}}\bigg{)}^{1/\sqrt{\Lambda}}}\rightarrow\infty$
Consequently, not even light can reach $r=0$ in a finite amount of time. But
what about the inertial behavior of massive particles in these systems? At
first glance, (24) and (25) appear to be divergent for $t=0$. However, to
evaluate all of these velocity expressions for $t=0$, we return to symmetry
equations (19) and (20):
$E=\sqrt{\Lambda}r\frac{dt}{d\sigma}\bigg{|}_{t=0}\indent{\rm
and}\indent\Omega=\sqrt{\Lambda}r^{2}\cos{\theta}\frac{dt}{d\sigma}\bigg{|}_{t=0}$
Therefore,
$\frac{E}{r}\cos{\theta}=\frac{\Omega}{r^{2}}\bigg{|}_{t=0}$ (33)
Plugging into (23),
$\displaystyle\sqrt{}|_{dt/d\sigma}|_{t=0}=\bigg{\\{}\bigg{(}\frac{E}{r}\sin^{2}{\theta}+\frac{E}{r}\cos^{2}{\theta}\bigg{)}^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\frac{E}{r}\bigg{)}^{2}+\bigg{(}\frac{E}{r}\cos^{2}{\theta}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
$\displaystyle=\sqrt{\bigg{(}\frac{E}{r}\bigg{)}^{2}-\bigg{(}\frac{E}{r}\bigg{)}^{2}}=0$
which one can plug back into (22) to find consistency with our expressions for
$dt/d\sigma|_{t=0}$ above. Yet we see from (27), (28), and (29) that
$\displaystyle\sqrt{}|_{dr/dt}=\frac{r}{E}\cdot\sqrt{}|_{dt/d\sigma}$ (34)
$\displaystyle\sqrt{}|_{d\theta/dt}=\frac{r^{2}}{\Omega}\cdot\sqrt{}|_{dt/d\sigma}$
(35) $\displaystyle\sqrt{}|_{d\phi/dt}=r^{2}\cdot\sqrt{}|_{dt/d\sigma}$ (36)
which implies for $t=0$ that all of these terms vanish. Then our spatial
velocity terms for $t=0$ become
$\displaystyle\frac{dr}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}+\cos^{2}{\theta}}-1\bigg{]}=\frac{0}{0}$
$\displaystyle\frac{d\theta}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}}{\sin{\theta}\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{\cos{\theta}}{\sin^{2}{\theta}+\cos^{2}{\theta}}-\cos{\theta}\bigg{]}=\frac{0}{0}$
$\displaystyle\frac{d\phi}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}}\bigg{[}\frac{1}{Er\sin^{2}{\theta}+Er\cos^{2}{\theta}}\bigg{]}=\frac{\sqrt{\Lambda}L}{Er\sin^{2}{\theta}}$
where we have used (33) in these limit expressions. So we see that our
velocity terms are not necessarily divergent for $t=0$. However, we’ll address
the issue of motion for small $\sqrt{\Lambda}t$ later when we relate
Einstein’s special relativity to our theory of inertial centers. One must also
keep in mind that expressions (24), (25), and (26) represent a set of complex
differential equations that we unfortunately will not be able to solve in this
paper. The purpose of the following portion of this section is in fact to
evaluate the large $t$ behavior of all spatial velocities where it is not
explicitly apparent how to evaluate this limit if one were to work in
Minkowski coordinates while keeping in mind the notion that he/she must relate
back to the radial Rindler chart for inertial time as $d\chi^{2}\neq
c^{2}d\tau^{2}$ (we’ll elaborate further on the term “inertial time” in our
next section). It does appear easier to proceed in this manner of working in
Minkowski and relating back to radial Rindler for solely radial motion as we
shall do later in this section.
Yet, we return to our velocity expressions from Noether’s theorem in order to
examine the general expression for $dr/dt$ as $t\rightarrow\infty$. First, we
determine the limiting value of $\sqrt{}|_{dr/dt}$:
$\lim_{t\rightarrow\infty}{\sqrt{}|_{dr/dt}}=\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$
Then for massive particles ($k=1$) assuming $\theta\neq 0$ or $\pi$,
$\lim_{t\rightarrow\infty}{\frac{dr}{dt}}=-\sqrt{\Lambda}r$ (37)
which is just the equation for a massless particle traveling radially inward.
When $\theta=0$ or $\pi$, we must return to conservation equations (20) and
(21). We find $L=0$ and
$\frac{dt}{d\sigma}\bigg{|}_{\theta=0,\pi}=\frac{\Omega}{\sqrt{\Lambda}r^{2}\cos{\theta}}$
Plugging in (19) and solving when $t\rightarrow\infty$, we again find (37). We
now understand that eventually all massive particles move toward $r=0$. Yet as
the object approaches the center, its speed decreases as well and will only
stop moving inward when it reaches this inertial center point in an infinite
amount of time. Thus, with this large $t$ behavior, we apparently inherit the
ability to model the observed anomalous effects of ‘dark flow’ [3]. In our
next section, we will provide an interpretation for the physical significance
of our coordinate time in our theory of inertial centers, relating $t$ back to
the rate at which physical clocks are observed to tick.
However, we progress onward and look at the large $t$ limits for both
$d\theta/dt$ and $d\phi/dt$. Beginning with the former, we find
$\lim_{t\rightarrow\infty}{\sqrt{}|_{d\theta/dt}}=\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$
Plugging into $d\theta/dt$ and examining for massive particles,
$\lim_{t\rightarrow\infty}{\frac{d\theta}{dt}}=\frac{-\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\cot{\theta}$
(38)
And for $\theta\neq 0$ or $\pi$,
$\lim_{t\rightarrow\infty}{\frac{d\theta}{dt}}\bigg{|}_{\theta\neq 0,\pi}=0$
(39)
Solving for $d\theta/dt$ when $\theta=0$ or $\pi$ using (25) and $L=0$,
$\frac{d\theta}{dt}\bigg{|}_{\theta=0,\pi}=\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\tan{\theta}=0$
At the poles, particles have no angular velocity in $\theta$ nor angular
momentum in $\phi$ ($d\theta/d\sigma,L=0$). Lastly for our large $t$ limits,
we have $d\phi/dt$:
$\lim_{t\rightarrow\infty}{\sqrt{}|_{d\phi/dt}}=Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$
Plugging into $d\phi/dt$,
$\lim_{t\rightarrow\infty}\frac{d\phi}{dt}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)(1\pm\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}})}\bigg{]}$
For massive particles and assuming $\theta\neq 0$ or $\pi$,
$\lim_{t\rightarrow\infty}\frac{d\phi}{dt}=0$ (40)
We have a clearer picture of the inertial trajectories of massive particles
over time in the context of our redefined inertial reference frames. As time
progresses, massive objects will eventually move radially inward losing
angular velocity in $\theta$ and angular momentum in $\phi$, slowing down in
radial velocity as they approach the center point about which they orbit.
Looking back at our expression for $dr/dt$, we ask ourselves the question: for
what values of $\theta$ is $dr/dt$ most positive? For positive $dr/dt$, we
have particles moving radially outward, and maximizing this expression with
respect to $\theta$ provides us with the easiest possible path to be ejected
away from our inertial center. Examining particles with large radial ‘proper
velocities’ relative to their own angular ‘proper velocities’ which from (19),
(20), (21) implies $Er\gg\Omega,L$ since $dr/d\sigma\gg
r\cosh(\sqrt{\Lambda}t)\cdot d\theta/d\sigma$ and $dr/d\sigma\gg
r\cosh(\sqrt{\Lambda}t)\cdot d\phi/d\sigma$ in this limit:
$\displaystyle\frac{dr}{dt}\bigg{|}_{Er\gg\Omega,L}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}\pm\sqrt{\sin^{4}{\theta}-\frac{\sin^{2}{\theta}}{\cosh^{2}(\sqrt{\Lambda}t)}(1+k\frac{\sinh^{2}(\sqrt{\Lambda}t)}{E^{2}})}}-\cosh^{2}(\sqrt{\Lambda}t)\bigg{]}$
But the largest positive value of $dr/dt|_{Er\gg\Omega,L}$ occurs if we
minimize the denominator of the first term in brackets with respect to
$\theta$. Clearly, this term needs to be re-evaluated when $\theta=0$ or
$\pi$. Returning to conservation equations (19) and (20), we solve for the
radial motion of a particle through the poles by plugging into (6) ($\theta=0$
or $\pi$ and $d\theta/d\sigma=0$):
$\displaystyle\frac{dr}{dt}\bigg{|}_{\theta=0,\pi}=\frac{\sqrt{\Lambda}r}{1+\frac{k}{E^{2}}\sinh^{2}(\sqrt{\Lambda}t)}\cdot\bigg{\\{}-\frac{k}{E^{2}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\pm\bigg{[}\bigg{(}\frac{k}{E^{2}}\bigg{)}^{2}\cosh^{2}(\sqrt{\Lambda}t)\sinh^{2}(\sqrt{\Lambda}t)$
$\displaystyle+\bigg{(}1+\frac{k}{E^{2}}\sinh^{2}(\sqrt{\Lambda}t)\bigg{)}\bigg{(}1-\frac{k}{E^{2}}\cosh^{2}(\sqrt{\Lambda}t)\bigg{)}\bigg{]}^{1/2}\bigg{\\}}$
For large proper radial motion, we assume $E^{2}\gg
k\cosh^{2}(\sqrt{\Lambda}t)$ (as $E^{2}\geq 1$ is our analogue of the rest
mass condition from Einstein’s special relativity). Then, our expression for
radial motion through the poles reduces to
$\frac{dr}{dt}\bigg{|}_{\theta=0,\pi}\approx\pm\sqrt{\Lambda}r$ (41)
We see that massive particles can travel at speeds near that of photons
through the poles, and therefore it appears that the easiest way for particles
to be ejected radially outward away from an inertial center would be through
the poles of the inertial system. If we imagine a supernova occurring near the
center point of an inertial system, we find that a simple potential scenario
for the occurrence of relativistic jets [28] in this reference frame would be
the expulsion of stellar remnants through the poles. Consequently, if we use
this logic to provide an alternative for relativistic jet production, we must
then require that each of our inertial frames have a particular orientation
governed by the location of these poles and embodied mathematically by the
spatial positions for which particular metric components vanish. In other
words, when describing a particular inertial frame, these are the $\theta$
values for which $\sin{\theta}=0$ previously referred to as “coordinate
singularities” (e.g. see Chapter 5.1 of [29]) but taken here as a physical
attribute of the inertial system reflecting the idea that the radial Rindler
chart is the “natural” coordinate system for an inertial reference frame in
flat space-time. Thus we must ask ourselves the following question. How is
this orientation established in the theory of inertial centers? As we shall
mention later in our paper, this is an open question which we will have to
address in future work.
Back to our circular orbit analysis, we solve for the radius at which light
can have circular paths in a particular inertial system for possibly both
$d\theta/dt=0$, $d\phi/dt=\pm\sqrt{\Lambda}/\cosh(\sqrt{\Lambda}t)$ and
$d\theta/dt=\pm\sqrt{\Lambda}/\cosh(\sqrt{\Lambda}t)$, $d\phi/dt=0$. For the
two, we obtain from (19)
$\frac{dt}{d\sigma}\bigg{|}_{k=0,{\rm
circular}}=\frac{E}{\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)}$ (42)
In the former situation ($\theta=\pi/2$, $\Omega=0$), we substitute (42) and
(21) into (6) and arrive at
$r_{\theta=\pi/2,{\rm circular}}=\frac{L}{E}$ (43)
Whereas for the latter case ($\phi=\phi_{0}$, $L=0$), we plug (42) and (20)
into (6) to find
$r=\frac{\Omega\cosh(\sqrt{\Lambda}t)}{E[\cos^{2}{\theta}\pm\sin{\theta}\sinh(\sqrt{\Lambda}t)]}$
which is not constant. Consequently, in our inertial systems, light can travel
in circular orbits only in the equatorial plane with angular velocity given by
(9) at a radius given by (43). The type of lensing expected from a black hole
or ‘dark matter’[30] is evidently reproduced in a similar manner by light
traveling with angular velocity about an inertial center point. Although in
the analysis above, we studied circular orbits where light remains at a
constant $r$, the logic applies similarly for the case where the photon has
both radial and angular velocity components.
We come to the redshift factor for light traveling radially. Before we begin
with this analysis, we must refer back to our procedure for determining the
observed wavelength of a photon when operating under the assumptions of
special and general relativity. In general relativity, the observed frequency
$f$ of a photon with momentum $p^{a}$ ($p^{a}=\hbar k^{a}$) emitted/received
by an observer traveling with proper velocity in component form given by
$u^{\mu}=dx^{\mu}/d\tau$ is (see Chapter 6.3 of [24])
$-2\pi f=k_{a}u^{a}\bigg{|}_{P}$
where $P$ is the location in space-time at which the event in question occurs
(i.e. emission/absorption). Dividing through by the Minkowski constant for the
speed of light $c$, we have
$-\frac{2\pi}{\lambda}=\sum_{\mu}k_{\mu}\cdot\frac{1}{c}\frac{dx^{\mu}}{d\tau}$
where $c=\lambda f$ and $\lambda$ is the wavelength of the photon
emitted/received by our observer. Since we require that our theory in flat
space-time reduce to special relativity within a localized region of our
respective inertial system (i.e. $d\chi^{2}\rightarrow c^{2}d\tau^{2}$ in this
localized region), it appears necessary to assume that, in our theory of
inertial centers, the wavelength of a photon with wave-vector $k^{a}$
emitted/received by an observer with ‘proper velocity’ $U^{a}$ is given by
$-\frac{2\pi}{\lambda}=k_{a}U^{a}\bigg{|}_{P}$
where we emphasize to the reader that in our theory the component form of the
‘four-velocity’ for our observer is affinely parametrized by $\chi$ (i.e.
$U^{\mu}=dx^{\mu}/d\chi$), in direct contrast to special and general
relativity for which the four-velocity of an observer would be affinely
parametrized by proper time $\tau$. Proceeding with our radial treatment, the
wave-vector for this photon is of the form,
$k^{\mu}\rightarrow\langle k^{t},k^{r},0,0\rangle$
And the wavelength observed by a radially traveling individual is given by
$-\frac{2\pi}{\lambda}=k^{a}U_{a}=-\Lambda
r^{2}k^{t}\frac{dt}{d\chi}+k^{r}\frac{dr}{d\chi}$
Using the Killing vector field in (16), we obtain the conservation law
($-\rho_{0}=k^{a}\rho_{a}$):
$\rho_{0}=\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)k^{t}+\sinh(\sqrt{\Lambda}t)k^{r}$
But for a photon, $0=k^{a}k_{a}\Longrightarrow k^{r}=\pm\sqrt{\Lambda}rk^{t}$.
So,
$k^{t}=\frac{\rho_{0}}{\sqrt{\Lambda}r[\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)]}$
where the positive root corresponds to light traveling away from $r=0$ and
negative to light traveling inward. Solving for the motion of the observer in
this particular inertial reference frame,
$-1=-\Lambda
r^{2}\bigg{(}\frac{dt}{d\chi}\bigg{)}^{2}+\bigg{(}\frac{dr}{d\chi}\bigg{)}^{2}$
And for an observer nearly at rest with respect to the inertial center about
which he/she orbits,
$\frac{dt}{d\chi}=\pm\frac{1}{\sqrt{\Lambda}r}$
Taking time to move forward, we find that
$\frac{2\pi}{\lambda}=\frac{\rho_{0}}{\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)}$
But from our earlier analysis, we found that a radially traveling photon
abides by the equation,
$r(t)=r_{0}\textrm{exp}(\pm\sqrt{\Lambda}t)=r_{0}[\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)]$.
Thus,
$\lambda\propto r$ (44)
Then for a light signal sent between two observers at rest in this inertial
frame, the redshift factor $z$ is given by the expression:
$z=\frac{\lambda_{\rm absorber}-\lambda_{\rm emitter}}{\lambda_{\rm
emitter}}=\frac{r(t_{{\rm absorber}})}{r(t_{{\rm emitter}})}-1$ (45)
Consequently, we see large shifts from emitters much closer to the center of
the system (assuming the absorber position remains the same).
Suppose, within the framework of this theory, we examine light propagating at
the scale of the inertial reference frame associated with our observable
universe. Then analogous to the manner in which the expression for the Hubble
parameter [13] is derived in the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric[31][32][33], we set the Hubble constant $H_{0}$ equal to
$H_{0}=\bigg{|}\frac{\dot{r}}{r}\bigg{|}=\sqrt{\Lambda}$ (46)
where $\dot{r}=dr/dt$. We notice that
$\frac{\ddot{r}}{r}=\Lambda>0$ (47)
producing a positive value for the acceleration of cosmological redshift and
therefore replicating the observed effects assumed to be the result of ‘dark
energy’ [12]. Thus, for any particular inertial reference frame, we should see
a shift in wavelength similar to the Hubble constant for the radial motion of
photons. We’ll use this conclusion later when we take the Pioneer anomaly as
support for our theory in the context of the inertial system associated with
the Milky Way. However, we should be concerned with our expression for
$\dot{r}/r$,
$\frac{\dot{r}}{r}=\pm\sqrt{\Lambda}$
as this theory then requires that we also have blueshifted objects if the
absorber is in fact closer than the emitter to the center of the inertial
frame within which the light signal in question propagates (i.e. negative
values for $\dot{r}/r$ and $z$). Nevertheless, if we apply our analysis to
objects at the scale of the Local Group [34][35] as in Table 1, we would
require an alternative interpretation for the observed significant blueshifts.
Whereas in current models, this blueshift would be interpreted as the Doppler
effect and thus for example as Andromeda (Messier 31) moving with velocity
toward the Milky Way [36], in our theory of inertial centers one could
interpret a portion of this blueshift (we say portion as the motion of our
observers within an inertial system also affects wavelength) as the
possibility that Andromeda is farther away from the inertial center associated
with the Local Group than we are. In support of these observations, we refer
to Table 2 where there appears to be an orientation associated with our
redshift values. For similar values of right ascension ($\pm 2$ h), we see a
steady change in wavelength shift from blue ($-$) to red ($+$) as one proceeds
from large positive values of declination to large negative values of
declination. In our theory, we would still need to consider differences in
radial distance associated with these objects and not just spatial
orientation. However, given that our distance modulus values are very much
similar for most of these entries ($\approx 24$ mag), it seems that this
interpretation for an orientation to the Local Group should be taken into
consideration. On the other hand, even if there does appear to be an
orientation associated with the Local Group, we must question why we have not
seen significant blueshifts at much larger scales. We will come back to these
ideas later in our work.
Until now, we have assumed that our coordinate time can take values between
$-\infty<t<\infty$ without explicitly examining the motion of particles in the
$t<0$ region. Reducing our analysis to solely radial motion away from the
poles, we analyze geodesic paths in Minkowski coordinates ($T,R$) first for
simplicity. However, we must be very clear that under our assumptions $T$ does
not represent inertial time as previously stated and in our theory of inertial
centers corresponds to an “unnatural” time coordinate for flat space-time
combining both physically observable clock time and spatial distance as
$cT=r\sinh(\sqrt{\Lambda}t)$. Then our equations of motion reduce to
$0=\frac{d^{2}T}{d{\sigma}^{2}}\indent\indent{\rm and}\indent\indent
0=\frac{d^{2}R}{d{\sigma}^{2}}$
leading to the straight lines that we expect in Minkowski coordinates:
$R=v\cdot(cT)+R_{0}$ (48)
where $v$ is a constant bounded by $|v|\leq 1$. We leave the physical
interpretation of the Minkowski constant $c$ in this theory of inertial
centers for the next section. However, using our transformation equations, we
find in radial Rindler coordinates
$r(t)=\frac{R_{0}}{\cosh(\sqrt{\Lambda}t)-v\sinh(\sqrt{\Lambda}t)}$ (49)
One immediately notices that for massive particles ($|v|<1$), both limiting
cases of $t\rightarrow\pm\infty$ result in the particle heading inward toward
the $r=0$ center point of the inertial system. This produces a scenario for
inertial motion of massive objects beginning at a center point in the far
past, coming to a maximum radial distance away at a later time, and then
heading back inward to eventually return to the same center point. In other
words, classically, all particles must also originate from the $r=0$ center
point of the particular inertial frame in question (see Figure 1).
### Reduction to special relativity
Taking the differential of both Rindler transformation equations:
$cdT=dr\sinh(\sqrt{\Lambda}t)+\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)dt$
$dR=dr\cosh(\sqrt{\Lambda}t)+\sqrt{\Lambda}r\sinh(\sqrt{\Lambda}t)dt$
where
$\sinh(\sqrt{\Lambda}t)=\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\frac{(\sqrt{\Lambda}t)^{5}}{5!}+\ldots$
$\cosh(\sqrt{\Lambda}t)=1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\frac{(\sqrt{\Lambda}t)^{4}}{4!}+\ldots$
Plugging in these expressions above, we find that
$cdT=dr\bigg{(}\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\ldots\bigg{)}+\sqrt{\Lambda}rdt\bigg{(}1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\ldots\bigg{)}$
$dR=dr\bigg{(}1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\ldots\bigg{)}+\sqrt{\Lambda}rdt\bigg{(}\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\ldots\bigg{)}$
If we localize our view of space-time such that all differential terms
$\mathcal{O}(\Lambda)=0$ (50)
we will then have
$cdT\approx\sqrt{\Lambda}\bigg{(}tdr+rdt\bigg{)}$ $dR\approx dr$
Further, we require for this local patch of space-time that
$tdr\ll rdt$ (51)
and our transformation equations reduce to
$\displaystyle cT\approx\sqrt{\Lambda}rt$ (52) $\displaystyle R\approx r$ (53)
with differential expressions
$\displaystyle cdT\approx\sqrt{\Lambda}rdt$ (54) $\displaystyle dR\approx dr$
(55)
For the observer remaining a radial distance $r=r_{0}$ away from the center of
his/her reference frame, the radial Rindler chart will be accurately
approximated by Minkowski coordinates under conditions (50) and (51) as $R=r$
and $T\propto t$ in this small $\sqrt{\Lambda}t$ limit. If one takes
$\sqrt{\Lambda}$ to be a fundamental property of each inertial system in
question, it must be that the measured Minkowski value for the speed of light
constant $c$ is a byproduct of the reference frame we wish to locally
approximate. In other words, in the Minkowski approximation for the radial
Rindler chart, an observer, located a radial distance $r=r_{0}$ away from the
inertial center point about which he/she orbits at $t=0$, will find:
$c=\sqrt{\Lambda}r_{0}$ (56)
If we treat $t=0$ as the point at which we determine the initial conditions
for the particle that we are observing (i.e. boundary conditions for position
and velocity), then our object will appear to move along straight line
geodesics for small values of $\sqrt{\Lambda}t$, but as we continue to observe
for longer periods of time, the properties of the radial Rindler chart which
we are approximating become more and more relevant.
In order for us to relate our theory of inertial centers to special
relativity, we must require that coordinate time in the radial Rindler chart
progress at the same rate as the proper time of an observer stationary
relative to the center of the inertial system within which we are analyzing
events (i.e. inertial time). In other words, $dt/d\tau=1$ for stationary
observers located at any particular radial distance $r=r_{0}$ away from an
inertial center. However, keep in mind that stationary observers do not follow
along geodesic paths from equation (7). Then, for observers which we can
consider as nearly stationary relative to the center of a particular inertial
system (i.e. $r={\rm constant}$), we have $d\chi^{2}\rightarrow
c^{2}d\tau^{2}$ where $c$ is given by (56), effectively ensuring that our
coordinate time $t$ progresses at the same rate as the proper time $\tau$ of a
stationary observer. Consequently, we find under (50) and (51) in addition to
our stationary observer assumption that our line element can be treated
approximately as
$-c^{2}d\tau^{2}=-c^{2}dT^{2}+dR^{2}+R^{2}d\Omega^{2}$
And therefore in this “stationary” limit (relative to the inertial center),
when not operating about the poles, we come upon time- and spatial-
translational invariance within our local region where the origin of our
coordinate system is located at the inertial center of this reference frame.
Because it appears that we have now recovered time- and spatial-translational
invariance in this limit, we can naively assume that we have the ability to
translate our coordinate system in any way we prefer (e.g. moving the center
of our reference frame away from the inertial center). In other words, we can
approximate when our motion is not near the poles of our global inertial
system with the metric:
$-c^{2}d\tau^{2}=-c^{2}dT^{2}+dX^{2}+dY^{2}+dZ^{2}$
where $X=R\sin{\theta}\cos{\phi}+X_{0}$, $Y=R\sin{\theta}\sin{\phi}+Y_{0}$,
and $Z=R\cos{\theta}+Z_{0}$ for $R>0$ and $X_{0}$, $Y_{0}$, and $Z_{0}$ are
constants (see Figure 2). Thus, this local stationary approximation reduces
our theory to special relativity (see Chapter 4.2 of [24]). Additionally, we
see from these transformation equations that the Minkowski chart is not able
to cover all of space-time in our theory of inertial centers (i.e. $r<0$
values are neglected by the Minkowski chart). We will come back to this idea
later in our work. Physically, our localization conditions require that the
time-scale constant $\sqrt{\Lambda}$ for our inertial systems be small enough
such that we as observers here on Earth would observe only the stationary
limit in our “everyday lives”. Of course, this statement also assumes that we
are nearly stationary relative to the inertial center about which we orbit
taken in our next section to be the center of the Milky Way. Yet, given our
redshift analysis, it appears that the Hubble constant provides the necessary
scale [20] for this requirement from (46).
Furthermore, it is clear that in order to express $d\chi^{2}$ in terms of the
proper time of the observer whose motion we wish to analyze in the relevant
inertial frame (i.e. the particular system in which the observer can be
treated as a point-particle orbiting an inertial center point) and still have
our invariant interval reduce to $c^{2}d\tau^{2}$ in our stationary limit
where the observer’s distance to the inertial center point about which he/she
orbits is very nearly constant, we must have
$d\chi^{2}=\Lambda r^{2}d\tau^{2}$ (57)
where $r$ represents the physical distance to the center point of the inertial
system in question. In addition, according to our theory of inertial centers,
the value that we use in special and general relativity for the constant $c$
in our massive geodesic equations relies on the particular inertial reference
frame in which we can regard the object whose local behavior we wish to
examine as a point-particle orbiting an inertial center point (in special and
general relativity, $g_{ab}u^{a}u^{b}=-c^{2}$ where $u^{\nu}=dx^{\nu}/d\tau$).
Therefore, the local Minkowski constant that we measure for the speed of light
is dependent upon our position in our most local inertial reference frame
(i.e. the frame in which we can be treated as a point-particle orbiting an
inertial center). As well, for two different stationary observers orbiting
about the same inertial center point, the clock of the observer located closer
to their shared inertial center will appear to run faster when examined from
the perspective of the more distant observer. Meaning, not only do observable
clock rates differ due to the relative velocity of individuals as in special
relativity, but they also differ due to the difference in distance of each
individual from the inertial center about which each orbits. Thus, initially
synchronized clocks that are stationary relative to the shared inertial center
about which both orbit do not remain synchronized if they are located at
different distances from this inertial center.
### Application to a local gravitational system
Before we present the approximations of this section, it seems necessary to
provide remarks as to how gravitation fits into the theory of inertial
centers. The formulation of our theory of inertial centers detailed in
previous sections deals with the structure of flat space-time ignoring
possible issues with curvature. So what we are really asking is the following.
How does an object move in flat space-time when absolutely no external forces,
fields, etc. are present to affect said object? Nevertheless, we still assume
within our model that all objects cause curvature in space-time due to their
intrinsic energy-momentum content, but this curvature we take to be a local
effect within the far larger inertial system that we are attempting to
redefine in this work. However, as presented in the previous section
“Reduction to special relativity”, we claim that locally the structure of flat
space-time within our redefined inertial reference frames reduces to the flat
space-time of Einstein’s theory of special relativity as long as the observer
remains at very nearly the same distance away from the inertial center about
which he/she orbits. As discussed above, we term this the “local stationary
approximation”, and in this approximation the observer sees space-time locally
within a region located at the same radial distance as the observer away from
the inertial center about which he/she orbits as approximately special
relativistic, where the speed of light in this confined region of space-time
is given by (56) and our affine parameter reduces to $\chi=c\cdot\tau$. If one
then considers the influence of an object on the structure of space-time in
this local region where special relativity approximately holds, we assume in
the theory of inertial centers that this object will bend space-time locally
according to Einstein’s general relativity. Meaning, gravity remains a
consequence of local curvature in space-time in the theory of inertial
centers. However, when we take a perspective far away from our massive object
so that the curvature this object induces in space-time looks approximately
insignificant for the purpose of examining motion at these larger scales, we
claim that we can treat the object very nearly as a particle in a flat space-
time inertial reference frame as formulated above, where the inertial motion
of the object is dictated by the geodesics of our model. But again, if we
focus our attention on the local region around the massive object while
disregarding the existence of the larger inertial system, we will still
observe the effects resulting from the curvature the object induces in space-
time and thus the gravitational effects it has on other objects around it
(i.e. general relativity holds locally).
In the following, we give an approximate method under our stationary
localization conditions as described in the previous section for determining a
potential implication of our theory of inertial centers with regard to the
observables of a local gravitational system. We take the view that the
Schwarzschild metric [37] applies in the small $\sqrt{\Lambda}t$ limit within
confined regions of our inertial reference frame for observers nearly
stationary with respect to the inertial center about which they orbit as one
would expect from the well-established accuracy of general relativity[38].
Below, our “mixing” of the Schwarzschild metric with the radial Rindler chart
is an approximate way of expressing the fact that locally in the inertial
reference frame of our theory the observer can treat the speed of light as
nearly constant if one were to remove the massive object and work in flat
space-time (i.e. set $M=0$ in the Schwarschild metric) as well as the idea
that general relativity holds locally. But the observer must always keep in
mind that the inertial frame of Einstein’s special relativity is actually not
an inertial frame of the theory of inertial centers, and thus this local
system is located in the more globally relevant inertial system where the
speed of light is not constant. Then if we no longer take $M=0$ (i.e. return
the massive object to the local system), we should still expect gravitation as
formulated by Einstein when we examine locally and disregard the larger
inertial system from our model. In other words, the Schwarzschild metric still
applies locally in the theory of inertial centers when examining motion about
an uncharged non-rotating spherically symmetric massive object. However, if we
move our observer farther and farther away from the gravitational source, the
local limit will no longer apply since we have to take into consideration the
structure and properties of the larger inertial system as well as the fact
that in our theory objects move inertially along geodesics different from
those of Einstein’s theories of special and general relativity, even though
locally these different geodesics appear to be very similar (i.e.
$d\chi^{2}\rightarrow c^{2}d\tau^{2}$ in the local stationary limit).
Additionally, we need some approximate way to take into account the fact that
the speed of light is not constant throughout the inertial system while still
keeping in mind that locally the observer may experience gravitational effects
from a massive object nearby. We admit that the methods in this section are
rough at best, but it is our hope that in future work we will be able to model
far more accurately this transition from the local approximation of general
relativity to the more global application of the theory of inertial centers.
Our metric equation takes the form of the Schwarzschild solution:
$-d\chi^{2}=-Bc^{2}dT^{2}+\frac{1}{B}dR^{2}+R^{2}d\Omega_{l}^{2}$
where
$B(R)=1-\frac{2MG}{c^{2}R}\indent\indent{\rm and}\indent\indent
d{\Omega_{l}}^{2}=d{\theta_{l}}^{2}+d{\phi_{l}}^{2}\sin^{2}{\theta_{l}}$
($T,R,\theta_{l},\phi_{l}$) describe our local gravitational system and
($t,r,\theta,\phi$) refer to the global inertial reference frame within which
the local system is located. In other words, we assume that the observer takes
the coordinate transformations away from the inertial center to cover local
space-time in the same manner as outlined in our previous section. Meaning,
ignoring the existence of the massive object
$\displaystyle cT=r\sinh(\sqrt{\Lambda}t)$ $\displaystyle
X=r\cosh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi}+X_{0}$ $\displaystyle
Y=r\cosh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi}+Y_{0}$ $\displaystyle
Z=r\cosh(\sqrt{\Lambda}t)\cos{\theta}+Z_{0}$
where $X_{0}$, $Y_{0}$, and $Z_{0}$ are constants, $\sqrt{\Lambda}t$ is taken
to be small, and we only examine the $r>0$ region of the inertial system.
Thus, $T\approx t$ where we employ equation (56) for the local stationary
limit. Then taking into account the existence of this massive object in the
local region with $M=0$ flat space-time Minkowski coordinates given by
($T,X,Y,Z$) in the local stationary limit, we employ the Schwarzschild metric
noting that our affine parameter is approximately given by $\chi=c\cdot\tau$.
As well, $c$ refers to the speed of light in the local system at the point in
the global inertial frame where the observer and photon meet, and $M$ is the
mass of the object. Then we will proceed through a standard treatment of the
gravitational redshift for the Schwarzschild metric (see Chapter 6.3 of [24]).
However, we keep the Minkowski constant $c$ in all of our expressions as we
intend to investigate the implications of the variable nature of the speed of
light in flat space-time from our theory of inertial centers. For an observer
and photon both traveling radially in this local system
($U^{\mu}=dx^{\mu}/d\chi\rightarrow\langle U^{T},U^{R},0,0\rangle$,
$k^{\mu}\rightarrow\langle k^{T},k^{R},0,0\rangle$), we have
$-\frac{2\pi}{\lambda}=-Bc^{2}U^{T}k^{T}+\frac{1}{B}U^{R}k^{R}$
where $\lambda$ is the wavelength measured by our observer. Applying
conservation laws for $U^{a}$ and $k^{a}$ using the time-translationally
invariant Killing vector field for the Schwarzschild metric,
$\xi^{a}=(\partial/\partial T)^{a}$:
$U^{T}=\frac{E}{Bc^{2}}\indent\indent{\rm and}\indent\indent
k^{T}=\frac{\rho_{0}}{Bc^{2}}$
And taking into account the motion of the observer and photon ($0=k^{a}k_{a}$
and $-1=U^{a}U_{a}$)
$U^{R}=\pm\sqrt{\bigg{(}\frac{E}{c}\bigg{)}^{2}-B}\indent\indent{\rm
and}\indent\indent k^{R}=\pm\frac{\rho_{0}}{c}$
Plugging into our expression for the observed wavelength of the photon,
$\frac{2\pi}{\lambda}=\frac{E\rho_{0}}{B}\bigg{[}\frac{1}{c^{2}}-\bigg{(}\pm\bigg{)}\bigg{|}_{{\rm
photon}}\cdot\bigg{(}\pm\frac{1}{Ec}\sqrt{\bigg{(}\frac{E}{c}\bigg{)}^{2}-B}\bigg{)}\bigg{|}_{{\rm
observer}}\bigg{]}$ (58)
where $(\pm)|_{{\rm photon}}$ and $(\pm)|_{{\rm observer}}$ refer to the
photon/observer traveling radially outward/inward ($+/-$) in the local system
(in $R$). If we assume the observer to be nearly at rest in the local frame
($U^{T}\gg U^{R}$), then $B=(E/c)^{2}$ and expression (58) reduces to
$\lambda\propto c\sqrt{B}$
where in the following we approximate in the small $\sqrt{\Lambda}t$ limit
with our equation for the local speed of light in the inertial reference frame
(56). We employ this “trick” as the Schwarzschild metric is just an
approximation in our model valid under confined regions of the particular
inertial system within which the gravitational source is located. However, one
should be able to experimentally detect with an apparatus of the necessary
sensitivity that these photons progress along the geodesics of our theory of
inertial centers (and not straight lines) bent locally due to the curvature in
space-time caused by our massive object $M$. Therefore, we find a slight
modification to the Schwarzschild redshift factor:
$z=\frac{\lambda_{\rm absorber}-\lambda_{\rm emitter}}{\lambda_{\rm
emitter}}=\frac{r_{{\rm absorber}}}{r_{{\rm
emitter}}}\sqrt{\frac{1-\frac{2MG}{R_{{\rm absorber}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm absorber}}}}{1-\frac{2MG}{R_{{\rm emitter}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm emitter}}}}}-1$ (59)
where $r_{{\rm absorber/emitter}}$ refers to the radial position of the
absorber/emitter in the inertial reference frame (i.e. relative to the
inertial center) and $R_{{\rm absorber/emitter}}$ to the radial position
relative to the center of our massive object $M$ in the local gravitational
system. Consequently, we should see a modified redshift factor consisting of
the Schwarzschild expression (Chapter 6.3 of [24]) scaled by the solution
found in our flat space-time vacuum analysis.
Let us then apply this analysis to the case of a space probe traveling out of
our solar system where the $r_{\rm absorber}/r_{\rm emitter}$ factor should
have a larger impact on our observations. In our crude example, we treat both
the probe and the absorber as essentially stationary. Referring to expression
(59) for observers at rest, the absorber wavelength in terms of the emitter is
$\lambda_{{\rm absorber}}|_{{\rm Modified}}=\lambda_{{\rm
emitter}}\cdot\frac{r_{{\rm absorber}}}{r_{{\rm
emitter}}}\sqrt{\frac{1-\frac{2MG}{R_{{\rm absorber}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm absorber}}}}{1-\frac{2MG}{R_{{\rm emitter}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm emitter}}}}}$
where our $R$ values in this example refer to local radial distances away from
the center of the Sun and $r$ to distances away from the center of the Milky
Way. For the ratio between the Schwarzschild wavelength and the modified value
above assuming the term under the square root remains approximately the same
for small changes in $r$ relative to changes in $R$, we have
$\frac{\lambda_{{\rm absorber}}|_{{\rm Schw}}}{\lambda_{{\rm absorber}}|_{{\rm
Modified}}}\approx\frac{r_{{\rm emitter}}}{r_{{\rm absorber}}}$
where
$\lambda_{\rm absorber}|_{\rm Schw}=\lambda_{\rm
emitter}\cdot\sqrt{\frac{1-\frac{2MG}{R_{\rm
absorber}}\cdot\frac{1}{c^{2}_{\rm absorber}}}{1-\frac{2MG}{R_{\rm
emitter}}\cdot\frac{1}{c^{2}_{\rm emitter}}}}$
Since Pioneer 10 was on course to travel away from the center of the Milky Way
in the general direction of Aldebaran[21], we can approximate the path of our
photon as nearly a radial one in our galactic inertial reference frame.
Therefore, if we naively ignore the two-way nature of the Doppler residuals,
$r_{\rm absorber}\approx r_{\rm emitter}\cdot e^{-\sqrt{\Lambda}\Delta t}$
where $\Delta t$ is the time it takes the massless particle to travel from the
emitter to the absorber, assuming time measured by the emitter progresses at
nearly the same rate as that measured by the absorber in this short distance
calculation (i.e. $\tau_{a}\approx\tau_{e}=t$). Notice, these photons
travelled inward for Pioneer 10, so the root is negative. Plugging into our
expression above, the fractional difference in wavelength predicted here on
Earth is approximately
$\frac{\lambda_{{\rm absorber}}|_{{\rm Schw}}}{\lambda_{{\rm absorber}}|_{{\rm
Modified}}}=\frac{1}{e^{-\sqrt{\Lambda}t}}\approx 1+\sqrt{\Lambda}t$
to first order where we assume that our modified expression coincides with our
experimental values. Then the observed “time acceleration” reported in[39] and
[40] provides an estimate for the time-scale of our galaxy of
$\sqrt{\Lambda}|_{{\rm MW}}=2.92\times 10^{-18}$ s-1. The consistency of this
value with that of the Hubble constant[20] lends support to the argument that
the time-scale $\sqrt{\Lambda}$ is universal for all inertial reference frames
as we had implicitly assumed from our proposed form of the affine parameter
presented in our introduction. However, further experiment is necessary in
order to verify this claim.
Clearly, the two-way nature of the Doppler residuals of the Pioneer
experiments as well as the difference in clock rates for varying positions
within an inertial system in our theory will complicate our analysis further.
However, the purpose of this section is to illuminate to the reader the idea
that we may have evidence from experiments within our own solar system that
support the relevance of this theory of inertial centers and suggest that
possibly all inertial reference frames as defined within this theory abide by
the same fundamental time-scale constant $\sqrt{\Lambda}$. Nevertheless,
others have argued as in [41] that the Pioneer anomaly is a consequence of the
mechanics of the spacecrafts themselves instead of evidence of “new physics”.
Therefore, to gain more support for the theory of inertial centers, we must
address in future work not only the two-way nature of the Doppler residuals as
both Pioneer 10 and Pioneer 11 appear to report blueshifted wavelengths even
when they traveled in opposite directions with respect to the galactic
inertial frame of reference but also the possibility that our theory can
succinctly explain the other astrometric Solar System anomalies outlined in
[23] and [40].
### Quantization of a real scalar field
We begin our extension into quantum field theory from the covariant form of
the Klein-Gordon equation [42]:
$\nabla_{a}\nabla^{a}\phi-\mu^{2}\phi=0$
where $\nabla_{a}$ is the derivative operator compatible with the metric
$g_{ab}$ (i.e. $\nabla_{a}g_{bc}=0$), $\mu=mc/\hbar$, $m$ is the mass
associated with our field, and $\hbar$ is the reduced Planck constant. First,
we explore how one can intuitively arrive at this equation of motion given our
classical assumptions. In special relativity, we have
$-m^{2}c^{2}=p^{a}p_{a}=\sum_{\nu,\beta}\eta_{\nu\beta}p^{\nu}p^{\beta}$
where $p^{\nu}=m\cdot dx^{\nu}/d\tau$ and $\eta_{\nu\beta}$ refers to the
Minkowski metric components. Making the substitution
$p^{a}\rightarrow-i\hbar\nabla^{a}$, we come upon the Klein-Gordon equation
above for a scalar field.
However, in our theory of inertial centers, the equation of motion in terms of
‘momentum’ is given by
$-m^{2}=p^{a}p_{a}$
where now $p^{\nu}=m\cdot dx^{\nu}/d\chi$ and so we have a major difference in
our ‘momentum’ terms. In contrast with our experience in relativity, the
‘four-velocity’ for massive particles in our theory is parametrized by $\chi$
and not by proper time $\tau$. Unfortunately, there does not appear to be a
natural operator substitution for $dx^{\nu}/d\chi$. Yet, if we use expression
(57), we have a potential extension of the Klein-Gordon equation when
analyzing motion at a particular scale. It appears that one should substitute
$c\rightarrow\sqrt{\Lambda}r$ to find
$\Box\phi-\tilde{\mu}^{2}r^{2}\phi=0$ (60)
where $\tilde{\mu}=m\sqrt{\Lambda}/\hbar$ and $\Box=\nabla_{a}\nabla^{a}$ is
the Laplace-Beltrami operator. Notice that in our equation of motion we have
explicit reference to the particular inertial reference frame in which we are
analyzing the behavior of the field as opposed to the Klein-Gordon equation
which has no explicit reference to any inertial system. This seems to be
consistent with the idea that the proper time is not the invariant quantity
associated with our theory of inertial centers, and therefore our choice of
proper time reflects the choice of scale in which we must work to analyze the
progression of our field within this inertial system. One can also apply this
substitution in an analogous manner to other equations of motion/Lagrangians,
yet in the following we will only address the simple case of a free real
massive scalar field.
Then, as outlined in Chapter 4.2 of [43] and briefly reviewed in Appendix D,
we must “slice” our manifold $M$ into space-like hypersurfaces each indexed by
$t$ ($\Sigma_{t}$). For our radial Rindler chart, the future-directed unit
normal to each $\Sigma_{t}$ is given by
$n^{a}=\frac{1}{\sqrt{\Lambda}|r|}\bigg{(}\frac{\partial}{\partial
t}\bigg{)}^{a}$ (61)
where the absolute value is necessary to keep $n^{a}$ future-directed for all
values of $r:0<r^{2}<\infty$, allowing for positive and negative values. We
will interpret the physical significance of this relaxation on the domain
restrictions for our radial coordinate later in our analysis. We see that our
hypersurface can be decomposed into the union of two surfaces for each of the
Rindler wedges ($r<0$ and $r>0$), and thus the inner product of our Klein-
Gordon extension is given by
$\displaystyle(\phi_{1},\phi_{2})=-i\Omega([\bar{\phi}_{1},\bar{\pi}_{1}],[\phi_{2},\pi_{2}])$
$\displaystyle=-i\int_{\Sigma_{{\rm I}}\cup\Sigma_{{\rm
II}}}d^{3}x\sqrt{|h|}[\phi_{2}n^{a}\nabla_{a}\bar{\phi}_{1}-\bar{\phi}_{1}n^{a}\nabla_{a}\phi_{2}]$
(62)
where the bar symbol indicates complex conjugation (i.e. $\bar{\phi}_{i}$ is
the complex conjugate of $\phi_{i}$), $\Sigma_{0}=\Sigma_{{\rm
I}}\cup\Sigma_{{\rm II}}$ is the union of these two radial Rindler wedge
space-like hypersurfaces, $n^{a}$ is the unit normal to our space-like
hypersurface $\Sigma_{0}$, $h_{ab}$ is the induced Riemannian metric on
$\Sigma_{0}$ ($h=\mathrm{det}(h_{\nu\beta})$; $(h_{\nu\beta})$ denotes the
matrix associated with these Riemannian metric components), and $\Omega$
refers to the symplectic structure for our extension of the Klein-Gordon
equation.
We should be rather concerned considering the discontinuous nature of the
time-orientation of $n^{a}$ (the absolute value is not a smooth function) as
well as the undefined behavior of our unit normal for $r=0$, the location of
our inertial center. However, given the solutions we find below, it seems to
be an important question whether or not we are forced to treat each Rindler
wedge separately as its own globally hyperbolic space-time or the combination
of these wedges as the entire space-time over which we must analyze solutions
to our extension of the Klein-Gordon equation. The difference between these
two formulations will be that in the former we must define separate creation
and annihilation operators for each wedge as in the analysis of [44]. Whereas
in the latter, we have one set of creation and annihilation operators for all
values of $r$ over the range: $0<r^{2}<\infty$, where $r$ can take both
positive and negative values. It also seems likely that a greater
understanding of our inertial centers and their physical significance (i.e.
how are these inertial centers established?) will provide far more insight
into the proper way to treat this situation. In this paper, however, we assume
the latter approach requiring that we use all values of $r$ (positive and
negative) to cover our inertial reference frame and naively ignore the issues
with $r=0$ mentioned above. This approach seems to be far more consistent with
the idea implicit in our theory of inertial centers that the radial Rindler
chart covers the entire flat space-time manifold for the inertial system in
question, except of course for the location of each of our inertial centers
(i.e. $r=0$). We find that our inner product is given by
$\displaystyle(\phi_{1},\phi_{2})=-i\int_{-\infty}^{\infty}dr\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\bigg{[}r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin{\theta}\bigg{(}\frac{\phi_{2}}{\sqrt{\Lambda}|r|}\partial_{t}\bar{\phi}_{1}-\frac{\bar{\phi}_{1}}{\sqrt{\Lambda}|r|}\partial_{t}\phi_{2}\bigg{)}\bigg{]}\bigg{|}_{t=0}$
$\displaystyle=-\frac{i}{\sqrt{\Lambda}}\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\int_{0}^{\infty}rdr\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bigg{(}\phi_{2}\partial_{t}\bar{\phi}_{1}-\bar{\phi}_{1}\partial_{t}\phi_{2}\bigg{)}$
$\displaystyle-\int_{-\infty}^{0}rdr\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bigg{(}\phi_{2}\partial_{t}\bar{\phi}_{1}-\bar{\phi}_{1}\partial_{t}\phi_{2}\bigg{)}\bigg{]}\bigg{|}_{t=0}$
(63)
Our remaining task reduces to solving for solutions ($\phi_{i}$) to our
extension of the Klein-Gordon equation (60). From Appendix F which utilizes
[45], [46], [47], [48], and [49], we find
$\phi_{\alpha,l,m}=\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}\cdot\sqrt{1-\eta^{2}}\cdot
P_{l}^{-2i\alpha}(\eta)\cdot\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}\cdot
Y_{l}^{m}(\theta,\phi)$ (64)
where $\eta=\tanh(\sqrt{\Lambda}t)$ and $\rho=\sqrt{\tilde{\mu}}r$.
$Y_{l}^{m}$ is the spherical harmonic of degree $l$ and order $m$. We maintain
convention and use $m$ to denote the order of $Y_{l}^{m}$. However, this $m$
is a quantum number very different from the mass of our scalar field. The mass
term is contained solely in our expression for $\tilde{\mu}$. $K_{i\alpha}$ is
the Macdonald function (modified Bessel function) of imaginary order $\alpha$.
$P_{l}^{-2i\alpha}$ is the Legendre function of degree $l$ and imaginary order
$-2\alpha$.
Notice, we allow $K_{i\alpha}(\frac{\rho^{2}}{2})/\rho$ to have domain:
$0<\rho^{2}<\infty$ where $\rho$ can take both positive and negative values.
Physically, this interpretation requires the existence of the field in both
the negative and positive $r$ regions of the inertial system which brings us
back to the discussion earlier in this section of our concern with $n^{a}$.
From [50], the limiting behavior of $K_{i\alpha}$ expressed as
$\lim_{y\rightarrow
0^{+}}K_{i\alpha}(y)=-\bigg{(}\frac{\pi}{\alpha\sinh(\alpha\pi)}\bigg{)}^{1/2}\bigg{[}\sin(\alpha\ln(y/2)-\phi_{\alpha,0})+\mathcal{O}(y^{2})\bigg{]}$
where $\phi_{\alpha,0}=\arg\\{\Gamma(1+i\alpha)\\}$ and $\Gamma(z)$ is the
gamma function along with
$\lim_{y\rightarrow\infty}K_{i\alpha}(y)=\bigg{(}\frac{\pi}{2y}\bigg{)}^{1/2}e^{-y}\bigg{[}1+\mathcal{O}\bigg{(}\frac{1}{y}\bigg{)}\bigg{]}$
shows that $K_{i\alpha}(\frac{\rho^{2}}{2})/\rho$ oscillates for small
$|\rho|$ when $\alpha\neq 0$ and exponentially decays for large $|\rho|$. In
addition, from Figure 3, we see that our radial ‘wave function’ spreads out
away from $\rho=0$ for larger ‘momentum’ values of $\alpha$, allowing for
oscillatory behavior at larger values of $|\rho|$ and thus an increased
likelihood of observing quanta farther away from the inertial center of the
reference frame in question.
Our Heisenberg field operator can be expanded in the following manner (see
Chapters 3.1 and 3.2 of [43]):
$\displaystyle\hat{\Phi}(t,r,\theta,\phi)=\int_{0}^{\infty}d\alpha\sum_{l=0}^{\infty}\sum_{m=-l}^{l}[\phi_{\alpha,l,m}\hat{a}(\bar{\phi}_{\alpha,l,m})+\bar{\phi}_{\alpha,l,m}\hat{a}^{\dagger}(\phi_{\alpha,l,m})]$
$\displaystyle=\int_{0}^{\infty}d\alpha\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}\cdot\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}\cdot\sqrt{1-\eta^{2}}\cdot\bigg{[}\hat{a}(\bar{\phi}_{\alpha,l,m})P^{-2i\alpha}_{l}(\eta)Y_{l}^{m}(\theta,\phi)$
$\displaystyle+\hat{a}^{\dagger}(\phi_{\alpha,l,m})P^{2i\alpha}_{l}(\eta)\bar{Y}_{l}^{m}(\theta,\phi)\bigg{]}$
(65)
where the annihilation and creation operators in terms of our inner product
are
$\displaystyle\hat{a}(\bar{\phi}_{\alpha,l,m})=(\phi_{\alpha,l,m},\hat{\Phi})$
$\displaystyle\hat{a}^{\dagger}(\phi_{\alpha,l,m})=-(\bar{\phi}_{\alpha,l,m},\hat{\Phi})$
and $\\{\phi_{\alpha,l,m}\\}$ comprise an orthonormal basis of the “positive
frequency” solutions to the extended version of the Klein-Gordon equation for
the theory of inertial centers. For a real scalar field, these annihilation
and creation operators satisfy the commutation relations (bosonic statistics):
$[\hat{a}(\bar{\phi}_{\alpha,l,m}),\hat{a}^{\dagger}(\phi_{\alpha^{\prime},l^{\prime},m^{\prime}})]=(\phi_{\alpha,l,m},\phi_{\alpha^{\prime},l^{\prime},m^{\prime}})=\delta(\alpha-\alpha^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}$
A very important point for the reader to take away from our analysis in this
section is that our field operator as defined in (Quantization of a real
scalar field) exists in both the $r>0$ and $r<0$ portions of space-time. In
other words, we take space-time to be comprised of both the $r>0$ and $r<0$
regions of the inertial system, and thus the Minkowski chart is not able to
cover all of space-time in our theory. It then appears that a potential
explanation for the matter/antimatter asymmetry in our observable universe
within the framework of our theory of inertial centers would be that there
exists a parallel region of each inertial system embodied mathematically above
by the existence of our field operator in the hypothetical $r<0$ region of
space-time. Logically, if we exist in our region of space-time with an
imbalance toward matter, one would then assume that in this parallel region
there exists an imbalance in favor of antimatter as the total charge of the
field throughout all of space-time should be conserved. We are, of course,
operating under the assumption that the solutions to our equation of motion
extend in a similar manner as in special relativity when one allows for
complex fields of non-zero spin (e.g. solutions to a Dirac equation [51]
extension are also solutions to our Klein-Gordon extension) since we should
not worry about antiparticles with a real scalar field. Therefore, we must
extend our work on the theory of inertial centers to incorporate spin in order
to see the full significance of this possible explanation for the
matter/antimatter asymmetry in our observable universe.
To conclude our discussion, we assume throughout the rest of this section that
$\sqrt{\Lambda}$ is a universal constant for all inertial systems, taken to be
the Hubble constant as proposed in our introduction, and imagine that there
exists an observer very near to an inertial center point such that his/her
motion in this particular reference frame is approximately stationary (i.e.
spatial ‘four-velocities’ are very much outweighed by ‘velocity’ in time,
$dt/d\chi$). Then from our classical analysis of geodesic paths, our observer
experiences a radial acceleration according to (10) of
$\frac{d^{2}r}{dt^{2}}=-\Lambda r$
where $t$ coincides with the proper time $\tau$ for our nearly stationary
observer in this system. However, say we wish to understand our observer’s
motion not in terms of his/her proper time in this particular inertial frame
but instead in terms of his/her proper time in an external inertial frame of
reference where these two different systems do not share a common inertial
center point. We know that our invariant interval is given by
$-d\chi^{2}=-\Lambda r_{l}^{2}d\tau_{l}^{2}=-\Lambda r_{e}^{2}d\tau_{e}^{2}$
where the $e$ ($l$) subscript refers to quantities in the external (local)
inertial reference frame. Assuming our observer is nearly stationary in both
inertial systems (i.e. coordinate times for each system coincide with proper
times within each reference frame respectively), his/her clock in the local
frame progresses by
$\frac{dt_{l}}{dt_{e}}=\frac{c_{e}}{c_{l}}$ (66)
Thus,
$\frac{d}{dt_{l}}=\frac{c_{l}}{c_{e}}\cdot\frac{d}{dt_{e}}$
where the $c$’s refer to the Minkowski constants for each particular reference
frame (56). Plugging in above,
$\frac{d^{2}r_{l}}{dt_{e}^{2}}=-\Lambda_{{\rm eff}}\cdot r_{l}$ (67)
where $\sqrt{\Lambda_{{\rm eff}}}=\sqrt{\Lambda}\cdot c_{e}/c_{l}$.
According to Newtonian mechanics which is a good approximation here since we
assume our observer is nearly stationary in the local inertial system, one
would attribute this radial acceleration to a ‘force’ (even though we know
that there really is no force here), and associated with this ‘force’ is a
potential ($\vec{F}=-\vec{\nabla}V$; see Chapters 1 and 2 of [52]). So for the
acceleration above, one would assume while working in Newtonian mechanics that
there exists a potential causing this movement of the form:
$V=\frac{1}{2}m\Lambda_{{\rm eff}}\cdot r_{l}^{2}$ (68)
Then our Hamiltonian ($H=T+V$; see Chapter 8 of [52]) for this system is given
by
$H=\frac{p^{2}}{2m}+\frac{1}{2}m\Lambda_{{\rm eff}}\cdot r_{l}^{2}$ (69)
where $m$ is the mass of our observer and $T=p^{2}/2m$ is the kinetic energy
associated with his/her motion as observed in the external frame. If our
observer is on the order of $10^{-15}$ m[53] away from his/her local inertial
center and $c_{e}$ is found in the external frame to be $\approx 3.0\times
10^{8}$ m/s[54], we find $\sqrt{\Lambda_{{\rm eff}}}\sim 10^{23}$ s-1. We
remark for the reader less acquainted with nuclear theory that the Hamiltonian
above is referred to as the isotropic harmonic oscillator and was used as a
starting point for nuclear shell models due to its ability to reproduce the
“magic numbers” associated with stable configurations of nucleons within the
nucleus (see Chapter 4 of [55] and Chapter 3.7 of [56]). In addition, the
energy scale associated with the Hamiltonian above (i.e.
$\hbar\cdot\sqrt{\Lambda_{{\rm eff}}}\sim 10^{8}$ eV) is of a similar order as
the scale inputted into these isotropic harmonic oscillator models for the
magnitude of the nuclear ‘force’ [57]. Thus, our ability to replicate the same
features as those of the simplest nuclear shell model compels us to ask the
following question with regard to the theory of inertial centers: Is there an
inertial center point at the center of the nucleus of every atom?
## Limitations of the study, open questions, and future work
There is a plethora of data for us to critically investigate the validity of
this theory of inertial centers. Nevertheless, we have chosen to leave these
detailed investigations for future work as the purpose of this paper is to lay
out the theoretical foundations to illicit these types of rigorous comparisons
with experiment for all aspects of our model. As we have mentioned briefly at
certain points within our discussion, there are many open questions that must
be addressed. The most pressing of these appears to be how to explain the
cosmic microwave background (CMB) within our theory of inertial centers. One
may be tempted to immediately point to the Fulling-Davies-Unruh effect[58] as
the source of this cosmic radiation since the Unruh effect predicts that an
“accelerating” observer in Minkowski vacuum, who can be described by orbits of
constant spatial coordinate in the classic Rindler chart, detects black-body
radiation that appears to be nearly homogeneous and isotropic with predicted
anisotropies due to the orientation of this observer throughout his/her
“accelerated” path [59]. However, we must keep in mind that the scale
associated with the temperature of Unruh radiation [58][44]
$T=\frac{\hbar a}{2\pi k_{B}c}$
requires $a\sim 10^{20}$ m/s2 to produce a temperature on the order of the
CMB, $T\approx 2.7$ K [60], where $k_{B}\approx 1.38\times 10^{-23}$ J K-1 is
Boltzmann’s constant, $\hbar\approx 1.05\times 10^{-34}$ J s[54], and $a$ is
the proper acceleration of the observer. If we approximate the original
analysis of [58] by working in 1+1 space-time (i.e. 1 time and 1 spatial
dimension), the acceleration would be proportional to the inverse of $r=r_{0}$
for observers moving along orbits of constant $r$ [44]. This then requires
$r_{0}=c^{2}/a\sim 10^{-3}$ m for the CMB temperature scale, which clearly
makes no sense since we would be millimeters away from the center of our
observable universe. Nevertheless, the analysis used to derive the Unruh
effect implicitly operates under the assumption of the validity of special
relativity in flat space-time and therefore takes $d\chi^{2}=c^{2}d\tau^{2}$.
Yet, as we have emphasized repeatedly above, in our theory of inertial
centers, the invariant interval associated with the metric is given in terms
of proper time by $d\chi^{2}=\Lambda r^{2}d\tau^{2}$. Therefore, we must
extend these ideas to apply to our model where we are observers existing
within multiple inertial systems (universe $\rightarrow\ldots\rightarrow$
Local Group $\rightarrow$ Milky Way). In addition, for our situation, this
radiation would not be interpreted physically as due to the “acceleration” of
the observer as in the case of [58], but instead one would have to think of
this effect as simply the result of the restriction of the Minkowski vacuum to
each of the radial Rindler wedges (see Chapters 4.5 and 5.1 of [43]). We are
still encouraged that this course of action may result in a plausible
interpretation as experimental evidence of large-scale temperature anomalies
appears to suggest a significant orientation to the CMB [61].
At this point in our discussion, we offer a brief review of the literature
concerning both the Pioneer anomaly as well as the other known astrometric
anomalies within our own solar system. First, however, we mention other
theories which contrast with our own study but are relevant for the discussion
below. The authors of [62] and [63] investigate the potential effects of an
expanding universe which could be induced on objects within our solar system.
Furthermore, [64] attempts to model the consequences of an extra radial
acceleration on the orbital motion of a planet within our solar system. As
well, [65] provides an alternative model for gravitation resulting in an
additional “Rindler-like” term at large distances which the author claims can
potentially model the plateauing nature of observed orbital velocity curves.
We must stress that the model proposed in [65] is in fact very different from
the model that we have proposed above as our theory of inertial centers does
not attempt to reformulate gravity. As we emphasized earlier, our model is an
attempt to reformulate the motion of objects when no net external forces are
acting upon said objects in empty flat space-time. Nevertheless, [66], [67],
[68], [69], and [70] use these ideas of an additional “Rindler-like” term in
gravitation to examine the possible observable effects of the aforementioned
extension to general relativity. For a background reference concerning
phenomenology in the context of general relativity, we refer the reader to
[71] as preparation for our presentation of the known anomalies exhibited
within our own solar system.
Besides the Pioneer anomaly, there are experimental claims of possible
anomalies alluding to inconsistencies with our current model for the Solar
System. These include:
1\. An anomalous secular increase in the eccentricity of the orbit of the Moon
2\. The “flyby” anomaly
3\. An anomalous correction to the precession of the orbit of Saturn
4\. A secular variation of the gravitational parameter $GM_{\odot}$ where
$M_{\odot}$ is the mass of the Sun
5\. A secular variation of the astronomical unit (AU)
The anomalous secular increase in the eccentricity of the orbit of the Moon
was originally found in the experimental analysis of the Lunar Laser Ranging
(LLR) data in [72] and expanded upon in [73], [74], [75], and [76]. The
“flyby” anomaly refers to an anomalous shift in the Doppler residuals received
from spacecrafts when comparing signals before and after these spacecrafts
undergo gravitational assists about planets within the Solar System
[40][77][78]. The anomalous perihelion precession of Saturn appears to be a
more controversial claim as the work of [79] and [80] seems to suggest the
validity of this observation with further investigation in [81] and [82].
However, work such as [83], [84], and [85] seems to show that this reported
anomaly is an experimental artifact. Finally, the last two anomalies of a
secular variation in the product of the mass of our Sun and the gravitational
constant $G$ as well as the astronomical unit are more difficult claims to
understand in the context of our model as there are many complex mechanisms
which could affect our measurements of these quantities (e.g. rate of mass
accretion of the Sun from infalling objects versus depletion through expelled
radiation resulting from nuclear fusion) in addition to the fact that our
measurement of the AU is implicitly linked to our measurement of
$GM_{\odot}$[86]. Nevertheless, [86] and [87] are useful references for these
anomalies. Additionally, [23] provides a detailed summary of the majority of
the anomalies listed above.
Returning to the Pioneer anomaly, the reader may have concerns with our
earlier analysis as recent simulations such as [41] suggest that this anomaly
should be taken as a thermal effect from the spacecraft itself instead of
evidence linked to “new physics”. For a selection of work concerning the
possible thermal explanation of the Pioneer anomaly, see [41], [88], [89],
[90], [91], [92], and [93]. Nevertheless, this analysis still does not address
the asymmetric nature of the “flyby” anomaly [40][77] as well as the other
significant astrometric Solar System anomalies summarized in [23]. By
“asymmetric nature”, we are referring to the fact that the magnitude of the
“flyby” anomaly appears to depend upon the direction of approach of the space
probe toward Earth as well as the angle of deflection away after “flyby”.
Furthermore, as mentioned in [94], the “onset” of the Pioneer anomaly after
Pioneer 11’s encounter with Saturn is still of concern when explaining these
observables as the result of systemic thermal effects. While [41] briefly
addresses this “onset” in their conclusion, future analysis of the early data
points for Pioneer 11 near its gravitational assist about Saturn appears to be
of the utmost importance, especially considering before its encounter with
Saturn this spacecraft moved nearly tangentially to the direction of
Sagittarius A*, whereas after it traveled nearly toward the Milky Way center.
Thus, in the context of our own model, this “onset” has the potential to be
interpreted as the consequence of the spacecraft’s change in direction
relative to the inertial center associated with the center of the Milky Way,
similar to ideas we will have to explore for the asymmetric nature of the
Earth “flyby” anomalies (for potential connections between the Pioneer and
“flyby” anomalies, see [40]). Therefore, we choose not to rule out the
possibility that the Pioneer anomaly may be support for our theory of inertial
centers as this effect as modeled in our earlier analysis in fact must be
observed in order for our theory to have physical relevance. As mentioned
earlier, we will have to address in far more rigorous detail in future work
the dual nature of the Pioneer residuals in order to possibly explain the
blueshifts from both Pioneer 10 and Pioneer 11 data.
In addition, others such as [65], [66], and [67] have used a “Rindler-like
force” emanating from the center of a gravitational source to supplement
general relativistic gravity as a model that can potentially explain orbital
velocity curves as well as the Pioneer anomaly [68][69]. For a review of how
this and other gravitational supplements would impact current expectations for
the orbits of other major bodies in the Solar System, see [95], [79], [96],
[97], [98], [99], [100], [101], [102], [103], [67], [104], [105], [106],
[107], [108], and [109]. However, these supplements all require spherical
symmetry about the center of the gravitational source in question and are very
different from our reformulation of flat space-time where in our theory we do
not assume that there exists a gravitational source at the center of galaxies,
groups, clusters, etc. Recall that we are concerned with reformulating
inertial motion and inertial reference frames in flat space-time (i.e. our
description of the way in which objects move in flat space-time when subjected
to no net external forces). Additionally, we maintain that locally within
confined regions of the inertial system of our theory of inertial centers
Einstein’s version of gravitation seen as the consequence of space-time
curvature induced by the energy-momentum of a massive object in his theory of
general relativity still applies in the same manner. In other words, in our
theory of inertial centers, this observed deviation from assumed special
relativistic flat space-time geodesics arises from our redefinition of the
inertial system itself instead of some modification to gravitation.
Consequently, when attempting to explain these astrometric Solar System
anomalies in the context of our theory, we focus on the difference in
geodesics in the galactic inertial reference frame when compared to assumed
special relativistic geodesics for flat space-time and assume that all of the
objects in our Solar System including the Sun orbit about the inertial center
point associated with the center of the Milky Way (again, we assume that there
is no gravitational source at the center of our galaxy). Meaning, the Pioneer
anomaly is not taken to be a phenomenon due to gravity in the theory of
inertial centers. Instead the Pioneer anomaly and possibly the other
astrometric Solar System anomalies which we have listed above are taken to be
the result of our redefinition of inertial systems as well as the change in
our expectations for what constitutes inertial motion. Consequently, the
relative acceleration between massive objects in our solar system is nearly
unchanged from what one would expect from general relativity as all objects
within our solar system orbit about the center of the Milky Way along
relatively similar paths. Therefore, we are not modifying our expectations for
the interactions between objects within the Solar System. We are modifying our
expectations for the paths of all objects in the Solar System through the
Milky Way. While internally within our solar system the planets remain nearly
unchanged in their paths as they move slowly in the “Newtonian limit” (i.e.
their speeds are much less than that of light), light propagating between
these massive objects in our theory won’t behave as one would expect from
general relativity as at these speeds one must take into account the
properties of the larger inertial system associated with our galaxy.
One must bear in mind that these anomalies are linked to the propagation of
electromagnetic radiation throughout our solar system as our experimental
apparatuses use light for precision measurements. While the work of [62]
attributes the Pioneer anomaly to the local effects of light signal
propagation in an expanding universe as expressed by a “post-Friedmannian”
metric decomposition, these claims would not be able to explain the asymmetric
nature of the wavelength shift residuals in the “flyby” anomaly as the FLRW
metric requires homogeneous and isotropic expansion of space in all directions
[31]. However, there is no expansion in our theory of inertial centers and our
inertial reference frames do have an orientation. Therefore, we must take into
consideration, when comparing with our own model in future work, two important
ideas: in this theory of inertial centers, the speed of light is not constant
in flat space-time and objects follow inertial paths described by geodesics
about inertial centers in the radial Rindler chart, where we assume that the
inertial center associated with the Milky Way is in the direction of
Sagittarius A*. Thus, in our model, the observables associated with the
astrometric Solar System anomalies listed above do not necessarily reflect the
existence of an additional acceleration in the Solar System since our theory’s
radial acceleration would be imposed on all objects within the Solar System
including the Sun and in the same direction toward the center of the Milky Way
(10) with seemingly negligible difference in magnitude depending upon the
position of the massive object in question (i.e. changes in position within
our solar system are negligible relative to the distance of our solar system
from the center of the Milky Way when considering the motion of massive
satellites, planets, etc.). In other words, in sharp contrast with the
analysis in papers such as [65], [109], [97] and [95], we assume that there is
no additional acceleration associated with the Sun’s gravitational pull on
other objects within the Solar System, and thus the relative acceleration of a
satellite, planet, etc. with respect to the center of the Sun remains nearly
unaffected in our model when we compare with general relativity. Instead, it
appears that in the theory of inertial centers these anomalies should more
likely be interpreted as a consequence of the non-constant nature of the speed
of light within our galactic inertial system as well as of the expected shifts
in wavelength when light propagates between differing distances from an
inertial center point. Future experiments within the vicinity of our solar
system to test the validity of the theory of inertial centers could include
sending a spacecraft to the outer edges of our solar system along a closed
orbit about the Sun or using identical spacecrafts along open orbits in
different directions with respect to the galactic center (e.g. one travels
tangentially to the direction of the center of the Milky Way while another
moves directly toward/away from the center; for a hyperbolic orbit proposal,
see [110]). To test the positional dependence aspects for electromagnetic
radiation in this theory, these hypothetical missions should measure the
potential variations in wavelength shift and time delay for light signals sent
and received at different positions along these orbits with respect to the
center of the Milky Way. As well, future theoretical work will require us to
explicitly detail observational effects on our astrometric measurements of the
planetary ephemerides that are unique to the theory of inertial centers. One
could then potentially find these predicted deviations from current models
when comparing with the experimental work of [83] and [85].
Using the measured value for the speed of light on Earth ($c_{\rm
Earth}\approx 3.0\times 10^{8}$ m/s) and the value for the time-scale given
from the “time acceleration” in [40], we find that our distance to the center
of the Milky Way is approximately $r_{0}|_{{\rm MW}}\approx 1.03\times
10^{23}$ km. We see that the value obtained for our galactic radial distance
is far larger than the predicted value from models requiring a supermassive
black hole at the center of the Milky Way (intimidatingly, nearly six orders
of magnitude [111]). It is imperative then that we reconcile this calculated
value with observational data. Not only will this maintain consistency with
experiment but it will also provide accurate distance scales within our
galaxy. This will allow us to further understand the large observed wavelength
shifts near Sagittarius A* within the framework of our theory of inertial
centers and potentially explain the paradox of youth [112] through concrete
analysis of star formation near the Milky Way center.
Addressing our classical inertial motion analysis, one can immediately tell
from the theoretical approach in our discussion that this paper is limited by
the lack of necessary quantitative comparison with orbital velocity curves,
redshift surveys, and lensing observations. Future work will require modeling
using computer simulations of our equations of motion not only to produce
orbital velocity curves that will facilitate comparison with data but to also
give us a far more thorough understanding of classical inertial motion outside
of the limiting behavior examined in this paper. To implement, it appears that
we should use a finite difference method with the component form of our
geodesic equation parametrized in terms of the proper time of the object in
question within a particular inertial system as expanded upon at the end of
Appendix B. Furthermore, we will have to apply this same finite difference
method to our normalization condition for the ‘four-velocity’ but parametrized
in terms of the proper time in this inertial frame. We also have to attend to
a pressing issue with regard to the “Hubble behavior” associated with
wavelength shifts within our inertial system. As outlined earlier, this theory
requires that we observe both significant redshifts and blueshifts, yet on
scales larger than the Local Group, blueshifted emitters are reportedly
scarce. Thus, if our theory is to be considered seriously, we must provide an
explanation for why there is such an imbalance towards reported redshifted
emitters at the largest observable scales. Nevertheless, one apparent
resolution lies in the possible alternative “blueshift interpretation” of
spectroscopic profiles as mentioned and subsequently applied in [113], [114],
and [115] with possible support for the re-examination of spectroscopic
profiles in the blueshifted emission lines found in other work such as [116].
Proceeding to our quantum concerns, our seemingly shocking proposal that at
the center of the nucleus of every atom there could potentially exist an
inertial center point raises many more questions for our theory of inertial
centers. Of course, this type of claim requires thorough and rigorous
justification in both future theoretical work and even more importantly in
comparison with experiment. For example, a simple comparison with experiment
would be to determine how accurate of a fit our “n-particle amplitudes”
(reviewed in Appendix D) with individual solutions for quantum numbers
$(\alpha,l,m)$ given by (64) are with current experimental knowledge of the
nucleus. Nevertheless, we have chosen to mention these ideas in this paper in
order to highlight to the reader how much of a potential impact this
redefinition of inertial motion and inertial reference frames could possibly
have on our understanding of structure formation for all scales from the
largest to the smallest. As for questions: for one, can we reconcile these
claims with our current knowledge of the electronic and nuclear structure of
the atom when we factor in charge, spin, and electromagnetism? Additionally,
how much of our current model for the nucleus is affected by these ideas? It
also becomes ever more important to answer the following: What establishes one
of these inertial centers as well as the orientation of one of our inertial
systems?
## Conclusions
All of our assumptions within this work in one way or another are built upon
the idea that objects do not move in a straight line at a constant speed when
no external forces are acting upon them in empty flat space-time. In other
words, we assume that Newton’s first law does not give the correct
characterization of inertial motion. Therefore, we essentially “start from
scratch” and concentrate on how to incorporate all of the following observed
features into a revised understanding of inertial motion: accelerated
redshifts and the Hubble relation, plateauing orbital velocity curves at large
distances from a central point about which objects move, consistent velocity
“flow” on the largest of scales directed toward a central point, and an
orientation associated with each of these central points. We take an inertial
frame of reference to be the system within which objects follow these revised
inertial trajectories and begin our reformulation with the knowledge that our
theory of globally flat space-time must reduce to special relativity within
confined regions of our newly defined inertial systems. Consequently, it
appears natural to approach this reformulation from the notion that we should
have a metric theory of flat space-time, and within this metric theory objects
still follow along geodesic trajectories when no external forces are acting
upon them as in special and general relativity. However, in order to
distinguish our metric theory of flat space-time from special relativity, we
must require that our affine parameter not be proper time globally throughout
these reference frames. In addition, we find that we are able to reproduce the
previously listed features with the radial Rindler chart as the coordinate
parametrization of our flat space-time manifold, thereby assuming the physical
significance of special central points which we deem “inertial center points”
situated throughout all of space-time. As one would expect from their given
name, these inertial center points describe the centers of each of our
inertial systems, and our inertial trajectories are then assumed to be the
orbits of objects about these inertial centers. Meaning, inertial motion must
be thought of relative to both the center point and the orientation (i.e.
location of the poles) of each of these inertial reference frames.
Consequently, it is assumed that the observed motion of objects about central
points on the largest of scales (e.g. stars orbiting the center of a galaxy,
galaxies orbiting the center of a group/cluster, etc.) is not due to
gravitational effects but is instead a manifestation of inertial motion within
our theory of flat space-time, which we term our “Theory of Inertial Centers”.
This redefinition of inertial motion then allows us to no longer assume the
existence of ‘dark energy’, ‘dark matter’, and ‘dark flow’. Furthermore, as we
have the ability to model the Hubble relation within our theory, we do not
require the occurrence of a ‘Big-Bang’ event, and therefore we also do not
require ‘inflation’ nor an expanding universe (i.e. we do not operate under
the assumptions of $\Lambda$CDM). The cornerstone of our theory is embodied in
the statement that within our inertial systems, time and space are
fundamentally intertwined such that time- and spatial-translational invariance
are not inherent symmetries of flat space-time. Meaning, our invariant
interval associated with the metric incorporates both time and spatial
distance. Therefore, observable clock rates depend upon not only the relative
velocity of observers within these inertial systems but also on the difference
in distance of each observer from an inertial center, expressed mathematically
by relation (57). Given this relation, we find that our theory of globally
flat space-time in fact reduces to special relativity for observers which we
can consider as nearly stationary with respect to the inertial center point
about which they orbit (i.e. the local stationary limit). As well, our ideas
then require that the local speed of light which we measure within a confined
region of these newly defined inertial systems is linearly dependent upon our
distance away from the inertial center about which we orbit (56). Thus, the
speed of light throughout each of these redefined inertial systems in flat
space-time is not constant.
With these theoretical foundations presented, we proceeded by examining the
local consequences of our theory for a gravitational system located within one
of these inertial systems as an observer should be able to measure with a
detector of the necessary sensitivity the deviation of an object’s
(specifically light’s) inertial path in flat space-time away from special
relativistic geodesics and into the geodesics of our theory as outlined in the
local stationary limit. Thus, within the framework of the theory of inertial
centers, we interpret the Pioneer anomaly as an observable consequence of our
revised ideas on inertial motion. However, as mentioned later in our paper,
there are many open questions that must be answered with regard to the
propagation of light signals within our solar system in the context of our
theory. Specifically, can our revision of inertial motion and inertial
reference frames explain the other known astrometric Solar System anomalies
(i.e. “flyby” anomaly, the anomalous increase in the eccentricity of the Moon,
and the variation in the AU)? And, can we explain the blueshifted nature of
both Pioneer 10 and Pioneer 11 Doppler data once we factor in the two-way
nature of these residuals as well as the change in clock rates for observers
located at different distances from the center of the Milky Way in our model?
Furthermore, after quantizing for a real massive scalar field, we came upon a
potential explanation for the asymmetry between matter and antimatter in our
observable universe within the context of our theory of inertial centers. If
we allow for the possibility that our field exists in both radial Rindler
wedges (i.e. $r>0$ and $r<0$), it appears that a logical explanation for the
observable imbalance toward matter would be that our antimatter counterparts
are located in the “other” radial Rindler wedge for each of our inertial
systems, as the charge of each field in these systems should be conserved
(e.g. abundance of electrons in one wedge should imply an abundance of
positrons in the “other” wedge). Nevertheless, this logic relies on the
consistency of our extension for a real scalar field to complex fields with
spin. Thus, in future work, we will have to address the validity of this
interpretation when we extend our analysis (e.g. Dirac spinors). In addition,
we concluded our discussion by examining the nearly stationary limit for
particles close to an inertial center point. Using expression (10), we chose
to work naively under Newton’s assumptions and take this acceleration on our
observer to be the result of a Newtonian force derived from a conservative
potential. Then, the stationary Hamiltonian associated with this Newtonian
approximation would take the form of the isotropic harmonic oscillator. Taking
the perspective of an observer exterior to the inertial system in question
(i.e. the external observer orbits a different inertial center), we found the
observed oscillator energy scale using relation (57) while operating under the
assumption that the time-scale for each inertial system is a universal
constant and therefore the same for each. A simple potential explanation for
the ability of the isotropic harmonic oscillator to explain the “magic
numbers” associated with stable arrangements of nucleons within the nucleus of
an atom then arose in the context of our model. Since both the form of our
stationary Hamiltonian as well as the determined energy scale match that of
the starting point for our nuclear shell models, it appears that we must
seriously consider the possibility that there exists an inertial center point
at the center of the nucleus of every atom when working under the assumptions
of our theory of inertial centers as, in our stationary limit, the
acceleration of each particle within the inertial system mimics what one would
find if he/she naively assumed a Newtonian Hamiltonian of the form of the
isotropic harmonic oscillator. In other words, within the context of our
theory, the ability of the isotropic harmonic oscillator to model the simplest
nuclear configurations would be interpreted as a consequence of the physical
existence of an inertial center located at the center of the nucleus of every
atom, where these simple configurations of nucleons arise from the stationary
limit for objects very near to an inertial center. Although these claims are
radical in nature, we are still compelled to question whether or not the
nuclear ‘force’ is even really a force within the framework of our model.
Future theoretical and experimental work will be required in order to fully
understand the nature of these ideas.
## Acknowledgments
I would like to thank several anonymous reviewers as well as both editors for
useful comments and critiques which very much helped to improve the clarity of
this work.
## Appendix A Affine connection terms, Ricci and Riemann tensors
Following [24] for our general expressions below, we work in the metric
$ds^{2}=-\Lambda
r^{2}dt^{2}+dr^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)[d\theta^{2}+d\phi^{2}\sin^{2}{\theta}]$
Our affine connection tensor for this choice of coordinates is given by the
expression:
$\Gamma^{c}_{ab}=\frac{1}{2}g^{cd}[\partial_{a}g_{db}+\partial_{b}g_{da}-\partial_{d}g_{ab}]$
where in component form $\partial_{\mu}=\partial/\partial x^{\mu}$ is our
ordinary partial derivative and in radial Rindler coordinates our metric
components gathered in matrix form are given by
$\displaystyle(g_{\mu\nu})=\begin{pmatrix}-\Lambda r^{2}&0&0&0\\\ 0&1&0&0\\\
0&0&r^{2}\cosh^{2}(\sqrt{\Lambda}t)&0\\\
0&0&0&r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}\end{pmatrix}$ (70)
$\displaystyle(g^{\mu\nu})=\begin{pmatrix}-\frac{1}{\Lambda r^{2}}&0&0&0\\\
0&1&0&0\\\ 0&0&\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}&0\\\
0&0&0&\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}}\end{pmatrix}$
(71)
with $(g^{\mu\nu})$ corresponding to the inverse of the matrix associated with
$(g_{\mu\nu})$ ($g_{tt}=-\Lambda
r^{2},g_{rr}=1,g_{\theta\theta}=r^{2}\cosh^{2}(\sqrt{\Lambda}t),g_{\phi\phi}=r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}$).
For the reader who may be unfamiliar with abstract index notation, we look for
each affine connection term associated with the tensor above by examining this
expression in component form:
$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}\sum_{\rho}g^{\lambda\rho}[\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}]$
where we use greek indices (e.g. $\lambda,\mu,\nu$) for components and latin
indices (e.g. $a,b,c$) for the tensor itself. Just as an example, we look for
the component $\Gamma^{t}_{tr}$:
$\Gamma^{t}_{tr}=\frac{1}{2}\sum_{\beta}g^{t\beta}[\partial_{t}g_{\beta
r}+\partial_{r}g_{\beta t}-\partial_{\beta}g_{tr}]$
where we sum over like indices for $\beta=t$, $r$, $\theta$, $\phi$. Then,
$\displaystyle\Gamma^{t}_{tr}=\frac{1}{2}\bigg{\\{}g^{tt}[\partial_{t}g_{tr}+\partial_{r}g_{tt}-\partial_{t}g_{tr}]+g^{tr}[\partial_{t}g_{rr}+\partial_{r}g_{rt}-\partial_{r}g_{tr}]+g^{t\theta}[\partial_{t}g_{\theta
r}+\partial_{r}g_{\theta t}-\partial_{\theta}g_{tr}]$
$\displaystyle+g^{t\phi}[\partial_{t}g_{\phi r}+\partial_{r}g_{\phi
t}-\partial_{\phi}g_{tr}]\bigg{\\}}$
However, from (71), we know that $g^{tr}=g^{t\theta}=g^{t\phi}=0$. Therefore,
our expression reduces to
$\Gamma^{t}_{tr}=\frac{1}{2}g^{tt}[\partial_{t}g_{tr}+\partial_{r}g_{tt}-\partial_{t}g_{tr}]$
Yet from (70), we see that $g_{tr}=0$. This leaves us with
$\Gamma^{t}_{tr}=\frac{1}{2}g^{tt}\partial_{r}g_{tt}=\frac{1}{2}\bigg{(}-\frac{1}{\Lambda
r^{2}}\bigg{)}\cdot\partial_{r}(-\Lambda r^{2})=\frac{1}{r}$
For the reader who wishes to derive the rest of these affine connection
components, we notice from (70) that $g_{\mu\nu}=0$ for $\mu\neq\nu$ (our
metric is diagonal in radial Rindler coordinates). Then the expression for our
affine connection terms reduces to
$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\lambda}[\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}]$
Yet, because $\Gamma^{c}_{ab}$ is symmetric in $a\Leftrightarrow b$ (i.e. if
we swap $a$ and $b$ indices, our tensor remains the same as one can see above
since $g_{ab}$ is also symmetric under the same exchange by definition), our
possibilities for the affine connection terms are limited to three cases:
$\lambda=\nu\neq\mu$; $\lambda=\nu=\mu$; $\mu=\nu\neq\lambda$. For
$\lambda=\nu\neq\mu$,
$\Gamma^{\lambda}_{\mu\lambda}=\frac{1}{2}g^{\lambda\lambda}[\partial_{\mu}g_{\lambda\lambda}+\partial_{\lambda}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\lambda}]=\frac{1}{2}g^{\lambda\lambda}\partial_{\mu}g_{\lambda\lambda}\indent(\lambda\neq\mu)$
where we used the diagonal property of our metric parametrization in the last
equality. Applying similar logic to our other two cases, we obtain
$\Gamma^{\lambda}_{\lambda\lambda}=\frac{1}{2}g^{\lambda\lambda}\partial_{\lambda}g_{\lambda\lambda}\indent{\rm
and}\indent\Gamma^{\lambda}_{\mu\mu}=-\frac{1}{2}g^{\lambda\lambda}\partial_{\lambda}g_{\mu\mu}\indent(\lambda\neq\mu)$
Using these identities, one finds that our non-zero affine connection terms
are:
$\displaystyle\Gamma^{t}_{tr}=\frac{1}{r}\indent\Gamma^{t}_{\theta\theta}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\indent\Gamma^{t}_{\phi\phi}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\sin^{2}{\theta}$
$\displaystyle\Gamma^{r}_{tt}=\Lambda
r\indent\Gamma^{r}_{\theta\theta}=-r\cosh^{2}(\sqrt{\Lambda}t)\indent\Gamma^{r}_{\phi\phi}=-r\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}$
$\displaystyle\Gamma^{\theta}_{\theta t}=\Gamma^{\phi}_{\phi
t}=\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\indent\Gamma^{\theta}_{\theta
r}=\Gamma^{\phi}_{\phi
r}=\frac{1}{r}\indent\Gamma^{\theta}_{\phi\phi}=-\sin{\theta}\cos{\theta}\indent\Gamma^{\phi}_{\phi\theta}=\cot{\theta}$
We define the curvature tensor by the action of the linear map
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})$ on a dual vector field
$\omega_{c}$ (for more information on vector fields, see Chapter 2 of [24]):
$\nabla_{a}\nabla_{b}\omega_{c}-\nabla_{b}\nabla_{a}\omega_{c}={R_{abc}}^{d}\omega_{d}$
where $\nabla_{a}$ is the derivative operator compatible with our metric (or
covariant derivative; $\nabla_{a}g_{bc}=0$) and we refer to ${R_{abc}}^{d}$ as
the Riemann curvature tensor. Our Riemann curvature tensor can be expressed in
terms of the affine connection associated with a particular choice of
coordinate chart:
${R_{abc}}^{d}=\partial_{b}\Gamma^{d}_{ac}-\partial_{a}\Gamma^{d}_{bc}+\Gamma^{e}_{ca}\Gamma^{d}_{be}-\Gamma^{e}_{cb}\Gamma^{d}_{ae}$
And in component form,
${R_{\mu\nu\rho}}^{\sigma}=\partial_{\nu}\Gamma^{\sigma}_{\mu\rho}-\partial_{\mu}\Gamma^{\sigma}_{\nu\rho}+\sum_{\lambda}\bigg{[}\Gamma^{\lambda}_{\rho\mu}\Gamma^{\sigma}_{\nu\lambda}-\Gamma^{\lambda}_{\rho\nu}\Gamma^{\sigma}_{\mu\lambda}\bigg{]}$
Using our affine connection terms, we discover ${R_{\mu\nu\rho}}^{\sigma}=0$
$\forall\mu,\nu,\rho,\sigma$ as we should expect since the radial Rindler
chart is just a coordinate transformation away from the Minkowski chart (the
geometric properties of the manifold are independent of coordinate
parametrization). Therefore,
$R_{\mu\beta}=\sum_{\lambda}{R_{\mu\lambda\beta}}^{\lambda}=0$ and
$\sum_{\lambda,\mu,\nu,\beta}R^{\lambda\mu\nu\beta}R_{\lambda\mu\nu\beta}=0$.
The metric satisfies the Einstein field equations in vacuum without a
cosmological constant [6] and represents flat space-time.
## Appendix B Equations of motion
$0=U^{a}\nabla_{a}U^{b}$ (72)
where $\nabla_{a}$ is the derivative operator compatible with our metric (or
covariant derivative; $\nabla_{a}g_{bc}=0$). For the reader who may be
unfamiliar with concepts in differential geometry, the action of this
derivative operator on an arbitrary vector field (a vector field is an
assignment of a vector at each point on the manifold) can be expressed in
terms of our more familiar partial derivatives through the affine connection
tensor associated with a particular coordinate system. When our derivative
operator acts on an arbitrary vector field $v^{a}$, we have
$\nabla_{a}v^{b}=\partial_{a}v^{b}+\Gamma^{b}_{ac}v^{c}$
and for this same derivative operator acting upon a dual vector field,
$\nabla_{a}v_{b}=\partial_{a}v_{b}-\Gamma^{c}_{ab}v_{c}$
Without going into further detail with regard to vector spaces, the reader may
feel more informed to know that we can relate a vector with its dual space
counterpart through the metric:
$v_{a}=g_{ab}v^{b}$
In addition, a vector $v^{a}$ given at each point on a curve $C$ is said to be
parallelly transported as one moves along this curve if
$t^{a}\nabla_{a}v^{b}=0$
where $t^{a}$ refers to the tangent vector to the curve. We then define a
geodesic to be a curve whose tangent denoted $U^{a}$ satisfies (72) (for more
on parallel transport, see Chapter 3.3 of [24]) and assume that our particles
travel along these curves when subjected to no net external forces.
Additionally, a parametrization of a curve which yields (72) is called an
affine parametrization, and thus by definition a geodesic is required to be
affinely parametrized.
For a geodesic along which one of our particles moves denoted
$x^{\mu}(\sigma)$ in our particular coordinate system and parametrized in
terms of the affine parameter $\sigma$, our tangent vector to this curve in
component form is given by $U^{\mu}=dx^{\mu}/d\sigma$ (where $\sigma=\chi$ for
massive particles) and is said to be the ‘proper velocity’ or ‘four-velocity’
of this particle. We also define
$\chi=\int(-g_{ab}T^{a}T^{b})^{1/2}dt$
where $T^{a}$ is the tangent vector to any particular time-like (i.e.
$g_{ab}T^{a}T^{b}<0$) curve and $t$ is an arbitrary parametrization of this
curve. Thus, along a time-like geodesic affinely parameterized by $\chi$, we
have
$g_{ab}U^{a}U^{b}=-1$
Applying all of the above concepts to expand our equation of motion (72) in a
particular coordinate system,
$0=U^{a}\bigg{[}\partial_{a}U^{b}+\Gamma^{b}_{ac}U^{c}\bigg{]}$
Using our expression for the ‘proper velocity’ in component form, we come upon
the geodesic equation of motion for particles in terms of our affine
connection terms:
$\displaystyle
0=\sum_{\alpha}\frac{dx^{\alpha}}{d\sigma}\cdot\frac{\partial}{\partial
x^{\alpha}}(\frac{dx^{\nu}}{d\sigma})+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$
$\displaystyle=\frac{d^{2}x^{\nu}}{d\sigma^{2}}+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$
Therefore, for our radial Rindler metric, we can plug in the affine connection
terms found in Appendix A where in addition $x^{\mu}(\sigma)\rightarrow\langle
t(\sigma),r(\sigma),\theta(\sigma),\phi(\sigma)\rangle$. This notation for our
vector components signifies
$x^{a}=t(\sigma)\bigg{(}\frac{\partial}{\partial
t}\bigg{)}^{a}+r(\sigma)\bigg{(}\frac{\partial}{\partial
r}\bigg{)}^{a}+\theta(\sigma)\bigg{(}\frac{\partial}{\partial\theta}\bigg{)}^{a}+\phi(\sigma)\bigg{(}\frac{\partial}{\partial\phi}\bigg{)}^{a}$
where $(\partial/\partial t)^{a}$, $(\partial/\partial r)^{a}$,
$(\partial/\partial\theta)^{a}$, and $(\partial/\partial\phi)^{a}$ are
linearly independent tangent vectors which span the tangent spaces at each
point on the manifold. For example, we take our equation of motion for
$t(\sigma)$:
$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\sum_{\mu,\rho}\Gamma^{t}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$
$\displaystyle=\frac{d^{2}t}{d\sigma^{2}}+\Gamma^{t}_{tr}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\Gamma^{t}_{rt}\frac{dr}{d\sigma}\frac{dt}{d\sigma}+\Gamma^{t}_{\theta\theta}\frac{d\theta}{d\sigma}\frac{d\theta}{d\sigma}+\Gamma^{t}_{\phi\phi}\frac{d\phi}{d\sigma}\frac{d\phi}{d\sigma}$
However, we know from our work in Appendix A that
$\Gamma^{t}_{tr}=\Gamma^{t}_{rt}$. Consequently, we find after plugging in for
each affine connection term
$0=\frac{d^{2}t}{d\sigma^{2}}+\frac{2}{r}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
Applying similar logic for $\nu=r$, $\theta$, and $\phi$:
$\displaystyle 0=\frac{d^{2}r}{d\sigma^{2}}+\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\frac{d\theta}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}\bigg{]}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
$\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$
As briefly mentioned above, if one evaluates the norm of the ‘proper
velocity’, he/she will find:
$U^{a}U_{a}=\sum_{\mu,\nu}g_{\mu\nu}U^{\mu}U^{\nu}=\sum_{\mu,\nu}g_{\mu\nu}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\nu}}{d\sigma}=\left\\{\begin{array}[]{l
l}0&\quad\textrm{null geodesics}\\\ -1&\quad\textrm{time-like geodesics}\\\
\end{array}\right.$
In a relatively simple way, one can see this from our line element where
$-d\chi^{2}=\sum_{\mu,\nu}g_{\mu\nu}dx^{\mu}dx^{\nu}$. Massless particles
travel along null geodesics (i.e. our norm vanishes) whereas massive particles
travel along time-like geodesics. For comparison with special relativity and
general relativity, we express the component form of our time-like geodesics
where $\sigma=\chi$ in terms of the physically observable elapsed time as
measured by a clock carried along the given curve in a particular inertial
system, $\tau$:
$0=\frac{d^{2}\tau}{d\chi^{2}}\frac{dx^{\nu}}{d\tau}+\bigg{(}\frac{d\tau}{d\chi}\bigg{)}^{2}\cdot\bigg{[}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\tau}\frac{dx^{\rho}}{d\tau}\bigg{]}$
One immediately notices that the term in brackets represents the component
form of the geodesic equation for special and general relativity and would be
set equal to zero in both of these theories. However, since in the theory of
inertial centers $d^{2}\tau/{d\chi^{2}}\neq 0$ as
$d\chi/d\tau=\sqrt{\Lambda}\cdot r(\tau)$, the term in brackets is not
necessarily zero for our theory, and thus the observed inertial motion of
massive objects in our model characterized by the equation above is in fact
very different from inertial motion as seen in special and general relativity.
## Appendix C Killing vector fields
As in our previous appendices, we provide a summary of [24] with regard to the
more general statements below (see Appendix C and Chapter 2 of [24]). In order
to understand the relevance of Killing vector fields with respect to inherent
symmetries associated with our manifold, we must begin with a brief
introduction to isometries and Lie derivatives. For two manifolds $M$ and $N$,
let $\phi$ be a smooth map from $M$ to $N$ ($\phi:M\rightarrow N$) and $f$ be
a function from $N$ to the reals ($f:N\rightarrow\mathbb{R}$). Then the
composition of $f$ with $\phi$, $f\circ\phi$, produces a function from
$M\rightarrow\mathbb{R}$ and $\phi$ is said to “pull back” $f$. In addition
$\phi$ “carries along” tangent vectors at a particular point $p\in M$ to
tangent vectors at $\phi(p)\in N$, and therefore defines a map
$\phi^{\star}:V_{p}\rightarrow V_{\phi(p)}$ in the following manner:
$(\phi^{\star}v)(f)=v(f\circ\phi)$
where $v\in V_{p}$, $\phi^{\star}v\in V_{\phi(p)}$, and $V_{p}$ denotes the
tangent vector space at $p$. One can also use $\phi$ to “pull back” dual
vectors at $\phi(p)$ to dual vectors at $p$ by defining a map
$\phi_{\star}:V^{\star}_{\phi(p)}\rightarrow V^{\star}_{p}$ requiring for all
$v^{a}\in V_{p}$
$(\phi_{\star}\mu)_{a}v^{a}=\mu_{a}(\phi^{\star}v)^{a}$
where $V^{\star}_{p}$ denotes the dual vector space at $p$. If
$\phi:M\rightarrow N$ is a diffeomorphism (i.e. a smooth function that is one-
to-one, onto, and its inverse is also smooth), then for an arbitrary tensor
$T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}$ of type $(k,l)$ at $p$ (type
$(k,l)$ refers to the number of dual vector “slots” and vector “slots”,
respectively), the tensor $(\phi^{\star}T)^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}$ at $\phi(p)$ is defined by
$(\phi^{\star}T)^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}(\mu_{1})_{b_{1}}\cdots(\mu_{k})_{b_{k}}(t_{1})^{a_{1}}\cdots(t_{l})^{a_{l}}=T^{b_{1}\ldots
b_{k}}{}_{a_{1}\ldots
a_{l}}(\phi_{\star}\mu_{1})_{b_{1}}\cdots([\phi^{-1}]^{\star}t_{l})^{a_{l}}$
as $(\phi^{-1})^{\star}:V_{\phi(p)}\rightarrow V_{p}$. If $\phi:M\rightarrow
M$ is a diffeomorphism and $T$ is a tensor field on $M$, then we refer to
$\phi$ as a symmetry transformation for the tensor field $T$ if
$\phi^{\star}T=T$. In addition, if
$(\phi^{\star}g)_{ab}=g_{ab}$
we refer to $\phi$ as an isometry.
To introduce the notion of Lie derivatives, we come back to diffeomorphisms
and define a one-parameter group of diffeomorphisms $\phi_{t}$ as a smooth map
from $\mathbb{R}\times M\rightarrow M$ such that for fixed $t\in\mathbb{R}$,
$\phi_{t}:M\rightarrow M$ is a diffeomorphism. As well, for all
$t,s\in\mathbb{R}$, $\phi_{t}\circ\phi_{s}=\phi_{t+s}$. In particular, this
requires $\phi_{t=0}$ to be the identity map. A vector field $v^{a}$ can be
thought of as the infinitessimal generator of a one-parameter group of finite
transformations of $M$ in the following manner. For fixed $p\in M$, we refer
to the curve $\phi_{t}(p):\mathbb{R}\rightarrow M$ as an orbit of $\phi_{t}$
which passes through $p$ at $t=0$. $v|_{p}$ is defined to be the tangent to
this curve at $t=0$. We also define the Lie derivative with respect to $v^{a}$
by
$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}=\lim_{t\rightarrow 0}\bigg{\\{}\frac{\phi^{\star}_{-t}T^{b_{1}\ldots
b_{k}}{}_{a_{1}\ldots a_{l}}-T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}}{t}\bigg{\\}}$
where all tensors above are evaluated at a point $p$. $\mathfrak{L}_{v}$ is
then a linear map from smooth tensor fields of type $(k,l)$ to smooth tensor
fields of type $(k,l)$ and satisfies the Leibniz rule on outer products of
tensors. Since $v^{a}$ is tangent to the integral curves of $\phi_{t}$, for
functions $f:M\rightarrow\mathbb{R}$
$\mathfrak{L}_{v}(f)=v(f)$
In addition, if $\phi_{t}$ is a symmetry transformation for $T$, we have
$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}=0$. Furthermore,
it is found that the Lie derivative with respect to $v^{a}$ of a vector field
$w^{a}$ is given by the commutator:
$\mathfrak{L}_{v}w^{a}=[v,w]^{a}$
where
$[v,w]^{a}=v^{b}\nabla_{b}w^{a}-w^{b}\nabla_{b}v^{a}$
and for a dual vector,
$\mathfrak{L}_{v}\mu_{a}=v^{b}\nabla_{b}\mu_{a}+\mu_{b}\nabla_{a}v^{b}$
The more general action of a Lie derivative with respect to $v^{a}$ on a
general tensor field $T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}$ is given by
$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}=v^{c}\nabla_{c}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}-\sum_{i=1}^{k}T^{b_{1}\ldots c\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}\nabla_{c}v^{b_{i}}+\sum_{j=1}^{l}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
c\ldots a_{l}}\nabla_{a_{j}}v^{c}$
where $\nabla_{a}$ is our derivative operator compatible with the metric
$g_{ab}$ (i.e. $\nabla_{c}g_{ab}=0$). Then a Killing vector field $\xi^{a}$ is
defined to be the vector field which generates a one-parameter group of
isometries $\phi_{t}:M\rightarrow M$ of the metric,
$(\phi^{\star}_{t}g)_{ab}=g_{ab}$. As remarked earlier, the necessary
condition for $\phi_{t}$ to be a group of isometries is
$\mathfrak{L}_{\xi}g_{ab}=0$. Using the expression above for the action of a
Lie derivative on a tensor field,
$\displaystyle\mathfrak{L}_{\xi}g_{ab}=\xi^{c}\nabla_{c}g_{ab}+g_{cb}\nabla_{a}\xi^{c}+g_{ac}\nabla_{b}\xi^{c}$
$\displaystyle=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$
Thus, we come upon Killing’s equation:
$\nabla_{a}\xi_{b}+\nabla_{a}\xi_{b}=0$
For any particular Killing vector field $\xi^{a}$, along a geodesic $\gamma$
with tangent vector $U^{a}$ one finds
$\displaystyle
U^{b}\nabla_{b}(\xi_{a}U^{a})=U^{b}U^{a}\nabla_{b}\xi_{a}+\xi^{a}U^{b}\nabla_{b}U^{a}$
$\displaystyle=\frac{1}{2}U^{a}U^{b}[\nabla_{b}\xi_{a}+\nabla_{a}\xi_{b}]+\xi^{a}U^{b}\nabla_{b}U^{a}=0$
where the first term vanishes by Killing’s equation and the second by the
geodesic equation (72). Meaning, along $\gamma$, $\xi^{a}U_{a}$ is constant
(Noether’s theorem).
Using our affine connection component terms, Killing’s equation takes the form
$\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}=2\sum_{\rho}\Gamma^{\rho}_{\mu\nu}\xi_{\rho}$
which gives for each pair ($\mu,\nu$),
$\displaystyle(t,t):\partial_{t}\xi_{t}=\Lambda
r\xi_{r}\indent(t,r):\partial_{t}\xi_{r}+\partial_{r}\xi_{t}=\frac{2}{r}\xi_{t}\indent(t,\theta):\partial_{t}\xi_{\theta}+\partial_{\theta}\xi_{t}=2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\xi_{\theta}$
$\displaystyle(t,\phi):\partial_{t}\xi_{\phi}+\partial_{\phi}\xi_{t}=2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\xi_{\phi}\indent(r,r):\partial_{r}\xi_{r}=0\indent(r,\theta):\partial_{\theta}\xi_{r}+\partial_{r}\xi_{\theta}=\frac{2}{r}\xi_{\theta}$
$\displaystyle(r,\phi):\partial_{\phi}\xi_{r}+\partial_{r}\xi_{\phi}=\frac{2}{r}\xi_{\phi}\indent(\theta,\phi):\partial_{\theta}\xi_{\phi}+\partial_{\phi}\xi_{\theta}=2\cot{\theta}\xi_{\phi}$
$\displaystyle(\theta,\theta):\partial_{\theta}\xi_{\theta}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\xi_{t}-r\cosh^{2}(\sqrt{\Lambda}t)\xi_{r}$
$\displaystyle(\phi,\phi):\partial_{\phi}\xi_{\phi}=\sin^{2}{\theta}\bigg{[}\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\xi_{t}-r\cosh^{2}(\sqrt{\Lambda}t)\xi_{r}\bigg{]}-\sin{\theta}\cos{\theta}\xi_{\theta}$
Immediately we notice from the $(t,t)$ equation that
$\xi_{t}=0\Longrightarrow\xi_{r}=0$ and from the $(\theta,\theta)$ equation
that $\xi_{r}=\xi_{\theta}=0\Longrightarrow\xi_{t}=0$. For
$\xi_{t}=\xi_{r}=0$, we find the three rotational Killing vector fields:
$\displaystyle\Omega^{\mu}_{1}\rightarrow\langle
0,0,\cos{\phi},-\cot{\theta}\sin{\phi}\rangle$
$\displaystyle\Omega^{\mu}_{2}\rightarrow\langle
0,0,\sin{\phi},\cot{\theta}\cos{\phi}\rangle$
$\displaystyle\psi^{\mu}\rightarrow\langle 0,0,0,1\rangle$
For $\xi_{\theta}=\xi_{\phi}=0$, we have a time and radial Killing vector
field:
$\rho^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}r}\cosh(\sqrt{\Lambda}t),-\sinh(\sqrt{\Lambda}t),0,0\rangle$
For $\xi_{r}=\xi_{\phi}=0$, a time and $\theta$ Killing vector field:
$\Theta^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}}\cos{\theta},0,-\sin{\theta}\tanh(\sqrt{\Lambda}t),0\rangle$
For $\xi_{\phi}=0$, a time, radial, and $\theta$ Killing vector field:
$\xi^{\mu}_{(t,r,\theta)}=\langle-\frac{\sinh(\sqrt{\Lambda}t)\cos{\theta}}{\sqrt{\Lambda}r},\cosh(\sqrt{\Lambda}t)\cos{\theta},-\frac{\sin{\theta}}{r\cosh(\sqrt{\Lambda}t)},0\rangle$
In addition, for only $\xi_{r}=0$, we find two Killing vector fields:
$\displaystyle\xi^{\mu}_{(t,\theta,\phi),1}\rightarrow\langle\frac{1}{\Lambda}\sin{\theta}\sin{\phi},0,\frac{1}{\sqrt{\Lambda}}\tanh(\sqrt{\Lambda}t)\cos{\theta}\sin{\phi},\frac{1}{\sqrt{\Lambda}\sin{\theta}}\tanh(\sqrt{\Lambda}t)\cos{\phi}\rangle$
$\displaystyle\xi^{\mu}_{(t,\theta,\phi),2}\rightarrow\langle-\frac{1}{\Lambda}\sin{\theta}\cos{\phi},0,-\frac{1}{\sqrt{\Lambda}}\tanh(\sqrt{\Lambda}t)\cos{\theta}\cos{\phi},\frac{1}{\sqrt{\Lambda}\sin{\theta}}\tanh(\sqrt{\Lambda}t)\sin{\phi}\rangle$
Finally, taking all components to be non-zero, we have the last two Killing
vector fields:
$\displaystyle\xi^{\mu}_{(t,r,\theta,\phi),1}\rightarrow\langle-\frac{1}{\sqrt{\Lambda}r}\sinh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi},\cosh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi},\frac{1}{r\cosh(\sqrt{\Lambda}t)}\cos{\theta}\sin{\phi},$
$\displaystyle\frac{1}{r\cosh(\sqrt{\Lambda}t)\sin{\theta}}\cos{\phi}\rangle$
$\displaystyle\xi^{\mu}_{(t,r,\theta,\phi),2}\rightarrow\langle-\frac{1}{\sqrt{\Lambda}r}\sinh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi},\cosh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi},\frac{1}{r\cosh(\sqrt{\Lambda}t)}\cos{\theta}\cos{\phi},$
$\displaystyle-\frac{1}{r\cosh(\sqrt{\Lambda}t)\sin{\theta}}\sin{\phi}\rangle$
Summarizing, we have ten linearly independent Killing vector fields for this
metric.
## Appendix D Symplectic structure
Within this appendix, we’ll briefly address and apply the concepts presented
in [43] for the formulation of a quantum field theory of a real scalar field
with a general background metric, where we shall not concern ourselves with
the interaction between matter and space-time at the quantum level and instead
treat the metric as non-dynamic (hence the term “background”). However, we
strongly encourage the reader to review [43] in order to fully understand all
the material presented below.
The information associated with the dynamical evolution of a physical system
can be conveyed within the symplectic structure $\Omega$ for a particular
action $S$ given by
$\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])=\int_{\Sigma_{0}}d^{3}x\bigg{(}\pi_{1}\phi_{2}-\pi_{2}\phi_{1}\bigg{)}$
where $\Omega$ is a non-degenerate antisymmetric bilinear map from the
solutions of the equation of motion associated with our action to the real
numbers. In addition, a point in phase-space (Hamiltonian formalism)
corresponds to the specification of our field solution $\phi$ and its
conjugate momentum $\pi=\partial\mathcal{L}/\partial\dot{\phi}$ on a space-
like hypersurface $\Sigma_{0}$ associated with our “initial value”
configuration ($\mathcal{L}$ is the Lagrangian density associated with the
action $S$ which we’ll give below). The fundamental Poisson brackets in
classical theory can then be expressed as
$\\{\Omega([\phi_{1},\pi_{1}],\cdot),\Omega([\phi_{2},\pi_{2}],\cdot)\\}=-\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])$
where $\Omega(y,\cdot)$ is a linear function assuming our choice of $y$ does
not vary (our input argument is only ‘$\cdot$’). If we arbitrarily choose
$[\phi_{1},\pi_{1}]=[0,f_{1}]$ and $[\phi_{2},\pi_{2}]=[f_{2},0]$, our
classical Poisson brackets reduce to
$\bigg{\\{}\int d^{3}xf_{1}(x)\phi(x),\int d^{3}yf_{2}(y)\pi(y)\bigg{\\}}=\int
d^{3}xf_{1}(x)f_{2}(x)$
which we can think of as the more familiar canonical relations
$\\{\phi(x),\pi(y)\\}=\delta(x-y)$
Then to construct our quantum theory of a scalar field, we extend the
functions $\Omega([\phi,\pi],\cdot)$ to operators
$\hat{\Omega}([\phi,\pi],\cdot)$ satisfying the commutation relations
$[\hat{\Omega}([\phi_{1},\pi_{1}],\cdot),\hat{\Omega}([\phi_{2},\pi_{2}],\cdot)]=-i\hbar\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])\hat{\mathrm{I}}$
($\hat{\mathrm{I}}$ denotes the identity operator) and introduce the inner
product associated with this system
$(\psi^{+},\chi^{+})=-i\Omega(\bar{\psi}^{+},\chi^{+})$
where $\bar{\psi}^{+}$ represents the complex conjugate of $\psi^{+}$ and we
have decomposed our full solutions $\psi,\chi\in\mathcal{S}$ of the equation
of motion for our action into
$\psi=\psi^{+}+\psi^{-}$
such that our inner product with respect to these “positive frequency”
solutions $\psi^{+},\chi^{+}$ is positive-definite. We denote the solution
space spanned by these “positive frequency” parts as
$\mathcal{S}^{\mathbb{C}+}$. In addition, we have expressed our inner product
only in terms of solutions to our equation of motion $\psi\in\mathcal{S}$ as
for each solution there corresponds a point in phase space
$[\psi,\pi_{\psi}]$. One proceeds to “Cauchy-complete” in the norm defined by
this inner product to obtain our complex Hilbert space $\mathcal{H}$ (see
Chapter 3.2 and Appendix A.1 of [43]). Thus, we represent our classical
observables $\Omega(\psi,\cdot)$ for each solution $\psi$ by the operator
$\hat{\Omega}(\psi,\cdot)=i\hat{a}(\bar{\mathrm{K}\psi})-i\hat{a}^{\dagger}(\mathrm{K}\psi)$
where $\mathrm{K}:\mathcal{S}\rightarrow\mathcal{H}$ is a map from the full
solutions to our complex Hilbert space. As well, $\hat{a}(\cdot)$ and
$\hat{a}^{\dagger}(\cdot)$ denote the annihilation and creation operators,
respectively, which act on a general state $\Psi$ in the symmetric Fock space
$\mathcal{F}_{s}(\mathcal{H})$ in the following manner. For a general state
$\Psi=\langle\psi,\psi^{a_{1}},\psi^{a_{1}a_{2}},\ldots,\psi^{a_{1}\ldots
a_{n}},\dots\rangle$ representing our “n-particle amplitudes” where for scalar
theory $\psi^{a_{1}\ldots a_{n}}=\psi^{(a_{1}\ldots a_{n})}$ $\forall n$
(round parantheses denote symmetrization when dealing with abstract indices
here and below) and $\xi^{a}\in\mathcal{H}$,
$\bar{\xi}_{a}\in\bar{\mathcal{H}}$, we have
$\displaystyle\hat{a}(\bar{\xi})\Psi=\langle\bar{\xi}_{a}\psi^{a},\sqrt{2}\bar{\xi}_{a}\psi^{aa_{1}},\sqrt{3}\bar{\xi}_{a}\psi^{aa_{1},a_{2}},\ldots\rangle$
$\displaystyle\hat{a}^{\dagger}(\xi)\Psi=\langle
0,\psi\xi^{a_{1}},\sqrt{2}\xi^{(a_{1}}\psi^{a_{2})},\sqrt{3}\xi^{(a_{1}}\psi^{a_{2}a_{3})},\ldots\rangle$
Indices on $\bar{\xi}_{a}$ and $\xi^{a}$ are dropped in our expressions on the
left-hand side of these equations for notational convenience. In this paper,
we operate under the assumption that the norms of these two expressions are
finite. In addition, the inner product of two vectors $\xi,\eta\in\mathcal{H}$
is denoted by
$(\xi,\eta)=\bar{\xi}_{a}\eta^{a}$
In this notation, $\psi\in\otimes^{n}\mathcal{H}$ is denoted
$\psi^{a_{1}\ldots a_{n}}$ and $\bar{\psi}\in\otimes^{n}\bar{\mathcal{H}}$ as
$\bar{\psi}_{a_{1}\ldots a_{n}}$ where
$\otimes^{n}\mathcal{H}=\mathcal{H}_{1}\otimes\ldots\otimes\mathcal{H}_{n}$
for $\mathcal{H}_{1}=\ldots=\mathcal{H}_{n}=\mathcal{H}$ (n-fold tensor
product space). As well,
$\mathcal{F}_{s}(\mathcal{H})=\oplus_{n=0}^{\infty}(\otimes^{n}_{s}\mathcal{H})$
where $\otimes^{n}_{s}\mathcal{H}$ is the symmetric n-fold tensor product
space and $\otimes^{0}\mathcal{H}$ is defined to be the complex numbers
$\mathbb{C}$ (see Appendix A of [43]).
To clarify further with regard to Fock space notation, we relate back to Dirac
“bra-ket” notation:
$|0\rangle_{\hat{a}}\equiv\langle 1,0,0,\ldots\rangle$
where
$\hat{a}(\bar{\xi})|0\rangle_{\hat{a}}=0$
for some general $\xi^{a}$ given $|0\rangle_{\hat{a}}$ denotes the vacuum
state associated with the creation and annihilation operators on our Fock
space (i.e. the $\hat{a}$’s). Given our general state $\Psi$, the probability
of finding only a single ‘$\hat{a}$ particle’ in state $\beta\in\mathcal{H}$
is taken to be $|\bar{\beta}_{a}\psi^{a}|^{2}$ (i.e. $\psi^{a}$ is the “one-
particle amplitude”, $\psi^{a_{1}a_{2}}$ is the “two-particle amplitude”,
etc.). Here, the term ‘particle’ really refers to an excitation of the
particular field associated with our $\hat{a},\hat{a}^{\dagger}$ operators
(quanta). Therefore, one can think of $\hat{a}(\bar{\xi})$ as an operator
annihilating a quantum of state $\xi^{a}$ from each of the “n-particle” states
in the general state $\Psi$, and analogously $\hat{a}^{\dagger}(\xi)$ as
creating a quantum in each. Our annihilation and creation operators also
satisfy the commutation relation:
$[\hat{a}(\bar{\xi}),\hat{a}^{\dagger}(\eta))]=\bar{\xi}_{a}\eta^{a}\hat{\mathrm{I}}$
Note that our use of abstract index notation in this paragraph does not refer
to the metric. In other words, when working with Hilbert space vectors, we
always assume contraction occurs over the inner product of the respective
Hilbert space as defined earlier in this appendix and not with regard to the
metric.
Then for our theory of inertial centers, the action associated with the
equation of motion for our Klein-Gordon extension in a particular inertial
system (60) takes the form
$S=-\frac{1}{2}\int
d^{4}x\sqrt{|g|}\bigg{(}\nabla^{a}\phi\nabla_{a}\phi+{\tilde{\mu}}^{2}r^{2}\phi^{2}\bigg{)}$
(73)
where one can verify this by extremizing the action ($\delta S=0$) to obtain
our equation of motion. We proceed with our formulation by “slicing” our
manifold $M$ into space-like hypersurfaces each indexed by a time parameter
$t$ ($\Sigma_{t}$). Then, we introduce a vector field on $M$ associated with
our time evolution and defined by $t^{a}\nabla_{a}t=1$, which we can decompose
in the following manner:
$t^{a}=Nn^{a}+N^{a}$
(in contrast with our previous paragraph, abstract index notation here employs
the metric). $n^{a}$ is the future-directed unit normal vector field to our
space-like hypersurfaces $\Sigma_{t}$ (future-directed in the sense that
$n^{a}$ lies in the same direction as $t^{a}$), and $N^{a}$ represents the
remaining tangential portion of $t^{a}$ to $\Sigma_{t}$. In addition, we
introduce coordinates $t,x^{1},x^{2},x^{3}$ such that $t^{a}\nabla_{a}x^{i}=0$
for $i=1,2,3$ which allows $t^{a}=(\partial/\partial t)^{a}$. Our action in
(73) can then be rewritten in terms of the integral of a Lagrangian density
$\mathcal{L}$ over our time parameter $t$ and our space-like hypersurface
$\Sigma_{t}$:
$S=\int dt\int_{\Sigma_{t}}d^{3}x\mathcal{L}$
with
$\mathcal{L}=\frac{1}{2}N\sqrt{|h|}\bigg{(}(n^{a}\nabla_{a}\phi)^{2}-h^{ab}\nabla_{a}\phi\nabla_{b}\phi-{\tilde{\mu}}^{2}r^{2}\phi^{2}\bigg{)}$
where $h_{ab}$ is the induced Riemannian metric on $\Sigma_{t}$ and
$h=\mathrm{det}(h_{\beta\nu})$. Yet, since
$n^{a}\nabla_{a}\phi=\frac{1}{N}(t^{a}-N^{a})\nabla_{a}\phi=\frac{1}{N}\dot{\phi}-\frac{1}{N}N^{a}\nabla_{a}\phi$
where $\dot{\phi}=t^{a}\nabla_{a}\phi$, we find that our conjugate momentum
density on $\Sigma_{t}$ takes the form
$\pi=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=(n^{a}\nabla_{a}\phi)\sqrt{|h|}$
as it does with our original Klein-Gordon action. Consequently, our symplectic
structure for a free scalar field in the theory of inertial centers is given
by
$\displaystyle\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])=\int_{\Sigma_{0}}d^{3}x(\pi_{1}\phi_{2}-\pi_{2}\phi_{1})$
$\displaystyle=\int_{\Sigma_{0}}d^{3}x\sqrt{|h|}[\phi_{2}n^{a}\nabla_{a}\phi_{1}-\phi_{1}n^{a}\nabla_{a}\phi_{2}]$
where $\Sigma_{0}$ is the space-like hypersurface associated with our “initial
value” configuration at $t=0$.
## Appendix E Divergence of a vector field
Following Chapter 3.4 of [24], the divergence of a vector field $v^{a}$ is
given by
$\nabla_{a}v^{a}=\partial_{a}v^{a}+\Gamma^{a}_{ab}v^{b}$
where we have used our knowledge from Appendix B to expand this expression.
However, in component form
$\Gamma^{a}_{a\nu}=\sum_{\mu}\Gamma^{\mu}_{\mu\nu}=\frac{1}{2}\sum_{\mu,\rho}g^{\mu\rho}[\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}]$
Yet the first and last of these terms cancel as we are summing over both $\mu$
and $\rho$ and $g^{ab}=g^{ba}$. This leaves us with
$\Gamma^{a}_{a\nu}=\frac{1}{2}\sum_{\mu,\rho}g^{\mu\rho}\partial_{\nu}g_{\mu\rho}$
but if we think in terms of the matrix form of our components, $(g_{\mu\nu})$,
we have
$\sum_{\mu,\rho}g^{\mu\rho}\partial_{\nu}g_{\mu\rho}=\frac{\partial_{\nu}g}{g}$
where $g=\mathrm{det}(g_{\mu\nu})$. Therefore,
$\Gamma^{a}_{a\nu}=\frac{1}{2}\frac{\partial_{\nu}g}{g}=\partial_{\nu}\ln{\sqrt{|g|}}$
Plugging in above for our divergence term,
$\nabla_{a}v^{a}=\sum_{\mu}\bigg{[}\partial_{\mu}v^{\mu}+v^{\mu}\partial_{\mu}\ln{\sqrt{|g|}}\bigg{]}=\sum_{\mu}\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}v^{\mu})$
Then for a scalar field $f$,
$\nabla_{a}\nabla^{a}f=\sum_{\mu}\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}g^{\mu\nu}\partial_{\nu}f)$
as $\nabla_{a}f=\partial_{a}f$.
## Appendix F Scalar field solutions
We look for solutions ($\phi_{i}$) to our extension of the Klein-Gordon
equation:
$(\nabla_{a}\nabla^{a}-\tilde{\mu}^{2}r^{2})\phi_{i}=0$
As shown in Appendix E, for a real scalar field
$\nabla_{a}\nabla^{a}\phi_{i}=\sum_{\nu,\beta}\frac{1}{\sqrt{|g|}}\partial_{\nu}(\sqrt{|g|}g^{\nu\beta}\partial_{\beta}\phi_{i})$
where for our purposes $g_{\nu\beta}$ refers to the radial Rindler metric
components and
$\sqrt{|g|}=\sqrt{\Lambda}r^{3}\cosh^{2}(\sqrt{\Lambda}t)\sin{\theta}$.
Expanding (60),
$\displaystyle 0=-\tilde{\mu}^{2}r^{2}\phi_{i}-\frac{1}{\Lambda
r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}\phi_{i})+\frac{1}{r^{3}}\partial_{r}(r^{3}\partial_{r}\phi_{i})$
$\displaystyle+\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin{\theta}}\partial_{\theta}(\sin{\theta}\partial_{\theta}\phi_{i})+\frac{1}{\sin^{2}{\theta}}\partial^{2}_{\phi}\phi_{i}\bigg{]}$
(74)
We look for separable solutions of the form, $\phi_{i}=Z_{i}\cdot g(t)\cdot
h(r)\cdot Y_{l}^{m}(\theta,\phi)$, where $Z_{i}$ is a normalization constant
and the $Y_{l}^{m}$ are spherical harmonics satisfying
$\frac{1}{\sin{\theta}}\partial_{\theta}(\sin{\theta}\partial_{\theta}Y_{l}^{m})+\frac{1}{\sin^{2}{\theta}}\partial^{2}_{\phi}Y_{l}^{m}=-l(l+1)Y_{l}^{m}$
(75)
where
$Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos{\theta})\cdot
e^{im\phi}$ (76)
with $l$ as a non-negative integer, $|m|\leq l$, and $m$ also as an integer
(see Chapter 3.6 of [56] or Chapter 15.5 of [45]). $P_{l}^{m}$ is an
associated Legendre function which satisfies the differential equation
$\bigg{[}(1-x^{2})\partial^{2}_{x}-2x\partial_{x}+\bigg{(}l[l+1]-\frac{m^{2}}{1-x^{2}}\bigg{)}\bigg{]}P^{m}_{l}(x)=0$
and can be expressed in terms of Rodrigues’ formula [46]:
$P_{l}^{m}(x)=\frac{(-1)^{m}}{2^{l}l!}(1-x^{2})^{m/2}\frac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}$
with
$P_{l}^{-m}=(-1)^{m}\frac{(l-m)!}{(l+m)!}P_{l}^{m}$
The spherical harmonics $Y_{l}^{m}$ as expressed above obey the orthogonality
relation:
$\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi
Y_{l}^{m}\bar{Y}_{l^{\prime}}^{m^{\prime}}=\delta_{ll^{\prime}}\delta_{mm^{\prime}}$
Plugging in and dividing by $\phi_{i}$,
$0=-\tilde{\mu}^{2}r^{2}-\frac{1}{g\Lambda
r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\frac{1}{hr^{3}}\partial_{r}(r^{3}\partial_{r}h)-\frac{l(l+1)}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}$
Multiplying through by $r^{2}$ and grouping functions of $t$ and $r$:
$\frac{1}{hr}\partial_{r}(r^{3}\partial_{r}h)-\tilde{\mu}^{2}r^{4}=\frac{1}{g\Lambda\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\frac{l(l+1)}{\cosh^{2}(\sqrt{\Lambda}t)}=-(4\alpha^{2}+1)$
where $\alpha$ is a constant. Thus, we have two differential equations:
$\displaystyle
0=r^{2}\partial^{2}_{r}h+3r\partial_{r}h+\bigg{[}(4\alpha^{2}+1)-\tilde{\mu}^{2}r^{4}\bigg{]}h$
(77) $\displaystyle
0=\frac{1}{\Lambda\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\bigg{[}\frac{l(l+1)}{\cosh^{2}(\sqrt{\Lambda}t)}+(4\alpha^{2}+1)\bigg{]}g$
(78)
Focusing on our radial equation first, we set $\rho=\sqrt{\tilde{\mu}}r$:
$0=\rho^{2}\partial^{2}_{\rho}h+3\rho\partial_{\rho}h+\bigg{[}(4\alpha^{2}+1)-\rho^{4}\bigg{]}h$
(79)
Letting $h(\rho)=z(\rho)/\rho$
$0=\rho^{2}\partial^{2}_{\rho}z+\rho\partial_{\rho}z+\bigg{[}4\alpha^{2}-\rho^{4}\bigg{]}z$
and setting $y=\rho^{2}/2$, we find
$0=y^{2}\partial^{2}_{y}z+y\partial_{y}z+[\alpha^{2}-y^{2}]z$
But this is just the modified Bessel equation of pure imaginary order (see
Chapter 3 of [47]). Choosing the physically realistic solution (we expect $h$
to decay for large $r$ since in our classical analysis massive objects are
“confined” to motion about their inertial centers), our full expression takes
the form
$h_{\alpha}(\rho)=\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}$ (80)
where $K_{i\alpha}$ is the Macdonald function (modified Bessel function) of
imaginary order $\alpha$ given in integral form (for $y>0$):
$K_{i\alpha}(y)=\int_{0}^{\infty}d\eta\cos(\alpha\eta)e^{-y\cosh{\eta}}$
and $\alpha$ is restricted to the range: $0\leq\alpha<\infty$ (see Chapter
4.15 of [48] and [42] for its application to quantization in the classic
Rindler case with the Klein-Gordon equation).
The Macdonald function of imaginary order obeys an orthogonality relation
which will be useful to us for determining part of our normalization constant.
From [49],
$\int_{0}^{\infty}dy\frac{K_{i\alpha}(y)K_{i\alpha^{\prime}}(y)}{y}=\frac{\pi^{2}}{2\alpha\sinh(\pi\alpha)}\delta(\alpha-\alpha^{\prime})$
Later we’ll need:
$\displaystyle\int_{0}^{\infty}d\rho\frac{K_{i\alpha}(\frac{\rho^{2}}{2})K_{i\alpha^{\prime}}(\frac{\rho^{2}}{2})}{\rho}=\int_{0}^{\infty}d\eta\int_{0}^{\infty}d\eta^{\prime}\cos(\alpha\eta)\cos(\alpha^{\prime}\eta^{\prime})\int_{0}^{\infty}\frac{d\rho}{\rho}e^{-(\rho^{2}/2)(\cosh{\eta}+\cosh{\eta^{\prime}})}$
$\displaystyle=\int_{0}^{\infty}d\eta\int_{0}^{\infty}d\eta^{\prime}\cos(\alpha\eta)\cos(\alpha^{\prime}\eta^{\prime})\int_{0}^{\infty}\frac{dy}{2y}e^{-y(\cosh{\eta}+\cosh{\eta^{\prime}})}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}dy\frac{K_{i\alpha}(y)K_{i\alpha^{\prime}}(y)}{y}=\frac{\pi^{2}}{4\alpha\sinh(\pi\alpha)}\delta(\alpha-\alpha^{\prime})$
Examining our second differential equation, we let
$\eta=\tanh(\sqrt{\Lambda}t)$:
$0=(1-\eta^{2})^{2}\partial^{2}_{\eta}g+[l(l+1)(1-\eta^{2})+(4\alpha^{2}+1)]g$
(81)
where $\eta^{2}<1$. Before proceeding any further, we must make one remark
that will be crucial for evaluating our inner product. Taking the complex
conjugate of (81)
$0=(1-\eta^{2})^{2}\partial^{2}_{\eta}\bar{g}+[l(l+1)(1-\eta^{2})+(4\alpha^{2}+1)]\bar{g}$
Multiplying (81) by $\bar{g}$ and subtracting by $g$ times the complex
conjugate of (81), we find
$\bar{g}\partial^{2}_{\eta}g-g\partial^{2}_{\eta}\bar{g}=0$
or
$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}={\rm constant}$ (82)
Returning to (81), we divide through by $1-\eta^{2}$ and make the substitution
$g(\eta)=\sqrt{1-\eta^{2}}\cdot p(\eta)$:
$0=(1-\eta^{2})\sqrt{1-\eta^{2}}\bigg{[}\partial^{2}_{\eta}p-\frac{2\eta}{1-\eta^{2}}\partial_{\eta}p-\frac{p}{(1-\eta^{2})^{2}}\bigg{]}+\bigg{[}l(l+1)+\frac{4\alpha^{2}}{1-\eta^{2}}+\frac{1}{1-\eta^{2}}\bigg{]}\sqrt{1-\eta^{2}}\cdot
p$
which reduces to
$0=(1-\eta^{2})\partial^{2}_{\eta}p-2\eta\partial_{\eta}p+\bigg{[}l(l+1)+\frac{4\alpha^{2}}{1-\eta^{2}}\bigg{]}p$
But the solution to this differential equation is the Legendre function of the
first kind [46] (since our domain is restricted to $\eta^{2}<1$) which can be
expressed in the following manner:
$P^{\pm 2i\alpha}_{l}(\eta)=\frac{1}{\Gamma(1\mp
2i\alpha)}\bigg{[}\frac{1+\eta}{1-\eta}\bigg{]}^{\pm
i\alpha}\,_{2}F_{1}(-l,l+1;1\mp 2i\alpha,\frac{1-\eta}{2})$ (83)
where $\,{}_{2}F_{1}$ is the hypergeometric function which for our parameters
can take the form
$\,{}_{2}F_{1}(-l,l+1;1\mp 2i\alpha,\frac{1-\eta}{2})=\frac{\Gamma(1\mp
2i\alpha)}{\Gamma(-l)\Gamma(l+1)}\sum_{k=0}^{\infty}\frac{\Gamma(k-l)\Gamma(k+l+1)}{k!\cdot\Gamma(k+1\mp
2i\alpha)}\bigg{(}\frac{1-\eta}{2}\bigg{)}^{k}$ (84)
and $\Gamma(z)$ is the gamma function which can be written in integral form
for $\Re(z)>0$ where $z$ is a complex variable as
$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$
Note that this Legendre function, $P_{\nu}^{\mu}$, is in fact a generalization
of the Legendre function used earlier for our angular dependence where the
parameters $\mu$, $\nu$ here are allowed to be complex numbers instead of
solely real integers. Our full expression for $g$ is then
$g(\eta)=\sqrt{1-\eta^{2}}\cdot P_{l}^{-2i\alpha}(\eta)$ (85)
where as we’ll see below, our choice of $-2i\alpha$ is necessary in order to
ensure that our inner product is positive-definite with respect to our
solutions for $\phi_{i}$ so that we may properly construct our field operator
(i.e. we take the “positive frequency” solutions; see Chapter 3.2 of [43]).
To find our normalization constant, we’ll need to evaluate (82). For the rest
of our analysis in this appendix, we use Chapter 8 of [46] as a reference for
our general expressions. Beginning with $\partial_{\eta}g$:
$\partial_{\eta}g=-\frac{\eta}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\sqrt{1-\eta^{2}}\partial_{\eta}P_{l}^{-2i\alpha}$
But
$\partial_{\eta}P_{\nu}^{\mu}=-\frac{\nu\eta}{1-\eta^{2}}P_{\nu}^{\mu}+\frac{\mu+\nu}{1-\eta^{2}}P_{\nu-1}^{\mu}$
Plugging in above
$\displaystyle\partial_{\eta}g=-\frac{\eta}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\sqrt{1-\eta^{2}}\bigg{(}-\frac{l\eta}{1-\eta^{2}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{1-\eta^{2}}P_{l-1}^{-2i\alpha}\bigg{)}$
$\displaystyle=-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{\sqrt{1-\eta^{2}}}P_{l-1}^{-2i\alpha}$
Then
$\displaystyle\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=\sqrt{1-\eta^{2}}\bigg{[}\bar{P}_{l}^{-2i\alpha}\bigg{(}-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{\sqrt{1-\eta^{2}}}P_{l-1}^{-2i\alpha}\bigg{)}$
$\displaystyle-
P_{l}^{-2i\alpha}\bigg{(}-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}\bar{P}_{l}^{-2i\alpha}+\frac{l+2i\alpha}{\sqrt{1-\eta^{2}}}\bar{P}_{l-1}^{-2i\alpha}\bigg{)}\bigg{]}$
$\displaystyle=(l-2i\alpha)\bar{P}_{l}^{-2i\alpha}P_{l-1}^{-2i\alpha}-(l+2i\alpha)P_{l}^{-2i\alpha}\bar{P}_{l-1}^{-2i\alpha}$
However, as one can tell from (83),
$\bar{P}_{l}^{-2i\alpha}=P_{l}^{2i\alpha}$. Therefore,
$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=(l-2i\alpha)P_{l}^{2i\alpha}P_{l-1}^{-2i\alpha}-(l+2i\alpha)P_{l}^{-2i\alpha}P_{l-1}^{2i\alpha}={\rm
constant}$
Yet since this expression must be constant, we can evaluate for any particular
value of $\eta$. Because we have expressions for the Legendre functions at
$\eta=0$, we’ll make this convenient choice where
$P_{\nu}^{\mu}(0)=2^{\mu}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(\nu+\mu)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{1}{2}\nu+\frac{1}{2}\mu)}{\Gamma(1+\frac{1}{2}\nu-\frac{1}{2}\mu)}$
For $\eta=0$:
$\displaystyle P_{l-1}^{-2i\alpha}(0)\cdot
P_{l}^{2i\alpha}(0)=2^{-2i\alpha}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{l-1}{2}-i\alpha)}{\Gamma(1+\frac{l-1}{2}+i\alpha)}$
$\displaystyle\cdot
2^{2i\alpha}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{l}{2}+i\alpha)}{\Gamma(1+\frac{l}{2}-i\alpha)}$
$\displaystyle=\frac{1}{\pi}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}\frac{\Gamma(\frac{l}{2}-i\alpha)}{\Gamma(1+\frac{l}{2}-i\alpha)}$
But from properties of the gamma function (see Chapter 6 of [46]),
$\Gamma(1+z)=z\Gamma(z)$
Using this property above,
$\displaystyle P_{l-1}^{-2i\alpha}(0)\cdot
P_{l}^{2i\alpha}(0)=\frac{1}{\pi(\frac{l}{2}-i\alpha)}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}$
And with the expressions
$\displaystyle\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}=\frac{1}{2}\bigg{[}e^{i(\pi/2)(l-1-2i\alpha)}+e^{-i(\pi/2)(l-1-2i\alpha)}\bigg{]}$
$\displaystyle\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}=\frac{1}{2}\bigg{[}e^{i(\pi/2)(l+2i\alpha)}+e^{-i(\pi/2)(l+2i\alpha)}\bigg{]}$
we have
$\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}=\frac{1}{4}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha+1/2)}+e^{-i\pi(2i\alpha+1/2)}\bigg{]}$
Thus,
$(l-2i\alpha)P_{l-1}^{-2i\alpha}P_{l}^{2i\alpha}|_{\eta=0}=\frac{1}{2\pi}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha+1/2)}+e^{-i\pi(2i\alpha+1/2)}\bigg{]}$
Applying similar logic to the second term in our expression above, we find
$(l+2i\alpha)P_{l}^{-2i\alpha}P_{l-1}^{2i\alpha}|_{\eta=0}=\frac{1}{2\pi}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha-1/2)}+e^{-i\pi(2i\alpha-1/2)}\bigg{]}$
Then
$\displaystyle\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=\frac{1}{2\pi}\bigg{[}e^{i\pi/2-2\pi\alpha}+e^{-i\pi/2+2\pi\alpha}-e^{i\pi/2+2\pi\alpha}-e^{-i\pi/2-2\pi\alpha}\bigg{]}$
$\displaystyle=\frac{1}{2\pi}\bigg{[}e^{i\pi/2}\bigg{(}e^{-2\pi\alpha}-e^{2\pi\alpha}\bigg{)}+e^{-i\pi/2}\bigg{(}e^{2\pi\alpha}-e^{-2\pi\alpha}\bigg{)}\bigg{]}$
$\displaystyle=-\frac{(e^{2\pi\alpha}-e^{-2\pi\alpha})(e^{i\pi/2}-e^{-i\pi/2})}{2\pi}$
But
$e^{i\pi/2}-e^{-i\pi/2}=2i\sin(\pi/2)=2i$
and
$e^{2\pi\alpha}-e^{-2\pi\alpha}=2\sinh(2\pi\alpha)=4\sinh(\pi\alpha)\cosh(\pi\alpha)$
Therefore, plugging in above
$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=-\frac{4i}{\pi}\sinh(\pi\alpha)\cosh(\pi\alpha)$
Addressing our inner product where our solutions are of the form
$\phi_{i}=Z_{i}\cdot g\cdot h\cdot Y_{l}^{m}$
$\displaystyle(\phi_{1},\phi_{2})=-\frac{i}{\sqrt{\Lambda}}Z_{1}Z_{2}\cosh^{2}(\sqrt{\Lambda}t)[g_{2}\partial_{t}\bar{g}_{1}-\bar{g}_{1}\partial_{t}g_{2}]|_{t=0}\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bar{Y}_{l_{1}}^{m_{1}}Y_{l_{2}}^{m_{2}}$
$\displaystyle\cdot\bigg{[}\int_{0}^{\infty}r\bar{h}_{\alpha_{1}}h_{\alpha_{2}}dr-\int_{-\infty}^{0}r\bar{h}_{\alpha_{1}}h_{\alpha_{2}}dr\bigg{]}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}\cdot\bigg{[}\int_{0}^{\infty}dr\frac{K_{i\alpha_{1}}(\frac{\tilde{\mu}r^{2}}{2})K_{i\alpha_{2}}(\frac{\tilde{\mu}r^{2}}{2})}{\tilde{\mu}r}$
$\displaystyle-\int_{-\infty}^{0}dr\frac{K_{i\alpha_{1}}(\frac{\tilde{\mu}r^{2}}{2})K_{i\alpha_{2}}(\frac{\tilde{\mu}r^{2}}{2})}{\tilde{\mu}r}\bigg{]}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}\cdot\frac{2}{\tilde{\mu}}\int_{0}^{\infty}d\rho\frac{K_{i\alpha_{1}}(\frac{\rho^{2}}{2})K_{i\alpha_{2}}(\frac{\rho^{2}}{2})}{\rho}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\cdot[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}$
$\displaystyle=iZ_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\cdot[\bar{g}_{1}\partial_{\eta}g_{1}-g_{1}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}$
$\displaystyle=iZ_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\bigg{[}-\frac{4i\sinh(\pi\alpha_{1})\cosh(\pi\alpha_{1})}{\pi}\bigg{]}$
$\displaystyle=Z_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{2\pi\cosh(\pi\alpha_{1})}{\tilde{\mu}\alpha_{1}}$
where we keep in mind the fact that
$\frac{\partial}{\partial t}=\frac{\partial\eta}{\partial
t}\frac{\partial}{\partial\eta}=\frac{\sqrt{\Lambda}}{\cosh^{2}(\sqrt{\Lambda}t)}\cdot\frac{\partial}{\partial\eta}$
Thus, our normalization constant is found to be
$Z_{\alpha}=\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}$ (86)
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## Figure Legends
Figure 1: Geodesic paths in Minkowski coordinates Figure 2: Local
approximation of the inertial frame of reference Figure 3: Plots of
$h_{\alpha}(\rho)$ for $\alpha=0,1,5,$ and $20$
## Tables
Table 1: Redshift from objects within the Local Group
Object | RA (J2000.0) | Dec (J2000.0) | Redshift | Distance Mod (mag)
---|---|---|---|---
Andromeda V | 01h10m17.10s | +47d37m41.0s | -0.001344 | 24.52
Andromeda I | 00h45m39.80s | +38d02m28.0s | -0.001228 | 24.46
Andromeda VI | 23h51m46.30s | +24d34m57.0s | -0.001181 | 24.58
Andromeda III | 00h35m33.78s | +36d29m51.9s | -0.001171 | 24.38
IC 0010 | 00h20m17.34s | +59d18m13.6s | -0.001161 | 24.57
Andromeda VII | 23h26m31.74s | +50d40m32.6s | -0.001024 | 24.7
MESSIER 031 | 00h42m44.35s | +41d16m08.6s | -0.001001 | 24.46
Draco Dwarf | 17h20m12.39s | +57d54m55.3s | -0.000974 | 19.61
Pisces I | 01h03m55.00s | +21d53m06.0s | -0.000956 | 24.5
UMi Dwarf | 15h09m08.49s | +67d13m21.4s | -0.000824 | 19.3
MESSIER 110 | 00h40m22.08s | +41d41m07.1s | -0.000804 | 24.5
IC 1613 | 01h04m47.79s | +02d07m04.0s | -0.000781 | 24.33
NGC 0185 | 00h38m57.97s | +48d20m14.6s | -0.000674 | 24.13
MESSIER 032 | 00h42m41.83s | +40d51m55.0s | -0.000667 | 24.42
NGC 0147 | 00h33m12.12s | +48d30m31.5s | -0.000644 | 24.3
Andromeda II | 01h16m29.78s | +33d25m08.8s | -0.000627 | 24.03
Pegasus Dwarf | 23h28m36.25s | +14d44m34.5s | -0.000612 | 26.34
MESSIER 033 | 01h33m50.89s | +30d39m36.8s | -0.000597 | 24.69
Aquarius dIrr | 20h46m51.81s | -12d50m52.5s | -0.00047 | 26
WLM | 00h01m58.16s | -15d27m39.3s | -0.000407 | 25.09
Cetus Dwarf Spheroidal | 00h26m11.03s | -11d02m39.6s | -0.00029 | 24.51
SagDIG | 19h29m59.58s | -17d40m51.3s | -0.000264 | 25.03
NGC 6822 | 19h44m57.74s | -14d48m12.4s | -0.00019 | 23.41
Leo A | 09h59m26.46s | +30d44m47.0s | 0.000067 |
Fornax Dwarf Spheroidal | 02h39m59.33s | -34d26m57.1s | 0.000178 | 20.7
Phoenix Dwarf | 01h51m06.34s | -44d26m40.9s | 0.000187 | 23.08
Leo B | 11h13m28.80s | +22d09m06.0s | 0.000264 | 21.67
Sculptor Dwarf Elliptical | 01h00m09.36s | -33d42m32.5s | 0.000367 | 19.67
Sagittarius Dwarf Spheroidal | 18h55m19.50s | -30d32m43.0s | 0.000467 | 17.17
Small Magellanic Cloud | 00h52m44.78s | -72d49m43.0s | 0.000527 | 18.95
Tucana Dwarf | 22h41m49.60s | -64d25m10.0s | 0.000647 | 24.74
Sextans Dwarf Spheroidal | 10h13m02.96s | -01d36m52.6s | 0.000747 | 19.73
Carina Dwarf | 06h41m36.69s | -50d57m58.3s | 0.000764 | 20.02
Large Magellanic Cloud | 05h23m34.53s | -69d45m22.1s | 0.000927 | 18.46
Leo I | 10h08m28.10s | +12d18m23.0s | 0.000951 | 21.91
Data retrieved from NASA/IPAC Extragalactic Database (NED):
http://ned.ipac.caltech.edu
Table 2: Redshift from galaxies within $\approx\pm 2$ h of 0 h in RA within
Local Group
Object | RA (J2000.0) | Dec (J2000.0) | Redshift | Distance Mod (mag)
---|---|---|---|---
Andromeda V | 01h10m17.10s | +47d37m41.0s | -0.001344 | 24.52
Andromeda I | 00h45m39.80s | +38d02m28.0s | -0.001228 | 24.46
Andromeda VI | 23h51m46.30s | +24d34m57.0s | -0.001181 | 24.58
Andromeda III | 00h35m33.78s | +36d29m51.9s | -0.001171 | 24.38
IC 0010 | 00h20m17.34s | +59d18m13.6s | -0.001161 | 24.57
Andromeda VII | 23h26m31.74s | +50d40m32.6s | -0.001024 | 24.7
MESSIER 031 | 00h42m44.35s | +41d16m08.6s | -0.001001 | 24.46
Pisces I | 01h03m55.00s | +21d53m06.0s | -0.000956 | 24.5
MESSIER 110 | 00h40m22.08s | +41d41m07.1s | -0.000804 | 24.5
IC 1613 | 01h04m47.79s | +02d07m04.0s | -0.000781 | 24.33
NGC 0185 | 00h38m57.97s | +48d20m14.6s | -0.000674 | 24.13
MESSIER 032 | 00h42m41.83s | +40d51m55.0s | -0.000667 | 24.42
NGC 0147 | 00h33m12.12s | +48d30m31.5s | -0.000644 | 24.3
Andromeda II | 01h16m29.78s | +33d25m08.8s | -0.000627 | 24.03
Pegasus Dwarf | 23h28m36.25s | +14d44m34.5s | -0.000612 | 26.34
MESSIER 033 | 01h33m50.89s | +30d39m36.8s | -0.000597 | 24.69
WLM | 00h01m58.16s | -15d27m39.3s | -0.000407 | 25.09
Cetus Dwarf Spheroidal | 00h26m11.03s | -11d02m39.6s | -0.00029 | 24.51
Fornax Dwarf Spheroidal | 02h39m59.33s | -34d26m57.1s | 0.000178 | 20.7
Phoenix Dwarf | 01h51m06.34s | -44d26m40.9s | 0.000187 | 23.08
Sculptor Dwarf Elliptical | 01h00m09.36s | -33d42m32.5s | 0.000367 | 19.67
Small Magellanic Cloud | 00h52m44.78s | -72d49m43.0s | 0.000527 | 18.95
Data retrieved from NASA/IPAC Extragalactic Database (NED):
http://ned.ipac.caltech.edu
|
arxiv-papers
| 2013-11-10T03:23:30 |
2024-09-04T02:49:54.822209
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael R. Feldman",
"submitter": "Michael Feldman",
"url": "https://arxiv.org/abs/1312.1182"
}
|
1312.1208
|
# Fundamental groups of clique complexes of random graphs
Armindo Costa, Michael Farber and Danijela Horak
(November 15, 2014)
###### Abstract
We study fundamental groups of clique complexes associated to random Erdös -
Rényi graphs $\Gamma$. We establish thresholds for a number of properties of
fundamental groups of these complexes $X_{\Gamma}$. In particular, if
$p=n^{\alpha}$, we show that
$\begin{array}[]{lll}{\rm gdim}(\pi_{1}(X_{\Gamma}))={\rm
cd}(\pi_{1}(X_{\Gamma}))=1,&\mbox{if}&\alpha<-1/2,\\\ {\rm
gdim}(\pi_{1}(X_{\Gamma}))={\rm
cd}(\pi_{1}(X_{\Gamma}))=2,&\mbox{if}&-1/2<\alpha<-11/30,\\\ {\rm
gdim}(\pi_{1}(X_{\Gamma}))={\rm
cd}(\pi_{1}(X_{\Gamma}))=\infty,&\mbox{if}&-11/30<\alpha<-1/3,\end{array}$
a.a.s., where ${\rm gdim}$ and ${\rm cd}$ denote the geometric dimension and
cohomological dimension correspondingly. It is known that the fundamental
group $\pi_{1}(X_{\Gamma})$ is trivial for $\alpha>-1/3$. We prove that for
$-11/30<\alpha<-1/3$ the fundamental group $\pi_{1}(X_{\Gamma})$ has 2-torsion
but has no $m$-torsion for any given prime $m\geq 3$. We also prove that
aspherical subcomplexes of the random clique complex $X_{\Gamma}$ satisfy the
Whitehead Conjecture, i.e. all their subcomplexes are also aspherical,
a.a.s.111The symbol a.a.s. stands for “asymptotically almost surely” which
means that the probability that the corresponding statement holds tends to 1
as $n$ tends to infinity.
## 1 Introduction
A clique in a graph $\Gamma$ is a set of vertices of $\Gamma$ such that any
two of them are connected by an edge. The family of cliques of $\Gamma$ forms
a simplicial complex $X_{\Gamma}$ with the vertex set $V(X_{\Gamma})$ equal
the vertex set $V(\Gamma)$ of $\Gamma$. The complex $X_{\Gamma}$ is called the
clique complex (or the flag complex) of $\Gamma$. Clearly, the 1-skeleton of
$X_{\Gamma}$ is the graph $\Gamma$ itself.
In this paper we consider the clique complexes $X_{\Gamma}$ of random Erdős -
Rényi graphs $\Gamma\in G(n,p)$. Recall that $G(n,p)$ is the probability space
of all subgraphs $\Gamma$ of the complete graph on $n$ vertices satisfying
$V(\Gamma)=\\{1,\dots,n\\}$, where the probability of a graph $\Gamma$ equals
${\mathbb{P}}(\Gamma)\,=\,p^{e(\Gamma)}(1-p)^{{\binom{n}{2}}-e(\Gamma)}.$
Here $p\in(0,1)$ is a probability parameter, which in general is a function of
$n$, and $e(\Gamma)$ denotes the number of edges in $\Gamma$. The complex
$X_{\Gamma}$, where $\Gamma\in G(n,p)$, is a random simplicial complex. One is
interested in topological properties of $X_{\Gamma}$ which are satisfied with
high probability when the number of vertices $n$ tends to infinity.
Topology of clique complexes of random graphs were studied by M. Kahle et al.
in a series of papers [20], [21], [22], [23]. A recent survey is given in
[24]. The following result is stated in a simplified form.
Theorem: [See M. Kahle [20], Theorems 3.5 and 3.6] Consider the clique complex
$X_{\Gamma}$ of a random graph $\Gamma\in G(n,p)$ where $p=n^{\alpha}$. Let
$k>0$ be a fixed integer. Then
1. (a)
If $\alpha<-1/k$ then $H_{k}(X_{\Gamma};{\mathbf{Z}})=0$ a.a.s.
2. (b)
If $-1/k<\alpha<-1/(k+1)$ then $H_{k}(X_{\Gamma};{\mathbf{Q}})\not=0$, a.a.s.
One knows (see Theorem 3.4 from [20]) that for $p=n^{\alpha}$ the random
clique complex $X_{\Gamma}$ is $k$-connected a.a.s. if
$\alpha>-(2k+1)^{-1}.$
In particular, the clique complex $X_{\Gamma}$ is connected for $\alpha>-1$
and it is simply connected for $\alpha>-1/3$, a.a.s.
In this paper we are interested in the properties of the fundamental group of
a random clique complex and therefore (see above) we shall restrict our
attention to the regime $\alpha<-1/3$ where $p=n^{\alpha}$. In a recent
preprint [5] E. Babson proved that for $\epsilon>0$ and
$n^{\epsilon-1/2}<p<n^{n-\epsilon-1/3}$ the fundamental group
$\pi_{1}(X_{\Gamma})$ is nontrivial and is hyperbolic in the sense of Gromov
[16].
In this paper we use the notation $f\ll g$ to indicate that $f/g\to 0$ as
$n\to\infty$.
The main results of this paper are as follows: Theorem A: [See Theorem 3.1] If
$\displaystyle p\ll n^{-1/2}$ (1)
then, with probability tending to 1 as $n\to\infty$, the clique complex
$X_{\Gamma}$ is simplicially collapsible to a graph. In particular under the
above assumption the fundamental group $\pi_{1}(X_{\Gamma},x_{0})$ of a random
clique complex $X_{\Gamma}$, where $\Gamma\in G(n,p)$, is free, for any choice
of the base point $x_{0}\in X_{\Gamma}$, a.a.s. Moreover,each connected
component of the 2-skeleton $X_{\Gamma}^{(2)}$ is homotopy equivalent to a
wedge of circles and 2-spheres, a.a.s. Note that in the range (1) the
dimension of $X_{\Gamma}$ is $\leq 3$ and the 2-skeleton $X_{\Gamma}^{(2)}$
contains the tetrahedron and its subdivision having 5 vertices, a.a.s. Hence,
the 2-skeleton $X_{\Gamma}^{(2)}$ is not collapsible to a graph.
Note also that for $p\gg n^{-1/2}$ the fundamental group $\pi_{1}(X_{\Gamma})$
ceases to be free. This follows from a theorem of M. Kahle [21] which states
that for
$p^{2}\geq(3/2+\epsilon)\cdot n^{-1}\cdot\log n$
the fundamental group $\pi_{1}(X_{\Gamma})$ has property (T) and thus its
cohomological dimension is $\geq 2$, a.a.s. In the following theorem we
describe the range in which the cohomological dimension of
$\pi_{1}(X_{\Gamma})$ equals 2.
We wish to mention a recent preprint [3] where random triangular groups are
studied; this class of random groups is different from the class of
fundamental groups of random clique complexes although these two classes of
random groups share several common features. The main result of [3] states
that there exists an interval in which the random triangular group is neither
free nor possesses the property T. We expect that such intermediate regime
exists in the model which we study in this paper. We shall address this issue
elsewhere.
Theorem B: [See Theorem 7.1] Assume that
$\displaystyle p\ll n^{-11/30}.$ (2)
Then the fundamental group $\pi_{1}(X_{\Gamma})$ of the clique complex of a
random graph $\Gamma\in G(n,p)$ satisfies
$\displaystyle{\rm gdim}(\pi_{1}(X_{\Gamma}))={\rm
cd}(\pi_{1}(X_{\Gamma}))\leq 2,$ (3)
and in particular $\pi_{1}(X_{\Gamma})$ is torsion free, a.a.s. Moreover, if
for some $\epsilon>0$ one has
$\left((3/2+\epsilon)\cdot n^{-1}\cdot\log
n\right)^{1/2}\,\leq\,p\,\ll\,n^{-11/30}$
then
$\displaystyle{\rm gdim}(\pi_{1}(X_{\Gamma}))={\rm
cd}(\pi_{1}(X_{\Gamma}))=2,$ (4)
a.a.s.
Recall that geometric dimension ${\rm gdim}(G)$ of a discrete group $G$ is
defined as the minimal dimension of an aspherical CW-complex having $G$ as its
fundamental group. The cohomological dimension ${\rm cd}(G)$ is the shortest
length of a free resolution of ${\mathbf{Z}}$ viewed as a
${\mathbf{Z}}[G]$-module. In general ${\rm cd}(G)\leq{\rm gdim}(G)$ and a
classical theorem of Eilenberg and Ganea [14] states that ${\rm cd}(G)={\rm
gdim}(G)$, except of three low-dimensional cases. At present it is known that
the equality ${\rm cd}(G)={\rm gdim}(G)$ holds, except possibly for the case
when ${\rm cd}(G)=2$ and ${\rm gdim}(G)=3$. The Eilenberg–Ganea Conjecture
states that ${\rm cd}(G)=2$ implies ${\rm gdim}(G)=2$.
The following theorem states that 2-torsion appears in the fundamental group
of a random clique complex when we cross the threshold 11/30:
Theorem C: [See Theorem 7.2] Assume that
$\displaystyle n^{-11/30}\ll p\ll n^{-1/3-\epsilon}$ (5)
where $0<\epsilon<1/30$ is fixed. Then the fundamental group
$\pi_{1}(X_{\Gamma})$ has 2-torsion and thus its cohomological dimension and
geometric dimension are infinite, a.a.s.
Surprisingly, odd torsion does not appear in fundamental groups of random
clique complexes until the triviality threshold $p=n^{-1/3}$:
Theorem D: [See Theorem 8.1] Let $m\geq 3$ be a fixed prime. Assume that
$\displaystyle p\ll n^{-1/3-\epsilon}$ (6)
where $\epsilon>0$ is fixed. Then a random graph $\Gamma\in G(n,p)$ with
probability tending to 1 has the following property: the fundamental group of
any subcomplex $Y\subset X_{\Gamma}$ has no $m$-torsion.
Surprisingly, we see that for all the assumptions on the probability parameter
$p$ considered in Theorems A, B, C, the fundamental groups of random clique
complexes have cohomological dimension $1,2$ or $\infty$, which implies that
probabilistically the Eilenberg–Ganea conjecture is satisfied. Note also that
in the complementary range, when $p=n^{\alpha}$ with $\alpha>-1/3$, the clique
complex $X_{\Gamma}$ of a random graph $\Gamma$ is simply connected a.a.s. (by
Theorem 3.4 from [20]) and hence the Eilenberg–Ganea conjecture is also
probabilistically satisfied. We may also mention here that any finitely
presented group appears as the fundamental group of a clique complex
$X_{\Gamma}$ for a graph $\Gamma\in G(n,p)$ with any $n$ large enough.
Next we state a result in the direction of the Whitehead conjecture.
Recall that a connected simplicial complex $Y$ is said to be aspherical if
$\pi_{i}(Y)=0$ for all $i\geq 2$; this is equivalent to the requirement that
the universal cover of $Y$ is contractible. For 2-dimensional complexes $Y$
the asphericity is equivalent to the vanishing of the second homotopy group
$\pi_{2}(Y)=0$, or equivalently, that any continuous map $S^{2}\to Y$ is
homotopic to a constant map. Random aspherical 2-complexes could be helpful
for testing probabilistically the open problems of two-dimensional topology,
such as the Whitehead conjecture. This conjecture stated by J.H.C. Whitehead
in 1941 claims that a subcomplex of an aspherical 2-complex is also
aspherical. Surveys of results related to the Whitehead conjecture can be
found in [7], [26].
Theorem E: [See Corollary 6.2] Assume that $p\ll n^{-1/3-\epsilon}$, where
$\epsilon>0$ is fixed. Then, for a random graph $\Gamma\in G(n,p)$, the clique
complex $X_{\Gamma}$ has the following property with probability tending to 1
as $n\to\infty$: any aspherical subcomplex $Y\subset X_{\Gamma}^{(2)}$
satisfies the Whitehead Conjecture, i.e. any subcomplex $Y^{\prime}\subset Y$
is also aspherical.
Thus we see that probabilistically, for large finite simplicial complexes the
Whitehead conjecture holds for all their aspherical subcomplexes.
We also remark that, as is well known, any finite simplicial complex is
homeomorphic to a clique complex $X_{\Gamma}$ with $\Gamma\in G(n,p)$ for any
$n$ large enough; thus every 2-dimensional finite simplicial complex appears
up to homeomorphism with positive probability for large $n$.
Recall that a well known result of Bestvina and Brady [6] states that either
the Whitehead conjecture or the Eilenberg–Ganea conjecture must be false.
However, Bestvina and Brady consider these two conjectures in a larger class
of infinite simplicial complexes and not necessarily finitely presented
groups.
We should also point out the limitations of our approach. We have no results
in the direction of the Eilenberg–Ganea conjecture near the two critical
values of the probability parameter $p=n^{-11/30}$ and $p=n^{-1/3}$. Besides,
we do not know the validity of the probabilistic version of the Whitehead
conjecture (the analogue of Theorem E) for $p\gg n^{-1/3}$.
A few words about terminology we use in this paper. By a 2-complex we
understand a finite simplicial complex of dimension $\leq 2$. The
$i$-dimensional simplexes of a 2-complex are called vertices (for $i=0$),
edges (for $i=1$) and faces (for $i=2$).
A 2-complex is said to be pure if every vertex and every edge are incident to
a face.
The pure part of a 2-complex is the closure of the union of all faces.
The degree of an edge $e$ of $X$ is the number of faces containing $e$.
The boundary $\partial X$ of a 2-complex $X$ is the union of all edges of
degree one. We say that a 2-complex $X$ is closed if $\partial X=\emptyset$.
We denote by $V(X),E(X),F(X)$ the sets of vertices, edges and faces of $X$,
correspondingly. We also use the notations $v(X)=|V(X)|,$ $e(X)=|E(X)|,$
$f(X)=|F(X)|$.
A connected 2-complex $X$ is strongly connected if $X-V(X)$ is connected.
We use the notations $P^{2}$ for the real projective plane.
The authors thank the referee for making useful critical remarks.
## 2 The containment problem
In this section we collect some known results which we shall use in this
paper. The only new result here is Theorem 2.8 which describes properties of
clean triangulations of surfaces.
Let $S$ be a 2-complex. We have already introduced the notations
$\displaystyle\nu(S)=\frac{v(S)}{e(S)},\quad\quad\tilde{\nu}(S)=\min_{S^{\prime}\subset
S}\nu(S^{\prime})$ (7)
where $v(S)$ and $e(S)$ denote the number of vertices and edges in $S$.
Although the numbers $\nu(S)$ and $\tilde{\nu}(S)$ depend only on the
1-skeleton of $S$, it is convenient to think about $\nu(S)$ and
$\tilde{\nu}(S)$ as being associated to the whole 2-complex $S$ due to the
following formula
$\displaystyle\nu(S)=\frac{1}{3}+\frac{3\chi(S)+L(S)}{3e(S)}.$ (8)
Here
$\displaystyle L(S)$ $\displaystyle=$
$\displaystyle\sum_{e}\left[2-\deg(e)\right]$ $\displaystyle=$ $\displaystyle
2e(S)-3f(S).$
In the definition of $L(S)$ the sum is over all the edges $e$ of $S$ and
$\deg(e)$ is the number of faces containing $e$. Note that $L(S)\leq 0$
assuming that $S$ is closed, i.e. if $\deg(e)\geq 2$ for every edge $e$ of
$S$.
The embeddability of $S$ into $X_{\Gamma}$ is equivalent to the embeddability
of the 1-skeleton of $S$ into $\Gamma$. The following result follows from the
well known subgraph containment problem in random graph theory, see [19],
Theorem 3.4 on page 56.
###### Theorem 2.1.
Let $S$ be a fixed finite simplicial complex. Consider the clique complex
$X_{\Gamma}$ associated to a random Erdős - Rényi graph $\Gamma\in G(n,p)$.
Then:
1. (A)
If $p\ll n^{-\tilde{\nu}(S)}$ then the probability that $S$ admits a
simplicial embedding into $X_{\Gamma}$ tends to $0$ as $n\to\infty$;
2. (B)
If $p\gg n^{-\tilde{\nu}(S)}$ the the probability that $S$ admits a simplicial
embedding into $X_{\Gamma}$ tends to $1$ as $n\to\infty$;
###### Definition 2.2.
A graph $\Gamma$ is said to be balanced if for any proper subgraph
$\Gamma^{\prime}\subset\Gamma$ one has $\nu(\Gamma)\leq\nu(\Gamma^{\prime})$.
A graph $\Gamma$ is said to be strictly balanced if for any proper subgraph
$\Gamma^{\prime}\subset\Gamma$ one has $\nu(\Gamma)<\nu(\Gamma^{\prime})$.
###### Definition 2.3.
A simplicial 2-complex $S$ is said to be $\nu$-balanced (or strictly $\nu$\-
balanced) if its 1-skeleton is balanced (or strictly balanced,
correspondingly).
###### Definition 2.4.
A simplicial 2-complex $S$ is called clean if it coincides with the 2-skeleton
of the clique complex of its 1-skeleton.
In other words, a triangulation is clean if any clique consisting of three
vertices spans a 2-simplex.
###### Example 2.5.
Let $K_{r+1}$ be the complete graph on $r+1$ vertices. It is easy to see that
it is strictly $\nu$-balanced and
$\tilde{\nu}(K_{r+1})=\nu(K_{r+1})=\frac{2}{r}.$
As a corollary of Theorem 2.1 we obtain:
###### Corollary 2.6.
If the probability parameter $p$ satisfies
$n^{-2/r}\ll p\ll n^{-2/(r+1)}$
where $r\geq 2$ is an integer, then the dimension $\dim X_{\Gamma}$ of a
random clique complex $X_{\Gamma}$ equals $r$, a.a.s.
###### Example 2.7.
Consider a triangulated surface $S$ having a vertex $x$ of degree 3. Clearly
such triangulation is not clean. Assume that either $S$ is orientable and has
genus $>1$ or it is non-orientable and has genus $>2$. If $\Gamma$ denotes the
1-skeleton of $S$ then $\nu(S)=\nu(\Gamma)<1/3$. Removing the vertex $x$ and
the three incident to it edges we obtain a graph
$\Gamma^{\prime}\subset\Gamma$ with $v(\Gamma^{\prime})=v(\Gamma)-1$ and
$e(\Gamma^{\prime})=e(\Gamma)-3$. Since $\nu(\Gamma)<1/3$ we see that
$\nu(\Gamma^{\prime})=\frac{v(\Gamma)-1}{e(\Gamma)-3}<\nu(\Gamma),$
i.e. $\Gamma$ is not $\nu$-balanced.
The following Theorem is analogous to Theorem 27 from [10].
###### Theorem 2.8.
Any clean triangulation of a closed connected surface $S$ with $\chi(S)\geq 0$
is $\nu$-balanced. Moreover, if $\chi(S)>0$ then any clean triangulation of
$S$ is strictly $\nu$-balanced.
###### Proof.
Let $\Gamma$ be a graph such that $S$ is the clique complex $S=X_{\Gamma}$.
Let $\Gamma^{\prime}\subset\Gamma$ be a proper subgraph and let
$S^{\prime}=X_{\Gamma^{\prime}}$ denotes the clique complex of
$\Gamma^{\prime}$. Without loss of generality we may assume that
$\Gamma^{\prime}$ is connected. Due to formula (8), the inequality
$\nu(S)<\nu(S^{\prime})$ would follow from
$\displaystyle 3\chi(S^{\prime})+L(S^{\prime})\geq 3\chi(S),$ (9)
since $L(S)=0$, $e(S)>e(S^{\prime})$ and $\chi(S)\geq 0$. From now on all
homology and cohomology group will have coefficient group ${\mathbf{Z}}_{2}$
which will be omitted from the notation. Besides, we will use the symbol
$b_{i}^{\prime}(X)$ to denote $\dim_{{\mathbf{Z}}_{2}}H_{i}(X)$, the $i$-th
Betti number with ${\mathbf{Z}}_{2}$ coefficients. Consider the exact sequence
$\displaystyle 0\to H_{2}(S)\to H_{2}(S,S^{\prime})\stackrel{{\scriptstyle
j_{\ast}}}{{\to}}H_{1}(S^{\prime})\to H_{1}(S)\to H_{1}(S,S^{\prime})\to 0.$
(10)
Here we used that $H_{2}(S^{\prime})=0$ (since $S^{\prime}$ is a proper
subcomplex of $S$) and $H_{2}(S)={\mathbf{Z}}_{2}$. By Poincaré duality, the
dimension of $H_{2}(S,S^{\prime})$ equals $\dim H^{0}(S-S^{\prime})=k$, the
number of path-connected components of the complement $S-S^{\prime}$, see
Proposition 3.46 from [18]. Thus, (10) implies the inequality
$\displaystyle b_{1}^{\prime}(S)\geq b_{1}^{\prime}(S^{\prime})-k+1.$ (11)
Substituting $\chi(S)=2-b_{1}^{\prime}(S)$,
$\chi(S^{\prime})=1-b_{1}^{\prime}(S^{\prime})$ into (9) we see that (9) would
follows from (11) once we show that $L(S^{\prime})\geq 3k$. Note that
$L(S^{\prime})=e_{1}(S^{\prime})+2e_{2}(S^{\prime})$ where $e_{i}(S^{\prime})$
denotes the number of edges of $S^{\prime}$ which have degree $i$, where
$i=0,1$. If $C_{1},\dots,C_{k}$ denote the boundary circles of the connected
components of $S-S^{\prime}$ then one has
$\sum_{j=1}^{k}|C_{j}|=e_{1}(S^{\prime})+2e_{0}(S^{\prime})=L(S^{\prime}),$
since each edge of $S^{\prime}$ having degree one belong to exactly one of the
circles $C_{j}$ and each edge of degree zero belongs to two circles $C_{j}$.
Clearly, $|C_{j}|\geq 3$ for each $C_{j}$ and the inequality
$L(S^{\prime})\geq 3k$ follows. ∎
For a triangulation $S$ of a compact orientable surface $\Sigma_{g}$ of genus
$g$ one has using the formula (8),
$\displaystyle\nu(S)=\frac{1}{3}+\frac{2-2g}{e(S)}.$ (12)
Similarly, for a triangulation $S$ of a compact non-orientable surface $N_{g}$
of genus $g$ one has
$\displaystyle\nu(S)=\frac{1}{3}+\frac{2-g}{e(S)}.$ (13)
Thus we see that $\nu(S)<1/3$ if $S$ is orientable and $g>1$ or if $S$ is non-
orientable and $g>2$.
###### Remark 2.9.
It is easy to show that the assumption $\chi(S)\geq 0$ of Theorem 2.8 is
necessary. More specifically, any closed surface with $\chi(S)<0$ admits a
non-$\nu$-balanced clean triangulation.
Indeed, let $S$ be a clean triangulation of a surface with $\chi(S)<0$; then
$\nu(S)<1/3$ (by (12) and (13)). Let $X\subset S$ be the subcomplex obtained
from $S$ by removing an edge $e\subset S$ and the interiors of two adjacent to
$e$ 2-simplexes. Then
$\nu(X)=\frac{v(S)}{e(S)-1}=\frac{e(S)/3+\chi(S)}{e(S)-1}\leq\frac{e(S)/3-1}{e(S)-1}<\frac{1}{3}$
Let $D$ be a clean triangulated disc with $r$ interior vertices and whose
boundary is a simplicial circle with 4 vertices and 4 edges. For any
triangulated disc we have (using the Euler - Poincare formula),
$e(D)=2v(D)+r-3$
and since $v(D)=r+4$ we obtain
$e(D)=3r+5.$
Let $S^{\prime}$ be the result of gluing $D$ to $X$ with the identification
$\partial D=\partial X$. Obviously $S^{\prime}$ is homeomorphic to $S$. One
has
$v(S^{\prime})=v(S)+r\quad\mbox{and}\quad e(S^{\prime})=e(X)+e(D)-e(\partial
D)=e(X)+3r+1=e(S)+3r.$
Hence
$\nu(S^{\prime})=\frac{v(S)+r}{e(S)+3r}\to\frac{1}{3}$
tends to $1/3$ as $r\to\infty$. Thus, by taking $r$ large enough we shall have
$\nu(S^{\prime})>\nu(X)$. The obtained triangulation $S^{\prime}$ is clean and
unbalanced since $X$ is a subcomplex of $S^{\prime}$.
###### Remark 2.10.
Theorem 27 from [10] (which is similar to Theorem 2.8) is valid under an
additional assumption $\chi(S)\geq 0$ which is missing in its statement. The
assumption $\chi(S)\geq 0$ is essential since any closed surface with negative
Euler characteristic $\chi(S)<0$ admits a not $\mu$-balanced triangulation.
## 3 Threshold for collapsibility to a graph
In this section we prove Theorem A which we restate below:
###### Theorem 3.1.
If
$\displaystyle p\ll n^{-1/2}$ (14)
then, with probability tending to 1 as $n\to\infty$, the clique complex
$X_{\Gamma}$ is simplicially collapsible to a graph, a.a.s. In particular the
fundamental group $\pi_{1}(X_{\Gamma},x_{0})$ of a random clique complex
$X_{\Gamma}$, where $\Gamma\in G(n,p)$, is free, for any choice of the base
point $x_{0}\in X_{\Gamma}$. Moreover, under the above assumptions each
connected component of the 2-skeleton $X_{\Gamma}^{(2)}$ is homotopy
equivalent to a wedge of circles and 2-spheres, a.a.s.
The proof of Theorem 3.1 uses a deterministic combinatorial assertion
described below as Theorem 3.2. In its statement we use the notation
$\displaystyle\nu(S)=\frac{v(S)}{e(S)}$ (15)
where $S$ is a simplicial 2-complex and $v(S)$ and $e(S)$ denote the number of
its vertices and edges. We will also use the invariant
$\displaystyle\tilde{\nu}(X)=\min_{S\subset X}\nu(S),$ (16)
where $S$ runs over all subcomplexes of $X$.
We shall denote by ${\cal S}_{1}$ the tetrahedron (the 2-complex homeomorphic
to the sphere $S^{2}$ and having 4 vertices, 6 edges and 4 faces) and by
${\cal S}_{2}$ the triangulation of $S^{2}$ having 5 vertices, 9 edges and 6
faces. Clearly, $\nu({\cal S}_{1})=2/3>1/2$ and $\nu({\cal S}_{2})=5/9>1/2$.
The complexes ${\cal S}_{1}$ and ${\cal S}_{2}$ play a special role in our
study: Theorem 3.2 below implies that any closed 2-complex $X$ satisfying
$\tilde{\nu}(X)>1/2$ contains either ${\cal S}_{1}$ or ${\cal S}_{2}$ as a
simplicial subcomplex.
###### Theorem 3.2.
There exists an infinite set $\cal L$ of isomorphism types of finite
simplicial 2-complexes satisfying the following properties:
1. (1)
for any $S\in\cal L$ one has $\nu(S)\leq 1/2$;
2. (2)
the set $\cal L$ has at most exponential size in the following sense: for an
integer $E$ let ${\cal L}_{E}$ denote the set $\\{S\in{\cal L};e(S)\leq E\\}$.
Then for some positive constants $A$ and $B$ one has
$|{\cal L}_{E}|\leq A\cdot B^{E},$
where $A$ and $B$ are independent of $E$;
3. (3)
any closed pure 2-complex $X$ contains a simplicial subcomplex isomorphic to
some $S\in{\cal L}\cup\\{{{\cal S}}_{1},{{\cal S}}_{2}\\}$.
Property (3) is the main universal feature of the set $\cal L$.
###### Proof of Theorem 3.2.
We start with a few remarks:
For a triangulated 2-disc $X$ having $v$ vertices such that among them there
are $v_{i}$ internal vertices, one has
$\displaystyle\nu(X)=\frac{v}{2v+v_{i}-3}.$ (17)
Thus one has $\nu(X)=1/2$ for $v_{i}=3$ and $\nu(X)>1/2$ only for
$v_{i}=0,1,2$. Formula (17) follows from the relations $3f=2e-v_{\partial}$
and $v-e+f=1$ where $v_{\partial}=v-v_{i}$ is the number of vertices on the
boundary.
Figure 1: External edge $E$.
The operation of adding an external edge to a simplicial complex $X$ gives a
simplicial complex $X^{\prime}=X\cup E$ where $E$ is a arc (i.e. a space
homeomorphic to $[0,1]$) and $X\cap E=\partial E$, see Figure 1. Clearly
$\nu(X^{\prime})<\nu(X)$.
If $X$ is obtained from a triangulated disc with $v_{i}$ internal vertices by
adding $c$ external edges, then
$\displaystyle\nu(X)=\frac{v}{2v+v_{i}+c-3}$ (18)
and therefore we see that $\nu(X)\leq 1/2$ if and only if $v_{i}+c\geq 3$.
We denote by ${\cal L}$ the set of isomorphism types of finite simplicial
2-complexes $S$ having the following properties: the pure part $S_{0}$ of $S$
admits a surjective simplicial map
$f:S^{\prime}\to S_{0}$ where:
(a) $S^{\prime}$ is a triangulated disc with one internal vertex;
(b) $f$ is bijective on the set of faces;
(c) the image of any edge of $S^{\prime}$ is an edge of $S_{0}$;
(d) $v(S^{\prime})-v(S_{0})\leq 2;$
(e) the complex $S$ is obtained from its pure part $S_{0}$ by adding at most
$2$ external edges;
(f) and finally we require that
$\displaystyle\nu(S)\leq 1/2.$ (19)
Typical examples of complexes from $\mathcal{L}$ are given below.
If $v(S^{\prime})-v(S_{0})=a$ (“the vertex defect”) and
$e(S^{\prime})-e(S_{0})=b$ (“the edge defect”) then using the inequality
$e(S^{\prime})=2v(S^{\prime})-2$ (which follows from (17)) we obtain that (19)
is equivalent to
$\displaystyle 2a+c\geq b+2,$ (20)
where $c=0,1,2$ denotes the number of external edges in $S$. Since $a\leq 2$
and $c\leq 2$, the total number of solutions $(a,b,c)$ to (20) is 19.
Next we show that the set $\cal L$ satisfies property (2) of Theorem 3.2.
According to W. Brown [8], the number of isomorphism types of triangulations
of the disc $S^{\prime}$ with $v$ vertices having one internal vertex is less
than or equal to
$\frac{2v-5}{v-1}\cdot\binom{2v-6}{v-2}\leq 2\cdot 2^{2v-6}<4^{v};$
here we use formula (4.7) from [8] with $v=m+4$ and $n=1$. This implies that
the number of isomorphism types of triangulations of the disc $S^{\prime}$
with at most $v$ vertices and one internal vertex is less than or equal to
$1+4+\dots+4^{v}<4/3\cdot 4^{v}.$
We want to estimate above the number of elements $S\in\cal L$ satisfying
$e(S)\leq E$. For $S\in\cal L$ with $e(S)\leq E$, let $f:S^{\prime}\to S_{0}$
be a surjective simplicial map as in the definition of $\mathcal{L}$. Here
$S_{0}$ is the pure part of $S$ and $S$ is obtained from $S_{0}$ by adding
$c=\,0,\,1,\,2$ edges. Then using (19), we find $v(S_{0})\leq e(S_{0})/2+1\leq
E/2$ and
$v(S^{\prime})\leq v(S_{0})+2=v(S)+2\leq E/2+2.$
The complex $S_{0}$ is obtained from $S^{\prime}$ by identifying at most 2
pairs of vertices or by identifying a triple of vertices; the identification
of vertices determines the identification of edges. As we noted above, there
are 19 types of quotients. Hence we obtain (assuming that $E\geq 6$)
$\displaystyle|{\cal L}_{E}|\leq 4/3\cdot 4^{E/2+2}\cdot
19\cdot(E/2+2)^{4}\cdot(E/2+2)^{4}.$
In the above inequality the first factor $(E/2+2)^{4}$ accounts for the ways
of doing identifications of vertices and the second factor $(E/2+2)^{4}$
accounts for the ways to add 2 additional edges. Since $(E/2+2)^{4}\leq
4^{E/2+2}$ we see that
$|{\cal L}_{E}|\leq\frac{4^{7}\cdot 19}{3}\cdot 8^{E}.$
This proves that the set $\cal L$ has property (2) of Theorem 3.2.
Below we show that the set $\cal L$ has property (3) of Theorem 3.2. We start
by describing examples of complexes from $\mathcal{L}$.
Example 1: Triangulated disc with one internal point and two added external
edges (i.e. $v_{i}=1$ and $c=2$).
Example 2: Triangulated disc with two internal points and one added external
edge (i.e. $v_{i}=2$ and $c=1$).
Note that a triangulated disc with $k$ internal points may be obtained as a
quotient of a triangulated disc with $k-1$ internal points by identifying two
vertices and two adjacent edges on the boundary. This fact is illustrated by
Figure 2.
Figure 2: Disc with 3 internal points as a quotient of a disc with no internal
points;
3 pairs of adjacent edges are identified.
Example 3: Consider a simplicial surjective map $f:X^{\prime}\to X$ where
$X^{\prime}$ is a triangulated disc and $f$ is bijective on faces and every
edge of $X^{\prime}$ is mapped to an edge of $X$ and such that
$v(X^{\prime})-v(X)=1$ and $e(X^{\prime})-e(X)=1$, i.e. exactly two vertices
and two (adjacent) edges are identified. If $X^{\prime}$ has $i$ internal
vertices then we call such an $X$ a scroll with $i$ internal points.
Figure 3: Example of a scroll without internal points.
A scroll with two internal points is an element of $\mathcal{L}$. In
particular, every triangulated disc with 3 internal points belongs to
$\mathcal{L}$.
Example 4: A scroll with one internal point and with one external edge added
is an element of $\mathcal{L}$.
Example 5: As above, consider a simplicial surjective map $f:X^{\prime}\to X$
where $X^{\prime}$ is a triangulated disc and $f$ is bijective on faces and
every edge of $X^{\prime}$ is mapped to an edge of $X$. Assume that exactly
two pairs of vertices and two pairs of adjacent edges are identified, i.e.
$v(X^{\prime})-v(X)=2$ and $e(X^{\prime})-e(X)=2$. If $X^{\prime}$ has one
internal vertex then $X\in\mathcal{L}$. We shall call such an $X$ disc with
one internal point and with two scrolls.
Now we show that the set $\cal L$ has property (3) of Theorem 3.2. We shall
assume the negation of property (3) and arrive to a contradiction. Hence,
below we assume that there exists a closed pure 2-complex $X$ which contains
no subcomplexes isomorphic to any $S\in{\cal L}\cup\\{{\cal S}_{1},{\cal
S}_{2}\\}$.
Consider a vertex $v\in X$ and let ${\rm{Lk}}_{X}(v)$ be the link of $v$ in
$X$; it is a graph having no univalent vertices (since $X$ is closed) and
hence each connected component of ${\rm{Lk}}_{X}(v)$ contains a simple cycle
$C\subset{\rm{Lk}}_{X}(v)$. The cone $D=vC\subset X$ with base $C$ and apex
$v$ is a disc with one internal point.
There may exist at most one external edge, i.e. an edge $e\subset X$ such that
$e\not\subset D$ and $\partial e\subset D$ (since otherwise the union of $D$
and of two such edges would be isomorphic to an element of $\mathcal{L}$, see
Example 1). Consider a vertex $w\in C=\partial D$ which is not incident to an
external edge (such point exists since $C$ has at least 3 vertices). Let
$w^{\prime},w^{\prime\prime}\in C$ be the two neighbours of $w$ along $C$. The
link ${\rm{Lk}}_{X}(w)$ of the vertex $w$ in $X$ is a graph without univalent
vertices and the link $\alpha={\rm{Lk}}_{D}(w)$ is an arc connecting the
points $w^{\prime}$ and $w^{\prime\prime}$. It is obvious that the arc
$\alpha$ is contained in a subgraph $\Gamma\subset{\rm{Lk}}_{X}(w)$ which is
homeomorphic either to the circle or to one of the two graphs shown in Figure
4 (the graph $\Gamma$ can be obtained by extending $\alpha$ in
${\rm{Lk}}_{X}(w)$ until the extension “hits itself”).
Figure 4: Graphs containing the arc $\alpha$.
Thus the complex $X$ contains the cone $w\Gamma$ over $\Gamma$ with apex $w$.
The intersection $w\Gamma\cap D$ clearly contains $w\alpha$, the cone over the
arc $\alpha$. Any vertex $u$ of $(w\Gamma\cap D)-w\alpha$ corresponds to an
edge $e\subset X$ such that $\partial e=\\{w,u\\}$ and $e\not\subset D$. By
construction, we know that there are no such external edges. Thus, we see that
the set of vertices of $w\Gamma\cap D$ coincides with the set of vertices of
$w\alpha$.
In the case when $\Gamma$ is homeomorphic to one of the graphs shown in Figure
4 the union $w\Gamma\cup D\subset X$ is a disc with one internal point and
with two scrolls which is impossible due to Example 5. Thus the only remaining
possibility is that $\Gamma$ is a simple circle. The union $w\Gamma\cup D$ can
be the tetrahedron ${\cal S}_{1}$ (iff $\Gamma-\rm Int(\alpha)$ is a single
edge contained in $\partial D$); otherwise the union $w\Gamma\cup D$ is a
disc. The first possibility contradicts our assumptions (we know that $X$ does
not contain ${\cal S}_{1}$ as a subcomplex), therefore the union
$D_{1}=w\Gamma\cup D\subset X$ is a disc with two internal points $v,w$.
Next we repeat the above arguments applied to $D_{1}\subset X$ instead of
$D\subset X$. Consider a point $w_{1}\in\partial D_{1}$ and its two neighbours
$w^{\prime}_{1},w^{\prime\prime}_{1}\in C_{1}=\partial D_{1}$. The link
${\rm{Lk}}_{X}(w_{1})$ is a graph without univalent vertices and
$\alpha_{1}={\rm{Lk}}_{D}(w_{1})$ is an arc connecting the points
$w^{\prime}_{1},w^{\prime\prime}_{1}$. The arc $\alpha_{1}$ is contained in a
subgraph $\Gamma\subset{\rm{Lk}}_{X}(w_{1})$ which is homeomorphic either to
the circle or to one of the graphs shown in Figure 4. The set of vertices of
$\Gamma$ contained in $D_{1}$ coincides with the set of vertices of
$\alpha_{1}$ (since otherwise $X$ would contain a disc with two internal
points and with one external edge which contradicts Example 2). In the case
when $\Gamma$ is homeomorphic to one of the graphs of Figure 4 the union
$w_{1}\Gamma\cup D_{1}\subset X$ contains a scroll with two internal points
which is impossible because of Example 3. If $\Gamma$ is a simple circle then
the union $w_{1}\Gamma\cup D_{1}$ is either ${\cal S}_{2}$ (iff $\Gamma-\rm
Int(\alpha_{1})$ is a single edge contained in $\partial D_{1}$), or the union
$D_{2}=w_{1}\Gamma\cup D_{1}$ is a disc with 3 internal points $v,w,w_{1}$.
Both these possibilities contradict our assumptions concerning $X$. This
completes the proof.
∎
###### Proof of Theorem 3.1.
Consider a random graph $\Gamma\in G(n,p)$ and its clique complex
$X_{\Gamma}$. Clearly, $X_{\Gamma}$ is connected if and ony if $\Gamma$ is
connected. Since $p\ll n^{-1/2}$ we know that $\dim X_{\Gamma}\leq 3$ a.a.s.;
see Corollary 2.6. The 3-simplexes of $X_{\Gamma}$ are in one-to-one
correspondence with the embedding of the complete graph $K_{4}$ into $\Gamma$.
Let us show that each 3-simplex of $X_{\Gamma}$ has at least three free faces.
Indeed, assume that there is a 3-simplex in $X_{\Gamma}$ with less than three
free faces. Then the complex $S$ formed as the union $S=S_{1}\cup S_{2}\cup
S_{3}$ of three tetrahedra $S_{1},S_{2},S_{3}$, where the intersections
$S_{1}\cap S_{2}$ and $S_{1}\cap S_{3}$ are 2-simplexes and $S_{2}\cap S_{3}$
is an edge, would be embeddable into $X_{\Gamma}$; however this is impossible
due to Theorem 2.1 since $\nu(S)=6/12=1/2$ and $p\ll n^{-1/2}$. Choosing a
free face in each 3-simplex and performing collapse $X_{\Gamma}\searrow
X^{\prime}_{\Gamma}$ we obtain a 2-complex $X^{\prime}_{\Gamma}$. Clearly,
$X^{\prime}_{\Gamma}$ does not contain ${\cal S}_{1}$ and ${\cal S}_{2}$ as
subcomplexes.
Next we perform a sequence of simplicial collapses
$X^{\prime}_{\Gamma}\searrow X^{\prime\prime}_{\Gamma}\searrow
X^{\prime\prime\prime}_{\Gamma}\searrow\dots$
where on each step we collapse all free faces of 2-simplexes. After finitely
many such collapses we obtain a complex $X_{\Gamma}^{\infty}$ which is either
(a) a graph, or (b) a closed 2-dimensional simplicial complex. We know that
$X_{\Gamma}^{\infty}$ contains neither ${\cal S}_{1}$ nor ${\cal S}_{2}$ as a
subcomplex, and besides,
$\pi_{1}(X_{\Gamma},x_{0})=\pi_{1}(X_{\Gamma}^{\infty},x_{0})$ for any base
point $x_{0}$.
We claim that option (b) happens with probability tending to zero as
$n\to\infty$; in other words, $X_{\Gamma}^{\infty}$ is a graph, a.a.s. Indeed,
by Theorem 3.2 if $X^{\infty}_{\Gamma}$ is not a graph then it admits a
simplicial embedding $S\to X_{\Gamma}^{\infty}$ of some $S\in\cal L$. However
for a fixed $S\in\cal L$ one has
$\mathbb{P}(S\subset X_{\Gamma}^{\infty})\leq\mathbb{P}(S\subset
X_{\Gamma})\leq n^{v(S)}p^{e(S)}\leq\left(n^{1/2}p\right)^{e(S)}$
and therefore (using Theorem 3.2) the probability that $X_{\Gamma}^{\infty}$
is not a graph is less than or equal to
$\displaystyle\sum_{S\in{\cal L}}\mathbb{P}(S\subset X_{\Gamma}^{\infty})$
$\displaystyle\leq$ $\displaystyle\sum_{S\in{\cal
L}}\left(n^{1/2}p\right)^{e(S)}$ $\displaystyle=$ $\displaystyle\sum_{E\geq
1}|{\cal L}_{E}|\left(n^{1/2}p\right)^{E}\leq A\sum_{E\geq
1}\left(bn^{1/2}p\right)^{E}\to 0$
as $n\to\infty$ since we assume that $pn^{1/2}\to 0$.
∎
## 4 Uniform hyperbolicity
Let $X$ be a finite simplicial complex. For a simplicial loop $\gamma:S^{1}\to
X^{(1)}\subset X$ we denote by $|\gamma|$ the length of $\gamma$. If $\gamma$
is null-homotopic, $\gamma\sim 1$, we denote by $A_{X}(\gamma)$ the area of
$\gamma$, i.e. the minimal number of triangles in any simplicial filling $V$
for $\gamma$. A simplicial filling (or a simplicial Van Kampen diagram) for a
loop $\gamma$ is defined as a pair of simplicial maps
$S^{1}\stackrel{{\scriptstyle i}}{{\to}}V\stackrel{{\scriptstyle b}}{{\to}}X$
such that $\gamma=b\circ i$ and the mapping cylinder of $i$ is a disc with
boundary $S^{1}\times 0$, see [4].
Clearly $I(X)=I(X^{(2)})$ i.e. the isoperimetric constant $I(X)$ depends only
on the 2-skeleton $X^{(2)}$.
Define the following invariant of $X$
$I(X)=\inf\left\\{\frac{|\gamma|}{A_{X}(\gamma)};\quad\gamma:S^{1}\to
X^{(1)},\gamma\sim 1\quad\mbox{in $X$}\right\\}\,\in\,{\mathbf{R}}.$
The inequality $I(X)\geq a$ means that for any null-homotopic loop $\gamma$ in
$X$ one has the isoperimetric inequality $A_{X}(\gamma)\leq
a^{-1}\cdot|\gamma|$. The inequality $I(X)<a$ means that there exists a null-
homotopic loop $\gamma$ in $X$ with $A_{X}(\gamma)>a^{-1}\cdot|\gamma|$, i.e.
$\gamma$ is null-homotopic but does not bound a disk of area less than
$a^{-1}\cdot|\gamma|$.
It is well known that $I(X)>0$ if and only if $\pi_{1}(X)$ is hyperbolic in
the sense of M. Gromov [16].
###### Example 4.1.
For $X=T^{2}$ one has $I(X)=0$.
It is known that the number $I(X)$ coincides with the infimum of the ratios
${|\gamma|}\cdot{A_{X}(\gamma)}^{-1}$ where $\gamma$ runs over all null-
homotopic simplicial prime loops in $X$, i.e. such that their lifts to the
universal cover $\tilde{X}$ of $X$ are simple. Note that any simplicial
filling $S^{1}\stackrel{{\scriptstyle i}}{{\to}}V\stackrel{{\scriptstyle
b}}{{\to}}X$ for a prime loop $\gamma:S^{1}\to X$ has the property that $V$ is
a simplicial disc and $i$ is a homeomorphism $i:S^{1}\to\partial V$. Hence for
prime loops $\gamma$ the area $A_{X}(\gamma)$ coincides with the minimal
number of 2-simplexes in any simplicial spanning disc for $\gamma$.
The following Theorem 4.2 gives a uniform isoperimetric constant for random
complexes $X_{\Gamma}$ where $\Gamma\in G(n,p)$. It is a slightly stronger
statement than simply hyperbolicity of the fundamental group of $Y$.
###### Theorem 4.2.
Suppose that for some $\epsilon>0$ the probability parameter $p$ satisfies
$\displaystyle p\ll n^{-1/3-\epsilon}.$ (21)
Then there exists a constant $c_{\epsilon}>0$ depending only on $\epsilon$
such that the clique complex $X_{\Gamma}$ of a random graph $\Gamma\in
G(n,p)$, with probability tending to 1 as $n\to\infty$, has the following
property: any subcomplex $Y\subset X_{\Gamma}$ satisfies $I(Y)\geq
c_{\epsilon}$; in particular, for any subcomplex $Y\subset X_{\Gamma}$ the
fundamental group $\pi_{1}(Y)$ is hyperbolic, a.a.s.
The proof of Theorem 4.2 is given in the Appendix at the end of the paper.
## 5 Topology of minimal cycles with $\tilde{\nu}(Z)>1/3$
We start with the following Lemma which describes 2-complexes $S$ with
$b_{2}(S)=0$ and $\nu(S)>1/3$.
###### Lemma 5.1.
Let $S$ be a closed strongly connected pure 2-complex with $b_{2}(S)=0$. If
$\nu(S)>1/3$ then $S$, as a simplicial complex, is either a triangulated
projective plane $P^{2}$ or a simplicial quotient $P^{\prime}$ of a
triangulated projective plane $P^{2}$ where two vertices of $P^{2}$ and two
adjacent edges are identified, i.e. $v(P^{\prime})=v(P^{2})-1$,
$e(P^{\prime})=e(P^{2})-1$, and $f(P^{\prime})=f(P^{2})$.
###### Proof.
Since $\chi(S)=1-b_{1}(S)$, using formula (8), we see that the assumption
$\nu(S)>1/3$ implies that
$\displaystyle 3(1-b_{1}(S))+L(S)>0.$ (22)
In particular, we have
$L(S)\geq-2\quad\mbox{and}\quad b_{1}(S)=0.$
Since $S$ is closed we have $L(S)\leq 0$ and therefore there are 3
possibilities: $L(S)=0,-1,-2$.
If $L(S)=0$ then each edge has degree 2 and $S$ is a pseudo-surface. Using
Corollary 2.1 from [12] we obtain that $S$ is a genuine triangulated surface
without singularities and the only surface satisfying $b_{1}(S)=b_{2}(S)=0$ is
the projective plane.
The case $L(S)=-1$ is impossible. Indeed, if $L(S)=-1$ then there is a single
edge of degree 3 and all other edges have degree 2. The link of a vertex
incident to the edge of degree 3 will be a graph with all vertices of degree 2
and one vertex of degree 3 which is impossible.
Assume now that $L(S)=-2$. There are two possibilities: (a) either there are
two edges of degree 3 and all other edges have degree 2, or (b) there is a
single edge of degree 4 and all other edges have degree 2.
The possibility (a) cannot happen. Indeed, if $e,e^{\prime}$ are two edges of
degree 3 and $v$ is a vertex incident to $e$ but not to $e^{\prime}$ then the
link of $v$ is a graph with all vertices of degree 2 and one vertex of degree
3 which is impossible.
Consider now the case (b). Let $e$ be the edge of degree 4 and let $v,w$ be
the endpoints of $e$. Repeating the arguments of the proof of Theorem 2.4 from
[12] (see Case C in [12]) we see that $S$ is obtained from a pseudo-surface
$S^{\prime}$ by identifying two adjacent edges. Since $S^{\prime}$ and $S$ are
homotopy equivalent, we obtain that $b_{1}(S^{\prime})=b_{2}(S^{\prime})=0$.
Using Corollary 2.1 from [12] and classification of surfaces we see that
$S^{\prime}$ is homeomorphic to the projective plane; therefore $S$ is
isomorphic to a simplicial complex of type $P^{\prime}$ as explained above. ∎
###### Corollary 5.2.
Let $S$ be a connected 2-complex with $b_{2}(S)=0$. If $\tilde{\nu}(S)>1/3$
then $S$ is homotopy equivalent to a wedge of circles and projective planes.
###### Definition 5.3.
A finite pure 2-complex $Z$ is said to be a minimal cycle if $b_{2}(Z)=1$ and
for any proper subcomplex $Z^{\prime}\subset Z$ one has $b_{2}(Z^{\prime})=0$.
Any minimal cycle is closed and strongly connected.
###### Example 5.4.
Let $Z$ be the union of two subcomplexes $Z=A\cup B$ where each $A$ and $B$ is
a triangulated projective plane and the intersection $C=A\cap B$ is a circle
which is not null-homotopic in both $A$ and $B$.
###### Definition 5.5.
A minimal cycle $Z$ is said to be of type A if it has no proper closed
$2$-dimensional subcomplexes. If $Z$ contains a proper closed $2$-dimensional
subcomplex then we say that $Z$ is a minimal cycle of type B.
###### Lemma 5.6.
Let $Z$ be a minimal cycle of type A satisfying $\tilde{\nu}(Z)>1/3$. Then $Z$
is homotopy equivalent either to $S^{2}$ or to the wedge $S^{2}\vee S^{1}$.
Moreover, for any face $\sigma\subset Z$ the boundary $\partial\sigma$ is
null-homotopic in $Z-\rm Int(\sigma)$.
###### Proof.
Let $\sigma\subset Z$ be an arbitrary face. Starting with the complex $Z-\rm
Int(\sigma)$ and collapsing subsequently faces across the free edges we shall
arrive to a connected graph $\Gamma$ (due to our assumption about the absence
of closed subcomplexes). Let us show that $b_{1}(\Gamma)\leq 1$. The
inequality $\nu(Z)>1/3$ is equivalent to $3\chi(Z)+L(Z)>0$ (see formula (8))
where $L(Z)\leq 0$ (since $Z$ is closed) and hence $\chi(Z)\geq 1$. Therefore
$\chi(\Gamma)=\chi(Z)-1\geq 0$ which implies $b_{1}(\Gamma)\leq 1$. Hence,
$\Gamma$ is either contractible or it is homotopy equivalent to the circle. In
the first case, $Z$ is homotopy equivalent to $S^{2}$. In the second case, $Z$
is homotopy equivalent to the result of attaching a 2-cell to the circle,
$S^{1}\cup_{f}e^{2}$. Since $b_{2}(Z)=1$ we obtain that $\deg(f)=0$, and hence
$Z$ is homotopy equivalent to $S^{1}\vee S^{2}$. We see that the inclusion
$\partial\sigma\to Z-\rm Int(\sigma)\simeq\Gamma$ is homotopically trivial in
both cases. ∎
###### Lemma 5.7.
Let $Z$ be a minimal cycle of type B such that $\tilde{\nu}(Z)>1/3$. Suppose
that any edge $e$ of $Z$ has degree $\leq 3$. Then $Z$ is isomorphic (as a
simplicial complex) to the union $P^{2}\cup D^{2}$, where $P^{2}$ and $D^{2}$
are triangulated projective plane and the disc, $P^{2}\cap D^{2}=\partial
D^{2}=P^{1}\subset P^{2}$, and the loop $\partial D^{2}$ has either $3,4$ or
$5$ edges. Here $P^{1}\subset P^{2}$ denotes a simple homotopically nontrivial
simplicial loop on the projective plane. In particular, $Z$ is homotopy
equivalent to $S^{2}$ and for any face $\sigma\subset P^{2}\subset Z$ the
boundary $\partial\sigma$ is null-homotopic in $Z-\rm Int(\sigma)$.
###### Proof.
Let $Z^{\prime}$ be a strongly connected proper closed 2-dimensional
subcomplex of $Z$. Since any edge of $Z^{\prime}$ has degree $\leq 3$ in
$Z^{\prime}$, it follows from Lemma 5.1 that $Z^{\prime}$ is homeomorphic to
$P^{2}$.
Denote $Z^{\prime\prime}=\overline{(Z-Z^{\prime})}$ and let $G$ be the graph
$G=Z^{\prime}\cap Z^{\prime\prime}$. Let $\Gamma$ be the subgraph of the
1-skeleton $Z^{(1)}$ of $Z$ formed by the edges of degree $3$ in $Z$. Clearly
$G\subset\Gamma$. By definition of $\Gamma$ and the assumptions of the Lemma,
any edge of $\Gamma$ has degree $3$ in $Z$ and every edge of $Z$ which is not
in $\Gamma$ must have degree $2$ in $Z$. In particular one has that
$L(Z)=-e(\Gamma)$.
The graph $G$ (and therefore $\Gamma$) must contain a cycle, since otherwise
$Z$ is homotopy equivalent to $Z^{\prime}\vee Z^{\prime\prime}$ and thus
$b_{2}(Z^{\prime\prime})=1$, contradicting the minimality of $Z$. In
particular $e(G)\geq 3$. Moreover, $\Gamma$ has at most 5 edges since
$\tilde{\nu}(Z)>1/3$ implies $L(Z)\geq-5$ (using formula (8) and
$L(Z)=-e(\Gamma)\leq-e(G)$). Hence $\Gamma$ either contains exactly one cycle
(of length 3, 4 or 5) or $\Gamma$ is a square with one diagonal.
Figure 5: Graph $\Gamma$.
Let us show that the latter case is impossible. Indeed, suppose that $\Gamma$
is a square with one diagonal. Let $v_{0}$ be one of the vertices of degree
$3$ in $\Gamma$. Then $v_{0}$ is incident to exactly three odd degree edges in
$Z$ (corresponding to the three neighbours $v_{1},v_{2},v_{3}$ of $v_{0}$ in
$\Gamma$). In particular the link ${\rm{Lk}}_{Z}(v_{0})$ would have an odd
number of odd degree vertices which is impossible. We conclude that
$b_{1}(\Gamma)=b_{1}(G)=1$.
We now show that $\Gamma$ is a cycle and therefore $G=\Gamma$. Suppose that
$\Gamma$ contains an edge $e$ with a free vertex $v$. Then the link
${\rm{Lk}}_{Z}(v)$ is a graph with exactly one vertex of degree $3$ and all
other vertices of degree $2$. This contradicts the fact that every graph has
an even number of odd degree vertices.
We have shown that $G=\Gamma$ is a cycle of length 3, 4 or 5 and that all
edges of $G$ have degree $3$ in $Z$ and all edges of $Z$ which are not in $G$
have degree $2$. Recall that $Z=Z^{\prime}\cup_{G}Z^{\prime\prime}$ where
$Z^{\prime}$ is a triangulated projective plane. Since for any edge $e\in G$,
one has $deg_{Z^{\prime\prime}}(e)=deg_{Z}(e)-deg_{Z^{\prime}}(e)=1$ it
follows that $Z^{\prime\prime}$ is a pseudo-surface with boundary. Moreover,
since $\chi(G)=0$ and $\chi(Z^{\prime})=1$ we obtain
$2=\chi(Z)=\chi(Z^{\prime})+\chi(Z^{\prime\prime})$, i.e.
$\chi(Z^{\prime\prime})=1$. Hence $Z^{\prime\prime}$ is a disk. Besides,
$G=Z^{\prime}\cap Z^{\prime\prime}$ is not null-homotopic in $Z^{\prime}$
since otherwise $G$ bounds a disc $A^{2}\subset Z^{\prime}$ and
$b_{2}(Z)=b_{2}(Z^{\prime})+b_{2}(Z^{\prime\prime}\cup A)$ implying
$b_{2}(Z^{\prime\prime}\cup A)=1$ which would contradict the minimality of
$Z$. Hence we see that $Z$ is homotopy equivalent to $S^{2}$ and any 2-simplex
$\sigma\subset Z^{\prime}$ has the required property.
∎
###### Lemma 5.8.
Let $Z$ be a minimal cycle of type B such that $\tilde{\nu}(Z)>1/3$ and such
that an edge $e$ of $Z$ has degree $\geq 4$. Then $Z$ is isomorphic (as a
simplicial complex) to the quotient $q:\hat{Z}=P^{2}\cup D^{2}\to Z$ of a
minimal cycle $\tilde{Z}$ of type B with $\tilde{\nu}(\hat{Z})>1/3$ and such
that all edges of $\tilde{Z}$ have degree $\leq 3$ (as described in the
previous Lemma); the map $q$ identifies two vertices and two adjacent edges.
In particular, $Z$ is homotopy equivalent to $S^{2}$ and for any face
$\sigma\subset q(P^{2})\subset Z$ the boundary $\partial\sigma$ is
contractible in $Z-\rm Int(\sigma)$.
###### Proof.
Let $\Gamma$ be the subgraph of the 1-skeleton of $Z$ which is the union of
the edges of degree $\geq 3$. As in the proof of the previous lemma, the
inequality $\tilde{\nu}(Z)>1/3$ implies $L(Z)\geq-5$ and using our assumption
that at least one edge of $\Gamma$ has degree $\geq 4$ we obtain $-5\leq
L(Z)\leq-e(\Gamma)-1$, i.e. $\Gamma$ has at most 4 edges. On the other hand
$e(\Gamma)\geq 3$ since $\Gamma$ must contain a cycle as follows from the
argument used in the proof of Lemma 5.7. Thus we have consider the cases
$e(\Gamma)$ equals 3 or 4.
Define $\Gamma_{\rm odd}$ to be the subgraph of $\Gamma$ formed by the edges
of odd degree in $Z$. The graph $\Gamma_{\rm odd}$ is non-empty; indeed, since
$e(\Gamma)\geq 3$ and every edge of $\Gamma$ with even degree must have degree
$\geq 4$, the assumption $\Gamma_{\rm odd}=\emptyset$ would imply
$L(Z)\leq-2e(\Gamma)\leq-6$ contradicting $L(Z)\geq-5$. Furthermore the graph
$\Gamma_{\rm odd}$ may not have a free vertex. If $\Gamma_{\rm odd}$ contained
an edge $e$ with a free vertex $v$ then the link ${\rm{Lk}}_{Z}(v)$ would be
graph with exactly one vertex of odd degree contradicting the fact that every
graph has an even number of odd degree vertices. We obtain in particular that
$e(\Gamma_{\rm odd})\geq 3$ and $b_{1}(\Gamma_{\rm odd})\geq 1$.
We can now describe the graph $\Gamma$.
If $e(\Gamma)=3$, then all edges of $\Gamma$ must have odd degree in $Z$, i.e.
$\Gamma=\Gamma_{\rm odd}$. Furthermore, since $L(Z)\geq-5$ and $Z$ has at
least one edge of degree $>3$, it follows that $\Gamma$ is a cycle formed by
two edges of degree 3 and one edge of degree 5. In particular, $L(Z)=-5$.
Denote the edge of degree 5 by $e$. Let $v$ be a vertex of $e$. Then the link
${\rm{Lk}}_{Z}(v)$ is a graph with exactly two vertices of odd degree. One of
these vertices has degree 3 in the link ${\rm{Lk}}_{Z}(v)$ and the other
vertex has degree 5.
The link ${\rm{Lk}}(v)$ is connected since otherwise we would have
$b_{1}(Z)\geq 1$ (by Corollary 2.1 from [12]) implying $\chi(Z)\leq 1$ and
$L(Z)\geq-2$, a contradiction. Hence, the link ${\rm{Lk}}(v)$ is a connected
graph with one vertex of degree 3, one vertex of degree 5 and all other
vertices of degree 2. There are two possibilities for ${\rm{Lk}}(v)$ which are
shown in Figure 6.
Figure 6: Links of a vertex incident to an edge of degree 4.
A neighbourhood of the point $v$ is the cone $v\cdot{\rm{Lk}}(v)$ over the
link ${\rm{Lk}}(v)$. We may represent ${\rm{Lk}}(v)$ as the union $A\cup B$
where $A$ is a circle and the intersection $A\cap B$ is one point, the vertex
of degree 5. We may cut $Z$ from the vertex $v$ and along the edge $e$
introducing instead of $v$ two new vertices $v_{1}$ and $v_{2}$ end two edges
(of degree 3 and 2) instead of $e$. Formally we replace the cone
$v\cdot{\rm{Lk}}(v)$ by the union of two cones $(v_{1}\cdot A)\cup(v_{2}\cdot
B)$ as shown in Figure 7. The obtained 2-complex $\hat{Z}$ is a minimal cycle
$\chi(\hat{Z})=\chi(Z)=2$ and $L(\hat{Z})=-3$. To apply Lemma 5.7 we want to
show that $\tilde{\nu}(\hat{Z})>1/3$. The negation $\tilde{\nu}(\hat{Z})\leq
1/3$ means that there exists a subgraph $H\subset\hat{Z}^{(1)}$ with
$\nu(H)\leq 1/3$. Identifying two adjacent edges of $H$ we obtain a subgraph
$H^{\prime}$ of the 1-skeleton $Z^{(1)}$ with $v(H^{\prime})=v(H)-1$,
$e(H^{\prime})=e(H)-1$ and now the inequality $e(H)\geq 3v(H)$ implies
$e(H^{\prime})\geq 3v(H^{\prime})$, and therefore $\tilde{\nu}(Z)\leq 1/3$
which contradicts our assumption $\tilde{\nu}({Z})>1/3$.
From Lemma 5.7 we know that $\hat{Z}$ is isomorphic to $P^{2}\cup D^{2}$ where
the intersection $P^{2}\cap D^{2}=P^{1}\subset P^{2}$ has length 3 (since
$L(\hat{Z})=-3$). Therefore, we obtain that $Z$ can be obtained from
$P^{2}\cup D^{2}$ by identifying two adjacent edges.
Figure 7: Resolving the cone.
Consider now the remaining case $e(\Gamma)=4$. Then $\Gamma$ has three edges
of degree 3 and one edge of degree 4. Besides, the edges of degree 3 form a
cycle since $b_{1}(\Gamma_{\rm odd})\geq 1$. Suppose $e(\Gamma)=4$, i.e.
$\Gamma=\Gamma_{\rm odd}\cup e$ where $\Gamma_{\rm odd}$ is a cycle of length
3 with all edges of degree 3, and where $e$ is the edge of degree 4. Then $e$
contains a free vertex $v$ in $\Gamma$. Since $\deg_{Z}(e)=4$, we see that the
link ${\rm{Lk}}_{Z}(v)$ is topologically a wedge of two circles. A
neighbourhood of $v$ in $Z$ is a cone $v\cdot{\rm{Lk}}(v)$ and representing
the link ${\rm{Lk}}(v)$ and a union of two circles $A\cup B$ intersecting at
the vertex of degree 4, we may replace the cone $v\cdot{\rm{Lk}}(v)$ by the
union of two cones $(v_{1}\cdot A)\cup(v_{2}\cdot B)$ where $v_{1}$ and
$v_{2}$ are two new vertices. We obtain a simplicial complex $\hat{Z}$ such
that $Z$ is obtained from $\hat{Z}$ by identifying two adjacent edges. Clearly
$\hat{Z}$ is a minimal cycle of type $B$ with all edges of degree $\leq 3$. As
in the case $e(\Gamma)=3$ considered above one shows that
$\tilde{\nu}(\hat{Z})>1/3$. Thus, we see that $\hat{Z}$ is a minimal cycle
satisfying conditions of Lemma 5.7 and $Z$ is obtained from $\hat{Z}$ by
identifying two adjacent edges. ∎
###### Corollary 5.9.
Let $X$ be a connected 2-complex satisfying $\tilde{\nu}(X)>1/3$. Then $X$ is
homotopy equivalent to a wedge of circles, 2-spheres and real projective
planes. Besides, there exists a subcomplex $X^{\prime}\subset X$ containing
the 1-skeleton of $X$ and having the homotopy type of a wedge of circles and
real projective planes and such that $\pi_{1}(X^{\prime})\to\pi_{1}(X)$ is an
isomorphism. In particular, the fundamental group of $X$ is a free product of
several copies of ${\mathbf{Z}}$ and ${\mathbf{Z}}_{2}$ and hence it is
hyperbolic.
This Corollary is equivalent to Theorem 1.2 from [5]. The proof given below is
independent of the arguments of [5]. Our proof is based on the classification
of minimal cycles described above in the this section. This classification of
minimal cycles is not only useful for the proof of Corollary 5.9 but it is
also plays an important role in the proofs of many results presented in this
paper.
###### Proof.
We will act by induction on $b_{2}(X)$.
If $b_{2}(X)=0$ and $\tilde{\nu}(X)>1/3$ then using Corollary 5.2 we see that
the complex $X$ is homotopy equivalent to a wedge of circles and projective
planes. In this case one sets $X^{\prime}=X$ and the result follows.
Assume now that Corollary 5.9 was proven for all connected 2-complexes $X$
satisfying $\tilde{\nu}(X)>1/3$ and $b_{2}(X)<k$.
Consider a 2-complex $X$ satisfying $b_{2}(X)=k>0$ and $\tilde{\nu}(Z)>1/3$.
Find a minimal cycle $Z\subset X$. Then the homomorphism
$H_{2}(Z;{\mathbf{Z}})={\mathbf{Z}}\to H_{2}(X;{\mathbf{Z}})$ is an injection.
Let $\sigma\subset Z$ be a simplex given by Lemmas 5.6, 5.7, 5.8. Then
$Y=X-\rm Int(\sigma)$ satisfies $b_{2}(Y)=k-1$. Indeed,
$H_{2}(X,Y)={\mathbf{Z}}$ and in the exact sequence
$0\to H_{2}(Y)\to H_{2}(X)\to
H_{2}(X,Y)\stackrel{{\scriptstyle\partial_{\ast}}}{{\to}}H_{1}(Y)\to\dots$
the homomorphism $\partial_{\ast}=0$ is trivial since the curve
$\partial\sigma$ is contractible in $Y$. Since $\tilde{\nu}(Y)>1/3$, by the
induction hypothesis there exists a subcomplex $Y^{\prime}\subset Y$ such that
$\pi_{1}(Y^{\prime})\to\pi_{1}(Y)$ is an isomorphism and $Y^{\prime}$ is
homotopy equivalent to a wedge of circles and projective planes. However $X$
is homotopy equivalent to $Y\vee S^{2}$ and the result follows (with
$X^{\prime}=Y^{\prime}$).
∎
## 6 The Whitehead Conjecture
If $p\ll n^{-1/3}$ then $\dim X_{\Gamma}\leq 5$ a.a.s. (see Corollary 2.6). We
consider below the 2-dimensional skeleton $X_{\Gamma}^{(2)}$ which can be
viewed as a random 2-complex. In this section we shall examine the validity of
the Whitehead Conjecture for aspherical subcomplexes of $X_{\Gamma}^{(2)}$.
Recall that for any simplicial complex $K$, its first barycentric subdivision
$K^{\prime}$ is a clique complex. Thus, if there exists a counterexample to
the Whitehead Conjecture then there exists a counterexample of the form
$X^{(2)}_{\Gamma}$ for certain graph $\Gamma$.
###### Theorem 6.1.
Assume that $p\ll n^{-1/3-\epsilon}$, where $\epsilon>0$ is fixed. Then, for a
random graph $\Gamma\in G(n,p)$ the 2-skeleton $X^{(2)}_{\Gamma}$ of the
clique complex $X_{\Gamma}$ has the following property with probability
tending to 1 as $n\to\infty$: a subcomplex $Y\subset X^{(2)}_{\Gamma}$ is
aspherical if and only if every subcomplex $S\subset Y$ having at most
$2\epsilon^{-1}$ edges is aspherical.
Intuitively, this statement asserts that a subcomplex $Y\subset
X_{\Gamma}^{(2)}$ is aspherical iff it has no “small bubbles” where by a
“bubble” we understand a subcomplex $S\subset Y$ with $\pi_{2}(S)\neq 0$ and
“a bubble is small” if it satisfies the condition $e(S)\leq 2\epsilon^{-1}$.
###### Corollary 6.2.
Assume that $p\ll n^{-1/3-\epsilon}$, where $\epsilon>0$ is fixed. Then, for a
random graph $\Gamma\in G(n,p)$, the clique complex $X_{\Gamma}$ has the
following property with probability tending to 1 as $n\to\infty$: any
aspherical subcomplex $Y\subset X_{\Gamma}^{(2)}$ satisfies the Whitehead
Conjecture, i.e. any subcomplex $Y^{\prime}\subset Y$ is also aspherical.
Here is another interesting statement about the local structure of aspherical
subcomplexes of $X_{\Gamma}^{(2)}$.
###### Corollary 6.3.
Assume that $p\ll n^{-1/3-\epsilon}$, where $\epsilon>0$ is fixed. Then, for a
random graph $\Gamma\in G(n,p)$ the clique complex $X_{\Gamma}$ has the
following property with probability tending to 1 as $n\to\infty$: for any
aspherical subcomplex $Y\subset X^{(2)}_{\Gamma}$ any subcomplex $S\subset Y$
with $e(S)\leq 2\epsilon^{-1}$ is collapsible to a graph.
We now start preparations for the proofs of theorems 6.1 and 6.3 which appear
below in this section. Corollary 6.2 obviously follows from Theorem 6.1.
Let $Y$ be a simplicial complex with $\pi_{2}(Y)\not=0$. As in [12], we define
a numerical invariant $M(Y)\in{\mathbf{Z}}$, $M(Y)\geq 4$, as the minimal
number of faces in a 2-complex $\Sigma$ homeomorphic to the sphere $S^{2}$
such that there exists a homotopically nontrivial simplicial map $\Sigma\to
Y$.
We define $M(Y)=0$, if $\pi_{2}(Y)=0$.
###### Lemma 6.4 (See Corollary 5.3 in [12]).
Let $Y$ be a 2-complex with $I(Y)\geq c>0$. Then
$M(Y)\leq\left(\frac{16}{c}\right)^{2}.$
Combining this with Theorem 4.2 we obtain:
###### Lemma 6.5.
Assume that the probability parameter satisfies $p\ll n^{-1/3-\epsilon}$ where
$\epsilon>0$ is fixed. Then there exists a constant $C_{\epsilon}>0$ such that
for a random graph $\Gamma\in G(n,p)$ the clique complex $X_{\Gamma}$ has the
following property with probability tending to one: for any subcomplex
$Y\subset X^{(2)}_{\Gamma}$ one has $M(Y)\leq C_{\epsilon}$.
Clearly, Lemma 6.5 follows from Theorem 4.2 and from Lemma 6.4.
###### Proof of Theorem 6.1.
Let $\Gamma$ be a random graph, $\Gamma\in G(n,p)$, $p\ll n^{-1/3-\epsilon}$,
and let $Y\subset X_{\Gamma}^{(2)}$ be a 2-dimensional subcomplex. Suppose
that $\pi_{2}(Y)\not=0$, i.e. $Y$ is not aspherical. Using Lemma 6.5 we have
$M(Y)\leq C_{\epsilon}$ a.a.s. where $C_{\epsilon}>0$ is a constant depending
on $\epsilon$. There is a homotopically nontrivial simplicial map $\phi:S\to
Y$ where $S$ is a triangulation of the sphere $S^{2}$ having at most
$C_{\epsilon}$ faces. Hence, $Y$ must contain as a subcomplex a simplicial
quotient $S^{\prime}=\phi(S)$ of a triangulation $S$ of the sphere $S^{2}$
having at most $C_{\epsilon}$ faces and such that
$\phi_{\ast}:\pi_{2}(S)\to\pi_{2}(S^{\prime})$ is nonzero.
Consider the set of isomorphism types $\mathcal{L}=\\{S^{\prime}\\}$ of pure
2-complexes $S^{\prime}$ having at most $C_{\epsilon}$ faces and such that
$\pi_{2}(S^{\prime})\not=0$; clearly the list $\mathcal{L}$ is finite. By
Theorem 2.1 any $S^{\prime}\in\mathcal{L}$ satisfying
$\tilde{\nu}(S^{\prime})<1/3+\epsilon$ is not embeddable into $Y$, a.a.s. Next
we show that any $S^{\prime}\in\mathcal{L}$ with $\tilde{\nu}(S^{\prime})\geq
1/3+\epsilon$ contains a small bubble, i.e. a non-aspherical subcomplex with
at most $2\epsilon^{-1}$ edges. If $b_{2}(S^{\prime})=0$ then by Lemma 5.1 we
see that $S^{\prime}$ contains a subcomplex $S^{\prime\prime}\subset S$ which
is either a triangulation of $P^{2}$ or a triangulation of $P^{2}$ with 2
adjacent edges identified. In both cases one has
$\pi_{2}(S^{\prime\prime})\not=0$. Now we use the inequality
$\nu(S^{\prime\prime})\geq 1/3+\epsilon$
to show that $e(S^{\prime\prime})\leq\epsilon^{-1}$. Indeed, in the first case
one has
$\nu(S^{\prime\prime})=1/3+\frac{1}{e(S^{\prime})}$
implying $e(S^{\prime\prime})\leq\epsilon^{-1}$ and in the second case
$\nu(S^{\prime\prime})=1/3+\frac{1}{3e(S^{\prime})}$
implying $e(S^{\prime\prime})\leq(3\epsilon)^{-1}\leq\epsilon^{-1}.$
Consider now that case when $b_{2}(S^{\prime})>0$. Then $S^{\prime}$ contains
a minimal cycle $S^{\prime\prime}\subset S^{\prime}$. By Lemma 5.6
$\pi_{2}(S^{\prime\prime})\not=0$ and we need to show that
$e(S^{\prime\prime})\leq 2\epsilon^{-1}$. Indeed, we know that
$\nu(S^{\prime\prime})\geq 1/3+\epsilon$ and $\chi(S^{\prime\prime})\leq 2$.
Hence
$\frac{1}{3}+\frac{2}{e(S^{\prime\prime})}\geq\nu(S^{\prime\prime})=\frac{1}{3}+\frac{3\chi(S^{\prime\prime})+L(S^{\prime\prime})}{3e(S^{\prime\prime})}\geq\frac{1}{3}+\epsilon$
implying $e(S^{\prime\prime})\leq 2\epsilon^{-1}$.
Let us now prove the inverse implication, i.e. that the random complex
$X_{\Gamma}$ with probability tending to 1 as $n\to\infty$ has the following
property: if a subcomplex $Y\subset X_{\Gamma}^{(2)}$ contains a small bubble
$S\subset Y$, $\pi_{2}(S)\not=0$, $e(S)\leq 2\epsilon^{-1}$, then $Y$ is not
aspherical. There are finitely isomorphism types of 2-complexes $S$ with at
most $2\epsilon^{-1}$ edges. Therefore, by Theorem 2.1 we may conclude that a
random complex $X_{\Gamma}^{(2)}$ may contain as a subcomplex only the bubbles
$S$, $\pi_{2}(S)\not=0$, $e(S)\leq 2\epsilon^{-1}$satisfying
$\tilde{\nu}(S)\geq 1/3+\epsilon$.
If $b_{2}(S)>0$ then there is a minimal cycle $S^{\prime}\subset S$,
$\tilde{\nu}(S^{\prime})\geq 1/3+\epsilon$. By Lemma 5.6 the Hurewicz map
$h:\pi_{2}(S^{\prime})\to H_{2}(S^{\prime})$ is an epimorphism. Since
$H_{2}(S^{\prime})\to H_{2}(Y)$ is injective, we see that $H_{2}(Y)$ contains
a spherical homology class and hence $\pi_{2}(Y)\not=0$.
If $b_{2}(S)=0$ then by Lemma 5.1 there is a subcomplex $K\subset S$ which is
homotopy equivalent to the real projective plane $P^{2}$. By a Theorem of
Crockfort [9], see also [1], an aspherical complex cannot contain such $K$ as
a subcomplex; hence $\pi_{2}(Y)\not=0$, i.e. $Y$ is not aspherical.
∎
###### Proof of Corollary 6.3.
Let $X_{\Gamma}$ be the clique complex of a random graph $\Gamma\in G(n,p)$
where $p\ll n^{-1/3-\epsilon}$. By Theorem 6.1, for any aspherical subcomplex
$Y\subset X_{\Gamma}^{(2)}$, any subcomplex $S\subset Y$ with $e(S)\leq
2\epsilon^{-1}$ is aspherical, a.a.s. We shall also assume (using Theorem 2.1
and the finiteness of the set of isomorphism types of 2-complexes satisfying
$e(S)\leq 2\epsilon^{-1}$) that any subcomplex $S\subset Y\subset
X_{\Gamma}^{(2)}$ has the property $\tilde{\nu}(S)>1/3$.
We want to show that each $S\subset Y\subset X_{\Gamma}^{(2)}$, $e(S)\leq
2\epsilon^{-1}$ is collapsible to a graph. Indeed, performing all possible
simplicial collapses on $S$ we either obtain a graph or a closed 2-dimensional
complex $S^{\prime}$ with $e(S^{\prime})\leq 2\epsilon^{-1}$ and
$\tilde{\nu}(S^{\prime})>1/3$. If $b_{2}(S^{\prime})>0$ then $S^{\prime}$
contains a minimal cycle $Z\subset S^{\prime}$, $\tilde{\nu}(Z)>1/3$ and using
Lemma 5.6 we see that $S^{\prime}$ is not aspherical - a contradiction. If
$b_{2}(S^{\prime})=0$ then by Lemma 5.1 we see that $S^{\prime}$ contains a
subcomplex $X\subset S^{\prime}$ homotopy equivalent to $P^{2}$ and
$S^{\prime}$ is not aspherical by a theorem of Cockcroft [9]. Hence the only
possibility is that $S$ is collapsible to a graph. ∎
## 7 2-torsion in fundamental groups of random clique complexes
###### Theorem 7.1.
Assume that
$\displaystyle p\ll n^{-11/30}.$ (23)
Then the fundamental group $\pi_{1}(X_{\Gamma})$ of the clique complex of a
random graph $\Gamma\in G(n,p)$ has geometric dimension and cohomological
dimension at most $2$, and in particular $\pi_{1}(X_{\Gamma})$ is torsion
free, a.a.s. Moreover, if
$n^{-1/2}\ll p\ll n^{-11/30}$
then the geometric dimension and the cohomological dimension of
$\pi_{1}(X_{\Gamma})$ equal two.
###### Theorem 7.2.
Assume that
$\displaystyle n^{-11/30}\ll p\ll n^{-1/3-\epsilon}$ (24)
where $0<\epsilon<1/30$ is fixed. Then the fundamental group
$\pi_{1}(X_{\Gamma})$ has 2-torsion and its cohomological dimension is
infinite, a.a.s.
###### Proof of Theorem 7.1.
Consider the set ${\mathcal{C}}_{60}$ of isomorphism types of simplicial
complexes having at most
$60+\frac{3}{2}(3c^{-1}_{\epsilon}-1)$
edges, where $c_{\epsilon}>0$ is the constant given by Theorem 4.2 for
$\epsilon=1/30$. This set is clearly finite. For any $n$, consider the set
$\mathcal{X}_{n}$ of graphs $\Gamma\in G(n,p)$ such that the corresponding
clique complex $X_{\Gamma}$ does not contain as a subcomplex complexes
$S\in{\mathcal{C}}_{60}$ satisfying $\nu(S)\leq 11/30$ and such that for any
subcomplex $Y\subset X_{\Gamma}$ one has $I(Y)\geq c_{\epsilon}$. From Theorem
2.1 and Theorem 4.2 we know that, under our assumption $p\ll n^{-11/30}$, the
probability of this set $\mathcal{X}_{n}$ of graphs tends to one as
$n\to\infty$.
To prove the first part of Theorem 7.1 we shall construct, for any
$\Gamma\in\mathcal{X}_{n}$, a subcomplex $Y_{\Gamma}\subset X_{\Gamma}^{(2)}$
which is aspherical $\pi_{2}(Y_{\Gamma})=0$ and has the same fundamental
group, $\pi_{1}(Y_{\Gamma})=\pi_{1}(X_{\Gamma})$. The existence of such
$Y_{\Gamma}$ implies that
${\rm gdim}(\pi_{1}(X_{\Gamma}))={\rm cd}(\pi_{1}(X_{\Gamma}))\leq 2.$
Here we use the results of Eilenberg and Ganea [14] in conjunction with the
theorem of Swan [27] stating that a group of cohomological dimension one is a
free group.
The equality ${\rm cd}(\pi_{1}(X_{\Gamma}))=2$ under the assumptions
$n^{-1/2}\ll p\ll n^{-11/30}$ follows from the result of [21], Theorem 1.2
which states that for
$p\geq\left(\frac{(3/2+\epsilon)\log n}{n}\right)^{1/2}$
the fundamental group $\pi_{1}(X_{\Gamma})$ has property T (a.a.s.) implying
${\rm cd}(\pi_{1}(X_{\Gamma}))>1$.
Consider the minimal cycles $Z\in\mathcal{C}_{60}$ and their all possible
embeddings $Z\subset X_{\Gamma}$ where $\Gamma\in\mathcal{X}_{n}$. By Lemma
5.6 each such $Z$ contains a 2-simplex $\sigma$ such that $\partial\sigma$ is
null-homotopic in $Z-\rm Int(\sigma)$. We remove subsequently one such
2-simplex from each of the minimal cycles $Z\subset X_{\Gamma}$. The union of
the 1-skeleton of $X_{\Gamma}$ and the remaining 2-simplexes is a 2-complex
which we denote by $Y_{\Gamma}$. Clearly
$\pi_{1}(Y_{\Gamma})=\pi_{1}(X_{\Gamma})$. To show that $Y_{\Gamma}$ is
aspherical we shall apply Theorem 6.1. We need to show that any subcomplex
$S\subset Y_{\Gamma}$, where $S\in\mathcal{C}_{60}$, is aspherical. By the
above construction we know that $S\subset Y_{\Gamma}$ cannot contain minimal
cycles, and therefore $b_{2}(S)=0$. Without loss of generality we may assume
that $S$ is closed, pure and strongly connected; then Lemma 5.1 implies that
$S$ must contain a triangulation of the projective plane or its quotient with
two adjacent edges are identified.
We know that for any triangulation $S$ of $P^{2}$ one has
$\tilde{\nu}(S)=\nu(S)=1/3+1/e(S).$
We obtain that only triangulations $S$ of $P^{2}$ having less than $30$ edges,
$e(S)<30$, are embeddable into $X_{\Gamma}$ where $\Gamma\in\mathcal{X}_{n}$.
Recall that a triangulation of a 2-complex is called clean if for any clique
of three vertices $\\{v_{0},v_{1},v_{2}\\}$ the complex contains also the
simplex $(v_{0}v_{1}v_{2})$. We shall use the following fact: any clean
triangulation of the projective plane $P^{2}$ contains at least $11$ vertices
and $30$ edges, see [17]. The minimal clean triangulation is shown in Figure
8; the antipodal points of the circle must be identified.
Figure 8: The minimal clean triangulation of $P^{2}$, according to [17].
Any triangulation $S$ of $P^{2}$ containing less than $30$ edges is not clean,
i.e. it contains a cycle of length 3 which is not filled by a triangle. If
this cycle is null-homologous than we may split $S$ into two smaller surfaces
one of which is a disk and another is a projective plane with smaller number
of edges. Continuing by induction, we obtain that for any triangulation $S$ of
$P^{2}$ containing less than $30$ edges there is a cycle of length 3
representing a non-contractible loop in $S$.
We claim that $Y_{\Gamma}$ contains no subcomplexes $S$ with $e(S)\leq 60$
which are triangulations of $P^{2}$. Indeed, if $S$ is embedded into
$Y_{\Gamma}$, where $\Gamma\in\mathcal{X}_{n}$, then a nontrivial cycle of $S$
bounds a triangle in $X_{\Gamma}$. In particular, the inclusion $S\to
X_{\Gamma}$ induces a trivial homomorphism of the fundamental groups
$\pi_{1}(S)\to\pi_{1}(X_{\Gamma})$. Since the inclusion induces an isomorphism
$\pi_{1}(Y_{\Gamma})\to\pi_{1}(X_{\Gamma})$ we obtain that the inclusion
$S\subset Y_{\Gamma}$ also induces a trivial homomorphism
$\pi_{1}(S)\to\pi_{1}(Y_{\Gamma})$ however now the length 3 cycle of $S$ may
bound a larger disc and not a simple 2-simplex. We may apply Theorem 4.2 about
uniform hyperbolicity to estimate the size of the minimal bounding disc for
this cycle. Since $I(Y_{\Gamma})\geq c_{\epsilon}$ where $\epsilon=1/30$, we
see that the area of the bounding disc is $\leq 3c_{\epsilon}^{-1}$. We obtain
that there exists a subcomplex $S\subset L\subset Y_{\Gamma}$ such that
$\pi_{1}(S)\to\pi_{1}(L)$ is trivial and
$\displaystyle e(L)\leq 60+\frac{3}{2}(3c_{\epsilon}^{-1}-1).$ (25)
Since $\Gamma\in\mathcal{X}_{n}$ we see that $\tilde{\nu}(L)>1/3$. By
construction, $L$ (as well as $Y_{\Gamma}$) may not contain minimal cycles
since any minimal cycle $Z$ satisfying $\tilde{\nu}(Z)>11/30$ must have at
most $60$ edges; therefore $b_{2}(L)=0$. We may assume that $L$ is strongly
connected and pure. Then by Lemma 5.1 we see that each strongly connected pure
component of $L$ must be isomorphic either to the projective plane or to its
quotient, and in both cases we obtain a contradiction to the homomorphism
$\pi_{1}(S)\to\pi_{1}(L)$ being trivial.
Similarly, one shows that $Y_{\Gamma}$ contains no subcomplexes $S^{\prime}$
isomorphic to the quotients of a triangulation of $P^{2}$ with two adjacent
edges identified and with $e(S^{\prime})\leq 60$. One has
$\tilde{\nu}(S^{\prime})=\nu(S^{\prime})=\frac{1}{3}+\frac{1}{3e(S^{\prime})}$
and for $\Gamma\in\mathcal{X}_{n}$ we shall find subcomplexes
$S^{\prime}\subset X_{\Gamma}$ only if $e(S^{\prime})<10$. Thus, using the
result of [17], we obtain that that if $S^{\prime}$ is embedded into
$X_{\Gamma}$, where $\Gamma\in\mathcal{X}_{n}$, then there is a cycle of
length 3 in $S^{\prime}$ which is not null-homotopic in $S^{\prime}$; this
cycle bounds a triangle in $X_{\Gamma}$ and as a result the inclusion
$S^{\prime}\to X_{\Gamma}$ induces a trivial homomorphism of the fundamental
groups $\pi_{1}(S^{\prime})\to\pi_{1}(X_{\Gamma})$. Repeating the arguments of
the preceding paragraph we find a subcomplex $S^{\prime}\subset L\subset
Y_{\Gamma}$ such that $\pi_{1}(S^{\prime})\to\pi_{1}(L)$ is trivial and $L$
satisfies (25). As above we find that $\tilde{\nu}(L)>1/3$, $b_{2}(L)=0$ and
therefore $L$ is an iterated wedge of projective planes or projective planes
with two adjacent edges identified; this contradicts the fact that
$\pi_{1}(S^{\prime})\to\pi_{1}(L)$ is trivial.
∎
### 7.1 Proof of Theorem 7.2
#### 7.1.1 The number of combinatorial embeddings
Consider two 2-complexes $S_{1}\supset S_{2}$. Denote by $v_{i}$ and $e_{i}$
the numbers of vertices and faces of $S_{i}$. We have $v_{1}\geq v_{2}$ and
$e_{1}\geq e_{2}$. We will assume that $e_{1}>e_{2}$.
Let $\nu(S_{1},S_{2})$ denote the ratio
$\nu(S_{1},S_{2})=\frac{v_{1}-v_{2}}{e_{1}-e_{2}}.$
Clearly, $\nu(S_{1},S_{2})$ depends only on the 1-skeleta of $S_{1}$ and
$S_{2}$, however it will be convenient to think of this quantity as being a
function of the 2-complexes $S_{1},S_{2}$. If $\nu(S_{1})<\nu(S_{2})$ then
$\displaystyle\nu(S_{1},S_{2})<\nu(S_{1})<\nu(S_{2}).$ (26)
If $\nu(S_{1})>\nu(S_{2})$ then
$\displaystyle\nu(S_{1},S_{2})>\nu(S_{1})>\nu(S_{2}).$ (27)
These two observations can be summarised by saying that $\nu(S_{1})$ always
lies in the interval connecting $\nu(S_{2})$ and $\nu(S_{1},S_{2})$. One has
the following formula
$\displaystyle\nu(S_{1},S_{2})=\frac{1}{3}+\frac{3(\chi(S_{1})-\chi(S_{2}))+L(S_{1})-L(S_{2})}{3(e_{1}-e_{2})},$
(28)
which follows from the equation $3v_{i}=e_{i}+3\chi(S_{i})+L(S_{i})$; the
latter is equivalent to (8).
###### Lemma 7.3.
Let $S_{1}$ be closed, i.e. $\partial S_{1}=\emptyset$, and $S_{2}$ be a
pseudo-surface such that $\chi(S_{1})\leq\chi(S_{2})$. Then
$\nu(S_{1},S_{2})<1/3$.
###### Proof.
Since $S_{1}$ is closed, $L(S_{1})\leq 0$. Besides, $L(S_{2})=0$ since $S_{2}$
is a pseudo-surface. The result now follows from the above formula. ∎
###### Theorem 7.4.
Let $S_{1}\supset S_{2}$ be two fixed 2-complexes and222The assumption (29) is
meaningful iff $\nu(S_{1},S_{2})<\tilde{\nu}(S_{2})\leq\nu(S_{2})$ which, as
follows from (26) and (27), implies that $\nu(S_{1})<\nu(S_{2})$. Thus, if
Theorem 7.4 is applicable, then $\nu(S_{1})<\nu(S_{2})$.
$\displaystyle n^{-\tilde{\nu}(S_{2})}\ll p\ll n^{-\nu(S_{1},S_{2})}.$ (29)
Then the number of embeddings of $S_{1}$ into the clique complex $X_{\Gamma}$
of a random graph $\Gamma\in G(n,p)$ is smaller than the number of embeddings
of $S_{2}$ into $X_{\Gamma}$, a.a.s. In particular, under the assumptions
(29), with probability tending to one, there exists an embedding $S_{2}\to
X_{\Gamma}$ which does not extend to an embedding $S_{1}\to X_{\Gamma}$.
###### Proof.
Let $T_{i}:G(n,p)\to{\mathbf{Z}}$ be the random variable counting the number
of embeddings of $S_{i}$ into $X_{\Gamma}$, $i=1,2$ (where by an embedding we
understand a simplicial injective map $S_{i}\to X_{\Gamma}$). We know that
$\mathbb{E}(T_{i})=\binom{n}{v_{i}}v_{i}!p^{e_{i}}\sim n^{v_{i}}p^{e_{i}}.$
Our goal is to show that $T_{1}<T_{2}$, a.a.s. We have
$\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{2})}\sim
n^{v_{1}-v_{2}}p^{e_{1}-e_{2}}=\left[n^{\nu(S_{1},S_{2})}p\right]^{e_{1}-e_{2}}\to
0$
tends to zero, under our assumption (29) (right).
Find $t_{1},t_{2}>0$ such that
$t_{1}+t_{2}=\mathbb{E}(T_{2})-\mathbb{E}(T_{1})$
and $\mathbb{E}(T_{1})/t_{1}\to 0$ while $\mathbb{E}(T_{2})/t_{2}$ is bounded.
Then
$P(T_{1}<T_{2})\geq
1-P(T_{1}>\mathbb{E}(T_{1})+t_{1})-P(T_{2}<\mathbb{E}(T_{2})-t_{2}).$
By Markov’s inequality
$P(T_{1}>\mathbb{E}(T_{1})+t_{1})<\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{1})+t_{1}}=\frac{\frac{\mathbb{E}(T_{1})}{t_{1}}}{1+\frac{\mathbb{E}(T_{1})}{t_{1}}}\to
0$
while by Chebyschev’s inequality
$P(T_{2}<\mathbb{E}(T_{2})-t_{2})<\frac{{\rm{Var(T_{2})}}}{t_{2}^{2}}.$
It is known (see [10], proof of Theorem 15) that under our assumptions (29)
the ratio $\frac{{\rm{Var(T_{2})}}}{\mathbb{E}(T_{2})^{2}}$ tends to zero.
We take $t_{1}=\sqrt{\mathbb{E}(T_{1})\mathbb{E}(T_{2})}$ and
$t_{2}=\mathbb{E}(T_{2})-\mathbb{E}(T_{1})-t_{1}$. Then
$\frac{\mathbb{E}(T_{1})}{t_{1}}=\sqrt{\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{2})}}\to
0$
and
$\frac{\mathbb{E}(T_{2})}{t_{2}}=\frac{1}{1-\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{2})}-\sqrt{\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{2})}}}\to
1$
is bounded. ∎
###### Theorem 7.5.
Let
$S_{j}\supset S,\quad j=1,\dots,N,$
be a finite family of 2-complexes containing a given 2-complex $S$ and
satisfying $\nu(S_{j})<\nu(S)$. Assume that
$\displaystyle n^{-\tilde{\nu}(S)}\ll p\ll n^{-\nu(S_{j},S)},\quad\mbox{for
any}\quad j=1,\dots,N.$ (30)
Then, with probability tending to one, for the clique complex $X_{\Gamma}$ of
a random graph $\Gamma\in G(n,p)$ there exists an embedding $S\to X_{\Gamma}$
which does not extend to an embedding $S_{j}\to X_{\Gamma}$, for any
$j=1,\dots,N$.
###### Proof.
Let $T_{1,j}:G(n,p)\to{\mathbf{Z}}$ denote the random variable counting the
number of embeddings of $S_{j}$ into $X_{\Gamma}$. Denote
$T_{1}=\sum_{j=1}^{N}T_{1,j}.$ Besides, Let $T_{2}:G(n,p)\to{\mathbf{Z}}$
denote the number of embeddings of $S$ into a random clique complex
$X_{\Gamma}$. One has
$\frac{\mathbb{E}(T_{1})}{\mathbb{E}(T_{2})}=\sum_{j=1}^{N}\frac{\mathbb{E}(T_{1,j})}{\mathbb{E}(T_{2})}\to
0$
thanks to our assumption (30) (right). Taking
$t_{1}=\sqrt{\mathbb{E}(T_{1})\mathbb{E}(T_{2})}$ and
$t_{2}=\mathbb{E}(T_{2})-\mathbb{E}(T_{1})-t_{1}$ (as in the proof of the
previous theorem) one has $t_{1}+t_{2}=\mathbb{E}(T_{2})-\mathbb{E}(T_{1})$
and $\mathbb{E}(T_{1})/t_{1}\to 0$ while $\mathbb{E}(T_{2})/t_{2}$ is bounded.
Repeating the arguments used in the proof of the previous theorem we see that
$T_{1}>T_{2}$ a.a.s. Since every embedding $S_{j}\to X_{\Gamma}$ determines
(by restriction) an embedding $S\to X_{\Gamma}$, the inequality
$T_{1}(\Gamma)>T_{2}(\Gamma)$ implies that there there are embeddings $S\to
X_{\Gamma}$ which admit no extensions to $S_{j}\to X_{\Gamma}$, for any
$j=1,\dots,N$. ∎
### 7.2 Projective planes in clique complexes of random graphs
Recall that a connected subcomplex $S\subset X$ is said to be essential if the
induced homomorphism $\pi_{1}(S)\to\pi_{1}(X)$ is injective.
###### Theorem 7.6.
Let $S$ be a clean triangulation of the real projective plane $P^{2}$ having
11 vertices, 30 edges and 20 faces. Assume that $0<\epsilon<1/30$ and
$n^{-11/30}\ll p\ll n^{-1/3-\epsilon}.$
Then the clique complex $X_{\Gamma}$ of a random graph $\Gamma\in G(n,p)$
contains $S$ as an essential subcomplex, a.a.s. In particular, the fundamental
group $\pi_{1}(X_{\Gamma})$ has an element of order two and hence its
cohomological dimension is infinite.
###### Proof.
Consider the set $\mathcal{S}_{\epsilon}$ of isomorphism types of pure
connected closed 2-complexes $X$ satisfying the following conditions:
1. (a)
$X$ contains $S$ as a subcomplex;
2. (b)
The inclusion $S\to X$ induces a trivial homomorphism
$\pi_{1}(S)\to\pi_{1}(X)$;
3. (c)
For any subcomplex $S\subset X^{\prime}\subset X$, $X^{\prime}\not=X$, the
homomorphism $\pi_{1}(S)\to\pi_{1}(X^{\prime})$ is nontrivial;
4. (d)
$X$ has at most
$20+4c_{\epsilon}^{-1}$
faces, where $c_{\epsilon}$ is the constant given by Theorem 4.2.
5. (e)
$\tilde{\nu}(X)>1/2+\epsilon$;
The set $\mathcal{S}_{\epsilon}$ is finite due to the condition (d).
Let us show that for any $X\in{\mathcal{S}}_{\epsilon}$ one has
$\displaystyle\nu(X,S)\leq 1/3,$ (31)
which is equivalent to the inequality
$\displaystyle 3(\chi(X)-\chi(S))+L(X)-L(S)\leq 0$ (32)
by formula (28). From the exact sequence of the pair $(X,S)$
$0\to H_{2}(X)\to H_{2}(X,S)\to H_{1}(S)={\mathbf{Z}}_{2}\to 0$
we see (since the middle group has no torsion) that $b_{2}(X)=b_{2}(X,S)\geq
1$. Let $Z$ be a minimal cycle in $X$. Let us show that the assumption that
$Z$ is of type A leads to a contradiction. Clearly, $Z\not\subset S$ and let
$\sigma$ be a simplex of $Z-S$. Then $\partial\sigma$ is null-homotopic in
$Z-\rm Int(\sigma)$ and hence in $X-\rm Int(\sigma)$, and therefore
$\pi_{1}(X-\rm Int(\sigma))\to\pi_{1}(X)$ is an isomorphism and
$\pi_{1}(S)\to\pi_{1}(X-\rm Int(\sigma))$ is injective violating (c). Thus $Z$
is a minimal cycle of type B, i.e. there exists a proper pure and strongly
connected closed subcomplex $Z^{\prime}\subset Z$. Since
$\tilde{\nu}(Z^{\prime})>1/3$ we see (using Lemma 5.1) that $Z^{\prime}$ is a
triangulated $P^{2}$ (or quotient its quotient with two adjacent edges
identified). If $Z^{\prime}\not\subset S$ then we can again remove any
2-simplex of $Z^{\prime}-S$ to find a contradiction with (c), similarly to the
argument given above. Thus $Z^{\prime}\subset S$ implying that $Z^{\prime}=S$.
However, $\pi_{1}(S)\to\pi_{1}(Z)$ is trivial (by Lemma 5.6) and now the
property (c) gives $X=Z$. Therefore, we obtain $b_{2}(X)=1.$ This also implies
that $L(X)$ is either $-3,-4$ or $-5$. Now we apply formula (28) with
$\chi(X)=2$, $\chi(S)=1$, $L(S)=0$ and $L(X)\leq-3$ to obtain (32).
Applying Theorem 7.5 we find that for $n^{-11/30}\ll p\ll n^{-1/3-\epsilon}$,
with probability tending to 1, there exist embedding $S\to X_{\Gamma}$ (where
$\Gamma\in G(n,p)$ is random) which cannot be extended to an embedding of
$X\to X_{\Gamma}$ for any $X\in{\mathcal{S}}_{\epsilon}$. Let us show that any
such embedding $S\subset X_{\Gamma}$ induces a monomorphism
$\pi_{1}(S)\to\pi_{1}(X_{\Gamma})$. If $S\subset X_{\Gamma}$ is not essential
then the central cycle $\gamma$ of $S$ (of length 4) bounds in $X_{\Gamma}$ a
simplicial disc. Under the assumption $p\ll n^{-1/3-\epsilon}$, using Theorem
4.2, we find that the circle $\gamma$ bounds in $X_{\Gamma}$ a simplicial disk
$b:D^{2}\to X_{\Gamma}$ of area $\leq 4c_{\epsilon}^{-1}$ where
$c_{\epsilon}>0$ in the constant of Theorem 4.2 which depends only on the
value of $\epsilon$. Consider the union $Y=S\cup b(D^{2})$. This is a
subcomplex of $X_{\Gamma}$ satisfying properties (a), (b), (d). We may assume
that $\Gamma\in G(n,p)$ is such that any subcomplex $T\subset
X_{\Gamma}^{(2)}$ with at most $20+3c_{\epsilon}^{-1}$ faces satisfies
$\tilde{\nu}(T)>1/3+\epsilon$; the set of such graphs $\Gamma\in G(n,p)$ has
probability tending to one as $n\to\infty$ according to Theorem 2.1. Hence, we
see that $Y$ satisfies the property (e) as well. However the property (c) can
be violated. In this case we find a minimal subcomplex $S\subset X\subset Y$
which satisfies all the properties (a)-(e). Hence, if $S\subset X_{\Gamma}$ is
not essential then there would exist a complex $X\in{\mathcal{S}}_{\epsilon}$
such that the embedding $S\subset X_{\Gamma}$ extends to an embedding
$X\subset X_{\Gamma}$ contradicting our construction. ∎
## 8 Absence of odd torsion
In this section we prove the following statement complementing Theorems 7.1
and 7.2.
###### Theorem 8.1.
Let $m\geq 3$ be a fixed prime. Assume that
$\displaystyle p\ll n^{-1/3-\epsilon}$ (33)
where $\epsilon>0$ is fixed. Then a random graph $\Gamma\in G(n,p)$ with
probability tending to 1 has the following property: the fundamental group of
any subcomplex $Y\subset X_{\Gamma}$ has no $m$-torsion.
Let $\Sigma$ be a simplicial 2-complex homeomorphic to the Moore surface
$M({\mathbf{Z}}_{m},1)=S^{1}\cup_{f_{m}}e^{2},\quad\mbox{where}\quad\quad
m\geq 3;$
it is obtained from the circle $S^{1}$ by attaching a 2-cell via the degree
$m$ map $f_{m}:S^{1}\to S^{1}$, $f_{m}(z)=z^{m}$, $z\in S^{1}$. The 2-complex
$\Sigma$ has a well defined circle $C\subset\Sigma$ (called the singular
circle) which is the union of all edges of degree $m$; all other edges of
$\Sigma$ have degree $2$. Clearly, the homotopy class of the singular circle
generates the fundamental group $\pi_{1}(\Sigma)\simeq{\mathbf{Z}}_{m}$.
As in [12], define an integer $N_{m}(Y)\geq 0$ associated to any connected
2-complex $Y$. If $\pi_{1}(Y)$ has no $m$-torsion we set $N_{m}(Y)=0.$ If
$\pi_{1}(Y)$ has elements of order $m$ we consider homotopically nontrivial
simplicial maps $\gamma:C_{r}\to Y$, where $C_{r}$ is the simplicial circle
with $r$ edges, such that
1. (a)
$\gamma^{m}$ is null-homotopic (as a free loop in $Y$);
2. (b)
$r$ is minimal: for $r^{\prime}<r$ any simplicial loop
$\gamma:C_{r^{\prime}}\to Y$ satisfying (a) is homotopically trivial.
Any such simplicial map $\gamma:C_{r}\to Y$ can be extended to a simplicial
map $f:\Sigma\to Y$ of a triangulation $\Sigma$ of the Moore surface, such
that the singular circle $C$ of $\Sigma$ is isomorphic to $C_{r}$ and
$f|C=\gamma$. We shall say that a simplicial map $f:\Sigma\to Y$ is
$m$-minimal if it satisfies (a), (b) and the number of 2-simplexes in $\Sigma$
is the smallest possible. Now, we denote by
$N_{m}(Y)\in{\mathbf{Z}}$
the number of 2-simplexes in a triangulation of the Moore surface $\Sigma$
admitting an $m$-minimal map $f:\Sigma\to Y$.
###### Lemma 8.2.
Let $Y$ be a 2-complex satisfying $I(Y)\geq c>0$. Let $m\geq 3$ be an odd
prime. Then one has
$N_{m}(Y)\leq\left(\frac{6m}{c}\right)^{2}.$
This is Lemma 4.7 from [12]; we refer the reader to [12] for a proof.
###### Theorem 8.3.
Assume that the probability parameter $p$ satisfies $p\ll n^{-1/3-\epsilon}$
where $\epsilon>0$ is fixed. Let $m\geq 3$ be an odd prime. Then there exists
a constant $C_{\epsilon}>0$ such that a random graph $\Gamma\in G(n,p)$ with
probability tending to 1 has the following property: for any subcomplex
$Y\subset X_{\Gamma}$ one has
$\displaystyle N_{m}(Y)\leq C_{\epsilon}.$ (34)
###### Proof.
We know from Theorem 4.2 that, with probability tending to 1, a random
2-complex $X_{\Gamma}$ has the following property: for any subcomplex
$Y\subset X_{\Gamma}$ one has $I(Y)\geq c_{\epsilon}>0$ where $c_{\epsilon}>0$
is the constant given by Theorem 4.2. Then, setting
$C=\left(\frac{6m}{c_{\epsilon}}\right)^{2}$, the inequality (34) follows from
Lemma 8.2. ∎
###### Proof of Theorem 8.1.
Let $c_{\epsilon}>0$ be the number given by Theorem 4.2. Consider the finite
set of all isomorphism types of triangulations $\mathcal{S}_{m}=\\{\Sigma\\}$
of the Moore surface $M({\mathbf{Z}}_{m},1)$ having at most
$\left(\frac{6m}{c_{\epsilon}}\right)^{2}$ two-dimensional simplexes. Let
$\mathcal{X}_{m}$ denote the set of isomorphism types of images of all
surjective simplicial maps $\Sigma\to X$ inducing injective homomorphisms
$\pi_{1}(\Sigma)={\mathbf{Z}}_{m}\to\pi_{1}(X)$, where
$\Sigma\in\mathcal{S}_{m}$. The set $\mathcal{X}_{m}$ is also finite.
From Theorem 8.3 we obtain that, with probability tending to one, for any
subcomplex $Y\subset X_{\Gamma}$, either $\pi_{1}(Y)$ has no $m$-torsion, or
there exists an $m$-minimal map $f:\Sigma\to Y$ with $\Sigma$ having at most
$\left(\frac{6m}{c_{\epsilon}}\right)$ simplexes of dimension 2; in the second
case the image $X=f(\Sigma)$ is a subcomplex of $Y^{\prime}$ and $f:\Sigma\to
X$ induces a monomorphism $\pi_{1}(\Sigma)\to\pi_{1}(X)$, i.e.
$X\in{\mathcal{X}}_{m}$.
From Corollary 5.9 we know that the fundamental group of any 2-complex
satisfying $\tilde{\nu}(X)>1/3$ is a free product of several copies of
${\mathbf{Z}}$ and ${\mathbf{Z}}_{2}$ and has no $m$-torsion, as we assume
that $m\geq 3$. Since the fundamental group of any $X\in\mathcal{X}_{m}$ has
$m$-torsion, where $m\geq 3$, one has $\tilde{\nu}(X)\leq 1/3$ for any
$X\in\mathcal{X}_{m}$. Hence, using the finiteness of $\mathcal{X}_{m}$ and
the results on the containment problem (Theorem 2.1) we see that for $p\ll
n^{-1/3-\epsilon}$ the probability that a random complex $X_{\Gamma}$, where
$\Gamma\in G(n,p)$, contains a subcomplex isomorphic to one of the complexes
$X\in\mathcal{X}_{m}$ tends to $0$ as $n\to\infty$. Hence, we obtain that
(a.a.s.) any subcomplex $Y\subset X_{\Gamma}$ does not contain
$X\in\mathcal{X}_{m}$ as a subcomplex and therefore the fundamental group of
$Y$ has no $m$-torsion. ∎
## Appendix A Appendix: Proof of Theorem 4.2
In this Appendix we give a complete and self-contained proof of Theorem 4.2
which plays a key role in this paper. As we mentioned above, this statement is
closely related to Theorem 1.1 from [5]. The proof of Theorem 4.2 given below
is similar to the arguments of [4], [5] and [12] and is based on two auxiliary
results: (1) the local-to-global principle of Gromov [16] and on (2) Theorem
A.2 giving uniform isoperimetric constants for complexes satisfying
$\tilde{\nu}(X)\geq 1/3+\epsilon$.
The local-to-global principle of Gromov can be stated as follows:
###### Theorem A.1.
Let $X$ be a finite 2-complex and let $C>0$ be a constant such that any pure
subcomplex $S\subset X$ having at most $(44)^{3}\cdot C^{-2}$ two-dimensional
simplexes satisfies $I(S)\geq C$. Then $I(X)\geq C\cdot 44^{-1}$.
Let $X$ be a 2-complex satisfying $\tilde{\nu}(X)>1/3$. Then by Corollary 5.9
the fundamental group of $X$ is hyperbolic as it is a free product of several
copies of cyclic groups ${\mathbf{Z}}$ and ${\mathbf{Z}}_{2}$. Hence,
$I(X)>0$. The following theorem gives a uniform lower bound for the numbers
$I(X)$.
###### Theorem A.2.
Given $\epsilon>0$ there exists a constant $C_{\epsilon}>0$ such that for any
finite pure 2-complex $X$ with $\tilde{\nu}(X)\geq 1/3+\epsilon$ one has
$I(X)\geq C_{\epsilon}$.
This Theorem is equivalent to Lemma 3.6 from [5]. The key ingredient of the
proof is the classification of minimal cycles (given by Lemmas 5.6, 5.7, 5.8
and Corollary 5.9). We do not use webs (as in [4], [5]) and operate with
simplicial complexes.
###### Proof of Theorem 4.2 using Theorem A.1 and Theorem A.2.
Let $C_{\epsilon}$ be the constant given by Theorem A.2. Consider the set
$\mathcal{S}$ of isomorphism types of all pure 2-complexes having at most
$44^{3}\cdot C_{\epsilon}^{-2}$ faces. In particular, all complexes in
$\mathcal{S}$ have at most $3^{-1}\cdot 44^{3}\cdot C_{\epsilon}^{-2}$ edges.
Clearly, the set $\mathcal{S}$ is finite. We may present it as the disjoint
union $\mathcal{S}=\mathcal{S}_{1}\sqcup\mathcal{S}_{2}$ where any
$S\in\mathcal{S}_{1}$ satisfies $\tilde{\nu}(S)\geq 1/3+\epsilon$ while for
$S\in\mathcal{S}_{2}$ one has $\tilde{\nu}(S)<1/3+\epsilon$. By Theorem 2.1, a
random complex $X_{\Gamma}$ contain as subcomplexes of $X_{\Gamma}^{(2)}$
complexes $S\in\mathcal{S}_{2}$ with probability tending to zero as
$n\to\infty$. Hence, $X_{\Gamma}$ may contain as subcomplexes of
$X_{\Gamma}^{(2)}$ only complexes $S\in\mathcal{S}_{1}$, a.a.s. By Theorem
A.2, any $S\in\mathcal{S}_{1}$ satisfies $I(S)\geq C_{\epsilon}$. Hence we see
that with probability tending to one, any subcomplex $S$ of $Y$ having at most
$44^{3}\cdot C_{\epsilon}^{-2}$ faces satisfies $I(S)\geq C_{\epsilon}$. Now
applying Theorem A.1 we obtain $I(Y^{\prime})\geq C_{\epsilon}\cdot
44^{-1}=c_{\epsilon}$, for any subcomplex $Y^{\prime}\subset Y$, a.a.s. ∎
### Proof of Theorem A.2
###### Definition A.3.
[12] We will say that a finite 2-complex $X$ is tight if for any proper
subcomplex $X^{\prime}\subset X$, $X^{\prime}\not=X$, one has
$I(X^{\prime})>I(X).$
Clearly, one has
$\displaystyle I(X)\geq\min\\{I(Y)\\}$ (35)
where $Y\subset X$ is a proper tight subcomplex. Since
$\tilde{\nu}(Y)\geq\tilde{\nu}(X)$ for $Y\subset X$, it is obvious from (35)
that it is enough to prove Theorem A.2 under the additional assumption that
$X$ is tight.
###### Remark A.4.
Suppose that $X$ is pure and tight and suppose that $\gamma:S^{1}\to X$ is a
simplicial loop with the ratio $|\gamma|\cdot A_{X}(\gamma)^{-1}$ less than
the minimum of the numbers $I(X^{\prime})$ where $X^{\prime}\subset X$ is a
proper subcomplex. Let $b:D^{2}\to X$ be a minimal spanning disc for $\gamma$;
then $b(D^{2})=X,$ i.e. $b$ is surjective. Indeed, if the image of $b$ does
not contain a 2-simplex $\sigma$ then removing it we obtain a subcomplex
$X^{\prime}\subset X$ with $A_{X^{\prime}}(\gamma)=A_{X}(\gamma)$ and hence
$I(X^{\prime})\leq I(X)\leq|\gamma|\cdot A_{X}(\gamma)^{-1}$ contradicting the
assumption on $\gamma$.
###### Lemma A.5.
If $X$ is a tight complex with $\tilde{\nu}(X)>1/3$ then $b_{2}(X)=0$.
###### Proof.
Assume that $b_{2}(X)\not=0$. Then there exists a minimal cycle $Z\subset X$
satisfying $\tilde{\nu}(Z)>1/3$. Hence, by Lemmas 5.6, 5.7 and 5.8 we may find
a 2-simplex $\sigma\subset Z\subset X$ such that $\partial\sigma$ is null-
homotopic in $Z-\sigma\subset X-\sigma=X^{\prime}$. Note that
$X^{\prime(1)}=X^{(1)}$ and a simplicial curve $\gamma:S^{1}\to X^{\prime}$ is
null-homotopic in $X^{\prime}$ if and only if it is null-homotopic in $X$.
Besides, $A_{X}(\gamma)\leq A_{X^{\prime}}(\gamma)$ and hence
$\frac{|\gamma|}{A_{X}(\gamma)}\geq\frac{|\gamma|}{A_{X^{\prime}}(\gamma)},$
which implies that $I(X)\geq I(X^{\prime})>I(X)$ – contradiction. ∎
###### Lemma A.6.
Given $\epsilon>0$ there exists a constant $C^{\prime}_{\epsilon}>0$ such that
for any finite pure tight connected 2-complex with $\tilde{\nu}(X)\geq
1/3+\epsilon$ and $L(X)\leq 0$ one has $I(X)\geq C^{\prime}_{\epsilon}$.
This Lemma is similar to Theorem A.2 but it has an additional assumption that
$L(X)\leq 0$. It is clear from the proof that the assumption $L(X)\leq 0$ can
be replaced, without altering the proof, by any assumption of the type
$L(X)\leq 1000$, i.e. by any specific upper bound.
###### Proof.
We show that the number of isomorphism types of complexes $X$ satisfying the
conditions of the Lemma is finite; hence the statement of the Lemma follows by
setting $C^{\prime}_{\epsilon}=\min I(X)$ and using Corollary 5.9 which gives
$I(X)>0$ (since $\pi_{1}(X)$ is hyperbolic) and hence
$C^{\prime}_{\epsilon}>0$. The inequality
$\nu(X)=\frac{1}{3}+\frac{3\chi(X)+L(X)}{3e(X)}\geq\frac{1}{3}+\epsilon$
is equivalent to
$e(X)\leq\epsilon^{-1}\cdot(3\chi(X)+L(X)/2),$
where $e(X)$ denotes the number of 1-simplexes in $X$. By Lemma A.5 we have
$\chi(X)=1-b_{1}(X)\leq 1$ and using the assumption $L(X)\leq 0$ we obtain
$e(X)\leq\epsilon^{-1}.$ This implies the finiteness of the set of possible
isomorphism types of $X$ and the result follows. ∎
We will use a relative isoperimetric constant $I(X,X^{\prime})\in{\mathbf{R}}$
for a pair consisting of a finite 2-complex $X$ and its subcomplex
$X^{\prime}\subset X$; it is defined as the infimum of all ratios
${|\gamma|}\cdot{A_{X}(\gamma)}^{-1}$ where $\gamma:S^{1}\to X^{\prime}$ runs
over simplicial loops in $X^{\prime}$ which are null-homotopic in $X$.
Clearly, $I(X,X^{\prime})\geq I(X)$ and $I(X,X^{\prime})=I(X)$ if
$X^{\prime}=X$. Below is a useful strengthening of Lemma A.6.
###### Lemma A.7.
Given $\epsilon>0$, let $C^{\prime}_{\epsilon}>0$ be the constant given by
Lemma A.6. Then for any finite pure tight connected 2-complex with
$\tilde{\nu}(X)\geq 1/2+\epsilon$ and for a connected subcomplex
$X^{\prime}\subset X$ satisfying $L(X^{\prime})\leq 0$ one has
$I(X,X^{\prime})\geq C^{\prime}_{\epsilon}$.
###### Proof.
We show below that under the assumptions on $X$, $X^{\prime}$ one has
$\displaystyle I(X,X^{\prime})\geq\min_{Y}I(Y)$ (36)
where $Y$ runs over all subcomplexes $X^{\prime}\subset Y\subset X$ satisfying
$L(Y)\leq 0$. Clearly, $\tilde{\nu}(Y)\geq 1/3+\epsilon$ for any such $Y$. By
Lemma A.5 we have that $b_{2}(X)=0$ which implies that $b_{2}(Y)=0$. Besides,
without loss of generality we may assume that $Y$ is connected. The arguments
of the proof of Lemma A.6 now apply (i.e. $Y$ may have finitely many
isomorphism types, each having a hyperbolic fundamental group) and it follows
that $\min_{Y}I(Y)\geq C^{\prime}_{\epsilon}$ where $C^{\prime}_{\epsilon}>0$
is a constant that only depends on $\epsilon$. Hence if (36) holds we have
$I(X,X^{\prime})\geq\min_{Y}I(Y)\geq C^{\prime}_{\epsilon}$ and the result
follows.
Suppose that inequality (36) is false, i.e. $I(X,X^{\prime})<\min_{Y}I(Y)$,
and consider a simplicial loop $\gamma:S^{1}\to X^{\prime}$ satisfying
$\gamma\sim 1$ in $X$ and $|\gamma|\cdot A_{X}(\gamma)^{-1}<\min_{Y}I(Y).$ Let
$\psi:D^{2}\to X$ be a simplicial spanning disc of minimal area. It follows
from the arguments of Ronan [25], that $\psi$ is non-degenerate in the
following sense: for any 2-simplex $\sigma$ of $D^{2}$ the image
$\psi(\sigma)$ is a 2-simplex and for two distinct 2-simplexes
$\sigma_{1},\sigma_{2}$ of $D^{2}$ with $\psi(\sigma_{1})=\psi(\sigma_{2})$
the intersection $\sigma_{1}\cap\sigma_{2}$ is either $\emptyset$ or a vertex
of $D^{2}$. In other words, we exclude foldings, i.e. situations such that
$\psi(\sigma_{1})=\psi(\sigma_{2})$ and $\sigma_{1}\cap\sigma_{2}$ is an edge.
Consider $Z=X^{\prime}\cup\psi(D^{2})$. Note that $L(Z)\leq 0$. Indeed, since
$L(Z)=\sum_{e}(2-\deg_{Z}(e)),$
where $e$ runs over the edges of $Z$, we see that for $e\subset X^{\prime}$,
$\deg_{X^{\prime}}(e)\leq\deg_{Z}(e)$ and for a newly created edge
$e\subset\psi(D^{2})$, clearly $\deg_{Z}(e)\geq 2$. Hence, $L(Z)\leq
L(X^{\prime})\leq 0$. On the other hand, $A_{X}(\gamma)=A_{Z}(\gamma)$ and
hence $I(Z)\leq|\gamma|\cdot A_{X}(\gamma)^{-1}<\min_{Y}I(Y)$, a
contradiction. ∎
The main idea of the proof of Theorem A.2 in the general case is to find a
planar complex (a “singular surface”) $\Sigma$, with one boundary component
$\partial_{+}\Sigma$ being the initial loop and such that “the rest of the
boundary” $\partial_{-}\Sigma$ is a “product of negative loops” (i.e. loops
satisfying Lemma A.7). The essential part of the proof is in estimating the
area (the number of 2-simplexes) of such $\Sigma$.
###### Proof of Theorem A.2.
Consider a connected tight pure 2-complex $X$ satisfying
$\displaystyle\tilde{\nu}(X)\geq\frac{1}{3}+\epsilon$ (37)
and a simplicial prime loop $\gamma:S^{1}\to X$ such that the ratio
$|\gamma|\cdot A_{X}(\gamma)^{-1}$ is less than the minimum of the numbers
$I(X^{\prime})$ for all proper subcomplexes $X^{\prime}\subset X$. Consider a
minimal spanning disc $b:D^{2}\to X$ for $\gamma=b|_{\partial D^{2}}$; here
$D^{2}$ is a triangulated disc and $b$ is a simplicial map. As we showed in
Remark A.4, the map $b$ is surjective. As explained in the proof of Lemma A.7,
due to arguments of Ronan [25], we may assume that $b$ has no foldings.
For any integer $i\geq 1$ we denote by $X_{i}\subset X$ the pure subcomplex
generated by all 2-simplexes $\sigma$ of $X$ such that the preimage
$b^{-1}(\sigma)\subset D^{2}$ contains $\geq i$ two-dimensional simplexes. One
has $X=X_{1}\supset X_{2}\supset X_{3}\supset\dots.$ Each $X_{i}$ may have
several connected components and we will denote by $\Lambda$ the set labelling
all the connected components of the disjoint union $\sqcup_{i\geq 1}X_{i}$.
For $\lambda\in\Lambda$ the symbol $X_{\lambda}$ will denote the corresponding
connected component of $\sqcup_{i\geq 1}X_{i}$ and the symbol
$i=i(\lambda)\in\\{1,2,\dots\\}$ will denote the index $i\geq 1$ such that
$X_{\lambda}$ is a connected component of $X_{i}$, viewed as a subset of
$\sqcup_{i\geq 1}X_{i}$. We endow $\Lambda$ with the following partial order:
$\lambda_{1}\leq\lambda_{2}$ iff $X_{\lambda_{1}}\supset X_{\lambda_{2}}$
(where $X_{\lambda_{1}}$ and $X_{\lambda_{2}}$ are viewed as subsets of $X$)
and $i(\lambda_{1})\leq i(\lambda_{2})$.
Next we define the sets
$\Lambda^{-}=\\{\lambda\in\Lambda;L(X_{\lambda})\leq 0\\}$
and
$\Lambda^{+}=\\{\lambda\in\Lambda;\mbox{for any $\mu\in\Lambda$ with
$\mu\leq\lambda$, }\,L(X_{\mu})>0\\}.$
Finally we consider the following subcomplex of the disk $D^{2}$:
$\displaystyle\Sigma^{\prime}=D^{2}-\bigcup_{\lambda\in\Lambda^{-}}{\rm{Int}}(b^{-1}(X_{\lambda}))$
(38)
and we shall denote by $\Sigma$ the connected component of $\Sigma^{\prime}$
containing the boundary circle $\partial D^{2}$.
Recall that for a 2-complex $X$ the symbol $f(X)$ denotes the number of
2-simplexes in $X$. We have
$\displaystyle f(D^{2})=\sum_{\lambda\in\Lambda}f(X_{\lambda}),$ (39)
and
$\displaystyle f(\Sigma)\leq
f(\Sigma^{\prime})=\sum_{\lambda\in\Lambda^{+}}f(X_{\lambda}).$ (40)
Formula (39) follows from the observation that any 2-simplex of $X=b(D^{2})$
contributes to the RHS of (26) as many units as its multiplicity (the number
of its preimages under $b$). Formula (40) follows from (39) and from the fact
that for a 2-simplex $\sigma$ of $\Sigma$ the image $b(\sigma)$ lies always in
the complexes $X_{\lambda}$ with $L(X_{\lambda})>0$.
###### Lemma A.8.
One has the following inequality
$\displaystyle\sum_{\lambda\in\Lambda^{+}}L(X_{\lambda})\leq|\partial D^{2}|.$
(41)
See [12], Lemma 6.8 for the proof.
Now we continue with the proof of Theorem A.2. Consider a tight pure 2-complex
$X$ satisfying (37) and a simplicial loop $\gamma:S^{1}\to X$ as above. We
will use the notation introduced earlier. The complex $\Sigma$ is a connected
subcomplex of the disk $D^{2}$; it contains the boundary circle $\partial
D^{2}$ which we will denote also by $\partial_{+}\Sigma$. The closure of the
complement of $\Sigma$,
$N=\overline{D^{2}-\Sigma}\subset D^{2}$
is a pure 2-complex. Let $N=\cup_{j\in J}N_{j}$ be the strongly connected
components of $N$. Each $N_{j}$ is PL-homeomorphic to a disc and we define
$\partial_{-}\Sigma=\cup_{j\in J}\partial N_{j},$
the union of the circles $\partial N_{j}$ which are the boundaries of the
strongly connected components of $N$. It may happen that $\partial_{+}\Sigma$
and $\partial_{-}\Sigma$ have nonempty intersection. Also, the circles forming
$\partial_{-}\Sigma$ may not be disjoint.
We claim that for any $j\in J$ there exists $\lambda\in\Lambda^{-}$ such that
$b(\partial N_{j})\subset X_{\lambda}$. Indeed, let
$\lambda_{1},\dots,\lambda_{r}\in\Lambda^{-}$ be the minimal elements of
$\Lambda^{-}$ with respect to the partial order introduced earlier. The
complexes $X_{\lambda_{1}},\dots,X_{\lambda_{r}}$ are connected and pairwise
disjoint and for any $\lambda\in\Lambda^{-}$ the complex $X_{\lambda}$ is a
subcomplex of one of the sets $X_{\lambda_{i}}$, where $i=1,\dots,i$. From our
definition (38) it follows that the image of the circle $b(\partial N_{j})$ is
contained in the union $\cup_{i=1}^{r}X_{\lambda_{i}}$ but since $b(\partial
N_{j})$ is connected it must lie in one of the sets $X_{\lambda_{i}}$.
We may apply Lemma A.7 to each of the circles $\partial N_{j}$. We obtain that
each of the circles $\partial N_{j}$ admits a spanning discs of area $\leq
K_{\epsilon}|\partial N_{j}|$, where $K_{\epsilon}=C^{\prime-1}_{\epsilon}$ is
the inverse of the constant given by Lemma A.7. Using the minimality of the
disc $D^{2}$ we obtain that the circles $\partial N$ bound in $D^{2}$ several
discs with the total area $A\leq K_{\epsilon}\cdot|\partial_{-}\Sigma|.$
For $\lambda\in\Lambda^{+}$ one has $L(X_{\lambda})\geq 1$ and
$\chi(X_{\lambda})\leq 1$ (since $b_{2}(X_{\lambda})=0$); in particular,
$e(X_{\lambda})\geq f(X_{\lambda})$. Hence we have
$4L(X_{\lambda})\geq 3\chi(X_{\lambda})+L(X_{\lambda})\geq 3\epsilon
e(X_{\lambda})\geq 3\epsilon f(X_{\lambda})$
where on the second last inequality we used the inequality
$\nu(X_{\lambda})\geq 1/3+\epsilon$. Summing up we get
$f(\Sigma)\leq\sum_{\lambda\in\Lambda^{+}}f(X_{\lambda})\leq\frac{4}{3\epsilon}\sum_{\lambda\in\Lambda^{+}}L(X_{\lambda})\leq\frac{4}{3\epsilon}|\partial
D^{2}|.$
The rightmost inequality is given by Lemma A.8.
Next we observe, that
$\displaystyle|\partial_{-}\Sigma|\leq 2f(\Sigma)+|\partial_{+}\Sigma|.$ (42)
Therefore, we obtain
$\displaystyle f(D^{2})$ $\displaystyle\leq$ $\displaystyle
f(\Sigma)+A\,\leq\,\frac{4}{3\epsilon}|\gamma|+K_{\epsilon}\cdot 2\cdot
f(\Sigma)+K_{\epsilon}|\gamma|$ $\displaystyle\leq$
$\displaystyle\left(\frac{4}{3\epsilon}(1+2K_{\epsilon})+K_{\epsilon}\right)\cdot|\gamma|,$
implying
$\displaystyle I(X)\geq\frac{3\epsilon}{4+8K_{\epsilon}+3\epsilon
K_{\epsilon}}.$ (43)
This completes the proof of Theorem A.2. ∎
## References
* [1] J. F. Adams, A new proof of a theorem of W. H. Cockcroft, J. London Math. Soc. 30 (1955), 482 488.
* [2] L. Aronshtam, N. Linial, T. Łuczak, R. Meshulam, Vanishing of the top homology of a random complex, Discrete & Computational Geometry 49(2013), pp 317–334.
* [3] S. Antoniuk, T. Łuczak, J. Świa̧tkowski, Random triangular groups at density 1/3, preprint arXiv:1308.5867v2.
* [4] E. Babson, C. Hoffman, M. Kahle, The fundamental group of random $2$-complexes, J. Amer. Math. Soc. 24 (2011), 1-28. See also the latest archive version arXiv:0711.2704 revised on 20.09.2012.
* [5] E. Babson, Fundamental groups of random clique complexes, arXiv:1207.5028v2
* [6] M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445 -470.
* [7] W. A. Bogley, J.H.C. Whitehead’s asphericity question, in: ”Two-dimensional Homotopy and Combinatorial Group Theory”, eds. C. Hog-Angeloni, A. Sieradski and W. Metzler, LMS Lecture Notes 197, Cambridge Univ Press (1993), 309-334.
* [8] W. Brown, Enumeration of triangulations of the disk, Proc. London Math. Soc. (3)14 (1964), 746-768.
* [9] W.H. Cockcroft, On two-dimensional aspherical complexes, Proc. London Math. Soc. 4(1954), 375-384.
* [10] D. Cohen, A.E. Costa, M. Farber, T. Kappeler, Topology of random 2-complexes, Journal of Discrete and Computational Geometry, 47(2012), 117-149.
* [11] A.E. Costa, M. Farber, The asphericity of random 2-dimensional complexes, to appear in Random Structures and Algorithms, preprint arXiv:1211.3653v1
* [12] A.E. Costa, M. Farber, Geometry and topology of random 2-complexes, arXiv:1307.3614.
* [13] P. Erdős, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61.
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* [15] M. Gromov, Filling Riemannian manifolds, J. Differential Geometry, 18(1983), 1–147.
* [16] M. Gromov, Hyperbolic groups, in Essays in group theory, ed. S. M. Gersten, Springer (1987), 75-265.
* [17] N. Hartdfield and G. Ringel, Clean triangulations, Combinatorica 11(2)(1991), pp. 145 - 155.
* [18] A. Hatcher, Alegbraic Topology, Cambridge University Press, Cambridge 2002.
* [19] S. Janson, T. Łuczak, A. Ruciński, Random graphs, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley-Interscience, New York, 2000.
* [20] M. Kahle, Topology of random clique complexes, Discrete Math. 309 (2009), no. 6, 1658 – 1671. MR MR2510573
* [21] M. Kahle, Sharp vanishing thresholds for cohomology of random flag complexes, Ann. of Math. (2) 179 (2014), no. 3, 1085 1107.
* [22] M. Kahle and E. Meckes, Limit theorems for Betti numbers of random simplicial complexes, Homology Homotopy Appl. 15 (2013), no. 1, 343 – 374.
* [23] M. Kahle and B. Pittel, Inside the critical window for cohomology of random k-complexes, To appear in Random Structures Algorithms, arXiv:1301.1324.
* [24] M. Kahle, Topology of random simplicial complexes: a survey, To appear in AMS Contemporary Volumes in Mathematics. Nov 2014. arXiv:1301.7165.
* [25] M.A. Ronan, On the second homotopy group of certain simplicial complexes and some combinatorial applications, Quart. J. Math. 32(1981), 225 - 233.
* [26] S. Rosenbrock, The Whitehead Conjecture - an overview, Siberian Electronoc Mathematical Reports, 4(2007), 440-449.
* [27] R. G. Swan, “Groups of cohomological dimension one”. Journal of Algebra 12 (1969), 585 610
A. Costa and M. Farber:
School of Mathematical Sciences, Queen Mary University of London, London E1
4NS
D. Horak: Theoretical Systems Biology, Impreial College, London, SW7 2AZ
|
arxiv-papers
| 2013-12-04T15:31:23 |
2024-09-04T02:49:54.847072
|
{
"license": "Public Domain",
"authors": "Armindo Costa, Michael Farber and Danijela Horak",
"submitter": "Michael Farber",
"url": "https://arxiv.org/abs/1312.1208"
}
|
1312.1216
|
# Complete non-compact gradient Ricci solitons with nonnegative Ricci
curvature
Yuxing Deng and Xiaohua $\text{Zhu}^{*}$ Yuxing Deng
School of Mathematical Sciences, Peking University, Beijing, 100871, China
Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
[email protected]
###### Abstract.
In this paper, we give a delay estimate of scalar curvature for a complete
non-compact expanding (or steady) gradient Ricci soliton with nonnegative
Ricci curvature. As an application, we prove that any complete non-compact
expanding (or steady) gradient Kähler-Ricci solitons with positively pinched
Ricci curvature should be Ricci flat. The result answers a question in case of
Kähler-Ricci solitons proposed by Chow, Lu and Ni in a book.
###### Key words and phrases:
Ricci soliton, Ricci flow, pinched Ricci curvature
###### 2000 Mathematics Subject Classification:
Primary: 53C25; Secondary: 53C55, 58J05
* Partially supported by the NSFC Grants 11271022 and 11331001
##
## 1\. Introduction
Ricci soliton plays an important role in the study of Hamilton’s Ricci flow,
in particular in the singularities analysis of Ricci flow [15], [3], [21]. In
case of shrinking gradient Ricci solitons with positive curvature, Hamilton
proved that the solitons should be isometric to a standard sphere in
$\mathbb{R}^{3}$ in two dimensional case [15]. Perelman generalized Hamilton’s
result to three dimensional case [21]. Later on, Nabor proved that any four-
dimensional shrinking gradient Ricci soliton with positive bounded curvature
operator should be a standard sphere in $\mathbb{R}^{5}$ [17]. On the other
hand, Perelman and Brendle proved that any steady gradient Ricci soliton with
nonnegative sectional curvature should be a Bryant’s soliton in case of
2-dimension and 3-dimension, respectively [21], [6], [2], [1]. However, to
author’s acknowledge, there is rarely understanding in case of expanding
gradient Ricci solitons even for lower dimensional manifolds. For example, how
to classify complete non-compact gradient expanding (or steady) Ricci solitons
under a suitable curvature condition. The purpose of this paper is to give a
rigidity theorem for a class of expanding (or steady) gradient Kähler-Ricci
solitons with nonnegative Ricci curvature.
###### Definition 1.1.
A complete Riemannian metric $g$ on $M$ is called a gradient Ricci soliton if
there exists a smooth function $f$ ( which is called a defining function) on
$M$ such that
(1.1) $R_{ij}+\rho g_{ij}=\nabla_{i}\nabla_{j}f,$
where $\rho\in\mathbb{R}$ is a constant. The gradient Ricci soliton is called
expanding, steady and shrinking according to the sign $\rho>,=,<0$,
respectively.
For simplicity, we normalize $\rho=1,0,-1$. In addition, $g$ is a Kähler
metric on a complex manifold $M$, we call $g$ is a Kähler-Riccoi soliton.
Since $\overline{\partial}f$ induces a holomorphic vector field on $M$, (1.1)
was usually written in a complex version,
(1.2) $R_{i\bar{j}}+\rho g_{i\bar{j}}=\nabla_{i}\nabla_{\bar{j}}f,$
A gradient soliton $(M,g,f)$ is called complete if $g$ and $\nabla f$ are both
complete. It is known that the completeness of $(M,g)$ implies the
completeness of $\nabla f$ [26]. Throughout this paper, we always assume the
soliton is complete. If there is a point $o\in M$ such that $\nabla f(o)=0$,
we call $o$ an equilibrium point of $(M,g)$. By studying the existence of
equilibrium points, we prove the boundedness of scalar curvature of $g$.
###### Theorem 1.2.
Let $(M,g)$ be a complete non-compact expanding gradient Ricci soliton with
nonnegative Ricci curvature or a complete non-compact steady gradient Kähler-
Ricci soliton with nonnegative bisectional curvature and positive Ricci
curvature. Then the scalar curvature of $g$ is bounded and it attains the
maximum at the unique equilibrium point.
The proof of Theorem 1.2 will be given in case of expanding Ricci solitons in
next section. For the steady Ricci solitons, the proof for the existence of
equilibrium points is a bit different, although the boundedness of scalar
curvature is directly from an identity (4.2). We will use a result of local
convergence for Kähler-Ricci flow by Chau and Tam to prove the existence in
Section 5 [11].
Theorem 1.2 will be applied to prove the following rigidity theorem for
Kähler-Ricci solitons with nonnegative Ricci curvature.
###### Theorem 1.3.
Let $(M^{n},g)$ be a complete non-compact gradient Kähler-Ricci soliton with
non-negative Ricci curvature. Suppose that there exists a point $p\in M$ such
that the scalar curvature $R$ of $g$ satisfies
(1.3) $\displaystyle\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dr}\leq\frac{\varepsilon(r)}{1+r^{2}},~{}\text{if}~{}g~{}\text{ is
expending};$
or
(1.4) $\displaystyle\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dr}\leq\frac{\varepsilon(r)}{1+r},~{}\text{if}~{}g~{}\text{ is steady},$
where $\varepsilon(r)\rightarrow 0$ as $r\to\infty$. Then $g$ is Ricci-flat.
Moreover, $(M,g)$ is isometric to $\mathbb{C}^{n}$ if $g$ is expending.
We note that under the condition of nonnegative sectional curvature (or
nonnegative holomorphic bisectional curvature for Kähler manifolds) several
rigidity theorems were obtained in [15], [19], [22], etc. For Ricci solitons,
we are able to use the Ricci flow to weaken the condition of curvature to
nonnegative Ricci curvature.
As a corollary, we obtain a version of Theorem 1.3 under the pointed-wise
Ricci decay condition.
###### Theorem 1.4.
Let $(M^{n},g)$ be a complete non-compact gradient Kähler-Ricci soliton with
non-negative Ricci curvature. Suppose that $g$ satisfies
(1.5) $\displaystyle
R(x)\leq\frac{\varepsilon(r(x))}{1+r(x)^{2}}~{}(\varepsilon(r)\rightarrow
0,\text{ as}~{}r\to\infty),~{}\text{if}~{}g~{}\text{ is expending};$
or
(1.6) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{2n+\epsilon}}~{}\text{for
some}~{}C,\epsilon>0,~{}\text{if}~{}g~{}\text{ is steady}.$
Then $g$ is Ricci-flat. Moreover, $(M,g)$ is isometric to $\mathbb{C}^{n}$ if
$g$ is expending.
In case of steady solitons in Theorem 1.4, if we assume that $(M,g)$ has
nonnegative bisectional curvature instead of nonnegative Ricci curvature, then
the condition (1.6) can be weakened as
(1.7) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{1+\epsilon}}.$
In fact, we can prove that $(M,g)$ is isometric to $\mathbb{C}^{n}$ by Theorem
1.2, see Proposition 5.1. Proposition 5.1 is an analogy of Hamilton’s result
for Kähler manifolds [13].
A Riemannian metric is called with property of positively pinched Ricci
curvature if there is a uniform constant $\delta>0$ such that
$\text{Ric}(g)\geq\delta Rg$ [13], [15]. It was proved that the scalar
curvature of complete non-compact expanding (or steady) gradient Ricci
solitons with positively pinched Ricci curvature has exponential decay (cf.
Theorem 9.56, [6]). Thus as a direct consequence of Theorem 1.3, we obtain
###### Corollary 1.5.
Non-trivial complete non-compact expanding or steady gradient Kähler-Ricci
soliton with positively pinched Ricci curvature doesn’t exist for $n\geq 2$.
Corollary 1.5 answers a question in case of Kähler-Ricci solitons proposed by
Chow, Lu and Ni in their book [6] (cf. page 390). They asked whether there
exists an expanding gradient Ricci soliton with positively pinched Ricci
curvature when $n\geq 3$.
Theorem 1.3 and 1.4 will be proved in Section 4 and Section 5 according to
expending or steady solitons, respectively.
## 2\. Boundedness of scalar curvature–I
In this section, we prove the boundedness of scalar curvature in case of
expending Ricci solitons. Let $(M^{n},g)$ be a Riemannian manifold. In local
coordinates $(x^{1},x^{2},\cdots,x^{n})$, curvature tensor Rm of $g$ is
defined by
${\rm Rm}(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}})\frac{\partial}{\partial x^{k}}\triangleq\sum
R^{l}_{ijk}\frac{\partial}{\partial x^{l}}$
and $R_{ijkl}\triangleq\sum g_{lm}R_{ijk}^{m}$. Then the Ricci curvature is
given by
$R_{jk}=\sum R_{ijk}^{i}.$
Thus by the commutation formula,
(2.1)
$\displaystyle(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i})\alpha_{k_{1}\cdots
k_{r}}=-\sum_{l=1}^{r}R^{m}_{ijk_{l}}\alpha_{k_{1}\cdots k_{l-1}mk_{l+1}\cdots
k_{r}},$
we get from the Bianchi identity,
(2.2) $2\sum\nabla_{i}R_{ij}=\nabla_{j}R.$
Let $(M^{n},g,f)$ be an expanding gradient Ricci soliton and $\phi_{t}$ be a
family of diffeomorphisms generated by $-\nabla f$. Then the induced metrics
$g(t)=\phi_{t}^{*}g$ satisfy
(2.3) $\frac{\partial}{\partial t}g=-2\text{Ric}(g)-2g.$
(2.3) is equivalent to
(2.4) $R_{ij}(t)+g_{ij}(t)=\nabla_{i}\nabla_{j}f(t),$
where $f(t)=\phi_{t}^{*}f$ and $\nabla$ is taken w.r.t $g(t)$.
###### Lemma 2.1.
$\frac{\partial}{\partial t}R=2{\rm Ric}(\nabla f(t),\nabla f(t)).$
###### Proof.
Differentiating $(\ref{expanding-soliton})$ on both sides, we have
$\displaystyle\nabla_{k}R_{ij}=\nabla_{k}\nabla_{i}\nabla_{j}f.$
It follows from (2.1),
$\displaystyle\nabla_{i}R_{jk}-\nabla_{j}R_{ik}=-\sum R_{ijkl}\nabla_{l}f.$
Thus by (2.2), we get
(2.5) $\displaystyle\nabla_{j}R=-2R_{jl}\nabla_{l}f.$
Hence
$\displaystyle\frac{\partial}{\partial t}R(x,t)=\frac{\partial}{\partial
t}R(\phi_{t}(x),0)=-\langle\nabla R,\nabla f\rangle=2\text{Ric}(\nabla
f,\nabla f).$
∎
Let $B_{r}(o,t)$ be a $r$-geodesic ball centered at $o\in M$ w.r.t $g(t)$.
Then
###### Lemma 2.2.
Let $g(x,t)$ be a solution of (2.3) with nonnegative Ricci curvature for any
$t\in(0,\infty)$. Then for any $r>0$ and $\delta>0$, there exists a
$T_{0}=T_{0}(r,\delta)>0$ such that $B_{r}(o,0)\subset B_{\delta}(o,t)$ for
any $t\geq T_{0}$.
###### Proof.
By (2.3), it is easy to see that
$\displaystyle\frac{{\rm d}|v|^{2}_{t}}{{\rm
d}t}\leq-2|v|^{2}_{t},~{}\forall~{}t\geq 0,$
where $v\in T_{p}^{(1,0)}M$ for any $p\in M$. Then
$|v|_{t}^{2}\leq e^{-2t}|v|_{0}^{2}.$
Connecting $o$ and $p$ by a minimal geodesic curve $\gamma(s)$ with an arc-
parameter $s$ w.r.t the metric $g(x,0)$ in $B_{r}(o,0)$, we get
(2.6) $d_{t}(o,p)\leq\int_{0}^{l}|\gamma^{\prime}(s)|_{t}{\rm
ds}\leq\int_{0}^{l}|\gamma^{\prime}(s)|_{0}e^{-2t}{\rm ds}\leq re^{-2t},$
where $l$ is the length of $\gamma(s)$. Therefore, by taking $t$ large enough.
we see that $B_{r}(o,1)\subset B_{\delta}(o,t)$. ∎
Taking an integration along a geodesic curve on both sides of (1.1), on can
show that $f(x)\geq\frac{r(x)^{2}}{4}$ under the assumption of nonnegative
Ricci curvature. This implies that $f(x)$ attains the minimum at some point
$o\in M$. Thus $\nabla f(o)=0$. Moreover the equilibrium point $o$ is unique.
This is because, if there is another equilibrium point $p$, then
$\phi_{t}(o)=o$ and $\phi_{t}(p)=p$. In particular $d_{t}(o,p)=d_{0}(o,p)$. On
the other hand, by (2.6), $d_{t}(o,p)\leq e^{-2t}d_{0}(o,p),~{}\forall~{}t>0$.
Hence, $d_{0}(o,p)=0$, and consequently $o=p$.
Now we begin to prove Theorem 1.2.
###### Proof of Theorem 1.2 (the expanding case).
Let $o$ be the unique equilibrium point. Then by Lemma 2.2, for any
$r>\delta>0$, there exists $T_{0}$ such that
$B_{r}(o,0)\subset B_{\delta}(o,t),\mbox{\quad}\forall\mbox{\ }t\geq T_{0}.$
On the other hand, by Lemma 2.1, we see that $R(x,t)$ is nondecreasing in $t$.
Thus
(2.7) $\sup_{x\in B_{r}(o,0)}R(x,0)\leq\sup_{x\in
B_{\delta}(o,t)}R(x,t),\mbox{\quad}\forall\mbox{\ }r>0,\mbox{\ }\delta>0.$
Note that $\phi_{t}:(M^{n},g(t))\rightarrow(M^{n},g(0))$ are a family of
isometric deformations. It follows
(2.8) $\sup_{x\in B_{\delta}(o,0)}R(x,0)=\sup_{x\in
B_{\delta}(o,t)}R(x,t),\mbox{\quad}\forall\mbox{\ }\delta>0.$
Hence, combining $(\ref{T1-1})$ and $(\ref{T1-2})$, we get
$\sup_{x\in B_{r}(o,0)}R(x,0)=\sup_{x\in
B_{\delta}(o,0)}R(x,0),\mbox{\quad}\forall\mbox{\ }r>\delta>0.$
Let $\delta\rightarrow 0$, we derive
$\sup_{x\in B_{r}(o,0)}R(x,0)=R(o,0),\mbox{\quad}\forall\mbox{\ }r>0.$
This proves the theorem. ∎
## 3\. Expanding Kähler-Ricci solitons
In this section, we prove both Theorem 1.3 and Theorem 1.4 in case of
expanding Kähler-Ricci solitons. Theorem 1.4 is a consequence of Theorem 1.3
by the following lemma.
###### Lemma 3.1.
Let $(M,g)$ be an expanding gradient Kähler-Ricci soliton which satisfies
(1.5) in Theorem 1.4. Then there exists a function $\varepsilon^{\prime}(r)$ (
$\varepsilon^{\prime}(r)\to 0$ as $r\to\infty$) such that
$\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dv}\leq\frac{\varepsilon^{\prime}(r)}{1+r^{2}}.$
###### Proof.
Note that an expanding Ricci soliton with nonnegative Ricci curvature has
maximal volume growth (cf. [7] or [12]). Namely, there exists a uniform
constant $\delta>0$ such that
$\text{vol}(B_{r}(p))\geq\delta r^{2n}.$
On the other hand, by the volume comparison theorem, we have
$\text{vol}(\partial B_{r}(p))\leq n\frac{\text{vol}(B_{r}(p))}{r}\leq
Cr^{2n-1},$
where $C$ is a uniform constant. Thus
$\displaystyle\frac{1}{\text{vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm dv}=$
$\displaystyle\frac{1}{\text{vol}(B_{r}(p))}\int_{0}^{r}{\rm ds}\int_{\partial
B_{s}(p)}R{\rm d\sigma}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\delta
r^{2n}}\int_{0}^{r}\frac{\varepsilon(s)}{1+s^{2}}\text{vol}(\partial
B_{s}(p)){\rm ds}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\delta
r^{2n}}\int_{0}^{r}C(s+1)^{2n-3}\varepsilon(s){\rm ds}$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon^{\prime}(r)}{1+r^{2}},$
where $\varepsilon^{\prime}(r)\rightarrow 0$ as $r\rightarrow 0$. ∎
###### Proof of Theorem 1.3 (the expanding case).
Ricci-Flat: Let $\phi_{t}$ be a family of diffeomorphisms generated by
$-\nabla f$. Let $g(t)=\phi_{t}^{*}g$ and $\widehat{g}(\cdot,t)=tg(\cdot,\ln
t)$. Then $\widehat{g}(\cdot,t)$ satisfies
(3.1) $\left\\{\begin{aligned} \frac{\partial}{\partial
t}\widehat{g}_{i\bar{j}}(x,t)&=-\widehat{R}_{i\bar{j}}(x,t)\\\
\widehat{g}_{i\bar{j}}(x,1)&=g_{i\bar{j}}(x).\end{aligned}\right.$
Let
$F(x,t)=\ln\det(\widehat{g}_{i\bar{j}}(x,t))-\ln\det(\widehat{g}_{i\bar{j}}(x,1))$.
By $(\ref{kr-flow})$, it is easy to see
$\displaystyle F(x,t)=-\int_{1}^{t}\widehat{R}(x,s)\rm{ds}\leq 0.$
Since
$t\widehat{R}(o,t)=R(o,\ln t)=R(o,0),~{}\text{and}~{}t\widehat{R}(x,t)=R(x,\ln
t)\leq R(o,0),$
where $o$ is the equilibrium point of $M$, by Theorem 1.2, $F$ is uniformly
bounded on $x$. Moreover, we have
(3.2) $\displaystyle M(t)\doteq-\inf_{x\in M}F(x,t)=R(o,0)\ln t.$
In the following, we shall estimate the upper bound of $M(t)$ by using the
Green integration as in [24] (also see [18]).
By a direct computation, we have
(3.3)
$\Delta_{1}F(x,t)=\widehat{R}(x,1)-g^{i\bar{j}}(x,1)\widehat{R}_{i\bar{j}}(x,t)\leq\widehat{R}(x,1)+\frac{\partial}{\partial
t}e^{F(x,t)},$
where the Lapalace $\Delta_{1}$ is w.r.t $\widehat{g}(x,1)$. Let $G_{r}(x,y)$
be a positive Green’s function with zero boundary value w.r.t
$\widehat{g}(x,1)$ on $\hat{B}_{r}(x,1)$. Note that
$\int_{\hat{B}_{r}(x_{0},1)}\frac{\partial G_{r}(x_{0},y)}{\partial\nu}{\rm
ds}=-1\text{ and}~{}\frac{\partial G_{r}(x_{0},y)}{\partial\nu}\leq 0.$
By integrating (3.3) on both sides, we have
$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$
$\displaystyle\leq$ $\displaystyle\int_{1}^{t}{\rm
ds}\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(-\Delta_{1}F(x,s)){\rm dv}$
$\displaystyle+t\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ $\displaystyle=$
$\displaystyle\int_{1}^{t}\Big{(}F(x_{0},s)+\int_{\hat{B}_{r}(x_{0},1)}\frac{\partial
G_{r}(x_{0},y)}{\partial\nu}F(x,s){\rm dv}\Big{)}{\rm ds}$
$\displaystyle+t\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ (3.4) $\displaystyle\leq$ $\displaystyle
t\Big{(}M(t)+\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\Big{)}.$
On the other hand, by the Green function estimate (cf. Lemma 1.1 in [25]),
$G_{r}(x,y)\geq
C_{1}^{-1}\int_{d(x,y)}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x,1))}{\rm
ds},\mbox{\quad}\forall\mbox{\ }y\in\hat{B}_{\frac{r}{5}}(x,1),$
where $C_{1}$ is a uniform constant, we get
$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$
$\displaystyle\geq$ $\displaystyle
C_{1}^{-1}\int_{0}^{\frac{r}{5}}\Big{(}\int_{\tau}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}{\rm
ds}\Big{)}\Big{(}\int_{\partial\hat{B}_{\tau}(x_{0},1)}(1-e^{F(x,t)}){\rm
d\sigma}\Big{)}{\rm d\tau}$ $\displaystyle\geq$ $\displaystyle
C_{1}^{-1}\int_{0}^{\frac{r}{5}}\Big{(}\int_{\frac{r}{5}}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}{\rm
ds}\Big{)}\Big{(}\int_{\partial\hat{B}_{\tau}(x_{0},1)}(1-e^{F(x,t)}){\rm
d\sigma}\Big{)}{\rm d\tau}$ $\displaystyle\geq$
$\displaystyle\frac{C_{2}^{-1}r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(1-e^{F(x,t)}){\rm
dv}$ $\displaystyle\geq$
$\displaystyle\frac{eC_{2}^{-1}r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}\frac{-F(x,t)}{1-F(x,t)}{\rm
dv}.$
It follows
$\displaystyle\frac{r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ $\displaystyle\leq
C_{3}(1+M(t))\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$
Hence by (3), we derive
$\displaystyle\frac{r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ (3.5) $\displaystyle\leq
C_{3}t(1+M(t))\Big{(}M(t)+\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\Big{)}.$
By $(\ref{T2-1})$, we have $\Delta_{1}(-F(x,t))\geq-\hat{R}(x,1)$. Then by the
mean value inequality (cf. Lemma 2.1 of [18]), we see
$\displaystyle-F(x_{0},t)\leq\frac{C(n)}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ (3.6)
$\displaystyle+\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}G_{\frac{r}{5}}(x_{0},y)\widehat{R}(y,1){\rm
dv}.$
Hence, to get an upper bound of $-F(x_{0},t)$, by $(\ref{inequality-2})$ and
$(\ref{inequality-3})$, we shall estimate the integral
$\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}G_{\frac{r}{5}}(x_{0},y)\widehat{R}(y,1)\rm{dv}$.
Recall the Li-Yau’s estimate for the Green function: There exits a positive
Green’s function $G(x,y)$ such that (cf. Theorem 5.2 in [16])
(3.7) $\displaystyle C(n)^{-1}\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\leq G(x,y)\leq
C(n)\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}.$
Then
$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\leq\int_{0}^{r}{\rm
ds}\int_{\partial\hat{B}_{s}(x_{0},1)}G(x_{0},y)\widehat{R}(y,1){\rm d\sigma}$
$\displaystyle\leq$ $\displaystyle C(n)\int_{0}^{r}{\rm
ds}\Big{(}\int_{\partial\hat{B}_{s}(x_{0},1)}\widehat{R}(y,1){\rm
d\sigma}\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\Big{)}$ $\displaystyle=$
$\displaystyle C(n)\Big{(}\int_{r^{2}}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\int_{\hat{B}_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (3.8) $\displaystyle+$
$\displaystyle\int_{0}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}\int_{\hat{B}_{s}(x_{0},1)}\widehat{R}{\rm
dvds}\Big{)},\mbox{\quad}\forall\mbox{\ }r>0.$
Since $(M,g)$ has the maximal volume growth, there exists a uniform constant
$\delta>0$ such that
$\frac{\text{vol}(B_{s}(x))}{\text{vol}(B_{t}(x))}\geq\delta\Big{(}\frac{s}{t}\Big{)}^{2n},~{}\forall~{}s\geq
t\geq c_{0}.$
It follows
$\displaystyle\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(B_{\sqrt{t}}(x))}\leq
C_{4}\frac{d^{2}(x,y)}{\text{vol}(B_{d(x,y)}(x))}\mbox{,\quad}\forall\mbox{\
}d(x,y)\geq c_{0}.$
Hence, we get from (3),
$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ $\displaystyle\leq
C(n)\Big{(}C_{4}\frac{r^{2}}{\text{vol}(\hat{B}_{r}(x_{0},1)}\int_{\hat{B}_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (3.9)
$\displaystyle+\int_{0}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1)}\int_{\hat{B}_{s}(x_{0},1)}\widehat{R}{\rm
dvds}\Big{)},\mbox{\quad}\forall\mbox{\ }r>0.$
By the volume comparison theorem and the condition (1.5), we have
$\displaystyle\frac{1}{\text{vol}(B_{r}(o))}\int_{B_{r}(o)}R(x){\rm dv}\leq$
$\displaystyle\frac{\text{vol}(B_{r+d}(p))}{\text{vol}(B_{r}(o))}\cdot\frac{1}{\text{vol}(B_{r+d}(p))}\int_{B_{r+d}(p)}R(x)\rm{dv}$
$\displaystyle\leq$
$\displaystyle\frac{\text{vol}(B_{r+2d}(o))}{\text{vol}(B_{r}(o))}\cdot\frac{\varepsilon(r+d)}{1+(r+d)^{2}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{r+2d}{r}\Big{)}^{2n}\frac{\varepsilon(r+d)}{1+(r+d)^{2}},$
where $d=d(o,p)$. Then there exists another function $\varepsilon^{\prime}(r)$
($\varepsilon^{\prime}(r)\to 0$ as $r\to\infty$) such that
$\frac{1}{\text{vol}(B_{r}(o))}\int_{B_{r}(o)}R(x){\rm
dv}\leq\frac{\varepsilon^{\prime}(r)}{1+r^{2}}.$
Thus inserting the above inequality into (3), we derive at $x_{0}=o$,
(3.10) $\int_{\hat{B}_{r}(o,1)}G_{r}(o,y)\widehat{R}(y,1){\rm
dv}\leq\varepsilon^{\prime\prime}(r)+\varepsilon^{\prime\prime}(r)\ln(1+r^{2}),$
where $\varepsilon^{\prime\prime}(r)\to 0$ as $r\to\infty$.
Combining $(\ref{inequality-1})$, $(\ref{inequality-2})$ and
$(\ref{inequality-5})$, it is easy to see
$\displaystyle-F(o,t)\leq
r^{-2}C(n)t(M(t)+1)\Big{(}M(t)+C_{5}\varepsilon^{\prime\prime}(r)+C_{6}\varepsilon^{\prime\prime}(r)\ln(1+r^{2})\Big{)}$
$\displaystyle+\Big{(}C_{7}\varepsilon^{\prime\prime}(r)+C_{8}\varepsilon^{\prime\prime}(r)\ln(1+r^{2})\Big{)},\mbox{\quad}\forall\mbox{\
}r>0.$
Note that $-F(o,t)=M(t)=R(o,0)\ln t$. Then by taking $r=t$, we obtain
$\displaystyle R(o,0)\ln t\leq$ $\displaystyle
C_{9}\varepsilon^{\prime\prime}(t)+C_{10}\varepsilon^{\prime\prime}(t)\ln(1+t)+C_{11}\frac{\ln
t}{t},\mbox{\quad}\forall\mbox{\ }t\geq 1.$
Dividing by $\ln t$ on both sides of the above inequality and letting
$t\rightarrow\infty$, we deduce $R(o,0)=0$. Hence we prove that $g$ is Ricci
flat
Flatness: We shall further prove that $g$ is a flat metric on
$\mathbb{C}^{n}$. Note that
(3.11) $\displaystyle g=\text{hess}f$
since $g$ is Ricci flat. Then $f$ is strictly convex and $f$ attains the
minimum at $o$. By a direct computation, we have
$\displaystyle\langle\nabla f,X\rangle=(\nabla{\rm d}f)(\nabla
f,X)=\langle\nabla_{\nabla f}\nabla f,X\rangle,\mbox{\ }\forall\mbox{\
}X\in\Gamma^{\infty}(TM).$
It follows
$\nabla_{\nabla f}\nabla f=\nabla f.$
Thus
$\nabla_{\frac{\nabla f}{|\nabla f|}}\Big{(}\frac{\nabla f}{|\nabla
f|}\Big{)}=\frac{1}{|\nabla f|}(\frac{\nabla_{\nabla f}\nabla f}{|\nabla
f|}-\frac{\langle\nabla_{\nabla f}\nabla f,\nabla f\rangle}{|\nabla
f|^{3}}\nabla f)=0,~{}x\in M\setminus\\{o\\}.$
This implies that any integral curve generated by $\frac{\nabla f}{|\nabla
f|}$ ($x\in M\setminus\\{o\\}$) is geodesic.
Let $\phi_{t}$ and $\varphi_{t}$ be one-parameter diffeomorphisms groups
generated by $-\nabla f$ and $-\frac{\nabla f}{|\nabla f|}$, respectively.
Then as in the proof of (2.6), we have $d(\phi_{t}(x),o)=e^{-t}d(x,o)$. Thus
$\langle\nabla f,\nabla r\rangle=-\frac{{\rm d}}{{\rm dt}}d(\phi_{t}(x),o)>0$.
This shows that $\varphi_{s}(x)$ is a geodesic curve from $x$ to $o$. Let
$\gamma(s)=\varphi_{d(x,o)-s}(x)$. Then $\gamma(s)$ is a minimal geodesic
curve from $o$ to $x$ as long as $\text{dist}(o,x)\leq r_{0}<<1.$ Moreover, we
have
$\displaystyle\left\\{\begin{aligned} &\frac{{\rm d^{2}}}{{\rm
ds^{2}}}f(\gamma(s))=1,\\\ &\frac{{\rm d}}{{\rm ds}}f(\gamma(s))=|\nabla
f|\rightarrow 0,\mbox{\ as\ }s\rightarrow 0,\\\
&f(\gamma(0))=f(o)=0.\end{aligned}\right.$
Therefore, we deduce
$\displaystyle f(x)=\frac{1}{2}r^{2}(x),~{}\text{if}~{}r(x)\leq r_{0}.$
In particular,
(3.12) $\displaystyle|\nabla f(x)|=r.$
We claim that $g$ is flat on $B_{r_{0}}(o)$. Since $\partial B_{r_{0}}(o)$ is
diffeomorphic to $\mathbb{S}^{2n-1}$, we can choose an orthonormal basis
$\\{e_{1},\cdots,e_{2n-1}\\}$ on $\partial B_{r_{0}}(o)$. Let
$X_{i}(\varphi_{t}(x))=(\varphi_{t})_{*}e_{i}$ for $x\in\partial
B_{r_{0}}(o)$, $1\leq i\leq 2n-1$. Then $\\{\nabla r=\frac{\nabla f}{|\nabla
f|},X_{1},\cdots,X_{2n-1}\\}$ is a global frame on
$B_{r_{0}}(o)\setminus\\{o\\}$. Clearly, $\mathscr{L}_{\nabla r}X_{i}=0$,
$1\leq i\leq 2n-1$. Thus by (3.12) and (3.11), it follows
$\displaystyle\frac{\partial}{\partial r}\langle\nabla r,X_{i}\rangle=$
$\displaystyle\mathscr{L}_{\nabla r}\langle\nabla r,X_{i}\rangle$
$\displaystyle=$ $\displaystyle(\mathscr{L}_{\nabla r}g)(\nabla
r,X_{i})+\langle\mathscr{L}_{\nabla r}\nabla r,X_{i}\rangle+\langle\nabla
r,\mathscr{L}_{\nabla r}X_{i}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{r}{\rm Hess}f(\nabla r,X_{i})$ $\displaystyle=$
$\displaystyle\frac{2}{r}\langle\nabla r,X_{i}\rangle.$
Since $\langle\nabla r,X_{i}\rangle|_{r=r_{0}}=0$, we get $\langle\nabla
r,X_{i}\rangle=0$ for any $x\in B_{r_{0}}(o)\setminus\\{o\\}$. Similarly, we
have
(3.13) $\left\\{\begin{aligned} \frac{\partial}{\partial r}\langle
X_{i},X_{j}\rangle=&\frac{2}{r}\langle X_{i},X_{j}\rangle,\\\ \langle\nabla
r,X_{i}\rangle|_{r=r_{0}}&=\delta_{ij}.\end{aligned}\right.$
Consequently, $\langle X_{i},X_{j}\rangle=\frac{r^{2}}{r_{0}^{2}}\delta_{ij}$
for any $x\in B_{r_{0}}(o)\setminus\\{o\\}$. Hence,
$g={\rm dr}\otimes{\rm dr}+\frac{r^{2}}{r_{0}^{2}}\sum_{i,j=1}^{2n-1}{\rm
d\theta^{i}}\otimes{\rm d\theta^{j}},$
where $\\{dr,\theta^{1},\cdots,\theta^{2n-1}\\}$ are the corresponding coframe
of $\\{\nabla r=\frac{\nabla f}{|\nabla f|},X_{1},$
$\cdots,X_{2n-1}\\}$. This proves that $g$ is isometric to an Euclidean metric
on $B_{r_{0}}(o)$. Therefore, $g$ is flat on $B_{r_{0}}(o)$. The claim is
true.
At last, we show that $g$ is globally flat. Since $\phi_{t}$ is an isometric
diffeomorphism from $(B_{r_{0}}(o,t),g(x,t))$ to $(B_{r_{0}}(o,0),g(x,0))$. We
see that $(B_{r_{0}}(o,t),g(x,t))$ is flat by the above claim. On the other
hand, by the flow $(\ref{normalized-ricci-flow})$ and the fact that $g$ is
Ricci-flat, we have $g(x,t)=e^{-2t}g(x,0)$. Hence, $(B_{r_{0}}(o,t),g(x,0))$
is also flat. Since $M$ is exhausted by $B_{r_{0}}(o,t)$ as
$t\rightarrow\infty$ according to Lemma 2.2, we see that $g$ is globally flat
and $M$ is simply connected. As a consequence, $(M,g)$ is isometric to
$\mathbb{C}^{n}$. ∎
## 4\. Steady Kähler-Ricci solitons
In this section, we deal with steady gradient Ricci solitons $(M^{n},g,f)$. As
in Section 3, we let $\phi_{t}$ be a family of diffeomorphisms generated by
$-\nabla f$ and $g(\cdot,t)=\phi_{t}^{*}g$. Then $g(\cdot,t)$ satisfies
(4.1) $\frac{\partial}{\partial t}g_{ij}=-2R_{ij}.$
It turns
$R_{ij}(t)=\nabla_{i}\nabla_{j}f(t),$
where $f(t)=\phi_{t}^{*}f$ and $\nabla$ is taken w.r.t $g(t)$. Hence by the
Bianchi identity (2.2), one can obtain
(4.2) $R+|\nabla f|^{2}=\text{const}.$
This shows that the scalar curvature of $g$ is uniformly bounded.
Analogous to Lemma 2.1, we have
###### Lemma 4.1.
$\frac{\partial}{\partial t}R=2{\rm Ric}(\nabla f(t),\nabla f(t))$.
In general, we do not know whether a steady gradient Ricci soliton admits an
equilibrium point. However, we can still prove Rigidity Theorem 1.3 in case of
steady gradient Kähler-Ricci solitons by using the fact of boundedness of
scalar curvature.
###### Proof of Theorem 1.3 (the steady case).
Since $(M,g)$ is Kählerian, we may rewrite (4.1) as,
(4.3) $\left\\{\begin{aligned} \frac{\partial}{\partial
t}g_{i\bar{j}}(x,t)&=-R_{i\bar{j}}(x,t)\\\
g_{i\bar{j}}(x,0)&=g_{i\bar{j}}(x).\end{aligned}\right.$
In order to get the estimate for the Green function as in Section 3, we use a
trick in [24] to consider a product space $\widehat{M}=M\times\mathbb{C}^{2}$
with a product metric $\widehat{g}=g+dw^{1}\wedge d\bar{w}^{1}+dw^{2}\wedge
d\bar{w}^{2}$. Then $\widehat{g}(x,t)=g(x,t)+dw^{1}\wedge
d\bar{w}^{1}+dw^{2}\wedge d\bar{w}^{2}$ is a solution of (4.3) on
$\widehat{M}$ with the initial metric $\widehat{g}$.
It was proved by Shi that for any $s>t$ and $B_{t}(x)\subset
B_{s}(x)\subset(\widehat{M},\widehat{g})$ (cf. Section 6 in [24]), it holds
$\frac{\text{vol}(B_{s}(x))}{\text{vol}(B_{t}(x))}\geq
C_{0}^{-1}\Big{(}\frac{s}{t}\Big{)}^{4}.$
Then
$\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm dt}}{\text{vol}(B_{\sqrt{t}}(x))}\leq
C_{0}\frac{d^{2}(x,y)}{\text{vol}(B_{d(x,y)}(x))}\mbox{,\quad}\forall\mbox{\
}d(x,y)\geq c_{0}.$
Thus by the Li-Yau estimate [16], there exists a global Green’s function $G$
on $(\widehat{M},\widehat{g})$ which satisfies (3.7).
Let
$F(x,t)=\ln\det(\widehat{g}_{i\bar{j}}(x,t))-\ln\det(\widehat{g}_{i\bar{j}}(x,0))$.
By (4.3), it is easy to see
(4.4) $F(x,t)=-\int_{0}^{t}\widehat{R}(x,s){\rm ds}.$
Then by Lemma 4.1, we have
$t\widehat{R}(x,0)\leq-F(x,t)\leq C_{0}t,$
where $C_{0}=\sup_{\widehat{M}}\widehat{R}(x,t).$ Thus as in Section 3, to
prove that $R\equiv 0$, we shall give a growth estimate of $-F(x,t)$ on $t$.
Fix an arbitrary point $x_{0}\in M$. For convenience, we denote a $r$-geodesic
ball $B_{r}(x_{0},t)$ of $(\widehat{M},\widehat{g}(t))$ centered at
$(x_{0},0,0)$. As in Section 3, by using the Green formula, we can estimate
$\displaystyle-F(x_{0},t)\leq\frac{C_{1}t(1+M(t))}{r^{2}}$
$\displaystyle\Big{(}M(t)+\int_{B_{r}(x_{0},0)}G(x_{0},y)\widehat{R}(y,0){\rm
dv}\Big{)}$ (4.5)
$\displaystyle+\int_{B_{\frac{r}{5}}(x_{0},0)}G(x_{0},y)\widehat{R}(y,0){\rm
dv}.$
Moreover,
$\displaystyle\int_{B_{r}(x_{0},0)}G_{r}(x_{0},y)\widehat{R}(y,0){\rm dv}$
$\displaystyle\leq
C(n)\Big{(}\frac{r^{2}}{\text{vol}(B_{r}(x_{0},0))}\int_{B_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (4.6)
$\displaystyle+\int_{0}^{r}\frac{s}{\text{vol}(B_{s}(x_{0},0))}\int_{B_{s}(x_{0},0)}\widehat{R}\rm{dvds}\Big{)},~{}\forall~{}r>0.$
On the other hand, by the volume comparison together with $(\ref{condition-
steady})$, we have
(4.7) $\frac{1}{\text{vol}(B_{r}(x_{0},0)}\int_{B_{r}(x_{0},0)}\widehat{R}{\rm
dv}\leq\frac{C}{\text{vol}(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}R{\rm
dv}\leq\frac{\varepsilon_{1}(r)}{1+r},$
where the function $\varepsilon_{1}(r)\rightarrow 0$ as $r\rightarrow\infty$.
Thus combining (4) and (4.7), we get from (4),
$\displaystyle-F(x_{0},t)\leq
r^{-2}C(n)t(C^{\prime}t+1)(C^{\prime}t+r\varepsilon_{2}(r))+r\varepsilon_{2}(r),\mbox{\quad}\forall\mbox{\
}r>0.$
where $\varepsilon_{2}(r)\rightarrow 0$ as $r\rightarrow\infty$. Consequently,
(4.8) $\displaystyle tR(x_{0},0)\leq
r^{-2}C(n)t(C^{\prime}t+1)(C^{\prime}t+r\varepsilon_{2}(r))+r\varepsilon_{2}(r),\mbox{\quad}\forall\mbox{\
}r>0.$
Now we choose a monotonic $\varepsilon_{3}(r)$ such that
$\varepsilon_{3}(r)\rightarrow 0$ and
$\frac{\varepsilon_{2}(r)}{\varepsilon_{3}(r)}\rightarrow 0$ as
$r\rightarrow\infty$. Let $r=t\varepsilon_{3}^{-1}(t)$. Then by (4.8 ), we get
$tR(x_{0},0)\leq
C_{1}\varepsilon_{3}^{2}(t)(C^{\prime}t+t\frac{\varepsilon_{2}(t\varepsilon_{3}^{-1}(t))}{\varepsilon_{3}(t\varepsilon_{3}^{-1}(t))})+t\frac{\varepsilon_{2}(t\varepsilon_{3}^{-1}(t))}{\varepsilon_{3}(t\varepsilon_{3}^{-1}(t))},\mbox{\quad}\forall\mbox{\
}t\gg 1.$
By dividing by $t$ on both sides of the above inequality and then letting
$t\rightarrow\infty$, it is easy to see that $R(x_{0},0)=0$. Since $x_{0}$ is
an arbitrary point, we prove that $R(x)\equiv 0$. ∎
By Theorem 1.3 , we can finish the proof of Theorem 1.4.
###### Proof of Theorem 1.4 (the steady case).
Since $(M,g)$ is a complete non-
compact manifold with nonnegative Ricci curvature, the volume growth of $g$ is
at least linear. Then by (1.6), it is easy to see that the average curvature
condition (1.4) is satisfied in Theorem 1.3 as in the proof of Lemma 3.1.
Hence by Theorem 1.3, we get Theorem 1.4 immediately. ∎
## 5\. Boundedness of scalar curvature–II
In this section, we prove the existence and uniqueness of equilibrium point
for the steady Kähler-Ricci soliton $(M^{n},g,f)$ in Theorem 1.2. As a
consequence, the maximum of scalar curvature of $g$ can be attained.
###### Proof of Theorem 1.2 (the steady case).
Existence: Let $g(\cdot,t)=\phi_{t}^{*}g$ be a family of steady solitons
generated by $-\nabla f$. Then $g(\cdot,t)$ is an eternal solution of (4.3).
Since $g(\cdot,t)$ has uniformly positive holomorphic bisectional curvature in
space time $M\times(-\infty,\infty)$, we apply Theorem 2.1 in [11] to see that
there exists a sequence of solutions
$g_{\alpha}(\cdot,t)=g(\cdot,t_{\alpha}+t)$ on $\Phi_{\alpha}(D(r))(\subset
M)$ such that $\Phi_{\alpha}^{*}(g_{\alpha}(\cdot,t))$ converge to a smooth
solution $h(x,t)$ of (4.3) uniformly and smoothly on a compact subset $D(r)$
for any $t\in(-1,\infty)$, where $D(r)$ is an Euclidean ball centered at the
origin with radius $r$ and $\Phi_{\alpha}$ are local biholomorphisms from
$D(r)$ to $M$. Moreover, by using the Cao’s argument in [3], it was proved
that $h(x,t)$ is generated by a steady Kähler-Ricci soliton $(D(r),h,f^{h})$
with $\nabla f^{h}(o)=0$. Namely, $h(x,t)$ satisfies
$R_{i\bar{j}}(h(t))=\nabla_{i}\nabla_{\bar{j}}f^{h}(t),\mbox{\quad}\nabla_{i}\nabla_{j}f^{h}(t)=0,$
where $f^{h}(t)$ are induced functions of $f^{h}$ and $\nabla f^{h}(t)$ vanish
at the origin for any $t\in(-1,\infty)$.
On the other hand, similar to (2.5), we have for solitons $(M,g(t),f(t))$,
$R_{,i}(t)+R_{i\bar{j}}(t)\nabla_{\bar{j}}f(t)=0.$
Then $\nabla f(t)$ is determined by the curvature tensor. Define a sequence of
a family of holomorphic vector fields $V(t_{\alpha})$ on $D(r)$ by
$\Phi_{\alpha}^{*}R_{,i}(t_{\alpha})+\Phi_{\alpha}^{*}R_{i\bar{j}}(t_{\alpha})V(t_{\alpha})_{j}=0.$
Clearly, $(\Phi_{\alpha})_{*}V(t_{\alpha})=\nabla f(t_{\alpha})$. By the
convergence of $g_{\alpha}(\cdot,t)$, holomorphic vector fields
$V(t_{\alpha})$ converge to $\nabla f^{h}$ in $C^{\infty}$-topology on $D(r)$
for any $t\in(-1,\infty)$. Since the eigenvalues of ${\rm Ric}(h(t))$ are
positive at $0$ by Proposition 2.2 in [11], the integral curves of
$-\nabla^{h}f^{h}$ will converge to $0$ in $D(r)$ when $r$ is sufficiently
small by the soliton equation. By the convergence of $V(t_{\alpha})$, the
integral curve of $-V(t_{\alpha})$ will also converge to a point $q$ in
$D(r_{1})$ for some $r_{1}<r$ when $\alpha$ is large enough (cf. Page 9 of
[12]). As a consequence, $q$ is a zero point of $V(t_{\alpha})$ in $D(r_{1})$.
This proves that there exists a zero point of $\nabla f(t_{\alpha})$ in $M$
for each $\alpha$ since $\Phi_{\alpha}^{*}$ is a local biholomorphism.
Uniqueness: Suppose that $p$ and $q$ are two equilibrium points. Then
$d_{0}(p,q)=d_{t}(p,q)$. Choose $l>0$ such that $q\in B_{l}(p,0)$. Note that
$\phi_{t}:(M^{n},g(t))\rightarrow(M^{n},g(0))$ are a family of isometric
deformations. Thus
$C=\inf_{x\in B_{l}(p,t)}\mu_{1}(x,t)=\inf_{x\in
B_{l}(p,0)}\mu_{1}(x,0)>0,\mbox{\ }\forall x\in B_{l}(p,0),$
where $\mu_{1}(x,t)$ is the smallest eigenvalue of $\text{Ric}(x,t)$ w.r.t
$g(x,t)$. Since the metric is decreasing along the flow, we see that
$B_{l}(p,0)\subset B_{l}(p,t)$. Hence by (4.1), we get
$\displaystyle\frac{{\rm
d}|v_{x}|^{2}_{t}}{\rm{dt}}\leq-\mu_{1}(x,t)|v_{x}|^{2}_{t}\leq-C|v_{x}|^{2}_{t},~{}\forall~{}t\geq
0,$
where $x\in B_{l}(p,0)$ and $v_{x}\in T_{x}^{(1,0)}M$. Therefore, if we let
$\gamma(s)$ be a minimal geodesic curve connecting $p$ and $q$ with an arc-
parameter $s$ w.r.t the metric $g(x,0)$ in $B_{l}(p,0)$, we deduce
$\displaystyle d_{t}(p,q)\leq\int_{0}^{d}|\gamma^{\prime}(s)|_{t}{\rm
ds}\leq\int_{0}^{d}|\gamma^{\prime}(s)|_{0}e^{-Ct}{\rm ds}=d_{0}(p,q)e^{-Ct}.$
Letting $t\to\infty$, we see that $d_{t}(p,q)=d_{0}(p,q)=0$. This proves that
$p=q.$ ∎
###### Proposition 5.1.
Let $(M^{n},g,f)$ be a simply connected complete non-comp-act steady gradient
Kähler-Ricci soliton with nonnegative bisectional curvature. Suppose that $g$
satisfies
(5.1) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{1+\epsilon}},$
for some $\epsilon>0,C$. Then $(M,g)$ is isometric to $\mathbb{C}^{n}$.
###### Proof.
We suffice to show that $g$ is Ricci flat. On the contrary, we may assume that
the Ricci curvature of $g(\cdot,t)$ is positive everywhere by a dimension
reduction theorem of Cao for Kähler-Ricci flow on a simply connected complete
Kähler manifold with nonnegative bisectional curvature [4], where
$g(\cdot,t)=\phi_{t}^{*}g$ is the generated solution of (4.3) as in Section 4.
Let $o$ be the unique equilibrium point of $g$ according to Theorem 1.2. In
the following we use an argument of Hamilton in [13] to show that there exists
a pointedwise backward limit $g_{\infty}(x)$ of $g(x,t)$ on
$M\setminus\\{o\\}$ and $(M\setminus\\{o\\},g_{\infty})$ is a complete flat
Riemannian manifold.
Since
$R(x)+|\nabla f|^{2}=R(o),$
by (5.1), we see
$\lim_{d(x,o)\rightarrow\infty}|\nabla f|^{2}(x)=R(o)>0.$
Note the equilibrium point is unique. It follows
$C_{\delta}=\inf_{M\setminus B_{\delta}(o)}|\nabla
f|^{2}>0,\mbox{\quad}\forall\mbox{\ }\delta>0.$
This implies
$d(\phi_{t}(x),o)\geq C_{\delta}|t|\mbox{\quad}\forall\mbox{\ }x\in M\setminus
B_{\delta}(o)\mbox{,\ }t\leq 0.$
Hence by (5.1), we get from equation (4.3),
$\displaystyle 0$ $\displaystyle\leq-\frac{\partial}{\partial t}g(x,t)\leq
R(g(x,t))g(x.t)$
$\displaystyle\leq\frac{C}{1+d^{1+\epsilon}(\phi_{t}(x),o)}g(x,t)\leq\frac{C_{\delta}^{\prime}}{1+|t|^{1+\epsilon}}g(x,t).$
Therefore, we derive
(5.2) $g(x,0)\leq g(x,t_{1})\leq g(x,t_{2})\leq C_{\delta}^{\prime}g(x,0),$
for any $x\in M\setminus B_{\delta}(o,0)$ and $-\infty<t_{2}\leq t_{1}\leq 0$.
By (5.2) and Shi’s higher order estimate for curvatures, we see that $g(x,t)$
converge locally to a limit Kähler metric $g_{\infty}(x)$ on
$M\setminus\\{o\\}$ as $t\to-\infty$. Clearly, $g_{\infty}(x)$ is Ricci-flat
since
$0=\lim_{t\to-\infty}-\frac{\partial}{\partial
t}g(x,t)=\lim_{t\to-\infty}\text{Ric}(g(\cdot,t))=\text{Ric}(g_{\infty}).$
Consequently, $g_{\infty}$ is flat. Moreover, $g_{\infty}$ is a complete,
because
$\displaystyle\lim_{x^{\prime}\to
o}d_{g_{\infty}}(x,x^{\prime})=\lim_{t\rightarrow-\infty}d_{g(\cdot,t)}(x,o)=\lim_{t\rightarrow-\infty}d_{g}(\phi_{t}(x),o)=\infty,$
where $x\in M\setminus\\{o\\}$. On the other hand, it was proved by Chau and
Tam that $M$ is biholomorphic to $\mathbb{C}^{n}$ since $M$ is a a simply
connected complete non-compact steady gradient Kähler-Ricci soliton with
positive Ricci curvature [9]. Thus, $M\setminus\\{o\\}$ is simply connected.
Hence, $M\setminus\\{o\\}$ is also biholomorhic to $\mathbb{C}^{n}$. This is a
contradiction! Therefore, $g$ is Ricci flat and consequently, $(M,g)$ is
isometric to $\mathbb{C}^{n}$.
∎
## References
## References
* 1 Brendle, S., Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math. 194 No.3 (2013), 731-764.
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|
arxiv-papers
| 2013-12-04T15:45:07 |
2024-09-04T02:49:54.862473
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuxing Deng, and Xiaohua Zhu",
"submitter": "Xiaohua Zhu",
"url": "https://arxiv.org/abs/1312.1216"
}
|
1312.1217
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-218 LHCb-PAPER-2013-060 December 4, 2013
Measurement of the $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
and $\kern 3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow
D^{-}D^{+}_{s}$ effective lifetimes
The LHCb collaboration†††Authors are listed on the following pages.
The first measurement of the effective lifetime of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson in the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
is reported using a proton-proton collision dataset, corresponding to an
integrated luminosity of 3$\mbox{\,fb}^{-1}$, collected by the LHCb
experiment. The measured value of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
effective lifetime is $1.379\pm 0.026\pm 0.017$ ps, where the uncertainties
are statistical and systematic, respectively. This lifetime translates into a
measurement of the decay width of the light $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate of
$\Gamma_{\rm L}$ $=0.725\pm 0.014\pm 0.009$ ps-1. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime is also measured
using the flavor-specific $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay to be $1.52\pm 0.15\pm 0.01~{}{\rm ps}$.
Submitted to Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, A. Affolder51, Z. Ajaltouni5, J.
Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez
Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L.
Anderlini17,g, J. Anderson39, R. Andreassen56, M. Andreotti16,f, J.E.
Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli37, A.
Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,n, M. Baalouch5, S.
Bachmann11, J.J. Back47, A. Badalov35, V. Balagura30, W. Baldini16, R.J.
Barlow53, C. Barschel38, S. Barsuk7, W. Barter46, V. Batozskaya27, Th.
Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K.
Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S. Benson49, J.
Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A.
Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,i, P.M. Bjørnstad53, T. Blake47,
F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W.
Bonivento15,37, S. Borghi53, A. Borgia58, M. Borsato7, T.J.V. Bowcock51, E.
Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D.
Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39,
G. Busetto21,r, J. Buytaert37, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez35,p, A. Camboni35, P. Campana18,37, D. Campora
Perez37, A. Carbone14,d, G. Carboni23,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia49, L. Carson49, K. Carvalho Akiba2, G. Casse51, L. Castillo
Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles8, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco37, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, J. Dalseno45, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, S. Donleavy51, F. Dordei11, P.
Dorosz25,o, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
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Furfaro23,l, A. Gallas Torreira36, D. Galli14,d, M. Gandelman2, P. Gandini58,
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Golubkov30, A. Golutvin52,30,37, A. Gomes1,a, H. Gordon37, M. Grabalosa
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S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40,
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Heß60, A. Hicheur1, D. Hill54, M. Hoballah5, C. Hombach53, W. Hulsbergen40, P.
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M. Idzik26, P. Ilten55, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A.
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Jost37, N. Jurik58, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M.
Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20, S. Klaver53, O. Kochebina7,
I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40,
L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F.
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L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61National Research Centre Kurchatov Institute, Moscow, Russia, associated to
30
62KVI - University of Groningen, Groningen, The Netherlands, associated to 40
63Celal Bayar University, Manisa, Turkey, associated to 37
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
A central goal in quark-flavor physics is to test whether the Cabibbo-
Kobayashi-Maskawa (CKM) mechanism [1, 2] can fully describe all relevant weak
decay observables, or if physics beyond the Standard Model (SM) is needed. In
the neutral $B$ meson sector, the mass eigenstates do not coincide with the
flavor eigenstates as a result of $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ mixing. In addition to measurable mass
splittings between the mass eigenstates [3], the $B_{s}$ system also exhibits
a sizeable difference in the decay widths $\Gamma_{\rm L}$ and $\Gamma_{\rm
H}$, where the subscripts ${\rm L}$ and ${\rm H}$ refer to the light and heavy
mass eigenstates, respectively. This difference is due to the large decay
width to final states accessible to both $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$. In the absence of $C\\!P$
violation, the mass eigenstates are also eigenstates of $C\\!P$. The summed
decay rate of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to the $C\\!P$-even
$D^{+}_{s}D^{-}_{s}$ final state can be written as [4]
$\displaystyle\Gamma_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}(t)+\Gamma_{B^{0}_{s}\rightarrow
D^{+}_{s}D^{-}_{s}}(t)\propto(1+\cos\phi_{s})e^{-\Gamma_{\rm
L}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm H}t},$ (1)
where $\phi_{s}$ is the ($C\\!P$-violating) relative weak phase between the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing and
$b\rightarrow c\overline{}cs$ decay amplitudes.
The untagged decay rate in Eq. 1 provides a probe of $\phi_{s}$, $\Gamma_{\rm
L}$ and $\Gamma_{\rm H}$ in a way that is complementary to direct
determinations using $C\\!P$ violating asymmetries [5]. Approximating Eq. 1 by
a single exponential
$\displaystyle\Gamma_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}(t)+\Gamma_{B^{0}_{s}\rightarrow
D^{+}_{s}D^{-}_{s}}(t)\propto e^{-t/\tau^{\rm eff}_{\kern
0.89996pt\overline{\kern-0.89996ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}},$ (2)
defines the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ effective lifetime, which can be written as $\tau^{\rm
eff}_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}=\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}(1-y_{s}\cos\phi_{s}+{\cal{O}}(y_{s}^{2}))$
[4, 6], assuming no direct $C\\!P$ violation in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
decay. Here $y_{s}\equiv\Delta\Gamma_{s}/(2\Gamma_{s})$,
$\Delta\Gamma_{s}\equiv\Gamma_{\rm L}-\Gamma_{\rm H}$ and
$\Gamma_{s}=(\Gamma_{\rm H}+\Gamma_{\rm L})/2=1/\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}$, where $\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}$ is the flavor-specific $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime. Using the measured
value of $\phi_{s}=0.01\pm 0.07\pm 0.01$ rad [5], which is in good agreement
with the SM expectation of $-0.0363^{+0.0016}_{-0.0015}$ rad [7], it follows
that $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}\simeq\Gamma_{\rm L}^{-1}$.
The most precise measurement to date of the effective lifetime in a
$C\\!P$-even final state used $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}$ [8]
decays, and yielded a value $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}}=1.455\pm
0.046\mathrm{\,(stat)}\pm 0.006\mathrm{\,(syst)}$ ps. Loop contributions, both
within, and possibly beyond the SM, are expected to be significantly larger in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}$
than in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$. These contributions give rise to direct $C\\!P$ violation
in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
K^{+}K^{-}$ decay [9], which lead to differences between $\tau^{\rm eff}$ in
these two $C\\!P$ final state decays,making a comparison of their effective
lifetimes interesting. Measurements have also been made in $C\\!P$-odd modes,
such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ [10, 11] and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ [12]. The most precise value is from
the former, yielding $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)}=1.700\pm 0.040\mathrm{\,(stat)}\pm 0.026\mathrm{\,(syst)}$
ps [10]. Constraints from these measurements on the ($\Delta\Gamma_{s}$,
$\phi_{s}$) parameter space are given in Refs. [4, 13]. Improved precision on
the effective lifetimes will enable more stringent tests of the consistency
between the direct measurements of $\Delta\Gamma_{s}$ and $\phi_{s}$, and
those inferred using effective lifetimes.
In this Letter, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
time-dependent decay rate is normalized to the corresponding rate in the
$B^{-}\rightarrow D^{0}D^{-}_{s}$ decay, which has similar final state
topology and kinematic properties, and a precisely measured lifetime of
$\tau_{B^{-}}=1.641\pm 0.008$ ps [14]. As a result, many of the systematic
uncertainties cancel in the measured ratio. The relative rate is then given by
$\displaystyle\frac{\Gamma_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}(t)+\Gamma_{B^{0}_{s}\rightarrow
D^{+}_{s}D^{-}_{s}}(t)}{\Gamma_{B^{-}\rightarrow
D^{0}D^{-}_{s}}(t)+\Gamma_{B^{+}\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}D^{+}_{s}}(t)}\propto
e^{-\alpha_{su}t},$ (3)
where $\alpha_{su}=1/\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}-1/\tau_{B^{-}}$. A measurement of $\alpha_{su}$ therefore
determines $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}$.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson lifetime is
also measured using the flavor-specific, Cabibbo-suppressed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay. Its time-dependent rate is normalized to that of the $B^{0}\rightarrow
D^{-}D^{+}_{s}$ decay. In what follows, the symbol $B$ without a flavor
designation refers to either a $B^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson, and $D$ refers to
either a $D^{0}$, $D^{+}$ or $D^{+}_{s}$ meson. Unless otherwise indicated,
charge conjugate final states are included.
The measurements presented use a proton-proton ($pp$) collision data sample
corresponding to 3 $\mbox{\,fb}^{-1}$ of integrated luminosity,
1$\mbox{\,fb}^{-1}$ recorded at a center-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ and 2$\mbox{\,fb}^{-1}$ at
8$\mathrm{\,Te\kern-1.00006ptV}$, collected by the LHCb experiment. The LHCb
detector [15] is a single-arm forward spectrometer covering the pseudorapidity
range $2<\eta<5$, designed for the study of particles containing $b$ or $c$
quarks. The detector includes a high-precision tracking system consisting of a
silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip
detectors and straw drift tubes placed downstream. The combined tracking
system provides a momentum measurement with relative uncertainty that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20${\,\upmu\rm m}$ for tracks with large transverse momentum
($p_{\rm T}$). Ring-imaging Cherenkov detectors [16] are used to distinguish
charged hadrons, and photon, electron and hadron candidates are identified by
a calorimeter system consisting of scintillating-pad and preshower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers [17].
The trigger [18] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction [18, 19]. No specific requirement is made on the
hardware trigger decision. Of the $B$ meson candidates considered in this
analysis, about 60% are triggered at the hardware level by one or more of the
final state particles in the signal $B$ decay. The remaining 40% are triggered
due to other activity in the event. The software trigger requires a two-,
three- or four-track secondary vertex with a large sum of the transverse
momentum of the tracks and a significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference in $\chi^{2}$ of a given PV reconstructed with
and without the considered particle included. The signal candidates used in
this analysis are required to pass a multivariate software trigger selection
algorithm [19].
Proton-proton collisions are simulated using Pythia [20, *Sjostrand:2007gs]
with a specific LHCb configuration [22]. Decays of hadronic particles are
described by EvtGen [23], in which final state radiation is generated using
Photos [24]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [25,
*Agostinelli:2002hh] as described in Ref. [27].
Signal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ candidates are reconstructed using four final states: (i)
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+},~{}D^{-}_{s}\rightarrow
K^{-}K^{+}\pi^{-}$, (ii) $D^{+}_{s}\rightarrow
K^{+}K^{-}\pi^{+},~{}D^{-}_{s}\rightarrow\pi^{-}\pi^{+}\pi^{-}$, (iii)
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+},D^{-}_{s}\rightarrow
K^{-}\pi^{+}\pi^{-}$, and (iv)
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+},D^{-}_{s}\rightarrow\pi^{-}\pi^{+}\pi^{-}$.
In the normalization mode, $B^{-}\rightarrow D^{0}D^{-}_{s}$, only the final
state $D^{0}\rightarrow K^{-}\pi^{+},~{}D^{-}_{s}\rightarrow
K^{-}K^{+}\pi^{-}$ is used. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay and the corresponding $B^{0}$ normalization mode, the $D^{-}\rightarrow
K^{+}\pi^{-}\pi^{-},~{}D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ final state is
used. Loose particle identification (PID) requirements are imposed on kaon and
pion candidates, with efficiencies typically in excess of 95%. The $D$
candidates are required to have masses within
25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their known values [14] and to
have vertex separation from the $B$ vertex satisfying $\chi^{2}_{\rm VS}>2$.
Here $\chi^{2}_{\rm VS}$ is the increase in $\chi^{2}$ of the parent ($B$)
vertex fit when the ($D$ meson) decay products are constrained to come from
the parent vertex, relative to the nominal fit. To suppress the large
background from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ decays,
$D^{-}_{s}\rightarrow\pi^{-}\pi^{+}\pi^{-}$ candidates are required to have
$\chi^{2}_{\rm VS}>6$. As the signatures of $b$-hadron decays to double-charm
final states are similar, vetoes are employed to suppress the cross-feed
resulting from particle misidentification, following Ref. [28]. For the
$D^{+}_{s}\rightarrow K^{+}\pi^{-}\pi^{+}$ decay, an additional veto to
suppress cross-feed from $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ with double-
misidentification is employed, which renders this background negligible.
Potential background to $D^{+}_{s}$ decays from $D^{*+}\rightarrow
D^{0}\pi^{+}$ with $D^{0}\rightarrow K^{+}K^{-},~{}\pi^{+}\pi^{-}$ is also
removed by requiring the mass difference,
$M(D^{0}\pi^{+})-M(D^{0})>150$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
production point of each $B$ candidate is taken as the PV with the smallest
$\chi^{2}_{\rm IP}$ value. All $B$ candidates are refit taking both $D$ mass
and vertex constraints into account [29].
The efficiencies of the PID and veto requirements are evaluated using
dedicated $D^{*+}\rightarrow D^{0}\pi^{+},~{}D^{0}\rightarrow K^{-}\pi^{+}$
calibration samples collected at the same time as the data. The kinematic
distributions of kaons and pions from the calibration sample are reweighted
using simulation to match those of the $B$ decays under study. The combined
PID and veto efficiencies are 91.4% for $B^{-}\rightarrow D^{0}D^{-}_{s}$,
88.0% for $(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s},~{}B^{0})\rightarrow
D^{-}D^{+}_{s}$, and 86.5%, 90.8%, 86.6%, and 95.9% for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
final states (i)$-$(iv), respectively.
To further improve the signal-to-background ratio, a boosted decision tree
(BDT) [30, 31] algorithm using seventeen input variables is employed. Five
variables from the $B$ candidate are used, including $\chi^{2}_{\rm IP}$, the
vertex fit $\chi^{2}_{\rm vtx}$ (with $D$ mass, and vertex constraints), the
PV $\chi^{2}_{\rm VS}$, $p_{\rm T}$, and a $p_{\rm T}$ asymmetry variable
[32]. For each $D$ daughter, $\chi^{2}_{\rm IP}$, the flight distance from the
$B$ vertex normalized by its uncertainty, and the maximum distance between the
trajectories of any pair of particles in the $D$ decay, are used. Lastly, for
each $D$ candidate, the minimum $p_{\rm T}$, and both the smallest and largest
$\chi^{2}_{\rm IP}$, among the $D$ daughter particles are used. The BDT uses
simulated decays to emulate the signal and wrong-charge final states from data
with masses larger than 5.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the
background. Here, wrong-charge refers to $D_{s}^{\pm}D_{s}^{\pm}$,
$D^{\pm}D_{s}^{\pm}$, and $D^{0}D^{+}_{s}$ combinations, where in the latter
case we remove candidates within 30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the $B^{+}$ mass [14], to remove the small doubly-Cabibbo-suppressed decay
contribution to this final state. The selection requirement on the BDT output
is chosen to maximize the expected $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
signal significance, corresponding to signal and background efficiencies of
about 97% and 33%, respectively. More than one candidate per event is allowed,
but after all selections the fraction of events with multiple candidates is
below 0.25% for all modes.
For the lifetime analysis, we consider only $B$ candidates with reconstructed
decay time less than 9 ps. Signal efficiencies as functions of decay time are
determined using simulated decays after all selections, except those that
involve PID, as described above. The resulting $B^{-}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ relative efficiency as a
function of decay time is shown in Fig. 1, where six decay time bins with
widths ranging between 1 and 3 ps are used. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
decay, the efficiency used in the ratio is the weighted average of the
$D^{+}_{s}D^{-}_{s}$ final states (i)$-$(iv), where the weights are obtained
from the observed yields in data. The efficiency accounts for the migration
between bins, which is small since the resolution on the reconstructed time of
$\sim$50 fs is much less than the bin width. Moreover, the time resolution is
nearly identical for the signal and normalization modes, and is independent of
the reconstructed lifetime. The relative efficiency is consistent with being
independent of decay time, however, the computed bin-by-bin efficiencies are
used to correct the data.
Figure 1: Ratio of selection efficiencies for $B^{-}\rightarrow
D^{0}D^{-}_{s}$ relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
decays as a function of decay time. The uncertainties shown are due to finite
simulated sample sizes.
The mass distributions for the signal, summed over the four final states, and
the normalization modes are shown in Fig. 2, along with the results of binned
maximum likelihood fits. The $B$ signal shapes are each modeled using the sum
of two Crystal Ball (CB) functions [33] with a common mean. The shape
parameters are fixed from fits to simulated signal decays, with the exception
of the resolution parameter, which is found to be about 15% larger in data
than simulation. The shape of the low-mass background from partially
reconstructed decays, where either a photon or pion is missing, is obtained
from simulated decays, as are the cross-feed background shapes from
$B^{0}\rightarrow D^{-}D^{+}_{s}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}D^{-}_{s}$ decays
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ channel only). An additional peaking background due to
$B\rightarrow DK^{-}K^{+}\pi^{-}$ decays is also included in the fit. Its
shape is obtained from simulation and the yield is fixed to be 1% of the
signal yield from a fit to the $D$ mass sidebands. The combinatorial
background shape is described by an exponential function with the shape
parameter fixed to the value obtained from a fit to the mass spectrum of
wrong-charge candidates. All yields, except that of the $B\rightarrow
DK^{-}K^{+}\pi^{-}$, are freely varied in the fit to the full data sample.
In total, we observe 3499 $\pm$ 65 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
and 19,432 $\pm$ 140 $B^{-}\rightarrow D^{0}D^{-}_{s}$ decays.
Figure 2: Mass distributions and fits to the full data sample for (left)
$\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ and (right) $B^{-}\rightarrow D^{0}D^{-}_{s}$ candidates.
The points are the data and the curves and shaded regions show the fit
components.
The data are split into the time bins shown in Fig. 1, and each mass
distribution is fitted with the CB widths fixed to the values obtained from
the full fit. The independence of the signal shape parameters on decay time is
validated using simulated decays. The ratios of yields are then computed, and
corrected by the relative efficiencies shown in Fig. 1. Figure 3 shows the
efficiency-corrected yield ratios as a function of decay time. The data points
are placed at the average time within each bin assuming an exponential form
$e^{-t/(1.5\,{\rm ps})}$. Fitting an exponential function to the data yields
the result $\alpha_{su}=0.1156\pm 0.0139$ ps-1. The uncertainty in the fitted
slope due to using the value of 1.5 ps to get the average time in each bin is
negligible. Using the known $B^{-}$ lifetime, $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}}$
is determined to be $1.379\pm 0.026\mathrm{\,(stat)}$ ps.
As a cross-check, the full analysis is applied to the $B^{-}\rightarrow
D^{0}D^{-}_{s}$ and $B^{0}\rightarrow D^{-}D^{+}_{s}$ decays, treating the
former as the signal mode and the latter as the normalization mode. The fitted
value for $\alpha\equiv 1/\tau_{B^{0}}-1/\tau_{B^{-}}$ is $0.0500\pm 0.0076$
ps-1, in excellent agreement with the expected value of $0.0489\pm 0.0042$
[14]. This check indicates that the relative lifetime measurements are
insensitive to small differences in the number of charged particles or
lifetimes of the $D$ mesons in the final state. The $B^{0}\rightarrow
D^{-}D^{+}_{s}$ mode could have also been used as a normalization mode for the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ time-dependent rate measurement, but due to limited
simulated sample sizes it would have led to a larger systematic uncertainty.
Figure 3: Efficiency corrected yield ratio of $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
relative to $B^{-}\rightarrow D^{0}D^{-}_{s}$ as a function of decay time,
along with the exponential fit. The uncertainties are statistical only.
As the method for determining $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}}$
relies on ratios of yields and efficiencies, many systematic uncertainties
cancel. The robustness of the relative acceptance is tested by subdividing the
sample into mutually exclusive subsamples based on (i) center of mass energy,
(ii) $D^{-}_{s}D^{+}_{s}$ final states, and (iii) hardware trigger decision,
and searching for deviations larger than those expected from the finite sizes
of the samples. The results from all checks were found to be within one
standard deviation of the average. Based on the largest deviation, we assign a
0.010 ps systematic uncertainty due to the modeling of the relative
acceptance. The statistical precision on the relative acceptance, as obtained
from simulation, contributes an uncertainty of 0.011 ps. Using a different
signal shape to fit the data leads to 0.003 ps uncertainty. If the
combinatorial background shape parameter is allowed to freely vary in each
time bin fit, we find a deviation of 0.001 ps from the nominal value of
$\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}$, which is assigned as a systematic uncertainty. Due to
the presence of a non-trivial acceptance function, the result of fitting a
single exponential to the untagged $B^{0}_{s}$ decay time distribution does
not coincide precisely with the formal definition of the effective lifetime
[34]. The deviation between $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}}$
and the single exponential fit is at most 0.001 ps [34], which is assigned as
a systematic uncertainty. The precision on the $B^{-}$ lifetime leads to 0.008
ps uncertainty on the value of $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}$. Summing these deviations in quadrature, we obtain a
total systematic uncertainty of 0.017 ps. In converting to a measurement of
$\Gamma_{\rm L}$, an additional uncertainty due to a small $C\\!P$-odd
component of expected size $1-\cos\phi_{s}=(0.1\pm 3.2)\times 10^{-3}$ [5]
leads to a bias no larger than $-0.001$ ps-1. This is included in the
$\Gamma_{\rm L}$ systematic uncertainty.
The value of $\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}}$
and the corresponding decay width of the light $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate are determined
to be
$\displaystyle\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}}$
$\displaystyle=1.379\pm 0.026\pm 0.017~{}{\rm ps},$ $\displaystyle\Gamma_{\rm
L}$ $\displaystyle=0.725\pm 0.014\pm 0.009~{}{\rm ps}^{-1},$
where the first uncertainty is statistical and the second is systematic. These
are the first such measurements using the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
decay. The measured effective lifetime represents the most precise measurement
of the width of the light $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate, and is about
one standard deviation lower than the value obtained using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}$ decays
[8]. Compared to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
decay, which is dominated by tree-level processes, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}$ decay is
expected to have larger relative contributions from SM-loop amplitudes [35,
36, 4], and therefore one should not naively average the effective lifetimes
from these two decays. Moreover, if non-SM particles contribute additional
amplitudes, their effect is likely to be larger in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-}$ than in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}$ decays [37].
The value of $\Gamma_{\rm L}$ obtained in this analysis may be compared to the
value inferred from the time-dependent analyses of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ decays. Using the
values $\Gamma_{s}=0.661\pm 0.004\pm 0.006$ ps-1 and
$\Delta\Gamma_{s}=0.106\pm 0.011\pm 0.007$ ps-1 [5], we find $\Gamma_{\rm
L}=0.714\pm 0.010$ ps-1, in good agreement with the value obtained from
$\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}$.
The effective lifetime of the flavor-specific $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay is also measured, using the $B^{0}\rightarrow D^{-}D^{+}_{s}$ decay for
normalization. The technique is identical to that described above, with the
simplification that the relative efficiency equals one, since the final states
are identical. Effects due to the mass difference between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}$ mesons are
negligible. A tighter BDT selection is imposed to optimize the expected
signal-to-background ratio, which results in signal and background
efficiencies of 87% and 11%, respectively. The mass spectrum and the
corresponding fit are shown in Fig. 4, where the fitted components are
analogous to those described previously. A total of $230\pm 18$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$ and
21,195 $\pm$ 147 $B^{0}\rightarrow D^{-}D^{+}_{s}$ decays are obtained.
Figure 4: Mass distribution and fits to the full data sample for $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ and $B^{0}$ decays into the
$D^{-}D^{+}_{s}$ final state. The points are the data and the curves and
shaded regions show the fit components.
The time bins are the same as above, except the 6$-$9 ps bin is dropped, since
the yield in the signal mode beyond 6 ps is negligible. The relative decay
rate is fitted to an exponential form ${\cal{C}}e^{-\beta t}$, where
${\cal{C}}$ is a normalization constant. The fitted value of $\beta$ is
$0.000\pm 0.068$ ps-1. The systematic uncertainty due to the signal shape is
0.007 ${\rm\,ps}$, obtained by using a different signal shape function. The
exponential background shape is fixed in the nominal fit using
$D^{\pm}D_{s}^{\pm}$ candidates, and a systematic uncertainty of 0.010 ps is
determined by allowing its shape parameter to vary freely in the fit. In
determining the effective lifetime, an uncertainty of 0.007 ${\rm\,ps}$ due to
the limited precision of the $B^{0}$ lifetime [14] is also included. The
resulting effective lifetime in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$ mode
is
$\displaystyle\tau^{\rm eff}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}D^{+}_{s}}=1.52\pm 0.15\pm 0.01~{}{\rm ps}.$
This is the first measurement of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime using the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay. Its value is consistent with previous direct and indirect measurements
of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime in
other flavor-specific decays.
In summary, we report the first measurement of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}_{s}D^{+}_{s}$
effective lifetime and present the most precise direct measurement of the
width of the light $B_{s}$ mass eigenstate. Their values are $\tau^{\rm
eff}_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{-}_{s}D^{+}_{s}}=1.379\pm 0.026\pm 0.017~{}{\rm ps}$ and $\Gamma_{\rm
L}=0.725\pm 0.014\pm 0.009~{}{\rm ps}^{-1}$. The $\Gamma_{\rm L}$ result is
consistent with the value obtained from previously measured values of
$\Delta\Gamma_{s}$ and $\Gamma_{s}$ [5]. We also determine the average $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime to be $1.52\pm
0.15\pm 0.01~{}{\rm ps}$ using the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{-}D^{+}_{s}$
decay, which is consistent with other measurements.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-12-04T15:46:00 |
2024-09-04T02:49:54.872348
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid\n Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C.\n Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan,\n D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, J.\n Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De\n Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone,\n D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O.\n Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F. Dordei, P.\n Dorosz, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, P. Durante,\n R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, C. Farinelli, S. Farry,\n D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, E. Furfaro, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi,\n J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck,\n T. Gershon, Ph. Ghez, A. Gianelle, V. Gibson, L. Giubega, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n T.W. Hafkenscheid, S.C. Haines, S. Hall, B. Hamilton, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van\n Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N.\n Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R.\n Kenyon, T. Ketel, B. Khanji, S. Klaver, O. Kochebina, I. Komarov, R.F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev,\n K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai,\n D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T.\n Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A.\n Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, J.\n Luisier, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L. Pappalardo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch,\n A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker, B.\n Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven,\n S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards,\n K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, G.\n Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith,\n J. Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S.\n Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W.\n Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, D.\n Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu,\n F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning,\n M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C.\n Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N. Warrington, N.K.\n Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko,\n M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Steven R. Blusk",
"url": "https://arxiv.org/abs/1312.1217"
}
|
1312.1262
|
# The geometry of variations
in Batalin–Vilkovisky formalism
Arthemy V. Kiselev Johann Bernoulli Institute for Mathematics and Computer
Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The
Netherlands [email protected]
###### Abstract
We explain why no sources of divergence are built into the Batalin–Vilkovisky
(BV) Laplacian, whence there is no need to postulate any ad hoc conventions
such as “$\boldsymbol{\delta}(0)=0$” and “$\log\boldsymbol{\delta}(0)=0$”
within BV-approach to quantisation of gauge systems. Remarkably, the geometry
of iterated variations does not refer at all to the construction of Dirac’s
$\boldsymbol{\delta}$-function as a limit of smooth kernels. We illustrate the
reasoning by re-deriving –but not just ‘formally postulating’– the standard
properties of BV-Laplacian and Schouten bracket and by verifying their basic
inter-relations (e.g., cohomology preservation by gauge symmetries of the
quantum master-equation).
## Introduction
This is a paper about geometry of variations. We formulate definitions of the
objects and structures which are cornerstones of Batalin–Vilkovisky formalism
[5, 7, 20, 22, 55]. To confirm the intrinsic self-regularisation of BV-
Laplacian, we explain why there are no divergencies in it (such excessive
elements are traditionally encoded by using derivatives of Dirac’s
$\boldsymbol{\delta}$-distribution). Namely, we specify the geometry in which
the following canonical inter-relations between the variational Schouten
bracket $\lshad\,,\,\rshad$ and BV-Laplacian $\Delta$ are rigorously proven
for any BV-functionals $F,G,H$:
$\displaystyle\lshad F,G\cdot H\rshad$ $\displaystyle=\lshad F,G\rshad\cdot
H+(-)^{(|F|-1)\cdot|G|}G\cdot\lshad F,H\rshad,$ (1a)
$\displaystyle\Delta(F\cdot G)$ $\displaystyle=\Delta F\cdot G+(-)^{|F|}\lshad
F,G\rshad+(-)^{|F|}F\cdot\Delta G,$ (1b) $\displaystyle\Delta\bigl{(}\lshad
F,G\rshad\bigr{)}$ $\displaystyle=\lshad\Delta F,G\rshad+(-)^{|F|-1}\lshad
F,\Delta G\rshad,$ (1c) $\displaystyle\Delta^{2}$
$\displaystyle=0\qquad\Longleftrightarrow\qquad\text{Jacobi}\bigl{(}\lshad\,,\,\rshad\bigr{)}=0.$
(1d)
There is an immense literature on this subject’s intrinsic difficulties and
attempts of regularisation of apparent divergencies in it (e.g., see [12, 13,
25, 50, 51] vs [21]). While the BV-quantisation technique has advanced far
from its sources [7, 8], it is still admitted that it lacks sound mathematical
consistency ([22, §15] or [3, §3]). The calculus in this field is thus reduced
to formal operation with expressions which are expected to render the theory’s
main objects and structures. Several ad hoc techniques for cancellation of
divergencies, allowing one to strike through calculations and obtain
meaningful results, are adopted by repetition; we briefly review the plurality
of such tricks in what follows.
Our reasoning is independent from such conventional schemes for cancellation
of infinities or from other practised roundabouts for regularisation of terms
which are believed to be infinite (e.g., by erasing ‘infinite constants’
[11]). In particular, we do not pronounce the traditional password
$\boldsymbol{\delta}(0)\mathrel{{:}{=}}0$ (2)
which lets one enter the existing paradigm and use its quantum alchemistry for
operation with what remains from Dirac’s
$\boldsymbol{\delta}$-distribution.111Another convention is
$\log\boldsymbol{\delta}(0)=0$; we show that natural counterparts of the true
geometry of variations lead to this intuitive convention and simultaneously to
(2) — none of the two being actually required. Our message is this: we do not
propose to replace ‘bad slogans’ with ‘good slogans,’ which would mean that a
choice of conventions is still left to the one who attempts regularisation in
the BV-setup. Such deficiency would symptomise that the theory remains a
formal procedure. We now focus on the true sources of known difficulties. By
analysing the geometry of variations of functionals at a very basic level, we
prove the absence of apparently divergent essences. The intrinsically
regularised definitions of the BV-Laplacian $\Delta$ and Schouten bracket
$\lshad\,,\,\rshad$ are the main result of this paper.
The new understanding leaves intact but substantiates the bulk of results
which have been obtained by using various ad hoc techniques (that is,
explicitly or tacitly referring to the surreal equalities
$\boldsymbol{\delta}(0)=0$ and $\log\boldsymbol{\delta}(0)=0$); we refer to a
detailed review [3] for an account of early developments in BV-formalism. We
do not aim at a reformulation or reproduction of any old or recent
achievements, accomplishing here a different task.
In fact, we invent nothing new. It is the coupling of dual vector spaces which
ensures the intrinsic self-regularisation of BV-Laplacian and validity of
equalities (1), with (1c) in particular. Therefore, it would be redundant to
start developing any brand-new formalism (cf. [51]); on the other hand, we
prove properties (1) and not just postulate these assertions (cf. [21]).
We employ standard notions, constructions, and techniques from the geometry of
jet spaces [28, 40, 45]. Because the geometry of BV-objects is essentially
variational, it would be methodologically incomplete to handle them as if the
space-time, that is, the base manifold in the bundles of physical fields, were
just a point ([27, 48] or [39]). The language of jet spaces is extensively
used in the study of BV-models, see [3, 6, 21, 43]: the bundles of jets of
sections usually appear in such traditional contexts as calculation of
symmetries or conservation laws. In this paper we apply these geometric
techniques at a much more profound level and give rigorous definitions for BV-
objects. Let us emphasize that we do not aim at extending one’s ability to
write more formulas according to a regularly emended system of accepted
algorithms; we explicate the genuine nature of objects and their canonical
matchings, not taking any formulas for quasi-definitions.
This paper is structured as follows. Containing a brief overview of
traditional approaches to regularisation of the BV-formalism, this
introduction concludes with a parable; the line of our reasoning is
reminiscent to that of Lettres persanes by Montesquieu.
In section 1 we describe the true geometry of variations; we first reveal the
correspondence between action functionals and infinitesimal shifts of
classical trajectories or physical fields. An understanding of nontrivial
mechanism of such matching achieved for one variation, the picture of many
variations becomes clear. This approach resolves the obstructions for
regularisation of iterated variations in BV-formalism; we remark that Dirac’s
$\boldsymbol{\delta}$-function does not appear in section 2 at all.222We refer
to [19] for the theory of distributions. Let us specify that singular linear
integral operators which emerge in the course of our reasoning will not be
approached via parametric families of regular linear integral functionals with
piecewise continuous or smooth kernels (in which context the notation
“$\boldsymbol{\delta}(0)$” for Dirac’s function is used in the literature).
In section 2.1 we recall in proper detail the standard construction of
Batalin–Vilkovisky (BV) vector bundles with canonically conjugate pairs of
ghost parity-even and odd variables. In this specific setup we analyse the
construction of two distinct couplings of the BV-fibres’ ghost parity-
homogeneous vector subspaces with their respective duals. In particular, in
section 2.2 we focus on the rule of signs which determines the anti-
commutation of differential one-forms in the geometry at hand. Applying the
geometric concept of iterated variations in section 2.3, we represent the
left- and right variations of functionals in terms of left- or right-directed
singular linear integral operators; this framework ensures the intrinsic
regularisation of iterated variations. We then formulate in section 2.4 the
definitions of BV-Laplacian $\Delta$ and variational Schouten bracket
$\lshad\,,\,\rshad$ (or antibracket). We show that these definitions are
operational, amounting to natural, well-defined reconfigurations of the
geometry (but not to any hand-made algorithms for cancellation of divergent
terms; for those do not appear at all). Our main result, which is contained in
section 2.5, is an explicit proof – that is, starting from basic principles –
of relations (1). In other words, we neither postulate a validity of these
properties nor elaborate a cunning syllogism the aim of which would be to
convince why such assertions should hold provided that one knows when various
(derivatives of) Dirac’s $\boldsymbol{\delta}$-functions must be erased in the
course of so arguable a reasoning.
For consistency, we first apply the above theory to a standard derivation of
the quantum master-equation from the Schwinger–Dyson condition that
essentially eliminates a dependence on the unphysical, ghost parity-odd
dimensions (see section 3.1); we also recall here the construction of quantum
BV-differential. The point is that neither divergencies nor ad hoc
cancellations occur in the entire argument. On the same grounds we address in
section 3.2 the quantum BV-cohomology preservation by infinitesimal gauge
symmetries of the quantum master-equation. (We refer to [7, 8, 20, 22] and
also [1, 37, 51] in this context; several methodological comments, which
highlight our concept, are placed in section 3 along the lines of a well-known
reasoning.)
The paper concludes with a statement that an intrinsic regularisation in the
geometry of iterated variations relies on the principle of locality (which
manifests also through causality). We argue that a logical complexity of
geometric objects grows while they accumulate the (iterated) variations ; a
conversion of such composite-structure objects into maps which take physical
field configurations to numbers entails a decrease of the complexity via a
loss of information. Having motivated this claim in section 2, we prove that
the logic of analytic reasonings may not be interrupted ; for example, the
right-hand side of (1c) is not assembled from the would-be constituent blocks
$\Delta F$ and $\Delta G$ for which it is known in advance how they take field
configurations to numbers whenever the functionals $F$ and $G$ are given.
The paper explicitly answers the question what variations are — in particular,
what iterated variations are. Moreover, we tacitly describe a geometric
mechanism which is responsible for the anti-commutation of differential one-
forms ; such mechanism ensures that the results of calculations match empiric
data even if the exterior algebras of forms are introduced by hand. The roots
of this principle are none other that the ordering of dual vector spaces which
stem in the course of variations in models of nonlinear phenomena (this
picture is addressed in section 2.2).
We illustrate our approach with elementary starting section 1 in which we
inspect the matching of geometries –one for an action functional, the other
for a field’s test shift– in the course of derivation of Euler–Lagrange
equation of motion in field theory. The second example on pp. 2.4–33 clarifies
the idea specifically in the BV-setup of (anti)fields and (anti)ghosts. We
thus provide a pattern for all types of calculations which involve the
Schouten bracket and BV-Laplacian in any model.
### Historical context: an overview
There is a class of significant papers in which the BV-formalism is developed
under assumption that the space-time is a point. Indeed, such hypothesis is
equivalent to an agreement that the only admissible sections of bundles over
space-time are constant; this implies that even if their derivatives are
nominally present in some formulas, they are always equal to zero. The
calculus of variations then reduces to usual differential geometry on the
bundles’ fibres. It must be noted that publications containing the above
assumption did contribute to the subject and in many cases guided its further
development (we recall the respective comment in [51] and refer to [12, 22,
27, 39, 48, 53]). Moreover, the no-derivatives reduction sometimes allows one
to jump at conclusions which are correct; an integration by parts over the
base manifold $M^{n}$ is restored –whenever possible– at the end of the day.
Still this oversimplification is potentially dangerous because variational
calculus of integral functionals conceptually exceeds any classical
differential geometry on the fibres (see [33] for discussion and [28, 34]). In
the variational setup, the objects and their properties become geometrically
different from their analogues on usual manifolds even if the terminology is
kept unchanged. Here we recall for example that variational multivectors do
not split to wedge products of variational one-vectors and likewise, several
Leibniz rules are irreparably lost but this can not be noticed when all
derivatives equal zero. In fact, it is the abyss between classical geometry of
manifolds and geometry of variations for jet spaces of maps of manifolds which
motivated our earlier study [34]. Yet the misconception is still present in
active research, e.g., see [4, 27, 39, 44].
The fact of incompleteness of such heuristic analogies from usual geometry of
manifolds is signalled in [51]. Paradoxically, it is simultaneously not true
that a solution of the regularisation problem for BV-Laplacian has no
analogues in the case of ODE dynamics on manifolds. From section 1 below it is
readily seen that good old techniques persist in the finite-dimensional ODE
geometry at the level of standard linear algebra of dual vector spaces.333On
the other hand, the variational setup highlights the fundamental concept of a
physical field as a system with degrees of freedom attached at every point of
the space-time $M^{n}$; we focus on this aspect in what follows.
The article [51] is a considerable step towards a solution of the
regularisation problem in BV-formalism. A weighted, critical overview of
various inconsistencies, ad hoc practices, and roundabouts is summed up there.
The object of [51] was to formulate a self-contained analytic concept which
would make the variational calculus of functionals free from divergencies and
infinities. Still it remained unclear from [51] what the generality of
underlying geometry is and why such self-consistent formalism should actually
exist at the level of objects, i.e., beyond a mere ability to write formulas.
In particular, it remained unnoticed that the main motivating example –namely,
the canonical BV-setup– itself is the only class of geometries in which the
technique is grounded.444The integration of closed algebra of gauge symmetries
for the quantum master-equation to a group of transformations of the master-
action $S^{\hbar}$ remains a separate problem, which is also addressed in
[51]. Suppose that the standard cohomological obstructions to such integration
vanish (see section 3.2 below), whence (i) all infinitesimal transformations
of the functional $S^{\hbar}$ are exact, i.e., they are generated by odd
ghost-parity elements $F$, and also (ii) such transformations can be extended
from the master-action $S^{\hbar}$ to evolution of the observables
$\mathcal{O}$. We remark that, unlike it is claimed in [51], neither of the
two groups of functionals’ transformations is induced by any well-defined
change of BV-coordinates; of course, evolutionary vector fields are well-
defined objects in that geometry and one could study them regardless of these
functionals’ transformations. We shall recall in section 3.2 the standard
construction of automorphisms for quantum BV-cohomology groups; it illustrates
our concept because the notion of quantum gauge symmetries explicitly refers
to all basic properties of the BV-Laplacian and Schouten bracket, see (1) on
p. 1. A correctness but incompleteness of the approach in [51] means the
following in practice. Whenever a theorist refers to the formalism of loc.
cit., Nature immediately creates a new, principally inobservable essence –a
metric field which is denoted by $E(x_{1},\ldots,x_{n};\Gamma)$ in [51]– on
top of the electromagnetic and weak gauge connections, as well as the fields
for strong force, gravity, or any other gauge fields $\Gamma$. It is perhaps
this methodological difficulty which hints us why the approach of [51] is
considered “formal” by many experts; that conceptual paper remains scarcely
known to a wider community.555An attempt to interpret the formalism of [51] in
terms of the language of PDE geometry (particularly, in the context of [41],
see also [28, 40, 45]) was performed in [23] and published in abridged form in
[24]. The construction of Schouten bracket in [23] relies on the notion of
variational cotangent bundle [41] and on classical approach to the theory of
variations. On one hand, this ensures the validity of Jacobi identity for the
bracket (see the second half of Eq. (1d) but not the first one). But on the
other hand, we have showed by a counterexample in [35, §3] that the old
approach fails to relate by (1c) the Schouten bracket to BV-Laplacian. In
other words, the BV-Laplacian did not entirely generate the variational
Schouten bracket, making only Eq. (1b) but not (1c) possible in that geometry
(cf. [39]).
To demystify the notion of a “metric field $E(x_{1},\ldots,x_{n};\Gamma)$,” we
describe in this paper an elementary geometric mechanism for the long-expected
but still intuitively paradoxical analytic behaviour of variations. This
mechanism implies that Nature is not obliged to respond to the needs of a
theorist and create such multi-entry distributions upon request.
Another line of reasoning, which led to much progress in a revision of BV-
structures and regularisation of divergences, was pursued in [12, 13]. We
recall that the language of loc. cit. is functional analytic so that the
theory’s objects are viewed as (Dirac’s) distributions (and heat kernels are
implemented). According to [12, §1.8], the BV-Laplacian $\Delta$ which is used
in physical theories is ill-defined because for a given action $S$ over space-
time $M^{n}$ of positive dimension $n$ the object $\Delta S$ involves a
multiplication of singular distributions (and thus –a quotation from [12]
continues– $\Delta S$ has the same kind of singularities as appear in one-loop
Feynman diagrams). The regularisation technique proposed in [12, 13] stems
from analysis of the distributions’ limit behaviour as one approaches the
“physical” structures by using regular ones.
The resolution to apparent difficulties is that there are several distinct
geometric constructions which yield the same singular linear operators with
support on the diagonal (in what follows we study in detail on which space
such operators are defined).
We now discuss a peculiar, well-established domain, the very form of existence
of which could be hardly believed in. In that theory, there is a serious lack
of rigorous definitions for the most elementary objects; at the same time,
there is a rapidly growing number of monumental reviews. Whereas the theory’s
difficulties are clearly inherited from a deficit of boring rigour at the
initial stage, such hardships are proclaimed the theory’s immanent components.
At expert level it is mandatory to have a firm knowledge of the built-in
difficulties and readily classify the descriptive objects which those apparent
obstructions bring into the mathematical apparatus. (There is no firm
guarantee that the (un)necessary objects really exist beyond written
formulas.) The way of handling inconveniences largely amounts not to resolving
them by a thorough study of their origins but to some ad hoc methods for
hiding their presence. Doing research is thus substituted by practising a
ritual.
However, the community of experts who mature in operation with formulas (a
part of which are believed to express something objectively existing)
maintains a considerable pluralism about a proper way to mask the symptoms of
troubles:
* •
The radicals declare that undefined objects which seem to make trouble must be
set equal to zero.
* •
The revisionist approach prescribes a postfactum erasing of not the entire
objects (which are still undefined) but of undesirable elements in those
objects’ description.
* •
A diplomatic viewpoint is that there might be sources of trouble but their
contribution to final results is suppressed as soon as the objects’ desired
properties are postulated (regardless of the actual presence or absence of
such sources and one’s ability to substantiate those properties).
For an external observer, this state-of-the-art could seem atypical for a
consistent theory. Indeed, the reliability of its main pillar is a matter of
irrational belief.
## 1 The geometry of variations
Let us first analyse the basic geometry of variations of functionals; by
comprehending the full setup of a one-time variation, we shall understand the
geometry of many. Specifically, in this section we reveal the interrelation of
bundles in the course of integration by parts; we also explain a rigorous
construction of iterated variations.
The core of traditional difficulties in this domain is that a use of only
fibre bundles $\pi$ of physical fields, which are subjected to test shifts, is
insufficient. We argue that the tangent bundles $T\pi$ to the bundles $\pi$
may not be discarded (see Fig. 1).
${\boldsymbol{u}}$$W_{{\boldsymbol{x}}}=\text{fibre in }T\pi$$\delta
s({\boldsymbol{x}})\in
T_{s({\boldsymbol{x}})}\pi^{-1}({\boldsymbol{x}})$${\boldsymbol{x}}$$M^{n}$$\pi$$s$Fibrebundle:
Figure 1: The fibre bundle $\pi$ of fields $s$ and vector bundle $T\pi$ of
their variations $\delta s$.
For identities (1) to hold one must substantiate why higher-order variational
derivatives are (graded-)permutable whenever one inspects the response of a
given functional to shifts of its argument along several directions. To
resolve the difficulties, we properly enlarge the space of functionals and
adjust a description of the geometry for the functionals’ variations: in fact,
each variation brings its own copy of the base $M^{n}$ into the picture (see
Fig. 3 on p. 3).
### 1.1 Notation
We now fix some notations, in most cases matching that from [28] (for a more
detailed exposition of these matters, see for example [28, 40, 45]).
Let $\pi\colon E\to M$ be a smooth fibre bundle666Vector bundles are primary
examples but we do not actually use the linear vector space structure of their
fibres so that $\pi$ could be any smooth fibre bundle. with $m$-dimensional
fibres $\pi^{-1}({\boldsymbol{x}})$ over points ${\boldsymbol{x}}$ of a smooth
real oriented manifold $M$ of dimension $n$; we assume that all mappings,
including those which determine the smoothness class of manifolds, are
infinitely smooth.
We let $x^{i}$ denote local coordinates in a chart $U_{\alpha}\subseteq M^{n}$
and $u_{j}$ be the fibre coordinates. We denote by $[{\boldsymbol{u}}]$ a
differential dependence of the fibre variables (specifically in the BV-setup,
a differential dependence $[{\boldsymbol{q}}]$ on physical fields and other
ghost parity-even variables, and we denote by $[{\boldsymbol{q}}^{\dagger}]$
that of ghost parity-odd BV-variables).
###### Remark 1.1.
We suppose that the initially given bundle $\pi$ of physical fields is not
graded. In what follows, starting with $\pi$, we shall construct new bundles
whose fibres are endowed with the $\mathbb{Z}_{2}$-valued ghost parity
$\operatorname{gh}(\,\cdot\,)$. However, our reasoning remains valid for
superbundles $\pi^{(0|1)}$ over supermanifolds $M^{(n_{0}|n_{1})}$ ([10, 52])
and to a noncommutative setup of cyclic-invariant words (see [29, 32] and
references therein), cf. Fig. 8 below.
We take the infinite jet space $\pi_{\infty}\colon J^{\infty}(\pi)\to M$
associated with this bundle [15, 45]; a point from the jet space is then
$\theta=(x^{i},{\boldsymbol{u}},u^{k}_{x^{i}},u^{k}_{x^{i}x^{j}},\dots,{\boldsymbol{u}}_{\sigma},\dots)\in
J^{\infty}(\pi)$, where $\sigma$ is a multi-index and we put
${\boldsymbol{u}}_{\varnothing}\equiv{\boldsymbol{u}}$. If $s\in\Gamma(\pi)$
is a section of $\pi$, or a field, we denote by $j^{\infty}(s)$ its infinite
jet, which is a section $j^{\infty}(s)\in\Gamma(\pi_{\infty})$. Its value at
${\boldsymbol{x}}\in M$ is
$j^{\infty}_{\boldsymbol{x}}(s)=(x^{i},s^{\alpha}(x),\frac{\partial
s^{\alpha}}{\partial
x^{i}}(x),\dots,\frac{\partial^{|\sigma|}s^{\alpha}}{\partial
x^{\sigma}}(x),\dots)\in J^{\infty}(\pi)$.
We denote by $\mathcal{F}(\pi)$ the properly understood algebra of finite
differential order smooth functions on the infinite jet space
$J^{\infty}(\pi)$, see [28, 40] for details. The space of top-degree
horizontal forms on $J^{\infty}(\pi)$ is denoted by
$\overline{\Lambda}^{n}(\pi)$; let us also assume that at every
${\boldsymbol{x}}\in M$ a volume element
$\operatorname{dvol}({\boldsymbol{x}})$ is specified so that its pull-back
under $\pi^{*}_{\infty}$ is an $n$-th degree form in
$\overline{\Lambda}^{n}(\pi)$, cf. Remark 1.5 on p. 1.5.
The highest horizontal cohomology, i. e., the space of equivalence classes of
$n$-forms from $\overline{\Lambda}^{n}(\pi)$ modulo the image of the
horizontal exterior differential $\overline{{\mathrm{d}}}$ on
$J^{\infty}(\pi)$, is denoted by $\overline{H}^{n}(\pi)$; the equivalence
class of $\omega\in\overline{\Lambda}^{n}(\pi)$ is denoted by
$\int\omega\in\overline{H}^{n}(\pi)$. We assume that sections
$s\in\Gamma(\pi)$ are such that integration of functionals
$\Gamma(\pi)\to\Bbbk$ by parts is allowed and does not result in any boundary
terms (for example, the base manifold is closed, or the sections all have
compact support, or decay sufficiently fast towards infinity, or are
periodic).
### 1.2 Euler–Lagrange equations
A derivation of Euler–Lagrange equations ${\mathcal{E}}_{\text{{EL}}}$ for a
given action functional
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])\operatorname{dvol}({\boldsymbol{x}})$
is a model example which illustrates the correlation of two geometries:777An
arrow over a variational derivative indicates the direction along which the
shift $\delta s$ is transported left- or rightmost. While the objects are non-
graded commutative, this indication is not important. It becomes mandatory in
the $\mathbb{Z}_{2}$-graded commutative setup (see section 2): likewise, the
arrows are also mandatory and fix the direction of rotation for non-
commutative cyclic words [29, 32, 36]; note that our formalism is extended
verbatim to the variational calculus of such necklaces and their brackets. one
for “trajectories” $s\in\Gamma(\pi)$ and the other for shifts $\delta s$. It
is well known that the functional’s response to a test shift $\delta s$ of its
argument $s\in\Gamma(\pi)$ is described by the formula [2, §12]
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}S(s+\varepsilon\cdot\overleftarrow{\delta}\\!s)=\int_{M}\operatorname{dvol}({\boldsymbol{x}})\>\delta
s({\boldsymbol{x}})\cdot\left.\frac{\overleftarrow{\delta}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\delta{\boldsymbol{u}}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
(3)
We now claim that this one-step procedure is a correct consequence of
definitions but itself not a definition of the functional’s variation. The
above formula conceals a longer, nontrivial reasoning of which the right-hand
side in (3) is an implication — provided that the functional $S$ will not be
varied by using any other test shifts, i. e., if the correspondence
$S\mapsto{\mathcal{E}}_{\text{{EL}}}$ yields the object
${\mathcal{E}}_{\text{{EL}}}$ of further study (cf. [2, §13]). Indeed, we
notice that the left-hand side of (3) refers to three bundles (namely, the
fibre bundle $\pi$ for a section $s\in\Gamma(\pi)$ whose infinite jet is
$j^{\infty}(s)\in\Gamma(\pi_{\infty})$, the bundle $\pi_{\infty}$ for the
integral functional $S\in\overline{H}^{n}(\pi)$, and the tangent vector bundle
$T\pi$ such that $\delta s\in\Gamma(T\pi)$ at the graph of $s$ in $\pi$, see
Fig. 1. (In what follows, a reference to attachment points
$s({\boldsymbol{x}})\in\pi^{-1}({\boldsymbol{x}})$ will always be implicit in
the notation for $\delta s$: for a given section $s\in\Gamma(\pi)$, the base
manifold $M^{n}$ is the domain of definition for a test shift $\delta
s({\boldsymbol{x}},s({\boldsymbol{x}}))=\delta s({\boldsymbol{x}})$ that takes
values in $T_{s({\boldsymbol{x}})}\pi^{-1}({\boldsymbol{x}})$.) Let us figure
out how the domains of definition for the sections $s$ and $\delta s$ merge to
one copy of the manifold $M^{n}$ over which an integration is performed in the
right-hand side of (3). Strictly speaking, from (3) it is unclear whether the
variational derivative,
$\frac{\overleftarrow{\delta}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\delta{\boldsymbol{u}}}=\sum_{|\sigma|\geq
0}\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma}\frac{\vec{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}},$
stems from one (which would be false) or both (true!) copies of the base $M$.
To have a clear vision of the variations’ geometry and by this avoid an
appearance of phantoms in description, we now vary the action functional $S$
at $s\in\Gamma(\pi)$ along $\delta s\in\Gamma(T\pi)$, commenting on each step
we make. In fact, it suffices to figure out where the objects and structures
at hand belong to — in particular, we should explain the nature of binary
operation $\cdot$ in the right-hand side of conventional formula (3). The key
idea is to understand what we are actually doing but not what we have got used
to think we do in order to obtain an understandable result [2, §13]. The
discovery is that this “multiplication of functions” is a shorthand notation
for the canonically defined coupling between vectors and covectors from
(co)tangent spaces $W_{s({\boldsymbol{x}})}$ and
$W^{\dagger}_{s({\boldsymbol{x}})}$, respectively, at the points
$s({\boldsymbol{x}})$ of fibres $\pi^{-1}({\boldsymbol{x}})$ in the bundle
$\pi$.
To encode this linear-algebraic setup, let $i,j$ run from 1 to
$m=\dim(\pi^{-1}({\boldsymbol{x}}))=\operatorname{rank}(T\pi)$ and take a
local basis $\vec{e}_{i}({\boldsymbol{y}})$ in the tangent spaces
$W_{s({\boldsymbol{y}})}=T_{s({\boldsymbol{y}})}(\pi^{-1}({\boldsymbol{y}}))$
at $s({\boldsymbol{y}})$ over base points ${\boldsymbol{y}}\in M$. Introduce
the dual basis $\vec{e}^{{}\,\dagger j}({\boldsymbol{x}})$ in
$W^{\dagger}_{s({\boldsymbol{x}})}$ attached at $s({\boldsymbol{x}})$ over
${\boldsymbol{x}}\in M$. By construction, this means that the value
$\left\langle\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\right\rangle$ (4)
is equal to the Kronecker symbol $\delta_{i}^{j}$ if and only if
${\boldsymbol{x}}={\boldsymbol{y}}$ and the vector
$\vec{e}_{i}({\boldsymbol{y}})\in W_{p_{1}}$ and covector
$\vec{e}^{{}\,\dagger j}({\boldsymbol{x}})\in W^{\dagger}_{p_{2}}$ are
attached at the same point $p_{1}=p_{2}$ of the fibre
$\pi^{-1}({\boldsymbol{x}})$ over ${\boldsymbol{x}}={\boldsymbol{y}}\in M$.
The locality of this coupling is an absolute geometric postulate: the coupling
is not defined whenever ${\boldsymbol{x}}\neq{\boldsymbol{y}}$ or the values
$p_{1}=s_{1}({\boldsymbol{y}})$ and $p_{2}=s_{2}({\boldsymbol{x}})$ of two
local sections $s_{1},s_{2}\in\Gamma(\pi)$ are not equal at
${\boldsymbol{x}}={\boldsymbol{y}}$. Physically speaking, the coupling is then
not defined because there is no channel of information which would communicate
the value $\delta s^{i}({\boldsymbol{y}})\cdot\vec{e}_{i}({\boldsymbol{y}})$
of excitation of the physical field $s\in\Gamma(\pi)$ at a point
${\boldsymbol{y}}\in M$ to another point
${\boldsymbol{x}}\neq{\boldsymbol{y}}$ of the space-time $M$.
###### Remark 1.2.
Let us remember that the definition of coupling between sections of
(co)tangent bundles — i. e., (co)tangent to either a given manifold or a given
bundle $\pi$ which is the case here for Euler–Lagrange equations — forces the
congruence $\\{{\boldsymbol{x}}={\boldsymbol{y}},\
s_{1}({\boldsymbol{y}})=s_{2}({\boldsymbol{x}})\\}$ of the (co)vectors’
attachment points. We notice further that such congruence mechanism does not
refer to any limiting procedure for smooth distributed kernels and regular
linear operators on the space of (co)vector fields. Indeed, vectors couple
with their duals at a given point regardless of any phantom limiting procedure
which would grasp the (co)vector’s values at any other points of the
manifold.888We recall that a similar, purely local geometric principle, not
referring to the objects’ values at non-coinciding points, works in the
definition of Hirota’s bilinear derivative.
###### Remark 1.3.
The coupling is a matching between test-shift vector fields which are tangent
to the fibres of $\pi$ and, on the other hand, with the elements of
$\Gamma(T^{*}\pi)$ which are determined by the Lagrangian $\mathcal{L}$. This
binary operation yields the singular integral operator
$\int_{M}{\mathrm{d}}{\boldsymbol{y}}\,\langle\delta
s^{i}({\boldsymbol{y}})\vec{e}_{i}({\boldsymbol{y}})|$ with support on the
diagonal. Independently, the same operator can reappear as the limit in a
parametric family of regular integral operators with smooth, distributed
kernels. This shows that the same object is constructed by using several
algorithms. Yet the analytic behaviour of the limit is determined not only by
the limit itself but also by an algorithm how it is attained. Consequently,
the object’s analytic properties in the course of derivations could be (and
actually, indeed they are) drastically different for different scenarios. This
is the key point in a regularisation of the formalism; to achieve this goal,
we properly identify the objects which are de facto handled.
###### Remark 1.4.
Referring to a concept of locality of events, this definition of coupling
$\langle\,,\,\rangle$ ensures a very interesting analytic behaviour of the
value $\langle\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})\rangle$ of pairing for dual objects
$\vec{e}_{i}({\boldsymbol{y}})$ and $\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})$ at fixed $i$. Namely, this value is a constant scalar
field which equals unit $1\in\Bbbk$ at all points of the manifold $M$; the
scalar field’s partial derivatives with respect to $x^{j}$ or $y^{k}$, $1\leq
j,k\leq n$, vanish identically. We shall use this property in what follows
(see Remark 1.7 on p. 1.7). We also note that the logarithm of this coupling’s
unit value vanishes as well:
$\log\langle\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})\rangle=0$ whenever the coupling is well defined and
$1\leq i\leq m$.
Now let us return to the initial setup in context of Euler–Lagrange equation
${\mathcal{E}}_{\text{{EL}}}$ and one-step correspondence
$S\mapsto{\mathcal{E}}_{\text{{EL}}}$, see Fig. 1. We have that
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])\operatorname{dvol}({\boldsymbol{x}})$
is an integral functional; we let $s\in\Gamma(\pi)$ be a background section
(e. g., a sought-for solution of the Euler–Lagrange stationary point equation
$\left.\delta S\right|_{s}=0$) and $\delta s\in\Gamma(T\pi)$ be a test shift
of $s$. The linear term in a response of $S\colon\Gamma(\pi)\to\Bbbk$ to a
shift of its argument $s$ along $\delta s$ is (cf.(14) on p. 14)
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}S(s+\varepsilon\overleftarrow{\delta}\\!s)={}\\\
{}=\sum_{i,j}\sum_{|\sigma|\geq
0}\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle(\delta
s^{i})\left(\frac{\smash{\overleftarrow{\partial}}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\,\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u_{\sigma}^{j}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\rangle.$ (5)
###### Remark 1.5.
The rôles of two integral signs in (5) are different. Namely, the volume form
$\operatorname{dvol}({\boldsymbol{x}})$ at ${\boldsymbol{x}}\in M^{n}$ comes
from the integral functional $S\in\overline{H}^{n}(\pi)$; should a formal
choice of the volume form be different, the Euler–Lagrange equations would
also change.999There are natural classes of geometries in which the Lagrangian
$\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])$ in the action $S$ is a
well-defined top-degree differential form, e. g., if the unknowns
${\boldsymbol{u}}$ are differential one-forms (we recall the Yang–Mills or
Chern–Simons gauge theories in this context). Let us remember also that a
construction of $\mathcal{L}$ could refer to a choice of volume form
$\operatorname{dvol}({\boldsymbol{x}})$ on $M^{n}$. For instance, such is the
case when the Hodge structure $*$ is involved (the Yang–Mills Lagrangian
yields an example: $\mathcal{L}\sim F_{\mu\nu}*F^{\mu\nu}$ in standard
notation for the stress tensor). To avoid excessive case-study, we use a
uniform notation thus writing $\operatorname{dvol}({\boldsymbol{x}})$
explicitly. We recall further that the integration measure
$\operatorname{dvol}\bigl{(}{\boldsymbol{x}},s({\boldsymbol{x}})\bigr{)}={\sqrt{|\det\bigl{(}g_{\mu\nu}({\boldsymbol{x}},s)\bigr{)}|}}{\mathrm{d}}{\boldsymbol{x}}$
is field-dependent by virtue of Einstein’s general relativity equations which
–i̇n their right-hand sides – absorb the energy-momentum tensor of physical
fields $s\in\Gamma(\pi)$. The volume element will be denoted by
$\operatorname{dvol}({\boldsymbol{x}})$ in order to emphasize that the space-
time $M^{n}$ is unique: Namely, field-dependent objects interact at its points
only if the local geometry of underlying space-time is the same near
${\boldsymbol{x}}\in M^{n}$ for all objects (see Theorem 3 and Remark 2.11 on
p. 2.11 for a realisation of this principle for the smooth manifold $M^{n}$
endowed with metric tensor $g_{\mu\nu}$). At the same time, the other integral
sign $\int{\mathrm{d}}{\boldsymbol{y}}$ denotes the singular linear operator
$\Gamma(T^{*}\pi)\to\Bbbk$ with support on the diagonal [19]; in fact, this
notation means that a point ${\boldsymbol{y}}$ runs through the entire
integration domain $M$.
### 1.3 Integration by parts
The most interesting things start to happen when one integrates by parts over
the domain $M^{n}$ of test shifts $\delta s$. (By default, we let the supports
of local perturbations $\delta s$ be such that no boundary terms appear in the
course of integration by parts over $M$.)
For the sake of transparency let us first consider a model situation when
there is just one derivative falling on $\delta s$ at ${\boldsymbol{y}}$; all
higher-order cases are processed recursively. By the definition of a (partial)
derivative $\partial/\partial y^{i}$, we have that101010In the definition of
derivative, the calculation of length $|\Delta{\boldsymbol{y}}|$ in
denominators refers to the standard Euclidean metric in the linear vector
spaces which determine coordinate neighbourhoods near points of the manifold
$M$ at hand.
$\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle(\delta
s)\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}({\boldsymbol{y}})\,\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger}({\boldsymbol{x}})\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\rangle=\\\
=-\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s({\boldsymbol{y}})\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}}\left\\{\langle\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger}({\boldsymbol{x}})\rangle\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\\}.$
By using a definition of the partial derivative which falls on the comultiple
of $\delta s$, we obtain the difference111111Here and in the equalities below
we suppress the indexes $i$ running through $1,\dots,m$ at $\delta
s^{i}({\boldsymbol{y}})$ and $\vec{e}_{i}({\boldsymbol{y}})$ or
$\vec{e}^{{}\,\dagger i}({\boldsymbol{x}})$, or at $u^{i}_{\sigma}$ in the
derivative which acts on $\mathcal{L}$; we thus avoid an agglomeration of
formulas.
${}\stackrel{{\scriptstyle\text{def}}}{{=}}-\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s({\boldsymbol{y}})\cdot\lim_{|\Delta{\boldsymbol{y}}|\to
0}\frac{1}{|\Delta{\boldsymbol{y}}|}\Bigl{\\{}\left\langle\vec{e}({\boldsymbol{y}}+\Delta{\boldsymbol{y}}),\vec{e}({\boldsymbol{x}})\right\rangle\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}-\\\
-\left\langle\vec{e}({\boldsymbol{y}}),\vec{e}({\boldsymbol{x}})\right\rangle\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\Bigr{\\}}.$
The locality postulate for coupling between (co)vectors $\vec{e}$ and
$\vec{e}^{{}\,\dagger}$ forces the equality
${\boldsymbol{y}}+\Delta{\boldsymbol{y}}={\boldsymbol{x}}$ in the minuend,
which yields the two different points at which the restriction of Lagrangian
$\mathcal{L}$ to the jet $j^{\infty}(s)$ of section $s\in\Gamma(\pi)$ is
evaluated:
$=-\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s({\boldsymbol{y}})\cdot\langle\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger}({\boldsymbol{x}})\rangle\cdot\\\
\cdot\lim_{|\Delta{\boldsymbol{y}}|\to
0}\frac{1}{|\Delta{\boldsymbol{y}}|}\Bigl{\\{}\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}+\Delta{\boldsymbol{y}}}(s)}-\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\Bigr{\\}}.$
(Here we use the fact that the scalar product $\langle\,,\,\rangle$, whenever
defined, is the Kronecker symbol.) We continue the equality,
$\stackrel{{\scriptstyle\text{def}}}{{=}}-\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle\delta
s({\boldsymbol{y}})\,\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger}({\boldsymbol{x}})\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{x}}}\left(\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right)\right\rangle.$
We finally recall that the total derivative
${\mathrm{d}}/{\mathrm{d}}{\boldsymbol{x}}$ is defined121212By definition,
$(\vec{{\mathrm{d}}}f/{\mathrm{d}}x^{i}){\bigr{|}}_{j^{\infty}(s)}({\boldsymbol{x}})=\bigl{(}\vec{\partial}/\partial
x^{i}(f{\bigr{|}}_{j^{\infty}(s)})\bigr{)}({\boldsymbol{x}})$ for differential
functions $f$, see [28, 40, 45]. via an application of
$\partial/\partial{\boldsymbol{x}}$ to restriction to infinite jets
$j^{\infty}(s)$ of sections $s$ at base points ${\boldsymbol{x}}$. Therefore,
the above expression is equal to
$\stackrel{{\scriptstyle\text{def}}}{{=}}\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle\delta
s({\boldsymbol{y}})\,\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger}({\boldsymbol{x}})\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}_{\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\rangle.$
This shows that an integration by parts over the base $M$ in the geometry of
test shift $\delta s$ reappears as integration by parts in the bundle where
lives the background section $s\in\Gamma(\pi)$.
Repeating the integration by parts $|\sigma|\geq 0$ times in each term of the
sum in (5), we obtain the expression
$\sum_{i,j}\sum_{|\sigma|\geq
0}\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle\delta
s^{i}({\boldsymbol{y}})\,\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma}\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}^{j}_{\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\rangle.$
Let us recall once more that the coupling’s support is the diagonal in
$M\times M$, at points of which the value
$\langle\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\rangle$ is the Kronecker symbol $\delta_{i}^{j}$.
Consequently, we arrive at
${}=\sum_{i,j}\sum_{|\sigma|\geq
0}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s^{i}({\boldsymbol{x}})\cdot\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma}\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial{\boldsymbol{u}}^{i}_{\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
This is formula (3); it is familiar from any textbook on variational
principles of classical mechanics (e. g., see [2, §12–13]).
A standard reasoning shows that, whenever a response of the functional’s value
$S(s)$ to a test shift of $s$ along any direction $\delta s$ vanishes, the
Euler–Lagrange equation holds:
$\left.\frac{\overleftarrow{\delta}\\!S}{\delta{\boldsymbol{u}}}\right|_{j^{\infty}(s)}=0.$
(6)
Its left-hand side belongs to the space $\Gamma(T^{*}\pi)$ of sections of the
cotangent bundle to $\pi$.
###### Remark 1.6.
This conclusion tells us that traditional attempts of a brute-force labelling
of equations in a given system (6) by using the unknowns ${\boldsymbol{u}}$ is
not geometric. Indeed, the equations’ left-hand sides are sections of a vector
bundle, thus forming linear $\Bbbk$-vector spaces so that addition is well
defined for the equations within a system. On the other hand, the fibres in
the bundle $\pi$ can be smooth manifolds (i. e., not necessarily being vector
spaces) so that one may not add points of those fibres; for such operation is
in general not defined at all. Even if $\pi$ is a vector bundle, the fibres of
which are endowed with linear vector space structure, the two structures are
not related.
###### Remark 1.7.
The integration by parts transforms a derivative
$\partial/\partial{\boldsymbol{y}}$ along one copy of the base $M$ to the
minus derivative $-\partial/\partial{\boldsymbol{x}}$ along the other copy.
This produces no visible effect on the mechanism which ensures a restriction
onto the diagonal in $M\times M$, i. e., there appears no would-be third term
in the Leibniz rule for the product which is defined only on the diagonal. A
desperate prescription (2) was introduced in the literature in order to mimick
this paradoxical analytic behaviour of the coupling between elements of dual
bases.
### 1.4 Why are variations permutable ?
Having outlined the matching of geometries in the course of one sequence of
integrations by parts for one fixed pair $M\times
M\ni({\boldsymbol{y}},{\boldsymbol{x}})$ of copies of the base manifold, we
emphasize that such integrations must be performed last, i. e., only when the
objects at hand are finally viewed as maps $\Gamma(\pi)\to\Bbbk$.
Should one haste in absence of clear understanding of what is actually being
done and for which purpose, further calculation of higher-order variations
could predictably but uncontrollably lead to meaningless, manifestly erroneous
conclusions (e. g., compare left- and right-hand sides in (7) below).
Namely, there exist integral functionals which determine equal maps
$\Gamma(\pi)\to\Bbbk$ but, belonging to different spaces, behave differently
in the course of variations, should one attempt any. We say that such
functionals are synonyms; for instance, see Example 2.4 in the next section
for a nontrivial synonym $\Delta G$ of the zero functional (cf. Fig. 2).
Map:$\Gamma(\pi_{{\text{{BV}}}})\to\Bbbk$${}\neq
0.$Obj:$\overbrace{[\\![\underbrace{\ \ F\ \ }{},\underbrace{\ \ \Delta G\ \
}{}]\\!]}{}$$\int$0.$\int$ Figure 2: The synonyms $\Delta G$ of zero
functional yield constant maps $0\colon\Gamma(\pi_{{\text{{BV}}}})\to\Bbbk$
yet they can nontrivially contribute to larger structures such as $\lshad
F,\Delta G\rshad$, see Example 2.4 on p. 2.4.
Informally speaking, the composite structure objects with repeated integrals
over products $M\times M\times\ldots\times M$ of the base retain a kind of
memory of the way how they were obtained from primary objects such as the
action $S$. Let us illustrate these claims.
###### Example 1.1.
Let $\delta s_{1}\in\Gamma(T\pi)$ be a test shift at $s\in\Gamma(\pi)$ for an
integral functional
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])\operatorname{dvol}({\boldsymbol{x}})$
with density $\mathcal{L}$ of positive differential order. (That is, we
suppose that some positive-order derivatives are always present in densities
of all representatives of the cohomology class $S\in\overline{H}^{n}(\pi)$;
this assumption is not to any extent restrictive but it allows us to not take
into account $\overline{{\mathrm{d}}}$-exact terms whose orders may not be
bounded.) By using $S$, let us construct two new integral functionals. First,
we set
$F=\sum_{i}\sum_{|\sigma|\geq
0}\int\operatorname{dvol}({\boldsymbol{x}})\,\delta
s_{1}^{i}({\boldsymbol{x}})\cdot\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma}\left(\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{i}_{\sigma}}\right)\in\overline{H}^{n}(\pi),$
so that the mapping $F\colon\Gamma(\pi)\to\Bbbk$ is defined at
$s\in\Gamma(\pi)$ by restriction of the integrand to the jet $j^{\infty}(s)$
and then by actual integration over $M$.
Let the other functional $G\in\overline{H}^{2n}(\pi,T\pi)$ be such that its
value at the same section $s\in\Gamma(\pi)$ is
$G(s)=\sum_{i,j}\sum_{|\sigma|\geq
0}\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\left\langle(\delta
s_{1}^{i})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\,\vec{e}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\left.\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{j}_{\sigma}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\rangle.$
From the previous section it is clear that $F$ and $G$ are indistinguishable
as mappings to $\Bbbk$ for every $s\in\Gamma(\pi)$. Yet their variations, i.
e., the responses to an extra shift $\delta s_{2}\in\Gamma(T\pi)$, are
different. Indeed, they are equal to, first,
$\left(\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{2}}\right|_{\varepsilon_{2}=0}F\right)(s+\varepsilon_{2}\overleftarrow{\delta}\\!\\!s_{2})=\sum_{i_{1},i_{2}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq
0\\\ |\sigma_{2}|\geq
0\end{subarray}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s_{2}^{i_{2}}({\boldsymbol{x}})\delta s_{1}^{i_{1}}({\boldsymbol{x}})\cdot\\\
\cdot\left.\left\\{\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{2}}\frac{\overrightarrow{\partial}}{\partial
u^{i_{2}}_{\sigma_{2}}}\left(\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}}\frac{\overrightarrow{\partial}\\!\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{i_{1}}_{\sigma_{1}}}\right)\right\\}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
The above formula corresponds to a step-by-step calculation within a naïve
approach to the geometry of variations. However, the genuine value of second
variation of the integral functional $S$ along $\delta s_{1}$ and then $\delta
s_{2}$ at a section $s$ is
$\left(\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{2}}\right|_{\varepsilon_{2}=0}G\right)(s+\varepsilon_{2}\overleftarrow{\delta}\\!\\!s_{2})=\sum_{\begin{subarray}{c}i_{1},i_{2}\\\
j_{1},j_{2}\end{subarray}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\\\
\left\\{(\delta
s^{i_{2}}_{2})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{y}}_{2})\,\langle\vec{e}_{i_{2}}({\boldsymbol{y}}_{2}),\vec{e}^{{}\,\dagger
j_{2}}({\boldsymbol{x}})\rangle\cdot(\delta
s^{i_{1}}_{1})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{y}}_{1})\,\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}})\rangle\right\\}\\\
\cdot\left.\frac{\overrightarrow{\partial}^{2}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{j_{2}}_{\sigma_{2}}\partial
u^{j_{1}}_{\sigma_{1}}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
The analytic distinction between the operators
$\underbrace{\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{2}}\circ\frac{\overrightarrow{\partial}}{\partial
u^{i_{2}}_{\sigma_{2}}}\circ\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}}\circ\frac{\overrightarrow{\partial}}{\partial
u^{i_{1}}_{\sigma_{1}}}}_{\text{na\"{\i}ve approach}}\quad\text{ and
}\quad\underbrace{\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}\cup\sigma_{2}}\circ\frac{\overrightarrow{\partial}^{2}}{\partial
u^{i_{2}}_{\sigma_{2}}\partial u^{i_{1}}_{\sigma_{1}}}}_{\text{geometric
theory}}$ (7)
reveals why in positive-order Lagrangian models it is forbidden to haste,
which would imply that the derivatives along distinct copies of $M$ for
variations $\delta s_{1},\ \ldots,\ \delta s_{k}$ are too early transformed to
derivatives along the functional’s own base. Such a conceptual error would
repercuss with inexplicable, redundant terms in variations to-follow.
On the other hand, as soon as the product-bundle geometry of iterated
variations is properly realized — so that all restrictions to the diagonals
are postponed as late as possible, — the variations become
(graded-)permutable.131313An idea that iterated variations must be taken at
nominally different points ${\boldsymbol{x}}$ and ${\boldsymbol{y}}$ has been
in the air for a long time (let us refer to [38, §1] which contains due
credits to E. Witten). A somewhat less obvious fact is that those different
points belong to different copies of the manifold $M$ in the product bundle
${\pi\times T\pi\times\ldots\times T\pi}$ over ${M\times M\times\ldots\times
M}$. Namely, denote by $|u^{i}|$, $1\leq i\leq m$, the overall
$\mathbb{Z}_{2}$-valued parities of the fibre coordinates $u^{i}$; the ghost
parity $\operatorname{gh}(u^{i})$ or individual $\mathbb{Z}$\- or
$\mathbb{Z}_{2}$-valued gradings in the bundle $\pi$ contribute additively to
$|u^{i}|$ and then a residue modulo 2 is taken. Suppose that $\delta
s_{1}=(\delta s_{1}^{i_{1}})$ and $\delta s_{2}=(\delta s_{2}^{i_{2}})$ are
test shifts and
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])\operatorname{dvol}({\boldsymbol{x}})$
is an integral functional which maps a section $s\in\Gamma(\pi)$ to $\Bbbk$.
Then, _after_ the integrations by parts in the product-bundle geometry
$\pi\times T\pi\times T\pi$ which is described above, there remains
$\sum_{i_{1},i_{2}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s_{2}^{i_{2}}({\boldsymbol{x}})\delta
s_{1}^{i_{1}}({\boldsymbol{x}})\left.\left(\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}\cup\sigma_{2}}\frac{\overrightarrow{\partial}^{2}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{i_{2}}_{\sigma_{2}}\partial
u^{i_{1}}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\\\
=\sum_{i_{1},i_{2}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}(-)^{|u^{i_{1}}|\cdot|u^{i_{2}}|}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s_{1}^{i_{1}}({\boldsymbol{x}})\delta
s_{2}^{i_{2}}({\boldsymbol{x}})\left.\left(\left(-\frac{\overrightarrow{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}\cup\sigma_{2}}\frac{\overrightarrow{\partial}^{2}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{i_{1}}_{\sigma_{1}}\partial
u^{i_{2}}_{\sigma_{2}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
Likewise, higher-order iterated variations with $k\geq 2$ test shifts $\delta
s_{1},\dots,\delta s_{k}$ are (graded-)permutable with the same rule of signs
for permutations of order in which the (graded) partial derivatives
$\overrightarrow{\partial}/\partial
u^{i_{1}}_{\sigma_{1}},\dots,\overrightarrow{\partial}/\partial
u^{i_{k}}_{\sigma_{k}}$ fall from the left on the density $\mathcal{L}$ of the
functional $S$. (A case of $\mathbb{Z}_{2}$-graded base manifold
$M^{(n_{0}|n_{1})}$ would bring more signs which are also captured in a
standard way.)
Let there be $k\geq 2$ variations $\delta s_{1},\dots,\delta
s_{k}\in\Gamma(T\pi)$. We finally have that
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{k}}\right|_{\varepsilon_{k}=0}\circ\ldots\circ\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{1}}\right|_{\varepsilon_{1}=0}S(s+\varepsilon_{1}\overleftarrow{\delta}\\!\\!s_{1}+\ldots+\varepsilon_{k}\overleftarrow{\delta}\\!\\!s_{k})=\\\
=\sum_{\begin{subarray}{c}i_{1},\dots,i_{k}\\\
j_{1},\dots,j_{k}\end{subarray}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
\dots\\\ |\sigma_{k}|\geq
0\end{subarray}}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{k}\ldots\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\cdot{}\\\
\left\\{(\delta
s_{k}^{i_{k}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{k}}\right)^{\sigma_{k}}({\boldsymbol{y}}_{k})\left\langle\vec{e}_{i_{k}}({\boldsymbol{y}}_{k}),\vec{e}^{{}\,\dagger
j_{k}}({\boldsymbol{x}})\right\rangle\cdot\ldots\cdot(\delta
s_{1}^{i_{1}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{y}}_{1})\left\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}})\right\rangle\right\\}\\\
\cdot\left.\frac{\overrightarrow{\partial}^{k}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{u}}])}{\partial
u^{j_{k}}_{\sigma_{k}}\ldots\partial
u^{j_{1}}_{\sigma_{1}}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$ (8)
Whenever any $k-1$ variation(s) are fixed in the above formula, the co-
multiple $|\,\rangle$ of the remaining, $\ell$th variation $\delta
s_{\ell}=\langle\delta
s^{i_{\ell}}_{\ell}({\boldsymbol{y}}_{k})\vec{e}_{i_{\ell}}({\boldsymbol{y}}_{\ell})|$
is an element of the cotangent vector space
$T^{*}_{s({\boldsymbol{x}})}\pi^{-1}({\boldsymbol{x}})=V_{{\boldsymbol{x}}}^{\dagger}$
at the point $s({\boldsymbol{x}})$ in the fibre $\pi^{-1}({\boldsymbol{x}})$
over a base point ${\boldsymbol{x}}\in M^{n}$.
###### Remark 1.8.
The composite object in the left-hand side of equality (8) is an integral
functional in the bundle ${\pi\times T\pi\times\ldots\times T\pi}$ which
properly contains the geometry of $k$ variations from $\Gamma(T\pi)$, see Fig.
3.
${\boldsymbol{x}}_{i}$${\boldsymbol{y}}_{1}$${\boldsymbol{y}}_{2}$${\boldsymbol{y}}_{k}$
Figure 3: Each variation $\delta s_{1}$, $\ldots$, $\delta s_{k}$ brings its
own copy of the base $M^{n}\ni{\boldsymbol{y}}_{\ell}$ into the product bundle
$\pi\times T\pi\times\ldots\times T\pi$ over $M\times M\times\ldots\times M$.
This construction lives not on a Whitney sum
${\pi\mathbin{{\times}_{M}}T\pi\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}T\pi}$
over the base manifold $M$; that would force an untimely restriction to the
diagonal in the product ${M\times M\times\ldots\times M}$ of bases and hence
reproduce the old difficulties of the theory.
### 1.5 The spaces of functionals
The integral functionals $S\in\overline{H}^{n}(\pi)$, which we have been
dealing with until now, are building blocks in a wider class of mappings
$\Gamma(\pi)\to\Bbbk$. By viewing elements of $\Gamma(\pi)$ as “points” and
functionals from $\overline{H}^{n}(\pi)$ as “elementary functions” (see [40]
and references therein), we consider pointwise-defined (formal sums of)
products of such maps, e. g., we let
$(S_{1}\cdot S_{2})(s)\stackrel{{\scriptstyle\text{def}}}{{=}}S_{1}(s)\cdot
S_{2}(s)$
for any two already defined functionals $S_{1}$ and $S_{2}$; the binary
operation $\cdot$ for their values at $s\in\Gamma(\pi)$ is the usual
multiplication of $\Bbbk$-numbers ($\Bbbk=\mathbb{R}$ or $\mathbb{C}$). By
definition, we put
$\overline{\mathfrak{N}}^{n}(\pi,T\pi)=\bigoplus_{\ell=1}^{+\infty}\operatorname*{{\bigotimes\nolimits_{\Bbbk}}}_{i=1}^{\ell}\bigoplus_{k=0}^{+\infty}\overline{H}^{n(1+k)}(\pi\times\underbrace{T\pi\times\ldots\times
T\pi}_{k\text{ variations}}).$
This space contains the linear subspace of local functionals,
$\overline{\mathfrak{M}}^{n}(\pi)=\bigoplus_{\ell=1}^{+\infty}\operatorname*{{\bigotimes\nolimits_{\Bbbk}}}_{i=1}^{\ell}\overline{H}^{n}(\pi),$
for instance, such as the standard weight factor
$\exp(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar})$ in BV-models with quantum BV-
action $S^{\hbar}$ (see section 3.2 below, cf. [9]). The larger space
$\overline{\mathfrak{N}}^{n}(\pi,T\pi)\supsetneq\overline{\mathfrak{M}}^{n}(\pi)$
harbours local functionals _and_ their variations of arbitrarily high order.
The (products of) integral functionals in
$\overline{\mathfrak{M}}^{n}(\pi)\supset\overline{H}^{n}(\pi)$ could be viewed
as primary objects. In the course of variations, their descendants in
$\overline{\mathfrak{N}}^{n}(\pi,T\pi)$ absorb new test shifts and retain the
information about initial building blocks from $\overline{H}^{n}(\pi)$. This
memory governs the analytic behaviour of descendants in operations such as
calculation of the BV-Laplacian or taking the Schouten bracket; we also refer
to sections 1.4 above and 3.1 in what follows. The composite structure of the
bundle $\pi\times T\pi\times\ldots\times T\pi$ is crucial whenever one wants
to not only describe initial setup such as a given BV-model but to perform
rigorous calculations in it, handling higher-order variations of objects (e.
g., third-order variations occur in (1c) on p. 1c, see also Example 2.4 on p.
2.4 below, — and the order is equal to four in property (1d) for the BV-
Laplacian $\Delta$ to be a differential). The geometric approach to
(graded-)permutable variations of functionals makes such calculations well-
defined and proofs free from any ad hoc regularisation recipes.
## 2 The geometry of Batalin–Vilkovisky formalism
The geometry of variations which we analysed in the previous section was not
specific to a bundle $\pi$ of unknowns. In this section we first recall a
construction of the BV-superbundle whose fibres are endowed with
$\mathbb{Z}_{2}$-valued ghost parity. By definition, the BV-bundle
$\boldsymbol{\pi}_{{\text{{BV}}}}^{(0|1)}=\pi^{*}_{\infty}(\boldsymbol{\zeta}_{\infty}^{(0|1)})$
is induced from the Whitney sum
$\boldsymbol{\zeta}^{(0|1)}=\zeta_{0}\mathbin{{\times}_{M}}\zeta_{1}\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}\zeta_{\lambda}\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{0}\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{1}\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{\lambda}$
of some $\mathbb{Z}_{2}$-graded vector bundles over $M$ (in what follows we
sum up the construction of $\zeta_{0},\ldots,\zeta_{\lambda}$ and their
parity-reversed duals
$\Pi\widehat{\zeta}_{0},\ldots,\Pi\widehat{\zeta}_{\lambda}$) by the infinite
jet bundle $\pi_{\infty}\colon J^{\infty}(\pi)\to M$ associated with the
smooth fibre bundle $\pi$ of physical fields.141414A subtle point, which we
reconsider in section 2.1 (see also Remark 1.6), is that the _fibre_ bundle
$\pi$ is often identified with the _vector_ bundle component $\zeta_{0}$ in
$\boldsymbol{\zeta}^{(0|1)}$. Nevertheless, it is the construction of induced
bundle $\pi^{*}_{\infty}(\zeta_{0}\mathbin{{\times}_{M}}\ldots)$ by using
which the physical fields and their derivatives are remembered by the
Euler–Lagrange equations (referred to $\zeta_{0}$), Noether’s identities (in
$\zeta_{1}$), and higher geberations of syzygies from
$\zeta_{2},\ldots,\zeta_{\lambda}$ (if any).
### 2.1 The BV-zoo
Let a fibre bundle $\pi$ of physical fields over the base manifold $M^{n}$ be
given and denote by $\phi$ the fibre coordinates in it. Suppose that
${S_{0}=\int\mathcal{L}_{0}({\boldsymbol{x}},[\phi])\operatorname{dvol}({\boldsymbol{x}})\in\overline{H}^{n}(\pi)}$
is the action of a field model under study. By using the theory and techniques
from section 1 we know how one derives, via the stationary point condition
$\overleftarrow{\delta S}{\bigr{|}}_{s}=0$ at $s\in\Gamma(\pi)$ the
Euler–Lagrange equations of motion
$\mathcal{E}_{{\text{EL}}}=\\{\overleftarrow{\delta S_{0}}/\delta\phi=0\\}$
whose left-hand sides belong to the $C^{\infty}(J^{\infty}(\pi))$-module of
sections $P_{0}=\Gamma(\pi^{*}_{\infty}(\zeta_{0}))$ for the cotangent bundle
$\zeta_{0}$ to $\pi$ such that $\overleftarrow{\delta
S_{0}}/\delta\phi|_{j^{\infty}(s)}\cdot\operatorname{dvol}(\cdot)\in\Gamma(T^{*}\pi)\otimes_{C^{\infty}(M)}\Lambda^{n}(M)$
for any field configuration $s\in\Gamma(\pi)$.
We recall from Remark 1.6 that by following a misfortunate but long-
established tradition it is the unknowns $\phi$ in $\pi$ but not the global
coordinates ${\boldsymbol{F}}$ in the fibre of cotangent bundle $T^{*}\pi$ to
$\pi$ which are used to parametrise the equations within Euler–Lagrange system
$\mathcal{E}_{{\text{EL}}}$ at points of the graph of a section
$\phi\in\Gamma(\pi)$.
If the model at hand is gauge-invariant, then it admits an off-shell
differential dependence
$\boldsymbol{\Phi}({\boldsymbol{x}},[\phi];[{\boldsymbol{F}}])\equiv
0\in\Gamma((\pi_{\infty}\mathbin{{\times}_{M}}\zeta_{0,\infty})^{*}(\zeta_{1}))$
between the left-hand sides ${\boldsymbol{F}}$ of equations
$\mathcal{E}_{{\text{EL}}}$. We recall further that the dependence of
Noether’s identities $\boldsymbol{\Phi}$ on (the derivatives of)
${\boldsymbol{F}}$ is _linear_ for Euler–Lagrange systems
$\mathcal{E}_{{\text{EL}}}$; the generators
${\boldsymbol{p}}({\boldsymbol{x}},[\phi])\in\widehat{P}_{1}=\Gamma(\pi^{*}_{\infty}(\widehat{\zeta}_{1}))$
of Noether’s gauge symmetries for $S_{0}$ are sections of the bundle
$\widehat{\zeta}_{1}$ which is induced from the dual to $\zeta_{1}$ with
respect to the top-degree horizontal form-valued coupling
$\langle\,,\,\rangle$. Indeed, if
$0\equiv\left\langle{\boldsymbol{p}},\boldsymbol{\Phi}({\boldsymbol{x}},[\phi];[{\boldsymbol{F}}])\right\rangle$
and $\boldsymbol{\Phi}$ is linear in ${\boldsymbol{F}}$ or its finite-order
derivatives,
$\boldsymbol{\Phi}({\boldsymbol{x}},[\phi];[{\boldsymbol{F}}])=\ell^{({\boldsymbol{F}})}_{\boldsymbol{\Phi}}({\boldsymbol{F}})\equiv
0,$
then an integration by parts yields that
$0\cong\left\langle(\ell^{\,({\boldsymbol{F}})}_{\boldsymbol{\Phi}})^{\dagger}({\boldsymbol{p}}),\delta
S_{0}/\delta\phi\right\rangle\cong\vec{\partial}^{\,(\phi)}_{(\ell^{({\boldsymbol{F}})}_{\boldsymbol{\Phi}})^{\dagger}({\boldsymbol{p}})}(S_{0}).$
This shows that the evolutionary vector field
$\vec{\partial}^{\,(\phi)}_{A({\boldsymbol{p}})}$ with
$A=(\ell^{({\boldsymbol{F}})}_{\boldsymbol{\Phi}})^{\dagger}$ and
${\boldsymbol{p}}={\boldsymbol{p}}({\boldsymbol{x}},[\phi])$ is a Noether
symmetry of the action $S_{0}$. By reading the above equalities backwards, one
obtains the linear Noether relations
$\boldsymbol{\Phi}=A^{\dagger}({\boldsymbol{F}})$ between the Euler–Lagrange
equations of motion.
Likewise, there could in principle appear higher generations of linear
identities${\Psi_{2}({\boldsymbol{x}},[\phi],[{\boldsymbol{F}}];[\boldsymbol{\Phi}])\equiv
0}$, $\dots$,
${\Psi_{\lambda}({\boldsymbol{x}},[\phi],[{\boldsymbol{F}}],[\boldsymbol{\Phi}],\dots,[\Psi_{\lambda-2}];[\Psi_{\lambda-1}])\equiv
0}$ which hold for all $\phi$, sections ${\boldsymbol{F}}$ in $\zeta_{0}$, and
so on up to the coordinates $\Psi_{\lambda-2}$. Each $i$th generation of such
identities arises with the respective vector bundle $\zeta_{i}$ with fibre
dimension $m_{i}$; the total number of generations is bounded from above by a
constant $\lambda\in\mathbb{N}\cup\\{0\\}$ due to Hilbert’s theorem on
syzygies [16]: $0\leq i\leq\lambda\leq n$, where $n$ is the dimension of base
manifold $M^{n}$. For example, we have that $\lambda=1$ for Yang–Mills theory,
and $\lambda=2$ for gravity over a fourfold $M^{4}$.
We denote by ${\boldsymbol{F}}$ (alas! at once identifying this global
$m$-tuple in $\zeta_{0}$ for the equations with the local field variables
$\phi$), and by $\boldsymbol{\gamma}^{\dagger}$, ${\mathbf{c}}^{\dagger}$,
$\dots$, ${\mathbf{c}}_{\lambda}^{\dagger}$ the global fibre coordinates in
$\zeta_{1}$ for Noether’s identities, and so on up to $\zeta_{\lambda}$,
respectively (see Fig. 4).
$\begin{aligned} {\mathbf{c}}^{\dagger}&\leftrightarrow{\mathbf{c}}\\\
\boldsymbol{\gamma}^{\dagger}&\leftrightarrow\boldsymbol{\gamma}\\\
\underbrace{\phi\approx{\boldsymbol{F}}}_{{\boldsymbol{q}}}&\leftrightarrow\hbox
to0.0pt{$\displaystyle\underbrace{\phi^{\dagger}}_{{\boldsymbol{q}}^{\dagger}}$\hss}\end{aligned}\qquad\pi_{{\text{{BV}}}}^{(0|1)}\left\\{\text{
\begin{picture}(102.0,20.0)\put(-5.0,-20.0){\begin{picture}(0.0,0.0)\bezier{160}(5.0,10.0)(23.33,0.0)(40.0,10.0)\put(41.0,8.0){\makebox(0.0,0.0)[lb]{$M^{n}$}}
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\put(27.0,18.33){\makebox(0.0,0.0)[lb]{$\phi({\boldsymbol{x}})$}}
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}\right.$
Figure 4: The fibre bundle $\pi$ of physical fields $\phi$, the bundle
$\boldsymbol{\zeta}^{(0|1)}$ of BV-variables
$({\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})$, and the vector bundle
$T\boldsymbol{\zeta}^{(0|1)}$ of their variations
$\delta{\boldsymbol{s}}=(\delta s;\delta s^{\dagger})$.
In turn, each vector bundle $\zeta_{0}$, $\dots$, $\zeta_{\lambda}$ brings its
$\langle\,,\,\rangle_{i}$-dual $\widehat{\zeta}_{i}$ into the picture. (Note
that the equations $\overleftarrow{\delta S_{0}}{\bigr{|}}_{s}=0$ upon
$s\in\Gamma(\pi)$ for
$S_{0}=\int\mathcal{L}({\boldsymbol{x}},[\phi])\cdot\operatorname{dvol}({\boldsymbol{x}})$
and all equations’ linear-differential descendants retain the volume form
$\operatorname{dvol}({\boldsymbol{x}})$ from the model’s action $S_{0}$ at all
points ${\boldsymbol{x}}\in M^{n}$.)
We now reverse the parity of linear vector space fibres in
$\widehat{\zeta}_{0}$, $\dots$, $\widehat{\zeta}_{\lambda}$ by introducing the
$\mathbb{Z}_{2}$-valued ghost parity $\operatorname{gh}(\cdot)$ and
considering the odd neighbours $\Pi\widehat{\zeta}_{0}$, $\dots$,
$\Pi\widehat{\zeta}_{\lambda}$ of the dual vector bundles (see [34, 52] and
also Appendix A in [33] for discussion). Let us denote by $\phi^{\dagger}$,
$\boldsymbol{\gamma}$, ${\mathbf{c}}$, $\ldots$, ${\mathbf{c}}_{\lambda}$ the
ghost parity-odd global coordinates along linear vector space fibres in
$\Pi\widehat{\zeta}_{0}$, $\dots$, $\Pi\widehat{\zeta}_{\lambda}$,
respectively. These variables’ proper names are easily recognized from the
standard notation: $\phi$ replacing ${\boldsymbol{F}}$ are the fields and
$\phi^{\dagger}$ are odd-parity antifields, $\boldsymbol{\gamma}$ are the odd
ghosts and $\boldsymbol{\gamma}^{\dagger}$ are the parity-even antighosts,
whereas the canonically conjugate variables
${\mathbf{c}}\leftrightarrow{\mathbf{c}}^{\dagger}$, …,
${\mathbf{c}}_{\lambda}\leftrightarrow{\mathbf{c}}_{\lambda}^{\dagger}$ are
higher ghost-antighost pairs of opposite ghost parities (resp., odd and even).
We denote by ${\boldsymbol{q}}$ the agglomeration of ghost parity-even
variables and by ${\boldsymbol{q}}^{\dagger}$ their respective canonically
conjugate parity-odd neighbours.151515Consider Feynman’s path integral
$\int_{\Gamma(\zeta^{0})}[D{\boldsymbol{q}}]\,\mathcal{O}([{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])$
of an observable $\mathcal{O}$ over the space of ghost parity-even sections.
The BV-Laplacian $\Delta$ is the tool which ensures the integral’s effective
independence from the unphysical ghost parity-odd variables
${\boldsymbol{q}}^{\dagger}$, see section 3.1.
###### Remark 2.1.
Let us emphasize that by using the word “parity” we always refer to the ghost
parity $\operatorname{gh}(\,\cdot\,)$ of objects.161616By construction, the
ghost parities of canonically conjugate BV-variables are complementary modulo
2, that is, to each even-parity variable $q$ there corresponds its odd-parity
dual neighbour $q^{\dagger}$. Of course, there remains much freedom in a
choice of the integer ghost numbers followed by the group homomorphism
$(-)^{\operatorname{gh}(\,\cdot\,)}\colon\mathbb{Z}\to\mathbb{Z}_{2}$. For
example, let $(q,q^{\dagger})$ be a pair of conjugate BV-variables; then one
balances $\operatorname{gh}(q)=\operatorname{gh}(q^{\dagger})\pm 1$ or
$\operatorname{gh}(q)=-\operatorname{gh}(q^{\dagger})\pm 1$, or by using any
other integers such that one is even and the other is odd. Obviously any shift
by an even integer (e. g.,
$\operatorname{gh}(q)\mapsto-\operatorname{gh}(q)=\operatorname{gh}(q)-2\cdot\operatorname{gh}(q)$)
does not alter any values in the parity group $\mathbb{Z}_{2}$; this is no
more than another way to describe the same theory. In this paper we aim at
understanding the geometry of variations so that the graded arithmetic and
algebra of derivations play auxiliary rôles. However, as soon as the
interaction of geometries is properly fixed, their extension to a
$\mathbb{Z}_{2}$-graded setup of superbundle $\pi\colon
E^{(m_{0}+n_{0}|m_{1}+n_{1})}\to M^{(n_{0}|n_{1})}$ of physical fields
(possibly, over a base supermanifold $M^{(n_{0}|n_{1})})$ makes no conceptual
difficulty ([10], see also [22] and references therein). The theory then
becomes bi-graded: it involves (i) the $\mathbb{Z}_{2}$-grading $|\cdot|$ in
the ring of field coordinates, which echoes in the $\mathbb{Z}_{2}$-grading of
Euler–Lagrange equations of motion, Noether identities, etc., (the model’s
action functional $S_{0}$ has even grading by default), and (ii) the ghost
parity $\operatorname{gh}(\cdot)$, see [52].
The $\mathbb{Z}_{2}$-grading $|\cdot|$ and the ghost parity
$\operatorname{gh}(\cdot)$ are independent from each other. We denote by
${\boldsymbol{q}}={\boldsymbol{q}}^{(0|1)}$ the ghost parity-even BV-fibre
variables, which are then grouped in even- and odd-grading components.
Likewise, the ghost parity-odd BV-variables
${\boldsymbol{q}}^{\dagger}=({\boldsymbol{q}}^{\dagger})^{(0|1)}$ are arranged
in exactly the same way. By construction, the values of
$\mathbb{Z}_{2}$-gradings for canonically conjugate variables
$({\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})$ coincide: we have that
$|{\boldsymbol{q}}|=|{\boldsymbol{q}}^{\dagger}|$ and
$\operatorname{gh}({\boldsymbol{q}}^{\dagger})\equiv\operatorname{gh}({\boldsymbol{q}})+1\mod
2$.
Next, we take the Whitney sum
$\boldsymbol{\zeta}^{(0|1)}\stackrel{{\scriptstyle\text{def}}}{{=}}\zeta_{0}\mathbin{{\times}_{M}}\zeta_{1}\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}\zeta_{\lambda}\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{0}\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{1}\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}\Pi\widehat{\zeta}_{\lambda}$
of the double set of dual bundles with opposite ghost parities of fibre
coordinates. Finally, let us lift the Whitney sum of infinite jets of those
bundles, putting it over the bundle of physical fields by using a pull-back
under $\pi_{\infty}$. We denote the resulting bundle over the total space
$J^{\infty}(\pi)\to M$ by
$\pi_{{\text{{BV}}}}^{(0|1)}=\pi_{\infty}^{*}\bigl{(}\boldsymbol{\zeta}_{\infty}^{(0|1)}\bigr{)}.$
The fibre
$W_{{\boldsymbol{x}}}=V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}$ of $\boldsymbol{\zeta}^{(0|1)}$ admits the
canonical decomposition in two dual halves of opposite parities;171717To
highlight this duality between ghost parity-even vector space
$V_{{\boldsymbol{x}}}$ and ghost parity-odd subspace $\Pi
V_{{\boldsymbol{x}}}^{\dagger}$ in $W_{{\boldsymbol{x}}}$, we use the notation
$\widehat{\oplus}$ for their direct sum; whenever a coordinate in
$V_{{\boldsymbol{x}}}$ is rescaled by $\operatorname{const}$ times, the
respective conjugate variable in $\Pi V_{{\boldsymbol{x}}}^{\dagger}$ is
transformed inverse-proportionally by $\operatorname{const}^{-1}$ times, see
Remark 2.5 below. this is shown in Fig. 5.
$V_{x}$$W_{{\boldsymbol{x}}}={}$$\widehat{\bigoplus}$$(\Pi)V_{x}^{\dagger}$
Figure 5: The BV-fibre is a direct sum of dual vector spaces; one is parity-
even and the other is proclaimed ghost parity-odd.
Bearing in mind that the fields $\phi$ are artifically incorporated into the
newly built fibre by $\zeta_{0}$, we shall omit an ever-present reference to
points $({\boldsymbol{x}},\phi({\boldsymbol{x}}))$ of jets of sections of the
initial bundle $\pi$ when dealing with variations
$\delta{\boldsymbol{s}}=(\delta s;\delta s^{\dagger})$ for sections $s$ of
$\boldsymbol{\zeta}^{(0|1)}$ at $\phi({\boldsymbol{x}})$, see Fig. 4.
### 2.2 The signs convention in Nature
The construction of canonically conjugate pairs of global coordinates
$({\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})$ in the fibres
$W_{{\boldsymbol{x}}}=V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}$ refers to a choice of the smooth field of dual
bases in the two subspaces of even and odd ghost parity. Suppose that
$\vec{e}_{i}({\boldsymbol{x}})$ is a frame in $V_{{\boldsymbol{x}}}$ and
$\vec{e}^{{}\,\dagger i}({\boldsymbol{x}})$ is its dual in $\Pi
V_{{\boldsymbol{x}}}^{\dagger}$, where the index $i$ runs from 1 to the total
dimension of even- and odd-parity component in the fibre of
$\boldsymbol{\zeta}^{(0|1)}$; we denote by $N=m+m_{1}+\ldots+m_{\lambda}$ each
of the two dimensions so that the fibre of the Whitney sum
$\boldsymbol{\zeta}^{(0|1)}$ has superdimension $(N|N)$.
Let us recall that it is the parity of coordinates
${\boldsymbol{q}}^{\dagger}$ but not of the vectors $\vec{e}^{{}\,\dagger i}$
in a basis which is reversed by the operation $\Pi$. The odd-parity component
in the vector bundle $\boldsymbol{\zeta}^{(0|1)}$ is topologically
indistinguishable from
$\widehat{\zeta}_{0}\mathbin{{\times}_{M}}\ldots\mathbin{{\times}_{M}}\widehat{\zeta}_{\lambda}$
but the rules become new for arithmetic in the algebra of coordinate functions
on the total space. Therefore, we let the notation $\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})$ be identical for the same bases in
$V_{{\boldsymbol{x}}}^{\dagger}$ and $\Pi V_{{\boldsymbol{x}}}^{\dagger}$.
###### Remark 2.2.
The presence of _two_ dual vector spaces, $V_{{\boldsymbol{x}}}$ and
$(\Pi)V_{{\boldsymbol{x}}}^{\dagger}$, standardly implies that there are _two_
couplings,
$\langle\,,\,\rangle\colon
V_{{\boldsymbol{x}}}\times(\Pi)V_{{\boldsymbol{x}}}^{\dagger}\to\Bbbk\quad\text{and}\quad\langle\,,\,\rangle\colon(\Pi)V_{{\boldsymbol{x}}}^{\dagger}\times
V_{{\boldsymbol{x}}}\to\Bbbk;$ (9)
we denote both operations in the same way because the order of arguments
uniquely determines the choice. Let us remember also that it is not the linear
vector space fibres of the superbundle $\boldsymbol{\zeta}^{(0|1)}$ over the
bundle $\pi$ of physical fields but it is the tangent spaces
$T_{({\boldsymbol{x}},\phi({\boldsymbol{x}}),s({\boldsymbol{x}}))}\left(V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}\right)\cong
V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}$
to those fibres which harbour the variations $\delta{\boldsymbol{s}}=(\delta
s;\delta s^{\dagger})$ of sections $s$ of the BV-bundle.
A reason to study the geometry of variations in tangent spaces to the fibres
is clear from section 1. In fact, although we have substantiated in section
2.1 that Euler–Lagrange equations and their descendants do form linear vector
spaces, this structure is incidental for the BV-formalism while Feynman path
integration is not yet begun. The guiding geometric principle is that linear
vector spaces appear only in the course of inspection of functionals’
responses to infinitesimal test shifts of their arguments.
Couplings (9) are defined only if the linear vector spaces
$V_{{\boldsymbol{x}}}\ni\delta s({\boldsymbol{x}})$ and $\Pi
V_{{\boldsymbol{x}}}^{\dagger}\ni\delta s^{\dagger}({\boldsymbol{x}})$ are
located over the same point ${\boldsymbol{x}}\in M^{n}$ of the base manifold,
and over it they are attached as the two components of tangent space
$T_{s({\boldsymbol{x}})}\bigl{(}\boldsymbol{\zeta}^{(0|1)}\bigr{)}^{-1}\bigl{(}{\boldsymbol{x}},\phi({\boldsymbol{x}})\bigr{)}$,
at the same point
$s({\boldsymbol{x}})=s\left({\boldsymbol{x}},\phi({\boldsymbol{x}})\right)$ of
fibre in the superbundle $\boldsymbol{\zeta}^{(0|1)}$ over a point
$({\boldsymbol{x}},\phi({\boldsymbol{x}}))$ of the total space for the bundle
$\pi$ of physical fields (see Fig. 4).
A distinction between the vector space $V_{{\boldsymbol{x}}}$ and its parity-
reversed dual nontrivially determines the couplings’ values whenever they are
defined. Namely, each of the two finite-dimensional vector spaces is
reflexive,
$\left((V_{{\boldsymbol{x}}})^{\dagger}\right)^{\dagger}\cong
V_{{\boldsymbol{x}}}\quad\text{and}\quad\left((\Pi
V_{{\boldsymbol{x}}}^{\dagger})^{\dagger}\right)^{\dagger}\cong\Pi
V_{{\boldsymbol{x}}}^{\dagger},$ (10)
but these isomorphisms are not always identity mappings. We have that
$\left\langle\vec{e}_{i}({\boldsymbol{x}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\right\rangle=\boldsymbol{\delta}^{j}_{i}\quad\text{yet}\quad\left\langle\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}}),\vec{e}_{i}({\boldsymbol{x}}),\right\rangle=-\boldsymbol{\delta}^{j}_{i},$
(11)
where $\boldsymbol{\delta}^{j}_{i}$ is the Kronecker symbol whose value is the
unit iff $i=j$ and which is set equal to zero otherwise.
###### Remark 2.3.
We claim that this mechanism is responsible, in particular, for the skew-
symmetry of various Poisson brackets (e. g., of the parity-odd Schouten
bracket). Let us emphasize that this is a principle of order between geometric
objects; the concept is not restricted to the BV-setup which we study here.
Actually, Eq. (11) is the fundamental reason for differential $1$-forms to
anticommute181818That is, this argument reveals why a mathematical axiom that
differential forms do anticommute in the course of calculations leads to
verifiable and relevant theoretic predictions which match experimental data.
(in the class of geometries for which a coupling is defined between the linear
vector spaces of co-multiples under the wedge product $\wedge$; for instance,
such is the case of the Helmholtz criterion $\psi=\delta
S/\delta{\boldsymbol{q}}$ $\Leftrightarrow$
$\vec{\ell}_{\psi}^{\,({\boldsymbol{q}})}=\bigl{(}\vec{\ell}_{\psi}^{\,({\boldsymbol{q}})}\bigr{)}^{\dagger}$
for images of the variational derivative [28, 45]). Physically speaking, the
binary count by “a vector space,” “not the former, hence its dual,” and “not
the dual, but the initial space’s image under central symmetry” builds on the
notion of order and realizes the law of the excluded middle.
### 2.3 Left- and right-variations via operators
Suppose that
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\operatorname{dvol}({\boldsymbol{x}})$
is an integral functional $\Gamma(\pi_{{\text{{BV}}}})\to\Bbbk$. Let us focus
on the correspondence between test shifts $\delta{\boldsymbol{s}}=(\delta
s;\delta s^{\dagger})=\delta s^{i}\cdot\vec{e}_{i}+\delta
s_{i}^{\dagger}\cdot\vec{e}^{{}\,\dagger i}$ of BV-fields
$s\in\Gamma(\pi_{{\text{{BV}}}})$ and, on the other hand, left- or right-
acting linear singular integral operators
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ and
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$ which yield the functional’s
responses to shifts of its argument ${\boldsymbol{s}}$. By definition, we put
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}=\int_{M}{\mathrm{d}}{\boldsymbol{y}}\,\Bigl{\\{}(\delta
s^{i})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\cdot\left\langle(\vec{e}^{{}\,\dagger
i})^{\dagger}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}(\cdot)\right\rangle\frac{\overrightarrow{\partial}}{\partial
q^{j}_{\sigma}}+{}\\\ {}+(\delta
s^{\dagger}_{i})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\cdot\left\langle(\vec{e}_{i})^{\dagger}({\boldsymbol{y}}),\vec{e}_{j}(\cdot)\right\rangle\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j,\sigma}}\Bigr{\\}}$ (12a) and
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}=\int_{M}{\mathrm{d}}{\boldsymbol{y}}\,\Bigl{\\{}\frac{\overleftarrow{\partial}}{\partial
q^{j}_{\sigma}}\left\langle\vec{e}^{{}\,\dagger
j}(\cdot),{}^{\dagger}(\vec{e}^{{}\,\dagger
i})({\boldsymbol{y}})\right\rangle\left(\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}(\delta
s^{i})({\boldsymbol{y}})+{}\\\ {}+\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{j,\sigma}}\left\langle\vec{e}_{j}(\cdot),{}^{\dagger}(\vec{e}_{i})({\boldsymbol{y}})\right\rangle\left(\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}(\delta
s^{\dagger}_{i})({\boldsymbol{y}})\Bigr{\\}}.$ (12b)
The above formulas for directed operators
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$ and
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ contain new notation
$(\vec{e}_{i})^{\dagger},\ (\vec{e}^{{}\,\dagger i})^{\dagger}$ and
${}^{\dagger}(\vec{e}_{i}),\ {}^{\dagger}(\vec{e}^{{}\,\dagger i})$, also
referring to an important sign convention which fully determines those adjoint
objects. Namely, let us agree that over every ${\boldsymbol{x}}\in M^{n}$ the
covectors
$\left.\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})\left(\frac{\overrightarrow{\partial}}{\partial
q^{i}_{\sigma}}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}({\boldsymbol{s}})}+\left.\vec{e}_{i}({\boldsymbol{x}})\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{i,\sigma}}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}({\boldsymbol{s}})}$
and
$\left.\left(\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{i}_{\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}({\boldsymbol{s}})}\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})+\left.\left(\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{i,\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}({\boldsymbol{s}})}\vec{e}_{i}({\boldsymbol{x}})$
are expanded in the cotangent space
$T^{*}_{{\boldsymbol{s}}({\boldsymbol{x}})}W_{{\boldsymbol{x}}}\cong
V^{\dagger}_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}(TV^{\dagger}_{{\boldsymbol{x}}})^{\dagger}$
with respect to the original basis $(+\vec{e}^{{}\,\dagger i},+\vec{e}_{i})$;
note the signs (any other convention here would nohow alter the theory’s
content but it would (in)appropriately modify the signs in (13) below). The
normalization of left- and right-adjoint objects $(\vec{e}_{i})^{\dagger},\
(\vec{e}^{{}\,\dagger i})^{\dagger}$ and ${}^{\dagger}(\vec{e}_{i}),\
{}^{\dagger}(\vec{e}^{{}\,\dagger i})$ is immediate under assumption that the
couplings’ equations yield (5) and then (3) after integration by parts — no
extra sign factors appear in those formulas. This requirement determines the
table
$\begin{aligned} (\vec{e}^{{}\,\dagger
i})^{\dagger}&=\phantom{+}\vec{e}_{i},\\\
(\vec{e}_{i})^{\dagger}&=-\vec{e}^{{}\,\dagger
i},\end{aligned}\qquad\begin{aligned} {}^{\dagger}(\vec{e}^{{}\,\dagger
i})&=-\vec{e}_{i},\\\
{}^{\dagger}(\vec{e}_{i})&=\phantom{+}\vec{e}^{{}\,\dagger i},\end{aligned}$
(13)
so that the following defining relations hold:
$\left\langle(\vec{e}_{i})^{\dagger},\vec{e}_{i}\right\rangle=\left\langle\vec{e}_{i},{}^{\dagger}(\vec{e}_{i})\right\rangle=\left\langle(\vec{e}^{{}\,\dagger
i})^{\dagger},\vec{e}^{{}\,\dagger
i}\right\rangle=\left\langle\vec{e}^{{}\,\dagger
i},{}^{\dagger}(\vec{e}^{{}\,\dagger i})\right\rangle=+1.$
Let us notice that the left- and right-acting operation † provides the
analogue of left and right $\langle\,,\,\rangle$-dual in this ordered world;
the first column in (13) determines a clockwise rotation in the oriented plane
spanned by $\vec{e}_{i}\prec\vec{e}^{{}\,\dagger i}$, whereas taking the
adjoints ${}^{\dagger}(\cdot)\colon\vec{e}_{i}\mapsto\vec{e}^{{}\,\dagger i}$
and $\vec{e}^{{}\,\dagger i}\mapsto-\vec{e}_{i}$ induces the counterclockwise
rotation in that plane as shown in Fig. 6.
${}^{\dagger}(\vec{e}_{i})$$\vec{e}_{i}$${}^{\dagger}(^{\dagger}(^{\dagger}(\vec{e}_{i})))$${}^{\dagger}(^{\dagger}(\vec{e}_{i}))$$(((\vec{e}_{i})^{\dagger})^{\dagger})^{\dagger}$$\vec{e}_{i}$$(\vec{e}_{i})^{\dagger}$$(\vec{e}_{i})^{\dagger})^{\dagger}$
Figure 6: The orientation $\vec{e}_{i}\prec\vec{e}^{\,\dagger i}$ and
configuration of the left- and right- duals with respect to the couplings
$\langle\ ,\,\rangle$.
###### Example 2.1.
Identities (13) show up in the directed variations
$\left.\overleftarrow{\delta}\\!\\!S\right|_{s}^{\delta{\boldsymbol{s}}}=\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}(S)(s)$
and
$\left.\overrightarrow{\delta}\\!\\!S\right|_{s}^{\delta{\boldsymbol{s}}}=(S)\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}(s)$
of an integral functional
$S=\int\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\cdot\operatorname{dvol}({\boldsymbol{x}})$.
Namely, we have that
$\left.\overleftarrow{\delta}\\!\\!S\right|_{s}^{(\delta s,\delta
s^{\dagger})}=\\\
=\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\Biggl{\\{}(\delta
s^{i})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\left\langle\vec{e}_{i}({\boldsymbol{y}}),\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}})\right\rangle\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{j}_{\sigma}}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}+\\\
+(\delta
s^{\dagger}_{i})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}({\boldsymbol{y}})\left\langle-\vec{e}^{{}\,\dagger
i}({\boldsymbol{y}}),\vec{e}_{j}({\boldsymbol{x}})\right\rangle\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j,\sigma}}\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\Biggr{\\}}$
(14a) and $\left.\overrightarrow{\delta}\\!\\!S\right|_{s}^{(\delta s,\delta
s^{\dagger})}=\\\
=\int_{M}{\mathrm{d}}{\boldsymbol{y}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\Biggl{\\{}\left.\left(\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{j}_{\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\left\langle\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}}),-\vec{e}_{i}({\boldsymbol{y}})\right\rangle\left(\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}(\delta
s^{i})({\boldsymbol{y}})+{}\\\
+\left.\left(\mathcal{L}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{j,\sigma}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\left\langle\vec{e}_{j}({\boldsymbol{x}}),\vec{e}^{{}\,\dagger
i}({\boldsymbol{y}})\right\rangle\left(\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}}\right)^{\sigma}(\delta
s^{\dagger}_{i})({\boldsymbol{y}})\Biggr{\\}}.$ (14b)
The operators $\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$ and
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ act via ghost-parity graded
Leibniz’ rule on formal products of integral functionals (and on their inages
under other infinitesimal variation operators as well), so that the two
operators are defined on the entire space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$, see
section 2.4.2 below.
###### Remark 2.4.
A reversion
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}\rightleftarrows\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$
of the direction along which such an operator acts means that the initially
given operator (for definition, let it be
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ which acts to the left) is
_destroyed_ and in its place the other, opposite-direction operator is created
(here it would be $\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$). Note that
the variation $\delta{\boldsymbol{s}}\in\Gamma(T\pi)$ itself stays unchanged;
it is the two realizations of this object via
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ and then via
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$ which differ. (This concept of
test shifts as primary geometric objects which contain information about the
operators will be essential in Definition 2 of the variational Schouten
bracket.)
###### Remark 2.5.
The postulate of duality between $\vec{e}_{i}({\boldsymbol{x}})$ and
$\vec{e}^{{}\,\dagger i}({\boldsymbol{x}})$ correlates their transformation
laws under dilations: a rescaling
$\vec{e}_{i}\mapsto\operatorname{const}\cdot\vec{e}_{i}$ with
$\operatorname{const}\in\Bbbk\setminus\\{0\\}$ determines the inverse-
proportional mapping $\vec{e}^{{}\,\dagger
i}\mapsto\operatorname{const}^{-1}\cdot\vec{e}^{{}\,\dagger i}$ of respective
dual vectors. (Likewise, the coordinates in $V_{{\boldsymbol{x}}}$ and $\Pi
V_{{\boldsymbol{x}}}^{\dagger}$ are then rescaled by
$q^{i}\mapsto\operatorname{const}^{-1}\cdot q^{i}$ and
$q_{i}^{\dagger}\mapsto\operatorname{const}\cdot q_{i}^{\dagger}$.
respectively.)
Consider a variation $\delta{\boldsymbol{s}}=(\delta s;\delta
s^{\dagger})\in\Gamma(T\boldsymbol{\zeta}^{(0|1)})$ of a BV-section
$s\in\Gamma(\boldsymbol{\zeta}^{(0|1)})$ over a given field configuration
$\phi\in\Gamma(\pi)$ in the BV-bundle
$\boldsymbol{\pi}^{(0|1)}_{{\text{{BV}}}}$. The infinitesimal variation
vectors $\delta{\boldsymbol{s}}=(\delta s;\delta s^{\dagger})$ can be
naturally split to ghost parity-homogeneous components:
$\delta{\boldsymbol{s}}=(\delta s;0)+(0;\delta s^{\dagger}).$ (15)
Here we explicitly use the linear vector space structure in fibres of the
tangent bundle $T\boldsymbol{\zeta}^{(0|1)}$. Let us recall that the two
homogeneous variations
$\delta s({\boldsymbol{x}})=\delta
s^{i}({\boldsymbol{x}})\cdot\vec{e}_{i}({\boldsymbol{x}})\quad\text{and}\quad\delta
s^{\dagger}({\boldsymbol{x}})=\delta
s^{\dagger}_{i}({\boldsymbol{x}})\cdot\vec{e}^{{}\,\dagger
i}({\boldsymbol{x}})$
in the right-hand side of (15) are the canonically dual to each other.
Moreover, by Remark 2.5 it is then possible to have $\delta s$ and $\delta
s^{\dagger}$ normalized, for every $i$ running from 1 to the dimension $N$, by
the equalities
$\delta s^{i}({\boldsymbol{x}})\cdot\delta
s_{i}^{\dagger}({\boldsymbol{x}})\equiv+1$ (16)
at every ${\boldsymbol{x}}\in M^{n}$ where the smooth fields of dual bases
$\vec{e}_{i}$ and $\vec{e}^{\,\dagger i}$ are defined for the section $s$.
From now on, let us deal only with such normalized variations. This implies
that the coupling of these geometric objects are “invisible” but still the
order in which the co-multiples $\delta s$ and $\delta s^{\dagger}$ occur in
(11) does determine the signs in various formulas (e. g., in the definition of
Schouten bracket, see p. 2 below).
### 2.4 Definitions of the BV-Laplacian and Schouten bracket
We now combine the geometry of graded-permutable iterated variations, which we
explored in section 1 and which absorbs a new copy of the underlying base
manifold $M^{n}$ for each new infinitesimal test shift
$\delta{\boldsymbol{s}}({\boldsymbol{x}})\in
T_{s({\boldsymbol{x}})}W_{{\boldsymbol{x}}}$ of the functionals’ arguments at
${\boldsymbol{x}}\in M^{n}$, with the algebra of two couplings (9) between
ghost parity-homogeneous halves of infinitesimal variations in the BV-setup
$T_{s({\boldsymbol{x}})}W_{{\boldsymbol{x}}}\cong
V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}$; the absolute locality of such coupling events
is a fundamental principle.
To avoid an agglomeration of formulas and to match the notation with that in
section 1, we omit an explicit reference to field configuration
$\\{\phi({\boldsymbol{x}}),\,{\boldsymbol{x}}\in M^{n}\\}$, indicating only
the base points ${\boldsymbol{x}}\in M^{n}$. We also denote by
$\pi_{{\text{{BV}}}}$ the composite-structure superbundle over $M^{n}$ (see
Fig. 4) so that the notation for the vector bundle of BV-sections’
infinitesimal variations is $T\pi_{{\text{{BV}}}}$. However, let us remember
that only the linear BV-fibre variables
$({\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})$ but not the physical fields
$\phi$ are subjected to variations at points
$s({\boldsymbol{x}})\in\bigl{(}\boldsymbol{\zeta}^{(0|1)}\bigr{)}^{-1}({\boldsymbol{x}},s({\boldsymbol{x}}))$
over $({\boldsymbol{x}},\phi({\boldsymbol{x}}))\in\pi^{-1}({\boldsymbol{x}})$.
A brute force labelling of Euler–Lagrange equations by the respective unknowns
is an act of will by the one who writes formulas but it is not a prescription
from the model’s geometry.
This section contains rigorous, self-regularizing definitions of the BV-
Laplacian and Schouten bracket for integral functionals from
$\overline{H}^{n}(\pi_{{\text{{BV}}}})\subsetneq\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})\subsetneq\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$.
We shall extend the definition to the space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$ of
products of integral functionals, possibly with earlier-absorbed variations,
in the subsequent sections of this paper. We then establish the main
properties of these structures and prove relations between them. We note that
the definitions which we give here are operational: each of them is a surgery
for the couplings and their reconfiguration algorithm. (The locality postulate
ensures the restrictions onto diagonals in the product $M\times\ldots\times M$
so that those recombinations make sense at every point of $M$.)
#### 2.4.1 The BV-Laplacian $\Delta$
Let us first introduce some shorthand notation. Let $F=\int
f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\cdot\operatorname{dvol}({\boldsymbol{x}})$
be an integral functional and $\delta{\boldsymbol{s}}=(\delta s;0)+(0;\delta
s^{\dagger})$ be a variation’s splitting in two ghost parity-homogeneous
variations. From section 1 we know that each of the two is referred to its own
copy of the base: let it be $\delta s({\boldsymbol{y}}_{1})$ and $\delta
s^{\dagger}({\boldsymbol{y}}_{2})$ so that formula (5) defines the response of
$F$ to an infinitesimal shift of its argument along each of the two
directions.
###### Definition 1.
Let $\delta{\boldsymbol{s}}\in\Gamma(T\pi_{{\text{{BV}}}})$ be a test shift
normalized by (16) and then split to the sum $(\delta s;0)+(0;\delta
s^{\dagger})$ of ghost parity-homogeneous, $\langle\,,\,\rangle$-dual halves.
The BV-_Laplacian_ is the linear operator
$\Delta\colon\overline{H}^{n}(\pi_{{\text{{BV}}}})\to\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$;
for a ghost parity-homogeneous integral functional $F\in\overline{H}^{n}(\pi)$
and its argument $s$, the operator $\Delta$ is an algorithm for
reconfiguration of couplings in the second variation
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon^{\dagger}}\right|_{\varepsilon^{\dagger}=0}F(s+\varepsilon\cdot\overleftarrow{\delta}\\!\\!s+\varepsilon^{\dagger}\cdot\overleftarrow{\delta}\\!\\!s^{\dagger})=\sum_{\begin{subarray}{c}i_{1},i_{2}\\\
j_{1},j_{2}\end{subarray}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\\\
\left\\{\begin{matrix}\phantom{\hookrightarrow}(\delta
s^{i_{1}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{y}}_{1})\,\langle\phantom{+}\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}})\rangle\hookleftarrow\\\ \hookrightarrow(\delta
s^{\dagger}_{i_{2}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{y}}_{2})\langle-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2}),\vec{e}_{j_{2}}({\boldsymbol{x}})\rangle\phantom{\hookleftarrow}\end{matrix}\right\\}\cdot\left.\frac{\overrightarrow{\partial^{2}}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{j_{1}}_{\sigma_{1}}\partial
q^{\dagger}_{j_{2},\sigma_{2}}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\,.$
This second variation’s integrand contains the couplings
$\langle\,,\,\rangle$:
$T_{s({\boldsymbol{y}}_{1})}V_{{\boldsymbol{y}}_{1}}\times
T^{*}_{s({\boldsymbol{x}})}(\Pi)V_{{\boldsymbol{x}}}\to\Bbbk\quad\text{ and
}\quad T_{s({\boldsymbol{y}}_{2})}\Pi V_{{\boldsymbol{y}}_{2}}^{\dagger}\times
T^{*}_{s({\boldsymbol{x}})}(\Pi)V_{{\boldsymbol{x}}}^{\dagger}\to\Bbbk$
which are defined only if the attachment points coincide for these
(co)vectors; an optional presence of the parity reversion operator indicates a
possibility of having ghost parity-odd functional $F$.
At the moment when the object $\Delta F$ under construction – or a larger
object of which $\Delta F$ is an element, see (1c) – is evaluated at a section
$s\in\Gamma(\pi_{\text{{BV}}})$, the integrations by parts carry the
derivatives away from the variations’ components:
$\overleftarrow{\partial}/\partial{\boldsymbol{y}}_{i}\mapsto\overrightarrow{\partial}/\partial{\boldsymbol{y}}_{i}$
as explained in section 1.3. The third step in definition of $\Delta$ acting
on $F$ is a surgery algorithm for an on-the-diagonal reattachment of the
couplings, see Figure 7.
$\begin{array}[]{rrll}\langle\,{}^{1}\mars\,|&&|\,{}^{3}\venus\,\rangle&{}\\\
{}\hfil&\langle\,{}^{2}\venus\,|&&|\,{}^{4}\mars\,\rangle\end{array}\qquad\longmapsto\qquad\begin{array}[]{rlrl}\langle\,{}^{1}\mars\,|&&\langle\,{}^{3}\venus\,|&{}\\\
{}\hfil&|\,{}^{2}\venus\,\rangle&&|\,{}^{4}\mars\,\rangle\end{array}$ Figure
7: The on-the-diagonal coupling of variations versus taking the trace of bi-
linear form.
In other words, _after_ the integration by parts the surgery yields the
following:
$(\Delta
F)\Bigr{|}_{s}^{\delta{\boldsymbol{s}}}=\sum_{\begin{subarray}{c}i_{1},i_{2}\\\
j_{1},j_{2}\end{subarray}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\cdot\\\
\cdot\left\\{\delta
s^{i_{1}}({\boldsymbol{y}}_{1})\left(-\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}\underbrace{\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2})\rangle}_{-1}\cdot\delta
s^{\dagger}_{i_{2}}({\boldsymbol{y}}_{2})\left(-\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}\right\\}\cdot\\\
\cdot\left\\{\underbrace{\langle\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}}),\vec{e}_{j_{2}}({\boldsymbol{x}})\rangle}_{-1}\cdot\left.\frac{\vec{\partial^{2}}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{j_{1}}_{\sigma_{1}}\partial
q^{\dagger}_{j_{2},\sigma_{2}}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\\}\,.$
(17)
Note that the left-to-right order in
$\left\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}})\right\rangle\cdot\left\langle-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2}),\vec{e}_{j_{2}}({\boldsymbol{x}})\right\rangle$
is preserved by the respective couplings’ arguments in
$\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2})\rangle\cdot\langle\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}}),\vec{e}_{j_{2}}({\boldsymbol{x}})\rangle$, cf. Fig.
7.
###### Remark 2.6.
Until the moment when the integrations by parts are performed in $\Delta F$,
the derivatives $\partial/\partial{\boldsymbol{y}}_{1}$ and
$\partial/\partial{\boldsymbol{y}}_{2}$ refer to different copies of the
manifold $M^{n}$ in the base $M^{n}\times M^{n}\times M^{n}$ of the product
bundle $\pi_{{\text{{BV}}}}\times T\pi_{{\text{{BV}}}}\times
T\pi_{{\text{{BV}}}}$. This implies that the two variations of $F$ in the
definition of $\Delta$ are graded-permutable between each other and with all
other variations falling on
$f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])$ whenever
$\Delta F$ is a constituent element of a larger object (e. g., see (1c–1d) on
p. 1c).
###### Remark 2.7.
To keep track of multiple copies of the base $M^{n}$ for functionals and
variations (here ${\boldsymbol{x}}\in M^{n},\ {\boldsymbol{y}}_{1}\in M^{n},\
{\boldsymbol{y}}_{2}\in M^{n}$) in the course of integration by parts (see
section 1.3), we indicate the respective variations’ bases by explicitly
writing ${\boldsymbol{q}}({\boldsymbol{y}}_{1})$ and
${\boldsymbol{q}}^{\dagger}({\boldsymbol{y}}_{2})$ in the denominators _and_
we denote by $\partial/\partial{\boldsymbol{y}}_{1}$ and
$\partial/\partial{\boldsymbol{y}}_{2}$ the derivatives which now fall on the
functional’s density
$f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])$ — for
instance, we do so in Example 2.4 on p. 2.4 below. Namely, we put
$\displaystyle\left.\frac{\overleftarrow{\delta}\\!\\!{f}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta
q^{\alpha}({\boldsymbol{y}}_{1})}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}$
$\displaystyle=\sum\limits_{|\sigma_{1}|\geqslant
0}\Bigl{(}-\frac{\vec{\partial}}{\partial{\boldsymbol{y}}_{1}}\Bigr{)}^{\sigma_{1}}\left.\left(\frac{\vec{\partial}f(x,[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{\alpha}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}=$
(18a) $\displaystyle=\sum_{|\sigma_{1}|\geq
0}\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}\frac{\vec{\partial}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]}{\partial
q^{\alpha}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}$ and
$\displaystyle\left.\frac{\overleftarrow{\delta}\\!\\!{f}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta
q^{\dagger}_{\beta}({\boldsymbol{y}}_{2})}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}$
$\displaystyle=\sum\limits_{|\sigma_{2}|\geqslant
0}\Bigl{(}-\frac{\vec{\partial}}{\partial{\boldsymbol{y}}_{2}}\Bigr{)}^{\sigma_{2}}\left.\left(\frac{\vec{\partial}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{\dagger}_{\beta,\sigma_{2}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}=$
(18b) $\displaystyle=\sum_{|\sigma_{2}|\geq
0}\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}\frac{\vec{\partial}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]}{\partial
q^{\dagger}_{\beta,\sigma_{2}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}$
for the ghost parity-homogeneous components of variational derivative. At
every point $({\boldsymbol{x}},\phi({\boldsymbol{x}}),s({\boldsymbol{x}}))$ of
the total space for the bundle $\pi_{{\text{{BV}}}}$, and for a given
functional $F$ which is assumed ghost parity-homogeneous, we have that
$\left.\frac{\overleftarrow{\delta}\\!\\!{f}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta
q^{\alpha}({\boldsymbol{y}}_{1})}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\in
T^{*}_{s({\boldsymbol{x}})}(\Pi)V_{{\boldsymbol{x}}}\qquad\text{and}\qquad\left.\frac{\overleftarrow{\delta}\\!\\!{f}({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta
q^{\dagger}_{\beta}({\boldsymbol{y}}_{2})}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\in
T^{*}_{s({\boldsymbol{x}})}(\Pi)V^{\dagger}_{{\boldsymbol{x}}}.$
Let us remember that an attribution of denominators to ${\boldsymbol{y}}_{1}$
or ${\boldsymbol{y}}_{2}$ is a matter of notation in (18); whenever happening,
everything happens at ${\boldsymbol{x}}\in M^{n}$.
###### Lemma 1.
The BV-Laplacian $\Delta$ is independent of a choice of the variation
$\delta{\boldsymbol{s}}$ normalized by (16).
Indeed, whenever the integrations by parts are performed, products (16) of the
dual components are always the same at all points of the intersection of their
domains of definition.191919The assertion of Lemma 1 extends to the
variational Schouten bracket, which is a derivative structure with respect to
the BV-Laplacian (see Definition 2 on p. 2). Moreover, the independence of a
specific choice of variations implies that their coefficients $(\delta
s_{1},\delta s_{1}^{\dagger})$ and $(\delta s_{2},\delta s_{2}^{\dagger})$,
which are built into $\Delta$ and $\lshad\,,\,\rshad$, can be swapped, not
altering an object that contains these test shifts
$\delta{\boldsymbol{s}}_{1}$ and $\delta{\boldsymbol{s}}_{2}$ (see the proof
of Lemma 5 on p. 5). We illustrate the definition of BV-Laplacian $\Delta$ by
using Fig. 8;
$F$$\vec{\delta}s^{\dagger}(F)$$\vec{\delta}s\bigl{(}\vec{\delta}s^{\dagger}(F)\bigr{)}$$\int$$\langle
1,2\rangle\cdot\langle 3,4\rangle$$\Delta F$$\langle\delta s,\delta
s^{\dagger}\rangle=1$$\delta s^{\dagger}$$\delta s$$1=\delta s(x)\cdot\delta
s^{\dagger}(x)$ Figure 8: A variational update of the cyclic wor(l)d from
[36]: the (anti)words $\delta s$ and $\delta s^{\dagger}$ are pasted into a
necklace $F$ according to the graded Leibniz rule. Then they annihilate in
such a way that the respective loose ends of the string join, the cyclic order
of gems preserved; this yields $\Delta F$.
let us notice that it properly renders the assertion of Lemma 1 in a wider,
noncommutative setup of [36] and [29, 32] (see Remark 1.1 on p. 1.1).
###### Corollary 2.
In particular, we obtain the equality for immediate numeric value of $\Delta
F$ at $s$. Namely, we have that
$(\Delta F)(s)=\sum_{i_{1},i_{2}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq
0\\\ |\sigma_{2}|\geq
0\end{subarray}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\\\
\left.\left\\{\delta
s^{i_{1}}({\boldsymbol{y}}_{1})\cdot\boldsymbol{\delta}^{i_{2}}_{i_{1}}\cdot\delta
s^{\dagger}_{i_{2}}({\boldsymbol{y}}_{2})\cdot\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}\cup\sigma_{2}}\frac{\vec{\partial}^{2}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{i_{1}}_{\sigma_{1}}\partial
q^{\dagger}_{i_{2},\sigma_{2}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}\right\\}\right|_{\begin{subarray}{c}{\boldsymbol{y}}_{1}\,=\,{\boldsymbol{x}}\\\
{\boldsymbol{y}}_{2}\,=\,{\boldsymbol{x}}\end{subarray}}\in\Bbbk.$
By taking one sum containing Kronecker’s $\boldsymbol{\delta}$-symbol, one
arrives at a conventional formula with a summation over the diagonal:
$(\Delta F)(s)=\sum_{i=1}^{N}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq
0\end{subarray}}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\left.\left(\left(-\frac{\vec{{\mathrm{d}}}}{{\mathrm{d}}{\boldsymbol{x}}}\right)^{\sigma_{1}\cup\sigma_{2}}\frac{\vec{\partial}^{2}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\partial
q^{i}_{\sigma_{1}}\partial
q^{\dagger}_{i,\sigma_{2}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}(s)}}\mathrel{\stackrel{{\scriptstyle\text{def}}}{{=}}}{}\\\
{}\mathrel{\stackrel{{\scriptstyle\text{def}}}{{=}}}\sum_{i=1}^{N}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\frac{\overleftarrow{\delta^{2}}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta
q^{i}\delta q^{\dagger}_{i}}\,.$ (19)
We refer to footnote 13 on p. 13 in this context.
###### Remark 2.8.
The conventional formula
$\left.\frac{\overleftarrow{\delta^{2}}f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta{\boldsymbol{q}}({\boldsymbol{y}}_{1})\delta{\boldsymbol{q}}^{\dagger}({\boldsymbol{y}}_{2})}\right|_{\begin{subarray}{c}{\boldsymbol{y}}_{1}\,=\,{\boldsymbol{x}}\\\
{\boldsymbol{y}}_{2}\,=\,{\boldsymbol{x}}\end{subarray}}$
itself is not the definition of a density of the BV-Laplacian $\Delta F$ for
an integral functional $F=\int
f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\cdot\operatorname{dvol}({\boldsymbol{x}})$.
Not containing any built-in sources of divergence, the geometric definition
and its implication (19) yield identical results only when one calculates the
numeric value $(\Delta F)(s)\in\Bbbk$ — but not earlier: structurally
different objects (17) and (19) belong to non-isomorphic spaces (so that the
former contains more information then the latter), and their analytic
behaviour is also different, see Example 2.4 on p. 2.4.
The following two examples are quoted from [35]; they show that the structure
$\Delta$ defined above coincides – but only in the simplest situation– with
the one which is intuitively known from the literature. We refer to the main
Example 2.4 on p. 2.4 which illustrates the multiple-base geometry in a
logically more complex situation of (1c).
###### Example 2.2.
Take a compact, semisimple Lie group $G$ with Lie algebra $\mathfrak{g}$ and
consider the corresponding Yang–Mills theory. Write $A^{a}_{i}$ for the
(coordinate expression of) the gauge potential $A$ – a lower index $i$ because
$A$ is a one-form on the base manifold (i. e., a covector), and an upper index
$a$ because $A$ is a vector in the Lie algebra $\mathfrak{g}$ of the Lie group
$G$. Defining the field strength $\mathcal{F}$ by
$\mathcal{F}^{a}_{ij}=\partial_{i}A^{a}_{j}-\partial_{j}A^{a}_{i}+f^{a}_{bc}A^{b}_{i}A^{c}_{j}$
where $f^{a}_{bc}$ are the structure constants of the Lie algebra
$\mathfrak{g}$, the Yang–Mills action is202020The action functional
$S_{\text{YM}}$ is referred to Minkowski flat coordinates such that
$\operatorname{dvol}({\boldsymbol{x}})=\sqrt{|-1|}\,{\mathrm{d}}^{4}x$ in the
weak gauge field limit.
$S_{\text{YM}}=\tfrac{1}{4}\int\mathcal{F}^{a}_{ij}\mathcal{F}^{a,ij}\,{\mathrm{d}}^{4}x,$
and the full BV-action $S_{\text{{BV}}}$ is212121We denote by
$A_{a}^{i\dagger}$ the parity-odd antifields, by $\gamma^{a}$ the odd ghosts,
and by $\gamma^{\dagger}_{a}$ the parity-even antighosts.
$S_{\text{{BV}}}=S_{\text{YM}}+\int
A_{a}^{i\dagger}(\tfrac{{\mathrm{d}}}{{\mathrm{d}}x^{i}}\gamma^{a}+f_{bc}^{a}A_{i}^{b}\gamma^{c})\,{\mathrm{d}}^{4}x-\tfrac{1}{2}\int
f_{ab}^{c}\gamma^{a}\gamma^{b}\gamma^{\dagger}_{c}\,{\mathrm{d}}^{4}x.$
Let us calculate the BV-Laplacian of this functional. By Corollary 2, the only
terms which survive in $\Delta(S_{\text{BV}})$ are those which contain both
$A$ and $A^{\dagger}$, or both $\gamma$ and $\gamma^{\dagger}$. Therefore,
$\displaystyle\Delta(S_{\text{BV}})$
$\displaystyle=\int\left(\frac{\overleftarrow{\delta}}{\delta
A_{j}^{d}}\frac{\overleftarrow{\delta}}{\delta
A^{j\dagger}_{d}}(f_{bc}^{a}A_{a}^{i\dagger}A_{i}^{b}\gamma^{c})-\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\gamma^{\dagger}_{d}}\frac{\overleftarrow{\delta}}{\delta\gamma^{d}}(f^{c}_{ab}\gamma^{a}\gamma^{b}\gamma^{\dagger}_{c})\right){\mathrm{d}}^{4}x$
$\displaystyle=\int\left(\frac{\overleftarrow{\delta}}{\delta
A_{j}^{d}}(f_{bc}^{d}A_{j}^{b}\gamma^{c})-\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\gamma^{\dagger}_{d}}(f^{c}_{db}\gamma^{b}\gamma^{\dagger}_{c}-f^{c}_{ad}\gamma^{a}\gamma^{\dagger}_{c})\right){\mathrm{d}}^{4}x$
$\displaystyle=\int\left(f_{dc}^{d}\gamma^{c}-\tfrac{1}{2}\bigl{(}f^{d}_{db}\gamma^{b}-f^{d}_{ad}\gamma^{a}\bigr{)}\right){\mathrm{d}}^{4}x=0.$
Let us note also that, since the BV-action $S_{\text{BV}}$ is by construction
such that the horizontal cohomology class of
$\lshad{S_{\text{BV}},S_{\text{BV}}}\rshad$ is zero, as one easily checks by
using Definition 2 below, the functional $S_{\text{BV}}$ satisfies quantum
master-equation (40) tautologically: both sides are, by independent
calculations, equal to zero — should one inspect those values at any section
$s$ of the BV-bundle.
###### Example 2.3.
Consider the nonlinear Poisson sigma model introduced in [11]. Since its
fields are not all purely even, we have to generalize all of our reasoning so
far to a $\mathbb{Z}_{2}$-graded setup — which is, as noted in Remark 2.1,
tedious but straightforward. A verification that $\Delta(S_{\text{CF}})(s)=0$
for the BV-action $S_{\text{CF}}$ of this model and a section $s$ of the
respective BV-bundle would, up to minor differences in conventions and
notations, proceed just as it does in that paper itself, in section 3.2
thereof — except that no infinite constants or Dirac’s
$\boldsymbol{\delta}$-function appear.
###### Remark 2.9.
The BV-Laplacian $\Delta$ is extended by using Leibniz’ rule from the space
$\overline{H}^{n}(\pi_{{\text{{BV}}}})$ of building blocks in
$\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})$ to the space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$, see
Theorem 3 on p. 3. The couplings’ (re)attachment algorithm then results in
formula (1b) on p. 1, which is taken as a _definition_ of the variational
Schouten bracket $\lshad\,,\,\rshad$, see [39]. In turn, that structure’s
extention from
$\overline{H}^{n}(\pi_{{\text{{BV}}}})\times\overline{H}^{n}(\pi_{{\text{{BV}}}})$
to
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\times\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$
is immediate (see Theorem 4 below).
The correspondence between $\Delta$ and $\lshad\,,\,\rshad$ is furthered to an
equivalence between the property $\Delta^{2}=0$ of BV-Laplacian to be a
differential and, on the other hand, Jacobi’s identity for the variational
Schouten bracket. We emphasize that the latter can be verified within the old
approach [41] to geometry of variations. (We refer to [32] for a proof; its
crucial idea is that with evolutionary vector fields it does not matter under
“whose” total derivatives, ${\mathrm{d}}/{\mathrm{d}}{\boldsymbol{x}}$ or
${\mathrm{d}}/{\mathrm{d}}{\boldsymbol{y}}_{i}$, such fields dive.)
Nevertheless, the traditional paradigm fails to reveal that the operator
$\Delta$ is a differential because of a necessity to have the variations
graded-permutable and for that, to distinguish between the functionals’ and
variations’ domains of definition. Our geometric approach resolves that
obstruction and ensures the validity of identities (1c) and (1d) (see Theorems
6 and 8, respectively).
#### 2.4.2 The variational Schouten bracket $\lshad\,,\,\rshad$
The parity-odd Laplacian $\Delta$ is the parent object222222In particular, the
definition of BV-Laplacian logically precedes the construction of Schouten
bracket in BV-formalism (although such parity-odd variational Poisson bracket
is often introduced through postulated formula (25) in the context of
Hamiltonian dynamics and infinite-dimensional completely integrable systems
[14, 18, 26, 41, 42]). Indeed, the entire Schouten-bracket machinery of
(quantum) BV-cohomology groups and their automorphisms, which we consider in
secction 3.2, stems from quantum master-equation (40), see p. 40. which
induces the variational Schouten bracket $\lshad\,,\,\rshad$. Namely, the
bracket appears in the course of that operator’s extension from the space
$\overline{H}^{n}(\pi_{BV})\ni F$ to the space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\supseteq\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})$
of local functionals $F_{1}\cdot\ldots\cdot F_{\ell}$ (it is possible that
$F_{i}$’s already contain some normalized variations).
A distinction between _left_ and _right_ in the directed operators
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}$ and
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}$, the orientation
$\vec{e}_{i}\prec\vec{e}^{{}\,\dagger i}$ in the composite BV-fibres
$W_{{\boldsymbol{x}}}\cong V_{{\boldsymbol{x}}}\mathbin{\widehat{\oplus}}\Pi
V_{{\boldsymbol{x}}}^{\dagger}$ equipped with two couplings (9), and the
ordering of variations
$\delta{\boldsymbol{s}}_{1},\,\ldots,\,\delta{\boldsymbol{s}}_{k}$ specify the
logic of operational Definition 2, which is given in this section.
###### Remark 2.10.
For the sake of brevity, we extend the BV-Laplacian $\Delta$ from the space
$\overline{H}^{n}(\pi_{{\text{{BV}}}})$ of integral functionals
$F_{1},\,\dots\,,F_{\ell}$ to the space
$\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})$ of local functionals such
as $F_{1}\cdot\,\dots\,\cdot F_{\ell}$, the factors of which do not explicitly
contain any built-in variations. To further this extension verbatim onto the
full space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\supsetneq\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})$,
one must remember that it is forbidden to break the order in which the
directed variation operators
$\overrightarrow{\delta}\\!\\!{\boldsymbol{s}}_{k}$ and
$\overleftarrow{\delta}\\!\\!{\boldsymbol{s}}_{k}$ appear in the (ordered
collection of) objects at hand. (Such concept is illustrated by the third term
in (20) below.)
Likewise, we extend $\Delta$ to products of just two factors; in the case of
arbitrary number $\ell\geq 2$ of building blocks $F_{1},\,\dots\,,F_{\ell}$
one proceeds inductively by using the ghost parity-graded Leibniz rule, then
extending $\Delta$ onto the vector space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$ by
linearity.
Let $F=\int
f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\operatorname{dvol}({\boldsymbol{x}}_{1})$
and $G=\int
g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\operatorname{dvol}({\boldsymbol{x}}_{2})$
be integral functionals $\Gamma(\pi_{{\text{{BV}}}})\to\Bbbk$ and let
$\delta{\boldsymbol{s}}=(\delta s;\delta s^{\dagger})$ be a normalized test
shift of their product’s argument $s\in\Gamma(\pi_{{\text{{BV}}}})$. We now
define the operator $\Delta$ acting on the element $F\cdot G$ at
${\boldsymbol{s}}$ by variations first along $(0;\delta s^{\dagger})$ and then
along $(\delta s;0)$.
According to (14), the object to start with is
$\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\sum_{\begin{subarray}{c}i_{1},i_{2}\\\
j_{1},j_{2}\end{subarray}}\sum_{\begin{subarray}{c}|\sigma_{1}|\geq 0\\\
|\sigma_{2}|\geq 0\end{subarray}}\Biggl{\\{}(\delta
s^{i_{1}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{y}}_{1})\left\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),\vec{e}^{{}\,\dagger
j_{1}}(\cdot)\right\rangle\frac{\overrightarrow{\partial}}{\partial
q^{j_{1}}_{\sigma_{1}}}\circ\\\ \circ(\delta
s^{\dagger}_{i_{2}})\left(\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{y}}_{2})\left\langle-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2}),\vec{e}_{j_{2}}(\cdot)\right\rangle\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}}\Biggr{\\}}\\\ \left(\int
f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\operatorname{dvol}({\boldsymbol{x}}_{1})\cdot\int
g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\operatorname{dvol}({\boldsymbol{x}}_{2}\right)(s).$
Their order preserved, the directed operators $\overrightarrow{\delta}\\!\\!s$
and $\overrightarrow{\delta}\\!\\!s^{\dagger}$ spread over the two factors $F$
and $G$ by the binomial formula because of the Leibniz rule for graded
derivations $\overrightarrow{\partial}/\partial q^{j_{1}}_{\sigma_{1}}$ and
$\overrightarrow{\partial}/\partial q^{\dagger}_{j_{2},\sigma_{2}}$. Note that
whenever the ghost parity-odd object $\overrightarrow{\partial}/\partial
q^{\dagger}_{j_{2},\sigma_{2}}$ overtakes the density $f$ of ghost parity
$\operatorname{gh}(F)$, there appears an overall sign factor
$(-)^{\operatorname{gh}(F)}$. We thus obtain
$(\overrightarrow{\delta}\\!\\!s\circ\\!\overrightarrow{\delta}\\!\\!s^{\dagger})(F)+(-)^{\operatorname{gh}(F)}\overrightarrow{\delta}\\!\\!s(F)\overrightarrow{\delta}\\!\\!s^{\dagger}(G)+\overrightarrow{\delta}\\!\\!s\xrightarrow{{}\cdot\overrightarrow{\delta}\\!\\!s^{\dagger}(F)\cdot{}}G\\\
+(-1)^{\operatorname{gh}(F)}F\cdot(\overrightarrow{\delta}\\!\\!s\circ\\!\overrightarrow{\delta}\\!\\!s^{\dagger})(G).$
(20)
The next step is to push right through $F$ its single variations in the middle
two terms of the above expression. This yields the equality
${}=(\overrightarrow{\delta}\\!\\!s\circ\\!\overrightarrow{\delta}\\!\\!s^{\dagger})(F)\cdot
G+(-)^{\operatorname{gh}(F)}\left\\{(F)\overleftarrow{\delta}\\!\\!s\cdot\\!\overrightarrow{\delta}\\!\\!s^{\dagger}(G)+\overrightarrow{\delta}\\!\\!s\xrightarrow{{}\cdot(F)\overleftarrow{\delta}\\!\\!s^{\dagger}\cdot{}}G\right\\}\\\
{}+(-)^{\operatorname{gh}(F)}F\cdot(\overrightarrow{\delta}\\!\\!s\circ\\!\overrightarrow{\delta}\\!\\!s^{\dagger})(G).$
(21)
We emphasize that the operators $\overleftarrow{\delta}\\!\\!s$ and
$\overleftarrow{\delta}\\!\\!s^{\dagger}$ in the variations
$(F)\overleftarrow{\delta}\\!\\!s=\overrightarrow{\delta_{{\boldsymbol{q}}}F}$
and
$(F)\overleftarrow{\delta}\\!\\!s^{\dagger}=\overrightarrow{\delta_{{\boldsymbol{q}}^{\dagger}}F}$
are temporarily redirected to the left so that the middle terms in (21) are
$(-)^{\operatorname{gh}(F)}$ times
$\mathstrut\smash{\overrightarrow{\delta_{{\boldsymbol{q}}}F}\cdot\overleftarrow{\delta_{{\boldsymbol{q}}^{\dagger}}F}+\overleftarrow{\delta_{{\boldsymbol{q}}}}\xrightarrow{\overrightarrow{\delta_{{\boldsymbol{q}}^{\dagger}}F}\cdot}\overline{G}}\
;$ (22)
this is the input datum for a traditional definition of the variational
Schouten bracket (e. g., see [11] vs [39]). Let us remember that the BV-fibres
orientation ${\boldsymbol{q}}\prec{\boldsymbol{q}}^{\dagger}$ expressed by (9)
is built into the last term of (22) even if it is written as follows,
$(F)\overleftarrow{\delta_{{\boldsymbol{q}}}}\cdot\overrightarrow{\delta_{{\boldsymbol{q}}^{\dagger}}}(G)+(F)\overleftarrow{\delta_{{\boldsymbol{q}}^{\dagger}}}\cdot\overrightarrow{\delta_{{\boldsymbol{q}}}}(G).$
Should this be the notation for input, one then usually proclaims that
“differential 1-forms anticommute” so that
$\langle\delta{\boldsymbol{q}}^{\dagger}\wedge\delta{\boldsymbol{q}}\rangle=-\langle\delta{\boldsymbol{q}}\wedge\delta{\boldsymbol{q}}^{\dagger}\rangle=-1$
in $\lshad F,G\rshad=\langle\overrightarrow{\delta
F}\wedge\overleftarrow{\delta G}\rangle$.
We now are almost in a position to (re)configure the couplings in the four
terms of (21). The first term will of course become $\Delta F\cdot G$, and the
last will provide $(-)^{\operatorname{gh}(F)}F\cdot\Delta G$; one is here
allowed to integrate by parts (as explained in section 1.3) in order to shake
the derivatives off $\delta s^{i_{1}}$ and $\delta s^{\dagger}_{i_{2}}$ prior
to evaluation of couplings in the resulting object’s numeric value at its
argument $s$. Yet there remains one more logical step to be done with (22):
let us reverse back
$\overleftarrow{\delta}\\!\\!s\mapsto\overrightarrow{\delta}\\!\\!s$ and
$\overleftarrow{\delta}\\!\\!s^{\dagger}\mapsto\overrightarrow{\delta}\\!\\!s^{\dagger}$
so that on one hand, the vertical differentials fall on $F$ but on the other
hand, the normalization of the basis which stands near $\delta
s^{i}({\boldsymbol{y}}_{1})$ and $\delta
s^{\dagger}_{i}({\boldsymbol{y}}_{2})$ is the _first_ not second column in
(13). This yields the following integrand of
$(-)^{\operatorname{gh}(F)}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{1})\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{2})$,
with a summation over $i_{1},i_{2},j_{1},j_{2}$, and $|\sigma_{1}|\geq 0,\
|\sigma_{2}|\geq 0$,
$\displaystyle\left.\left(f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{j_{1}}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{1}}(s)}\langle\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}}_{1}),+\vec{e}_{i_{1}}({\boldsymbol{y}}_{1})\rangle\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}(\delta
s^{i_{1}})({\boldsymbol{y}}_{1})\cdot$ (23)
$\displaystyle{}\qquad{}\cdot(\delta
s^{\dagger}_{i_{2}})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{y}}_{2})\langle-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2}),\vec{e}_{j_{2}}({\boldsymbol{x}}_{2})\rangle\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{2}}(s)}+{}$
$\displaystyle{}+$
$\displaystyle\left.\left(f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{j_{1},\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{1}}(s)}\cdot{}$
$\displaystyle\Biggl{\\{}\begin{matrix}&(\delta
s^{i_{1}})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{2}}({\boldsymbol{y}}_{1})\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1})|&&|\vec{e}^{{}\,\dagger
j_{2}}({\boldsymbol{x}}_{2})\rangle\\\
\langle\vec{e}_{j_{1}}({\boldsymbol{x}}_{1})|&&|-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2})\rangle\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{1}}(\delta
s_{i_{2}}^{\dagger})({\boldsymbol{y}}_{2})\rangle&\end{matrix}\Biggr{\\}}\cdot$
$\displaystyle\mbox{\hbox to275.99173pt{{ }\hfil{
}}}\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{j_{2}}_{\sigma_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{2}}(s)}.$
The integrations by parts are performed and couplings are reconfigured at the
end of the day in exactly same manner as it has been done in Definition 1; let
us recall that we now define the BV-Laplacian on a larger space. Namely, the
variations couple with the dual variations whereas the differentials of
functionals’ densities attach to each other.
###### Definition 2.
The _variational Schouten bracket_ of two integral functionals
${F=\int
f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]\cdot\operatorname{dvol}({\boldsymbol{x}}_{1})}\qquad\text{and}\qquad{G=\int
g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]\cdot\operatorname{dvol}({\boldsymbol{x}}_{2})}$
is the on-the-diagonal couplings surgery which, by using a normalized test
shift $\delta{\boldsymbol{s}}=(\delta s;0)+(0;\delta
s^{\dagger})\in\Gamma(T\pi_{{\text{{BV}}}})$, yields the functional from
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$ whose
construction at a BV-section $s\in\Gamma(\pi_{{\text{{BV}}}})$ is232323Note
that the directions of $\partial/\partial{\boldsymbol{y}}_{i}$ are reversed so
that the minus signs appear. We emphasize that, prior to the evaluation of
reconfigured couplings, the (co)vectors at ${\boldsymbol{x}}_{j}$ channel the
partial derivatives to $f$ or $g$ according to the couplings’ old arrangement.
$\displaystyle\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}{\mathrm{d}}{\boldsymbol{x}}_{1}\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{2})\Biggl{[}\left.\left(f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{j_{1}}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{1}}(s)}$
$\displaystyle\Biggl{\\{}\begin{matrix}\left(-\frac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}\delta
s^{i_{1}}({\boldsymbol{y}}_{1})\overbrace{\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),-\vec{e}^{{}\,\dagger
i_{2}}({\boldsymbol{y}}_{2})\rangle}^{-1}\,\delta
s^{\dagger}_{i_{2}}({\boldsymbol{y}}_{2})\left(-\frac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}\\\
\underbrace{\langle\vec{e}^{{}\,\dagger
j_{1}}({\boldsymbol{x}}_{1})|\qquad\qquad\qquad\mathstrut,\mathstrut\qquad\qquad\qquad|\vec{e}_{j_{2}}({\boldsymbol{x}}_{2})\rangle}_{-1}\end{matrix}\Biggr{\\}}\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{2}}(s)}$
${}+\left.\left(f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{j_{1},\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{1}}(s)}\\\
{}{}\qquad\quad\Biggl{\\{}\begin{matrix}\delta
s^{i_{1}}({\boldsymbol{y}}_{1})\left(-\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{2}}\overbrace{\langle\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger
i_{2}})({\boldsymbol{y}}_{2})\rangle}^{-1}\cdot\left(-\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{1}}\delta
s^{\dagger}_{i_{2}}({\boldsymbol{y}}_{2})\\\
\underbrace{\langle\vec{e}_{j_{1}}({\boldsymbol{x}}_{1})|\qquad\qquad\qquad\qquad,\mathstrut\qquad\qquad\qquad\qquad|\vec{e}^{{}\,\dagger
j_{2}}({\boldsymbol{x}}_{2})\rangle}_{+1}\end{matrix}\Biggr{\\}}\\\
{}\mbox{\hbox to275.99173pt{{ }\hfil{
}}}\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{j_{2}}_{\sigma_{2}}}g({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{2}}(s)}\Biggr{]}.$
Note that the inner couplings between variations provide a restriction to the
diagonal ${\boldsymbol{y}}_{1}={\boldsymbol{y}}_{2}$ and yield the singular
integral operators which then act to the right via multiplication by $-1$ only
if ${\boldsymbol{y}}_{2}={\boldsymbol{x}}_{2}$ and
${\boldsymbol{y}}_{1}={\boldsymbol{x}}_{1}$, respectively. The outer coupling
then furnishes the main diagonal
${\boldsymbol{x}}_{1}={\boldsymbol{y}}_{1}={\boldsymbol{y}}_{2}={\boldsymbol{x}}_{2}$,
restricting the objects further to the same BV-fibre point in the total space
of the BV-bundle. This reveals why over each point of the base $M^{n}$ the
(derivatives of the) densities $f$ and $g$ are restricted to the infinite jet
of the same section $s$; this also means that, since the moment when the
couplings are reconfigured, the volume element
$\operatorname{dvol}({\boldsymbol{x}}_{1})$ is discarded because appears a new
singular linear integral operator with a standard sign
$\int{\mathrm{d}}{\boldsymbol{x}}_{1}$.
We conclude the reasoning and sum up the definitions and notations in the
following theorem.
###### Theorem 3.
The BV-Laplacian $\Delta$ is the linear operator
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\times\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\to\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$
which acts on products of functionals $F$ and
$G\in\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$ by
the rule
$\Delta(F\cdot G)=\Delta(F)\cdot G+(-)^{\operatorname{gh}(F)}\lshad
F,G\rshad+(-)^{\operatorname{gh}(F)}F\cdot\Delta G.$ (24)
The variational Schouten bracket $\lshad\,,\,\rshad$ measures the deviation
for the BV-Laplacian $\Delta$ from being a derivation.
$\bullet$ After integration by parts, Definition 2 implies the renouned
coordinate formula
$\lshad
F,G\rshad=\int\operatorname{dvol}({\boldsymbol{x}})\Biggl{(}\frac{\overrightarrow{\delta}\\!f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta{\boldsymbol{q}}}\cdot\frac{\overleftarrow{\delta}\\!g({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta{\boldsymbol{q}}^{\dagger}}-\\\
-\frac{\overrightarrow{\delta}\\!f({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta{\boldsymbol{q}}^{\dagger}}\cdot\frac{\overleftarrow{\delta}\\!g({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])}{\delta{\boldsymbol{q}}}\Biggr{)}.$
(25)
###### Remark 2.11.
Let us recall from Remark 1.5 that the building blocks of local functionals
are encoded by equivalence classes of their densities, whereas the underlying
integration manifold $M^{n}$ is endowed with the field-dependent volume
element $\operatorname{dvol}({\boldsymbol{x}},\phi)$. The variational Schouten
bracket transforms two given integral functionals $F$ and $G$ into $\lshad
F,G\rshad$. For every configuration of physical fields $\phi\in\Gamma(\pi)$,
the integration measure is the same in $F$, $G$, and $\lshad F,G\rshad$. This
is because the couplings are local over points
$\bigl{(}{\boldsymbol{x}},\phi({\boldsymbol{x}})\bigr{)}$ in the total space
of the bundle $\pi$ of physical fields, see Remark 2.2 on p. 2.2 ; the
equality of local sections $\phi$ at which all (derivatives of) functionals’
densities are evaluated ensures the equality of metric tensor elements
$g_{\mu\nu}$ in all functionals by virtue of Einstein’s general relativity
equations.
The operational definition of the antibracket $\lshad\,,\,\rshad$ determines
the way how this structure acts on the square
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\times\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$
of entire space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$
containing formal products of functionals.
###### Theorem 4.
Let $F$, $G$, and
$H\in\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$ be
ghost parity-homogeneous functionals. The variational Schouten bracket
$\lshad\,,\,\rshad\colon\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\times\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\to\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$
has the following properties:
* (i)
The value of $\lshad\,,\,\rshad$ at two arguments $F$ and $G\cdot H$ is
$\lshad F,G\cdot H\rshad=\lshad F,G\rshad\cdot
H+(-)^{(\operatorname{gh}(F)-1)\operatorname{gh}(G)}G\cdot\lshad F,H\rshad.$
(26)
This formula recursively extends to products of arbitrary finite number of
factors in the second argument.
* (ii)
The bracket $\lshad\,,\,\rshad$ is shifted-graded skew-symmetric:
$\lshad
F,G\rshad=-(-)^{(\operatorname{gh}(F)-1)\cdot(\operatorname{gh}(G)-1)}\lshad
G,F\rshad,$ (27)
which extends $\lshad\,,\,\rshad$ to products of arbitrary finite number of
factors taken as its first argument in (26).
* (iii)
The bracket $\lshad\,,\,\rshad$ satisfies the shifted-graded Jacobi identity
$(-)^{(\operatorname{gh}(F)-1)(\operatorname{gh}(H)-1)}\lshad F,\lshad
G,H\rshad\rshad+(-)^{(\operatorname{gh}(F)-1)(\operatorname{gh}(G)-1)}\lshad
G,\lshad H,F\rshad\rshad+{}\\\
{}+(-)^{(\operatorname{gh}(G)-1)(\operatorname{gh}(H)-1)}\lshad H,\lshad
F,G\rshad\rshad=0,$ (28)
which stems from graded Leibniz rule (36) for evolutionary vector fields
${\boldsymbol{Q}}^{F}$ defined by the rule
${\boldsymbol{Q}}^{F}(\cdot)\cong\lshad F,\,\cdot\,\rshad$ (here the
equivalence up to integration by parts is denoted by $\cong$ ).
Finally, the variational Schouten bracket extends by linearity to formal sums
of elements from
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})$.
###### Proof.
The bilinearity of $\lshad\,,\,\rshad$ is obvious. It is also clear that the
terms in $\lshad F,G\cdot H\rshad$ are grouped in two parts: those in which
the ghost-parity graded derivations
$\overrightarrow{\partial}/\partial{\boldsymbol{q}}^{\dagger}$ act on $G$ and
those for $H$; the former do not contribute with any extra sign factors
whereas the latter do — in a way which depends on the parity
$\operatorname{gh}(G)$. This means that $\lshad F,G\cdot H\rshad=\lshad
F,G\rshad\cdot H+\ldots$; to grasp the sign in front of the term which has
been omitted, let us swap the graded multiples $G$ and $H$. We have that
$G\cdot H=(-)^{\operatorname{gh}(G)\operatorname{gh}(H)}H\cdot G$, whence
$\lshad F,G\cdot H\rshad=(-)^{\operatorname{gh}(G)\operatorname{gh}(H)}\lshad
F,H\rshad\cdot G+\cdots$. By recalling that $\operatorname{gh}(\lshad
F,H\rshad)=\operatorname{gh}(F)+\operatorname{gh}(H)-1$, we conclude that
$\lshad F,G\cdot H\rshad=\lshad F,G\rshad\cdot
H+(-)^{\operatorname{gh}(G)\operatorname{gh}(H)}(-)^{(\operatorname{gh}(F)+\operatorname{gh}(H)-1)\cdot\operatorname{gh}(G)}G\cdot\lshad
F,H\rshad,$
which yields formula (26).
Proving (27) amounts to a count of signs whenever the bracket $\lshad
F,G\rshad$ of an ordered pair of ghost parity-graded objects is virtually
transformed into $\lshad G,F\rshad$. By using the rule of signs for odd-parity
coordinates, $q^{\dagger}_{\alpha,\sigma}\cdot
q^{\dagger}_{\beta,\tau}=-q^{\dagger}_{\beta,\tau}\cdot
q^{\dagger}_{\alpha,\sigma}$, we first note that
$\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}}g\bigl{(}{\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]\bigr{)}=(-)^{\operatorname{gh}(G)-1}\left(g\bigl{(}{\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]\bigr{)}\right)\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}},$
with a similar formula for the left- and right-acting graded derivative of
$f$. By swapping the (variational) derivatives of the densities $f$ and $g$,
we gain the signs $(-)^{\operatorname{gh}(F)\cdot(\operatorname{gh}(G)-1)}$
and $(-)^{(\operatorname{gh}(F)-1)\cdot\operatorname{gh}(G)}$ for the
respective terms in (23) on p. 23. Combined together, the two steps accumulate
equal factors
$(-)^{(\operatorname{gh}(F)+1)\cdot(\operatorname{gh}(G)-1)}=(-)^{(\operatorname{gh}(F)-1)\cdot(\operatorname{gh}(G)+1)}=(-)^{(\operatorname{gh}(F)-1)\cdot(\operatorname{gh}(G)-1)}$.
Thirdly, by comparing
$(-)^{(\operatorname{gh}(F)-1)\cdot(\operatorname{gh}(G)-1)}\,\lshad
F,G\rshad$ – in which the derivatives of $f$ and $g$ are interchanged and the
derivations’ directions are reversed – with $\lshad G,F\rshad$, we conclude
that the reconfiguration of couplings in the second term in (23) for the
former expression yields _minus_ the first term in $\lshad G,F\rshad$.
Likewise, the couplings reattachment in the first term of such (23) produces
minus the second term in $\lshad G,F\rshad$. This is because the (co)vectors
in the differentials of densities remain unswapped, now going in the ‘wrong’
order.
We now refer to [32, Proposition 3] for a proof of property (iii) in a wider,
non-commutative setup of cyclic words (cf. [29, 36, 46]). It is remarkable
that the reasoning persists within a naïve theory of variations, not referring
to our main idea that each test shift brings its own copy of the base $M^{n}$
into the picture. A key point in the proof is that the rule
${\boldsymbol{Q}}^{F}(\cdot)\cong\lshad F,\,\cdot\,\rshad$ naturally
associates with functionals $F$ the evolutionary fields ${\boldsymbol{Q}}^{F}$
on the infinite jet superbundles at hand, and with _evolutionary_ vector
fields it does not matter under ‘whose” total derivatives such fields dive,
obeying their defining property
$[{\boldsymbol{Q}}^{F},\overrightarrow{{\mathrm{d}}}/{\mathrm{d}}{\boldsymbol{x}}]=0$
(i. e., any integrations by parts, which transform the derivatives
$\overrightarrow{\partial}/\partial{\boldsymbol{y}}_{i}$ falling on test
shifts into total derivatives
$\overrightarrow{{\mathrm{d}}}/{\mathrm{d}}{\boldsymbol{x}}$ falling on the
functionals’ densities, do not mar the outcome even if one attempts to perform
such integrations ahead of time). ∎
### 2.5 Main result : the proof of properties (1c–1d)
We are ready to _prove_ the main interrelations between the BV-Laplacian
$\Delta$ and variational Schouten bracket $\lshad\,,\,\rshad$. Let us recall
that either a validity of these properties was postulated (see [21]) or an ad
hoc regularization technique was formally employed in the literature in order
to mask the seemingly present divergencies (which are actually not there), cf.
[22, §15].
Let us fix the terms. In what follows we refer to building blocks from
$\overline{H}^{n}(\pi_{{\text{{BV}}}})$ and their descendants – containing
reconfigured variations – from
$\overline{H}^{n(1+k)}(\pi_{{\text{{BV}}}}\times
T\pi_{{\text{{BV}}}}\times\ldots\times T\pi_{{\text{{BV}}}})$ as integral
functionals. Such objects will be used for bases of inductive proofs of Lemmas
5 and 7. We then extend the properties (1c) and $\Delta^{2}=0$ to the space
$\overline{\mathfrak{N}}^{n}(\pi_{{\text{{BV}}}},T\pi_{{\text{{BV}}}})\supseteq\overline{\mathfrak{M}}^{n}(\pi_{{\text{{BV}}}})$
of local functionals, that is, of formal sums of products of (varied
descendants of) building blocks.
###### Lemma 5.
Let $F\in\overline{H}^{n(1+k)}\bigl{(}\pi_{{\text{{BV}}}}\times
T\pi_{{\text{{BV}}}}\times\ldots\times T\pi_{{\text{{BV}}}}\bigr{)}$ and
$G\in\overline{H}^{n(1+\ell)}\bigl{(}\pi_{{\text{{BV}}}}\times
T\pi_{{\text{{BV}}}}\times\ldots\times T\pi_{{\text{{BV}}}}\bigr{)}$ be two
integral functionals; here $k,\ell\geqslant 0$. Then
$\Delta\bigl{(}\lshad{F,G}\rshad\bigr{)}=\lshad{\Delta
F,G}\rshad+(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta G}\rshad.$ (29)
###### Proof.
The key idea is that the structures $\Delta$ and $\lshad\,,\,\rshad$ yield
equivalence classes of integral functionals which, after an integration by
parts at the end of the day, are _independent_ of a choice of the built-in
test shifts normalized by (16). Consequently, the composite structure
$\Delta(\lshad{\cdot},{\cdot}\rshad)$ does not change under swapping $\delta
s_{1}^{\alpha}\rightleftarrows\delta s_{2}^{\beta}$, $\delta
s_{1,\alpha}^{\dagger}\rightleftarrows\delta s_{2,\beta}^{\dagger}$ of the
respective variations $\delta{\boldsymbol{s}}_{1}$ and
$\delta{\boldsymbol{s}}_{2}$ in $\Delta$ and $\lshad\,,\,\rshad$. Hence the
terms which are skew-symmetric under such exchange necessarily vanish.
For the sake of clarity, let us assume that $F=\int
f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\,\operatorname{dvol}({\boldsymbol{x}}_{1})$
and $G=\int
g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\,\operatorname{dvol}({\boldsymbol{x}}_{2})$
are just building blocks from the cohomology group
$\overline{H}^{n}(\pi_{{\text{{BV}}}})$; this simplification is legitimate
because new variations which come from $\Delta$ and $\lshad\,,\,\rshad$ do not
interfere with any other test shifts if those are already absorbed by the
densities $f$ and $g$. Suppose that $\delta{\boldsymbol{s}}_{1}$ and
$\delta{\boldsymbol{s}}_{2}$ are two normalized variations of a section
$s\in\Gamma(\pi_{{\text{{BV}}}})$. By definition, we have that242424To keep
track of their origin, we let the directed derivatives
$\partial/\partial{\boldsymbol{y}}_{i}$ or
$\partial/\partial{\boldsymbol{z}}_{j}$ remain falling on the respective
coefficients in $\delta{\boldsymbol{s}}_{1}$ and $\delta{\boldsymbol{s}}_{2}$;
the integration by parts is performed in a standard way prior to the
reconfigurations which are shown in the formula.
$\displaystyle\Delta\left(\lshad
F,G\rshad\right)(s)=\int_{M}{\mathrm{d}}{\boldsymbol{z}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{z}}_{2}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}{\mathrm{d}}{\boldsymbol{x}}_{1}\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{2})\
\cdot$ $\displaystyle\Biggl{\\{}(\delta
s_{1}^{\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\left\langle\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),-\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{z}}_{2})\right\rangle\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\langle\vec{e}^{{}\,\dagger\alpha}(\cdot),\vec{e}_{\alpha}(\cdot)\rangle\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}$
$\displaystyle\quad\smash{\Biggl{[}}f({\boldsymbol{x}}_{1}.[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\smash{\frac{\overleftarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}}\,\underline{\langle\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{1})|}\,\Bigl{\langle}\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}(\delta
s_{2}^{\beta})({\boldsymbol{y}}_{1})\,\vec{e}_{\beta}({\boldsymbol{y}}_{1}),$
$\displaystyle\mbox{\hbox to71.13188pt{{ }\hfil{
}}}{-}\vec{e}^{{}\,\dagger.\beta}({\boldsymbol{y}}_{2})\,(\delta
s^{\dagger}_{2,\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}({\boldsymbol{y}}_{2})\Bigr{\rangle}\,\underline{|\vec{e}_{\beta}({\boldsymbol{x}}_{2})\rangle}\,\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])+{}$
$\displaystyle{}\quad{}+f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\,\underline{\langle\vec{e}_{\beta}({\boldsymbol{x}}_{1})|}\,\Bigl{\langle}(\delta
s_{2}^{\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}({\boldsymbol{y}}_{1})\,\vec{e}_{\beta}({\boldsymbol{y}}_{1}),$
$\displaystyle\mbox{\hbox to71.13188pt{{ }\hfil{
}}}{-}\vec{e}^{{}\,\dagger\beta}({\boldsymbol{y}}_{2})\,\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}(\delta
s^{\dagger}_{2,\beta})({\boldsymbol{y}}_{2})\Bigr{\rangle}\,\underline{|\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{2})\rangle}\,\frac{\overrightarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\smash{\Biggr{]}\left.\Biggr{\\}}\right|_{\begin{subarray}{c}j^{\infty}(s)\\\
{\boldsymbol{x}}_{i}={\boldsymbol{y}}_{j}={\boldsymbol{z}}_{k}\end{subarray}}}.$
The partial derivatives $\overrightarrow{\partial}/\partial
q^{\alpha}_{\sigma_{1}}\circ\overrightarrow{\partial}/\partial
q^{\dagger}_{\alpha,\sigma_{2}}$ are distributed between the arguments $f$ and
$g$ by the graded Leibniz rule. Whenever _none_ of the two operators overtakes
the density of $F$, the reconfiguration yields $\lshad\Delta
F,G\rshad({\boldsymbol{s}})$. Likewise, if _both_ derivatives indexed by
$\alpha$ overtake $F$ and an old derivative that fell on $g$, then we obtain
$(-)^{\operatorname{gh}(F)-1}\lshad F,\Delta G\rshad({\boldsymbol{s}})$, which
is the second term in the right-hand side of (29). We claim that the remaining
four terms cancel out by virtue of independence of $\Delta$ and
$\lshad\,,\,\rshad$ from a choice of normalized variations. To prove this
claim, we consecutively inspect the behaviour of those four terms under a swap
$\delta{\boldsymbol{s}}_{1}\rightleftarrows\delta{\boldsymbol{s}}_{2}$ of
coefficients in the normalized test shifts.
The first and second terms sum up to the difference
$\left\langle(\delta
s^{\alpha}_{1})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\right\rangle\underbrace{\langle\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}_{2}),\vec{e}_{\alpha}({\boldsymbol{x}}_{1})\rangle}_{-1}\,\cdot{}\\\
\cdot\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}\,\underline{\bigl{\langle}\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{1})\bigr{|}}\Biggl{\langle}\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}(\delta
s_{2}^{\beta})({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\\\
(\delta
s^{\dagger}_{2,\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}({\boldsymbol{y}}_{2})\Biggr{\rangle}\underbrace{\bigl{|}\vec{e}_{\beta}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{-1}\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])+{}\\\
{}+\left\langle(\delta
s^{\alpha}_{1})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\right\rangle\underbrace{\langle\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}_{1}),\vec{e}_{\alpha}({\boldsymbol{x}}_{2})\rangle}_{-1}\,\cdot\\\
\cdot(-)^{\operatorname{gh}(F)-1}\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\underline{\bigl{\langle}\vec{e}_{\beta}({\boldsymbol{x}}_{1})\bigr{|}}\Biggl{\langle}(\delta
s_{2}^{\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\\\
\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}(\delta
s^{\dagger}_{2,\beta})({\boldsymbol{y}}_{2})\Biggr{\rangle}\underbrace{\bigl{|}\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{+1}\,\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\frac{\overrightarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]).$
(30)
Recalling that
$f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}=(-)^{\operatorname{gh}(F)-1}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]),$
let us swap the derivations which fall on $f$ from the left and right; this
eliminates the sign $(-)^{\operatorname{gh}(F)-1}$. We proceed likewise for
$g$ and then transport the variations $\delta{\boldsymbol{s}}_{1}$ and
$\delta{\boldsymbol{s}}_{2}$, exchanging their places (and their rôles with
respect to $\Delta$ and $\lshad\,,\,\rshad$). The second term in formula (30)
becomes
$\smash{\Biggl{\langle}(\delta
s_{2}^{\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\,\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}(\delta
s^{\dagger}_{2,\beta})({\boldsymbol{y}}_{2})\Biggr{\rangle}}\underbrace{\bigl{\langle}\vec{e}_{\beta}({\boldsymbol{x}}_{1}),\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{+1}\\\
\cdot\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}\underline{\bigl{\langle}\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}_{1})\bigr{|}}\Biggl{\langle}\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}(\delta
s^{\alpha}_{1})({\boldsymbol{z}}_{1})\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)({\boldsymbol{z}}_{2})\Biggr{\rangle}\\\
\underbrace{\bigl{|}\vec{e}_{\alpha}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{-1}\,\frac{\overrightarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]).$
It is now readily seen that the first term in (30) and this equivalent
expression of its second term are opposite to each other. Indeed, relabel the
summation indexes $\alpha\rightleftarrows\beta$, $\sigma\rightleftarrows\tau$
so that $\delta s^{\alpha}_{1}\rightleftarrows\delta s^{\beta}_{2}$, $\delta
s^{\dagger}_{1,\alpha}\rightleftarrows\delta s^{\dagger}_{2,\beta}$, and swap
the copies of base manifold $M^{n}$ by
${\boldsymbol{y}}\rightleftarrows{\boldsymbol{z}}$. Due to the second factors
in the products $(-1)\cdot(-1)\cdot(-1)\cdot(-1)=+1$ versus
$(-1)\cdot(+1)\cdot(-1)\cdot(-1)=-1$, the two terms in (30) cancel out after
the integration by parts and evaluation of the couplings in view of (16).
Next, the integrand of $\Delta\bigl{(}\lshad F,G\rshad\bigr{)}(s)$ contains a
restriction to the infinite jet $j^{\infty}(s)$ of the third term, which is
$\displaystyle\Biggl{\langle}(\delta
s_{1}^{\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\Biggr{\rangle}\underbrace{\bigl{\langle}\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}_{2}),\vec{e}_{\alpha}({\boldsymbol{x}}_{1})\bigr{\rangle}}_{-1}$
$\displaystyle\ {}\cdot\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\left(f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\right)$
$\displaystyle\quad\underline{\bigl{\langle}\vec{e}_{\beta}({\boldsymbol{x}}_{1})\bigr{|}}\Biggl{\langle}(\delta
s^{\beta}_{2})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\,\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}(\delta
s^{\dagger}_{2,\beta})({\boldsymbol{y}}_{2})\Biggr{\rangle}\underbrace{\bigl{|}\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{+1}$
$\displaystyle\mbox{\hbox to304.44447pt{{ }\hfil{
}}}\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]).$
Let the summation indexes be relabelled as above:
$\alpha\rightleftarrows\beta$, $\sigma\rightleftarrows\tau$, and
$\smash{\delta s^{\alpha}_{1}\rightleftarrows\delta s^{\beta}_{2}}$, $\delta
s^{\dagger}_{1,\alpha}\rightleftarrows\delta s^{\dagger}_{2,\beta}$ on top of
${\boldsymbol{y}}\rightleftarrows{\boldsymbol{z}}$. The transformation of
graded derivations falling from the left and right on $f$ is then
$\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\left(f\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\right)\longmapsto\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\left(f\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\right)=\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\left((-)^{\operatorname{gh}(F)-1}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}f\right)=\\\
=(-)^{\operatorname{gh}(F)-2}\cdot(-)^{\operatorname{gh}(F)-1}\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}f\right)\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}=-\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\left(f\frac{\overleftarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\right).$
This minus sign shows that the third term as it was written initially, and the
newly produced one in which the variations $\delta{\boldsymbol{s}}_{1}$ and
$\delta{\boldsymbol{s}}_{2}$ are interchanged have opposite signs. At the same
time, these integral functionals must be equal to each other due to
independence of $\Delta$ and $\lshad\,,\,\rshad$ of a choice of the test
shifts. Therefore, each of those expressions vanishes.
The fourth term is processed analogously; its integrand is
$\displaystyle(-)^{\operatorname{gh}(F)}\Biggl{\langle}(\delta
s_{1}^{\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\Biggr{\rangle}\cdot\underbrace{\bigl{\langle}\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}_{1}),\vec{e}_{\alpha}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{-1}$
$\displaystyle\ {}\cdot\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}f({\boldsymbol{x}}_{1},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\frac{\overleftarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}$
$\displaystyle\qquad\underline{\bigl{\langle}\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}_{1})\bigr{|}}\Biggl{\langle}\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}(\delta
s^{\beta}_{2})({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\,(\delta
s^{\dagger}_{2,\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}({\boldsymbol{y}}_{2})\Biggr{\rangle}\underbrace{\bigl{|}\vec{e}_{\beta}({\boldsymbol{x}}_{2})\bigr{\rangle}}_{-1}$
$\displaystyle\mbox{\hbox to304.44447pt{{ }\hfil{
}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}g({\boldsymbol{x}}_{2},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]).$
The very same procedure of two variations interchange and relabelling restores
an almost identical expression in which, however, the parity-odd derivations
go in the inverse order $\overrightarrow{\partial}/\partial
q^{\dagger}_{\beta,\tau_{2}}\circ\overrightarrow{\partial}/\partial
q^{\dagger}_{\alpha,\sigma_{2}}$. Equal to minus itself, the fourth term
vanishes. This concludes the proof. ∎
The following example illustrates the assertion of Lemma 5 (but not a
technique of its proof which itself accompanies Lemma 1). We use the
convention from Remark 2.7, denoting by
${\mathrm{d}}/{\mathrm{d}}{\boldsymbol{y}}_{i}$ or
${\mathrm{d}}/{\mathrm{d}}{\boldsymbol{z}}_{j}$ the total derivatives which
act on the functionals’ densities at points ${\boldsymbol{x}}_{k}$; this keeps
track of those derivatives origin and lets us indicate the couplings’ values
as they appear after the integrations by parts, contributing only with sign
factors $\pm 1$. For the sake of brevity we do not write the (co)vectors
$\vec{e}_{i}$ and $\vec{e}^{{}\,\dagger\,i}$ in the formulas below, referring
to the proofs in preceding sections. Likewise, we do not indicate the base
point congruences that occur due to the absolute locality of couplings.
An overall comment to Example 2.4 below is that, fully aware of the goal which
is to calculate $\Delta\left(\lshad F,G\rshad\right)$ or, respectively,
$\lshad\Delta F,G\rshad$ and $\lshad F,\Delta G\rshad$, we do not interrupt
the logic of our reasoning by attempting to view the intermediate objects
$\lshad F,G\rshad$ or $\Delta F$ and $\Delta G$ as mappings
$\Gamma(\pi_{{\text{{BV}}}})\to\Bbbk$, cf. Corollary 2 on p. 2. Such mappings
would not be elements of the structures which stand in the left- and right-
hand sides of the identity under examination. The slogan is that a step-by-
step evaluation is illegal; derivations of the end-product from input data
must not be interrupted at half-way.
We also emphasize that the example below is a prototype reasoning which is
equally well applicable to any other arguments $F$ and $G$ in (29); a choice
of the functionals is here not specific to any model. The point is that
equality (29) holds and does not require any manual regularization.
###### Example 2.4.
Consider the integral functionals
$F=\int q^{\dagger}qq_{x_{1}x_{1}}\,{\mathrm{d}}x_{1}\quad\text{and}\quad
G=\int q^{\dagger}_{x_{2}x_{2}}\cos q\,{\mathrm{d}}x_{2}.$
Let us show that equality (29) is satisfied for $F$ and $G$, that is,
$\Delta\left(\lshad F,G\rshad\right)=\lshad\Delta F,G\rshad+\lshad F,\Delta
G\rshad,\qquad\operatorname{gh}(F)=1,$ (31)
in the frames of product-bundle geometry of variations and operational
definitions of the BV-Laplacian $\Delta$ and variational Schouten bracket
$\lshad\,,\,\rshad$.
We have
$\lshad
F,G\rshad=\iiiint{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}{\mathrm{d}}y_{1}{\mathrm{d}}y_{2}\Bigl{\langle}\Bigl{(}\underbrace{q^{\dagger}q_{xx}+\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{1}^{2}}(q^{\dagger}q)}_{x_{1}}\Bigr{)}\cdot\underbrace{\langle\delta
s_{2}(y_{1}),\delta
s_{2}^{\dagger}(y_{2})\rangle}_{+1}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{\cos
q}_{x_{2}}\bigr{)}\Bigr{\rangle}\\\
{}+\iiiint{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}{\mathrm{d}}y_{1}{\mathrm{d}}y_{2}\Bigl{\langle}\bigl{(}\underbrace{qq_{xx}}_{x_{1}}\bigr{)}\cdot\underbrace{\langle\delta
s_{2}^{\dagger}(y_{2}),\delta
s_{2}(y_{1})\rangle}_{-1}\cdot\bigl{(}\underbrace{-q^{\dagger}_{xx}\,\sin
q}_{x_{2}}\bigr{)}\Bigr{\rangle}.$
Therefore, one side of the expected equality is
$\Delta\bigl{(}\lshad
F,G\rshad\bigr{)}=\int\\!\\!{\mathrm{d}}z_{1}\int\\!\\!{\mathrm{d}}z_{2}\int\\!\\!{\mathrm{d}}x_{1}\int\\!\\!{\mathrm{d}}x_{2}\int\\!\\!{\mathrm{d}}y_{1}\int\\!\\!{\mathrm{d}}y_{2}\underbrace{\langle\delta
s_{1}(z_{1}),\delta
s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot\underbrace{\langle\delta
s_{2}(y_{1}),\delta s_{2}^{\dagger}(y_{2})\rangle}_{+1}\cdot{}\\\
{}\cdot\Bigl{\langle}\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{1}^{2}}\underbrace{(1)}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{\cos
q}_{x_{2}}\bigr{)}+\underline{\underbrace{q_{xx}}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{-\sin
q}_{x_{2}}\bigr{)}}+\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{1}^{2}}\underbrace{(1)}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{\cos
q}_{x_{2}}\bigr{)}+\underline{\underline{\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{1}^{2}}\underbrace{(q)}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{-\sin
q}_{x_{2}}\bigr{)}}}\Bigr{\rangle}\\\
+\int\\!\\!{\mathrm{d}}z_{1}\int\\!\\!{\mathrm{d}}z_{2}\int\\!\\!{\mathrm{d}}x_{1}\int\\!\\!{\mathrm{d}}x_{2}\int\\!\\!{\mathrm{d}}y_{1}\int\\!\\!{\mathrm{d}}y_{2}\underbrace{\langle\delta
s_{1}(z_{1}),\delta
s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot\underbrace{\langle\delta
s_{2}^{\dagger}(y_{2}),\delta s_{2}(y_{1})\rangle}_{-1}\cdot{}\\\
\cdot\Bigl{\langle}\underline{\underbrace{q_{xx}}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\sin
q}_{x_{2}}\bigr{)}}+\underline{\underline{\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{1}^{2}}\underbrace{(q)}_{x_{1}}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\sin
q}_{x_{2}}\bigr{)}}}+\bigl{(}\underbrace{qq_{xx}}_{x_{1}}\bigr{)}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\cos
q}_{x_{2}}\bigr{)}\Bigr{\rangle}\ .$
The respective pairs of underlined terms cancel out and there remains only
${}=\int{\cdots}\int{\mathrm{d}}z_{1}\,{\mathrm{d}}z_{2}\,{\mathrm{d}}x_{1}\,{\mathrm{d}}x_{2}\,{\mathrm{d}}y_{1}\,{\mathrm{d}}y_{2}\underbrace{\langle\delta
s_{1}(z_{1}),\delta
s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot\bigl{\langle}\bigl{(}\underbrace{qq_{xx}}_{x_{1}}\bigr{)}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\cos
q}_{x_{2}}\bigr{)}\bigr{\rangle}\cdot\\\ \underbrace{\langle\delta
s_{2}^{\dagger}(y_{2}),\delta s_{2}(y_{1})\rangle}_{-1}.$ (32)
On the other hand, we obtain that
$\Delta
F=\iiint{\mathrm{d}}z_{1}{\mathrm{d}}z_{2}{\mathrm{d}}x_{1}\underbrace{\langle\delta
s_{1}(z_{1}),\delta
s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot\bigl{\langle}\underbrace{q_{xx}}_{x_{1}}+\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{1}^{2}}\underbrace{(q)}_{x_{1}}\bigr{\rangle},$
which yields
$\lshad\Delta
F,G\rshad=\int\\!\\!{\mathrm{d}}z_{1}\int\\!\\!{\mathrm{d}}z_{2}\int\\!\\!{\mathrm{d}}x_{1}\int\\!\\!{\mathrm{d}}x_{2}\int\\!\\!{\mathrm{d}}y_{1}\int\\!\\!{\mathrm{d}}y_{2}\,\underbrace{\langle\delta
s_{1}(z_{1}),\delta s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot{}\\\
\Bigl{\langle}\Bigl{(}\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{1}^{2}}\underbrace{(1)}_{x_{1}}+\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{1}^{2}}\underbrace{(1)}_{x_{1}}\Bigr{)}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}y_{2}^{2}}\bigl{(}\underbrace{\cos
q}_{x_{2}}\bigr{)}\Bigr{\rangle}\cdot\underbrace{\langle\delta
s_{2}(y_{1}),\delta s_{2}^{\dagger}(y_{2})\rangle}_{+1}=0.$
From the fact that the other BV-Laplacian,
$\Delta
G=\iiint\,{\mathrm{d}}z_{1}\,{\mathrm{d}}z_{2}\,{\mathrm{d}}x_{2}\underbrace{\langle\delta
s_{1}(z_{1}),\delta
s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot\bigl{\langle}\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\sin
q}_{x_{2}}\bigr{)}\bigr{\rangle},$
does not contain $q^{\dagger}$ so that the first half of the Schouten bracket
$\lshad F,\Delta G\rshad$ drops out, we deduce that
$\lshad F,\Delta
G\rshad=\int{\cdots}\int\,{\mathrm{d}}z_{1}\,{\mathrm{d}}z_{2}\,{\mathrm{d}}x_{1}\,{\mathrm{d}}x_{2}\,{\mathrm{d}}y_{1}\,{\mathrm{d}}y_{2}\underbrace{\langle\delta
s_{1}(z_{1}),\delta s_{1}^{\dagger}(z_{2})\rangle}_{+1}\cdot{}\\\
\bigl{\langle}\bigl{(}\underbrace{qq_{xx}}_{x_{1}}\bigr{)}\cdot\tfrac{{\mathrm{d}}^{2}}{{\mathrm{d}}z_{2}^{2}}\bigl{(}\underbrace{-\cos
q}_{x_{2}}\bigr{)}\bigr{\rangle}\cdot\underbrace{\langle\delta
s_{2}^{\dagger}(y_{2}),\delta s_{2}(y_{1})\rangle}_{-1}.$ (33)
Consequently, the two sides of (31), namely, $\Delta\bigl{(}\lshad
F,G\rshad\bigr{)}$ expressed by (32) and $\lshad\Delta F,G\rshad+\lshad
F,\Delta G\rshad$ accumulated in (33), match perfectly for the functionals $F$
and $G$ at hand.
###### Theorem 6.
Let $F$,
$G\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{{\text{{BV}}}})$ be
two functionals. The Batalin–Vilkovisky Laplacian $\Delta$ satisfies the
relation
$\Delta\bigl{(}\lshad{F,G}\rshad\bigr{)}=\lshad{\Delta
F,G}\rshad+(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta G}\rshad.$ (29)
In other words, the operator $\Delta$ is a graded derivation of the
variational Schouten bracket $\lshad{\,,\,}\rshad$.
###### Proof.
We prove this by induction over the number of building blocks in each argument
of the Schouten bracket in the left hand side of (29). If $F$ and $G$ both
belong to $\overline{H}^{*}(\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots\times T\pi_{\text{{BV}}})$, then Lemma 5 states
the assertion, which is the base of induction. To make an inductive step,
without loss of generality let us assume that the second argument of
$\lshad{\,,\,}\rshad$ in (29) is a product of two elements from
$\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{{\text{{BV}}}})$, each of
them containing less multiples from $\overline{H}^{*}(\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots T\pi_{\text{{BV}}})$ than the product. Denote
such factors by $G$ and $H$ and recall that by Theorem 4,
$\lshad{F,G\cdot H}\rshad=\lshad{F,G}\rshad\cdot
H+(-)^{(\operatorname{gh}(F)-1)\cdot\operatorname{gh}(G)}G\cdot\lshad{F,H}\rshad.$
Therefore, using Theorem 3 we have that
$\displaystyle\Delta(\lshad$ $\displaystyle F,G\cdot H\rshad)$
$\displaystyle={}$ $\displaystyle\Delta(\lshad{F,G}\rshad)\cdot
H+(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}[\\![\lshad{F,G}\rshad,H]\\!]+(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}\lshad{F,G}\rshad\cdot\Delta
H$
$\displaystyle+(-)^{(\operatorname{gh}(F)-1)\operatorname{gh}(G)}\left(\Delta
G\cdot\lshad{F,H}\rshad+(-)^{\operatorname{gh}(G)}[\\![G,\lshad{F,H}\rshad]\\!]+(-)^{\operatorname{gh}(G)}G\cdot\Delta(\lshad{F,H}\rshad)\right).$
Using the inductive hypothesis in the first and last terms of the right-hand
side in the above formula, we continue the equality and obtain
$\displaystyle={}$ $\displaystyle\lshad{\Delta F,G}\rshad\cdot
H+(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta G}\rshad\cdot
H+(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}[\\![\lshad{F,G}\rshad,H]\\!]$
$\displaystyle+(-)^{\operatorname{gh}(F)\operatorname{gh}(G)}[\\![G,\lshad{F,H}\rshad]\\!]+(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}\lshad{F,G}\rshad\cdot\Delta
H$ $\displaystyle+(-)^{(\operatorname{gh}(F)-1)\operatorname{gh}(G)}\Delta
G\cdot\lshad{F,H}\rshad+(-)^{\operatorname{gh}(F)\operatorname{gh}(G)}G\cdot\lshad{\Delta
F,H}\rshad$
$\displaystyle+(-)^{\operatorname{gh}(F)\operatorname{gh}(G)+\operatorname{gh}(F)-1}G\cdot\lshad{F,\Delta
H}\rshad.$ (34)
On the other hand, let us expand the formula
$\lshad{\Delta F,G\cdot
H}\rshad+(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta(G\cdot H)}\rshad,$
which is the right hand side of (29) in the inductive claim. We obtain
$\displaystyle={}$ $\displaystyle\lshad{\Delta F,G}\rshad\cdot
H+(-)^{(\operatorname{gh}(\Delta F)-1)\operatorname{gh}(G)}G\cdot\lshad{\Delta
F,H}\rshad$ $\displaystyle+(-)^{\operatorname{gh}(F)-1}[\\![F,\ \Delta G\cdot
H+(-)^{\operatorname{gh}(G)}\lshad{G,H}\rshad+(-)^{\operatorname{gh}(G)}G\cdot\Delta
H\ ]\\!]$ $\displaystyle={}$ $\displaystyle\lshad{\Delta F,G}\rshad\cdot
H+(-)^{\operatorname{gh}(F)\operatorname{gh}(G)}G\cdot\lshad{\Delta
F,H}\rshad+(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta G}\rshad\cdot H$ (35)
$\displaystyle+(-)^{\operatorname{gh}(F)-1}(-)^{(\operatorname{gh}(F)-1)(\operatorname{gh}(G)-1)}\Delta
G\cdot\lshad{F,H}\rshad+(-)^{\operatorname{gh}(F)-1}(-)^{\operatorname{gh}(G)}[\\![F,\lshad{G,H}\rshad]\\!]$
$\displaystyle+(-)^{\operatorname{gh}(F)-1}(-)^{\operatorname{gh}(G)}\lshad{F,G}\rshad\cdot\Delta
H+(-)^{\operatorname{gh}(F)-1}(-)^{\operatorname{gh}(G)}(-)^{(\operatorname{gh}(F)-1)\operatorname{gh}(G)}G\cdot\lshad{F,\Delta
H}\rshad.$
Comparing (35) with (34), which was derived from the inductive hypothesis, we
see that all terms match except for
$(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}[\\![\lshad{F,G}\rshad,H]\\!]+(-)^{\operatorname{gh}(F)\operatorname{gh}(G)}[\\![G,\lshad{F,H}\rshad]\\!]$
from (34) versus
$(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}[\\![F,\lshad{G,H}\rshad]\\!]$
from (35). However, these three terms constitute Jacobi’s identity (28) for
the variational Schouten bracket. Namely, we have that (cf. [32])
$[\\![F,\lshad{G,H}\rshad]\\!]=[\\![\lshad{F,G}\rshad,H]\\!]+(-)^{(\operatorname{gh}(F)-1)(\operatorname{gh}(G)-1)}[\\![G,\lshad{F,H}\rshad]\\!],$
(36)
so that by multiplying both sides of the identity by
$(-)^{\operatorname{gh}(F)+\operatorname{gh}(G)-1}$, we fully balance (34) and
(35). This completes the inductive step and concludes the proof. ∎
###### Lemma 7.
The linear operator
$\Delta\colon\overline{H}^{n(1+k)}\bigl{(}\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots\times
T\pi_{\text{{BV}}}\bigr{)}\longrightarrow\overline{H}^{n(2+k)}\bigl{(}\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots\times T\pi_{\text{{BV}}}\bigr{)}$
is a differential for every $k\geqslant 0$.
The proof of Lemma 7 is conceptually close to the second and third steps in
the proof of Lemma 5. Namely, two normalized variations are swapped in an
integral functional within the image of $\Delta^{2}$, which yields an
indistinguishable result of opposite sign.
###### Proof.
Let $\delta{\boldsymbol{s}}_{1}$ and $\delta{\boldsymbol{s}}_{2}$ be
normalized test shifts of a section $s\in\Gamma(\pi_{{\text{{BV}}}})$, and let
$H=\int
h({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\cdot\operatorname{dvol}({\boldsymbol{x}})$
be an integral functional. (It suffices to consider a simplified picture
$H\in\overline{H}^{n}(\pi_{{\text{{BV}}}})$, not taking into account any
built-in variations in the construction of $H$.) By definition, we have that
$\Delta(\Delta
H)(s)=\int_{M}{\mathrm{d}}{\boldsymbol{z}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{z}}_{2}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\int_{M}\operatorname{dvol}({\boldsymbol{x}})\cdot\\\
\cdot\Biggl{\\{}\left\langle(\delta
s^{\alpha}_{1})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{1}}\right)^{\sigma_{1}}({\boldsymbol{z}}_{1})\,\overbrace{\vec{e}_{\alpha}({\boldsymbol{z}}_{1}),(-\vec{e}^{{}\,\dagger\alpha})({\boldsymbol{z}}_{2})}^{-1}\,(\delta
s^{\dagger}_{1,\alpha})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{z}}_{2}}\right)^{\sigma_{2}}({\boldsymbol{z}}_{2})\right\rangle\underbrace{\left\langle\vec{e}^{{}\,\dagger\alpha}({\boldsymbol{x}}),\vec{e}_{\alpha}({\boldsymbol{x}})\right\rangle}_{-1}\\\
{}\quad\left\langle(\delta
s^{\beta}_{2})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\tau_{1}}({\boldsymbol{y}}_{1})\,\overbrace{\vec{e}_{\beta}({\boldsymbol{y}}_{1}),(-\vec{e}^{{}\,\dagger\beta})({\boldsymbol{y}}_{2})}^{-1}\,(\delta
s^{\dagger}_{2,\beta})\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\tau_{2}}({\boldsymbol{y}}_{2})\right\rangle\underbrace{\left\langle\vec{e}^{{}\,\dagger\beta}({\boldsymbol{x}}),\vec{e}_{\beta}({\boldsymbol{x}})\right\rangle}_{-1}\\\
\frac{\overrightarrow{\partial}}{\partial
q^{\alpha}_{\sigma_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\frac{\overrightarrow{\partial}}{\partial
q^{\beta}_{\tau_{1}}}\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}h({\boldsymbol{x}},[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\left.\Biggr{\\}}\right|_{j^{\infty}_{{\boldsymbol{x}}}(s)}.$
By exchanging the integrand’s upper two lines and then relabelling
$\alpha\rightleftarrows\beta$,
$\sigma\rightleftarrows\tau$ so that $\delta
s_{1}^{\alpha}\rightleftarrows\delta s_{2}^{\beta}$ and $\delta
s^{\dagger}_{1,\alpha}\rightleftarrows\delta s^{\dagger}_{2,\beta}$, and by
swapping the reference ${\boldsymbol{y}}\rightleftarrows{\boldsymbol{z}}$ to
copies of the base manifold $M^{n}$, we almost recover the initial expression
(which should be the case), yet the order in which the parity-odd partial
derivatives follow is inverse,
$\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\circ\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\longmapsto\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}\circ\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}=-\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\alpha,\sigma_{2}}}\circ\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{\beta,\tau_{2}}}.$
Therefore the integrand of functional $\Delta^{2}H$ vanishes, which proves the
assertion. ∎
###### Theorem 8.
The Batalin–Vilkovisky Laplacian $\Delta$ is a differential: for all
$H\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}}$, $T\pi_{{\text{{BV}}}})$
we have
$\Delta^{2}(H)=0.$
###### Proof.
We prove Theorem 8 by induction over the number of building blocks from
$\overline{H}^{*}\bigl{(}\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots\times T\pi_{\text{{BV}}}\bigr{)}$ in the
argument
$H\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{{\text{{BV}}}})$ of
$\Delta^{2}$. If $H\in\overline{H}^{*}\bigl{(}\pi_{\text{{BV}}}\times
T\pi_{\text{{BV}}}\times\ldots\times T\pi_{\text{{BV}}}\bigr{)}$ itself is an
integral functional, then by Lemma 7 there remains nothing to prove. Suppose
now that $H=F\cdot G$ for some
$F,G\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{{\text{{BV}}}})$.
Then Theorem 3 yields that
$\displaystyle\Delta^{2}$ $\displaystyle(F\cdot G)=\Delta\left(\Delta F\cdot
G+(-)^{\operatorname{gh}(F)}\lshad{F,G}\rshad+(-)^{\operatorname{gh}(F)}F\cdot\Delta
G\right).$ Using Theorem 3 again and also Theorem 6, we continue the equality:
$\displaystyle={}$ $\displaystyle\Delta^{2}F\cdot
G+(-)^{\operatorname{gh}(\Delta F)}\lshad{\Delta
F,G}\rshad+(-)^{\operatorname{gh}(\Delta F)}\Delta F\cdot\Delta G$
$\displaystyle{}+(-)^{\operatorname{gh}(F)}\lshad{\Delta
F,G}\rshad+(-)^{\operatorname{gh}(F)}(-)^{\operatorname{gh}(F)-1}\lshad{F,\Delta
G}\rshad$ $\displaystyle{}+(-)^{\operatorname{gh}(F)}\Delta F\cdot\Delta
G+(-)^{\operatorname{gh}(F)}(-)^{\operatorname{gh}(F)}\lshad{F,\Delta
G}\rshad+(-)^{\operatorname{gh}(F)}(-)^{\operatorname{gh}(F)}F\cdot\Delta^{2}G.$
By the inductive hypothesis, the first and last terms in the above formula
vanish; taking into account that $\operatorname{gh}(\Delta
F)=\operatorname{gh}(F)-1$ in $\mathbb{Z}_{2}$, the terms with $\Delta
F\cdot\Delta G$ cancel against each other, as do the terms containing
$\lshad{\Delta F,G}\rshad$ and $\lshad{F,\Delta G}\rshad$. The proof is
complete. ∎
## 3 The quantum master-equation
### 3.1 The Laplace equation
In this section we inspect the conditions upon functionals
$F\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$ under
which the Feynman path integrals
$\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds]\,F([s],[s^{\dagger}])$ are
(infinitesimally) independent of the unphysical anti-objects
$s^{\dagger}\in\Gamma(\boldsymbol{\zeta}^{1})$. The derivation of such a
condition (see equation (39) below) relies on an extra assumption of the
translation invariance of a measure in the path integral. It must be noted,
however, that we do not define Feynman’s integral here and do not introduce
that measure which essentially depends on the agreement about the classes of
‘admissible’ sections $\Gamma(\pi)$ or $\Gamma(\boldsymbol{\zeta}^{(0|1)})$.
Consequently, our reasoning is to some extent heuristic.
The basics of path integration, which we recall here for consistency, are
standard: they illustrate how the geometry of the BV-Laplacian works in
practice. We draw the experts’ attention only to the fact that in our notation
$\Psi$ is not the gauge fixing fermion $\boldsymbol{\Psi}$ such that the odd-
component’s section $s^{\dagger}\in\Gamma(\boldsymbol{\zeta}^{1})$ is the
restriction of $\delta\boldsymbol{\Psi}/\delta q$ to the jet of a section for
$\boldsymbol{\zeta}^{0}$ ; instead, we let $\Psi$ determine the infinitesimal
shift $\dot{q}^{\dagger}=\delta\Psi/\delta q$ of coordinates along the fibre’s
parity-odd half. We also note that the preservation of parity is not mandatory
here and thus an even-parity
$\Psi\in\overline{H}^{n}(\boldsymbol{\zeta}^{0})\hookrightarrow\overline{H}^{n}(\pi_{\text{{BV}}})$
is a legitimate choice.
Let $F=\int
f({\boldsymbol{x}},{\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})\,\operatorname{dvol}({\boldsymbol{x}})\in\overline{H}^{n}(\pi_{\text{{BV}}})$
be a functional; here and in what follows we proceed over the building blocks
of elements from
$\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$ by the
graded Leibniz rule. Let
$\Psi=\int\psi({\boldsymbol{y}},{\boldsymbol{q}})\,\operatorname{dvol}({\boldsymbol{y}})\in\overline{H}^{n}(\boldsymbol{\zeta}^{0})\hookrightarrow\overline{H}^{n}(\pi_{\text{{BV}}})$
be an integral functional which, by assumption, is constant along ghost
parity-odd variables:
$\Psi(s^{\alpha},s^{\dagger}_{\beta})=\Psi(s^{\alpha},t^{\dagger}_{\beta})$
for any sections $\\{s^{\alpha}\\}\in\Gamma({\boldsymbol{\zeta}}^{0})$ and
$\\{s^{\dagger}_{\beta}\\},\\{t^{\dagger}_{\beta}\\}\in\Gamma({\boldsymbol{\zeta}}^{1})$.
We investigate under which conditions the path integral
$\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\,F(s^{\alpha},s^{\dagger}_{\beta})\colon\Gamma({\boldsymbol{\zeta}}^{1})\to\Bbbk$
is infinitesimally independent of a choice of the anti-objects:
$\displaystyle\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon^{\dagger}}\right|_{\varepsilon^{\dagger}=0}\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\,F\Bigl{(}s^{\alpha},s^{\dagger}_{\beta}+\varepsilon^{\dagger}\,\frac{\vec{\delta}\psi}{\delta
q^{\beta}}\bigg{|}_{s^{\alpha}}\Bigr{)}=0\quad\text{for all
$s^{\dagger}\in\Gamma({\boldsymbol{\zeta}}^{1})$.}$ (37)
Note that this formula makes sense because the bundles
${\boldsymbol{\zeta}}^{0}$ and ${\boldsymbol{\zeta}}^{1}$ are dual so that a
variational covector in the geometry of ${\boldsymbol{\zeta}}^{0}$ acts as a
shift vector in the geometry of ${\boldsymbol{\zeta}}^{1}$. The left-hand side
of (37) equals
$\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\frac{\overrightarrow{\delta}\\!\psi}{\delta
q^{\beta}}({\boldsymbol{x}},{\boldsymbol{q}})\bigr{|}_{j^{\infty}_{\boldsymbol{x}}(s^{\alpha})}\cdot\frac{\overleftarrow{\delta}\\!f}{\delta
q^{\dagger}_{\beta}}({\boldsymbol{x}},{\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})\bigr{|}_{j^{\infty}_{\boldsymbol{x}}(s^{\alpha},s^{\dagger}_{\gamma})},\qquad
s^{\dagger}\in\Gamma({\boldsymbol{\zeta}}^{1}).$
Take any auxiliary section $\delta{\boldsymbol{s}}=(\delta s^{\alpha},\delta
s^{\dagger}_{\beta})\in\Gamma\bigl{(}T{\boldsymbol{\zeta}}^{(0|1)})\bigr{)}$
normalized by $\delta s^{\alpha}(x)\cdot\delta s^{\dagger}_{\alpha}(x)\equiv
1$ at every ${\boldsymbol{x}}\in M^{n}$ for each
$\alpha=1,\dots,m+m_{1}+\cdots+m_{\lambda}=N$ and blow up the scalar integrand
to a pointwise contraction of dual object taking their values in the fibres
$T_{({\boldsymbol{x}},\phi({\boldsymbol{x}}),s({\boldsymbol{x}}))}V_{\boldsymbol{x}}$
and
$T_{({\boldsymbol{x}},\phi({\boldsymbol{x}}),s^{\dagger}({\boldsymbol{x}}))}\Pi
V^{\dagger}_{{\boldsymbol{x}}}$ of $T(\pi_{\text{{BV}}})$ over $\phi(x)$: for
$s=(s^{\alpha},s^{\dagger}_{\beta})$ we have
$\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\left.\left(\frac{\overrightarrow{\delta}\\!\psi}{\delta
q^{\alpha}}\cdot\frac{\overleftarrow{\delta}\\!f}{\delta
q^{\dagger}_{\alpha}}\right)\right|_{j^{\infty}_{\boldsymbol{x}}(s)}\\\
{}=\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{1})\int_{M}\operatorname{dvol}({\boldsymbol{x}}_{2})\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{1}\int_{M}{\mathrm{d}}{\boldsymbol{y}}_{2}\left.\left(\psi({\boldsymbol{x}}_{1},{\boldsymbol{q}})\frac{\overleftarrow{\partial}}{\partial
q^{j_{1}}_{\sigma_{1}}}\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{1}}(s)}\mbox{\hbox
to56.9055pt{{ }\hfil{ }}}\\\ {}\cdot\underline{\langle\vec{e}^{\,\dagger
j_{1}}({\boldsymbol{x}}_{1})|}\,\bigl{\langle}\left(\tfrac{\overleftarrow{\partial}}{\partial{\boldsymbol{y}}_{1}}\right)^{\sigma_{1}}\delta
s^{i_{1}}({\boldsymbol{y}}_{1})\,\vec{e}_{i_{1}}({\boldsymbol{y}}_{1}),-\vec{e}^{\,\dagger
i_{2}}({\boldsymbol{y}}_{2})\,\delta
s^{\dagger}_{i_{2}}({\boldsymbol{y}}_{2})\left(\tfrac{\overrightarrow{\partial}}{\partial{\boldsymbol{y}}_{2}}\right)^{\sigma_{2}}\bigr{\rangle}\,\underline{|\vec{e}_{j_{2}}({\boldsymbol{x}}_{2})\rangle}\\\
{}\cdot\left.\left(\frac{\overrightarrow{\partial}}{\partial
q^{\dagger}_{j_{2},\sigma_{2}}}f({\boldsymbol{x}}_{2},{\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})\right)\right|_{j^{\infty}_{{\boldsymbol{x}}_{2}}(s)}.$
In fact, the integrand refers to a definition of the evolutionary vector field
${\boldsymbol{Q}}^{\Psi}$ such that
${\boldsymbol{Q}}^{\Psi}(F)\cong\lshad{\Psi,F}\rshad$ modulo integration by
parts in the building blocks of $F$, cf. [32]. Due to a special choice of the
dependence of $\Psi$ on $s$ only, this is indeed the Schouten bracket
$\lshad{\Psi,F}\rshad$.
To rephrase the indifference of the path integral to a choice of $\Psi$ in
terms of an equation upon the functional $F$ alone, we perform integration by
parts in Feynman’s integral. For this we employ the translation invariance
$[Ds]=[D(s-\mu\cdot\delta s)]$ of the functional measure.
###### Lemma 9.
Let $H=\int
h({\boldsymbol{x}},{\boldsymbol{q}},{\boldsymbol{q}}^{\dagger})\,\operatorname{dvol}({\boldsymbol{x}})\in\overline{H}^{n}(\pi_{\text{{BV}}})\subset\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
be an integral functional and $\delta
s\in\Gamma(T{\boldsymbol{\zeta}}^{0})\hookrightarrow\Gamma\bigl{(}T{\boldsymbol{\zeta}}^{(0|1)}\bigr{)}$
be a shift. Then we have that
$\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\int_{M}\operatorname{dvol}({\boldsymbol{x}})\,\delta
s^{\nu}({\boldsymbol{x}})\cdot\frac{\overleftarrow{\delta}\\!h}{\delta
q^{\nu}}\bigg{|}_{j^{\infty}_{\boldsymbol{x}}(s^{\alpha},s^{\dagger}_{\beta})}=0,$
where the section $s^{\dagger}\in\Gamma({\boldsymbol{\zeta}}^{1})$ is a
parameter.
###### Proof.
Indeed,
$\displaystyle 0$
$\displaystyle=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\,H(s^{\alpha},s^{\dagger}_{\beta}),$
because the integral contains no parameter $\mu\in\Bbbk$. We continue the
equality:
$\displaystyle=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[D(s^{\alpha}-\mu\,\delta
s^{\alpha})]\,H(s^{\alpha},s^{\dagger}_{\beta})$
$\displaystyle=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\,H(s^{\alpha}+\mu\,\delta
s^{\alpha},s^{\dagger}_{\beta})=\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}H(s^{\alpha}+\mu\,\delta
s^{\alpha},s^{\dagger}_{\beta}),$
which yields the helpful formula in the lemma’s assertion. ∎
Returning to the functionals $\Psi$ and $F$ and denoting
$G(s):=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\ell}\right|_{\ell=0}F(s+\ell\cdot\overleftarrow{\delta
s}^{\dagger})$, we use the Leibniz rule for the derivative of $H=\Psi\cdot G$:
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}(\Psi\cdot
G)(s+\mu\cdot\overleftarrow{\delta
s})=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}(\Psi)(s+\mu\cdot\overleftarrow{\delta
s})\cdot
G(s)+\Psi(s)\cdot\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\mu}\right|_{\mu=0}(G)(s+\mu\cdot\overleftarrow{\delta
s}).$
Because the path integral over $[Ds^{\alpha}]$ of the entire expression
vanishes by Lemma 9 in which we were ready to proceed by the Leibniz rule over
building blocks, we infer that the path integrals of the two terms are
opposite. Now take the traces over indexes in both variations. The integral of
the first term equals the initial expression for the path integral containing
$F$, i. e., the left-hand side of equation (37). Consequently, if
$\displaystyle\int_{\Gamma({\boldsymbol{\zeta}}^{0})}[Ds^{\alpha}]\,\Psi(s^{\alpha})\cdot\Delta
F(s^{\alpha},s^{\dagger}_{\beta})=0$ (38)
for $\\{s^{\dagger}_{\beta}\\}\in\Gamma({\boldsymbol{\zeta}}^{1})$ and for all
$\Psi\in\overline{H}^{n}({\boldsymbol{\zeta}}^{0})\hookrightarrow\overline{H}^{n}(\pi_{\text{{BV}}})$,
then the path integral over $F$ is infinitesimally independent of a section
$\\{s^{\dagger}_{\beta}\\}\in\Gamma({\boldsymbol{\zeta}}^{1})$.
The condition
$\Delta F=0$ (39)
is sufficient for equation (38), and therefore equation (37), to hold. By
specifying a class $\Gamma(\pi_{\text{{BV}}})$ of admissible sections of the
BV-bundle for a concrete field model, and endowing that space of sections with
a suitable metric, one could reinstate a path integral analogue of the main
lemma in the calculus of variations and then argue that the condition $\Delta
F=0$ is also necessary.
Summarizing, whenever equation (39) holds, one can assign arbitrary admissible
values to the odd-parity coordinates; for example, one can let
$s^{\dagger}_{\beta}({\boldsymbol{x}})=\delta\boldsymbol{\psi}/\delta
q^{\beta}\bigr{|}_{j^{\infty}_{\boldsymbol{x}}(s^{\alpha})}$ for a gauge-
fixing integral
$\boldsymbol{\Psi}=\int\boldsymbol{\psi}({\boldsymbol{x}},{\boldsymbol{q}})\,\operatorname{dvol}({\boldsymbol{x}})\in\overline{H}^{n}({\boldsymbol{\zeta}}^{0})$.
This choice is reminiscent of the substitution principle, see [32] and [45].
Laplace’s equation (39) ensures the infinitesimal independence from non-
physical anti-objects for path integrals of functionals over physical fields –
not only in the classical BV-geometry of the bundle $\pi_{\text{{BV}}}$, but
also in the quantum setup, whenever all objects are tensored with formal power
series $\Bbbk[[\hbar,\hbar^{-1}]]$ in the Planck constant $\hbar$. It is
accepted that each quantum field $s^{\hbar}$ contributes to the expectation
value of a functional $\mathcal{O}^{\hbar}$ with the factor
$\exp({{\boldsymbol{i}}}S^{\hbar}(s^{\hbar})/{\hbar})$, where $S^{\hbar}$ is
the quantum BV-action of the model. Solutions $\mathcal{O}^{\hbar}$ of the
equation
$\Delta\bigl{(}\mathcal{O}^{\hbar}\cdot\exp({{\boldsymbol{i}}}S^{\hbar}/{\hbar})\bigr{)}=0$
are the observables. In particular, the postulate that the unit $1\colon
s^{\hbar}\mapsto 1\in\Bbbk$ is averaged to unit by the Feynman integral of
$1\cdot\exp({{\boldsymbol{i}}}S^{\hbar}(s^{\hbar})/{\hbar})$ over the space of
quantum fields $s^{\hbar}$ normalizes the integration measure and constrains
the quantum BV-action by the quantum master-equation (see, e.g., [7, 8, 20,
22, 54]).
###### Proposition 10.
Let $S^{\hbar}$ be the even quantum BV-action (i. e., let it have a density
that has an even number of ghost parity-odd coordinates in each of its terms).
If the identity
$\Delta\left(\exp\bigl{(}\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\bigr{)}\right)=0$
holds, then $S^{\hbar}$ satisfies the quantum master-equation:
${\boldsymbol{i}}\hbar\,\Delta
S^{\hbar}=\tfrac{1}{2}\lshad{S^{\hbar},S^{\hbar}}\rshad.$ (40)
###### Proposition 11.
If an even functional $\mathcal{O}$ and the quantum BV-action $S^{\hbar}$ are
such that
$\Delta\bigl{(}\mathcal{O}\exp({\boldsymbol{i}}S^{\hbar}/\hbar)\bigr{)}=0$ and
$\Delta\bigl{(}\exp({\boldsymbol{i}}S^{\hbar}/\hbar)\bigr{)}=0$ hold,
respectively, then $\mathcal{O}$ satisfies
$\Omega^{\hbar}(\mathcal{O})\mathrel{{:}{=}}-{\boldsymbol{i}}\hbar\,\Delta\mathcal{O}+\lshad{S^{\hbar},\mathcal{O}}\rshad=0.$
(41)
We quote the standard proofs of Propositions 10 and 11 from [35] in A — yet
now we gain a deeper insight on a construction of the quantum BV-differential
$\Omega^{\hbar}$.
###### Remark 3.1.
A practical way to fix the signs which arise in the BV-Laplacian and Schouten
bracket from the ghost parity and a grading in the case of a superbundle
$\pi\colon E^{(m_{0}+n_{0}|m_{1}+n_{1})}\to M^{(n_{0}|n_{1})}$ of superfields
is by a re-derivation of the Laplace equation
$\Delta(\mathcal{O}\exp(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}))=0$ upon an
observable $\mathcal{O}$ starting from the Schwinger–Dyson condition,
$\vec{\partial}^{\,({\boldsymbol{q}}^{\dagger})}_{\vec{\delta}\Psi/\delta{\boldsymbol{q}}}\left(\int[D{\boldsymbol{q}}]\,\mathcal{O}([{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\bigl{(}[{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}]\bigr{)}\right)\right)=0,$
(42)
which postulates the Feynman path integral’s independence of the non-physical
BV-coordinates ${\boldsymbol{q}}^{\dagger}$ with odd ghost parity. Note that
the measure in the path integral involves only ghost parity-even objects
(whatever be their $\mathbb{Z}_{2}$-grading).
###### Theorem 12.
Let
$\mathcal{O}\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
be a functional and let the even functional
$S^{\hbar}\in\overline{H}^{n}(\pi_{\text{{BV}}})$ satisfy quantum master-
equation (40). Then the operator $\Omega^{\hbar}$, defined in (41), squares to
zero:
${(\Omega^{\hbar})}^{2}(\mathcal{O})=0.$
###### Proof.
We calculate, using Theorem 6,
$\displaystyle{(\Omega^{\hbar})}^{2}(\mathcal{O})$
$\displaystyle=\lshad{S^{\hbar},\lshad{S^{\hbar},\mathcal{O}}\rshad-{\boldsymbol{i}}\hbar\,\Delta\mathcal{O}}\rshad-{\boldsymbol{i}}\hbar\,\Delta\big{(}\lshad{S^{\hbar},\mathcal{O}}\rshad-{\boldsymbol{i}}\hbar\,\Delta\mathcal{O}\big{)}$
$\displaystyle=\lshad{S^{\hbar},\lshad{S^{\hbar},\mathcal{O}}\rshad}\rshad-{\boldsymbol{i}}\hbar\,\lshad{S^{\hbar},\Delta\mathcal{O}}\rshad-{\boldsymbol{i}}\hbar\,\lshad{\Delta
S^{\hbar},\mathcal{O}}\rshad+{\boldsymbol{i}}\hbar\,\lshad{S^{\hbar},\Delta\mathcal{O}}\rshad+({\boldsymbol{i}}\hbar)^{2}\Delta^{2}\mathcal{O}.$
The last term vanishes identically by Theorem 8, while the second term cancels
against the fourth term. Using Jacobi’s identity (28) for the Schouten bracket
on the first term, we obtain:
$\displaystyle{(\Omega^{\hbar})}^{2}(\mathcal{O})$
$\displaystyle=-{\boldsymbol{i}}\hbar\,\lshad{\Delta
S^{\hbar},\mathcal{O}}\rshad+{\textstyle\frac{1}{2}}\lshad{\lshad{S^{\hbar},S^{\hbar}}\rshad,\mathcal{O}}\rshad=\lshad{-{\boldsymbol{i}}\hbar\,\Delta
S^{\hbar}+{\textstyle\frac{1}{2}}\lshad{S^{\hbar},S^{\hbar}}\rshad,\mathcal{O}}\rshad.$
Now is the crucial moment in the entire proof. By the logic of our reasoning’s
objective, the theorem’s claim is that the operator $(\Omega^{\hbar})^{2}$
yields zero whenever acting on a functional $\mathcal{O}$. We accordingly
transform the variational Schouten bracket of two terms to the operator
realization,
$\displaystyle\cong\vec{{\boldsymbol{Q}}}^{-{\boldsymbol{i}}\hbar\,\Delta
S^{\hbar}+\frac{1}{2}\lshad S^{\hbar},S^{\hbar}\rshad}(\mathcal{O}),$
with the evolutionary derivation now acting on the argument. Let us emphasize
that a transition from the variational Schouten bracket – which increases the
number of bases $M\times\ldots\times M$ by construction – to the evolutionary
vector field chops a multiplication of geometries by uniquely fixing the
field’s generating section.252525It might happen otherwise that a co-multiple
of $\mathcal{O}$ under $\lshad\,,\,\rshad$ looks like zero as a map of the
space $\Gamma(\pi_{\text{{BV}}})$ yet the bracket with it could still be
nonzero, see, e. g., $\Delta G$ on p. 33 in Example 2.4. But by our initial
assumption, this generating section is zero by virtue of (40). Therefore the
image of $\mathcal{O}$ under such map vanishes, which proves the assertion. ∎
### 3.2 Gauge automorphisms of quantum BV-cohomology groups
By using the quantum BV-differential $\Omega^{\hbar}$, let us construct a
closed algebra of infinitesimal gauge symmetries for the quantum master-
equation (40).
###### Proposition 13.
Let $F\in\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$ be
an arbitrary odd-parity functional and $S^{\hbar}$ the quantum master-action
satisfying (40). Then the infinitesimal shift of the functional $S^{\hbar}$,
$\dot{S}^{\hbar}=\Omega^{\hbar}(F)\quad\Longleftrightarrow\quad
S^{\hbar}\mapsto
S^{\hbar}(\varepsilon)=S^{\hbar}+\varepsilon\cdot\Omega^{\hbar}(F)+\overline{o}(\varepsilon),\
\varepsilon\in\mathbb{R},$ (43)
is a symmetry of (40) so that
$\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}(\varepsilon)\right)\right)=\overline{o}(\varepsilon)$
in Peano’s notation.
$\bullet$ The algebra of infinitesimal gauge symmetries (43) of the quantum
master-equation is closed,
$\left.\left(\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{1}}\circ\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{2}}-\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{2}}\circ\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{1}}\right)\right|_{\varepsilon_{i}=0}(S^{\hbar})=\Omega^{\hbar}\bigl{(}\lshad
F_{1},F_{2}\rshad\bigr{)},$ (44)
i.e., the commutator of two even-parity symmetries with respective generators
$F_{1}$ and $F_{2}$ is the infinitesimal gauge symmetry whose generator is the
odd Poisson bracket of $F_{1}$ and $F_{2}$.
###### Remark 3.2.
The odd-parity generators
$F_{i}\in\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
never evolve in the course of a transformation which is induced by any
generator $F_{j}$ on the quantum BV-action functional $S^{\hbar}$.
###### Proof.
Assuming a smooth dependence of $S^{\hbar}(\varepsilon)$ on $\varepsilon$, we
obtain that262626This proof is standard: it originates from the cohomological
deformation theory for solutions of the Maurer–Cartan equation (e. g., of
(40)), see [37].
$\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}(\varepsilon)\right)\right)=\tfrac{{\boldsymbol{i}}}{\hbar}\dot{S}^{\hbar}\cdot\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}(\varepsilon)\right)\right)+\Omega^{\hbar}(\dot{S}^{\hbar})\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}(\varepsilon)\right).$
Because $(\Omega^{\hbar})^{2}=0$ by Theorem 12, for $\dot{S}^{\hbar}$ to be an
infinitesimal symmetry of the equation
$\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\right)=0$
it is sufficient that $S^{\hbar}=\Omega^{\hbar}(F)$ for some odd-parity
functional $F$.
Second, let
$\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{i}}(S^{\hbar})=-{\boldsymbol{i}}\hbar\,\Delta
F_{i}+\lshad S^{\hbar},F_{i}\rshad\qquad\text{for
}i=1,2,\qquad\varepsilon_{i}\in\mathbb{R},$
and postulate that
$\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon_{i}}(F_{j})\equiv 0$ for all $i$
and $j$. Then commutator (44) of even-parity infinitesimal transformations
(43) generated by the functionals $F_{1}$ and $F_{2}$ is
$\lshad-{\boldsymbol{i}}\hbar\,\Delta F_{1}+\lshad
S^{\hbar},F_{1}\rshad,F_{2}\rshad-\lshad-{\boldsymbol{i}}\hbar\,\Delta
F_{2}+\lshad S^{\hbar},F_{2}\rshad,F_{1}\rshad\\\
{}=-{\boldsymbol{i}}\hbar\,\left(\lshad\Delta F_{1},F_{2}\rshad-\lshad\Delta
F_{2},F_{1}\rshad\right)+\left(\lshad\lshad
S^{\hbar},F_{1}\rshad,F_{2}\rshad-\lshad\lshad
S^{\hbar},F_{2}\rshad,F_{1}\rshad\right).$
Because $F_{1}$ has odd parity, we swap the factors in $-\lshad\Delta
F_{2},F_{1}\rshad=\lshad F_{1},\Delta F_{2}\rshad$; likewise, $+\lshad
F_{1},\lshad S^{\hbar},F_{2}\rshad\rshad$ is the last term in the above
expression. From our main Theorem 6 and by Jacobi identity (28) we conclude
that the commutator is equal to
$-{\boldsymbol{i}}\hbar\,\Delta\bigl{(}\lshad F_{1},F_{2}\rshad\bigr{)}+\lshad
S^{\hbar},\lshad F_{1},F_{2}\rshad\rshad=\Omega^{\hbar}\bigl{(}\lshad
F_{1},F_{2}\rshad\bigr{)},$
that is, the Schouten bracket of $F_{1}$ and $F_{2}$ is the new gauge symmetry
generator. ∎
###### Remark 3.3.
(cf. [51, §5]). The transformation
$\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\mapsto\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}(\varepsilon)\right)$
for a finite $\varepsilon\in\mathbb{R}$ is determined by the operator
$\exp(\varepsilon[\Delta,F])$, where $[\ ,\ ]$ is the anticommutator of two
odd-parity objects. Indeed, by Theorem 3 we have that
$\displaystyle\Delta\bigl{(}$ $\displaystyle
F\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\bigr{)}+F\cdot\Delta\bigl{(}\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\bigr{)}$
$\displaystyle=\Delta
F\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)-\lshad
F,\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\rshad-F\cdot\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\right)+F\cdot\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\right)$
$\displaystyle=\tfrac{{\boldsymbol{i}}}{\hbar}(-{\boldsymbol{i}}\hbar\,\Delta
F+\lshad
S^{\hbar},F\rshad)\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)=\tfrac{{\boldsymbol{i}}}{\hbar}\dot{S}^{\hbar}\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)\right).$
Note that the Schouten bracket acts on
$\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right)$ by the Leibniz
rule (see Theorem 4) and we then use the equality $-\lshad
F,\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\rshad=\tfrac{{\boldsymbol{i}}}{\hbar}\lshad
S^{\hbar},F\rshad$ which holds by Theorem 4 again.
Let us now regard the full quantum BV-action as the generating functional for
ghost parity-even observables $\mathcal{O}$, see [54].
###### Lemma 14.
There are no observables $\mathcal{O}$, other than the identically zero
functional, which would be ghost parity-odd.
###### Proof.
Indeed, Eq. (42) implies that the path integral
$I=\int_{\Gamma(\zeta^{0})}[D{\boldsymbol{q}}]\,\mathcal{O}([{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}([{\boldsymbol{q}}],[{\boldsymbol{q}}^{\dagger}])\right)$
over the space of ghost parity-even BV-section components is effectively
independent of the ghost parity-odd BV-variables ${\boldsymbol{q}}^{\dagger}$.
Notice further that the ghost parity $\operatorname{gh}(I)$ of this constant
function $I([{\boldsymbol{q}}^{\dagger}])$ is equal to that of $\mathcal{O}$;
the quantum master-action $S^{\hbar}$ is parity-even. Under a (speculative)
assumption that an observable $\mathcal{O}$ could be ghost parity-odd, we
obtain an odd parity constant. Unless a possibility of their existence is
postulated by brute force, this odd-parity constant must be equal to zero,
whence the ghost parity-odd functional
$\mathcal{O}\in\overline{H^{n}}(\pi_{\text{{BV}}})\subseteq\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
itself is zero. ∎
In what follows we accept for transparency that there is no grading in the
initial geometry of physical fields, i.e., for sections of the bundle
$\pi\colon E^{n+m}\to M^{n}$. Let us focus on the standard cohomological
approach to quantum BV-models and to their gauge symmetries (cf. [37]).
###### Lemma 15.
Suppose that an infinitesimal shift $S^{\hbar}\mapsto
S^{\hbar}+\lambda\cdot\mathcal{O}+\overline{o}(\lambda)$ of the quantum BV-
action by using an even-parity functional
$\mathcal{O}\in\overline{H^{n}}(\pi_{\text{{BV}}})\subseteq\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
does not destroy the quantum master-equation,
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}\right|_{\lambda=0}\Delta\left(\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}(S^{\hbar}+\lambda\cdot\mathcal{O})\right)\right)=0.$
Then the observable $\mathcal{O}$ is $\Omega^{\hbar}$-closed:
$-{\boldsymbol{i}}\hbar\,\Delta\mathcal{O}+\lshad
S^{\hbar},\mathcal{O}\rshad=0.$
###### Proof.
The proof literally repeats that of Proposition 13. ∎
For a given odd-parity functional $F\in\overline{H^{n}}(\pi_{\text{{BV}}})$,
we organize the infinitesimal shift (43) of the master-functional $S^{\hbar}$
as follows:
$\displaystyle\dot{S}^{\hbar}$
$\displaystyle=-{\boldsymbol{i}}\hbar\,\Delta(F)+\lshad S^{\hbar},F\rshad,$
$\displaystyle\dot{\mathcal{O}}$ $\displaystyle=\lshad\mathcal{O},F\rshad.$
Note that, unless one has that $\Delta F=0$ incidentally, the transformation
of the integral _functional_ $S^{\hbar}$ is not induced by any infinitesimal
transformation of the BV-_variables_ , that is, by an evolutionary vector
field on the horizontal infinite jet space at hand. No earlier than the
transformation law $S^{\hbar}\mapsto S^{\hbar}(\varepsilon)$ is postulated, it
becomes an act of will to think that the functional $F$ is the generator of
parity-preserving evolutionary vector field
$\overleftarrow{Q}^{F}=\overrightarrow{Q}^{F}$ acting on the BV-variables so
that $\dot{\mathcal{O}}\cong\overrightarrow{Q}^{F}(\mathcal{O})$ for all
observables $\mathcal{O}$.
Furthermore, let us extend the deformation
$\mathcal{O}\mapsto\mathcal{O}(\varepsilon)$ of even-parity cocycles
$\mathcal{O}\in\ker\Omega^{\hbar}$ to the space of odd-parity functionals
$\xi\in\overline{H^{n}}(\pi_{\text{{BV}}})\subseteq\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$
which produce the coboundaries $\Omega^{\hbar}(\xi)$. Namely, we postulate
that
$\dot{\xi}=\lshad\xi,F\rshad$
for all such functionals $\xi$; here we denote by the dot over $\xi$ its
velocity in the course of the transformation generated by a given $F$. Let us
remember however that the law for evolution of the odd-parity functionals
$\xi$ which produce the $\Omega^{\hbar}$-coboundaries is different from our
earlier postulate (see Proposition 13) that the odd-parity generators $F_{i}$
of gauge symmetries do not evolve: $dF_{i}/d\varepsilon_{j}\equiv 0$ or, in
shorthand notation,
$\dot{F}\equiv 0.$ (45)
We claim that under these hypotheses, the structure of quantum BV-cohomology
group remains intact in the course of gauge symmetry transformations of the
quantum master-action, $S^{\hbar}\mapsto S^{\hbar}(\varepsilon)$, even though
the quantum BV-differential is modified,
$\Omega^{\hbar}\mapsto\Omega^{\hbar}(\varepsilon)$, and the cocycles and
coboundaries are also deformed.
###### Theorem 16.
An infinitesimal shift of the quantum BV-cohomology classes induced by (43),
(45), and
$\displaystyle\dot{\mathcal{O}}$ $\displaystyle=\lshad\mathcal{O},F\rshad,$
$\displaystyle\mathcal{O}$ $\displaystyle\in\ker\Omega^{\hbar},$
$\displaystyle\dot{\xi}$ $\displaystyle=\lshad\xi,F\rshad,$ $\displaystyle\xi$
$\displaystyle\in\overline{H^{n}}(\pi_{\text{{BV}}})\subseteq\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}}),\
\xi\text{ odd},$
yields an isomorphism of the $\Omega^{\hbar}$-cohomology group: under such
mapping, every $\Omega^{\hbar}$-closed, even-parity $\Omega^{\hbar}$-cocycle
$\mathcal{O}$ becomes $\Omega^{\hbar}(\varepsilon)$-closed, whereas the
transformation of an even-parity coboundary $\Omega^{\hbar}(\xi)$ produces an
$\Omega^{\hbar}(\varepsilon)$-coboundary:
$(\Omega^{\hbar}(\xi))(\varepsilon)=\Omega^{\hbar}(\varepsilon)\bigl{(}\xi(\varepsilon)\bigr{)}$.
###### Proof.
Let $\mathcal{O}\in\ker\Omega^{\hbar}$ be an even-parity observable and $F$ an
odd-parity generator of gauge transformation. Consider the equation
$\Omega^{\hbar}(\varepsilon)(\mathcal{O}(\varepsilon))=0$ which states that
the transformed functional $\mathcal{O}(\varepsilon)$ remains a coboundary.
The term which is proportional to $\varepsilon$ in this equation’s left-hand
side is equal to
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}\left(-{\boldsymbol{i}}\hbar\,\Delta\mathcal{O}(\varepsilon)+\lshad
S^{\hbar}(\varepsilon),\mathcal{O}(\varepsilon)\rshad\right)=\Omega^{\hbar}(\dot{\mathcal{O}})+\lshad\dot{S}^{\hbar},\mathcal{O}\rshad=\Omega^{\hbar}(\lshad\mathcal{O},F\rshad)+\lshad\Omega^{\hbar}(F),\mathcal{O}\rshad;$
recalling once again that
$\Omega^{\hbar}=-{\boldsymbol{i}}\hbar\,\Delta+\lshad
S^{\hbar},\,\cdot\,\rshad$, we continue the equality
$=-{\boldsymbol{i}}\hbar\,\Delta(\lshad\mathcal{O},F\rshad)+\lshad
S^{\hbar},\lshad\mathcal{O},F\rshad\rshad+\lshad-{\boldsymbol{i}}\hbar\,\Delta
F+\lshad S^{\hbar},F\rshad,\mathcal{O}\rshad.\mbox{\hbox to119.50157pt{{
}\hfil{ }}}$
Now by Theorem 6 we obtain that, the observable $\mathcal{O}$ being parity-
even,
$=-{\boldsymbol{i}}\hbar\,\lshad\Delta\mathcal{O},F\rshad+{\boldsymbol{i}}\hbar\,\lshad\mathcal{O},\Delta
F\rshad+\lshad
S^{\hbar},\lshad\mathcal{O},F\rshad\rshad-{\boldsymbol{i}}\hbar\,\lshad\Delta
F,\mathcal{O}\rshad+\lshad\mathcal{O},\lshad S^{\hbar},F\rshad\rshad=\\\
=\lshad\Omega^{\hbar}(\mathcal{O}),F\rshad\cong-\vec{{\boldsymbol{Q}}}^{F}\bigl{(}\Omega^{\hbar}(\mathcal{O})\bigr{)}=0,$
because $\lshad S^{\hbar},\lshad\mathcal{O},F\rshad\rshad=\lshad\lshad
S^{\hbar},\mathcal{O}\rshad,F\rshad-\lshad\mathcal{O},\lshad
S^{\hbar},F\rshad\rshad$ by Jacobi identity (28), because we are inspecting
the $\varepsilon$-linear term in the operator
$\Omega^{\hbar}(\varepsilon)\circ\bigl{(}\varepsilon=0\longmapsto\varepsilon\neq
0\bigr{)}$ applied to $\mathcal{O}$, and $\mathcal{O}$ is an
$\Omega^{\hbar}$-cocycle. Therefore, the zero initial condition
$\Omega^{\hbar}(\mathcal{O})=0$ evolves at zero velocity to the
$\Omega^{\hbar}(\varepsilon)$-cocycle equation
$\Omega^{\hbar}(\varepsilon)\bigl{(}\mathcal{O}(\varepsilon)\bigr{)}=0$ upon
$\mathcal{O}(\varepsilon)$.
Likewise, let $\Omega^{\hbar}(\xi)$ be a coboundary for some odd-parity
functional $\xi$ which evolves by $\dot{\xi}=\lshad\xi,F\rshad$. Then the
even-parity observable $\Omega^{\hbar}(\xi)\in\ker\Omega^{\hbar}$ evolves as
fast as $\lshad\Omega^{\hbar}(\xi),F\rshad$ but simultaneously we have that
the mapping $\Omega^{\hbar}$ and its argument $\xi$ change. We claim that the
two evolutions match so that $(\Omega^{\hbar}(\xi))(\varepsilon)$ is
$\Omega^{\hbar}(\varepsilon)$-exact. Indeed, we have that
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\right|_{\varepsilon=0}\bigl{(}\Omega^{\hbar}(\varepsilon)(\xi(\varepsilon))\bigr{)}=\Omega^{\hbar}\bigl{(}\lshad\xi,F\rshad\bigr{)}+\lshad\Omega^{\hbar}(F),\xi\rshad\\\
{}=-{\boldsymbol{i}}\hbar\,\lshad\Delta\xi,F\rshad\underline{{}-{\boldsymbol{i}}\hbar\,\lshad\xi,\Delta
F\rshad}+\lshad
S^{\hbar},\lshad\xi,F\rshad\rshad+\lshad\underline{-{\boldsymbol{i}}\hbar\,\Delta
F}+\lshad S^{\hbar},F\rshad,\underline{\xi}\rshad;$
by cancelling out the underlined Schouten brackets and then using the Jacobi
identity we obtain
$=\lshad-{\boldsymbol{i}}\hbar\,\Delta\xi,F\rshad+\lshad\lshad
S^{\hbar},\xi\rshad,F\rshad+\lshad\xi,\lshad
S^{\hbar},F\rshad\rshad-\lshad\xi,\lshad
S^{\hbar},F\rshad\rshad=\lshad\Omega^{\hbar}(\xi),F\rshad,$
which proves our claim.
Summarizing, we see that gauge symmetries of the quantum master-equation
induce automorphisms of the $\Omega^{\hbar}$-cohomology group. ∎
We conclude that it would be conceptually incorrect to say that the
infinitesimal gauge transformations of all functionals in a quantum BV-model
are induced by a canonical transformation, determined by the evolutionary
vector field $\overrightarrow{Q}^{F}$ acting on the BV-variables. Let us
remember that the even-parity quantum master-action
$S^{\hbar}\in\overline{H^{n}}(\pi_{\text{{BV}}})$ and its descendants, the
observables $\mathcal{O}$ evolve by
$\displaystyle\dot{S}^{\hbar}$ $\displaystyle=-{\boldsymbol{i}}\hbar\,\Delta
F+\lshad S^{\hbar},F\rshad=\Omega^{\hbar}(F),\qquad
F\in\overline{H^{n}}(\pi_{\text{{BV}}})\subseteq\overline{\mathfrak{N}^{n}}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}}),\quad
F\text{ odd},$ and $\displaystyle\dot{\mathcal{O}}$
$\displaystyle=\lshad\mathcal{O},F\rshad.$ We note that the evolution of the
generating functional $S^{\hbar}_{{\text{{BV}}}}$ is not determined by a
vector field on the space of BV-variables. Likewise, we recall that the odd-
parity arguments $\xi$ of $\Omega^{\hbar}$ for the coboundaries
$\Omega^{\hbar}(\xi)\sim 0$ do evolve, $\displaystyle\dot{\xi}$
$\displaystyle=\lshad\xi,F\rshad,$ whereas the generators $F$ of gauge
symmetries for (40) never change: symbolically, $\displaystyle\dot{F}$
$\displaystyle=0$
(see Eq. (45) above). In fact, one may think that each $F$ determines a
parity-preserving evolutionary vector field $\overrightarrow{Q}^{F}$ on the
space of BV-variables, but it is not the objects $\overrightarrow{Q}^{F}$ but
the full systems of four distinct evolution equations which encode the
deformation of respective functionals. Neither the functionals’ attribution to
the space of building blocks $\overline{H^{n}}(\pi_{\text{{BV}}})\ni
S^{\hbar}$, $\mathcal{O}$, $F$ nor a functional’s parity,
$\operatorname{gh}(S^{\hbar})=\operatorname{gh}(\mathcal{O})$ and
$\operatorname{gh}(F)=\operatorname{gh}(\xi)$, completely determines their
individual transformation laws.
###### Remark 3.4.
The supports of test shifts $\delta{\boldsymbol{s}}$ can be arbitrarily
small272727We recall that the smoothness class of variations
$\delta{\boldsymbol{s}}$ is determined by smoothness of the frame fields
$\vec{e}_{i}({\boldsymbol{x}}),\ \vec{e}^{{}\,\dagger i}({\boldsymbol{x}})$
and coefficient functions $\delta s^{i}({\boldsymbol{x}}),\ \delta
s_{i}^{\dagger}({\boldsymbol{x}})$. and they can be chosen in such a way that
all boundary terms vanish in the course of integration by parts within
equivalence classes from the horizontal cohomology groups
$\overline{H}^{n(1+k)}(\pi_{{\text{{BV}}}}\times
T\pi_{{\text{{BV}}}}\times\ldots\times T\pi_{{\text{{BV}}}})$. Let us note
also that these integrations by parts (see section 1.3) transport the
derivatives from one copy of the base manifold $M^{n}$ to another copy; this
reasoning stays local with respect to base points ${\boldsymbol{x}}$ and local
volume elements $\operatorname{dvol}({\boldsymbol{x}})$ because the geometric
mechanism of locality yields the diagonal in powers of the base manifold.
However, an integration by parts in functionals from
$\overline{H}^{n}(\pi_{{\text{{BV}}}})$ is a different issue. In fact, it
refers to the topology of $M^{n}$ or to a choice of the class
$\Gamma(\pi_{{\text{{BV}}}})$ of admissible sections (so that there appear no
boundary terms as well). Let us recall that the only place where such global,
de Rham cohomology aspect is explicitly used is the proof of Jacobi’s identity
for the variational Schouten bracket (see [32]). In turn, Theorems 8 and 12
relate these properties of the bracket $\lshad\,,\,\rshad$ to cohomology
generators $\Delta^{2}=0$ and $(\Omega^{\hbar})^{2}=0$. (The converse is also
true: Jacobi’s identity for $\lshad\,,\,\rshad$ stems from $\Delta^{2}=0$.)
This motivates why the de Rham and quantum BV-cohomologies are interrelated
(cf. [5]).
## Conclusion
Mathematical models are designed for description of phenomena of Nature ; a
construction of the models’ objects is not the same as their evaluation at
given configurations of the models, which would associate $\Bbbk$-numbers to
physical fields $\phi\in\Gamma(\pi)$ in terms of such objects. Namely,
consider an Euler–Lagrange model whose primary element is the action
functional $S\colon\Gamma(\pi)\to\Bbbk$. By definition, derivative objects are
obtained from $S$ by using natural operations such as $\smash{\vec{\delta}}$
or $\lshad\,,\,\rshad$ and $\Delta$. The derivative objects’ geometric
complexity is greater than that of $S$ because they absorb the domains of
definition for test shifts $\delta s_{1},\,\ldots\,,\delta s_{k}$ of field
configurations. We emphasize that such composite structure objects do not yet
become maps $\Gamma(\pi)\to\Bbbk$ which would suit well for their evaluation
at sections ${\boldsymbol{s}}\in\Gamma(\pi)$ yielding $\Bbbk$-numbers. The
intermediate objects can rather be used as arguments of $\lshad\,,\,\rshad$ or
$\Delta$ in a construction of larger, logically and geometrically more complex
objects ; we illustrate by Fig. 9
$S\colon\Gamma(\pi)\to\Bbbk$; obsevables $\mathcal{O}_{\mu}$ in
$S+\lambda_{\mu}\mathcal{O}_{\mu}$$\text{Object}\in\overline{H}^{n(1+k)}(\pi\times\underbrace{T\pi\times\ldots\times
T\pi}_{k})$$\text{Map}\colon\Gamma(\pi)\to\Bbbk$$\overleftarrow{\delta}\\!\\!,\
\lshad\,,\,\rshad,\ \Delta$by parts,surgery of $\langle\,,\,\rangle$ Figure 9:
The action $S$ as a generator of observables, building blocks of derivative
objects as horizontal cohomology classes in products of bundles over $M\times
M\times\ldots\times M$, and resulting mappings as the objects’ contractions
over Whitney’s sum of bundles.
the expansion of analytic structures and their shrinking in the course of
integration by parts and multiplication of normalized test shifts in
reconfigured couplings. Indeed, the derivative objects become multi-linear
maps with respect to $k$-tuples of the variations $\delta s_{1}$, $\ldots$,
$\delta s_{k}\in\Gamma(T\pi)$ only when the integrations by parts carry all
derivatives away from the test shifts, channelling the derivations to
densities of the object’s constituent blocks such as the Lagrangian in the
action functional. A surgery of couplings then contracts the values of
normalized test shifts by virtue of (16) at every point of the base manifold.
This is how maps $\Gamma(\pi)\to\Bbbk$ are obtained.
We conclude that a calculation of composite-structure object may not be
interrupted ahead of time. Otherwise speaking, every calculation stretches
from its input data to the end value at ${\boldsymbol{s}}$ ; independently
existing values at ${\boldsymbol{s}}$ for the resulting object’s constituent
elements not always contribute to the sought-for value of the large structure
(e. g., consider (1c) on p. 1c and Example 2.4 on p. 2.4 and try to calculate
consecutively the objects $\Delta F$, $\Delta G$, and their Schouten brackets
with $G$ and $F$, respectively, for that example’s functionals $F$ and $G$).
Summarizing, it is illegal to construct composite objects step by step,
redundantly inspecting the elements’ values at field configurations. One must
not deviate from a way towards the appointed end of logical reasoning.
In fact, it is us but not Nature who calculates (e. g., the left-hand sides of
equations of motion): Nature neither calculates nor evaluates ; for there is
no built-in mechanism for doing that.282828The probabilistic approach to
evolution of Nature suggests that maxima of transition (and correlation)
functions concentrate near the zero loci of such deterministic equations’
left-hand sides. At the same time, Noether symmetries of the action $S$ are
abundant in the models. Not referring to any actual transformation of a
system’s components, such symmetries reflect the model’s geometry. The
analytic machinery of self-regularizing structures yields the invariants – e.
g., cohomology classes as in section 3.2 – which constrain the probabilistic
laws of evolution. This implies that there is no ever-growing logical
complexity in a description of the Universe ; the flow of local, observer-
dependent time does not require any perpetual increase of the number
$k\geqslant 0$ of factors in the product-bundle location of objects over $k+1$
copies of the space-time. Conversely, there always remains a unique copy of
the space-time for all local functionals.
The space-time geometry of information transfer is very restrictive: its
pointwise locality of events of couplings between dual objects is an absolute
principle ; by weakening this hypothesis one could create a source of
difficulties through causality violation. Consequently, a count of space-time
points where the couplings with a given (co)vector occur makes the formalism
of singular linear integral operators truly adequate in mathematical models of
physical phenomena.292929We recall from Remark 1.5 on p. 1.5 that the volume
elements
$\operatorname{dvol}\bigl{(}{\boldsymbol{x}},\phi({\boldsymbol{x}})\bigr{)}=\sqrt{|\det(g_{\mu\nu})|}\,{\mathrm{d}}{\boldsymbol{x}}$
are present in the building blocks of composite-structure objects.Let us note
further that an association of the weight factors
$\operatorname{dvol}({\boldsymbol{x}})$ with point ${\boldsymbol{x}}\in M^{n}$
is intrinsically related to the structure of space-time $M^{n}$ as topological
manifold (cf. [31]). It is readily seen that a discrete tiling of space-time
converts the integrations over a measure on it to weighted sums over the
points which mark the quantum domains. This links the concept with loop
quantum gravity (see e. g. [17, 47, 49]).
We finally remark that the product-base approach of bundles $\pi\times
T\pi\times\ldots\times T\pi$ over $M\times M\times\ldots\times M$ to the
geometry of variations highlights the concept of physical field as infinite-
dimensional system with degrees of freedom which are attached at every point
of space-time. The locality principle for (co)vector interaction is the
mechanism which distinguishes between space-time points with respect to its
(non)Hausdorff topology.
### Discussion
Let us finally address two logical aspects of the geometry of variations.
#### Linear vector space structures
Nature is essentially nonlinear ; for there is no mechanism which would
realize – under a uniform time bound – an arbitrarily large number of
replications of an object. This is tautological for those physical fields
$\phi$ which take values in spaces without any linear structure. Moreover,
even if there is a brute force labelling of Euler–Lagrange equations by using
the fields $\phi$, a linear vector space pattern of the equations of motion is
not utilized (the same is true for the equations’ descendants such as the
antifields $\phi^{\dagger}$ or (anti)ghosts). Indeed, it is only their the
_tangent_ spaces whose linear structure is used, in particular, in order to
split the variations in ghost parity-homogeneous components. Objects are
linearized only in the course of variations under infinitesimal test shifts.
For example, this determines the distinction between finite offsets
$\Delta{\boldsymbol{x}}$ so that
$({\boldsymbol{x}},{\boldsymbol{x}}+\Delta{\boldsymbol{x}})\in M\times M$ and
infinitesimal test shifts
$\left.\mathstrut\delta{\boldsymbol{x}}\right|_{{\boldsymbol{x}}}\in
T_{{\boldsymbol{x}}}M$ which are mapped to the number field $\Bbbk$ by
covectors
$\left.\mathstrut{\mathrm{d}}{\boldsymbol{x}}\right|_{{\boldsymbol{x}}}\in
T^{*}_{{\boldsymbol{x}}}M$.
#### Annual reproduction rate for interspecimen breeding of cats and whales
An immediate comment on the title of this paragraph is as follows. One could
proclaim that the annual reproduction rate for interspecimen breeding of –
without loss of generality – cats and whales is equal to zero for a given
year. Alternatively, one should understand that such events never happen (not
that a given year brought no brood).
This grotesque illustration works equally well for the (co)tangent spaces to
fibres of the BV-zoo or, in broad terms, for a definition of Kronecker’s
symbol $\boldsymbol{\delta}_{i}^{j}$ by zero whenever the indices $i\neq j$ do
not match so that the couplings in (11) do not eventuate. We argue that, on
top of the absolute pointwise locality for couplings (9), a superficial
definition of $\langle\,,\,\rangle$ by zero for mismatching elements
$\vec{e}_{i}$ and $\vec{e}^{{}\,\dagger j}$ of dual bases is a mere act of
will ; in reality those evaluations do not occur. Consequently, the geometry
dictates that
$\log\left\langle\vec{e}_{i}({\boldsymbol{x}}),{}^{\dagger}(\vec{e}_{j})({\boldsymbol{x}})\right\rangle=\log
1=0\quad\text{and}\quad\log\left\langle\vec{e}^{{}\,\dagger
j}({\boldsymbol{x}}),{}^{\dagger}(\vec{e}^{{}\,\dagger
i})({\boldsymbol{x}})\right\rangle=\log 1=0.$
Combined with the geometric locality principle (4) realized by singular linear
integral operators (12), this argument finally resolves the paradoxical, ad
hoc conventions ${\boldsymbol{\delta}(0)=0}$ and
${\log\boldsymbol{\delta}(0)=0}$ for Dirac’s distribution.
The author thanks the Organizing committee of XXI International conference
‘Integrable systems & quantum symmetries’ (June 11 – 16, 2013; CVUT Prague,
Czech Republic) for cooperation and warm atmosphere during the meeting. These
notes follow the lecture course which was read by the author in October 2013
at the Taras Shevchenko National University and Bogolyubov Institute for
Theoretical Physics in Kiev, Ukraine; the author is grateful to BITP for
hospitality. The author thanks M. A. Vasiliev and A. G. Nikitin for helpful
discussions and constructive criticisms.
This research was supported in part by JBI RUG project 103511 (Groningen). A
part of this research was done while the author was visiting at the IHÉS
(Bures-sur-Yvette); the financial support and hospitality of this institution
are gratefully acknowledged.
## Appendix A Proof of Propositions 10 and 11
We need the following two lemmas.
###### Lemma 17.
Let $F\in\overline{H}^{n}(\pi_{\text{{BV}}})$ be an even integral functional,
let $G\in\overline{\mathfrak{N}}^{n}(\pi_{\text{{BV}}},T\pi_{\text{{BV}}})$ be
another functional, and let $n\in\mathbb{N}_{\geq 1}$. Then
$\lshad{G,F^{n}}\rshad=n\lshad{G,F}\rshad F^{n-1}.$
###### Proof.
We use induction on Theorem 4. Note that all signs vanish since $F$ is even,
meaning that whenever $F$ is multiplied with any other integral functional,
the factors may be freely swapped without this resulting in minus signs. For
$n=1$ the statement is trivial. Suppose the formula holds for some
$n\in\mathbb{N}_{>1}$, then $\lshad{G,F^{n+1}}\rshad={}$
$\lshad{G,F\cdot F^{n}}\rshad=\lshad{G,F}\rshad\cdot
F^{n}+F\cdot\lshad{G,F^{n}}\rshad=\lshad{G,F}\rshad\cdot
F^{n}+nF\cdot\lshad{G,F}\rshad F^{n-1}=(n+1)\lshad{G,F}\rshad\cdot F^{n},$
so that the statement also holds for $n+1$. ∎
###### Lemma 18.
Let $F\in\overline{H}^{n}(\pi_{\text{{BV}}})$ be an even integral functional,
and let $n\in\mathbb{N}_{\geq 2}$. Then
$\Delta(F^{n})=n(\Delta F)\cdot
F^{n-1}+\tfrac{1}{2}n(n-1)\lshad{F,F}\rshad\cdot F^{n-2}.$
###### Proof.
We use induction and the previous lemma. For $n=2$ the formula clearly holds
by Theorem 3. Suppose that it holds for some $n\in\mathbb{N}_{>2}$, then
$\displaystyle\Delta(F^{n+1})$ $\displaystyle=\Delta(F\cdot F^{n})=(\Delta
F)\cdot F^{n}+\lshad{F,F^{n}}\rshad+F\cdot\Delta(F^{n})$
$\displaystyle=(\Delta F)\cdot F^{n}+n\lshad{F,F}\rshad\cdot F^{n-1}+F\cdot
n(\Delta F)F^{n-1}+\tfrac{1}{2}n(n-1)F\cdot\lshad{F,F}\rshad F^{n-2}$
$\displaystyle=(n+1)(\Delta F)\cdot
F^{n}+\tfrac{1}{2}(n+1)n\,\lshad{F,F}\rshad\cdot F^{n-1},$
so that the statement also holds for $n+1$. ∎
###### Proof of Proposition 10.
For convenience, we denote $F=\frac{{\boldsymbol{i}}}{\hbar}S^{\hbar}$. Then
$\displaystyle 0$ $\displaystyle=\Delta(\exp
F)=\Delta\left(\sum_{n=0}^{\infty}\frac{1}{n!}F^{n}\right)=\sum_{n=0}^{\infty}\frac{1}{n!}\Delta(F^{n})$
$\displaystyle=\sum_{n=1}^{\infty}\frac{n}{n!}(\Delta F)\cdot
F^{n-1}+\sum_{n=2}^{\infty}\frac{1}{2n!}n(n-1)\lshad{F,F}\rshad\cdot F^{n-2}$
$\displaystyle=(\Delta
F)\cdot\sum_{n=1}^{\infty}\frac{1}{(n-1)!}F^{n-1}+\frac{1}{2}\lshad{F,F}\rshad\cdot\sum_{n=2}^{\infty}\frac{1}{(n-2)!}F^{n-2}$
$\displaystyle=\left(\Delta F+\tfrac{1}{2}\lshad{F,F}\rshad\right)\cdot\exp
F=\left(\frac{{\boldsymbol{i}}}{\hbar}\Delta
S^{\hbar}-\frac{1}{2\hbar^{2}}\lshad{S^{\hbar},S^{\hbar}}\rshad\right)\cdot\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right),$
from which the result follows. ∎
###### Proof of Proposition 11 (cf. Proposition 13 on p. 13).
Again, let us set $F=\frac{{\boldsymbol{i}}}{\hbar}S^{\hbar}$. We first
calculate, using Lemma 17,
$\lshad{\mathcal{O},\exp
F}\rshad=\sum_{n=0}^{\infty}\frac{1}{n!}\lshad{\mathcal{O},F^{n}}\rshad=\sum_{n=1}^{\infty}\frac{n}{n!}\lshad{\mathcal{O},F}\rshad
F^{n-1}=\lshad{\mathcal{O},F}\rshad\exp F.$
Then
$\displaystyle 0$ $\displaystyle=\Delta(\mathcal{O}\exp
F)=(\Delta\mathcal{O})\exp F+\lshad{\mathcal{O},\exp
F}\rshad+\mathcal{O}\cdot\Delta(\exp F)$
$\displaystyle=\big{(}\Delta\mathcal{O}+\lshad{\mathcal{O},F}\rshad\big{)}\exp
F=\left(\Delta\mathcal{O}+\tfrac{{\boldsymbol{i}}}{\hbar}\lshad{\mathcal{O},S^{\hbar}}\rshad\right)\exp\left(\tfrac{{\boldsymbol{i}}}{\hbar}S^{\hbar}\right),$
from which the assertion follows. ∎
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|
arxiv-papers
| 2013-12-04T17:55:15 |
2024-09-04T02:49:54.887735
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arthemy V. Kiselev",
"submitter": "Arthemy Kiselev",
"url": "https://arxiv.org/abs/1312.1262"
}
|
1312.1413
|
# Fast Subspace Approximation via Greedy Least-Squares
M. A. Iwen
Department of Mathematics, Michigan State University
Department of Electrical and Computer Engineering, Michigan State University
Email: [email protected]
Contact Author. Supported in part by NSA grant H98230-13-1-0275.
Felix Krahmer
Institute for Numerical and Applied Mathematics, University of Göttingen
Email: [email protected]
###### Abstract
In this note, we develop fast and deterministic dimensionality reduction
techniques for a family of subspace approximation problems. Let
$P\subset\mathbbm{R}^{N}$ be a given set of $M$ points. The techniques
developed herein find an $O(n\log M)$-dimensional subspace that is guaranteed
to always contain a near-best fit $n$-dimensional hyperplane $\mathcal{H}$ for
$P$ with respect to the cumulative projection error $\left(\sum_{{\bf x}\in
P}\|{\bf x}-\Pi_{\mathcal{H}}{\bf x}\|^{p}_{2}\right)^{1/p}$, for any chosen
$p>2$. The deterministic algorithm runs in
$\tilde{O}\left(MN^{2}\right)$-time, and can be randomized to run in only
$\tilde{O}\left(MNn\right)$-time while maintaining its error guarantees with
high probability. In the case $p=\infty$ the dimensionality reduction
techniques can be combined with efficient algorithms for computing the John
ellipsoid of a data set in order to produce an $n$-dimensional subspace whose
maximum $\ell_{2}$-distance to any point in the convex hull of $P$ is
minimized. The resulting algorithm remains $\tilde{O}\left(MNn\right)$-time.
In addition, the dimensionality reduction techniques developed herein can also
be combined with other existing subspace approximation algorithms for
$2<p\leq\infty$ – including more accurate algorithms based on convex
programming relaxations – in order to reduce their runtimes.
## 1 Introduction
Fitting a given point cloud with a low-dimensional affine subspace is a
fundamental computational task in data analysis. In this paper we consider
fast algorithms for approximating a given set of $M$ points,
$P\subset\mathbbm{R}^{N}$, with an $n$-dimensional affine subspace
$\mathcal{A}\subset\mathbbm{R}^{N}$ that is a near-best fit. Here the fitness
of $\mathcal{A}$ will be measured by
$d^{(p)}(P,\mathcal{A}):=\sqrt[p]{\sum_{{\bf x}\in P}\left(d({\bf
x},\mathcal{A})\right)^{p}}$, where $d({\bf x},\mathcal{A})$ is the Euclidean
distance from ${\bf x}$ to $\mathcal{A}$, and $p\in\mathbbm{R}^{+}$.
Similarly, when $p=\infty$ the fitness measure will be
$d^{(\infty)}(P,\mathcal{A}):=\max_{{\bf x}\in P}d({\bf x},\mathcal{A})$. An
$n$-dimensional affine subspace $\mathcal{A}\subset\mathbbm{R}^{N}$ is a near-
best fit for $P$ with respect to this fitness measure if there exists a small
constant $C\in\mathbbm{R}^{+}$ such that $d^{(p)}(P,\mathcal{A})\leq C\cdot
d^{(p)}(P,\mathcal{H})$ for all $n$-dimensional affine subspaces
$\mathcal{H}\subset\mathbbm{R}^{N}$.111The approximation constant $C$ may
depend (mildly) on both $p$ and $|P|=M$. In this paper we are interested in
calculating near-best fit affine subspaces for large and high-dimensional
point sets, $P\subset\mathbbm{R}^{N}$, as rapidly as possible.
In the case $p=2$ the problem above is the well known least-squares
approximation problem. Mathematically, a near-best fit $n$-dimensional least-
squares subspace can be obtained by computing the top $n$ eigenvectors of
$XX^{\rm T}$ for the matrix $X\in\mathbbm{R}^{N\times M}$ whose columns are
the points in $P$. Decades of progress related to the computational
eigenvector problem has resulted in many efficient numerical schemes for this
problem (see, e.g., [19, 7], and the references therein). The situation is
more difficult when $p\neq 2$. None the less, a good deal of work has been
done developing algorithms for other values of $p$ as well.
Examples include methods for approximately solving the case $p=1$, which has
been proposed as a means of reducing the effects of statistical outliers on an
approximating subspace (see, e.g., [15]). However, in this paper we are
primarily interested in $p>2$. In particular, we develop fast dimensionality
reduction techniques for the subspace approximation problem which can be used
in combination with existing solution methods for any $p>2$ [16, 2] in order
to reduce their runtimes. For the important case $p=\infty$ these new
dimensionality reduction methods yield a new fast approximation algorithm
guaranteed to find near-optimal solutions.
### 1.1 Results and Previous Work for the $p=\infty$ Case
The case $p=\infty$ is closely related to several fundamental computational
problems in convex geometry and has been widely studied (see, e.g., [6, 4, 8,
20, 1, 18], and references therein). Previous computational methods developed
for this case can be grouped into two general categories: methods based on
semi-definite programming relaxations (e.g., [20, 18]), and methods based on
core-set techniques (e.g., [8, 1]). Both approaches have comparative
strengths. The semidefinite programming approach leads to highly accurate
approximations. In particular, [18] demonstrates a randomized approach which
computes an $n$-dimensional subspace $\mathcal{A}$ that has
$d^{(\infty)}(P,\mathcal{A})\leq\sqrt{12\log M}\cdot
d^{(\infty)}(P,\mathcal{H})$ for all $n$-dimensional subspaces
$\mathcal{H}\subset\mathbbm{R}^{N}$ with high probability. Furthermore, the
approximation factor $\sqrt{12\log M}$ is shown to be close to the best
achievable in polynomial time. However, the method requires the solution of a
semi-definite program, and so has a runtime complexity that scales super-
linearly in both $M$ and $N$. This makes the technique intractable for large
sets of points in high dimensional space.
The core-set approach achieves better runtime complexities for small values of
$n$. In [1] a $\tilde{O}(MN2^{n})$-time randomized approximation algorithm is
developed for the $p=\infty$ case.222Herein, $\tilde{O}(\cdot)$-notation
indicates that polylogarithmic factors have been dropped from the associated
$O$-upper bounds for the sake of readability. This algorithm has the advantage
of being linear in both $M$ and $N$, but quickly becomes computationally
infeasible as the dimension of the approximating subspace, $n$, grows.
In this paper we develop an $\tilde{O}(MN^{2})$-time deterministic algorithm
which computes an $n$-dimensional subspace $\mathcal{A}$ that is guaranteed to
have $d^{(\infty)}(P,\mathcal{A})\leq C\sqrt{n\log M}\cdot
d^{(\infty)}(P,\mathcal{H})$ for all $n$-dimensional subspaces
$\mathcal{H}\subset\mathbbm{R}^{N}$. Here $C\in\mathbbm{R}^{+}$ is a small
universal constant (e.g., it can be made less than $10$). Furthermore, the
algorithm can be randomized to run in only $\tilde{O}(MNn)$-time while still
achieving the same accuracy guarantee with high probability. This improves on
the runtime complexities of existing core-set approaches while simultaneously
obtaining accuracies on the order of existing semi-definite programming
methods for small $n$.
The approximation algorithms for the $p=\infty$ case developed in this paper
are motivated by the following idea: The difficulty of approximating
$P\subset\mathbbm{R}^{N}$ with a subspace can be greatly reduced by first
approximating (the convex hull of) $P$ with an ellipsoid, and then
approximating the resulting ellipsoid with an $n$-dimensional subspace. In
fact, fast algorithms for approximating (the convex hull of) $P$ by an
ellipsoid are already known (see, e.g., [11, 14, 17]). And, it is
straightforward to approximate an ellipsoid optimally with an $n$-dimensional
subspace – one may simply use its $n$ largest semi-axes as a basis. The only
deficit in this simple approach is that the accuracy it guarantees is rather
poor. The resulting $n$-dimensional subspace $\mathcal{A}$ may have
$d^{(\infty)}(P,\mathcal{A})$ as large as $\sqrt{N}\cdot
d^{(\infty)}(P,\mathcal{H})$ for some other $n$-dimensional subspace
$\mathcal{H}\subset\mathbbm{R}^{N}$. This guarantee can be improved, however,
if $N$ (i.e., the dimension of the point set $P$) is reduced before the
approximating ellipsoid is computed. Motivated by this idea, we develop new
dimensionality reduction algorithms for the subspace approximation problem
below.
### 1.2 Dimensionality Reduction Results and Previous Work
An algorithm is a dimensionality reduction method for the subspace
approximation problem if, for any $P\subset\mathbbm{R}^{N}$, it finds a low-
dimensional subspace that is guaranteed to contain a near-best fit
$n$-dimensional hyperplane $\mathcal{H}$. Such dimensionality reduction
methods can be regarded as a “weak” approximate solution methods for the
subspace approximation problem in the following sense. They produce subspaces
whose dimensions are larger than $n$ (i.e., larger than the target dimension
of the desired best-fit hyperplane), but solving the problem restricted to
these subspaces will yield a near-optimal solution. Thus dimensionality
reduction methods – when sufficiently fast – allow the subspace approximation
problem to be simplified before more time intensive solution methods are
employed. For example, if a low-dimensional subspace has been found, which
still contains a near-best fit solution, high-dimensional data (i.e., with $N$
large) can be projected onto that subspace in order to reduce its complexity
before solving. Hence, fast dimensionality reduction algorithms can be used to
help speed up existing solutions methods for $p>2$ (e.g., by reducing the
input problem sizes for methods based on solving convex programs [2].)
Several dimensionality reduction techniques have been developed for the
subspace approximation problem over the past several years (see, e.g., [1, 3,
5] and references therein). These methods are all based on sampling techniques
and either have runtime complexities that scale exponentially in $n$, or
embedding subspace dimensions that scale exponentially in $p$. In [3], for
example, an $MNn^{O(1)}$-time randomized algorithm is given which is
guaranteed, with high probability, to return an
$\tilde{O}(n^{p+3})$-dimensional subspace that itself contains another
$n$-dimensional subspace, $\mathcal{A}$, whose fit, $d^{(p)}(P,\mathcal{A})$,
is the near-best possible for any $p\in[1,\infty)$. Although useful for small
$p$, these methods quickly become infeasible as $p$ increases.
In this paper a different dimensionality reduction approach is taken that
reduces the problem, for any $p\geq 2$, to a small number of least-squares
problems. The idea is to greedily approximate a large portion of the input
data $P$ with a fast least-squares method. It turns out that a large portion
of $P$ is always well-approximated, for any $p>2$, by $P$’s best-fit
$n$-dimensional least-squares subspace. Then, the previously worst-
approximated points in $P$ can be iteratively fit by least-squares subspaces
until all of $P$ has eventually been approximated well, with respect to any
desired $p>2$, by the union of $O(\log M)$ least-squares subspaces. Using this
idea, a deterministic $\tilde{O}(MN^{2})$-time algorithm can be developed
which is always guaranteed to return an $O(n\log M)$-dimensional subspace that
itself contains another $n$-dimensional subspace, $\mathcal{A}$, whose fit,
$d^{(p)}(P,\mathcal{A})$, is the near-best possible for any $p\in[2,\infty]$.
Furthermore, this algorithm can be randomized to run in only
$\tilde{O}(MNn)$-time while still achieving the same accuracy guarantees as
the deterministic variant with high probability.
### 1.3 Organization
The remainder of this paper is organized as follows: In Section 2 notation is
established and necessary theory is reviewed. Then, in Section 3, the
dimensionality reduction results are developed for any $p>2$. Finally, in
Section 4, our improved dimensionality reduction result for the case
$p=\infty$ is used to illustrate a fast and simple subspace approximation
algorithm for the $p=\infty$ subspace approximation problem.
## 2 Preliminaries: Notation and Setup
For any matrix $X\in\mathbbm{R}^{N\times M}$ we will denote the $j^{\rm th}$
column of $X$ by ${\bf X}_{j}\in\mathbbm{R}^{N}$. The transpose of a matrix,
$X\in\mathbbm{R}^{N\times M}$, will be denoted by $X^{\rm
T}\in\mathbbm{R}^{M\times N}$, and the singular values of any matrix
$X\in\mathbbm{R}^{N\times M}$ will always be ordered as
$\sigma_{1}(X)\geq\sigma_{2}(X)\geq\dots\geq\sigma_{\min(N,M)}(X)\geq 0.$ The
Frobenius norm of $X\in\mathbbm{R}^{N\times M}$ is defined as
$\|X\|_{F}:=\sqrt{\sum^{M}_{j=1}\sum^{N}_{i=1}|X_{i,j}|^{2}}=\sqrt{\sum^{\min(N,M)}_{l=1}\sigma^{2}_{l}(X)}.$
(1)
A key ingredient of our results is the following perturbation bounds for
singular values (see, e.g., [9]).
###### Theorem 1 (Weyl).
Let $A,B\in\mathbbm{R}^{M\times N}$, and $q=\min\\{M,N\\}$. Then,
$\sigma_{i+j-1}(A+B)\leq\sigma_{i}(A)+\sigma_{j}(B)$
holds for all $i,j\in\\{1,\dots,q\\}$ with $i+j\leq q+1$.
Given an $\tilde{n}$-dimensional subspace
$\mathcal{S}\subseteq\mathbbm{R}^{N}$, we will denote the set of all
$n$-dimensional affine subspaces of $\mathcal{S}$ by
$\Gamma_{n}\left(\mathcal{S}\right)$. Here, of course, we assume that
$N\geq\tilde{n}\geq n$. Given an affine subspace
$\mathcal{A}\in\Gamma_{n}\left(\mathcal{S}\right)$, we will denote the offset
of $\mathcal{A}$ by
${\bf a}_{\mathcal{A}}:=\operatorname*{arg\,min}_{{\bf x}\in\mathcal{A}}\|{\bf
x}\|_{2},$ (2)
and the $n$-dimensional subspace of $\mathcal{S}$ that is parallel to
$\mathcal{A}$ by
$\mathcal{S}_{\mathcal{A}}:={\mathcal{A}}-{\bf a}_{\mathcal{A}}:=\left\\{{\bf
x}-{\bf a}_{\mathcal{A}}~{}\big{|}~{}{\bf x}\in\mathcal{A}\right\\}.$ (3)
Note that ${\bf a}_{\mathcal{A}}\in\mathcal{S}_{\mathcal{A}}^{\perp}$. Thus,
we may define the projection operator onto $\mathcal{A}$,
$\Pi_{\mathcal{A}}:\mathbbm{R}^{N}\rightarrow\mathcal{A}$, by
$\Pi_{\mathcal{A}}{\bf x}:=\Pi_{\mathcal{S_{A}}}{\bf x}+{\bf
a}_{\mathcal{A}}.$ (4)
Here $\Pi_{\mathcal{S_{A}}}$ is the orthogonal projection onto
$\mathcal{S_{A}}$.
### 2.1 A Family of Distances
Given a subset $T\subset\mathbbm{R}^{N}$ and an affine subspace
$\mathcal{A}\in\Gamma_{n}(\mathcal{S})$ we will want to consider the
“distance” of $T$ from $\mathcal{A}$, defined by
$d^{(\infty)}(T,\mathcal{A}):=\sup_{{\bf x}\in T}\|{\bf
x}-\Pi_{\mathcal{A}}{\bf x}\|_{2}.$ (5)
Let $\mathcal{S}$ be an $\tilde{n}\geq n$ subspace of $\mathbbm{R}^{N}$. We
can now define the Euclidean Kolmogorov $n$-width of $T$ in this setting by
$d^{(\infty)}_{n}(T,{\mathcal{S}}):=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}d^{(\infty)}(T,\mathcal{A})=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}\sup_{{\bf
x}\in T}~{}\|{\bf x}-\Pi_{\mathcal{A}}{\bf x}\|_{2}.$ (6)
Finally, we note that there will always be (at least one) optimal affine
subspace, $\mathcal{A}_{\rm opt}\in\Gamma_{n}(\mathcal{S})$, with
$d^{(\infty)}(T,\mathcal{A}_{\rm opt})=d^{(\infty)}_{n}(T,{\mathcal{S}})$ (7)
when $T$ is “sufficiently nice” (e.g., when $T$ is either finite, or convex
and compact).333This follows from the fact that Stiefel manifolds are compact,
together with the fact that only offsets, ${\bf
a}_{\mathcal{A}}\in\mathbbm{R}^{N}$, contained in the ball of radius
$\sup_{{\bf x}\in T}\|x\|_{2}$ are ever relevant to minimizing
$d^{(\infty)}(T,\cdot)$. Thus, the set of relevant affine subspaces under
consideration is compact when $T$ is bounded. Finally,
$d^{(\infty)}(T,\cdot):\Gamma_{n}\left(\mathcal{S}\right)\rightarrow\mathbbm{R}^{+}$,
$T\subset\mathbbm{R}^{N}$ fixed, will be continuous when $T$ is sufficiently
well behaved (e.g., either finite, or compact and convex).
When $T=\\{{\bf t}_{1},\dots,{\bf t}_{M}\\}\subset\mathbbm{R}^{N}$ is finite,
we may define a vector ${\bf e}_{\mathcal{A}}\in\mathbbm{R}^{M}$ for any given
$\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$ by
$\left({\bf e}_{\mathcal{A}}\right)_{j}:=\left\|{\bf
t}_{j}-\Pi_{\mathcal{A}}{\bf t}_{j}\right\|_{2}.$ (8)
Thus, when $T$ is finite we can see that
$d^{(\infty)}_{n}(T,{\mathcal{S}})=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}\left\|{\bf
e}_{\mathcal{A}}\right\|_{\infty},$ (9)
and the least squares approximation error over all subspaces in
$\Gamma_{n}(\mathcal{S})$ is given by
$d^{(2)}_{n}(T,{\mathcal{S}})=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}\left\|{\bf
e}_{\mathcal{A}}\right\|_{2}.$ (10)
These two quantities can be seen as extreme instances of the infinite family
of approximation errors given by
$d^{(p)}_{n}(T,{\mathcal{S}}):=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}\left\|{\bf
e}_{\mathcal{A}}\right\|_{p},$ (11)
for any parameter $2\leq p\leq\infty$. Note that, analogously to (6), one has
$d^{(p)}_{n}(T,{\mathcal{S}}):=\inf_{\mathcal{A}\in\Gamma_{n}(\mathcal{S})}~{}d^{(p)}(T,\mathcal{A}),$
(12)
where
$d^{(p)}(T,\mathcal{A}):=\left\|{\bf e}_{\mathcal{A}}\right\|_{p}.$ (13)
Finally, as above, we note that there will always be at least one optimal
affine subspace, $\mathcal{A}_{\rm opt}\in\Gamma_{n}(\mathcal{S})$, with
$d^{(p)}(T,\mathcal{A}_{\rm opt})=d^{(p)}_{n}(T,{\mathcal{S}})$ (14)
when $T$ is finite.
### 2.2 Symmetry, Ellipsoids, and Properties of $n$-widths
Let $P=\\{{\bf p}_{1},\dots,{\bf p}_{M}\\}\subset\mathbbm{R}^{N}$, and define
$\bar{\bf p}:=\frac{1}{M}\cdot\sum^{M}_{j=1}{\bf p}_{j}.$ (15)
We will let $\bar{P}\subset\mathbbm{R}^{N}$ denote the following symmetrized
translation of $P$,
$\bar{P}:=(P-\bar{\bf p})\cup(\bar{\bf p}-P)\cup\\{{\bf 0}\\}:=\left\\{{\bf
p}_{j}-\bar{\bf p}~{}\big{|}~{}{\bf p}_{j}\in P\right\\}\cup\left\\{\bar{\bf
p}-{\bf p}_{j}~{}\big{|}~{}{\bf p}_{j}\in P\right\\}\cup\\{{\bf 0}\\}.$ (16)
We will say that $P$ is symmetric if and only if $P=\bar{P}$. Furthermore, we
will denote the convex hull of $P$ by ${\rm CH}(P)$. The following theorem due
to Fritz John [10] guarantees the existence of an ellipsoid that approximates
${\rm CH}\left(\bar{P}\right)$ well.
###### Theorem 2 (John).
Let $K\subset\mathbbm{R}^{N}$ be a compact and convex set with nonempty
interior that is symmetric about the origin (so that $K=-K$). Then, there is
an ellipsoid centered at the origin, $\mathcal{E}\subset\mathbbm{R}^{N}$, such
that $\mathcal{E}\subseteq K\subseteq\sqrt{N}\cdot\mathcal{E}$.
Given $P\subset\mathbbm{R}^{N}$, an ellipsoid which is nearly as good an
approximation to ${\rm CH}\left(\bar{P}\right)$ as the ellipsoid guaranteed by
Thoerem 2 can be computed in polynomial time (see, e.g., [11, 14, 17]). More
specifically, one can compute an ellipsoid $\mathcal{E}$ such that
$\mathcal{E}\subseteq{\rm
CH}\left(\bar{P}\right)\subseteq\sqrt{(1+\epsilon)N}\cdot\mathcal{E}$ in
$O(MN^{2}(\log N+1/\epsilon))$-time for any $\epsilon\in(0,\infty)$ [17].
Finally, in the following Lemma, we summarize a few facts concerning the
$n$-widths of finite sets, convex hulls, and ellipsoids that will be useful
for establishing our results (proofs are included in Appendix A for the sake
of completeness).
###### Lemma 1.
Let $P=\\{{\bf p}_{1},\dots,{\bf p}_{M}\\}\subset\mathbbm{R}^{N}$, and
$\mathcal{E}\subset\mathbbm{R}^{N}$ be the ellipsoid
$\left\\{{\bf x}\in\mathbbm{R}^{N}~{}\big{|}~{}{\bf x}^{T}Q{\bf x}\leq
1\right\\},$
where $Q\in\mathbbm{R}^{N\times N}$ is symmetric and positive definite. Then,
1. 1.
$d^{(\infty)}_{n}\left(P-{\bf
x},\mathbbm{R}^{N}\right)=d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$ for
all ${\bf x}\in\mathbbm{R}^{N}$, and $n=1,\dots,N$.
2. 2.
$\bar{P}$ will have an optimal $n$-dimensional subspace (i.e., with ${\bf
a}_{\mathcal{A}_{\rm opt}}={\bf 0}$) for all $n=1,\dots,N$.
3. 3.
$d^{(\infty)}_{n}\left(\bar{P},\mathbbm{R}^{N}\right)\leq 2\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$ for all $n=1,\dots,N$.
4. 4.
$d^{(\infty)}_{n}(B,\mathbbm{R}^{N})\leq d^{(\infty)}_{n}(C,\mathbbm{R}^{N})$
for all $B\subseteq C\subset\mathbbm{R}^{N}$, and $n=1,\dots,N$.
5. 5.
$d^{(\infty)}_{n}({\rm
CH}(P),\mathbbm{R}^{N})=d^{(\infty)}_{n}(P,\mathbbm{R}^{N})$ for all
$n=1,\dots,N$.
6. 6.
$d^{(\infty)}_{n}(\mathcal{E},\mathbbm{R}^{N})=\sqrt{\frac{1}{\sigma_{N-n+1}(Q)}}$
for all $n=1,\dots,N$. Consequently, an optimal $n$-dimensional subspace for
$\mathcal{E}$ is spanned by the eigenvectors of $Q$ associated with
$\sigma_{N}(Q),\dots,\sigma_{N-n+1}(Q)$.
We will assume hereafter, without loss of generality, that $P=\\{{\bf
p}_{0},\dots,{\bf p}_{M}\\}\subset\mathbbm{R}^{N}$ both spans
$\mathbbm{R}^{N}$ and is symmetric. If $P$ initially does not span
$\mathbbm{R}^{N}$, we will replace each element of $P$ with the coordinates of
its orthogonal projection into the span of $P$, reducing $N$ accordingly. Any
such change of basis for $P$ will lead to no loss of accuracy in our solution.
If $P$ is not symmetric we will approximate $\bar{P}$ by a subspace instead,
noting that a translation of our approximating subspace for $\bar{P}$ will
still approximate $P$ well by parts $(1)-(4)$ of Lemma 1. Finally, we will
assume hereafter that ${\bf p}_{0}={\bf 0}$.
## 3 Dimensionality Reduction Results
In this section we establish our main theorems regarding dimensionality
reduction. As we shall see, the main idea behind the proofs of both Theorems
45 and 50 below is to use fast existing least-squares methods in order to
quickly approximate the point set $P$ in a greedy fashion. To see how this
works, note that $P$’s best-fit least squares subspace will generally fail to
approximate all of $P$ to within
$d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)$-accuracy when $p>2$. However, it
will generally approximate a large fraction of $P$ sufficiently well.
Furthermore, we can easily tell which portion of $P$ is approximated best.
Hence, we may employ a divide-and-concur approach: we $(i)$ approximate $P$
with its best-fit least squares subspace, $(ii)$ identify the half of its
points fit the best, $(iii)$ remove them from $P$, and then $(iv)$ repeat the
process again on the remaining portion of $P$. After $O(\log M)$ repetitions
we end up with a collection of at most $O(\log M)$ least squares subspaces
whose collective span is guaranteed to contain a near-optimal $n$-dimensional
approximation to all of $P$ with respect to
$d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)$. We are now ready to begin proving
our results.
###### Lemma 2.
Let $P=\\{{\bf p}_{0}:={\bf 0},{\bf p}_{1},\dots,{\bf
p}_{M}\\}\subset\mathbbm{R}^{N}$ be symmetric, $n\in\\{1,\dots,N\\}$, and
$p\in(2,\infty]$. Then there is an $O\left(MN^{2}\right)$-time444We assume
here that $M\geq N\geq\log M$. We also note that this runtime complexity can
be improved substantially by utilizing randomized low-rank approximation
algorithms. See Remark 1 below. algorithm which outputs an $n$-dimensional
subspace $\mathcal{S}\subset\mathbbm{R}^{N}$ such that for
$m\in\\{1,\dots,M\\}$ one has
$\|{\bf p}_{l_{m}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{m}}\|^{2}_{2}\leq\frac{M^{1-\frac{2}{p}}}{M-m+1}\cdot\big{(}d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2},$
(17)
where the $\ell_{i}>0$, $i=1,\dots,M$, are chosen to satisfy
$0=\|{\bf p}_{0}-\Pi_{\mathcal{S}}{\bf p}_{0}\|_{2}\leq\|{\bf
p}_{l_{1}}-\Pi_{\mathcal{S}}{\bf p}_{l_{1}}\|_{2}\leq\|{\bf
p}_{l_{2}}-\Pi_{\mathcal{S}}{\bf p}_{l_{2}}\|_{2}\leq\dots\leq\|{\bf
p}_{l_{M}}-\Pi_{\mathcal{S}}{\bf p}_{l_{M}}\|_{2}.$ (18)
Proof: Denote the matrix whose columns are the points in $P$ by
$X\in\mathbbm{R}^{N\times M}$. That is, let
$X:=\left({\bf p}_{1},\dots,{\bf p}_{M}\right).$ (19)
Let $\mathcal{A}^{(p)}_{\rm opt}\in\Gamma_{n}(\mathbbm{R}^{D})$ be an optimal
$n$-dimensional subspace for $P$ satisfying
$d(P,\mathcal{A}^{(p)}_{\rm opt})=d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right).$
(20)
It is not difficult to see that we will have $X=Y+E$, where
$Y,E\in\mathbbm{R}^{N\times M}$ have the following properties: the column span
of $Y$ is contained in $\mathcal{A}^{(p)}_{\rm opt}$, and the vector $\bf{e}$
whose entries are the $\ell^{2}$-norms of the columns of $E$ has
$\ell^{p}$-norm at most $d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)$. It
follows from Hölder’s inequality that
$\sum^{\min(N,M)}_{l=1}\sigma^{2}_{l}(E)=\|E\|^{2}_{F}=\|{\bf{e}}\|^{2}_{2}\leq\|{\bf{e}}\|^{2}_{p}\|{\mathbb{I}}\|_{1+\frac{2}{p-2}}=M^{1-\frac{2}{p}}\cdot\big{(}d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2},$
(21)
where $\mathbb{I}\in\mathbbm{R}^{M}$ is the vector whose entries are all one.
Note that $Y$ has rank at most $n$ so that
$\sigma_{n+1}(Y)=\dots=\sigma_{\min(N,M)}(Y)=0.$ (22)
Applying Theorem 1 we now learn that
$\sigma_{n+l}(X)\leq\sigma_{l}(E)$ (23)
for all $l\in\\{1,\dots,N-n\\}$.
Let $X_{n}$ be the best rank $n$ approximation to $X$ with respect to
Frobenius norm,
$X_{n}:=\operatorname*{arg\,min}_{\begin{subarray}{c}L\in\mathbbm{R}^{N\times
M}\\\ {\rm rank}~{}L=~{}n\end{subarray}}\|X-L\|_{F}.$ (24)
Let $\mathcal{S}$ be the $n$-dimensional subspace spanned by the columns of
$X_{n}$. We have that
$\|X-X_{n}\|^{2}_{F}=\sum^{\min(N,M)}_{l=n+1}\sigma^{2}_{l}(X)\leq
M^{1-\frac{2}{p}}\cdot\big{(}d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2}$
(25)
due to (21) and (23). Thus, for each positive integer $k$ there can be at most
$k$ (nonzero) columns of $X$, ${\bf p}_{j}\in P$, with the property that
$\|{\bf p}_{j}-\Pi_{\mathcal{S}}{\bf
p}_{j}\|^{2}_{2}\geq\frac{M^{1-\frac{2}{p}}}{k}\cdot\big{(}d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2}.$
(26)
Setting $k=M-m+1$, we see that (17) must hold in order for
$\sum^{M}_{j=m}\|{\bf p}_{l_{j}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{j}}\|^{2}_{2}\leq\sum^{M}_{j=1}\|{\bf p}_{l_{j}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{j}}\|^{2}_{2}\leq\|X-X_{n}\|^{2}_{F}\leq
M^{1-\frac{2}{p}}\cdot\big{(}d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2}$
(27)
to hold (i.e., in order for (25) to hold) .
To finish, we note that the subspace $\mathcal{S}$ above is spanned by the $n$
left singular vectors of $X$ associated with its $n$ largest singular values.
These can be computed deterministically in
$O\left(NM\cdot\min\\{N,M\\}\right)$-time as part of the full singular value
decomposition of $X$, although significantly faster (randomized) approximation
algorithms exist (see, e.g., [19, 7]). The stated runtime complexity follows
given our assumption that $M\geq N\geq\log M$. ∎
###### Lemma 3.
Let $\xi\in(1,\infty)$, $P=\\{{\bf p}_{0}:={\bf 0},{\bf p}_{1},\dots,{\bf
p}_{M}\\}\subset\mathbbm{R}^{N}$ be symmetric, and $n\in\\{1,\dots,N\\}$.
Then, there is an $O\left(MN^{2}\right)$-time555Again, we assume that $M\geq
N\geq\log M$. algorithm which outputs both an $n$-dimensional subspace
$\mathcal{S}\subset\mathbbm{R}^{N}$, and a symmetric subset $P^{\prime}\subset
P$ with $|P^{\prime}|\geq\lceil(1-1/\xi)M\rceil+1$, such that
$d^{(\infty)}(P^{\prime},{\mathcal{S}})<\sqrt{\xi}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right).$ (28)
Proof: We first order the nonzero elements of $P$ according to (18), and then
set
$P^{\prime}:=\left\\{{\bf p}_{0},{\bf p}_{l_{1}},{\bf p}_{l_{2}},\dots,{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}\right\\}\subset P.$ (29)
If $P^{\prime}$ is not symmetric, continue to add additional points from $P$
until it is (i.e., by adding the negation of each current point in
$P^{\prime}$ to $P^{\prime}$). Applying Lemma 2 with
$m=\lceil(1-1/\xi)M\rceil$, we see that
$\|{\bf p}_{\lceil(1-1/\xi)M\rceil}-\Pi_{\mathcal{S}}{\bf
p}_{\lceil(1-1/\xi)M\rceil}\|^{2}_{2}\leq\xi\cdot\left(d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{2}.$
(30)
Thus there can be at most $\lfloor M/\xi\rfloor$ (nonzero) columns of $X$,
${\bf p}_{j}\in P$, with the property that
$\|{\bf p}_{j}-\Pi_{\mathcal{S}}{\bf
p}_{j}\|^{2}_{2}\geq\xi\cdot\left(d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{2}.$
(31)
By the ordering (18), the associated indices $j$ must be contained in
$\\{\ell_{\lceil(1-1/\xi)M\rceil+1},\dots,\ell_{M}\\}$, hence
$P^{\prime}\subset P$ will satisfy (28).
By Lemma 2, a suitable set $\mathcal{S}$ can be found in
$O\left(NM\cdot\min\\{N,M\\}\right)$-time. Having computed (the singular value
decomposition of) $X_{n}$, the ordering in (18) can then be determined in
$O(NM+M\log M)$-time. Finally, the symmetry of $P^{\prime}$ can be ensured in
$O(NM\log M)$-time by, e.g., ordering the points of $P^{\prime}$
lexicographically, and then performing a binary search for the negation of
each point in order to ensure its inclusion. The stated runtime complexity
follows given our assumption that $M\geq N\geq\log M$. ∎
###### Remark 1.
The runtime complexity quoted in Lemma 2 and consequently also Lemma 28 and
Lemma 33 is dominated by the time required to compute $X_{n}$ (24) via the
full singular value decomposition of $X$ (19). However, computing $X_{n}$ this
way is computationally wasteful when $n\ll\min\\{N,M\\}$. Note that it
suffices to find a $O(n)$-dimensional matrix,
$\tilde{X}_{n}\in\mathbbm{R}^{N\times M}$, with the property that
$\|X-\tilde{X}_{n}\|_{F}\leq C\cdot\|X-X_{n}\|_{F}$ (32)
for a suitably small constant $C$. Taking $\tilde{\mathcal{S}}$ to be the
column span of $\tilde{X}_{n}$ in the proof of Lemma 28 then produces a
similarly sized subset $P^{\prime}\subset P$ satisfying
$d^{(\infty)}(P^{\prime},\tilde{\mathcal{S}})\leq C\sqrt{\xi}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right).$ A tremendous number of
methods have been developed for rapidly computing an $\tilde{X}_{n}$ as above
(see, e.g., [19, 7]). In particular, we note here that there exists a modest
absolute constant $C\in\mathbbm{R}^{+}$ such that a randomly constructed
matrix $\tilde{X}_{n}$ of rank $\max\\{2n,7\\}$ will satisfy (32) with
probability $>0.9$.666See Theorem 10.7 from [7] for more details concerning
the constant $C$, etc.. Also, note that the probability of satisfying (32) can
be boosted as close to $1$ as desired by constructing several different
$\tilde{X}_{n}$ matrices independently, and then choosing the most accurate
one. Furthermore, this matrix can always be constructed in
$O(NMn+Nn^{2})$-time.
###### Lemma 4.
Let $p\in(2,\infty)$, $\xi\in\left(1,M/2\right]$, $P=\\{{\bf p}_{1},\dots,{\bf
p}_{M}\\}\subset\mathbbm{R}^{N}$ be symmetric, and $n\in\\{1,\dots,N\\}$.
Then, there is an $O\left(MN^{2}\right)$-time777Again, we assume that $M\geq
N\geq\log M$. algorithm which outputs both an $n$-dimensional subspace
$\mathcal{S}\subset\mathbbm{R}^{N}$, and a symmetric subset $P^{\prime}\subset
P$ with $|P^{\prime}|\geq\left\lceil\left(1-1/\xi\right)M\right\rceil+1$, such
that
$d^{(p)}(P^{\prime},{\mathcal{S}})\leq C\sqrt{\xi}\cdot
d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right).$ (33)
Proof: We again order the nonzero elements of $P$ according to (18), and then
define $P^{\prime}$ as above in (29). From Lemma 2 with
$m=\left\lceil\left(1-1/\xi\right)M\right\rceil$ we obtain that
$\displaystyle(d^{(p)}(P^{\prime},{\mathcal{S}}))^{p}$
$\displaystyle=\sum_{j=1}^{m}\|{\bf p}_{\ell_{j}}-\Pi_{\mathcal{S}}{\bf
p}_{\ell_{j}}\|^{p}_{2}$ (34)
$\displaystyle\leq\sum_{j=1}^{m}\left(\frac{M^{1-\frac{2}{p}}}{M-j+1}\cdot\left(d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\right)^{2}\right)^{p/2}$
(35)
$\displaystyle=M^{\frac{p}{2}-1}\left(d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right)\right)^{p}\sum^{M}_{j=M-m+1}j^{-p/2}$
(36) $\displaystyle\leq
M^{\frac{p}{2}-1}\left(d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{p}\int_{M-m}^{M}x^{-p/2}dx$
(37)
$\displaystyle=\frac{\left(1-\frac{m}{M}\right)^{1-\frac{p}{2}}-1}{\frac{p}{2}-1}\left(d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{p}.$
(38)
Set $\delta:=m/M-\left(1-1/\xi\right)<1/M$. It is not difficult to see that
$1/\xi-\delta\in(0,1)$ since $\xi\in\left(1,M/2\right]$. Thus,
$\left(\left(1/\xi\right)-\delta\right)^{1-\frac{p}{2}}\leq\left(\frac{\xi}{1-\xi/M}\right)^{\frac{p}{2}-1}\leq(2\xi)^{\frac{p}{2}-1},$
(39)
which now allows us to bound (38) as follows:
$(d^{(p)}(P^{\prime},{\mathcal{S}}))^{p}\leq\frac{\left(1-\frac{m}{M}\right)^{1-\frac{p}{2}}-1}{\frac{p}{2}-1}\left(d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{p}<\frac{(2\xi)^{\frac{p}{2}-1}-1}{\frac{p}{2}-1}\cdot\left(d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{p}.$
(40)
This last expression yields the first inequality in (33), as desired. For
large $p$, the lemma directly follows from the asymptotics, for $p\approx 2$
from l’Hospital’s rule. As the set $P^{\prime}$ is constructed in the same way
as in the proof of Lemma 28, the runtime analysis given there carries over
directly. ∎
###### Remark 2.
Note that the ordered distances (18) between the points in $P$ and the
subspace $\mathcal{S}$ from Lemma 28 satisfy
$\left\|{\bf p}_{l_{\lceil(1-1/\xi)M\rceil}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}\right\|_{2}\leq\sqrt{\xi}\cdot
d_{n}\left(P,\mathbbm{R}^{N}\right).$ (41)
We can use this information to bound $d_{n}\left(P,\mathbbm{R}^{N}\right)$
from above and below. Set
$\alpha:=\frac{\|{\bf p}_{l_{M}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{M}}\|_{2}}{\left\|{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}\right\|_{2}}.$ (42)
We now have
$d_{n}\left(P,\mathbbm{R}^{N}\right)\leq\|{\bf
p}_{l_{M-1}}-\Pi_{\mathcal{S}}{\bf p}_{l_{M-1}}\|_{2}=\alpha\cdot\left\|{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}-\Pi_{\mathcal{S}}{\bf
p}_{l_{\lceil(1-1/\xi)M\rceil}}\right\|_{2}\leq\alpha\sqrt{\xi}\cdot
d_{n}\left(P,\mathbbm{R}^{N}\right).$ (43)
Thus, computing $\alpha$ allows us to estimate
$d_{n}\left(P,\mathbbm{R}^{N}\right)$. If $\alpha$ is sufficiently small,
$\mathcal{S}$ will itself be a passible approximation to an optimal subspace
$\mathcal{A}_{\rm opt}$. Similarly, if $P^{\prime}\subset P$ and $\mathcal{S}$
from Lemma 33 satisfy
$d^{(p)}(P,\mathcal{S})\leq\alpha\cdot d^{(p)}(P^{\prime},\mathcal{S})$ (44)
for a modest $\alpha\in\mathbbm{R}^{+}$, then we may infer that $\mathcal{S}$
is a near-optimal subspace for $P$.
Lemmas 28 and 33 now allow us to establish the main results of this section.
###### Theorem 3.
Let $\xi\in(1,\infty)$, $P=\\{{\bf p}_{1},\dots,{\bf
p}_{M}\\}\subset\mathbbm{R}^{N}$ be symmetric, and $n\in\\{1,\dots,N\\}$.
Then, there is an $O\left(\frac{\xi}{\xi-1}\cdot MN^{2}+N\cdot
n^{2}\log^{2}_{\xi}M\right)$-time algorithm which outputs an at most
$(n\cdot\lceil\log_{\xi}M\rceil)$-dimensional subspace
$\mathcal{S}\subset\mathbbm{R}^{N}$ with
$d^{(\infty)}_{n}(P,{\mathcal{S}})\leq\left(1+\sqrt{\xi}\right)\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right).$ (45)
Proof: Let $\mathcal{S}\subset\mathbbm{R}^{D}$ be an $\tilde{n}$-dimensional
subspace with $\tilde{n}\geq n$, and
$\mathcal{A}\in\Gamma_{n}(\mathbbm{R}^{D})$. We have that
$d^{(\infty)}_{n}(P,\mathcal{S})\leq\max_{{\bf p}_{j}\in P}\|{\bf
p}_{j}-\Pi_{\mathcal{S}}\Pi_{\mathcal{A}}{\bf p}_{j}\|_{2}\leq\max_{{\bf
p}_{j}\in P}\|{\bf p}_{j}-\Pi_{\mathcal{S}}{\bf p}_{j}\|_{2}+\max_{{\bf
p}_{j}\in P}\|{\bf p}_{j}-\Pi_{\mathcal{A}}{\bf p}_{j}\|_{2}.$ (46)
The fact that this holds for all $\mathcal{A}\in\Gamma_{n}(\mathbbm{R}^{D})$
now immediately implies that
$d^{(\infty)}_{n}(P,\mathcal{S})\leq
d^{(\infty)}(P,\mathcal{S})+d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right).$
(47)
It remains to make a good choice for the subspace $\mathcal{S}$. More
precisely, we would like to find a subspace $\mathcal{S}$ with
$d^{(\infty)}(P,\mathcal{S})\leq\sqrt{\xi}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$ so that we can obtain (45)
from (47).
Appealing to Lemma 28, we note that we can find a sufficiently accurate
$n$-dimensional subspace, $\mathcal{S}^{1}$, for a large symmetric subset
$P^{\prime}\subset P$ with $|P^{\prime}|\geq\lceil(1-1/\xi)M\rceil+1$. It
remains to find a similarly accurate subspace for the rest of $P$. Set
$P_{2}:=P-P^{\prime}\cup\\{0\\}$, noting that $P_{2}$ will be a symmetric
point set with $|P_{2}|\leq M/\xi$. We may now apply Lemma 28 to $P_{2}$ in
order to find a second $n$-dimensional subspace, $\mathcal{S}^{2}$, which
approximates all but at most $M/\xi^{2}$ elements of $P_{2}$ to within the
desired $\sqrt{\xi}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$-accuracy. More generally, we
can see that iterating Lemma 28 at most $\lceil\log_{\xi}M\rceil$-times in
this fashion will produce a collection of at most $\lceil\log_{\xi}M\rceil$
different $n$-dimensional subspaces,
$\mathcal{S}^{1},\dots,\mathcal{S}^{\lceil\log_{\xi}M\rceil}$, which will
collectively approximate all of $P$ to the desired $\sqrt{\xi}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$-accuracy. We now set
$\mathcal{S}:={\rm
span}\left(\mathcal{S}^{1}\cup\dots\cup\mathcal{S}^{\lceil\log_{\xi}M\rceil}\right).$
(48)
It is not difficult to see that $\mathcal{S}$ will be at most
$(n\cdot\lceil\log_{\xi}M\rceil)$-dimensional. Furthermore, the at most
$\lceil\log_{\xi}M\rceil$ applications of Lemma 28 will induce a runtime of
complexity of
$O\left(\sum^{\lceil\log_{\xi}M\rceil-1}_{j=0}\frac{NM\cdot\min\\{N,M/\xi^{j}\\}}{\xi^{j}}\right)=O\left(\frac{\xi}{\xi-1}\cdot
MN^{2}\right).$ (49)
Finally, we note that an orthonormal basis for $\mathcal{S}$ can be computed
in $O\left(N\cdot n^{2}\log^{2}_{\xi}M\right)$-time via Gram–Schmidt. The
stated result follows.∎
###### Theorem 4.
Let $\xi\in(1,\infty)$, $P=\\{{\bf p}_{1},\dots,{\bf
p}_{M}\\}\subset\mathbbm{R}^{N}$ be symmetric, and $n\in\\{1,\dots,N\\}$.
Then, there is an $O\left(\frac{\xi}{\xi-1}\cdot MN^{2}+N\cdot
n^{2}\log^{2}_{\xi}M\right)$-time algorithm which outputs an at most
$(n\cdot\lceil\log_{\xi}M\rceil)$-dimensional subspace
$\mathcal{S}\subset\mathbbm{R}^{N}$ such that one has for an absolute constant
$C$, simultaneously for all $2<p<\infty$,
$d_{n}^{(p)}(P,{\mathcal{S}})\leq\left(1+C\lceil\log_{\xi}m\rceil^{1/p}\sqrt{\xi}\right)\cdot
d_{n}^{(p)}\left(P,\mathbbm{R}^{N}\right).$ (50)
Proof: Let $\mathcal{S}\subset\mathbbm{R}^{D}$ be an $\tilde{n}$-dimensional
subspace with $\tilde{n}\geq n$, and
$\mathcal{A}\in\Gamma_{n}(\mathbbm{R}^{D})$. We have that
$d_{n}^{(p)}(P,\mathcal{S})\leq\left(\sum_{{\bf p}_{j}\in P}\|{\bf
p}_{j}-\Pi_{\mathcal{S}}\Pi_{\mathcal{A}}{\bf
p}_{j}\|_{2}^{p}\right)^{1/p}\leq\left(\sum_{{\bf p}_{j}\in P}\|{\bf
p}_{j}-\Pi_{\mathcal{S}}{\bf p}_{j}\|_{2}^{p}\right)^{1/p}+\left(\sum_{{\bf
p}_{j}\in P}\|{\bf p}_{j}-\Pi_{\mathcal{A}}{\bf
p}_{j}\|^{p}_{2}\right)^{1/p}.$
The fact that this holds for all $\mathcal{A}\in\Gamma_{n}(\mathbbm{R}^{D})$
now again implies that
$d_{n}^{(p)}(P,\mathcal{S})\leq
d^{(p)}(P,\mathcal{S})+d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right).$ (51)
The subspace $\mathcal{S}$ is chosen in the same way as in the proof of
Theorem 45. That is, it is given as the union of $\lceil\log_{\xi}m\rceil$
recursively constructed subsets
$\mathcal{S}^{1},\dots,\mathcal{S}^{\lceil\log_{\xi}m\rceil}$. As both Lemma
28 and Lemma 33, the former of which motivates the construction of
$\mathcal{S}$, restrict $P$ to the same subset $P^{\prime}$, we can conclude
for the partition $P=\bigcup_{j=1}^{\lceil\log_{\xi}m\rceil}P_{i}$ of Theorem
45, that each $\mathcal{S}^{i}$ approximates $P_{i}$ also in the sense of
$d^{(p)}$. That is,
$d^{(p)}(P_{i},{\mathcal{S}}^{i})\leq C\sqrt{\xi}\cdot
d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right).$ (52)
Combining the contributions of the different $P_{i}$, we obtain using Lemma 33
$(d^{(p)}(P,{\mathcal{S}}))^{p}\leq\sum_{j=1}^{\lceil\log_{\xi}m\rceil}(d^{(p)}(P_{i},{\mathcal{S}}))^{p}\leq\sum_{j=1}^{\lceil\log_{\xi}m\rceil}(d^{(p)}(P_{i},{\mathcal{S}^{i}}))^{p}\leq
C^{p}\lceil\log_{\xi}m\rceil\xi^{p/2}\cdot\left(d^{(p)}_{n}\left(P,\mathbbm{R}^{N}\right)\right)^{p}.$
(53)
As the construction is the same, the runtime estimate of Theorem 45 carries
over. The stated result follows.∎
###### Remark 3.
Recalling Remark 1, we note that the runtime complexities quoted in both
Theorems 45 and 50 can be reduced by using faster randomized row-rank
approximation methods in Lemmas 28 and 33, respectively. Furthermore, we point
out that one can use the ideas from Remark 2 in order to guarantee a, e.g.,
$2\sqrt{\xi}\cdot d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$-accurate
approximation to $P$ with potentially fewer than $\lceil\log_{\xi}M\rceil$
applications of Lemma 28. This can be achieved by terminating the iterative
applications of Lemma 28 described in the proof of Theorem 45 once $\alpha$
from $\eqref{Defalpha}$ falls below $2$. Similarly, the iterative applications
of Lemma 33 described in the proof of Theorem 50 can be terminated without
seriously degrading accuracy as soon as
$\alpha:=d^{(p)}(P,\mathcal{S})/d^{(p)}(P^{\prime},\mathcal{S})$ falls below a
user prescribed threshold. Finally, it worth noting that the accuracy of
Theorem 45 (and Theorem 50) can be improved in practice by replacing
$P\setminus P^{\prime}$ with $\left(I-\Pi_{\mathcal{S}}\right)(P\setminus
P^{\prime})$ after each iteration of Lemma 28 (or Lemma 33). This allows
subsequent iterations to strictly improve on the progress made in previous
iterations.
## 4 A Fast Algorithm for $p=\infty$ Subspace Approximation
In this section we demonstrate that the dimensionality reduction results
developed above can be combined with computational techniques for computing
the John ellipsoid of a point set in order to produce a fast approximation
algorithm for the $p=\infty$ problem. The following result establishes the
speed and accuracy of this approach.
###### Theorem 5.
Let $P=\\{{\bf p}_{1},\dots,{\bf p}_{M}\\}\subset\mathbbm{R}^{N}$ be
symmetric, and $n\in\\{1,\dots,N\\}$. Then, one can calculate an
$\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$ with
$d^{(\infty)}\left(P,\mathcal{A}\right)\leq C\sqrt{n\cdot\log M}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$ (54)
in $O\left(MN^{2}+Mn^{2}\cdot\log^{2}M\cdot\log(n\log M)\right)$-time. Here
$C\in\mathbbm{R}^{+}$ is an absolute constant.
Proof: Choose $\epsilon\in(0,\infty)$, $\xi\in(1,\infty)$, and let
$m:=n\lceil\log_{\xi}M\rceil$. Use Theorem 45 to find
$\mathcal{S}\in\Pi_{\tilde{m}}\left(\mathbbm{R}^{N}\right)$ with
$n\leq\tilde{m}\leq m$, and let $B_{\mathcal{S}}$ be the associated
orthonormal basis of $\mathcal{S}$. Project $P$ onto $\mathcal{S}$ to obtain
$P^{\prime}:=\Pi_{\mathcal{S}}P\subset\mathbbm{R}^{N}$. We will also work with
$P^{\prime}$ expressed in terms of its $B_{\mathcal{S}}$ coordinates,
$P^{\prime\prime}\subset\mathbbm{R}^{\tilde{m}}$. Compute an ellipsoid
$\mathcal{E}:=\left\\{{\bf x}~{}\big{|}~{}{\bf x}^{T}Q{\bf x}\leq
1\right\\}\subset\mathbbm{R}^{\tilde{m}}$ such that $\mathcal{E}\subseteq{\rm
CH}\left(P^{\prime\prime}\right)\subseteq\sqrt{(1+\epsilon)m}\cdot\mathcal{E}$
in $O\left(Mm^{2}(\log m+1/\epsilon)\right)$-time [17]. Finally, let
$\mathcal{A^{\prime}_{E}}\subset\mathbbm{R}^{\tilde{m}}$ be the subspace
spanned by the $n$ eigenvectors of $Q$ associated with
$\sigma_{\tilde{m}}(Q),\dots,\sigma_{\tilde{m}-n+1}(Q)$, and
$\mathcal{A_{E}}\subset\mathcal{S}\ \subset\mathbbm{R}^{N}$ be
$\mathcal{A^{\prime}_{E}}$ re-expressed as an $n$-dimensional subspace of the
span of $B_{\mathcal{S}}$.
Choosing $\mathcal{A}^{\prime}_{\rm
opt}\in\Gamma_{n}\left(\mathbbm{R}^{\tilde{m}}\right)$ to satisfy
$d^{(\infty)}\left({\rm
CH}\left(P^{\prime\prime}\right),\mathcal{A}^{\prime}_{\rm
opt}\right)=d^{(\infty)}_{n}\left({\rm
CH}\left(P^{\prime\prime}\right),\mathbbm{R}^{\tilde{m}}\right)=d^{(\infty)}_{n}\left(P^{\prime\prime},\mathbbm{R}^{\tilde{m}}\right)$,
one can see that
$\displaystyle d^{(\infty)}\left(P^{\prime},\mathcal{A_{E}}\right)$
$\displaystyle=d^{(\infty)}\left(P^{\prime\prime},\mathcal{A^{\prime}_{E}}\right)\leq
d^{(\infty)}\left({\rm
CH}\left(P^{\prime\prime}\right),\mathcal{A^{\prime}_{E}}\right)\leq
d^{(\infty)}\left(\sqrt{(1+\epsilon)m}\cdot\mathcal{E},\mathcal{A^{\prime}_{E}}\right)$
(55) $\displaystyle=\sqrt{(1+\epsilon)m}\cdot
d^{(\infty)}\left(\mathcal{E},\mathcal{A^{\prime}_{E}}\right)\leq\sqrt{(1+\epsilon)m}\cdot
d^{(\infty)}\left(\mathcal{E},\mathcal{A}^{\prime}_{\rm opt}\right)$ (56)
$\displaystyle=\sqrt{(1+\epsilon)m}\cdot d^{(\infty)}\left({\rm
CH}\left(P^{\prime\prime}\right),\mathcal{A}^{\prime}_{\rm
opt}\right)=\sqrt{(1+\epsilon)m}\cdot
d^{(\infty)}_{n}\left(P^{\prime\prime},\mathbbm{R}^{\tilde{m}}\right).$ (57)
where the inequality in (55) follows from parts $(5)$ and $(6)$ of Lemma 1.
Finally, after noting that
$d^{(\infty)}_{n}\left(P^{\prime\prime},\mathbbm{R}^{\tilde{m}}\right)=d^{(\infty)}_{n}\left(P^{\prime},\mathcal{S}\right)$,
we can see that (55)-(57) imply that
$d^{(\infty)}\left(P^{\prime},\mathcal{A_{E}}\right)\leq\sqrt{(1+\epsilon)m}\cdot
d^{(\infty)}_{n}\left(P^{\prime},\mathcal{S}\right).$ (58)
Choose any $\mathcal{A}\in\Gamma_{n}\left(\mathcal{S}\right)$, thus ensuring
that $\Pi_{\mathcal{A}}\Pi_{\mathcal{S}}=\Pi_{\mathcal{A}}$, and then let
${\bf y}\in P$ be such that
$\left\|\Pi_{\mathcal{S}}{\bf y}-\Pi_{\mathcal{A}}{\bf
y}\right\|_{2}=\left\|\Pi_{\mathcal{S}}{\bf
y}-\Pi_{\mathcal{A}}\Pi_{\mathcal{S}}{\bf
y}\right\|_{2}=d^{(\infty)}\left(P^{\prime},\mathcal{A}\right)\geq
d^{(\infty)}_{n}\left(P^{\prime},\mathcal{S}\right).$ (59)
Choose any ${\bf x}\in P$. Combining (58) and (59), we can see that
$\left\|\Pi_{\mathcal{S}}{\bf x}-\Pi_{\mathcal{A_{E}}}\Pi_{\mathcal{S}}{\bf
x}\right\|^{2}_{2}=\left\|\Pi_{\mathcal{S}}{\bf x}-\Pi_{\mathcal{A_{E}}}{\bf
x}\right\|^{2}_{2}\leq(1+\epsilon)m\cdot\left\|\Pi_{\mathcal{S}}{\bf
y}-\Pi_{\mathcal{A}}{\bf y}\right\|_{2}^{2}$ (60)
which implies that
$\left\|\Pi_{\mathcal{S}}{\bf x}-\Pi_{\mathcal{A_{E}}}{\bf
x}\right\|^{2}_{2}+\left\|\Pi_{\mathcal{S}^{\perp}}{\bf
x}\right\|_{2}^{2}\leq(1+\epsilon)m\cdot\left(\left\|\Pi{\bf
y}-\Pi_{\mathcal{A}}{\bf
y}\right\|_{2}^{2}+\left\|\Pi_{\mathcal{S}^{\perp}}{\bf
y}\right\|_{2}^{2}\right)+\left(d^{(\infty)}\left(P,\mathcal{S}\right)\right)^{2}.$
(61)
Here we used that $\left\|\Pi_{\mathcal{S}^{\perp}}{\bf x}\right\|_{2}=\|{\bf
x}-\Pi_{\mathcal{S}}{\bf x}\|_{2}\leq d^{(\infty)}\left(P,\mathcal{S}\right)$.
Thus, again for arbitrary $\mathcal{A}\in\Gamma_{n}\left(\mathcal{S}\right)$,
$\displaystyle\left\|{\bf x}-\Pi_{\mathcal{A_{E}}}{\bf x}\right\|_{2}$
$\displaystyle\leq\sqrt{(1+\epsilon)m\cdot\left\|{\bf y}-\Pi_{\mathcal{A}}{\bf
y}\right\|_{2}^{2}+\left(d^{(\infty)}\left(P,\mathcal{S}\right)\right)^{2}}$
(62)
$\displaystyle\leq\sqrt{(1+\epsilon)m\cdot\left(d^{(\infty)}\left(P,\mathcal{A}\right)\right)^{2}+\left(d^{(\infty)}\left(P,\mathcal{S}\right)\right)^{2}}.$
(63)
Noting that (63) holds for all ${\bf x}\in P$ and
$\mathcal{A}\in\Gamma_{n}\left(\mathcal{S}\right)$, and recalling that
$\mathcal{S}$ was provided by Theorem 45, we obtain
$d^{(\infty)}\left(P,\mathcal{A_{E}}\right)\leq\sqrt{(1+\epsilon)m\cdot\big{(}d^{(\infty)}_{n}\left(P,\mathcal{S}\right)\big{)}^{2}+\xi\big{(}d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)\big{)}^{2}}.$
(64)
Appealing to the statement of Theorem 45 one last time yields (54).
The runtime complexity can be accounted for as follows: Computing
$\mathcal{S}$ via Theorem 45 can be accomplished in
$O\left(\frac{\xi}{\xi-1}\cdot MN^{2}+N\cdot
n^{2}\log^{2}_{\xi}M\right)$-time. Computing $P^{\prime\prime}$ from $P$ can
be done in $O(MN\cdot n\log_{\xi}M)$-time, after which
$\mathcal{A^{\prime}_{E}}$ can be found in $O\left(M\cdot
n^{2}\log^{2}_{\xi}M\cdot\left(\log(n\log_{\xi}M)+1/\epsilon\right)\right)$-time
via [17]. Finally, a basis for $\mathcal{A_{E}}$ can be computed in $O(N\cdot
n^{2}\log^{2}_{\xi}M)$-time once $\mathcal{A^{\prime}_{E}}$ is known. The
stated runtime complexity follows.∎
###### Remark 4.
The more precise accuracy bound in terms of the parameters $\epsilon$ and
$\xi$ derived in the proof of the theorem predicts that one can find a set
$\mathcal{A}$ that satisfies
$d^{(\infty)}\left(P,\mathcal{A}\right)\leq\Big{(}\sqrt{(1+\epsilon)\big{(}1+\sqrt{\xi}\big{)}^{2}n\lceil\log_{\xi}M\rceil+\xi}\Big{)}\cdot
d^{(\infty)}_{n}\left(P,\mathbbm{R}^{N}\right)$ (65)
in $O\big{(}\frac{\xi}{\xi-1}\cdot
MN^{2}+Mn^{2}\cdot\log^{2}_{\xi}M\cdot\left(\log(n\log_{\xi}M)+1/\epsilon\right)\big{)}$-time.
Choosing $\epsilon$ small and $\xi$ to minimize the accuracy bound to find
that one can achieve $C<10$. Finally, we note that the runtime complexity
quoted in Theorem 5 can be reduced, along the lines of Remark 1, by using a
fast randomized least-squares method instead of a deterministic SVD method.
## Acknowledgements
The authors would like to thank Kasturi Varadarajan for the helpful comments
and advice he kindly provided to us during the preparation of this manuscript.
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## Appendix A Proof of Lemma 1
We present the proof of each part below:
1. 1.
This follows directly from the fact that $d^{(\infty)}(P-{\bf
x},\mathcal{A})=d^{(\infty)}\left(P,\mathcal{A}-\Pi_{\mathcal{S}_{\mathcal{A}}^{\perp}}{\bf
x}\right)$ for all $\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$ and
${\bf x}\in\mathbbm{R}^{N}$.
2. 2.
Let $\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$ be such that
$d^{(\infty)}(\bar{P},\mathcal{A})=d^{(\infty)}_{n}(\bar{P},\mathbbm{R}^{N})$.
Suppose ${\bf a}_{\mathcal{A}}$ is nonzero. Partition $\bar{P}$ into three
parts:
1. (a)
$\bar{P}_{1}:=\left\\{{\bf p}\in\bar{P}~{}\big{|}~{}\langle{\bf p},{\bf
a}_{\mathcal{A}}\rangle=0\right\\}$
2. (b)
$\bar{P}_{2}:=\left\\{{\bf p}\in\bar{P}~{}\big{|}~{}\langle{\bf p},{\bf
a}_{\mathcal{A}}\rangle>0\right\\}$
3. (c)
$\bar{P}_{3}:=\left\\{{\bf p}\in\bar{P}~{}\big{|}~{}\langle{\bf p},{\bf
a}_{\mathcal{A}}\rangle<0\right\\}$
If ${\bf p}\in P_{1}$ then $\|{\bf p}-\Pi_{\mathcal{A}}{\bf
p}\|_{2}^{2}=\|{\bf p}-\Pi_{\mathcal{S}_{\mathcal{A}}}{\bf p}\|^{2}_{2}+\|{\bf
a}_{\mathcal{A}}\|^{2}_{2}$. This is minimized for all ${\bf p}\in P_{1}$ when
$\|{\bf a}_{\mathcal{A}}\|_{2}=0$. Next, note that ${\bf p}\in P_{3}$ if and
only if $-{\bf p}\in P_{2}$, and that ${\bf p}\in P_{3}$ means $\|{\bf
p}-\Pi_{\mathcal{A}}{\bf p}\|_{2}>\|(-{\bf p})-\Pi_{\mathcal{A}}(-{\bf
p})\|_{2}$. Thus, we can decrease $d^{(\infty)}(\bar{P},\mathcal{A})$ by
making ${\bf a}_{\mathcal{A}}$ shorter (a contradiction).
3. 3.
Let $\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$ be such that
$d^{(\infty)}(P,\mathcal{A})=d^{(\infty)}_{n}(P,\mathbbm{R}^{N})$. We have
that
$\displaystyle\|\bar{\bf p}-{\bf
p}_{j}-\Pi_{\mathcal{S}_{\mathcal{A}}}\left(\bar{\bf p}-{\bf
p}_{j}\right)\|_{2}$ $\displaystyle=\|{\bf p}_{j}-\bar{\bf
p}-\Pi_{\mathcal{S}_{\mathcal{A}}}\left({\bf p}_{j}-\bar{\bf
p}\right)\|_{2}=\left\|{\bf p}_{j}-\Pi_{\mathcal{S}_{\mathcal{A}}}{\bf
p}_{j}-\Pi_{\mathcal{S}^{\perp}_{\mathcal{A}}}\bar{\bf p}\right\|_{2}$ (66)
$\displaystyle\leq\|{\bf p}_{j}-\Pi_{\mathcal{A}}{\bf p}_{j}\|_{2}+\|\bar{\bf
p}-\Pi_{\mathcal{A}}\bar{\bf p}\|_{2}.$ (67)
Noting that $\|\bar{\bf p}-\Pi_{\mathcal{A}}\bar{\bf p}\|_{2}\leq
d^{(\infty)}(P,\mathcal{A})$ – see part five below for an analogous
calculation – concludes the proof.
4. 4.
This follows directly from the fact that $d^{(\infty)}(B,\mathcal{A})\leq
d^{(\infty)}(C,\mathcal{A})$ for all
$\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$.
5. 5.
Part four implies $d^{(\infty)}_{n}(P,\mathbbm{R}^{N})\leq
d^{(\infty)}_{n}({\rm CH}(P),\mathbbm{R}^{N})$ since $P\subseteq{\rm CH}(P)$.
To obtain the other inequality, we recall that every ${\bf x}\in{\rm CH}(P)$
has $\alpha_{j}\in[0,1]$, $j=1,\dots,M$, such that
${\bf x}=\sum^{M}_{j=1}\alpha_{j}\cdot{\bf p}_{j},$ (68)
and
$\sum^{M}_{j=1}\alpha_{j}=1.$ (69)
Hence, we can see that
$\|{\bf x}-\Pi_{\mathcal{A}}{\bf
x}\|_{2}=\left\|\sum^{M}_{j=1}\alpha_{j}\cdot\left({\bf
p}_{j}-\Pi_{\mathcal{S}_{\mathcal{A}}}{\bf p}_{j}-{\bf
a}_{\mathcal{A}}\right)\right\|_{2}~{}\leq~{}\sum^{M}_{j=1}\alpha_{j}\cdot\|{\bf
p}_{j}-\Pi_{\mathcal{A}}{\bf
p}_{j}\|_{2}~{}\leq~{}d^{(\infty)}(P,\mathcal{A})$ (70)
holds for all ${\bf x}\in{\rm CH}(P)$, and
$\mathcal{A}\in\Gamma_{n}\left(\mathbbm{R}^{N}\right)$. It now follows that
$d^{(\infty)}_{n}({\rm CH}(P),\mathbbm{R}^{N})\leq
d^{(\infty)}_{n}(P,\mathbbm{R}^{N})$.
6. 6.
Part two tells us that there will be an optimal subspace, since $\mathcal{E}$
is symmetric. Thus, standard results concerning the $n$-widths of ellipsoids
apply (see, e.g., [12, 13]).
|
arxiv-papers
| 2013-12-05T02:30:19 |
2024-09-04T02:49:54.916570
|
{
"license": "Public Domain",
"authors": "Mark Iwen and Felix Krahmer",
"submitter": "Mark Iwen",
"url": "https://arxiv.org/abs/1312.1413"
}
|
1312.1436
|
# Security flaw of counterfactual quantum cryptography in practical setting
Yan-Bing Li1,2,3, Qiao-yan Wen1, Zi-Chen Li2 1State Key Laboratory of
Networking and Switching Technology, Beijing University of Posts and
Telecommunications, Beijing, 100876, China
2Beijing Electronic Science and Technology Institute,Beijing 100070,China
3Department of Electrical Engineering and Computer Science, Northwestern
University, Evanston, Illinois 60208, USA [email protected]
###### Abstract
Recently, counterfactual quantum cryptography proposed by T. G. Noh [Phys.
Rev. Lett. 103, 230501 (2009)] becomes an interesting direction in quantum
cryptography, and has been realized by some researchers (such as Y. Liu et
al’s [Phys. Rev. Lett. 109, 030501 (2012)]). However, we find out that it is
insecure in practical high lossy channel setting. We analyze the secret key
rates in lossy channel under a polarization-splitting-measurement attack.
Analysis indicates that the protocol is insecure when the loss rate of the
one-way channel exceeds $50\%$.
## 1 Introduction
Quantum cryptography allows higher security than classical cryptography as it
is based on the laws of physics instead of the difficulty of solving
mathematical problems. Quantum key distribution (QKD)[1]-[3], which is to
provide secure means of distributing secret keys between the sender (Alice)
and the receiver (Bob), is often used to represent quantum cryptography as the
primary most important part. Now it has been researched and developed in both
theoretics and experiments. In theoretic, QKD could offer unconditional
security guaranteed by the laws of physics[4]. But due to the limitations of
real-life setting[5], such as the imperfect source, imperfect detector, loss
and noise in channel, practical QKD has security loopholes and has suffered
some attacks, such as photon number splitting (PNS) attack[6], Trojan-horse
attack[7], faked state attack[8]. On the other hand, some achievements, such
as decoy states mothod[9], measurement-device-independent QKD (MDI-QKD)
scheme[10] were made to let practical QKD be more secure.
Recently, counterfactual quantum cryptography proposed by Noh[11] has
attracted a lot of research, which allows participants to share secret
information using counterfactual quantum phenomena. It is believed that the
security is based on that quantum particles carrying secret information are
seemingly not transmitted through quantum channels. So far, some security
proof[12], improvements[13] and experimental demonstrations[14]-[18] of
counterfactual quantum cryptography have been proposed.
However, we find out the counterfactual quantum cryptography[11] is insecure
in practical long distance communication. The secret key rate will be $0$
under a polarization-splitting-measurement attack when the loss rate of the
one-way channel is no less than $50\%$. Namely, the eavesdropper (commonly
called Eve) can obtain all the secret information. Nevertheless, the cheat is
unknowable to Alice and Bob because its effect just likes a reasonable loss in
practical channel.
This paper is organized as follows. Sec. II reviews the counterfactual QKD
proposed in Ref.[11]. In Sec. III, We analyze the error rate of raw key in
lossy channel. A polarization-splitting-measurement attack is given in Sec.
IV. In Sec. V, we analyze the secret key rate under the attack. Finally, a
short conclusion are provided in Section VI.
## 2 Counterfactual QKD
Figure 1: (color online). The schematic of counterfactual QKD. For simpleness,
we have made some equivalent adjustments on the original one. Whole space is
divided by dotted line into three sub-spaces, Alice’s site, Bob’s site and
public space (i.e., Eve’s space). Alice sends the $i$th single-photon in state
$|V\rangle$ or $|H\rangle$, representing bit $0$ or $1$, to beam splitter
$BS_{1}$. Then the split pulses are transmitted into two paths $a$ which is
always in Alice’s site, and $b$ which is in public space toward Bob’s site.
Bob randomly uses $|V\rangle$ (representing $0$) or $|H\rangle$ (representing
$1$) $PBS$ to block the pulse in path $b$ when his bit is identical to
Alice’s, or let it pass when his bit is differ to Alice’s. When their bits are
different, detectors $D_{2}$ should always click since the interferometry
happens in $BS_{2}$. Else when their bits are same, the detectors $D_{1}$ and
$D_{2}$ and $D_{3}$ will click with some probabilities since interaction-free
measurement happens. Additional, it is assumed that all of $D_{1}$, $D_{2}$
and $D_{3}$ could detect the state’s polarization $|V\rangle$ or $|H\rangle$.
All the $D_{2}$’s and $D_{3}$’s clicks and a part of $D_{1}$’s clicks are used
to detect eavesdropping, and the rest of $D_{1}$’s clicks with correct
polarization are used as the raw key.
Fig.1 is the schematic of counterfactual QKD[11]. For simpleness, we have made
some equivalent adjustments on it. Alice triggers the single-photon source
$S$, which emits a short optical pulse containing a single photon at a certain
time interval. She randomly chooses the photon polarization in $|V\rangle$
representing the bit value 0 , or $|H\rangle$ representing the bit value 1 .
Thereafter, the photon enters a beam splitter $BS_{1}$ and is split to two
wave pulses $s_{a}$ and $s_{b}$. Then the system state evolves into one of the
following states:
$\displaystyle|\phi_{0}\rangle=\sqrt{R}|0\rangle_{a}|V\rangle_{b}+\sqrt{T}|V\rangle_{a}|0\rangle_{b},$
(1)
$\displaystyle|\phi_{1}\rangle=\sqrt{R}|0\rangle_{a}|H\rangle_{b}+\sqrt{T}|H\rangle_{a}|0\rangle_{b}.$
(2)
where subscripts $a$ and $b$ represent the path towards Alice’s site and the
path toward Bob’s site, respectively, and $|0\rangle$ denotes the vacuum state
in the path $a$ or $b$. $R$ and $T=1-R$ are the reflectivity and
transmissivity of both $BS_{1}$ and $BS_{2}$, respectively.
Bob has two polarizing beam splitter (PBS), $|V\rangle$ PBS (representing the
bit value 0 ) and $|H\rangle$ PBS (representing the bit value 1 ), where
$|V\rangle$ (or $|H\rangle$) PBS means it addresses the state $|V\rangle$ (or
$|H\rangle$) towards detector $D_{3}$, while the state $|H\rangle$ (or
$|V\rangle$) is sent towards the beam splitter $BS_{2}$. He randomly chooses
to use the $|V\rangle$ PBS or $|H\rangle$ PBS as his device $PBS_{1}$.
If Alice and Bob’s bits are different, the pulse on path $b$ will be reflected
by Bob and combined again at Alice’s device $BS_{2}$. The case just likes an
interferometry with a single photon. In the ideal setting, detector $D_{2}$
will click with certainty. Else if Alice and Bob’s bits are identical, the
path $b$ will be blocked by Bob’s $PBS_{1}$. The case just likes an
interaction-free measurement with a single photon. Here the state
$|\phi_{0}\rangle$ will be collapsed to $|0\rangle_{a}|V\rangle_{b}$ or
$|V\rangle_{a}|0\rangle_{b}$, $|\phi_{1}\rangle$ will be collapsed to
$|0\rangle_{a}|H\rangle_{b}$ or $|H\rangle_{a}|0\rangle_{b}$. In the ideal
setting, detector $D_{1}$, $D_{2}$ and $D_{3}$ will click with probability
$RT$, $T^{2}$ and $R$, respectively.
So in the ideal setting, $D_{1}$ clicks means Alice’s source photon basis and
Bob’s PBS basis are identify. Then Alice and Bob have a certain amount of
identify bits, some of which could be used to check possible eavesdropping,
and the rest of with could be used as raw key bits. And some statistical laws
are between $D_{2}$’s, $D_{3}$’s clicks and Alice, Bob’s bits, which could be
used to check possible eavesdropping and judge error rate. Additional, it is
assumed that all the detectors $D_{1}$, $D_{2}$ and $D_{3}$ could detect the
state’s polarization $|V\rangle$ and $|H\rangle$, which also could be used to
check possible eavesdropping.
Since the raw key bits come from the events of $D_{1}$ clicks which means that
Bob’s measurement result is vacuum state, peoples feel that the participles
which carry secret information seemingly have not travelled between Alice and
Bob. In fact, its security is based on a type of noncloning principle for
orthogonal states[11]: if reduced density matrices of an available subsystem
are nonorthogonal and the other subsystem is not allowed access, it is
impossible to distinguish two orthogonal quantum states $|\phi_{0}\rangle$ and
$|\phi_{1}\rangle$ without disturbing them.
## 3 Users’ error raw key rate depend on lossy channel
Similar to other QKD schemes, the limitations of real-life setting will also
bring some troubles to counterfactual QKD. Specially, high lossy channel will
be a formidable difficulty to it. In this section, we will analyze the error
rate of users’ raw key pair, i.e., the different rate of Alice and Bob’s raw
key pair, depend on lossy channel. ( Besides the the loss in channel, some
other loss also appear in the source and the devices and some noise appear in
the source, channel and the devices, but they are not in the paper’s range.)
In the counterfactual QKD, the raw key rate is proportional to the single
detector click rate ( i.e., the rate of the case in which only one of
detectors $D_{1}$, $D_{2}$ and $D_{3}$ clicks), which will be affected by
source single photon rate $R_{single}$ and the loss rate. Symmetrically, we
suppose that both the loss rates in channel from Bob to Alice, and that from
Alice to Bob are $\eta$, i.e., the single photon will loss with probability
$\eta$ in one of the two channels. We recall that (1) Alice’s raw key bits are
generated from the source single photons’ bases. (2) Bob’s raw key bits are
generated from his $PBS_{1}$’s basis, i.e., state in which basis would be sent
from $PBS_{1}$ toward $D_{3}$.
Then we analyze the cases in which the raw key will be generated by Alice and
Bob. The analysis will be done on one single photon sent by Alice, which is in
state $|V\rangle$ or $|H\rangle$ with probability $1/2$ respectively. And we
suppose that the channel loss in the channel in public space and Bob’s site,
which is denoted as channel $c_{A\rightarrow B\rightarrow A}$ and could be
divided to two parts ( the channels from Alice to Bob $b_{A\rightarrow B}$ and
from Bob to Alice $b_{B\rightarrow A}$), is independent with state’s
polarization $|V\rangle$ and $|H\rangle$, i.e., all the possible wave pulse
will loss when channel loss happens. ( Note that the channel loss in
$c_{A\rightarrow B\rightarrow A}$ is different to the loss happens in Bob’s
PBS in which only one polarization is blocked. And also note that channel loss
in $c_{A\rightarrow B\rightarrow A}$ does not mean that the photon vanishes in
$c_{A\rightarrow B\rightarrow A}$ with certainly since it might go through
path $a$ probably.) Theses cases are divided by two elements (i) whether the
loss happens or not in the channel in public space and Bob’s site (then we
divide the channel $c_{A\rightarrow B\rightarrow A}$ to two parts, the
channels from Alice to Bob $b_{A\rightarrow B}$ and from Bob to Alice
$b_{B\rightarrow A}$) and (ii) if loss happens, whether it happens in the
channels $b_{A\rightarrow B}$ or $b_{B\rightarrow A}$.
_Case I._ Channel loss does not happen either on $b_{A\rightarrow B}$ or
$b_{B\rightarrow A}$.
This case just like the single photon has transmitted in a no-lossy channel.
Namely, there are not any blocks except the possible block from Bob’s PBS.
Case I will generate a raw key bit with probability $P_{1}=\frac{RT}{2}$ as
the reasons (1) Bob’s PBS blocks the special polarization with probability
$\frac{1}{2}$ (2) a raw key bit will be generated with probability $RT$ when
Bob’s PBS blocks the special polarization.
As both of the loss rates in the channels $b_{A\rightarrow B}$ and
$b_{B\rightarrow A}$ are $\eta$, channel loss will not happen on
$b_{A\rightarrow B}$ and $b_{B\rightarrow A}$ with probability $1-\eta$
respectively. So the Case I, channel loss does not happen either on
$b_{A\rightarrow B}$ or $b_{B\rightarrow A}$, will occur with probability
$P_{I}=(1-\eta)^{2}$. The raw key rate comes from Case I is
$R_{raw_{1}}^{AB}=P_{I}\cdot P_{1}\cdot
R_{single}=(1-\eta)^{2}\cdot\frac{RT}{2}\cdot R_{single}.$ (3)
Alice and Bob’s raw key are identify in this case.
_Case II._ Channel loss happens in the channel $b_{A\rightarrow B}$,
regardless of whether channel loss happens in the channel $b_{B\rightarrow A}$
or not.
When channel loss happened in $b_{A\rightarrow B}$, no wave pulse will pass
through $b_{B\rightarrow A}$, so we combine the cases that (II-1) channel loss
happens both in the channels $b_{A\rightarrow B}$ and $b_{B\rightarrow A}$
(II-2) channel loss only happens in the channel $b_{A\rightarrow B}$, not in
the channel $b_{B\rightarrow A}$ to Case II. Case II will generate an
additional raw key bit with probability $P_{2}=RT$ as following analysis.
Without loss of generality, we suppose the single photon Alice sent is
$|V\rangle$. After $BS_{1}$, the state could be described as Eq.(1a). When it
comes into Bob’s site, the state evolves to $|V\rangle_{a}|0\rangle_{b}$ with
probability $T$, or $|0\rangle_{a}|0\rangle_{b}$ with probability $R$ as the
possible pulse wave $|V\rangle_{b}$ lost in the channel $b_{A\rightarrow B}$.
The state $|0\rangle_{a}|0\rangle_{b}$ will not lead to any clicks, so no raw
key will be generated. But as the state $|V\rangle_{a}|0\rangle_{b}$, the
photon in path $a$ will fire detector $D_{1}$ and let Alice generate a raw key
bit with probability $R$, fire detector $D_{2}$ with probability $T$. After
Alice announced that $D_{1}$ clicked, Bob would generate an according raw key
bit based on his $PBS$’s basis, i.e., state in which basis is sent toward
$D_{3}$. So Case II will generate an additional raw key bit with probability
$T\cdot R$ as following analysis.
Case II will happen with probability $P_{II}=\eta$. Hence, with the loss in
channel from Alice to Bob, additional raw key bits are generated, the totally
rate of which is
$R_{raw_{2}}^{AB}=P_{II}\cdot P_{2}\cdot R_{single}=\eta\cdot RT\cdot
R_{single}.$ (4)
Since Bob has chosen his $PBS$’s basis randomly, his raw key bit will be
identify, and different with Alice’s with equal probability $1/2$. So both of
the correct and error raw key rates are $\frac{R_{raw_{2}}^{AB}}{2}$.
_Case III._ Channel loss does not happen in the channel $b_{A\rightarrow B}$,
but happens in the channel $b_{B\rightarrow A}$. Namely, a complete block is
in the channel $b_{B\rightarrow A}$ except the possible block from Bob’s PBS.
Case III will generate a raw key bit with probability $P_{3}=RT$ as following
analysis.
We still suppose the single photon Alice sent is $|V\rangle$. If Bob’s $PBS$
basis is same with Alice’s basis, Bob’s $PBS$ will send possible wave pulse
$|V\rangle_{b}$ toward $D_{3}$. On one hand, the system state evolves to
$|0\rangle_{a}|V\rangle_{b}$ with probability $R$, which means that the photon
went through path $b$, and it will be destroyed by detector $D_{3}$. So no
pulse wave will transmit from Bob to Alice. On the other hand, the system
state evolves to $|V\rangle_{a}|0\rangle_{b}$ with probability $T$, which
means that the photon went through path $a$, then it will fire detectors
$D_{1}$ and $D_{2}$ with probabilities $R$ and $T$ respectively. Bob will
generate a raw key bit which is identify with Alice’s after she announces that
$D_{1}$ clicked, whose probability is $T\cdot R$.
Else if Bob’s $PBS$ basis is different with Alice’s basis, Bob’s $PBS$ would
pass possible wave pulse $|V\rangle_{b}$, and send it back to Alice. After it
lost in the channel from Bob to Alice, the system state evolves to
$|0\rangle_{a}|0\rangle_{b}$ (with probability $R$) which means that it is
destroyed by the lossy channel, or $|V\rangle_{a}|0\rangle_{b}$ (with
probability $T$) which means that it will fire $D_{1}$ or $D_{2}$ with
probabilities $R$ and $T$, respectively. Bob will generate a raw key bit which
is identify with Alice’s after she announces that $D_{1}$ clicked, whose
probability is $T\cdot R$.
So regardless Bob’s $PBS$ basis is $|V\rangle$ or $|H\rangle$, this case will
generate a raw key bit with probability $RT$. But Alice’s and Bob’s bits are
same and different with equal probability $\frac{1}{2}$.
The case will happen with probability $P_{III}=(1-\eta)\cdot\eta$ as channel
loss does not happen in the channel $b_{A\rightarrow B}$ with probability
$1-\eta$, happens in the channel $b_{B\rightarrow A}$ with probability $\eta$.
Hence, with the loss in channel from Bob to Alice, additional raw key bits is
generated, the totally rate of which is
$R_{raw_{3}}^{AB}=P_{III}\cdot P_{3}\cdot R_{single}=(1-\eta)\cdot\eta\cdot
RT\cdot R_{single}.$ (5)
Both of the same and different raw key rates are $\frac{R_{raw_{3}}}{2}$.
All in all, the raw key rate is
$\begin{array}[]{ll}R_{raw}^{AB}=R_{raw_{1}}^{AB}+R_{raw_{2}}^{AB}+R_{raw_{3}}^{AB}\\\
\hskip 25.60747pt=\frac{1+2\eta-\eta^{2}}{2}\cdot TR\cdot
R_{single}.\end{array}$ (6)
The probability of that Alice’s and Bob’s raw keys in a same order are
identify is
$\begin{array}[]{ll}P_{raw}^{AB\\_same}=\frac{R_{raw_{1}}^{AB}+\frac{R_{raw_{2}}^{AB}}{2}+\frac{R_{raw_{3}}^{AB}}{2}}{R_{raw}}\\\
\hskip 45.5244pt=\frac{1}{1+2\eta-\eta^{2}},\end{array}$ (7)
the probability of that they are different is
$\begin{array}[]{ll}P_{raw}^{AB\\_diff}=\frac{\frac{R_{raw_{2}}^{AB}}{2}+\frac{R_{raw_{3}}^{AB}}{2}}{R_{raw}}\\\
\hskip 42.67912pt=\frac{2\eta-\eta^{2}}{1+2\eta-\eta^{2}}.\end{array}$ (8)
Namely, in users’ raw key pair, the error rate is $P_{raw}^{AB\\_diff}$ which
should be correct by some following classical postprocessing such as
information reconciliation.
The error in users’ raw key pairs will give a lot of chances to Eve to perform
some attacks. But to Eve, the first aim is that her attacks should not be
detected by the users. Following polarization-splitting-measurement attack is
one of the attacks.
## 4 Polarization-splitting-measurement attack
Usually, we assume that Eve has unlimited technological, which is only limited
by the laws of nature. So Eve could replace the lossy channel by a perfect
quantum channel, and use the excess power for her mischievous purposes. In
this section, we first give an attack method which can cheat the raw key bits
and be concealed by the practical lossy channel with loss rate $\frac{1}{2}$,
then give the special cheat strategies according to special loss rate range
for cheating maximal information.
In the _attack method_ , polarization-splitting and measurement will be used
to cheat secret information from channel $b_{A\rightarrow B}$ (shown in
fig.2). Eve first replaces the lossy channel $b_{A\rightarrow B}$ by a perfect
quantum channel. She also has two polarizing beam splitters, $|V\rangle$ PBS
representing the bit value $0$ and $|H\rangle$ PBS representing the bit value
$1$. She randomly chooses the $|V\rangle$ or $|H\rangle$ PBS for the $i$th
order, and inserts it in front of Bob’s site.
If Eve’s $i$th bit is identical with Alice’s $i$th bit, the detector $D_{4}$
will click with probability $R$, else if her $i$th bit is differ to Alice’s
$i$th bit, the detector $D_{4}$ will not click. In other words, the case that
$D_{4}$ clicks means that Eve’s bit is identical to Alice’s $i$th bit, and the
case that $D_{4}$ does not click means that Eve is uncertain about Alice’s
$i$th bit now. We recall that Alice and Bob’s raw key pair will product from
these uncertain bits corresponding to the case that detector $D_{4}$ does not
click. So Eve cannot make sure of the raw key bit. However, Eve could easily
extract the raw key bit according to what Alice and Bob will announce in the
following processing.
Figure 2: (color online). The schematic of polarization-splitting-measurement
attack on counterfactual QKD. Eve performs attack on channel $b_{A\rightarrow
B}$ in front of Bob’s site. Eve replaces the lossy channel $b_{A\rightarrow
B}$ by a perfect quantum channel. Then she randomly uses $|V\rangle$
(representing $0$) or $|H\rangle$ (representing $1$) $PBS$ to block the pulse
in path $b_{A\rightarrow B}$ when her bit is identical to Alice’s, or let it
pass when her bit is differ to Alice’s. What she dose just like a reasonable
loss in path $b_{A\rightarrow B}$.
Without loss of generality, we consider the case of Eve chooses $0$, i.e., she
inserts a $|V\rangle$ PBS. When Alice’s bit is $0$, two possible cases are
here. (1) When Eve’s detector $D_{4}$ clicked, the system state has been
collapsed to $|0\rangle_{a}|V\rangle_{b}$ which means Alice’s bit is identical
to Eve’s bit $0$, and vacuum state will go into Bob’s site. (2) Else when
Eve’s detector $D_{4}$ did not click, the system has been collapsed to
$|V\rangle_{a}|0\rangle_{b}$, and vacuum state still will go into Bob’s site.
Altogether, vacuum state (i.e., nothing) always will go into Bob’s site when
Eve and Alice’s bits are same, which likes the pulse in path $b$ has lost
completely by the lossy channel.
On the other hand, when Alice’s bit is $1$, the pulse in path $b$ will pass
Eve’s $PBS_{2}$ completely, so the system state still is
$|\phi_{1}\rangle$($=\sqrt{R}|0\rangle_{a}|H\rangle_{b}+\sqrt{T}|H\rangle_{a}|0\rangle_{b}$)
after Eve’s devices. The case is same to that Eve has done nothing, liking the
ideal setting. In the point view of Alice and Bob, all the following processes
will just like the normal processes. When $D_{1}$ clicks, the corresponding
bit will be chosen as a raw key bit by Alice followed by announcing its order.
Then Eve can always make sure that Alice’s raw key bit is $1$. In other words,
when Eve and Alice’s bits are different, a raw key bit will be produced with
probability. And the probability will be revealed with Alice’s announcement.
Since the raw key bit is generated from the inverse of Eve’s $PBS_{2}$’s
basis, Eve can not only know the raw key bits, but also decide its value with
some probability.
Since Eve chooses bit $0$ or $1$ randomly, her bit will be same and different
with Alice’s bit with probability $1/2$ respectively. The complete loss will
happen when their bits are same, and the ideal setting will happen when their
bits are different. Totally, the cheat method likes a loss of rate
$\frac{1}{2}$ happens in the channel $b_{A\rightarrow B}$. The cheat method
could be used on every photon to cheat the secret information when
$\eta=\frac{1}{2}$ and will not be detected (the analysis will be given in the
following). To other value of $\eta$, more complex strategies should be
designed for optimal cheating.
We suppose the amount of Alice sent single photons is $n$. Using the above
attack method, Eve could simulate practical loss channel with loss rate
$0\leq\eta\leq 1$ and cheat raw key bits with following strategies.
_Cheat strategy (I)_ When $0\leq\eta<\frac{1}{2}$, Eve performs the attack
method on $2\eta\cdot n$ single photons randomly, and fills the raw key orders
which she has not attacked in with random bits.
_Cheat strategy (II)_ When $\frac{1}{2}\leq\eta\leq 1$, Eve performs the
attack method on $2(1-\eta)\cdot n$ single photons randomly, and blocks the
remaining $(2\eta-1)\cdot n$ single photons. After Alice announced in which
orders the remaining single photons have fired detector $D_{1}$, she fills
these raw key orders in with random bits.
Like the loss in practical channel, what Eve did has brought some errors to
the protocol (we will analyze the details in next section). For instance, some
$D_{1}$’s clicks happened not only when Alice and Bob’s bits were same, but
also when they were different as long as Eve blocked the channel. However,
since the error rate is same as that brought by practical lossy channel, it
will be judged as a legal case by the protocol’s detection process. The basis
reason is that, the system state under the above cheat strategies is same to
the system state transmitted from a practical channel. We will analyze it as
follows.
We suppose the photon Alice sent is $|V\rangle$. If Eve’s $PBS$ past wave
pulse $|V\rangle$ to Bob’s site, the density matrix of system state is
$\begin{array}[]{ll}\rho_{1}^{attack}=|\phi_{0}\rangle\langle\phi_{0}|,\end{array}$
(9)
when it comes into Bob’s site. If Eve’s $PBS$ blocked wave pulse $|V\rangle$,
the system state is a mixed state with density matrix
$\begin{array}[]{ll}\rho_{2}^{attack}=R|0\rangle_{a}|0\rangle_{b}\langle
0|_{b}\langle 0|_{a}+T|V\rangle_{a}|0\rangle_{b}\langle 0|_{b}\langle
V|_{a},\end{array}$ (10)
when it comes into Bob’s site.
So after the strategy (I), the system state is a mixed state with density
matrix
$\begin{array}[]{ll}\rho_{I}^{attack}=(1-2\eta)\cdot|\phi_{0}\rangle\langle\phi_{0}|+\eta\cdot\rho_{1}^{attack}+\eta\cdot\rho_{2}^{attack}\\\
\hskip
31.29802pt=(1-\eta)\cdot\rho_{1}^{attack}+\eta\cdot\rho_{2}^{attack},\end{array}$
(11)
where $0\leq\eta<\frac{1}{2}$. After the strategy (II), the system state is a
mixed state with density matrix
$\begin{array}[]{ll}\rho_{II}^{attack}=(1-\eta)\cdot\rho_{1}^{attack}+[(1-\eta)+(2\eta-1)]\cdot\rho_{2}^{attack}\\\
\hskip
31.29802pt=(1-\eta)\cdot\rho_{1}^{attack}+\eta\cdot\rho_{2}^{attack},\end{array}$
(12)
where $\frac{1}{2}\leq\eta\leq 1$.
Now we analyze the system state in practical lossy channel without the attack
strategies. If the wave pulse in channel $b_{A\rightarrow B}$ has not lost,
the density matrix of the system state is
$\begin{array}[]{ll}\rho_{1}^{loss}=|\phi_{0}\rangle\langle\phi_{0}|,\end{array}$
(13)
when it come into Bob’s site. If the wave pulse in channel $b_{A\rightarrow
B}$ has lost, the system state will be a mixed state with density matrix
$\begin{array}[]{ll}\rho_{2}^{loss}=R|0\rangle_{a}|0\rangle_{b}\langle
0|_{b}\langle 0|_{a}+T|V\rangle_{a}|0\rangle_{b}\langle 0|_{b}\langle
V|_{a}\end{array}$ (14)
when it come into Bob’s site.
Since the loss rate is $\eta$ on the practical lossy channel $b_{A\rightarrow
B}$, the general system state is a mixed state with density matrix
$\begin{array}[]{ll}\rho^{loss}=(1-\eta)\cdot\rho_{1}^{loss}+\eta\cdot\rho_{2}^{loss}\\\
\hskip
22.76219pt=(1-\eta)\cdot\rho_{1}^{attack}+\eta\cdot\rho_{2}^{attack},\end{array}$
(15)
when it goes into Bob’s site, which is same with $\rho_{I}^{attack}$ when
$0\leq\eta<\frac{1}{2}$, $\rho_{II}^{attack}$ when $\frac{1}{2}\leq\eta\leq
1$.
So the states are same either when the protocol suffers a lossy channel or
when it is under the cheat strategies. The conclusion is still tenable when
the photon Alice sent is $|H\rangle$. Consequently, Alice and Bob could not
distinguish between the practical lossy channel and the cheat strategies.
## 5 Secret key rate under the cheat strategies in lossy channel
In this section, we will analyze the protocol in lossy channel with the secret
key rate $R_{QKD}$[19, 20], a convenient and commonly used quantitate measure
of protocol security.
Secret key rate $R_{QKD}$ is the product of the raw key rate $R_{raw}$ and the
secret fraction $r_{\infty}$. The secret fraction represents the fraction of
secure bits that may be extracted from the raw key. Formally, we have
$R_{QKD}=R_{raw}\cdot r_{\infty}.$ (16)
The expression for the secret fraction extractable[19, 21] using one-way
classical postprocessing reads
$r_{\infty}=I(A;B)-\min(I_{EA},I_{EB}),$ (17)
where $I(A;B)$ is Alice and Bob’s mutual information,
$I_{EA}=\max_{Eve}I(A;E)$, $I_{EB}=\max_{Eve}I(B;E)$. Since Alice and Bob’s
each raw key pair is randomly in $\\{0,1\\}$, it should be $H(A)=H(B)=1$. We
also have $P(A=0,B=0)=P(A=1,B=1)=P_{raw}^{AB\\_same}/2$,
$P(A=0,B=1)=P(A=1,B=0)=P_{raw}^{AB\\_diff}/2$. Combined with Eqs.(6) and (7),
it should be that
$\begin{array}[]{ll}I(A;B)=H(A)+H(B)-H(A,B)\\\ \hskip
34.1433pt=1+1+\sum_{A\in\\{0,1\\},B\in\\{0,1\\}}p(A,B)\log p(A,B)\\\ \hskip
34.1433pt=2+2\cdot\frac{1}{2(1+2\eta-\eta^{2})}\log\frac{1}{2(1+2\eta-\eta^{2})}\\\
\hskip
45.5244pt+2\cdot\frac{2\eta-\eta^{2}}{2(1+2\eta-\eta^{2})}\log\frac{2\eta-\eta^{2}}{2(1+2\eta-\eta^{2})}.\end{array}$
(18)
Then we analyze the secret key rate under the cheat strategies (I) and (II)
respectively depend on the loss rate $\eta$ by calculating
$\min(I_{EA},I_{EB})$.
We recall that (1) Alice’s raw key bits are generated from the source single
photons’ bases. (2) Bob’s raw key bits are generated from his $PBS_{1}$’s
basis, i.e., state in which basis would be sent from $PBS_{1}$ toward $D_{3}$.
(3) Eve’s raw key bits are generated from the inverse of her $PBS_{2}$’s
basis, i.e., state in which basis would be sent from $PBS_{2}$ toward Bob’s
site. Now we analyze the cases in which the raw key will be cheated by Eve
when she cheats in the channel from Alice to Bob. And we still suppose the
single photon Alice sent is $|V\rangle$.
### 5.1 Secret key rate under the cheat strategy (I) in lossy channel
We first analyze the cases in _cheat strategy (I)_ , i.e., the strategy with
$0\leq\eta<\frac{1}{2}$. We recall _Cheat strategy (I)_ : When
$0\leq\eta<\frac{1}{2}$, Eve performs the attack method on $2\eta\cdot n$
single photons randomly, and fills the raw key orders which she has not
attacked in with random bits. We divide the cases with elements (i) whether
Eve performs the attack method or not and (ii) if Eve performs the attack
method, whether her $PBS$ basis is same with Alice’s basis or not.
_Cheated raw key I._ The cheated raw key when Eve does not perform the attack
method.
For $0\leq\eta<\frac{1}{2}$, Eve does not perform the attack method on
$(1-2\eta)\cdot n$ source single photons, in which raw key bits will be
generated as the _case I_ and _case III_ (shown in Sec.III). Due to that the
loss rate in the channel $b_{A\rightarrow B}$ is $\eta$, _case I_ will happen
with probability $(1-2\eta)\cdot(1-\eta)$, _case III_ will happen with
probability $(1-2\eta)\cdot\eta$. The totally rate of these raw key is
$\begin{array}[]{ll}R_{raw_{1}}^{E}=((1-2\eta)\cdot(1-\eta)\cdot
P_{1}+(1-2\eta)\cdot\eta\cdot P_{3})\cdot R_{single}\\\ \hskip
31.29802pt=\frac{1-\eta-2\eta^{2}}{2}\cdot RT\cdot R_{single}.\end{array}$
(19)
Eve will guess these raw key bits, so the correct probability is
$\frac{1}{2}$. So compared to Alice’s and Bob’s raw keys, both of Eve’s same
and different raw key rates in this case are
$\begin{array}[]{ll}R_{raw_{1}}^{EA\\_same}=R_{raw_{1}}^{EA\\_diff}=R_{raw_{1}}^{EB\\_same}=R_{raw_{1}}^{EB\\_diff}\\\
\hskip 48.36967pt=\frac{1-\eta-2\eta^{2}}{4}\cdot RT\cdot
R_{single}.\end{array}$ (20)
_Cheated raw key II._ The cheated raw key when Eve performs the attack method,
and her $PBS$ basis is same with Alice’s basis.
When Eve’s $PBS$ basis is same with Alice’s basis (namely, Eve’s $PBS$ will
send wave pulse $|V\rangle$ toward $D_{4}$), raw key bits will be generated as
the _case II_(shown in Sec.III). It will happen with probability $\eta$. So
the totally rate of these raw key is
$\begin{array}[]{ll}R_{raw_{2}}^{E}=\eta\cdot P_{2}\cdot R_{single}\\\ \hskip
31.29802pt=\eta\cdot RT\cdot R_{single}.\end{array}$ (21)
Since Eve always generates her raw key bit as the inverse of her $PBS_{2}$’s
basis, all her raw key bits are different to Alice’s, and different to Bob’s
with probability $\frac{1}{2}$. Compared to Alice’s raw key, Eve’s same and
different raw key rates are
$\begin{array}[]{ll}R_{raw_{2}}^{EA\\_same}=0,\\\
R_{raw_{2}}^{EA\\_diff}=\eta\cdot RT\cdot R_{single},\end{array}$ (22)
respectively. Compared to Bob’s raw key, Eve’s same and different raw key
rates are
$\begin{array}[]{ll}R_{raw_{2}}^{EB\\_same}=R_{raw_{2}}^{EB\\_diff}=\frac{\eta}{2}\cdot
RT\cdot R_{single},\end{array}$ (23)
_Cheated raw key III._ The cheated raw key when Eve performs the attack
method, and her $PBS$ basis is different with Alice’s basis.
When Eve’s $PBS$ basis is different with Alice’s basis, Eve’s $PBS$ will send
wave pulse $|V\rangle$ toward Bob’s site. It will happen with probability
$\frac{1}{2}\cdot 2\eta=\eta$. And raw key bits will be generated as the _case
I_ and _case III_. Due to that the loss rate in channel from Bob to Alice is
$\eta$, _case I_ will happen with probability $\eta\cdot(1-\eta)$, _case III_
will happen with probability $\eta\cdot\eta$. So the totally rate of these raw
key is
$\begin{array}[]{ll}R_{raw_{3}}^{E}=[\eta\cdot(1-\eta)\cdot
P_{1}+\eta\cdot\eta\cdot P_{3}]\cdot R_{single}\\\ \hskip
31.29802pt=\frac{\eta+\eta^{2}}{2}\cdot RT\cdot R_{single}.\end{array}$ (24)
Since Eve always generates her raw key bit as the inverse of her $PBS_{2}$’s
basis, all her raw key bits are identify with Alice’s. Compared to Alice’s raw
key, Eve’s same and different raw key rates are
$\begin{array}[]{ll}R_{raw_{3}}^{EA\\_same}=\frac{\eta+\eta^{2}}{2}\cdot
RT\cdot R_{single},\\\ R_{raw_{3}}^{EA\\_diff}=0.\end{array}$ (25)
Compared to Bob’s raw key, Eve’s same and different raw key rates are
$\begin{array}[]{ll}R_{raw_{3}}^{EB\\_same}=\frac{\eta\cdot(1-\eta)}{2}\cdot
RT\cdot R_{single},\\\ R_{raw_{3}}^{EB\\_diff}=\eta\cdot\eta\cdot RT\cdot
R_{single}.\end{array}$ (26)
All in all, the raw key rate Eve cheated is
$\begin{array}[]{ll}R_{raw}^{E}=R_{raw_{1}}^{E}+R_{raw_{2}}^{E}+R_{raw_{3}}^{E}\\\
\hskip 25.60747pt=\frac{1+2\eta-\eta^{2}}{2}RT\cdot R_{single},\end{array}$
(27)
which is same as users’ raw key rate. The probabilities of that Eve and
Alice’s raw key bits are same and different are
$\begin{array}[]{ll}P_{raw}^{EA\\_same}=\frac{P_{raw_{1}}^{EA\\_same}+P_{raw_{2}}^{EA\\_same}+P_{raw_{3}}^{EA\\_same}}{R_{raw}^{E}}\\\
\hskip 45.5244pt=\frac{1+\eta}{2(1+2\eta-\eta^{2})},\\\ \end{array}$ (28)
$\begin{array}[]{ll}P_{raw}^{EA\\_diff}=\frac{P_{raw_{1}}^{EA\\_diff}+P_{raw_{2}}^{EA\\_diff}+P_{raw_{3}}^{EA\\_diff}}{R_{raw}^{E}}\\\
\hskip 42.67912pt=\frac{1+3\eta-2\eta^{2}}{2(1+2\eta-\eta^{2})}.\end{array}$
(29)
The probabilities of that Eve and Bob’s raw key bits are same and different
are
$\begin{array}[]{ll}P_{raw}^{EB\\_same}=\frac{P_{raw_{1}}^{EB\\_same}+P_{raw_{2}}^{EB\\_same}+P_{raw_{3}}^{EB\\_same}}{R_{raw}^{E}}\\\
\hskip 45.5244pt=\frac{1+3\eta-4\eta^{2}}{2(1+2\eta-\eta^{2})},\\\
\end{array}$ (30)
and
$\begin{array}[]{ll}P_{raw}^{EB\\_diff}=\frac{P_{raw_{1}}^{EB\\_diff}+P_{raw_{2}}^{EB\\_diff}+P_{raw_{3}}^{EB\\_diff}}{R_{raw}^{E}}\\\
\hskip 42.67912pt=\frac{1+\eta+2\eta^{2}}{2(1+2\eta-\eta^{2})}.\end{array}$
(31)
. In fact, Eve’s error rate will not be larger than $50\%$ by using a simple
way[22].
Similar to the calculation of $I(A;B)$, combined with Eqs.(27-30) it should be
$\begin{array}[]{ll}I(E;A)^{i}=H(E)+H(A)-H(E,A)\\\ \hskip
39.83385pt=1+1+\sum_{E\in\\{0,1\\},A\in\\{0,1\\}}p(E,A)\log p(E,A)\\\ \hskip
39.83385pt=2+2\cdot\frac{1+\eta}{4(1+2\eta-\eta^{2})}\log\frac{1+\eta}{4(1+2\eta-\eta^{2})}\\\
\hskip
48.36967pt+2\cdot\frac{1+3\eta-2\eta^{2}}{4(1+2\eta-\eta^{2})}\log\frac{1+3\eta-2\eta^{2}}{4(1+2\eta-\eta^{2})},\end{array}$
(32)
and
$\begin{array}[]{ll}I(E;B)^{i}=H(E)+H(B)-H(E,B)\\\ \hskip
39.83385pt=1+1+\sum_{E\in\\{0,1\\},B\in\\{0,1\\}}p(E,B)\log p(E,B)\\\ \hskip
39.83385pt=2+2\cdot\frac{1+3\eta-4\eta^{2}}{4(1+2\eta-\eta^{2})}\log\frac{1+3\eta-4\eta^{2}}{4(1+2\eta-\eta^{2})}\\\
\hskip
48.36967pt+2\cdot\frac{1+\eta+2\eta^{2}}{4(1+2\eta-\eta^{2})}\log\frac{1+\eta+2\eta^{2}}{4(1+2\eta-\eta^{2})},\end{array}$
(33)
Then the secret fraction is
$\begin{array}[]{ll}r_{\infty}^{i}=I(A;B)-\min(I_{EA}^{i},I_{EB}^{i}),\end{array}$
(34)
where $0\leq\eta<\frac{1}{2}$.
For simpleness, we set $R=T=\frac{1}{2}$. Then secret key rate is
$\begin{array}[]{ll}R_{QKD}=R_{raw}\cdot r_{\infty}^{i}\\\ \hskip
31.29802pt=\frac{1+2\eta-\eta^{2}}{8}\cdot r_{\infty}^{i}\cdot
R_{single},\end{array}$ (35)
where $0\leq\eta<\frac{1}{2}$.
### 5.2 Secret key rate under the cheat strategy (II) in lossy channel
Now we analyze the cases in which the raw key will be cheated by Eve using
_cheat strategy (II)_ , i.e., the strategy with $\frac{1}{2}\leq\eta\leq 1$.
We recall _Cheat strategy (II)_ : When $\frac{1}{2}\leq\eta\leq 1$, Eve
performs the attack method on $2(1-\eta)\cdot n$ single photons randomly, and
blocks the remaining $(2\eta-1)\cdot n$ single photons. After Alice announced
in which orders the remaining single photons have fired detector $D_{1}$, she
fills these raw key orders in with random bits.
In the strategy, the attack is performed with probability $2(1-\eta)$
replacing the probability $2\eta$ in _cheat strategy (I)_. So the amount of
raw key rate generated by the attack is
$\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{E}+R_{raw_{3}}^{E})$.
In addition, Eve blocks the remaining $(2\eta-1)\cdot n$ wave pulses in the
channel $b_{A\rightarrow B}$ followed by guessing the possible raw key bits.
This just likes _case II_. It will generate raw key bits whose amount is
$(2\eta-1)\cdot P_{2}\cdot R_{single}$. And both of the probabilities of them
are same and different with Alice (and Bob’s) are $\frac{1}{2}$.
Hence, the raw key rate is
$\begin{array}[]{ll}R_{raw}^{E}=[\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{E}+R_{raw_{3}}^{E})+(2\eta-1)\cdot
P_{2}]\cdot R_{single}\\\ \hskip 25.60747pt=\frac{1+2\eta-\eta^{2}}{2}RT\cdot
R_{single},\end{array}$ (36)
which is same as users’ raw key rate. The probabilities of that Eve’s and
Alice’s raw key bits are same and different are
$\begin{array}[]{ll}P_{raw}^{EA\\_same}=\frac{\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{EA\\_same}+R_{raw_{3}}^{EA\\_same})+\frac{(2\eta-1)}{2}\cdot
P_{2}}{R_{raw}^{E}}\\\ \hskip
45.5244pt=\frac{2\eta-\eta^{2}}{1+2\eta-\eta^{2}},\\\ \end{array}$ (37)
and
$\begin{array}[]{ll}P_{raw}^{EA\\_diff}=\frac{\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{EA\\_diff}+R_{raw_{3}}^{EA\\_diff})+\frac{(2\eta-1)}{2}\cdot
P_{2}}{R_{raw}^{E}}\\\ \hskip
45.5244pt=\frac{1}{1+2\eta-\eta^{2}}.\end{array}$ (38)
The probabilities of that Eve’s and Bob’s raw key bits are same and different
are
$\begin{array}[]{ll}P_{raw}^{EB\\_same}=\frac{\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{EB\\_same}+R_{raw_{3}}^{EB\\_same})+\frac{(2\eta-1)}{2}\cdot
P_{2}}{R_{raw}^{E}}\\\ \hskip
45.5244pt=\frac{1-\eta+\eta^{2}}{1+2\eta-\eta^{2}},\end{array}$ (39)
and
$\begin{array}[]{ll}P_{raw}^{EB\\_diff}=\frac{\frac{1-\eta}{\eta}\cdot(R_{raw_{2}}^{EB\\_diff}+R_{raw_{3}}^{EB\\_diff})+\frac{(2\eta-1)}{2}\cdot
P_{2}}{R_{raw}^{E}}\\\ \hskip
45.5244pt=\frac{3\eta-2\eta^{2}}{1+2\eta-\eta^{2}}.\end{array}$ (40)
Then we have
$\begin{array}[]{ll}I(E;A)^{ii}=H(E)+H(A)-H(E,A)\\\ \hskip
42.67912pt=1+1+\sum_{E\in\\{0,1\\},A\in\\{0,1\\}}p(E,A)\log p(E,A)\\\ \hskip
42.67912pt=2+2\cdot\frac{2\eta-\eta^{2}}{2(1+2\eta-\eta^{2})}\log\frac{2\eta-\eta^{2}}{2(1+2\eta-\eta^{2})}\\\
\hskip
51.21495pt+2\cdot\frac{1}{2(1+2\eta-\eta^{2})}\log\frac{1}{2(1+2\eta-\eta^{2})},\end{array}$
(41)
and
$\begin{array}[]{ll}I(E;B)^{ii}=H(E)+H(B)-H(E,B)\\\ \hskip
42.67912pt=1+1+\sum_{E\in\\{0,1\\},B\in\\{0,1\\}}p(E,B)\log p(E,B)\\\ \hskip
42.67912pt=2+2\cdot\frac{1-\eta+\eta^{2}}{2(1+2\eta-\eta^{2})}\log\frac{1-\eta+\eta^{2}}{2(1+2\eta-\eta^{2})}\\\
\hskip
51.21495pt+2\cdot\frac{3\eta-2\eta^{2}}{2(1+2\eta-\eta^{2})}\log\frac{3\eta-2\eta^{2}}{2(1+2\eta-\eta^{2})},\end{array}$
(42)
Then the secret fraction is
$\begin{array}[]{ll}r_{\infty}^{ii}=I(A;B)-\min(I_{EA}^{ii},I_{EB}^{ii}),\end{array}$
(43)
where $\frac{1}{2}\leq\eta\leq 1$.
For simpleness, we set $R=T=\frac{1}{2}$. Then secret key rate is
$\begin{array}[]{ll}R_{QKD}=R_{raw}\cdot r_{\infty}^{ii}\\\ \hskip
31.29802pt=\frac{1+2\eta-\eta^{2}}{8}\cdot r_{\infty}^{ii}\cdot
R_{single},\end{array}$ (44)
where $\frac{1}{2}\leq\eta\leq 1$.
### 5.3 Discuss of the secret key rate
Figure 3: (color online). $I(A;B)$ is Alice’s and Bob’s mutual information.
$min(I(E;A),I(E;B))$ is the minimum of Eve’s and Alice’s, Eve’s and Bob’s
mutual information. $r_{\infty}$ is the secret fraction. They are given
compared to loss rate $\eta$. The left figure is the whole show of them. In
the right figure, the ordinate scale is magnified. Figure 4: (color online).
The raw key rate $R_{raw}$ and the secret key rate $R_{QKD}$ compared to loss
rate $\eta$. Here the key rate is the key bit rate generated by one single
photon, and we set $T=R=1/2$. The left figure is the whole show of them. In
the right figure, the ordinate scale is magnified.
Fig.3 shows Alice and Bob’s mutual information $I(A;B)$, the minimum of Eve’s
and Alice’s, Eve’s and Bob’s mutual information $min(I(E;A),I(E;B))$ when Eve
uses the cheat strategies _(I)_ and _(II)_ , and the secret fraction
$r_{\infty}$($=I(A;B)-min(I(E;A),I(E;B))$) compared to the loss rate $\eta$.
It indicates that $r_{\infty}=0$ when $\frac{1}{2}\leq\eta\leq 1$ under the
cheat strategies.
We explain something about the data. When $\frac{1}{2}\leq\eta\leq 1$,
$min(I(E;A),I(E;B))=I(E;A)$, which is monotonic. But when
$0\leq\eta<\frac{1}{2}$, it will be $min(I(E;A),I(E;B))=I(E;B)$, which is not
monotonic. Specially, when $\eta=\frac{1}{3}$, minimal value $I(E;B)=0$ is
here with $P_{raw}^{EB\\_same}=P_{raw}^{EB\\_diff}$. The reason is that
information entropy is non-negative. With the increasing of disparity between
$\eta$ and the special value $\frac{1}{3}$, the disparity between
$P_{raw}^{EB\\_same}$ and $P_{raw}^{EB\\_diff}$ increases, consequently,
$I(E;B)$ increases. (Also see [22])
Fig.4 shows the counterfactual QKD’s raw key rate $R_{raw}$ and the secret key
rate $R_{QKD}$ compared to the loss rate $\eta$. It indicates that $R_{raw}$
increases with the increasing of $\eta$, $R_{QKD}$ decreases with the
increasing of $\eta$. Specially, $R_{QKD}$ will be equal to $0$ when
$\frac{1}{2}\leq\eta\leq 1$ under the cheat strategies, which means the
protocol is insecure.
As QKD applications, they usually need to distribute secret information over
long distance, so the high loss rate of channel is inevitable. For instance,
let us assume that the transmission line is a fiber-based channel, which is
always slightly lossy (about $0.2$ dB/km). If we want to use the cryptographic
system over reasonable distances, say up to $15$ km, transmission losses will
be as high as $3$dB, or about $50\%$. Then Eve could cheat all the secret
information using the cheat strategies proposed without leaving any clues.
## 6 Conclusion
In conclusion, we pointed out that counterfactual cryptography[11] is insecure
in practical high lossy channel. We proposed a polarization-splitting-
measurement attack and analyzed the secret key rate in lossy channel. The
analysis indicates that the protocol is insecure when the loss rate of the
channel from Alice to Bob is up to $50\%$. Since the attack’s effect just
likes a loss channel, it is invisible to the protocol’s participants. Maybe
the security flaw could be overcome by using nonorthogonal states as BB84
QKD[1], but the protocol will be more complex and lower efficient.
We are very grateful to Professor Horace P. Yuen for encouragement. This work
is supported by NSFC (Grant Nos. 61300181, 61272057, 61202434, 61170270,
61100203, 61121061, 61370188, and 61103210), Beijing Natural Science
Foundation (Grant No. 4122054), Beijing Higher Education Young Elite Teacher
Project, China scholarship council.
## References
* [1] Bennett C H and Brassard G 1984 Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing (Bangalore, India) p 175
* [2] Bennett C H 1992 Phys. Rev. Lett. 68 3121
* [3] Yuen H P 2001 in Proceedings of QCMC 00, Capri, edited by P. Tombesi and O. Hirota (Plenum Press, New York)
* [4] Lo H K and Chau H F 1999 Science 283 2050
* [5] Brassard G, Lütkenhaus N, Mor T, and Sanders B C 2000 Phys. Rev. Lett. 85 1330
* [6] Huttner B, Imoto N, Gisin N, and Mor T 1995 Phys. Rev. A 51 1863
* [7] Gisin N, Fasel S, Kraus B, Zbinden H, and Ribordy G 2006 Phys. Rev. A 73 022320
* [8] Makarov V, Anisimov A, and Skaar J 2006 Phys. Rev. A 74 022313
* [9] Hwang W Y 2003 Phys. Rev. Lett. 91 057901
* [10] Lo H K, Curty M, and Qi B 2012 Phys. Rev. Lett. 108 130503
* [11] Noh T G 2009 Phys. Rev. Lett. 103 230501
* [12] Yin Z Q, Li H W, Chen W, Han Z F, and Guo G C 2010 Phy. Rev. A 82 042335
* [13] Sun Y, and Wen Q Y 2010 Phys. Rev. A 82 052318
* [14] Ren M, Wu G, Wu E, and Zeng H 2011 Laser Phys. 21 755
* [15] Zhang S, Wang J , and Tang C J 2012 Chin. Phys. B 21 060303
* [16] Brida G, Cavanna A, Degiovanni I P, Genovese M, and Traina P 2012 Laser Phys. Lett. 9 247
* [17] Yin Z Q, Li H W, Yao Y, Zhang C M, Wang S, Chen W, Guo G C, and Han Z F 2012 Phys. Rev. A 86 022313
* [18] Liu Y, Ju L, Liang X L, Tang S B, Shen Tu G L., Zhou L, Peng C Z, Chen K, Chen T Y, Chen Z B, and Pan J W 2012 Phys. Rev. Lett. 109 030501
* [19] Scarani V, Bechmann-Pasquinucci H, Cerf N J, Dušek M, Lütkenhaus N , and Peev M 2009 Reviews of Modern Physics 81 1202
* [20] Abruzzo S, Bratzik S, Bernardes N K, Kampermann H, vanLoock P , and BrußD 2013 Phys. Rev. A 87 052315
* [21] It will be $I_{EB}<I_{EA}$ (or $I_{EB}>I_{EA}$) when $\eta$’s value is in some ranges. It means that in the counterfactual QKD, Bob’s (or Alice’s) raw key should be chosen as a reference raw key followed by classical postprocessing for more secure against Eve’s attack. So the mutual information Eve obtained is $I_{EB}$ (or $I_{EA}$), i.e., $\min(I_{EA},I_{EB})$.
* [22] It will be $R_{raw}^{EA\\_same}<R_{raw}^{EA\\_diff}$ ($R_{raw}^{EB\\_same}<R_{raw}^{EB\\_diff}$) when $\eta$’s value is in some ranges. In practical attack, Eve should reverse all her cheated bits for decreasing the error rate to less than $50\%$ based on the value of $\eta$. Then the information she obtained is corresponding to the result calculated from information theory. And what she did will not affect the calculation in information theory, so we have not emphasized this in the rest of this paper.
|
arxiv-papers
| 2013-12-05T05:12:06 |
2024-09-04T02:49:54.927730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan-Bing Li, Qiao-Yan Wen, Zi-Chen Li",
"submitter": "Li Yan-Bing",
"url": "https://arxiv.org/abs/1312.1436"
}
|
1312.1440
|
# A variational principle for discrete integrable systems
Sarah Lobb, Frank Nijhoff
###### Abstract
For multidimensionally consistent systems we can consider the Lagrangian as a
form, closed on the multidimensional equations of motion. For 2-dimensional
systems this allows us to define an action on a 2-dimensional surface embedded
in a higher dimensional space of independent variables. It is then natural to
propose that the system should be derived from a variational principle which
includes not only variations with respect to the dependent variables, but also
variations of the surface in the space of independent variables. Here we give
the resulting set of Euler-Lagrange equations firstly in 2 dimensions, and
show how they can specify equations on a single quad in the lattice. We give
the defining set of Euler-Lagrange equations also for 3-dimensional systems,
and in general for n-dimensional systems. In this way the variational
principle can be considered as supplying Lagrangians as solutions of a system
of equations, as much as the equations of motion themselves.
## 1 Introduction
It is a notion going back to the 1600s that a dynamical system should minimize
some quantity, i.e., that the equations of motion should arise as critical
points of some action functional. This action is a quantity depending in
principle on both the dependent and independent variables, and finding a
minimum, or more generally a critical point, specifies a path followed by the
dependent variable. The condition that specifies this path is the Euler-
Lagrange equation.
A discrete calculus of variations was first developed outside the scope of
integrable systems in the 1970s by Cadzow [6], Logan [11] and later Maeda [12,
13, 14]. Cadzow’s original motivation was the use of the digital computer in
modern systems and the solution of control problems, and it became clear that
the formulation of a discrete calculus of variations was important for
numerical methods, in optimization and engineering problems. In the discrete
realm instead of the action being an integral of a Lagrangian, it is a sum
over the independent variable(s).
In the case of multidimensionally consistent systems, we are able to embed the
system in a higher-dimensional space, with compatible systems living in each
subspace. Indeed, we may have an infinite number of compatible systems in an
infinite number of dimensions, and we do not need to restrict to any
particular subspace; we could have a system following a path through an
arbitrary number of dimensions. So we now have to consider not only the path
taken by the dependent variable(s) with respect to the independent
variable(s), but also the path through this space of independent variables.
Then it is natural to ask that the action be critical with respect to a change
in the path of independent variables.
This postulate was first put forward by the authors in [8], initially for
2-dimensional systems, both discrete and continuous. Requiring the action
functional to be invariant under small changes in the path (which for
2-dimensional systems is a surface) through the space of independent variables
leads to a condition on the Lagrangian, a closure relation, which was shown to
be satisfied for many examples of multidimensionally consistent systems [8, 9,
2, 10, 17, 5]. This serves as an answer to the question of how to encode an
entire multidimensionally consistent system in a variational principle.
An issue with the usual variational principle is that it is often impossible
to obtain the desired system of equations as Euler-Lagrange equations, but
only an integrated or derived form of those equations. This can be seen in the
continuous realm in the case of the potential Korteweg-de Vries (pKdV)
equation, where the Euler-Lagrange equation gives a derivative of the pKdV;
and it can be seen in the discrete realm in the case of quad equations, such
as those in the Adler-Bobenko-Suris (ABS) classification [1], where the Euler-
Lagrange equations give only consequences of the quad equations, which can be
considered as discrete derivatives. One does not obtain a quad equation
directly as an Euler-Lagrange equation on a fixed surface.
We show in this paper that the variational principle of [8], which considers
variations of the surface, provides a set of Euler-Lagrange equations,
specifying conditions on both the Lagrangian and Euler-Lagrange equations. In
the 2-dimensional discrete case this set is enough to specify an equation on a
single quad. The key point we wish to make is that the variational principle
should be considered as supplying Lagrangians as solutions of a system of
equations, as much as the equations of motion themselves. It is the latter
perspective, invited by the phenomenon of multidimensional consistency as the
defining aspect of integrability, that forms the main departure of our new
variational principle from any of the conventional variational theories.
The case of 1-dimensional systems was examined in [19, 18] and subsequently
from the Hamiltonian perspective in [16, 3]. Further work has also appeared
recently on 2-dimensional systems in [4].
This paper is concerned with discrete systems. In Section 2 we examine the
variational principle for 2-dimensional discrete systems: defining the action,
listing the Euler-Lagrange equations for the basic configurations in the
surface, and deriving quad equations as consequences of these Euler-Lagrange
equations. We give examples of H1 and H3 to serve as illustrations. In Section
3 we give the defining set of Euler-Lagrange equations for 3-dimensional
discrete systems, and show that these are compatible with the bilinear
discrete Kadomsev-Petviashvili (KP) equation. Section 4 provides some further
discussion and perspectives.
## 2 Discrete 2-dimensional systems
### 2.1 Defining the action
For a large class of equations defined on a quadrilateral, namely those in the
ABS list [1], we have Lagrangians involving 3 points of the quadrilateral.
Since the equations are multidimensionally consistent, we can think of them as
being defined on a surface embedded in arbitrary dimensions, instead of the
regular 2-dimensional lattice. And we can consider the action to be defined as
the sum of Lagrangian contributions from all elementary plaquettes in this
surface.
To this end, consider the surface $\sigma$ to be a connected configuration of
elementary plaquettes $\sigma_{ij}(\boldsymbol{n})$, where
$\sigma_{ij}(\boldsymbol{n})$ is specified by the position
$\boldsymbol{n}=(\boldsymbol{n},\boldsymbol{n}+\boldsymbol{e}_{i},\boldsymbol{n}+\boldsymbol{e}_{j})$
of one of its vertices in the lattice and the lattice directions given by the
base vectors $\boldsymbol{e}_{i},\boldsymbol{e}_{j}$, as in Figure 1. The
surface can be closed, or have a fixed boundary.
$\boldsymbol{e}_{i}$$\boldsymbol{e}_{j}$$\boldsymbol{n}$ Figure 1: Elementary
oriented plaquette.
Since the 3-point Lagrangians depend on two directions in the lattice, and
when embedded in a multidimensional lattice at each point can be associated
with an oriented plaquette $\sigma_{ij}(\boldsymbol{n})$, we can think of
these Lagrangians as defining a discrete 2-form
$\mathcal{L}_{ij}(\boldsymbol{n})$ whose evaluation on that plaquette is given
by the Lagrangian function as follows
$\mathcal{L}_{ij}(\boldsymbol{n})=\mathcal{L}(u(\boldsymbol{n}),u(\boldsymbol{n}+\boldsymbol{e}_{i}),u(\boldsymbol{n}+\boldsymbol{e}_{j});\alpha_{i},\alpha_{j}).$
(2.1)
The Lagrangians given in [8] are all antisymmetric with respect to the
interchange of lattice directions $i,j$, and so this is well-defined. Then the
action $S$ is also well-defined by
$S[u(\boldsymbol{n});\sigma]=\sum_{\sigma_{ij}(\boldsymbol{n})\in\sigma}{\mathcal{L}_{ij}(\boldsymbol{n})}.$
(2.2)
Note that in performing this sum we must be careful to take into account the
orientation of the plaquettes.
### 2.2 The Euler-Lagrange equations
To derive the set of Euler-Lagrange equations stemming from the action (2.2),
we look at what happens at a particular point $\boldsymbol{n}$ in the lattice.
For ease of notation we will suppress the dependence on $\boldsymbol{n}$,
writing $u=u(\boldsymbol{n})$, and make use of shift operators $T_{i}$,
writing
$T_{i}u=u(\boldsymbol{n}+\boldsymbol{e}_{i}),T_{j}u=u(\boldsymbol{n}+\boldsymbol{e}_{j}),T_{i}^{-1}u=u(\boldsymbol{n}-\boldsymbol{e}_{i})$,
etc.
The postulate is that the system lies at a critical point of the action, and
our point of view is that it lies at a critical point with respect to not only
the dependent variable $u$, but also the independent variables, i.e., the
surface $\sigma$. Since we are considering discrete surfaces here, the notion
of infinitesimal variations of the independent variables does not make sense,
and we can make only finite variations. Thus our postulate is that the action
is independent of $\sigma$ (keeping any boundary fixed) on solutions to the
system.
It suffices to consider a collection of fixed surfaces embedded in 3
dimensions, and compute variations with respect to $u$ on that surface. For an
action which is the sum of 3-point Lagrangians
$L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$, there are various possible
configurations involving the arbitrary point $u$. The first is the usual flat
2-dimensional configuration shown in Figure 2:
Figure 2: Usual configuration in 2 dimensions.
This corresponds to the Euler-Lagrange equation
$\frac{\partial}{\partial
u}\biggl{(}L(T_{i}^{-1}u,u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})+L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})+L(T_{j}^{-1}u,T_{i}T_{j}^{-1}u,u;\alpha_{i},\alpha_{j})\biggr{)}=0.$
(2.3)
The elementary configurations in 3 dimensions are shown in Figure 3; all other
configurations can be obtained as combinations of these. A statement to this
effect appears in [4].
Figure 3: Elementary configurations in 3 dimensions.
Note that in the final picture in Figure 3, only two plaquettes contribute,
because of the 3-point nature of the Lagrangians we are considering here.
Each of these pictures corresponds to a different Euler-Lagrange equation.
Since all surfaces in the lattice can be obtained by combining these
elementary configurations, the Euler-Lagrange equation for any surface can be
obtained by combining the Euler-Lagrange equations corresponding to the
respective elementary configurations.
###### Theorem 1
The following form a complete set of Euler-Lagrange equations for the
quadrilateral lattice system with action defined by (2.1) and (2.2).
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})+L(u,T_{j}u,T_{k}u;\alpha_{j},\alpha_{k})+L(u,T_{k}u,T_{i}u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}L(T_{i}^{-1}u,u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})-L(u,T_{j}u,T_{k}u;\alpha_{j},\alpha_{k})+L(T_{i}^{-1}u,T_{i}^{-1}T_{k}u,u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}L(T_{j}^{-1}u,u,T_{j}^{-1}T_{k}u;\alpha_{j},\alpha_{k})+L(T_{i}^{-1}u,T_{i}^{-1}T_{k}u,u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$
for all $i,j,k\in I$, where $I$ is the index set labelling the lattice
directions.
Proof: Consider the action of a closed surface. The smallest non-trivial
closed surface is a cube, for which the action is
$\displaystyle S[u;cube]$ $\displaystyle=$ $\displaystyle
L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})+L(u,T_{j}u,T_{k}u;\alpha_{j},\alpha_{k})+L(u,T_{k}u,T_{i}u;\alpha_{k},\alpha_{i})$
(2.5)
$\displaystyle-L(T_{k}u,T_{i}T_{k}u,T_{j}T_{k}u;\alpha_{i},\alpha_{j})-L(T_{i}u,T_{i}T_{j}u,T_{i}T_{k}u;\alpha_{j},\alpha_{k})$
$\displaystyle-L(T_{j}u,T_{j}T_{k}u,T_{i}T_{j}u;\alpha_{k},\alpha_{i}).$
We require variations of the action with respect to the dependent variables to
be zero. That is,
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})+L(u,T_{j}u,T_{k}u;\alpha_{j},\alpha_{k})+L(u,T_{k}u,T_{i}u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{(}L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})-L(T_{i}u,T_{i}T_{j}u,T_{i}T_{k}u;\alpha_{j},\alpha_{k})+L(u,T_{k}u,T_{i}u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
T_{j}T_{k}u}\biggl{(}L(T_{k}u,T_{i}T_{k}u,T_{j}T_{k}u;\alpha_{i},\alpha_{j})+L(T_{j}u,T_{j}T_{k}u,T_{i}T_{j}u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$
and cyclic permutations. Shifting these in the lattice we see that they are
equivalent to (LABEL:EL2)-(LABEL:EL4). Any closed surface can be constructed
from cubes, so at least away from any boundary all possible Euler-Lagrange
equations are consequences of (LABEL:a)-(LABEL:c).
$\square$
### 2.3 Consequences of the Euler-Lagrange equations
As in the previous subsection, we consider actions which are the sum of
3-point Lagrangians $\mathcal{L}(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$,
where the Lagrangians are anti-symmetric with respect to the interchange of
the lattice directions, so that the equations (LABEL:EL2)-(LABEL:EL4) hold.
The one further assumption we will make is that we may choose initial
conditions $u,T_{i}u,T_{j}u,T_{k}u$ independently and arbitrarily.
If we impose that the action remains invariant under perturbations of the
surface, then it is independent of the surface [8], and all of these equations
must hold simultaneously. Note that (2.3) is a consequence of
(LABEL:EL2)-(LABEL:EL4) and their cyclic permutations.
###### Theorem 2
Suppose $u,T_{i}u,T_{j}u,T_{k}u$ are independent and can be chosen
arbitrarily. The Euler-Lagrange equation (LABEL:EL2) implies that the anti-
symmetric Lagrangian $L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$ has the form
$L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=A(u,T_{i}u;\alpha_{i})-A(u,T_{j}u;\alpha_{j})+C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j}),$
(2.7)
for some functions $A$ and $C$.
Proof: Consider equation (LABEL:EL2). If all of $u,T_{i}u,T_{j}u,T_{k}u$ are
independent and can be chosen arbitrarily, then writing
$l(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=\frac{\partial}{\partial
u}L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j}),$ (2.8)
we must have
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{(}l(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})+l(u,T_{k}u,T_{i}u;\alpha_{k},\alpha_{i})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Rightarrow\quad
l(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=a(u,T_{i}u;\alpha_{i})+b(u,T_{j}u;\alpha_{i},\alpha_{j}),$
(2.9)
for some functions $a,b$. This, plus the various cyclic permutations of the
lattice directions, gives in fact
$l(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=a(u,T_{i}u;\alpha_{i})-a(u,T_{j}u;\alpha_{j}).$
(2.10)
Thus if $\partial A(u,T_{i}u;\alpha_{i})/\partial u=a(u,T_{i}u;\alpha_{i})$,
we should have
$L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=A(u,T_{i}u;\alpha_{i})-A(u,T_{j}u;\alpha_{j})+C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j}),$
(2.11)
for some function $C$, which is (2.7). Note that since
$L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$ is antisymmetric under the
interchange of lattice directions $i,j$, then the same must be true of
$C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$. $\quad\square$
###### Theorem 3
The Euler-Lagrange equations (LABEL:EL2)-(LABEL:EL4) determine the following
relation on each single quad:
$\frac{\partial}{\partial
u}\biggl{(}L(T_{i}^{-1}u,u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})\biggr{)}=\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})\biggr{)}-h(u),$ (2.12)
where $h(u)$ is an arbitrary function, which can be absorbed into $A$.
Proof: Substituting (2.7) into equations (LABEL:EL3) and (LABEL:EL4) gives
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}-A(T_{i}^{-1}u,T_{i}^{-1}T_{j}u;\alpha_{j})-A(u,T_{j}u;\alpha_{j})+A(u,T_{k}u;\alpha_{k})+A(T_{i}^{-1}u,T_{i}^{-1}T_{k}u;\alpha_{k})$
(2.13a)
$\displaystyle+C(u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})-C(u,T_{i}^{-1}T_{k}u;\alpha_{i},\alpha_{k})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(T_{j}^{-1}u,u;\alpha_{j})-A(T_{j}^{-1}u,T_{j}^{-1}T_{k}u;\alpha_{k})+A(T_{i}^{-1}u,T_{i}^{-1}T_{k}u;\alpha_{k})-A(T_{i}^{-1}u,u;\alpha_{i})$
(2.13b)
$\displaystyle+C(u,T_{j}^{-1}T_{k}u;\alpha_{j},\alpha_{k})-C(u,T_{i}^{-1}T_{k}u;\alpha_{i},\alpha_{k})\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$
where we have already cancelled some of the terms (provided we assume that
$T_{j}u$ and $T_{k}u$ can be independently chosen, so that they don’t depend
on $u$). We see that we can rewrite these in a suggestive way, isolating
dependence on particular lattice directions:
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})+A(T_{i}^{-1}u,T_{i}^{-1}T_{j}u;\alpha_{j})-C(u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})\biggr{)}$
$\displaystyle\quad=\frac{\partial}{\partial
u}\biggl{(}A(u,T_{k}u;\alpha_{k})+A(T_{i}^{-1}u,T_{i}^{-1}T_{k}u;\alpha_{k})-C(u,T_{i}^{-1}T_{k}u;\alpha_{i},\alpha_{k})\biggr{)},$
(2.14a) $\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(T_{j}^{-1}u,u;\alpha_{j})-A(T_{j}^{-1}u,T_{j}^{-1}T_{k}u;\alpha_{k})+C(u,T_{j}^{-1}T_{k}u;\alpha_{j},\alpha_{k})\biggr{)}$
$\displaystyle\quad=\frac{\partial}{\partial
u}\biggl{(}A(T_{i}^{-1}u,u;\alpha_{i})-A(T_{i}^{-1}u,T_{i}^{-1}T_{k}u;\alpha_{k})+C(u,T_{i}^{-1}T_{k}u;\alpha_{i},\alpha_{k})\biggr{)},$
(2.14b)
and of course this must be true for all $i,j,k$. Thus we must have
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})+A(T_{i}^{-1}u,T_{i}^{-1}T_{j}u;\alpha_{j})-C(u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})\biggr{)}$
$\displaystyle=f(\dots,T_{i}^{-1}u,u,T_{i}u,\dots;\alpha_{i}),$ (2.15a)
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(T_{i}^{-1}u,u;\alpha_{i})-A(T_{i}^{-1}u,T_{i}^{-1}T_{j}u;\alpha_{j})+C(u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})\biggr{)}$
$\displaystyle=g(\dots,T_{j}^{-1}u,u,T_{j}u,\dots;\alpha_{j}).$ (2.15b)
for some $f,g$ depending on $u$ and its shifts in only one lattice direction.
Adding these expressions together, we deduce that
$\displaystyle f(\dots,T_{i}^{-1}u,u,T_{i}u,\dots;\alpha_{i})$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(T_{i}^{-1}u,u;\alpha_{i})\biggr{)}+h(u),$ (2.16a) $\displaystyle
g(\dots,T_{j}^{-1}u,u,T_{j}u,\dots;\alpha_{j})$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})\biggr{)}-h(u),$ (2.16b)
for some function $h$. Thus we obtain the relation on a single quad
$\frac{\partial}{\partial
u}\biggl{(}A(T_{i}^{-1}u,u;\alpha_{i})-A(T_{i}^{-1}u,T_{i}^{-1}T_{j}u;\alpha_{j})+C(u,T_{i}^{-1}T_{j}u;\alpha_{i},\alpha_{j})\biggr{)}=\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})\biggr{)}-h(u),$ (2.17)
which is in fact exactly (2.12). It is easy to check that (2.7) and (2.12) are
enough to satisfy all Euler-Lagrange equations (LABEL:EL2)-(LABEL:EL4). Of
course, so far $A$ is only defined up to an arbitrary function of $u$, so
$h(u)$ can w.l.o.g. be chosen to be zero.
$\square$
Therefore, in fact, we can rewrite the Euler-Lagrange equations (or rather,
the consequences thereof) in the following two equivalent ways:
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{\\{}A(u,T_{i}u;\alpha_{i})-A(T_{i}u,T_{i}T_{j}u;\alpha_{j})+C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})\biggr{\\}}$
$\displaystyle=$ $\displaystyle 0,$ (2.18a)
$\displaystyle\frac{\partial}{\partial
T_{j}u}\biggl{\\{}A(T_{j}u,T_{i}T_{j}u;\alpha_{i})-A(u,T_{j}u;\alpha_{j})+C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})\biggr{\\}}$
$\displaystyle=$ $\displaystyle 0.$ (2.18b)
By construction, on solutions to the equations, the Lagrangians satisfy a
closure relation
$\Delta_{i}\mathcal{L}_{jk}+\Delta_{j}\mathcal{L}_{ki}+\Delta_{k}\mathcal{L}_{ij}=0,$
(2.19)
where $\Delta_{i}$ is a difference operator defined by $\Delta_{i}=T_{i}-id$.
### 2.4 Example: H1
If we consider the example of H1, the Lagrangian (which was first given in
[7]) evaluated on a plaquette in the $(i,j)$-direction has the form
$L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})=(T_{i}u-T_{j}u)u-(\alpha_{i}-\alpha_{j})\ln(T_{i}u-T_{j}u).$
(2.20)
Then the usual Euler-Lagrange equation (2.3) coming from an action on a flat
2-d surface is
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}(T_{i}u-T_{j}u+T_{i}^{-1}u-T_{j}^{-1}u)u-(\alpha_{i}-\alpha_{j})\ln(u-T_{i}^{-1}T_{j}u)-(\alpha_{i}-\alpha_{j})\ln(T_{i}T_{j}^{-1}u-u)\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Rightarrow\quad\quad
T_{i}u-T_{j}u+T_{i}^{-1}u-T_{j}^{-1}u-\frac{\alpha_{i}-\alpha_{j}}{u-T_{i}^{-1}T_{j}u}+\frac{\alpha_{i}-\alpha_{j}}{T_{i}T_{j}^{-1}u-u}$
$\displaystyle=$ $\displaystyle 0,$ which consists of 2 shifted copies of H1
lying on a 7-point configuration, i.e., a consequence of H1. The Euler-
Lagrange equations on non-flat surfaces (LABEL:EL2)-(LABEL:EL4) are
respectively $\frac{\partial}{\partial u}\biggl{(}0\biggr{)}=0,$ (2.21b)
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}u(-T_{j}u+T_{k}u)-(\alpha_{i}-\alpha_{j})\ln(u-T_{i}^{-1}T_{j}u)-(\alpha_{k}-\alpha_{i})\ln(T_{i}^{-1}T_{k}u-u)\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Rightarrow\quad\quad-
T_{j}u+T_{k}u-\frac{\alpha_{i}-\alpha_{j}}{u-T_{i}^{-1}T_{j}u}+\frac{\alpha_{k}-\alpha_{i}}{T_{i}^{-1}T_{k}u-u}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}u(T_{j}^{-1}u-T_{i}^{-1}u)-(\alpha_{j}-\alpha_{k})\ln(u-T_{j}^{-1}T_{k}u)-(\alpha_{k}-\alpha_{i})\ln(T_{i}^{-1}T_{k}u-u)\biggr{)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Rightarrow\quad\quad
T_{j}^{-1}u-T_{i}^{-1}u-\frac{\alpha_{j}-\alpha_{k}}{u-T_{j}^{-1}T_{k}u}+\frac{\alpha_{k}-\alpha_{i}}{T_{i}^{-1}T_{k}u-u}$
$\displaystyle=$ $\displaystyle 0.$
In fact, (2.21) is a consequence of (2.21) and its copies under permutation of
lattice directions. Also equation (2.12) with $h$ taken to be zero is
$\displaystyle T_{i}^{-1}u-\frac{\alpha_{i}-\alpha_{j}}{u-T_{i}^{-1}T_{j}u}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}A(u,T_{j}u;\alpha_{j})\biggr{)}$ $\displaystyle\Rightarrow\quad
u-\frac{\alpha_{i}-\alpha_{j}}{T_{i}u-T_{j}u}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{(}A(T_{i}u,T_{i}T_{j}u;\alpha_{j})\biggr{)}.$ (2.22)
Of course by swapping the lattice directions we also get
$u-\frac{\alpha_{i}-\alpha_{j}}{T_{i}u-T_{j}u}=\frac{\partial}{\partial
T_{j}u}\biggl{(}A(T_{j}u,T_{i}T_{j}u;\alpha_{i})\biggr{)}.$ (2.23)
Combining (2.22) and (2.23) we get
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{(}A(T_{i}u,T_{i}T_{j}u;\alpha_{j})\biggr{)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{j}u}\biggl{(}A(T_{j}u,T_{i}T_{j}u;\alpha_{i})\biggr{)}$ (2.24)
$\displaystyle=$ $\displaystyle f(T_{i}T_{j}u),$ (2.25)
for some function $f$. This implies
$A(u,T_{j}u;\alpha_{j})=uf(T_{j}u)+g(T_{j}u;\alpha_{j}).$ (2.26)
Here, we know that
$A(u,T_{i}u;\alpha_{i})-A(u,T_{j}u;\alpha_{j})=(T_{i}u-T_{j}u)u,$ (2.27)
and so we must have
$A(u,T_{i}u;\alpha_{i})=u(T_{i}u+\lambda)+\mu,$ (2.28)
where $\lambda,\mu$ are arbitrary constants. Then the Euler-Lagrange equation
is
$(u+\lambda-T_{i}T_{j}u)(T_{i}u-T_{j}u)-\alpha_{i}+\alpha_{j}=0,$ (2.29)
which is consistent around a cube for arbitrary $\lambda$.
### 2.5 Example: H3
The Lagrangian evaluated on a plaquette in the $(i,j)$-direction has the form
$\displaystyle L(u,T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$ $\displaystyle=$
$\displaystyle\ln(\alpha_{i}^{2})\ln{u}-{\rm
Li}_{2}\biggl{(}-\frac{uT_{i}u}{\alpha_{i}\delta}\biggr{)}-\ln(\alpha_{j}^{2})\ln{u}-{\rm
Li}_{2}\biggl{(}-\frac{uT_{j}u}{\alpha_{j}\delta}\biggr{)}$ (2.30)
$\displaystyle+{\rm
Li}_{2}\biggl{(}\frac{\alpha_{j}T_{i}u}{\alpha_{i}T_{j}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}\frac{\alpha_{i}T_{i}u}{\alpha_{j}T_{j}u}\biggr{)}+\ln(\alpha_{j}^{2})\ln\biggl{(}\frac{T_{i}u}{T_{j}u}\biggr{)},$
where $\delta$ is an arbitrary constant parameter, so for an as yet
unspecified function $f$ we see
$\displaystyle A(u,T_{i}u;\alpha_{i})$ $\displaystyle=$
$\displaystyle\ln(\alpha_{i}^{2})\ln{u}-{\rm
Li}_{2}\biggl{(}-\frac{uT_{i}u}{\alpha_{i}\delta}\biggr{)}+f(u),$ (2.31)
$\displaystyle C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})$ $\displaystyle=$
$\displaystyle{\rm
Li}_{2}\biggl{(}\frac{\alpha_{j}T_{i}u}{\alpha_{i}T_{j}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}\frac{\alpha_{i}T_{i}u}{\alpha_{j}T_{j}u}\biggr{)}+\ln(\alpha_{j}^{2})\ln\biggl{(}\frac{T_{i}u}{T_{j}u}\biggr{)}.$
(2.32)
The equation (2.18a) is then
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{\\{}A(u,T_{i}u;\alpha_{i})-A(T_{i}u,T_{i}T_{j}u;\alpha_{j})+C(T_{i}u,T_{j}u;\alpha_{i},\alpha_{j})\biggr{\\}}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{\\{}\ln(\alpha_{i}^{2})\ln{u}-{\rm
Li}_{2}\biggl{(}-\frac{uT_{i}u}{\alpha_{i}\delta}\biggr{)}+fu-\ln(\alpha_{j}^{2})\ln{T_{i}u}+{\rm
Li}_{2}\biggl{(}-\frac{T_{i}uT_{i}T_{j}u}{\alpha_{j}\delta}\biggr{)}-f(T_{j}u)$
$\displaystyle+{\rm
Li}_{2}\biggl{(}\frac{\alpha_{j}T_{i}u}{\alpha_{i}T_{j}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}\frac{\alpha_{i}T_{i}u}{\alpha_{j}T_{j}u}\biggr{)}+\ln(\alpha_{j}^{2})\ln\biggl{(}\frac{T_{i}u}{T_{j}u}\biggr{)}\biggr{\\}}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\Rightarrow\frac{1}{T_{i}u}\ln\biggl{(}1+\frac{uT_{i}u}{\alpha_{i}\delta}\biggr{)}-\frac{1}{T_{i}u}\ln(\alpha_{j}^{2})-\frac{1}{T_{i}u}\ln\biggl{(}1+\frac{T_{i}uT_{i}T_{j}u}{\alpha_{j}\delta}\biggr{)}-f^{\prime}(T_{i}u)$
$\displaystyle-\frac{1}{T_{i}u}\ln\biggl{(}1-\frac{\alpha_{j}T_{i}u}{\alpha_{i}T_{j}u}\biggr{)}+\frac{1}{T_{i}u}\ln\biggl{(}1+\frac{\alpha_{i}T_{i}u}{\alpha_{j}T_{j}u}\biggr{)}+\frac{1}{T_{i}u}\ln(\alpha_{j}^{2})$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\Rightarrow\frac{1}{T_{i}u}\ln\biggl{(}\frac{uT_{i}u+\alpha_{i}\delta}{T_{i}uT_{i}T_{j}u+\alpha_{j}\delta}\cdot\frac{\alpha_{j}T_{j}u-\alpha_{i}T_{i}u}{\alpha_{i}T_{j}u-\alpha_{j}T_{i}u}\biggr{)}-f^{\prime}(T_{i}u)$
$\displaystyle=$ $\displaystyle 0,$
where $f^{\prime}(z)=df/dz$. If we define
$t_{i}=\exp\left\\{T_{i}uf^{\prime}(T_{i}u)\right\\},$ (2.34)
then
$\alpha_{i}(uT_{i}u+t_{i}T_{j}uT_{i}T_{j}u)-\alpha_{j}(uT_{j}u+t_{i}T_{i}uT_{i}T_{j}u)+\delta(\alpha_{i}^{2}-t_{i}\alpha_{j}^{2})+\alpha_{i}\alpha_{j}\delta\frac{T_{j}u}{T_{i}u}(t_{i}-1)=0.$
(2.35)
Note from (2.18b) that we also have the equation
$\alpha_{i}(uT_{i}u+t_{j}T_{j}uT_{i}T_{j}u)-\alpha_{j}(uT_{j}u+t_{j}T_{i}uT_{i}T_{j}u)+\delta(\alpha_{i}^{2}-t_{j}\alpha_{j}^{2})-\alpha_{i}\alpha_{j}\delta\frac{T_{i}u}{T_{j}u}(t_{j}-1)=0,$
(2.36)
and so we must have
$(\alpha_{i}T_{j}u-\alpha_{j}T_{i}u)T_{i}T_{j}u(t_{i}-t_{j})+\delta\left[\alpha_{j}(t_{i}-1)\left(\alpha_{i}\frac{T_{j}u}{T_{i}u}-\alpha_{j}\right)+\alpha_{i}(t_{j}-1)\left(\alpha_{j}\frac{T_{i}u}{T_{j}u}-\alpha_{i}\right)\right]=0.$
(2.37)
Therefore $t_{i}=t_{j}$, and if $\delta\neq 0$ then $t_{i}=t_{j}=1$, and we
have the usual equation H3. If on the other hand $\delta=0$ we have a little
more freedom, and we can let $t_{i}=t_{j}=t$ for some arbitrary constant $t$.
In that case, the equation is
$\alpha_{i}(uT_{i}u+tT_{j}uT_{i}T_{j}u)-\alpha_{j}(uT_{j}u+tT_{i}uT_{i}T_{j}u)=0,$
(2.38)
and this equation is also consistent around the cube.
## 3 Discrete 3-dimensional systems
### 3.1 Defining the action
A Lagrangian for a 3-dimensional system can be defined on an elementary cube
$\nu_{ijk}(\boldsymbol{n})$, where $\nu_{ijk}(\boldsymbol{n})$ is specified by
the position
$\boldsymbol{n}=(\boldsymbol{n},\boldsymbol{n}+\boldsymbol{e}_{i},\boldsymbol{n}+\boldsymbol{e}_{j},\boldsymbol{n}+\boldsymbol{e}_{k})$
of one of its vertices in the lattice and the lattice directions given by the
base vectors $\boldsymbol{e}_{i},\boldsymbol{e}_{j},\boldsymbol{e}_{k}$, as in
Figure 4.
$\boldsymbol{e}_{j}$$\boldsymbol{e}_{i}$$\boldsymbol{e}_{k}$$\boldsymbol{n}$
Figure 4: Elementary oriented cube.
The Lagrangian can depend in principle on the fields at all 8 vertices of the
elementary cube:
$\mathcal{L}_{ijk}(\boldsymbol{n})=\mathcal{L}(u(\boldsymbol{n}),u(\boldsymbol{n}+\boldsymbol{e}_{i}),u(\boldsymbol{n}+\boldsymbol{e}_{j}),u(\boldsymbol{n}+\boldsymbol{e}_{k}),u(\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}),u(\boldsymbol{n}+\boldsymbol{e}_{j}+\boldsymbol{e}_{k}),u(\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{k}),u(\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}+\boldsymbol{e}_{k})).$
(3.1)
The action can then be defined as a connected configuration $\nu$ of these
elementary cubes,
$S[u(\boldsymbol{n});\nu]=\sum_{\nu_{ijk}(\boldsymbol{n})\in\nu}{\mathcal{L}_{ijk}(\boldsymbol{n})}.$
(3.2)
This action is of course still perfectly valid if the Lagrangian doesn’t
depend on the fields at all vertices of the cube. For example, in the case of
the bilinear discrete KP equation, one can write a Lagrangian depending on
fields at 6 vertices.
### 3.2 The Euler-Lagrange equations
The Euler-Lagrange equation in the usual 3-dimensional space is
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\dfrac{\partial}{\partial
u}\biggl{(}\mathcal{L}_{ijk}+T_{i}^{-1}\mathcal{L}_{ijk}+T_{j}^{-1}\mathcal{L}_{ijk}+T_{k}^{-1}\mathcal{L}_{ijk}\biggr{.}$
(3.3)
$\displaystyle\quad\quad\quad\biggl{.}+T_{i}^{-1}T_{j}^{-1}\mathcal{L}_{ijk}+T_{j}^{-1}T_{k}^{-1}\mathcal{L}_{ijk}+T_{k}^{-1}T_{i}^{-1}\mathcal{L}_{ijk}+T_{i}^{-1}T_{j}^{-1}T_{k}^{-1}\mathcal{L}_{ijk}\biggr{)},$
where we take into account all Lagrangian contributions that involve the field
$u$. We have suppressed the dependence on the variables, writing
$\mathcal{L}_{ijk}=\mathcal{L}_{ijk}(u,T_{i}u,T_{j}u,T_{k}u,T_{i}T_{j}u,T_{j}T_{k}u,T_{i}T_{k}u,T_{i}T_{j}T_{k}u).$
Note that any point in $\mathbb{Z}^{3}$ belongs to 8 cubes, so we have in
principle 8 terms in the above equation. This is the analogue of the “flat”
equation (2.3) we had in 2 dimensions.
Embed the system in 4 dimensions. In 3 dimensions, the smallest closed
2-dimensional space is a cube, consisting of 6 faces; in 4 dimensions, the
smallest closed 3-dimensional space is a hypercube, consisting of 8 cubes. The
action on the elementary hypercube will have the form
$S(u;hypercube)=\Delta_{l}\mathcal{L}_{ijk}-\Delta_{i}\mathcal{L}_{jkl}+\Delta_{j}\mathcal{L}_{kli}-\Delta_{k}\mathcal{L}_{lij}.$
(3.4)
Because of the symmetry, we need only take derivatives with respect to
$u,T_{i}u$, $T_{i}T_{j}u$, $T_{i}T_{j}T_{k}u$ and $T_{i}T_{j}T_{k}T_{l}u$, and
the other equations will follow by cyclic permutation of the lattice
directions. Then we have the set of equations
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}-\mathcal{L}_{ijk}+\mathcal{L}_{jkl}-\mathcal{L}_{kli}+\mathcal{L}_{lij}\biggr{)},$
(3.5a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{i}u}\biggl{(}-\mathcal{L}_{ijk}-T_{i}\mathcal{L}_{jkl}-\mathcal{L}_{kli}+\mathcal{L}_{lij}\biggr{)},$
(3.5b) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{i}T_{j}u}\biggl{(}-\mathcal{L}_{ijk}-T_{i}\mathcal{L}_{jkl}+T_{j}\mathcal{L}_{kli}+\mathcal{L}_{lij}\biggr{)},$
(3.5c) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{i}T_{j}T_{k}u}\biggl{(}-\mathcal{L}_{ijk}-T_{i}\mathcal{L}_{jkl}+T_{j}\mathcal{L}_{kli}-T_{k}\mathcal{L}_{lij}\biggr{)},$
(3.5d) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
T_{i}T_{j}T_{k}T_{l}u}\biggl{(}T_{l}\mathcal{L}_{ijk}-T_{i}\mathcal{L}_{jkl}+T_{j}\mathcal{L}_{kli}-T_{k}\mathcal{L}_{lij}\biggr{)},$
(3.5e)
along with the equivalent shifted versions
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
u}\biggl{(}-\mathcal{L}_{ijk}+\mathcal{L}_{jkl}-\mathcal{L}_{kli}+\mathcal{L}_{lij}\biggr{)},$
(3.6a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}-T_{i}^{-1}\mathcal{L}_{ijk}-\mathcal{L}_{jkl}-T_{i}^{-1}\mathcal{L}_{kli}+T_{i}^{-1}\mathcal{L}_{lij}\biggr{)},$
(3.6b) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}-T_{i}^{-1}T_{j}^{-1}\mathcal{L}_{ijk}-T_{j}^{-1}\mathcal{L}_{jkl}+T_{i}^{-1}\mathcal{L}_{kli}+T_{i}^{-1}T_{j}^{-1}\mathcal{L}_{lij}\biggr{)},$
(3.6c) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}-T_{i}^{-1}T_{j}^{-1}T_{k}^{-1}\mathcal{L}_{ijk}-T_{j}^{-1}T_{k}^{-1}\mathcal{L}_{jkl}+T_{i}^{-1}T_{k}^{-1}\mathcal{L}_{kli}-T_{i}^{-1}T_{j}^{-1}\mathcal{L}_{lij}\biggr{)},$
(3.6d) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}\biggl{(}T_{i}^{-1}T_{j}^{-1}T_{k}^{-1}\mathcal{L}_{ijk}-T_{j}^{-1}T_{k}^{-1}T_{l}^{-1}\mathcal{L}_{jkl}+T_{i}^{-1}T_{k}^{-1}T_{l}^{-1}\mathcal{L}_{kli}-T_{i}^{-1}T_{j}^{-1}T_{l}^{-1}\mathcal{L}_{lij}\biggr{)}.$
### 3.3 Example: bilinear discrete KP
The Lagrangian for the bilinear discrete KP equation was first given in [9],
and in 3-dimensional space gives as Euler-Lagrange equations 12 copies of the
bilinear discrete KP equation itself, on 6 elementary cubes. The Lagrangian
$\mathcal{L}_{ijk}$ depends on the six fields
$T_{i}u,T_{j}u,T_{k}u,T_{i}T_{j}u,T_{j}T_{k}u,$ and $T_{i}T_{k}u$, and has the
following explicit form:
$\displaystyle\mathcal{L}_{ijk}$ $\displaystyle=$
$\displaystyle\;\;\;\ln\biggl{(}\frac{T_{k}uT_{i}T_{j}u}{T_{j}uT_{k}T_{i}u}\biggr{)}\ln\biggl{(}-\frac{A_{ki}T_{j}u}{A_{jk}T_{i}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}-\frac{A_{ij}T_{k}uT_{i}T_{j}u}{A_{ki}T_{j}uT_{k}T_{i}u}\biggr{)}$
(3.7)
$\displaystyle+\ln\biggl{(}\frac{T_{i}uT_{j}T_{k}u}{T_{k}uT_{i}T_{j}u}\biggr{)}\ln\biggl{(}-\frac{A_{ij}T_{k}u}{A_{ki}T_{j}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}-\frac{A_{jk}T_{i}uT_{j}T_{k}u}{A_{ij}T_{k}uT_{i}T_{j}u}\biggr{)}$
$\displaystyle+\ln\biggl{(}\frac{T_{j}uT_{k}T_{i}u}{T_{i}uT_{j}T_{k}u}\biggr{)}\ln\biggl{(}-\frac{A_{jk}T_{i}u}{A_{ij}T_{k}u}\biggr{)}-{\rm
Li}_{2}\biggl{(}-\frac{A_{ki}T_{j}uT_{k}T_{i}u}{A_{jk}T_{i}uT_{j}T_{k}u}\biggr{)}$
$\displaystyle-\frac{1}{2}\bigl{(}\bigl{(}\ln\bigl{(}T_{i}T_{j}u\bigr{)}\bigr{)}^{2}+\bigl{(}\ln\bigl{(}T_{j}T_{k}u\bigr{)}\bigr{)}^{2}+\bigl{(}\ln\bigl{(}T_{k}T_{i}u\bigr{)}\bigr{)}^{2}$
$\displaystyle\;\;\;\;\;\;\;\;-\bigl{(}\ln\bigl{(}T_{i}u\bigr{)}\bigr{)}^{2}-\bigl{(}\ln\bigl{(}T_{j}u\bigr{)}\bigr{)}^{2}-\bigl{(}\ln\bigl{(}T_{k}u\bigr{)}\bigr{)}^{2}$
$\displaystyle\;\;\;\;\;\;\;\;-\ln\bigl{(}T_{i}T_{j}u\bigr{)}\ln\bigl{(}T_{j}T_{k}u\bigr{)}-\ln\bigl{(}T_{j}T_{k}u\bigr{)}\ln\bigl{(}T_{k}T_{i}u\bigr{)}-\ln\bigl{(}T_{k}T_{i}u\bigr{)}\ln\bigl{(}T_{i}T_{j}u\bigr{)}$
$\displaystyle\;\;\;\;\;\;\;\;+\ln\bigl{(}T_{i}u\bigr{)}\ln\bigl{(}T_{j}u\bigr{)}+\ln\bigl{(}T_{j}u\bigr{)}\ln\bigl{(}T_{k}u\bigr{)}+\ln\bigl{(}T_{k}u\bigr{)}\ln\bigl{(}T_{i}u\bigr{)}$
$\displaystyle\;\;\;\;\;\;\;\;-A_{ij}^{2}-A_{jk}^{2}-A_{ki}^{2}+A_{ij}A_{jk}+A_{jk}A_{ki}+A_{ki}A_{ij}\bigr{)}.$
Here the $A_{ij}$ are constants which are antisymmetric with respect to
swapping the indices.
If we introduce the quantity $C_{ijk}$, defined by
$C_{ijk}=\frac{A_{ij}T_{k}uT_{i}T_{j}u+A_{jk}T_{i}uT_{j}T_{k}u}{A_{ki}T_{j}uT_{k}T_{i}u},$
(3.8)
then the bilinear discrete KP equation itself can be written $C_{ijk}=1$. The
usual Euler-Lagrange equation is
$0=\frac{1}{u}\ln\biggl{\\{}\frac{T_{k}^{-1}C_{kij}T_{i}^{-1}T_{j}^{-1}C_{kij}}{T_{j}^{-1}C_{kij}T_{k}^{-1}T_{i}^{-1}C_{kij}}\cdot\frac{T_{i}^{-1}C_{ijk}T_{j}^{-1}T_{k}^{-1}C_{ijk}}{T_{k}^{-1}C_{ijk}T_{i}^{-1}T_{j}^{-1}C_{ijk}}\cdot\frac{T_{j}^{-1}C_{jki}T_{k}^{-1}T_{i}^{-1}C_{jki}}{T_{i}^{-1}C_{jki}T_{j}^{-1}T_{k}^{-1}C_{jki}}\biggr{\\}}.$
(3.9)
The Euler-Lagrange equations (3.5a) and (3.5e) are trivial in this case, while
(3.5b)-(3.5d) are
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{T_{i}u}\ln\left\\{\frac{C_{ijk}C_{jli}C_{ikl}}{C_{jki}C_{ijl}C_{kli}}\right\\},$
(3.10a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{T_{i}T_{j}u}\ln\left\\{\frac{C_{kij}C_{ijl}T_{i}C_{jkl}T_{j}C_{kli}}{C_{ijk}C_{lij}T_{i}C_{klj}T_{j}C_{ikl}}\right\\},$
(3.10b) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{T_{i}T_{j}T_{k}u}\ln\left\\{\frac{T_{i}C_{ljk}T_{k}C_{lij}T_{j}C_{ikl}}{T_{i}C_{jkl}T_{k}C_{ijl}T_{j}C_{lik}}\right\\}.$
(3.10c)
## 4 Summary and conclusions
Multidimensionally consistent systems can be considered as critical points of
an action: critical with respect to the dependent variable, and also with
respect to the curve or surface in the space of independent variables. In the
case of discrete systems, this means the action is required to be independent
of the curve or surface on which it is defined, whilst keeping any boundary it
may have fixed. This leads to a set of Euler-Lagrange equations, corresponding
to basic configurations of points in a surface, which should be satisfied
simultaneously.
In the case of 2-dimensional discrete systems, we have shown that the set of
Euler-Lagrange equations arising from this variational principle specify
firstly a particular form of the Lagrangian, and furthermore quad equations
themselves, whereas previously only a weaker form of the equations could be
derived. Starting from known examples of Lagrangians, we can show that the
resulting quad equations are compatible with previous results.
It would be interesting to see if the results of this paper can be extended to
higher than 3 dimensions where we will have a Lagrangian function evaluated on
an n-dimensional object, in particular on an n-dimensional cube. Embedding
this in higher dimensions, we consider an action on the smallest closed
n-dimensional surface in (n+1) dimensions, a hypercube. Then the minimal set
of Euler-Lagrange equations are obtained by demanding that the derivative of
this action with respect to each variable is zero.
As we pointed out earlier, the set of Euler-Lagrange equations could, and
maybe should, be viewed as a system of equations for the Lagrangian itself.
This constitutes a significant departure from the conventional point of view
where the Lagrangian is a given object (usually obtained from considerations
of physics) and the main issue is to derive the equations of the motion of the
system from a variational approach. In the integrable case of Lagrangian
multiforms, the Lagrangians themselves are part of the solution of the
extended system of equations obtained from varying not only the field
variables on a given space-time of independent variables, but by also varying
the geometry of space-time itself. It would be of interest to see whether
Lagrangians associated with descriptions of known physical processes could be
obtained from such a novel variational theory.
## Acknowledgements
The authors would like to thank James Atkinson for helpful comments and
suggestions. SBL was supported by Australian Laureate Fellowship Grant
#FL120100094 from the Australian Research Council. FWN is partially supported
by the grants EP/I002294/1 and EP/I038683/1 of the Engineering and Physical
Sciences Research Council (EPSRC). FWN is grateful to the hospitality of the
Sophus Lie Center in Nordfjordeid (Norway) during the conference on ”Nonlinear
Mathematical Physics: Twenty Years of JNMP” (June 4-June 14, 2013) where a
(preliminary) account (joint with SBL) of the results of this paper was
presented [15].
## References
* [1] Adler, V.E., A.I. Bobenko and Yu.B. Suris. Classification of Integrable Equations on Quad-Graphs, the Consistency Approach. _Communications in Mathematical Physics_ , 2003: 233, pp.513-543.
* [2] Bobenko, A.I., and Yu.B. Suris. On the Lagrangian structure of integrable quad-equations. _Letters in Mathematical Physics_ , 2010: 92, pp. 17-31.
* [3] Boll, R., M. Petrera and Yu.B. Suris. Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems. _Journal of Physics A: Mathematical and Theoretical_ , 2013: 46 275204\.
* [4] Boll, R., M. Petrera and Yu.B. Suris. What is integrability of discrete variational systems? arXiv:1307.0523 [math-ph].
* [5] Boll, R., and Yu.B. Suris. On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations. _Journal of Physics A: Mathematical and Theoretical_ , 2012: 45 115201\.
* [6] Cadzow, J.A. Discrete Calculus of Variations. _International Journal of Control_ , 1970: 11(3), pp.393-407.
* [7] Capel, H.W., F.W. Nijhoff and V.G. Papageorgiou. Complete Integrability of Lagrangian Mappings and Lattices of KdV Type. _Physics Letters A_ , 1991: 155, pp.377-387.
* [8] Lobb, S.B., and F.W. Nijhoff. Lagrangian multiforms and multidimensional consistency. _Journal of Physics A: Mathematical and Theoretical_ , 42 (2009) 454013.
* [9] Lobb, S.B., F.W. Nijhoff and G.R.W. Quispel. Lagrangian multiform structure for the lattice KP system. _Journal of Physics A: Mathematical and Theoretical_ , 42 (2009) 472002.
* [10] Lobb, S.B., and F.W. Nijhoff. Lagrangian multiform structure for the lattice Gel’fand-Dikii hierarchy. _Journal of Physics A: Mathematical and Theoretical_ , 43 (2010) 072003.
* [11] Logan, J.D. First Integrals in the Discrete Variational Calculus. _Aequationes Mathematicae_ , 1973: 9, pp.210-220.
* [12] Maeda, S. Canonical Structure and Symmetries for Discrete Systems. _Mathematica Japonica_ , 1980: 25(4), pp.405-420.
* [13] Maeda, S. Extension of discrete Noether theorem. _Mathematica Japonica_ , 1981: 26, pp.85-90.
* [14] Maeda, S. Lagrangian formulation of discrete systems and concept of difference space. _Mathematica Japonica_ , 1982: 27, pp.345-356.
* [15] Nijhoff, F.W. _New variational principle for integrable systems_. Nonlinear Mathematical Physics: Twenty Years of JNMP, Norway, 4-14 June 2013. Slides available from: http://staff.www.ltu.se/ norbert/JNMP-Conference-2013/JNMP-conference-2013.html
* [16] Suris, Yu. B. Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms. arXiv:1212.3314v2 [math-ph].
* [17] Xenitidis, P., F.W. Nijhoff and S.B. Lobb. On the Lagrangian formulation of multidimensionally consistent systems. _Proceedings of the Royal Society A_ , A, 467 # 2135 (2011) 3295-3317, [published online before print July 13, 2011, doi:10.1098/rspa.2011.0124].
* [18] Yoo-Kong, S. _Calogero-Moser type systems, associated KP systems, and Lagrangian structures_. Ph.D. thesis, University of Leeds, 2011.
* [19] Yoo-Kong, S., and F.W. Nijhoff. Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure. arXiv:1112.4576 [nlin.SI].
|
arxiv-papers
| 2013-12-05T05:43:02 |
2024-09-04T02:49:54.937445
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Sarah B. Lobb, Frank W. Nijhoff",
"submitter": "Frank W. Nijhoff",
"url": "https://arxiv.org/abs/1312.1440"
}
|
1312.1445
|
# Bayesian Machine Learning via Category Theory
Jared Culbertson and Kirk Sturtz
###### Abstract
From the Bayesian perspective, the category of conditional probabilities (a
variant of the Kleisli category of the Giry monad, whose objects are
measurable spaces and arrows are Markov kernels) gives a nice framework for
conceptualization and analysis of many aspects of machine learning. Using
categorical methods, we construct models for parametric and nonparametric
Bayesian reasoning on function spaces, thus providing a basis for the
supervised learning problem. In particular, stochastic processes are arrows to
these function spaces which serve as prior probabilities. The resulting
inference maps can often be analytically constructed in this symmetric
monoidal weakly closed category. We also show how to view general stochastic
processes using functor categories and demonstrate the Kalman filter as an
archetype for the hidden Markov model.
Keywords: Bayesian machine learning, categorical probability, Bayesian
probability
###### Contents
1. 1 Introduction
2. 2 The Category of Conditional Probabilities
1. 2.1 (Weak) Product Spaces and Joint Distributions
2. 2.2 Constructing a Joint Distribution Given Conditionals
3. 2.3 Constructing Regular Conditionals given a Joint Distribution
3. 3 The Bayesian Paradigm using $\mathcal{P}$
4. 4 Elementary applications of Bayesian probability
5. 5 The Tensor Product
1. 5.1 Graphs of Conditional Probabilities
2. 5.2 A Tensor Product of Conditionals
3. 5.3 Symmetric Monoidal Categories
6. 6 Function Spaces
1. 6.1 Stochastic Processes
2. 6.2 Gaussian Processes
3. 6.3 GPs via Joint Normal Distributions161616This section is not required for an understanding of subsequent material but only provided for purposes of linking familiar concepts and ideas with the less familiar categorical perspective.
7. 7 Bayesian Models for Function Estimation
1. 7.1 Nonparametric Models
1. 7.1.1 Noise Free Measurement Model
2. 7.1.2 Gaussian Additive Measurement Noise Model
2. 7.2 Parametric Models
8. 8 Constructing Inference Maps
1. 8.1 The noise free inference map
2. 8.2 The noisy measurement inference map
3. 8.3 The inference map for parametric models
9. 9 Stochastic Processes as Points
1. 9.1 Markov processes via Functor Categories
2. 9.2 Hidden Markov Models
10. 10 Final Remarks
11. 11 Appendix A: Integrals over probability measures.
12. 12 Appendix B: The weak closed structure in $\mathcal{P}$
## 1 Introduction
Speculation on the utility of using categorical methods in machine learning
(ML) has been expounded by numerous people, including by the denizens at the
n-category cafe blog [5] as early as 2007. Our approach to realizing
categorical ML is based upon viewing ML from a probabilistic perspective and
using categorical Bayesian probability. Several recent texts (e.g., [2, 19]),
along with countless research papers on ML have emphasized the subject from
the perspective of Bayesian reasoning. Combining this viewpoint with the
recent work [6], which provides a categorical framework for Bayesian
probability, we develop a category theoretic perspective on ML. The
abstraction provided by category theory serves as a basis not only for an
organization of ones thoughts on the subject, but also provides an efficient
graphical method for model building in much the same way that probabilistic
graphical modeling (PGM) has provided for Bayesian network problems.
In this paper, we focus entirely on the supervised learning problem, i.e., the
regression or function estimation problem. The general framework applies to
any Bayesian machine learning problem, however. For instance, the unsupervised
clustering or density estimation problems can be characterized in a similar
way by changing the hypothesis space and sampling distribution. For
simplicity, we choose to focus on regression and leave the other problems to
the industrious reader. For us, then, the Bayesian learning problem is to
determine a function $f:X\rightarrow Y$ which takes an input $\mathbf{x}\in
X$, such as a feature vector, and associates an output (or class)
$f(\mathbf{x})$ with $\mathbf{x}$. Given a measurement $(\mathbf{x},y)$, or a
set of measurements $\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$ where each
$y_{i}$ is a labeled output (i.e., training data), we interpret this problem
as an estimation problem of an unknown function $f$ which lies in $Y^{X}$, the
space of all measurable functions111Recall that a $\sigma$-algebra
$\Sigma_{X}$ on $X$ is a collection of subsets of $X$ that is closed under
complements and countable unions (and hence intersections); the pair
$(X,\Sigma_{X})$ is called a measurable space and any set $A\in\Sigma_{X}$ is
called a measurable set of $X$. A measurable function $f\colon X\to Y$ is
defined by the property that for any measurable set $B$ in the
$\sigma$-algebra of $Y$, we have that $f^{-1}(B)$ is in the $\sigma$-algebra
of $X$. For example, all continuous functions are measurable with respect to
the Borel $\sigma$-algebras. from $X$ to $Y$ such that
$f(\mathbf{x}_{i})\approx y_{i}$. When $Y$ is a vector space the space $Y^{X}$
is also a vector space that is infinite dimensional when $X$ is infinite. If
we choose to allow all such functions (every function $f\in Y^{X}$ is a valid
model) then the problem is nonparametric. On the other hand, if we only allow
functions from some subspace $V\subset Y^{X}$ of _finite_ dimension $p$, then
we have a parametric model characterized by a measurable map
$i:\mathbb{R}^{{}^{p}}\rightarrow Y^{X}$. The image of $i$ is then the space
of functions which we consider as valid models of the unknown function for the
Bayesian estimation problem. Hence, the elements
$\mathbf{a}\in\mathbb{R}^{{}^{p}}$ completely determine the valid modeling
functions $i(\mathbf{a})\in Y^{X}$. Bayesian modeling splits the problem into
two aspects: (1) specification of the hypothesis space, which consist of the
“valid” functions $f$, and (2) a noisy measurement model such as
$y_{i}=f(\mathbf{x}_{i})+\epsilon_{i}$, where the noise component
$\epsilon_{i}$ is often modeled by a Gaussian distribution. Bayesian reasoning
with the hypothesis space taken as $Y^{X}$ or any subspace $V\subset Y^{X}$
(finite or infinite dimensional) and the noisy measurement model determining a
sampling distribution can then be used to efficiently estimate (learn) the
function $f$ without over fitting the data.
We cast this whole process into a graphical formulation using category theory,
which like PGM, can in turn be used as a modeling tool itself. In fact, we
view the components of these various models, which are just Markov kernels, as
interchangeable parts. An important piece of the any solving the ML problem
with a Bayesian model consists of choosing the appropriate parts for a given
setting. The close relationship between parametric and nonparametric models
comes to the forefront in the analysis with the measurable map
$i:\mathbb{R}^{{}^{p}}\rightarrow Y^{X}$ connecting the two different types of
models. To illustrate this point suppose we are given a normal distribution
$P$ on $\mathbb{R}^{{}^{p}}$ as a prior probability on the unknown parameters.
Then the push forward measure222A measure $\mu$ on a measurable space
$(X,\Sigma_{X})$ is a nonnegative real-valued function $\mu\colon
X\to{\mathbb{R}}_{\geq 0}$ such that $\mu(\emptyset)=0$ and
$\mu(\cup_{i=1}^{\infty}A_{i})=\sum_{i=1}^{\infty}\mu(A_{i})$. A probability
measure is a measure where $\mu(X)=1$. In this paper, all measures are
probability measures and the terminology “distribution” will be synonymous
with “probability measure.” of $P$ by $i$ is a Gaussian process, which is a
basic tool in nonparametric modeling. When composed with a noisy measurement
model, this provides the whole Bayesian model required for a complete analysis
and an inference map can be analytically constructed.333The inference map need
not be unique. Consequently, given any measurement $(\mathbf{x},y)$ taking the
inference map conditioned at $(\mathbf{x},y)$ yields the updated prior
probability which is another normal distribution on $\mathbb{R}^{{}^{p}}$.
The ability to do Bayesian probability involving function spaces relies on the
fact that the category of measurable spaces, $\mathcal{M}eas$, has the
structure of a symmetric monoidal closed category (SMCC). Through the
evaluation map, this in turn provides the category of conditional
probabilities $\mathcal{P}$ with the structure of a symmetric monoidal
_weakly_ closed category (SMwCC), which is necessary for modeling stochastic
processes as probability measures on function spaces. On the other hand, the
ordinary product $X\times Y$ with its product $\sigma$-algebra is used for the
Bayesian aspect of updating joint (and marginal) distributions. From a
modeling viewpoint, the SMwCC structure is used for carrying along a parameter
space (along with its relationship to the output space through the evaluation
map). Thus we can describe training data and measurements as ordered pairs
$(\mathbf{x}_{i},y_{i})\in X\otimes Y$, where $X$ plays the role of a
parameter space.
##### A few notes on the exposition.
In this paper our intended audience consists of (1) the practicing ML engineer
with only a passing knowledge of category theory (e.g., knowing about objects,
arrows and commutative diagrams), and (2) those knowledgeable of category
theory with an interest of how ML can be formulated within this context. For
the ML engineer familiar with Markov kernels, we believe that the presentation
of $\mathcal{P}$ and its applications can serve as an easier introduction to
categorical ideas and methods than many standard approaches. While some
terminology will be unfamiliar, the examples should provide an adequate
understanding to relate the knowledge of ML to the categorical perspective. If
ML researchers find this categorical perspective useful for further
developments or simply for modeling purposes, then this paper will have
achieved its goal.
In the categorical framework for Bayesian probability, Bayes’ equation is
replaced by an integral equation where the integrals are defined over
probability measures. The analysis requires these integrals be evaluated on
arbitrary measurable sets and this is often possible using the three basic
rules provided in Appendix A. Detailed knowledge of measure theory is not
necessary outside of understanding these three rules and the basics of
$\sigma$-algebras and measures, which are used extensively for evaluating
integrals in this paper. Some proofs require more advanced measure-theoretic
ideas, but the proofs can safely be avoided by the unfamiliar reader and are
provided for the convenience of those who might be interested in such details.
For the category theorist, we hope the paper makes the fundamental ideas of ML
transparent, and conveys our belief that Bayesian probability can be
characterized categorically and usefully applied to fields such as ML. We
believe the further development of categorical probability can be motivated by
such applications and in the final remarks we comment on one such direction
that we are pursuing.
These notes are intended to be tutorial in nature, and so contain much more
detail that would be reasonable for a standard research paper. As in this
introductory section, basic definitions will be given as footnotes, while more
important definitions, lemmas and theorems Although an effort has been made to
make the exposition as self-contained as possible, complete self-containment
is clearly an unachievable goal. In the presentation, we avoid the use of the
terminology of _random variables_ for two reasons: (1) formally a random
variable is a measurable function $f:X\rightarrow Y$ and a probability measure
$P$ on $X$ gives rise to the distribution of the random variable
$f_{\star}(P)$ which is the push forward measure of $P$. In practice the
random variable $f$ itself is more often than not impossible to characterize
functionally (consider the process of flipping a coin), while reference to the
random variable using a binomial distribution, or any other distribution, is
simply making reference to some probability measure. As a result, in practice
the term “random variable” is often not making reference to any measurable
function $f$ and the pushforward measure of some probability measure $P$ at
all but rather is just referring to a probability measure; (2) the term
“random variable” has a connotation that, we believe, should be de-emphasized
in a Bayesian approach to modeling uncertainty. Thus while a random variable
can be modeled as a push forward probability measure within the framework
presented we feel no need to single them out as having any special relevance
beyond the remark already given. In illustrating the application of
categorical Bayesian probability we do however show how to translate the
familiar language of random variables into the unfamiliar categorical
framework for the particular case of Gaussian distributions which are the most
important application for ML since Gaussian Processes are characterized on
finite subsets by Gaussian distributions. This provides a particularly nice
illustration of the non uniqueness of conditional sampling distribution and
inference pairs given a joint distribution.
##### Organization.
The paper is organized as follows: The theory of Bayesian probability in
$\mathcal{P}$ is first addressed and applied to elementary problems on finite
spaces where the detailed solutions to inference, prediction and decision
problems are provided. If one understands the “how and why” in solving these
problems then the extension to solving problems in ML is a simple step as one
uses the same basic paradigm with only the hypothesis space changed to a
function space. Nonparametric modeling is presented next, and then the
parametric model can seen as a submodel of the nonparametric model. We then
proceed to give a general definition of stochastic process as a special type
of arrow in a functor category $\mathcal{P}^{X}$, and by varying the category
$X$ or placing conditions on the projection maps onto subspaces one obtains
the various types of stochastic processes such as Markov processes or GP.
Finally, we remark on the area where category theory may have the biggest
impact on applications for ML by integrating the probabilistic models with
decision theory into one common framework.
The results presented here derived from a categorical analysis of the ML
problem(s) will come as no surprise to ML professionals. We acknowledge and
thank our colleagues who are experts in the field who provided assistance and
feedback.
## 2 The Category of Conditional Probabilities
The development of a categorical basis for probability was initiated by
Lawvere [16], and further developed by Giry [14] using monads to characterize
the adjunction given in Lawvere’s original work. The Kleisli category of the
Giry monad $\mathcal{G}$ is what Lawvere called the category of probabilistic
mappings and what we shall refer to as the category of conditional
probabilities.444Monads had not yet been developed at the time of Lawvere’s
work. However the adjunction construction he provided was the Giry monad on
measurable spaces. Further progess was given in the unpublished dissertation
of Meng [18] which provides a wealth of information and provides a basis for
thinking about stochastic processes from a categorical viewpoint. While this
work does not address the Bayesian perspective it does provide an alternative
“statistical viewpoint” toward solving such problems using generalized
metrics. Additional interesting work on this category is presented in a
seminar by Voevodsky, in Russian, available in an online video [22]. The
extension of categorical probability to the Bayesian viewpoint is given in the
paper [6], though Lawvere and Peter Huber were aware of a similar approach in
the 1960’s.555In a personal communication Lawvere related that he and Peter
Huber gave a seminar in Zurich around 1965 on “Bayesian sections.” This refers
to the existence of inference maps in the Eilenberg–Moore category of
$\mathcal{G}$-algebras. These inference maps are discussed in Section 3,
although we discuss them only in the context of the category $\mathcal{P}$.
Coecke and Speckens [4] provide an alternative graphical language for Bayesian
reasoning under the assumption of finite spaces which they refer to as
standard probability theory. In such spaces the arrows can be represented by
stochastic matrices [13]. More recently Fong [12] has provided further
applications of the category of conditional probabilities to Causal Theories
for Bayesian networks.
Much of the material in this section is directly from [6], with some
additional explanation where necessary. The category666A category is a
collection of (1) objects and (2) morphisms (or arrows) between the objects
(including a required identity morphism for each object), along with a
prescribed method for associative composition of morphisms. of conditional
probabilities, which we denote by $\mathcal{P}$, has countably generated777A
space $(X,\Sigma_{X})$ is countably generated if there exist a countable set
of measurable sets $\\{A_{i}\\}_{i=1}^{\infty}$ which generated the
$\sigma$-algebra $\Sigma_{X}$. measurable spaces $(X,\Sigma_{X})$ as objects
and an arrow between two such objects
$(X,\Sigma_{X})$$(Y,\Sigma_{Y})$$T$
is a Markov kernel (also called a _regular_ conditional probability) assigning
to each element $x\in X$ and each measurable set $B\in\Sigma_{Y}$ the
probability of $B$ given $x$, denoted $T(B\mid x)$. The term “regular” refers
to the fact that the function $T$ is conditioned on points rather than
measurable sets $A\in\Sigma_{X}$. When $(X,\Sigma_{X})$ is a countable set
(either finite or countably infinite) with the discrete $\sigma$-algebra then
every singleton $\\{x\\}$ is measurable and the term “regular” is unnecessary.
More precisely, an arrow $T\colon X\rightarrow Y$ in $\mathcal{P}$ is a
function $T\colon~{}\Sigma_{Y}\times X~{}\rightarrow~{}[0,1]$ satisfying
1. 1.
for all $B\in\Sigma_{Y}$, the function $T(B\mid\cdot)\colon X\rightarrow[0,1]$
is measurable, and
2. 2.
for all $x\in X$, the function $T(\cdot\mid
x)\colon\Sigma_{Y}\rightarrow[0,1]$ is a perfect probability measure888A
perfect probability measure $P$ on $Y$ is a probability measure such that for
any measurable function $f:Y\rightarrow\mathbb{R}$ there exist a real Borel
set $E\subset f(Y)$ satisfying $P(f^{-1}(E))=1$. on $Y$.
For technical reasons it is necessary that the probability measures in (2)
constitute an equiperfect family of probability measures to avoid pathological
cases which prevent the existence of inference maps necessary for Bayesian
reasoning.999Specifically, the subsequent Theorem 1 is a constructive
procedure which requires perfect probability measures. Corollary 2 then gives
the inference map. Without the hypothesis of perfect measures a pathological
counterexample can be constructed as in [9, Problem 10.26]. The paper by Faden
[11] gives conditions on the existence of conditional probabilities and this
constraint is explained in full detail in [6]. Note that the class of perfect
measures is quite broad and includes all probability measures defined on
Polish spaces.
The notation $T(B\mid x)$ is chosen as it coincides with the standard notation
“$p(H\mid D)$” of conditional probability theory. For an arrow
$T\colon(X,\Sigma_{X})\rightarrow(Y,\Sigma_{Y})$, we occasionally denote the
measurable function $T(B\mid\cdot)\colon\Sigma_{Y}\rightarrow[0,1]$ by $T_{B}$
and the probability measure $T(\cdot\mid x)\colon\Sigma_{Y}\rightarrow[0,1]$
by $T_{x}$. Hereafter, for notational brevity we write a measurable space
$(X,\Sigma_{X})$ simply as $X$ when referring to a generic $\sigma$-algebra
$\Sigma_{X}$.
Given two arrows
$X$$Y$$Z$$T$$U$
the composition $U\circ T\colon\Sigma_{Z}\times X\rightarrow[0,1]$ is
_marginalization over $Y$_ defined by
$(U\circ T)(C\mid x)=\int_{y\in Y}U(C\mid y)\,dT_{x}.$
The integral of any real valued measurable function $f\colon
X\rightarrow\mathbb{R}$ with respect to any measure $P$ on $X$ is
$\mathbb{E}_{P}[f]=\int_{x\in X}f(x)\,dP,$ (1)
called the _$P$ -expectation of $f$_. Consequently the composite $(U\circ
T)(C\mid x)$ is the $T_{x}$-expectation of $U_{C}$,
$(U\circ T)(C\mid x)=\mathbb{E}_{T_{x}}[U_{C}].$
Let $\mathcal{M}eas$ denote the category of measurable spaces where the
objects are measurable spaces $(X,\Sigma_{X})$ and the arrows are measurable
functions $f\colon X\rightarrow Y$. Every measurable mapping $f\colon
X\rightarrow Y$ may be regarded as a $\mathcal{P}$ arrow
$X$$Y$$\delta_{f}$
defined by the Dirac (or one point) measure
$\begin{array}[]{lclcl}\delta_{f}&:&X\times\Sigma_{Y}&\rightarrow&[0,1]\\\
&:&(B\mid x)&\mapsto&\left\\{\begin{array}[]{c}1\quad\textrm{ If }f(x)\in B\\\
0\quad\textrm{If }f(x)\notin B.\end{array}\right.\end{array}$
The relation between the dirac measure and the characteristic (indicator)
function $\mathbb{1}$ is
$\delta_{f}(B\mid x)=\mathbb{1}_{f^{-1}(B)}(x)$
and this property is used ubiquitously in the analysis of integrals.
Taking the measurable mapping $f$ to be the identity map on $X$ gives for each
object $X$ the morphism
$X\stackrel{{\scriptstyle\delta_{Id_{X}}}}{{\longrightarrow}}X$ given by
$\delta_{Id_{X}}(B\mid x)=\left\\{\begin{array}[]{lcl}1&\textrm{ if }x\in B\\\
0&\textrm{ if }x\notin B\end{array}\right.$
which is the identity morphism for $X$ in $\mathcal{P}$. Using standard
notation we denote the identity mapping on any object $X$ by
$1_{X}=\delta_{Id_{X}}$, or for brevity simply by $1$ if the space $X$ is
clear from the context. With these objects and arrows, law of composition,
associativity, and identity, standard measure-theoretic arguments show that
$\mathcal{P}$ forms a category.
There is a distinguished object in $\mathcal{P}$ that play an important role
in Bayesian probability. For any set $Y$ with the indiscrete $\sigma$-algebra
$\Sigma_{Y}=\\{Y,\emptyset\\}$, there is a unique arrow from any object $X$ to
$Y$ since any arrow $P\colon X\rightarrow Y$ is completely determined by the
fact that $P_{x}$ must be a probability measure on $Y$. Hence $Y$ is a
_terminal_ object, and we denote the unique arrow by $!_{X}:X\rightarrow Y$.
Up to isomorphism, the canonical terminal object is the one-element set which
we denote by $1=\\{\star\\}$ with the only possible $\sigma$-algebra. It
follows that any arrow $P:1\rightarrow X$ from the terminal object to any
space $X$ is an (absolute) probability measure on $X$, i.e., it is an
“absolute” probability measure on $X$ because there is no variability
(conditioning) possible within the singleton set $1=\\{\star\\}$.
$1$$X$$P$ Figure 1: The representation of a probability measure in
$\mathcal{P}$.
We refer to any arrow $P\colon 1\rightarrow X$ with domain $1$ as either a
probability measure or a distribution on $X$. If $X$ is countable then $X$ is
isomorphic in $\mathcal{P}$ to a discrete space
$\mathbf{m}=\\{0,1,2,\ldots,m-1\\}$ with the discrete $\sigma$-algebra where
the integer $m$ corresponds to the number of atoms in the $\sigma$-algebra
$\Sigma_{X}$. Consequently every finite space is, up to isomorphism, just a
discrete space and therefore every distribution $P\colon 1\rightarrow X$ is of
the form $P=\sum_{i=0}^{m-1}p_{i}\delta_{i}$ where $\sum_{i=0}^{m-1}p_{i}=1$.
### 2.1 (Weak) Product Spaces and Joint Distributions
In Bayesian probability, determining the joint distribution on a “product
space” is often the problem to be solved. In many applications for which
Bayesian reasoning in appropriate, the problem reduces to computing a
particular marginal or conditional probability; these can be obtained in a
straightforward way if the joint distribution is known. Before proceeding to
formulate precisely what the term “product space” means in $\mathcal{P}$, we
describe the categorical construct of a _finite product space_ in any
category.
Let $\mathcal{C}$ be an arbitary category and $X,Y\in_{ob}\mathcal{C}$. We say
the product of $X$ and $Y$ exists if there is an object, which we denote by
$X\times Y$, along with two arrows $p_{X}\colon X\times Y\rightarrow X$ and
$p_{Y}\colon X\times Y\rightarrow Y$ in $\mathcal{C}$ such that given any
other object $T$ in $\mathcal{C}$ and arrows $f:T\rightarrow X$ and
$g:T\rightarrow Y$ there is a _unique_ $\mathcal{C}$ arrow $\langle
f,g\rangle\colon T\rightarrow X\times Y$ that makes the diagram
$T$$X$$Y$$X\times Y$$f$$g$$\langle f,g\rangle$$p_{X}$$p_{Y}$ (2)
commute. If the given diagram is a product then we often write the product as
a triple $(X\times Y,p_{X},p_{Y})$. We must not let the notation deceive us;
the object $X\times Y$ could just as well be represented by $P_{X,Y}$. The
important point is that it is an object in $\mathcal{C}$ that we need to
specify in order to show that binary products exist. Products are an example
of a universal construction in categories. The term “universal” implies that
these constructions are unique up to a unique isomorphism. Thus if
$(P_{X,Y},p_{X},p_{y})$ and $(Q_{X,Y},q_{X},q_{Y})$ are both products for the
objects $X$ and $Y$ then there exist unique arrows $\alpha\colon
P_{X,Y}\rightarrow Q_{X,Y}$ and $\beta\colon Q_{X,Y}\rightarrow P_{X,Y}$ in
$\mathcal{C}$ such that $\beta\circ\alpha=1_{P_{X,Y}}$ and
$\alpha\circ\beta=1_{Q_{X,Y}}$ so that the objects $P_{X,Y}$ and $Q_{X,Y}$ are
isomorphic.
If the product of all object pairs $X$ and $Y$ exist in $\mathcal{C}$ then we
say binary products exist in $\mathcal{C}$. The existence of binary products
implies the existence of arbitrary finite products in $\mathcal{C}$. So if
$\\{X_{i}\\}_{i=1}^{N}$ is a finite set of objects in $\mathcal{C}$ then there
is an object which we denote by $\prod_{i=1}^{N}X_{i}$ (in general, this need
not be the cartesian product) as well as arrows
$\\{p_{X_{j}}:\prod_{i=1}^{N}X_{i}\rightarrow X_{j}\\}_{j=1}^{N}$. Then if we
are given an arbitrary $T\in_{ob}C$ and a family of arrows $f_{j}:T\rightarrow
X_{j}$ in $\mathcal{C}$ there exists a unique $\mathcal{C}$ arrow $\langle
f_{1},\ldots,f_{N}\rangle$ such that for every integer
$j\in\\{1,2,\ldots,N\\}$ the diagram
$T$$X_{j}$$\displaystyle{\prod_{i=1}^{N}}X_{i}$$f_{j}$$\langle
f_{1},\ldots,f_{N}\rangle$$p_{X_{j}}$
commutes. The arrows $p_{X_{i}}$ defining a product space are often called the
projection maps due to the analogy with the cartesian products in the category
of sets, $\mathcal{S}et$.
In $\mathcal{S}et$, the product of two sets $X$ and $Y$ is the cartesian
product $X\times Y$ consisting of all pairs $(x,y)$ of elements with $x\in X$
and $y\in Y$ along with the two projection mappings $\pi_{X}\colon X\times
Y\rightarrow X$ sending $(x,y)\mapsto x$ and $\pi_{Y}\colon X\times
Y\rightarrow Y$ sending $(x,y)\mapsto y$. Given any pair of functions $f\colon
T\rightarrow X\times Y$ and $g\colon T\rightarrow X\times Y$ the function
$\langle f,g\rangle\colon T\rightarrow X\times Y$ sending
$t\mapsto(f(t),g(t))$ clearly makes Diagram 2 commute. But it is also the
unique such function because if $\gamma\colon T\rightarrow X\times Y$ were any
other function making the diagram commute then the equations
$(p_{X}\circ\gamma)(t)=f(t)\quad\textrm{ and }\quad(p_{Y}\circ\gamma)(t)=g(t)$
(3)
would also be satisfied. But since the function $\gamma$ has codomain $X\times
Y$ which consist of ordered pairs $(x,y)$ it follows that for each $t\in T$
that $\gamma(t)=\langle\gamma_{1}(t),\gamma_{2}(t)\rangle$ for some functions
$\gamma_{1}\colon T\rightarrow X$ and $\gamma_{2}\colon T\rightarrow Y$.
Substituting $\gamma=\langle\gamma_{1},\gamma_{2}\rangle$ into equations 3 it
follows that
$\begin{array}[]{c}f(t)=(p_{X}\circ(\langle\gamma_{1},\gamma_{2}\rangle))(t)=p_{X}(\gamma_{1}(t),\gamma_{2}(t))=\gamma_{1}(t)\\\
g(t)=(p_{Y}\circ(\langle\gamma_{1},\gamma_{2}\rangle))(t)=p_{Y}(\gamma_{2}(t),\gamma_{2}(t))=\gamma_{2}(t)\end{array}$
from which it follows $\gamma=\langle\gamma_{1},\gamma_{2}\rangle=\langle
f,g\rangle$ thereby proving that there exist at most one such function
$T\rightarrow X\times Y$ making the requisite Diagram 2 commute. If the
requirement of the uniqueness of the arrow $\langle f,g\rangle$ in the
definition of a product is dropped then we have the definition of a _weak
product_ of $X$ and $Y$.
Given the relationship between the categories $\mathcal{P}$ and
$\mathcal{M}eas$ it is worthwhile to examine products in $\mathcal{M}eas$.
Given $X,Y\in_{ob}\mathcal{M}eas$ the product $X\times Y$ is the cartesian
product $X\times Y$ of sets endowed with the smallest $\sigma$-algebra such
that the two set projection maps $\pi_{X}\colon X\times Y\rightarrow X$
sending $(x,y)\mapsto x$ and $\pi_{Y}\colon X\times Y\rightarrow Y$ sending
$(x,y)\mapsto y$ are measurable. In other words, we take the smallest subset
of the powerset of $X\times Y$ such that for all $A\in\Sigma_{X}$ and for all
$B\in\Sigma_{Y}$ the preimages $\pi_{X}^{-1}(A)=A\times Y$ and
$\pi_{Y}^{-1}(B)=X\times B$ are measurable. Since a $\sigma$-algebra requires
that the intersection of any two measurable sets is also measurable it follows
that $\pi_{X}^{-1}(A)\cap\pi_{Y}^{-1}(B)=A\times B$ must also be measurable.
Measurable sets of the form $A\times B$ are called rectangles and _generate_
the collection of all measurable sets defining the $\sigma$-algebra
$\Sigma_{X\times Y}$ in the sense that $\Sigma_{X\times Y}$ is equal to the
intersection of all $\sigma$-algebras containing the rectangles. When the
$\sigma$-algebra on a set is determined by the a family of maps
$\\{p_{k}\colon X\times Y\rightarrow Z_{k}\\}_{k\in K}$, where $K$ is some
indexing set such that all of these maps $p_{k}$ are measurable we say the
$\sigma$-algebra is induced (or generated) by the family of maps
$\\{p_{k}\\}_{k\in K}$.101010The terminology _initial_ is also used in lieu of
induced. The cartesian product $X\times Y$ with the $\sigma$-algebra induced
by the two projection maps $\pi_{X}$ and $\pi_{Y}$ is easily verified to be a
product of $X$ and $Y$ since given any two measurable maps $f\colon
Z\rightarrow X$ and $g\colon Z\rightarrow Y$ the map $\langle f,g\rangle\colon
Z\rightarrow X\times Y$ sending $z\mapsto(f(z),g(z))$ is the unique measurable
map satisfying the defining property of a product for $(X\times
Y,\pi_{X},\pi_{Y})$. This $\sigma$-algebra induced by the projection maps
$\pi_{X}$ and $\pi_{Y}$ is called the product $\sigma$-algebra and the use of
the notation $X\times Y$ in $\mathcal{M}eas$ will imply the product
$\sigma$-algebra on the set $X\times Y$.
Having the product $(X\times Y,\pi_{X},\pi_{Y})$ in $\mathcal{M}eas$ and the
fact that every measurable function $f\in_{ar}\mathcal{M}eas$ determines an
arrow $\delta_{f}\in_{ar}\mathcal{P}$, it is tempting to consider the triple
$(X\times Y,\delta_{\pi_{X}},\delta_{\pi_{Y}})$ as a potential product in
$\mathcal{P}$. However taking this triple fails to be a product space of $X$
and $Y$ in $\mathcal{P}$ because the uniqueness condition fails; given two
probability measures $P\colon 1\rightarrow X$ and $Q\colon 1\rightarrow Y$
there are many joint distributions $J$ making the diagram
$1$$X$$Y$$X\times Y$$P$$Q$$J$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$ (4)
commute. In particular, the tensor product measure defined on rectangles by
$(P\otimes Q)(A\times B)=P(A)Q(B)$ extends to a joint probability measure on
$X\times Y$ by
$(P\otimes Q)(\varsigma)=\int_{y\in
Y}P(\Gamma_{\overline{y}}^{-1}(\varsigma))\,dQ\quad\forall\varsigma\in\Sigma_{X\times
Y}$ (5)
or equivalently,
$(P\otimes Q)(\varsigma)=\int_{x\in
X}Q(\Gamma_{\overline{x}}^{-1}(\varsigma))\,dP\quad\forall\varsigma\in\Sigma_{X\times
Y}.$ (6)
Here $\overline{x}\colon Y\to X$ is the constant function at $x$ and
$\Gamma_{\overline{x}}\colon Y\to X\times Y$ is the associated graph function,
with $\overline{y}$ and $\Gamma_{\overline{y}}$ defined similarly. The fact
that $Q\otimes P=P\otimes Q$ is Fubini’s Theorem; by taking a rectangle
$\varsigma=A\times B\in\Sigma_{X\times Y}$ the equality of these two measures
is immediate since
$\begin{array}[]{lcl}(P\otimes Q)(A\times B)&=&\int_{y\in
Y}P(\underbrace{\Gamma_{\overline{y}}^{-1}(A\times
B)}_{=\left\\{\begin{array}[]{ll}A&\textrm{ iff }y\in B\\\ \emptyset&\textrm{
otherwise }\end{array}\right.})\,dQ\\\ &=&\int_{y\in B}P(A)\,dQ\\\
&=&P(A)\cdot Q(B)\\\ &=&\int_{x\in A}Q(B)\,dP\\\ &=&\int_{x\in
X}Q(\Gamma_{\overline{x}}^{-1}(A\times B))\,dP\\\ &=&(Q\otimes P)(A\times
B)\end{array}$ (7)
Using the fact that every measurable set $\varsigma$ in $X\times Y$ is a
countable union of rectangles, Fubini’s Theorem follows.
It is clear that in $\mathcal{P}$ the uniqueness condition required in the
definition of a product of $X$ and $Y$ will always fail unless at least one of
$X$ and $Y$ is a terminal object $1$, and consequently only weak products
exist in $\mathcal{P}$. However it is the nonuniqueness of products in
$\mathcal{P}$ that makes this category interesting. Instead of referring to
weak products in $\mathcal{P}$ we shall abuse terminology and simply refer to
them as products with the understanding that all products in $\mathcal{P}$ are
weak.
### 2.2 Constructing a Joint Distribution Given Conditionals
We now show how marginals and conditionals can be used to determine joint
distributions in $\mathcal{P}$. Given a conditional probability measure
$h\colon X\to Y$ and a probability measure $P_{X}\colon 1\to X$ on $X$,
consider the diagram
$1$$X$$Y$$X\times Y$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{h}$$h$ (8)
where $J_{h}$ is the uniquely determined joint distribution on the product
space $X\times Y$ defined on the rectangles of the $\sigma$-algebra
$\Sigma_{X}\times\Sigma_{Y}$ by
$J_{h}(A\times B)=\int_{A}h_{B}\,dP_{X}.$ (9)
The marginal of $J_{h}$ with respect to $Y$ then satisfies
$\delta_{\pi_{Y}}\circ J_{h}=h\circ P_{X}$ and the marginal of $J_{h}$ with
respect to $X$ is $P_{X}$. By a symmetric argument, if we are given a
probability measure $P_{Y}$ and conditional probability $k\colon Y\to X$ then
we obtain a unique joint distribution $J_{k}$ on the product space $X\times Y$
given on the rectangles by
$J_{k}(A\times B)=\int_{B}k_{A}\,dP_{Y}.$
However if we are given $P_{X},P_{Y},h,k$ as indicated in the diagram
$1$$X$$Y$,$X\times
Y$$P_{Y}$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{k}$$J_{h}$$h$$k$ (10)
then we have that $J_{h}=J_{k}$ if and only if the compatibility condition is
satisfied on the rectangles
$\int_{A}h_{B}\,dP_{X}=J(A\times B)=\int_{B}k_{A}\,dP_{Y}\quad\forall
A\in\Sigma_{X},\forall B\in\Sigma_{Y}.$ (11)
In the extreme case, suppose we have a conditional $h\colon X\to Y$ which
factors through the terminal object $1$ as
$X$$Y$$1$$h$$!$$Q$
where $!$ represents the unique arrow from $X\to 1$. If we are also given a
probability measure $P\colon 1\to X$, then we can calculate the joint
distribution determined by $P$ and $h=Q\circ!$ as
$\begin{array}[]{lcl}J(A\times B)&=&\int_{A}(Q\circ!)_{B}\,dP\\\ &=&P(A)\cdot
Q(B)\end{array}$
so that $J=P\otimes Q$. In this situation we say that the marginals $P$ and
$Q$ are _independent_. Thus in $\mathcal{P}$ independence corresponds to a
special instance of a conditional—one that factors through the terminal
object.
### 2.3 Constructing Regular Conditionals given a Joint Distribution
The following result is the theorem from which the inference maps in Bayesian
probability theory are constructed. The fact that we require equiperfect
families of probability measures is critical for the construction.
###### Theorem 1.
Let $X$ and $Y$ be countably generated measurable spaces and $(X\times
Y,\Sigma_{X\times Y})$ the product in $\mathcal{M}eas$ with projection map
$\pi_{Y}$. If $J$ is a joint distribution on $X\times Y$ with marginal
$P_{Y}=\delta_{\pi_{Y}}\circ J$ on $Y$, then there exists a $\mathcal{P}$
arrow $f$ that makes the diagram
$1$$Y$$X\times Y$$P_{Y}$$J$$\delta_{\pi_{Y}}$$f$ (12)
commute and satisfies
$\int_{A\times B}{\delta_{\pi_{Y}}}_{C}\,dJ=\int_{C}f_{A\times B}\,dP_{Y}.$
Moreover, this $f$ is the unique $\mathcal{P}$-morphism with these properties,
up to a set of $P_{Y}$-measure zero.
###### Proof.
Since $\Sigma_{X}$ and $\Sigma_{Y}$ are both countably generated, it follows
that $\Sigma_{X\times Y}$ is countably generated as well. Let $\mathcal{G}$ be
a countable generating set for $\Sigma_{X\times Y}$. For each
$A\in\mathcal{G}$, define a measure $\mu_{A}$ on $Y$ by
$\mu_{A}(B)=J(A\cap\pi_{Y}^{-1}B).$
Then $\mu_{A}$ is absolutely continuous with respect to $P_{Y}$ and hence we
can let $\widetilde{f}_{A}=\frac{d\mu_{A}}{dP_{Y}}$, the Radon–Nikodym
derivative. For each $A\in\mathcal{G}$ this Radon–Nikodym derivative is unique
up to a set of measure zero, say $\hat{A}$. Let
$N=\cup_{A\in\mathcal{A}}\hat{A}$ and $E_{1}=N^{c}$. Then
$\widetilde{f}_{A}|_{E_{1}}$ is unique for all $A\in\mathcal{A}$. Note that
$f_{X\times Y}=1$ and $f_{\emptyset}=0$ on $E_{1}$. The condition
$\widetilde{f}_{A}\leq 1$ on $E_{1}$ for all $A\in\mathcal{A}$ then follows.
For all $B\in\Sigma_{Y}$ and any countable union $\cup_{i=1}^{n}A_{i}$ of
disjoint sets of $\mathcal{A}$ we have
$\begin{array}[]{lcl}\int_{B\cap
E_{1}}\widetilde{f}_{\cup_{i=1}^{n}A_{i}}dP_{Y}&=&J\left((\cup_{i=1}^{n}A_{i})\cap\pi_{Y}^{-1}B\right)\\\
&=&\sum_{i=1}^{n}J(A_{i}\cap\pi_{Y}^{-1}B)\\\ &=&\int_{B\cap
E_{1}}\sum_{i=1}^{n}\widetilde{f}_{A_{i}}dP_{Y},\end{array}$
with the last equality following from the Monotone Convergence Theorem and the
fact that all of the $\widetilde{f}_{A_{i}}$ are nonnegative. From the
uniqueness of the Radon–Nikodym derivative it follows
$\widetilde{f}_{\cup_{i=1}^{n}A_{i}}=\sum_{i=1}^{n}\widetilde{f}_{A_{i}}\quad
P_{Y}\text{-a.e.}$
Since there exist only a countable number of finite collection of sets of
$\mathcal{A}$ we can find a set $E\subset E_{1}$ of $P_{Y}$-measure one such
that the normalized set function
$\widetilde{f}_{\cdot}(y)\colon\mathcal{A}\rightarrow[0,1]$ is finitely
additive on $E$.
These facts altogether show there exists a set $E\in\Sigma_{Y}$ with
$P_{Y}$-measure one where for all $y\in E$,
1. 1.
$0\leq\widetilde{f}_{A}(y)\leq 1\quad\forall A\in\mathcal{A}$,
2. 2.
$\widetilde{f}_{\emptyset}(y)=0$ and $\widetilde{f}_{X\times Y}(y)=1$, and
3. 3.
for any finite collection $\\{A_{i}\\}_{i=1}^{n}$ of disjoint sets of
$\mathcal{A}$ we have
$\widetilde{f}_{\cup_{i=1}^{n}A_{i}}(y)=\sum_{i=1}^{n}\widetilde{f}_{A_{i}}(y)$.
Thus the set function $\widetilde{f}\colon E\times\mathcal{A}\rightarrow[0,1]$
satisfies the condition that $\widetilde{f}(y,\cdot)$ is a probability measure
on the algebra $\mathcal{A}$. By the Caratheodory extension theorem there
exist a unique extension of $\widetilde{f}(y,\cdot)$ to a probability measure
$\hat{f}(y,\cdot)\colon\Sigma_{X\times Y}\rightarrow[0,1]$. Now define a set
function $f\colon Y\times\Sigma_{X\times Y}\to[0,1]$ by
$f(y,A)=\left\\{\begin{array}[]{ll}\hat{f}(y,A)&\textrm{if $y\in E$}\\\
J(A)&\textrm{if $y\notin E$}\end{array}\right..$
Since each $A\in\Sigma_{X\times Y}$ can be written as the pointwise limit of
an increasing sequence $\\{A_{n}\\}_{n=1}^{\infty}$ of sets
$A_{n}\in\mathcal{A}$ it follows that
$f_{A}=\lim_{n\rightarrow\infty}f_{A_{n}}$ is measurable. From this we also
obtain the desired commutativity of the diagram
$\begin{array}[]{lcl}f\circ
P_{Y}(A)&=&\int_{Y}f_{A}dP_{Y}=\int_{E}f_{A}dP_{Y}=\lim_{n\rightarrow\infty}\int_{E}\widetilde{f}_{A_{n}}dP_{Y}\\\
&=&\lim_{n\rightarrow\infty}\int_{Y}\widetilde{f}_{A_{n}}dP_{Y}\\\
&=&\lim_{n\rightarrow\infty}J(A_{n})\\\ &=&J(A)\end{array}$
∎
We can use the result from Theorem 1 to obtain a broader understanding of the
situation.
###### Corollary 2.
Let $X$ and $Y$ be countably generated measurable spaces and $J$ a joint
distribution on $X\times Y$ with marginal distributions $P_{X}$ and $P_{Y}$ on
$X$ and $Y$, respectively. Then there exist $\mathcal{P}$ arrows $f$ and $g$
such that the diagram
$1$$Y$$X$$X\times
Y$$P_{Y}$$P_{X}$$J$$\delta_{\pi_{Y}}$$\delta_{\pi_{X}}$$\delta_{\pi_{X}}\circ
f$$f$$g$$\delta_{\pi_{Y}}\circ g$
commutes and
$\int_{U}(\delta_{\pi_{Y}}\circ g)_{V}\,dP_{X}=J(U\times
V)=\int_{V}(\delta_{\pi_{X}}\circ f)_{U}\,dP_{Y}.$
###### Proof.
From Theorem 1 there exist a $\mathcal{P}$ arrow $Y\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X\times Y$ satisfying $J=f\circ P_{Y}$. Take the
composite $\delta_{\pi_{X}}\circ f$ and note $(\delta_{\pi_{X}}\circ
f)_{U}(y)=f_{y}(U\times Y)$ giving
$\begin{array}[]{lcl}\int_{V}(\delta_{\pi_{X}}\circ
f)_{U}dP_{Y}&=&\int_{V}f_{U\times Y}dP_{Y}\\\ &=&J(U\times
Y\cap\pi_{Y}^{-1}V)\\\ &=&J(U\times V)\end{array}$
Similarly using a $\mathcal{P}$ arrow $X\stackrel{{\scriptstyle
g}}{{\longrightarrow}}X\times Y$ satisfying $J=g\circ P_{X}$ gives
$\int_{U}(\delta_{\pi_{Y}}\circ g)_{V}dP_{X}=J(U\times V).$
∎
Note that if the joint distribution $J$ is _defined_ by a probability measure
$P_{X}$ and a conditional $h\colon X\rightarrow Y$ using Diagram 8, then using
the above result and notation it follows $h=\delta_{\pi_{Y}}\circ g$.
## 3 The Bayesian Paradigm using $\mathcal{P}$
The categorical paradigm of Bayesian probability can be compactly summarized
with as follows. Let $D$ and $H$ be measurable spaces, which model a data and
hypothesis space, respectively. For example, $D$ might be a Euclidean space
corresponding to some measurements that are being taken and $H$ a
parameterization of some decision that needs to be made.
$1$$H$$D$$P_{H}$$\mathcal{S}$$\mathcal{I}$ Figure 2: The generic Bayesian
model.
The notation $\mathcal{S}$ is used to emphasize the fact we think of
$\mathcal{S}$ as a _sampling distribution_ on $D$. In the context of Bayesian
probability the (perfect) probability measure $P_{H}$ is often called a _prior
probability_ or, for brevity, just a _prior_. Given a prior $P$ and sampling
distribution $\mathcal{S}$ the joint distribution $J\colon 1\rightarrow
H\times D$ can be constructed using Definition 9. Using the marginal
$P_{D}=\mathcal{S}\circ P_{H}$ on $D$ it follows by Corollary 2.2 there exist
an arrow $f\colon D\rightarrow H\times D$ satisfying $J=f\circ P_{D}$.
Composing this arrow $f$ with the coordinate projection $\delta_{\pi_{H}}$
gives an arrow $\mathcal{I}=\delta_{\pi_{H}}\circ f\colon D\rightarrow H$
which we refer to as the inference map, and it satisfies
$\int_{B}\mathcal{I}_{A}\,dP_{D}=J(A\times
B)=\int_{A}\mathcal{S}_{B}\,dP_{H}\quad\forall A\in\Sigma_{H},\textrm{ and
}\forall B\in\Sigma_{D}$ (13)
which is called the product rule.
With the above in mind we formally define a Bayesian model to consist of
1. (i)
two measurable spaces $H$ and $D$ representing hypotheses and data,
respectively,
2. (ii)
a probability measure $P_{H}$ on the $H$ space called the prior probability,
3. (iii)
a $\mathcal{P}$ arrow $\mathcal{S}\colon H\rightarrow D$ called the sampling
distribution,
The sampling distribution $\mathcal{S}$ and inference map $\mathcal{I}$ are
often written as $P_{D\mid Y}$ and $P_{H\mid D}$, respectively, although using
the notation $P_{\cdot\mid\cdot}$ for all arrows in the category which are
necessarily conditional probabilities is notationally redundant and
nondistinguishing (requiring the subscripts to distinguish arrows).
Given this model and a measurement $\mu$, which is often just a point mass on
$D$ (i.e., $\mu=\delta_{d}\colon 1\to D$), there is an update procedure that
incorporates this measurement and the prior probability. Thus the measurement
$\mu$ can itself be viewed as a probability measure on $D$, and the
“posterior” probability measure can be calculated as
$\hat{P}_{H}=\mathcal{I}\circ\mu$ on $H$ provided the measurement $\mu$ is
absolutely continuous with respect to $P_{D}$, which we write as $\mu\ll
P_{D}$. Informally, this means that the observed measurement is considered
“possible” with respect to prior assumptions.
Let us expand upon this condition $\mu\ll P_{D}$ more closely. We know from
Theorem 1 that the inference map $\mathcal{I}$ is uniquely determined by
$P_{H}$ and $\mathcal{S}$ up to a set of $P_{D}$-measure zero. In general,
there is no reason a priori that an arbitrary (perfect) probability
measurement $\mu\colon 1\to D$ is required to be absolutely continuous with
respect to $P_{D}$. If $\mu$ is not absolutely continuous with respect to
$P_{D}$, then a different choice of inference map $\mathcal{I}^{\prime}$ could
yield a different posterior probability—i.e., we could have
$\mathcal{I}\circ\mu\neq\mathcal{I}^{\prime}\circ\mu$. Thus we make the
assumption that measurement probabilities on $D$ are absolutely continuous
with respect to the prior probability $P_{D}$ on $D$.
In practice this condition is often not met. For example the probability
measure $P_{D}$ may be a normal distribution on $\mathbb{R}$ and consequently
$P_{D}(\\{y\\})=0$ for any point $y\in\mathbb{R}$. Since Dirac measurements do
not satisfy $\delta_{y}\ll P_{D}$, this could create a problem. However, it is
clear that the Dirac measures can be approximated arbitrarily closely by a
limiting process of sharply peaked normal distributions which do satisfy this
absolute continuity condition. Thus while the absolute continuity condition
may not be satisfied precisely the error in approximating the measurement by
assuming a Dirac measure is negligible. Thus it is standard to assume that
measurements belong to a particular class of probability measures on $D$ which
are broad enough to approximate measurements and known to be absolutely
continuous with respect to the prior.
In summary, the Bayesian process works in the following way. Given a prior
probability $P_{H}$ and sampling distribution $\mathcal{S}$ one determines the
inference map $\mathcal{I}$. (For computational purposes the construction of
the entire map $\mathcal{I}$ is in general not necessary.) Once a measurement
$\mu\colon 1\to D$ is taken, we then calculate the posterior probability by
$\mathcal{I}\circ\mu$. This updating procedure can be characterized by the
diagram
$1$$H$$D$$P_{H}$$\mu$$\mathcal{S}$$\mathcal{I}$$\mathcal{I}\circ\mu$ (14)
where the solid lines indicate arrows given a priori, the dotted line
indicates the arrow determined using Theorem 1, and the dashed lines indicate
the updating after a measurement. Note that if there is no uncertainty in the
measurement, then $\mu=\delta_{\\{x\\}}$ for some $x\in D$, but in practice
there is usually some uncertainty in the measurements themselves. Consequently
the posterior probability must be computed as a composite - so the _posterior
probability_ of an event $A\in\Sigma_{H}$ given a measurement $\mu$ is
$(\mathcal{I}\circ\mu)(A)=\int_{D}\mathcal{I}_{A}(x)\,d\mu$.
Following the calculation of the posterior probability, the sampling
distribution is then updated, if required. The process can then repeat: using
the posterior probability and the updated sampling distribution the updated
joint probability distribution on the product space is determined and the
corresponding (updated) inference map determined (for computational purposes
the “entire map” $\mathcal{I}$ need not be determined if the measurements are
deterministic). We can then continue to iterate as long as new measurements
are received. For some problems, such as with the standard urn problem with
replacement of balls, the sampling distribution does not change from iterate
to iterate, but the inference map is updated since the posterior probability
on the hypothesis space changes with each measurement.
###### Remark 3.
Note that for countable spaces $X$ and $Y$ the compatibility condition reduces
to the standard Bayes equation since for any $x\in X$ the singleton
$\\{x\\}\in\Sigma_{X}$ and similarly any element $y\in Y$ implies
$\\{y\\}\in\Sigma_{Y}$, so that the joint distribution $J\colon 1\rightarrow
X\times Y$ on $\\{x\\}\times\\{y\\}$ reduces to the equation
$\mathcal{S}(\\{y\\}\mid
x)P_{X}(\\{x\\})=J(\\{x\\}\times\\{y\\})=\mathcal{I}(\\{x\\}\mid
y)P_{Y}(\\{y\\})$ (15)
which in more familiar notation is the Bayesian equation
$P(y\mid x)P(x)=P(x,y)=P(x\mid y)P(y).$ (16)
## 4 Elementary applications of Bayesian probability
Before proceeding to show how the category $\mathcal{P}$ can be can be applied
to ML where the unknowns are functions, we illustrate its use to solve
inference, prediction, and decision processes in the more familiar setting
where the unknown parameter(s) are real values. We present two elementary
problems illustrating basic model building using categorical diagrams, much
like that used in probabilistic graphical models for Bayesian networks, which
can serve to clarify the modeling aspect of any probabilistic problem.
To illustrate the inference-sampling distribution relationship and how we make
computations in the category $\mathcal{P}$, we consider first an urn problem
where we have discrete $\sigma$-algebras. The discreteness condition is not
critical as we will eventually see - it only makes the analysis and
_computational_ aspect easier.
###### Example 4.
Million dollar draw.111111 This problem is taken from Peter Green’s tutorial
on Bayesian Inference which can be viewed at
http://videolectures.net/mlss2011_green_bayesian.
RBRBRUrn 1Urn 2RBBB
You are given two draws and if you pull out a red ball you win a million
dollars. You are unable to see the two urns so you don’t know which urn you
are drawing from and the draw is done without replacement. The $\mathcal{P}$
diagram for both inference and calculating sampling distributions is given by
$1$$U$$B$$P_{U}$$\mathcal{S}$$\mathcal{I}$$P_{B}$
where the dashed arrows indicate morphisms to be calculated rather than
morphisms determined by modeling,
$\begin{array}[]{l}U=\\{u_{1},u_{2}\\}=\textrm{\\{Urn 1, Urn 2\\}}\\\
B=\\{b,r\\}=\textrm{\\{blue, red\\}}\end{array}$
and
$P_{U}=\frac{1}{2}\delta_{u_{1}}+\frac{1}{2}\delta_{u_{2}}.$
The sampling distribution is the binomial distribution given by
$\begin{array}[]{ll}\mathcal{S}(\\{b\\}\mid
u_{1})=\frac{2}{5}&\mathcal{S}(\\{r\\}\mid u_{1})=\frac{3}{5}\\\
\mathcal{S}(\\{b\\}\mid u_{2})=\frac{3}{4}&\mathcal{S}(\\{r\\}\mid
u_{2})=\frac{1}{4}.\end{array}$
Suppose that on our first draw, we draw from one of the urns (which one is
unknown) and draw a blue ball. We ask the following questions:
1. 1.
(Inference) What is the probability that we made the draw from Urn 1 (Urn 2)?
2. 2.
(Prediction) What is the probability of drawing a red ball on the second draw
(from the same urn)?
3. 3.
(Decision) Given you have drawn a blue ball on the first draw should you
switch urns to increase the probability of drawing a red ball?
To solve these problems, we implicitly or explicitly construct the joint
distribution $J$ via the standard construction given $P_{U}$ and the
conditional $\mathcal{S}$
$1$$U$$B$$U\times B$$P_{B}=\mathcal{S}\circ
P_{U}$$\delta_{\pi_{B}}$$P_{U}$$\delta_{\pi_{U}}$$J$$\mathcal{S}$
and then construct the inference map by requiring the compatibility condition,
i.e., the integral equation
$\int_{u\in
U}\mathcal{S}(\mathcal{B}|u)dP_{U}=J(\mathcal{B}\times\mathcal{H})=\int_{c\in
B}\mathcal{I}(\mathcal{H}|c)dP_{B}\quad\forall\mathcal{B}\in\Sigma_{B}\quad\forall\mathcal{H}\in\Sigma_{U}$
(17)
is satisfied. Since our problem is discrete the integral reduces to a sum.
Our first step is to calculate the prior on $B$ which is the composite
$P_{B}=\mathcal{S}\circ P_{U}$, from which we calculate
$\begin{array}[]{lcl}P_{B}(\\{b\\})&=&(\mathcal{S}\circ P_{U})(\\{b\\})\\\
&=&\int_{v\in U}\mathcal{S}(\\{b\\}|v)dP_{U}\\\ &=&\int_{v\in
U}\mathcal{S}(\\{b\\}|v)d(\frac{1}{2}\delta_{u_{1}}+\frac{1}{2}\delta_{u_{2}})\\\
&=&\mathcal{S}(\\{b\\}|u_{1})\cdot
P_{U}(\\{u_{1}\\})+\mathcal{S}(\\{b\\}|u_{2})\cdot P_{U}(\\{u_{2}\\})\\\
&=&\frac{2}{5}\cdot\frac{1}{2}+\frac{3}{4}\cdot\frac{1}{2}\\\
&=&\frac{23}{40}\end{array}$
and similarly
$P_{B}(\\{r\\})=\frac{17}{40}.$
To solve the _inference_ problem, we need to compute the values of the
inference map $\mathcal{I}$ using equation 17. This amounts to computing the
joint distribution on all possible measurable sets,
$\begin{array}[]{l}\int_{\\{u_{1}\\}}\mathcal{S}(\\{b\\}|u)dP_{U}=J(\\{u_{1}\\}\times\\{b\\})=\int_{\\{b\\}}\mathcal{I}(\\{u_{1}\\}|c)dP_{B}\\\
\int_{\\{u_{2}\\}}\mathcal{S}(\\{b\\}|u)dP_{U}=J(\\{u_{2}\\}\times\\{b\\})=\int_{\\{b\\}}\mathcal{I}(\\{u_{2}\\}|c)dP_{B}\\\
\int_{\\{u_{1}\\}}\mathcal{S}(\\{r\\}|u)dP_{U}=J(\\{u_{1}\\}\times\\{r\\})=\int_{\\{r\\}}\mathcal{I}(\\{u_{1}\\}|c)dP_{B}\\\
\int_{\\{u_{2}\\}}\mathcal{S}(\\{r\\}|u)dP_{U}=J(\\{u_{2}\\}\times\\{r\\})=\int_{\\{r\\}}\mathcal{I}(\\{u_{2}\\}|c)dP_{B}\end{array}$
which reduce to the equations
$\begin{array}[]{l}\mathcal{S}(\\{b\\}|u_{1})\cdot
P_{U}(\\{u_{1}\\})=\mathcal{I}(\\{u_{1}\\}|b)\cdot P_{B}(\\{b\\})\\\
\mathcal{S}(\\{b\\}|u_{2})\cdot
P_{U}(\\{u_{2}\\})=\mathcal{I}(\\{u_{2}\\}|b)\cdot P_{B}(\\{b\\})\\\
\mathcal{S}(\\{r\\}|u_{1})\cdot
P_{U}(\\{u_{1}\\})=\mathcal{I}(\\{u_{1}\\}|r)\cdot P_{B}(\\{r\\})\\\
\mathcal{S}(\\{r\\}|u_{2})\cdot
P_{U}(\\{u_{2}\\})=\mathcal{I}(\\{u_{2}\\}|r)\cdot P_{B}(\\{r\\}).\\\
\end{array}$
Substituting values for $\mathcal{S}$, $P_{B}$, and $P_{I}$ one determines
$\begin{array}[]{ll}\mathcal{I}(\\{u_{1}\\}|b)=\frac{8}{23}&\mathcal{I}(\\{u_{2}\\}|b)=\frac{15}{23}\\\
\\\
\mathcal{I}(\\{u_{1}\\}|r)=\frac{12}{17}&\mathcal{I}(\\{u_{2}\\}|r)=\frac{5}{17}\end{array}$
which answers question (1). The odds that one drew the blue ball from Urn 1
relative to Urn 2 are $\frac{8}{15}$, so it is almost twice as likely that one
made the draw from the second urn.
The Prediction Problem. Here we implicitly (or explicitly) need to construct
the product space $U\times B_{1}\times B_{2}$ where $B_{i}$ represents the
$i^{th}$ drawing of a ball from the same (unknown) urn. To do this we use the
basic construction for joint distributions using a regular conditional
probability, $\mathcal{S}_{2}$, which expresses the probability of drawing
either a red or a blue ball _from the same urn_ as the first draw. This
conditional probability is given by
$\begin{array}[]{ll}\mathcal{S}_{2}(\\{b\\}|(u_{1},b))=\frac{1}{4}&\mathcal{S}_{2}(\\{r\\}|(u_{1},b))=\frac{3}{4}\\\
\mathcal{S}_{2}(\\{b\\}|(u_{2},b))=\frac{2}{3}&\mathcal{S}_{2}(\\{r\\}|(u_{2},b))=\frac{1}{3}\\\
\mathcal{S}_{2}(\\{b\\}|(u_{1},r))=\frac{1}{2}&\mathcal{S}_{2}(\\{r\\}|(u_{1},r))=\frac{1}{2}\\\
\mathcal{S}_{2}(\\{b\\}|(u_{2},r))=1&\mathcal{S}_{2}(\\{r\\}|(u_{2},r))=0.\end{array}$
Now we construct the joint distribution $K$ on the product space $(U\times
B_{1})\times B_{2}$
$1$$U\times B_{1}$$B_{2}.$$U\times B_{1}\times
B_{2}$$P_{B_{2}}=\mathcal{S}_{2}\circ
J$$\delta_{\pi_{B_{2}}}$$J$$\delta_{\pi_{U\times B_{1}}}$$K$$\mathcal{S}_{2}$
To answer the prediction question we calculate the odds of drawing a red
versus a blue ball. Thus
$K(U\times\\{b\\}\times\\{r\\})=\int_{U\times\\{b\\}}\mathcal{S}_{2}({\\{r\\}}|(u,\beta))dJ,$
(18)
where the right hand side follows from the definition (construction) of the
iterated product space $(U\times B_{1})\times B_{2}$. The computation of the
expression 18 yields
$\begin{array}[]{lcl}K(U\times\\{b\\}\times\\{r\\})&=&\int_{U\times\\{b\\}}\mathcal{S}_{2}({\\{r\\}}|(u,\beta))dJ\\\
&=&\underbrace{\mathcal{S}(\\{r\\}|(u_{1},b))}_{=\frac{3}{4}}\cdot\underbrace{J(\\{u_{1}\\}\times\\{b\\})}_{=\frac{1}{5}}+\underbrace{\mathcal{S}(\\{r\\}|(u_{2},b))}_{=\frac{1}{3}}\cdot\underbrace{J(\\{u_{2}\\}\times\\{b\\})}_{=\frac{3}{8}}\\\
&=&\frac{11}{40}.\end{array}$
Similarly $K(U\times\\{b\\}\times\\{b\\})=\frac{12}{40}$. So the odds are
$\frac{r}{b}=\frac{11}{12}\quad Pr(\\{r\\}|\\{b\\})=\frac{11}{23}.$
The Decision Problem To answer the decision problem we need to consider the
conditional probability of switching urns on the second draw which leads to
the conditional
$U\times B_{1}$$B_{2}$$\hat{\mathcal{S}}_{2}$
given by
$\begin{array}[]{ll}\hat{\mathcal{S}}_{2}(\\{b\\}|(u_{1},b))=\frac{3}{4}&\hat{\mathcal{S}}_{2}(\\{r\\}|(u_{1},b))=\frac{1}{4}\\\
\hat{\mathcal{S}}_{2}(\\{b\\}|(u_{2},b))=\frac{2}{5}&\hat{\mathcal{S}}_{2}(\\{r\\}|(u_{2},b))=\frac{3}{5}\\\
\hat{\mathcal{S}}_{2}(\\{b\\}|(u_{1},r))=\frac{3}{4}&\hat{\mathcal{S}}_{2}(\\{r\\}|(u_{1},r))=\frac{1}{4}\\\
\hat{\mathcal{S}}_{2}(\\{b\\}|(u_{2},r))=\frac{2}{5}&\hat{\mathcal{S}}_{2}(\\{r\\}|(u_{2},r))=\frac{3}{5}.\end{array}$
Carrying out the same computation as above we find the joint distribution
$\hat{K}$ on the product space $(U\times B_{1})\times B_{2}$ constructed from
$J$ and $\hat{\mathcal{S}}_{2}$ yields
$\begin{array}[]{lcl}\hat{K}(U\times\\{b\\}\times\\{r\\})&=&\int_{U\times\\{b\\}}\hat{\mathcal{S}}_{2}(\\{r\\}|(u,\beta))dJ\\\
&=&\hat{\mathcal{S}_{2}}(\\{r\\}|(u_{1},b))J(\\{u_{1}\\}\times\\{b\\})+\hat{\mathcal{S}_{2}}(\\{r\\}|(u_{2},b))J(\\{u_{2}\\}\times\\{b\\})\\\
&=&\frac{1}{4}\cdot\frac{1}{5}+\frac{3}{5}\cdot\frac{3}{8}\\\
&=&\frac{11}{40},\end{array}$
which shows that it doesn’t matter whether you switch or not - you get the
same probability of drawing a red ball.
The probability of drawing a blue ball is
$\hat{K}(U\times\\{b\\}\times\\{b\\})=\frac{12}{40}=K(U\times\\{b\\}\times\\{b\\}),$
so the odds of drawing a blue ball outweigh the odds of drawing a red ball by
the ratio $\frac{12}{11}$. The odds are against you.
Here is an example illustrating that the regular conditional probabilities
(inference or sampling distributions) are defined only up to sets of measure
zero.
###### Example 5.
We have a rather bland deck of three cards as shown
Card 1Card 2Card 3FrontBack$R$$R$$R$$G$$G$$G$
We shuffle the deck, pull out a card and expose one face which is red.121212
This problem is taken from David MacKays tutorial on Information Theory which
can be viewed at $http://videolectures.net/mlss09uk\\_mackay\\_it/$. The
prediction question is
What is the probability the other side of the card is red?
To answer this note that this card problem is identical to the urn problem
with urns being cards and balls becoming the colored sides of each card. Thus
we have an analogous model in $\mathcal{P}$ for this problem. Let
$\begin{array}[]{l}C(ard)=\\{1,2,3\\}\\\ F(ace\,Color)=\\{r,g\\}.\end{array}$
We have the $\mathcal{P}$ diagram
$1$$C$$F$$P_{C}$$\mathcal{S}$$\mathcal{I}$$P_{F}$
with the sampling distribution given by
$\begin{array}[]{lcl}\mathcal{S}(\\{r\\}|1)=1&\mathcal{S}(\\{g\\}|1)=0\\\
\mathcal{S}(\\{r\\}|2)=\frac{1}{2}&\mathcal{S}(\\{g\\}|2)=\frac{1}{2}\\\
\mathcal{S}(\\{r\\}|3)=0&\mathcal{S}(\\{g\\}|3)=1.\\\ \end{array}$
The prior on $C$ is
$P_{C}=\frac{1}{3}\delta_{1}+\frac{1}{3}\delta_{2}+\frac{1}{3}\delta_{3}$.
From this we can construct the joint distribution on $C\times F$
$1$$C$$F.$$C\times F$$P_{F}=\mathcal{S}\circ
P_{C}$$\delta_{\pi_{F}}$$P_{C}$$\delta_{\pi_{C}}$$J$$\mathcal{S}$
Using
$J(A\times B)=\int_{n\in A}\mathcal{S}(B|n)dP_{C},$
we find
$\begin{array}[]{lcl}J(\\{1\\}\times\\{r\\})=\frac{1}{3}&J(\\{1\\}\times\\{g\\})=0\\\
J(\\{2\\}\times\\{r\\})=\frac{1}{6}&J(\\{2\\}\times\\{g\\})=\frac{1}{6}\\\
J(\\{3\\}\times\\{r\\})=0&J(\\{3\\}\times\\{g\\})=\frac{1}{3}.\\\ \end{array}$
Now, like in the urn problem, to predict the next draw (flip of the card), it
is necessary to add another measurable set $F_{2}$ and conditional probability
$\mathcal{S}_{2}$ and construct the product diagram and joint distribution $K$
$1$$C\times F_{1}$$F_{2}$.$C\times F_{1}\times
F_{2}$$P_{F_{2}}=\mathcal{S}_{2}\circ
J$$\delta_{\pi_{F_{2}}}$$J$$\delta_{\pi_{C\times F_{1}}}$$K$$\mathcal{S}_{2}$
The twist now arises in that the conditional probability $\mathcal{S}_{2}$ is
not uniquely defined - what are the values
$\mathcal{S}_{2}(\\{r\\}|(1,g))=~{}?\quad\mathcal{S}_{2}(\\{g\\}|(1,g))=~{}?$
The answer is it doesn’t matter what we put down for these values since they
have measure $J(\\{1\\}\times\\{g\\})=0$. We can still compute the desired
quantity of interest proceeding forth with these arbitrarily chosen values on
the point sets of measure zero. Thus we choose
$\begin{array}[]{ll}\mathcal{S}_{2}(\\{g\\}|(1,r))=0&\mathcal{S}_{2}(\\{r\\}|(1,r))=1\\\
\mathcal{S}_{2}(\\{g\\}|(1,g))=1&\mathcal{S}_{2}(\\{r\\}|(1,g))=0\quad\textrm{doesn't
matter}\\\
\mathcal{S}_{2}(\\{g\\}|(2,r))=1&\mathcal{S}_{2}(\\{r\\}|(2,r))=0\\\
\mathcal{S}_{2}(\\{g\\}|(2,g))=0&\mathcal{S}_{2}(\\{r\\}|(2,g))=1\\\
\mathcal{S}_{2}(\\{g\\}|(3,r))=0&\mathcal{S}_{2}(\\{r\\}|(3,r))=1\quad\textrm{doesn't
matter}\\\
\mathcal{S}_{2}(\\{g\\}|(3,g))=1&\mathcal{S}_{2}(\\{r\\}|(3,g))=0.\end{array}$
We chose the arbitrary values such that $\mathcal{S}_{2}$ is a deterministic
mapping which seems appropriate since flipping a given card uniquely
determined the color on the other side.
Now we can solve the prediction problem by computing the joint measure values
$\begin{array}[]{lcl}K(C\times\\{r\\}\times\\{r\\})&=&\int_{C\times\\{r\\}}(\mathcal{S}_{2})_{\\{r\\}}(n,c)dJ\\\
&=&\mathcal{S}_{2}(\\{r\\}|(1,r))\cdot
J(\\{1\\}\times\\{r\\})+\mathcal{S}_{2}(\\{r\\}|(2,r))\cdot
J(\\{2\\}\times\\{r\\})\\\ &=&1\cdot\frac{1}{3}+0\cdot\frac{1}{6}\\\
&=&\frac{1}{3}\end{array}$
and
$\begin{array}[]{lcl}K(C\times\\{r\\}\times\\{g\\})&=&\int_{C\times\\{r\\}}\mathcal{S}_{2}(\\{g\\}|(n,c))dJ\\\
&=&\mathcal{S}_{2}(\\{g\\}|(1,r))\cdot
J(\\{1\\}\times\\{r\\})+\mathcal{S}_{2}(\\{g\\}|(2,r))\cdot
J(\\{2\\}\times\\{r\\})\\\ &=&0\cdot\frac{1}{3}+1\cdot\frac{1}{6}\\\
&=&\frac{1}{6},\end{array}$
so it is twice as likely to observe a red face upon flipping the card than
seeing a green face. Converting the odds of $\frac{r}{g}=\frac{2}{1}$ to a
probability gives $Pr(\\{r\\}|\\{r\\})=\frac{2}{3}$.
To test one’s understanding of the categorical approach to Bayesian
probability we suggest the following problem.
###### Example 6.
The Monty Hall Problem. You are a contestant in a game show in which a prize
is hidden behind one of three curtains. You will win a prize if you select the
correct curtain. After you have picked one curtain but befor the curtain is
lifted, the emcee lifts one of the other curtains, revealing a goat, and asks
if you would like to switch from your current selection to the remaining
curtain. How will your chances change if you switch?
There are three components which need modeled in this problem:
$\begin{array}[]{l}D(oor)=\\{1,2,3\\}\quad\textrm{The prize is behind this
door.}\\\ C(hoice)=\\{1,2,3\\}\quad\textrm{The door you chose.}\\\
O(penddoor)=\\{1,2,3\\}\quad\textrm{The door Monty Hall opens}\end{array}$
The prior on $D$ is
$P_{D}=\frac{1}{3}\delta_{d_{1}}+\frac{1}{3}\delta_{d_{2}}+\frac{1}{3}\delta_{d_{3}}$.
Your selection of a curtain, say curtain $1$, gives the deterministic measure
$P_{C}=\delta_{C_{1}}$. There is a conditional probability from the product
space $D\times C$ to $O$
$1$$D\times C$$O$$(D\times C)\times O$$P_{O}=\mathcal{S}\circ P_{D}\otimes
P_{C}$$\delta_{\pi_{O}}$$P_{D}\otimes P_{C}$$\delta_{\pi_{D\times
C}}$$J$$\mathcal{S}$
where the conditional probability $\mathcal{S}((i,j),\\{k\\})$ represents the
probability that Monty opens door $k$ given that the prize is behind door $i$
and you have chosen door $j$. If you have chosen curtain $1$ then we have the
partial data given by
$\begin{array}[]{lll}\mathcal{S}((1,1),\\{1\\})=0&\mathcal{S}((1,1),\\{2\\})=\frac{1}{2}&\mathcal{S}((1,1),\\{2\\})=\frac{1}{2}\\\
\mathcal{S}((2,1),\\{1\\})=0&\mathcal{S}((2,1),\\{2\\})=0&\mathcal{S}((2,1),\\{3\\})=1\\\
\mathcal{S}((3,1),\\{1\\})=0&\mathcal{S}((3,1),\\{2\\})=1&\mathcal{S}((3,1),\\{3\\})=0.\\\
\end{array}$
Complete the table, as necessary, to compute the inference conditional,
$D\times C\stackrel{{\scriptstyle\mathcal{I}}}{{\longleftarrow}}O$, and
conclude that if Monty opens either curtain $2$ or $3$ it is in your best
interest to switch doors.
## 5 The Tensor Product
Given any function $f\colon X\rightarrow Y$ the graph of $f$ is defined as the
set function
$\begin{array}[]{ccccc}\Gamma_{f}&\colon&X&\longrightarrow&X\times Y\\\
&\colon&x&\mapsto&(x,f(x)).\end{array}$
By our previous notation $\Gamma_{f}=\langle Id_{X},f\rangle$. If $g\colon
Y\rightarrow X$ is any function we also refer to the set function
$\begin{array}[]{ccccc}\Gamma_{g}&\colon&Y&\longrightarrow&X\times Y\\\
&\colon&y&\mapsto&(g(y),y)\end{array}$
as a graph function.
Any fixed $x\in X$ determines a constant function $\overline{x}\colon
Y\rightarrow X$ sending every $y\in Y$ to $x$. These functions are always
measurable and consequently determine “constant” graph functions
$\Gamma_{\overline{x}}\colon Y\rightarrow X\times Y$. Similarly, every fixed
$y\in Y$ determines a constant graph function $\Gamma_{\overline{y}}\colon
X\rightarrow X\times Y$. Together, these constant graph functions can be used
to define a $\sigma$-algebra on the set $X\times Y$ which is finer (larger)
than the product $\sigma$-algebra $\Sigma_{X\times Y}$. Let $X\otimes Y$
denote the set $X\times Y$ endowed with the largest $\sigma$-algebra structure
such that all the constant graph functions $\Gamma_{\overline{x}}\colon
X\rightarrow X\otimes Y$ and $\Gamma_{\overline{y}}\colon Y\rightarrow
X\otimes Y$ are measurable. We say this $\sigma$-algebra $X\otimes Y$ is
_coinduced_ by the maps $\\{\Gamma_{\overline{x}}\colon X\rightarrow X\times
Y\\}_{x\in X}$ and $\\{\Gamma_{\overline{y}}\colon Y\rightarrow X\times
Y\\}_{y\in Y}$. Explicitly, this $\sigma$-algebra is given by
$\Sigma_{X\otimes Y}=\bigcap_{x\in
X}{\Gamma_{\overline{x}}}_{\ast}\Sigma_{Y}\cap\bigcap_{y\in
Y}{\Gamma_{\overline{y}}}_{\ast}\Sigma_{X},$ (19)
where for any function $f\colon W\to Z$,
$f_{\ast}\Sigma_{W}=\\{C\in 2^{Z}\mid f^{-1}(C)\in\Sigma_{W}\\}.$ (20)
This is in contrast to the smallest $\sigma$-algebra on $X\times Y$, defined
in Section 2.1 so that the two projection maps $\\{\pi_{X}\colon X\times
Y\rightarrow X,\pi_{Y}\colon X\times Y\rightarrow Y\\}$ are measurable. Such a
$\sigma$-algebra is said to be _induced_ by the projection maps, or simply
referred to as the _initial_ $\sigma$-algebra.
The following result on coinduced $\sigma$-algebras is used repeatedly.
###### Lemma 7.
Let the $\sigma$-algebra of $Y$ be coinduced by a collection of maps
$\\{f_{i}\colon X_{i}\rightarrow Y\\}_{i\in I}$. Then any map $g\colon
Y\rightarrow Z$ is measurable if and only if the composition $g\circ f_{i}$ is
measurable for each $i\in I$.
###### Proof.
Consider the diagram
$X_{i}$$Y$$Z$$f_{i}$$g$$g\circ f_{i}$
If $B\in\Sigma_{Z}$ then $g^{-1}(B)\in\Sigma_{Y}$ if and only if
$f_{i}^{-1}(g^{-1}(B))\in\Sigma_{X}$. ∎
This result is used frequently when $Y$ in the above diagram is replaced by a
tensor product space $X\otimes Y$. For example, using this lemma it follows
that the projection maps $\pi_{Y}\colon X\otimes Y\rightarrow Y$ and
$\pi_{X}\colon X\otimes Y\rightarrow X$ are both measurable because the
diagrams in Figure 3 commute.
$X$$Y$$X\otimes
Y$$\overline{y}$$\Gamma_{\overline{y}}$$\pi_{Y}$$Y$$X$$X\otimes
Y$$\overline{x}$$\Gamma_{\overline{x}}$$\pi_{X}$ Figure 3: The commutativity
of these diagrams, together with the measurability of the constant functions
and constant graph functions, implies the projection maps $\pi_{X}$ and
$\pi_{Y}$ are measurable.
By the measurability of the projection maps and the universal property of the
product, it follows the identity mapping on the set $X\times Y$ yields a
measurable function
$X\otimes Y$$X\times Y$$id$
called the _restriction of the $\sigma$-algebra_. In contrast, the identity
function $X\times Y\rightarrow X\otimes Y$ is not necessarily measurable.
Given any probability measure $P$ on $X\otimes Y$ the restriction mapping
induces the pushforward probability measure $\delta_{id}\circ
P=P(id^{-1}(\cdot))$ on the product $\sigma$-algebra.
### 5.1 Graphs of Conditional Probabilities
The tensor product of two probability measures $P\colon 1\rightarrow X$ and
$Q\colon 1\rightarrow Y$ was defined in Equations 5 and 6 as the joint
distribution on the product $\sigma$-algebra by either of the expressions
$(P\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\ltimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\ltimes$\hfil\cr}}}Q)(\varsigma)=\int_{y\in
Y}P(\Gamma_{\overline{y}}^{-1}(\varsigma))\,dQ\quad\forall\varsigma\in\Sigma_{X\times
Y}$
and
$(P\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}Q)(\varsigma)=\int_{x\in
X}Q(\Gamma_{\overline{x}}^{-1}(\varsigma))\,dP\quad\forall\varsigma\in\Sigma_{X\times
Y}$
which are equivalent on the product $\sigma$-algebra. Here we have introduced
the new notation of left tensor $\textstyle\bigcirc$ $\textstyle\ltimes$ and
right tensor $\textstyle\bigcirc$ $\textstyle\rtimes$ because we can extend
these definitions to be defined on the tensor $\sigma$-algebra though in
general the equivalence of these two expressions may no longer hold true.
These definitions can be extended to conditional probability measures $P\colon
Z\rightarrow X$ and $Q\colon Z\rightarrow Y$ trivially by conditioning on a
point $z\in Z$,
$(P\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\ltimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\ltimes$\hfil\cr}}}Q)(\varsigma\mid
z)=\int_{y\in
Y}P(\Gamma_{\overline{y}}^{-1}(\varsigma))\,dQ_{z}\quad\forall\varsigma\in\Sigma_{X\otimes
Y}$ (21)
and
$(P\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}Q)(\varsigma\mid
z)=\int_{x\in
X}Q(\Gamma_{\overline{x}}^{-1}(\varsigma))\,dP_{z}\quad\forall\varsigma\in\Sigma_{X\otimes
Y}$ (22)
which are equivalent on the product $\sigma$-algebra but not on the tensor
$\sigma$-algebra. However in the special case when $Z=X$ and $P=1_{X}$, then
Equations 21 and 22 do coincide on $\Sigma_{X\otimes Y}$ because by Equation
21
$\begin{array}[]{lcl}(1_{X}\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\ltimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\ltimes$\hfil\cr}}}Q)(\varsigma\mid
x)&=&\int_{y\in
Y}\underbrace{\delta_{x}(\Gamma_{\overline{y}}^{-1}(\varsigma))}_{=\left\\{\begin{array}[]{ll}1&\textrm{
iff }(x,y)\in\varsigma\\\ 0&\textrm{ otherwise
}\end{array}\right.}\,dQ_{x}\quad\forall\varsigma\in\Sigma_{X\otimes Y^{X}}\\\
&=&\int_{y\in Y}\chi_{\Gamma_{\overline{x}}^{-1}(\varsigma)}(y)\,dQ_{x}\\\
&=&Q_{x}(\Gamma_{\overline{x}}^{-1}(\varsigma)),\end{array}$ (23)
while by Equation 22
$\begin{array}[]{lcl}(1_{X}\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}Q)(\varsigma\mid
x)&=&\int_{u\in
X}Q_{x}(\Gamma_{\overline{u}}^{-1}(\varsigma))\,d\underbrace{(\delta_{Id_{X}})_{x}}_{=\delta_{x}}\quad\forall\varsigma\in\Sigma_{X\otimes
Y^{X}}\\\ &=&Q_{x}(\Gamma_{\overline{x}}^{-1}(\mathcal{U})).\end{array}$ (24)
In this case we denote the common conditional by $\Gamma_{Q}$, called _the
graph of $Q$_ by analogy to the graph of a function, and this map gives the
commutative diagram in Figure 4.
$X$$X$$Y$$X\otimes
Y$$1_{X}$$Q$$\Gamma_{Q}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$ Figure 4: The
tensor product of a conditional with an identity map in $\mathcal{P}$.
The commutativity of the diagram in Figure 4 follows from
$\begin{array}[]{lcl}(\delta_{\pi_{X}}\circ\Gamma_{Q})(A\mid
x)&=&\int_{(u,v)\in X\otimes
Y}\delta_{\pi_{X}}(A\mid(u,v))\,d\underbrace{(\Gamma_{Q})_{x}}_{=Q\Gamma_{\overline{x}}^{-1}}\\\
&=&\int_{v\in Y}\delta_{\pi_{X}}(A\mid\Gamma_{\overline{x}}(v))\,dQ_{x}\\\
&=&\int_{v\in Y}\delta_{x}(A)dQ_{x}\\\ &=&\delta_{x}(A)\int_{Y}dQ_{x}\\\
&=&1_{X}(A\mid x)\end{array}$ (25)
and
$\begin{array}[]{lcl}(\delta_{\pi_{Y}}\circ\Gamma_{Q})(B\mid
x)&=&\int_{(u,v)\in X\otimes
Y}\delta_{\pi_{Y}}(B\mid(u,v))\,d((\Gamma_{Q})_{x})\\\ &=&\int_{v\in
Y}\delta_{\pi_{Y}}(B\mid(x,v))\,dQ_{x}\\\ &=&\int_{v\in
Y}\chi_{B}(v)\,dQ_{x}\\\ &=&Q(A\mid x).\end{array}$ (26)
### 5.2 A Tensor Product of Conditionals
Given any conditional $P\colon Z\rightarrow Y$ in $\mathcal{P}$ we can define
a tensor product $1_{X}\otimes P$ by
$(1_{X}\otimes
P)(\mathcal{A}\mid(x,z))=P(\Gamma_{\overline{x}}^{-1}(\mathcal{A})\mid
z)\quad\quad\forall\mathcal{A}\in\Sigma_{X\otimes Y}$
which makes the diagram in Figure 5 commute and justifies the notation
$1_{X}\otimes P$ (and explains also why the notation $\Gamma_{Q}$ for the
graph map was used to distinguish it from this map).
$X\otimes Z$$X$$Z$$X\otimes Y$$X$$Y$$1_{X}\otimes
P$$\delta_{\pi_{X}}$$\delta_{\pi_{Z}}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$1_{X}$$P$
Figure 5: The tensor product of conditional $1_{X}$ and $P$ in $\mathcal{P}$.
This tensor product $1_{X}\otimes P$ essentially comes from the diagram
$Z$$Y$$X\otimes Y$,$P$$\delta_{\Gamma_{\overline{x}}}$
where given a measurable set $\mathcal{A}\in\Sigma_{X\otimes Y}$ one pulls it
back under the constant graph function $\Gamma_{\overline{x}}$ and then
applies the conditional $P$ to the pair
$(\Gamma_{\overline{x}}^{-1}(\mathcal{A})\mid z)$.
### 5.3 Symmetric Monoidal Categories
A category $\mathcal{C}$ is said to be a monoidal category if it possesses the
following three properties:
1. 1.
There is a bifunctor
$\begin{array}[]{lcccc}\square&\colon&\mathcal{C}\times\mathcal{C}&\rightarrow&\mathcal{C}\\\
&\colon_{ob}&(X,Y)&\mapsto&X\square Y\\\
&\colon_{ar}&(X,Y)\stackrel{{\scriptstyle(f,g)}}{{\longrightarrow}}(X^{\prime},Y^{\prime})&\mapsto&X\square
Y\stackrel{{\scriptstyle(f\square g)}}{{\longrightarrow}}X^{\prime}\square
Y^{\prime}\end{array}$
which is associative up to isomorphism,
$\square(\square\times
Id_{\mathcal{C}})\cong\square(Id_{\mathcal{C}}\times\square)\colon\mathcal{C}\times\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$
where $Id_{\mathcal{C}}$ is the identity functor on $\mathcal{C}$. Hence for
every triple $X,Y,Z$ of objects, there is an isomorphism
$a_{X,Y,Z}\colon(X\square Y)\square Z\longrightarrow X\square(Y\square Z)$
which is natural in $X,Y,Z$. This condition is called the associativity axiom.
2. 2.
There is an object $I\in\mathcal{C}$ such that for every object
$X\in_{ob}\mathcal{C}$ there is a left unit isomorphism
$l_{X}\colon 1\square X\longrightarrow X.$
and a right unit isomorphism
$r_{X}\colon X\square 1\longrightarrow X.$
These two conditions are called the unity axioms.
3. 3.
For every quadruple of objects $X,Y,W,Z$ the diagram
$((X\square Y)\square W)\square Z$$(X\square(Y\square W))\square
Z$$X\square((Y\square W)\square Z)$$(X\square Y)\square(W\square
Z)$$X\square(Y\square(W\square Z))$$a_{X\square Y,W,Z}$$Id_{X}\square
a_{Y,W,Z}$$a_{X,Y,W\square Z}$$a_{X\square Y,W,Z}$$a_{X,Y\square W,Z}$
commutes. This is called the associativity coherence condition.
If $\mathcal{C}$ is a monoidal category under a bifunctor $\square$ and
identity $1$ it is denoted $(\mathcal{C},\square,1)$. A monoidal category
$(\mathcal{C},\square,1)$ is symmetric if for every pair of objects $X,Y$
there exist an isomorphism
$s_{X,Y}\colon X\square Y\longrightarrow Y\square X$ (27)
which is natural in $X$ and $Y$, and the three diagrams in Figure 6 commute.
$(X\square Y)\square Z$$X\square(Y\square Z)$$Y\square(Z\square X)$$(Y\square
X)\square Z$$Y\square(X\square Z)$$Y\square(Z\square X)$$s_{X,Y}\square
Id_{Z}$$a_{Y,Z,X}$$a_{Y,X,Z}$$a_{X,Y,Z}$$s_{X,Y\square Z}$$Id_{Y}\otimes
s_{X,Z}$$X\square I$$I\square X$$X$$s_{X,1}$$r_{X}$$l_{X}$$X\square
Y$$Y\square X$$X\square Y$$s_{X,Y}$$s_{Y,X}$$Id_{X}$ Figure 6: The additional
conditions required for a symmetric monoidal category.
The main example of a symmetric monoidal category is the category of sets,
$Set$, under the cartesian product with identity the terminal object
$1=\\{\star\\}$. Similarly, for the categories $\mathcal{M}eas$ and
$\mathcal{P}$, the tensor product $\otimes$ along with the terminal object $1$
acting as the identity element make both $(\mathcal{M}eas,\otimes,1)$ and
$(\mathcal{P},\otimes,1)$ symmetric monoidal categories with the above
conditions straightforward to verify. This provides a good exercise for the
reader new to categorical methods.
## 6 Function Spaces
For $X,Y\in_{ob}\mathcal{M}eas$ let $Y^{X}$ denote the set of all measurable
functions from $X$ to $Y$ endowed with the $\sigma$-algebra induced by the set
of all point evaluation maps $\\{ev_{x}\\}_{x\in X}$, where
$\begin{array}[]{ccc}Y^{X}&\stackrel{{\scriptstyle
ev_{x}}}{{\longrightarrow}}&Y\\\ f&\mapsto&f(x).\end{array}$
Explicitly, the $\sigma$-algebra on $Y^{X}$ is given by
$\Sigma_{Y^{X}}=\sigma\left(\bigcup_{x\in X}ev_{x}^{-1}\Sigma_{Y}\right),$
(28)
where for any function $f\colon W\to Z$ we have
$f^{-1}\Sigma_{Z}=\\{B\in 2^{W}\mid\exists C\in\Sigma_{Z}\text{ with
}f^{-1}(C)=B\\}$ (29)
and $\sigma(\mathcal{B})$ denotes the $\sigma$-algebra generated by any
collection $\mathcal{B}$ of subsets.
Formally we should use an alternative notation such as $\ulcorner f\urcorner$
to distinguish between the measurable function $f\colon X\rightarrow Y$ and
the point $\ulcorner f\urcorner\colon 1\rightarrow Y^{X}$ of the function
space $Y^{X}$.131313Having defined $Y^{X}$ to be the set of all measurable
functions $f\colon X\rightarrow Y$ it seems contradictory to then define
$ev_{x}$ as acting on “points” $\ulcorner f\urcorner\colon 1\rightarrow Y^{X}$
rather than the functions $f$ themselves! The apparent self contradictory
definition arises because we are interspersing categorical language with set
theory; when defining a set function, like $ev_{x}$, it is implied that it
acts on points which are defined as “global elements” $1\rightarrow Y^{X}$. A
global element is a map with domain $1$. This is the categorical way of
defining points rather than using the _elementhood_ operator “$\in$”. Thus, to
be more formal, we could have defined $ev_{x}$, where $x\colon 1\rightarrow X$
is any global element, by $ev_{x}\circ\ulcorner f\urcorner=\ulcorner
f(x)\urcorner\colon 1\rightarrow Y$, where $f(x)=f\circ x$. However, it is
common practice to let the context define which arrow we are referring to and
we shall often follow this practice unless the distinction is critical to
avoid ambiguity or awkward expressions.
An alternative notation to $Y^{X}$ is $\prod_{x\in X}Y_{x}$ where each $Y_{x}$
is a copy of $Y$. The relationship between these representations is that in
the former we view the elements as functions $f$ while in the latter we view
the elements as the indexed images of a function, $\\{f(x)\\}_{x\in X}$.
Either representation determines the other since a function is uniquely
specified by its values.
Because the $\sigma$-algebra structure on tensor product spaces was defined
precisely so that the constant graph functions were all measurable, it follows
that in particular the constant graph functions $\Gamma_{\overline{f}}\colon
X\rightarrow X\otimes Y^{X}$ sending $x\mapsto(x,f)$ are measurable. (The
graph function symbol $\Gamma_{\cdot}$ is overloaded and will need to be
specified directly (domain and codomain) when the context is not clear.)
Define the evaluation function
$\begin{array}[]{ccc}X\otimes Y^{X}&\stackrel{{\scriptstyle
ev_{X,Y}}}{{\longrightarrow}}&Y\\\ (x,f)&\mapsto&f(x)\end{array}$ (30)
and observe that for every $\ulcorner f\urcorner\in Y^{X}$ the right hand
$\mathcal{M}eas$ diagram in Figure 7 is commutative as a set mapping,
$f=ev_{X,Y}\circ\Gamma_{\overline{f}}$.
$X\cong X\otimes 1$$X\otimes Y^{X}$$Y$$1$$Y^{X}$$\Gamma_{\overline{f}}\cong
Id_{X}\otimes\ulcorner f\urcorner$$f$$ev_{X,Y}$$\ulcorner f\urcorner$ Figure
7: The defining characteristic property of the evaluation function $ev$ for
graphs.
By rotating the diagram in Figure 7 and also considering the constant graph
functions $\Gamma_{\overline{x}}$, the right hand side of the diagram in
Figure 8 also commutes for every $x\in X$.
$X$$Y^{X}$$X\otimes
Y^{X}$$Y$$\Gamma_{\overline{f}}$$f$$\Gamma_{\overline{x}}$$ev_{x}$$ev_{X,Y}$
Figure 8: The commutativity of both triangles, the measurability of $f$ and
$ev_{x}$, and the induced $\sigma$-algebra of $X\otimes Y^{X}$ implies the
measurability of $ev$.
Since $f$ and $\Gamma_{\overline{f}}$ are measurable, as are $ev_{x}$ and
$\Gamma_{\overline{x}}$, it follows by Lemma 7 that $ev_{X,Y}$ is measurable
since the constant graph functions generate the $\sigma$-algebra of $X\otimes
Y^{X}$. More generally, given any measurable function $f\colon X\otimes
Z\rightarrow Y$ there exists a unique measurable map $\tilde{f}\colon
Z\rightarrow Y^{X}$ defined by $\tilde{f}(z)=\ulcorner
f(\cdot,z)\urcorner\colon 1\rightarrow Y^{X}$ where $f(\cdot,z)\colon
X\rightarrow Y$ sends $x\mapsto f(x,z)$. This map $\tilde{f}$ is measurable
because the $\sigma$-algebra is generated by the _point evalutation_ maps
$ev_{x}$ and the diagram
$X\otimes Z$$Y^{X}$$Y$$Z$$ev_{x}$$\tilde{f}$$\Gamma_{\overline{x}}$$f$
commutes so that
$\tilde{f}^{-1}(ev_{x}^{-1}(B))=(f\circ\Gamma_{\overline{x}})^{-1}(B)\in\Sigma_{Z}$.
Conversely given any measurable map $g\colon Z\rightarrow Y^{X}$, it follows
the composite
$ev_{X,Y}\circ(Id_{X}\otimes g)$
is a measurable map. This sets up a bijective correspondence between
measurable functions denoted by
$Z$$Y^{X}$$X\otimes Z$$Y$$\tilde{f}$$f$
or the diagram in Figure 9.
$X\otimes Z$$X\otimes
Y^{X}$$Y$$Z$$Y^{X}$$Id_{X}\otimes\tilde{f}$$f$$ev_{X,Y}$$\tilde{f}$ Figure 9:
The evaluation function $ev$ sets up a bijective correspondence between the
two measurable maps $f$ and $\tilde{f}$.
The measurable map $\tilde{f}$ is called the adjunct of $f$ and vice versa, so
that $\tilde{\tilde{f}}=f$. Whether we use the tilde notation for the map
$X\otimes Z\rightarrow Y$ or the map $Z\rightarrow Y^{X}$ is irrelevant, it
simply indicates it’s the map uniquely determined by the other map.
The map $ev_{X,Y}$, which we will usually abbreviate to simply $ev$ with the
pair $(X,Y)$ obvious from context, is called a universal arrow because of this
property; it mediates the relationship between the two maps $f$ and
$\tilde{f}$. In the language of category theory using functors, for a fixed
object $X$ in $\mathcal{M}eas$, the collection of maps
$\\{ev_{X,Y}\\}_{Y\in_{ob}\mathcal{M}eas}$ form the components of a natural
transformation $ev_{X,-}\colon(X\otimes\cdot)\circ\\_^{X}\rightarrow
Id_{\mathcal{M}eas}$. In this situation we say the pair of functors
$\\{X\otimes\\_,\\_^{X}\\}$ forms an adjunction denoted
$X\otimes\\_\dashv\\_^{X}$. This adjunction $X\otimes\\_\dashv\\_^{X}$ is the
defining property of a closed category. We previously showed $\mathcal{M}eas$
was symmetric monoidal and combined with the closed category structure we
conclude that $\mathcal{M}eas$ is a symmetric monoidal closed category (SMCC).
Subsequently we will show that $\mathcal{P}$ satisfies a weak version of SMCC,
where uniqueness cannot be obtained.
##### The Graph Map.
Given the importance of graph functions when working with tensor spaces we
define the graph map
$\begin{array}[]{ccccc}\Gamma_{\cdot}&\colon&Y^{X}&\rightarrow&(X\otimes
Y)^{X}\\\ &\colon&\ulcorner
f\urcorner&\mapsto&\ulcorner\Gamma_{f}\urcorner.\end{array}$
Thus $\Gamma_{\cdot}(\ulcorner f\urcorner)=\ulcorner\Gamma_{f}\urcorner$ gives
the name of the graph
$X$$X\otimes Y$.$\Gamma_{f}$
The measurability of $\Gamma_{\cdot}$ follows in part from the commutativity
of the diagram in Figure 10, where the map $\hat{ev}_{x}\colon(X\otimes
Y)^{X}\rightarrow X\otimes Y$ denotes the standard point evaluation map
sending $g\mapsto(x,g(x))$.
$Y^{X}$$(X\otimes Y)^{X}$$X\otimes
Y$$Y$$\Gamma_{\cdot}$$\hat{ev}_{x}$$\langle\overline{x},ev_{x}\rangle$$ev_{x}$$\Gamma_{\overline{x}}$
Figure 10: The relationship between the graph map, point evaluations, and
constant graph maps.
We have used the notation $\hat{ev}_{x}$ simply to distinguish this map from
the map $ev_{x}$ which has a different domain and codomain. The
$\sigma$-algebra of $(X\otimes Y)^{X}$ is determined by these point evaluation
maps $\hat{ev}_{x}$ so that they are measurable. The maps $ev_{x}$ and
$\Gamma_{\overline{x}}$ are both measurable and hence their composite
$\Gamma_{\overline{x}}\circ ev_{x}=\langle\overline{x},ev_{x}\rangle$ is also
measurable.
To prove the measurability of the graph map we use the dual to Lemma 7
obtained by reversing all the arrows in that lemma to give
###### Lemma 8.
Let the $\sigma$-algebra of $Y$ be induced by a collection of maps
$\\{g_{i}\colon Y\rightarrow Z_{i}\\}_{i\in I}$. Then any map $f\colon
X\rightarrow Y$ is measurable if and only if the composition $g_{i}\circ f$ is
measurable for each $i\in I$.
###### Proof.
Consider the diagram
$X$$Y$$Z_{i}$$f$$g_{i}$$g_{i}\circ f$
The necessary condition is obvious. Conversely if $g_{i}\circ f$ is measurable
for each $i\in I$ then $f^{-1}(g_{i}^{-1}(B))\in\Sigma_{X}$. Because the
$\sigma$-algebra $\Sigma_{Y}$ is generated by the measurable sets
$g_{i}^{-1}(B)$ it follows that every measurable $U\in\Sigma_{Y}$ also
satisfies $f^{-1}(U)\in\Sigma_{X}$ so $f$ is measurable. ∎
Applying this lemma to the diagram in Figure 10 with the maps $g_{i}$
corresponding to the point evaluation maps $ev_{x}$ and the map $f$ being the
graph map $\Gamma_{\cdot}$ proves the graph map is indeed measurable.
The measurability of both of the maps $ev$ and $\Gamma_{\cdot}$ yield
corresponding $\mathcal{P}$ maps $\delta_{ev}$ and $\delta_{\Gamma_{\cdot}}$
that play a role in the construction of sampling distributions defined on any
hypothesis spaces that involves function spaces.
### 6.1 Stochastic Processes
Having defined function spaces $Y^{X}$, we are now in a position to define
stochastic processes using categorical language. The elementary definition
given next suffices to develop all the basic concepts one usually associates
with traditional ML and allows for relatively elegant proofs. Subsequently,
using the language of functors, a more general definition will be given and
for which the following definition can be viewed as a special instance.
###### Definition 9.
A stochastic process is a $\mathcal{P}$ map
$1$$Y^{X}$$P$
representing a probability measure on the function space $Y^{X}$. A
_parameterized_ stochastic process is a $\mathcal{P}$ map
$Z$$Y^{X}$$P$
representing a family of stochastic processes parameterized by $Z$.
Just as we did for the category $\mathcal{M}eas$, we seek a bijective
correspondence between two $\mathcal{P}$ maps, a stochastic process $P$ and a
corresponding conditional probability measure $\overline{P}$. In the
$\mathcal{P}$ case, however, the two morphisms do not uniquely determine each
other, and we are only able to obtain a symmetric monoidal weakly closed
category (SMwCC).
In Section 5.2 the tensor product $1_{X}\otimes P$ was defined, and by
replacing the space “$Y$” in that definition to be a function space $Y^{X}$ we
obtain the tensor product map
$1_{X}\otimes P\colon X\otimes Z\rightarrow X\otimes Y^{X}$
given by (using the same formula as in Section 5.2)
$(1_{X}\otimes
P)(\mathcal{U}\mid(x,z))=P(\Gamma_{\overline{x}}^{-1}(\mathcal{U})\mid z)$
For a given parameterized stochastic process $P\colon Z\rightarrow Y^{X}$ we
obtain the tensor product $1_{X}\otimes P$, and composing this map with the
deterministic $\mathcal{P}$ map determined by the evaluation map we obtain the
composite $\overline{P}$ in the diagram in Figure 11.
$X\otimes Z$$X\otimes Y^{X}$$Y$$Z$$Y^{X}$$1_{X}\otimes
P$$\overline{P}$$\delta_{ev}$$P$ Figure 11: The defining characteristic
property of the evaluation function $ev$ for tensor products of conditionals
in $\mathcal{P}$.
Thus
$\begin{array}[]{lcl}\overline{P}(B\mid(x,z))&=&\int_{(u,f)\in X\otimes
Y^{X}}{(\delta_{ev})}_{B}(u,f)\,d(1_{X}\otimes P)_{(x,z)}\\\ &=&\int_{f\in
Y^{X}}\delta_{ev}(B\mid\Gamma_{\overline{x}}(f))\,dP_{z}\\\ &=&\int_{f\in
Y^{X}}\chi_{B}(ev_{x}(f))\,dP_{z}\\\ &=&P(ev_{x}^{-1}(B)\mid z)\end{array}$
and every parameterized stochastic process determines a conditional
probability
$\overline{P}\colon X\otimes Z\rightarrow Y.$
Conversely, given a conditional probability $\overline{P}\colon X\otimes Z\to
Y$, we wish to define a parameterized stochastic process $P\colon Z\to Y^{X}$.
We might be tempted to define such a stochastic process by letting
$P(ev_{x}^{-1}(B)\mid z)=\overline{P}(B\mid(x,z)),$ (31)
but this does not give a well-defined measure for each $z\in Z$. Recall that a
probability measure cannot be unambiguously defined on an arbitrary generating
set for the $\sigma$-algebra. We can, however, uniquely define a measure on a
$\pi$-system141414A $\pi$-system on $X$ is a nonempty collection of subsets of
$X$ that is closed under finite intersections. and then use Dynkin’s
$\pi$-$\lambda$ theorem to extend to the entire $\sigma$-algebra (e.g., see
[10]). This construction requires the following definition.
###### Definition 10.
Given a measurable space $(X,\Sigma_{X})$, we can define an equivalence
relation on $X$ where $x\sim y$ if $x\in A\Leftrightarrow y\in A$ for all
$A\in\Sigma_{X}$. We call an equivalence class of this relation an atom of
$X$. For an arbitrary set $A\subset X$, we say that $A$ is
1. $\bullet$
separated if for any two points $x,y\in A$, there is some $B\in\Sigma_{X}$
with $x\in B$ and $y\notin B$
2. $\bullet$
unseparated if $A$ is contained in some atom of $X$.
This notion of separation of points is important for finding a generating set
on which we can define a parameterized stochastic process. The key lemma which
we state here without proof151515This lemma and additional work on symmetric
monoidal weakly closed structures on $\mathcal{P}$ will appear in a future
paper. is the following.
###### Lemma 11.
The class of subsets of $Y^{X}$
$\mathcal{E}=\emptyset\cup\left\\{\bigcap_{i=1}^{n}ev^{-1}_{x_{i}}(A_{i})\quad\middle\mid\quad\begin{matrix}\\{x_{i}\\}_{i=1}^{n}\text{
is separated in }X,\\\ A_{i}\in\Sigma_{Y}\text{ is nonempty and
proper}\end{matrix}\right\\}$
is a $\pi$-system which generates the evaluation $\sigma$-algebra on $Y^{X}$.
We can now define many parameterized stochastic processes “adjoint” to
$\overline{P}$, with the only requirement being that Equation 31 is satisfied.
This is not a deficiency in $\mathcal{P}$, however, but rather shows that we
have ample flexibility in this category.
###### Remark 12.
Even when such an expression does provide a well-defined measure as in the
case of finite spaces, it does not yield a unique $P$. Appendix B provides an
elementary example illustrating the failure of the bijective correspondence
property in this case. Also observe that the proposed defining Equation 31 can
be extended to
$P(\cap_{i=1}^{n}ev_{x_{i}}^{-1}(B_{i})\mid
z)=\prod_{i=1}^{n}\overline{P}(B_{i}\mid(x_{i},z))$
which does provide a well-defined measure by Lemma 11. However it still does
not provide a bijective correspondence which is clear as the right hand side
implies an independence condition which a stochastic process need not satisfy.
However it does provide for a bijective correspondence if we _impose_ an
additional independence condition/assumption. Alternatively, by imposing the
additional condition that for each $z\in Z$, $P_{z}$ is a Gaussian Processes
we can obtain a bijective correspondence. In Section 6.3 we illustrate in
detail how a joint normal distribution on a finite dimensional space gives
rise to a stochastic process, and in particular a GP.
Often, we are able to exploit the weak correspondence and use the conditional
probability $\overline{P}\colon X\rightarrow Y$ rather than the stochastic
process $P\colon 1\to Y^{X}$. While carrying less information, the conditional
probability is easier to reason with because of our familiarity with Bayes’
rule (which uses conditional probabilities) and our unfamiliarity with
measures on function spaces.
Intuitively it is easier to work with the conditional probability
$\overline{P}$ as we can represent the graph of such functions. In Figure 12
the top diagram shows a prior probability $P\colon
1\rightarrow\mathbb{R}^{[0,10]}$, which is a stochastic process, depicted by
representing its adjunct illustrating its expected value as well as its
$2\sigma$ error bars on each coordinate. The bottom diagram in the same figure
illustrates a parameterized stochastic process where the parameterization is
over four measurements. Using the above notation, $Z=\prod_{i=1}^{4}(X\times
Y)_{i}$ and $\overline{P}(\cdot\mid\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{4})$ is
a posterior probability measure given four measurements
$\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{4}$. These diagrams were generated under
the hypothesis that the process is a GP.
Figure 12: The top diagram shows a (prior) stochastic process represented by
its adjunct $\overline{P}\colon[0,10]\rightarrow\mathbb{R}$ and characterized
by its expected value and covariance. The bottom diagram shows a parameterized
stochastic process (the same process), also expressed by its adjunct, where
the parameterization is over four measurements.
### 6.2 Gaussian Processes
To further explicate the use of stochastic processes we consider the special
case of a stochastic process that has proven to be of extensive use for
modeling in ML problems. To be able to compute integrals, notably
expectations, we will assume hereafter that $Y=\mathbb{R}$ and
$X=\mathbb{R}^{n}$ for some integer $n$, or a compact subset thereof with the
standard Borel $\sigma$-algebras. We use the bold notation $\mathbf{x}$ to
denote a vector in $X$. Because ML applications often simply stress scalar
valued functions, we have take $Y=\mathbb{R}$ and write elements in $Y$ as
$y$. At any rate, the generalization to an arbitrary Euclidean space amounts
to carrying around vector notation and using vector valued integrals in the
following.
For any finite subset $X_{0}\subset X$ the set $X_{0}$ can be given the
subspace $\sigma$-algebra which is the induced $\sigma$-algebra of the
inclusion map $\iota\colon X_{0}\hookrightarrow X$. Given any measurable
$f\colon X\rightarrow Y$ the restriction of $f$ to $X_{0}$ is
$f|_{X_{0}}=f\circ\iota$ and “substitution” of an element $\mathbf{x}\in X$
into $f|_{X_{0}}$ is precomposition by the point $\mathbf{x}\colon
1\rightarrow X$ giving the commutative $\mathcal{M}eas$ diagram in Figure 13,
where the composite $f|_{X_{0}}(\mathbf{x})=f\circ\iota\circ\mathbf{x}$ is
equivalent to the map $ev_{\mathbf{x}}(\ulcorner f\urcorner)\colon
1\rightarrow Y$.
$1$$X_{0}$$X$$Y$$\iota$$\mathbf{x}$$f$$f(\mathbf{x})=f\circ\iota\circ\mathbf{x}=ev_{\mathbf{x}}(\ulcorner
f\urcorner)$$f|_{X_{0}}$ Figure 13: The substitution/evaluation relation.
Thus the inclusion map $\iota$ induces a measurable map
$\begin{array}[]{lclcl}Y^{\iota}&\colon&Y^{X}&\rightarrow&Y^{X_{0}}\\\
&\colon&\ulcorner f\urcorner&\mapsto&\ulcorner
f\circ\iota\urcorner,\end{array}$
which in turn induces the deterministic map $\delta_{Y^{\iota}}\colon
Y^{X}\rightarrow Y^{X_{0}}$ in $\mathcal{P}$. For any probability measure $P$
on the function space $Y^{X}$, we have the composite of $\mathcal{P}$ arrows
shown in the left diagram of Figure 14. For a singleton set
$X_{0}=\\{\mathbf{x}\\}$ this diagram reduces to the diagram on the right in
Figure 14.
$1$$Y^{X}$$Y^{X_{0}}$$P$$\delta_{Y^{\iota}}$$P\iota^{-1}$$1$$Y^{X}$$Y$$P$$\delta_{ev_{\mathbf{x}}}$$Pev_{\mathbf{x}}^{-1}$
Figure 14: The defining property of a Gaussian Process is the commutativity of
a $\mathcal{P}$ diagram.
Given $m\in Y^{X}$ and $k$ a bivariate function $k\colon X\times
X\rightarrow\mathbb{R}$, let $m|_{X_{0}}=m\circ\iota\in Y^{X_{0}}$ denote the
restriction of $m$ to $X_{0}$ and similiarly let
$k|_{X_{0}}=k\circ(\iota\times\iota)$ denote the restriction of $k$ to
$X_{0}\times X_{0}$.
###### Definition 13.
A Gaussian process on $Y^{X}$ is a probability measure $P$ on the function
space $Y^{X}$, denoted $P\sim\mathcal{G}\mathcal{P}(m,k)$, such that _for all
finite subsets_ $X_{0}$ of $X$ the push forward probability measure
$P\iota^{-1}$ is a (multivariate) Gaussian distribution denoted
$P\iota^{-1}\sim\mathcal{N}(m|_{X_{0}},k|_{X_{0}})$.
A bivariate function $k$ satisfying the condition in the definition is called
the _covariance function_ of the Gaussian process $P$ while the function $m$
is the _expected value_. A Gaussian process is completely specified by its
mean and covariance functions. These two functions are defined _pointwise_ by
$m(\mathbf{x})\triangleq\mathbb{E}_{P}[ev_{\mathbf{x}}]=\int_{f\in
Y^{X}}(ev_{\mathbf{x}})(\ulcorner f\urcorner)\,dP=\int_{f\in
Y^{X}}f(\mathbf{x})\,dP$ (32)
and by the vector valued integral
$\begin{array}[]{lcl}k(\mathbf{x},\mathbf{x}^{\prime})&\triangleq&\mathbb{E}_{P}[(ev_{\mathbf{x}}-\mathbb{E}_{P}[ev_{\mathbf{x}}])(ev_{\mathbf{x}^{\prime}}-\mathbb{E}_{P}[ev_{\mathbf{x}^{\prime}}])]\\\
&=&\displaystyle{\int_{f\in
Y^{X}}}\left(f(\mathbf{x})-m(\mathbf{x})\right)^{T}\left(f(\mathbf{x}^{\prime})-m(\mathbf{x}^{\prime})\right)\,dP.\end{array}$
(33)
Abstractly, if $P$ is given, then we could determine $m$ and $k$ by these two
equations. However in practice it is the two functions, $m$ and $k$ which are
used to specify a GP $P$ rather than $P$ determining $m$ and $k$. For general
stochastic processes higher order moments
$\mathbb{E}_{P}[ev_{\mathbf{x}}^{j}]$, with $j>1$, are necessary to
characterize the process.
For the covariance function $k$ we make the following assumptions for all
$\mathbf{x},\mathbf{z}\in X$,
1. 1.
$k(\mathbf{x},\mathbf{z})\geq 0$,
2. 2.
$k(\mathbf{x},\mathbf{z})=k(\mathbf{z},\mathbf{x})$, and
3. 3.
$k(\mathbf{x},\mathbf{x})k(\mathbf{z},\mathbf{z})-k(\mathbf{x},\mathbf{z})^{2}\geq
0$.
### 6.3 GPs via Joint Normal Distributions161616This section is not required
for an understanding of subsequent material but only provided for purposes of
linking familiar concepts and ideas with the less familiar categorical
perspective.
A simple illustration of a GP as a probability measure on a function space can
be given by consideration of a joint normal distribution. Here we relate the
familiar presentation of multivariate normal distributions as expressed in the
language of random variables into the categorical framework and language, and
illustrate that the resulting conditional distributions correspond to a GP.
Let X and Y represent two vector valued real random variables having a joint
normal distribution
$J=\left[\begin{array}[]{c}\textbf{X}\\\
\textbf{Y}\end{array}\right]\sim\mathcal{N}\left(\left[\begin{array}[]{c}\mu_{1}\\\
\mu_{2}\end{array}\right],\left[\begin{array}[]{cc}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\Sigma_{22}\end{array}\right]\right)$
with $\Sigma_{11}$ and $\Sigma_{22}$ nonsingular.
Represented categorically, these random variables X and Y determine
distributions which we represent by $P_{1}$ and $P_{2}$ on two measurable
spaces $X=\mathbb{R}^{{}^{m}}$ and $Y=\mathbb{R}^{{}^{n}}$ for some finite
integers $m$ and $n$, and the various relationships between the $\mathcal{P}$
maps is given by the diagram in Figure 15.
$1$$X\times
Y$$X$$Y$$J$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$P_{1}\sim\mathcal{N}(\mu_{1},\Sigma_{11})$$P_{2}\sim\mathcal{N}(\mu_{2},\Sigma_{22})$$\overline{\mathcal{S}}$$\overline{\mathcal{I}}$
Figure 15: The categorical characterization of a joint normal distribution.
Here $\overline{\mathcal{S}}$ and $\overline{\mathcal{I}}$ are the conditional
distributions
$\displaystyle\overline{\mathcal{S}}_{x}\sim\mathcal{N}\left(\mu_{2}+\Sigma_{21}\Sigma_{11}^{-1}(x-\mu_{1}),\Sigma_{22}-\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}\right)$
$\displaystyle\overline{\mathcal{I}}_{y}\sim\mathcal{N}\left(\mu_{1}+\Sigma_{12}\Sigma_{22}^{-1}(y-\mu_{2}),\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\right)$
and the overline notation on the terms “$\overline{\mathcal{S}}$” and
“$\overline{\mathcal{I}}$” is used to emphasize that the transpose of both of
these conditionals are GPs given by a bijective correspondence in Figure 16.
$X\otimes 1$$X\otimes
Y^{X}$$Y$$1$$Y^{X}$$\Gamma_{\mathcal{S}}$$\overline{\mathcal{S}}$$ev_{X,Y}$$\mathcal{S}$
Figure 16: The defining characteristic property of the evaluation function
$ev$ for graphs.
In the random variable description, these conditionals
$\overline{\mathcal{S}}_{\mathbf{x}}$ and $\overline{\mathcal{I}}_{y}$ are
often represented simply by
$\mu_{\textbf{Y}|\textbf{X}}=\mu_{2}+\Sigma_{21}\Sigma_{11}^{-1}(x-\mu_{1})\quad\mu_{\textbf{X}|\textbf{Y}}=\mu_{1}+\Sigma_{12}\Sigma_{22}^{-1}(y-\mu_{2})$
and
$\Sigma_{\textbf{Y}|\textbf{X}}=\Sigma_{22}-\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}\quad\quad\Sigma_{\textbf{X}|\textbf{Y}}=\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}$
It is easily verified that this pair $\\{\mathcal{S},\mathcal{I}\\}$ forms a
sampling distribution/inference map pair; i.e., the joint distribution can be
expressed in terms of the prior X and sampling distribution
$\overline{\mathcal{S}}$ or in terms of the prior Y and inference map
$\overline{\mathcal{I}}$. It is clear from this example that what one calls
the sampling distribution and inference map depends upon the perspective of
what is being estimated.
In subsequent developments, we do not assume a joint normal distribution on
the spaces $X$ and $Y$. If such an assumption is reasonable, then the
following constructions are greatly simplified by the structure expressed in
Figure 15. As noted previously, it is knowledge of the relationship between
the distributions $P_{1}$ and $P_{2}$ which characterize the joint and, is the
main modeling problem. Thus the two perspectives on the problem are to find
the conditionals, or equivalently, find the prior on $Y^{X}$ which specifies a
function $X\rightarrow Y$ along with the noise model which is “built into” the
sampling distribution.
## 7 Bayesian Models for Function Estimation
We now have all the necessary tools to build several Bayesian models, both
parametric and nonparametric, which illustrate the model building process for
ML using CT. To say we are building Bayesian models means we are constructing
the two $\mathcal{P}$ arrows, $P_{H}$ and $\mathcal{S}$, corresponding to (1)
the prior probability, and (2) the sampling distribution of the diagram in
Figure 2. The sampling distribution will generally be a composite of several
simple $\mathcal{P}$ arrows. We start with the nonparametric models which are
in a modeling sense more basic than the parametric models involving a fixed
finite number of parameters to be determined. The inference maps $\mathcal{I}$
for all of the models will be constructed in Section 8.
### 7.1 Nonparametric Models
In estimation problems where the unknown quantity of interest is a function
$f:X\rightarrow Y$, our hypothesis space $H$ will be the function space
$Y^{X}$. However, simply expressing the hypothesis space as $Y^{X}$ appears
untenable because, in supervised learning, we never measure $Y^{X}$ directly,
but only measure a finite number of sampling points
$\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$ satisfying some measurement model
such as $y_{i}=f(\mathbf{x}_{i})+\epsilon$ where $f$ is an “ideal” function we
seek to determine.
With precise knowledge of the input state $\mathbf{x}$ and assuming a generic
stochastic process $P\colon 1\rightarrow Y^{X}$, we are led to propose either
the left
$\delta_{\mathbf{x}}\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\ltimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\ltimes$\hfil\cr}}}P$
or right
$\delta_{\mathbf{x}}\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}P$
tensor product as a prior on the hypothesis space $X\otimes Y^{X}$. However,
when one of the components in a left or right tensor product is a Dirac
measure, then both the left and right tensors coincide and the choice of right
or left tensor is irrelevant. In this case, we denote the common probability
measure by $\delta_{\mathbf{x}}\otimes P$. Moreover, a simple calculation
shows the prior $\delta_{\mathbf{x}}\otimes
P=\Gamma_{P}(\cdot\mid\mathbf{x})$, the graph of $P$ at $\mathbf{x}$. Thus our
proposed model, in analogy to the generic Bayesian model, is given by the
diagram in Figure 17.181818It would be interesting to analyze the more general
case where there is uncertainty in the input state also and take the prior as
$Q\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}P$
or
$Q\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\ltimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\ltimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\ltimes$\hfil\cr}}}P$
for some measure $Q$ on $X$.
$1$$X\otimes Y^{X}$$X\otimes Y$$d$ is measurement
data$\Gamma_{P}(\cdot\mid\mathbf{x})$$d$$\mathcal{S}$ Figure 17: The generic
nonparametric Bayesian model for stochastic processes.
By a _nonparametric_ (Bayesian) model, we mean any model which fits into the
scheme of Figure 17. For all of our analysis purposes we take
$P\sim\mathcal{G}\mathcal{P}(m,k)$. A data measurement $d$, corresponding to a
collection of sample data $\\{\mathbf{x}_{i},y_{i}\\}$ is, in ML applications,
generally taken as a Dirac measure, $d=\delta_{(\mathbf{x},y)}$. As in all
Bayesian problems, the measurement data $\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{N}$
can be analyzed either sequentially or as a single batch of data. For analysis
purpose in Section 8, we consider the data one point at a time (sequentially).
#### 7.1.1 Noise Free Measurement Model
In the noise free measurement model, we make the hypothesis that the data we
observe—consisting of input output pairs $(\mathbf{x}_{i},y_{i})\in X\times
Y$—satisfies the condition that $y_{i}=f(\mathbf{x}_{i})$ where $f$ is the
unknown function we are seeking to estimate. While the actual measured data
will generally not satisfy this hypothesis, this model serves both as an
idealization and a building block for the subsequent noisy measurement model.
Using the fundamental maps $\Gamma_{\cdot}\colon Y^{X}\rightarrow(X\otimes
Y)^{X}$ and $ev\colon X\otimes(X\otimes Y)^{X}\rightarrow X\otimes Y$ gives a
sequence of measurable maps which determine corresponding deterministic
$\mathcal{P}$ maps. This composite, shown in Figure 18, is our noise free
sampling distribution.
$X\otimes Y^{X}$$X\otimes(X\otimes Y)^{X}$$X\otimes
Y$$1\otimes\delta_{\Gamma}$$\delta_{ev}$$\mathcal{S}_{nf}$ = composite Figure
18: The noise free sampling distribution $\mathcal{S}_{nf}$.
This deterministic sampling distribution is given by the calculation of the
composition, i.e., evaluating the integral
$\begin{array}[]{lcl}\mathcal{S}_{nf}(U\mid(\mathbf{x},f))&=&\int_{(\mathbf{u},g)\in
X\otimes(X\otimes
Y)^{X}}(\delta_{ev})_{U}(\mathbf{u},g)\,d(1\otimes\delta_{\Gamma})_{(\mathbf{x},f)}\quad\textrm{for
}U\in\Sigma_{X\otimes Y}\\\ &=&(\delta_{ev})_{U}(\mathbf{x},\Gamma_{f})\\\
&=&\delta_{\Gamma_{f}(\mathbf{x})}(U)\\\
&=&\delta_{(\mathbf{x},f(\mathbf{x}))}(U)\\\
&=&\delta_{(\Gamma_{\overline{\mathbf{x}}}(ev_{\mathbf{x}}(\ulcorner
f\urcorner)))}(U)\quad\textrm{ because
}(\mathbf{x},f(\mathbf{x}))=\Gamma_{\overline{x}}(ev_{\mathbf{x}}(f))\\\
&=&\chi_{U}(\Gamma_{\overline{\mathbf{x}}}(ev_{\mathbf{x}}(f)))\\\
&=&\chi_{ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(U))}(f).\end{array}$
Using the commutativity of Figure 10, the noise free sampling distribution can
also be written as
$\mathcal{S}_{nf}(U\mid(\mathbf{x},f))=\chi_{\Gamma_{\cdot}^{-1}\left(\hat{ev}_{\mathbf{x}}^{-1}(U)\right)}(f)$.
Precomposing the sampling distribution with this prior probability measure the
composite
$\begin{array}[]{lcl}(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))(U)&=&\int_{(\mathbf{u},f)\in
X\otimes
Y^{X}}\mathcal{S}_{nf}(U\mid(\mathbf{u},f))\,d(\underbrace{\Gamma_{P}(\cdot\mid\mathbf{x})}_{=P\Gamma_{\overline{\mathbf{x}}}^{-1}})\quad\textrm{
for }U\in\Sigma_{X\otimes Y}\\\ &=&\int_{f\in
Y^{X}}\mathcal{S}_{nf}(U\mid\Gamma_{\overline{\mathbf{x}}}(f))\,dP\\\
&=&\int_{f\in
Y^{X}}\chi_{\Gamma_{\cdot}^{-1}\left(\hat{ev}_{\mathbf{x}}^{-1}(U)\right)}(f)\,dP\\\
&=&P(\Gamma^{-1}_{\cdot}(\hat{ev}_{\mathbf{x}}^{-1}(U)))\end{array}$ (34)
By the relation $\Gamma_{\overline{\mathbf{x}}}\circ
ev_{\mathbf{x}}=\hat{ev}_{\mathbf{x}}\circ\Gamma_{\cdot}$ this can also be
written as
$(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))(U)=P(ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(U))).$
Given that the probability measure $P$ is specified as a Gaussian process
(which is defined in terms of how it restricts to finite subspaces
$X_{0}\subset X$), for computational purposes we need to consider the push
forward probability measure of $P$ on $Y^{X}$ to $Y^{X_{0}}$ as in Figure 14.
Taking the special case with $X_{0}=\\{\mathbf{x}\\}$, the pushforward
corresponds to composition with the deterministic projection map
$\delta_{ev_{\mathbf{x}}}$. Starting with the diagram of Figure 10,
precomposing with $P$ and postcomposition with the deterministic map
$\delta_{\pi_{Y}}\circ\delta_{\iota}$ gives the diagram in Figure 19. Then we
can use the fact $P$ projected onto any coordinate is a Gaussian distribution
to compute the likelihood that a measurement will occur in a measurable set
$B\subset Y$.
$1$$Y^{X}$$(X\otimes Y)^{X}$$X\otimes
Y$$Y$$P\sim\mathcal{G}\mathcal{P}(m,k)$$\Gamma_{\cdot}$$\hat{ev}_{\mathbf{x}}$$\delta_{\pi_{Y}}$$\delta_{ev_{\mathbf{x}}}$$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$
Figure 19: The distribution $P\sim\mathcal{G}\mathcal{P}(m,k)$ can be
evaluated on rectangles $U=A\times B$ by projecting onto the given $x$
coordinate.
Under this assumption $P\sim\mathcal{G}\mathcal{P}(m,k)$ the expected value of
the probability measure
$(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))$ on the real vector
space $X\otimes Y$ is
$\begin{array}[]{lcl}\mathbb{E}_{(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[Id_{X\otimes
Y}]&=&\int_{(\mathbf{u},\mathbf{v})\in X\otimes
Y}(\mathbf{u},\mathbf{v})\,d(P(\Gamma_{\cdot}^{-1}\hat{ev}_{\mathbf{x}}^{-1})\\\
&=&\int_{g\in(X\otimes
Y)^{X}}\hat{ev}_{\mathbf{x}}(g)\,d(P\Gamma_{\cdot}^{-1})\\\ &=&\int_{f\in
Y^{X}}\hat{ev}_{\mathbf{x}}(\Gamma(f))\,dP\\\ &=&\int_{f\in
Y^{X}}(\mathbf{x},f(\mathbf{x}))\,dP\\\
&=&(\mathbf{x},m(\mathbf{x})),\end{array}$
where the last equation follows because on the two components of the vector
valued integral, $\int_{f\in Y^{X}}f(\mathbf{x})\,dP=m(\mathbf{x})$ and
$\int_{f\in Y^{X}}\mathbf{x}\,dP=\mathbf{x}$ as the integrand is constant. The
variance is191919The squaring operator in the variance is defined component
wise on the vector space $X\otimes Y$.
$\mathbb{E}_{(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[(Id_{X\otimes
Y}-\mathbb{E}_{(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[Id_{X\otimes
Y}])^{2}]=\mathbb{E}_{(\mathcal{S}_{nf}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[(Id_{X\otimes
Y}-(\mathbf{x},m(\mathbf{x})))^{2}],$
which when expanded gives
$\begin{array}[]{lcl}&=&\int_{(\mathbf{u},v)\in X\otimes Y}(Id_{X\otimes
Y}-(\mathbf{x},m(\mathbf{x})))^{2}(\mathbf{u},v)\,d(P(\Gamma_{\cdot}^{-1}ev_{\mathbf{x}}^{-1}))\\\
&=&\int_{f\in Y^{X}}(Id_{X\otimes
Y}-(\mathbf{x},m(\mathbf{x})))^{2}\underbrace{(ev_{\mathbf{x}}(\Gamma_{\cdot}(f)))}_{=(\mathbf{x},f(\mathbf{x}))}\,dP\\\
&=&\int_{f\in
Y^{X}}\left((\mathbf{x}-\mathbf{x})^{2},\left(f(\mathbf{x})-m(\mathbf{x})\right)^{2}\right)\,dP\\\
&=&(\mathbf{0},k(\mathbf{x},\mathbf{x})).\end{array}$
Consequently this sampling distribution, together with the prior distribution
$\delta_{\mathbf{x}}\mathchoice{{\ooalign{$\displaystyle\bigcirc$\cr\hfil$\displaystyle\rtimes$\hfil\cr}}}{{\ooalign{$\textstyle\bigcirc$\cr\hfil$\textstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptstyle\bigcirc$\cr\hfil$\scriptstyle\rtimes$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\bigcirc$\cr\hfil$\scriptscriptstyle\rtimes$\hfil\cr}}}P=\Gamma_{P}(\cdot\mid\mathbf{x})$,
provide what we expect of such a model.
#### 7.1.2 Gaussian Additive Measurement Noise Model
Additive noise measurement models are often expressed by the simple expression
$z=y+\epsilon$ (35)
where $y$ represents the state while the $\epsilon$ term itself represents a
normally distributed random variable with zero mean and variance $\sigma^{2}$.
In categorical terms this expression corresponds to the map in Figure 20.
$1$$Y$$M_{y}\sim\mathcal{N}(y,\sigma^{2})$ Figure 20: The additive Gaussian
noise measurement model.
Because the state $y$ in Equation 35 is arbitrary, this additive noise model
is representative of the $\mathcal{P}$ map $Y\stackrel{{\scriptstyle
M}}{{\longrightarrow}}Y$ defined by
$M(B\mid y)=M_{y}(B)\quad\forall y\in Y,\,\forall B\in\Sigma_{Y}.$
Given a GP $P\sim\mathcal{G}\mathcal{P}(f,k)$ on $Y^{X}$, it follows that for
any $\mathbf{x}\in X$,
$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(f(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$
and for any $B\in\Sigma_{Y}$, the composition
$1$$Y$$Y$$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(f(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$$M$
is
$\begin{array}[]{lcl}(M\circ Pev_{\mathbf{x}}^{-1})(B)&=&\int_{u\in
Y}M_{B}(u)\,d(Pev_{\mathbf{x}}^{-1})\\\ &=&\int_{u\in
Y}\left(\frac{1}{\sqrt{2\pi}\sigma}\int_{v\in
B}e^{-\frac{(v-u)^{2}}{2\sigma^{2}}}\,dv\right)d(Pev_{\mathbf{x}}^{-1})\\\
&=&\frac{1}{\sqrt{2\pi k(f(\mathbf{x}),f(\mathbf{x}))}}\,\int_{u\in
Y}\left(\frac{1}{\sqrt{2\pi}\sigma}\int_{v\in
B}e^{-\frac{(v-u)^{2}}{2\sigma^{2}}}\,dv\right)e^{-\frac{(u-f(\mathbf{x}))^{2}}{2\cdot
k(f(\mathbf{x}),f(\mathbf{x}))}}du\\\
&=&\frac{1}{2\pi\cdot\sigma\cdot\sqrt{k(f(\mathbf{x}),f(\mathbf{x}))}}\,\int_{v\in
B}\int_{u\in
Y}e^{-\frac{(v-u)^{2}}{2\sigma^{2}}}\,e^{-\frac{(u-f(\mathbf{x}))^{2}}{2\cdot
k(f(\mathbf{x}),f(\mathbf{x}))}}\,du\,dv\\\
&=&\frac{1}{\sqrt{2\pi(k(\mathbf{x},\mathbf{x})+\sigma^{2})}}\int_{v\in
B}e^{-\frac{(v-f(\mathbf{x}))^{2}}{2(k(\mathbf{x},\mathbf{x})+\sigma^{2})}}\,dv.\end{array}$
Thus this composite is the normal distribution
$1$$Y$$M\circ
Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(f(\mathbf{x}),k(\mathbf{x},\mathbf{x})+\sigma^{2})$
(36)
More generally we have the commutative $\mathcal{P}$ diagram given in Figure
21, where, for all $f\in Y^{X}$,
$N_{f}\sim\mathcal{G}\mathcal{P}(f,k_{N})\quad\quad
k_{N}(\mathbf{x},\mathbf{x}^{\prime})=\left\\{\begin{array}[]{ll}\sigma^{2}&\textrm{
iff }\mathbf{x}=\mathbf{x}^{\prime}\\\ 0&\textrm{ otherwise.
}\end{array}\right.$ (37)
$1$$Y^{X}$$Y^{X}$$Y$$Y$$P\sim\mathcal{G}\mathcal{P}(f,k)$$N$$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(f(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$$M$$\delta_{ev_{\mathbf{x}}}$$\delta_{ev_{\mathbf{x}}}$
Figure 21: Construction of the generic Markov kernel $N$ for modeling the
Gaussian additive measurement noise.
The commutativity of the right hand square in Figure 21 follows from
$\begin{array}[]{lcl}(\delta_{ev_{\mathbf{x}}}\circ N)(B\mid f)&=&\int_{g\in
Y^{X}}(\delta_{ev_{\mathbf{x}}})_{B}(g)\,dN_{f}\\\
&=&N_{f}(ev_{\mathbf{x}}^{-1}(B))\\\ &=&N_{f(\mathbf{x})}(B)\\\ &=&\int_{y\in
Y}M_{B}(y)\,d(\underbrace{\delta_{ev_{\mathbf{x}}})_{f}}_{=\delta_{f(\mathbf{x})}}\\\
&=&(M\circ\delta_{ev_{\mathbf{x}}})(B\mid f).\end{array}$
With this Gaussian additive noise measurement model $N$ our sampling
distribution $\mathcal{S}_{nf}$ can easily be modified by incorporating the
additional map $N$ into the sequence in Figure 18 to yield the Gaussian
additive noise sampling distribution model $\mathcal{S}_{n}$ shown in Figure
22.
$X\otimes Y^{X}$$X\otimes Y^{X}$$X\otimes(X\otimes Y)^{X}$$X\otimes
Y$$1_{X}\otimes
N$$1_{X}\otimes\delta_{\Gamma_{\cdot}}$$\delta_{ev}$$\mathcal{S}_{n}$ =
composite Figure 22: The sampling distribution model in $\mathcal{P}$ with
additive Gaussian noise.
Here $1_{X}\otimes N$ is, by the definition given in Section 5.2,
$(1_{X}\otimes
N)\left(U,(\mathbf{x},f)\right)=N(\Gamma_{\overline{\mathbf{x}}}^{-1}(U)\mid
f)$
so the nondeterministic noisy sampling distribution is given by
$\begin{array}[]{lcl}\mathcal{S}_{n}(U\mid(\mathbf{x},f))&=&\left(\mathcal{S}_{nf}\circ(1\otimes
N)\right)(U\mid(\mathbf{x},f))\quad\textrm{for }U\in\Sigma_{X\otimes Y}\\\ \\\
&=&\int_{(\mathbf{u},g)\in X\otimes
Y^{X}}(\mathcal{S}_{nf})_{U}(\mathbf{u},g)\,d(N(\Gamma_{\overline{\mathbf{x}}}^{-1}(\cdot)\mid
f)\\\ &=&\int_{g\in
Y^{X}}(\mathcal{S}_{nf})_{U}(\Gamma_{\overline{\mathbf{x}}}(g))\,dN(\cdot\mid
f)\\\ &=&\int_{g\in
Y^{X}}(\mathcal{S}_{nf})(U\mid(\mathbf{x},g))\,dN(\cdot\mid f)\\\
&=&\int_{g\in
Y^{X}}\chi_{\Gamma_{\cdot}^{-1}\left(\hat{ev}_{\mathbf{x}}^{-1}(U)\right)}(g)dN(\cdot\mid
f)\\\
&=&N\left(\Gamma_{\cdot}^{-1}\left(\hat{ev}_{\mathbf{x}}^{-1}(U)\right)\mid
f\right)\\\
&=&N\left(ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(U))\mid
f\right)\\\
&=&N_{f}ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(U)).\end{array}$
(38)
Just as we did for the GP $P\colon 1\rightarrow Y^{X}$ in Figure 19, each GP
$N_{f}$ can be analyzed by its push forward measures onto any coordinate
$\mathbf{x}\in X$ to obtain the diagram in Figure 23.
$1$$Y^{X}$$(X\otimes Y)^{X}$$X\otimes
Y$$Y$$N_{f}\sim\mathcal{G}\mathcal{P}(f,k_{N})$$\delta_{\Gamma_{\cdot}}$$\delta_{\hat{ev}_{\mathbf{x}}}$$\delta_{\pi_{Y}}$$\delta_{ev_{\mathbf{x}}}$$N_{f}\left(ev_{\mathbf{x}}^{-1}(\cdot)\right)\sim\mathcal{N}(f(\mathbf{x}),\sigma^{2})$
Figure 23: The GP $N_{f}$ can be evaluated on rectangles $U=A\times B$ by
projecting onto the given $x$ coordinate.
Taking $U$ as a rectangle, $U=A\times B$, with $A\in\Sigma_{X}$ and
$B\in\Sigma_{Y}$, the likelihood that a measurement will occur in the
rectangle conditioned on $(\mathbf{x},f)$ is given by
$\begin{array}[]{lcl}\mathcal{S}_{n}(A\times
B\mid(\mathbf{x},f))&=&N_{f}ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(A\times
B))\\\ &=&\delta_{\mathbf{x}}(A)\cdot N_{f}ev_{\mathbf{x}}^{-1}(B)\\\
&=&\delta_{\mathbf{x}}(A)\,\cdot\,\frac{1}{\sqrt{2\pi}\sigma}\int_{y\in
B}e^{-\frac{(y-f(\mathbf{x}))^{2}}{2\sigma^{2}}}\,dy.\end{array}$
Using the associativity property of categories, from Figure 22 with a prior
$\Gamma_{P}(\cdot\mid\mathbf{x})$ on $X\otimes Y^{X}$, the composite
$\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x})$ can be decomposed as
$\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x})=\mathcal{S}_{nf}\circ((1_{X}\otimes
N)\circ\Gamma_{P}(\cdot\mid\mathbf{x}))$
while the term $((1_{X}\otimes
N)\circ\Gamma_{P}(\cdot\mid\mathbf{x}))=\Gamma_{N\circ
P}(\cdot\mid\mathbf{x})$ follows from the commutativity of the diagram in
Figure 24, where, as shown in Equation 36, $M\circ
Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(m,k(\mathbf{x},\mathbf{x})+\sigma^{2})$
which implies $N\circ P\sim\mathcal{G}\mathcal{P}(m,k+k_{N})$.
$1$$X$$X$$Y^{X}$$Y^{X}$$X\otimes Y^{X}$$X\otimes Y^{X}$$1$$X\otimes
Y^{X}$$\delta_{\mathbf{x}}$$1_{X}$$P$$N$$\Gamma_{P}(\cdot\mid\mathbf{x})$$1_{X}\otimes
N$$\delta_{\pi_{X}}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y^{X}}}$$\delta_{\pi_{Y^{X}}}$$\Gamma_{N\circ
P}(\cdot\mid\mathbf{x})$ Figure 24: The composite of the prior and noise
measurement model is the graph of a GP at $\mathbf{x}$.
Using the fact
$\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x})=\mathcal{S}_{nf}\circ\Gamma_{N\circ
P}(\cdot\mid\mathbf{x})$, the expected value of the composite
$\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x})$ is readily shown to be
$\mathbb{E}_{(\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[Id_{X\otimes
Y}]=(\mathbf{x},m(\mathbf{x}))$
while the variance is
$\mathbb{E}_{(\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[(Id_{X\otimes
Y}-\mathbb{E}_{(\mathcal{S}_{n}\circ\Gamma_{P}(\cdot\mid\mathbf{x}))}[Id_{X\otimes
Y}])^{2}]=(\mathbf{0},k(\mathbf{x},\mathbf{x})+\sigma^{2}).$
### 7.2 Parametric Models
A parametric model can be though of as carving out a subset of $Y^{X}$
specifying the form of functions which one wants to consider as valid
hypotheses. With this in mind, let us define a $p$-dimensional _parametric
map_ as a measurable function
$i\colon\mathbb{R}^{{}^{p}}\longrightarrow Y^{X}$
where $\mathbb{R}^{{}^{p}}$ has the product $\sigma$-algebra with respect to
the canonical projection maps onto the measurable space $\mathbb{R}$ with the
Borel $\sigma$-algebra. Note that $i(\mathbf{a})\in Y^{X}$ corresponds (via
the SMwCC structure) to a function $\overline{i(\mathbf{a})}\colon
X\rightarrow Y$.202020Note that the function $i(\mathbf{a})$ is unique by our
construction of the transpose of the function $i(\mathbf{a})\in Y^{X}$. The
non-uniqueness aspect of the SMwCC structure only arises in the other
direction - given a conditional probability measure there may be multiple
functions satisfying the required commutativity condition. This parametric map
$i$ determines the deterministic $\mathcal{P}$ arrow
$\delta_{i}\colon\mathbb{R}^{{}^{p}}\rightarrow Y^{X}$, which in turn
determines the deterministic tensor product arrow
$1_{X}\otimes\delta_{i}\colon X\otimes\mathbb{R}^{{}^{p}}\longrightarrow
X\otimes Y^{X}$. This arrow serves as a bridge connecting the two forms of
Bayesian models, the parametric and nonparametric models.
A parametric model consists of a parametric mapping combined with a
nonparametric noisy measurement model $\mathcal{S}_{n}$ with prior
$(1_{X}\otimes\delta_{i})\circ\Gamma_{P}(\cdot\mid\mathbf{x})$ to give the
diagram in Figure 25 and we define a _parametric Bayesian model_ as any model
which fits into the scheme of Figure 25.
$1$$X\otimes\mathbb{R}^{{}^{p}}$$X\otimes Y^{X}$$X\otimes
Y$$\Gamma_{P}(\cdot\mid\mathbf{x})$$1_{X}\otimes\delta_{i}$$d$$\mathcal{S}_{n}$
Figure 25: The generic parametric Bayesian model.
In the ML literature, one generally assumes complete certainty with regards to
the input state $\mathbf{x}\in X$. However, there are situations in which
complete knowledge of the input state $\mathbf{x}$ is itself uncertain. This
occurs in object recognition problems where $\mathbf{x}$ is a feature vector
which may be only partially observed because of obscuration and such data is
the only training data available.
For real world modeling applications there must be a noise model component
associated with a parametric model for it to make sense. For example we could
estimate an unknown function as a constant function, and hence have the $1$
parameter model $i\colon\mathbb{R}\rightarrow Y^{X}$ given by
$i(a)=\overline{a}$, the constant function on $X$ with value $a$. Despite how
crude this approximation may be, we can still obtain a “best” such Bayesian
approximation to the function given measurement data where “best” is defined
in the Bayesian probabilistic sense - given a prior and a measurement the
posterior gives the best estimate under the given modeling assumptions.
Without a noise component, however, we cannot even account for the fact our
data is different than our model which, for analysis and prediction purposes,
is a worthless model.
###### Example 14.
Affine Parametric Model Let $X=\mathbb{R}^{{}^{n}}$ and $p=n+1$. The affine
parametric model is given by considering the valid hypotheses to consist of
affine functions
$\begin{array}[]{lclcl}F_{\mathbf{a}}&:&X&\rightarrow&Y\\\
&:&\mathbf{x}&\mapsto&\sum_{j=1}^{n}a_{j}x_{j}+a_{n+1}\end{array}$ (39)
where $\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\in X$, the ordered $(n+1)-tuple$
$\mathbf{a}=(a_{1},\ldots,a_{n},a_{n+1})\in\mathbb{R}^{{}^{n+1}}$ are fixed
parameters so $F_{\mathbf{a}}\in Y^{X}$ and the parametric map
$\begin{array}[]{lclcl}i&:&\mathbb{R}^{n+1}&\longrightarrow&Y^{X}\\\
&:&\mathbf{a}&\mapsto&i(\mathbf{a})=\overline{F_{\mathbf{a}}}\end{array}$
specifies the subset of all possible affine models $F_{\mathbf{a}}$.
In particular, if $n=2$ and the test data consist of two data classes, say
with labels $-1$ and $1$, which is separable then the coefficients
$\\{a_{1},a_{2},a_{3}\\}$ specify the hyperplane separating the data points as
shown in Figure 26.
Figure 26: An affine model suffices for separable data.
In this particular example where the class labels are integer valued, the
resulting function we are estimating will not be integer valued but, as usual,
approximated by real values.
Such parametric models are useful to avoid over fitting data because the
number of parameters are finite and fixed with respect to the number of
measurements in contrast to nonparametric methods in which each measurement
serves as a parameter defining the updated probability measure on $Y^{X}$.
More generally, for any parametric map $i$ take the canonical basis vectors
$\mathbf{e}_{j}$, which are the $j^{th}$ unit vector in $\mathbb{R}^{{}^{p}}$,
and let the image of the basis elements $\\{\mathbf{e}_{j}\\}_{j=1}^{p}$ under
the parametric map $i$ be $i(\mathbf{e}_{j})=f_{j}\in Y^{X}$. Because $Y^{X}$
forms a real vector space under pointwise addition and scalar multiplication,
$(f+g)(\mathbf{x})=f(\mathbf{x})+g(\mathbf{x})$ and $(\alpha
f)(\mathbf{x})=\alpha(f(\mathbf{x}))$ for all $f,g\in Y^{X},\mathbf{x}\in X$,
and $\alpha\in\mathbb{R}$, we observe that the “image carved out” by the
parametric map $i$ is just the span of the image of the basis elements
$\\{e_{j}\\}_{j=1}^{p}$. In the above example $f_{j}=\pi_{j}$, for $j=1,2$
where $\pi_{j}$ is the canonical projection map
$\mathbb{R}^{2}\rightarrow\mathbb{R}$, and $f_{3}=\overline{1}$, the constant
function with value $1$ on all points $\mathbf{x}\in X$. Thus the image is as
specified by the Equation 39.
###### Example 15.
Elliptic Parametric Model When the data is not linearly separable as in the
previous example, but rather of the form shown in Figure 27, then a higher
order parametric model is required.
Figure 27: An elliptic parametric model suffices to separate the data.
Taking $X=\mathbb{R}^{n}$ and $p=n^{2}+n+1$, the elliptic parametric model is
given by considering the valid hypotheses to consist of all elliptic functions
$\begin{array}[]{lclcl}F_{\mathbf{a}}&:&X&\rightarrow&Y\\\
&:&\mathbf{x}&\mapsto&\sum_{j=1}^{n}a_{j}x_{j}+\sum_{j=1}^{n}\sum_{k=1}^{n}a_{n+n(j-1)+k}x_{j}x_{k}+a_{n^{2}+n+1}\end{array}$
(40)
where $\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\in X$, the ordered
$(n^{2}+n+1)-tuple$
$\mathbf{a}=(a_{1},\ldots,a_{n^{2}+n+1})\in\mathbb{R}^{n^{2}+n+1}$ are fixed
parameters so $F_{\mathbf{a}}\in Y^{X}$ and the parametric map
$\begin{array}[]{lclcl}i&:&\mathbb{R}^{{}^{n^{2}+n+1}}&\longrightarrow&Y^{X}\\\
&:&\mathbf{a}&\mapsto&i(\mathbf{a})=\overline{F_{\mathbf{a}}}\end{array}$
specifies the subset of all possible elliptic models $F_{\mathbf{a}}$.
With this model the linearly nonseparable data becomes separable. This is the
basic idea behind support vector machines (SVMs): simply embed the data into a
higher order space where it can be (approximately) separated by a higher order
parametric model.
Returning to the general construction of the Bayesian model for the parametric
model we take the Gaussian additive noise model, Equation 38, and expand the
diagram in Figure 25 to the diagram in Figure 28, where the parametric model
sampling distribution can be readily determined on rectangles $A\times
B\in\Sigma_{X\otimes Y}$ by
$\begin{array}[]{lcl}\mathcal{S}(A\times
B\mid(\mathbf{x},\mathbf{a}))&=&N\left(ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(A\times
B))\mid\underbrace{i(\mathbf{a})}_{=F_{\mathbf{a}}}\right)\\\
&=&N_{F_{\mathbf{a}}}ev_{\mathbf{x}}^{-1}(\Gamma_{\overline{\mathbf{x}}}^{-1}(A\times
B))\\\ &=&\delta_{\mathbf{x}}(A)\cdot\frac{1}{\sqrt{2\pi}\sigma}\int_{y\in
B}e^{-\frac{(y-F_{\mathbf{a}}(\mathbf{x}))^{2}}{2\sigma^{2}}}\,dy.\end{array}$
$1$$X\otimes\mathbb{R}^{{}^{p}}$$X\otimes Y^{X}$$X\otimes
Y^{X}$$X\otimes(X\otimes Y)^{X}$$X\otimes
Y$$\Gamma_{P}(\cdot\mid\mathbf{x})$$1_{X}\otimes\delta_{i}$$1_{X}\otimes
N$$1_{X}\otimes\delta_{\Gamma}$$\delta_{ev}$$\mathcal{S}$$\mathcal{S}_{n}$
Figure 28: The parametric model sampling distribution as a composite of four
components.
Here we have used the fact
$N_{F_{\mathbf{a}}}ev_{\mathbf{x}}^{-1}\sim\mathcal{N}(F_{\mathbf{a}}(\mathbf{x}),\sigma^{2})$
which follows from Equation 37 and the property that a GP evaluated on any
coordinate is a normal distribution with the mean and variance evaluated at
that coordinate.
## 8 Constructing Inference Maps
We now proceed to construct the inference maps $\mathcal{I}$ for each of the
models specified in the previous section. This construction permits the
updating of the GP prior distributions $P$ for the nonparametric models and
the normal priors $P$ on $\mathbb{R}^{k}$ for the parametric models through
the relation that the posterior measure is given by $\mathcal{I}\circ d$,
where $d$ is a data measurement. The resulting analysis produces the familiar
updating rules for the mean and covariance functions characterizing a GP.
### 8.1 The noise free inference map
Under a prior probability of the form $\delta_{\mathbf{x}}\otimes
P=\Gamma_{P}(\cdot\mid\mathbf{x})$ on the hypothesis space $X\otimes Y^{X}$,
which is a one point measure with respect to the component $X$, the sampling
distribution $\mathcal{S}_{nf}$ in Figure 18 can be viewed as a _family_ of
deterministic $\mathcal{P}$ maps—one for each point $\mathbf{x}\in X$.
$Y^{X}$$Y$$\mathcal{S}^{\mathbf{x}}=\delta_{ev_{\mathbf{x}}}$ Figure 29: The
noise free sampling distributions $\mathcal{S}^{\mathbf{x}}$ given the prior
$\delta_{\mathbf{x}}\otimes P$ with the dirac measure on the $X$ component.
Using the property that $\delta_{ev_{\mathbf{x}}}(B\mid
f)=\mathbb{1}_{ev_{\mathbf{x}}^{-1}(B)}(f)$ for all $B\in\Sigma_{Y}$ and $f\in
Y^{X}$, the resulting deterministic sampling distributions (one for each
$\mathbf{x}\in X$) are given by
$\mathcal{S}^{\mathbf{x}}(B\mid f)=\mathbb{1}_{ev_{\mathbf{x}}^{-1}(B)}(f).$
(41)
This special case of the prior $\delta_{\mathbf{x}}\otimes P$, which is the
most important one for many ML applications and the one implicitly assumed in
ML textbooks, permits a complete mathematical analysis.
Given the probability measure $P\sim\mathcal{G}\mathcal{P}(m,k)$ and
$\mathcal{S}^{\mathbf{x}}=\delta_{ev_{\mathbf{x}}}$, it follows the composite
is the pushforward probability measure
$\mathcal{S}^{\mathbf{x}}\circ P=Pev_{\mathbf{x}}^{-1},$ (42)
which is the special case of Figure 14 with $X_{0}=\\{\mathbf{x}\\}$. Using
the fact that $P$ projected onto any coordinate is a normal distribution as
shown in Figure 30, it follows that the expected mean is
$\begin{array}[]{lcl}\mathbb{E}_{Pev_{\mathbf{x}}^{-1}}(Id_{Y})&=&\mathbb{E}_{P}(ev_{\mathbf{x}})\\\
&=&m(\mathbf{x})\end{array}$
while the expected variance is
$\begin{array}[]{lcl}\mathbb{E}_{Pev_{\mathbf{x}}^{-1}}(Id_{Y}-\mathbb{E}_{Pev_{\mathbf{x}}^{-1}}(Id_{Y}))^{2})&=&\mathbb{E}_{P}(ev_{\mathbf{x}}-\mathbb{E}_{P}(ev_{\mathbf{x}}))^{2})\\\
&=&k(\mathbf{x},\mathbf{x}).\end{array}$
These are precisely specified by the characterization
$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$.
$1$$Y^{X}$$Y$$P\sim\mathcal{G}\mathcal{P}(m,k)$$\mathcal{S}^{\mathbf{x}}=\delta_{ev_{\mathbf{x}}}$$\mathcal{I}^{\mathbf{x}}$$Pev_{\mathbf{x}}^{-1}\sim\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$
Figure 30: The composite of the prior distribution
$P\sim\mathcal{G}\mathcal{P}(m,k)$ and the sampling distribution
$\mathcal{S}^{\mathbf{x}}$ give the coordinate projections as priors on $Y$.
Recall that the corresponding inference map $\mathcal{I}^{\mathbf{x}}$ is any
$\mathcal{P}$ map satisfying the necessary and sufficient condition of
Equation 11, i.e., for all $\mathcal{A}\in\Sigma_{Y^{X}}$ and
$B\in\Sigma_{Y}$,
$\int_{f\in\mathcal{A}}\mathcal{S}^{\mathbf{x}}(B\mid f)\,dP=\int_{y\in
B}\mathcal{I}^{\mathbf{x}}(\mathcal{A}\mid y)\,d(Pev_{\mathbf{x}}^{-1}).$ (43)
Since the $\sigma$-algebra of $Y^{X}$ is generated by elements
$ev_{\mathbf{z}}^{-1}(A)$, for $\mathbf{z}\in Y$ and $A\in\Sigma_{Y}$, we can
take $\mathcal{A}=ev_{\mathbf{z}}^{-1}(A)$ in the above expression to obtain
the equivalent necessary and sufficient condition on
$\mathcal{I}^{\mathbf{x}}$ of
$\int_{f\in ev_{\mathbf{z}}^{-1}(A)}\mathcal{S}^{\mathbf{x}}(B\mid
f)\,dP=\int_{y\in B}\mathcal{I}^{\mathbf{x}}(ev_{\mathbf{z}}^{-1}(A)\mid
y)\,d(Pev_{\mathbf{x}}^{-1}).$
From Equation 41, $\mathcal{S}^{\mathbf{x}}(B\mid
f)=\mathbb{1}_{ev_{\mathbf{x}}^{-1}(B)}(f)$, so substituting this value into
the left hand side of this equation reduces that term to
$P(ev_{\mathbf{x}}^{-1}(B)\cap ev_{\mathbf{z}}^{-1}(A))$. Rearranging the
order of the terms it follows the condition on the inference map
$\mathcal{I}^{\mathbf{x}}$ is
$\int_{y\in B}\mathcal{I}^{\mathbf{x}}(ev_{\mathbf{z}}^{-1}(A)\mid
y)\,d(Pev_{\mathbf{x}}^{-1})=P(ev_{\mathbf{x}}^{-1}(B)\cap
ev_{\mathbf{z}}^{-1}(A)).$
Since the left hand side of this expression is integrated with respect to the
pushforward probability measure $Pev_{\mathbf{x}}^{-1}$ it is equivalent to
$\begin{array}[]{lcl}\int_{y\in
B}\mathcal{I}^{\mathbf{x}}(ev_{\mathbf{z}}^{-1}(A)\mid
y)\,d(Pev_{\mathbf{x}}^{-1})&=&\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\mathcal{I}^{\mathbf{x}}(ev_{\mathbf{z}}^{-1}(A)\mid
ev_{\mathbf{x}}(f))\,dP\\\
&=&\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\mathcal{I}^{\mathbf{x}}ev_{\mathbf{z}}^{-1}(A\mid
ev_{\mathbf{x}}(f))\,dP.\end{array}$
In summary, if $\mathcal{I}^{\mathbf{x}}$ is to be an inference map for the
prior $P$ and sampling distribution $\mathcal{S}^{\mathbf{x}}$, then it is
necessary and sufficient that it satisfy the condition
$\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\mathcal{I}^{\mathbf{x}}ev_{\mathbf{z}}^{-1}(A\mid
ev_{\mathbf{x}}(f))\,dP=P(ev_{\mathbf{x}}^{-1}(B)\cap
ev_{\mathbf{z}}^{-1}(A)).$
Given a (deterministic) measurement212121Meaning the arrow $d=\delta_{y}$ in
Figure 17. In general it is unnecessary to assume deterministic measurements
in which case the composite $\mathcal{I}^{\mathbf{x}}\circ d$ represents the
posterior. at $(\mathbf{x},y)$, the stochastic process
$\mathcal{I}^{\mathbf{x}}(\cdot\mid y):1\rightarrow Y^{X}$ is the posterior of
$P\sim\mathcal{G}\mathcal{P}(m,k)$. This posterior, denoted
$P^{1}_{Y^{X}}\triangleq\mathcal{I}^{\mathbf{x}}(\cdot\mid y)$, is generally
not unique. However we can require that the posterior $P^{1}_{Y^{X}}$ be a GP
specified by updated mean and covariance functions $m^{1}$ and $k^{1}$
respectively, which depend upon the conditioning value $y$, so
$P^{1}_{Y^{X}}\sim\mathcal{G}\mathcal{P}(m^{1},k^{1})$. To determine
$P^{1}_{Y^{X}}$, and hence the desired inference map
$\mathcal{I}^{\mathbf{x}}$, we make a hypothesis about the updated mean and
covariance functions $m^{1}$ and $k^{1}$ characterizing $P^{1}_{Y^{X}}$ given
a measurement at the pair $(\mathbf{x},y)\in X\times Y$. Let us assume the
updated mean function is of the form
$m^{1}(\mathbf{z})=m(\mathbf{z})+\frac{k(\mathbf{z},\mathbf{x})}{k(\mathbf{x},\mathbf{x})}(y-m(\mathbf{x}))$
(44)
and the updated covariance function is of the form
$k^{1}(\mathbf{w},\mathbf{z})=k(\mathbf{w},\mathbf{z})-\frac{k(\mathbf{w},\mathbf{x})k(\mathbf{x},\mathbf{z})}{k(\mathbf{x},\mathbf{x})}.$
(45)
To prove these updated functions suffice to specify the inference map
$\mathcal{I}^{\mathbf{x}}(\cdot\mid
y)=P^{1}_{Y^{X}}\sim\mathcal{G}\mathcal{P}(m^{1},k^{1})$ satisfying the
necessary and sufficient condition we simply evaluate
$\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\mathcal{I}^{\mathbf{x}}ev_{\mathbf{z}}^{-1}(A\mid
ev_{\mathbf{x}}(f))\,dP$
by substituting $\mathcal{I}^{\mathbf{x}}(\cdot\mid
f(\mathbf{x}))=P^{1}(m^{1},k^{1})$ and verify that it yields
$P(ev_{\mathbf{x}}^{-1}(B)\cap ev_{\mathbf{z}}^{-1}(A))$. Since
$\mathcal{I}^{\mathbf{x}}ev_{\mathbf{z}}^{-1}(\cdot\mid
f(\mathbf{x}))=P^{1}_{Y^{X}}ev_{\mathbf{x}}^{-1}$ is a normal distribution of
mean
$m^{1}(\mathbf{z})=m(\mathbf{x})+\frac{k(\mathbf{z},\mathbf{x})}{k(\mathbf{x},\mathbf{x})}\cdot(f(\mathbf{x})-m(\mathbf{x}))$
and covariance
$k^{1}(\mathbf{z},\mathbf{z})=k(\mathbf{z},\mathbf{z})-\frac{k(\mathbf{z},\mathbf{x})^{2}}{k(\mathbf{x},\mathbf{x})}$
it follows that
$\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\mathcal{I}^{\mathbf{x}}ev_{\mathbf{z}}^{-1}(A\mid
ev_{\mathbf{x}}(f))\,dP=\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\left(\frac{1}{\sqrt{2\pi
k^{1}(\mathbf{z},\mathbf{z})}}\int_{v\in
A}e^{\frac{-(m^{1}(\mathbf{z})-v)^{2}}{2k^{1}(\mathbf{z},\mathbf{z})}}dv\right)dP$
which can be expanded to
$\int_{f\in\,ev_{\mathbf{x}}^{-1}(B)}\left(\frac{1}{\sqrt{2\pi
k^{1}(\mathbf{z},\mathbf{z})}}\int_{v\in
A}e^{\frac{-(m(\mathbf{z})+\frac{k(\mathbf{z},\mathbf{x})}{k(\mathbf{x},\mathbf{x})}(f(\mathbf{x})-m(\mathbf{x}))-v)^{2}}{2k^{1}(\mathbf{z},\mathbf{z})}}dv\right)dP$
and equals
$\int_{y\in B}\left(\frac{1}{\sqrt{2\pi
k^{1}(\mathbf{z},\mathbf{z})}}\int_{v\in
A}e^{\frac{-(m(\mathbf{z})+\frac{k(\mathbf{z},\mathbf{x})}{k(\mathbf{x},\mathbf{x})}(y-m(\mathbf{x}))-v)^{2}}{2k^{1}(\mathbf{z},\mathbf{z})}}dv\right)dPev_{\mathbf{x}}^{-1}.$
Using
$P_{Y^{X}}ev_{\mathbf{x}}^{-1}\sim\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$
we can rewrite the expression as
$\frac{1}{\sqrt{2\pi}\mid\Omega\mid}\int_{y\in B}\int_{v\in
A}e^{-\frac{1}{2}(\mathbf{u}-\overline{\mathbf{u}})^{T}\Omega^{-1}(\mathbf{u}-\overline{\mathbf{u}})}dv\,dy$
where
$\mathbf{u}=\left(\begin{array}[]{c}y\\\
v\end{array}\right)\quad\quad\overline{\mathbf{u}}=\left(\begin{array}[]{c}m(\mathbf{x})\\\
m(\mathbf{z})\end{array}\right)$
and
$\Omega=\left(\begin{array}[]{cc}k[\mathbf{x},\mathbf{x}]&k[\mathbf{x},\mathbf{z}]\\\
k[\mathbf{z},\mathbf{x}]&k[\mathbf{z},\mathbf{z}]\end{array}\right),$
which we recognize as a normal distribution
$\mathcal{N}(\overline{\mathbf{u}},\Omega)$.
On the other hand, we claim that
$1$$Y_{\mathbf{x}}\times Y_{\mathbf{z}}$,$P(ev_{\mathbf{x}}^{-1}(\cdot)\cap
ev_{\mathbf{z}}^{-1}(\cdot))$
where $Y_{\mathbf{x}}$ and $Y_{\mathbf{z}}$ are two copies of $Y$, is also a
normal distribution of mean $\overline{u}=(m(\mathbf{x}),m(\mathbf{z}))$ with
covariance matrix $\Omega$.222222Formally the arguments should be numbered in
the given probability measure as $P(ev_{\mathbf{x}}^{-1}(\\#1)\cap
ev_{\mathbf{z}}^{-1}(\\#2))$ because $ev_{\mathbf{x}}^{-1}(A)\cap
ev_{\mathbf{z}}^{-1}(B)\neq ev_{\mathbf{x}}^{-1}(B)\cap
ev_{\mathbf{z}}^{-1}(A)$. However the subscripts can be used to identify which
component measurable sets are associated with each argument. To prove our
claim consider the $\mathcal{P}$ diagram in Figure 31 where
$X_{0}=\\{\mathbf{x},\mathbf{z}\\}$, $\iota:X_{0}\hookrightarrow X$ is the
inclusion map referenced in Section 6.2, and $ev_{\mathbf{x}}\times
ev_{\mathbf{z}}$ is an isomorphism between the two different representations
of the set of all measurable functions $Y^{X_{0}}$ alluded to in the second
paragraph of Section 6.
$1$$Y^{X}$$Y_{\mathbf{x}}$$Y_{\mathbf{z}}$$Y^{X_{0}}$$Y_{\mathbf{x}}\times
Y_{\mathbf{z}}$$P$$\delta_{Y^{\iota}}$$\delta_{ev_{\mathbf{x}}}$$\delta_{ev_{\mathbf{z}}}$$\delta_{\pi_{Y_{\mathbf{x}}}}$$\delta_{\pi_{Y_{\mathbf{z}}}}$$\delta_{ev_{\mathbf{x}}}$$\delta_{ev_{\mathbf{z}}}$$\delta_{ev_{\mathbf{x}}\times
ev_{\mathbf{z}}}$$\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}))$$\mathcal{N}(m(\mathbf{z}),k(\mathbf{z},\mathbf{z}))$
Figure 31: Proving the joint distribution $\delta_{ev_{\mathbf{x}}\times
ev_{\mathbf{z}}}\circ\delta_{Y^{\iota}}\circ
P=P(ev_{\mathbf{x}}^{-1}(\cdot)\cap ev_{\mathbf{z}}^{-1}(\cdot)))$ is a normal
distribution $\mathcal{N}(\overline{\mathbf{u}},\Omega)$.
The diagram in Figure 31 commutes because
$\delta_{\pi_{Y_{\mathbf{x}}}}\circ\delta_{ev_{\mathbf{x}}\times
ev_{\mathbf{z}}}=\delta_{\mathbf{x}}$ and
$\delta_{\pi_{Y_{\mathbf{z}}}}\circ\delta_{ev_{\mathbf{x}}\times
ev_{\mathbf{z}}}=\delta_{\mathbf{z}}$ while, using $(ev_{\mathbf{x}}\times
ev_{\mathbf{z}})\circ Y^{\iota}=(ev_{\mathbf{x}},ev_{\mathbf{z}})$,
$\begin{array}[]{lcl}(\delta_{ev_{\mathbf{x}}\times
ev_{\mathbf{z}}}\circ\delta_{Y^{\iota}}\circ P)(A\times B)&=&\int_{f\in
Y^{X}}\underbrace{\delta_{(ev_{\mathbf{x}},ev_{\mathbf{z}})}(A\times B\mid
f)}_{=(\mathbb{1}_{A\times B})(ev_{\mathbf{x}},ev_{\mathbf{z}})(f)}\,dP\\\
&=&\int_{f\in
Y^{X}}(\mathbb{1}_{ev_{\mathbf{x}}^{-1}(A)}\cdot\mathbb{1}_{ev_{\mathbf{z}}^{-1}(B)})(f)\,dP\\\
&=&\int_{f\in Y^{X}}\mathbb{1}_{ev_{\mathbf{x}}^{-1}(A)\cap
ev_{\mathbf{z}}^{-1}(B)}(f)\,dP\\\ &=&P(ev_{\mathbf{x}}^{-1}(A)\cap
ev_{\mathbf{z}}^{-1}(B)).\end{array}$
Moreover, the covariance $k$ of $P(ev_{\mathbf{x}}^{-1}(\cdot)\cap
ev_{\mathbf{z}}^{-1}(\cdot)))$ is represented by the matrix $\Omega$ because
by definition of $P$, in terms of $m$ and $k$, its restriction to
$Y^{X_{0}}\cong Y_{\mathbf{x}}\times Y_{\mathbf{z}}$ has covariance
$k\mid_{X_{0}}\cong\Omega$.
Consequently the necessary and sufficient condition for
$\mathcal{I}^{\mathbf{x}}=P^{1}_{Y^{X}}\sim\mathcal{G}\mathcal{P}(m^{1},k^{1})$
to be an inference map is satisfied by the projection of $P_{Y^{X}}^{1}$ onto
any single coordinate $\mathbf{z}$ which corresponds to the restriction of
$P_{Y^{X}}^{1}$ via the deterministic map $Y^{\iota}:Y^{X}\rightarrow
Y^{X_{0}}$ with $X_{0}=\\{\mathbf{z}\\}$ as in Figure 14. But this procedure
immediately extends to all finite subsets $X_{0}\subset X$ using matrix
algebra and consequently we conclude that the necessary and sufficient
condition for $\mathcal{I}^{\mathbf{x}}$ to be an inference map for the prior
$P$ and the noise free sampling distribution $\mathcal{S}^{\mathbf{x}}$ is
satisfied.
Writing the prior GP as $P\sim\mathcal{G}\mathcal{P}(m^{0},k^{0})$ the
recursive updating equations are
$m^{i+1}\left(\mathbf{z}\mid(\mathbf{x}_{i},y_{i})\right)=m^{i}(\mathbf{z})+\frac{k^{i}(\mathbf{z},\mathbf{x}_{i})}{k^{i}(\mathbf{x}_{i},\mathbf{x}_{i})}(y_{i}-m^{i}(\mathbf{x}_{i}))\quad\textrm{
for }i=0,\ldots,N-1$ (46)
and
$k^{i+1}((\mathbf{w},\mathbf{z})\mid(\mathbf{x}_{i},y_{i}))=k^{i}(\mathbf{w},\mathbf{z})-\frac{k^{i}(\mathbf{w},\mathbf{x}_{i})k^{i}(\mathbf{x}_{i},\mathbf{z})}{k^{i}(\mathbf{x}_{i},\mathbf{x}_{i})}\quad\textrm{
for }i=0,\ldots,N-1$ (47)
where the terms on the left denote the posterior mean and covariance functions
of $m^{i}$ and $k^{i}$ given a new measurement $(\mathbf{x}_{i},y_{i})$. These
expressions coincide with the standard formulas written for $N$ arbitrary
measurements $\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N-1}$, with
$X_{0}=(\mathbf{x}_{0},\ldots,\mathbf{x}_{N-1})$ a finite set of independent
points of $X$ with corresponding measurements
$\mathbf{y}^{T}=(y_{0},y_{1},\ldots,y_{N-1})$,
$\tilde{m}(\mathbf{z}\mid
X_{0})=m(\mathbf{z})+K(\mathbf{z},X_{0})K(X_{0},X_{0})^{-1}(\mathbf{y}-m(X_{0}))$
(48)
where $m(X_{0})=(m(\mathbf{x}_{0}),\ldots,m(\mathbf{x}_{N-1}))^{T}$, and
$\tilde{k}((\mathbf{w},\mathbf{z})\mid
X_{0})=k(\mathbf{w},\mathbf{z})-K(\mathbf{w},X_{0})K(X_{0},X_{0})^{-1}K(X_{0},\mathbf{z})$
(49)
where $K(\mathbf{w},X_{0})$ is the row vector with components
$k(\mathbf{w},\mathbf{x}_{i})$, $K(X_{0},X_{0})$ is the matrix with components
$k(\mathbf{x}_{i},\mathbf{x}_{j})$, and $K(X_{0},\mathbf{z})$ is a column
vector with components $k(\mathbf{x}_{i},\mathbf{z})$.232323When the points
are not independent then one can use a perturbation method or other procedure
to avoid degeneracy. The notation $\tilde{m}$ and $\tilde{k}$ is used to
differentiate these standard expressions from ours above. Equations 48 and 49
are a computationally efficient way to keep track of the updated mean and
covariance functions. One can easily verify the recursive equations determine
the standard equations using induction.
A review of the derivation of $P^{1}_{Y^{X}}$ indicates that the posterior
$P^{1}_{Y^{X}}\sim\mathcal{G}\mathcal{P}(m^{1},k^{1})$ is actually
parameterized by the measurement $(\mathbf{x}_{1},y_{1})$ because the above
derivation holds for any measurement $(\mathbf{x}_{1},y_{1})$ and this pair of
values uniquely determines $m^{1}$ and $k^{1}$ through the Equations 46 and
47, or equivalently Equations 48 and 49, for a single measurement.
By the SMwCC structure of $\mathcal{P}$ each parameterized GP $P_{Y^{X}}^{1}$
can be put into the bijective correspondence shown in Figure 32, where
$\begin{array}[]{lcl}\overline{P_{Y^{X}}^{1}}(B\mid(z,(\mathbf{x},y)))&=&P_{Y^{X}}^{1}(ev_{\mathbf{z}}^{-1}(B)\mid(\mathbf{x},y))\quad\quad\forall
B\in\Sigma_{Y},\mathbf{z}\in X,y\in Y\\\
&=&P_{Y^{X}}^{1}ev_{\mathbf{z}}^{-1}(B\mid(\mathbf{x},y))\\\
&=&\frac{1}{\sqrt{2\pi k^{1}(\mathbf{z},\mathbf{z})}}\int_{v\in
B}e^{-\frac{(v-m^{1}(z))^{2}}{2k^{1}(\mathbf{z},\mathbf{z})}}\,dv\\\
&=&\frac{1}{\sqrt{2\pi\frac{k(\mathbf{x},\mathbf{x})k(\mathbf{z},\mathbf{z})-k(\mathbf{x},\mathbf{z})^{2}}{k(\mathbf{x},\mathbf{x})}}}\int_{v\in
B}e^{-\frac{(v-(m(z)+\frac{k(\mathbf{z},\mathbf{x})}{k(\mathbf{x},\mathbf{x})}(y-\mathbf{x}))^{2}}{2\frac{k(\mathbf{x},\mathbf{x})k(\mathbf{z},\mathbf{z})-k(\mathbf{x},\mathbf{z})^{2}}{k(\mathbf{x},\mathbf{x})}}}\,dv\end{array}$
which is a probability measure on $Y$ conditioned on $\mathbf{z}$ and
parameterized by the pair $(\mathbf{x},y)$. Iterating this process we obtain
the viewpoint that the parameterized process
$P_{Y^{X}}(ev_{\mathbf{z}}^{-1}(B)\mid\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N})$
is a posterior conditional probability parameterized over $N$ measurements.
$X\otimes Y$$Y^{X}$$X\otimes(X\otimes
Y)$$Y$$P_{Y^{X}}^{1}$$\overline{P_{Y^{X}}^{1}}$ Figure 32: Each GP
$P_{Y^{X}}^{1}$, which is parameterized by a measurement $(\mathbf{x},y)\in
X\otimes Y$, determines a conditional $\overline{P_{Y^{X}}^{1}}$.
### 8.2 The noisy measurement inference map
When the measurement model has additive Gaussian noise which is iid on each
slice $\mathbf{x}\in X$, the resulting inference map is easily given by
observing that from Equation 36, the composite $\delta_{ev_{\mathbf{x}}}\circ
N\circ
P\sim\mathcal{N}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x})+k_{N}(\mathbf{x},\mathbf{x}))$.
Thus, the noisy sampling distribution along with the prior
$P\sim\mathcal{G}\mathcal{P}(m,k)$ can be viewed as a noise free distribution
$P\sim\mathcal{G}\mathcal{P}(m,\kappa)$ on $Y^{X}$, where $\kappa\triangleq
k+k_{N}$, and $k_{N}$ is given by equation 37. This is clear from the
composite of Figure 24 with the Dirac measure $\delta_{\mathbf{x}}$ on the $X$
component. Now the noisy measurement inference map for the Bayesian model with
prior $P$ and sampling distribution
$\mathcal{S}^{\mathbf{x}}=\delta_{\mathbf{x}}\circ N$, as shown in Figure 33,
can be determined by decomposing it into two simpler Bayesian problems whose
inference maps are (1) trivial (the identity map) and (2) already known.
$1$$Y^{X}$$Y^{X}$$Y$$P\sim\mathcal{N}(m,k)$$N$$N\circ
P$$\delta_{\mathbf{x}}\circ N\circ
P\sim\mathcal{N}(m(\mathbf{x}),\underbrace{k(\mathbf{x},\mathbf{x})+k_{N}(\mathbf{x},\mathbf{x})}_{=\kappa(\mathbf{x},\mathbf{x})})$$\delta_{ev_{\mathbf{x}}}$$\mathcal{S}^{\mathbf{x}}$$\Downarrow$
Decomposition$1$$Y^{X}$$Y^{X}$$P\sim\mathcal{N}(m,k)$$N$$N\circ
P$$\mathcal{I}_{*}$$1$$Y^{X}$$Y$$N\circ
P$$\delta_{\mathbf{x}}$$\delta_{\mathbf{x}}\circ N\circ P$$\mathcal{I}_{nf}$
Figure 33: Splitting the Gaussian additive noise Bayesian model (top diagram)
into two separate Bayesian models (bottom two diagrams) and composing the
inference maps for these two simple Bayesian models gives the inference map
for the original Gaussian additive Bayesian model.
Observe that the composition of the two bottom diagrams is the top diagram.
The bottom diagram on the right is a noise free Bayesian model with GP prior
$N\circ P$ and sampling distribution $\delta_{\mathbf{x}}$ whose inference map
$\mathcal{I}_{nf}$ we have already determined analytically in Section 8.1.
Given a measurement $y\in Y$ at $\mathbf{x}\in X$, the inference map is given
by the updating Equations 44 and 45 for the mean and covariance functions
characterizing the GP on $Y^{X}$. The resulting posterior GP on $Y^{X}$ can
then be viewed as a measurement on $Y^{X}$ for the bottom left diagram, which
is a Bayesian model with prior $P$ and sampling distribution $N$. The
inference map $\mathcal{I}_{\star}$ for this diagram is the identity map on
$Y^{X}$, $\mathcal{I}_{\star}=\delta_{Id_{Y^{X}}}$. This is easy to verify
using Bayes product rule (Equation 13), $\int_{a\in A}N(B\mid
a)\,dP=\int_{f\in B}\delta_{Id_{Y^{X}}}(A\mid f)\,d(N\circ P)$, for any
$A,B\in\Sigma_{Y^{X}}$. Composition of these two inference maps,
$\mathcal{I}_{nf}$ and $\mathcal{I}_{\star}$ then yields the resulting
inference map for the Gaussian additive noise Bayesian model.
With this observation both of the recursive updating schemes given by
Equations 46 and 47 are valid for the Gaussian additive noise model with $k$
replaced by $\kappa$. The corresponding standard expressions for the noisy
model are then
$\tilde{m}(\mathbf{z}\mid
X_{0})=m(\mathbf{z})+K(\mathbf{z},X_{0})K(X_{0},X_{0})^{-1}(\mathbf{y}-m(X_{0}))$
and
$\tilde{\kappa}((\mathbf{w},\mathbf{z})\mid
X_{0})=\kappa(\mathbf{w},\mathbf{z})-K(\mathbf{w},X_{0})K(X_{0},X_{0})^{-1}K(X_{0},\mathbf{z}),$
where the quantities like $K(\mathbf{w},X_{0})$ are as defined previously
(following Equation 49) except now $k$ is replaced by $\kappa$. For
$\mathbf{w}\neq\mathbf{z}$ and neither among the measurements $X_{0}$ these
expressions, upon substituting in for $\kappa$, reduce to the familiar
expressions
$\tilde{m}(\mathbf{z}\mid
X_{0})=m(\mathbf{z})+K(\mathbf{z},X_{0})(K(X_{0},X_{0})+\sigma^{2}\textit{I})^{-1}(\mathbf{y}-m(X_{0}))$
and
$\tilde{k}((\mathbf{w},\mathbf{z})\mid
X_{0})=k(\mathbf{w},\mathbf{z})-K(\mathbf{w},X_{0})(K(X_{0},X_{0})+\sigma^{2}\textit{I})^{-1}K(X_{0},\mathbf{z}),$
which provide for a computationally efficient way to compute the mean and
covariance of a GP given a finite number of measurements.
### 8.3 The inference map for parametric models
Under the prior $\delta_{\mathbf{x}}\otimes P$ on the hypothesis space in the
parametric model, Figure 28, the parametric sampling distribution model can be
viewed as a family of models, one for each $\mathbf{x}\in X$, given by the
diagram in Figure 34.
$\mathbb{R}^{{}^{p}}$$Y^{X}$$Y^{X}$$Y$$\delta_{i}$$N$$\delta_{ev_{\mathbf{x}}}$$\mathcal{S}^{\mathbf{x}}_{p}$
Figure 34: The Gaussian additive noise parametric sampling distributions
$\mathcal{S}^{\mathbf{x}}_{p}$ viewed as a family of sampling distributions,
one for each $\mathbf{x}\in X$.
The sampling distribution can be computed as
$\begin{array}[]{lcl}\mathcal{S}^{\mathbf{x}}_{p}(B\mid\mathbf{a})&=&(\delta_{ev_{\mathbf{x}}}\circ
N\circ\delta_{i})(B\mid\mathbf{a})\\\ &=&\int_{f\in
Y^{X}}(\delta_{ev_{\mathbf{x}}}\circ N)(B\mid
f)\,d\underbrace{(\delta_{i})_{\mathbf{a}}}_{\delta_{F_{\mathbf{a}}}}\\\
&=&(\delta_{ev_{\mathbf{x}}}\circ N)(B\mid F_{\mathbf{a}})\\\
&=&N(ev_{\mathbf{x}}^{-1}(B)\mid F_{\mathbf{a}}).\end{array}$
Because $N_{F_{\mathbf{a}}}\sim GP(F_{\mathbf{a}},k_{N})$, it follows that
$N(ev_{\mathbf{x}}^{-1}(\bullet)\mid
F_{\mathbf{a}})=N_{F_{\mathbf{a}}}ev_{\mathbf{x}}^{-1}\sim\mathcal{N}(F_{\mathbf{a}}(\mathbf{x}),\sigma^{2})$
and consequently
$\mathcal{S}^{\mathbf{x}}_{p}(B\mid\mathbf{a})=\frac{1}{\sqrt{2\pi}\sigma}\int_{B}e^{-\frac{(y-F_{\mathbf{a}}(\mathbf{x}))^{2}}{2\sigma^{2}}}\,dy.$
Taking the prior $P:1\rightarrow\mathbb{R}^{{}^{p}}$ as a normal distribution
with mean $\mathbf{m}$ and covariance function $k$, it follows that the
composite $\mathcal{S}^{\mathbf{x}}_{p}\circ
P\sim\mathcal{N}(F_{\mathbf{m}}(\mathbf{x}),k(\mathbf{x},\mathbf{x})+\sigma^{2})$
while the inference map $\mathcal{I}^{\mathbf{x}}_{p}$ satisfies, for all
$B\in\Sigma_{Y}$ and all $\mathcal{A}\in\Sigma_{\mathbb{R}^{p}}$,
$\int_{\mathbf{a}\in\mathcal{A}}\mathcal{S}^{\mathbf{x}}_{p}(B\mid\mathbf{a})\,dP=\int_{y\in
B}\mathcal{I}^{\mathbf{x}}_{p}(\mathcal{A}\mid
y)\,d(\mathcal{S}^{\mathbf{x}}_{p}\circ P).$
To determine this inference map $\mathcal{I}^{\mathbf{x}}_{p}$ it is necessary
to require the parametric map
$\begin{array}[]{lclcl}i&:&\mathbb{R}^{{}^{p}}&\longrightarrow&Y^{X}\\\
&:&\mathbf{a}&\mapsto&i_{\mathbf{a}}\end{array}$
be an injective linear homomorphism. Under this condition, which can often be
achieved simply by eliminating redundant modeling parameters, we can
explicitly determine the inference map for the parameterized model, denoted
$\mathcal{I}^{\mathbf{x}}_{p}$, by decomposing it into two inference maps as
displayed in the diagram in Figure 35.
$1$$\mathbb{R}^{{}^{n}}$$Y^{X}$$Y$$\mathcal{S}^{\mathbf{x}}_{p}=\mathcal{S}^{\mathbf{x}}_{n}\circ\delta_{i}$$Pi^{-1}$$P$$\delta_{i}$$\mathcal{I}_{\star}$$\mathcal{S}^{\mathbf{x}}_{n}$$\mathcal{I}^{\mathbf{x}}_{n}$$\mathcal{S}^{\mathbf{x}}_{n}\circ
Pi^{-1}$$\mathcal{I}^{\mathbf{x}}_{p}$ Figure 35: The inference map for the
parametric model is a composite of two inference maps.
We first show the stochastic process $Pi^{-1}$ is a GP and by taking the
sampling distribution
$\mathcal{S}^{\mathbf{x}}_{n}=\delta_{ev_{\mathbf{x}}}\circ N$ as the noisy
measurement model we can use the result of the previous section to provide us
with the inference map $\mathcal{I}^{\mathbf{x}}_{n}$ in Figure 35.
###### Lemma 16.
Let $\mathbf{k}$ be the matrix representation of the covariance function $k$.
The the push forward of $P~{}\mathcal{N}(\mathbf{m},k)$ by $i$ is a GP
$Pi^{-1}\sim\mathcal{G}\mathcal{P}(i_{\mathbf{m}},\hat{k})$, where
$\hat{k}(\mathbf{u},\mathbf{v})=\mathbf{u}^{T}\mathbf{k}\mathbf{v}$.
###### Proof.
We need to show that the push forward of $Pi^{-1}$ by the restriction map
$Y^{\iota}:Y^{X}\longrightarrow Y^{X_{0}}$ is a normal distribution for any
finite subspace $\iota:X_{0}\hookrightarrow X$. Consider the commutate diagram
in Figure 36, where $Y_{\mathbf{x}}$ is a copy of $Y$,
$X_{0}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n^{\prime}})$, and
$ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n^{\prime}}}:Y^{X_{0}}\rightarrow\prod_{\mathbf{x}\in
X_{0}}Y_{\mathbf{x}}$
is the canonical isomorphism.
$1$$\mathbb{R}^{{}^{n}}$$Y^{X}$$Y^{X_{0}}$$\prod_{\mathbf{x}\in
X_{0}}Y_{\mathbf{x}}$$Pi^{-1}$$P$$\delta_{i}$$\delta_{Y^{\iota}}$$\delta_{Y^{\iota}\circ
i}$$\delta_{ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n^{\prime}}}}$ Figure 36: The restriction of $Pi^{-1}$.
The composite of the measurable maps
$\left((ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n^{\prime}}})\circ Y^{\iota}\circ
i\right)(\mathbf{a})=(i_{\mathbf{a}}(\mathbf{x}_{1}),\ldots,i_{\mathbf{a}}(\mathbf{x}_{n^{\prime}}))$
(50)
from which it follows that the composite map
$\delta_{ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n^{\prime}}}}\circ\delta_{Y^{\iota}}\circ
Pi^{-1}\sim\mathcal{N}(X_{0}^{T}\mathbf{m},X_{0}^{T}\mathbf{k}X_{0})$. ∎
Now the diagram
$1$$\mathbb{R}^{{}^{n}}$$Y^{X}$$Pi^{-1}$$P$$\delta_{i}$$\mathcal{I}_{\star}$
with the sampling distribution for this Bayesian problem as $\delta_{i}$. Let
$\mathbf{e}_{j}^{T}=(0,\ldots,0,1,0,\ldots,0)$ be the $j^{th}$ unit vector of
$\mathbb{R}^{{}^{p}}$ and let $i_{\mathbf{e}_{j}}=f_{j}\in Y^{X}$. The
elements $\\{f_{j}\\}_{j=1}^{p}$ form the components of a basis for the image
of $i$ by the assumed injective property of $i$. Let this finite basis have a
dual basis $\\{f_{j}^{*}\\}_{j=1}^{p}$ so that
$f_{k}^{*}(f_{j})=\delta_{k}(j)$.
Consider the measurable map
$\begin{array}[]{lclcl}f_{1}^{*}\times\ldots\times
f_{p}^{*}&:&Y^{X}&\rightarrow&\mathbb{R}^{{}^{p}}\\\
&;&g&\mapsto&(f^{*}_{1}(g),\ldots,f_{p}^{*}(g)),\end{array}$
Using the linearity of the parameter space $\mathbb{R}^{{}^{p}}$ it follows
$\mathbf{a}=\sum_{i=1}^{p}a_{i}\mathbf{e}_{i}$ and consequently
$\begin{array}[]{lcl}\left((f_{1}^{*}\times\ldots\times f_{p}^{*})\circ
i\right)(\mathbf{a})&=&(f_{1}^{*}(i_{\mathbf{a}}),\ldots,f_{p}^{*}(i_{\mathbf{a}}))\\\
&=&\mathbf{a}\quad\textrm{ using
}f_{j}^{*}(i_{\mathbf{a}})=f_{j}^{*}(\sum_{k=1}^{p}a_{k}f_{k})=a_{j}\end{array}$
and hence $(f_{1}^{*}\times\ldots\times f_{p}^{*})\circ
i=id_{\mathbb{R}^{{}^{p}}}$ in $\mathcal{M}eas$. Now it follows the
corresponding inference map
$\mathcal{I}_{\star}=\delta_{f_{1}^{*}\times\ldots\times f_{p}^{*}}$ because
the necessary and sufficient condition for $\mathcal{I}_{\star}$ is given, for
all $ev_{\mathbf{z}}^{-1}(B)\in\Sigma_{Y^{X}}$ (which generate
$\Sigma_{Y^{X}}$) and all $\mathcal{A}\in\Sigma_{\mathbb{R}^{{}^{n}}}$, by
$\int_{\mathbf{a}\in\mathcal{A}}\delta_{i}(ev_{\mathbf{z}}^{-1}(B)\mid\mathbf{a})\,dP=\int_{g\in
ev_{\mathbf{z}}^{-1}(B)}\mathcal{I}_{\star}(\mathcal{A}\mid g)\,dPi^{-1}$ (51)
with the left hand term reducing to the expression
$\int_{\mathbf{a}\in\mathcal{A}}\mathbb{1}_{i^{-1}(ev_{\mathbf{z}}^{-1}(B))}(\mathbf{a})\,dP=P(i^{-1}(ev_{\mathbf{z}}^{-1}(B))\cap\mathcal{A}).$
On the other hand, using
$\mathcal{I}_{\star}=\delta_{f_{1}^{*}\times\ldots\times f_{p}^{*}}$, the
right hand term of Equation 51 also reduces to the same expression since
$\begin{array}[]{lcl}\int_{g\in
ev_{\mathbf{z}}^{-1}(B)}\delta_{f_{1}^{*}\times\ldots\times
f_{p}^{*}}(\mathcal{A}\mid g)\,d(Pi^{-1})&=&\int_{\mathbf{a}\in
i^{-1}(ev_{\mathbf{z}}^{-1}(B))}\mathbb{1}_{((f_{1}^{*}\times\ldots\times
f_{p}^{*})\circ i)^{-1}(\mathcal{A})}(\mathbf{a})\,dP\\\
&=&\int_{\mathbf{a}\in
i^{-1}(ev_{\mathbf{z}}^{-1}(B))}\mathbb{1}_{\mathcal{A}}(\mathbf{a})\,dP\\\
&=&P(i^{-1}(ev_{\mathbf{z}}^{-1}(B))\cap\mathcal{A})\end{array}$
thus proving $\mathcal{I}_{\star}=\delta_{f_{1}^{*}\times\ldots\times
f_{p}^{*}}$.
Taking
$\mathcal{I}^{\mathbf{x}}_{p}=\mathcal{I}_{\star}\circ\mathcal{I}^{\mathbf{x}}_{n},$
it follows that for a given measurement $(\mathbf{x},y)$ that the composite is
$\mathcal{I}^{\mathbf{x}}_{p}=\mathcal{I}^{\mathbf{x}}_{n}((f_{1}^{*}\times\ldots\times
f_{p}^{*})^{-1}(\cdot)\mid y)$ (52)
which is the push forward measure of the GP
$\mathcal{I}^{\mathbf{x}}_{n}(\cdot\mid
y)\sim\mathcal{G}\mathcal{P}(i_{\mathbf{m}}^{1},\kappa^{1})$ where (as defined
previously) $\kappa=k+k_{N}$ and
$i_{\mathbf{m}}^{1}(\mathbf{z})=i_{\mathbf{m}}(\mathbf{z})+\frac{\kappa(\mathbf{z},\mathbf{x})}{\kappa(\mathbf{x},\mathbf{x})}(y-i_{\mathbf{m}}(\mathbf{x}))$
(53)
and
$\kappa^{1}(\mathbf{u},\mathbf{v})=\kappa(\mathbf{u},\mathbf{v})-\frac{\kappa(\mathbf{u},\mathbf{x})\kappa(\mathbf{x},\mathbf{v})}{\kappa(\mathbf{x},\mathbf{x})}.$
(54)
This GP projected onto any finite subspace $\iota:X_{0}\hookrightarrow X$ is a
normal distribution and, for
$X_{0}=\\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}\\}$, it follows
that
$1$$Y^{X}$$Y^{X_{0}}$$\prod_{i=1}^{n}Y_{i}\cong\mathbb{R}^{{}^{n}}$$\delta_{Y^{\iota}}$$\delta_{ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n}}}\mid_{Y^{X_{0}}}$$\delta_{ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n}}}$$\mathcal{I}^{\mathbf{x}}_{n}(\bullet\mid
y)\sim\mathcal{G}\mathcal{P}(i_{\mathbf{m}}^{1},\kappa^{1})$$\mathcal{I}_{p}(\bullet\mid
y)\sim\mathcal{N}((i_{\mathbf{m}}^{1}(\mathbf{x}_{1}),\ldots,i_{\mathbf{m}}^{1}(\mathbf{x}_{n}))^{T},\kappa^{1}\mid_{X_{0}})$
where $Y_{i}$ is a copy of $Y=\mathbb{R}$ and the restriction
$\delta_{ev_{\mathbf{x}_{1}}\times\ldots\times
ev_{\mathbf{x}_{n}}}\mid_{Y^{X_{0}}}$ is an isomorphism. The inference map
$\mathcal{I}_{p}(\bullet\mid y)$ is the updated normal distribution on
$\mathbb{R}^{{}^{n}}$ given the measurement $(\mathbf{x},y)$ which can be
rewritten as
$\mathcal{I}_{p}(\bullet\mid
y)\sim\mathcal{N}(\mathbf{m}+K(X_{0},\mathbf{x})\kappa(\mathbf{x},\mathbf{x})^{-1}(y-\mathbf{m}^{T}\mathbf{x}),\kappa^{1}\mid_{X_{0}}),$
where $X_{0}$ is now viewed as the ordered set
$X_{0}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and $K(X_{0},\mathbf{x})$ is
the $n$-vector with components $\kappa(\mathbf{x}_{j},\mathbf{x})$.
Iterating this updating procedure for $N$ measurements
$\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$ the $N^{th}$ posterior coincides with
the analogous noisy measurment inference updating Equations 48 and 49 with
$\kappa$ in place of $k$.
## 9 Stochastic Processes as Points
Having defined stochastic processes we would be remiss not to mention the
Markov process—one of the most familiar type of processes used for modeling.
Many applications can be approximated by Markov models and a familiar example
is the Kalman filter which we describe below as it is the archetype. While
Kalman filtering is not commonly viewed as a ML problem, it is useful to put
it into perspective with respect to the Bayesian modeling paradigm.
By looking at Markov processes we are immediately led to a generalization of
the definition of a stochastic process which is due to Lawvere and Meng [18].
To motivate this we start with the elementary idea first before giving the
generalized definition of a stochastic process.
### 9.1 Markov processes via Functor Categories
Here we assume knowledge of the definition of a functor, and refer the
unfamiliar reader to any standard text on category theory. Let $T$ be any set
with a total (linear) ordering $\leq$ so for every $t_{1},t_{2}\in T$ either
$t_{1}\leq t_{2}$ or $t_{2}\leq t_{1}$. (Here we have switched from our
standard “$X$” notation to “$T$” as we wish to convey the image of a space
with properties similar to time as modeled by the real line.) We can view
$(T,\leq)$ as a category with the objects as the elements and the set of
arrows from one object to another as
$hom_{T}(t_{1},t_{2})=\left\\{\begin{array}[]{ll}\star\textrm{ iff }t_{1}\leq
t_{2}\\\ \emptyset\textrm{ otherwise }\end{array}\right.$
The functor category $\mathcal{P}^{T}$ has as objects functors
$\mathcal{F}:(T,\leq)\rightarrow\mathcal{P}$ which play an important role in
the theory of stochastic processes, and we formally give the following
definition.
###### Definition 17.
A _Markov transformation_ is a functor
$\mathcal{F}:(T,\leq)\rightarrow\mathcal{P}$.
From the modeling perspective we look at the image of the functor
$\mathcal{F}\in_{ob}\mathcal{P}^{T}$ in the category $\mathcal{P}$ so given
any sequence of ordered points $\\{t_{i}\\}_{i=1}^{\infty}$ in $T$ their image
under $\mathcal{F}$ is shown in Figure 37, where
$\mathcal{F}_{t_{i},t_{i+1}}=\mathcal{F}(\leq)$ is a $\mathcal{P}$ arrow.
$\mathcal{F}(t_{1})$$\mathcal{F}(t_{2})$$\mathcal{F}(t_{3})$$\ldots$$\mathcal{F}_{t_{1},t_{2}}$$\mathcal{F}_{t_{2},t_{3}}$$\mathcal{F}_{t_{3},t_{4}}$
Figure 37: A Markov transformation as the image of a $\mathcal{P}$ valued
Functor.
By functoriality, these arrows satisfy the conditions
1. 1.
$\mathcal{F}_{t_{i},t_{i}}=id_{t_{i}}$, and
2. 2.
$\mathcal{F}_{t_{i},t_{i+2}}=\mathcal{F}_{t_{i+1},t_{i+2}}\circ\mathcal{F}_{t_{i},t_{i+1}}$
Using the definition of composition in $\mathcal{P}$ the second condition can
be rewritten as
$\mathcal{F}_{t_{i},t_{i+2}}(B\mid
x)=\int_{u\in\mathcal{F}(t_{i+1})}\mathcal{F}_{t_{i+1},t_{i+2}}(B\mid
u)\,d\mathcal{F}_{t_{i},t_{i+1}}(\cdot\mid x)$
for $x\in\mathcal{F}(t_{i})$ (the “state” of the process at time $t_{i}$) and
$B\in\Sigma_{F(t_{i+2})}$. This equation is called the Chapman-Kolomogorov
relation and can be used, in the non categorical characterization, to define a
Markov process.
The important aspect to note about this definition of a Markov model is that
the measurable spaces $\mathcal{F}(t_{i})$ can be distinct from the other
measurable spaces $\mathcal{F}(t_{j})$, for $j\neq i$, and of course the
arrows $\mathcal{F}_{t_{i},t_{i+1}}$ are in general distinct. This simple
definition of a Markov transformation as a functor captures the property of an
evolving process being “memoryless” since if we know where the process
$\mathcal{F}$ is at $t_{i}$, say $x\in\mathcal{F}(t_{i})$, then its
expectation at $t_{i+1}$ (as well as higher order moments) can be determined
without regard to its “state” prior to $t_{i}$.
The arrows of the functor category $\mathcal{P}^{T}$ are natural
transformations $\eta:\mathcal{F}\rightarrow\mathcal{G}$, for
$\mathcal{F},\mathcal{G}\in_{ob}\mathcal{P}^{T}$, and hence satisfy the
commutativity relation given in Figure 38 for every $t_{1},t_{2}\in T$ with
$t_{1}\leq t_{2}$.
$\mathcal{F}(t_{1})$$\mathcal{F}(t_{2})$$\mathcal{G}(t_{1})$$\mathcal{G}(t_{2})$$\mathcal{F}_{t_{1},t_{2}}$$\eta_{t_{1}}$$\eta_{t_{2}}$$\mathcal{G}_{t_{1},t_{2}}$
Figure 38: An arrow in $\mathcal{P}^{T}$ is a natural transformation.
The functor category $\mathcal{P}^{T}$ has a terminal object $\mathbf{1}$
mapping $t\mapsto 1$ for every $t\in T$ and this object
$\mathbf{1}\in_{ob}\mathcal{P}^{T}$ allows us to generalize the definition of
a stochastic process.242424The elementary definition of a stochastic process,
Definition 9, as a probability measure on a function space suffices for what
we might call standard ML. For more general constructions, such as Markov
Models and Hierarchical Hidden Markov Models (HHMM) the generalized definition
is required.
###### Definition 18.
Let $X$ be _any category_. A _stochastic process_ is a point in the category
$\mathcal{P}^{X}$, i.e., a $\mathcal{P}^{X}$ arrow
$\eta:\mathbf{1}\rightarrow\mathcal{F}$ for some
$\mathcal{F}\in_{ob}\mathcal{P}^{X}$.252525In _any category_ with a terminal
object $1$ an arrow whose domain is $1$ is called a point. So an arrow
$x:1\rightarrow X$ is called a point of $X$ whereas $f:X\rightarrow Y$ is
sometimes referred to as a generalized element to emphasize that it “varies”
over the domain. It is constructive to consider what this means in the
category of Sets and why the terminology is meaningful.
Different categories $X$ correspond to different types of stochastic
processes. Taking the simplest possible case let $X$ be a set considered as a
discrete category—the objects are the elements $x\in X$ while there are no
nonidentity arrows in $X$ viewed as a category. This case generalizes
Definition 9 because, for $Y$ a fixed measurable space we have the functor
$\hat{Y}:X\rightarrow\mathcal{P}$ mapping each object $x\in_{ob}X$ to a copy
$Y_{x}$ of $Y$ and this special case corresponds to Definition 9.
Taking $X=T$, where $T$ is a totally ordered set (and subsequently viewed as a
category with one arrow between any two elements), and looking at the image of
$t_{1}$$t_{2}$$t_{3}$$\ldots$$\leq$$\leq$$\leq$
under the stochastic process $\mu:\mathbf{1}\rightarrow\mathcal{F}$ gives the
commutative diagram in Figure 39.
$1$$\mathcal{F}(t_{1})$$\mathcal{F}(t_{2})$$\mathcal{F}(t_{3})$$\ldots$$\mathcal{F}_{t_{1},t_{2}}$$\mathcal{F}_{t_{2},t_{3}}$$\mathcal{F}_{t_{3},t_{4}}$$\mu_{t_{1}}$$\mu_{t_{2}}$$\mu_{t_{3}}$$\mu_{t_{4}}$
Figure 39: A Markov model as the image of a stochastic process.
From this perspective a stochastic process $\mu$ can be viewed as a family of
probability measures on the measurable spaces $\mathcal{F}(t_{i})$, and the
stochastic process $\mu$ coupled with a $\mathcal{P}^{T}$ arrow
$\eta:\mathcal{F}\rightarrow\mathcal{G}$ maps one Markov model to another
$1$$\mathcal{F}(t_{1})$$\mathcal{F}(t_{2})$$\mathcal{F}(t_{3})$$\ldots$$\mathcal{G}(t_{1})$$\mathcal{G}(t_{2})$$\mathcal{G}(t_{3})$$\ldots$$\mathcal{F}_{t_{1},t_{2}}$$\mathcal{F}_{t_{2},t_{3}}$$\mathcal{F}_{t_{3},t_{4}}$$\mathcal{G}_{t_{1},t_{2}}$$\mathcal{G}_{t_{2},t_{3}}$$\mathcal{G}_{t_{3},t_{4}}$$\mu_{t_{1}}$$\mu_{t_{2}}$$\mu_{t_{3}}$$\mu_{t_{4}}$$\eta_{t_{1},t_{2}}$$\eta_{t_{2},t_{3}}$$\eta_{t_{3},t_{4}}$
One can also observe that GPs can be defined using this generalized definition
of a stochastic process. For $X$ a measurable space it follows for any finite
subset $X_{0}\subset X$ we have the inclusion map $\iota:X_{0}\hookrightarrow
X$ which is a measurable function, using the subspace $\sigma$-algbra for
$X_{0}$, and we are led back to Diagram 14 with the stochastic process
$P:\textbf{1}\rightarrow\hat{Y}$, where $\hat{Y}$ is as defined in the
paragraph above following Definition 18, which satisfies the appropriate
restriction property defining a GP.
These simple examples illustrate that different stochastic processes can be
obtained by either varying the structure of the category $X$ and/or by placing
additional requirements on the projection maps, e.g., requiring the
projections be normal distributions on finite subspaces of the exponent
category $X$.
### 9.2 Hidden Markov Models
To bring in the Bayesian aspect of Markov models it is necessary to consider
the measurement process associated with a sequence as in Figure 39. In
particular, consider the standard diagram
$1$$\mathcal{F}(t_{1})$$Y_{t_{1}}$$\mu_{t_{1}}$$\mathcal{S}_{t_{1}}$$d_{t_{1}}$
which characterizes a Bayesian model, where $Y_{t_{1}}$ is a copy of a $Y$
which is a data measurement space, $\mathcal{S}_{t_{1}}$ is interpreted as a
measurement model and $d_{t_{1}}$ is an actual data measurement on the “state”
space $\mathcal{F}(t_{1})$. This determines an inference map
$\mathcal{I}_{t_{1}}$ so that given a measurement $d_{t_{1}}$ the posterior
probability on $\mathcal{F}(t_{1})$ is $\mathcal{I}_{t_{1}}\circ d_{t_{1}}$.
Putting the two measurement models together with the Markov transformation
model $\mathcal{F}$ we obtain the following diagram in Figure 40.
$1$$\mathcal{F}(t_{1})$$\mathcal{F}(t_{2})$$Y_{t_{1}}$$Y_{t_{2}}$$\mu_{t_{1}}$$\mathcal{F}_{t_{1},t_{2}}$$\mathcal{S}_{t_{1}}$$\mathcal{I}_{t_{1}}$$d_{t_{1}}$$\hat{\mu}_{t_{1}}=\mathcal{I}\circ
d_{t_{1}}$$\mathcal{F}_{t_{1},t_{2}}\circ\hat{\mu}_{t_{1}}$$\mathcal{S}_{t_{2}}$$\mathcal{I}_{t_{2}}$
Figure 40: The hidden Markov model viewed in $\mathcal{P}$.
This is the hidden Markov process in which given a prior probability
$\mu_{t_{1}}$ on the space $\mathcal{F}_{t_{1}}$ we can use the measurement
$d_{t_{1}}$ to update the prior to the posterior
$\hat{\mu}_{t_{1}}=\mathcal{I}_{t_{1}}\circ d_{t_{1}}$ on
$\mathcal{F}(t_{1})$. The posterior then composes with
$\mathcal{F}_{t_{1},t_{2}}$ to give the prior
$\mathcal{F}_{t_{1},t_{2}}\circ\hat{\mu}_{t_{1}}$ on $\mathcal{F}(t_{2})$, and
now the process can be repeated indefinitely. The Kalman filter is an example
in which the Markov map $\mathcal{F}_{t_{1},t_{2}}$ describe the linear
dynamics of some system under consideration (as in tracking a satellite),
while the sampling distributions $\mathcal{S}_{t_{1}}$ model the noisy
measurement process which for the Kalman filter is Gaussian additive noise. Of
course one can easily replace the linear dynamic by a nonlinear dynamic and
the Gaussian additive noise model by any other measurement model, obtaining an
extended Kalman filter, and the above form of the diagram does not change at
all, only the $\mathcal{P}$ maps change.
## 10 Final Remarks
In closing, we would like to make a few comments on the use of category theory
for ML, where the largest potential payoff lies in exploiting the abstract
framework that categorical language provides. This section assumes a basic
familiarity with monads and should be viewed as only providing conceptual
directions for future research which we believe are relevant for the
mathematical development of learning systems. Further details on the theory of
monads can be found in most category theory books, while the basics as they
relate to our discussion below can be found in our previous paper [6], in
which we provide the simplest possible example of a decision rule on a
discrete space.
Seemingly all aspects of ML including Dirichlet distributions and unsupervised
learning (clustering) can be characterized using the category $\mathcal{P}$.
As an elementary example, mixture models can be developed by consideration of
the space of all (perfect) probability measures $\mathscr{P}X$ on a measurable
space $X$ endowed with the coarsest $\sigma$-algebra such that the evaluation
maps $ev_{B}:\mathscr{P}X\to[0,1]$ given by $ev_{B}(P)=P(B)$, for all
$B\in\Sigma_{X}$, are measurable. This actually defines the object mapping of
a functor $\mathscr{P}:\mathcal{P}\to\mathcal{M}eas$ which sends a measurable
space $X$ to the space $\mathscr{P}X$ of probability measures on $X$. On
arrows, $\mathscr{P}$ sends the $\mathcal{P}$-arrow $f:X\to Y$ to the
measurable function $\mathscr{P}f:\mathscr{P}X\to\mathscr{P}Y$ defined
pointwise on $\Sigma_{Y}$ by
$\mathscr{P}f(P)(B)=\int_{X}f_{B}\,dP.$
This functor is called the Giry monad, denoted $\mathcal{G}$, and the Kleisli
category $K(\mathcal{G})$ of the Giry monad is equivalent to
$\mathcal{P}$.262626See Giry[14] for the basic definitions and equivalence of
these categories. The reason we have chosen to present the material from the
perspective of $\mathcal{P}$ rather that $K(\mathcal{G})$ is that the existing
literature on ML uses Markov kernels rather than the equivalent arrows in
$K(\mathcal{G})$. The Giry monad determines the nondeterministic $\mathcal{P}$
mapping
$X$$\mathscr{P}X$$\varepsilon_{X}$
given by $\varepsilon_{X}(P,B)=ev_{B}(P)=P(B)$ for all $P\in\mathscr{P}(X)$
and all $B\in\Sigma_{X}$. Using this construction, any probability measure $P$
on $\mathscr{P}X$ then yields a mixture of probability measures on $X$ through
the composite map
$1$$X$$\mathscr{P}X$$P$$\varepsilon_{X}$$\varepsilon_{X}\circ P=$ A mixture
model.
We have briefly introduced the Kleisli category
$K(\mathcal{G})\,(\cong\mathcal{P})$ because it is a subcategory $\mathcal{D}$
of the Eilenberg–Moore category of $\mathcal{G}$-algebras, which we call the
category of decision rules,272727Doberkat [8] has analyzed the Kleisli
category under the condition that the arrows are not only measurable but also
continuous. This is an unnecessary assumption, resulting in all finite spaces
having no decision rules, though his considerable work on this category
$K(\mathcal{G})$ provides much useful insight as well as applications of this
category. because the objects of this category are $\mathcal{M}eas$ arrows
$r:\mathscr{P}X\rightarrow X$ sending a probability measure $P$ on $X$ to an
actual element of $X$ satisfying some basic properties including
$r(\delta_{x})=x$. Thus $r$ acts as a decision rule converting a probability
measure on $X$ to an actual element of $X$ and, if $P$ is deterministic, takes
that measure to the point $x\in X$ of nonzero measure.282828Measurable spaces
are defined only up to isomorphism, so that if two elements $x,y\in X$ are
nondistinguishable in terms of the $\sigma$-algebra, meaning there exist no
measurable set $A\in\Sigma_{X}$ such that $x\in A$ and $y\not\in A$, then
$\delta_{x}=\delta_{y}$ and we also identify $x$ with $y$. Decision theory is
generally presented from the perspective of taking probability measures on $X$
and, usually via a family of loss functions $\theta:X\rightarrow\mathbb{R}$,
making a selection among a family of possible choices $\theta\in\Theta$ where
$\Theta$ is some measurable space rather than $X$. However, it can clearly be
viewed from this more basic viewpoint.
The largest potential payoff in using category theory for ML and related
applications appears to be in integrating decision theory with probability
theory, expressed in terms of the category $\mathcal{D}$, which would provide
a basis for an automated reasoning system. While the Bayesian framework
presented in this paper can fruitfully be exploited to construct estimation of
unknown functions it still lacks the ability to _make decisions_ of any kind.
Even if we were to invoke a list of simple rules to make decisions the
category $\mathcal{P}$ is too restrictive to implement these rules. By working
in the larger category of decision rules $\mathcal{D}$, it is possible to
implement both the Bayesian reasoning presented in this work as well as
decision rules as part of larger reasoning system. Our perspective on this
problem is that Bayesian reasoning in general is inadequate—not only because
it lacks the ability to make decisions—but because it is a _passive_ system
which “waits around” for additional measurement data. An automated reasoning
system must take self directed action as in commanding itself to “swivel the
camera $45$ degrees right to obtain necessary additional information”, which
is a (decision) command and control component which can be integrated with
Bayesian reasoning. An intelligent system would in addition, based upon the
work of Rosen [21], in which he employed categorical ideas, possess an
anticipatory component. While he did not use the language of SMCC it is clear
this aspect was his intention and critical in his method of modeling
intelligent systems, and within the category $\mathcal{D}$ this additional
aspect can also be modeled.
## 11 Appendix A: Integrals over probability measures.
The following three properties are the only three properties used throughout
the paper to derive the values of integrals defined over probability measures.
1. 1.
The integral of any measurable function $f:X\rightarrow\mathbb{R}$ with
respect to a dirac measure satisfies
$\int_{u\in X}f(u)\,d\delta_{x}=f(x).$
This is straightforward to show using standard measure theoretic arguments.
2. 2.
Integration with respect to a push forward measure can be pulled back. Suppose
$f:X\rightarrow Y$ is any measurable function, $P$ is a probability measure on
$X$, and $\phi:Y\rightarrow\mathbb{R}$ is any measurable function. Then
$\int_{y\in Y}\phi(y)\,d(Pf^{-1})=\int_{x\in X}\phi(f(x))\,dP$
To prove this simply show that it holds for $\phi=\mathbb{1}_{B}$, the
characteristic function at $B$, then extend it to any simple function, and
finally use the monotone convergence theorem to show it holds for any
measurable function.
3. 3.
Suppose $f:X\rightarrow Y$ is any measurable function and $P$ is a probability
measure on $X$. Then
$\int_{x\in X}\delta_{f}(B\mid x)\,dP=\int_{x\in
X}\mathbb{1}_{B}(f(x))\,dP=P(f^{-1}(B))$
This is a special case of case (2) with $\phi=\mathbb{1}_{B}$.
## 12 Appendix B: The weak closed structure in $\mathcal{P}$
Here is a simple illustration of the weak closed property of $\mathcal{P}$
using finite spaces. Let $X=2=\\{0,1\\}$ and $Y=\\{a,b,c\\}$, both with the
powerset $\sigma$-algebra. This yields the powerset $\sigma$-algebra on
$Y^{X}$ and each function can be represented by an ordered pair, such as
$(b,c)$ denoting the function $f(1)=b$ and $f(2)=c$. Define two probability
measures $P,Q$ on $Y^{X}$ by
$\begin{array}[]{lcccl}P(\\{(b,c)\\})&=&.5&=&P(\\{(c,b)\\})\\\
Q(\\{(b,b)\\})&=&.5&=&Q(\\{(c,c)\\})\end{array}$
and both measures having a value of $0$ on all other singleton measurable
sets. Both of these probability measures on $Y^{X}$ yield the same conditional
probability measure
$(X,\Sigma_{X})$$(Y,\Sigma_{Y})$$\overline{P}=\overline{Q}$
since
$\begin{array}[]{lcccl}\overline{P}(\\{a\\}|1)&=&0&=&\overline{Q}(\\{a\\}|1)\\\
\overline{P}(\\{b\\}|1)&=&.5&=&\overline{Q}(\\{b\\}|1)\\\
\overline{P}(\\{c\\}|1)&=&.5&=&\overline{Q}(\\{c\\}|1)\end{array}$
and
$\begin{array}[]{lcccl}\overline{P}(\\{a\\}|2)&=&0&=&\overline{Q}(\\{a\\}|2)\\\
\overline{P}(\\{b\\}|2)&=&.5&=&\overline{Q}(\\{b\\}|2)\\\
\overline{P}(\\{c\\}|2)&=&.5&=&\overline{Q}(\\{c\\}|2)\end{array}$
Since $P\neq Q$ the uniqueness condition required for the closedness property
fails and only the existence condition is satisfied.
## References
* [1] S. Abramsky, R. Blute, and P. Panangaden, Nuclear and trace ideals in tensored-$\ast$ categories. Journal of Pure and Applied Algebra, Vol. 143, Issue 1-3, 1999, pp 3-47.
* [2] David Barber, Bayesian Reasoning and Machine Learning, Cambridge University Press, 2012.
* [3] N. N. Cencov, Statistical decision rules and optimal inference, Volume 53 of Translations of Mathematical Monographs, American Mathematical Society, 1982.
* [4] Bob Coecke and Robert Speckens, Picturing classical and quantum Bayesian inference, Synthese, June 2012, Volume 186, Issue 3, pp 651-696. http://link.springer.com/article/10.1007/s11229-011-9917-5
* [5] David Corfield, Category Theory in Machine Learning, $n$-category cafe blog. http://golem.ph.utexas.edu/category/2007/09/category_theory_in_machine_lea.html
* [6] Jared Culbertson and Kirk Sturtz, A Categorical Foundation for Bayesian Probability, Applied Categorical Structures, 2013. http://link.springer.com/article/10.1007/s10485-013-9324-9.
* [7] R. Davis, Gaussian Processes, http://www.stat.columbia.edu/~rdavis/papers/VAG002.pdf
* [8] E.E. Doberkat, Kleisi morphism and randomized congruences for the Giry monad, J. Pure and Applied Algebra, Vol. 211, pp 638-664, 2007.
* [9] R.M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, no. 74, Cambridge University Press, 2002
* [10] R. Durrett, Probability: Theory and Examples, 4th ed., Cambridge University Press, New York, 2010.
* [11] A.M. Faden. The Existence of Regular Conditional Probabilities: Necessary and Sufficient Conditions. The Annals of Probability, 1985, Vol. 13, No. 1, 288-298.
* [12] Brendan Fong, Causal Theories: A Categorical Perspective on Bayesian Networks. Preprint, April 2013. http://arxiv.org/pdf/1301.6201.pdf
* [13] Tobias Fritz, A presentation of the category of stochastic matrices, 2009. http://arxiv.org/pdf/0902.2554.pdf
* [14] M. Giry, A categorical approach to probability theory, in Categorical Aspects of Topology and Analysis, Vol. 915, pp 68-85, Springer-Verlag, 1982.
* [15] E.T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, 2003.
* [16] F.W. Lawvere, The category of probabilistic mappings. Unpublished seminar notes 1962.
* [17] F.W. Lawvere, Bayesian Sections, private communication, 2011.
* [18] X. Meng, Categories of convex sets and of metric spaces, with applications to stochastic programming and related areas, Ph.D. Thesis, State University of New York at Buffalo, 1988.
* [19] Kevin Murphy, Machine Learning: A Probabilistic Perspective, MIT Press, 2012.
* [20] C.E. Rasmussen and C.K.I.. Williams, Gaussian Processes for Machine Learning. MIT Press, 2006.
* [21] Robert Rosen, Life Itself, Columbia University Press, 1991.
* [22] V.A. Voevodskii, Categorical probability, Steklov Mathematical Institute Seminar, Nov. 20, 2008. http://www.mathnet.ru/php/seminars.phtml?option_lang=eng&presentid=259.
Jared Culbertson | Kirk Sturtz
---|---
RYAT, Sensors Directorate | Universal Mathematics
Air Force Research Laboratory, WPAFB | Vandalia, OH 45377
Dayton, OH 45433 | [email protected]
[email protected]
|
arxiv-papers
| 2013-12-05T06:38:05 |
2024-09-04T02:49:54.949725
|
{
"license": "Public Domain",
"authors": "Jared Culbertson and Kirk Sturtz",
"submitter": "Jared Culbertson",
"url": "https://arxiv.org/abs/1312.1445"
}
|
1312.1634
|
arxiv-papers
| 2013-12-05T18:04:26 |
2024-09-04T02:49:54.991375
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Xiaoxian Yan, Chang Huai, Hui Xing, James P. Parry, Yusen Yang,\n Guoxiong Tang, Chao Yao, Guohan Hu, Renat Sabirianov, and Hao Zeng",
"submitter": "Hao Zeng",
"url": "https://arxiv.org/abs/1312.1634"
}
|
|
1312.1638
|
# PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems
on Streaming Architectures using the Flux Reconstruction Approach
F. D. Witherden111Corresponding author; e-mail
[email protected]., A. M. Farrington, P. E. Vincent
Department of Aeronautics, Imperial College London, SW7 2AZ
###### Abstract
High-order numerical methods for unstructured grids combine the superior
accuracy of high-order spectral or finite difference methods with the
geometric flexibility of low-order finite volume or finite element schemes.
The Flux Reconstruction (FR) approach unifies various high-order schemes for
unstructured grids within a single framework. Additionally, the FR approach
exhibits a significant degree of element locality, and is thus able to run
efficiently on modern streaming architectures, such as Graphical Processing
Units (GPUs). The aforementioned properties of FR mean it offers a promising
route to performing affordable, and hence industrially relevant, scale-
resolving simulations of hitherto intractable unsteady flows within the
vicinity of real-world engineering geometries. In this paper we present PyFR,
an open-source Python based framework for solving advection-diffusion type
problems on streaming architectures using the FR approach. The framework is
designed to solve a range of governing systems on mixed unstructured grids
containing various element types. It is also designed to target a range of
hardware platforms via use of an in-built domain specific language based on
the Mako templating engine. The current release of PyFR is able to solve the
compressible Euler and Navier-Stokes equations on grids of quadrilateral and
triangular elements in two dimensions, and hexahedral elements in three
dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented
for various benchmark flow problems, single-node performance is discussed, and
scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The
software is freely available under a 3-Clause New Style BSD license (see
www.pyfr.org).
_Keywords:_ High-order; Flux reconstruction; Parallel algorithms;
Heterogeneous computing
## Program Description
Authors
F. D. Witherden, A. M. Farrington, P. E. Vincent
Program title
PyFR v0.1.0
Licensing provisions
New Style BSD license
Programming language
Python, CUDA and C
Computer
Variable, up to and including GPU clusters
Operating system
Recent version of Linux/UNIX
RAM
Variable, from hundreds of megabytes to gigabytes
Number of processors used
Variable, code is multi-GPU and multi-CPU aware through a combination of MPI
and OpenMP
External routines/libraries
Python 2.7, numpy, PyCUDA, mpi4py, SymPy, Mako
Nature of problem
Compressible Euler and Navier-Stokes equations of fluid dynamics; potential
for any advection-diffusion type problem.
Solution method
High-order flux reconstruction approach suitable for curved, mixed,
unstructured grids.
Unusual features
Code makes extensive use of symbolic manipulation and run-time code generation
through a domain specific language.
Running time
Many small problems can be solved on a recent workstation in minutes to hours.
## Nomenclature
Throughout we adopt a convention in which dummy indices on the right hand side
of an expression are summed. For example
$C_{ijk}=A_{ijl}B_{ilk}\equiv\sum_{l}A_{ijl}B_{ilk}$ where the limits are
implied from the surrounding context. All indices are assumed to be zero-
based.
Functions.
---
$\delta_{ij}$ | Kronecker delta
$\det\bm{\mathsf{A}}$ | Matrix determinant
$\dim\bm{\mathsf{A}}$ | Matrix dimensions
Indices.
$e$ | Element type
$n$ | Element number
$\alpha$ | Field variable number
$i,j,k$ | Summation indices
$\rho,\sigma,\nu$ | Summation indices
Domains.
$\mathbf{\Omega}$ | Solution domain
$\mathbf{\Omega}_{e}$ | All elements in $\mathbf{\Omega}$ of type $e$
$\hat{\mathbf{\Omega}}_{e}$ | A _standard_ element of type $e$
$\partial\hat{\mathbf{\Omega}}_{e}$ | Boundary of $\hat{\mathbf{\Omega}}_{e}$
$\mathbf{\Omega}_{en}$ | Element $n$ of type $e$ in $\mathbf{\Omega}$
$\left\lvert\mathbf{\Omega}_{e}\right\rvert$ | Number of elements of type $e$
Expansions.
$\wp$ | Polynomial order
$N_{D}$ | Number of spatial dimensions
$N_{V}$ | Number of field variables
$\ell_{e\rho}$ | Nodal basis polynomial $\rho$ for element type $e$
$x,y,z$ | Physical coordinates
$\tilde{x},\tilde{y},\tilde{z}$ | Transformed coordinates
$\bm{\mathcal{M}}_{en}$ | Transformed to physical mapping
Adornments and suffixes.
$\tilde{\square}$ | A quantity in transformed space
$\hat{\square}$ | A vector quantity of unit magnitude
$\square^{T}$ | Transpose
$\square^{(u)}$ | A quantity at a solution point
$\square^{(f)}$ | A quantity at a flux point
$\square^{(f_{\perp})}$ | A normal quantity at a flux point
Operators.
$\mathfrak{C}_{\alpha}$ | Common solution at an interface
$\mathfrak{F}_{\alpha}$ | Common normal flux at an interface
## 1 Introduction
There is an increasing desire amongst industrial practitioners of
computational fluid dynamics (CFD) to undertake high-fidelity scale-resolving
simulations of transient compressible flows within the vicinity of complex
geometries. For example, to improve the design of next generation unmanned
aerial vehicles (UAVs), there exists a need to perform simulations—at Reynolds
numbers $10^{4}$–$10^{7}$ and Mach numbers $M\sim 0.1$–$1.0$—of highly
separated flow over deployed spoilers/air-brakes; separated flow within
serpentine intake ducts; acoustic loading in weapons bays; and flow over
entire UAV configurations at off-design conditions. Unfortunately, current
generation industry-standard CFD software based on first- or second-order
accurate Reynolds Averaged Navier-Stokes (RANS) approaches is not well suited
to performing such simulations. Henceforth, there has been significant
interest in the potential of high-order accurate methods for unstructured
mixed grids, and whether they can offer an efficient route to performing
scale-resolving simulations within the vicinity of complex geometries. Popular
examples of high-order schemes for unstructured mixed grids include the
discontinuous Galerkin (DG) method, first introduced by Reed and Hill [1], and
the spectral difference (SD) methods originally proposed under the moniker
‘staggered-gird Chebyshev multidomain methods’ by Kopriva and Kolias in 1996
[2] and later popularised by Sun et al. [3]. In 2007 Huynh proposed the flux
reconstruction (FR) approach [4]; a unifying framework for high-order schemes
for unstructured grids that incorporates both the nodal DG schemes of [5] and,
at least for a linear flux function, any SD scheme. In addition to offering
high-order accuracy on unstructured mixed grids, FR schemes are also compact
in space, and thus when combined with explicit time marching offer a
significant degree of element locality. As such, explicit high-order FR
schemes are characterised by a large degree of structured computation.
Over the past two decades improvements in the arithmetic capabilities of
processors have significantly outpaced advances in random access memory.
Algorithms which have traditionally been compute bound—such as dense matrix-
vector products—are now limited instead by the bandwidth to/from memory. This
is epitomised in Figure 1. Whereas the processors of two decades ago had
FLOPS-per-byte of ${\sim}0.2$ more recent chips have ratios upwards of
${\sim}4$. This disparity is not limited to just conventional CPUs. Massively
parallel accelerators and co-processors such as the NVIDIA K20X and Intel Xeon
Phi 5110P have ratios of $5.24$ and $3.16$, respectively.
Figure 1: Trends in the peak floating point performance (double precision) and
memory bandwidth of sever-class Intel processors from 1994–2013. The quotient
of these two measures yields the FLOPS-per-byte of a processor. Data courtesy
of Jan Treibig.
A concomitant of this disparity is that modern hardware architectures are
highly dependent on a combination of high speed caches and/or shared memory to
maintain throughput. However, for an algorithm to utilise these efficiently
its memory access pattern must exhibit a degree of either spatial or temporal
locality. To a first-order approximation the spatial locality of a method is
inversely proportional to the amount of memory indirection. On an unstructured
grid indirection arises whenever there is coupling between elements. This is
potentially a problem for discretisations whose stencil is not compact.
Coupling also arises in the context of implicit time stepping schemes.
Implementations are therefore very often bound by memory bandwidth. As a
secondary trend we note that the manner in which FLOPS are realised has also
changed. In the early 1990s commodity CPUs were predominantly scalar with a
single core of execution. However in 2013 processors with eight or more cores
are not uncommon. Moreover, the cores on modern processors almost always
contain vector processing units. Vector lengths up to 256-bits, which permit
up to four double precision values to be operated on at once, are not
uncommon. It is therefore imperative that compute-bound algorithms are
amenable to both multithreading and vectorisation. A versatile means of
accomplishing this is by breaking the computation down into multiple,
necessarily independent, streams. By virtue of their independence these
streams can be readily divided up between cores and vector lanes. This leads
directly to the concept of _stream processing_. We will refer to architectures
amenable to this form of parallelisation as streaming architectures.
A corollary of the above discussion is that compute intensive discretisations
which can be formulated within the stream processing paradigm are well suited
to acceleration on current—and likely future—hardware platforms. The FR
approach combined with explicit time stepping is an archetypical of this.
Our objective in this paper is to present PyFR, an open-source Python based
framework for solving advection-diffusion type problems on streaming
architectures using the FR approach. The framework is designed to solve a
range of governing systems on mixed unstructured grids containing various
element types. It is also designed to target a range of hardware platforms via
use of an in-built domain specific language derived from the Mako templating
engine. The current release of PyFR is able to solve the compressible Euler
and Navier-Stokes equations on unstructured grids of quadrilateral and
triangular elements in two-dimensions, and unstructured grids of hexahedral
elements in three-dimensions, targeting clusters of CPUs, and NVIDIA GPUs. The
paper is structured as follows. In section 2 we provide a overview of the FR
approach for advection-diffusion type problems on mixed unstructured grids. In
section 3 we proceed to describe our implementation strategy, and in section 4
we present the Euler and Navier-Stokes equations, which are solved by the
current release of PyFR. The framework is then validated in section 5, single-
node performance is discussed in section 6, and scalability of the code is
demonstrated on up to 104 NVIDIA M2090 GPUs in section 7. Finally, conclusions
are drawn in section 8.
## 2 Flux Reconstruction
A brief overview of the FR approach for solving advection-diffusion type
problems is given below. Extended presentations can be found elsewhere [4, 6,
7, 8, 9, 10, 11, 12, 13, 14].
Consider the following advection-diffusion problem inside an arbitrary domain
$\mathbf{\Omega}$ in $N_{D}$ dimensions
$\frac{\partial u_{\alpha}}{\partial
t}+\bm{\nabla}\cdot\mathbf{f}_{\alpha}=0,$ (1)
where $0\leq\alpha<N_{V}$ is the _field variable_ index,
$u_{\alpha}=u_{\alpha}(\mathbf{x},t)$ is a conserved quantity,
$\mathbf{f}_{\alpha}=\mathbf{f}_{\alpha}(u,\bm{\nabla}u)$ is the flux of this
conserved quantity and $\mathbf{x}=x_{i}\in\mathbb{R}^{N_{D}}$. In defining
the flux we have taken $u$ in its unscripted form to refer to all of the
$N_{V}$ field variables and $\bm{\nabla}u$ to be an object of length
$N_{D}\times N_{V}$ consisting of the gradient of each field variable. We
start by rewriting Equation 1 as a first order system
$\displaystyle\frac{\partial u_{\alpha}}{\partial
t}+\bm{\nabla}\cdot\mathbf{f}_{\alpha}(u,\mathbf{q})$ $\displaystyle=0,$ (2a)
$\displaystyle\mathbf{q}_{\alpha}-\bm{\nabla}u_{\alpha}$ $\displaystyle=0,$
(2b)
where $\mathbf{q}$ is an auxiliary variable. Here, as with $\bm{\nabla}u$, we
have taken $\mathbf{q}$ in its unsubscripted form to refer to the gradients of
all of the field variables.
Take $\mathcal{E}$ to be the set of available element types in
$\mathbb{R}^{N_{D}}$. Examples include quadrilaterals and triangles in two
dimensions and hexahedra, prisms, pyramids and tetrahedra in three dimensions.
Consider using these various elements types to construct a conformal mesh of
the domain such that
$\mathbf{\Omega}=\bigcup_{e\in\mathcal{E}}\mathbf{\Omega}_{e}\qquad\text{and}\qquad\mathbf{\Omega}_{e}=\bigcup_{n=0}^{|\mathbf{\Omega}_{e}|-1}\mathbf{\Omega}_{en}\qquad\text{and}\qquad\bigcap_{e\in\mathcal{E}}\bigcap_{n=0}^{|\mathbf{\Omega}_{e}|-1}\mathbf{\Omega}_{en}=\emptyset,$
where $\mathbf{\Omega}_{e}$ refers to all of the elements of type $e$ inside
of the domain, $\left\lvert\mathbf{\Omega}_{e}\right\rvert$ is the number of
elements of this type in the decomposition, and $n$ is an index running over
these elements with $0\leq n<\left\lvert\mathbf{\Omega}_{e}\right\rvert$.
Inside each element $\mathbf{\Omega}_{en}$ we require that
$\displaystyle\frac{\partial u_{en\alpha}}{\partial
t}+\bm{\nabla}\cdot\mathbf{f}_{en\alpha}$ $\displaystyle=0,$ (3a)
$\displaystyle\mathbf{q}_{en\alpha}-\bm{\nabla}u_{en\alpha}$
$\displaystyle=0.$ (3b)
It is convenient, for reasons of both mathematical simplicity and
computational efficiency, to work in a transformed space. We accomplish this
by introducing, for each element type, a standard element
$\mathbf{\hat{\Omega}}_{e}$ which exists in a transformed space,
$\tilde{\mathbf{x}}=\tilde{x}_{i}$. Next, assume the existence of a mapping
function for each element such that
$\displaystyle x_{i}$ $\displaystyle=\mathcal{M}_{eni}(\tilde{\mathbf{x}}),$
$\displaystyle\mathbf{x}$
$\displaystyle=\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}),$
$\displaystyle\tilde{x}_{i}$
$\displaystyle=\mathcal{M}^{-1}_{eni}(\mathbf{x}),$
$\displaystyle\tilde{\mathbf{x}}$
$\displaystyle=\bm{\mathcal{M}}^{-1}_{en}(\mathbf{x}),$
along with the relevant Jacobian matrices
$\displaystyle\bm{\mathsf{J}}_{en}=J_{enij}$
$\displaystyle=\frac{\partial\mathcal{M}_{eni}}{\partial\tilde{x}_{j}},$
$\displaystyle J_{en}$ $\displaystyle=\det\bm{\mathsf{J}}_{en},$
$\displaystyle\bm{\mathsf{J}}^{-1}_{en}=J^{-1}_{enij}$
$\displaystyle=\frac{\partial\mathcal{M}^{-1}_{eni}}{\partial x_{j}},$
$\displaystyle J^{-1}_{en}$
$\displaystyle=\det\bm{\mathsf{J}}^{-1}_{en}=\frac{1}{J_{en}}.$
These definitions provide us with a means of transforming quantities to and
from standard element space. Taking the transformed solution, flux, and
gradients inside each element to be
$\displaystyle\tilde{u}_{en\alpha}$
$\displaystyle=\tilde{u}_{en\alpha}(\tilde{\mathbf{x}},t)=J_{en}(\tilde{\mathbf{x}})u_{en\alpha}(\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}),t),$
(4a) $\displaystyle\tilde{\mathbf{f}}_{en\alpha}$
$\displaystyle=\tilde{\mathbf{f}}_{en\alpha}(\tilde{\mathbf{x}},t)=J_{en}(\tilde{\mathbf{x}})\bm{\mathsf{J}}^{-1}_{en}(\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}))\mathbf{f}_{en\alpha}(\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}),t),$
(4b) $\displaystyle\tilde{\mathbf{q}}_{en\alpha}$
$\displaystyle=\tilde{\mathbf{q}}_{en\alpha}(\tilde{\mathbf{x}},t)=\bm{\mathsf{J}}^{T}_{en}(\tilde{\mathbf{x}})\mathbf{q}_{en\alpha}(\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}),t),$
(4c)
and letting $\tilde{\bm{\nabla}}=\partial/\partial\tilde{x}_{i}$, it can be
readily verified that
$\displaystyle\frac{\partial u_{en\alpha}}{\partial
t}+J^{-1}_{en}\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}}_{en\alpha}$
$\displaystyle=0,$ (5a)
$\displaystyle\tilde{\mathbf{q}}_{en\alpha}-\tilde{\bm{\nabla}}u_{en\alpha}$
$\displaystyle=0,$ (5b)
as required. We note here the decision to multiply the first equation through
by a factor of $J^{-1}_{en}$. Doing so has the effect of taking
$\tilde{u}_{en}\mapsto u_{en}$ which allows us to work in terms of the
physical solution. This is more convenient from a computational standpoint.
We next proceed to associate a set of solution points with each standard
element. For each type $e\in\mathcal{E}$ take
$\set{\tilde{\mathbf{x}}^{(u)}_{e\rho}}$ to be the chosen set of points where
$0\leq\rho<N^{(u)}_{e}(\wp)$. These points can then be used to construct a
nodal basis set $\set{\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}})}$ with the
property that
$\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(u)}_{e\sigma})=\delta_{\rho\sigma}$.
To obtain such a set we first take $\set{\psi_{e\sigma}(\tilde{\mathbf{x}})}$
to be any basis which spans a selected order $\wp$ polynomial space defined
inside $\hat{\mathbf{\Omega}}_{e}$. Next we compute the elements of the
generalised Vandermonde matrix
$\mathcal{V}_{e\rho\sigma}=\psi_{e\rho}(\tilde{\mathbf{x}}^{(u)}_{e\sigma})$.
With these a nodal basis set can be constructed as
$\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}})=\mathcal{V}^{-1}_{e\rho\sigma}\psi_{e\sigma}(\tilde{\mathbf{x}})$.
Along with the solution points inside of each element we also define a set of
flux points on $\partial\hat{\mathbf{\Omega}}_{e}$. We denote the flux points
for a particular element type as $\set{\tilde{\mathbf{x}}^{(f)}_{e\rho}}$
where $0\leq\rho<N^{(f)}_{e}(\wp)$. Let the set of corresponding normalised
outward-pointing normal vectors be given by
$\set{\hat{\tilde{\mathbf{n}}}^{(f)}_{e\rho}}$. It is critical that each flux
point pair along an interface share the same coordinates in physical space.
For a pair of flux points $e\rho n$ and $e^{\prime}\rho^{\prime}n^{\prime}$ at
a non-periodic interface this can be formalised as
$\bm{\mathcal{M}}_{en}(\tilde{\mathbf{x}}^{(f)}_{e\rho})=\bm{\mathcal{M}}_{e^{\prime}n^{\prime}}(\tilde{\mathbf{x}}^{(f)}_{e^{\prime}\rho^{\prime}})$.
A pictorial illustration of this can be seen in Figure 2.
Figure 2: Solution points (blue circles) and flux points (orange squares) for
a triangle and quadrangle in physical space. For the top edge of the
quadrangle the normal vectors have been plotted. Observe how the flux points
at the interface between the two elements are co-located.
The first step in the FR approach is to go from the discontinuous solution at
the solution points to the discontinuous solution at the flux points
$u^{(f)}_{e\sigma n\alpha}=u^{(u)}_{e\rho
n\alpha}\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma}),$ (6)
where $u^{(u)}_{e\rho n\alpha}$ is an approximate solution of field variable
$\alpha$ inside of the $n$th element of type $e$ at solution point
$\tilde{\mathbf{x}}^{(u)}_{e\rho}$. This can then be used to compute a _common
solution_
$\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}u^{(f)}_{\vphantom{\widetilde{e\rho n}\alpha}e\rho
n\alpha}=\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}u^{(f)}_{\widetilde{e\rho
n}\alpha}=\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}(u^{(f)}_{\vphantom{\widetilde{e\rho n}\alpha}e\rho
n\alpha},u^{(f)}_{\widetilde{e\rho n}\alpha}),$ (7)
where $\mathfrak{C}_{\alpha}(u_{L},u_{R})$ is a scalar function that given two
values at a point returns a common value. Here we have taken $\widetilde{e\rho
n}$ to be the element type, flux point number and element number of the
adjoining point at the interface. Since grids in FR are permitted to be
unstructured the relationship between $e\rho n$ and $\widetilde{e\rho n}$ is
indirect. This necessitates the use of a lookup table. As the common solution
function is permitted to perform upwinding or downwinding of the solution it
is in general the case that
$\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}(u^{(f)}_{\vphantom{\widetilde{e\rho n}\alpha}e\rho
n\alpha},u^{(f)}_{\widetilde{e\rho
n}\alpha})\neq\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}(u^{(f)}_{\widetilde{e\rho
n}\alpha},u^{(f)}_{\vphantom{\widetilde{e\rho n}\alpha}e\rho n\alpha})$.
Hence, it is important that each flux point pair only be visited _once_ with
the same common solution value assigned to both
$\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}u^{(f)}_{\vphantom{\widetilde{e\rho n}\alpha}e\rho n\alpha}$
and $\mathfrak{C}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\rho
n}\alpha}\alpha}u^{(f)}_{\widetilde{e\rho n}\alpha}$.
Further, associated with each flux point is a vector correction function
$\mathbf{g}^{(f)}_{e\rho}(\tilde{\mathbf{x}})$ constrained such that
$\hat{\tilde{\mathbf{n}}}^{(f)}_{e\sigma}\cdot\mathbf{g}^{(f)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma})=\delta_{\rho\sigma},$
(8)
with a divergence that sits in the same polynomial space as the solution.
Using these fields we can express the solution to Equation 5b as
$\tilde{\mathbf{q}}^{(u)}_{e\sigma
n\alpha}=\bigg{[}\hat{\tilde{\mathbf{n}}}^{(f)}_{e\rho}\cdot\tilde{\bm{\nabla}}\cdot\mathbf{g}^{(f)}_{e\rho}(\tilde{\mathbf{x}})\left\\{\mathfrak{C}^{\vphantom{(f)}}_{\alpha}u^{(f)}_{e\rho
n\alpha}-u^{(f)}_{e\rho n\alpha}\right\\}+u^{(u)}_{e\nu
n\alpha}\tilde{\bm{\nabla}}\ell^{(u)}_{e\nu}(\tilde{\mathbf{x}})\bigg{]}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(u)}_{e\sigma}},$
(9)
where the term inside the curly brackets is the ‘jump’ at the interface and
the final term is an order $\wp-1$ approximation of the gradient obtained by
differentiating the discontinuous solution polynomial. Following the
approaches of Kopriva [15] and Sun et al. [3] we can now compute physical
gradients as
$\displaystyle\mathbf{q}^{(u)}_{e\sigma n\alpha}$
$\displaystyle=\bm{\mathsf{J}}^{-T\,(u)}_{e\sigma
n}\tilde{\mathbf{q}}^{(u)}_{e\sigma n\alpha},$ (10)
$\displaystyle\mathbf{q}^{(f)}_{e\sigma n\alpha}$
$\displaystyle=\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma})\mathbf{q}^{(u)}_{e\rho
n\alpha},$ (11)
where $\bm{\mathsf{J}}^{-T\,(u)}_{e\sigma
n}=\bm{\mathsf{J}}^{-T}_{en}(\tilde{\mathbf{x}}^{(u)}_{e\sigma})$. Having
solved the auxiliary equation we are now able to evaluate the transformed flux
$\tilde{\mathbf{f}}^{(u)}_{e\rho n\alpha}=J^{(u)}_{e\rho
n}\bm{\mathsf{J}}^{-1\,(u)}_{e\rho n}\mathbf{f}_{\alpha}(u^{(u)}_{e\rho
n},\mathbf{q}^{(u)}_{e\rho n}),$ (12)
where $J^{(u)}_{e\rho
n}=\det\bm{\mathsf{J}}_{en}(\tilde{\mathbf{x}}^{(u)}_{e\rho})$. This can be
seen to be a collocation projection of the flux. With this it is possible to
compute the normal transformed flux at each of the flux points
$\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha}=\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma})\hat{\tilde{\mathbf{n}}}^{(f)}_{e\sigma}\cdot\tilde{\mathbf{f}}^{(u)}_{e\rho
n\alpha}.$ (13)
Considering the physical normals at the flux points we see that
$\mathbf{n}^{(f)}_{e\sigma n}=n^{(f)}_{e\sigma
n}\hat{\mathbf{n}}^{(f)}_{e\sigma n}=\bm{\mathsf{J}}^{-T\,(f)}_{e\sigma
n}\hat{\tilde{\mathbf{n}}}^{(f)}_{e\sigma},$ (14)
which is the outward facing normal vector in physical space where
$n^{(f)}_{e\sigma n}>0$ is defined as the magnitude. As the interfaces between
two elements conform we must have
$\hat{\mathbf{n}}^{(f)}_{\vphantom{\widetilde{e\sigma n}}e\sigma
n}=-\hat{\mathbf{n}}^{(f)}_{\widetilde{e\sigma n}}$. With these definitions we
are now in a position to specify an expression for the _common normal flux_ at
a flux point pair as
$\mathfrak{F}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}f^{(f_{\perp})}_{\vphantom{\widetilde{e\sigma n}}e\sigma
n\alpha}=-\mathfrak{F}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}f^{(f_{\perp})}_{\widetilde{e\sigma
n}\alpha}=\mathfrak{F}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}(u^{(f)}_{\vphantom{\widetilde{e\sigma n}}e\sigma
n},u^{(f)}_{\widetilde{e\sigma
n}},\mathbf{q}^{(f)}_{\vphantom{\widetilde{e\sigma n}}e\sigma
n},\mathbf{q}^{(f)}_{\widetilde{e\sigma
n}},\hat{\mathbf{n}}^{(f)}_{\vphantom{\widetilde{e\sigma n}}e\sigma n}).$ (15)
The relationship $\mathfrak{F}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}f^{(f_{\perp})}_{\vphantom{\widetilde{e\sigma n}}e\sigma
n\alpha}=-\mathfrak{F}^{\vphantom{(f)}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}f^{(f_{\perp})}_{\widetilde{e\sigma n}\alpha}$ arises from the
desire for the resulting numerical scheme to be conservative; a net outward
flux from one element must be balanced by a corresponding inward flux on the
adjoining element. It follows that that
$\mathfrak{F}_{\alpha}(u_{L},u_{R},\mathbf{q}_{L},\mathbf{q}_{R},\hat{\mathbf{n}}_{L})=-\mathfrak{F}_{\alpha}(u_{R},u_{L},\mathbf{q}_{R},\mathbf{q}_{L},-\hat{\mathbf{n}}_{L})$.
The common normal fluxes in Equation 15 can now be taken into transformed
space via
$\displaystyle\mathfrak{F}^{\vphantom{(f_{\perp})}}_{\alpha}\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha}$ $\displaystyle=J^{(f)}_{e\sigma n}n^{(f)}_{e\sigma
n}\mathfrak{F}_{\alpha}f^{(f_{\perp})}_{e\sigma n\alpha},$ (16)
$\displaystyle\mathfrak{F}^{\vphantom{(f_{\perp})}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}\tilde{f}^{(f_{\perp})}_{\widetilde{e\sigma n}\alpha}$
$\displaystyle=J^{(f)}_{\widetilde{e\sigma n}}n^{(f)}_{\widetilde{e\sigma
n}}\mathfrak{F}^{\vphantom{(f_{\perp})}}_{\vphantom{\widetilde{e\sigma
n}}\alpha}f^{(f_{\perp})}_{\widetilde{e\sigma n}\alpha},$ (17)
where $J^{(f)}_{e\sigma
n}=\det\bm{\mathsf{J}}_{en}(\tilde{\mathbf{x}}^{(f)}_{e\sigma})$.
It is now possible to compute an approximation for the divergence of the
_continuous_ flux. The procedure is directly analogous to the one used to
calculate the transformed gradient in Equation 9
$(\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}})^{(u)}_{e\rho
n\alpha}=\bigg{[}\tilde{\bm{\nabla}}\cdot\mathbf{g}^{(f)}_{e\sigma}(\tilde{\mathbf{x}})\left\\{\mathfrak{F}^{\vphantom{(f)}}_{\alpha}\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha}-\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha}\right\\}+\tilde{\mathbf{f}}^{(u)}_{e\nu
n\alpha}\cdot\tilde{\bm{\nabla}}\ell^{(u)}_{e\nu}(\tilde{\mathbf{x}})\bigg{]}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(u)}_{e\rho}},$
(18)
which can then be used to obtain a semi-discretised form of the governing
system
$\frac{\partial u^{(u)}_{e\rho n\alpha}}{\partial t}=-J^{-1\,(u)}_{e\rho
n}(\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}})^{(u)}_{e\rho n\alpha},$ (19)
where $J^{-1\,(u)}_{e\rho
n}=\det\bm{\mathsf{J}}^{-1}_{en}(\tilde{\mathbf{x}}^{(u)}_{e\rho})=1/J^{(u)}_{e\rho
n}$.
This semi-discretised form is simply a system of ordinary differential
equations in $t$ and can be solved using one of a number of schemes, e.g. a
classical fourth order Runge-Kutta (RK4) scheme.
## 3 Implementation
### 3.1 Overview
PyFR is a Python based implementation of the FR approach described in section
section 2. It is designed to be compact, efficient, and platform portable. Key
functionality is summarised in table Table 1.
Table 1: Key functionality of PyFR. Dimensions | 2D, 3D
---|---
Elements | Triangles, Quadrilaterals, Hexahedra
Spatial orders | Arbitrary
Time steppers | Euler, RK4, DOPRI5
Precisions | Single, Double
Platforms | CPUs via C/OpenMP, Nvidia GPUs via CUDA
Communication | MPI
Governing Systems | Euler, Compressible Navier-Stokes
The majority of operations within an FR step can be cast in terms of matrix-
matrix multiplications, as detailed in Appendix A. All remaining operations
(e.g. flux evaluations) are point-wise, concerning themselves with either a
single solution point inside of an element or two collocating flux points at
an interface. Hence, in broad terms, there are five salient aspects of an FR
implementation, specifically i.) definition of the constant operator matrices
detailed in Appendix A, ii.) specification of the state matrices detailed in
Appendix A, iii.) implementation of matrix multiply kernels, iv.)
implementation of point-wise kernels, and finally v.) handling of distributed
memory parallelism and scheduling of kernel invocations. Details regarding how
each of the above were achieved in PyFR are presented below.
### 3.2 Definition of Constant Operator Matrices
Setup of the seven constant operator matrices detailed in Appendix A requires
evaluation of various polynomial expressions, and their derivatives, at
solution/flux points within each type of standard element. Although
conceptually simple, such operations can be cumbersome to code. To keep the
codebase compact PyFR makes extensive use of symbolic manipulation via SymPy
[16], which brings computer algebra facilities similar to those found in Maple
and Mathematica to Python. SymPy has built-in support for most common
polynomials and can readily evaluate such expressions to arbitrary precision.
Efficiency of the setup phase is not critical, since the operations are only
performed once at start-up. Since efficiency is not critical, platform
portability is effectively achieved by running such operations on the host CPU
in all cases.
### 3.3 Specification of State Matrices
In specifying the state matrices detailed in Appendix A there is a degree of
freedom regarding how the field variables of each element are packed along a
row. The packing of field variables can be characterised by considering the
distance, $\Delta j$ (in columns) between two subsequent field variables for a
given element. The case of $\Delta j=1$ corresponds to the array of structures
(AoS) packing whereas the choice of $\Delta
j=\left\lvert\mathbf{\Omega}_{e}\right\rvert$ leads to the structure of arrays
(SoA) packing. A hybrid approach wherein $\Delta j=k$ with $k$ being constant
results in the AoSoA($k$) approach. An implementation is free to chose between
any of these counting patterns so long as it is consistent. For simplicity
PyFR uses the SoA packing order across all platforms.
### 3.4 Matrix Multiplication Kernels
PyFR defers matrix multiplication to the GEMM family of sub-routies provided a
suitable Basic Linear Algebra Subprograms (BLAS) library. BLAS is available
for virtually all platforms and optimised versions are often maintained by the
hardware vendors themselves (e.g. cuBLAS for Nvidia GPUs). This approach
greatly facilitates development of efficient and platform portable code. We
note, however, that the matrix sizes encountered in PyFR are not necessarily
optimal from a GEMM perspective. Specifically, GEMM is optimised for the
multiplication of large square matrices, whereas the constant operator
matrixes in PyFR are ‘small and square’ with $10$–$100$ rows/columns, and the
state matrices are ‘short and fat’ with $10$–$100$ rows and
$10\,000$–$100\,000$ columns. Moreover, we note that the constant operator
matrices are know a priori, and do not change in time. This a priori knowledge
could, in theory, be leveraged to design bespoke matrix multiply kernels that
are more efficient than GEMM. Development of such bespoke kernels will be a
topic of future research - with results easily integrated into PyFR as an
optional replacement for GEMM.
### 3.5 Point-Wise Kernels
Point-wise kernels are specified using a domain specific language implemented
in PyFR atop of the Mako templating engine [17]. The templated kernels are
then interpreted at runtime, converted to low-level code, compiled, linked and
loaded. Currently the templating engine can generate C/OpenMP to target CPUs,
and CUDA (via the PyCUDA wrapper [18]) to target Nvidia GPUs. Use of a domain
specific language avoids implementation of each point-wise kernel for each
target platform; keeping the codebase compact and platform portable. Runtime
code generation also means it is possible to instruct the compiler to emit
binaries which are optimised for the current hardware architecture. Such
optimisations can result in anything up to a fourfold improvement in
performance when compared with architectural defaults.
As an example of a point-wise kernel we consider the evaluation of the right
hand side of Equation 19, which reads $-J^{-1\,(u)}_{e\rho
n}(\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}})^{(u)}_{e\rho n\alpha}$. The
operation consists of a point-wise multiplication between the negative
reciprocal of the Jacobian and the transformed divergence of the flux at each
solution point. Figure 3 shows how such a kernel can be expressed in the
domain specific language of PyFR. There are several points of note. Firstly,
the kernel is purely scalar in nature. This is by design; in PyFR point-wise
kernels need only prescribe the point-wise operation to be applied. Important
choices such as how to vectorise a given operation or how to gather data from
memory are all delegated to templating engine. Secondly, we note it is
possible to utilise Python when generating the main body of kernels. This
capability is showcased on lines four, five and six where it is used to unroll
a for loop over each of the field variables. Finally, we also highlight the
use of an abstract data type _fpdtype_t_ for floating point variables which
permits a single set of kernels to be used for both single and double
precision operation. Generated CUDA source for this kernel can be seen in
Figure 4, and the equivalent C kernel can be found in Figure 5.
Figure 3: An example of an extrinsic kernel in PyFR. The template variable
_nvars_ is taken to be the number of field variables, $N_{v}$. The kernel
arguments _tdivtconf_ and _rcpdjac_ correspond to
$\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}}$ and $J^{-1}$ respectively with
the operation being performed in-place. Figure 4: Generated CUDA source for
the template in Figure 3 for when $N_{V}=4$. Figure 5: Generated OpenMP
annotated C source code for the template in Figure 3 for when $N_{V}=4$. The
somewhat unconventional structure is necessary to ensure that the kernel is
properly vectorised across a range of compilers.
### 3.6 Distributed Memory Parallelism and Scheduling
PyFR is capable of operating on heigh performance computing clusters utilising
distributed memory parallelism. This is accomplished through the Message
Passing Interface (MPI). All MPI functionality is implemented at the Python
level through the mpi4py [19] wrapper. To enhance the scalability of the code
care has been taken to ensure that all requests are persistent, point-to-point
and non-blocking. Further, the format of data that is shared between ranks has
been made backend independent. It is therefore possible to deploy PyFR on
heterogeneous clusters consisting of both conventional CPUs and accelerators.
The arrangement of kernel calls required to solve an advection-diffusion
problem can be seen in Figure 6. Our primary objective when scheduling kernels
was to maximise the potential for overlapping communication with computation.
In order to help achieve this the common interface solution,
$\mathfrak{C}_{\alpha}$, and common interface flux, $\mathfrak{F}_{\alpha}$,
kernels have been broken apart into two separate kernels; suffixed in the
figure by int and mpi. PyFR is therefore able to perform a significant degree
of rank-local computation while the relevant ghost states are being exchanged.
Figure 6: Flow diagram showing the stages required to compute
$-\bm{\nabla}\cdot\mathbf{f}$. Symbols correspond to those of Appendix A. For
simplicity arguments referencing constant data have been omitted. Memory
indirection is indicated by red underlines. Synchronisation points are
signified by black horizontal lines. Dotted lines correspond to data reuse.
Our secondary objective when scheduling kernels was to minimise the amount of
temporary storage required during the evaluation of
$-\bm{\nabla}\cdot\mathbf{f}$. Such optimisations are critical within the
context of accelerators which often have an order of magnitude less memory
than a contemporary platform. In order to help achieve this
$\bm{\mathsf{U}}^{(u)}$, $\tilde{\bm{\mathsf{R}}}^{(u)}$, and
$\bm{\mathsf{R}}^{(u)}$ are allowed to alias. By permitting the same storage
location to be used for both the inputted solution and the outputted flux
divergence it is possible to reduce the storage requirements of the RK
schemes. Another opportunity for memory reuse is in the transformed flux
function where the incoming gradients, $\bm{\mathsf{Q}}^{(u)}$, can be
overwritten with the transformed flux, $\tilde{\bm{\mathsf{F}}}^{(u)}$. A
similar approach can be used in the common interface flux function whereby
$\bm{\mathsf{U}}^{(f)}$ can updated in-place with the entires of
$\tilde{\bm{\mathsf{D}}}^{(f)}$ which holds the transformed common normal
flux. Moreover, $\bm{\mathsf{C}}^{(f)}$ is also able to utilise the same
storage as the somewhat larger $\bm{\mathsf{Q}}^{(f)}$ array. These
optimisations allow PyFR to process over $100\,000$ curved, unstructured,
hexahedral elements at $\wp=3$ inside of a $5\,\text{GiB}$ memory footprint.
## 4 Governing Systems
### 4.1 Overview
PyFR is a framework for solving various advection-diffusion type problems. In
the current release of PyFR two specific governing systems can be solved,
specifically the Euler equations for inviscid compressible flow, and the
compressible Navier-Stokes equations for viscous compressible flow. Details
regarding both are given below.
### 4.2 Euler Equations
Using the framework introduced in section 2 the three dimensional Euler
equations can be expressed in conservative form as
$u=\begin{Bmatrix}\rho\\\ \rho v_{x}\\\ \rho v_{y}\\\ \rho v_{z}\\\
E\end{Bmatrix},\qquad\mathbf{f}=\mathbf{f}^{(\mathrm{inv})}=\begin{Bmatrix}\rho
v_{x}&\rho v_{y}&\rho v_{z}\\\ \rho v_{x}^{2}+p&\rho v_{y}v_{x}&\rho
v_{z}v_{x}\\\ \rho v_{x}v_{y}&\rho v_{y}^{2}+p&\rho v_{z}v_{y}\\\ \rho
v_{x}v_{z}&\rho v_{y}v_{z}&\rho v_{z}^{2}+p\\\
v_{x}(E+p)&v_{y}(E+p)&v_{z}(E+p)\end{Bmatrix},$ (20)
with $u$ and $\mathbf{f}$ together satisfying Equation 1. In the above $\rho$
is the mass density of the fluid, $\mathbf{v}=(v_{x},v_{y},v_{z})^{T}$ is the
fluid velocity vector, $E$ is the total energy per unit volume and $p$ is the
pressure. For a perfect gas the pressure and total energy can be related by
the ideal gas law
$E=\frac{p}{\gamma-1}+\frac{1}{2}\rho\|\mathbf{v}\|^{2},$ (21)
with $\gamma=C_{p}/C_{v}$.
With the fluxes specified all that remains is to prescribe a method for
computing the common normal flux, $\mathfrak{F}_{\alpha}$, at interfaces as
defined in Equation 15. This can be accomplished using an approximate Riemann
solver for the Euler equations. There exist a variety of such solvers as
detailed in [20]. A description of those implemented in PyFR can be found in
Appendix B.
### 4.3 Compressible Navier-Stokes Equations
The compressible Navier-Stokes equations can be viewed as an extension of the
Euler equations via the inclusion of viscous terms. Within the framework
outlined above the flux now takes the form of
$\mathbf{f}=\mathbf{f}^{(\text{inv})}-\mathbf{f}^{(\text{vis})}$ where
$\mathbf{f}^{(\mathrm{vis})}=\begin{Bmatrix}0&0&0\\\
\mathcal{T}_{xx}&\mathcal{T}_{yx}&\mathcal{T}_{zx}\\\
\mathcal{T}_{xy}&\mathcal{T}_{yy}&\mathcal{T}_{zy}\\\
\mathcal{T}_{xz}&\mathcal{T}_{yz}&\mathcal{T}_{zz}\\\
v_{i}\mathcal{T}_{ix}+\Delta\partial_{x}T&v_{i}\mathcal{T}_{iy}+\Delta\partial_{y}T&v_{i}\mathcal{T}_{iz}+\Delta\partial_{z}T\end{Bmatrix}.$
(22)
In the above we have defined $\Delta=\mu C_{p}/P_{r}$ where $\mu$ is the
dynamic viscosity and $P_{r}$ is the Prandtl number. The components of the
stress-energy tensor are given by
$\mathcal{T}_{ij}=\mu(\partial_{i}v_{j}+\partial_{j}v_{i})-\frac{2}{3}\mu\delta_{ij}\bm{\nabla}\cdot\mathbf{v}.$
(23)
Using the ideal gas law the temperature can be expressed as
$T=\frac{1}{C_{v}}\frac{1}{\gamma-1}\frac{p}{\rho},$ (24)
with partial derivatives thereof being given according to the quotient rule.
Since the Navier-Stokes equations are an advection-diffusion type system it is
necessary to both compute a common solution ($\mathfrak{C}_{\alpha}$ of
Equation 7) at element boundaries and augment the inviscid Riemann solver to
handle the viscous part of the flux. A popular approach is the LDG method as
presented in [5, 13]. In this approach the common solution is given
$\forall\alpha$ according to
$\mathfrak{C}(u_{L},u_{R})=(\tfrac{1}{2}-\beta)u_{L}+(\tfrac{1}{2}+\beta)u_{R},$
(25)
where $\beta$ controls the degree of upwinding/downwinding. The common normal
interface flux is then prescribed, once again $\forall\alpha$, according to
$\mathfrak{F}(u_{L},u_{R},\mathbf{q}_{L},\mathbf{q}_{R},\hat{\mathbf{n}}_{L})=\mathfrak{F}^{(\text{inv})}-\mathfrak{F}^{(\text{vis})},$
(26)
where $\mathfrak{F}^{(\text{inv})}$ is a suitable inviscid Riemann solver (see
Appendix B) and
$\mathfrak{F}^{(\text{vis})}=\hat{\mathbf{n}}^{\vphantom{(\text{vis})}}_{L}\cdot\left\\{(\tfrac{1}{2}+\beta)\mathbf{f}^{(\text{vis})}_{L}+(\tfrac{1}{2}-\beta)\mathbf{f}^{(\text{vis})}_{R}\right\\}+\tau(u_{L}^{\vphantom{(\text{vis})}}-u_{R}^{\vphantom{(\text{vis})}}),$
(27)
with $\tau$ being a penalty parameter,
$\mathbf{f}^{(\text{vis})}_{L}=\mathbf{f}^{(\text{vis})}_{\vphantom{L}}(u^{\vphantom{(\text{vis})}}_{L},\mathbf{q}^{\vphantom{(\text{vis})}}_{L})$,
and
$\mathbf{f}^{(\text{vis})}_{R}=\mathbf{f}^{(\text{vis})}_{\vphantom{R}}(u^{\vphantom{(\text{vis})}}_{R},\mathbf{q}^{\vphantom{(\text{vis})}}_{R})$.
We observe here that if the common solution is upwinded then the common normal
flux will be downwinded. Generally, $\beta=\pm 1/2$ as this results in the
numerical scheme having a compact stencil and $0\leq\tau\leq 1$.
#### 4.3.1 Presentation in Two Dimensions
The above prescriptions of the Euler and Navier-Stokes equations are valid for
the case of $N_{D}=3$. The two dimensional formulation can be recovered by
deleting the fourth rows in the definitions of $u$,
$\mathbf{f}^{(\text{inv})}$ and $\mathbf{f}^{(\text{vis})}$ along with the
third columns of $\mathbf{f}^{(\text{inv})}$ and $\mathbf{f}^{(\text{vis})}$.
Vectors are now two dimensional with the velocity being given by
$\mathbf{v}=(v_{x},v_{y})^{T}$.
## 5 Validation
### 5.1 Euler Equations: Euler Vortex Super Accuracy
Various authors [4, 10] have shown FR schemes exhibit so-called ‘super
accuracy’ (an order of accuracy greater than the expected $\wp+1$). To confirm
PyFR can achieve super accuracy for the Euler equations a square domain
$\mathbf{\Omega}=[-20,20]^{2}$ was decomposed into four structured quad meshes
with spacings of $h=1/3$, $h=2/7$, $h=1/4$, and $h=2/9$. Initial conditions
were taken to be those of an isentropic Euler vortex in a free-stream
$\displaystyle\rho(\mathbf{x},t=0)$
$\displaystyle=\left\\{1-\frac{S^{2}M^{2}(\gamma-1)\exp
2f}{8\pi^{2}}\right\\}^{\frac{1}{\gamma-1}},$ (28)
$\displaystyle\mathbf{v}(\mathbf{x},t=0)$ $\displaystyle=\frac{Sy\exp{f}}{2\pi
R}\hat{\mathbf{x}}+\left\\{1-\frac{Sx\exp{f}}{2\pi
R}\right\\}\hat{\mathbf{y}},$ (29) $\displaystyle p(\mathbf{x},t=0)$
$\displaystyle=\frac{\rho^{\gamma}}{\gamma M^{2}},$ (30)
where $f=(1-x^{2}-y^{2})/2R^{2}$, $S=13.5$ is the strength of the vortex,
$M=0.4$ is the free-stream Mach number, and $R=1.5$ is the radius. All meshes
were configured with periodic boundary conditions along boundaries of constant
$x$. Along boundaries of constant $y$ the dynamical variables were fixed
according to
$\displaystyle\rho(\mathbf{x}=x\hat{\mathbf{x}}\pm 20\hat{\mathbf{y}},t)$
$\displaystyle=1,$ $\displaystyle\mathbf{v}(\mathbf{x}=x\hat{\mathbf{x}}\pm
20\hat{\mathbf{y}},t)$ $\displaystyle=\hat{\mathbf{y}},$ $\displaystyle
p(\mathbf{x}=x\hat{\mathbf{x}}\pm 20\hat{\mathbf{y}},t)$
$\displaystyle=\frac{1}{\gamma M^{2}},$
which are simply the limiting values of the initial conditions. Strictly
speaking these conditions, on account of the periodicity, result in the
modelling of an infinite array of coupled vortices. The impact of this is
mitigated by the observation that the exponentially decaying vortex has a
characteristic radius which is far smaller than the extent of the domain.
Neglecting these effects the analytic solution of the system is a time $t$ is
simply a translation of the initial conditions.
Using the analytical solution we can define an $L^{2}$ error as
$\sigma(t)^{2}=\int_{-2}^{2}\int_{-2}^{2}\Bigl{[}\rho^{\delta}(\mathbf{x}+\Delta_{y}(t)\hat{\mathbf{y}},t)-\rho(\mathbf{x},t=0)\Bigr{]}^{2}\,\mathrm{d}^{2}\mathbf{x},$
(31)
where $\rho^{\delta}(\mathbf{x},t)$ is the numerical mass density,
$\rho(\mathbf{x},t=0)$ the analytic mass density, and $\Delta_{y}(t)$ is the
ordinate corresponding to the centre of the vortex at a time $t$ and accounts
for the fact that the vortex is translating in a free stream velocity of unity
in the $y$ direction. Restricting the region of consideration to a small box
centred around the origin serves to further mitigate against the effects of
vortices coupling together. The initial mass density along with the
$[-2,-2]\times[2,2]$ region used to evaluate the error can be seen in Figure
7. At times, $t_{c}$, when the vortex is centred on the box the error can be
readily computed by integrating over each element inside the box and summing
the residuals
$\sigma(t_{c})^{2}=\iint_{\hat{\mathbf{\Omega}}_{e}}\Bigl{[}\rho^{\delta}_{i}(\tilde{\mathbf{x}},t_{c})-\rho(\bm{\mathcal{M}}_{i}(\tilde{\mathbf{x}}),0)\Bigr{]}^{2}J_{i}(\tilde{\mathbf{x}})\,\mathrm{d}^{2}\tilde{\mathbf{x}},$
(32)
where, $\rho^{\delta}_{i}(\tilde{\mathbf{x}},t_{c})$ is the approximate mass
density inside of the $i$th element, and $J_{i}(\tilde{\mathbf{x}})$ the
associated Jacobian. These integrals can be approximated by applying Gaussian
quadrature
$\displaystyle\sigma(t_{c})^{2}$ $\displaystyle\approx
J_{i}(\tilde{\mathbf{x}}_{j})\Bigl{[}\rho^{\delta}_{i}(\tilde{\mathbf{x}}_{j},t_{c})-\rho(\bm{\mathcal{M}}_{i}(\tilde{\mathbf{x}}_{j}),0)\Bigr{]}^{2}\omega_{j}$
(33)
$\displaystyle=\frac{h^{2}}{4}\Bigl{[}\rho^{\delta}_{i}(\tilde{\mathbf{x}}_{j},t_{c})-\rho(\bm{\mathcal{M}}_{i}(\tilde{\mathbf{x}}_{j}),0)\Bigr{]}^{2}\omega_{j},$
where $\set{\tilde{\mathbf{x}}_{j}}$ are abscissa and $\set{\omega_{j}}$ the
weights of a rule determined for integration inside of a standard
quadrilateral. So long as the rule used is of a suitable strength then this
will be a very good approximation of the true $L^{2}$ error.
Figure 7: Initial density profile for the vortex in $\mathbf{\Omega}$. The
black box shows the region where the error is calculated.
Following [10] the initial conditions were laid onto the mesh using a
collocation projection with $\wp=3$. The simulation was then run with three
different flux reconstruction schemes: DG, SD, and HU as defined in [10].
Solution points were placed at a tensor product construction of Gauss-Legendre
quadrature points. Common interface fluxes were computed using a Rusanov
Riemann solver. To advance the solutions in time a classical fourth order
Runge-Kutta method (RK4) was used. The time step was taken to be $\Delta
t=0.00125$ with $t=0..1800$ with solutions written out to disk every $32\,000$
steps. The order of accuracy of the scheme at a particular time can be
determined by plotting $\log\sigma$ against $\log h$ and performing a least-
squares fit through the four data points. The order is given by the gradient
of the fit. A plot of order of accuracy against time for the three schemes can
be seen in Figure 8. We note that the order of accuracy changes as a function
of time. This is due to the fact that the error is actually of the form
$\sigma(t)=\sigma_{\text{p}}+\sigma_{\text{so}}(t)$ where $\sigma_{\text{p}}$
is a constant projection error and $\sigma_{\text{so}}$ is a time-dependent
spatial operator error. The projection error arises as a consequence of the
forth order collocation projection of the initial conditions onto the mesh.
Over time the spatial operator error grows in magnitude and eventually
dominates. Only when $\sigma_{\text{so}}(t)\gg\sigma_{\text{p}}$ can the true
order of the method be observed. The results here can be seen to be in
excellent agreement with those of [10].
Figure 8: Spatial super accuracy observed for a $\wp=3$ simulation using DG,
SD and HU as defined in [10].
### 5.2 Compressible Navier-Stokes Equations: Couette Flow
Consider the case in which two parallel plates of infinite extent are
separated by a distance $H$ in the $y$ direction. We treat both plates as
isothermal walls at a temperature $T_{w}$ and permit the top plate to move at
a velocity $v_{w}$ in the $x$ direction with respect to the bottom plate. For
simplicity we shall take the ordinate of the bottom plate as zero. In the case
of a constant viscosity $\mu$ the Navier-Stokes equations admit an analytical
solution in which
$\displaystyle\rho(\phi)$
$\displaystyle=\frac{\gamma}{\gamma-1}\frac{2p}{2C_{p}T_{w}+P_{r}v_{w}^{2}\phi(1-\phi)},$
(34) $\displaystyle\mathbf{v}(\phi)$
$\displaystyle=v_{w}\phi\hat{\mathbf{x}},$ (35) $\displaystyle p$
$\displaystyle=p_{c},$ (36)
where $\phi=y/H$ and $p_{c}$ is a constant pressure. The total energy is given
by the ideal gas law of Equation 21. On a finite domain the Couette flow
problem can be modelled through the imposition of periodic boundary
conditions. For a two dimensional mesh periodicity is enforced in $x$ whereas
for three dimensional meshes it is enforced in both $x$ and $z$. To validate
the Navier-Stokes solver in PyFR we take $\gamma=1.4$, $P_{r}=0.72$,
$\mu=0.417$, $C_{p}=$1005\text{\,}\mathrm{J}\text{\,}{\mathrm{K}}^{-1}$$,
$H=$1\text{\,}\mathrm{m}$$, $T_{w}=$300\text{\,}\mathrm{K}$$,
$p_{c}=$1\text{\times}{10}^{5}\text{\,}\mathrm{Pa}$$, and
$v_{w}=$69.445\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$. These values
correspond to a Mach number of 0.2 and a Reynolds number of 200. The plates
were modelled as no-slip isothermal walls as detailed in subsection C.4 of
Appendix C. A plot of the resulting energy profile can be seen in Figure 9.
Constant initial conditions are taken as
$\rho=\big{\langle}\,\rho(\phi)\,\big{\rangle}$,
$\mathbf{v}=v_{w}\hat{\mathbf{x}}$, and $p=p_{c}$. Using the analytical
solution we again define an $L^{2}$ error as
$\displaystyle\sigma(t)^{2}$
$\displaystyle=\int_{\mathbf{\Omega}}\left[E^{\delta}(\mathbf{x},t)-E(\mathbf{x})\right]^{2}\,\mathrm{d}^{N_{D}}\mathbf{x}$
(37)
$\displaystyle=\int_{\mathbf{\Omega}_{ei}}\left[E^{\delta}_{ei}(\tilde{\mathbf{x}},t)-E(\bm{\mathcal{M}}_{ei}(\tilde{\mathbf{x}}))\right]^{2}J_{ei}(\tilde{\mathbf{x}})\,\mathrm{d}^{N_{D}}\tilde{\mathbf{x}}$
(38)
$\displaystyle\approx\left[E^{\delta}_{ei}(\tilde{\mathbf{x}}_{ej},t)-E(\bm{\mathcal{M}}_{ei}(\tilde{\mathbf{x}}_{ej}))\right]^{2}J_{ei}(\tilde{\mathbf{x}}_{ej})\omega_{ej},$
(39)
where $\mathbf{\Omega}$ is the computational domain,
$E^{\delta}(\mathbf{x},t)$ is the numerical total energy, and $E(\mathbf{x})$
the analytic total energy. In the third step we have approximated each
integral by a quadrature rule with abscissa $\set{\tilde{\mathbf{x}}_{ej}}$
and weights $\set{\omega_{ej}}$ inside of an element type $e$. Couette flow is
a steady state problem and so in the limit of $t\rightarrow\infty$ the
numerical total energy should converge to a solution. Starting from a constant
initial condition the $L^{2}$ error was computed every $0.1$ time units. The
simulation was said to have converged when $\sigma(t)/\sigma(t+0.1)\leq 1.01$
where $\sigma$ is the $L^{2}$ error. We will denote the time at which this
occurs by $t_{\infty}$.
Once the system has converged for a range of meshes it is possible to compute
the order of accuracy of the scheme. For a given $\wp$ this is the slope (plus
or minus a standard error) of a linear least squares fit of $\log
h\sim\log\sigma(t_{\infty})$ where $h$ is an approximation of the
characteristic grid spacing. The expected order of accuracy is $\wp+1$. In all
simulations inviscid fluxes were computed using the Rusanov approach and the
LDG parameters were taken to be $\beta=1/2$ and $\tau=0.1$. All simulations
were performed with DG correction functions and at double precision. Inside
tensor product elements Gauss-Legendre solution and flux points were employed.
Triangular elements utilised Williams-Shunn solution points and Gauss-Legendre
flux points.
Figure 9: Converged steady state energy profile for the two dimensional
Couette flow problem.
##### Two dimensional unstructured mixed mesh.
For the two dimensional test cases the computational domain was taken to be
$[-1,1]\times[0,1]$. This domain was then meshed with both triangles and
quadrilaterals at four different refinement levels. The Couette flow problem
described above was then solved on each of these meshes. Experimental $L^{2}$
errors and orders of accuracy can be seen in Table 2. We note that in all
cases the expected order of accuracy was obtained.
(a)
(b)
(c)
(d)
Figure 10: Unstructured mixed element meshes used for the two dimensional Couette flow problem. Table 2: $L^{2}$ energy error and orders of accuracy for the Couette flow problem on four mixed meshes. The mesh spacing was approximated as $h\sim N_{E}^{-1/2}$ where $N_{E}$ is the total number of elements in the mesh. | | $\sigma(t_{\infty})\,/\,$\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$$
---|---|---
Tris | Quads | $\wp=1$ | $\wp=2$ | $\wp=3$ | $\wp=4$
2 | 8 | $1.26\times 10^{2}$ | $5.77\times 10^{-1}$ | $5.54\times 10^{-3}$ | $6.62\times 10^{-5}$
6 | 22 | $3.56\times 10^{1}$ | $1.40\times 10^{-1}$ | $6.72\times 10^{-4}$ | $3.91\times 10^{-6}$
10 | 37 | $2.08\times 10^{1}$ | $4.35\times 10^{-2}$ | $2.54\times 10^{-4}$ | $8.16\times 10^{-7}$
16 | 56 | $1.46\times 10^{1}$ | $3.52\times 10^{-2}$ | $1.09\times 10^{-4}$ | $4.62\times 10^{-7}$
Order | $2.21\pm 0.12$ | $2.99\pm 0.32$ | $3.97\pm 0.05$ | $5.20\pm 0.38$
##### Three dimensional extruded hexahedral mesh.
For this three dimensional case the computational domain was taken to be
$[-1,1]\times[0,1]\times[0,1]$. Meshes were constructed through first
generating a series of unstructured quadrilateral meshes in the $x$-$y$ plane.
A three layer extrusion was then performed on this meshes to yield a series of
hexahedral meshes. Experimental $L^{2}$ errors and orders of accuracy for
these meshes can be seen in Table 3.
Table 3: $L^{2}$ energy errors and orders of accuracy for the Couette flow problem on three extruded hexahedral meshes. On account of the extrusion $h\sim N^{-1/2}_{E}$ where $N_{E}$ is the total number of elements in the mesh. | $\sigma(t_{\infty})\,/\,$\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$$
---|---
Hexes | $\wp=1$ | $\wp=2$ | $\wp=3$
78 | $3.35\times 10^{1}$ | $5.91\times 10^{-2}$ | $7.28\times 10^{-4}$
195 | $1.23\times 10^{1}$ | $1.87\times 10^{-2}$ | $1.15\times 10^{-4}$
405 | $6.15\times 10^{0}$ | $5.49\times 10^{-3}$ | $2.72\times 10^{-5}$
Order | $2.06\pm 0.08$ | $2.87\pm 0.24$ | $3.99\pm 0.03$
##### Three dimensional unstructured hexahedral mesh.
As a further test a domain of dimension $[0,1]^{3}$ was considered. This
domain was meshed using completely unstructured hexahedra. Three levels of
refinement were used resulting in meshes with 96, 536 and 1004 elements. A
cutaway of the most refined mesh can be seen in Figure 11. Experimental
$L^{2}$ errors and the resulting orders of accuracy are presented in Table 4.
Despite the fully unstructured nature of the mesh the expected order of
accuracy was again obtained in all cases. We do, however, note the higher
standard errors associated with these results.
Figure 11: Cutaway of the unstructured hexahedral mesh with 1004 elements. Table 4: $L^{2}$ energy errors and orders of accuracy for the Couette flow problem on three unstructured hexahedral meshes. Mesh spacing was taken as $h\sim N^{-1/3}_{E}$ where $N_{E}$ is the total number of elements in the mesh. | $\sigma(t_{\infty})\,/\,$\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$$
---|---
Hexes | $\wp=1$ | $\wp=2$ | $\wp=3$
96 | $1.91\times 10^{1}$ | $4.32\times 10^{-2}$ | $5.83\times 10^{-4}$
536 | $8.20\times 10^{0}$ | $9.11\times 10^{-3}$ | $6.89\times 10^{-5}$
1004 | $3.82\times 10^{0}$ | $3.22\times 10^{-3}$ | $2.04\times 10^{-5}$
Order | $1.93\pm 0.46$ | $3.19\pm 0.48$ | $4.16\pm 0.44$
### 5.3 Compressible Navier-Stokes Equations: Flow Over a Cylinder
In order to demonstrate the ability of PyFR to solve the unsteady Navier-
Stokes equations flow over a cylinder at Reynolds number 3900 and Mach number
$M=0.2$ was simulated. A cylinder of radius $1/2$ was placed at $(0,0)$ inside
of a domain of dimension $[-18,30]\times[-10,10]\times[0,3.2]$. This domain
was then meshed in the $x$-$y$ plane with 4661 quadratically curved
quadrilateral elements. Next, this grid was extruded along the $z$-axis to
yield a total of 46610 hexahedra. The grid, which can be seen in Figure 12,
was partitioned into four pieces. Along surfaces of $y=\pm 10$ and $x=-18$ the
inflow boundary condition of subsection C.2 in Appendix C was imposed. Along
the surface of $x=30$ the outflow condition of subsection C.3 in Appendix C
was used. Periodic conditions were imposed in the $z$ direction. On the
surface of the cylinder the no-slip isothermal wall condition of subsection
C.4 in Appendix C was imposed. The free-stream conditions were taken to be
$\rho=1$, $\mathbf{v}=\hat{\mathbf{x}}$, and $p=1/\gamma M^{2}$. These were
also used as the initial conditions for the simulation. DG correction
functions were used with the LDG parameters being $\beta=1/2$ and $\tau=0.1$.
The ratio of specific heats was taken as $\gamma=1.4$ and the Prandtl number
as $P_{r}=0.72$.
Figure 12: Cross section in the $x$-$y$ plane of the cylinder mesh. Colours
indicate the partition to which the elements belong.
The simulation was run with $\wp=4$ with four NVIDIA K20c GPUs. It contained
some $29\times 10^{6}$ degrees of freedom. Isosurfaces of density captured
after the turbulent wake had fully developed can be seen in Figure 13.
Figure 13: Isosurfaces of density around the cylinder.
## 6 Single Node Performance
The single node performance of PyFR has been evaluated on an NVIDIA M2090 GPU.
This accelerator has a theoretical peak double precision floating point
performance of $665\,\text{GFLOP/s}$, and when ECC is disabled the theoretical
peak memory bandwidth is $177\,\text{GB/s}$. As points of reference we observe
that cuBLAS (CUDA 5.5) is able to obtain $407\,\text{GFLOP/s}$ when
multiplying a pair of $4096\times 4096$ matrices on this hardware, and the
maximum device bandwidth obtainable by the bandwidth test application included
with the CUDA SDK is $138.9\,\text{GiB/s}$ when ECC is disabled. We shall
refer to these values as _realisable peaks_.
To conduct the evaluation a fully periodic cuboidal domain was meshed with
$50\,176$ hexahedral elements. The double precision Navier-Stokes solver of
PyFR was then run on this mesh at orders $\wp=2,3,4$ with $\beta=1/2$. In
conducting the analysis kernels were grouped into one of three categories:
matrix multiplications (DGEMM), point-wise kernels with direct memory access
patterns (PD) and point-wise kernels with some level of indirect memory access
(PI). Indirection arises in the computation of $\mathfrak{C}_{\alpha}$ in
Equation 7 and $\mathfrak{F}_{\alpha}$ in Equation 15 and occurs as a
consequence of the unstructured nature of PyFR. The resulting breakdowns of
wall-clock time, memory bandwidth and floating point operations can be seen in
Table 5. It is clear that he majority of floating point operations are
concentrated inside the calls to DGEMM with the point-wise operations are
heavily memory bandwidth bound. Of this bandwidth some ${\sim}15\%$ was
ascribed to register spillage above and beyond that which can be absorbed by
the L1 cache.
Table 5: Single GPU performance of PyFR for the Navier-Stokes equations when run on an NVIDIA M2090 with ECC disabled. As the memory bandwidth requirements of DGEMM are dependent on the accumulation strategy adopted by the implementation these values have been omitted. | | Order
---|---|---
| | $\wp=2$ | $\wp=3$ | $\wp=4$
Wall time / % | | | |
| DGEMM | $55.7$ | $66.2$ | $81.4$
| PD | $24.9$ | $21.5$ | $12.8$
| PI | $19.4$ | $12.3$ | $5.8$
Bandwidth / GiB/s | | | |
| PD | $125.5$ | $125.0$ | $124.8$
| PI | $124.8$ | $124.3$ | $124.2$
Arithmetic / GFLOP/s | | | |
| DGEMM | $205.3$ | $368.1$ | $305.4$
| PD | $0.7$ | $0.7$ | $0.7$
| PI | $0.9$ | $0.8$ | $0.9$
The high fraction of peak bandwidth obtained by the indirect kernels can be
attributed to three factors. Firstly, the constant data required for
calculations at ????, such as $\hat{\mathbf{n}}^{(f)}_{e\sigma n}$ and
$J^{(f)}_{e\sigma n}n^{(f)}_{e\sigma n}$, is ordered to ensure direct
(coalesced) access. Secondly, at start-up PyFR attempts to determine an
iteration ordering over the various flux-point pairs that will minimise the
number of cache misses.
Many of the memory accesses are therefore are near-coalesced. Thirdly and
finally we highlight the impressive latency-hiding capabilities of the CUDA
programming model.
In line with expectations the proportion of time spent performing matrix-
matrix multiplications increases as a function of order. When going from
$\wp=2$ to $\wp=3$ a significant portion of the additional compute is offset
by the improved performance of cuBLAS. However, when $\wp=4$ the performance
of these kernels in absolute terms can be seen to regress slightly. This
contributes to the greatly increased fraction of wall-clock time spent inside
of these kernels. Nevertheless, the achieved rate of $305.4\text{GFLOP/s}$ is
still over $75\%$ of the realisable peak. Also in line with expectations is
the invariance of the arithmetic performance of the point-wise kernels with
respect to order. As the order is varied all that changes is the number of
points to be processed with the operation itself remaining identical.
## 7 Scalability
The scalability of PyFR has been evaluated on the Emerald GPU cluster. It is
housed at the STFC Rutherford Appleton Laboratory and based around 60 HP SL390
nodes with three NVIDIA M2090 GPUs and 24 HP SL390 nodes with eight NVIDIA
M2090 GPUs. Nodes are connected by QDR InfiniBand.
For simplicity all runs herein were performed on the eight GPU nodes. As a
starting point a domain of dimension $[-16,16]\times[-16,16]\times[0,1.75]$
was meshed isotropically with $N_{E}=114\,688$ structured hexahedral elements.
The mesh was configured with completely periodic boundary conditions. When run
with the Navier-Stokes solver in PyFR with $\wp=3$ the mesh gives a working
set of ${\sim}4720\,\text{MiB}$. This is sufficient to 90% load an M2090 which
when ECC is enabled has ${\sim}5250\,\text{MiB}$ memory available to the user.
When examining the scalability of a code there are two commonly used metrics.
The first of these is weak scalability in which the size of the target problem
is increased in proportion to the number of ranks $N$ with $N_{E}\propto N$.
For a code with perfect weak scalability the runtime should remain unchanged
as more ranks are added. The second metric is strong scalability wherein the
problem size is fixed and the speedup compared to a single rank is assessed.
Perfect strong scalability implies that the runtime scales as $1/N$.
For the domain outlined above weak scalability was evaluated by increasing the
dimensions of the domain according to $[-16,16]\times
N[-16,16]\times[0,1.75]$. This extension permitted the domain to be trivially
decomposed along the $y$-axis. The resulting runtimes for $1\leq N\leq 104$
can be seen in Table 6. We note that in the $N=104$ case that the simulation
consisted of some $3.8\times 10^{9}$ degrees of freedom with a working set of
${\sim}485\,\text{GiB}$.
Table 6: Weak scalability of PyFR for the Navier-Stokes equations with $\wp=3$. Runtime is normalised with respect to a single NVIDIA M2090 GPU. # M2090s | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 104
---|---|---|---|---|---|---|---|---
Runtime | 1.00 | 1.00 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01
To study the strong scalability the initial domain was partitioned along the
$x$\- and $y$-axes. Each partition contained exactly $N_{E}/N$s. The resulting
speedups for $1\leq N\leq 32$ can be seen in Table 7. Up to eight GPUs
scalability can be seen to be near perfect. Beyond this the relationship
begins to break down. When $N=32$ an improvement of 26 can be observed.
However, in this case each GPU is loaded to less than 3% and so the result is
to be expected.
Table 7: Strong scalability of PyFR for the Navier-Stokes equations with $\wp=3$. The speedup is relative to a single NVIDIA M2090 GPU. # M2090s | 1 | 2 | 4 | 8 | 16 | 32
---|---|---|---|---|---|---
Speedup | 1.00 | 2.03 | 3.96 | 7.48 | 14.07 | 26.18
## 8 Conclusions
In this paper we have described PyFR, an open source Python based framework
for solving advection-diffusion type problems on streaming architectures. The
structure and ethos of PyFR has been explained including our methodology for
targeting multiple hardware platforms. We have shown that PyFR exhibits
spatial super accuracy when solving the 2D Euler equations and the expected
order of accuracy when solving Couette flow problem on a range of grids in 2D
and 3D. Qualitative results for unsteady 3D viscous flow problems on curved
grids have also been presented. Performance of PyFR has been validated on an
NVIDIA M2090 GPU in three dimensions. It has been shown that the compute bound
kernels are able to obtain between $50\%$ and $90\%$ of realisable peak FLOP/s
whereas the bandwidth bound point-wise kernels are, across the board, able to
obtain in excess of $89\%$ realisable peak bandwidth. The scalability of PyFR
has been demonstrated in the strong sense up to 32 NVIDIA M2090s and in the
weak sense up to 104 NVIDIA M2090s when solving the 3D Navier-Stokes
equations.
## Acknowledgements
The authors would like to thank the Engineering and Physical Sciences Research
Council for their support via two Doctoral Training Grants and an Early Career
Fellowship (EP/K027379/1). The authors would also like to thank the
e-Infrastructure South Centre for Innovation for granting access to the
Emerald supercomputer, and NVIDIA for donation of three K20c GPUs.
## Appendix A Matrix Representation
It is possible to cast the majority of operations in an FR step as matrix-
matrix multiplications of the form
$\bm{\mathsf{C}}\leftarrow
c_{1}\bm{\mathsf{A}}\bm{\mathsf{B}}+c_{2}\bm{\mathsf{C}},$ (40)
where $c_{1,2}\in\mathbb{R}$ are constants, $\bm{\mathsf{A}}$ is a constant
operator matrix, and $\bm{\mathsf{B}}$ and $\bm{\mathsf{C}}$ are state
matrices. To accomplish this we start by introducing the following constant
operator matrix
$\displaystyle\big{(}\bm{\mathsf{M}}^{0}_{e}\big{)}_{\sigma\rho}$
$\displaystyle=\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma}),$
$\displaystyle\dim\bm{\mathsf{M}}^{0}_{e}$ $\displaystyle=N_{e}^{(f)}\times
N_{e}^{(u)},$
and the following state matrices
$\displaystyle\big{(}\bm{\mathsf{U}}^{(u)}_{e}\big{)}_{\rho(n\alpha)}$
$\displaystyle=u^{(u)}_{e\rho n\alpha},$
$\displaystyle\dim\bm{\mathsf{U}}^{(u)}_{e}$ $\displaystyle=N_{e}^{(u)}\times
N_{V}|\mathbf{\Omega}_{e}|,$
$\displaystyle\big{(}\bm{\mathsf{U}}^{(f)}_{e}\big{)}_{\sigma(n\alpha)}$
$\displaystyle=u^{(f)}_{e\sigma n\alpha},$
$\displaystyle\dim\bm{\mathsf{U}}^{(f)}_{e}$ $\displaystyle=N_{e}^{(f)}\times
N_{V}|\mathbf{\Omega}_{e}|.$
In specifying the state matrices there is a degree of freedom associated with
how the $N_{V}$ field variables for each element are packed along a row of the
matrix, with the possible packing choices being discussed in subsection 3.3.
Using these matrices we are able to reformulate Equation 6 as
$\bm{\mathsf{U}}^{(f)}_{e}=\bm{\mathsf{M}}^{0}_{e}\bm{\mathsf{U}}^{(u)}_{e}.$
(41)
In order to apply a similar procedure to Equation 9 we let
$\displaystyle\big{(}\bm{\mathsf{M}}^{4}_{e}\big{)}_{\rho\sigma}$
$\displaystyle=\big{[}\tilde{\bm{\nabla}}\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}})\big{]}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(u)}_{e\sigma}},$
$\displaystyle\dim\bm{\mathsf{M}}^{4}_{e},$
$\displaystyle=N_{D}N_{e}^{(u)}\times N_{e}^{(u)},$
$\displaystyle\big{(}\bm{\mathsf{M}}^{6}_{e}\big{)}_{\rho\sigma}$
$\displaystyle=\big{[}\hat{\tilde{\mathbf{n}}}^{(f)}_{e\rho}\cdot\tilde{\bm{\nabla}}\cdot\mathbf{g}^{(f)}_{e\rho}(\tilde{\mathbf{x}})\big{]}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(f)}_{e\sigma}},$
$\displaystyle\dim\bm{\mathsf{M}}^{6}_{e},$
$\displaystyle=N_{D}N_{e}^{(u)}\times N_{e}^{f},$
$\displaystyle\big{(}\bm{\mathsf{C}}^{(f)}_{e}\big{)}_{\rho(n\alpha)}$
$\displaystyle=\mathfrak{C}_{\alpha}u^{(f)}_{e\rho n\alpha},$
$\displaystyle\dim\bm{\mathsf{C}}^{(f)}_{e}$ $\displaystyle=N^{(f)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
$\displaystyle\big{(}\tilde{\bm{\mathsf{Q}}}^{(u)}_{e}\big{)}_{\sigma(n\alpha)}$
$\displaystyle=\tilde{\mathbf{q}}^{(u)}_{e\sigma n\alpha},$
$\displaystyle\dim\tilde{\bm{\mathsf{Q}}}^{(u)}_{e}$
$\displaystyle=N_{D}N^{(u)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
Here it is important to qualify assignments of the form
$\bm{\mathsf{A}}_{ij}=\mathbf{x}$ where $\mathbf{x}$ is a $N_{D}$ component
vector. As above there is a degree of freedom associated with the packing.
With the benefit of foresight we take the stride between subsequent elements
of $\mathbf{x}$ in a matrix column to be either $\Delta i=N^{(u)}_{e}$ or
$\Delta i=N^{(f)}_{e}$ depending on the context. With these matrices Equation
9 reduces to
$\displaystyle\tilde{\bm{\mathsf{Q}}}^{(u)}_{e}$
$\displaystyle=\bm{\mathsf{M}}^{6}_{e}\big{\\{}\bm{\mathsf{C}}^{(f)}_{e}-\bm{\mathsf{U}}^{(f)}_{e}\big{\\}}+\bm{\mathsf{M}}^{4}_{e}\bm{\mathsf{U}}^{(u)}_{e}$
(42)
$\displaystyle=\bm{\mathsf{M}}^{6}_{e}\big{\\{}\bm{\mathsf{C}}^{(f)}_{e}-\bm{\mathsf{M}}^{0}_{e}\bm{\mathsf{U}}^{(u)}_{e}\big{\\}}+\bm{\mathsf{M}}^{4}_{e}\bm{\mathsf{U}}^{(u)}_{e}$
$\displaystyle=\bm{\mathsf{M}}^{6}_{e}\bm{\mathsf{C}}^{(f)}_{e}+\big{\\{}\bm{\mathsf{M}}^{4}_{e}-\bm{\mathsf{M}}^{6}_{e}\bm{\mathsf{M}}^{0}_{e}\big{\\}}\bm{\mathsf{U}}^{(u)}_{e}.$
Applying the procedure to Equation 11 we take
$\displaystyle\bm{\mathsf{M}}^{5}_{e}$
$\displaystyle=\operatorname{diag}(\bm{\mathsf{M}}^{0}_{e},\ldots,\bm{\mathsf{M}}^{0}_{e})$
$\displaystyle\dim{\bm{\mathsf{M}}^{5}_{e}}$
$\displaystyle=N_{D}N^{(f)}_{e}\times N_{D}N^{(u)}_{e},$
$\displaystyle\big{(}\bm{\mathsf{Q}}^{(u)}_{e}\big{)}_{\sigma(n\alpha)}$
$\displaystyle=\mathbf{q}^{(u)}_{e\sigma n\alpha},$
$\displaystyle\dim{\bm{\mathsf{Q}}^{(u)}_{e}}$
$\displaystyle=N_{D}N^{(u)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
$\displaystyle\big{(}\bm{\mathsf{Q}}^{(f)}_{e}\big{)}_{\sigma(n\alpha)}$
$\displaystyle=\mathbf{q}^{(f)}_{e\sigma n\alpha},$
$\displaystyle\dim{\bm{\mathsf{Q}}^{(f)}_{e}}$
$\displaystyle=N_{D}N^{(f)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
hence
$\bm{\mathsf{Q}}^{(f)}_{e}=\bm{\mathsf{M}}^{5}_{e}\bm{\mathsf{Q}}^{(u)}_{e},$
(43)
where we note the block diagonal structure of $\bm{\mathsf{M}}^{5}_{e}$. This
is a direct consequence of the above choices for $\Delta i$. Finally, to
rewrite Equation 18 we write
$\displaystyle\big{(}\bm{\mathsf{M}}^{1}_{e}\big{)}_{\rho\sigma}$
$\displaystyle=\big{[}\tilde{\bm{\nabla}}\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}})\big{]}^{T}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(u)}_{e\sigma}},$
$\displaystyle\dim\bm{\mathsf{M}}^{1}_{e}$ $\displaystyle=N^{(u)}_{e}\times
N_{D}N^{(u)}_{e},$
$\displaystyle\big{(}\bm{\mathsf{M}}^{2}_{e}\big{)}_{\rho\sigma}$
$\displaystyle=\big{[}\ell^{(u)}_{e\rho}(\tilde{\mathbf{x}}^{(f)}_{e\sigma})\hat{\tilde{\mathbf{n}}}^{(f)}_{e\sigma}\big{]}^{T},$
$\displaystyle\dim\bm{\mathsf{M}}^{2}_{e}$ $\displaystyle=N^{(f)}_{e}\times
N_{D}N^{(u)}_{e},$
$\displaystyle\big{(}\bm{\mathsf{M}}^{3}_{e}\big{)}_{\rho\sigma}$
$\displaystyle=\big{[}\tilde{\bm{\nabla}}\cdot\mathbf{g}^{(f)}_{e\sigma}(\tilde{\mathbf{x}})\big{]}_{\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{(u)}_{e\rho}},$
$\displaystyle\dim\bm{\mathsf{M}}^{3}_{e}$ $\displaystyle=N^{(u)}_{e}\times
N^{(f)}_{e},$
$\displaystyle\big{(}\tilde{\bm{\mathsf{D}}}^{(f)}_{e}\big{)}_{\sigma(n\alpha)}$
$\displaystyle=\mathfrak{F}^{\vphantom{(f_{\perp})}}_{\alpha}\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha},$ $\displaystyle\dim\tilde{\bm{\mathsf{D}}}^{(f)}_{e}$
$\displaystyle=N^{(f)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
$\displaystyle\big{(}\tilde{\bm{\mathsf{F}}}^{(u)}_{e}\big{)}_{\rho(n\alpha)}$
$\displaystyle=\tilde{\mathbf{f}}^{(u)}_{e\rho n\alpha},$
$\displaystyle\dim\tilde{\bm{\mathsf{F}}}^{(u)}_{e}$
$\displaystyle=N_{D}N^{(u)}_{e}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
$\displaystyle\big{(}\tilde{\bm{\mathsf{R}}}^{(u)}_{e}\big{)}_{\rho(n\alpha)}$
$\displaystyle=(\tilde{\bm{\nabla}}\cdot\tilde{\mathbf{f}})^{(u)}_{e\rho
n\alpha},$ $\displaystyle\dim\tilde{\bm{\mathsf{R}}}^{(u)}_{e}$
$\displaystyle=N_{e}^{(u)}\times
N_{V}\left\lvert\mathbf{\Omega}_{e}\right\rvert,$
and after substitution of Equation 13 for $\tilde{f}^{(f_{\perp})}_{e\sigma
n\alpha}$ obtain
$\displaystyle\tilde{\bm{\mathsf{R}}}^{(u)}_{e}$
$\displaystyle=\bm{\mathsf{M}}^{3}_{e}\big{\\{}\tilde{\bm{\mathsf{D}}}^{(f)}_{e}-\bm{\mathsf{M}}^{2}_{e}\tilde{\bm{\mathsf{F}}}^{(u)}_{e}\big{\\}}+\bm{\mathsf{M}}^{1}_{e}\tilde{\bm{\mathsf{F}}}^{(u)}_{e}$
(44)
$\displaystyle=\bm{\mathsf{M}}^{3}_{e}\tilde{\bm{\mathsf{D}}}^{(f)}_{e}+\big{\\{}\bm{\mathsf{M}}^{1}_{e}-\bm{\mathsf{M}}^{3}_{e}\bm{\mathsf{M}}^{2}_{e}\big{\\}}\tilde{\bm{\mathsf{F}}}^{(u)}_{e}.$
## Appendix B Approximate Riemann Solvers
### B.1 Overview
In the following section we take $u_{L}$ and $u_{R}$ to be the two
discontinuous solution states at an interface and $\hat{\mathbf{n}}_{L}$ to be
the normal vector associated with the first state. For convenience we take
$\mathbf{f}^{(\text{inv})}_{L}=\mathbf{f}^{(\text{inv})}_{\vphantom{L}}(u^{\vphantom{(\text{inv})}}_{L})$,
and
$\mathbf{f}^{(\text{inv})}_{R}=\mathbf{f}^{(\text{inv})}_{\vphantom{R}}(u^{\vphantom{(\text{inv})}}_{R})$
with inviscid fluxes being prescribed by Equation 20.
### B.2 Rusanov
Also known as the local Lax-Friedrichs method a Rusanov type Riemann solver
imposes inviscid numerical interface fluxes according to
$\mathfrak{F}^{(\text{inv})}=\frac{\hat{\mathbf{n}}_{L}}{2}\cdot\left\\{\mathbf{f}^{(\text{inv})}_{L}+\mathbf{f}^{(\text{inv})}_{R}\right\\}+\frac{s}{2}(u_{L}-u_{R}),$
(45)
where $s$ is an estimate of the maximum wave speed
$s=\sqrt{\frac{\gamma(p_{L}+p_{R})}{\rho_{L}+\rho_{R}}}+\frac{1}{2}\big{|}\hat{\mathbf{n}}_{L}\cdot(\mathbf{v}_{L}+\mathbf{v}_{R})\big{|}.$
(46)
## Appendix C Boundary Conditions
### C.1 Overview
To incorporate boundary conditions into the FR approach we introduce a set of
boundary interface types $b\in\mathcal{B}$. At a boundary interface there is
only a single flux point: that which belongs to the element whose edge/face is
on the boundary. Associated with each boundary type are a pair of functions
$\mathfrak{C}^{(b)}_{\alpha}(u_{L})$ and
$\mathfrak{F}^{(b)}_{\alpha}(u_{L},\mathbf{q}_{L},\hat{\mathbf{n}}_{L})$ where
$u_{L}$, $\mathbf{q}_{L}$, and $\hat{\mathbf{n}}_{L}$ are the solution,
solution gradient and unit normals at the relevant flux point. These functions
prescribe the common solutions and normal fluxes, respectively.
Instead of directly imposing solutions and normal fluxes it is oftentimes more
convenient for a boundary to instead provide ghost states. In its simplest
formulation
$\mathfrak{C}^{(b)}_{\alpha}=\mathfrak{C}_{\alpha}(u_{L},\mathfrak{B}^{(b)}u_{L})$
and
$\mathfrak{F}^{(b)}_{\alpha}=\mathfrak{F}_{\alpha}(u_{L},\mathfrak{B}^{(b)}u_{L},\mathbf{q}_{L},\mathfrak{B}^{(b)}\mathbf{q}_{L},\hat{\mathbf{n}}_{L})$
where $\mathfrak{B}^{(b)}u_{L}$ is the ghost solution state and
$\mathfrak{B}^{(b)}\mathbf{q}_{L}$ is the ghost solution gradient. It is
straightforward to extend this prescription to allow for the provisioning of
different ghost solution states for $\mathfrak{C}_{\alpha}$ and
$\mathfrak{F}_{\alpha}$ and to permit $\mathfrak{B}^{(b)}\mathbf{q}_{L}$ to be
a function of $u_{L}$ in addition to $\mathbf{q}_{L}$.
### C.2 Supersonic Inflow
The supersonic inflow condition is parameterised by a free-stream density
$\rho_{f}$, velocity $\mathbf{v}_{f}$, and pressure $p_{f}$.
$\displaystyle\mathcal{B}^{(\text{inv})}u_{L}=\mathcal{B}^{(\text{ldg})}u_{L}$
$\displaystyle=\begin{Bmatrix}\rho_{f}\\\ \rho_{f}\mathbf{v}_{f}\\\
p_{f}/(\gamma-1)+\frac{\rho_{f}}{2}\|\mathbf{v}_{f}\|^{2}\end{Bmatrix},$ (47)
$\displaystyle\mathcal{B}^{(\text{ldg})}\mathbf{q}_{L}$ $\displaystyle=0,$
(48)
### C.3 Subsonic Outflow
Subsonic outflow boundaries are parameterised by a free-stream pressure
$p_{f}$.
$\displaystyle\mathcal{B}^{(\text{inv})}u_{L}=\mathcal{B}^{(\text{ldg})}u_{L}$
$\displaystyle=\begin{Bmatrix}\rho_{L}\\\ \rho_{L}\mathbf{v}_{L}\\\
p_{f}/(\gamma-1)+\frac{\rho_{L}}{2}\|\mathbf{v}_{L}\|^{2}\end{Bmatrix},$ (49)
$\displaystyle\mathcal{B}^{(\text{ldg})}\mathbf{q}_{L}$ $\displaystyle=0,$
(50)
### C.4 No-slip Isothermal Wall
The no-slip isothermal wall condition depends on the wall temperature
$C_{p}T_{w}$ and the wall velocity $\mathbf{v}_{w}$. Usually
$\mathbf{v}_{w}=0$.
$\displaystyle\mathcal{B}^{(\text{inv})}u_{L}$
$\displaystyle=\rho_{L}\begin{Bmatrix}1\\\ 2\mathbf{v}_{w}-\mathbf{v}_{L}\\\
C_{p}T_{w}/\gamma+\frac{1}{2}\left\lVert
2\mathbf{v}_{w}-\mathbf{v}_{L}\right\rVert^{2}\end{Bmatrix},$ (51)
$\displaystyle\mathcal{B}^{(\text{ldg})}u_{L}$
$\displaystyle=\rho_{L}\begin{Bmatrix}1\\\ \mathbf{v}_{w}\\\
C_{p}T_{w}/\gamma+\frac{1}{2}\left\lVert\mathbf{v}_{w}\right\rVert^{2}\end{Bmatrix},$
(52) $\displaystyle\mathcal{B}^{(\text{ldg})}\mathbf{q}_{L}$
$\displaystyle=\mathbf{q}_{L},$ (53)
## References
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* [2] David A Kopriva and John H Kolias. A conservative staggered-grid Chebyshev multidomain method for compressible flows. Journal of computational physics, 125(1):244–261, 1996.
* [3] Yuzhi Sun, Zhi Jian Wang, and Yen Liu. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Communications in Computational Physics, 2(2):310–333, 2007.
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* [7] P Castonguay, PE Vincent, and A Jameson. A new class of high-order energy stable flux reconstruction schemes for triangular elements. Journal of Scientific Computing, 2011.
* [8] Patrice Castonguay, PE Vincent, and Antony Jameson. Application of high-order energy stable flux reconstruction schemes to the Euler equations. In 49th AIAA Aerospace Sciences Meeting, volume 686, 2011.
* [9] A Jameson, PE Vincent, and P Castonguay. On the non-linear stability of flux reconstruction schemes. Journal of Scientific Computing, 50(2):434–445, 2011.
* [10] PE Vincent, P Castonguay, and A Jameson. Insights from von Neumann analysis of high-order flux reconstruction schemes. Journal of Computational Physics, 230(22):8134–8154, 2011.
* [11] Patrice Castonguay, Peter E Vincent, and Antony Jameson. A new class of high-order energy stable flux reconstruction schemes for triangular elements. Journal of Scientific Computing, 51(1):224–256, 2012.
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|
arxiv-papers
| 2013-12-05T18:39:40 |
2024-09-04T02:49:54.997223
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Freddie D Witherden and Antony M Farrington and Peter E Vincent",
"submitter": "Freddie Witherden",
"url": "https://arxiv.org/abs/1312.1638"
}
|
1312.1643
|
Charles University in Prague
Faculty of Mathematics and Physics
DOCTORAL THESIS
Ivana Ebrová
Shell galaxies:
kinematical signature of shells, satellite galaxy disruption and dynamical
friction
Astronomical Institute
of the Academy of Sciences of the Czech Republic
Supervisor of the doctoral thesis: RNDr. Bruno Jungwiert, Ph.D.
Study program: Physics
Specialization: Theoretical Physics, Astronomy and Astrophysics
Prague 2013
This research has made use of NASA’s Astrophysics Data System, micronised
purified flavonoid fraction, and a lot of iso-butyl-propanoic-phenolic acid.
Typeset in LYX, an open source document processor. For graphical presentation,
we used Gnuplot, the PGPLOT (a graphics subroutine library written by Tim
Pearson) and scripts and programs written by Miroslav Křížek using Python and
matplotlib. Calculations and simulations have been carried out using Maple 10,
Wolfram Mathematica 7.0, and own software written in programming language
FORTRAN 77, Fortran 90 and Fortran 95. The software for simulation of shell
galaxy formation using test particles are based on the source code of the
MERGE 9 (written by Bruno Jungwiert, 2006; unpublished); kinematics of shell
galaxies in the framework of the model of radial oscillations has been studied
using the smove software (written by Lucie Jílková, 2011; unpublished); self-
consistent simulations have been done by Kateřina Bartošková with GADGET-2
(Springel, 2005).
We acknowledge support from the following sources: grant No. 205/08/H005 by
Czech Science Foundation; research plan AV0Z10030501 by Academy of Sciences of
the Czech Republic; and the project SVV-267301 by Charles University in
Prague. This work has been done with the support for a long-term development
of the research institution RVO67985815.
I declare that I carried out this doctoral thesis independently, and only with
the cited sources, literature and other professional sources.
I understand that my work relates to the rights and obligations under the Act
No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that
the Charles University in Prague has the right to conclude a license agreement
on the use of this work as a school work pursuant to Section 60 paragraph 1 of
the Copyright Act.
In Prague, 19. 8. 2013Ivana Ebrová
Title: Shell galaxies: kinematical signature of shells, satellite galaxy
disruption and dynamical friction
Author: Ivana Ebrová
Department / Institute: Astronomical Institute of the Academy of Sciences of
the Czech Republic
Supervisor of the doctoral thesis: RNDr. Bruno Jungwiert, Ph.D., Astronomical
Institute of the Academy of Sciences of the Czech Republic
Abstract: Stellar shells observed in many giant elliptical and lenticular as
well as a few spiral and dwarf galaxies presumably result from radial minor
mergers of galaxies. We show that the line-of-sight velocity distribution of
the shells has a quadruple-peaked shape. We found simple analytical
expressions that connect the positions of the four peaks of the line profile
with the mass distribution of the galaxy, namely, the circular velocity at the
given shell radius and the propagation velocity of the shell. The analytical
expressions were applied to a test-particle simulation of a radial minor
merger, and the potential of the simulated host galaxy was successfully
recovered. Shell kinematics can thus become an independent tool to determine
the content and distribution of dark matter in shell galaxies up to
$\sim$$100$ kpc from the center of the host galaxy. Moreover we investigate
the dynamical friction and gradual disruption of the cannibalized galaxy
during the shell formation in the framework of a simulation with test
particles. The coupling of both effects can considerably redistribute
positions and luminosities of shells. Neglecting them can lead to significant
errors in attempts to date the merger in observed shell galaxies.
Keywords: galaxies: kinematics and dynamics, galaxies: interactions, galaxies:
evolution, methods: analytical and numerical
###### ?contentsname?
1. 1 Objectives and motivation
2. I Introduction
1. 2 Shell galaxies in brief
2. 3 Observational knowledge of shell galaxies
1. 3.1 Observational history
2. 3.2 Occurrence of shell galaxies
3. 3.3 Appearance of the shells
4. 3.4 Colors
5. 3.5 Gas and dust
6. 3.6 Radio and infrared emission
7. 3.7 Other features of host galaxies
3. 4 Summary of shell characteristics
4. 5 Scenarios of shells’ origin
1. 5.1 Gas dynamical theories
2. 5.2 Weak Interaction Model (WIM)
5. 6 Merger model
1. 6.1 Phase wrapping
2. 6.2 Cannibalized galaxy
3. 6.3 Ellipticity of the host galaxy
4. 6.4 Radial distribution of shells
5. 6.5 Radiality of the merger
6. 6.6 Major mergers
7. 6.7 Simulations with gas
8. 6.8 Merger model and observations
6. 7 Measurements of gravitational potential in galaxies
1. 7.1 Insight into methods
2. 7.2 Use of shells
3. II Shell kinematics
1. 8 Preliminary provisions
1. 8.1 Host galaxy potential model
2. 8.2 Terminology
3. 8.3 Quantities
2. 9 Model of radial oscillations
1. 9.1 Turning point positions and their velocities
2. 9.2 Real shell positions and velocities
3. 9.3 Appearance of the shells
4. 9.4 Kinematics of shell stars
5. 9.5 Characteristics of spectral peaks
6. 9.6 Equations of LOSVD
7. 9.7 Shell-edge density distribution and LOSVD
8. 9.8 Nature of the quadruple-peaked profile
3. 10 Stationary shell
1. 10.1 Motion of stars in a shell system
2. 10.2 Constant acceleration
3. 10.3 LOSVD
4. 10.4 Comparison with the model of radial oscillations
4. 11 Constant acceleration and shell velocity
1. 11.1 Motion of a star in a shell system
2. 11.2 Approximative LOSVD
3. 11.3 Radius of maximal LOS velocity
4. 11.4 Approximative maximal LOS velocity
5. 11.5 Slope of the LOSVD intensity maxima
6. 11.6 Comparison of approaches
7. 11.7 Projection factor approximation
5. 12 Higher order approximation
1. 12.1 Motion of a star in a shell system
2. 12.2 Comparison of approximations
3. 12.3 $\boldsymbol{a{}_{1}}$
6. 13 Test-particle simulation
1. 13.1 Parameters of the simulation
2. 13.2 Comparison of the simulation with models
3. 13.3 Recovering the potential from the simulated data
4. 13.4 Notes about observation
7. 14 Shell density
1. 14.1 Projected surface density of the shell edge
2. 14.2 Time evolution
3. 14.3 Volume density
4. 14.4 Projected surface density
8. 15 Discussion
4. III Dynamical friction and gradual disruption
1. 16 Motivation
2. 17 Description of simulation
1. 17.1 Configuration
2. 17.2 Plummer sphere
3. 17.3 Velocity dispersion in Plummer potential
4. 17.4 Velocity dispersion in a double Plummer sphere
5. 17.5 Standard set of parameters
3. 18 Dynamical friction
4. 19 Multiple Three-Body Algorithm (MTBA)
1. 19.1 Principle and characteristics
2. 19.2 Merger parameters
3. 19.3 Results of simulations
5. 20 Comparison with self-consistent simulations
1. 20.1 Altering GADGET-2 computational setting
2. 20.2 Comparison of methods
6. 21 Tidal disruption
1. 21.1 Massloss of the secondary
2. 21.2 Deformation of the secondary galaxy
7. 22 Simulations of shell structure
1. 22.1 Dynamical friction and tidal disruption
2. 22.2 Dark halo
3. 22.3 Self-consistent versus test-particle simulations
8. 23 Discussion
5. IV Conclusions
6. V Appendix
1. A Units and conversions
2. B List of abbreviations
3. C Initial velocity distribution
4. D Introduction to dynamical friction
1. D.1 A thermodynamic meditation
2. D.2 Chandrasekhar formula
3. D.3 What a wonderful universe
4. D.4 Why does it work?
5. E Our method
1. E.1 Avoiding some approximations
2. E.2 Back to Chandrasekhar formula
3. E.3 Incorporation of the friction in the simulation
6. F Tidal radius
7. G Expressions for the tidal radius
8. H Videos
### 1 Objectives and motivation
The most successful theory of the evolution of the Universe so far seems to be
the theory of the hierarchical formation based on the assumption of the
existence of cold dark matter, significantly dominating the baryonic one. In
such a universe, large galaxies are formed by merging of small galaxies,
protogalaxies and diffuse accretion of surrounding matter. Galactic
interaction and dark matter play thus a crucial role in the life of every
galaxy.
But the determination of both the dark matter content and the merger history
of a galaxy is difficult. Firstly, the cold dark matter interacts only
gravitationally (and possibly via the weak interaction) and thus the mapping
of its distribution in galaxies is tricky. Secondly, the nature disallows us
to see individual galaxies from different angles, thus our knowledge of their
spatial properties is degenerate. Thirdly, it is non-trivial to determine
anything about the history of a given galaxy as the whole existence of
humanity presents only a snapshot in the evolution of the Universe. Yet this
knowledge is important to confirm or disprove theories of the creation and
evolution of the Universe, improve their accuracy and to understand how the
Universe we live in actually looks.
The deal of the galactic astronomy is to try to circumvent these obstacles.
One of the possibilities is to use tidal features left by the galactic
interactions. They act as dynamical tracers of the potential of their host
galaxies and as hints left behind by the accreted galaxies in the past. The
special case is that of arc-like fine structures found in shell galaxies.
Their unique kinematics carries both qualitative and quantitative information
on the distribution of the dark matter, the shape of the potential of the host
galaxy and its merger history. Moreover, shell galaxies have their own
mysteries that call for an explanation.
?figurename? 1: Shell galaxy M89.
Some shells need to be discovered using deep photometry, e.g., Duc et al.
(2011), whereas others can be today captured using amateur technology. The
photography of galaxy M89 in Fig. 1 was taken by a member of our research
group Michal Bílek using his own amateur equipment (taking 4.4 hours of
exposure with an 8", f/4 Schmidt-Newton telescope equipped with a CCD at a
site about 50 km from Prague). Faint structures were first identified by Malin
(1979) and Xu et al. (2005) who concluded that the galaxy possibly possesses a
low-luminosity active galactic nucleus. Michal’s image shows fairly well the
shell at bottom left, the jet at bottom right and a less prominent shell at
top right.
However all the information is hidden so deep in the structure and kinematics
of shell galaxies that it is not clear that they could be practically
unraveled. Certainly, a lot of effort and invention is required. In this work
we focus mainly on the possibility to deduce the potential of the host galaxy
using shell kinematics (Part II). We aim at creating equations and algorithms
applicable to observed data. Now comes the era when the instrumental equipment
begins to allow us to actually obtain such kind of data and that requires
deeper theoretical understanding of the topic. Having no such data yet at
hand, we apply our methods to simulated data. This method requires that the
shell is formed by stars on mainly radial orbits. According to present state
of knowledge, shells in one galaxy are probably bound by common origin in a
radial minor merger. Reproducing their overall structure is nevertheless
complicated by physical processes such as the dynamical friction and the
gradual decay of the cannibalized galaxy. We deal with these phenomena in Part
III.
Self-consistent simulations allow us to simulate many physical processes at
once. Some of them are difficult or outright impossible to reproduce by
analytical or semi-analytical methods. At the same time, the manifestation of
these processes in self-consistent simulations is difficult to separate and
sometimes they may even be confused with non-physical outcomes of used
methods. Moreover, self-consistent simulations with high resolution necessary
to analyze delicate tidal structures such as the shells are demanding on
computation time. This demand is even larger if we want to explore a
significant part of the parameter space.
Attempts to date a merger from observed positions of shells have been made in
previous works. Recently, Canalizo et al. (2007) presented HST/ACS
observations of spectacular shells in a quasar host galaxy (Fig. 3) and, by
simulating the position of the outermost shell by means of restricted $N$-body
simulations, attempted to put constraints on the age of the merger. They
concluded that it occurred a few hundred Myr to $\sim 2$ Gyr ago, supporting a
potential causal connection between the merger, the post-starburst ages in
nuclear stellar populations, and the quasar. A typical delay of 1–2.5 Gyr
between a merger and the onset of quasar activity is suggested by both
$N$-body simulations by Springel et al. (2005) and observations by Ryan et al.
(2008). It might therefore appear reassuring to find a similar time lag
between the merger event and the quasar ignition in a study of an individual
spectacular object. In Part III we explore the options for inclusion of the
dynamical friction and the gradual decay of the cannibalized galaxy in test-
particle simulations and we look at what these simulations tell us about the
potential and merger history of shell galaxies.
In Appendix A, we show the conversion of units used in the thesis to SI units.
List of abbreviations can be found in Appendix B. Videos mostly illustrating
the formation and evolution of shell structures are part of the electronic
attachment of the thesis. Their description can be found in Appendix H and the
videos can be downloaded at: galaxy.asu.cas.cz/$\sim$ivaana/phd
## ?partname? I Introduction
### 2 Shell galaxies in brief
Shell galaxies, like e.g. the beautiful and renowned NGC 3923 in Fig. 2, are
galaxies containing fine structures. These structures are made of stars and
form open, concentric arcs that do not cross each other. The term shells has
spread throughout the literature, gradually superseding the competing term
ripples. According to the knowledge gained over the past more than thirty
years, their origin lies in the interactions between galaxies.
?figurename? 2: NGC 3923 from Malin and Carter (1983) made from UK Shmidt
IIIa-J plates. The bottom row shows more central parts of the galaxy. All
images were processed (unsharp masking) to emphasize the shell structure. 10
′′ roughly corresponds to 1 kpc in the galaxy.
### 3 Observational knowledge of shell galaxies
This section is mostly based on the review of literature presented in Ebrová
(2007).
#### 3.1 Observational history
It was Halton Arp, who first noticed the shell galaxies in his Atlas of
Peculiar Galaxies (Arp, 1966a) and the accompanying article Arp (1966b). He
used the term “shells” to describe the structures associated with galaxy Arp
230. The Atlas contains 338 objects, divided into several subgroups. Shell
galaxies are found under “concentric rings” (Arp numbers 227 to 231), but many
other objects are in fact shell galaxies (Arp 92, 103, 104, 153–155, 171, 215,
223, 226 and probably others).
To date, the only (at least partial) list of shell galaxies is “A catalogue of
elliptical galaxies with shells” from Malin and Carter (1983). The authors
present a catalogue of 137 galaxies (with declination south of
$-17\text{\textdegree}$) that exhibit shell or ripple features at large
distances from the galaxy or in the outer envelope. Some further work has been
done on this set of galaxies: Wilkinson et al. (1987a, b) examined these shell
galaxies to find radio and infrared sources, Wilkinson et al. (1987c) carried
out two-color CCD photometry of 66 Malin-Carter galaxies, Carter et al. (1988)
obtained nuclear spectra for 100 of the galaxies in the catalogue. In a series
of articles, Longhetti et al. (1998a, b); Rampazzo et al. (1999); Longhetti et
al. (2000, 1999) (the fifth part surprisingly preceding the fourth) examined
star formation history in 21 catalogued shell galaxies. Forbes et al. (1994)
were searching for secondary nuclei in 29 shell galaxies. Larger samples of
shell galaxies were studied for example by Schweizer (1983), Thronson et al.
(1989), Forbes and Thomson (1992) or Colbert et al. (2001). Their results will
be mentioned in the following chapters. Unsurprisingly, many observational
studies have been carried out over decades for smaller samples or many
individual shell galaxies.
#### 3.2 Occurrence of shell galaxies
Originally (Arp, 1966b; Malin and Carter, 1983), shells were discovered
basically in galaxies of E, E/S0 or S0 morphological type. Schweizer and
Seitzer (1988) revealed that they can be found also in S0/Sa and Sa galaxies
(NGC 3032, NGC 3619, NGC 4382, NGC 5739, and a Seyfert galaxy NGC 5548) and
even one Sbc galaxy (NGC 3310) was found likely to contain a shell. In fact,
Schweizer and Seitzer were against the term “shells”, supporting the term
“ripples” being more descriptive and not forcing a particular geometric
interpretation. NGC 2782 (Arp 215) is probably a spiral galaxy with shells
which Arp misclassified as spiral arms rather than as shells. NGC 7531, NGC
3521, and NGC 4651 (Martínez-Delgado et al., 2010) are examples of some other
lesser known cases of spiral galaxies with shells. The last of them, NGC 4651
and also M31 (Fardal et al., 2007, 2012) are the only spiral galaxies where a
multiple shell system has been discovered. Coleman (2004) and Coleman et al.
(2004) reported a shell, immediately followed by another one (Coleman and Da
Costa, 2005; Coleman et al., 2005) in Fornax dwarf spheroidal galaxy and it
became the only shell galaxy of this type.
The realistic estimate of the relative abundance of shell galaxies (Schweizer,
1983; Schweizer and Ford, 1985; cited in Hernquist and Quinn, 1988, and Malin
and Carter, 1983) is about 10% in early-type galaxies.111We use the term
early-type galaxies to denote all the Hubble types E, E/S0, and S0 (elliptical
and lenticular galaxies), because many galaxies gradually wander between these
classes according to different classifications or simply in time (not
physically, of course, e.g. because of better or other observations). Malin
and Carter (1983) state a surface brightness detection limit
$\mu_{\mathrm{max}}=26.5$ mag$/$arcsec2 in B filter222It is interesting to
note that according to van Dokkum (2005), galaxy surveys in blue filters would
miss the majority of faint features in their sample even if they met the same
surface brightness limit.. Schweizer and Seitzer (1988) quoted similar results
for their sample of more than a hundred of galaxies, with the abundance of 6%
for S0 and 10% for E type galaxies, but with significantly lower number among
spirals (around 1%). Weil and Hernquist (1993a) state that Seitzer and
Schweizer (1990) found 56% and 32% of 74 E and S0 type galaxies respectively
posses ripples.
In a complete sample of 55 elliptical galaxies at distances 15–50 Mpc and
luminosity cut of M${}_{\mbox{B}}$$<-20$ with detection limit
$\mu_{\mathrm{max}}=27.7$ mag$/$arcsec2 in V band, at least 22% of galaxies
have shells, making them the most common interaction signature identified by
Tal et al. (2009). Shells are also the most commonly detected feature in a
sample of radio galaxies of Ramos Almeida et al. (2011) with
$\mu_{\mathrm{max}}\sim 26$ mag$/$arcsec2 in V filter.
On the contrary, in ATLAS3D sample of 260 early-type galaxies Krajnović et al.
(2011) found only 9 (3.5%) galaxies with shells at the limiting surface
brightness $\mu_{\mathrm{max}}\sim 26$ mag$/$arcsec2 in r band. Kim et al.
(2012) examined a sample of 65 early types drawn from the Spitzer Survey of
Stellar Structure in Galaxies (S4G) and identified 4 shell galaxies (6%).
Their detection limit was 25.2 mag$/$arcsec2 for newly obtained S4G data and
26.5 mag$/$arcsec2 for some Spitzer archival images, both at 3.6 $\mu$m, which
correspond to 26.9 and 28.2 mag$/$arcsec2 in B band, respectively. But they
failed to detect some previously known shells in at least three cases: NGC
2974 and NGC 5846 (Tal et al., 2009) and NGC 680 (Duc et al., 2011) – these
three galaxies alone increase the percentage of shell galaxies in their sample
to 11%. Atkinson et al. (2013) found shells in 6% of blue galaxies and around
14% in red galaxies.333Red and blue galaxies are defined based on position in
the color-magnitude diagram in order to discriminate between systems on the
red sequence and blue cloud. It corresponds to a morphological segregation as
well. Vast majority of the red sequence galaxies are early-type galaxies,
while the blue sequence represents the late-type galaxies (Coupon et al.,
2009). The survey concerns 1781 luminous galaxies with the redshift range
$0.04<z<0.2$ and detection limit 27.7 mag$/$arcsec2 in $\textrm{g}^{\prime}$
filter.
The occurrence of tidal features of any kind in galaxies is quite high: 73% in
the sample of Tal et al. (2009); 53% in a sample of 126 red galaxies at a
median redshift of $z=0.1$ and $\mu_{\mathrm{max}}\sim 28$ mag$/$arcsec2 using
B, V, and R filters (van Dokkum, 2005); 71% in the subsample of 86 color- and
morphology- selected bulge-dominated early-type galaxies of the previous
sample; about 24% in s sample of 474 close to edge-on early-type galaxies
using the Sloan Digital Sky Survey DR7 archive with $\mu_{\mathrm{max}}\sim
26$ mag$/$arcsec2 using $\mathrm{\textrm{u}^{\prime}}$,
$\mathrm{\textrm{g}^{\prime}}$, $\textrm{r}^{\prime}$,
$\mathrm{\textrm{i}^{\prime}}$, $\mathrm{\textrm{z}^{\prime}}$ bends
(Miskolczi et al., 2011); 12–26% (according to confidence level of a feature
identification) in the sample of Atkinson et al. (2013). The lower detection
rate in Atkinson et al. (2013) is explained by authors by assertion that the
majority of tidal features in early-type galaxies are seen at surface
brightness near (or below) 28 mag$/$arcsec2. Since shells are generally low
surface brightness features, the abundance of shell galaxies will probably
rise with deeper photometric observations.
Another important piece of information from the above mentioned studies is the
environmental dependence of occurrence of shell structures. They are seen
about five times more often in isolated galaxies than in galaxies in clusters.
Malin and Carter (1983) explored 137 shell galaxies – 65 (47.5%) are isolated,
42 (30.9%) occur in loose groups (of these 13% have one or two close
companions), only 5 (3.6%) occur in clusters or rich groups, and the remaining
25 (18%) occur in groups of two to five galaxies. Taking into account only
isolated galaxies, the relative abundance of shell galaxies increases to 17%.
Similar result was reached more recently by Colbert et al. (2001) – they
detected shell/tidal features in nine of the 22 isolated galaxies (41%), but
only one of the twelve (8%) group early-type galaxies shows evidence for
shells. Reduzzi et al. (1996) presented their result that 4% of 54 pairs of
galaxies (pairs are located in low-density environments) and 16% of 61
isolated early-type galaxies exhibit shells. Adams et al. (2012) found
abundance of tidal features about 3% in a sample of 54 galaxy clusters
($0.04<z<0.15$) containing 3551 early-type galaxies, $\mu_{\mathrm{max}}=26.5$
mag$/$arcsec2 in $\textrm{r}^{\prime}$ filter.
Schweizer and Ford (1985) have investigated an unbiased sample of 36 isolated
giant ellipticals, in order to study their fine morphology. They found that 16
of them (44%) possess ripples (some of them very weak, as Schweizer and Ford
note). In contrast to this, Marcum et al. (2004) did not find a single shell
galaxy in their sample of nine early-type galaxies previously verified to
exist in extremely isolated environments, even though, according to the
prognosis, at least four shell galaxies should have been present. The
probability of this (a sample of nine early-type galaxies from regions of low
galaxy density with no shell) is about 1% if we assume that 40% of galaxies in
low-density environments have shells.
However, the true abundance of shell galaxies can still be different from what
has been summarized here. It crucially depends on which galaxies we classify
as shell galaxies and on our ability to detect faint shells in otherwise
innocent looking galaxies.
#### 3.3 Appearance of the shells
Shells have been detected in various numbers, appearance and distributions.
Rich systems like NGC 3923 (Fig. 2) or NGC 5982 (Sikkema et al., 2007) show
about 30 shells, but it is rather an exception among shell galaxies. A large
fraction of the Malin-Carter catalogue (1983) consists of galaxies with less
then 4 shells. It is in fact difficult to make statements about numbers of
shells in galaxies, because the detection of all of them (sometimes even the
proof of their existence) is a delicate matter. Shells actually contain only a
fraction of total luminosity of the host galaxy, mostly from 3 to 6% (e.g., it
is 5% for the famous NGC 3923; Prieur, 1988). Shell surface brightness
contrast is very low, about 0.1–0.2 mag (Dupraz and Combes, 1986). Schweizer
(1986) states that on the brightness profiles of host galaxies, ripples appear
as minor steps of about 1–10% in the local light distribution.
To enhance or detect shells and other fine structures in galaxies, some more
or less sophisticated techniques are often used, like unsharp masking for
photographic images (Malin, 1977), digital masking (Schweizer and Ford, 1985)
or structure map (Pogge and Martini, 2002; based on the probabilistic image-
restoration method of Richardson and Lucy; Richardson, 1972). Host galaxy
subtraction was used to process the image of a shell galaxy in Fig. 3.
Shells are stellar structures that form arcs in galaxies (circular or slightly
elliptical) that either lie within a specific double cone on opposite sides of
the galaxy, or encircle the galaxy almost all around. In general, they tend to
have sharp outer boundaries, but many of them are faint and diffuse. Prieur
(1990) and Wilkinson et al. (1987c) recognized three different morphological
categories of shell galaxies.
?figurename? 3: Top: Very deep ACS/WFC image (total integration time of 11432
s) of a formerly unknown shell galaxy, the host galaxy of the quasar MC2
1635+119 (Canalizo et al., 2007; Bennert et al., 2007; the three images shown
here are unpublished and were kindly provided by G. Canalizo and N. Bennert).
The shell structure is already visible in this final reduced but otherwise
unaltered image. The image size is 10 ′′$\times$10 ′′. The residual image is
shown in the bottom left panel and was obtained by subtracting a model –
fitted using GALFIT (Peng et al., 2002) – for the host galaxy light (bottom
right) from the original data (top). Acknowledgment: NASA, STScI.
?figurename? 4: Galaxy-subtracted image of the type I shell galaxy NGC 7600
from Turnbull et al. (1999). North is up and east is to the left. The dark
oval shape is an artifact of the subtraction process. The easternmost shell
lies 215 ′′ away from the galaxy center. The field of view is 9 ′.
Acknowledgment: The Isaac Newton Group of Telescopes and the Royal
Astronomical Society.
* •
Type I (Cone) – shells are interleaved in radius. That is, the next outermost
shell is usually on the opposite side of the nucleus. They are well-aligned
with the major axis of the galaxy. Shell separation increases with radius.
Prominent examples are NGC 3923 (Fig. 2), NGC 5982, NGC 1344 but also NGC 7600
in Fig. 4.
* •
Type II (Randomly distributed arcs) – shell systems that exhibit arcs which
are randomly distributed all around a rather circular galaxy. A typical
example of this kind is NGC 474 in Fig. 5.
* •
Type III (Irregular) – shell systems that have more complex structure or have
too few shells to be classified.
Prieur (1990) has found all three types in approximately the same fraction.
Dupraz and Combes (1986) state that the angular distribution of the shells is
strongly related to the eccentricity of the galaxy. When the elliptical is
nearly E0, the structures are randomly spread around the galactic center. On
the contrary, when the galaxy appears clearly flattened ($>$E3), the shell
system tends to be aligned with its major axis. In this case, shells are also
interleaved on both sides of the center. Their ellipticity is in general low,
but neatly correlated to the eccentricity of the elliptical. Nearly E0
galaxies are surrounded by circular shells, while the ellipticity of the
shells is of about 0.15 for E3–E4 galaxies.
When we define the radial range of the shell system as the ratio between the
distance from the galactic center to the outermost and the innermost shells,
then this range of radii, over which shells are found, is large. The value
reaches over 60 for type I galaxy NGC 3923 (the innermost shell is less than 2
kpc from center and the outermost one $\sim$100 kpc; Prieur, 1988), but in
most systems, a ratio of 10 or less would be more typical. The range is lower
than 5 for systems where only a few shells are detected (Dupraz and Combes,
1986).
In their sample of three shell galaxies, Fort et al. (1986) found that the
characteristic thicknesses of shells are of the order of 10% or less of their
distance from the center of the galaxy.
Wilkinson et al. (1987c) probed 66 of the 74 galaxies in the range from 01h
40m to 13h 46m in the Malin and Carter (1983) catalogue. They found that
shells commonly occur close to the nucleus. In roughly 20% of the systems
these innermost shells have spiral morphology.
?figurename? 5: Galaxy-subtracted image of the type II shell galaxy NGC 474
from Turnbull et al. (1999). North is up and east is to the left. The
easternmost shell is 202 ′′ from the galaxy center. NGC 470 is located just
off the frame, $\sim$300 ′′ west. The field of view is 9 ′. Acknowledgment:
The Isaac Newton Group of Telescopes and the Royal Astronomical Society.
#### 3.4 Colors
At the beginning of the research on shell galaxies, it was widely believed
that shells are rather bluer than the underlying galaxy (Athanassoula and
Bosma, 1985). But it was rather difficult to obtain relevant data for shells
with only several percent of galaxy’s luminosity and the uncertainty was
probably huge.
Carter et al. (1982) presented broad-band optical and near-IR photometry of
NGC 1344. The color indices derived suggest that the shell comprises a stellar
population, perhaps bluer than the main body of the galaxy. The first CCD
photometric observations of shell galaxies were made in April 1983 at the CFHT
(Canada-France-Hawaii Telescope) by Fort et al. (1986) for their three objects
(NGC 2865, NGC 5018, and NGC 3923). Unlike the shells of NGC 2865 and NGC 5018
which were found bluer than the galaxy itself, the shells of NGC 923 had
similar color indices to those of the galaxy. The results were obtained from
the outer shells of the galaxies.
Pence (1986) got the same result for NGC 3923 and in addition for NGC 3051 as
well. On the other hand, McGaugh and Bothun (1990) found both redder and
slightly bluer systems of shells among their three shell galaxies (Arp 230,
NGC 7010, and Arp 223 = NGC 7585). Multicolor photometry of NGC 7010 shows a
color trend between the center and the galaxy periphery, red in the center and
blue further out.
Recent observations, using the ever-improving observational capabilities may
turn the old myth of blue shells over. Sikkema et al. (2007) wrote: “To date,
observations give a confusing picture on shell colors. Examples are found of
shells that are redder, similar, or bluer, than the underlying galaxy. In some
cases, different authors report opposite color differences (shell minus
galaxy) for the same shell. Color even seems to change along some shells;
examples are NGC 2865 (Fort et al., 1986), NGC 474 (Prieur, 1990), and NGC
3656 (Balcells, 1997). Errors in shell colors are very sensitive to the
correct modeling of the underlying light distribution. HST images allow for a
detailed modeling of the galaxy light distribution, especially near the
centers, and should provide increased accuracy in the determination of shell
colors.” In their sample of central parts of six galaxies (NGC 1344, NGC 3923,
NGC 5982, NGC 474, NGC 2865, and NGC 7626) they find only one shell (in NGC
474) with blue color. All other shells have similar or redder colors – what is
just contrary to the results of Fort et al. in 80’s for NGC 2865 and Carter et
al. (1982) for NGC 1344. Sikkema et al. attribute the red color to dust which
is physically connected to the shell (see Sect. 3.5).
Forbes et al. (1995) measured shell colors of shell galaxy IC 1459 and found
them to be similar to the underlying galaxy. In their study of the shell
galaxies NGC 474 and NGC 7600, Turnbull et al. (1999) found inner shells
redder than the outer ones. For the first shells, colors seem to follow those
of the galaxy, for NGC 7600 three outermost shells are bluer than the galaxy.
In Liu et al. (1999) it is said that a preliminary reduction of the shell
sample shows that most of the shells have colors that are similar to the
elliptical. The shell colors in the shell galaxy MC 0422-476444The reference
name of object derived from the 1950 coordinates. The last digit is a decimal
fraction of degree, truncated. Notation used in Malin and Carter (1983)
catalogue (MC). are scattered around the underlying galaxy value (Wilkinson et
al., 2000). Pierfederici and Rampazzo (2004) inspected another sample of five
galaxies with shells (NGC 474, NGC 6776, NGC 7010, NGC 7585, and IC 1575) and
found the color of the shells being similar to or slightly redder than that of
the host galaxy with the exception of one of the outer shells in NGC 474, the
only interacting galaxy in the sample.
#### 3.5 Gas and dust
Athanassoula and Bosma (1985) found that shells are not a good indicator of
the presence of dust. Shell galaxies (64 items) of Wilkinson et al. (1987b)
have rather higher dust contain than normal elliptical. Sikkema et al. (2007)
detected central dust features out of dynamical equilibrium in all of their
six shell galaxies. Using HST archival data, about half of all elliptical
galaxies exhibit visible dust features (Lauer et al. 2005: 47% of 177 in field
galaxies). On the other hand, Colbert et al. (2001) found evidence for dust
features in approximately 75% of both the isolated and group galaxies (17 of
22 and 9 of 12, respectively). But in their sample also all of the galaxies
that display shell/tidal features contain dust. Also Rampazzo et al. (2007)
found all of their three shell galaxies to show evidence of dust features in
their center.
Moreover, Sikkema et al. (2007) discovered that the shells contain more dust
per unit stellar mass than the main body of the galaxy. This could explain
redder color of shells which is observed in many cases (Sect. 3.4).
Observational evidence for significant amounts of dust residing in a shell was
also found in NGC 5128 (Stickel et al., 2004).
In general, both the ionized and neutral gas contents of shell galaxies are
thus comparable to those of normal early-type galaxies (Dupraz and Combes,
1986) or rather higher (Wilkinson et al., 1987b). However, arcs of H I have
been discovered (Schiminovich et al., 1994, 1995) lying parallel to but
outside of the outer stellar arcs in a few shell systems (Cen A = NGC 5128 and
NGC 2865). In Centaurus A, gas has the same arc-like curvature but is
displaced 1 ′ ($\sim 1$ kpc) to the outside of the stellar shells. A similar
discovery has been made by Balcells et al. (2001) in NGC 3656. The shell, at 9
kpc from the center, has traces of H I with velocities bracketing the stellar
velocities, providing evidence for a dynamical association of H I and stars at
the shell. Petric et al. (1997) found an off-centered H I ring in NGC 1210. A
short report about H I in shell galaxies has been done by (Schiminovich et
al., 1997).
Charmandaris et al. (2000) reveal the presence of dense molecular gas in the
shells of NGC 5128 (Cen A). Cen A, the closest active galaxy, is a giant
elliptical with jets and strong radio lobes on both sides of a prominent dust
lane which is aligned with the minor axis of the galaxy (van Gorkom et al.,
1990; Clarke et al., 1992; Hesser et al., 1984). A significant amount of gas
and dust is situated predominantly in an equatorial disk where vigorous star
formation is occurring (Dufour et al., 1979). Charmandaris et al. detected CO
emission from two of the fully mapped optical shells with associated H I
emission, indicating the presence of $4.3\times 10^{7}$ M⊙ of
H${}_{\text{2}}$, assuming the standard CO to H${}_{\text{2}}$ conversion
ratio.
About $5\times 10^{8}$ M⊙ of molecular gas is located in the inner 2 ′ ($\sim
13$ kpc) of the NGC 1316 (Fornax A) and is mainly associated with the dust
patches along the minor axis (Horellou et al., 2001). In addition, the four H
I detections in the outer regions are all far outside the main body of NGC
1316 and lie at or close to the edge of the faint optical shells and X-ray
emission of NGC 1316. The location and velocity structure of the H I are
reminiscent of other shell galaxies such as Cen A.
Around $8\times 10^{7}$ M⊙ of neutral hydrogen, and some $10^{9}$ M⊙ of
molecular hydrogen have been previously found in NGC 3656 by Balcells and
Sancisi (1996). Roughly 10% of the total gas content, one third of the neutral
hydrogen, lies in an extension to the south, what is also similar to Cen A.
NGC 3656 also contains a prominent central dust line (Leeuw et al., 2007).
These galaxies seem to form up an interesting category of shell galaxies –
aside from the shells, they also contain a prominent central dust line, good
amount of gas (usually both H I and CO detected), and are usually strong radio
sources with jets and active nucleus. Galaxies with these features are
suspected of cannibalization of a gas-rich companion. Some examples of this
group are NGC 5128 (Centaurus A), NGC 1316 (Fornax A), NGC 3656, NGC 1275
(Perseus A; massive network of dust, active nucleus; Carlson et al., 1998), IC
1575 (active nucleus in the center drives the jet orthogonally to the strong
central dust lane, producing the two radio lobes; Pierfederici and Rampazzo,
2004), and possibly IC 51 (Schiminovich et al., 2013), NGC 5018 (Rampazzo et
al., 2007), and NGC 7070A (Rampazzo et al., 2003).
Pellegrini (1999) found that the softer X-ray component which likely comes
from hot gas, is not as large as expected for a global inflow, in a galaxy of
an optical luminosity as high as that of NGC 3923. Sansom et al. (2000) find
that early-type galaxies with fine structure (e.g. shells) are exclusively
X-ray underluminous and, therefore, deficient in hot gas.
Rampazzo et al. (2003) analyzed the warm gas kinematics in five shell
galaxies. They found that stars and gas appear to be decoupled in most cases.
Rampazzo et al. (2007) , Marino et al. (2009), and Trinchieri et al. (2008)
investigated star formation histories and hot gas content using the NUV and
FUV Galaxy Evolution Explorer (GALEX) observations (and in the latter case
also X-ray ones) in a few shell galaxies.
#### 3.6 Radio and infrared emission
Wilkinson et al. (1987a) surveyed a subset of 64 galaxies of the Malin &
Carter catalogue at 20 and 6 cm with the VLA. Apart from Fornax A, only two
galaxies of their set contained obvious extended radio sources. 42% of the
galaxies were detected, down to a 6-cm flux density limit of about 0.6 mJy.
This detection rate does not differ significantly from normal early-type
galaxies. In a complete sample of 46 southern 2 Jy radio galaxies at
intermediate redshifts ($0.05<z<0.7$) of Ramos Almeida et al. (2011), 35% of
galaxies have shells.
A more interesting discovery was made by Wilkinson et al. (1987b). Eight of
the previous sample of 64 shell galaxies plus two from Sadler (1984) sample of
E and S0 galaxies were detected by IRAS. And here comes the discovery: All of
these galaxies are also radio sources with 6-cm flux densities $\geq 0.6$ mJy.
They noted that according to the binomial distribution, the probability of
finding all 10 galaxies at both wavelengths by chance would be 0.1%. From non-
shell galaxies which are detected in the IRAS survey, only 58% are radio
sources. So, there is a strong radio-infrared correlation for shell galaxies.
In the tree-dimensional radio-infrared-shell space, no significant correlation
is seen in any two dimensions, but a correlation is apparently found if all
three are taken together.
Thronson et al. (1989) investigated infrared color-color diagram of early-type
galaxies. On average, shell galaxies appear to have broadband mid- and far-
infrared energy distributions very similar to those of normal S0 galaxies,
although many of them were classified as ellipticals.
#### 3.7 Other features of host galaxies
From their sample of 100 shell galaxies, Carter et al. (1988) derived that
about 15–20% of shell galaxies have nuclear post-starburst spectra. Ramos
Almeida et al. (2011) found shells in 15 out of 33 (45%) of the non-starburst
systems, but in only 1 out of 13 (8%) of the starburst systems. All their
objects are powerful radio galaxies (PRGs) and quasars.
Longhetti et al. (2000) have studied star formation history in a sample of 21
shell galaxies and 30 early-type galaxies that are members of pairs, located
in very low density environments. The last star formation event (which
involved different percentages of mass) that happened in the nuclear region of
shell galaxies is statistically old (age of the burst from 0.1 to several Gyr)
with respect to the corresponding one in the sub-sample of the interacting
galaxies (age of the burst $<0.1$ Gyr or ongoing). This distinction has been
possible only using diagrams involving newly calibrated “blue” indices.
Assuming that stellar activity is somehow related to the shell formation,
shells have to be long lasting structures.
There is an obvious strong association between kinematically
distinct/decoupled cores (“KDC” or “KDCs”) and shell galaxies. First example
of an elliptical galaxy with a KDC was NGC 5813 (Efstathiou et al., 1982).
These galaxies are characterized by a rotation curve that shows a decoupling
in rotation between the outer and inner parts of the galaxy. In some
spectacular cases, the core can be spinning rapidly in the opposite direction
to the outer part of the galaxy (e.g. IC 1459). It was found by Forbes (1992;
cited in Hau et al., 1999) that all of the nine well-established KDCs and a
further four out of the six “possible KDCs” possess shells.
Some galaxies are known to contain multiple nuclei (e.g. NGC 4936, NGC 7135,
MC 0632-629, MC 0632-629). Forbes et al. (1994) conducted the first systematic
search for secondary nuclei in a sample of 29 known shell galaxies. They find
six (20%) galaxies with a possible secondary nucleus, what they concluded to
be a probable upper limit to the true fraction of secondary nuclei. In the
sample of radio galaxies of Ramos Almeida et al. (2011), five galaxies have
more than one nucleus while also having shells detected. That makes 20% of
their shell galaxies containing the secondary nucleus. Thereof one double
nucleus is uncertain (PKS 1559+02) and one galaxy has triple nucleus indicated
(PKS 0117-15). On the other hand, Longhetti et al. (1999) in their sample of
21 shell galaxies found only one (ESO 240-100) to be characterized by the
presence of a double nucleus.
According to Wilkinson et al. (1987c), shell galaxies have an enormous
diversity of central surface brightness. In addition, Wilkinson et al. (1987a)
found a wide variety of optical appearances, suggesting that shell galaxies
are not a homogeneous class with uniform physical characteristics.
### 4 Summary of shell characteristics
1. 1.
Shells are observed in at least 10% of early-type galaxies (E and S0) and
$\sim$1% of spirals.
2. 2.
Shell galaxies occur markedly most often in regions of low galaxy density.
3. 3.
The number of shells in a galaxy ranges from 1 to $\sim$30.
4. 4.
The shells contain at most a few per cent of the overall brightness of the
galaxy.
5. 5.
Surface brightness contrast of the shells is very low, about 0.1–0.2 mag.
6. 6.
Shells are of stellar nature.
7. 7.
For type I shell galaxies (see in Sect. 3.3), shells are interleaved in radius
and their separation increases with radius.
8. 8.
Shells appear to be aligned with the galaxy’s major axis and slightly
elliptical for flattened galaxies, and randomly spread around the galactic
center for nearly E0 galaxies.
9. 9.
The radial range of shells (the ratio of the radii of the outermost and the
innermost shells) is typically less then 10 but can reach over 60.
10. 10.
Shells commonly occur close to the nucleus.
11. 11.
In roughly 20% of the systems, the innermost shells have spiral morphology.
12. 12.
Shells can have any color, perhaps they are rather similar to or slightly
redder than the host galaxy.
13. 13.
The colors of shells are different even in the same galaxy, tend to be red in
the center and bluer further out.
14. 14.
It seems that galaxies with shells also contain central dust features.
15. 15.
An increased amount of dust has been observed in shells.
16. 16.
Slightly displaced arcs of H I, with respect to the stellar shells, have been
discovered in some galaxies.
17. 17.
Molecular gas associated with shells was detected in several galaxies.
18. 18.
The detection rate of radio emission of shell galaxies is similar to other
early-type galaxies.
19. 19.
There is probably a strong radio-infrared correlation for galaxies which
possess shells.
20. 20.
15–20% of shell galaxies have nuclear post-starburst spectra.
21. 21.
There is a strong association between kinematically distinct/decoupled cores
and shells in galaxies.
22. 22.
The shell galaxies have an enormous diversity of central surface brightness
and a wide variety of optical appearances.
### 5 Scenarios of shells’ origin
In the eighties and nineties several theories of formation of shell galaxies
were proposed. They can be divided into three categories:
* •
Gas dynamical theories (Sect. 5.1) – The first truly developed theories
connect star formation and the formation of shells. These theories, however,
seem to be contradicted by observation and now they are not usually taken into
consideration.
* •
Weak Interaction Model (WIM, Sect. 5.2) – According to this model, shells are
density waves induced in a thick disk population of dynamically cold stars by
a weak interaction with another galaxy. WIM has nice explanations for many
phenomena related to the shells but suffers from some deficiencies and
obscurities.
* •
Merger model – The most widely accepted theory is based on the idea that the
stars in shells come from a cannibalized galaxy. The entire Sect. 6 is devoted
to this model.
For a more detailed review, see Ebrová (2007).
#### 5.1 Gas dynamical theories
The first theory of shell formation has been proposed by Fabian et al. (1980),
who suggested that shells are regions of recent star formation in a shocked
galactic wind. Gas produced by the evolution of stars in an elliptical galaxy
and driven out of the galaxy in a wind powered by supernovae would be heated
and compressed as it passes through a shock. As the gas cools, star formation
can occur. This scenario was expanded by Bertschinger (1985) and Williams and
Christiansen (1985). In the Williams and Christiansen (1985) model, shells are
initiated in a blast wave expelled during an active nucleus phase early in the
history of the galaxy, sweeping the interstellar medium in a gas shell, in
which successive bursts of star formation occur, leading to the formation of
several stellar shells.
This scenario was inspired by the supposedly bluer color of the shells, but as
time and the measurements have shown, shells are composed mostly of old
populations of stars (see Sect. 3.4). As Williams and Christiansen mention,
star formation is a subject only to local conditions and is a stochastic
process. This is in conflict with the observed interleaving of shells in many
shell galaxies. Further, there is the failure to detect either ionized or
neutral gas associated with the shells except in a very few cases. Dupraz and
Combes (1986) argued that the mechanism of star formation in such a galactic
wind is not known; the galaxy should have possessed a very large amount of
interstellar matter in order to produce stellar mass of a typical shell
system; and the supernovae explosions might rapidly dispel the wind which
would exclude that as much as 20–25 shells form around some shell galaxies.
Loewenstein et al. (1987) reconciled previous models with the last
observations at that time. Only a modest outburst is demanded by the authors
to cause a period of star-formation in an outward-moving disturbance from the
galactic core. The newly-formed stars occupy a small volume in the orbital
phase-space of the underlying galaxy. The shells were produced in the same
phase-wrapping mechanism as in the merger model (Sect. 6.1) producing an
interleaved shell system (point 7 in Sect. 4). The model does not exclude the
merger hypothesis, since a merger can lead to a burst of star formation in the
galactic core that is the precursor of the initial blast wave. The inner
shells are older than the outer ones in this scenario. This could lead to the
color gradient which seems to be observed in some cases (point 13 in Sect. 4)
and which was not known at the time.
All these arguments are sound, but other observed aspects of shell galaxies
seem to exclude the model of Loewenstein et al. anyway. Aside from the already
mentioned points, Colbert et al. (2001) discovered a consistency of the colors
of the isolated galaxies with and without shells and it argues against the
picture in which shells are caused by asymmetric star formation. Again the
failure to detect gas in shells argues against this scenario. Finally, the
lack of signs of recent star formation in the shells is the most fatal reality
for the model discussed here.
A rather different scenario was proposed by Umemura and Ikeuchi (1987), and
was quickly forgotten for its clumsiness and only a little agreement with
observations. They tentatively considered a hot supernova-driven galactic wind
as a process which produces both extended multiple stellar shells and hot
X-ray coronae which have been detected around a number of early-type galaxies.
Few of them also have shells (NGC 1316, NGC 1395, NGC 3923, and NGC 5128).
This scenario suffers from much the same diseases as the former ones.
Moreover, it gives no explanation for the increasing separation of shells with
radius, since the distribution of shells is variable with the lapse of time in
this scenario. As previously mentioned, early-type galaxies with fine
structure are X-ray underluminous, thus deficient in hot gas (Sect. 3.5).
However, this theory seems to be primarily out of game because of the observed
systematic interleaving of shells.
All the models mentioned above more or less fell in condemnation and oblivion
before they even started to try explaining more detailed characteristics
observed in shell galaxies.
#### 5.2 Weak Interaction Model (WIM)
Thomson and Wright (1990) came up with an elegant and revolutionary model of
shell formation in elliptical/lenticular galaxies which is still in the game
today. According to them, shells are density waves induced in a thick disk
population of dynamically cold stars by a weak interaction with another galaxy
– whence the name, the Weak Interaction Model (WIM). A year later, this
hypothesis was further developed and supported by new simulations of Thomson
(1991).
To support their theory, the authors state that Thronson et al. (1989) pointed
out that most of the elliptical galaxies with shells catalogued by Malin and
Carter (1983) are classified elsewhere as S0s. As such, a significant
population of dynamically cold stars moving on nearly circular orbits could be
present in these systems. They also note that faint thick disks could be
present in many elliptical galaxies without detection. The authors noted that
a thick-disk population which makes up only a few per cent of the total mass
of a galaxy is required to explain the faint features seen in most shell
galaxies. But the disk must by heavy enough to produce shells which form a few
per cent of the overall brightness of the galaxy (point 4 in Sect. 4).
Wilkinson et al. (2000) looked for such a disk in the shell galaxy MC 0422-476
and found no sign of an exponential disk, or any thick disk additional to the
short-axis tube orbits already expected within an oblate ellipsoidal
potential.
The WIM has always been simulated with the parabolic encounter of the
secondary galaxy, since more circular orbits would decay rapidly during a
close encounter, resulting in a merger scenario, while more hyperbolic orbits
would result in encounters too quick to be effective. This fact can also
account for the less frequent occurrence of shell galaxies in clusters than in
the field (point 2 in Sect. 4).
Required mass of the secondary is about 0.05–0.2 of the primary mass and
orbital inclination 45° or less with respect to the thick disk. The total time
of the shell structure’s visibility is typically around 10 Gyr in Thomson and
Wright (1990). But in the simulations of Thomson (1991), the shells are
visible for only about 3 Gyr.
Possibly, the age of the shell system can be deduced from its appearance and
thus the presence of a suitable secondary galaxy at an appropriate distance
could be checked. But e.g. around NGC 3610 no surrounding galaxies were found
(Silva and Bothun, 1998).
In WIM, the host galaxy is an oblate555An _oblate_ ellipsoid is rotationally
symmetric around its shortest axis, whereas for a _prolate_ ellipsoid the axis
of symmetry is the longest one. A _triaxial_ ellipsoid has no rotational
symmetry at all. spheroid, and shells are readily formed as spiral density
waves in the thick disk which is symmetric about the plane of symmetry of the
galaxy. The model also gives the correct relative frequency of two types of
shell galaxies (i.e. 1:1, Sect. 3.3), since the systems appear as type II
shell galaxies when viewed at inclination angles less than approximately 60°
(0° is face-on). At inclination angles larger than 60°, the systems appear as
type I. As we change the viewing angle, the observed ellipticity changes from
E0 (for 0°) to E4 (90°), where E4 may be the true ellipticity of the galaxy,
since Prieur (1990), cited in Thomson (1991), found a strong peak at this
value in the type I ellipticity histogram. However, implications of this would
be somewhat strange – either all elliptical galaxies are E4 type oblate
spheroids seen from different angles, or shells do occur only in E4 galaxies,
what would be probably in contradiction to their relatively frequent
occurrence.
Prieur (1988) pointed out that the shells in NGC 3923 are much rounder than
the underlying galaxy and have an ellipticity which is similar to the inferred
equipotential surfaces. This idea was originally put forward by Dupraz and
Combes (1986) who found such a relationship for their merger simulations
(Sect. 6). The same effect can be seen in the simulations presented by Thomson
(1991).
Another advantage of the WIM lies in its ability to explain the occurrence of
the shells over a broad range of radii (point 9 in Sect. 4) and close to the
nucleus (point 10), since shells are formed in the thick disk that is required
to be already present in the galaxy.
In his study of the shell galaxy NGC 3923, Prieur (1988) discussed varying
distribution of the shells – interleaved in outer region and roughly symmetric
in inner parts. According to this model, in the outer region of the galaxy,
the simulations show a predominantly one-armed trailing spiral density wave
which, when viewed edge-on, gives rise to the interleaving of the outer
shells, naturally aligned with the major axis. Inside the perigalactic radius
of the path of the intruder, the tidal forces produced during the encounter
induce a bi-symmetric kinematic density wave in the thick disk. Thomson has
achieved an almost breathtaking agreement with the observation of radial shell
distribution, except for the innermost shells that have not appeared at all in
his simulations. But he believes it could be remedied by shrinking the core
radius of primary galaxy.
The WIM for shells does not predict the existence of a kinematically distinct
nucleus (KDC, point 21 in Sect. 4). Hau and Thomson (1994) proposed a
mechanism whereby a counter-rotating core could be formed by the retrograde
passage of a massive galaxy past a slowly rotating elliptical with a pre-
existing rapidly rotating central disk. In their study of the shell galaxy NGC
2865, Hau et al. (1999) state that the requirement of the WIM for the nuclear
disk to be primordial is in conflict with the observed absorption line
indices. It is also unlikely that a passing galaxy can transfer a large amount
of orbital angular momentum over a period longer than 0.5 Gyr without being
captured or substantially disrupted, as NGC 2865 has an extended massive dark
halo (Schiminovich et al., 1995). Thus a purely interaction induced origin for
the shells and KDC in NGC 2865 is ruled out.
The observation by Pence (1986) shows that the surface brightness of shells in
NGC 3923 is a “surprisingly constant” fraction ($\sim$3–5%) of the surface
brightness of the underlying galaxy. The WIM produces shells with the correct
surface brightness, since they are formed in a thick disk which has the same
surface brightness profile as the underlying galaxy. However, further
observations (Prieur, 1988; Sikkema et al., 2007) revealed more shells in NGC
3923 that defy this rule. And there are more disobedient shell galaxies: NGC
474 and NGC 7600 (Turnbull et al., 1999) and MC 0422-476 (Wilkinson et al.,
2000). Similarly for NGC 2865, the WIM origin is in conflict with the
existence of bright outer shells, their blue colors, and their chaotic
distribution (Fort et al., 1986).
Furthermore, Carter et al. (1998) revealed a minor axis rotation of the famous
NGC 3923 what suggests a prolate or triaxial potential, and challenges the
requirement of an oblate potential by the WIM. They noted that it is difficult
to induce minor axis rotation in an oblate potential without inducing any
corresponding major axis rotation that has not been observed.
Silva and Bothun (1998) note that the spectacular morphological fine structure
of the shell galaxy NGC 3610 leads to the natural conclusion that this galaxy
has undergone a recent merger event. This scenario is supported by the
existence of a centrally concentrated intermediate-age stellar population
which is a prediction of the dissipative gas infall models. Furthermore, the
central stellar structure could have been formed by this infalling gas. It
seems unlikely that the structures were formed by a non-merging tidal
interaction since there is no nearby galaxy.
It is interesting that nobody has ever noticed any general one-armed spiral in
the outer shells of type II shell galaxies nor any bi-symmetric spiral in
inner regions. Only Wilkinson et al. (1987c) probed 66 shell galaxies and
found that in roughly 20% of the systems these innermost shells have spiral
morphology. But they did not specify which galaxies they were nor what spiral
morphology has been found. Thomson (1991) explains: “The broken appearance of
the shells is actually an interference pattern formed by the leading and
trailing density waves induced during the encounter”, and he adds that the
faint residual one-armed leading spiral feature seen at the end of some of the
simulations is probably an $m$ = 1 kinematic density wave666Here, a common
method of decomposition of a 2D density or potential to Fourier modes in the
azimuthal direction (that is, Fourier transforming in the angle separately for
every radius) is used. The potential is decomposed as
$\phi(R,\theta)=\phi_{0}(R)+\sum_{m=1}^{\infty}\phi_{m}(R)\cos[m(\theta-\theta_{m}(R))],$
what means a sum of harmonics with different amplitudes and phase shifts for
every R. The $\phi_{0}$ (_m_ = 0) mode is the axisymmetric part of the
potential, the _m_ = 1 mode has an azimuthal period of 360°, the _m_ = 2 mode
has 180° and so on. It is most frequently used for spiral galaxies. The _m_ =
1 mode corresponds to one spiral arm ($\theta_{1}$ is dependent on _R_) or a
closed structure (an ellipse when $\theta_{1}$is a constant) not concentric
with the galaxy. The _m_ = 2 mode is the most common, being either a bar
(constant $\theta_{2}$) or two spiral arms. In the WIM case, the _m_ = 2 mode
(bi-symmetric spiral density wave) is important for the inner parts of the
disk.. The relative importance of this mode for the shell forming process is
not fully understood, but it does play an important role in determining the
shell morphology produced by the more massive encounters.
Wilkinson et al. (2000) found many arguments for and against the WIM in their
study of the shell galaxy MC 0422-476.
Longhetti et al. (1999) favor the WIM, since they derived that in shell
galaxies, the age of the last star forming event ranges from 0.1 to several
Gyr. If the last burst of stellar activity that affects the absorption line
strength indices, correlates with the dynamical mechanism forming the shell
features, these shells are long lasting phenomena. The WIM predicts such a
long life for the shells, whereas for the merger model of Quinn (1984), Sect.
6, guessed a shorter lifetime due to the initial dispersion of velocities that
the stars of the shell inherited. But for example, in the framework of the
merger model, Dupraz and Combes (1986) happily simulated shell systems for 10
Gyr.
A consequence of the WIM is that the stars which make up the shells must be in
nearly circular orbits. That is almost opposite to the conclusions of the
merger model (Sect. 6). It could be thus decided from measurements of the
shell velocity fields which model is favored, but this is indeed a formidable
task, as the shells contain at most a few per cent of the overall brightness
of the host galaxy. Some attempts have been already carried out (Balcells and
Sancisi, 1996), but as far as we know, the results are inconclusive.
To conclude, the WIM has nice explanations for many phenomena related to the
shells (inner shells, shell distribution, symmetry of inner shells, etc.), for
which the competing merger model (Sect. 6) seeks explanations with
difficulties or has none at all. On the other side, the WIM suffers from some
deficiencies and obscurities (thick disk, KDC, shells brightness, etc.).
Generally, it seems to lack observational confirmation of phenomena specific
to the model.
?figurename? 6: Time evolution of a cloud of test particles falling into a one
dimensional Plummer potential $v-x$ space (upper row), particle radial density
(lower row). The $x$ axis is centered with the center of the potential and
scaled so that 1 on the axis is the Plummer radius.
### 6 Merger model
In this section we introduce the merger origin scenario of the shell galaxies
that we consider for the rest of the thesis. For a more detailed (but slightly
outdated) review, see Ebrová (2007).
#### 6.1 Phase wrapping
The idea of a connection between mergers and shells was first published by
Schweizer (1980) in his study of the shell galaxy NGC 1316 (Fornax A). The
presence of shells (or “ripples” as Schweizer calls them) deep within NGC 1316
and a surprising number of galaxies with ripples but no companions fosters his
belief that Fornax A, too, has been shaken by a recent intruder rather than by
any of the present neighbors. Schweizer imagined that the ripples represent a
milder version of the strong response that occurs in the disk of a galaxy when
an intruder of comparable mass free-falls through the center: A circular
density wave runs outward, followed sometimes by minor waves, and give the
galaxy the appearance of a ring (Lynds and Toomre, 1976; Toomre, 1978).
Quinn (1983, 1984) took up the idea of a merger origin of shells, but showed
it in a slightly different spirit. When a small galaxy (secondary) enters the
scope of influence of a big elliptical galaxy (primary) on a radial or close
to a radial trajectory, it splits up and its stars begin to oscillate in the
potential of the big galaxy which itself remains unaffected. In their turning
points, the stars have the slowest speed and thus tend to spend most of the
time there, they pile up and produce arc-like structures in the luminosity
profile of the host galaxy. Quinn modeled the formation of shell galaxies
using test-particle and restricted $N$-body codes, much as many other did
later (e.g, Hernquist and Quinn, 1987b, 1988, 1989; Dupraz and Combes, 1986)
and as we will do in this work as well. It should be also noted that already
Lynden-Bell (1967) described something like a pig-trough dynamics in violent
relaxation in stellar systems.
?figurename? 7: Surface brightness density from the simulation of a radial
minor merger. Top row: both primary and secondary galaxy are displayed. Bottom
row: only the surface density of particles originally belonging to the
secondary is displayed. Panels show an area of $300\times 300$ kpc. Time-
stamps mark the time since the release of the star in the center of the host
galaxy. For parameters of the simulation, see Appendix H point 1.
The mechanism is illustrated on the one dimensional example in Fig. 6. The
density maxima occur near the turnaround points of the particle orbits. The
maximal radial position of the orbit is first reached by the most tightly
bound particles, but as more distant particles stop and turn around, the
density wave propagates slowly in radius to the outermost turning point set by
the least bound particle. The particles in phase space form a characteristic
structure, for which this mechanism of shell formation is often called “phase
wrapping”.
In an idealized case, the edges in density are the caustics of the mapping of
the phase density of particles into physical space (Nulsen, 1989). As a
natural consequence, the shells are interleaved in radius and their separation
increases with radius (point 7 in Sect. 4). Furthermore, the range of the
number of shells present around ellipticals is a simple consequence of the age
of the event. More shells will imply that a longer time has passed since the
merger event. A more detailed explanation and some equations can be found in
Sect. 9.1. The best insight on the shell formation is provided by video
1-shells.avi, which is a part of the electronic attachment. Five snapshots
related to the video can be seen in Fig. 7. For the description, see Appendix
H point 1.
#### 6.2 Cannibalized galaxy
The choice of the type of the secondary galaxy initially felt on a disk
galaxy. The authors were probably led to it by two aspects. Firstly,
dynamically cold systems promised to be better in shell formation, since they
occupy a smaller phase volume than velocity dispersion supported galaxies of
comparable masses. In such a process of non-colliding stars we can assume
phase volume conservation according to the Liouville’s theorem. This means
that a system with an initially small phase volume keeps this property and
forms sharper shells. So, the visibility of the shell system is expected to be
lower for an elliptical companion than for a spiral companion of the same
mass, since the velocity dispersion is greater for the elliptical. Secondly,
the observations seemed to suggest that the stars in shells have the color
indices of late-type galaxies (see Sect. 3.4). Later observations have shown
that the shells are not that blue (see also Sect. 3.4), but even before that
the simulations showed that the shell systems can be formed by a disk as well
as an elliptical companion (Dupraz and Combes, 1986; Hernquist and Quinn,
1988).
Hernquist and Quinn (1988) examined among others the influence of the phase
volume and velocity dispersion of a spherical companion on shell formation. As
was already mentioned above, higher dispersion means higher blur of resulting
shells through the increase of the phase volume (velocity dispersion is
proportional to the square root of mass of the accreted companion). Another
effect brought in by higher dispersion is that the material can be captured
into more tightly bound orbits, so shells are produced more rapidly, since the
shell production rate is indirectly proportional to the shortest period of
stellar oscillations. This means that for the same potential of the primary
galaxy, we can easily get different shell systems by changing some parameters
of the accreted galaxy, what constituted one of several serious problems of
the idea to explore the potential of the host galaxy through its shell system.
The disk-like secondary galaxy has some extra options that the spherical one
lacks. By accreting differently inclined disks we can get different peculiar
structures. The resulting configuration of sharp-edged features is
considerably more complex and disordered than for a spherical companion. For a
very flat system, there is also the possibility of forming caustics through
spatial wrapping. That is to say, as the sheet of particles moves and folds in
three-dimensional space, sharp edges can be formed in its two-dimensional
projection onto the plane of the sky. Projection effects become critical in
this context, as evidenced by the different viewing angles, see Hernquist and
Quinn (1988). This effect was evident already in the simulations by Quinn
(1984).
#### 6.3 Ellipticity of the host galaxy
Dupraz and Combes (1986) tried to explain the observed characteristics of
shell morphology (point 8 in Sect. 4) with the encounter of a disk galaxy with
a prolate or oblate primary E-galaxy. The secondary galaxy falls into the
prolate galaxy around its symmetry axis and into the oblate galaxy
perpendicularly to its symmetry axis (the symmetry axis is the major axis when
the E-galaxy is prolate, minor axis when oblate). The disk of the secondary
galaxy is always oriented in the direction of the collision. In the prolate
case, the companion stars achieve pendular motion along the major axis of the
E-galaxy. The shells form consequently along this axis, alternatively on one
side and the other (type I shell galaxy, see Sect. 3.3). On the contrary, in
the oblate case, the shell system does not possess any symmetry, since there
is no privileged major axis here. The shells appear randomly spread around the
center of the E-galaxy (type II shell galaxy).
Dupraz and Combes (1986) state that a shell system is found aligned with the
major axis of an elliptical galaxy, only when the E-galaxy is prolate and the
impact angle is likely to be lower than 60°. A shell system is found aligned
with the minor axis of an E-galaxy, only when the latter is oblate and the
impact angle is lower than $\sim$30°. It is interesting to note that no such
system, with the shell aligned with the minor axis, is known.
However, all this results were negated by Hernquist and Quinn (1989), who also
simulated an ellipsoidal potential of the primary galaxy. Their result is that
if the potential well maintains the same shape at all radii as in the
simulations of Dupraz and Combes, then the shape of the dark matter halo, as
well as that of the central galaxy, is responsible for aligning and confining
the shells. If, on the other hand, the potential is allowed to become
spherical at large radii, the shell alignment and angular extent are less
sensitive to the properties of the potential at small radii. This means that
two primaries, one oblate and the other prolate, can have similar projected
shapes and similar outer shells if the outer isophotentials in each case
become spherical. Hence the shape of the potential at large as well as small
radii needs to be considered when examining the shell extent and alignment.
Even the same authors formerly tried to get some information about the
potential of several chosen shell galaxies (Hernquist and Quinn, 1987b), but
for those reasons and the reasons stated in Sects. 6.2 and 6.4, they were left
with nothing to say but: “The shell morphology is sensitive to the shape of
the primary at large and small radii as well as to the detailed structure of
the companion. This would imply that it is difficult, if not impossible, to
infer the form of the primary from the shell geometry alone. In this
conclusion, we disagree with Dupraz and Combes (1986).”
#### 6.4 Radial distribution of shells
The radial distribution of shells was always probably the most watched aspect
of the merger model. From Sect. 6.1 we already know how easily the merger
model reproduces the interleaving in radii. The shell formation is closely
connected to the period of radial oscillation in the host galaxy potential,
what is in any case an increasing function of radius, see Sect. 9. The shells
as density waves receding from the center, composed in every moment of
different stars, are the older the further from the center they are. With
time, the frequency of the shells increases, thus the distances between shells
decrease towards the center, what is also in agreement with observations (see
Sect. 3.3).
The above-mentioned facts suggest a connection of shell distribution and the
potential of the underlying galaxy. But already Quinn (1984) discovered that
the radial distribution of shells derived from the potential inferred from the
observed luminous matter distribution cannot agree with the observed reality.
Quinn (1984) derived that the potential of the shell galaxy NGC 3923 must be
less centrally condensed at radii $1<r/r_{\text{e}}<4$ (where $r_{\text{e}}$
is the half-mass radius) than the luminous matter observations predict. This
discovery was reflected by Dupraz and Combes (1986); Hernquist and Quinn
(1987b) as they added an extensive dark matter halo in their simulations and
then they were able to better reproduce the observed shape of the shell
distribution. But immediately after that, Dupraz and Combes (1987) synthesized
successfully a similar radial distribution taking into account the dynamical
friction instead of dark matter. Moreover, in spite of the simplicity of their
model, they synthesized a wide variety of shapes for the shell distribution by
varying only the two parameters: mass ratio of primary and secondary and
impact parameter. It all leads to the conclusion that the shell system is not
suitable to study the potential of a host galaxy.
Note that in the eighties only photometric data were considered. Merrifield
and Kuijken (1998) suggested methods of measurement of the potential using
shell kinematics (Sect. 7.2). The method relies on the stars, which form the
shell, to be on the close-to-radial orbits and it is insensitive to the
details of the merger such as the type of cannibalized galaxy and dynamic
friction.
The cornerstone of the merger theory is also the huge range of radii in which
the shells occur. A simple merger simulation, as of Quinn (1984) (see Sect.
6.1), is not able to produce shells simultaneously on large and small radii.
The presence of shells deep within the host galaxy (and thus the presence of
deeply bound stars that once were part of the secondary galaxy) was mysterious
from the very beginning. But because at that time the merger model had no
direct competition, it was felt more as a challenge than a flaw. However, the
advent of the WIM (Sect. 5.2) that does not have any problems explaining this
phenomenon, challenges the merger model more seriously.
Quinn (1984) suggested three possible explanations: First, the infall velocity
of the disk may have been small and hence the disk was initially strongly
bound to the elliptical. Second, the mass ratio may have been closer to unity,
and hence energy could have been transferred from orbital motion to internal
velocity dispersion. But as the most probable explanation he promoted the idea
that the disruption process is a gradual one and that the center-of-mass
motion of the disk is subject to dynamical friction.
Another effect that no one predicted was found by Heisler and White (1990).
They self-consistently simulated the secondary galaxy and left the primary as
a rigid potential. During the disruption event there is a substantial transfer
of energy between the various parts of the satellite. Stars which lead the
main body through the encounter are braked and later form the inner shell
system. Stars which lag the main body are accelerated and turn into an
escaping tail. This transfer is asymmetric and, for the encounters they have
studied, the surviving core suffers a net loss of orbital energy which can
shrink the apocenter of its orbit by a large factor. All these transfer
effects increase with the mass of the satellite. It should be emphasized that
this energy transfer happens only within the original secondary galaxy and no
dynamical friction from the stars of the primary galaxy is accounted for in
this case.
This scenario also allows the shell formation in a larger spread of radii. If
the core of the cannibalized galaxy survives the merger, new generations of
shells are added during each successive passage. This was predicted by Dupraz
and Combes (1987) and successfully reproduced by Bartošková et al. (2011) in
self-consistent simulations. Further, the combination of the loss of orbital
energy in this way and the dynamical friction could bring new results, if
properly modeled. This was also mentioned by Seguin and Dupraz (1996), who
also simulated the formation of shell galaxies in a radial merger in a self-
consistent manner, although without any dark matter halo in the primary
galaxy.
#### 6.5 Radiality of the merger
The assumption of a radial merger is the most awkward and criticized point of
Quinn’s model of shell formation. In his work, Quinn (1984) has shown that if
the center-of-mass motion of the infalling disk is predominantly non-radial,
the merger produces confused, often overlapping shells which appear enclosing.
This does not correspond to what we see in real shell galaxies.
On the other hand, A. Toomre modeled an off-axis release of a non-rotating,
inclined disk into a fixed spherical force field (shown in Schweizer, 1983)
and his results resemble the observed shapes. The model was similar to that of
Quinn in that the disk was released as a set of test particles with identical
subparabolic velocities. The shells are created via the mass transfer from the
secondary galaxy flying by on a parabolic trajectory. The captured part forms
a complex structure around the primary galaxy. In this case, a complete merger
is not necessary to produce the shells. Hernquist and Quinn (1988) present
examples of objects from the Arp atlas (Arp, 1966a) that may well have
resulted from such non-merging encounters – Arp 92 (NGC 7603), 103, 104 (NGC
5216 + NGC 5218), and 171 (NGC 5718 + IC 1042) all show evidence of
interactions as well as diffuse shell-like features surrounding the more
luminous galaxy. Hernquist and Quinn (1988) also note that, as in the strictly
planar case, the term "shell" can occasionally be a misnomer since the stars
near the vicinity of a sharp edge are not necessarily distributed on a three-
dimensional surface in space.
However, the requirement of a fairly radial encounter stays valid to produce
type I shell galaxies (Sect. 3.3) as NGC 3923 or NGC 7600 that we have already
seen in Fig. 2 and Fig. 4, respectively. A strictly radial merger of galaxies
is improbable, but now cosmological $N$-body simulations tell us that
satellites are preferentially accreted on very eccentric orbits (Wang et al.,
2005; Benson, 2005; Khochfar and Burkert, 2006).
Dupraz and Combes (1987) considered that the shell distribution, from the
parabolic encounter with dynamical friction, remains unchanged for a (small
but) significant range of impact parameters. The more massive the secondary
galaxy is (compared with the primary), the larger range is allowed. González-
García and van Albada (2005a, b) carried out $N$-body simulations of
encounters between spherical galaxies with and without a dark halo with $\sim
10^{4}$ particles. Shells are rather a byproduct of their work, but they were
able to get them even for impact parameters enclosing 95% of the total mass of
the primary. Even earlier, Barnes (1989) examined the evolution of a compact
group of six disk galaxies in a self-consistent simulation of 65,536
particles. The result was a giant elliptical galaxy containing the shells. The
shells were created during the final infall of the last galaxy into the merged
body of all other galaxies. The initial distribution of the disk galaxies and
their inclinations were by no means special, and Barnes did not specifically
try to get the shells. This simulation may mean that during the evolution of a
compact group, the shell galaxies are indeed formed in the final stage of the
merger. Similarly, recently Cooper et al. (2011) found shell galaxies as a
product of galaxy formation in Milky Way-mass dark halo in two from six
simulated halos from the Aquarius project (Springel et al., 2008), which
builds upon large-scale cosmological simulations. Furthermore, it is supported
by the observed high occurrence of shells in isolated giant galaxies (Sect.
3.2).
#### 6.6 Major mergers
Hernquist and Spergel (1992) published results of their simulation of a major
merger which creates shells. Two identical galaxies with self-gravitating
disks and halos merged following a close collision from a parabolic orbit. The
plane of each disk initially coincides with the orbital plane. When plotted in
phase space, the remnant exhibits more than 10 clearly defined phase-wraps
which can be identified with shells. Shells also occur near the nucleus and
appear to be aligned with the major axis of the resulting galaxies.
González-García and Balcells (2005) examined the creation of elliptical
galaxies from mergers of disks. They used disk-bulge-halo or bulge-less, disk-
halo models with mass ratios of the participants of 1:1, 1:2, and 1:3 and
various impact parameters. As a result of those mergers, shells which could be
identified in phase space occurred sometimes. They found out that the models
without bulges with the mass ratio of 1:2 or 1:3 lead to more prominent
shells. But these were always shell systems of type II (all-round) or type III
(irregular). González-García and Balcells note the lack of shells in remnants
of equal-mass mergers and on all prograde mergers. This contrasts with the
shell system presented by Hernquist and Spergel (1992), a prograde merger of
two equal-mass, bulge-less disks. The perfect alignment of the disk spins with
the orbital angular momentum may have favored the formation of shells in their
model.
González-García and van Albada (2005a, b) have also carried out simulations of
encounters between spherical galaxies (see Sect. 6.5): In their first paper
without a dark halo and in the second one with a dark halo (with mass ratios
of 1:1, 1:2, and 1:4). The sharpness of the occurring shells was higher in
models with a halo. A head-on collision for a run with mass a ratio 4:1 showed
the shells even after 5 Gyr from the first encounter of the galaxy centers.
But the shells showed up also in the merger with 1:2 mass ratio and a nonzero
impact parameter. In any case, the shells are formed from particles of the
less massive galaxy through the same phase wrapping that was established by
Quinn (1984).
To summarize, shells can be formed via a merger even in the cases when the
mass ratios are not as dramatic as it has been simulated in the 80s (the big
mass of the secondary galaxy could influence the alignment of shells with the
major axis of the host galaxy, but no one has so far explored it). It is
probably not common to have shells when two disk galaxies of comparable masses
merge. Hernquist and Spergel (1992) got shells in their model maybe only
thanks to the very special conditions of the collision they have chosen.
Furthermore, the interleaving structure and more generally the distribution of
shells is not known for such cases. Some authors have guessed a major-merger
origin for the shell galaxies in their observational studies (Schiminovich et
al., 1995; Balcells et al., 2001; Goudfrooij et al., 2001; Serra et al.,
2006).
#### 6.7 Simulations with gas
Only a few works have been dedicated to modeling the formation of shell
galaxies in the presence of gas, all of them in the framework of the minor-
merger model. Weil and Hernquist (1993b) used a variant of the TREESPH code
but self-gravity was strictly ignored. The primary galaxy was treated as a
rigid spherically symmetric potential. They performed four runs – two radial
and two non-radial; two of them were prograde with the disk inclined by 45°.
Isothermal processes were assumed (T = $10^{4}$ K) except for one run where
radiative cooling was allowed, and at the end 94% particles had temperature
6,000–10,000 K. Main results are that in all cases gaseous and stellar debris
segregated and gas forms dense rings around the nucleus of the primary galaxy
where massive star formation may occur. Furthermore the diameter of the ring
depends on the impact parameter (the total angular momentum in the ring is 50%
of the initial value for those particles); radial and inclined encounter forms
a s-shaped ring and a counterrotating core; and about a half of all the gas
particles is captured in these rings.
A completely different conclusion was reached by Kojima and Noguchi (1997).
They used the sticky particle method (after collision, the radial velocity
component of the particle is halved and the sign reversed) and performed four
runs of simulation – radial (twice), prograde, retrograde (all with zero
inclination). Both galaxies were self-gravitating systems. Star formation was
modeled as a probability of a change of a gas particle to a stellar based on
local gas density. They found definitely no significant segregation of gas and
stars; star formation was mainly reduced because of scattering on the deep
potential well of the primary (radial and retrograde runs); for slightly
prograde orbit, the inner part of the secondary galaxy survives, a small
stellar bar of the secondary is created which causes bar-driven gas inflow and
a strong starburst. In the radial run with a less concentrated primary, a
larger part of the secondary survives and the oscillating remnant destroys the
shells. They state that the “poststarburst” nature of shell galaxies is due to
the cessation of star formation in the disk galaxies caused by the merger (no
massive star formation is caused by the encounter itself).
The model of Combes and Charmandaris (1999, 2000); Charmandaris and Combes
(2000) was based on the belief in two components of galactic gas – diffuse H I
gas ends in center of primary, while the small and dense gas clouds have an
intermediate behavior between stars and H I. They took into account the
dynamical friction and a proper treatment of the dissipation of the gas (using
cloud-cloud collision code). The gaseous component was liberated first since
it was less bound than stars. Then stars lose their energy due to the
dynamical friction what causes some displacement of the gaseous and stellar
shells. That was really observed in some shell galaxies, see Sect. 3.5.
#### 6.8 Merger model and observations
Merger models can well explain the interleaving of shells and their increasing
separation with radius (point 7 in Sect. 4) and the number of shells increases
with time. The observed brightness of shells puts a lower limit to the mass of
the original secondary galaxy that is usually several per cent of the primary
(point 4 in Sect. 4). The question of an alignment of shells with the major
axis of the host galaxy and the correlation between the type of the shell
galaxy and ellipticity (point 8 in Sect. 4) remains unsettled for the merger
model. The merger model has also problems explaining the large range of radii
where the shells are found and their occurrence at low radii (points 9 and 10
in Sect. 4). Mergers of different secondary galaxies can explain different
colors of shells and their possible difference from the color of the
underlying galaxy (point 12 in Sect. 4).
A merger origin of shell systems is supported by many observations, a list of
which would be lengthy. It seems that all the shell galaxies that have been so
far examined in detail contain dust close to the nucleus (point 14 in Sect.
4). These dust features are often found to be out of dynamical equilibrium
(Sect. 3.5), what clearly points to their external origin. Shell galaxies
contain even more characteristics believed to be the results of a merger,
including tidal tails, multiple nuclei or nuclear post-starburst spectra.
It seems that about 20% of shell galaxies could contain a second nucleus
(Sect. 3.7) – a characteristic that one would expect in a galaxy after a
merger event. Forbes et al. (1994) calculate that this could be an expected
frequency due to the short lifetime of the nucleus of the secondary galaxy as
opposed to the long-living shells. They note that it is also the expected
frequency for the WIM origin of shell galaxies – the galaxies with the double
nuclei would be those we see at the moment when the secondary galaxy just
passes through the primary.
A large support for the merger theories comes from the kinematically distinct
cores (KDCs). Even before it was recognized that all known galaxies with KDCs
in 1992 are shell galaxies, (point 21 in Sect. 4, see also Sect. 3.7), the
origin of KDCs from mergers of galaxies has been independently anticipated.
Already Kormendy (1984) proposed this mechanism for the formation of
counterrotating cores in elliptical galaxies and Balcells and Quinn (1990)
investigated this using self-consistent numerical simulations of mergers
between elliptical galaxies of unequal mass, and found that the core
kinematics in the remnant depend mostly upon the orbital angular momentum at a
late stage of the merger, whereas the kinematics of the outer regions is
largely the original kinematics of the primary. Thus, in retrograde encounters
a counter-rotating core can form. Hernquist and Barnes (1991), cited in
Turnbull et al. (1999), demonstrated the formation of a counterrotating
central gas disk in a merger of two gas-rich disk galaxies of equal mass. But
this model is less widely accepted than the previous one. Hau and Thomson
(1994) suggested a model that would comply with the WIM, but it is probably
even less popular.
Enormous diversity of central surface brightness (point 22 in Sect. 4) and
other characteristic show that shell galaxies are otherwise not a compact or
privileged group of galaxies – so to say, the secondary cannot choose on what
it falls. Still some selection effect seems to be there, because shell
galaxies are much more often seen in regions with low galactic density (point
2 in Sect. 4). That can be explained with velocities in galaxy clusters being
too high for one galaxy to be captured by another, or the influence of the
surrounding galaxies breaks the shells structure or even prevents it from
forming; or both.
Simulations show (Sect. 6.7) that in the framework of the merger model of
shells’ creation, diffuse gas is introduced into the center of the host galaxy
(point 20 in Sect. 4), while dense gas clouds form slightly displaced shells
with respect to the stellar shells (points 16 and 17 in Sect. 4). Both are in
agreement with the observations.
As the observations show, shells in galaxies are fairly common (point 1 in
Sect. 4, see also Sect. 3.2). It means that in fact they occur even more
frequently because from the three-dimensional shape of the shells as
introduced by Quinn (1984), Sect. 6, we can easily understand that we see
shells only when looking from angles close to the plane perpendicular to the
line of the collision. But it is not that improbable as the shells in mergers
are formed in a much larger range of impact parameters than it was originally
believed (see Sect. 6.6) and interactions between galaxies are quite a common
matter.
### 7 Measurements of gravitational potential in galaxies
Before we present our original results, we introduce the reader shortly to the
topic of measuring galactic potentials, particularly in the case of elliptical
and shell galaxies.
#### 7.1 Insight into methods
The issue of the determination of the overall potential and distribution of
the dark matter in galaxies is among the most prominent in galactic
astrophysics. In disk galaxies, where stars and gas move on near-circular
orbits, we can derive the potential (at least in the disk plane) directly up
to several tens of kiloparsecs from the center of the galaxy in question.
Early-type galaxies lack such kinematical beacons.
Several different methods have been used to measure the potentials and the
potential gradients of elliptical galaxies, including strong gravitational
lensing (e.g., Koopmans et al., 2006, 2009; Auger et al., 2010), weak
gravitational lensing (e.g., Mandelbaum et al., 2008), X-ray observations of
hot gas in the massive gas-rich galaxies (e.g., Fukazawa et al., 2006;
Churazov et al., 2008; Das et al., 2010), rotational curves from detected
disks and rings of neutral hydrogen (e.g., Weijmans et al., 2008), stellar-
dynamical modeling from integrated light spectra (e.g., Thomas et al., 2011),
as well using tracers such as planetary nebulae (e.g., Coccato et al., 2009),
globular clusters (e.g., Norris et al., 2012) and satellite galaxies (e.g.,
Nierenberg et al., 2011; Deason et al., 2012).
All the methods have various limits, e.g., the redshift of the observed
object, the luminosity profile, gas content, and so forth. In particular, the
use of stellar dynamical modeling is plausible in the wide range of galactic
masses, as far as spectroscopic data are available. However, it becomes more
challenging past few optical half-light radii. Moreover, the situation is made
complex by our insufficient knowledge of the anisotropy of spatial velocities.
Another complementary gravitational tracers or techniques are required to
derive mass profiles in outer parts of the galaxies. While comparing
independent techniques for the same objects at the similar galactocentric
radii, the discrepancies in the estimated circular velocity777The concept of
circular velocity is commonly used even in elliptical galaxies where none or
small amount of the matter is expected to move on circular orbits. It is a
quantity which says what speed would move the body launched into a circular
orbit. Provided spherical symmetry of the galaxy, it simply denotes the
quantity $\sqrt{r\phi^{\prime}(r)}$ , where $\phi^{\prime}(r)$ is the first
derivative of the galactic potential with respect to the galactocentric radius
$r$. curves were revealed together with several interpretations (e.g.,
Churazov et al., 2010; Das et al., 2010). The compared techniques usually
employ modeling the X-ray emission of the hot gas (assuming hydrostatic
equilibrium) and dynamical modeling of the optical data in the massive early-
type galaxies. Therefore, even for the most massive galaxies with X-ray
observations at disposal, there is a need for other methods to independently
constrain the gravitational potential at various radii.
#### 7.2 Use of shells
Using the radial distribution of shells to derive the potential of the host
galaxy seems tempting, but it insofar generally failed due to reasons
discussed in Sect. 6.4. The question remains whether it is better to use the
outer shells that are less affected by the dynamical friction and possible
later generations of shells, or if we could, by careful modeling of all the
relevant physical processes, reproduce the whole observed shell distribution
for a suitable potential.
An alternative hypothetical use of shells to determine the dark matter content
of galaxies is proposed by Sanderson et al. (2012). The increased
concentration of matter and its low velocity dispersion in the shells is
favorable for indirect detection of dark matter via gamma-ray emission from
dark matter self-annihilation due to the Sommerfeld effect.
A slightly less exotic, though not less bold method has been proposed by
Merrifield and Kuijken (1998). The method uses shells to constrain the form of
the gravitational potential in the case of validity of the Quinn (1984) merger
model (described in Sect. 6.1). They studied theoretically the kinematics of a
stationary shell, a monoenergetic spherically symmetric system of stars
oscillating on radial orbits in a spherically symmetric potential. They
predicted that spectral line profiles of such a system exhibit two clear
maxima, which provide a direct measure of the gradient of the gravitational
potential at the shell radius.
In practice, the situation is far more complex and the shells themselves are
faint structures in a bright galaxy, so the fulfillment of this program seems
almost impossible. However, the authors state that they have carried out
signal-to-noise ratio calculations for some of the brighter shell galaxies
such as NGC 3923, and have ascertained that data of the requisite quality
could be obtained with a couple of nights integration using a 4-m telescope.
Now comes the era when the instrumental equipment begins to allow us to
actually obtain such kind of data and that requires deeper theoretical
understanding of the topic. In Part II, we extend the work of Merrifield and
Kuijken (1998) and we develop methods to better reproduce parameters of the
potential of the host galaxy from measured data.
The first attempt to analyze the kinematical imprint of a shell
observationally was made by Romanowsky et al. (2012), who used globular
clusters as shell tracers in the early-type galaxy M87, the central galaxy in
the Virgo cluster. They obtained wide-field (0–200 kpc from the center) high-
precision (median velocity uncertainties: 14 km$/$s) spectroscopic data for
488 globular clusters. They found signatures of a cold stream (about 15
globular clusters at 150 kpc) and a large shell-like pattern (about 30
globular clusters between 50 and 100 kpc) and verified the presence of these
features using statistical tests. These features are the first large stellar
substructure with a clear kinematical detection in any type of galaxy beyond
the Local Group. The stream is associated with a known stellar filament but
there is no photometric shell visible in the galaxy. Typical surface
brightness in the region of the shell-like pattern is $\mu_{\mathrm{V}}\sim
27$ mag$/$arcsec2. Following the calculations of Merrifield and Kuijken
(1998), Romanowsky et al. (2012) derived circular velocity at the shell radius
$v\mathrm{{}_{c}\sim 270}$ km$/$s while X-ray data indicate
$v\mathrm{{}_{c}\sim}$650–900 km$/$s in the same region. Further analysis done
by the authors suggests that for such a shell to be created, the host galaxy
would have to accrete a large group of dwarf galaxies or a single giant
elliptical or a lenticular galaxy (about 5 times bigger than the entire Milky
Way system).
Fardal et al. (2012) obtained radial velocities (median error 3 km$/$s) of 363
red giant branch stars in the region of the so-called Western Shelf in M31,
the Andromeda galaxy. The Western Shelf, located about 25 kpc from the center
of the galaxy, is one of several features in the stellar halo of M31. In the
space of line-of-sight velocity velocity versus projected radius, the data
they obtained show a wedge-like pattern. This is consistent with the previous
finding of Fardal et al. (2007) who reproduced main photometric structures in
the stellar halo using a simulation of an accretion of a dwarf satellite
within the accurate M31 potential model. They inferred that the Western Shelf
is a shell from the third orbital wrap888If we considered the remains of the
accreted satellite to be a shell system, we would assign number 2 to this
shell, see Sect. 9.1. of a tidal debris stream. Using similar simulation,
Fardal et al. (2012) derived that the Western Shelf moves with phase velocity
of 40 km$/$s and that the wedge pattern has a global offset -20 km$/$s with
respect to the systemic velocity due to the angular momentum.
## ?partname? II Shell kinematics
A lot of useful information about the shell galaxies can be extracted from the
kinematics of the stars forming the shell system. That it is by measuring the
line-of-sight velocity distribution (LOSVD) near the edge of the shell. Now
comes the era when the instrumental equipment begins to allow us to actually
obtain such kind of data and that requires deeper theoretical understanding of
the topic. First attempts to analyze such kind of data have been already made,
see Sect. 7.2. The idea to use shell kinematics, has been proposed by
Merrifield and Kuijken (1998), hereafter MK98, and we further developed it in
papers Jílková et al. (2010) and Ebrová et al. (2012), Appendices
LABEL:apx:clanek-lucka and LABEL:apx:clanek-huevo, respectively.
?figurename? 8: Potential of the host galaxy. The potential is modeled as a
double Plummer sphere with parameters listed in Table 2.
### 8 Preliminary provisions
First we introduce several useful notions to aid the reader.
#### 8.1 Host galaxy potential model
In this part of the thesis, we will often need to illustrate the shell
kinematics using specific examples. For this purpose, the potential of the
host galaxy is modeled as a double Plummer sphere with parameters presented in
Table 1, unless specified otherwise. This model has properties consistent with
observed massive early-type (and even shell) galaxies (Auger et al., 2010;
Nagino and Matsushita, 2009; Fukazawa et al., 2006). The forms of the
potential and density for the chosen model are shown in Figs. 8 and 9,
respectively.
| Plummer radius | total mass
---|---|---
| kpc | M⊙
luminous component | 5 | $2\times 10^{11}$
dark halo | 100 | $1.2\times 10^{13}$
?tablename? 1: Parameters of the potential of the host galaxy used in Part II.
The potential is modeled as a double Plummer sphere.
The potential of a Plummer sphere can be expressed as
$\phi(r)=-\frac{\mathrm{G}\,M}{\sqrt{r^{2}+\varepsilon^{2}}},$ (1)
where G is the gravitational constant, $M$ is the total mass of the galaxy,
$r$ is the distance from the center of the galaxy and $\varepsilon$ is the
Plummer radius. The radial density then reads
$\rho(r)=\rho_{0}\frac{1}{(1+r^{2}/\varepsilon^{2})^{5/2}},$ (2)
where $\rho_{0}=3M/(4\pi\varepsilon^{3})$ is the central density. The
interested reader can find more on the Plummer potential in Sects. 17.2–17.4.
Let us note that such a choice of the potential of the host galaxy represents
a whole class of models. For example, we can express all distances in the
terms of the Plummer radius of the luminous component and all masses in the
terms of the total mass of the luminous component and then choose these two
parameters at will. For clarity, we nevertheless keep the specific values
noted below.
?figurename? 9: Density of the host galaxy. The potential is modeled as a
double Plummer sphere with parameters listed in Table 2.
#### 8.2 Terminology
In this section, we briefly introduce terms used in next sections.
* •
Model of radial oscillations – through Part II, the word model is assigned to
the concept described in Sect. 9 and used for modeling of shell kinematics.
The model assumes that shells are made by stars on strictly radial orbits
released at one moment in the center of the host galaxy. The potential of the
host galaxy is chosen to represent real galaxies reasonably well. In our work,
we restrict ourselves to a double Plummer sphere introduced in Sect. 8.1.
* •
Approximation of constant acceleration and shell velocity (Sect. 11) – it is
basically the model of radial oscillations but the value of acceleration in
the host galaxy as well as the value of the shell phase velocity are always
constant. The approximation is assumed to be valid only in the vicinity of the
shell edge. In the framework of this approximation, the position of line-of-
sight velocity maxima are calculated using either of the following three
methods: the approximative LOSVD (Sect. 11.2); the approximative maximal LOS
velocities (Sect. 11.4); and and the method using the slope of the LOSVD
intensity maxima (Sect. 11.5). Differences between these methods are
summarized in Sect. 11.6.
* •
Higher order approximation (Sect. 12) – similarly as previous, but this time
we allow the value of acceleration in the host galaxy to change linearly with
galactocentric radius.
* •
Simulation – in this part, we only use this term when we model shell galaxies
in the simulation of a radial minor merger of galaxies using test particles
(Sect. 13).
#### 8.3 Quantities
* $t$
time; usually indicates the time since the release of stars at the center of
the host galaxy
* $\mathbf{r}=(x,y,z)$
vector of Cartesian coordinates that are oriented so that the origin is at the
center of the host galaxy; $x-y$ is the projected plane (“the sky”) and the
$z$ direction coincides with the line of sight (LOS); $x$-axis is also the
collision axis although in the model of radial oscillations it is just a
virtual concept, since no collision is actually modeled
* $X,Y$
coordinates of the projected plane
* $r$
galactocentric radius, distance from center of the galaxy;
$r=\sqrt{x^{2}+y^{2}+z^{2}}$
* $R$
projected radius, the projection of $r$ into the $x-y$ plane;
$R=\sqrt{x^{2}+y^{2}}$
* $\phi(r)$
potential of the host galaxy; in this part, we use a spherically symmetric
potential introduced in Sect. 8.1; parameters of the potential are the total
mass $M_{*}$, $M_{\mathrm{DM}}$ and the scale radius $\varepsilon_{*}$,
$\varepsilon_{\mathrm{DM}}$ of the luminous and dark component, respectively
* $\rho(r)$
spatial density (in a spherically symmetric system)
* $v_{\mathrm{c}}$
circular velocity; provided spherical symmetry of the galaxy, it simply
denotes the quantity $\sqrt{r\phi^{\prime}(r)}$ , where $\phi^{\prime}(r)$ is
the first derivative of the galactic potential with respect to the
galactocentric radius $r$.
* $a$
acceleration in the host galaxy; $a_{0}$ is the constant term and $a_{1}$ is
the coefficient of a linear term of the expansion of the acceleration around
the shell edge
* $T(r)$
period of radial motion at the galactocentric radius $r$ in the host galaxy
potential; Eq. (4)
* $n$
serial number of a shell; shells are traditionally numbered from the outermost
to the innermost ones; Sect. 9.1
* $r_{\mathrm{TP}}$
current turning point, i.e. the radius where the stars are located in their
apocenters at a given moment (the moment of measurement); Eq. (3)
* $v_{\mathrm{TP}}$
phase velocity of a current turning point; Eq. (5)
* $r_{\mathrm{*}}$
position of a star at a given time $t$ since the release of the star in the
center of the host galaxy; Eqs. (6) and (7); often plain $r$ also denotes the
position of stars but the meaning is clear from the context
* $r_{\mathrm{ac}}$
position of the apocenter of a star (uniquely related to the energy of the
star for radial orbits); Eqs. (6) and (7)
* $v_{\mathrm{r}}$
stellar velocity at the galactocentric radius $r$; in the model of radial
oscillations the stellar velocity is always in the radial direction
* $r_{\mathrm{s}}$
position of the edge of a shell, a function of time $r_{\mathrm{s}}(t)$; Sect.
9.2, Eq. (8)
* $r_{\mathrm{s0}}$
position of the shell edge at the moment of measurement
* $v_{\mathrm{s}}$
phase velocity of a shell edge; approximately equal to $v_{\mathrm{TP}}$; Eq.
(9)
* $t_{\mathrm{s}}$
time when a star currently at radius $r$ will or did reach the corresponding
edge of the shell; Sect. 11.1
* $v_{\mathrm{los}}$
line-of-sight velocity; the projection of the stellar velocity into $z$
direction; $v_{\mathrm{los}}=v_{r}z/r$
* $v_{\mathrm{los,max}}$
the maximal absolute value of the LOS velocity
* $r_{v\mathrm{max}}$
radius of maximal LOS velocity, radius from which comes the contribution to
the LOSVD at the maximal speed $v_{\mathrm{los,max}}$; Sect. 11.3
* $z_{v\mathrm{max}}$
spot at the line of sight from which comes the contribution to the LOSVD at
the maximal speed; $z_{v\mathrm{max}}=\pm\sqrt{r_{v\mathrm{max}}^{2}-R^{2}}$,
Sect. 9.8
* $F(v_{\mathrm{los}})$
line-of-sight velocity distribution (LOSVD); Eq. (11)
* $\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$
shell-edge density distribution; Eq. (13), Sects. 9.6, 9.7, and 9.8
* $\Sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$
discrete equivalents of $\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$;
Eq. (17)
* $\Sigma_{\mathrm{los}}\left(R\right)$
projected surface density, the projection of spacial density into the $x-y$
plane
### 9 Model of radial oscillations
If we approximate the shell system with a simplified model, we can describe
its evolution completely depending only on the potential of the host galaxy.
The approximation lies in the numerical integration of radial trajectories of
stars in a spherically symmetric potential. Stars behave as if they were
released in the center of the host galaxy at the same time and their
distribution of energies is continuous. Usually we demand that the
distribution is continuous at least in such a range that stars with apocentra
10–30 kpc around the edge of the observed shell are present. Moreover we need
that their density in this region does not go sharply to zero. In some cases,
we need to know the distribution of energies explicitly. We express it in
terms of the shell-edge density distribution (Sects. 9.6), which is a quantity
more suitable for our situation and which can be unambiguously converted to
the distribution of energies or the initial velocity distribution (Appendix
C). We show that the particular choice of the function does not affect the
results presented in this work (Sects. 9.7).
We call this model the model of radial oscillations, and it corresponds to the
notion that the cannibalized galaxy came along a radial path and disintegrated
in the center of the host galaxy. As a result the stars were released at one
moment in the center and began to oscillate freely on radial orbits. This
approach was first used by Quinn (1984), followed by Dupraz and Combes (1986,
1987) and Hernquist and Quinn (1987a); Hernquist and Quinn (1987b).
This model uses the exact knowledge of the chosen potential of the host
galaxy, but requires it to be spherically symmetric. The potential can be
given analytically or numerically and the stellar trajectories are usually
integrated numerically. It differs from the real shell galaxies in several
aspects but it is still the most exact analytical model that we can easily
construct. We will show that in this model, the LOSVD of shells exhibits four
intensity maxima and how the position of these maxima are connected with the
parameters of the host galaxy potential. All the following approximations will
be compared to the model of radial oscillations. Later we will show that the
model agrees very well with results of test-particle simulations of the
formation of the shell galaxies (Sect. 13).
#### 9.1 Turning point positions and their velocities
In shell galaxies, the shells are traditionally numbered according to the
serial number of the shell, $n$, from the outermost to the innermost (which in
the model of radial oscillations for a single-generation shell system
corresponds to the oldest and the youngest shell, respectively). If the
cannibalized galaxy comes from the right side of the host galaxy, stars are
released in the center of the host galaxy. After that, they reach their
apocenters for the first time. But a shell does not form here yet, because the
stars are not sufficiently phase wrapped. We call this the zeroth oscillation
(the zeroth turning point) as we try to match the number of oscillations with
the customary numbering scheme of the shells. We label the first shell that
occurs on the right side (the same side from which the cannibalized galaxy
approached) with $n=1$. Shell no. 2 appears on the left side of the host
galaxy, no. 3 on the right, and so forth.
In the model of radial oscillations, the shells occur close to the radii where
the stars are located in their apocenters at a given moment (the current
turning point, $r_{\mathrm{TP}}$, in our notation). The shell number $n$
corresponds to the number of oscillations that the stars near the shell have
completed or are about to complete. The current turning point
$r_{\mathrm{TP}}$ must follow the equation
$t=(n+1/2)T(r_{\mathrm{TP}}),$ (3)
where $t$ is the time elapsed since stars were released in the center of the
host galaxy. $T(r)$ is the period of radial motion at a galactocentric radius
$r$ in the host galaxy potential $\phi(r)$:
$T(r)=\sqrt{2}\int_{0}^{r}\left[\phi(r)-\phi(r^{\prime})\right]^{-1/2}\mathrm{d}r^{\prime}.$
(4)
The radial period is defined as the time required for a star to travel from
apocenter to pericenter and back (Binney and Tremaine, 1987).
The position of the current turning point evolves in time with a velocity
given by the derivative of Eq. (3) with respect to radius
$v_{\mathrm{TP}}(r;n)=\mathrm{d}r/\mathrm{d}t=\frac{1}{\mathrm{d}t/\mathrm{d}r}=\frac{1}{n+1/2}\left(\mathrm{d}T(r)/\mathrm{d}r\right)^{-1}.$
(5)
We can clearly see from this relation, which was first derived by Quinn
(1984), that any further turning point (turning point with higher $n$) at the
same radius moves more slowly than the former one. Thus causes a gradual
densification of the space distribution of the shell system with time.
Technically, the reason for this densification is that the time difference
between the moments when two stars with similar energy reach their turning
points is cumulative. Let $\bigtriangleup t$ be the difference in periods at
two different radii $r_{\mathrm{a}}$ and $r_{\mathrm{b}}$ (with
$r_{\mathrm{a}}<r_{\mathrm{b}}$, on the right). The radius where stars
complete the first oscillation moves from $r_{\mathrm{a}}$ to $r_{\mathrm{b}}$
in $\bigtriangleup t$. But in the second orbit on the left, the stars from
$r_{\mathrm{b}}$ will already have a lag of $\bigtriangleup t$ behind those
from $r_{\mathrm{a}}$ and will just be getting a second one, so the third one
(the second on the same side) reaches $r_{\mathrm{b}}$ from $r_{\mathrm{a}}$
in $3\times\bigtriangleup t$. Every $n$th completed oscillation on the right
side, then moves $n$ times more slowly than the first one. The situation is
similar on the left side, and the shell system is getting denser. Moreover,
the turning point has an additional lag of $1/2T(r_{\mathrm{TP}})$, because
the stars were released in the center of the host galaxy before their zeroth
oscillation. This is the source of the factor $(n+1/2)$ in Eqs. (3) and (4).
#### 9.2 Real shell positions and velocities
Even in the framework of the model of radial oscillations, the position and
velocity of the true edge of the shell cannot be expressed in a
straightforward manner. Photometrically, shells appear as a step in the
luminosity profile of the galaxy with a sharp outer cut-off. This is because
the stars of the cannibalized galaxy occupy a limited volume in the phase
space. With time, the shape of this volume gets thinner, more elongated, and
wrapped around invariant surfaces defined by the trajectories of the stars in
the phase space, increasing its coincidence with these surfaces. A shell
appears close to the points where the invariant surface is perpendicular to
the plane of the sky (Nulsen, 1989). For the $n$th shell, this is the largest
radius where stars about to complete their $n$th oscillation are currently
located. This radius corresponds to the shell edge (Sect. 9.3) and it is
always larger than that of the current turning point of the stars that are
completing their $n$th oscillation. Thus, the shell edge consists of outward-
moving stars about to complete their $n$th oscillation.
Dupraz and Combes (1986) state that the stars forming the shell move with the
phase velocity of the shell. While we show that this holds only roughly, we
use this approximation in Sect. 11 to derive the relation between the shell
kinematics and the potential of the host galaxy.
The position of a star, $r_{\mathrm{*}}$, at a given time $t$ since the
release of the star in the center of the host galaxy is given by an implicit
equation for $r_{\mathrm{*}}$ and is a function of the star energy, or
equivalently the position of its apocenter $r_{\mathrm{ac}}$.999We denote the
apocenter of the star corresponding to its energy as $r_{\mathrm{ac}}$,
whereas $r_{\mathrm{TP}}$ (the current turning point) is the radius at which
the stars reach their apocenters at the time of measurement. For stars with
the integer part of $t/[2T(r_{\mathrm{ac}})]$ odd, the equation reads:
$\begin{array}[]{rcl}t=(n+1)\sqrt{2}&\int_{0}^{r_{\mathrm{ac}}}&\left[\phi(r_{\mathrm{ac}})-\phi(r^{\prime})\right]^{-1/2}\mathrm{d}r^{\prime}-\\\
-&\int_{0}^{r_{\mathrm{*}}}&\left[2(\phi(r_{\mathrm{ac}})-\phi(r^{\prime}))\right]^{-1/2}\mathrm{d}r^{\prime}.\end{array}$
(6)
For stars that have completed an even number of half-periods (only such stars
are found on the shell edge), the equation is
$\begin{array}[]{rcl}t=n\sqrt{2}&\int_{0}^{r_{\mathrm{ac}}}&\left[\phi(r_{\mathrm{ac}})-\phi(r^{\prime})\right]^{-1/2}\mathrm{d}r^{\prime}+\\\
+&\int_{0}^{r_{\mathrm{*}}}&\left[2(\phi(r_{\mathrm{ac}})-\phi(r^{\prime}))\right]^{-1/2}\mathrm{d}r^{\prime}.\end{array}$
(7)
The first term in Eq. (7) corresponds to $n$ radial periods for the star’s
energy ($n$ is maximal so that $nT(r_{\mathrm{ac}})<t$), while the other term
corresponds to the time that it takes to reach radius $r_{\mathrm{*}}$ from
the center of the galaxy. Even for the simplest galactic potentials, these
equations are not analytically solvable and must be solved numerically.
The position of the $n$th shell $r_{\mathrm{s}}$ equals the maximal radius
$r_{\mathrm{*}}$ that solves Eq. (7) for the given $n$.101010In the
approximation of a constant shell velocity, $v_{\mathrm{s}}$, and a constant
galactocentric acceleration, $a_{0}$ (Sect. 11), the distance between the
current turning points and the shell radius is
$r_{\mathrm{s}}-r_{\mathrm{TP}}=-v_{\mathrm{s}}^{2}/(2a_{0})$. In symbolic
notation
$r_{\mathrm{s}}=\mathrm{max}\\{r_{\mathrm{*}}(r_{\mathrm{ac}});\left\lfloor
t/T(r_{\mathrm{ac}})\right\rfloor=n-1\\},$ (8)
where $r_{\mathrm{*}}(r_{\mathrm{ac}})$ is an implicit function given by Eq.
(7). Simultaneously, we require $r_{\mathrm{ac}}$ to satisfy the equation
$\left\lfloor t/T(r_{\mathrm{ac}})\right\rfloor=n-1$, where $\left\lfloor
x\right\rfloor$ indicates the integer part of $x$, so that $\left\lfloor
t/T(r_{\mathrm{ac}})\right\rfloor$ is the number of periods completed by the
star since the release of the star in the center of the host galaxy. Radial
period $T(r_{\mathrm{ac}})$ is defined by Eq. (4) and $n$ is the serial number
of the shell for which we want to find the edge radius $r_{\mathrm{s}}$.
Such a radius is actually identical to the step in projected surface density
that corresponds to the shell edge (Sect. 9.3). For a shell with nonzero phase
velocity the shell edge is always further from the center than the current
turning point, $r_{\mathrm{TP}}<r_{\mathrm{s}}$. On the other hand, the
apocenter $r_{\mathrm{ac}}$ of a star currently located at the shell edge is
obviously further from the center than the current shell edge position.
The shell velocity $v_{\mathrm{s}}$ is obtained from the numerical derivative
of a set of values of $r_{\mathrm{s}}$ for several close values of $t$
$v_{\mathrm{s}}=\mathrm{d}r_{\mathrm{s}}/\mathrm{d}t.$ (9)
The stellar velocity at the shell edge, $v_{r}(r_{\mathrm{s}})$, is obtained
by inserting $r_{\mathrm{s}}$ with its corresponding111111By corresponding we
mean that the pair of values $r_{\mathrm{s}}=r_{*}$ and $r_{\mathrm{ac}}$
solves Eq. (7) for a given time $t$ since the release of the star in the
center of the host galaxy, a given serial number $n$ of a shell and a given
potential of the host galaxy $\phi(r)$. $r_{\mathrm{ac}}$ into:
$v_{r}(r_{\mathrm{*}})=\pm\sqrt{2[\phi(r_{\mathrm{ac}})-\phi(r_{\mathrm{*}})]}.$
(10)
For the stars following Eq. (7), the velocity will be positive; for the rest,
it will be negative. The positive velocity means that the stars are moving
outward. The edge of a shell is exclusively made up of stars with positive
velocities. Recall that the star moves along radial trajectories.
It is clear that $v_{r}(r_{\mathrm{s}})\leq v_{\mathrm{s}}$. Actually,
$v_{r}(r_{\mathrm{s}})$ is lower than the phase velocity of the shell (Table
2) but the difference between the values of these velocities is small. At the
same time, the position of the shell for a given time is not far from the
current turning point, and their separation changes slowly in galactic
potentials. Thus, the velocity of the turning points given in Eq. (5) is a
good approximation for the shell velocity (Fig. 14). Eq. (5) is not generally
solvable analytically either, but the numerical calculation of
$v_{\mathrm{TP}}$ is much easier than determining the true shell velocity
$v_{\mathrm{s}}$. The procedure to calculate $v_{\mathrm{s}}$ is described in
this section.
?figurename? 10: Projected surface density of shells in the host galaxy, with
potential introduced in Sect. 8.1, 2.2 Gyr after the release of the stars in
the center. The scale bar is logarithmic in arbitrary units.
#### 9.3 Appearance of the shells
The model of radial oscillations is primarily used for calculating the
positions of LOSVD maxima. Nevertheless, we can also use it to derive the
spatial and projected surface density of the stars that form the shell
($\rho(r)$ and $\Sigma_{\mathrm{los}}(R)$, respectively) and the shape of the
LOSVD itself. We do not aim to produce these quantities with such a precision
that would be required for comparison with observation within this model. But
we can still have a look at them to obtain qualitative insight, although their
exact shape is not important for our work.
To do that, it is not sufficient to know the kinematics as described in Sect.
9.2 but we need to add an assumption about the radial dependence of the shell-
edge density distribution $\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$.
We chose this to correspond to a constant number of stars at the edge of the
shell; for more details, see Sects. 9.6, 9.7, and 9.8. Furthermore we assume
that the density of stars on the shells has uniform angular distribution. In
most cases, we follow the shell kinematics only between
$0.9r_{\mathrm{s}}-r_{\mathrm{s}}$ and thus an opening angle of at least
$51.7\text{\textdegree}$ is sufficient.
Fig. 10 shows the projected surface density of the five outermost shells at
2.2 Gyr after the release of the stars in the center of the host galaxy (for
parameters of the potential, see Sect. 8.1). Projected surface density of the
host galaxy itself is not displayed. The opening angle of the shells is chosen
to be the full $180\text{\textdegree}$. Shells with an odd serial number are
to the right, those with an even number to the left, corresponding to the
cannibalized galaxy flying in from the right hand side of the host galaxy. The
whole picture is analogical to the results of the N-particle simulation
analyzed in Sect. 13.2.
In practice, such a projected surface density depends only on the projected
radius $R$ and it is shown also in Fig. 11. Jumps in the density do indeed
correspond to the radius $r_{\mathrm{s}}$ in the sense in which it is
introduced in Sect. 9.2, Eq. (8).
?figurename? 11: Projected surface density of shells with respect to the
projected radius, the same as in Fig 10.
#### 9.4 Kinematics of shell stars
In the model of radial oscillations, we can also describe the LOSVD of a shell
at a given time $t$, for a given potential of the host galaxy $\phi(r)$. Eqs.
(6) and (7) determine the current star position $r_{\mathrm{*}}$ and the shell
number $n$ for any apocenter $r_{\mathrm{ac}}$ in a range of energies. The
radial velocity of a star on the particular radius is given by inserting the
corresponding pair of $r_{\mathrm{ac}}$ a $r_{\mathrm{*}}$ in Eq. (10).
Naturally, the projections of these velocities to the selected line of sight
(LOS) form the LOSVD, which can be formally expressed by Eq. (15). To
reconstruct the LOSVD, we have to add an assumption about the radial
dependence of the shell-edge density distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$. We chose this to
correspond to a constant number of stars at the edge of the shell,
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto
1/r_{\mathrm{s}}^{2}$. In Sects. 9.6, 9.7, and 9.8, we deal with this function
in detail and show that the particular choice does not matter much. Here we
concisely describe the LOSVD at the projected radius $R$ which is less than
the position of current turning points, $R<r_{\mathrm{TP}}$. The other case
($r_{\mathrm{TP}}<R<r_{\mathrm{s}}$) is discussed in Sect. 9.5.
?figurename? 12: Left: Scheme of the kinematics of a shell with radius
$r_{\mathrm{s}}$ and phase velocity $v_{\mathrm{s}}$. The shell is composed of
stars on radial orbits with radial velocity $v_{\mathrm{r}}$ and LOS velocity
$v_{\mathrm{los}}$. Right: The LOSVD at projected radius
$R=0.9r_{\mathrm{s}}$, where $r_{\mathrm{s}}=120$ kpc (parameters of the shell
are highlighted in bold in Table 2), in the framework of the model of radial
oscillations. The profile does not include stars of the host galaxy, which are
not part of the shell system, and is normalized, so that the total flux equals
one. (a) The LOSVD showing separate contributions from inward and outward
stars; (b) the same profile, separated for contributions from the near and far
half of the host galaxy.
Mr. Eggy measures the LOSVD of stars in the shell, which is composed of inward
and outward stars on radial trajectories as illustrated in Fig. 12. The stars
near the edge of the shell move slowly. But it is clear from the geometry that
contributions add up from different galactocentric distances, where the stars
are either still traveling outwards to reach the shell or returning from their
apocenters to form a nontrivial LOSVD.
For every galactocentric distance $r$ intersected by the line of sight $z$,
there is a different radial stellar velocity $v_{\mathrm{r}}$ and a different
projection factor $z/r$. The maximal/minimal LOS velocity comes from stars at
two particular locations along the line of sight (A and B), both of which are
at the same galactocentric distance for outward or inward stars (the radii of
maximal LOS velocity, Sects. 9.8 and 9.8;
$r_{\mathrm{A}}^{\mathrm{outward}}=r_{\mathrm{B}}^{\mathrm{outward}}\equiv
r_{v\mathrm{max}}^{\mathrm{outward}}$;
$r_{\mathrm{A}}^{\mathrm{inward}}=r_{\mathrm{B}}^{\mathrm{inward}}\equiv
r_{v\mathrm{max}}^{\mathrm{inward}}$). For inward stars, points A and B are
closer to the center of the host galaxy than for outward stars
($r_{v\mathrm{max}}^{\mathrm{inward}}<r_{v\mathrm{max}}^{\mathrm{outward}}$)
as indicated in Fig. 12 on the left. This will be discussed more precisely in
Sect. 11.6 (see also Fig. 21). The maximal/minimal LOS velocity corresponds to
the intensity maximum of the LOSVD, as can be seen in the right-hand panels of
Fig. 12. The nature of this correspondence is explained in Sect. 9.8.
The edge of the shell moves outwards with velocity $v_{\mathrm{\mathrm{s}}}$.
At any given instant, the stars that move inwards are returning from a point
where the shell edge was at some earlier time, and so their apocenter is
inside the current shell radius $r_{\mathrm{s}}$. Similarly, the stars that
move outwards will reach the shell edge in the future. Consequently, the stars
that move inwards are always closer to their apocenter than those moving
outwards at the same radius, and their velocity is thus smaller. The inward
stars move toward Mr. Eggy in the farther of the two points (A) and away from
them in the nearer point (B), while the stars moving outwards behave in the
opposite manner. Together, there are four possible velocities with the maximal
contribution to the LOSVD, resulting in its symmetrical quadruple shape shown
in Fig. 12. In the picture, the intensity maxima coincide with velocity
extremes for separate contributions to the LOSVD (for more details, see Sect.
9.8).
?figurename? 13: Locations of peaks of the LOSVDs in the framework of the
model of radial oscillations: (a) for the first shell at different radii, (b)
for the first to the fourth shell at the radius of 120 kpc. Parameters of all
shells are shown in Table 2. For parameters of the host galaxy potential, see
Sect. 8.1.
#### 9.5 Characteristics of spectral peaks
In this section we describe and demonstrate the characteristics of the LOSVD
maxima in the model of radial oscillations using a particular host galaxy
model. We model the potential of the host galaxy as a double Plummer sphere,
as described in Sect. 8.1.
The separation between peaks of the LOSVD for a given projected radius $R$ is
given by the distance of $R$ from the edge of the shell $r_{\mathrm{s}}$. The
profile shown in Fig. 12 corresponds to projected radius
$R=0.9r_{\mathrm{s}}$. The closer to the shell edge, the narrower the profile
is. The separation of the peaks at a given $R$ depends on the phase velocity
of the specific shell, near which we observe the LOSVD. This velocity is, for
a fixed potential, given by the shell radius and its serial number (Sect.
9.1). These effects are illustrated in Fig. 13, where we show the positions of
the LOSVD peaks for the first shell at different radii $r_{\mathrm{s}}$ and
for a shell at 120 kpc with different serial numbers $n$. Note that the higher
the serial number $n$ at a given radius, the smaller is the difference in the
phase velocity between the two shells with consecutive serial numbers and thus
in the positions of the respective peaks. Parameters of the corresponding
shells can be found in Table 2.
$t$ | $n$ | $r_{\mathrm{s}}$ | $r_{\mathrm{TP}}$ | $v_{\mathrm{s}}$ | $v_{r}(r_{\mathrm{s}})$ | $v_{\mathrm{TP}}$ | $v_{\mathrm{c}}$
---|---|---|---|---|---|---|---
Myr | | kpc | kpc | km$/$s | km$/$s | km$/$s | km$/$s
215 | 1 | 15 | 14.5 | 63.5 | 57.5 | 61.2 | 245
416 | 1 | 30 | 28.3 | 90.3 | 82.6 | 81.0 | 261
634 | 1 | 60 | 53.9 | 165.8 | 151.5 | 151.8 | 362
1006 | 1 | 120 | 113.9 | 142.4 | 133.3 | 141.8 | 450
1722 | 2 | 120 | 117.9 | 84.7 | 79.4 | 84.7 | 450
2428 | 3 | 120 | 118.9 | 60.3 | 54.6 | 60.3 | 450
3130 | 4 | 120 | 119.3 | 46.8 | 42.6 | 47.0 | 450
?tablename? 2: Parameters of shells for which the LOSVD intensity maxima are
shown in Fig. 13. $t$: time since the release of stars at the center of the
host galaxy, in which the shell has reached its current radius calculated in
the framework of the model of radial oscillations; $n$: serial number of a
shell (Sect. 9.1); $r_{\mathrm{s}}$: shell radius; $v_{\mathrm{s}}$: shell
phase velocity according to the method described in Sect. 9.2;
$r_{\mathrm{TP}}$: galactocentric radius of the current turning points of the
stars at this time, given by Eq. (3); $v_{r}(r_{\mathrm{s}})$: radial velocity
of the stars at the shell edge; $v_{\mathrm{TP}}$: phase velocity of the
current turning point according Eq. (5); $v_{\mathrm{c}}$: circular velocity
at the shell-edge radius. For parameters of the host galaxy, see Sect. 8.1.
The shell that is used in Figs. 12, 15–18, and 21–25 is highlighted in bold.
?figurename? 14: Dependence of the phase velocity of the turning points on the
galactocentric radius for the first four shells according to Eq. (5). For
parameters of the host galaxy potential, see Sect. 8.1. Black crosses show the
true velocity of the first shell calculated for several radii according to the
method described in Sect. 9.2.
The radial dependence of the phase velocity of the first four shells in the
whole host galaxy is shown in Fig. 14. Using Eq. (5), we see that the velocity
of each subsequent shell differs from the first one only by a factor of
$3/(1+2n)$. The large interval of the galactocentric radii where the shell
velocity increases is caused by the presence of the halo with a large scaling
parameter. In fact, we do not show the shell velocity, but the velocity of the
turning points at the same radius. Nevertheless, these are sufficiently close.
Black crosses show the true velocity of the first shell calculated for several
radii according to the method described in Sect. 9.2. For shells of higher
$n$, these differences between the phase velocity of a shell and the
corresponding turning point with consecutive serial numbers are even smaller.
The edge of a moving shell is at the radius, which is always slightly further
from the center than the current turning points. Between these radii
($r_{\mathrm{TP}}<R<r_{\mathrm{s}}$), there is an intricate zone, where all
the stars of a given shell move outwards. As shown in Fig. 15, when the LOS
radius from lower radii gets near to the turning points of the stars, the
inner maxima of the LOSVD approach each other until they merge and finally
disappear. We actually see a minimum in the middle of the LOSVD closer to the
shell edge than the current turning points. The intricate zone is much larger
for the first shell. For the shell radius of 120 kpc in our host galaxy
potential, it occupies 6 kpc for the first shell, 2 kpc for the second one,
and less than one kpc for the fourth shell (Table 2).
?figurename? 15: Evolution of the LOSVD near the shell edge for the second
shell at $r_{\mathrm{s}}=120$ kpc (parameters of the shell are highlighted in
bold in Table 2) for the projected radius 116, 117, 118, and 119 kpc in the
framework of the model of radial oscillations. In this model, the current
turning points of stars in the shell are at $r_{\mathrm{TP}}=117.9$ kpc. For
$R>r_{\mathrm{TP}}$ the inner maxima disappear. Profiles do not include stars
of the host galaxy, which are not part of the shell system and are normalized
so that the total flux equals one. For parameters of the host galaxy
potential, see Sect. 8.1.
#### 9.6 Equations of LOSVD
We want to investigate the LOSVD, $F(v_{\mathrm{los}})$, on a given projected
radius $R$ for one particular shell. Assuming cylindrical symmetry of the
shell system,
$F^{\mathrm{near}}(v_{\mathrm{los}})=F^{\mathrm{far}}(-v_{\mathrm{los}})$,
where the superscripts indicate the near and far half of the galaxy. The total
LOSVD is obtained adding the two contributions together.
$F^{\mathrm{far}}(v_{\mathrm{los}})$ form the far half of the galaxy is given
by the integral of the distribution of shell stars
$f(\mathbf{r},v_{\mathrm{los}})$ along the line of sight
$F^{\mathrm{far}}(v_{\mathrm{los}})=\int_{0}^{z_{\mathrm{fin}}}f(\mathbf{r},v_{\mathrm{los}})\mathrm{d}z.$
(11)
In the model of radial oscillations, we assume spherical symmetry of the shell
system and thus the distribution function depends only on galactocentric
radius $r$. Moreover, in this model, stars are located on a three-dimensional
hypersurface in the six-dimensional phase space as they move as if they were
released all at once in the center of the galaxy. In this case
$z_{\mathrm{fin}}=\sqrt{r_{\mathrm{s}}^{2}-R^{2}}$. Furthermore, for a given
$r$, in each moment there are only two possible values for the radial
velocity, $v_{r1}$ and $v_{r2}$, therefore only two possible values for its
projection to the line of sight, thus
$f(\mathbf{r},v_{\mathrm{los}})=\rho_{1}(r)\delta[v_{\mathrm{los}}-\frac{z}{r}v_{r1}]+\rho_{2}(r)\delta[v_{\mathrm{los}}-\frac{z}{r}v_{r2}],$
(12)
where $\delta$ is the Dirac delta function; and $\rho_{1}(r)$ and
$\rho_{2}(r)$ are the densities of stars with the velocities $v_{r1}$ and
$v_{r2}$, respectively. The values $v_{r1}$ and $v_{r2}$ are taken from Eq.
(10), into which we put both pairs $[r;r_{\mathrm{ac}1}]$ and
$[r;r_{\mathrm{ac}2}]$, that solve Eqs. (6) and (7) in Sect 9.2 for given
galactic potential $\phi(r)$, time $t$ since the release of the star, and
serial number $n$ of a shell. In Eqs. (6) and (7) $r$ is substituted for
$r_{*}$ and $r_{\mathrm{ac}1}$ or $r_{\mathrm{ac}2}$ for $r_{\mathrm{ac}}$.
To evaluate the density, $\rho(r)$, let us first define
$N\left(r_{\mathbf{s}}\right)$ as the probability density for stars to have
their shell radius within an interval
$(r_{\mathbf{s}},r_{\mathbf{s}}+\mathrm{d}r_{\mathbf{s}})$. Then we can define
the distribution $\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ as
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)=m\frac{N\left(r_{\mathbf{s}}\right)}{r_{\mathbf{s}}^{2}},$
(13)
where $m$ is the (average) mass of a star. We call
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ the shell-edge density
distribution. In this case, $r_{\mathbf{s}}$ is a function of the stellar
energy, $r_{\mathbf{s}}(r_{\mathrm{ac}})$, and stands for the value of the
shell edge radius at the moment when the star with the corresponding energy is
at the shell edge.
The radial dependence of $\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$
determines the time evolution of the projected surface density of a shell,
Sect. 14. The shell-edge density distribution also determines what the
distribution of stellar velocities was at the time of their release in the
center of the host galaxy, see Appendix C.
The spatial density $\rho(r)$ is given by
$\rho(r)=\sum_{i=1}^{2}\frac{r_{\mathrm{s}i}^{2}(r)}{r^{2}}\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}i}(r)\right)\frac{\mathrm{d}r_{\mathrm{s}i}(r)}{\mathrm{d}r},$
(14)
where $r_{\mathrm{s}}(r)$ is the location where the stars, currently situated
at the radius $r$, will or did reach their respective shell edge, and
$r_{\mathrm{s}}(r)$ has two solutions, $r_{\mathrm{s1}}(r)$ and
$r_{\mathrm{s2}}(r)$, for one $r$, where $0<r<r_{\mathrm{s}}$.
Eq. (14) is easy to understand: the first fraction,
$r_{\mathrm{s}i}^{2}(r)/r^{2}$, corresponds to the geometrical dilution of the
number of stars during radial movement and the last fraction,
$\mathrm{d}r_{\mathrm{s}i}(r)/\mathrm{d}r$, converts the somewhat ephemeral
distribution function in an artificially chosen parameter (shell radius) into
a coordinate density. The final formal expression for the LOSVD then reads
$F^{\mathrm{far}}(v_{\mathrm{los}})=\int_{0}^{z_{\mathrm{fin}}}\sum_{i=1}^{2}\frac{r_{\mathrm{s}i}^{2}(r)}{r^{2}}\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}i}(r)\right)\frac{\mathrm{d}r_{\mathrm{s}i}(r)}{\mathrm{d}r}\delta[v_{\mathrm{los}}-\frac{z}{r}v_{ri}]\mathrm{d}z.$
(15)
We call this expression “formal”, because – at least in the model of radial
oscillations – we are not able to obtain closed analytical expression for
almost any of the terms involved.
#### 9.7 Shell-edge density distribution and LOSVD
For us, the modeling of the shape of the LOSVD is of peripheral importance, as
we will eventually need to know only the positions of the LOSVD maxima. The
peaks occur at the edge of the distribution (Sect 9.8). The determination of
the location of the line-of-sight velocity extremes does not require the
knowledge of stellar density profile. We do not even aim to qualitatively
model the shape of the LOSVD, but we can still show it to obtain a qualitative
insight.
?figurename? 16: LOSVD of the second shell at $r_{\mathrm{s}}=120$ kpc
(parameters of the shell are highlighted in bold in Table 2) for the projected
radius 108 kpc in the framework of the model of radial oscillations, where the
shell-edge density distribution is
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto r_{\mathrm{s}}^{2}$
for the blue curve and
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto 1/r_{\mathrm{s}}^{2}$
for the red one. The profiles do not include stars of the host galaxy, which
are not part of the shell system, and are normalized, so that the total flux
equals one.
If we want to obtain the full LOSVD, we have to choose the radial dependence
of the shell-edge density distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$. In the framework of the
radial-minor-merger origin of shell galaxies,
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ depends on the parameters
of the merger that has produced the shells. It is determined by the energy
distribution of stars of the cannibalized galaxy in the instant of its decay
in the center of the host galaxy. But the energy distribution is principle
unknown for real shell galaxies and it can be very different for various
collisions even if we consider only radial mergers. Thus you need to choose
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ somehow arbitrary.
For simplicity, we choose the shell-edge density distribution to be
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto
1/r_{\mathrm{s}}^{2},$ (16)
corresponding to a shell containing the same number of stars at each moment.
It turns out that no reasonable choice of this function has an effect on the
general characteristics of the LOSVD and the principles of its formation that
we describe in Sect. 9.8.
For illustration, we demonstrate the LOSVD of $\sigma_{\mathrm{sph}}$
increasing as $r^{2}$ and $\sigma_{\mathrm{sph}}$ decreasing as $1/r^{2}$ in
Fig. 16. For the profiles shown, the ratio of the inner and outer peaks
changes with the change of the $\sigma_{\mathrm{sph}}$, but the peak positions
are unaffected and the overall shape of the profile does not change
significantly. For shells that were created in a radial minor merger, we can
expect the shell-edge density distribution to rise in the inner part of the
host galaxy, followed by an extensive area of its decrease. The fact that the
main features of the LOSVD do not depend on the choice of
$\sigma_{\mathrm{sph}}$ means that our method of measuring the potential of
shell galaxies is not sensitive to the details of the decay of the
cannibalized galaxy It also means that, for the purposes of the modeling the
LOSVD of shells, we can safely pick $\sigma_{\mathrm{sph}}$ of our choice.
#### 9.8 Nature of the quadruple-peaked profile
Now we will show, why the LOSVD is so insensitive to the choice of the radial
dependence of the shell-edge density distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$. Fig. 17 shows the
formation of the quadruple-peaked profile for the far half of the galaxy (that
is, for positive values of $z$) at particular projected radius $R$. The inner
peak is located to the left, the outer one to the right (Fig. 17 – lower
panels). For the near half of the galaxy, the graph is simply reflected along
the axis $v_{\mathrm{los}}=0$. To help visualize the problem, we show the
individual contributions to the LOSVD from stars with different shell radii
that correspond to different points along the line of sight. To allow that, we
discretize their continuous distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ to a set of equidistant
spheres. Each of the spheres carries a density of stars obtained by
integration of the distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ over a small range in shell
radii as follows:
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})=\int_{r_{\mathrm{s}}-\Delta
r_{\mathrm{s}}/2}^{r_{\mathrm{s}}+\Delta
r_{\mathrm{s}}/2}\sigma_{\mathrm{sph}}(r)\mathrm{d}r$ (17)
to represent the given part of the distribution. To each of the spheres, we
associate the weight
$I=(r_{\mathrm{s}}/r)^{2}\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})\frac{r}{\sqrt{r^{2}-R^{2}}},$
(18)
which shows its contribution to the LOSVD. Similarly to Eq. (14), the term
$(r_{\mathrm{s}}/r)^{2}$ simply takes into account the geometric dilution of
the sphere with radius. The factor $r/\sqrt{r^{2}-R^{2}}$ reflects the fact
that spheres with different radii are intersected by the line of sight under
different angles. The color of the point encodes the weight $I$ for each
contributing sphere – the upper panels of both figures (a) and (b) in Fig. 17.
Note that to each value of $z$ we can assign the corresponding
$r=\sqrt{z^{2}+R^{2}}$.
To evaluate which spheres contribute to the observed shell profile, we let
them evolve (either backwards or forwards) from the point in time when they
will reach or have reached their shell radii to the time of the observation
and we place them on the exact locations they reach after this evolution. This
operation is a discrete analog of the term
$\mathrm{d}r_{\mathrm{s}}(r)/\mathrm{d}r$ in Eq. (14) which transfers the
distribution in $r_{\mathrm{s}}$ into the distribution in actual positions at
the time of observation. In the figures, we can see its effects as dilution
and thickening of the distribution of the colored points in different parts of
the plane. The points are located at a curve in the $v_{\mathrm{los}}-z$
plane. The shape of the curve is determined by the $\delta$ functions in Eq.
(12). Finally, we count the spheres in bins of $v_{\mathrm{los}}$ irrespective
of their $z$ coordinate to obtain the LOSVD (the lower panels of both figures
(a) and (b) in Fig. 17).
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ and
$\Sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ are different quantities,
but from Eq. (17) it is clear that once we choose the radial dependence of one
of them, the other has to have the same dependence. In Fig. 17 (a), this
function is chosen to be $\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})\propto
1/r_{\mathrm{s}}^{2}$, which is the formula we generally use for
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ or
$\Sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ unless specifically noted
otherwise. In Fig. 17 (b) we show that the quadruple-peaked shape appears even
for a completely reversed density function
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})\propto r_{\mathrm{s}}^{2}$. The
densities are calculated relative to the density at the radius of current
turning points, $\Sigma_{\mathrm{sph}}(r_{\mathrm{TP}})=1$.
?figurename? 17: The LOSVD and its different contributions along the line of
sight $z$ for the far half of the host galaxy for the second shell at
$r_{\mathrm{s}}=120$ kpc (parameters of the shell are highlighted in bold in
Table 2) for the projected radius 108 kpc in the framework of the model of
radial oscillations. Graphs (a) and (b) differ in the choice of
$\Sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$: (a)
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})\propto 1/r_{\mathrm{s}}^{2}$, (b)
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})\propto r_{\mathrm{s}}^{2}$.
The bottom panels of both figures in Fig. 17 show the LOSVD itself. Although
the weights of every point are different for the different choices of
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}})$, the dominant effect is the bending of
the curve in the $v_{\mathrm{los}}-z$ plane around $z_{v\mathrm{max1/2}}$ at
the LOS velocity extremes and thus the points around these extremes are much
denser for a unit of the $v_{\mathrm{los}}$ than in the inner part of the
distribution. This effect is completely the same for both (a) and (b). The
change of the weight causes relative differences in the heights of the LOSVD
peaks, but in no way casts any doubts over their existence at the extremes of
the projected velocity.
?figurename? 18: LOSVD and its individual contributions along the line of
sight for the far half of the host galaxy for the second shell at
$r_{\mathrm{s}}=120$ kpc (parameters of the shell are highlighted in bold in
Table 2) for the projected radius 108 kpc (light blue curve in the lower
panel) and 119 kpc (dark blue curve in the lower panel) in the framework of
the model of radial oscillations.
The points $z_{v\mathrm{max1/2}}$ correspond to the radii of maximal LOS
velocity $r_{v\mathrm{max1/2}}$ (points A and B from Sect. 9.4) through the
equation $r_{v\mathrm{max}}=\sqrt{z_{v\mathrm{max}}^{2}+R^{2}}$. If the
density in the vicinity of these points quickly dropped towards zero, the
peaks could disappear. This should certainly not happen at all projected radii
around the shell edge, because then there would be no shell at all. Moreover,
such a gap has no physical foundation for shells of radial-minor-merger
origin. On the other hand, if the shell has rather stream-like nature, the
stars may be present in only one half of the galaxy. Then just one inner and
one outer peak would be observable (i.e. the inner peak at negative velocities
and the outer at positive or vice versa). This is probable the case of the so-
called Western Shelf in the Andromeda galaxy (Fardal et al., 2012).
The only case of disappearance of peaks, which is natural to the model of
radial oscillations, occurs for the inner peaks in the zone between the
current turning points $r_{\mathrm{TP}}$ and the shell edge. The reason is
evident from Fig. 18 where we show the contributions along the line of sight
at projected radii $R=108$ kpc and $R=119$ kpc, while the edge of the shell is
at $r_{\mathrm{s}}=120$ kpc and the current turning points at
$r_{\mathrm{TP}}=118$ kpc. The color code in this case encodes the positions
of the apocenters of the stars contributing to the respective LOSVD. The
location of the apocenters $r_{\mathrm{ac}}$ roughly corresponds to the radii
$r_{\mathrm{s}}(r)$ where the stars will or have been located during their
passage through the edge of the shell. The radius $r_{\mathrm{s}}(r)$ is
obviously always slightly closer to the center of the host galaxy than the
apocenters of the respective stars. For the shell that we show (the second
shell at $r_{\mathrm{s}}=120$ kpc) the difference of these radii is (for the
chosen potential of the host galaxy) approximately
$r_{\mathrm{ac}}-r_{\mathrm{s}}(r)=2$ kpc.121212In the approximation of a
constant shell velocity, $v_{\mathrm{s}}$, and a constant galactocentric
acceleration, $a_{0}$ (Sect. 11), the following holds:
$r_{\mathrm{ac}}-r_{\mathrm{s}}(r)=-v_{\mathrm{s}}^{2}/(2a_{0})$. This is an
expression for the difference of the radius of apocenter of a star and the
radius of the passage of the very same star through the edge of the shell.
Incidentally (and only in this approximation), the same expression holds for
the difference of the current turning point and the shell radius
$r_{\mathrm{TP}}-r_{\mathrm{s}}$ even though the current turning point
represents the apocenter for stars that have already been on the shell edge.
### 10 Stationary shell
MK98 studied the kinematics of a stationary shell – a monoenergetic
spherically symmetric system of stars oscillating on radial orbits in a
spherically symmetric potential. They derived an analytic approximation for
the LOSVD in the vicinity of the shell edge, predicting a double-peaked
spectral-line profile, where the locations of these peaks are connected via a
simple relation to the gradient of the potential of the host galaxy at the
shell edge.
As our work expands the analysis of MK98, we also show the derivation of their
results. In Sect. 13, we apply also their method to the simulated data and
compare the results with the results of our methods. Furthermore, the
approximation of a stationary shell allows some calculations that prove
impossible for a moving shell, such as the calculation of an explicit
analytical shape of the LOSVD.
The stationary shell differs qualitatively from the model of radial
oscillations, because it requires stars to appear at all radii between $R$ and
$r_{\mathrm{s}}$, where $R$ is the projected radius at which we observe the
LOSVD. But because all the stars in this system have the same energy, it is
impossible to create such a situation by releasing all of the stars at one
time from one point.
#### 10.1 Motion of stars in a shell system
Let the shell edge be again $r_{\mathrm{s}}$. Stars at this radius are in
their apocenters and thus stationary. We assume following:
* •
stars are on strictly radial orbits
* •
all stars have the same energy
* •
stars are near the shell edge, so $1-r/r_{\mathrm{s}}\ll 1$
The radial velocity of stars at a given galactocentric radius $r$ is then
given by the difference of the host galaxy potential $\phi$ at this radius and
at the edge of the shell
$v_{r\pm}=\pm\sqrt{2\left[\phi(r_{\mathrm{s}})-\phi(r)\right]}.$ (19)
The velocity projected to the line of sight is
$v_{\mathrm{los}}^{2}=\left(1-R^{2}/r^{2}\right)2\sqrt{\phi(r_{\mathrm{s}})-\phi(r)}.$
(20)
Expanding this function around $r=r_{\mathrm{s}}$ we obtain
$\begin{array}[]{rcl}v_{\mathrm{los}}^{2}=&&-2\left(r-r_{\mathrm{s}}\right)\phi^{\prime}(r_{\mathrm{s}})\left(1-R^{2}/r_{\mathrm{s}}^{2}\right)-\\\
&&-\left(r-r_{\mathrm{s}}\right)^{2}\frac{1}{r_{\mathrm{s}}^{3}}\left[4R^{2}\phi^{\prime}(r_{\mathrm{s}})+r_{\mathrm{s}}\left(r_{\mathrm{s}}^{2}-R^{2}\right)\phi^{\prime\prime}(r_{\mathrm{s}})\right]+\\\
&&+o\left[\left(r-r_{\mathrm{s}}\right)^{3}\right],\end{array}$ (21)
where $\phi^{\prime}(r_{\mathrm{s}})$ and
$\phi^{\prime\prime}(r_{\mathrm{s}})$ are the first and the second derivative
of the potential of the host galaxy with respect to the radius at
$r_{\mathrm{s}}$. Near the edge of the shell
($\left|R-r_{\mathrm{s}}\right|\ll r_{\mathrm{s}}$), the following holds:
$\left(1-R^{2}/r_{\mathrm{s}}^{2}\right)\simeq
2\frac{r_{\mathrm{s}}-R}{r_{\mathrm{s}}}.$ (22)
Using Eq. (22) and neglecting all terms of the order
$o\left[\left(R-r_{\mathrm{s}}\right)^{3}\right]$, Eq. (21) takes the form
$v_{\mathrm{los}}^{2}\simeq
4\left(r_{\mathrm{s}}-r\right)\left(r-R\right)\frac{\phi^{\prime}(r_{\mathrm{s}})}{r_{\mathrm{s}}}.$
(23)
The derivative of this expression is zero when
$r=\frac{1}{2}(R+r_{\mathrm{s}}).$ (24)
thus the extremes of the projected velocity, $v_{\mathrm{los,max}\pm}$, must
follow
$v_{\mathrm{los,max}\pm}=\pm v_{\mathrm{c}}(1-R/r_{\mathrm{s}}),$ (25)
where $v_{\mathrm{c}}=\sqrt{r_{\mathrm{s}}\phi^{\prime}(r_{\mathrm{s}})}$ is
the circular velocity in the potential of the host galaxy at the radius of the
shell. If we call $\bigtriangleup
v_{\mathrm{los}}=2\left|v_{\mathrm{los,max}\pm}\right|$ the difference between
the minimal and maximal LOS velocity at the given galactocentric radius, the
derivative of this variable directly gives the derivative of the gravitational
potential of the galaxy at the radius of the shell edge (equation (7) in
MK98):
$\frac{\mathrm{d}(\bigtriangleup
v_{\mathrm{los}})}{\mathrm{d}R}=-2\frac{v_{\mathrm{c}}}{r_{\mathrm{s}}}.$ (26)
#### 10.2 Constant acceleration
Alternatively, we may assume that the stars move in a gravitational field of a
constant acceleration $a_{0}=-\phi^{\prime}(r_{\mathrm{s}})$. In such a case,
the radial velocity $v_{r}$ of a star at radius $r$ will by given by
$v_{r\pm}=\pm\sqrt{2a_{0}(r-r_{\mathrm{s}})}$ (27)
and its projection to the line of sight
$v_{\mathrm{los}}^{2}=\left(v_{r\pm}z/r\right)^{2}=-2a_{0}(r_{\mathrm{s}}-r)\left(1-R^{2}/r^{2}\right),$
(28)
where $R$ and $z$ denote the projected radius and the distance along the line
of sight, respectively. The center of the host galaxy is located at $R=0$ and
$z=0$. Comparing Eq. (23) and Eq. (28) , we obtain an approximative relation
for the projection factor $z/r$ near the edge of the shell
$z/r=\sqrt{1-R^{2}/r^{2}}\simeq\sqrt{2(r/r_{\mathrm{s}}-R/r_{\mathrm{s}})}.$
(29)
We use this relation in Sect. 11.7 in order to calculate the extremes of the
LOS velocity in the approximation of a shell with a constant phase velocity.
#### 10.3 LOSVD
Eq. (26) shows, that by measuring the width of the projected velocity
distribution at different radii near the shell edge we can easily obtain the
gradient of the potential of the host galaxy at the shell edge. Measuring the
extremes of the LOS velocity may prove very difficult in practice,
particularly because of the contamination of the signal from the shell by the
light of the host galaxy. For the stationary shell, we can however calculate
the shape of the LOSVD explicitly and it turns out that the extremes of the
LOS velocity correspond to the maxima of the intensity in the LOSVD, as shown
below in this section.
?figurename? 19: LOSVD of the stationary shell at four projected radii
according to Eq. (35).
MK98 derived the analytical form of LOSVD, $F(v_{\mathrm{los}})$, in the
approximation for the projected radius close to the edge of a stationary shell
$r_{\mathrm{s}}$. For the construction of the LOSVD, we start with Eq. (11) –
the integration of the stellar distribution function in the shell along the
line of sight at the chosen projected radius $R$. The problem is again
spherically symmetric, thus the distribution depends only on the radius $r$.
Moreover, for a stationary shell, the spatial density near the shell edge is
proportional to $\rho(r)\varpropto(v_{r}r^{2})^{-1}$, thus it is useful to
express the distribution function in radial velocity
$f(\mathbf{r},v_{\mathrm{los}})=f(r,v_{r})\frac{\mathrm{d}v_{r}}{\mathrm{d}v_{\mathrm{los}}}.$
(30)
It follows from Eq. (23) that a particular value of the projected velocity can
be found only at two specific galactocentric radii $r_{\pm}$ along the line of
sight
$r_{\pm}=r_{\mathrm{s}}/2\sqrt{R/r_{\mathrm{s}}+1\pm\left[(1-R/r_{\mathrm{s}})^{2}-\left(v_{\mathrm{los}}/v_{\mathrm{c}}\right)^{2}\right]}.$
(31)
Note that at a particular galactocentric radius, the value of the radial
velocity is fully determined in the case of a stationary shell, see Eq. (27).
Thus
$f(r,v_{r})=\frac{k}{v_{r}r^{2}}\delta(v_{r}-v_{r\pm}),$ (32)
where $\delta$ is the Dirac delta function and $k$ is a constant of
proportionality of the density at the given shell radius. The LOSVD then take
the form
$F(v_{\mathrm{los}})=\int\frac{k}{v_{r}r^{2}}\delta(v_{r}-v_{r\pm})\frac{\mathrm{d}z}{\mathrm{d}v_{\mathrm{los}}}\mathrm{d}v_{r}$
(33)
yielding after the integration
$F(v_{\mathrm{los}})=\frac{kr_{\mathrm{s}}^{2}\left|v_{\mathrm{los}}\right|}{2v_{\mathrm{c}}}\left[\frac{1}{r_{+}z_{+}v_{r+}\left|R+r_{\mathrm{s}}-2r_{+}\right|}+\frac{1}{r_{-}z_{-}v_{r-}\left|R+r_{\mathrm{s}}-2r_{-}\right|}\right],$
(34)
where $z_{\pm}=(r_{\pm}^{2}-R^{2})$. Eq. (34) can be further simplified for
$r_{\pm}$ near $r_{\mathrm{s}}$ and assuming $1-R/r_{\mathrm{s}}\ll 1$ to
obtain a final relation (equation (15) in MK98)
$F(v_{\mathrm{los}})\propto
1/\left[r_{\mathrm{s}}\sqrt{(1-R/r_{\mathrm{s}})^{2}-\left(v_{\mathrm{los}}/v_{\mathrm{c}}\right)^{2}}\right].$
(35)
The function $F(v_{\mathrm{los}})$ has a clear double-peaked profile,
symmetric around zero (or rather the overall velocity of the system). Examples
of such a profile are shown in Fig. 19.
#### 10.4 Comparison with the model of radial oscillations
The approximation of the stationary model differs qualitatively from the model
of radial oscillations in that there is only a double-peaked profile (instead
of a quadruple-peaked one). If the real shell galaxies are of radial-minor-
merger origin, they would rather exhibit a profile with four peaks (Sect.
9.4). Nevertheless, we can compare the locations of the two peaks of the
stationary shell with the model (Sect. 9.4) in Fig. 20. We have inserted the
values of the shell radius $r_{\mathrm{s}}=120$ kpc and the circular velocity
at the edge of the shell in the chosen potential $v_{\mathrm{c}}=450$ km$/$s
(for parameters of the host galaxy potential, see Sect. 8.1) into Eq. (25).
On the other hand the model of radial oscillations uses the complete knowledge
of the potential and the velocity of the shell at different times derived from
it. The higher is the number of the shell, the lower is its velocity and the
closer are the peaks of the quadruple-peaked profile to each other and to the
green line of the stationary shell. However this holds only near the edge of
the shell. For lower radii, the approximation of a stationary shell causes the
positions of the peaks to diverge from the model of the radial oscillations.
?figurename? 20: LOSVD peak locations for the stationary shell at the radius
of 120 kpc according to Eq. (25) (green dashed lines); and for the first four
shells at the radius of 120 kpc (parameters of the shells are listed in Table
2) according to the model of radial oscillations (Sect. 9.4). The upper panel
shows the whole range of radii, the lower zooms in on the edge of the shell.
### 11 Constant acceleration and shell velocity
Now we will leave the stationary case and look at the kinematics of a moving
shell. The nonzero velocity of the shell complicates the kinematics of shells
in two aspects. Due to the energy difference between inward and outward stars
at the same radius, the LOSVD peak is split into two, see Fig. 12, and the
shell edge is not at the radius of the current turning point, but slightly
further from the center of the host galaxy. In this section, we describe the
LOSVD of such a shell using the assumption of a locally constant galactic
acceleration together with the assumption of a locally constant shell phase
velocity. We call it the approximation of constant acceleration and shell
velocity. In addition, we assume that the velocity of stars at the edge of the
shell is equal to the phase velocity of the shell.
This approximation is nothing but a modification of the model of radial
oscillations for a constant acceleration and shell velocity and thus the
concept that the stars behave as if they were released in the center of the
host galaxy at the same time and their distribution of energies is continuous
is still valid in this approximation.
#### 11.1 Motion of a star in a shell system
The galactocentric radius of the shell edge is a function of time,
$r_{\mathrm{s}}(t)$, where $t=0$ is the moment of measurement and
$r_{\mathrm{s}}(0)=r_{\mathrm{s0}}$ is the position of the shell edge at this
time. We assume following:
* •
stars are on strictly radial orbits
* •
locally constant value of the radial acceleration $a_{0}$ in the host galaxy
potential131313By locally constant we mean that we apply one constant value of
radial acceleration or shell velocity to the calculation of the stellar
kinematics for one shell in the whole range of radii of interest.
Nevertheless, we use a different value for different shells, even when
considering stars at the same radii. Moreover note, that for stars that give
the highest contribution to the LOSVD peaks, the range $0-r_{\mathrm{s0}}$ in
projected radii corresponds approximately to
$1/2r_{\mathrm{s0}}-r_{\mathrm{s0}}$ in galactocentric radii.
* •
a locally constant velocity of the shell edge $v_{\mathrm{s}}$13
* •
stars at the shell edge have the same velocity as the shell141414In Sect. 9.2
we have discussed that the stars at the shell edge in fact do not have the
same velocity as the shell, but in Table 2 we show using examples that these
velocities are very similar.
The galactocentric radius of each star is at any time $r(t)$, while
$t_{\mathrm{s}}$ is the time when the star could be found at the shell edge
$r_{\mathrm{s}}(t_{\mathrm{s}})$. Then the equation of motion and the initial
conditions for the star near a given shell radius are
$\frac{\mathrm{d}^{2}r(t)}{\mathrm{d}t^{2}}=a_{0},$ (36)
$\left.\frac{\mathrm{d}r(t)}{\mathrm{d}t}\right|_{t=t_{\mathrm{s}}}=v_{\mathrm{s}},$
(37)
$r(t_{\mathrm{s}})=r_{\mathrm{s}}(t_{\mathrm{s}})=v_{\mathrm{s}}t_{\mathrm{s}}+r_{\mathrm{s0}}.$
(38)
The solution of these equations is
$r(t)=a_{0}(t-t_{\mathrm{s}})^{2}/2+v_{\mathrm{s}}(t-t_{\mathrm{s}})+r_{\mathrm{s}}(t_{\mathrm{s}}),$
(39)
$v_{r}(t)=v_{\mathrm{s}}+a_{0}(t-t_{\mathrm{s}}),$ (40)
and the actual position of the star $r(0)$ and its radial velocity $v_{r}(0)$
at time of measurement ($t=0$) are
$r(0)=t_{\mathrm{s}}^{2}a_{0}/2+r_{\mathrm{s0}},$ (41)
$v_{r}(0)=v_{\mathrm{s}}-a_{0}t_{\mathrm{s}}.$ (42)
Eliminating $t_{\mathrm{s}}$ from the two previous equations, we get
$v_{r}(0)_{\pm}=v_{\mathrm{s}}\pm
v_{\mathrm{c}}\sqrt{2\left(1-r(0)/r_{\mathrm{s0}}\right)},$ (43)
where $v_{\mathrm{c}}=\sqrt{-a_{0}r_{\mathrm{s0}}}$ is the circular velocity
at the shell-edge radius.
#### 11.2 Approximative LOSVD
The projection of the velocity given by Eq. (43) to the LOS at a projected
radius $R$ will be
$\begin{array}[]{rcl}v_{\mathrm{los}\pm}&=&\sqrt{1-R^{2}/\left(r\left(0\right)\right)^{2}}v_{r}(0)_{\pm}=\\\
&=&\sqrt{1-R^{2}/\left(r\left(0\right)\right)^{2}}\left[v_{\mathrm{s}}\pm
v_{\mathrm{c}}\sqrt{2\left(1-r(0)/r_{\mathrm{s0}}\right)}\right].\end{array}$
(44)
Using this expression, we can model the LOSVD at a given projected radius for
a given shell. For the proper choice of a pair of values $v_{\mathrm{c}}$ and
$v_{\mathrm{s}}$, we can find a match with observed and modeled peaks of the
LOSVD. When we use this approach, we call it the approximative LOSVD.
To model the approximative LOSVD by Eq. (44), we have to add an assumption
about the radial dependence of the shell-edge density distribution, Eq. (13).
We chose this function in the same manner as in the model of radial
oscillations that is in a way that corresponds to constant number of stars at
the edge of the shell. In Sect. 9.7 we have shown in the model of radial
oscillations that a different choice of the radial dependence of the shell
brightness changes neither the quadruple-peaked shape of the LOSVD of the
shells, nor the positions of the maximal/minimal velocity which corresponds to
the peaks of the LOSVD. This holds also for the approximative LOSVD, because
the approximative LOSVD is very close to the LOSVD from the model of the
radial oscillations, see Fig. 23. For the approximative LOSVD also holds that
the inner peaks of the LOSVD disappear in the zone between the current turning
points and the edge of the shell.
#### 11.3 Radius of maximal LOS velocity
MK98 proved that near the edge of a stationary shell, $r_{s}$, the maximum
intensity of the LOSVD is at the edge of the distribution. They also proved
that the maximal absolute value of the LOS velocity $v_{\mathrm{los,max}}$
comes from stars at the galactocentric radius
$r_{v\mathrm{max}}=\frac{1}{2}(R+r_{\mathrm{s0}}),$ (45)
at each projected radius $R$.
For a moving shell, analogous equations are significantly more complex and a
similar relation cannot be easily proven. Nevertheless, when we apply both
results of MK98 we can show in examples (Figs. 22, 23, and others) that their
use is valid, even for nonstationary shells. In the framework of the radial
oscillations model (Sect. 9.4), we have shown that the peaks of the LOSVD
occur at the edges of distributions of the near or the far half of the galaxy
(Sect. 9.8). The inner peak corresponds to inward-moving stars and the outer
one to outward-moving ones. This approach is used in the equations in Sect.
11.4. The maximal LOS velocity corresponds to the outer peak and the minimal
to the inner one. Reasons and justification for use of Eq. (45) for
$r_{v\mathrm{max}}$ are discussed in Sect. 11.6, point 2 (see also Fig. 21).
#### 11.4 Approximative maximal LOS velocity
Using the results of MK98, we derive an expression for the maxima/minima of
the LOS velocity corresponding to locations of the LOSVD peaks in observable
quantities (i.e., the maxima/minima of the LOS velocity, the projected radius,
and the shell radius) by substituting $r_{v\mathrm{max}}$ given by Eq. (45)
for $r(0)$ in Eq. (44)
$\begin{array}[]{rcl}v_{\mathrm{los,max}\pm}\\!&=&\\!\left(v_{\mathrm{s}}\pm
v_{\mathrm{c}}\sqrt{1-R/r_{\mathrm{s0}}}\right)\,\times\\\
&&\times\sqrt{1-4\left(R/r_{\mathrm{s0}}\right)^{2}\left(1+R/r_{\mathrm{s0}}\right)^{-2}}.\end{array}$
(46)
For the measured locations of the LOSVD peaks $v_{\mathrm{los,max}+}$,
$v_{\mathrm{los,max}-}$, projected radius $R$, and shell-edge radius
$r_{\mathrm{s0}}$, we can express the circular velocity $v_{\mathrm{c}}$ at
the shell-edge radius and the current shell velocity $v_{\mathrm{s}}$ by using
inverse equations:
$v_{\mathrm{c}}=\frac{\left|v_{\mathrm{los,max}+}-v_{\mathrm{los,max}-}\right|}{2\sqrt{\left(1-R/r_{\mathrm{s0}}\right)\left[1-4\left(R/r_{\mathrm{s0}}\right)^{2}\left(1+R/r_{\mathrm{s0}}\right)^{-2}\right]}},$
(47)
$v_{\mathrm{s}}=\frac{v_{\mathrm{los,max}+}+v_{\mathrm{los,max}-}}{2\sqrt{1-4\left(R/r_{\mathrm{s0}}\right)^{2}\left(1+R/r_{\mathrm{s0}}\right)^{-2}}}.$
(48)
We call this approach the approximative maximal LOS velocity.
#### 11.5 Slope of the LOSVD intensity maxima
Alternatively, the value of the circular velocity $v_{\mathrm{c}}$ at the
shell-edge radius could be inferred from measurements of positions of peaks at
two or more different projected radii for the same shell: let $\bigtriangleup
v_{\mathrm{los}}=v_{\mathrm{los,max}+}-v_{\mathrm{los,max}-}$, where
$v_{\mathrm{los,max}\pm}$ satisfy Eq. (46). Then, in the vicinity of the shell
edge,
$\begin{array}[]{rcl}\bigtriangleup
v_{\mathrm{los}}&=&2v_{\mathrm{c}}\sqrt{\left(R/r_{\mathrm{s0}}-1\right)\left[1-4\left(R/r_{\mathrm{s0}}\right)^{2}\left(1+R/r_{\mathrm{s0}}\right)^{-2}\right]}\simeq\\\
&\simeq&2(1-R/r_{\mathrm{s0}})v_{\mathrm{c}},\end{array}$ (49)
and taking the derivative with respect to the projected radius
$\frac{\mathrm{d}(\bigtriangleup
v_{\mathrm{los}})}{\mathrm{d}R}=-2\frac{v_{\mathrm{c}}}{r_{\mathrm{s0}}},$
(50)
which happens to be the same expression as Eq. 26 (equation (7) in MK98).
Nevertheless, for a stationary shell, $\bigtriangleup v_{\mathrm{los}}$ is the
distance between the two LOSVD intensity maxima of a stationary shell, whereas
in this framework, it is the distance between the outer peak for positive
velocities and the inner peak for negative velocities or vice versa. This
equation allows us to measure the circular velocity in shell galaxies using
the slope of the LOSVD intensity maxima in the $R\times v_{\mathrm{los}}$
diagram.
When we use this approach, we call it the use of the slope of the LOSVD
intensity maxima. It requires us to measure the LOSVD for at least two
different projected radii. In exchange, as we show in Sect. 13.3, that it
promises a more accurate derivation of $v_{\mathrm{c}}$. However it does not
allow the derivation of the shell velocity $v_{s}$. For this purpose, we can
use Eq. (46) to derive a hybrid relation between the positions of the LOSVD
peaks, the circular velocity at the shell-edge radius $v_{\mathrm{c}}$, and
the shell velocity:
$v_{\mathrm{s}}^{2}=v_{\mathrm{c}}^{2}(1-R/r_{\mathrm{s0}})+\frac{v_{\mathrm{los,max}+}v_{\mathrm{los,max}-}}{4\left(R/r_{\mathrm{s0}}\right)^{2}\left(1+R/r_{\mathrm{s0}}\right)^{-2}-1}.$
(51)
If we insert the value of $v_{\mathrm{c}}$ derived from the measurement of the
LOSVD intensity maxima into this equation, we can expect a better estimate of
the phase velocity of the shell.
#### 11.6 Comparison of approaches
The approximation of a constant radial acceleration in the host galaxy
potential and shell phase velocity (Sect. 11) splits into three different
analytical and semi-analytical approaches for obtaining values of the circular
velocity $v_{\mathrm{c}}$ at the shell-edge radius and the shell phase
velocity $v_{\mathrm{s}}$. Different approaches/models have a different color
assigned. This color is used in Figs. 21–27 and 31–33 to represent the output
of the corresponding approach. Here we summarize differences, advantages and
disadvantages in these three approaches:
1. 1.
The approximative LOSVD (purple curves): For the given shell at the chosen
projected radius, Eq. (44) is a function of only two parameters –
$v_{\mathrm{c}}$ and $v_{\mathrm{s}}$. Assuming a radial dependence of the
shell-edge density distribution, Eq. (44) allows us to plot the whole LOSVD
(Sect. 11.2). However, computing the LOSVD and the positions of peaks requires
a numerical approach in this framework. When deriving $v_{\mathrm{c}}$ and
$v_{\mathrm{s}}$ from the observed LOSVD, we need to find a numerical solution
to Eq. (44) and to search for a pair of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$,
which matches the (simulated) data best.
2. 2.
The approximative maximal LOS velocities (orange curves): Eq. (46) supplies
the positions of the peaks directly. It differs from the previous
approximation in the assumption about the galactocentric radius
$r_{v\mathrm{max}}$, from which comes the contribution to the LOSVD at the
maximal speed. The assumption is that $r_{v\mathrm{max}}$ is given by Eq.
(45), which was derived by MK98 for a stationary shell. This equation is
actually only very approximate (see Fig. 21), but allows us to analytically
invert Eq. (46) to obtain formulae for the direct calculation of
$v_{\mathrm{c}}$ and $v_{\mathrm{s}}$ from the measured peak positions in the
spectrum of the shell galaxy near the shell edge – Eqs. (47) and (48).
Nevertheless, when measuring in the zone between the radius of the current
turning points and the shell radius, we can expect very bad estimates of
$v_{\mathrm{c}}$ and $v_{\mathrm{s}}$.
3. 3.
Using the slope of the LOSVD intensity maxima in the $R\times
v_{\mathrm{los}}$ diagram: Eq. (50) cannot be used to draw theoretical LOSVD
maxima for the given potential of the host galaxy, because it connects only
the circular velocity in the host galaxy and the difference of the slopes of
the LOSVD maxima. Moreover, the difference of the slopes alone does not allow
us to determine the shell velocity, but we can use Eq. (51) as it is described
in Sect. 11.5. Nevertheless it is this approach that gives the most accurate
estimate of $v_{\mathrm{c}}$ when applied to simulated data, Sect. 13.3.
These methods can be compared with the model of radial oscillations described
in Sect. 9.4 (plotted with light blue curves in the relevant figures). The
model of radial oscillations uses thorough knowledge of the potential of the
host galaxy. From it we extract the circular velocity at the shell-edge radius
and the current shell velocity and we put them in the approximative relations
derived in Sect. 11. We apply all the three approximations to the simulated
data in Sect. 13.3.
?figurename? 21: Galactocentric radii $r_{v\mathrm{max}}$ that contribute to
the LOSVD maxima according to Eq. (45), which was used in the derivation of
the approximative maximal/minimal LOS velocities (Sect. 11.6, point 2) –
orange curve, according to the approximative LOSVD (Sect. 11.6, point 1) –
purple curves, and according to the model of radial oscillations (Sect. 9.4) –
light blue curves for the second shell at 120 kpc (parameters of the shell are
highlighted in bold in Table 2). For parameters of the host galaxy potential,
see Sect. 8.1.
Fig. 21 shows a comparison of the radii that contribute to the LOSVD maxima
according to the model of radial oscillations, the approximative LOSVD, and
the approximative maximal LOS velocities. For the first two methods, the
radius corresponding to the inner maxima of the LOSVD (which are the maxima
created by the inward stars) is lower than that for the outer maxima, whereas
Eq. (45) assumes the same $r_{v\mathrm{max}}$ for both inward and outward
stars.
?figurename? 22: LOSVD peak locations for the second shell at the radius of
120 kpc (parameters of the shell are highlighted in bold in Table 2) according
to the approximative maximal LOS velocities (Sect. 11.6, point 2) given by Eq.
(46) (orange curves); the approximative LOSVD (Sect. 11.6, point 1) given by
Eq. (44) (purple curves); and the model of radial oscillations (Sect. 9.4)
(light blue curves which almost merged with the purple ones near the shell
edge). The red line shows the position of the LOSVD from Fig. 23, the black
one shows the position of the current turning points. The upper panel shows
the whole range of radii, the lower zooms in on the edge of the shell. For
parameters of the host galaxy potential, see Sect. 8.1.
Fig. 22 shows locations of the LOSVD peaks for the second shell at the radius
of 120 kpc near the shell-edge radius. The purple curve is calculated using
the approximative LOSVD (Sect. 11.6, point 1) given by Eq. (44), into which we
inserted the velocity of the second shell according to the model of radial
oscillations and the circular velocity in the potential of the host galaxy
(see Sect. 8.1 for parameters of the potential). The purple curve does not
differ significantly from the light blue curve calculated in the model of
radial oscillations (Sect. 9.4). The more important deviations in the orange
curve of the approximative maximal LOS velocities (Sect. 11.6, point 2) given
by Eq. (46), are caused by Eq. (45) for $r_{v\mathrm{max}}$. With this
assumption, approximative maximal LOS velocities (the orange curve) predict
that around the zone between the current turning point and the shell edge, the
inner peaks change signs. This means that for the part of the galaxy closer to
the observer, both inner and outer peaks will fall into negative values of the
LOS velocity and vice versa. However, from the model of the radial
oscillations, we know that the signal from the inner peak in a given (near or
far) part of the galaxy is always zero or has the opposite sign to that of the
outer peak.
The model of the radial oscillations and the approximative LOSVD given by Eq.
(44) were also used to construct the LOSVD for the second shell located at 120
kpc, at the projected radius of 108 kpc in Fig. 23. The graph also shows the
locations of the peaks using the approximative maximal LOS velocities given by
Eq. (46).
To wrap up, all three approaches give a good agreement with the model of
radial oscillations. The first approach is practically identical to this model
in the vicinity of the shell edge, but it requires numerical solution of
equations. The second approach is more approximative and gives worse results
particularly in the zone between the current turning point and the shell edge,
but allows direct expression of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$. The
third approach gives only the relation between the slopes of the LOSVD maxima
and $v_{\mathrm{c}}$, but we have already announced that it gives the best
results when calculating $v_{\mathrm{c}}$ from the simulated data.
?figurename? 23: LOSVD of the second shell at $r_{\mathrm{s}}=120$ kpc
(parameters of the shell are highlighted in bold in Table 2) for the projected
radius $R=0.9r_{\mathrm{s}}=108$ kpc according to the approximative LOSVD
(Sect. 11.6, point 1) given by Eq. (44) (purple curve) and the model of radial
oscillations (Sect. 9.4) (light blue curve almost merged with the purple one).
Locations of peaks as given by the approximative maximal LOS velocities (Sect.
11.6, point 2) given by Eq. (46) are plotted with orange lines. Profiles do
not include stars of the host galaxy that are not part of the shell system and
are normalized, so that the total flux equals to one. For parameters of the
host galaxy potential see Sect. 8.1.
#### 11.7 Projection factor approximation
In Sect. 10.2 we have derived an approximative relation for the factor $z/r$
that projects the galactocentric velocity of the stars at radial trajectories
to the line of sight, Eq. (29), which has been already used by Fardal et al.
(2012) to derive the relation for $v_{\mathrm{los,max}\pm}$. Inserting this
equation to the expression for the projected velocity of the stars of the
shell, Eq. (44) in Sect. 11.2, we get
$v_{\mathrm{los}\pm}(r)\simeq\sqrt{2(r/r_{\mathrm{s0}}-R/r_{\mathrm{s0}})}\left[v_{\mathrm{s}}\pm
v_{\mathrm{c}}\sqrt{2\left(1-r/r_{\mathrm{s0}}\right)}\right].$ (52)
?figurename? 24: Galactocentric radii $r_{v\mathrm{max\pm}}$ that contribute
to the LOSVD maxima for the second shell at 120 kpc (parameters of the shell
are highlighted in bold in Table 2) according to Eq. (53) – red curves. For
comparison, we show the radii $r_{v\mathrm{max}}$ according to the model of
radial oscillations (Sect. 9.4) – light blue curves – and according to the
approximation of Sect. 11.6 (orange and purple curves). See also Fig. 21.
The derivative of this expression is zero for $r=r_{v\mathrm{max}\pm}$, where
$r_{v\mathrm{max\pm}}=r_{\mathrm{s0}}\left(\frac{v_{\mathrm{s}}}{4v_{\mathrm{c}}}\right)^{2}\left[\frac{1}{2}\left(\frac{4v_{\mathrm{c}}}{v_{\mathrm{s}}}\right)^{2}\left(1+\frac{R}{r_{\mathrm{s0}}}\right)-1\pm\sqrt{\left(\frac{4v_{\mathrm{c}}}{v_{\mathrm{s}}}\right)^{2}\left(1-\frac{R}{r_{\mathrm{s0}}}\right)+1}\right].$
(53)
Near the edge of the shell, the values $r_{v\mathrm{max\pm}}$ are in good
coincidence with the galactocentric radii that contribute to the LOSVD maxima
according to the model of radial oscillations (Sect. 9.4), whereas at lower
radii they differ substantially, Fig. 24.
The position of the outer LOSVD peaks is expressed as the function
$v_{\mathrm{los+}}(r_{v\mathrm{max+}})$, the position of the inner peaks as
$v_{\mathrm{los-}}(r_{v\mathrm{max-}})$, Fig. 25. The equations have a
solution only for $r_{v\mathrm{max}}<R$. The radius, where
$r_{v\mathrm{max-}}=R$, is the radius of the current turning point
$r_{\mathrm{TP}}$ in this approximation and for $R>r_{\mathrm{TP}}$ the inner
peaks disappear. Eq. (53) implies
$r_{\mathrm{TP}}=r_{\mathrm{s0}}\left[1-\frac{1}{2}\left(\frac{v_{\mathrm{s}}}{v_{\mathrm{c}}}\right)^{2}\right].$
(54)
?figurename? 25: LOSVD peak locations for the second shell at the radius of
120 kpc (parameters of the shell are highlighted in bold in Table 2). The red
curves show the values of the functions
$v_{\mathrm{los+}}(r_{v\mathrm{max+}})$ and
$v_{\mathrm{los-}}(r_{v\mathrm{max-}})$, where $v_{\mathrm{los\pm}}(r)$ is
given by Eq. (52) and $r_{v\mathrm{max\pm}}$ follows Eq. (53). The light blue
curves are LOSVD peak locations according to the model of radial oscillations
(Sect. 9.4). The left panel shows the whole range of radii, the right zooms in
on the edge of the shell. For parameters of the host galaxy potential, see
Sect. 8.1.
The functions $v_{\mathrm{los+}}(r_{v\mathrm{max+}})$ and
$v_{\mathrm{los-}}(r_{v\mathrm{max-}})$ are a good approximation to the LOSVD
peak locations near the edge of the shell, as can be seen in Fig. 25. Using
these functions are a better way to calculate these than the approximative
LOSVD (Sect. 11.6, point 1), because their values are given analytically.
Nevertheless they are such a complicated function of the circular velocity
$v_{\mathrm{c}}$ at the shell-edge radius and the current shell velocity
$v_{\mathrm{s}}$ that they do not allow the expression of these variables as a
simple function of observable quantities, unlike the approximative
maximal/minimal LOS velocities (Sect. 11.6, point 2). Thus we will not use
these function in the following and show them only for the sake of
completeness and comparison with Fardal et al. (2012).
### 12 Higher order approximation
The approximation of a locally constant galactic acceleration $a_{0}$ and
shell phase velocity $v_{\mathrm{s}}$, Sect. 11, describes the positions of
the LOSVD peaks fairly well and allows a good determination of the parameters
of the potential of the host galaxy. Nevertheless we try to have a look
outside the realm of constant $a_{0}$ and $v_{\mathrm{s}}$ using the same
concept that stars behave as if they were released in the center of the host
galaxy at the same time and their distribution of energies is continuous.
#### 12.1 Motion of a star in a shell system
The galactocentric radius of the shell edge is a function of time,
$r_{\mathrm{s}}(t)$, where $t=0$ is the moment of measurement and
$r_{\mathrm{s}}(0)=r_{\mathrm{s0}}$ is the position of the shell edge at this
time. Let us define a new coordinate system $s$, where the radial coordinate
is the distance from the edge of the shell, in the same direction as the
galactocentric radius
$s(t)=r(t)-r_{\mathrm{s0}}.$ (55)
The position of the stars of the given shell in this system is always
negative.
We assume the following:
* •
stars are on strictly radial orbits
* •
radial acceleration in the potential of the host galaxy is given as
$a(s)=a_{0}+a_{1}s$, where $a_{0}$ and $a_{1}$ are constant for a given shell
* •
position of the shell edge is (insofar) a general function of time
$s_{\mathrm{s}}(t)$
* •
stars at the shell edge have the same velocity as the shell
The position of each star is at any time $s(t)$, while $t_{\mathrm{s}}$ is the
time when the star could be found at the shell edge
$s_{\mathrm{s}}(t_{\mathrm{s}})$. Then the equation of motion and the initial
conditions for the star near a given shell radius are
$\frac{\mathrm{d}^{2}s(t)}{\mathrm{d}t^{2}}=a_{0}+a_{1}s,$ (56)
$\left.\frac{\mathrm{d}s(t)}{\mathrm{d}t}\right|_{t=t_{\mathrm{s}}}=v_{\mathrm{s}},$
(57)
$s(t_{\mathrm{s}})=s_{\mathrm{s}}(t_{\mathrm{s}}).$ (58)
The solution to these equation differs for negative and positive values of
$a_{1}$. The position of a star in a general time $t$ is given by
$s(t,a_{1}>0)=\frac{\left[a_{1}s_{\mathrm{s}}(t_{\mathrm{s}})+a_{0}\right]\cosh\left[\sqrt{a_{1}}\left(t-t_{\mathrm{s}}\right)\right]+\sqrt{a_{1}}v_{\mathrm{s}}\sinh\left[\sqrt{a_{1}}\left(t-t_{\mathrm{s}}\right)\right]-a_{0}}{a_{1}},$
(59)
$s(t,a_{1}<0)=\frac{\left[\left|a_{1}\right|s_{\mathrm{s}}(t_{\mathrm{s}})-a_{0}\right]\cos\left[\sqrt{\left|a_{1}\right|}\left(t-t_{\mathrm{s}}\right)\right]+\sqrt{\left|a_{1}\right|}v_{\mathrm{s}}\sin\left[\sqrt{\left|a_{1}\right|}\left(t-t_{\mathrm{s}}\right)\right]+a_{0}}{\left|a_{1}\right|},$
(60)
where $\sinh(x)=1/2\left[\exp(x)-\exp(-x)\right]$ and
$\cosh(x)=1/2\left[\exp(x)+\exp(-x)\right]$. For $a_{1}=0$, the solution of
Sect. 11.1 holds. At the time of the measurement $t=0$ we obtain two pairs of
equations for the position of the star $s(0)$ and its radial velocity
$v_{r}(0)=\left.\mathrm{d}s(t)/\mathrm{d}t\right|_{t=0}$, depending on the
sign of $a_{1}$
$\begin{array}[]{rcl}s(0,a_{1}>0)&=&1/a_{1}\left\\{\left[a_{1}s_{\mathrm{s}}(t_{\mathrm{s}})+a_{0}\right]\cosh\left(t_{\mathrm{s}}\sqrt{a_{1}}\right)-\sqrt{a_{1}}v_{\mathrm{s}}\sinh\left(t_{\mathrm{s}}\sqrt{a_{1}}\right)-a_{0}\right\\},\\\
v_{r}(0,a_{1}>0)&=&1/\sqrt{a_{1}}\left\\{\sqrt{a_{1}}v_{\mathrm{s}}\cosh\left(t_{\mathrm{s}}\sqrt{a_{1}}\right)-\left[a_{1}s_{\mathrm{s}}(t_{\mathrm{s}})+a_{0}\right]\sinh\left(t_{\mathrm{s}}\sqrt{a_{1}}\right)\right\\},\end{array}$
(61)
$\begin{array}[]{rcl}s(0,a_{1}<0)&=&1/\left|a_{1}\right|\left\\{\left[\left|a_{1}\right|s_{\mathrm{s}}(t_{\mathrm{s}})-a_{0}\right]\cos\left(t_{\mathrm{s}}\sqrt{\left|a_{1}\right|}\right)-\sqrt{\left|a_{1}\right|}v_{\mathrm{s}}\sin\left(t_{\mathrm{s}}\sqrt{\left|a_{1}\right|}\right)+a_{0}\right\\},\\\
v_{r}(0,a_{1}<0)&=&1/\sqrt{\left|a_{1}\right|}\left\\{\sqrt{a_{1}}v_{\mathrm{s}}\cos\left(t_{\mathrm{s}}\sqrt{\left|a_{1}\right|}\right)+\left[\left|a_{1}\right|s_{\mathrm{s}}(t_{\mathrm{s}})-a_{0}\right]\sin\left(t_{\mathrm{s}}\sqrt{\left|a_{1}\right|}\right)\right\\}.\end{array}$
(62)
For galactic potentials, one value of $s(0)$ will yield solutions for two
different values of $t_{\mathrm{s}}$ and correspondingly two values of
$v_{r}(0)$ and its projection to the line of sight. The minimal and maximal
LOS velocities show the positions of LOSVD peaks.
#### 12.2 Comparison of approximations
Now we compare this higher order approximation with the approximation of a
constant acceleration (Sect. 11) and the model of radial oscillations (Sect.
9.4). For higher accuracy, we can obviously introduce the acceleration of the
shell $a{}_{\mathrm{s}}$ and express the shell position as
$s_{\mathrm{s}}(t_{\mathrm{s}})=v_{\mathrm{s}}t_{\mathrm{s}}+a{}_{\mathrm{s}}t_{\mathrm{s}}^{2}/2$.
However, for observation data it would mean to fit 4 parameters ($a_{0}$,
$a_{1}$, $v_{\mathrm{s}}$, and $a{}_{\mathrm{s}}$), what could prove difficult
in practice. To compare the approximations, we thus restrict ourselves to a
shell of constant velocity, that is
$s_{\mathrm{s}}(t_{\mathrm{s}})=v_{\mathrm{s}}t_{\mathrm{s}}$, like in Sect.
11.
?figurename? 26: Comparison of LOSVD peak locations in different
approximations for the second shell at the radius of 120 kpc, $a_{1}=1.2\times
10^{-5}$ Myr-2. The upper panel shows the whole range of radii, the lower
zooms in on the edge of the shell. For parameters of the host galaxy
potential, see Sect. 8.1.
?figurename? 27: Comparison of LOSVD peak locations in different
approximations for the first shell at the radius of 10 kpc, $a_{1}=8.5\times
10^{-4}$ Myr-2. The upper panel shows the whole range of radii, the lower
zooms in on the edge of the shell. For parameters of the host galaxy
potential, see Sect. 8.1.
Besides the usual second shell at 120 kpc showed in Fig. 26, we show also the
first shell at 10 kpc in Fig. 27. In our case, the value of $a_{1}$ at the
galactocentric distance of 10 kpc is almost two orders of magnitude larger
than the corresponding value at 120 kpc (see Fig. 28). For the approximations,
we have used values of parameters calculated from the potential of the host
galaxy (for parameters of the host galaxy potential, see Sect. 8.1). The model
of radial oscillations (thick light-blue curves) requires the knowledge of the
potential at all radii. The maxima/minima of the LOS velocities (that
correspond to the locations of the peaks of the LOSVD) are shown in purple for
the approximation of a constant acceleration (or, as we call it, using the
"approximative LOSVD" by Eq. (44)), and in dark blue for a LOS projection of
the solution of Eq. (61) with a nonzero $a_{1}$, which is positive for both
shells. At the edge of the shell, both approximations are almost identical to
the model of radial oscillations. On the other hand, at lower galactocentric
radii, only the approximation with a nonzero $a_{1}$ follows the model of
radial oscillations reasonably well. In general, the shell will be difficult
to observe in real galaxies at lower projected radii, but for the case of
observations of individual stars, star clusters and planetary nebulae, the
kinematical imprint of the shell could be observed considerably far from its
edge.
The purple and blue curves are calculated by finding maxima/minima of the LOS
velocities at each projected radius. It is possible to obtain these in a much
easier, but less accurate manner using the approximation for the radius of
maximal LOS velocity $r_{v\mathrm{max}}=\frac{1}{2}(R+r_{\mathrm{s0}})$, as
described in Sect. 11.3. The orange and red curves in Fig. 26 and Fig. 27 show
the result of this procedure in the approximation of a constant acceleration
(the "approximative maximal LOS velocity", Eq. (46)) and in the approximation
with a nonzero value of $a_{1}$, respectively. Again, both approximations
merge near the edge of the shell. For lower projected radii, the two curves
separate again, but taking into account their overall difference from the
model of radial oscillations, we cannot in this case consider the
approximation of a nonzero $a_{1}$ to be a significant improvement. The
approximative maximal LOS velocity with constant acceleration has the
advantage that it allows a direct expression of basic variables (the circular
velocity $v_{\mathrm{c}}$ at the shell-edge radius and shell phase velocity
$v_{\mathrm{s}}$) in terms of observable quantities, facilitating and easy
application to measured data. The same cannot be done in the approximation
with a nonzero value of $a_{1}$.
#### 12.3 $\boldsymbol{a{}_{1}}$
The assumption about the function $a(r)$ in the host galaxy is in fact an
assumption on the radial dependence of the density of the host galaxy, by
$a(r)=\frac{4\pi\mathrm{G}}{r^{2}}\int_{0}^{r}\rho(r^{\prime})r^{\prime
2}\mathrm{d}r^{\prime},$ (63)
where $\rho(r)$ is the density in the host galaxy and G is the gravitational
constant. For the case of constant acceleration $a=a_{0}$ the derivative of
Eq. (63) with respect to $r$ shows that the density goes to zero for large $r$
as
$\rho(r)=\frac{a_{0}}{2\pi\mathrm{G}}r^{-1},$ (64)
whereas for $a=a_{0}+a_{1}(r-r_{\mathrm{s0}})$ the density goes to
$\frac{3a_{1}}{4\pi\mathrm{G}}$ for large $r$ as
$\rho(r)=\frac{3a_{1}}{4\pi\mathrm{G}}+\frac{a_{0}+a_{1}r_{\mathrm{s0}}}{2\pi\mathrm{G}}r^{-1}.$
(65)
It is important to note that this approximation of the acceleration is applied
only locally, although this word may sometimes mean a fairly large span of
radii. The parameter $a_{1}$ may, in real galaxies, assume both positive and
negative values. In Fig. 28 we show the radial dependence of $a_{1}$ in the
host galaxy modeled as a double Plummer sphere (for parameters of the host
galaxy potential, see Sect. 8.1).
?figurename? 28: The radial dependence of $a_{1}$ in the host galaxy. For
parameters of the host galaxy potential, see Sect. 8.1.
### 13 Test-particle simulation
We performed a simplified simulation of formation of shells in a radial minor
merger of galaxies. Both merging galaxies are represented by smooth potential.
Millions of test particles were generated so that they follow the distribution
function of the cannibalized galaxy at the beginning of the simulation. The
particles then move according to the sum of the gravitational potentials of
both galaxies. When the centers of the galaxies pass through each other, the
potential of the cannibalized galaxy is suddenly switched off and the
particles continue to move only in the fixed potential of the host galaxy. We
use the simulation to demonstrate the validity of our methods of recovering
the parameters of the host galaxy potential by measuring151515By measuring, we
mean that the data measured are the output of our simulation. the positions of
the peaks in the LOSVD of simulated data.
?figurename? 29: Snapshots from our test-particle simulation of the radial
minor merger, leading to the formation of shells. Each panel covers
300$\times$300 kpc and is centered on the host galaxy. Only the surface
density of particles originally belonging to the satellite galaxy is
displayed. The density scale varies between frames, so that the respective
range of densities is optimally covered. Time-stamps mark the time since the
release of the star in the center of the host galaxy.
In all cases, we look at the galaxy from the view perpendicular to the axis of
collision, so that the cannibalized galaxy originally flew in from the
right.161616We use the term cannibalized galaxy even before and during the
merger process. Information on details of the simulation process can be found
in Sect. 17.1.
#### 13.1 Parameters of the simulation
The potential of the host galaxy is the same as the one described in Sect.
8.1. Let us only recall that it is a double Plummer sphere with respective
masses $M_{*}=2\times 10^{11}$ M⊙ and $M_{\mathrm{DM}}=1.2\times 10^{13}$ M⊙ ,
and Plummer radii $\varepsilon_{*}=5$ kpc and $\varepsilon_{\mathrm{DM}}=100$
kpc for the luminous component and the dark halo, respectively. The potential
of the cannibalized galaxy is chosen to be a single Plummer sphere with the
total mass $M=2\times 10^{10}$ M⊙ and Plummer radius $\varepsilon_{*}=2$ kpc.
The details of the simulations are described in Sect. 17.1. In the simulations
that we present in this part, neither the gradual decay of the cannibalized
galaxy nor the dynamical friction is included. The cannibalized galaxy is
released from rest at a distance of 100 kpc from the center of the host
galaxy. When it reaches the center of the host galaxy in 306.4 Myr, its
potential is switched off and its particles begin to oscillate freely in the
host galaxy. The shells start appearing visibly from about 50 kpc of
galactocentric distance and disappear at around 200 kpc, as there are very few
particles with apocenters outside these radii, Fig. 29. Video from the
simulation is part of the electronic attachment. For the description of the
video, see Appendix H point 2 and 3.
#### 13.2 Comparison of the simulation with models
In the simulations, some of the assumptions that we used earlier (the model of
radial oscillations, Sect. 9) are not fulfilled. First, the particles do not
move radially, but on more general trajectories, which are, in the case of a
radial merger, nevertheless very eccentric. Second, not all the particles are
released from the cannibalized galaxy right in the center of the host galaxy;
when the potential is switched off, the particles are located in the broad
surroundings of the center and some are even released before the decay of the
galaxy. These effects cause a smearing of the kinematical imprint of shells,
as the turning points are not at a sharply defined radius, but rather in some
interval of radii for a given time.
?figurename? 30: Simulated shell structure 2.2 Gyr after the decay of the
cannibalized galaxy. Only the particles originally belonging to the
cannibalized galaxy are taken into account. Top: surface density map; middle:
the LOSVD density map of particles in the $\pm 1$ kpc band around the
collision axis; bottom: histogram of galactocentric distances of particles.
The angle between the radial position vector of the particle and the $x$-axis
(the collision axis) is less than 90$\degr$ for the blue curve and less than
45$\degr$ for the red curve. The horizontal axis corresponds to the projected
distance $X$ in the upper panel, to the projected radius $R$ in the middle
panel, and to the galactocentric distance $r$ in the lower panel.
$r_{\mathrm{s}}$ | $n$ | $r_{\mathrm{TP,model}}$ | $v_{\mathrm{s,sim}}$ | $v_{\mathrm{s,model}}$ | $v\mathrm{{}_{c,model}}$
---|---|---|---|---|---
kpc | | kpc | km$/$s | km$/$s | km$/$s
48.8 | 5 | 48.5 | 38.7$\pm$2.1 | 38.7 | 326
$-$70.6 | 4 | $-$69.9 | 59.8$\pm$1.6 | 54.3 | 390
105.0 | 3 | 103.9 | 68.1$\pm$1.9 | 63.5 | 441
$-$157.8 | 2 | $-$155.7 | 74.3$\pm$1.2 | 72.4 | 450
257.4 | 1 | 251.0 | 97.5$\pm$1.4 | 95.7 | 406
?tablename? 3: Parameters of the shells in a simulation 2.2 Gyr after the
decay of the cannibalized galaxy. The shell positions $r_{\mathrm{s}}$ are
taken from the simulation. The values of $r_{\mathrm{TP,model}}$ and
$v_{\mathrm{s,model}}$ are calculated for the shell position $r_{\mathrm{s}}$
and its corresponding serial number $n$ according to the model of radial
oscillations (Sect. 9). The shell velocity $v_{\mathrm{s,sim}}$ is derived
from 20 positions between the times 2.49–2.51 Gyr for each shell. The value
$v\mathrm{{}_{c,model}}$ corresponds to the circular velocity at the shell-
edge radius $r_{\mathrm{s}}$ for the chosen potential of the host galaxy
(Sect. 8.1). ?figurename? 31: LOSVD map of the simulated shell structure 2.2
Gyr after the decay of the cannibalized galaxy (middle panel in Fig. 30).
Light blue curves show locations of the maxima according to the model of
radial oscillations (Sect. 9.4) for shell radius $r_{\mathrm{s}}$,
corresponding serial number $n$, and the known potential of the host galaxy
(Sect. 8.1). Orange curves are derived from the approximative maximal LOS
velocities (Sect. 11.6, point 2) given by Eq. (46) for $r_{\mathrm{s}}$,
$v_{\mathrm{s,model}}$, and $v\mathrm{{}_{c,model}}$. Parameters of the shells
are shown in Table 3. Black lines mark the location at $0.9r_{\mathrm{s}}$ for
each shell. The LOSVD for these locations are shown in Fig. 32. The map
includes only stars originally belonging to the cannibalized galaxy.
The model of radial oscillations presented in Sect. 9 predicts that 2.2 Gyr
after the decay of the cannibalized galaxy, five outermost shells should lie
at the radii of 257.3, $-$157.8, 105.1, $-$70.5, and 48.8 kpc. The negative
radii refer to the shell being on the opposite side of the host galaxy with
respect to the direction from which the cannibalized galaxy flew in. These
radii agree surprisingly well with the radii of the shells
measured171717Recall that by measuring, we mean that the data measured are the
output of our simulation. in the simulation 2.2 Gyr after the decay of the
cannibalized galaxy, see Fig. 30 and Table 3. The position of the shell edge
$r_{\mathrm{s}}$ in the simulation was determined as the position of a sudden
decrease of the projected surface density (see Figs. 41 and 42). These values
are shown in Table 3.
In the simulation, the first shell at 257.4 kpc is composed of only a few
particles, and therefore we will not consider it (its parameters are listed in
Table 3 for completeness). Thus, the outermost relevant shell in the system
lies at $-$157.8 kpc and has a serial number $n=2$. Also, the shell at 48.8
kpc suffers from lack of particles, but we will include it nevertheless.
?figurename? 32: LOSVDs of four shells at projected radii $0.9r_{\mathrm{s}}$
(indicated as the title of each plot) 2.2 Gyr after the decay of the
cannibalized galaxy (parameters of the shells are shown in Table 3). The
simulated data are shown in green, the LOSVDs according to the approximative
LOSVD (Sect. 11.6, point 1) given by Eq. (44) in purple, and LOSVDs according
to the model of radial oscillations (Sect. 9.4) in light blue. The graph also
shows the locations of the peaks using the approximative maximal LOS
velocities (Sect. 11.6, point 2) given by Eq. (46) by orange lines. Profiles
do not include stars of the host galaxy, which are not part of the shell
system. The theoretical profiles are scaled so that the intensity of their
highest peak approximately agrees with the highest peak of the simulated data.
LOSVD is given in relative units, so maxima of the profiles have values of
about 0.9.
Fig. 31 shows the comparison between the LOSVD in the simulation, the peaks of
the LOSVD computed in the model of radial oscillations (light blue curves),
and the approximative maximal LOS velocities – Eq. (46) (orange curves). To
evaluate the approximative maximal LOS velocities, we obtained the shell
velocity $v_{\mathrm{\mathrm{s,model}}}$ from the model of radial oscillations
(Sect. 9) for the respective serial number $n$ of the shell and circular
velocity $v\mathrm{{}_{c,model}}$ at the shell-edge radius, using our
knowledge of the potential of the host galaxy. The values of all the
respective shell quantities are listed in Table 3. Within the resolution of
Fig. 31, the theoretical positions of the LOSVD maxima agree very well with
the simulated data, even further from the shell edge than the usual limit of
$0.9r_{\mathrm{s}}$.
Fig. 31 also shows the locations that correspond to the radii of
$0.9r_{\mathrm{s}}$ for each individual shell (black lines). The LOSVD for
these locations is shown in Fig. 32. The data are taken from an area spanning
$0.5\times 2$ kpc centered at $(R,0)$ in the projected $X-Y$ plane, where $R$
is the number indicated above the corresponding panel in Fig. 32. The
positions of simulated LOSVD peaks largely agree with the approaches of the
approximation of constant acceleration and shell velocity described in Sect.
11.6 and with the model of radial oscillations (Sect. 9).
?figurename? 33: Fits for circular velocity $v_{\mathrm{c}}$ and shell
velocity $v_{\mathrm{s}}$ using the approximative LOSVD (Sect. 11.6, point 1)
given by Eq. (44) for four shells ($r_{\mathrm{s}}$ indicated in bottom right
corner of each plot) in the simulation 2.2 Gyr after the decay of the
cannibalized galaxy. The best fit is the purple curve, and its parameters are
shown in Tables 4 and 5 in the columns labeled $v_{\mathrm{c,fit}}$ and
$v_{\mathrm{s,fit}}$. The green crosses mark the measured maxima in the LOSVD,
and the light blue curves show the locations of the theoretical maxima derived
from the host galaxy potential according to the model of radial oscillations
(Sect. 9.4). Note that the values of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$
used in the approximative LOSVD for the purple line were obtained by fitting
the parameters to the simulated data, whereas in Figs. 22, 23, and 32, the
values are known from the model of the host galaxy potential.
#### 13.3 Recovering the potential from the simulated data
We used a snapshot from our simulation, which 2.2 Gyr after the decay of the
cannibalized galaxy, as a source of the simulated data and tried to
reconstruct the parameters of the potential of the host galaxy from the
locations of the LOSVD peaks measured from the simulated data by using the the
approximation of constant acceleration and shell velocity (Sect. 11).
For a given host galaxy, the signal-to-noise (S$/$N) ratio in the simulated
data is a function of the number of simulated particles, the age of the shell
system, the distribution function of the cannibalized galaxy, and the impact
velocity. For a given radius in the simulated data, we can obtain arbitrarily
good or bad S$/$N ratios by tuning these parameters. Thus, we adopted the
universal criteria: 1) the LOSVD of each shell is observed down to 0.9 times
its radius; 2) we measured the positions of the LOSVD peaks in different
locations within the shell, sampled by 1 kpc steps. These criteria give us
between 7 and 15 measurements for a shell. Each measurement contains two
values: the positions of the outer and inner peaks, $v_{\mathrm{los,max}+}$
and $v_{\mathrm{los,max}-}$, respectively, for each projected radius $R$ (see
green crosses in Fig. 33).
?figurename? 34: Comparison of velocity of the shell as a function of radius
from the model and the simulated data. Velocity for the first shell ($n=1$) in
the host galaxy model is shown by the black line. Red crosses show
$v_{\mathrm{s,eq(\ref{eq:vs-vc})-slope}}$ (Table 5) as they result from the
analysis of the simulated LOSVD. Values are corrected for shell number $n$ by
the factor $3/(2n+1)$, so they correspond to velocity of the first shell,
e.g., Eq. (5).
We do not estimate the errors, since the real data will be dominated by other
sources, such as the contamination of the signal from the light of the host
galaxy and the accuracy of the subtraction of this background light, night-sky
background in the case of ground-based telescopes, detector noise,
instrumental dispersion, accuracy in the determination of the systemic
velocity and so forth. So we decided to quote only the mean square deviation
and the standard error of the linear regression.
First we used the approximative maximal LOS velocities given by Eqs. (47) and
(48) for a direct calculation of the circular velocity
$v_{\mathrm{c,eq(\ref{eq:vc,obs})}}$ at the shell-edge radius $r_{\mathrm{s}}$
and the current shell velocity $v_{\mathrm{s,eq(\ref{eq:vs,obs})}}$. These
equations are the inverse of Eq. (46), which corresponds to the model shown in
orange lines in pictures throughout the text (Sect. 11.6, point 2). Mean
values from all the measurements for each shell are shown in Tables 4 and 5 in
the end of the section.
Compared with the approximative maximal LOS velocities, we obtain a better
agreement with the circular velocity of our host galaxy potential when using
the slope of the LOSVD intensity maxima (Sect. 11.6, point 3) given by Eq.
(50), where we fit the linear function of the measured distance between the
outer and the inner peak on the projected radius ($v_{\mathrm{c,slope}}$ in
Table 4 and in Fig. 35). To estimate the shell velocity, we use a hybrid
relation Eq. (51) between the positions of the LOSVD peaks, the circular
velocity at the shell-edge radius $v_{\mathrm{c}}$, and the shell velocity. We
substitute the values of $v_{\mathrm{c,slope}}$ derived from the measurements
(that we know better describe the real circular velocity of the host galaxy)
into this relation, thus obtaining the improved measured shell velocity
$v_{\mathrm{s,eq(\ref{eq:vs-vc})-slope}}$ (Table 5 and Fig. 34).
In the zone between the current turning points and the shell edge, the inner
peaks coalesce and gradually disappear (Fig. 15). The simulated data do not
show a disappearance of the inner peaks as abrupt and clear as the theoretical
LOSVD profiles predict, so that in this zone, we can usually measure one inner
peak at 0 km$/$s. The information from these measurements is degenerate, and
thus we defined a subsample of simulated measurements with all four clear
peaks in the LOSVD (in the columns labeled SS in Tables 4 and 5). The spread
of the values derived using the approximative maximal LOS velocities given by
Eqs. (47) and (48) is significantly lower for the subsample
($v_{\mathrm{c,eq(\ref{eq:vs,obs})}}^{\mathrm{SS}}$ and
$v_{\mathrm{s,eq(\ref{eq:vc,obs})}}^{\mathrm{SS}}$) due to the exclusion of
areas where these equations do not hold well. On the contrary, the slope of
the linear regression in Eq. (50) using the slope of the LOSVD intensity
maxima gives a worse result (with a larger error) for the subsample
$v_{\mathrm{c,slope}}^{\mathrm{SS}}$ than the approximative maximal LOS
velocities.
?figurename? 35: Circular velocity of the model and values derived from the
simulated data: $v\mathrm{{}_{c,model}}$ of the host galaxy model is shown by
the black line; blue and red points show values of circular velocity as they
result from the analysis of the simulated LOSVD (see Sect. 13.2 and Table 4
for the numbers).
The third option to derive the circular velocity $v_{\mathrm{c}}$ at the
shell-edge radius $r_{\mathrm{s}}$ and shell velocity $v_{\mathrm{s}}$ from
the simulated data is to use the approximative LOSVD given by Eq. (44), which
corresponds to the model shown in purple lines in pictures throughout the text
(Sect. 11.6, point 1). However, this requires a numerical solution of the
equation for a given pair of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$. We have
calculated two sums of squared differences between
$v_{\mathrm{los,max}}(v_{\mathrm{c}},v_{\mathrm{s}})$ as given by the
approximative LOSVD and the simulated data. One for
$v_{\mathrm{los,max-}}(v_{\mathrm{c}},v_{\mathrm{s}})$ and a second one for
$v_{\mathrm{los,max+}}(v_{\mathrm{c}},v_{\mathrm{s}})$ . Then we have searched
for the minimum of the sum of these two values to obtain best fitted values
$v_{\mathrm{c,fit}}$ and $v_{\mathrm{s,fit}}$ (see Tables 4 and 5 for the
results). Errors were estimated using the ordinary least squared minimization
as if the functions
$v_{\mathrm{los,max}+}(v_{\mathrm{c,fit}},v_{\mathrm{s,fit}})$ and
$v_{\mathrm{los,max}-}(v_{\mathrm{c,fit}},v_{\mathrm{s,fit}})$ were fitted
separately; quoted is the larger of the two errors.
$r_{\mathrm{s}}$ | $v\mathrm{{}_{c,model}}$ | $N$ | $N^{\mathrm{SS}}$ | $v_{\mathrm{c,eq(\ref{eq:vc,obs})}}$ | $v_{\mathrm{c,eq(\ref{eq:vc,obs})}}^{\mathrm{SS}}$ | $v_{\mathrm{c,slope}}$ | $v_{\mathrm{c,slope}}^{\mathrm{SS}}$ | $v_{\mathrm{c,fit}}$ | $v_{\mathrm{c,slope(MK98)}}$
---|---|---|---|---|---|---|---|---|---
kpc | km$/$s | | | km$/$s | km$/$s | km$/$s | km$/$s | km$/$s | km$/$s
48.8 | 326 | 5 | 4 | 346$\pm$130 | 340$\pm$94 | 322$\pm$19 | 314$\pm$32 | 318$\pm$51 | 449$\pm$26
$-$70.6 | 390 | 7 | 5 | 394$\pm$85 | 390$\pm$53 | 391$\pm$5 | 392$\pm$11 | 368$\pm$60 | 570$\pm$23
105.0 | 441 | 11 | 8 | 478$\pm$144 | 452$\pm$64 | 440$\pm$5 | 447$\pm$7 | 427$\pm$28 | 632$\pm$9
$-$157.8 | 450 | 15 | 10 | 497$\pm$236 | 472$\pm$79 | 462$\pm$8 | 484$\pm$14 | 460$\pm$32 | 671$\pm$11
?tablename? 4: Circular velocity at the shell-edge radius $r_{\mathrm{s}}$ derived from the measurement of the simulated data 2.2 Gyr after the decay of the cannibalized galaxy. $r_{\mathrm{s}}$ and $v\mathrm{{}_{c,model}}$ have the same meaning as in Table 3. $N$: number of measurements for each shell; $v_{\mathrm{c,eq(\ref{eq:vc,obs})}}$: the mean of values derived from the approximative maximal LOS velocities given by Eq. (47) with its mean square deviation; $v_{\mathrm{c,slope}}$: a value derived from the linear regression using the slope of the LOSVD intensity maxima given by Eq. (50) and its standard error (see also Fig. 35); $v_{\mathrm{c,fit}}$: a value derived by fitting a pair of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$ in the approximative LOSVD given by Eq. (44) (Sect. 11.6, point 1 and Fig. 33); $v_{\mathrm{c,slope(MK98)}}$: the mean of values derived from the slope of the LOSVD intensity maxima given by Eq. (50) with its standard error (see also Fig. 35). In the equation, however, $\bigtriangleup v_{\mathrm{los}}$ is substituted with the distance between the two outer peaks of the LOSVD intensity maxima in order to mimic the measurement as originally proposed by MK98 for double-peaked profile. The quantities with the superscript SS correspond to the subsample, where only measurements with two discernible inner peaks in the LOSVD are used. $r_{\mathrm{s}}$ | $v_{\mathrm{s,model}}$ | $v_{\mathrm{s,sim}}$ | $v_{\mathrm{s,eq(\ref{eq:vs,obs})}}$ | $v_{\mathrm{s,eq(\ref{eq:vs,obs})}}^{\mathrm{SS}}$ | $v_{\mathrm{s,eq(\ref{eq:vs-vc})-slope}}$ | $v_{\mathrm{s,eq(\ref{eq:vs-vc})-slope}}^{\mathrm{SS}}$ | $v_{\mathrm{s,fit}}$
---|---|---|---|---|---|---|---
kpc | km$/$s | km$/$s | km$/$s | km$/$s | km$/$s | km$/$s | km$/$s
48.8 | 38.7 | 38.7$\pm$2.1 | 50.7$\pm$2.3 | 51.7$\pm$1.1 | 44.2$\pm$6.5 | 44.9$\pm$6.3 | 53$\pm$16
$-$70.6 | 54.3 | 59.8$\pm$1.6 | 60.8$\pm$9.8 | 65.6$\pm$2.0 | 60.7$\pm$10.8 | 66.0$\pm$2.9 | 66$\pm$19
105.0 | 63.5 | 68.1$\pm$1.9 | 74.8$\pm$4.6 | 76.5$\pm$1.4 | 68.0$\pm$8.9 | 71.3$\pm$2.5 | 79$\pm$9
$-$157.8 | 72.4 | 74.3$\pm$1.2 | 84.4$\pm$5.4 | 86.7$\pm$2.0 | 78.7$\pm$10.5 | 82.$\pm$3.5 | 85$\pm$14
?tablename? 5: Velocity of the shell at the radius $r_{\mathrm{s}}$ derived
from the measurement of the simulated data 2.2 Gyr after the decay of the
cannibalized galaxy. $r_{\mathrm{s}}$, $v_{\mathrm{s,model}}$, and
$v_{\mathrm{s,sim}}$ have the same meaning as in Table 3.
$v_{\mathrm{s,eq(\ref{eq:vs,obs})}}$: the mean of values derived from the
approximative maximal LOS velocities given by Eq. (48) with its mean square
deviation; $v_{\mathrm{s,eq(\ref{eq:vs-vc})-slope}}$: the mean of values
derived from the hybrid relation given by Eq. (51) with its mean square
deviation (see also Fig. 34); $v_{\mathrm{s,fit}}$: a value derived by fitting
a pair of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$ in the approximative LOSVD
given by Eq. (44) (Sect. 11.6, point 1 and Fig. 33). The quantities with the
superscript SS correspond to the subsample, where only measurements with two
discernible inner peaks in the LOSVD are used. Number of measurements is the
same as in Table 4 for each shell.
The LOSVD intensity maxima resulting from this procedure are plotted in Fig.
33, together with the fitted data and the maxima given by the model of radial
oscillations (Sect. 9.4). All three agree fairly well. The remaining two
methods (the approximative maximal LOS velocities and using the slope of the
LOSVD intensity maxima) use only equations to derive $v_{\mathrm{c}}$ and
$v_{\mathrm{s}}$ and thus we do not show them in the plot. On the other hand,
in Figs. 34 and 35, we show the comparison of values extracted from the
simulated data with model values only for the most successful approach – using
the slope of the LOSVD intensity maxima.
For the sake of comparison with the method of MK98, we calculated the circular
velocity $v_{\mathrm{c,slope(MK98)}}$ at the shell-edge radius
$r_{\mathrm{s}}$ using the slope of the LOSVD intensity maxima given by Eq.
(50). To mimic the measurement of the circular velocity according to the Eq.
(26), which was derived for the double-peaked profile, we assume
$\bigtriangleup v_{\mathrm{los}}$ is the distance between the two outer peaks
of the LOSVD intensity maxima. In Table 4 and Fig. 35, we can easily see that
the values $v_{\mathrm{c,slope(MK98)}}$ differ from the actual circular
velocity of the host galaxy $v\mathrm{{}_{c,model}}$ by a factor of 1.3–1.5.
The main message of this section is that in order to obtain the value of the
circular velocity $v_{\mathrm{c}}$ at the shell-edge radius and shell phase
velocity $v_{\mathrm{s}}$ from kinematical data near the shell edge, the best
approach to use is the method based on the slope of the LOSVD intensity maxima
given by Eq. (50) without limiting the data to a subsample.
#### 13.4 Notes about observation
This work is a theoretical one, dealing with simulations and models. Obtaining
and analyzing real data requires preparation, knowledge and experience that
are beyond the goals we have set in this research. Nevertheless, we will make
some remarks regarding potential observation of shell kinematics.
?figurename? 36: Line profiles of four shells at projected radii
$0.9r_{\mathrm{s}}$ (indicated as the title of each plot, same as in Fig. 32)
2.2 Gyr after the decay of the cannibalized galaxy: gray lines show the LOSVDs
for the host galaxy at a given radius (except for the radius of 44 kpc the
signal of the host galaxy is negligible comparing to the signal from the
cannibalized galaxy); green lines show the total LOSVDs from the host and the
cannibalized galaxy together; red, blue, and yellow lines show convolutions of
the total simulated data with different Gaussians representing the
instrumental profiles having the FWHM 10, 30, and 60 km$/$s, respectively.
Scaling is relative, similar as in Fig. 32.
When it comes to real observational data, there will be additional issues to
deal with, night-sky background, detector noise, instrumental dispersion and
so forth. MK98 estimated the data of the requisite quality could be obtained
with a couple of nights integration using a 4-m telescope.
The situation gets more complex when the LOSVD assumes the quadruple-peaked
profile instead of a double-peaked one. Not only becomes the intensity of a
single peak smaller, but a higher spectral resolution is also needed to
distinguish all four peaks. The instrumental dispersion naturally smooths
features of the spectral profile. In Fig. 36, we show the LOSVDs from the
simulated data smoothed with different Gaussians representing the instrumental
profiles having the full width at half maximum (FWHM) of 10, 30, and 60
km$/$s. It is obvious that relatively high spectral resolution is necessary
for observing an imprint of shell peaks in line profiles.
We have done our own simplified estimations of the observability of the LOSVD
of shells. First, we used archival data of long-slit spectroscopy of the
outermost shell in NGC 3923. The data were taken in July 2001 (about 10 hours
of exposure time) and in March 2005 (about 20 hours) with FORS2 instrument at
the Very Large Telescope (VLT, 8.2 meter diameter) of the European Southern
Observatory. We processed a part of the data from 2005 using the FORS
pipeline.181818The procedure was done mostly by Lucie Jílková, Ivana Orlitová,
and Tereza Skalická The spectra are generally of a very low signal-to-noise
ratio (S$/$N). We were particularly looking for the magnesium triplet around
5200 Å (taken into account the redshift of NGC 3923, about 30 Å) and we found
no sign of it, so the analysis of kinematics was not possible. We conclude
that the estimate of MK98 was probably a bit of an understatement.
Furthermore, we used exposure time calculators to determine expected S$/$N at
available instruments (VLT/FORS2, VLT/FLAMES, Calar Alto/PPAK) assuming the
exposure time 20 hours and the surface brightness of shells between 25 and 28
mag$/$arcsec2 in V filter. The resulting S$/$N ranges from $\sim 0.3$ to $\sim
4.4$. This is not very satisfactory but using the integral field spectroscopy
or the multi object spectroscopy, S$/$N could be increased by a factor of up
to $\sim 10$ by summing the signal from all fibers. Moreover, one can use some
kind of a cross-correlation technique (e.g., Simkin, 1974; Tonry and Davis,
1979) which allows to extract more accurate kinematic measurements than the
actual resolution of the data is or extract more information from data with
low S$/$N. Eventually, the situation should be much better with the next
generation of telescopes, like the European Extremely Large Telescope or the
James Webb Space Telescope.
Another important issue is the background light of the host galaxy. It is
possible to model the LOSVD of the host galaxy, subtract it from the overall
LOSVD and obtain the clear quadruple-peaked profile, but it may not be even
necessary, because the velocity dispersion of the stars in the host galaxy
would be likely significantly broader than the distance between the peaks and
thus the peaks should be clearly visible already in the overall LOSVD.
Moreover, for shells at large radii, the contribution from the stars of the
host galaxy becomes negligible – and it is exactly the shells at large radii
that are the most interesting because our knowledge of the potential of the
host galaxy is the worst in the outer parts of the galaxy, where the potential
is expected to be dominated by the dark matter. In our simulated data, the
host galaxy light is negligible already for the shell at 70 kpc, see Fig. 36.
The surface brightness of observed shells goes from 24.5 mag$/$arcsec2 (in V
filter) up to the current detection limit of the deepest photometric
observation $\sim 29$ mag$/$arcsec2 (McGaugh and Bothun, 1990; Turnbull et
al., 1999; Pierfederici and Rampazzo, 2004). The surface brightness of giant
elliptical galaxies at $\sim 100$ kpc (the position of the outermost shell in
NGC 3923) is 28–30 mag$/$arcsec2 (in g and r filters; Tal and van Dokkum,
2011).
A category on its own is the measurement of LOS velocities of individual
objects, such as globular clusters, planetary nebulae and individual giant
stars (Fardal et al., 2012; Romanowsky et al., 2012), where the result is
dependent only on the accuracy of the measurement and the number of measured
objects.
The positions of LOSVD maxima should be symmetric around the systemic velocity
which we can measure or assume to be in the middle between the peaks. We also
need photometric data to find the center of the host galaxy and to measure the
distance of the point of the spectroscopic observation and the shell edge from
the center. As soon as we measure the locations of the LOSVD peaks
$v_{\mathrm{los,max}+}$, $v_{\mathrm{los,max}-}$, the projected radius $R$ of
the measurement, and the shell-edge radius $r_{\mathrm{s0}}$, we can calculate
the value of the circular velocity $v_{\mathrm{c}}$ at the shell-edge radius
and shell phase velocity $v_{\mathrm{s}}$ using one of the three approaches
described in Sect. 11.6. Using the simulated data (Sect. 13.3), we found the
derived $v_{\mathrm{c}}$ to be the most accurate when using the slope of the
LOSVD intensity maxima given by Eq. (50), which requires the peak locations to
be measured at several different radii. When a measurement from only one
projected radius is available, Eqs. (47) and (48) can be used to derive
$v_{\mathrm{c}}$ and $v_{\mathrm{s}}$ , respectively.
### 14 Shell density
In this section we take an apparent detour from the shell kinematics to
explore the projected and volume densities of a shell. In Sect. 14.1 we
express the projected surface density of the shell edge
$\Sigma_{\mathrm{los}}(r_{\mathrm{s}})$ (that is, the projected surface
density at the projected radius $R=r_{\mathrm{s}}$) as a function of
$\Sigma_{\mathrm{sph}}$ (Sects. 9.6, 9.7, and 9.8) and the shell-edge radius
$r_{\mathrm{s}}$. In Sect. 14.2 we investigate the evolution of
$\Sigma_{\mathrm{los}}(r_{\mathrm{s}})$ as a function of time, as the position
of the shell edge is a function of time. In Sect. 14.3 we show the volume
density of a shell at a frozen moment and finally in Sect. 14.4, we explore
the projected surface density of shells near the shell edge at a given time as
a function of the projected radius $R$.
#### 14.1 Projected surface density of the shell edge
Each time we needed to model an LOSVD, we have used the assumption that the
shell-edge density distribution
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ or rather
$\Sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$ decreases as
$1/r_{\mathrm{s}}^{2}\left(t\right)$, see Sects. 9.6, 9.7, and 9.8. Now we
show how is this value related to an observable quantity, the projected
surface density of the shell edge $\Sigma_{\mathrm{los}}(r_{\mathrm{s}})$. If
we knew or assumed the mass-to-light ratio, $\Sigma_{\mathrm{los}}$ could be
easily converted to the projected surface brightness.
?figurename? 37: Schema for the calculation of the projected surface density.
Consider a thin sphere of mass with a uniform spatial density $\rho$ and
radius $r_{s}$, Fig. 37. When observed along the line of sight $z$, the amount
of light registered from a point with a projected radius $R$ in the sphere’s
image is proportional to the expression
$\rho\Delta
z=\rho\left(\sqrt{r_{\mathrm{s}}^{2}-R^{2}}-\sqrt{\left(r_{\mathrm{s}}-\Delta
r\right)^{2}-R^{2}}\right),$ (66)
which for an infinitesimally thin sphere ($\Delta r\rightarrow 0$) reduces to
$\rho\Delta
z\rightarrow\frac{r_{\mathrm{s}}\Sigma_{\mathrm{sph}}}{\sqrt{r_{\mathrm{s}}^{2}-R^{2}}}.$
(67)
This expression diverges when the sphere is observed tangentially to its
surface, that is on the shell edge – thus to talk about the projected surface
density of the shell edge, we have to integrate the flux over a small
observation area. As the shape of the area is irrelevant for infinitesimal
sizes, we choose an area that is the easiest to integrate over in spherical
coordinates that are convenient for a radially-symmetric density. Note that
the angular size of the area is approximately $2\Delta R/r_{s}$ and thus the
integrated flux is
$\Sigma_{\mathrm{los}}=\frac{2}{S}\Sigma_{\mathrm{sph}}r_{\mathrm{s}}\intop_{0}^{\frac{\Delta
R}{r_{\mathrm{s}}}}\intop_{r_{\mathrm{s}}-\Delta
R}^{r_{\mathrm{s}}}\frac{R}{\sqrt{r_{\mathrm{s}}^{2}-R^{2}}}\mathrm{d}R\mathrm{d}\phi,$
(68)
where $S=2\Delta R^{2}+o(\Delta R^{3})$ is the size of the integration area.
Since
$\intop_{a}^{b}\frac{x}{\sqrt{r^{2}-x^{2}}}\mathrm{d}x=\sqrt{r^{2}-b^{2}}-\sqrt{r^{2}-a^{2}}$,
the integral reads
$\Sigma_{\mathrm{los}}\simeq\Sigma_{\mathrm{sph}}\sqrt{\left(2r_{\mathrm{s}}-\Delta
R\right)/\Delta R}\propto r_{\mathrm{s}}^{1/2}\Sigma_{\mathrm{sph}}.$ (69)
?figurename? 38: Time evolution of the projected surface density of the shell
edge (0.01 kpc) in the approximation of a constant radial acceleration in the
host galaxy potential and shell phase velocity (Sect. 11) – yellow curve, in
arbitrary units. The red curve represents a function $r^{-3/2}$ normalized so
that it has the same value at $R=60$ kpc as the yellow curve. For the
parameters of the host galaxy potential, see Sect. 8.1. ?figurename? 39:
Histogram of apocenters of particles in the simulation used in Sect. 13.
#### 14.2 Time evolution
The radial dependence of $\Sigma_{\mathrm{sph}}$ is chosen, as usual, as
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}}(t))\propto 1/r_{\mathrm{s}}^{2}(t)$.
Then from Eq. (69) it follows that
$\Sigma_{\mathrm{los}}(r_{\mathrm{s}}(t))\propto r_{\mathrm{s}}^{-3/2}(t).$
(70)
However, the calculation leading to Eq. (69) assumes that all the stars are
located at the sphere with the radius of the shell. We have thus examined the
time evolution of the projected surface density of the shell edge in the
framework of the approximation of a constant radial acceleration in the host
galaxy potential and shell phase velocity (Sect. 11, in this section, Sect.
14, hereafter the approximation) – Fig. 38. For each shell radius we calculate
the motion of stars under a constant acceleration, but we update this
acceleration for different shell radii according to the chosen potential of
the host galaxy (for the parameters of the potential, see Sect. 8.1). The time
evolution of the projected surface density of the shell edge in this
approximation does not depend on its velocity and thus on its serial number,
see Sect. 14.4. In this approximation, stars are present at all radii,
0–$r_{\mathrm{s}}$, in contrast to the calculation that lead us to Eq. (70),
where we assumed the stars to be located only at the shell radius (in a given
time). Nevertheless, the time evolution of
$\Sigma_{\mathrm{sph}}(r_{\mathrm{s}}(t))$, Fig. 38, turns out to be
essentially identical when calculated by either of these approaches.
Both the calculation of Eq. (70), and the approximation assume
$\Sigma_{\mathrm{sph}}$ to decrease as $1/r_{\mathrm{s}}^{2}\left(t\right)$,
corresponding to constant number of stars at the edge of the shell,
$N\left(r_{\mathbf{s}}\right)$. Fig. 39 shows the distribution of apocenters
of particles in the simulation from Sect. 13, which is a good approximation to
real $N\left(r_{\mathbf{s}}\right)$. We have to honestly admit that this
function is anything but constant, but it is difficult to devise any
approximation as the shape of the distribution significantly varies with
parameters of the collision. Moreover, we do apply this function usually only
in a small range of radii and as we have already shown, the character of the
LOSVD does not depend much on its choice (Sects. 9.7 and 9.8). Converting the
histogram of apocenters of the particles to the shell brightness is not
straightforward as, both in the simulation and real shell galaxies, the
distribution of particles is not uniform in azimuth, contrary to what he
assumed in modeling the LOSVD both in the approximation and in the model of
radial oscillations (Sect. 9.4).
#### 14.3 Volume density
The calculation in Sect. 14.1 assumes that stars are at each moment located
only on a sphere with the radius of the shell. Nevertheless it gives good
results when compared to the approximation (Fig. 38), where this assumption
does not hold. The reason is that the volume density decreases quickly inward
from the shell edge (it obviously decreases outward in a jump, but that is not
of concern at the moment). In their work, Hernquist and Quinn (1988) recall
that Arnold (1984) states that for phase wrapped shells, that are just
caustics in the mapping of the particle density from phase space into three-
dimensional space, it holds that the density behind a caustic should scale as
$(r_{\mathrm{s}}-r)^{-1/2}$. This behavior should be independent of the used
potential of the host galaxy. In Fig. 40 we have compared the volume density
near the shell edge in the approximation with this function and they indeed
show a pretty good agreement.
?figurename? 40: Volume density for the third shell at $105$ kpc in the
approximation of a constant radial acceleration in the host galaxy potential
and shell phase velocity (Sect. 11) – yellow curve, in arbitrary units. The
red curve represents a function $(r_{\mathrm{s}}-r)^{-1/2}$ normalized so that
at $r_{\mathrm{s}}-r=1.1$ kpc it has the same value as the yellow curve. For
the parameters of the host galaxy potential, see Sect. 8.1.
For a stationary shell, the volume density near the shell edge holds
$\rho(r)=\frac{k}{v_{r}r^{2}},$ (71)
where $k$ is a constant for the given shell and $v_{r}$ is the radial velocity
of the shell. In a field of constant acceleration $a_{0}$ Eq. (27) holds –
$v_{r}=\sqrt{2a_{0}(r-r_{\mathrm{s}})},$ thus the volume density is
$\rho(r)\propto\frac{1}{r^{2}\sqrt{r-r_{\mathrm{s}}}}.$ (72)
In the vicinity of the shell, the term $(r_{\mathrm{s}}-r)^{-1/2}$ dominates.
For a moving shell it is difficult to make such analysis, but we have seen on
an example, in Fig. 40, that this holds even in such case.
#### 14.4 Projected surface density
Finally we reach a really observable quantity that is the projected surface
density on the sky for a shell in a given time. For volume density following
Eq. (72) the projected surface density turns out to be constant after
integration. Thus we can assume constant projected surface density/brightness
immediately behind the shell. The sharp-edged appearance of shells is caused
by the abrupt decrease of their brightness outside the shell radius, as we
already demonstrated in Sect. 9.3.
?figurename? 41: Surface brightness profile for two shells from simulation
used in Sect. 13 – green curve; for equivalent shells using the approximation
(Sect. 11) – yellow curve, and the model of radial oscillations (Sect. 9) –
red curve. The curves are normalized so that they coincide and assume unit
value at 50 and 80 kpc for shells with radii 70 and 105 kpc, respectively.
Fig. 41 shows the projected surface density profile for two shells from the
simulation (Sect. 13) and for shells on same radii (70 and 105 kpc) using the
approximation and the model of radial oscillations (Sect. 9). The
approximation departs from the model of radial oscillations slightly only in
the vicinity of the center of the host galaxy. In the approximation, the
current location of a star for different $t_{\mathrm{s}}$ does not depend on
the shell velocity, see Eq. (41), where $t_{\mathrm{s}}$ is the time where the
star was or will be at the shell edge. Thus even the projected surface density
calculated in the approximation does not depend on the serial number of the
shell. The character of the profile immediate behind the shell is however
slowly rising toward the center of the host galaxy, rather than constant. The
shapes of the profile from the simulation and the approximation or the model
of the radial oscillations coincide fairly well, even though the approximation
and the model of radial oscillations assume uniform azimuthal distribution of
particles which is obviously not valid in the simulation (see e.g. Figs. 30 or
29).
?figurename? 42: Surface brightness profile near the shell edge for the outer
shell from simulation used in Sect. 13 – green curve; and for equivalent
shells using the approximation – yellow curve, and the model of radial
oscillations – red curve. The curves are normalized so that they coincide and
assume unit value at 100 kpc.
On the other hand, no agreement at all is found for the outermost shell from
the simulation at 158 kpc near its edge with the approximation or the model of
radial oscillations, Fig. 42. The simulated shell even significantly decreases
in brightness just at its edge. The reason for this is that the shell is
nearing its demise and stars to arrive at higher radii are missing (see Figs.
30). Another factor is the azimuthal development of brightness, as the shell
is the brightest near the axis of the merger and at higher angles (measured
from the axis of the merger) the number of stars decreases. That, together
with a large shell radius causes a decrease in the projected surface density
at radii lower than the shell radius. A universal profile of the projected
surface density/brightness for phase wrapped shells thus does not exist, but
in general a rather constant or rising behavior can be expected for the inner
shells, whereas the outer shell can show decrease toward the center of the
host galaxy.
All the profiles of the projected surface density have been drawn for a band
$\pm 1$ kpc around the merger axis in the projected plane perpendicular to the
merger axis.
### 15 Discussion
In this part of the thesis, we developed a method to measure the potential of
shell galaxies from kinematical data, extending the work of MK98, assuming a
constant shell phase velocity and a constant radial acceleration in the host
galaxy potential for each shell. The method splits into three different
analytical and semi-analytical approaches (Sect. 11.6) for obtaining the
circular velocity in the host galaxy, $v_{\mathrm{c}}$, and the current shell
phase velocity, $v_{\mathrm{s}}$ – the approximative LOSVD, the approximative
maximal LOS velocities, and the slope of the LOSVD intensity maxima. In Sect.
11.6, the first two approaches are compared to the model of radial
oscillations (numerical integration of radial trajectories of stars in the
host galaxy potential, Sect. 9). All three approaches are then applied to data
for the four shells obtained from a test-particle simulation and compared to
the theoretical values (Sect. 13.2).
The approximative LOSVD requires a numerical solution to Eq. (44) and the
search for a pair of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$, which matches the
(simulated) data best. Although this approach is not limited by any
assumptions about the radius of the maximal LOS velocity (Sect. 11.3), it does
not give a better estimate of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$ for our
simulated shell galaxy than the other two methods. The deviation from the real
value of $v_{\mathrm{c}}$ is between 2 % and 6 %.
Using the approximative maximal LOS velocities results in simple analytical
relations and is the only one that can in principle be used for a LOSVD
measured at only one projected radius. Nevertheless, when measuring in the
zone between the radius of the current turning points and the shell radius, we
can expect very bad estimates of $v_{\mathrm{c}}$ and $v_{\mathrm{s}}$. The
mean value from more measurements of the LOSVD peaks for each shell of our
simulated shell galaxy has similar accuracy to those of the approximative
LOSVD, provided that we include only the measurements outside the zone between
the radius of the current turning points and the shell radius.
The best method for deriving the circular velocity in the potential of the
host galaxy seems to be to use the slope of the LOSVD intensity maxima, with a
typical deviation in the order of units of km$/$s when fitting a linear
function over all the measured positions of the LOSVD peaks for each shell.
This circular velocity is then used in the hybrid relation, Eq. (51), to
obtain the best estimate of the shell phase velocity.
All the approaches, however, derive the shell phase velocity systematically
larger than the prediction of the model of radial oscillations
$v_{\mathrm{s,model}}$ and the value derived from positions between the times
2.49–2.51 Gyr in the simulation $v_{\mathrm{s,sim}}$ (Table 5). This is
because the simulated LOSVD peaks lie too far out (for the outer peaks) or too
far in (for the inner peaks) when compared to the model of radial
oscillations. That can be caused by nonradial trajectories of the stars of the
cannibalized galaxy or by poor definition of the shell radius in the
simulation.
Nevertheless, the shell phase velocity depends, even in the simplified model
of an instant decay of the cannibalized galaxy in a spherically symmetric host
galaxy (Sec. 9), on the serial number of the shell $n$ and on the whole
potential from the center of the galaxy up to the shell radius, Eq. (5). A
comparison of its measured velocity to theoretical predictions is possible
only for a given model of the potential of the host galaxy and the presumed
serial number of the observed shells. In such a case, however, it can be used
to exclude some parameters or models of the potential that would otherwise fit
the observed circular velocity.
The first shell has a serial number equal to one. A higher serial number means
a younger shell. On the same radius, the velocity of each shell is always
smaller than that of the previous one. In practice, it is difficult to
establish whether the outermost observed shell is the first one created, or
whether the first shell (or even the first couple of shells) is already
unobservable. Here, we can use the potential derived from our method or a
completely different one in a reverse way: to determine the velocity of the
first shell on the given radius and to compare it to the velocity derived from
the positions of the LOSVD peaks. Knowing the serial number of the outermost
shell and its position allows us then to determine the time from the merger
and the impact direction of the cannibalized galaxy. Moreover, the measurement
of shell velocities can theoretically distinguish the shells from different
generations, which can be present in a shell galaxy (Bartošková et al., 2011).
Our method for measuring the potential of shell galaxies has several
limitations. Theoretical analyses were conducted over spherically symmetric
shells, while the test-particle simulation was run for a strictly radial
merger and analyzed in a projection plane parallel to the axis of the merger.
In addition, both analytical analyses and simulations assume spherical
symmetry of the potential of the host galaxy. In reality, the regular shell
systems with higher number of shells in a single host galaxy are more often
connected to galaxies with significant ellipticity (Dupraz and Combes, 1986).
Moreover, in cosmological simulations with cold dark matter, halos of galaxies
are described as triaxial ellipsoids (e.g., Jing and Suto, 2002; Bailin and
Steinmetz, 2005; Allgood et al., 2006). However, the effect of the ellipticity
of the isophotes of the host galaxy on the shell kinematics need not be
dramatic, as the shells have the tendency to follow equipotentials that are in
general less elliptical than the isophotes. Dupraz and Combes (1986) concluded
that while the ellipticity of observed shells is generally low, it is neatly
correlated to the eccentricity of the host galaxy. Prieur (1988) pointed out
that the shells in NGC 3923 are much rounder than the underlying galaxy and
have an ellipticity that is similar to the inferred equipotential surfaces.
This idea was originally put forward by Dupraz and Combes (1986), who found
such a relationship for their merger simulations. Our method is in principle
applicable even to shells spread around the galactic center, which are usually
connected to rounder elliptical galaxies if they were created in a close-to-
radial merger. Nevertheless, the combination of the effects of the projection
plane, merger axis, and ellipticity of the host galaxy can modify our results
and require further analyses.
Because the kinematics of the stars that left the cannibalized galaxy is in
the first approximation a test-particle problem, they should not be much
affected by self-gravity of the cannibalized galaxy and the dynamical friction
that this galaxy undergoes during the merger, both of which have been
neglected in Part II.
Another complication is that the spectral resolution required to distinguish
all four peaks is probably quite high (Sect. 13.4 and Fig. 36) and the shell
contrast is usually small. The higher order approximation, Sect. 12, is
sensible only when kinematical data are available to larger distances from the
shell edge. In the application to simulated data, we considered a shell that
is observable down to 0.9 shell radii. Nevertheless, there is the possibility
to measure shell kinematics using the LOS velocities of individual globular
clusters, planetary nebulae, and, in the Local group of galaxies, even of
individual stars. It is even possible that the shell kinematics will be
detectable in H I and CO emission, see Sects. 3.5 and 6.7.
We have also explored the projected surface density of shells, Sect. 14.4. In
the model of radial oscillations, the shells show constant projected surface
density near the shell edge, whereas outside the shell radius, there is a
step-like decrease of the density, creating the sharp-edged feature of the
shells. This behavior can be expected from shells with a large development in
azimuth and a sufficient supply of stars at different energies. Already in our
simple simulation of a radial minor merger with test particles and
instantaneous decay of the cannibalized galaxy, we can observe a shell with a
projected surface density that defies this description. We can assume that the
self-gravity and gradual decay of the cannibalized galaxy can disrupt the
observed profile even further. Moreover, we worked only in strictly spherical
potentials and any non-zero ellipticity of the host galaxy can play a
significant role. For the moment, all we can say is how the projected surface
density of shells looks within the model of radial oscillations – any stronger
statement would require more detailed simulations.
## ?partname? III Dynamical friction and gradual disruption
In the same spirit as in Part II, we will consider the formation of the shell
structure during a radial minor merger. This time, we will try to get closer
to real shell galaxies by introducing into the test-particle simulations the
gradual decay of the secondary galaxy as well as its braking by dynamical
friction against the primary.191919In this section, we use the terms secondary
or satellite, rather than cannibalized galaxy. The host galaxy will be usually
referred to as primary. In related papers, one may also find the notation
dwarf or small galaxy for the secondary and giant elliptical or big galaxy for
the primary.
### 16 Motivation
In Sect. 9.1 we have shown how are the positions of the shells related to the
potential of the host galaxy at different times from the merger that created
the shells. In practice, nevertheless, it has proven difficult to reproduce
the space distribution of the shells in the observed shell galaxies using
sensible potentials (Sect. 6). The main suspects of making the relation more
complex are the dynamical friction and the possibility to have shells from
multiple generations. In the case when the measurement of shell kinematics is
not available, we can pose a goal less ambitions than the derivation of the
potential of the host galaxy, that is to determine the age of the shell system
(the time of the merger). To this end, the measurement of the position of the
outermost shell could be sufficient, as this shell is the one which is the
least effected by those additional effects.
As we have mentioned in Sect. 1, this is the approach chosen by Canalizo et
al. (2007). They presented observations of shells in a quasar host galaxy and,
by simulating the position of the outermost shell by means of restricted
$N$-body simulations, attempted to put constraints on the age of the merger.
They concluded that it occurred a few hundred Myr to $\sim 2$ Gyr ago,
supporting a potential causal connection between the merger, the post-
starburst ages in nuclear stellar populations, and the quasar. A typical delay
of 1–2.5 Gyr between a merger and the onset of quasar activity is suggested by
both $N$-body simulations by Springel et al. (2005) and observations by Ryan
et al. (2008). It might therefore appear reassuring to find a similar time lag
between the merger event and the quasar ignition in a study of an individual
spectacular object.
The issue here is that noone has studied in detail the effects assumed to
complicate the shell distribution (the dynamical friction and the gradual
decay of the secondary galaxy) and thus it is not clear how exactly they
change the shell structure and how they influence the position of the
outermost shell. We try to include the dynamical friction and the gradual
decay of the cannibalized galaxy in test-particle simulations. The
manifestation of these processes in self-consistent simulations is difficult
to separate and sometimes they may even be confused with non-physical outcomes
of used methods. Test-particle simulations helped us to separate and better
understand the roles of the dynamical friction and gradual tidal decay in the
shell formation. Moreover, self-consistent simulations become demanding on
computation time when we want to explore a significant part of the parameter
space.
We look at what these enhanced test-particle simulations tell us about the
potential and merger history of shell galaxies with the focus on the
plausibility of the use of the outermost shell for dating the merger.
### 17 Description of simulation
In this and the previous part we show results of test-particle simulations and
in this section we describe the procedure of their calculation in detail.
#### 17.1 Configuration
The test (i.e. mass-less) particles of the secondary galaxy are generated
(usually in counts from $10^{4}$ to $10^{7}$) so that they follow the density
profile of the secondary galaxy. The particles then move according to a smooth
gravitational potential of both galaxies, which move with respect to each
other based on their masses, shape of potentials, positions and velocities;
Eq. (77). Figures and videos are generally oriented so that the secondary
galaxy approaches originally from the right hand side.
In the simplest case, when the centers of the galaxies pass through each
other, the potential of the secondary galaxy is suddenly switched off and the
particles continue to move only in the fixed potential of the primary galaxy.
This approach is applied in simulation in Part II and in some simulations in
Part III. In the simulations with dynamical friction and gradual disruption,
the smooth potential of the secondary galaxy is kept for whole time and its
mass is progressively lowered during each successive passage. The dynamical
friction is added in the form of an (semi-)analytical prescription into the
equations of motion of galaxies.
All the simulations in the thesis are, for the sake of simplicity, carried out
for spherical galaxies, i.e. elliptical galaxies with zero ellipticity. The
secondary (cannibalized) galaxy is always modeled as a single Plummer sphere.
The primary (host) galaxy is modeled as a single or double Plummer sphere in
Part III, while in Part II its potential has always two components, both
Plummer spheres.
For the numerical integration of the motion of the test particles and the
galaxies, the Leapfrog method was chosen. In this method, velocities derived
for a time half step earlier (or later) than the current position are used to
update the position. Conversely, to update the half-step velocity one step
forward, the positions for the round position in between are used. In so doing
the velocities can be seen to “leapfrog” over the current time step. This
simple enterprise improves the accuracy of the numerical computation by an
order compared to when the position $x$ and velocities $v$ are taken
simultaneously. The error is of the order of $(\bigtriangleup t)^{3}$, where
$\Delta t$ is the time step. For the longest time step used in our simulations
(1 Myr), the error for the trial circular motion was only 11 revolutions after
10,000 (compared to the simple analytical solution that is available in this
case), what is only 1 per mille.
#### 17.2 Plummer sphere
The gravitational potential of each of the galaxies in this part, Part III, is
modeled with the Plummer profile with varying parameters in different
simulations:
$\phi(r)=-\frac{\mathrm{G}\,M}{\sqrt{r^{2}+\varepsilon^{2}}},$ (73)
where G is the gravitational constant, $M$ is the total mass of the galaxy,
$r$ is the distance from the center of the galaxy and $\varepsilon$ is the
Plummer radius – a scale parameter that determines the compactness of the
galaxy. For $\varepsilon$ = 0 the Eq. (73) represents a simple potential of a
point mass. The Plummer radius corresponds to the effective radius202020The
effective radius is the radius at which one half of the total light of the
galaxy is emitted interior to this radius. of the galaxy.
While the Plummer model follows the profile of the real spherical galaxies
only approximately, we use it here – as was the case of numerous other studies
of galaxies – because of its simple expressions of dynamical quantities. It
was first used by Plummer (1911) to fit the observations of globular clusters
and now is often used as a stellar distribution model in simulations.
From the Poisson equation $\bigtriangleup\phi=4\pi\mathrm{G}\rho$, we can
easily infer the radial density distribution $\rho$ that acts as the source
for the Plummer potential:
$\rho(r)=\rho_{0}\frac{1}{(1+r^{2}/\varepsilon^{2})^{5/2}},$ (74)
where $\rho_{0}=3M/(4\pi\varepsilon^{3})$ is the central density. About
$\sqrt{2}/4$ (approx. 35%) of the total mass of the galaxy is enclosed inside
the $r=\varepsilon$ radius.
The force $F(r)$ acting on a test particle (of a mass m) is calculated from
the potential by the equation $F(r)=-\bigtriangledown\phi(r)$, what reads in
Plummer potential as:
$F(r)=-\mathrm{G}\,M\,m\frac{r}{(r^{2}+\varepsilon^{2})^{3/2}}.$ (75)
The particles in our model then move according to an acceleration
$\mathbf{a}(\mathbf{r})$ given by the potentials of both galaxies
$\mathbf{a}(\mathbf{r})=-\mathrm{G}\sum_{i}\frac{M_{i}\mathbf{r}_{i}}{(r_{i}^{2}+\varepsilon_{i}^{2})^{3/2}},$
(76)
where the summation goes over pres quantities corresponding to the secondary
galaxy, and one or two components of the primary galaxy. In simulations where
the potential of secondary galaxy is switched off, the particles continue to
move only in the fixed potential of the primary galaxy. $\mathbf{r}_{i}$ is
the vector of distance between the center of the primary or secondary galaxy
and the particle:
$\mathbf{r}_{i}=\mathbf{r_{\mathrm{particle}}-r_{\mathrm{galaxy}}}$, where
$\mathbf{r_{\mathrm{particle}}}$ is a position vector of the particle and
$\mathbf{r_{\mathrm{galaxy}}}$ is the position vector of the center of the
primary or the secondary galaxy.
The action of two Plummer spheres on each other is a little more intricate.
The non-zero radius reduces their attraction compared to two point masses.
This interaction cannot be appropriately described by simple means, but we
approximate the attraction by keeping the form of the Plummer potential and by
defining a common softening parameter in order to fulfill the law of the
action and reaction. The definition of the common softening parameter is
derived from both Plummer radii and then we use it in the equation of motion
as:
$F(r)=-\mathrm{G}\,M_{\mathrm{p}}M_{\mathrm{s}}\frac{r}{(r^{2}+\varepsilon_{\mathrm{p}}^{2}+\varepsilon_{\mathrm{s}}^{2})^{3/2}},$
(77)
where $r$ is the relative distance of centers of masses of galaxies. The
indexes $\mathrm{p}$ and $\mathrm{s}$ mark the quantities corresponding to the
primary and the secondary galaxy. The common softening parameter is then
$\varepsilon_{\mathrm{common}}=\sqrt{\varepsilon_{\mathrm{p}}^{2}+\varepsilon_{\mathrm{s}}^{2}}$.
In the case of a two-component primary galaxy, we use in Eq. (77) with
$M_{\mathrm{p}}=M_{*}+M_{\mathrm{DM}}$ and
$\varepsilon_{\mathrm{p}}^{2}=\varepsilon_{*}^{2}+\varepsilon_{\mathrm{DM}}^{2}$,
where $*$ stands for luminous component and $\mathrm{DM}$ for the dark halo.
#### 17.3 Velocity dispersion in Plummer potential
For computation of dynamic friction we will need to know the velocity
dispersion in the Plummer potential, so let’s derive it briefly now. Applying
the Jeans equations (see Binney and Tremaine, 1987, Ch. 4.2) to our
spherically symmetric galaxy without any systematical movement, we get
$\frac{\partial\left(\rho(r)\sigma^{2}(r)\right)}{\partial
r}=-\rho(r)\frac{\partial\phi(r)}{\partial r},$ (78)
where $\sigma$ stands for the velocity dispersion, which is assumed isotropic
at any given $r$. Applying the assumption $\sigma(\infty)=0$ we get the
solution:
$\sigma^{2}(r)=\frac{1}{\rho(r)}\intop_{r}^{\infty}\rho(r^{\prime})\frac{\mathrm{d}\phi(r^{\prime})}{\mathrm{d}r^{\prime}}\mathrm{d}r^{\prime}.$
(79)
The density $\rho$ and potential $\phi$ of the Plummer sphere are given by the
Eq. (74) and Eq. (73), respectively. The final formula for the velocity
dispersion of the galaxy with mass _$M$_ and Plummer radius $\varepsilon$ is
thus
$\sigma^{2}(r)=\frac{\mathrm{G}\,M}{6\,\sqrt{\varepsilon^{2}+r^{2}}}.$ (80)
?figurename? 43: The radial dependence of the velocity dispersion in a Plummer
sphere galaxy extending to infinity (red line) and a galaxy having the same
Plummer profile truncated in 10 times its scale radius (green line). The
distance is in multiples of the scale and the velocity dispersion in the units
of the dispersion in the center $\sigma_{0}$ ($\sigma_{0}$ differs negligibly
between the two cases).
For the galaxies in our model, we use the Eq. (80) in a slightly modified
from, because in the previous derivation we considered an isolated Plummer
sphere extending to the infinity. In reality, the size of a single galaxy is
limited (by tidal forces) and so we assume that at some distance
$R_{\mathrm{tc}}$ it ends and here, $\sigma(R_{\mathrm{tc}})=0$. With this
assumption we get:
$\sigma^{2}(r)=\frac{\mathrm{G}\,M}{6\,\varepsilon}(1+r^{2}/\varepsilon^{2})^{5/2}\left[\frac{1}{(1+r^{2}/\varepsilon^{2})^{3}}-\frac{1}{(1+R_{\mathrm{tc}}^{2}/\varepsilon^{2})^{3}}\right].$
(81)
The radial dependence of the velocity dispersion for the truncated and the
infinite galaxy are compared in Fig. 43.
#### 17.4 Velocity dispersion in a double Plummer sphere
For a galaxy modeled as two Plummer spheres – one for the luminous component
and another one for the dark halo – the situation with the velocity dispersion
is more complex. The presence of one component influences the dispersion in
the other one and vice versa. Eq. (79) changes to
$\sigma_{1}^{2}(r)=\frac{1}{\rho_{1}(r)}\intop_{r}^{\infty}\rho_{1}(r^{\prime})\frac{\mathrm{d}\left[\phi_{1}(r^{\prime})+\phi_{2}(r^{\prime})\right]}{\mathrm{d}r^{\prime}}\mathrm{d}r^{\prime}.$
(82)
Using Eq. (74) and Eq. (73) and after a partial integration, we obtain
$\sigma_{1}^{2}(r)=\frac{\mathrm{G}\,M_{1}}{6\,\sqrt{\varepsilon_{1}^{2}+r^{2}}}+\frac{\mathrm{G}\,M_{2}}{\varepsilon_{2}^{3}}\left(1+r^{2}/\varepsilon_{1}^{2}\right)^{5/2}I(r,\varepsilon_{1},\varepsilon_{2}),$
(83)
where the first term is identical to the dispersion of the first component
without in the absence of the second one. The integral
$I(r,\varepsilon_{1},\varepsilon_{2})$ is solved as follows
$I(r,\varepsilon_{1},\varepsilon_{2})=\intop_{r}^{\infty}\frac{r^{\prime}}{\left(1+r^{\prime
2}/\varepsilon_{1}^{2}\right)^{5/2}\left(1+r^{\prime
2}/\varepsilon_{2}^{2}\right)^{3/2}}\mathrm{d}r^{\prime}=$ (84)
$=\frac{1}{3\left(\varepsilon_{2}^{2}-\varepsilon_{1}^{2}\right)}\left[\frac{1}{\left(r^{2}+\varepsilon_{1}^{2}\right)^{3/2}\left(r^{2}+\varepsilon_{2}^{2}\right)^{1/2}}+\frac{4}{\left(\varepsilon_{2}^{2}-\varepsilon_{1}^{2}\right)^{2}}\left(2-\sqrt{\frac{r^{2}+\varepsilon_{2}^{2}}{r^{2}+\varepsilon_{1}^{2}}}-\sqrt{\frac{r^{2}+\varepsilon_{1}^{2}}{r^{2}+\varepsilon_{2}^{2}}}\right)\right].$
Fig. 44 illustrates the effect of the presence of the other component on the
velocity dispersion of a Plummer sphere.
?figurename? 44: An illustration of the effect of a second component on the
dispersion of a Plummer sphere ($M_{*}=3.2\times 10^{11}$ M⊙ ,
$\varepsilon_{*}=7$ kpc). Red: the dispersion of the isolated sphere, green:
additional dispersion caused by the presence of a second component of large
mass ($M_{\mathrm{DM}}=6.4\times 10^{12}$ M⊙ ) and large Plummer radius
($\varepsilon_{\mathrm{DM}}=60$ kpc), blue: the sum of the two. The dispersion
is normalized so that the dispersion in the center of the first component in
the absence of the second one (181 km$/$s) equals 1.
#### 17.5 Standard set of parameters
For the future reference, let us define the standard set of parameters for
simulations (used in this Part) as the following set of values:
The mass of the primary galaxy: $M_{\mathrm{p}}=3.2\times 10^{11}$ M⊙
The secondary to primary mass ratio: 0.02
Plummer radius of the primary galaxy: $\varepsilon_{\mathrm{p}}=20$ kpc
The cut-off diameter for the primary galaxy: $R_{\mathrm{tc}}=200$ kpc
Plummer radius of the secondary galaxy: $\varepsilon_{\mathrm{s}}=2$ kpc
The initial radial distance of the secondary galaxy: 180 kpc
The initial velocity of the secondary galaxy: $125$ km$/$s, the escape
velocity for the initial distance
These values are used as the usual setup of the presented simulations and we
will refer to them often, so we do not have to repeat them.
Let us only remark that the escape velocity, $v_{\mathrm{esc}}$, is computed
only approximately, on the same grounds as the force between the galaxy (see
Eq. 77), i.e. we put
$v_{\mathrm{esc}}=\sqrt{\frac{2\,G\,(M_{\mathrm{p}}+M_{\mathrm{s}})}{\sqrt{r^{2}+\varepsilon_{\mathrm{common}}^{2}}}.}$
(85)
The results of our simulation show that, in the relevant range of radii, its
difference from reality is negligible.
### 18 Dynamical friction
Dynamical friction is a braking force of gravitational origin acting on a body
that moves through the field of stars or any other matter. We will be
interested in the dynamical friction incurred on the secondary galaxy by the
stars and dark matter of the primary galaxy. We encourage the reader to
consult Appendix D – Introduction to dynamical friction – which is a modified
chapter from Ebrová (2007). It explains in detail the nature of this
phenomenon and it is likely to be of interest even to a reader already
familiar with the topic.
Appendix D also contains a derivation of the Chandrasekhar formula (Sect.
D.2). Chandrasekhar formula is an analytical expression derived by
Chandrasekhar (1943) that is still often used to calculate the dynamic
friction. The formula is a good approximation for the dynamical friction and
is easy to use in test-particle simulations.
There are several different simplifications done during its derivation (see
Sect. E.1). One is the assumption of homogeneity of the stellar field around
the braked body (both density and velocity dispersion are taken as constants).
This leads to a relatively simple expression that contains the so-called
Coulomb logarithm. The exact value of this logarithm is unknown and is usually
roughly estimated and taken as a constant.
In Ebrová (2007), we have devised an alternative way to calculate the
dynamical friction in radial mergers (that are the most likely to produce
shell structures). We call it our modification of the Chandrasekhar formula
and a detailed description and derivation can be found in Appendix E. Here we
summarize only the main ideas.
The homogeneity of density and velocity dispersion is not assumed during the
derivation of the Chandrasekhar formula. Instead, a more realistic stellar
distribution function is used, varying both the density and velocity
dispersion based on the chosen model of the host galaxy. Using the radial
symmetry, the originally 5-dimensional problem is reduced to a 2-dimensional
one, Eq. (104), which is analytically insolvable and so numerical integration
is used to calculate the final result for the dynamical friction. In this
approach, no estimated values are needed as an input, only the distribution
function chosen for the galaxy determines the friction.
In Sect. E.2, we compare the result of Eq. (104) to the Chandrasekhar formula.
It is shown that using a constant as the Coulomb logarithm is completely
inadequate for the problem at hand.
### 19 Multiple Three-Body Algorithm (MTBA)
We now investigate another alternative method to calculate the dynamical
friction in radial minor merger. The method is described in the paper Seguin
and Dupraz (1994) and it is also suitable for test-particle simulations.
#### 19.1 Principle and characteristics
Seguin and Dupraz (1994) used restricted tree-body simulations to examine
dynamical friction in head-on encounter. They adopted the Multiple Three-Body
Algorithm which was originally proposed by Borne (1984). The basis of the
method is to calculate the motion of the satellite galaxy from the
gravitational influence of the particles in the primary galaxy. However, it is
not a self-consistent simulation, as the particles are otherwise treated as
test particles – their motion is calculated as the motion of massless
particles in the sum of the gravitational potentials of both galaxies, in the
same manner as in our simulations of the creation of the shell structure
(Sect. 13 and Sect. 22). In the case of the MTBA, the particles are generated
so that they follow the distribution function of the primary galaxy. Only when
the motion of the secondary galaxy is calculated, these particles are used as
if each of them had a mass of $m=M_{\mathrm{p}}/N$, where $M_{\mathrm{p}}$ is
the total mass of the primary galaxy and $N$ is the total number of particles
used. The force/acceleration acting upon the secondary galaxy in each step is
fully determined by the action of all particles in the primary galaxy upon a
chosen smooth potential of the secondary galaxy. Having also the potential of
the satellite act on these particles naturally perturbs their trajectories and
from their force exerted back on the satellite galaxy the dynamical friction
naturally arises.
To summarize, this method to calculate the dynamical friction requires a model
for the potentials of the primary and the secondary galaxy and the use of
particles in the primary galaxy. The particles are treated in two different
ways: as massless when their motion is calculated and as massive when the
motion of the secondary galaxy is calculated.
Seguin and Dupraz (1994) have directly compared the results of a MTBA
simulation with the coupled solution of the linearized Poison and
collisionless Boltzmann equations for the first passage of the satellite. They
found MTBA to be equivalent to the analytical method. Compared to their
analytical method, the MTBA has the advantage of easier and faster
calculation. Moreover the MTBA is more flexible so it can follow the whole
process until a complete merger. Both these methods show that the dynamical
friction in radial merger is not strictly proportional to the local density –
contrary to what is assumed in the Chandrasekhar formula. Moreover, it is a
time-dependent process which depends on the full past history of the merger,
contrary to a satellite on a circular orbit in the co-rotating frame. This
observation cannot be reproduced in any modification of the Chandrasekhar
formula (including ours) which is fundamentally local.
In Seguin and Dupraz (1996) the MTBA has been compared with a self-consistent
Particle-Mesh simulation. The MTBA gives an accurate estimate of the decay
rate of orbital energy of the satellite, within 10% of the $N$-body simulation
during the first orbit. But it fails to reproduce the ultimate phase of the
merger.
#### 19.2 Merger parameters
To compare different methods for the calculation of the dynamical friction, we
have modeled the secondary as a point mass (eventually with a very small
softening – 0.01 kpc) and have chosen the following parameters of the
collision:
The mass of the primary galaxy: $M_{\mathrm{p}}=10^{12}$ M⊙
The secondary to primary mass ratio: 0.01
Plummer radius of the primary galaxy: $\varepsilon_{\mathrm{p}}=10$ kpc
The cut-off diameter for the primary galaxy: $R=200$ kpc
The initial radial distance of the secondary galaxy: 100 kpc
The initial velocity of the secondary galaxy: 0 km$/$s
#### 19.3 Results of simulations
It turns out that for a successful application of the MTBA it is necessary to
use a high enough number of particles in the primary galaxy and a small enough
time step of integration. The simulation for the chosen set of parameters
(Sect. 19.2) stabilizes for about 100,000 particles with time step of 0.01
Myr, but even then there are noticeable differences mainly in the later part
of the merger as we further increase the number of particles and decrease the
time step, see Fig. 45 and Fig. 46. On the other hand, the introduction of the
slight softening in the interaction of the secondary does not influence the
results provided that enough particles and a small enough time step are used.
?figurename? 45: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary. The motion was calculated using the MTBA
with 100,000 particles for time steps of 0.001–1 Myr. Parameters of the
collision are given in Sect. 19.2.
?figurename? 46: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary.The motion was calculated using the MTBA
with time step 0.01 Myr for 1,000-1,000,000 particles. Parameters of the
collision are given in Sect. 19.2.
### 20 Comparison with self-consistent simulations
To compare the calculation of the dynamical friction using the methods
mentioned earlier (Appendix E and Sect. 19) with the self-consistent
simulations, we use the simulations performed by Kateřina Bartošková using
GADGET-2. GADGET-2 is free software, distributed under the GNU General Public
License. The code can be used for studies of isolated systems, or for
simulations that include the cosmological expansion of space. It computes
gravitational forces with a hierarchical tree algorithm (optionally in
combination with a particle-mesh scheme for long-range gravitational forces)
and represents fluids by means of smoothed particle hydrodynamics (SPH). Both
the force computation and the time stepping are fully adaptive. The code is
written in highly portable C and uses a spatial domain decomposition to map
different parts of the computational domain to individual processors. GADGET-2
was publicly released in 2005 (Springel, 2005) and presently is the most
widely employed code for the cosmic structure formation.
#### 20.1 Altering GADGET-2 computational setting
The parameters of the collision have been set the same as in the previous
case, Sect. 19.2, but with no cut-off diameter. $10^{5}$ particles have been
used to represent the primary galaxy. The results differ for different
settings of computational parameters in GADGET-2. Here we present results of
five simulations that differ in settings for three chosen parameters and in
the accuracy of variables during the calculation.
During the calculation of the gravitation force, spline softening is used.
$SoftPar$ is the magnitude of the softening used for mutual interactions of
the particles of the primary galaxy. $SoftSec$ is the softening for the
secondary and in an interaction between the secondary and a particle of the
primary galaxy, the larger value from $SoftPar$ and $SoftSec$ is used. $ETIA$
(ErrorTolIntAccuracy) influences the accuracy of the integration method. It is
used in the estimation of the adaptive integration step $\Delta t$
$\Delta t=\sqrt{\frac{2\,ETIA\,SoftPar}{a}},$ (86)
where $a$ is the amount of acceleration the particle has been subjected to in
the previous step. Thus the smaller $ETIA$ we choose, the shorter will be the
time step. $Precision$ refers to the type of the floating-point precision used
during numerical calculations.
The values we have used in the five different simulations and the labels of
the simulations are shown in Table 6. The orbital decay of the satellite for
all the runs is shown in Fig. 47. Run D has been calculated with the highest
precision and we thus use it as a reference in the following section.
run | $ETIA$ | $SoftPar$ | $SoftSec$ | $Precision$
---|---|---|---|---
| | kpc | kpc |
A | 0.002 | 0.21 | 0.05 | Single
B | 0.008 | 0.05 | 0.05 | Single
C | 0.04 | 0.01 | 0.01 | Single
D | 0.04 | 0.01 | 0.01 | Double
E | 0.002 | 0.05 | 0.05 | Single
?tablename? 6: The settings for the GADGET-2 simulations. The meaning of the
parameters is explained in Sect. 20.1.
?figurename? 47: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary.The motion has been calculated using
GADGET-2. The parameters of the collision are given in Sect. 19.2, the
settings for each simulation in Table 6.
#### 20.2 Comparison of methods
Fig. 48 shows the orbital decay of the secondary in the merger with parameters
given in Sect. 19.2 for three different methods of calculation of dynamical
friction. Our modification of Chandrasekhar formula adds to the equations of
motion of the secondary the dynamical friction calculated using a numerically
integrated analytical formula as described in Appendix E. The MTBA method
(Sect. 19) is represented by a simulation with 100,000 particles and time step
of 0.01 Myr. From the self-consistent simulation with GADGET-2 we show run D
(see Sect. 20.1).
?figurename? 48: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary in three different methods: our
modification of Chandrasekhar formula (Sect. E.1, red curve); inconsistent
simulation with GADGET-2 (Sect. 20.1, green curve); and MTBA (Sect. 19.1, blue
curve). Parameters of the collision are given in Sect. 19.2.
Our modification of Chandrasekhar formula gives by far the fastest loss of the
orbital energy of the satellite, but even the MTBA gives a significantly
larger value of the dynamical friction than the self-consistent simulation.
In Sect. 22 we will however use our modification of Chandrasekhar formula for
the calculation of the dynamical friction, as we have conducted a sizable
number of simulation using this method before we became familiar with the
MTBA. The MTBA is also more computationally demanding. It requires a small
enough time step and the inclusion of test particles in the primary galaxy,
which are otherwise of no interest for us. Our modification of Chandrasekhar
formula, on the contrary, gives the same results for the motion of the
secondary galaxy for the time step of 1 Myr as for any shorter step.
Doing self-consistent simulations is not an option because of the number of
different simulations required for this study (most of which we do not show
explicitly in this thesis). Because it seems that our modification of
Chandrasekhar formula significantly overestimates the real value of the
friction, the results have to be considered an upper bound for the influence
of the dynamical friction on the shell structure. At the end, it turns out
that the differences in the shell structure related to the choice of a method
to calculate the dynamical friction is smaller than the uncertainty in the
models of the tidal decay of the secondary galaxy (Sect. 21).
### 21 Tidal disruption
Together with the dynamical friction, the tidal disruption is another effect
that is important for the galactic merger. The tidal disruption gradually
lowers the mass of the cannibalized galaxy and thus mitigates the effect of
the dynamical friction. During shell formation, it is of particular
importance, because the gradual release of stars from the secondary galaxy has
an important effect on the growing shell structure. The introduction of the
tidal disruption into test-particle simulation is nevertheless a difficult
task.
#### 21.1 Massloss of the secondary
In the context of the tidal disruption of an object in the gravitation field
of another body, the notion of the _tidal radius_ is frequently introduced.
This is an approximative approach to the tidal forces, assuming that under the
tidal radius the matter is still bound to the disrupted body, but it is not
the case anymore outside the tidal radius. The reader may find more details on
the concept in Appendix F. Here we will only show how we used it in our test-
particle simulations.
?figurename? 49: The purely analytical approach to the decay of the secondary
galaxy during the first passage for the standard set of parameters (Sect.
17.5). Left: the evolution of the mass of the secondary galaxy. Rights:
Distance of the secondary from the center of the primary galaxy (blue curve)
and tidal radius of the secondary (red curve).
First we have implemented a purely analytical approach, where we calculate the
current tidal radius in every step using Eq. (107) and update the mass of the
secondary galaxy accordingly to the mass of a Plummer sphere with the original
parameters of the secondary galaxy but restricted to the tidal radius. But
this leads to us only lowering the satellite mass during the first passage
through the center of the primary galaxy, see Fig. 49. Particles are released
in limited amount also during further passages, but this mechanism obviously
does not reflect the real situation for multiple passages.
?figurename? 50: Gradual decay of the secondary galaxy calculated using test
particles. Top: distance between the centers of the primary and the secondary
galaxy. Bottom: the number of particles bound to the secondary galaxy. Blue
curves show the development for the simulation where we consider as bound
particles those inside the sphere of the tidal radius, the red curves
correspond to keeping particles with lower than escape velocity. Both
simulations are carried out for the standard set of parameters (Sect. 17.5),
the dynamical friction is calculated using our modification of the
Chandrasekhar formula (Appendix E).
To describe the decay of the satellite during further passages, we have
included in its calculation the test particles of the secondary galaxy. We
count particles that we still consider bound with the satellite galaxy. The
ratio between their number and the number of particles that we have put in the
secondary galaxy at the beginning of the simulation determines its current
mass. As a criterion for bound particles we consider that 1) the distance of
the particle from the center of the secondary galaxy is lower than the current
tidal radius; 2) the velocity of the particle with respect to the secondary
galaxy does not exceed the escape velocity for its given distance from the
center of the secondary galaxy. Fig. 50 shows how these two approaches differ
for otherwise identical initial conditions.
The use of the tidal radius causes large fluctuations of the number of bound
particles near the passage of the secondary galaxy through the center of the
primary galaxy, when many particles suddenly find themselves outside the tidal
radius. When later the secondary galaxy retreats from the center of the
primary, the tidal radius quickly increases and more particles are included.
Some of them eventually escape before the secondary reaches its apocenter, but
still more particles stay bound to the secondary than there were during its
passage through the center of the primary. In the other simulation the loss of
particles is more monotonous, the orbital decay slightly faster, and more
particles are caught in the center of the host galaxy.
?figurename? 51: Development of the distance between the primary and the
secondary galaxy (top) and the Plummer radius of the secondary galaxy
(bottom). The simulations carried out for the standard set of parameters
(Sect. 17.5), the dynamical friction is calculated using our modification of
the Chandrasekhar formula (Appendix E). The radial density of the secondary
galaxy at the beginning of the simulation and in 5 Gyr is shown Fig. 52.
The use of the two different methods to model the tidal disruption of the
secondary does not have a dramatic impact on the merer. Nevertheless, the
times of the passages of the secondary through the center of the host galaxy
and the volume of particles released in each passage differ between the two
models, mainly in the later phases of the merger. This may have a noticeable
impact on the appearance of the shell system in different time, that is the
positions of the shells, their number, brightness, opening angle and so forth.
The problem is that we have no hint as to which of the methods is a better
approximation for the true decay of the secondary galaxy. If we were to
compare the results with self-consistent simulations, we would likely get
different results depending mainly on the configuration of the merger. Thus we
compare the test-particle simulations done with different methods for the
tidal disruption of the secondary galaxy and focus on features of the shell
system that are independent of the method used (Sect. 22.1).
#### 21.2 Deformation of the secondary galaxy
Another thing going on during the merger that is difficult to reproduce in
test-particle simulations is the deformation if the cannibalized galaxy. We
model components of galaxies with spherically symmetric Plummer spheres. Thus
we have tried at least to change the profile of the sphere of the secondary
galaxy during the simulation.
The mean value of the radial distance of a particle $\left\langle
r\right\rangle$ in a Plummer sphere is given as
$\left\langle r\right\rangle=\frac{\intop_{0}^{R_{\mathrm{tc}}}r^{\prime
3}\rho(r^{\prime})dr^{\prime}}{\intop_{0}^{R_{\mathrm{tc}}}r^{\prime\prime
2}\rho(r^{\prime\prime})dr^{\prime\prime}},$ (87)
where $\rho(r^{\prime})$ is the density of the Plummer sphere Eq. (74) and we
express the cut-off in multiplies of the Plummer radius
$R_{\mathrm{tc}}=p\varepsilon$. The mean value of the radial distance is then
$\left\langle
r\right\rangle=\varepsilon\frac{2\left(1+p^{2}\right)^{3/2}-2-3p^{2}}{p^{3}}.$
(88)
Thus if we calculate the mean value radial distance from the center of the
secondary galaxy for the particles that we consider bound to it in the
simulation
$\left\langle r\right\rangle=\sum_{i=1}^{N}r_{i}/N,$ (89)
we can easily convert it to a new Plummer radius for the secondary galaxy
$\varepsilon_{\mathrm{s}}$. Fig. 51 shows the development of the Plummer
radius of the secondary galaxy in a simulation with the standard set of
parameters (Sect. 17.5). The Plummer radius is calculated using Eq. (88),
where $\left\langle r\right\rangle$ is the mean radial distance of particles
under the current tidal radius. The radial density of the secondary galaxy at
the beginning of the simulation and in 5 Gyr is shown in Fig. 52. It is
important to keep in mind that the density is calculated only from radial
distances from the center of the satellite even though the spherical symmetry
was surely broken during the simulation.
?figurename? 52: The radial density of the secondary galaxy at the beginning
of the simulation and in 5 Gyr for the standard set of parameters (Sect.
17.5). In blue is the density calculated from the test particles of the
secondary galaxy, in green the model of the secondary chosen at the start of
the simulation and in red the density of the Plummer sphere that corresponds
to the changing Plummer radius which is calculated from the distribution of
the test particles. The density is normalized so that the central density of
the initially chosen Plummer sphere of the secondary galaxy is one.
?figurename? 53: Snapshots of simulations. For description of all runs see
text in Sect. 22.1. Time 0 is defined as the moment then the secondary galaxy
reaches the center of the primary galaxy for the first time, which is (for all
three runs) almost exactly 1 Gyr after it has been released from the distance
of 180 kpc with escape velocity. Only the surface density of particles
originally belonging to the satellite galaxy is displayed corresponding to the
subtraction of the profile of the primary galaxy. Each box, centered on the
host galaxy, shows 300$\times$300 kpc. Radial histogram of particles in 5 Gyr
is shown in Fig. 54.
### 22 Simulations of shell structure
Now we finally show the combined effect that the inclusion of both the
dynamical friction and gradual decay of the secondary galaxy in the
simulations has on the shell formation. The simulations are carried out using
the method described in Sect. 17, i.e. millions of test particles were
generated so that they follow the distribution function of the secondary
galaxy at the beginning of the simulation. The particles then move according
to the sum of the gravitational potentials of both galaxies that are both
represented by a smooth potential. The galaxies move with respect to each
other as dictated by their masses, shape of potentials, positions and
velocities.
The dynamical friction, when included, is calculated using our modification of
the Chandrasekhar formula, see Appendix E, and the gradual decay of the
secondary galaxy, when included, is calculated using some of the methods from
Sect. 21.1. In Sect. 22.2, we have added the dark halo to the primary galaxy
and Sect. 22.3 shows the shell formation in a self-consistent simulations. All
the outputs are oriented so that the secondary originally approached the
primary galaxy from the right hand side.
#### 22.1 Dynamical friction and tidal disruption
We have compared three simulations, all of them for the standard set of
parameters (Sect. 17.5).
* •
Run 1 – without dynamical friction and with instant disruption of the
secondary.
* •
Run 2 – dynamical friction is calculated using our modification of the
Chandrasekhar formula and the tidal disruption using the analytical approach
based on the tidal radius as described at the beginning Sect. 21.1.
* •
Run 3 – dynamical friction is again calculated using our modification of the
Chandrasekhar formula, the tidal disruption is based on the counting of
particles inside/outside the current tidal radius. Additionally, the Plummer
radius of the secondary galaxy is constantly recalculated as described in
Sect. 21.2.
Snapshot from all the runs for two different times are shown in Fig. 53,
radial histograms of particles in Fig. 54. Video from run 1 and run 2 is part
of the electronic attachment. For the description of the video, see Appendix H
point 4.
We compare a simple simulation (Run 1) with a pair of simulations (Runs 2 &
3), where the tidal decay of the secondary galaxy is modeled using two
different methods. However, we can see a qualitative shift in the same
direction between both Runs 2 & 3 and the simple simulation. The result of
both Runs 2 & 3 is a multi-generation shell system, whereas Run 1 can in
principle give rise only to one generation of shells.
For both Runs 2 & 3 there were more particles trapped in the gravitational
field of the host galaxy and a large part of them has been transported to the
vicinity of the center of the host galaxy. The outer shells (of the first
generation) are more diffuse and significantly less luminous when compared to
Run 1, whereas their positions remain essentially the same. On the other hand,
in the later generations, there are brighter shells, some of which can overlap
with the first-generation shells. The shells of the later generations appear
on smaller radii and are often bright, whereas in Run 1 shells at small radii
are completely missing. The evolution of the shell brightness in Run 1 is
somehow calmer, whereas the shells of the later generations in Runs 2 & 3 have
a tendency to reach very high brightness in certain small range of radii.
?figurename? 54: Radial histogram of stars of the secondary galaxy, centered
on the primary 5 Gyr after the first passage of the secondary galaxy through
the center of the primary galaxy for the three different simulations – run 1
(red), run 2 (green) and run 3 (blue). For description of all runs, see text
in Sect. 22.1.
Runs 2 & 3 are more consistent with observations (Sect. 3) in the sense that
their contain shells on both small and large radii. An important thing to
notice is that within our model, any subsequent passage of the secondary
galaxy through the center of the primary galaxy does not lead to a complete
destruction of the shells from the previous passages. Towards the center of
the host galaxy, we find shells with larger surface brightness, also a feature
found in real shell galaxies. At the same time, in Runs 2 & 3 we can find
faint shells surrounded by brighter ones from both sides, another effect
observed in real galaxies and impossible to reproduce in a simple simulation.
The main difference between Run 2 and Run 3 lies in the positions of the
shells from the later generations – those shells that dominate the system in
later times thanks to their brightness. The timing of the second passage of
the secondary galaxy through the center of the host galaxy is very similar for
Run 2 and Run 3 but the difference in energy, mass and decay of the secondary
galaxy is sufficient to produce shells at different radii. Run 3 also differs
significantly from Run 2 (and also Run 1) in that a bright shells system
persist even a long time after the first approach of the secondary galaxy (7
Gyr). However, we cannot say whether it is Run 2 or Run 3 that better
describes the real merger of two galaxies under given initial conditions. This
indicates that quantitative modeling of a shell system using test-particle
simulation is very difficult or even impossible.
In spite of the difficulties, we dare to state qualitative conclusions
independently on the method chosen for the tidal decay of the secondary
galaxy: the introduction of the dynamical friction and the gradual decay to
our simulations dramatically changes the appearance of shell structures. Only
the outermost shell of the first generation is not overlayed by later,
brighter generations of shells added during next passages of the satellite
through the center of the primary. While the position of the outermost shell
is not much affected by the dynamical friction, its brightness is rapidly
lowered due to the many particles staying trapped in the weakened but
remaining potential of the small galaxy.
#### 22.2 Dark halo
To be even more realistic, we present a two-component model of the galaxy – a
luminous component with a dark halo. The velocity dispersion of each component
is under the influence of the other (Sect. 17.4). The velocity dispersion is
an important parameter of the dynamical friction a thus values of the friction
induced by each component slightly differ (the amount depends on parameters)
from the values we get when the component is isolated (Sect. E.1).
We performed three simulations with parameters listed in Table 7. In all the
cases, the mass of the secondary galaxy is 0.02 of the total mass of the
primary; and the secondary approaches with escape velocity. Dynamical friction
is calculated using our modification of the Chandrasekhar formula (Appendix
E). The mass of the secondary galaxy was gradually lowered during the
simulation according to the number of test particles under the current tidal
radius (Sect. 21.1) and its Plummer radius was being adjusted according to the
method described in Sect. 21.2.
run | $\varepsilon_{*}$ | $M_{*}$ | $\varepsilon_{\mathrm{DM}}$ | $M_{\mathrm{DM}}$ | $\varepsilon_{\mathrm{s}}$ | $M_{\mathrm{s}}$ | $D_{\mathrm{ini}}$ | $v_{\mathrm{ini}}$
---|---|---|---|---|---|---|---|---
| kpc | M⊙ | kpc | M⊙ | kpc | M⊙ | kpc | km$/$s
M0B0 | 7 | $3.2\times 10^{11}$ | - | - | 2 | $6.4\times 10^{9}$ | 180 | 125
M2B6 | 7 | $3.2\times 10^{11}$ | 60 | $6.4\times 10^{12}$ | 2 | $1.344\times 10^{11}$ | 300 | 443
M6B10 | 7 | $3.2\times 10^{11}$ | 100 | $1.92\times 10^{13}$ | 2 | $3.904\times 10^{11}$ | 300 | 756
?tablename? 7: Parameters of simulations. The potentials of the galaxies are
modeled as a single Plummer sphere for the secondary galaxy in all runs and
the primary galaxy in the run M0B0; and as a double Plummer sphere for the
primary in runs M2B6 and M6B10. Indices *, DM and S refer to the luminous and
dark components of the primary galaxy and the secondary galaxy, respectively.
$\varepsilon$ is Plummer radius, $M$ total mass of the Plummer sphere,
$D_{\mathrm{ini}}$ initial distance between centers of the secondary and
primary galaxies and $v_{\mathrm{ini}}$ their mutual velocity.
?figurename? 55: Evolution of the merger for three different configurations of
the dark halo of the primary galaxy – distance between galaxies, number of
particles bound to the secondary galaxy and its Plummer radius. For the
parameters of the mergers, see Table 7. ?figurename? 56: Snapshots from three
simulations, for the parameters of the mergers, see Table 7. Time stamps refer
to the time elapsed since the first passage of the secondary galaxy through
the center of the primary galaxy. Each panel covers 300$\times$300 kpc and is
centered on the host galaxy. Only the surface density of particles originally
belonging to the satellite galaxy is displayed. The density scale varies
between frames, so that the respective range of densities is optimally
covered.
Fig. 55 illustrates the evolution of the distance between the galaxies and the
gradual decay of the secondary galaxy. Time stamps of each run have been
shifted so that in each case the secondary galaxy reaches the center of the
primary galaxy at time 0\. In the first case (run M0B0 without any halo), the
secondary galaxy lost all particles during the first passage and this
simulation is rather equivalent to simulations with instant disruption. In the
configurations that include the halo (runs M2B6 and M6B10), the velocity is
such on the other hand that the primary galaxy catches only very few particles
in the first passage and a significant growth of the shell structure is
observed only in later phases of the merger.
Snapshots for three different times are shown in Fig. 56 and radial histograms
for time 3 Gyr in Fig. 57. For the simulation with a heavy halo (run M6B10)
the particles cover the largest span of energies (apocenters) and in both
simulations with a halo (runs M2B6 and M6B10), new shells on lower radii are
created in further passages of the secondary through the center of the primary
galaxy and many particles end up being caught in the center of the primary. In
the simulation without a halo (run M0B0) the secondary decays in the first
passage, but the particles have mostly sizable energies at that time and thus
have apocenters at larger galactocentric radii or outright escape the system.
The positions of the shells in a given time are obviously different for
different potentials of the primary galaxy.
?figurename? 57: Radial histogram of stars of the secondary galaxy, centered
on the primary 3 Gyr after the first passage of the secondary galaxy through
the center of the primary for three different simulations. The meaning of
colors is the same as in Fig. 55 (red: M0B0 – without halo, green: M2B6 – halo
20 times more massive than the luminous component, blue: M6B10 – 60 times more
massive). For the parameters of the mergers, see Table 7.
The main effect of the halo on the shell system is probably in that its
presence (through the increased mass of the primary galaxy) allows for a
faster development of shells at larger radii, despite the secondary releasing
in our case only a small part of its stars during its first passage through
the center of the host galaxy. Meanwhile, there are additional shells created
in the following passages, creating the high radial range of shells observed
in some galaxies which has continuously proven difficult to reproduce in
simulations.
The increased total mass of the host galaxy is apparently more important than
the difference in the dynamical friction caused by the differences in local
density and velocity dispersion for different halo configurations. The more
massive halo accelerates the secondary galaxy more, reducing the loss of its
energy via the dynamical friction and increasing the time before a subsequent
return of the secondary galaxy. But the higher energy/velocity of the
secondary galaxy allows the existence of shells at larger radii - while it is
important to note that in our simulations, we see shells at 200 to 300 kpc
from the center of the host galaxy, which is a distance where noone ever
observed (or even looked for) shells in real galaxies.
?figurename? 58: Comparison of histograms of radial distances of shells‘
particles in the self-consistent (green) and test-particle (red) simulations
at two different time steps.
#### 22.3 Self-consistent versus test-particle simulations
In this section, we compare two simulations with the same initial conditions,
one conducted in a self-consistent manner using GADGET-2 by Kateřina
Bartošková, the other one with test particles. Originally we intended to keep
the parameters of the primary galaxy, but a two-component (luminous+dark
matter) Plummer sphere is not a consistent system for an arbitrary choice of
parameters, particularly for those we have used so far. The system is
consistent when each physically distinct component has a positive distribution
function (Ciotti, 1996). Thus we have chosen the following parameters for the
merger:
The potential of the primary galaxy is a double Plummer sphere with respective
masses $M_{*}=2\times 10^{11}$ M⊙ and $M_{\mathrm{DM}}=8\times 10^{12}$ M⊙ ,
and Plummer radii $\varepsilon_{*}=8$ kpc and $\varepsilon_{\mathrm{DM}}=20$
kpc for the luminous component and the dark halo, respectively. The potential
of the secondary galaxy is chosen to be a single Plummer sphere with the total
mass $M=2\times 10^{10}$ M⊙ and Plummer radius $\varepsilon_{*}=2$ kpc. The
cannibalized galaxy is released from the distance of 200 kpc from the center
of the host galaxy with the initial velocity 102 km$/$s in the radial
direction (as always).
Snapshots from several times for both of the simulations are shown in Fig. 59,
radial histograms for the chosen times in Fig. 58. Video from the self-
consistent simulation is part of the electronic attachment. For the
description, see Appendix H point 5.
?figurename? 59: Snapshots from a test-particle simulation (left) and from the
corresponding self-consistent simulation (right). Time equal zero corresponds
to the passage of the secondary galaxy through the center of the primary
galaxy. Each panel covers 400$\times$400 kpc and is centered on the host
galaxy. Only the surface density of particles originally belonging to the
satellite galaxy is displayed. The density scale varies between frames, so
that the respective range of densities is optimally covered.
Unfortunately it turns out that for this choice of parameters, our method of
including the gradual decay of the secondary galaxy (Sect. 21.1) does not lead
to a very gradual decay at all. In the case of simulation with test particles,
the secondary galaxy loses all its particles near its first passage through
the center of the primary galaxy. Thus we use the model with instant
disruption of the secondary instead. To make the comparison even worse, the
self-consistent simulations behaves in yet another way: the core of the
secondary galaxy survives the first two passages through the center of the
primary galaxy and for some reason dissolves close to its apocenter.
However, despite these significant differences, the results are surprisingly
similar. Most importantly, the radii of the outermost shells differ by less
than 10%. In comparison with our enhanced test-particle simulations (e.g.,
Sect. 22.1 – Runs 2 & 3), the self-consistent simulation does not show a
significant transport of particles of the secondary galaxy to the area around
the center of the host galaxy, neither does it produce shells at low radii.
Where on the other hand the self-consistent simulation resembles more the
enhanced test-particle simulations than the simple test-particle simulation
(with instantaneous disruption of the secondary galaxy and no dynamical
friction) is the dramatic decrease of the brightness of the outermost shell on
large radii (compare with Sect. 22.1 – Fig. 54).
### 23 Discussion
Our goal in this Part of the work was to include the dynamical friction and
the gradual decay of the secondary galaxy in the test-particle simulations. It
has been previously pointed out that coupling of these phenomena is a key
effect in the shell structure formation but it was never modeled in much
detail so far. Using these simulations, we aimed to asses the plausibility of
timing the shell-creating merger using the outermost observed shell in a shell
galaxy.
For the dynamical friction we used our own modification of the Chandrasekhar
formula for radial trajectories, Appendix E, which is more faithful to the
true stellar distribution function of the host galaxy. The dynamical friction
calculated in this way is fully determined by the distribution function of the
host galaxy and the mass and velocity of the secondary, thus is contains no
free parameters. Comparison between our modification and the commonly used
form of the Chandrasekhar formula, Sect. E.2, shows that the use of a constant
Coulomb logarithm is completely inappropriate for radial mergers. But when
compared with the self-consistent simulations, our method is found to
significantly overestimate the friction, Sect. 20.2. In reality, the dynamical
friction in a radial merger depends on the whole merger history and thus can
be hardly reproduced by any modification of the Chandrasekhar formula, Sect.
19.1. Our simulations thus have to be understood as the upper estimate on the
true effect of the dynamical friction on the shell formation.
Including the tidal disruption of the secondary galaxy in test-particle
simulation is even more complicated. We have tried several methods, Sect.
21.1, and none of them is a priori better than any other. Moreover we tried to
reflect on the change of the shape of the gravitational potential of the
cannibalized galaxy during the merger using a variable Plummer radius, Sect.
21.2. We have carried out several simulations using different methods for the
decay of the secondary galaxy, focusing on qualitative effects in which these
simulations differ from simple simulations that assume instantaneous breakdown
of the secondary galaxy and no dynamical friction. We can believe that effects
that are independent of the method used are more likely to participate in
shell forming process in reality.
One such effect is that while the position of the outermost shells of the
first generation is not much affected by the inclusion of the gradual decay
and dynamical friction in the simulations, its brightness is drastically
lowered. The same effect is observed in our self-consistent simulation, Sect.
22.3. Even easily inferring the age of the collision is rendered impossible
(as already pointed out by Dupraz and Combes, 1987). The shell systems in Fig.
54 (Sect. 22.1), all having the outermost shell at $+150$ kpc, are seen 5 Gyr
after the first passage of the cannibalized galaxy through the center of the
host galaxy. If we observationally identify the leftmost shell (around $-80$
kpc in Fig. 54) as being the outermost one, we would mistakenly estimate the
merger age to be only $\sim 2.5$ Gyr. We would also wrongly determine the
direction from which the secondary galaxy came: assuming the classical picture
(based on simulations without friction and with instantaneous disruption), the
outermost shell would be located on the side from which the satellite came, so
we would conclude it went from the left while the opposite is true.
Furthermore, with respect to the simple simulation, in the simulations with
gradual decay of the secondary, we observe the creation of new generations of
shell during every passage of the remnant of the secondary galaxy through the
center of the primary. In the consecutive generations, shells are created at
lower radii and with higher brightness. It is important to see that, in our
simulations, the subsequent passages of the secondary galaxy do not
significantly disturb the existing shell structure of the previous generation
and thus a shell system with a large range of radii is created. The radial
range of shells observed in some real shell galaxies is truly impressive and
it is impossible to reproduce in a simple simulation. It is also worth noting
that in simulations with the gradual decay of the secondary, a large part of
the mass of the secondary ends up in the proximity of the center of the
primary galaxy.
Presence of a dark matter halo in the primary galaxy, Sect. 22.2, changes not
only the dependence of the period of radial oscillations on radius (Sect. 9),
but also the range of stellar energies through the change of the velocity of
the accreted satellite. The halo allows for a faster development of shells at
larger radii. A more massive halo creates a larger range of shell radii in our
simulations than a less massive one. The increased total mass of the host
galaxy is more important than the difference in the dynamical friction caused
by the differences in local density and velocity dispersion for different halo
configurations. The more massive halo accelerates the secondary galaxy more,
reducing the loss of its energy via the dynamical friction and increasing the
time before a subsequent return of the secondary galaxy. The higher velocity
of the secondary galaxy also means that the primary galaxy catches only very
few particles in the first passage and a significant growth of the shell
structure is observed only in later phases of the merger.
In general, it seems that test-particle simulations are not suitable for a
quantitative reproduction of observed shell systems. There is no reliable
(semi-)analytical method to calculate the dynamical friction in radial and
close-to-radial minor mergers. Apparently even more importantly, there is no
universal method to model the tidal decay of the cannibalized galaxy in test-
particle simulations. Unfortunately, it turns out that it is exactly the
details of the decay of the secondary galaxy that affect significantly the
overall shell structure. In two simulations, with apparently small differences
in the loss of mass and energy of the secondary galaxy during the first
passage and the time of the second passage, shells of the second generations
were created at different radii with respect to the shells from the first
generation (which are otherwise very similar between the simulations).
Moreover, the brightness of these shells differs and with each farther passage
of the secondary galaxy, the difference in the appearance of the shell system
increases and the observability of shells in the host galaxy changes by whole
gigayears. Overall, an accurate reproduction of a shell galaxy is a very
delicate matter, as in practice we do not know an exact distribution of mass
in the host galaxy, the original trajectory of the secondary galaxy, nor its
own mass distribution and our simulations suggest that the shell structure is
very sensitive even to small details in these quantities.
Nevertheless even despite the simplicity of the models we used, it turned out
that our test-particle simulations with gradual disruption and dynamical
friction of the secondary galaxy do better than the simple simulations in
reproducing observed features in real shell galaxies. We thus conclude that
also in real galaxies, these features are the result of combined effects of
the gradual decay and dynamical friction.
At the end, we shall stress that while all these details have a large effect
on the overall appearance of the shell system, they are not very important for
the application of the method to measure the host galaxy potential from
kinematical data that we have introduced in Part II. This method relies only
on the assumption that the stars that form one particular shell are moving
along radial trajectories and were released in the center of the primary
galaxy together at some moment in the past. Within the framework the radial-
minor-merger model, neither the gradual decay of the secondary galaxy nor the
dynamical friction do not in principle have a large influence on the radiality
of the stellar trajectories. Also, even when these effects are present, stars
are being released in short time intervals when the secondary galaxy passes
through the center of the primary galaxy, however these intervals are slightly
larger than zero, which would be the case for the instantaneous decay of the
secondary galaxy. This fact causes the shells to be slightly more diffuse and
can interfere with an effort to determine the positions of the spectral peaks
and the shell edge. Nevertheless, in principle the measurement of the
potential should be still possible.
## ?partname? IV Conclusions
In Part I we have summarized observational and theoretical knowledge about the
shell galaxies according to the available literature. Shell galaxies are
mostly elliptical galaxies containing fine structures which are made of stars
and form open, concentric arcs that do not cross each other. The most
prominent observational characteristics of shells are summarized in 22 points
in Sect. 4. In Sect. 5, we introduce all proposed scenarios of origin of shell
galaxies. The most widely accepted theory, supported by a multitude of
observational evidence, is the close-to-radial minor merger of galaxies
introduced by Quinn (1984). In the framework of this model, Merrifield and
Kuijken (1998) suggested using shell kinematics to measure the potential of
the host galaxy. The issue of the determination of the overall potential and
distribution of the dark matter in galaxies is among the most prominent in
galactic astrophysics since the most successful theory of the evolution of the
Universe so far seems to be the theory of the hierarchical formation based on
the assumption of the existence of cold dark matter, significantly dominating
the baryonic one. Thus, independent measurement of the dark matter content in
galaxies is highly desirable. Measurement of galactic potential is
particularly difficult in elliptical galaxies at large distances from the
center of the galaxy. Incidentally, shells are found mainly in elliptical
galaxies and they do occur in distances up to 100 kpc from the center.
The method of Merrifield and Kuijken (1998) is based on the approximation of a
stationary shell. Using positions of peaks in the line-of-sight velocity
distribution (LOSVD), it allows the calculation of the gradient of the
potential near the shell edge. We have developed this method further in Part
II assuming validity of the radial-minor-merger model and spherical symmetry
of the host galaxy. Using both analytical calculations and test-particle
simulations, we have shown that the LOSVD has a quadruple shape in this
situation. Assuming a constant shell phase velocity and a constant radial
acceleration in the host galaxy potential for each shell, we have developed
three different analytical and semi-analytical approaches (Sect. 11.6) for
obtaining the circular velocity in the host galaxy and the current shell phase
velocity from the positions of the peaks of the maxima of the LOSVD.
The applicability of our different approaches varies with the character of
measured data. As obtaining suitable data is at the very limit of current
observational tools and thus no such data is yet available for analysis, we
have applied our methods to results of a simulation of a radial minor merger.
We were able to reproduce the circular velocity at shell radii to within $\sim
1$ % from the actual value. Applying the method of Merrifield and Kuijken
(1998) to the simulated data, we have derived a circular velocity larger by
40–50% than the true value.
All our approaches, however, derive the shell phase velocity systematically
larger, 7–30%, than the real velocity is. That can be caused by nonradial
trajectories of the stars of the cannibalized galaxy or by poor definition of
the shell radius in the simulation. The method of Merrifield and Kuijken
(1998) does not allow to derive the shell phase velocity at all since it is
based on the approximation of a stationary shell.
In the case of spherical symmetry, the value of the circular velocity directly
determines the amount of mass enclosed under the given radius, thus
determining the dark matter content of the galaxy. On the other hand, the
shell velocity depends on the serial number of the shell and on the whole
potential from the center of the galaxy up to the shell radius and thus its
interpretation is less straightforward. A comparison of its measured velocity
to theoretical predictions is possible only for a given model of the potential
of the host galaxy and the presumed serial number of the observed shells. In
such a case, however, it can be used to exclude some parameters or models of
the potential that would otherwise fit the observed circular velocity.
Moreover, the measurement of shell velocities can theoretically decide whether
the outermost observed shell is the first one created; determine the time from
the merger and the impact direction of the cannibalized galaxy; and reveal the
shells from different generations, which can be present in a shell galaxy
(Bartošková et al., 2011).
In Part II we have examined effects of the gradual decay and dynamical
friction of the cannibalized (secondary) galaxy on the appearance of the shell
structure. Our goal was to asses the plausibility of timing the shell-creating
merger using the outermost observed shell in a shell galaxy. Attempts to date
a merger from observed positions of shells, using simple test-particle
simulations, have been made in previous work of Canalizo et al. (2007)
supporting a potential causal connection between the merger, the post-
starburst ages in nuclear stellar populations, and the quasar.
We have searched for a method to include the gradual decay and dynamical
friction of the secondary galaxy into the test-particle simulations. While
these effects are (along with many other physical processes) naturally
included in self-consistent simulations, using these has also some serious
drawbacks when compared to test-particle simulations. For example, some
effects seen in self-consistent simulations are difficult or outright
impossible to reproduce by analytical or semi-analytical methods. At the same
time, their manifestation in self-consistent simulations is difficult to
separate and sometimes they may even be confused with non-physical outcomes of
used methods. Moreover, self-consistent simulations with high resolution
necessary to analyze delicate tidal structures such as the shells are
demanding on computation time. This demand is even larger if we want to
explore a significant part of the parameter space.
For the dynamical friction we used our own modification of the Chandrasekhar
formula for radial trajectories, Appendix E. The dynamical friction calculated
in this way is fully determined by the distribution function of the host
galaxy and the mass and velocity of the secondary, thus is contains no free
parameters. But when compared with the self-consistent simulations, our method
is found to significantly overestimate the friction, Sect. 20.2. Our
simulations thus have to be understood as the upper estimate on the true
effect of the dynamical friction on the shell formation.
We have tried several methods for including the tidal disruption and
deformation of the secondary galaxy, Sect. 21, and none of them is a priori
better than any other. In our simulations it turns out that the resulting
shell system is very sensitive to small differences during the decay of the
cannibalized galaxy and thus the test-particle simulations are not suitable
for a quantitative reproduction of observed shell systems. We have thus
focused on qualitative effects in which our enhanced simulations differ from
simple simulations that assume instantaneous breakdown of the secondary galaxy
and no dynamical friction. It turned out that these enhanced test-particle
simulations do better than the simple simulations in reproducing observed
features in real galaxies, including features that the simple simulations
cannot show at all. We thus conclude that also in real galaxies, these
features are the result of combined effects of the gradual decay and dynamical
friction.
One effect found commonly in all the enhanced test-particle simulations is
that while the position of the outermost shells of the first generation is not
much affected by the inclusion of the gradual decay and dynamical friction in
the simulations, its brightness is drastically lowered. The same effect is
observed in our self-consistent simulation, Sect. 22.3. Even just inferring
the age of the collision is thus tricky: if we observationally miss the
weakened outermost shell, which should be clearly visible according to simple
simulations, we would underestimate the merger age by a factor of 2. At the
same time, we would also wrongly determine the direction from which the
secondary galaxy came.
Ideally, for systems with multiple shells we would like to combine
measurements of shell kinematics and their radial distribution, possibly also
with measurements of surface brightness profile (Sect. 14.4). The kinematical
measurements supply us with the magnitude of acceleration at the shell edge
and an estimate of the phase shell velocity, which allows us to separate the
shells in different generations, if these are present. Simulations with the
dynamical friction and gradual decay of the secondary galaxies that reproduce
the kinematic and photometric data will then constrain other parameters of the
merger such as its age and the trajectory and nature of the satellite galaxy.
A similar result has been obtained for M31, Fardal et al. (2007, 2008, 2012),
whereas for the other shell galaxies, obtaining the kinematical data is a
great challenge for the next generation of astronomical instruments.
## ?partname? V Appendix
### ?appendixname? A Units and conversions
When dealing with galaxies, we need to describe objects and time spans
incommensurable with our daily experience that defines the standard sets of
units, such as SI. Throughout the text we thus use a set of units adapted for
this task – we measure the mass in M⊙ the length in kpc and the time in Myr.
Although their meaning is clear, they sometimes give rise to rather awkward
derived units. We will briefly list the most prominent of them (together with
the basic ones) and give their relation to the SI and cgs units.
* Time:
1 Myr = $10^{6}$ yr = $3.156\times 10^{13}$ s
* Distance:
1 kpc = 3 262 ly = $3.086\times 10^{19}$ m = $3.086\times 10^{21}$ cm
* Mass:
1 M⊙ = $1.989\times 10^{30}$ kg = $1.989\times 10^{33}$ g
* Velocity:
1 kpc$/$Myr = 977.8 km$/$s = $9.778\times 10^{7}$ cm$/$s (the roundness of
this value allows for an easy conversion for most of our plots)
* Acceleration:
1 kpc$/$Myr2= $3.098\times 10^{-8}$ m$/$s2 = $3.098\times 10^{-6}$ cm$/$s2
* Density:
1 M⊙$/$kpc3= $6.768\times 10^{-29}$ kg$/$m3 = $6.768\times 10^{-32}$ g$/$cm3
* Grav. unit:
1 kpc3$/$Myr2$/$M⊙ = 14.83 m3$/$s2$/$kg = 14 830 cm3$/$s2$/$g –
thus G = $6.674\times 10^{-11}$ m3$/$s2$/$kg = $4.500\times 10^{-12}$
kpc3$/$Myr2$/$M⊙
### ?appendixname? B List of abbreviations
AU
arbitrary unit, a relative placeholder unit for when the actual value of a
measurement is unknown or unimportant
DM
dark matter
FWHM
full width at half maximum, parameter of Gaussian function
GADGET-2
free software used for self-consistent simulations, see Sect. 20
KDC
kinematically distinct/decoupled cores of galaxies, see Sect. 3.7
LOS
line-of-sight
LOSVD
line-of-sight velocity distribution
MK98
paper about measuring gravitational potential using shell kinematics
Merrifield and Kuijken (1998)
MTBA
Multiple Three-Body Algorithm, a method used by Seguin and Dupraz (1994) to
study dynamical friction in head-on galaxy collisions, see Sect. 19
S$/$N
signal-to-noise ratio
WIM
Weak Interaction Model of origin of shell galaxies by Thomson and Wright
(1990), see Sect. 5.2
### ?appendixname? C Initial velocity distribution
The shell-edge density distribution,
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$, is defined by Eq. (13).
Note that, since, in the model of the radial oscillations, all stars at the
shell edge have the same energy, the function $N\left(r_{\mathbf{s}}\right)$
determines the distribution of stellar apocenters, the radial dependence of
which differs just slightly from
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$.
Let $f(r_{\mathrm{ac}})$ and $g(v_{0})$ be the distribution function of the
stellar apocenters and the initial velocities, respectively, then
$g(v_{0})=f(r_{\mathrm{ac}})\frac{\mathrm{d}r_{\mathrm{ac}}}{\mathrm{d}v_{0}}.$
(90)
In almost all cases in the thesis
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto
1/r_{\mathbf{s}}^{2}$, so the distribution function of the stellar apocenters
is a constant function $f(r_{\mathrm{ac}})=A$.
Initially, all stars are at the center of the host galaxy, so
$v_{0}=\sqrt{-2\left[\phi(r_{\mathrm{ac}})-\phi(0)\right]},$ (91)
where $\phi(r)$ is the spherically symmetric potential of the host galaxy.
Then
$g(v_{0})=A\left.\frac{\mathrm{d}\phi(v)}{\mathrm{d}v}\right|_{v=v_{0}}v_{0},$
(92)
where $\phi(v)$ is inverse function to $v_{0}\left(\phi\right)$ given by Eq.
(91).
The correspondence between the shell-edge density distribution and initial
velocity distribution is one-to-one, unlike for example the one between the
spatial (or projected) density and the shell-edge density distribution, Eq.
(14), as the density at one radius receives contributions from particles with
two distinct velocities.
### ?appendixname? D Introduction to dynamical friction
Appendix D was, with some adjustments, adapted from the master thesis Ebrová
(2007).
#### D.1 A thermodynamic meditation
The dynamical friction is a braking force of gravitational origin, caused by
the sole fact that the area, through which the secondary galaxy (or, in
general, any object passing through a galaxy or another extended object) flies
is not an empty space filled with a smooth potential, but a large sea of
individual stars.
Thinking deeper, we can easily see that some slowdown of the secondary galaxy
is inevitable. Every system, where energy transfer is possible tends to
temperature equilibrium. In a system of at least three gravitating bodies such
a transfer is indeed possible and frequently happens. The relatively fast and
heavy secondary galaxy possesses a decent amount of kinetic energy and as such
it is just a hot piece thrown into a colder sea of the stars of the primary
galaxy. The slowdown of the intruder that cannot be accounted for in the
fixed-potential model, is the way of leveling the temperatures. The kinetic
energy transfers to the primary’s stars – the same effect causes the heating
of the cold disk population in the week interaction model, as mentioned in
Sect. 5.2.
The reality of this process can be grasped from a different point of view. The
relatively massive secondary galaxy attracts the primary’s stars and thus
creates an area of a higher density of stars behind itself. The passing galaxy
is attracted backwards by this condensation, lowering its speed towards the
primary.
#### D.2 Chandrasekhar formula
An analytical derivation of such a braking force is based on the following
thought: In a distant encounter with just one star, the velocity of an object
cannot be changed, instead it is only deflected from the original direction
and thus enriched with a component of speed perpendicular to the original
direction. For a very massive body, as our secondary galaxy is, the magnitude
of this perpendicular component will not be large, neither will be the loss of
the velocity in the original direction. But when it undergoes many such
encounters, the contributions add. The contributions in the perpendicular
directions will have randomly scattered azimuthal angles and thus add to zero
(except for the overall action of the smooth potential). On the other hand,
the contributions to the original direction of the velocity will always be
opposite to it, resulting in the braking of the galaxy.
The Chandrasekhar formula was originally derived by Chandrasekhar (1943). Here
we present a short version of the presentation of the chapter 7.1 in the bible
of the galactic astronomy, “Galactic Dynamics” by Binney and Tremaine (1987).
To start, let us imagine the encounter of our object of interest with a single
star. When two bodies meet, energy is not transferred, but the direction of
velocity of our object changes. It is a matter of a simple mechanics and as a
result, the change of the component of velocity parallel to its original
direction, $\mid\mathbf{\bigtriangleup v}_{M\parallel}\mid$ between the times
$t=-\infty$ and $t=\infty$ is given by (Eq. 7-10b in Binney and Tremaine,
1987; see its derivation there):
$\mid\bigtriangleup\mathbf{v}_{M\parallel}\mid=\frac{2\,m\,V_{0}}{M+m}\left[1+\frac{b^{2}V_{0}^{4}}{G^{2}(M+m)^{2}}\right]^{-1},$
(93)
where $M$ is our object’s (the secondary galaxy) mass, $m$ is the mass of the
star, $b$ the impact parameter (the length of $\mathbf{b}$, the vector
indicating the position of the star in a plane perpendicular to the original
velocity of the galaxy) and $\mathbf{V}_{0}$ is the difference between the
original velocity of the star $\mathbf{v}_{m}$ and velocity of our object
$\mathbf{v}_{M}$, so $\mathbf{V}_{0}=\mathbf{v}_{m}-\mathbf{v}_{M}$. The bold
typeface indicates vectors, and their length is indicated by the same symbol
in normal type.
For an object flying through a field of stars with the phase-space number
density of stars $f(\mathbf{v}_{m},\mathbf{b})$, the change in the parallel
component of velocity $\mathrm{d}\mathbf{v}_{M\parallel}$ in an infinitesimal
time $\mathbf{\mathrm{d}}t$ will be given by the integration of Eq. (93)
multiplied by the density $f(\mathbf{v}_{m},\mathbf{b})$ over the plane of
$\mathbf{b}$ and the space $\mathbf{v}_{m}$. For $\mathbf{b}$ is measured from
a given point in a plane, it is advantageous to use the polar coordinates
$(b,\varphi)$:
$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=\intop\intop\intop
f(\mathbf{v}_{m},b,\varphi)\frac{2m\,V_{0}(\mathbf{v}_{m})\,\mathbf{V}_{0}(\mathbf{v}_{m})}{(M+m)\left[1+\frac{b^{2}V_{0}^{4}(\mathbf{v}_{m})}{G^{2}(M+m)^{2}}\right]}\mathrm{d}^{3}\mathbf{v}_{m}\,b\mathrm{d}b\,\mathrm{d}\varphi.$
(94)
To derive the Chandrasekhar formula we further assume the homogeneity of the
field of stars, so as the distribution function of the stars does not depend
on $\mathbf{b}$. The remaining $\mathbf{b}$-dependent part is of the following
form a can be easily integrated from 0 to some $b_{\mathrm{max}}$:
$\intop_{0}^{b_{\mathrm{max}}}\frac{b\mathrm{d}b}{1+c^{2}b^{2}}=\left[\frac{\ln(1+c^{2}b^{2})}{2\,c^{2}}\right]_{b=0}^{b=b_{\mathrm{max}}},$
(95)
where in our case $c=V_{0}^{2}/[G(M+m)]$. It is conventional to introduce the
notation
$\Lambda=\frac{b_{\mathrm{max}}V_{0}^{2}}{G(M+m)}.$ (96)
A typical value of $\Lambda$ would be of the order of $10^{3}$, thus we can
neglect the one and put $\frac{1}{2}\ln(1+\Lambda^{2})\cong\ln(\Lambda)$. This
factor is often called the Coulomb logarithm. Furthermore we assume that we do
not err too much when replacing $V_{0}$ in $\Lambda$ by $v_{\mathrm{typ}}$, a
typical speed. Then the Coulomb logarithm does not depend on $\mathbf{v}_{m}$,
and still $V_{0}=\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid$ and the whole Eq. (94)
goes to
$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=4\pi\ln(\Lambda)G^{2}m(M+m)\intop
f(\mathbf{v}_{m})\frac{\mathbf{v}_{m}-\mathbf{v}_{M}}{\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid^{3}}\mathrm{d}^{3}\mathbf{v}_{m}.$
(97)
The integral is of exactly the same form as in the Newton’s law of gravity and
if the stars move isotropically, the density distribution is spherical and by
Newton‘s first theorem (see Binney and Tremaine, 1987; chapter 2), the total
acceleration of our object by dynamical friction is:
$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=-16\pi^{2}\ln(\Lambda)G^{2}m(M+m)\frac{\intop_{0}^{v_{M}}f(v_{m})v_{m}\mathrm{d}v_{m}}{v_{M}^{3}}\mathbf{v}_{M}$
(98)
i.e., only stars moving slower then our object contribute to the force and
this force always opposes the motion. Eq. (98) is usually called the
Chandrasekhar dynamical friction formula.
If $f(v_{m})$ is Maxwellian with dispersion $\sigma$
$f=\frac{n_{0}}{(2\pi\sigma^{2})^{3/2}}\exp(-\frac{1}{2}v^{2}/\sigma^{2}),$
(99)
we can integrate Eq. (98). The density of the stars is $\rho_{0}=n_{0}\,m$ and
for $M\gg m$, what happens to be our case, we can put $(M+m)\cong M$, and then
Eq. (98) reads:
$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=-\frac{4\pi\ln(\Lambda)G^{2}\rho_{0}M}{v_{M}^{3}}\left[\mathrm{erf}(X)-\frac{2\,X}{\sqrt{\pi}}\mathrm{e}^{-X^{2}}\right]\mathbf{v}_{M},$
(100)
where $\Lambda$ is given by Eq. (96), $X\equiv v_{M}/(\sigma\sqrt{2})$ and
erf($X$) is the error function given by
$\mathrm{erf}(X)\equiv\frac{2}{\sqrt{\pi}}\intop_{0}^{X}\mathrm{e}^{-t^{2}}\mathrm{d}t$
(101)
for which we can obtain tabulated values, or we can pre-generate them
numerically with an arbitrary precision.
#### D.3 What a wonderful universe
Giving it a deeper thought, one can consider the validity of the Chandrasekhar
formula almost a miracle. We have by the way disclosed that it works, at least
approximately – the confrontation with numerical simulations of flybys through
a galaxy or a cluster has been carried out by e.g. Lin and Tremaine (1983);
Bontekoe and van Albada (1987), who proved that the analytical solution (given
by the Chandrasekhar formula) is in a good agreement with the simulations in a
relatively wide range of situations. The analytical solutions has some freedom
in the Coulomb logarithm which is not completely well-defined. Its correct
choice can help to better reproduce the numerical results and compensate other
drawbacks of the formula – anyway, the freedom is small when we demand the
Coulomb logarithm to stay constant.
?figurename? 60: The path and velocity changes of the objects undergoing
encounters with individual stars. The absolute value of the velocity remains
unchanged.
But back to our astonishment. When the secondary galaxy deviates from its
course, its speed in the original direction is reduced. But after meeting
another star that compensates the deviation, it also gets back the original
velocity in this direction, as is shown in Fig. 60.
The point is that the Chandrasekhar formula evaluates the change of the
parallel component of the velocity after the flyby from infinity to infinity
for every single star with the same initial conditions and then adds these
changes and applies them to the secondary galaxy in one moment, the moment of
the closest approach with these stars, see Fig. 61. The change of the parallel
component of the velocity and the compensation of the changes in the
perpendicular direction then happen somehow at the same time, although the
magnitude of their effect is calculated as if they happen consecutively – and
by some wonder, it works.
?figurename? 61: A schematic depiction of the change of the velocity of the
secondary galaxy after three steps. In every moment, only the influence of the
stars lying in one plane perpendicular to the motion of the galaxy is taken
into account.
Let us just remark that the fact that we account for the influence of the
stars in the moment of the closest approach is not so strong neglection.
During an encounter of two bodies, roughly one half of the velocity change
takes place around the point of the closest approach on the scale of the
impact parameter. For the encounter of the galaxy with two stars, it is
confirmed in the right panel of Fig. 62.
#### D.4 Why does it work?
We can see the mechanism of the dynamical friction in action even in a simple
model of a “galaxy” interacting with two “stars”, results of which are seen in
Fig. 62. Although the model is a very simple one, it allows us to see in
practice that yet in the system of three bodies (in contrary to two) the
permanent energy and momentum transfer is possible. The symmetry of the
configuration ensures that the galaxy will keep a straight line and thus any
change of velocity it undergoes will be a change in the magnitude of the
velocity. According to the idea of an infinite sea of stars, we take into
account only the interaction between the stars and the galaxy, not mutually
between the stars.
?figurename? 62: The result of the simulation. A large body (with the mass of
200 M⊙, straight black line) moves in the direction of the $x$-axis (with the
velocity of only 100 m$/$s – this and the other unrealistic values have been
chosen only to make the picture more illustrative in a linear and uniform
scale) and encounters a pair of stars (2 M⊙ each) that are initially place
symmetrically with respect to its track (0.1 pc from the track). The mutual
gravitational attraction of the stars is neglected. The right panel shows the
development of the velocity of the large body during the closest approach. The
blue line represents its original velocity, thus its path if the stars were
not present.
It is clear that due to the galaxy’s gravity, the stars begin to move towards
its track (meanwhile also moving towards the galaxy along the track, but let
us not care for a moment). While the stars move towards the track, the
attraction accumulates and they gather speed. When they cross the galaxy’s
track, the galaxy starts pulling them back (at least when we speak about the
perpendicular component of the velocity) and they slow down. Anyway, thanks to
the fact that they cross the track _after_ the galaxy’s passage, they spend
more time in the phase where their perpendicular velocity component is
increased than otherwise and finally they retain some speed in this direction.
But it means they have gathered kinetic energy, what must be at the expense of
the galaxy’s kinetic energy and so the speed of the galaxy must have decreased
(that is the dynamical friction) – even though it has moved much faster than
before during the closest approach of the encounter. In reality, the situation
is a little more complex, because apart from the energy, the momentum has to
be also conserved – the momentum of the galaxy has decreased and so the stars
must have also a non-zero parallel component of the velocity, to maintain this
component of momentum.
In accordance with the derivation of the Chandrasekhar formula, we use Eq.
(93) just multiplied by two to derive the analytical formula for the change of
the galaxy’s velocity. For the impact parameter $b$ we obviously put the
original distance between the stars and the galaxy’s track. Our numerical
tests for various values of parameters (masses, initial velocity of the
galaxy, impact parameter) show that the analytical results obtained this way
tend to overestimate the decrease in the velocity, typically by about 15 per
cent.
It could be anticipated that the numerical and analytical results will differ,
as the analytical formula counts with two separated encounters from infinity
to infinity. In such a case the galaxy follows a curved trajectory and thus
its interaction with the star is slightly different than when both encounters
happen at the same time and the galaxy is forced to stay on a straight line.
Let us remark that we have tested the model by removing one of the stars and
then the results for the change of the parallel component of the velocity
differ from the prediction in fractions of per mille.
In reality, the situation is even more complex, there are many stars in the
game and they also mutually interact and undergo the influence of all the
surrounding stars that do not take part in the dynamical friction directly.
### ?appendixname? E Our method
In Ebrová (2007) we have introduced our method to calculate the dynamical
friction in restricted $N$-body simulations during the radial merger. In this
section we remind the reader of its characteristics and derivation as
introduced in the master thesis.
#### E.1 Avoiding some approximations
The Chandrasekhar formula contains two kinds of inaccuracies. The first of
them is the principal one, namely the fact that the change in the parallel
component of the velocity from any individual star is added instantaneously at
the point of the closest approach (of the secondary galaxy) to it. We have
already shown that it is not too wrong, but what is worse, the influence of
the star is taken to be such as if the galaxy passed it from infinity to
infinity and there was nothing in the universe but the star and the galaxy.
Sects. D.2–D.4 for details.
The second source of inaccuracy lies in all the approximation that have been
done when passing from Eq. (94) to Eq. (98). These will concern us in this
section, leaving aside the assumptions of the Maxwellian velocity distribution
and the negligence of the masses of the stars compared to that of the
secondary galaxy, that led us from Eq. (98) to Eq. (100), which we use in the
simulations and keeping the “principal inaccuracy” mentioned above.
The first approximations that allowed us to integrate Eq. (94) over the plane
of the impact parameter was the assumed homogeneity of the star field, i.e.
that the distribution function does not depend on position. Then we have taken
the Coulomb logarithm to be independent of velocity of the stars
$\mathbf{v}_{m}$ (it obviously isn’t, but it varies slowly) and this has
allowed us to simplify the $\mathbf{v}_{m}$-integral and given a suitable
choice of the distribution function we could even carry out the integration
(see Sect. D.2). Both steps are only approximate even in the simple case of
the spherical galaxy with the Plummer profile, as both the density – Eq. (74)
and the velocity dispersion – Eq. (80) of the Plummer sphere do depend on the
radius.
If we wish to avoid these simplification, we have to turn back to Eq. (94) and
put in e.g. the Maxwellian distribution, Eq. (99), for
$f(\mathbf{v}_{m},b,\varphi)$, together with putting $n_{0}m=\rho$, where
$\rho$ is the density of the primary at a given point – keeping in mind that
the radius _$r$_ (the distance of a point from the center of the primary
galaxy) on which the formulae depend is a function of $b$_,_ $\varphi$ and in
fact also of the direction of motion of the braked body (the secondary
galaxy). When dealing with the radial mergers, this direction points towards
the center of the primary galaxy and $r$ becomes a particularly simple
function of $b$:
$r=\sqrt{d^{2}+b^{2}},$ (102)
_where_ $d$ is (also in the following) the distance between the centers of the
primary and the secondary galaxy. There is no $\varphi$-dependence in the
radial case and the integration gives a trivial factor of $2\pi$. For
simplicity, we put the Eq. (80) for the velocity dispersion, as the friction
is essentially negligible for both the simple and the cut-off dispersion in
the areas where they significantly differ (see Fig. 43). Furthermore, during
the multiple passages that occur in the simulations (where the friction
becomes significant) the secondary galaxy does not reach these areas at all.
Using Eq. (80) for the cut-off velocity dispersion would thus unnecessarily
complicate the already complex formulae.
Putting all this together, we get
$\displaystyle\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}$
$\displaystyle=$ $\displaystyle\frac{3^{5/2}\varepsilon_{\mathrm{p}}^{2}}{(\pi
G)^{3/2}M_{\mathrm{p}}^{1/2}M_{\mathrm{s}}}\intop\intop\frac{\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid(\mathbf{v}_{m}-\mathbf{v}_{M})}{(b^{2}+d^{2}+\varepsilon_{\mathrm{p}}^{2})^{7/4}}\times$
(103)
$\times\left[1+\frac{b^{2}(\mathbf{v}_{m}-\mathbf{v}_{M})^{4}}{G^{2}M_{\mathrm{s}}^{2}}\right]^{-1}\mathrm{\exp\left[-\frac{3\mathbf{v}_{\mathit{m}}}{\mathit{GM_{\mathrm{p}}}}\sqrt{\mathit{b}^{2}+\mathit{d}^{2}+\varepsilon_{\mathrm{p}}^{2}}\right]}b\mathrm{d}b\,\mathbf{\mathrm{d^{3}}v}_{m},$
where the meaning of the variables is the same as when we derived the
Chandrasekhar formula in Sect. D.2. The indexes $p$ and $s$ again stand for
the parameters of the primary and the secondary galaxy, respectively.
First, we shift the integration variable to
$\mathbf{v}_{m}^{\prime}=\mathbf{v}_{m}-\mathbf{v}_{M}$ and immediately rename
it back $\mathbf{v}_{m}^{\prime}\rightarrow\mathbf{v}_{m}$. We then perform
the scalar product with the unit vector $\mathbf{v}_{M}/v_{M}$ on both sides,
getting the projection of the friction acceleration to the direction of the
velocity of the secondary galaxy. This is by symmetry its only nonzero
component in the radial case and it will be advantageous to deal with a
scalar-valued integral. The negative value means that the friction acts in the
direction opposite to the motion of the braked body, what is the only feasible
situation in any setup with an isotropic velocity distribution in the primary
galaxy.
Transforming to the spherical coordinates (taking the $z$-axis parallel with
the velocity of the secondary galaxy), we have
$\mathbf{v}_{m}\cdot\mathbf{v}_{M}=v_{m}v_{M}\cos\theta$ and again no
dependence on the azimuthal angle, leaving us with the obligatory factor of
$2\pi$. The $\theta$-integral then can be carried out in the form that could
be with some effort put on mere three lines:
$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}\cdot\frac{\mathbf{v}_{M}}{v_{M}}=\frac{\sqrt{3\,M_{\mathrm{p}}}G^{3/2}M_{\mathrm{s}}\varepsilon_{\mathrm{p}}^{2}}{2\sqrt{\pi}v_{M}^{2}}\intop_{0}^{\sqrt{R^{2}-d^{2}}}\intop_{0}^{\infty}\frac{b\mathrm{d}b\,v_{m}^{2}\mathrm{d}v_{m}}{\left(\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}\right)^{11/4}\left(G^{2}M_{\mathrm{s}}^{2}+b^{2}v_{m}^{4}\right)}\times$
(104)
$\times$[$\mathrm{e}^{-3\,\frac{\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\left(v_{m}-v_{M}\right)^{2}}{GM_{\mathrm{p}}}}\left(\mathrm{G}M_{\mathrm{p}}-6\,v_{m}v_{M}\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\right)-$
$-\mathrm{e}^{-3\,\frac{\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\left(v_{M}+v_{m}\right)^{2}}{GM_{\mathrm{p}}}}\left(\mathrm{G}M_{\mathrm{p}}+6\,v_{m}v_{M}\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\right)$]
,
where $R$ is the considered cut-off of the primary galaxy. We cannot proceed
analytically with the integration (not even in one of the variables), instead
we have solved it numerically in Maple for chosen values of the parameters.
We have come to a formula for the dynamical friction Eq. (104) that is
physically more accurate than the Chandrasekhar formula, but it is valid only
for a radially moving body in the Plummer sphere. It is also only more
accurate in the sense of avoiding the approximation used between Eq. (98) and
Eq. (100) but it is still built atop the “principal inaccuracies” described
above.
?figurename? 63: The value of the integrand (including all the constants) from
Eq. (104) in the dependence on the integration variables (the impact parameter
and the relative velocity between the secondary galaxy and the stars) for the
standard set of parameters (see __ Sect. _17.5_) and the distance of the
braked body (the secondary galaxy) from the center of the primary of 70 kpc.
The velocity of the body is taken to be 0.2 kpc$/$Myr (1 kpc$/$Myr $\doteq$
1000 km$/$s, see Appendix A). This value is also indicated in the graph by a
red marker – it not surprising to find it near the peak, because there is a
strong contribution from the stars that are in rest with respect to the center
of the primary galaxy, as the Maxwellian distribution peaks in zero.
The reader who considers a formula to be the best figure can enjoy Eq. (104)
and who considers a figure to be the best formula can explore Fig. 63, where
the integrand of Eq. (104) is shown in dependence of both integration
variables for a chosen set of parameters. It is clear that far most of the
acceleration comes from a close neighborhood of the braked body both in the
plane of the impact parameter and the velocity space. However, the maximum of
the integrand does not exactly coincide with the actual speed of the body, as
there is no reason for it to be so, but it is very close.
For a primary galaxy made of two Plummer spheres – one for the luminous
component and one for the dark matter halo – the equivalent of Eq. (104)
becomes much more complicated. It can be obtain in much the same manner as
described in this chapter, only using Eq. (83) instead of Eq. (80) for the
velocity dispersion. But the angular integration is not possible to
analytically, and the resulting three-dimensional integral cannot be written
in a couple of lines’ worth of space. A numerical solution is necessary for
specific values of parameters.
#### E.2 Back to Chandrasekhar formula
We have examined how the braking force according to Eq. (104) differs from
that calculated using the Chandrasekhar formula. The Coulomb logarithm is in
some sense a free parameter of the formula, thus we have adjusted it to
maximize the agreement between the two methods of calculation of the friction.
For further details, see Ebrová (2007).
?figurename? 64: The logarithmic and linear plots of the time dependence of
the dynamical friction for multiple passages of the secondary galaxy for the
standard set of parameters (Sect. 17.5), using $b_{\mathrm{max}}=10$ kpc
together with the lower limit of the Coulomb logarithm
$\ln\Lambda_{\mathrm{crit}}=2$. Red values are computed in the model, blue
values are numerical solution of Eq. (104).
Using a constant value of the Coulomb logarithm we did not obtain a good
agreement between the friction calculated using the Chandrasekhar formula and
using our method. The best option seems to be to calculate the value of the
Coulomb logarithm in every step from the actual value of the velocity of the
secondary galaxy. The $V_{0}$ in the definition Eq. (96) for $\Lambda$ is the
difference between the velocities of the stars and the secondary galaxy. As
the stellar velocities are isotropic, the average value is just the velocity
of the secondary galaxy with respect to the center of the primary.
There is a uncertainty in the parameter $b_{\mathrm{max}}$ in the same
equation – it should be theoretically equal to the distance between the center
of the secondary and the outer boundary of the primary measured in the plane
perpendicular to the motion of the secondary. But Eq. (100) assumes a
homogeneous field of stars across all this distance, what is obviously not
true. As the plane of the impact parameter is the plane perpendicular to the
radial motion of the secondary galaxy, the density of the primary galaxy is
always the highest in its center and decreases outwards. Thus it may seem that
the $b_{\mathrm{max}}$ should be smaller than the normal distance to the edge
of the primary galaxy, but the approximation of the $V_{0}$ with the velocity
of the galaxy and other circumstances make the situation more complex. The
value of $b_{\mathrm{max}}$ must be chosen in a trial-and-error method for the
chosen parameters of collision so that the magnitude of the friction agrees
best with the numerical solution of the integral Eq. (104).
The adaptive version of the Coulomb logarithm with a suitable chosen
$b_{\mathrm{max}}$ fits nicely in the high velocity regime. The problem
appears when the satellite gets close to its apocenter and also mainly in the
late parts of the merger when the velocity of the satellite is much lower than
during its first passage through the center of the primary galaxy. Here the
adaptive version of the Coulomb logarithm with the chosen $b_{\mathrm{max}}$
significantly underestimates the friction when compared with the numerical
solution of the integral Eq. (104). So we use the adaptive Coulomb logarithm
until its value drops under a certain limit $\ln\Lambda_{\mathrm{crit}}$, then
we put this limit for the Coulomb logarithm instead. With this modification of
the Chandrasekhar formula, we can achieve a reasonable agreement, see Fig. 64.
$b_{\mathrm{max}}$ and the lower limit for the Coulomb logarithm are free
parameters and they depend on the parameters of the radial merger – the
initial mutual velocity of the galaxies, their masses and Plummer radii.
#### E.3 Incorporation of the friction in the simulation
The question of incorporation of the dynamical friction in the simulations of
the shell formation is tricky. In a fully self-consistent simulation, the
dynamical friction would be automatically included, but such a simulation
would be extremely demanding on resources – for the friction to be really well
simulated, the number of particles of primary galaxy should not be several
orders of magnitude smaller than the true amount of stars in the galaxies.
Joining the stars in a smaller amount of more massive objects systematically
overcounts the friction. Peñarrubia et al. (2004) remarked that Prugniel and
Combes (1992); Wahde and Donner (1996) have indeed shown that the dynamical
friction is artificially increased if the particle number is small. Using the
analytical formula for the friction is not devoid of problems, but in some
respects it could be more accurate than some of the self-consistent
simulations.
On the other hand, the number of the particles of the secondary is an
important quantity for the visibility of the shells in the simulations. And
for the large number of required test particles $(\sim 10^{6})$ that represent
just the secondary galaxy, even our “simple” simulations take hours of
computation on a contemporary desktop computer. Furthermore, to explore the
parameter space we have to run a lot of simulations, so we can really use a
handy (semi-)analytical formula. We can easily add the acceleration calculated
by Eq. (104) into the equation of motion of the galaxies.
It is worth mentioning that we departed in two aspects from the potential that
we chose to model the merging galaxies. We assumed Maxwell velocity
distribution, Eq. (99). This is not exactly true for the Plummer sphere, but
the difference is small and the true velocity distribution in real galaxies is
not known, so we cannot do much better, or say exactly how big mistake do we
make.
The secondary galaxy is here treated as a point mass what artificially
increases the friction, because the extended character of the galaxy softens
the force (Sect. 17.2). Specially the stars with a small impact parameter with
respect to the center of the secondary galaxy fly straight through it and
their effect is significantly reduced compared to the Chandrasekhar formula
for the point mass. The overestimation of the dynamical friction is not a
crucial problem as we want to estimate how much the shell system is influenced
by it – we can assume that the reality is not worse than our results and we
get the upper bound on the effect.
### ?appendixname? F Tidal radius
For starters, let us remind the reader of the derivation of the tidal radius,
as presented in Ebrová (2007). The tidal forces acting on an object are often
derived using the following picture: A massive body (secondary galaxy) as a
whole follows the force acting on it in its center of mass. But the force
acting on outer parts of the body is different, as it is at different
distances of the source (the primary galaxy). If this difference is larger
than the binding force with the secondary for a given star, it is stripped
off.
The tidal radius $r_{\mathrm{tidal}}$ is then defined as the distance (from
the center of the secondary), where the difference of the force of the primary
from its force in the center of mass of the secondary is just equal to the
force from the secondary:
$F_{\mathrm{p}}(d-r_{\mathrm{tidal}})-F_{\mathrm{p}}(d)=F_{\mathrm{s}}(r_{\mathrm{tidal}}),$
(105)
where _$d$_ is the separation between the centers of the galaxies and
$F_{\mathrm{p}}(r)$ and $F_{\mathrm{s}}(r)$ is the force from the primary and
the secondary for a given test particle (its mass is immediately canceled out
from the equation).
For two point-like bodies (with masses $M_{\mathrm{p}}$ and $M_{\mathrm{s}}$),
we can write Eq. (105) as:
$\frac{G\,M_{\mathrm{p}}}{d^{2}(1-r_{\mathrm{tidal}}/d)^{2}}-\frac{G\,M_{\mathrm{p}}}{d^{2}}=\frac{G\,M_{\mathrm{s}}}{r_{\mathrm{tidal}}^{2}}.$
(106)
Assuming further $r_{\mathrm{tidal}}\ll d$ we can use the Taylor expansion
$(1-x)^{-2}\cong 1+2x$ for $x=r_{\mathrm{tidal}}/d$ as it is then a small
quantity. under this assumption we get a simple formula for the tidal radius:
$r_{\mathrm{tidal}}=d\sqrt[3]{\frac{M_{\mathrm{s}}}{2\,M_{\mathrm{p}}}}.$
(107)
However, for two point masses we can get an exact result for the tidal radius.
Not making any approximation in Eq. (106) we can cast it as a fourth-order
polynomial
$X^{4}-2\,X^{3}+q\,X^{2}-2\,q\,X+q=0,$ (108)
where $X=r_{\mathrm{tidal}}/d$ and $q=M_{\mathrm{s}}/M_{\mathrm{p}}$. A
polynomial with an order less than five can be always solved. In our case,
where _$q$_ is positive, there are two real roots, from which we take the one
that gives $r_{\mathrm{tidal}}<d$ and thus $X<1$. The second real root
corresponds to a point of the other side of the primary galaxy that is not of
interest for us. The expression for this root does not give much insight, but
an interested reader can find it in Appendix G.
?figurename? 65: Tidal radius for two point masses: the approximate solution,
Eq. (107), is shown in blue, the exact solutions in red (the outer one in
light red, the inner in dark red). The shows y-axis $X=r_{\mathrm{tidal}}/d$,
the $x$-axis shows the secondary-to-primary mass ratio.
Eq. (105) gives the tidal radius for the particles on the line connecting the
centers of the two bodies – we call it the inner tidal radius. Similarly we
can write an equation for the particles on the other side of the secondary
than the center of primary lies:
$F_{\mathrm{p}}(d)-F_{\mathrm{p}}(d+r_{\mathrm{tidal}})=F_{\mathrm{s}}(r_{\mathrm{tidal}}).$
(109)
It again leads to a fourth-order polynomial for which we can obtain the root
that we call the outer tidal radius. The approximate solution Eq. (107) is the
same for both equations, Eq. (105) and Eq. (109). Let us remark that the tidal
radius is in any case just proportional to _$d$_ as there is no other scale in
the problem. Fig. 65 shows the dependence of the three radii on the mass ratio
of the bodies. We can see that for all relevant ratios the approximate formula
is just between the inner and the outer tidal radius.
The tidal radius for a point mass is in some sense an oxymoron, as these
objects have zero proportions by definition. For spherically symmetric bodies
we can write Eq. (106) as
$\frac{G\,M_{\mathrm{p}}(d-r_{\mathrm{tidal}})}{(d-r_{\mathrm{tidal}})^{2}}-\frac{G\,M_{\mathrm{p}}(d)}{d^{2}}=\frac{G\,M_{\mathrm{s}}(r_{\mathrm{tidal}})}{r_{\mathrm{tidal}}^{2}},$
(110)
where $M(r)$ is the mass enclosed in the radius _$r$._ Particularly for the
Plummer sphere we get this value integrating Eq. (74) over the sphere with the
radius _$r$_ :
$M(r)=\frac{M}{(1+\varepsilon^{2}/r^{2})^{3/2}},$ (111)
where _$M$_ is the overall mass of the body and $\varepsilon$ is the Plummer
radius. Unfortunately this makes the equation too complex to be easily solved.
Let us compare graphically the tidal radii for point masses and Plummer
spheres of the same overall masses just for one particular case – Fig. 66.
The figure (or a simple thought) shows that the notion of the tidal radius in
a general potential makes sense only when the force grows with the distance.
Otherwise the tidal force acts in the same direction as the gravitation of the
secondary and thus cannot strip off any mass. In the Plummer potential the
force reaches its maximum in $\sqrt{2}\,\varepsilon/2$, so the tidal radius is
not defined under this radius, whereas for the point masses it is defined
everywhere.
The idea of the tidal radius is just an approximation to the complex processes
during encounters of two extended bodies. It also does not define a sphere
around the center of the secondary galaxy, but as we have seen, it is
different for various locations, with the lowest value towards the center of
the primary galaxy and the highest on the opposite side. For these reasons it
is not really useful to improve its evaluation and so we have used the
approximate Eq. (107) that as we have seen gives the values somewhere in the
middle between the two extreme values of the tidal radius.
?figurename? 66: The outer and inner tidal radii (marked with circles) for the
point masses and Plummer spheres with the secondary-to-primary mass ratio of
0.02. In the Plummer case, the Plummer radius of the primary is 0.5 of the
distance between the bodies and the Plummer radius of the secondary is 0.1 of
the same quantity. Blue lines (light blue for the point mass, dark blue for
the Plummer sphere) show the gravitational force of the primary in arbitrary
units, red lines (light red for the point mass, dark red for the Plummer
sphere) show the difference between the gravitational force of the primary in
a given point and its value in 1, where the center of the secondary is. The
tidal radii are the points of intersection of corresponding curves.
### ?appendixname? G Expressions for the tidal radius
Here we give the analytical formulae for the tidal radii in the system of two
point masses as discussed in Appendix F. For the inner tidal radius we have:
$\frac{r}{d}=\frac{1}{2}+\frac{\sqrt{3}}{6}\left(\frac{\sqrt{y}}{\sqrt[6]{z}}-\sqrt{6-4\,q-\sqrt[3]{qz}-\sqrt[3]{\frac{q^{5}}{z}}+6\,\sqrt[6]{z}\sqrt{\frac{3}{y}}(q+1)}\,\right),$
(112)
where
$y=(3-2\,q)\sqrt[3]{z}+\sqrt[3]{qz^{2}}+q^{5/3}$ (113)
$z=54+q^{2}+6\,\sqrt{81+3\,q^{2}}$ (114)
and for the outer tidal radius we get similar expressions:
$\frac{r}{d}=\frac{1}{2}+\frac{\sqrt{3}}{6}\left(\frac{\sqrt{u}}{\sqrt[6]{v}}+\sqrt{6+4\,q-\sqrt[3]{qv}-\sqrt[3]{\frac{q^{5}}{v}}+6\,\sqrt[6]{v}\sqrt{\frac{3}{u}}(q-1)}\,\right),$
(115)
where
$u=(3+2\,q)\sqrt[3]{v}+\sqrt[3]{qv^{2}}+q^{5/3}$ (116)
$v=-54-q^{2}+6\,\sqrt{81+3\,q^{2}}$ (117)
and in all the expressions we use
$q=\frac{M_{\mathrm{s}}}{M_{\mathrm{p}}}.$ (118)
### ?appendixname? H Videos
Several videos are also part of the electronic attachment of the thesis. Here
we present their description. Information on details of the simulation process
can be found in Sect. 17. The videos can be also downloaded at:
galaxy.asu.cas.cz/$\sim$ivaana/phd
1. 1.
1-shells.avi – Video from a simulation of a shell-producing radial minor
merger from a perspective perpendicular to the axis of the merger. The bottom
three panels show an area of $60\times 60$ kpc centered on the primary which
is the zoomed part of the upper panels of size $300\times 300$ kpc. The first
column shows the surface density of both the primary and the secondary galaxy,
the second only the surface density of the particles originally belonging to
the secondary galaxy (corresponding to the host galaxy subtraction, a
technique used in processing real galaxy images). The third column shows the
surface density of particles originally belonging to the secondary galaxy
divided by the surface density of the primary galaxy (also corresponding to an
observational technique). The parameters of the merger are the following: the
mass of the primary is $3\times 10^{11}$ M⊙ , the secondary-to-primary mass
ratio is 0.02, the Plummer radius of the primary is 7.6 kpc, of the secondary
0.76 kpc. The initial relative velocity of the galaxies was equal to the
escape velocity of the secondary and the separation of their centers was 90
kpc. When the centers of the galaxies pass through each other, the potential
of the secondary is suddenly switched off.
2. 2.
2-shells.mpg – Video from a simulation of a shell-producing radial minor
merger used in Sect. 13. The top panel ($300\times 300$ kpc centered on
primary) shows the surface density of the particles originally belonging to
the secondary galaxy from a perspective perpendicular to the axis of the
merger; the bottom panel shows the density of the particles originally
belonging to the secondary in the space of radial velocity (vertical axis)
versus galactocentric distance (horizontal axis). The potential of the host
galaxy is the same as the one described in Sect. 8.1. Primary is modeled as a
double Plummer sphere with respective masses $M_{*}=2\times 10^{11}$ M⊙ and
$M_{\mathrm{DM}}=1.2\times 10^{13}$ M⊙ , and Plummer radii $\varepsilon_{*}=5$
kpc and $\varepsilon_{\mathrm{DM}}=100$ kpc for the luminous component and the
dark halo, respectively. The potential of the cannibalized galaxy is chosen to
be a single Plummer sphere with the total mass $M=2\times 10^{10}$ M⊙ and
Plummer radius $\varepsilon_{*}=2$ kpc. The cannibalized galaxy is released
from rest at a distance of 100 kpc from the center of the host galaxy. When it
reaches the center of the host galaxy in 306.4 Myr, its potential is switched
off and its particles begin to oscillate freely in the host galaxy.
3. 3.
3-projection.mpg – Video shoes the simulation from point 2 (used in Sect. 13)
at the time 2.2 Gyr after the decay of the cannibalized galaxy (2.5 Gyr of the
simulation time) from different perspectives. Angle of 0 degrees corresponds
to the perspective perpendicular to the axis of the merger.
4. 4.
4-friction.avi – Surface density of the particles originally belonging to the
secondary galaxy from two simulation of a radial minor merger from Sect. 22.1
(run 1 – right panels and run 2 – left panels). The first column corresponds
to the simulation with dynamical friction and gradual decay of the secondary;
the other corresponds to the simulation without friction and with the instant
disruption of the secondary near the center of the primary galaxy. The bottom
panels show an area of $60\times 60$ kpc centered on the primary which is the
zoomed part of the upper panels of size $300\times 300$ kpc. The video covers
8 Gyr since the release of the secondary galaxy from distance of 180 kpc from
the center of the primary with the escape velocity. Both simulations were
executed for the the standard set of parameters (Sect. 17.5): the mass of the
primary is $3.2\times 10^{11}$ M⊙ , the secondary-to-primary mass ratio is
0.02, the Plummer radius of the primary is 20 kpc, of the secondary 2 kpc.
5. 5.
5-selfconsistent.avi – Video from self-consistent simulation of a radial minor
merger from Sect. 22.3. The bottom panel ($400\times 400$ kpc centered on
primary) shows the surface density of the particles originally belonging to
the secondary galaxy from a perspective perpendicular to the axis of the
merger; the top panel shows the density of the particles originally belonging
to the secondary in the space of radial velocity (vertical axis) versus
galactocentric distance (horizontal axis). The potential of the primary galaxy
is a double Plummer sphere with respective masses $M_{*}=2\times 10^{11}$ M⊙
and $M_{\mathrm{DM}}=8\times 10^{12}$ M⊙ , and Plummer radii
$\varepsilon_{*}=8$ kpc and $\varepsilon_{\mathrm{DM}}=20$ kpc for the
luminous component and the dark halo, respectively. The potential of the
secondary galaxy is chosen to be a single Plummer sphere with the total mass
$M=2\times 10^{10}$ M⊙ and Plummer radius $\varepsilon_{*}=2$ kpc. The
cannibalized galaxy is released from the distance of 200 kpc from the center
of the host galaxy with the initial velocity 102 km$/$s.
Videos 2–4 were made from simulated data by Miroslav Křížek.
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|
arxiv-papers
| 2013-12-05T18:45:38 |
2024-09-04T02:49:55.016773
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ivana Ebrova",
"submitter": "Ivana Ebrova",
"url": "https://arxiv.org/abs/1312.1643"
}
|
1312.1798
|
Utilitarian opacity model for aggregate particles in protoplanetary nebulae
and exoplanet atmospheres
Jeffrey N. Cuzzi1,∗, Paul R. Estrada2, and Sanford S. Davis1
1Space Science Division, Ames Research Center, NASA; 2SETI Institute
∗[email protected]
Abstract
As small solid grains grow into larger ones in protoplanetary nebulae, or in
the cloudy atmospheres of exoplanets, they generally form porous aggregates
rather than solid spheres. A number of previous studies have used highly
sophisticated schemes to calculate opacity models for irregular, porous
particles with size much smaller than a wavelength. However, mere growth
itself can affect the opacity of the medium in far more significant ways than
the detailed compositional and/or structural differences between grain
constituents once aggregate particle sizes exceed the relevant wavelengths.
This physics is not new; our goal here is to provide a model that provides
physical insight and is simple to use in the increasing number of
protoplanetary nebula evolution, and exoplanet atmosphere, models appearing in
recent years, yet quantitatively captures the main radiative properties of
mixtures of particles of arbitrary size, porosity, and composition. The model
is a simple combination of effective medium theory with small-particle closed-
form expressions, combined with suitably chosen transitions to geometric
optics behavior. Calculations of wavelength-dependent emission and Rosseland
mean opacity are shown and compared with Mie theory. The model’s fidelity is
very good in all comparisons we have made, except in cases involving pure
metal particles or monochromatic opacities for solid particles with size
comparable to the wavelength.
Astrophysical Journal Supplement; submitted June 14 2013, accepted December 3,
2013
## 1 Introduction
The total extinction opacity of a medium $\kappa_{e}$ (cm2g-1) is the ratio of
its volume extinction coefficient (cm-1 of path) to its volume mass density.
Equivalently, it is the cross-section per unit mass along a path. In
protoplanetary nebulae, the relatively small fraction (by mass) of solid
particles provides the bulk of the opacity (D’Alessio et al 1999, 2001) unless
the temperature is so large (above 1500K) that common geological solids
evaporate and molecular species dominate (Nakamoto and Nakagawa 1994, Ferguson
et al 2005). In exoplanet atmospheres, which are generally much denser, the
particle contributions are more situation-dependent (Marley et al 1999,
Sudarsky et al 2000, 2003; Ackerman and Marley 2001, Tsuji 2002, Currie et al
2011, de Kok et al 2011, Madhusudhan et al 2011, Morley et al 2012, 2013;
Vasquez et al 2013).
While typical interstellar grain size distributions (Mathis et al 1977, Draine
and Lee 1984) have been assumed in many nebula and planetary atmospheric
opacity models (Pollack et al 1985, 1994; D’Alessio et al 1999, 2001;
Bodenheimer et al 2000, Dullemond et al 2002), coagulation models in the
nebula context, going back to Weidenschilling (1988, 1997), and most recently
Ormel and Okuzumi (2013), suggest that such small grains grow to 100$\mu$m
size or larger on short timescales. This is because small and/or fluffy grains
collide at very low relative velocity and stick readily. Other recent studies
of grain growth in the nebula context include Brauer et al (2008), Zsom et al
(2010), Schraepler et al (2012), and Birnstiel et al (2010, 2012) to give only
a few examples. In cloud formation models for gas giant planets (Movshovitz et
al 2010) and for brown dwarfs and exoplanets (Helling et al 2001, Kaltenegger
et al 2007, Kitzmann et al 2010, Zsom et al 2012). A review by Helling et al
(2008) shows how exoplanet and brown dwarf clouds appear at different
altitudes and with different particle sizes, formed from a variety of
constituents from volatile ices to silicate or even iron metal. In a recent
review, Marley et al (2013) illustrate how some materials condense as liquids
(forming droplets that are appropriate for the usual Mie theory models), while
other materials condense as solids, forming irregular, porous aggregates as
they grow. To our knowledge, no current exoplanet radiative transfer models
have explored the implications of porous aggregate cloud particles. In this
paper we present a simple way of accounting for radiative transfer involving
aggregates of arbitrary size and porosity. Some giant planet atmosphere models
(Podolak 2003, Movshovitz and Podolak 2008, Movshovitz et al 2010, and Helled
and Bodenheimer 2011) and one other application (Amit and Podolak 2009) have
already incorporated the opacity model we describe here. Using these opacity
models, Movshovitz et al (2010) showed that realistic grain coagulation leads
to smaller opacities, and shorter formation times, for gas giant planets than
previously appreciated in the context of the Core-instability scenario (see
section 5.4).
Miyake and Nakagawa (1993), Pollack et al (1994), and D’Alessio et al (2001)
give a few examples of the dramatic effect of particle growth on opacity
across a broad range of sizes of interest in the nebula case. Because the
aggregating constituents are not always liquid, the growing particles can have
a fairly open, highly porous, nature (Meakin and Donn 1988, Donn 1990, Dominik
and Tielens 1997, Beckwith et al 2000, Dominik et al 2007, Blum 2010). Growth
beyond mm-cm sizes in protoplanetary nebulae remains an active area of study
(see reviews by Cuzzi and Weidenschilling 2006 and Dominik et al 2007, and
models by Brauer et al 2008, Zsom et al 2010, Hughes and Armitage 2012,
Birnstiel et al 2010, 2012; and others), yet none of these latter models
actually incorporates equally realistic and self-consistent treatments of
opacity either in their radiative transfer or in predictions of Spectral
Energy Distributions, and only Pollack et al (1985, 1994) and Miyake and
Nakagawa (1993) have addressed the issue of large particle porosity in the
implications for thermal radiative transfer. In both nebula and atmospheric
cases, the vertical temperature structure is determined by the heating source
(external and/or internal) and the opacity - usually expressed as a
temperature-weighted mean such as the Rosseland mean opacity $\kappa_{R}$
(Pollack et al 1985, 1994; Henning and Stognienko 1996, Semenov et al 2003);
the Planck mean can also be used for optically thin applications and is even
simpler to calculate (Nakamoto and Nakagawa 1994).
Interpretation of Spectral Energy Distributions, or intensity as a function of
wavelength, requires an understanding of the wavelength-dependent opacity as
well as the detailed vertical temperature distribution of the nebula or
planet’s atmosphere. Traditionally, regions sampled are thought to be
optically thin, and particles poor scatterers being much smaller than the
wavelength, so only the absorption part of the extinction opacity is needed
(Miyake and Nakagawa 1993). Nebula gas masses have often been inferred by
combining the mass of particulates (inferred from the observed emission and
theoretical opacity) with the canonical mass ratio of roughly one percent for
solids to total mass (eg. Andrews and Williams 2007, Williams and Cieza 2011).
Theoretical analyses of observed mm-cm wavelength opacities at least since
D’Alessio et al (2001) (eg. review by Natta et al 2007; also Isella et al
2009, Birnstiel et al 2010, Ricci et al 2010, and others) use full Mie
theoretical analyses of particle opacities, which we show in section 5.3 is,
in fact, essential when the particle size and wavelength are comparable. de
Kok et al (2011) modeled emission spectra of exoplanets, where the primary
spectral variation is that of the gas, but where the more gradually varying
extinction properties of cloud particles still provide important variations
across the near-IR passband because the thermal wavelengths are not extremely
large compared to the particle size.
Overview: Below we present a very simple, but general and flexible, model
which captures the essential physics of opacity in ensembles of aggregate
particles of arbitrary size and porosity, and can be easily made part of
nebula or planetary atmosphere thermal structure and/or evolution models. We
give some examples of how growth and porosity affect the opacity of realistic
aggregate particles. The opacity model is simple to combine with any grain
coagulation/aggregation model, and can be used to study the temporal/spatial
dependence of the SED and Rosseland mean opacity $\kappa_{R}$ on varying
particle size, density, porosity, and composition, or to calculate thermal
equilibrium profiles. The model is not ideally suited to interpreting mm-cm
wavelength spectral slopes in protoplanetary disks, where the dominant size is
often found to be comparable to the wavelength.
## 2 Prior Work and overview of paper
Draine and Lee (1984) studied small, independent particles ($r\ll\lambda$,
where $r$ is particle radius and $\lambda$ is wavelength) in the dipole
approximation; Pollack et al (1985) modeled various size distributions of
mineralogically realistic, spherical, solid grains using exhaustively compiled
refractive indices and Mie scattering; Wright (1987) modeled grains of various
fractal dimension (basically, porosity) using the Discrete Dipole
Approximation or DDA (Purcell and Pennypacker 1973, Draine and Goodman 1993).
Wright (1987) showed that nonsphericity could cause a significant enhancement
of opacity, if grains had fairly large refractive indices. More recent work
has elaborated upon this (Fabian et al 2001, Min et al 2008, Lindsay et al
2013), showing how this caveat applies to detailed spectral analysis of highly
nonspherical particles (elongated, flattened, or even having sharp edges)
through strong absorption bands. Bohren and Huffman (1983, p. 140) also give
constraints on how particle size and refractive index must both be considered,
in cases where the refractive index is large. Our model is most suitable for
roughly equidimensional particles with moderate refractive indices (high
porosity and realistic compositional mixtures usually preclude high average
indices).
Ossenkopf (1991) studied the effect of metallic inclusions on equidimensional
porous particles, emphasizing small particles and the role of randomly
connected iron grains in leading to effective dipole-like structures. For the
low volume densities of iron grains in protoplanetary nebula aggregates, at
least for the abundances assumed here, the effect is small. Pollack et al
(1985, 1994) assumed materials which are cosmically abundant (with an eye
towards molecular cloud and protoplanetary nebula applications); of these,
water ice, silicates, and refractory organics have refractive indices that are
too small for nonsphericity effects to be especially important in the regime
where particle size is smaller than a wavelength. Other common materials,
specifically iron metal and iron sulfide (Troilite, FeS) do have refractive
indices that are high enough to show an effect, but nonsphericity was not
treated by Pollack et al (1985, 1994).
Regarding distributions of larger particles, Pollack et al (1985, 1994) and
Miyake and Nakagawa (1993) calculated both monochromatic opacities and
Rosseland Mean opacities, and D’Alessio et al (2001) calculated monochromatic
opacities and employed them in a wide range of nebula structure models.
Pollack et al (1985) showed selected results in the separate limits of large
and small particles, but did not connect them. Mizuno et al (1988) used a
combination of Rayleigh scattering for small particles and diffraction-
augmented geometric optics ($Q_{e}=2$) for large ones. Miyake and Nakagawa
(1993) used a combination of full Mie calculations and geometrical optics,
along with an ad hoc powerlaw particle size distribution, to demonstrate the
effects of particle growth. To match confidently with the geometric optics
limit, they carried out Mie calculations to impressively large values of
$r/\lambda\sim 10^{5}$, with implications for array size and computational
time that would be hard for the typical user to achieve even today, in
multidimensional and/or evolutionary nebula or exoplanet atmosphere models.
D’Alessio et al (2001) also seem to have used full Mie calculations for all
their particles, up to 10cm radius, treating each material as a separate
species. Pollack et al. (1985), Mizuno et al (1988), Miyake and Nakagawa
(1993), and D’Alessio et al (2001) all assumed solid objects; Pollack et al
(1994) looked briefly at particles of higher porosity, using an Effective
Medium Theory (EMT; see section 4) and Mie calculations. Rannou et al (1999)
used a semi-empirical model of their own design to capture the properties of
aggregates in certain size and compositional regimes.
In this paper, we build on and generalize this prior work by showing how
arbitrary size, porosity, and compositional distributions can be handled
easily by users wanting to explore their own choices for grain properties, and
simply enough to incorporate within evolutionary models or in iterative
analysis of observed Spectral Energy Distributions for applications where
grain growth into sizeable aggregates has occurred. We show how porosity
trades off with size in determining emergent Spectral Energy Distributions and
determining Rosseland mean opacities. We use accurate material properties for
a realistic, temperature-dependent, compositional suite, and simplified but
realistic scattering and absorption efficiencies which avoid the need for
numerical Mie calculations. In this sense the approach is similar in spirit to
some previous studies, but emphasizes a utilitarian approach and ease of
general applicability. The physics and simplifications are described in
section 3 below; in section 4 we show some validation tests of the model, and
in section 5 we describe the behavior of porous aggregates. Some basic
derivations are presented in Appendices. The code itself is quite simple,
taking negligible cpu time compared to a Mie code, and a version is available
online.
## 3 Opacity model
In this section we describe the theoretical basis for our calculations of
particulate opacity. Our goal is to capture all of the significant physics in
the simplest fashion possible, so the model can be incorporated into
evolutionary models at little computational cost. Realistic material
refractive indices are included for a cosmic abundance suite of likely nebula
solids: water ice, silicates, refractory organics, iron sulfide, and metallic
iron. These refractive indices, and relative abundances as a function of
temperature are taken from Pollack et al (1994; see Table). We chose these
specific values to allow better validation and comparison with previous work,
but they are widely used; alternate tabulations are easily incorporated, such
as found in Draine and Lee (1984) or Henning et al (1999).
material | density | $\alpha_{j}$ | $T_{evap}$(K) | $\beta_{j}$ $<$160K | $\beta_{j}$ $<$425K | $\beta_{j}$ $<$680K | $\beta_{j}$ $<$1500K
---|---|---|---|---|---|---|---
water ice | 0.9 | 5.55e-3 | 160 | 6.11e-1 | 0 | 0 | 0
organics | 1.5 | 4.13e-3 | 425 | 2.73e-1 | 7.00e-1 | 0 | 0
troilite | 4.8 | 7.68e-4 | 680 | 1.58e-2 | 4.07e-2 | 1.36e-1 | 0
silicates | 3.4 | 3.35e-3 | 1500 | 9.93e-2 | 2.55e-1 | 8.51e-1 | 9.84e-1
iron ($<$680K) | 7.8 | 1.26e-4 | 1500 | 1.60e-3 | 4.11e-3 | 1.37e-2 | -
iron ($>$680K) | 7.8 | 6.15e-4 | 1500 | - | - | 0 | 1.58e-2
Table 1: Our assumed compositional mixture, adapted from Pollack et al (1994);
for simplicity we have merged their two kinds of silicates and their two kinds
of organics. When Troilite (FeS) decomposes at 680K, we follow Pollack et al
in assuming the liberated iron adds to the existing iron metal and the S
remains in the gas phase. The parameters $\alpha_{j}$ and $\beta_{j}$ define
the fractional mass of a compositional species, and the fractional number of
its particles if particles are segregated compositionally. For their
definitions see section 4 and equations 48 (or 19) respectively.
Our emphasis is on how size and porosity effects can be dealt with simply and
seamlessly as grains aggregate and grow into a size regime where details of
composition and refractive indices are perhaps secondary. Our basic approach
will be to separate particles having an arbitrary size distribution into two
regimes which both have closed-form solutions in practical cases. We use the
fact that particle interactions with radiation of any wavelength $\lambda$ can
be systematically addressed in terms of the optical size of the particle
$x=2\pi r/\lambda$, where $r$ is the particle radius. The model presented here
treats “small” particles ($x\ll 1$) as volume absorbers/scatterers and “large”
particles ($x\gg 1$) as geometrical optics absorbers/scatterers. We will
occasionally refer to Van de Hulst (1957; henceforth VDH), Hansen and Travis
(1974; henceforth HT), and Bohren and Huffman (1983).
### 3.1 Extinction, absorption, and scattering efficiencies
Radiation is removed from any beam at a rate:
$I=I_{0}e^{-\kappa_{e}\rho l}$ (1)
where $\rho$ is the volume mass density of the gas and dust mixture,
$\kappa_{e}$ (cm2 g-1) is the total opacity, and $l$ is the path length. The
monochromatic opacity, for a gas-particle mixture containing particles of
radius $r$ with number density $n(r)$, is formally defined as:
$\kappa_{e,\lambda}=\frac{1}{\rho_{g}}\int\pi r^{2}n(r)Q_{e}(r,\lambda)dr,$
(2)
where $Q_{e}$ is the extinction efficiency for a particle of radius $r$, at
wavelength $\lambda$. The Rosseland mean opacity, which controls the flow of
energy through an optically thick medium, is derived by an appropriate
weighting of $\kappa_{e,\lambda}$ over wavelength (see Appendix A). Extinction
is due to a combination of pure absorption (reradiated as heat) and scattering
(redirection of the incident beam). Rigorously, these components are additive;
that is $Q_{e}=Q_{a}+Q_{s}$, where $Q_{a}$ is the absorption efficiency and
$Q_{s}$ the scattering efficiency. The single-scattering albedo of the
particle is $\varpi=Q_{s}/Q_{e}$. To the extent that the particles do scatter
radiation without absorbing it ($Q_{s}\neq 0$), the angular distribution of
the scattered component (the phase function $P(\Theta)$) is relevant. The
first moment of the phase function $g$ describes the degree of forward
scattering:
$g=\left<{\rm cos}\Theta\right>={\int P(\Theta){\rm cos}\Theta{\rm sin}\Theta
d\Theta\over\int P(\Theta){\rm sin}\Theta d\Theta}$ (3)
Isotropic scattering results in $g=0$; very small particles (Rayleigh
scatterers) have a phase function proportional to cos${}^{2}\Theta$, which
also leads to $g=0$. This is why scattering by tiny particles is often
approximated as isotropic. Large particles have a strong forward scattering
lobe due to diffraction; for such particles $g$ may approach unity (cf. HT)
and can even be approximated as unscattered (Irvine 1975). In may cases of
thermal emission when $r\ll\lambda$ (section 5.3), only the
absorption/emission component is important:
$\kappa_{a,\lambda}=\frac{1}{\rho_{g}}\int\pi r^{2}n(r)Q_{a}(r,\lambda)dr,$
(4)
Analytical expressions for $Q_{a}$ and $Q_{s}$ have been known for over a
century in certain asymptotic regimes; the well-known Rayleigh scattering
regime for particles much smaller than the wavelength is one example.
Similarly, scattering properties of very large objects have long been well
understood in terms of Lambertian and related scattering laws. Considerable
amounts of computation have been devoted in recent years to determination of
efficiencies and phase functions in the intermediate Mie scattering regime,
where particle size is comparable to the wavelength.
One important point for our purposes, that is demonstrated by HT, is that many
of the exotic fluctuations in scattering and absorption properties which
characterize the Mie regime vanish if one averages over a broad distribution
of particle sizes. Further smoothing occurs given a combination of realistic
particle nonsphericity and random particle orientations. Furthermore, a large
number of experimental studies (see also Bohren and Huffman 1983 and Pollack
and Cuzzi 1980 for references) have shown that integrated parameters such as
$Q$ and $g$ are much less affected by shape effects than, for example, the
phase function itself, and the parameters of interest for thermal emission and
absorption ($Q_{a},Q_{s}$, and $g$) vary smoothly between the “Rayleigh” and
“geometric optics” regimes. This observed behavior provides a certain comfort
level for the simple assumptions we make here. Our model does not treat
$P(\Theta)$ at all, but constrains $g$ directly based on a fairly well-behaved
dependence on the size and composition of the scatterer (section 3.2). We use
$g$ to correct $Q_{a}$ and $Q_{s}$ for energy which is primarily scattered
forward, and thus does not participate significantly in directional
redistribution of energy. Thus, our model would not be appropriate for
applications where scattering dominates absorption and the directional
distribution of scattered energy is of primary interest. Fortunately,
primitive aggregate particles made of ice, silicates, carbon-rich organics and
tiny metal grains are good absorbers and poor scatterers across most size
ranges, but $Q_{s}$ does contribute to $\kappa_{\lambda}$ in some regimes.
### 3.2 The model: Calculation of efficiencies
A good exposition of the form of the particle absorption and scattering
coefficients is presented by Draine and Lee (1984; henceforth DL); our
expressions may be derived directly from those published by DL, and we will
not repeat much of their presentation. DL describe the radiative properties in
terms of absorption and scattering cross sections $C_{a,s}=Q_{a,s}\pi r^{2}$,
which are directly related to particle volume in the limit $r\ll\lambda$. The
efficiencies may then be expressed in terms of the electric polarizability
(VDH p. 73), which is a function of the (complex) particle dielectric constant
$\epsilon=\epsilon_{1}+i\epsilon_{2}$ (DL eqn 3.11). The particle refractive
index $m=n_{r}+in_{i}$ is the square root of the dielectric constant. In
general, $Q_{a}$ and $Q_{s}$ depend on the particle shape and orientation as
well as the material refractive indices. The “small dipole” limit, in which
the particles are both highly nonspherical and have extremely large refractive
indices, has been discussed by DL, Wright (1987), Fabian et al (2001), and Min
et al (2008). Lindsay et al (2013) show that even the shape of tiny solid
grains can be diagnostic in high spectral resolution observations of strong
absorption bands (section 2). After a small amount of algebra, DL equations
3.3, 3.6, and 3.11 lead directly to values of $Q_{ak}=C_{ak}/\pi r^{2}$ and
$Q_{sk}=C_{sk}/\pi r^{2}$, where subscript $k$ refers to alternate
orientations of a nonspherical particle relative to the wave vector:
$Q_{ak}={2\over r^{2}\lambda}{V\over
L_{k}^{2}}{\epsilon_{2}\over(L_{k}^{-1}+\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}}=\frac{4}{3L_{k}^{2}}\left(\frac{2\pi
r}{\lambda}\right){\epsilon_{2}\over(L_{k}^{-1}+\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}}$
(5)
and
$Q_{sk}={8\over 3r^{2}}\left({2\pi\over\lambda}\right)^{4}\left({V\over 4\pi
L_{k}}\right)^{2}{(\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}\over(L_{k}^{-1}+\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}}=\frac{8}{27L_{k}^{2}}\left(\frac{2\pi
r}{\lambda}\right)^{4}{(\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}\over(L_{k}^{-1}+\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}},$
(6)
where $V$ is the particle volume and
$\epsilon_{1}=n_{r}^{2}-n_{i}^{2};\hskip 36.135pt\epsilon_{2}=2n_{r}n_{i}.$
(7)
Only the expression for $Q_{a}$ is explicitly given in DL (compare our
equation 5 with their equation 3.12). For the case at hand, however, once any
significant amount of particle accumulation occurs, aggregates are likely to
be roughly equidimensional and, especially if the particles are porous, the
average refractive indices are not large (see below), so we will treat $Q_{a}$
and $Q_{s}$ as independent of particle orientation and, setting $L_{k}=1/3$,
obtain
$Q_{a}=12\left(\frac{2\pi
r}{\lambda}\right){\epsilon_{2}\over(\epsilon_{1}+2)^{2}+\epsilon_{2}^{2}}$
(8)
and
$Q_{s}=\frac{8}{3}\left(\frac{2\pi
r}{\lambda}\right)^{4}{(\epsilon_{1}-1)^{2}+\epsilon_{2}^{2}\over(\epsilon_{1}+2)^{2}+\epsilon_{2}^{2}},$
(9)
(VDH pp. 71, DL84; see also equation (14) of Miyake and Nakagawa 1993).
Neither Miyake and Nakagawa (1993) or Pollack et al (1994) discuss $Q_{s}$,
but it becomes important for porous, weakly absorbing particles in the
transition regimes we wish to bridge with our model, where wavelength and
particle size are not extremely different. In the geometrical optics regime,
nonspherical shapes can increase the surface area per unit mass, and thus the
extinction efficiency, but for moderate nonsphericity the effect is only some
tens of percent (Pollack and Cuzzi 1980). This effect is also neglected for
the present. For plausible aggregates, magnetic effects are not important (see
however Appendix B).
In the limit where $n_{i}\ll n_{r}-1$, which is reasonable for ices,
silicates, organics, and their ensemble aggregates, equations 8 and 9 reduce
to the simpler and more familiar forms:
$Q_{a}={24xn_{r}n_{i}\over(n_{r}^{2}+2)^{2}}$ (10)
and
$Q_{s}={8x^{4}\over 3}{(n_{r}^{2}-1)^{2}\over(n_{r}^{2}+2)^{2}},$ (11)
where $x=2\pi r/\lambda$. These are essentially the classical expressions
given by VDH (p. 70).
In our code we actually use the full equations 8 and 9 above for $Q_{a}$ and
$Q_{s}$ for the radiative behavior of particles much smaller than the
geometric optics regime; these equations are valid for arbitrary $m$ (DL).
Equation 8 for $Q_{a}$ (linear in particle radius) is used for all sizes.
However, in the Mie transition region (in which the particle size is
comparable to a wavelength) we modeled $Q_{s}$ using a function valid for
$n_{r}$ of order unity (the Rayleigh-Gans regime; VDH p132-133, p.182). That
is, at a transition value $x_{0}$ we change from the DL expression (equation
9) to the Rayleigh-Gans regime expression (VDH p. 182 and final equation of
section 11.23):
$Q_{s}=\frac{1}{2}(2x)^{2}(n_{r}-1)^{2}\left(1+\left({n_{i}\over
n_{r}-1}\right)^{2}\right),$ (12)
The transition value $x_{0}=1.3$ comes from setting the two expressions equal
to each other. This transition from $x^{4}$ to $x^{2}$ dependence bridges the
region between the Rayleigh and geometric optics regimes. It is shown in VDH
(p 177) that a coupled dependence on the so-called “phase shift” $2x(n_{r}-1)$
(for $n_{i}<<n_{r}-1$), which includes both $n$ and $r$, captures the peak and
shape of $Q_{s}$. These expressions are valid as long as $n_{r}-1$ is not much
larger than $1/x$ (Bohren and Huffman 1983, p.140); this is the expected
regime for aggregates and mixtures of plausible materials. Our expressions do
not capture the Fresnel-like oscillatory behavior of $Q$ shown by the the more
complete series expansions in VDH and the full Mie theory, but do capture the
correct small- and large-$x$ behavior, and the proper transition
size/wavelength. This is an increasingly good approach as the particle size
distribution gets broader and the particles become more absorbing (HT), in
which case the resonance behavior near $2x(n_{r}-1)\sim$ several diminishes
and indeed vanishes (see, eg., Cuzzi and Pollack 1978). Aggregate particles
considered here are very good absorbers in general, having very broad size
distributions, but we show below that the technique works surprisingly well in
even more challenging regimes.
A key simplifying assumption of our model is that, while $Q_{s}$ and $Q_{e}$
grow from small values as $x$ increases, following equations (8) and (9 or
12), they are truncated at values representing the geometrical optics limit,
as discussed below. See for instance figure 1 left, where in the geometric
optics limit $2\pi r/\lambda\rightarrow\infty$, $Q_{e}=Q_{s}\rightarrow 2$.
The figure represents lossless particles; if absorption is present, $Q_{s}$
(which includes diffraction) trends downwards and asymptotes to a value closer
to unity (HT figure 9). Note that the exotic ripples are primarily seen for
monodispersions and narrow size distributions, and vanish as broader size
distributions are used.
Figure 1: Left: Scattering efficiency $Q_{s}$ of a particle with refractive
indices $n_{r}=1.33,\,n_{i}=0$ as a function of its optical size $x=2\pi
a/\lambda$ using Mie Theory (figure reproduced directly from Hansen and Travis
1974, by permission, where particle radius = $a$). The parameter $b$ varies
the width of the size distribution; as the size distribution gets broad,
exotic ripples dues to interference average out and vanish. Right: dependence
on the optical size and material properties of the scattering asymmetry
parameter $g=\left<{\rm cos}\Theta\right>$, for narrow size distributions of
different optical size. Note there are essentially two well-defined regimes in
$n_{i}$. Figure 2 gives more detail on how $g$ varies with $x$.
Although the full radiative transfer problem must be solved when the angular
variation of the intensity is of primary concern, so that the phase function
$P(\Theta)$ is important, certain valuable simplifications are common when
only angle-integrated quantities such as energy or flux are of concern. In
such cases, one often merely corrects the extinction efficiency for the
overall degree of forward- or back-scattering. We will adopt a common scaling
(e.g., Van de Hulst 1980) sometimes referred to as the radiation pressure
scaling, and adjust our efficiencies to take account of nonisotropic
scattering as follows:
$Q^{\prime}_{e}=Q_{e}-gQ_{s}=Q_{e}(1-\varpi_{0}g)=Q_{a}+Q_{s}(1-g).$ (13)
That is, to the degree that scattered radiation is concentrated into the
forward direction, it may be regarded as unremoved from the beam. For
instance, even a large scattering efficiency contributes nothing to the
extinction if the scattering is purely forward directed ($g=1$). Negative
values of $g$ (preferential backscattering) increase the extinction
efficiency; in this case, it is even harder for radiation to escape than for
thermalized or isotropically scattered radiation. This same correction is
applied by Pollack et al (1985, 1994); see section 5.4 for more discussion.
Modeling efforts relying to any degree on the Eddington approximation, in
which scattered radiation is nearly isotropic, are perhaps better conducted
using extinction efficiencies that are scaled in the way shown above, instead
of using the formal Mie calculations (see Irvine 1975). For instance, de Kok
et al (2011) show how forward scattering by plausible exoplanet cloud
particles can influence emergent near infrared fluxes; the above treatment of
forward scattering partially accounts for this behavior by truncating away
strong forward scattering and allowing the non-truncated scattered component
to be treated as isotropic. In the approach presented here, the normal optical
depth is defined using a value of $\kappa_{\lambda}$ that has been adjusted
for forward scattering effects following equations 2 and 13 (and 17 if
appropriate):
$\tau_{\lambda}(z)=\int_{z}^{\infty}\kappa_{\lambda}(z)\rho_{g}(z)dz.$ (14)
Instead of performing full Mie calculations to obtain the scattering asymmetry
parameter $g=\left<{\rm cos}\Theta\right>$, we make use of the crudely
partitioned behavior illustrated by HT (reproduced in figure 1 (right)). We
have explored several simple ways of doing this, and have chosen the following
determination of $g$, consistent with figures 1 and 2 ( notice that figure
1(right) is for a fairly low $n_{r}=1.33$):
${\rm for}\,\,n_{i}<1:g=0.7(x/3)^{2}\hskip 7.22743pt{\rm if}\hskip
7.22743ptx<3,\,\,{\rm and}\,\,g\approx 0.7\hskip 7.22743pt{\rm if}\hskip
7.22743ptx>3,$ (15) ${\rm for}\,\,n_{i}>1:g\approx-0.2\hskip 7.22743pt{\rm
if}\hskip 7.22743ptx<3,\,\,{\rm and}\,\,g\approx 0.5\hskip 7.22743pt{\rm
if}\hskip 7.22743ptx>3.$ (16)
The value of the $g$-asymptote at large $x$, and the transition value of $x$,
vary slightly with $n_{r}$ (figure 2); the values we selected apply to the EMT
values of $n_{r}\sim 1.7-2.1$ and $n_{i}\ll n_{r}$, most relevant to our
mixed-aggregate applications. We have not attempted to fine-tune either the
asymptote or the transition value, and leave the large-$n_{i}$ definition of
$g$ in its crude bimodal form (the large-$n_{i}$ recipe has almost no effect
on the results). Further fine-tuning is somewhat beyond what we expect of a
simple utilitarian model, but could be done easily.
Figure 2: Variation of $g=\left<{\rm cos}\Theta\right>$ ($\alpha=\Theta$ and
$a=r$ in the notation of HT), and $x=2\pi r/\lambda$, for a range of $n_{r}$.
It is easy to show that the dependence in the steep transition region is
$g\propto x^{2}$, while the large-$x$ asymptote and transition point depend on
$n_{r}$. With an eye towards the Garnett EMT value of $n_{r}$ for our cosmic
mixture (1.7-2.1), we have chosen the single function $g=0.7(x/3)^{2}$ to
represent all particles and materials. Figure reproduced from HT figure 12, by
permission.
The final piece of the model involves matching the growing values of $Q_{s}$
and $Q_{a}$ in the Rayleigh and/or Rayleigh-Gans regime to constant values in
the geometric optics regime, simply by limiting their magnitudes:
$Q_{a}<1\hskip 14.45377pt{\rm and}\hskip 14.45377ptQ_{s}(1-g)<1.$ (17)
This can be thought of as limiting the absorption cross section, and the
diffraction-corrected scattering cross section, to the physical cross section.
D’Alessio et al (2001) did not adjust their $Q$, albedo, or $\kappa$ values
for $g$ as do we and Pollack et al (1994), instead taking them straight from
their Mie code. If one then carries on to solve the radiative transfer
equations in an Eddington-like approximation (that is, as in their equation
(3), assuming isotropic scattering, or otherwise without detailed treatment of
the angular dependence of often strongly forward-scattered radiation), indeed
it would be better to first perform the truncation following equation 13. A
complete multidimensional treatment can capture all this behavior of course.
D’Alessio et al (2006) compared forward scattering with isotropic scattering,
showing very little difference (their figure 7)111Some of the curves in the
lower left panel of Dalessio et al 2006 seem to be mislabeled, but the trends
are fairly clear., but at their longer wavelengths $r\ll\lambda$ so scattering
is negligible in the first place, and at shorter wavelengths the emission
arises from the photosphere of an opaque disk. A detailed discussion of this
issue is beyond the scope of this paper.
One final detail must be mentioned, relevant for applications where particles
of pure metal are important. Equations (8) and (9 or 12) include only the
electric dipole interaction terms that are appropriate for everything but
metals, and thus for all of our aggregate particle models where refractive
indices are not extremely large. However, when we compare our model with the
heterogeneous-composition grain mixture of Pollack et al (1994), in which pure
iron metal grains play a role, we have used a crude correction to $Q_{a}$ for
magnetic dipole terms as well, following DL84, Pollack et al (1994), and
others (see Appendix B); these calculations are all compared with the full Mie
theory below.
Figure 3: Real (left) and imaginary (right) indices as used in our code,
representing a number of common materials, taken directly from Pollack et al
(1994). Black long dash: water ice; Cyan dash-dot: silicates; Green short
dash: Iron Sulfide; Blue dotted: Iron metal; Red long dash-dot: organics. For
simplicity we have used only one of Pollack et al’s two varieties of silicates
(pyroxene) as they both have similar properties, and only one variety of
organics (the two of Pollack et al (1994) had the same refractive indices but
two different evaporation temperatures were assumed).
### 3.3 Material Properties
For ease of comparison with previously published results, we adopt refractive
indices, material abundances, and stability regimes for the condensible
constituents as published by Pollack et al (1994). We have merged the two
different silicates (olivine and pyroxene) and the two different organics
(higher and lower volatility) of Pollack et al (1994) into a single population
of each for simplicity. The exact composition of protoplanetary nebulae is,
after all, poorly known, but these are consistent with known properties of
grains in primitive meteorites and comets and cover an appropriate range of
evaporation temperature. The components, their assumed relative abundances,
and their evaporation temperatures are shown in the Table. We will show that
particle growth leads to at best a weak dependence on detailed stipulations of
composition. It would be simple for anyone to select an alternate mixture.
Our refractive indices are shown in figure 3. The materials can be classified
into two groups - the water ice, silicate, and organic material have real
refractive index $n_{r}$ on the order of unity, and imaginary refractive index
$n_{i}$ much less than unity. On the other hand, metallic iron and iron
sulfide particles have both refractive indices larger (often much larger) than
unity.
## 4 Mixtures of grains with pure composition
In our primary intended application to protoplanetary nebulae, particles are
porous composites or granular mixtures of much smaller grains of all
compositions that are solid at local temperature $T$, and we will calculate
their ensemble refractive indices $(n_{r},n_{i})$ using the Garnett Effective
Medium Theory (EMT) as described in Appendix C and section 5.1 below. However,
in some applications, such as exoplanet atmospheres, grains of a particular
composition - even perhaps grains of iron metal - might be isolated within
their own cloud layer.
To assess our approach in this regime, we first calculate opacities for a
compositionally heterogeneous ensemble of micron-size grains adopted by
Pollack et al (1985, 1994) in their exact Mie scattering calculations. That
is, distinct populations of non-porous grains with distinct compositions and
refractive indices are assumed, and the opacities so determined are averaged
as described below. This is a more challenging regime than a situation in
which aggregates are of mixed composition and thus do not have extreme values
of refractive index (section 5.1).
For particles of species $j$, the monochromatic opacity at wavelength
$\lambda$ (relative to the gas mass density $\rho_{g}$) is:
$\kappa_{e,j,\lambda}={\beta_{j}\over\rho_{g}}\int n(r)\pi
r^{2}Q^{\prime}_{e,j}(r,\lambda)dr$ (18)
where $n(r)=\sum n_{j}(r)$ and $n_{j}(r)=\beta_{j}n(r)$; $\beta_{j}$ is the
fraction by number of particles of species $j$; that is, as assumed by Pollack
et al (1994) for this particular case, all species are assumed to have the
same functional form for their size distribution. The Pollack size
distribution used is a differential powerlaw for the particle number volume
density in some radius bin $dn(r,r+dr)=n(r)dr$ where $n(r)=n_{o}r^{-p}$,
$p=-3.5$ for 0.05$\mu$m$<r<$1$\mu$m, $p=-5.5$ for 1$\mu$m$<r<$5$\mu$m, and
zero otherwise. Note that this size distribution is actually quite narrow: it
covers roughly one decade in radius nominally (the piece from 1-5$\mu$m
contains very little mass or area), and is sufficiently steep for most of the
surface area to be provided by the smallest members. In Appendix C we show
that
$\beta_{j}={\alpha_{j}/\rho_{j}\over\sum(\alpha_{j}/\rho_{j})}.$ (19)
where $\alpha_{j}$ is the fractional mass of constituent $j$ relative to the
gas, and $\rho_{j}$ is the density of solid constituent $j$. Note that
$\beta_{j}$ is the fractional number of particles of species $j$, and is
subtly different in general from the fractional volume weighting factor
$f_{j}$ that is used when compositions are mixed together in the same particle
as in our EMT approach (for compact or non-porous particles they are the
same). The above definition of $\beta_{j}$ differs slightly from that given in
Pollack et al (1985); their equations (2)-(4) capture the right essence of the
solution, but are not rigorously correct as written. By stepping back and
incorporating the $Q$ factors (including the correction for scattering
asymmetry) inside the integrals of their equation (4), the same result can be
obtained:
$\kappa_{e,\lambda}=\frac{1}{\rho_{g}}\sum_{j}\beta_{j}\int n(r)\pi
r^{2}Q^{\prime}_{e,j}(r,\lambda)dr={1\over\rho_{g}}\int n(r)\pi
r^{2}(\sum\beta_{j}Q^{\prime}_{e,j}(r,\lambda))dr,$ (20)
where $Q^{\prime}_{e,j}$ includes the forward scattering correction (equation
13). This is probably the most challenging test we could pose - an ensemble of
heterogeneous particles, including some with very large refractive indices.
Some are good absorbers and some are very good and rather anisotropic
scatterers.
Figure 4: A comparison with Mie theory of our mean extinction efficiency
$Q^{\prime}_{ej}(\lambda)$ for five pure materials, averaged and weighted over
the Pollack et al (1994) size distribution (see section 4.1). Our model
results are shown in solid curves and Mie calculations are shown in dashed
curves. The lower right panel is the abundance-weighted average value
$Q^{\prime}_{e}(\lambda)$. The solid curve in the iron panel includes an
(imperfect) correction for magnetic dipole effects (Appendix B), and the
dotted ($Q_{a}$) and dot-dash ($Q_{s}$) curves do not. Actually, all the solid
curves incorporate this correction term, but it makes a difference only for
iron.
### 4.1 Wavelength-dependent opacities
The fundamental calculation is of the effective, forward-scattering-corrected
(or “radiation”) extinction efficiency for each compositional type
$Q^{\prime}_{ej}(r,\lambda)$ (equation 13, using equations 8 and 9 or 12). If
the situation calls for a heterogeneous mixture of grains with distinct
compositions, $Q^{\prime}_{ej}(r,\lambda)$ must be calculated separately for
each composition.
We have compared our model calculations directly with Mie calculations for
five separate compositions, in each case averaged over the Pollack et al
(1994) particle size distribution (which is fairly narrow) as weighted by
particle number density and geometric area. That is:
$Q^{\prime}_{ej}(\lambda)={\int n(r)\pi
r^{2}Q^{\prime}_{ej}(r,\lambda)dr\over\int n(r)\pi r^{2}dr}.$ (21)
These results are shown in figure 4. The model does surprisingly well overall,
except for metallic particles (FeS has a near-metallic behavior at short
wavelengths). Note the different model curves in the iron metal panel; the
solid curve attempts to correct for the effects of magnetic dipole
interactions with the DL84 approximation also used by Pollack et al (1994)
(section 3.2). This term is only the leading term in an expansion, here used
outside its formal realm of validity (see Appendix B). We would not advocate
use of our simple model in cases where pure metal particle clouds might be
encountered, such as in some exoplanet atmospheres (Lodders and Fegley 2002,
Marley et al 2013). However, it does improve agreement (especially at long
wavelengths) with the full Mie calculations of similarly size-and-abundance
averaged values of both $Q^{\prime}_{e}(\lambda)$ (bottom right panel of
figure 4) and $\kappa_{R}$ (figure 5 below), for the Pollack et al (1994)
heterogeneous grain mixtures. In fact, all panels in figure 4 include
calculations done with and without magnetic dipole terms, but the differences
are insignificant for all materials except iron metal (there is a small effect
for FeS), and in the averaged values (lower right panel). These individual
compositional efficiencies and/or cross-sections are then combined into an
overall opacity (equation 20), or simply an average efficiency, as weighted by
their abundances:
$Q^{\prime}_{e}(\lambda)=\sum\beta_{j}Q^{\prime}_{ej}(\lambda).$ (22)
A plot of $Q^{\prime}_{e}(\lambda)$ is shown in the bottom right panel of
figure 4, also compared with the similarly summed Mie-derived values.
The more plausible case for protoplanetary nebulae, where all the particles
are single (mixed) composition aggregates, with refractive index given by the
Garnett EMT (see Appendix C and section 5 below), is even simpler. A single
efficiency $Q^{\prime}_{e}(r,\lambda)$, associated with the EMT refractive
indices, is simply integrated over the size distribution to give some
wavelength-dependent opacity $Q_{e}(\lambda)$. Whether determined from a
compositionally heterogeneous or homogeneous mixture,
$Q^{\prime}_{e}(\lambda)$ is the basis of extinction calculations such as
Rosseland opacities (equations 1 and 2).
Figure 5: A comparison between the Rosseland mean opacities as determined
here, with much more elaborate full Mie scattering calculations (see text for
discussion). The Pollack model has silicates, organics, and water ice
evaporating at slightly different temperatures than assumed here (in fact they
are a function of pressure); these are easily adjusted if desired.
### 4.2 Rosseland Mean opacities
We initially apply our models to calculate Rosseland mean opacities
$\kappa_{R}$ (Appendix A) for the standard Pollack et al (1994) heterogeneous
grain size distribution. Figure 5 shows that our results agree quite well with
the exact Mie scattering calculations of Pollack et al (1994), especially
considering the simplicity of our approach. The Pollack et al size
distribution is an ISM distribution, with a steep powerlaw $n(r)$ between
radii of .005 - 1.0$\mu$m and a steeper powerlaw from 1.0-5.0$\mu$m (section 4
above). Pollack et al (1994) introduce two types of moderately refractory
organics (with the same refractive indices but different abundances and
evaporation temperatures) to provide greatly increased opacity between 160K
and 425K, which we combine into a single component; thus we lack one
evaporation boundary at about 260K; a second difference is that we have only
used a single silicate component. In view of these differences in detail, and
for a situation like this (five distinct species including iron metal, with
grain sizes in the resonance regime at the shorter wavelengths at high
temperatures) which is far more challenging than our primary intended
application (well-mixed aggregates of moderate refractive index, many in the
geometrical optics limit), the agreement is surprisingly good.
## 5 Aggregate particles, each of mixed composition
In the remainder of the paper, we explore the radiative properties of more
mature and (at least for protoplanetary nebula applications) probably more
realistic particle porosity and size distributions between a micron and
arbitrarily large sizes. Evidence from meteorite and interplanetary dust
particle samples indicates that accumulated particles of these larger sizes
are heterogeneous aggregates of all candidate materials, with individual
submicron-to-micron size “elements” being composed of one mineral or other.
Even mm-size chondrules are each a mixture of 1-10$\mu$m size mineral grains
of silicate, iron, and iron sulfide, and the ubiquitous meteorite matrix is a
mix of smaller grains of all these materials, perhaps originally aggregates.
So, for our aggregates, we assume only one (mixed) grain composition; the
assumption may be shaky at the very smallest sizes which may be mono-
mineralic, but coagulation models show that the tiniest particles are quickly
consumed by growing aggregates (Ormel and Okuzumi 2013). Then, we are back to
equation 2:
$\kappa_{e,\lambda}=\frac{1}{\rho_{g}}\int n_{0}(r)\pi
r^{2}Q^{\prime}_{e}(r,\lambda)dr.$ (23)
As the gas density (and grain number density) increases, the collision rate
increases accordingly. Grains of micron size are well coupled to the gas, and
have fairly low collision velocities relative to each other (most recently
Ormel and Cuzzi 2007). The low relative velocities imply that sticking will be
fairly efficient at least until mm-cm sizes are reached (Dominik et al 2007,
Güttler et al 2010, Zsom et al 2010, Birnstiel et al 2010,2012). Several
laboratory simulations of this process have shown that the resulting
aggregates are of fairly low density (Donn 1990, Beckwith et al 2000). It has
been noted that aggregates of this sort are fractals in the sense that their
volume depends on their mass to an arbitrary power. Normally this relationship
is expressed as $m\propto r^{s}$, where $m$ is an individual particle mass and
$s$ is the fractal dimension (Wright 1987, Beckwith et al 2000, Dominik et al
2007). If the dimension is less than 3, their internal density decreases as
the mass increases. Experimental results suggest that the internal densities
and volume fractions of such particles are likely to be quite small, with
porosities approaching 70% even after they begin compacting each other at
increasing relative velocities (Weidenschilling 1997, Ormel et al 2008 and
references therein). Growth of this sort is certain to be robust in giant
planet atmospheres as well (Ackerman and Marley 2001, Helling et al 2008,
Marley et al 2013 and references therein).
### 5.1 Refractive indices of aggregate particles
We envision each particle as compositionally heterogeneous - made up of much
smaller constituents of specific composition and/or mineralogy, as seen in
meteorites, interplanetary dust particles (IDP’s), and cometary dust. That is,
our particles are aggregates of subelements of species $j$, each smaller (and
usually much smaller) than any relevant wavelength. The average particle
refractive indices can then be calculated using an Effective Medium Theory
(EMT) approach. The two best known EMTs are due to Garnett, and Bruggeman
(Bohren and Huffman 1983). In the Garnett model, there is presumed to be one
pervasive “matrix” in which distinct grains of other materials are embedded.
In the Bruggeman theory, there is no structural distinction between domains of
different refractive index. Bohren and Huffman (1983) believe that the Garnett
rule is fundamentally to be preferred for aggregate particles where there is a
well-defined matrix and it is vacuum (even, in principle, as the porosity gets
very small). As this is our application, we use the Garnett EMT. The
expressions and some additional discussion are given in Appendix C, and a set
of effective refractive indices for each temperature range (each ensemble of
condensed solids) is shown in figure 6.
Figure 6: Left: Refractive indices of aggregate particles, obtained using the
Garnett EMT, as functions of wavelength for four different temperature regimes
in which all materials are present (black), water ice has evaporated (red),
organics have evporated (green) and troilite has evaporated (blue). The
temperatures at which these transitions occur are shown in the Table. Right:
ratio of solid/porous particle EMT refractive indices in the four temperature
regions, for a porosity of 90%. For porosities in this range, these refractive
indices could simply be used in our basic equations 8 and (9 or 12), along
with equation 13 as constrained by equations 15-17. These results would be
used in our model to calculate particle opacities at temperatures where one or
more of the five basic constituents has evaporated or decomposed (this is, of
course, implicit in calculation of Rosseland mean opacities as a function of
temperature, but the monochromatic opacities shown here are for a temperature
where all candidate solids are condensed). Tabulated values available from the
authors and will be posted online.
In the limit where all of the refractive indices are of order unity (this
holds away from absorption bands for all species but the iron and troilite),
an intuitively simple linear volume average captures the sense of the effect
and might serve acceptably well in some regimes:
$n_{i}=\frac{1}{f}\sum_{j}f_{j}n_{ij}$ (24)
and
$n_{r}=1+\frac{1}{f}\sum_{j}f_{j}(n_{rj}-1).$ (25)
Here, $f$ is the volume fraction of all solids in a given particle, and
$f_{j}$ is the volume fraction for each species $j$. For low mass density
particles such as some of those seen in our model distributions, the average
imaginary index can get quite small and the real index can approach unity.
However, in Appendix C and section 5.2 below, we demonstrate that such a
simple volume average greatly overestimates the contribution of even a small
volume fraction of high-refractive-index grains (ie., iron and troilite) in
the mixture; for the purpose of this paper only the full Garnett theory
(described in detail in Appendix C) is used.
Figure 7: A comparison of the wavelength-dependent mean extinction efficiency
$Q^{\prime}_{e}(\lambda)$, weighted and summed over the Pollack et al (1994)
particle size distribution, for two particle structure assumptions. The
heterogeneous mixture in which each composition $j$ is treated as a separate
pure material with the same size distribution, and the various
$Q^{\prime}_{ej}(\lambda)$ then combined using their appropriate abundances,
is shown in black for our model and green for Mie theory. Also shown (red) is
the Garnett EMT treatment of the same size distribution, but with each
particle an identical aggregate of the five materials, along with (blue) the
linear mixture “approximation” to EMT. Note how badly the linear approximation
performs, as also discussed in the Appendix. On the other hand, for these
small particles, the heterogeneous mixture and the Garnett EMT results are
very similar.
### 5.2 Wavelength dependent opacities
In figure 7 we compare weighted averages of $Q^{\prime}_{e}(\lambda)$ over
heterogeneous mixtures, as described in section 4, with values obtained for
the same size distribution of aggregates in which the same material is mixed
within each particle, and refractive indices are determined using the Garnett
EMT (section 5.1 and Appendix C). As expected for particles much smaller than
the wavelength (section 3.2), and as found by other investigators in the past,
there is very little difference in the wavelength-dependent efficiency between
a calculation in which each material is treated separately, and one in which
they are merged together and treated with the EMT. The Pollack size
distribution, while formally extending to 5$\mu$m radius, is very steep and in
reality has nearly all the cross section and mass at radii smaller than, if
not much smaller than, 1 $\mu$m (section 4).
The extremely high degree of agreement between the Garnett EMT and the
heterogeneous mixture shown in figure 7 was initially a surprise to us, given
the presence of materials of high refractive index (see Appendix C). However,
we believe this agreement actually validates not only the Garnett EMT itself
(which hardly needs more validation) but also our numerical implementation of
it. Consider the fact that for tiny particles with $r\ll\lambda$, as described
in section 3.2, $Q_{e}$ becomes proportional to the total volume or mass of
material, regardless of the specific grain size distribution. This suggests
that $Q_{e}$ also becomes independent of the configuration of the grains; they
can be dispersed or combined into clumps, as long as the clumps themselves
remain much smaller than the wavelength, without affecting the result. Indeed
this is the configuration implicit in the Garnett-averaged results for the
Pollack size distribution; tiny particles are rearranged into aggregates of
still-tiny particles. The good agreement testifies to the validity of the
Garnett EMT in calculating a single set of effective refractive indices (at
each wavelength) from which to calculate $Q_{e}$. That this is not trivial is
demonstrated by the poor agreement obtained from the intuitively simpler
linear volume mixing approximation to the mean refractive index (Appendix C;
blue curve in Figure 7). Figure 12 in Appendix C shows in another way how
badly volume mixing performs when even small amounts of high-refractive-index
material are involved, as they are here. Of course, as particles grow larger,
this equivalence will fail (a hint of divergence is seen at short wavelength
in figure 7). In this regime we will continue to rely on the Garnett EMT.
Figure 8: Monochromatic opacity in pure absorption (eg. using only $Q_{a}$),
such as would be relevant to interpreting mm-cm wavelength observations of
disk fluxes. Two powerlaw distributions are shown, all of the form
$n(r)=n_{o}r^{-s}$, where $n(r)$ is a particle number density per unit
particle radius, with three different values of $r_{max}$. The particles are
aggregates of mixed cosmic composition at 100K. Left: solid particles
(porosity =0); right: porous particles (porosity =0.9).
### 5.3 Monochromatic opacities and disk masses
Perhaps the hardest challenge for our model is calculating monochromatic
opacities for particle size distributions dominated by sizes comparable to the
wavelength. Applications to wavelengths which are either much longer than the
particle sizes in question (eg. most of figures 4 or 7), or much shorter
(geometrical optics), are very reliable and straightforward. To better
determine the limitations of our model, we explore monochromatic opacity for
broad size distributions where many of the particles are comparable to the
wavelengths of greatest interest to mm-cm observations of disks. The results
(figure 8) are comparable to those shown by Miyake and Nakagawa (1993; MN93)
and D’Alessio et al (2001; D01). Because MN93 and D01 each adopt somewhat
different choices for refractive indices and compositional regimes, we do not
compare our results directly to theirs, but instead conduct our own Mie
calculations using the Pollack et al (1994) refractive indices. D01 assumed a
heterogeneous particle distribution (for instance, particles of pure ice, pure
silicate, or pure troilite occur up to the mm-cm maximum sizes), whereas MN93
more typically assumed heterogeneous mixtures of aggregate particles with
variable porosity, which we feel is more plausible under realistic conditions
and also assume below.
In figure 8 we compare monochromatic profiles of the true absorption opacity
$\kappa_{a,\lambda}$ from our model (equation 4) with full Mie calculations.
MN93 have argued that scattering (which is important for extinction and
thermal equilibrium modeling) is negligible in observations of this type and
that pure absorption dominates thermal emission, and D01 agree.
Results are shown for differential size distributions $n(r)=n_{o}r^{-s}$, with
$s$=2.0 and 3.1, with smallest radius of 1$\mu$m and largest radius $r_{max}$.
The value $s=2$ (heavy curves) allows the larger particles to dominate the
area and mass, while $s=3.1$ (light curves) gives a more equitable
distribution of area across the range of sizes. For each powerlaw slope,
$r_{max}$ is varied from 10$\mu$m to 1cm (the Mie calculations bog down for
larger sizes). In the left panel, we show results for solid particles
(internal density = 1.38 g/cm3). In the right panel, we show particles with
90% porosity. The solid curves are from our model and the dashed curves are
full Mie calculations.
The agreement for porous particles is extremely good for the full range of
sizes and size distributions. For solid particles (figure 8 left), the Mie
calculations exhibit an enhanced $Q_{a}$ in the resonance region, covering
perhaps a decade of wavelength around $2\pi r/\lambda\sim 1$ (see eg HT). This
effect is difficult for a model lacking complete physical optics to reproduce.
Such “bumps” in $\kappa$ are visible in figure 5 of MN93, for the solid
particle case, but lacking in their figure 8 (for 90% porous particles), in
good agreement with our results where all combinations of $r_{max}$ and
powerlaw slope reach the same long-wavelength asymptote rather quickly. The
same “bump” is seen, for instance, in figure 3 of Ricci et al (2010), where
the particles are actually not very porous ($\sim$ 40%?). It is this
contribution which carries through at shorter wavelengths, leaving our
opacities 20% low or so relative to Mie values; this small difference, in a
wavelength-independent regime, is probably insignificant for most purposes
when the maximum particle size is probably close to the specific observing
wavelength and the porosity is not large, as discussed more below.
The more shallow 2.0 powerlaw, dominated in area and mass by particles at or
near the upper size cutoff, abruptly transitions to wavelength-independent
opacity at the wavelength where, roughly, $Q_{a}(r_{max},\lambda)=1$. At
longer wavelengths, a universal curve is followed, characterized by the total
mass in the system. Notice that, for the 2.0 powerlaws, the inflection point
moves to shorter wavelengths and the short-wavelength opacity increases as the
upper radius cutoff decreases or as the porosity increases. This is a direct
implication of equations 8 and 7, which together imply that $Q_{a}\propto
rn_{i}$, and in most cases, in spite of the imperfection of the linear volume
mixing model, $n_{i}$ is roughly proportional to $(1-\phi)$ (figure 12).
Meanwhile, for wavelengths shorter than the $Q_{a}(r_{max},\lambda)=1$
turnover, each particle or radius $r$ is less massive than a solid particle by
the factor $(1-\phi)$, so the number of particles is larger (for a given total
mass) by a factor $1/(1-\phi)$ (see also section 5.4 below). That is, fluffy
cm-size particles have ten times the opacity of solid particles of the same
radius (in the short-wavelength limit) because their 10 times lower mass per
particle allows there to be ten times as many of them than the solid particles
of the same size. Their mass per particle is still 100 times larger than that
of a mm-size solid particle, but their cross-sectional area per particle is
100 times larger, so the curve for porous, cm-radius particles lies on top of
the curve for solid, mm-radius particles even at short wavelengths. This
behavior can be seen tracing the differences between the heavy red and black
curves, through solid, dashed, and dotted manifestations.
The lightweight curves, for the steeper 3.1 powerlaw size distribution having
the same radius limits, contain more small particles and thus show more
spectral signatures at mid-infrared wavelengths. They also show broader slope
transitions in $\kappa^{a}_{\lambda}$ around $Q_{a}(r_{max},\lambda)=1$. Note
the envelope of slope for all combinations of powerlaw, upper size limit, and
porosity, for which $Q_{a}(r_{max},\lambda)=1$ has not been reached. Opacities
(fluxes) at these wavelengths are independent of particle size, and thus
capture all the mass even if particles have grown to cm size. Different
powerlaw slopes or other details of the size distributions could lead to
slightly different functional forms in the transition region, and can thus be
constrained by high-quality observations. These comparisons of our model
monochromatic opacities with full Mie theory provide further support for the
Rosseland mean opacities which are based on them, certainly for porous
particles, and illustrate the extent of the limitations for solid particles.
In applications such as mm-cm monochromatic SED slope analysis, with the goal
of determining largest particle sizes, full Mie theory should be used, and are
not burdensome here because the wavelengths of interest are not much smaller
than the particle sizes of interest.
Figure 9: Monochromatic opacity, comparing Mie calculations for solid
particles (dashed lines) with our model for porous particles (solid lines).
Two powerlaw distributions are shown: $s$=2.0 (heavy lines) and $s$=3.1 (light
lines), and the upper size is varied from 1mm to 100cm (the 10 and 100cm size
results are not shown for the Mie case because the mm-wavelength slopes would
be too flat and the calculations are onerous). The widely used opacity law of
Beckwith et al (1990) is the dashed line labeled B90. In principle, meter-
size, very porous aggregates could be compatible with the slope of the mm-
wavelength opacities; however as discussed in the text we believe that solid,
or nearly solid, cm-size particles are more realistic.
These scalings with size and porosity are further illustrated in figure 9,
comparing our runs for porous particles, which agree well with Mie
calculations but are easier to extend to large sizes, with actual Mie
calculations for solid particles. In figure 9 we show a widely used opacity
rule (Beckwith et al 1990, Williams and Cieza 2011):
$\kappa_{\nu}=0.1(\nu/10^{12}{\rm Hz})^{\beta}$, with $\beta$=1. Note that the
canonical Beckwith et al (1990) opacity could be fit by a suite of porous
particles extending to 100 cm radius, especially with a slope of 3.1 (or
perhaps slightly steeper). As noted by previous authors, cm-size solid
particles also provide a fairly good match to the slope. All models that match
the slope have opacity quantitatively smaller than the canonical B90 opacity,
however, by a factor of at least several (also found by Birnstiel et al 2010)
possibly suggesting larger inferred disk masses.
In outer disks where gas densities are very low, particles of even mm size and
low-to-moderate porosity are dynamic radial migrators under the influence of
gas drag (Takeuchi and Lin 2005, Brauer et al 2008, Hughes and Armitage 2010,
2012, Birnstiel et al 2010, 2012) and observations apparently show radial
segregation of gas and particles in more than one outer disk (Andrews et al
2012, Pérez et al 2012). Moreover, such particles also couple to the large,
high-velocity eddies and would be expected to have fairly high collision
velocities. In this regime, large particles, even with 90% porosity, would
collide at significant velocities, inconsistent with retaining their
postulated high porosities. This is because the aerodynamic coupling of a
particle to turbulence is determined by the product of its radius and density
(see, eg, Völk et al l980, Cuzzi and Hogan 2003, Dominik et al 2007, Ormel and
Cuzzi 2007). Thus it seems more plausible that the particles matching the B90
opacity in shape (or something like it) are indeed moderately compact, cm-
radius, objects, and not meter-size, high-porosity puffballs; thus full Mie
theory is needed for analyses of these observations.
### 5.4 Rosseland mean opacity: effects of size and porosity
In this section we continue to assume well-mixed aggregates of material in
each particle and extend our calculations of Rosseland mean opacities to
particles with a wider range of size and porosity. First we show how growth
affects the Rosseland mean opacities introduced in section 3.1 and the
Appendix. Clearly, for particle size much larger than the wavelength, the
opacity varies as the ratio of cross section to mass, or as $1/r$ (see
Appendix of Pollack et al 1994 for a longer exposition). As shown by Pollack
et al (1985) and Miyake and Nakagawa (1993), growth from microns to
centimeters implies a decrease in $\kappa_{R}$ by four orders of magnitude,
dwarfing any uncertainties regarding the actual composition of the particles
and rendering the small differences seen in figures 4 \- 7 somewhat moot. We
illustrate the effect using figures 10 and 11. The size distributions in
figure 10 are monodispersions, which we have cautioned about previously, but
in these calculations the large range of wavelengths over which efficiencies
are integrated mimics the effects of a broad size distribution.
Particle growth from the ISM distribution of Pollack et al (containing many
submicron grains which are effective at blocking shortwave radiation
characteristic of the higher temperatures shown), rapidly decreases the
Rosseland opacity, even when growth is only to radius of 10$\mu$m (solid black
curve). The decrease continues as particles grow to 100$\mu$m (dashed black)
and 1mm (dotted black, multiplied by 10) radii. The opacity curves for the
larger particles have a characteristic stepped appearance, with the steps
representing changes in solid mass fraction at different evaporation
temperatures. The opacity is nearly constant within a step, because particles
much larger than a wavelength are in the constant-$Q_{e}$ regime independent
of wavelength. Comparing the black dashed and dotted curves, where the dotted
curve is 10 times the value of $\kappa_{R}$ for 1 mm radius particles, shows
that the $1/r$ scaling is nearly exact for these sizes. The scaling is only
slightly less easily explained between the 10$\mu$ m and 100$\mu$m radius
solid particles. At the lower temperatures (longer wavelengths) the 10$\mu$m
particles have not yet reached the geometrical optics limit, which occurs
roughly for temperatures higher than 300K where the blackbody peak wavelength
is around 10$\mu$m. Thus, the scaling between these two sizes is less than a
factor of ten by an amount that depends on temperature (wavelength).
Figure 10: Rosseland mean opacities for various particle sizes and porosities,
all calculated using our approach. Green: heterogeneous ISM distribution of
Pollack et al (1994); black curves: solid aggregate particles of no porosity
with the same mass; solid: 10 $\mu$m radius monodispersion; Dashed: 100 $\mu$m
radius monodispersion; dotted: 1000 $\mu$m radius monodispersion (multiplied
by 10). Also shown are red: 100 $\mu$m radius monodispersion with the same
mass but porosity of 70%; blue: 100 $\mu$m radius monodispersion with the same
mass but porosity of 90% (discussed in section 5.2).
Figure 10 also shows the expected effects of porosity (section 5.3); the red
and green curves, for 100$\mu$m radius monodispersions of particles having
porosities of 70% and 90% respectively, show good agreement with the expected
scaling by $1/(1-\phi)$; that is, to conserve mass if the particle density
decreases to $(1-\phi)\overline{\rho}$, we must increase their number density
by a factor of $1/(1-\phi)$. In this regime where the particles are already
much larger than the most heavily weighted wavelengths in question (except at
the far left of the plot), the $Q_{e}$ per particle does not change, so the
net opacity of the ensemble increases by a factor of $1/(1-\phi)$. So, the
particle growth and porosity effects are robust, easily explained using simple
physical arguments, and captured by the model.
The Rosseland mean opacities for several wide powerlaw size distributions are
shown in figure 11. The behavior is similar overall to behavior seen
previously and explained above. These powerlaws have a smallest particle
radius (1 $\mu$m) which is slightly larger than the typical size on the
Pollack et al size distribution, so $\kappa_{R}$ is somewhat larger at low
temperatures (long effective wavelengths) but slightly lower and flatter at
high temperature (short effective wavelengths). Powerlaws extending to larger
size reduce the overall opacity at higher temperature, at least, by tying up
more mass in larger particles. Porosity increases the overall opacity
everywhere for the powerlaws extending to the larger sizes, because the
dominant wavelengths involved are generally tens of microns or less. However,
some wrinkles in the details, and differences between figures 10 and 11, arise
from the scattering terms due to $Q_{s}(1-g)$ in $\kappa_{R}$, which become
important for particles with low $n_{i}$ when the particle size and wavelength
are comparable.
It is these drastic decreases in opacity with grain growth (Movshovitz and
Podolak 2008) that led Movshovitz et al (2010) to conclude that gas giant
formation could have happened much earlier than previous estimates which had
already assumed an arbitrary 50x cut in opacity relative to the Pollack et al
(1994) baseline (Hubickyj et al 2005, Lissauer et al 2009). The reason is that
growth in the most important part of the radiative zone proceeds to cm-size,
implying opacities even 10x smaller there than the smallest (for solid mm-size
particles) shown in figures 10 and 11. If the particles are porous, their
opacity could be increased, however, as shown in those figures. This would
seem to be an example of how the destiny of the great can be determined by the
behavior of the small.
Figure 11: Rosseland mean opacities for the same powerlaw size and porosity
distributions as used in figure 8. Both linear and logarithmic forms are
presented for the same results. Green: heterogeneous ISM distribution of
Pollack et al (1994); black curves: solid aggregate particles of no porosity
with the same mass and powerlaw size distributions of differential slope 3.1;
solid: 1-10 $\mu$m radius; Dashed: 1-100 $\mu$m radius; dotted: 1$\mu$m -1mm
radius; dot-dash: 1$\mu$m -1cm radius (shown are red: 1$\mu$m -1cm radius with
porosity of 90%.
## 6 Conclusions
We outline a very simple, closed-form radiative transfer model which
incorporates and connects well-understood asymptotic behavior for particles
smaller than and larger than the wavelength. The approach is simple enough to
provide good physical insight and to include in evolutionary models requiring
radiative transfer. The model is easily adapted to arbitrary combinations of
particle size, composition, and porosity across the range of plausible
protoplanetary nebula and exoplanet cloud particle properties (excepting
highly elongated particles), and yields values of Rosseland mean opacity which
are in good agreement with more sophisticated but more time consuming Mie or
DDA calculations. Planck opacities are even simpler to calculate (they are
straight Blackbody-weighted means over wavelength). We illustrate the
significant roles of particle growth and porosity in determining opacity. The
model is not recommended even for Rosseland opacities in cases where ensembles
of large, pure metal particles are expected. The method even gives very good
approximations to monochromatic opacities unless the particles are solid and
the specific wavelength of interest is comparable to the dominant particle
size, where the model is unable to track “resonance” behavior of the
absorption efficiency. Thus for detailed spectral index analysis of mm-cm
wavelength protoplanetary nebula emission spectra, in cases where particles in
the mm-cm radius range might be solid and contribute significant mass and
area, full Mie theory should be used. It appears that canonical mm-wavelength
spectral slopes are more plausibly explained by solid, cm-size particles than
larger, but more porous, aggregates.
To summarize, the model consists of equations 8 and (9 or 12), along with
equation 13 as constrained by equations 15-17. These equations lead to a
monochromatic opacity $\kappa_{e,\lambda}$, which is then used to calculate a
Rosseland mean $\kappa_{R}$ using equation 30. For calculation of mm-cm SEDs,
only $\kappa^{a}_{R}$ should be used (equation 4). Our treatment can assume
either a heterogeneous “salt and pepper” particle size and compositional mix,
as in Pollack et al 1994 or D’Alessio et al 2001 (with equations 18-20), or a
distribution of internally-mixed, homogeneous aggregate particles having
arbitrary porosity (using equations 37 \- 42 in equations 8-17, and
integrating with equation 23), as for instance modeled by Miyake and Nakagawa
(1993).
Acknowledgements: JNC benefitted greatly from many discussions of radiative
transfer with Jim Pollack over the years. We thank Ted Roush for providing
tabular values of the refractive indices used by Pollack et al (1994). We
thank Pat Cassen, Ke Chang, Tom Greene, Lee Hartmann, Stu Weidenschilling, and
Diane Wooden for helpful conversations. We thank Kees Dullemond and Naor
Movshovitz for encouragement to make this work more widely available. We thank
our reviewer for helpful comments. JNC and PRE were supported under a grant
from the Origins of Solar Systems Program. SSD was partially supported by
Grants from the LASER and Astrobiology programs.
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## Appendix A: Rosseland Mean Opacity
While derivations of the Rosseland mean opacity can be found in various
sources, we provide here a quick derivation of $\kappa_{R}$ because it is
surprisingly difficult to find good ones, and we make reference to some of its
specific aspects. Essentially one derives an expression for the monochromatic
flux in a high-opacity thermal radiation field, then integrates over
frequency, and from this identifies the associated mean opacity. Start with
the radiative transfer equation for monochromatic intensity $I_{\nu}$ in a
plane layered medium: $\mu dI_{\nu}/d\tau=S_{\nu}-I_{\nu}$ where $S_{\nu}$ is
the source function and $\tau_{\nu}$ is monochromatic optical depth measured
normal to the layer, and $\mu={\rm cos}\theta$ where $\theta$ here is the
angle from the layer normal and $d\tau_{\nu}=\kappa_{\nu}dz$ where $dz$ is an
increment of thickness in the layer. In a medium of high optical depth where
thermal radiation dominates everything else, $I_{\nu}\approx S_{\nu}\approx
B_{\nu}$ where $B_{\nu}$ is the Planck function. Then we can rewrite the above
equation to first order as $I_{\nu}=B_{\nu}-dB_{\nu}/d\tau$. We then determine
the local energy flux through the layer $F_{\nu}=2\pi\int_{-1}^{1}I_{\nu}\mu
d\mu$. In the presence of the gradient derived above, substituting for
$I_{\nu}$:
$F_{\nu}=2\pi\left[\int_{-1}^{1}B_{\nu}\mu d\mu-\int_{-1}^{1}{\mu
dB_{\nu}\over\kappa_{\nu}dz}\mu d\mu\right].$ (26)
The first integral vanishes because $B_{\nu}$=constant; the flux becomes
$F_{\nu}=-\frac{2\pi}{\kappa_{\nu}}\frac{dB_{\nu}}{dz}\int_{-1}^{1}\mu^{2}d\mu=-\frac{4\pi}{3\kappa_{\nu}}\frac{dB_{\nu}}{dz}.$
(27)
The standard trick is to set $dB_{\nu}/dz=(dB_{\nu}/dT)(dT/dz)$. The
monochromatic flux is then integrated over frequency, after rearranging terms:
$F=\int
F_{\nu}d\nu=-\frac{4\pi}{3}\frac{dT}{dz}\int\frac{1}{\kappa_{\nu}}\frac{dB_{\nu}}{dT}d\nu.$
(28)
One then merely asserts that the frequency integrated flux can be written in
the same form, except with a mean opacity $\kappa_{R}$:
$F=-\frac{4\pi}{3}\frac{dT}{dz}\frac{1}{\kappa_{R}}\int\frac{dB_{\nu}}{dT}d\nu;$
(29)
and after setting the two expressions equal, we obtain the definition of
$\kappa_{R}$:
$\frac{1}{\kappa_{R}}=\frac{\int\frac{1}{\kappa_{\nu}}\frac{dB_{\nu}}{dT}d\nu}{\int\frac{dB_{\nu}}{dT}d\nu};$
(30)
essentially, we are obtaining the weighted average of $1/\kappa_{\nu}$,
thereby emphasizing spectral regions where energy “leaks through”, and where
the integral of the weighting function $dB_{\nu}/dT$ in the denominator may,
if we like, be further simplified as $d/dT(\int
B_{\nu}d\nu)=4\sigma_{SB}T^{3}$, where $\sigma_{SB}$ is the Stefan-Bolzmann
constant.
## Appendix B: Metal Particles
The small particle expansions of DL84, leading to our primary equations 8 and
9, incorporate only electric dipole terms. Similar expressions can be derived
for magnetic dipole terms, which are more cumbersome in their full glory (see
eg Tanner 1984 or Ossenkopf 1991) and are usually approximated. For instance,
DL84 (equation 3.27), citing Landau and Lifshitz (1960 sections 45, 72, and
73), give a handy first-order correction factor for $Q_{a}$ with the caveat
that it is valid for small $x$, but without specific guidance as to what is
“small”. The correction is simply
$Q^{\prime}_{a}(r,\lambda)=Q_{a}(r,\lambda)(1+F)$, where
$F=\left(\frac{2\pi
r}{\lambda}\right)^{2}{(\epsilon_{1}+2)^{2}+\epsilon_{2}^{2}\over 90},$ (31)
and $\epsilon_{1}=n_{r}^{2}-n_{i}^{2}$ and $\epsilon_{2}=2n_{r}n_{i}$
(equation 7). Similar expressions appear in Ossenkopf (1991) and Tanner
(1984), and surely elsewhere. The factor $F$ gets very large when the
refractive indices are large. In the case of iron metal, where $n_{r}$ and
$n_{i}$ are nearly proportional to $\lambda$, the spectral behavior of
equation 8 is flattened from $x^{-3}$ nearly to $x^{-1}$, as can be seen by
expanding terms in the large-refractive-index limit.
This magnetic dipole correction is insignificant in the mixed-material,
aggregate particle case, as it only enters when the refractive indices of a
particle are $\gg\lambda/r$, so readers interested only in aggregate grains
need not be concerned further with this term. Yet, for those who might be
interested in clouds of metallic particles, the correction might be of
interest. Unfortunately, it is formally only valid for particles that are
small compared to the wavelength inside the particle, or equivalently the skin
depth of the wave, which is given by $x\sqrt{\epsilon}\ll 1$, and for the
large $\epsilon$ of metals (see figure 3) the allowed $x$ is much smaller than
the usual condition $x=2\pi r/\lambda\ll 1$. The correction is mentioned by
Pollack et al (1994) but it is unclear from that paper just how it was used;
their figure 2a shows results for metal particles having their standard size
distribution in a wavelength range where it is not formally valid, and they
give no comparison with Mie theory. We have found that if equation 31 is
blithely applied for $2\pi r/\lambda<1$, well out of its formal domain of
validity, and the ensuing $Q_{a}$ is subject to our overall constraint
$Q_{a}<1$), the agreement with Mie calculations for pure metal is improved
from nonexistent to marginal (figure 4). Tanner (1984) has found similar
behavior, in that use of the “approximation” outside of its formal domain of
validity gives surprisingly better agreement with observed behavior than
application of the more complete theory. This is not an argument for general
acceptance of the approximation, and in any application where metal particles
dominate the situation the complete Mie theory is probably required.
Nevertheless, we have employed it in our model.
## Appendix C: Garnett Effective Medium Theory
In this appendix, we give an overview of the Garnett theory for the average
complex dielectric constant ($\epsilon$) of an inhomogeneous medium. The
reader is referred to Bohren and Huffman (1983) for a more complete exposition
with background.
A number of exhaustive and sophisticated studies have compared several
different kinds of EMT models to rigorous, brute-force numerical Discrete
Dipole Approximation (DDA) models (Perrin and Lamy 1990, Ossenkopf et al 1991,
Stognienko et al 1995, Voshchinnikov et al 2005, 2006; see also Semenov et al
2003 and references therein); the differences are generally small and
composition-dependent for small particles which can be treated using DDA (and
much smaller than the huge differences due to growth which our model is
primarily intended to capture). For instance, figure 16 of Voshchinnikov et al
(2005) shows a nearly insignificant difference between Garnett and Bruggeman
models relative to DDA calculations, in the regime where all scattering
elements in the aggregates are truly small compared to the wavelength.
Interestingly, they show that a model of their own device does a better job
matching certain very porous aggregates containing monomers with a
distribution of sizes (at least, at long wavelengths). We believe that both
traditional EMT theories fail to match the “size distribution of inclusions”
DDA results of Voshchinnikov et al 2005 figure 16, because the aggregates
contain numerous embedded wavelength-sized monomers, which violate the
assumptions of both EMT models (some of the monomer inclusions have diameter
as large as the wavelength). For this situation to occur at wavelengths,
particles, and temperatures of interest for Rosseland mean opacities of our
paper would require monomers in the few-to-tens of micron size, that are in
turn embedded by assumption in much larger particles - which our model would
treat in the geometric optics limit in any case. In sections 5.3 and the
Conclusions, we note deviations from our model when a significant fraction of
the particles of interest are wavelength-sized (we anticipate this to be a
problem mostly for mm-cm wavelength observations). The behavior is similar to
that seen in figure 16 of Voshchinnikov et al (2005).
We derive the components of
$\epsilon=\epsilon^{\prime}+i\epsilon^{\prime\prime}$ and convert the results
to the perhaps more familiar complex refractive index
$n^{\prime}+in^{\prime\prime}$. We then assess the realm of validity for a
linear approximation of refractive index as a function of material density by
comparing the two expressions for a range of component materials.
Interactions among different constituents of an inhomogeneous medium make the
determination of an average dielectric constant problematic. The problem is
generally insoluble, save by brute force (DDA) or approximation methods, and
different approximations inevitably lead to different expressions. As an
example, one choice that can be found in the literature as far back as 1850 is
the Rayleigh expression that relates the density and dielectric constant of a
powder to that of the corresponding solid. This result follows from the
Clausius-Mosotti law that the quantity $\epsilon-1/\epsilon+2$ is proportional
to the density of a material:
$\frac{1}{\rho}\frac{\epsilon-1}{\epsilon+2}=\frac{1}{\rho_{o}}\frac{\epsilon_{o}-1}{\epsilon_{o}+2},$
(32)
where $\epsilon_{o},\rho_{o}$ are the complex dielectric constant and density
of the solid material, and $\epsilon,\rho$ are those of the powder. This
expression has been shown to be fairly accurate for powders made from various
geological materials (e.g. Campbell and Ulrichs 1969). A more complete
expression, which reduces to the Rayleigh formula in the case of one
component, was derived by Maxwell Garnett (see, e.g. Bohren and Hoffman 1983).
Garnett’s model is that of inclusions of dielectric constant $\epsilon_{o}$
embedded in a homogeneous medium of dielectric constant $\epsilon_{m}$. In the
simplest version, the inclusions are assumed to be identical in composition,
but may vary in shape, size, and orientation. Assuming that all the inclusions
are spherical, the expression for the average dielectric constant $\epsilon$
is given by
$\epsilon=\epsilon_{m}\left[1+3f\left(\frac{\epsilon_{o}-\epsilon_{m}}{\epsilon_{o}+2\epsilon_{m}}\right)\left(1-f\left(\frac{\epsilon_{o}-\epsilon_{m}}{\epsilon_{o}+2\epsilon_{m}}\right)\right)^{-1}\right],$
(33)
where $f$ is a mass fraction which is defined below. For $\epsilon_{m}=1$,
this reduces to the Rayleigh formula with $f=\rho/\rho_{o}$ (for a more
detailed derivation, including nonsphericity effects, see Bohren and Hoffman
1983). The advantage of the Garnett equation is that it can be generalized to
a multiple component medium. The general result can be cast into the same form
as (33) except now $f$ and $\epsilon_{o}$ become $f_{j}$ and $\epsilon_{j}$,
while the numerator and denominator of (33) are summed over $j$ species
(Bohren and Hoffman 1983, Sect. 8.5).
Noting that $\epsilon=\epsilon^{\prime}+i\epsilon^{\prime\prime}$, we expand
(33) into its complex parts, convert to fractional form, and separate the real
and imaginary components:
$\epsilon=\epsilon^{\prime}+i\epsilon^{\prime\prime}=\frac{1+2\sum_{j}f_{j}\sigma_{j}+i6\sum_{j}f_{j}\gamma_{j}}{1-\sum_{j}f_{j}\sigma_{j}-i3\sum_{j}f_{j}\gamma_{j}},$
(34)
where we have defined $\sigma_{j}$ and $\gamma_{j}$ to be
$\sigma_{j}=\frac{(\epsilon^{\prime}_{j}-1)(\epsilon^{\prime}_{j}+2)+\epsilon^{\prime\prime
2}_{j}}{(\epsilon^{\prime}_{j}+2)^{2}+\epsilon^{\prime\prime 2}_{j}},$ (35)
and
$\gamma_{j}=\frac{\epsilon^{\prime\prime}_{j}}{(\epsilon^{\prime}_{j}+2)^{2}+\epsilon^{\prime\prime
2}_{j}},$ (36)
respectively. We apply the complex conjugate to (34), determine the real and
imaginary parts of $\epsilon$, and combine like terms to get
$\epsilon^{\prime}=\frac{1+\sum_{j}f_{j}\sigma_{j}-2\sum_{i}\sum_{j}f_{i}f_{j}(\sigma_{i}\sigma_{j}+9\gamma_{i}\gamma_{j})}{1-2\sum_{j}f_{j}\sigma_{j}+\sum_{i}\sum_{j}f_{i}f_{j}(\sigma_{i}\sigma_{j}+9\gamma_{i}\gamma_{j})}=\frac{N_{R}}{D},$
(37)
for the real part, and
$\epsilon^{\prime\prime}=\frac{9\sum_{j}f_{j}\gamma_{j}}{1-2\sum_{j}f_{j}\sigma_{j}+\sum_{i}\sum_{j}f_{i}f_{j}(\sigma_{i}\sigma_{j}+9\gamma_{i}\gamma_{j})}=\frac{N_{I}}{D}$
(38)
for the imaginary part. A simple check will show that this generalized Garnett
formula reduces to equation (32), the Rayleigh formula, for $i=j=1$.
Conversion of equations (37) and (38) to an expression in terms of refractive
indices poses no great problem. Making the usual assumption that magnetic
permeability is of order unity (see however Appendix C) we define
$\epsilon^{\prime}+i\epsilon^{\prime\prime}=(n^{\prime}+in^{\prime\prime})^{2},$
(39)
where now, (35) and (36) become
$\sigma_{j}=\frac{(n^{\prime 2}_{j}-n^{\prime\prime 2}_{j}-1)(n^{\prime
2}_{j}-n^{\prime\prime 2}_{j}+2)+4n^{\prime 2}_{j}n^{\prime\prime
2}_{j}}{(n^{\prime 2}_{j}-n^{\prime\prime 2}_{j}+2)^{2}+4n^{\prime
2}_{j}n^{\prime\prime 2}_{j}},$ (40)
and
$\gamma_{j}=\frac{2n^{\prime}_{j}n^{\prime\prime}_{j}}{(n^{\prime
2}_{j}-n^{\prime\prime 2}_{j}+2)^{2}+4n^{\prime 2}_{j}n^{\prime\prime 2}_{j}}$
(41)
respectively. Finally, we express the average refractive index of the
inhomogeneous medium in terms of $\epsilon$ as
$n^{\prime}+in^{\prime\prime}=\left[\frac{\sqrt{\epsilon^{\prime
2}+\epsilon^{\prime\prime
2}}+\epsilon^{\prime}}{2}\right]^{1/2}+i\left[\frac{\sqrt{\epsilon^{\prime
2}+\epsilon^{\prime\prime 2}}-\epsilon^{\prime}}{2}\right]^{1/2}$ (42)
The Garnett formula defines the $f_{j}$ as the volume fraction of inclusions
of species $j$ within a particle of some overall volume $V$ and total mass
$M$; then the total solid volume fraction in the particle is
$f=\sum_{j}f_{j}=1-\phi$ where $\phi$ is the porosity of the particle. We can
determine $f_{j}$ in terms of mass fractions as follows: $v_{kj}$ is the $k$th
volume element of species $j$ so that the volume fraction of inclusions of
species $j$ is $f_{j}=\sum_{k}v_{kj}/V$. We define $\alpha_{j}$ as the mass of
all component $j$ per unit nebula gas mass, and $\alpha=\sum_{j}\alpha_{j}$ is
the mass fraction of all solids per unit nebula gas mass. With these
definitions in mind,
$f_{j}=\frac{v_{j}}{V}=\frac{m_{j}}{\rho_{j}V}=\frac{m_{j}\rho}{\rho_{j}M}=\frac{\rho}{\rho_{j}}\frac{\alpha_{j}}{\alpha}=\frac{(1-\phi)\overline{\rho}\alpha_{j}}{\rho_{j}\alpha}$
(43)
where $\rho$ is the average mass density of a composite particle; $\rho$ can
vary from nearly zero to the “solid” average value $\overline{\rho}$ at
$\phi=0$. Using the last expression of equation (43) in
$\sum_{j}f_{j}=f=1-\phi$ gives
$\overline{\rho}=\frac{\alpha}{\sum_{j}(\alpha_{j}/\rho_{j})}.$ (44)
Under certain conditions, a simple linear approximation for the refractive
index at a particular wavelength $\lambda$ can greatly simplify calculations
(see equations 22-23). Generally, such an approximation works well when $n$ is
of order unity (such as visible wavelengths).
Equations 22-23 follow from a simple, but approximate, extension of the so-
called “Wiener rule” (Voshchinnikov et al 2005): $\left<\epsilon\right>=\Sigma
f_{j}\epsilon_{j}$, where $\epsilon_{j}$ are the dielectric constants of the
different materials $j$ and the relation assumes lossless materials
($n_{i}=0)$. Eqns 22-23 are simply the analogous linear volume averages
applied to “moderate” refractive indices not far from unity, as is true for
most silicates and ices. Assuming that the Wiener rule can be applied to a
case where $n_{i}\neq 0$, equations 22-23 can be derived in an approximate
way, in the limit that $n_{i}\ll 1$ and $n_{r}=1+\delta$ with $\delta^{2}\ll
2\delta$. Given $\left<\epsilon\right>=\Sigma f_{j}\epsilon_{j}$, separate the
real and imaginary parts of the sum recalling that
$\epsilon=(n_{r}+in_{i})^{2}$, giving $\left<\epsilon_{i}\right>=\Sigma
f_{j}(2n_{i}n_{r})\sim 2\overline{n_{r}}\Sigma f_{j}n_{ji}$. The LHS of the
equation can also be approximated as $2\overline{n_{r}}\left<n_{i}\right>$, so
$\left<n_{i}\right>\sim\Sigma f_{j}n_{ji}$ as in equation (23). Similarly
$\left<\epsilon_{r}\right>=\left<(1+\overline{\delta})^{2}\right>\sim
1+2\overline{\delta}$, and on the RHS $\Sigma
f_{j}(1+\delta_{j})^{2}\sim\Sigma f_{j}(1+2\delta_{j})$. Separating the sum on
the RHS and noting that $\Sigma f_{j}=1$, we get
$\left<\delta\right>\sim\Sigma f_{j}\delta_{j}$ as in equation (22). Of
course, these are only approximations and valid only under the conditions
stated.
Figure (12) shows a direct comparison of both the linear approximation and the
generalized Garnett function, using the real and imaginary refractive indices
of several combinations of likely materials at $\lambda=100\,\mu$m as a
function of the density ratio $\rho/\overline{\rho}$. Curve (a) corresponds to
a single component of ice. Curves (b) and (c) are two-component mixtures of
ice/rock and ice/iron. The fraction of iron is considerably less than the ice
(see Table). Curve (d) is a three component mixture with the addition of
troilite to ice and rock. Troilite also has a fairly large refractive index at
this wavelength, which affects the linear approximation significantly. Curve
(e) is a five component mixture with the contents of (d) augmented by iron and
organics. The Garnett dielectric constant is unaffected by the minute amount
of iron, as has been found by others (Ossenkopf 1991), but the linear
approximation diverges considerably because of iron’s very high dielectric
constant. This shows that metals, in general, are well beyond the realm of
validity for the linear approximation.
Figure 12: Plots of the real (left)and imaginary (right) components of average
refractive index of a material of density $\rho$ as a function of the density
ratio $\rho/\overline{\rho}$. Curves marked (a) correspond to pure ice. Curves
(b,c) correspond to two component mixtures of ice and rock (b) and ice and
iron (c). Curve (d) corresponds to a three component mixtures of ice, rock,
and troilite, while curve (e) contains these three as well as organics and
iron. Note that the the linear and Garnett correlate very poorly for mixtures
that contain even small mixing ratios of elements with very high refractive
indices. This message is also conveyed by figure 7 and the associated
discussion.
The fractional mass and volume factors $\alpha_{j}$ and $f_{j}$ discussed
above are simply related to the particle number density weighting factors
$\beta_{j}$ of section 4 (eg. equation 16ff). Recall that $n(r)$ is the total
number density of particles of all species, and if all the species have the
same size distribution, the number density of particles of species $j$ is
$n_{j}(r)=\beta_{j}n(r)$, where $\sum\beta_{j}=1$. In the limit where the
scattering and absorption by each species needs to be separately treated
(physically, for particles small enough to be monomineralic),
$\rho_{j}(r)=\rho_{j}$ = constant. Then the total nebular mass density in
species $j$ is
$\rho_{pj}=\int\frac{4}{3}\pi
r^{3}\rho_{j}\beta_{j}n(r)dr=\rho_{j}\beta_{j}\int\frac{4}{3}\pi
r^{3}n(r)dr=\rho_{j}\beta_{j}\xi,$ (45)
where $\xi$ is the nebula volume fraction of all solids. Recall that
$\rho_{p}=\sum\rho_{pj}=\alpha\rho_{g}$, where $\alpha$ is the total mass
fraction solid material. Therefore,
$\beta_{j}={\rho_{pj}\over\xi\rho_{j}}={\alpha_{j}\rho_{g}\over\xi\rho_{j}}={\alpha_{j}\rho_{g}\overline{\rho}\over\rho_{p}\rho_{j}},$
(46)
where the last equality follows from
$\rho_{p}\equiv\alpha\rho_{g}=\overline{\rho}\xi$, since for the heterogeneous
particle case we assume the particle porosity $\phi=0$. We previously defined
the volume-averaged mass density of a compact particle as (44):
$\overline{\rho}=\frac{\alpha}{\sum_{j}(\alpha_{j}/\rho_{j})}=\frac{\rho_{p}}{\xi};$
(47)
then
$\beta_{j}=\frac{\alpha_{j}}{\rho_{j}}\frac{\rho_{g}}{\xi}=\frac{\alpha_{j}}{\rho_{j}}\frac{\rho_{p}}{\xi\alpha}=\frac{\alpha_{j}}{\rho_{j}}\frac{\overline{\rho}}{\alpha}=\frac{\alpha_{j}}{\alpha}\frac{\overline{\rho}}{\rho_{j}}={\alpha_{j}\over\rho_{j}\sum_{j}(\alpha_{j}/\rho_{j})}=\frac{f_{j}}{1-\phi},$
(48)
so it is simply verified that $\sum_{j}\beta_{j}=1$. Moreover, equations (44)
and (48) can be combined to give
$\alpha_{j}\overline{\rho}=\alpha\beta_{j}\rho_{j}$, and summing both sides
over $j$ leads to the intuitively obvious alternative expression
$\overline{\rho}=\sum_{j}\beta_{j}\rho_{j}$. All of the above quantities are
obtained from the presumed compositional makeup of the protoplanetary nebula.
|
arxiv-papers
| 2013-12-06T08:30:52 |
2024-09-04T02:49:55.084542
|
{
"license": "Public Domain",
"authors": "Jeffrey N. Cuzzi, Paul R. Estrada, and Sanford S. Davis",
"submitter": "Jeffrey Cuzzi",
"url": "https://arxiv.org/abs/1312.1798"
}
|
1312.1806
|
# Approximation in $AC(\sigma)$
Ian Doust and Michael Leinert Ian Doust, School of Mathematics and
Statistics, University of New South Wales, UNSW Sydney 2052 Australia
[email protected] Michael Leinert, Institut für Angewandte Mathematik,
Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg Germany
[email protected]
(Date: 28 November 2013)
###### Abstract.
In order to extend the theory of well-bounded operators to include operators
with nonreal spectrum, Ashton and Doust introduced definitions for two new
algebras of functions defined on a nonempty compact subset $\sigma$ of the
plane. These are the functions of bounded variation and the absolutely
continuous functions on $\sigma$. Proofs involving absolutely continuous
functions usually require that one first works with elements of a dense subset
and then take limits. In this paper we present some new theorems about
approximating absolutely continuous functions as well as providing missing
proofs for some important earlier results.
###### 2010 Mathematics Subject Classification:
26B30, 47B40
## 1\. Introduction
The algebra ${BV}(\sigma)$ of functions of bounded variation on a compact
subset of the plane was introduced by Ashton and Doust [1] in order to provide
a tool for analysing Banach space operators whose spectral expansions are of a
conditional rather than unconditional nature. The subalgebra of absolutely
continuous functions ${AC}(\sigma)$ was defined to be the closure of the set
of polynomials in two variables in ${BV}(\sigma)$. This algebra plays a
central role in the theory of ${AC}(\sigma)$ operators [3][4].
Working with ${AC}(\sigma)$ directly is often difficult and many of the
results in this area proceed by first using functions in a smaller dense
subset and then taking limits. In the classical case,
$\sigma=[a,b]\subseteq\mathbb{R}$, it is easy to show that the polynomials, as
well as $C^{1}[a,b]$ and the set of continuous piecewise linear functions, all
form dense subsets of ${AC}[a,b]$.
Analogues of these facts for general compact $\sigma\subset\mathbb{C}$ were
stated in [1] and these results were used extensively in developing the later
theory. In hindsight, some of the proofs given in [1] are somewhat cryptic and
some are certainly inadequate. It has also become clear that some of the
original definitions are more complicated than they need be and that some of
the results can be strengthened somewhat.
The aim of this paper is to redevelop some of the basic theory of
${AC}(\sigma)$ spaces based on a simpler set of definitions. We have been
careful in doing this not to rely on those results in [1]–[4] which were
inadequately justified. A major component of this paper however is formed by
the new results which give a much clearer picture of the properties of
absolutely continuous functions. Of particular importance will be the
interplay between local and global properties of such functions.
We begin in section 2 by examining the general one-dimensional case of compact
$\sigma\subseteq\mathbb{R}$. In this case absolute continuity is defined, as
in the classical case, by requiring that the function does not have positive
variation on small sets. Although it would be possible to work directly, the
easiest route to most facts about ${AC}(\sigma)$ in this case is to show that
every $f\in{AC}(\sigma)$ has a natural isometric extension to an element of
${AC}[a,b]$ where $\sigma\subseteq[a,b]$.
In section 3 we introduce the main definitions and results concerning
variation over a nonempty compact set $\sigma\subseteq\mathbb{C}$. Rather than
beginning with definitions based on the variation along curves in the plane,
we begin here with variation over a finite ordered list of points. In section
4 we show that every $C^{2}$ function on a rectangle is absolutely continuous,
a result stated essentially without proof in [1].
Section 5 contains the major new technical tool in the paper, the statement
that if a function is absolutely continuous on a compact neighbourhood of each
point in $\sigma$, then it is absolutely continuous on all of $\sigma$. This
‘Patching Lemma’ is then used in section 6 to show that every absolutely
continuous function can be approximated by continuous piecewise-planar
functions. Indeed, the space ${\mathrm{CTPP}}(\sigma)$ forms a dense subspace
of ${AC}(\sigma)$. In this section we also show that despite the fact than not
all Lipschitz functions are absolutely continuous in this context, it is
possible to relax the hypotheses of the earlier result and show that all
$C^{1}$ functions on a rectangle are absolutely continuous.
It is easy to produce a function $f$ such that
$f|\sigma_{1}\in{AC}(\sigma_{1})$ and $f|\sigma_{2}\in{AC}(\sigma_{2})$ but
$f\not\in{AC}(\sigma_{1}\cup\sigma_{2})$. This behaviour can be avoided by
placing suitable restrictions on the sets $\sigma_{1}$ and $\sigma_{2}$, and
in the final section we use the earlier theorems to prove some positive
results in this direction.
It should be noted that many somewhat different definitions of variation and
absolute continuity for functions of two variables have been given, arising in
areas such as Fourier analysis and partial differential equations. Already by
1933, Clarkson and Adams [6] had collected a list of seven such concepts and
many further definitions have been given since. The definition of absolute
continuity introduced in [1] was developed to have specific properties which
are appropriate for the proposed application to operator theory, namely:
1. (1)
it should apply to functions defined on the spectrum of a bounded operator,
that is, an arbitrary nonempty compact subset $\sigma$ of the plane,
2. (2)
it should agree with the usual definition if $\sigma$ is an interval in
$\mathbb{R}$;
3. (3)
${AC}(\sigma)$ should contain all sufficiently well-behaved functions;
4. (4)
if $\alpha,\beta\in\mathbb{C}$ with $\alpha\neq 0$, then the space
${AC}(\alpha\sigma+\beta)$ should be isometrically isomorphic to
${AC}(\sigma)$.
The present paper provides some more explicit detail about the third of these
properties. The interested reader may consult [2], [7] and [5] for some recent
papers discussing what is known about the relationships between some of the
different definitions.
## 2\. ${AC}(\sigma)$ for compact $\sigma\subseteq\mathbb{R}$
Functions of bounded variation and absolutely continuous functions on a
compact interval $J=[a,b]\subseteq\mathbb{R}$ are classical and well-studied
objects. Extending the notion of variation to a general compact subset
$\sigma\subseteq\mathbb{R}$ is completely straightforward. Absolute continuity
however requires more care. In this section we reproduce some of the results
from [1] with more straightforward (and in some cases, less flawed) proofs.
Suppose then that $\sigma\subseteq\mathbb{R}$ is non-empty and compact, and
that $f:\sigma\to\mathbb{C}$. The variation of $f$ over $\sigma$ is defined as
$\operatorname*{var}_{\sigma}f=\sup\sum_{i=1}^{n}|f(t_{i})-f(t_{i-1})|,$
where the supremum is taken over all finite increasing subsets
$t_{0}<t_{1}<\dots<t_{n}$ in $\sigma$. It was shown in [1] that
${BV}(\sigma)$, the set of all functions of bounded variation over $\sigma$,
is a Banach algebra under the norm
$\left\lVert f\right\rVert_{{BV}(\sigma)}=\left\lVert
f\right\rVert_{\infty}+\operatorname*{var}_{\sigma}f.$
(Here and throughout the paper, $\left\lVert f\right\rVert_{\infty}$ denotes
the supremum of $|f|$ rather than the $L^{\infty}$ norm.)
We say that $f$ is absolutely continuous over $\sigma$ if for all $\epsilon>0$
there exists $\delta>0$ such that for any finite collection of non-overlapping
intervals $\\{(s_{i},t_{i})\\}_{i=1}^{n}$ with $s_{i},t_{i}\in\sigma$ and
$\sum_{i=1}^{n}|t_{i}-s_{i}|<\delta$ we have
$\sum_{i=1}^{n}|f(t_{i})-f(s_{i})|<\epsilon$. Let ${AC}(\sigma)$ denote the
set of absolutely continuous functions on $\sigma$. It is clear that any
absolutely continuous function is continuous, and that the restriction of any
absolutely continuous function to a smaller compact set is absolutely
continuous.
Theorem 2.13 of [1] asserts that every absolutely continuous function
$f:\sigma\to\mathbb{C}$ extends to an absolutely continuous function on any
compact interval containing $\sigma$. Unfortunately the proof given in [1]
contains a flaw so we give here a self-contained demonstration of this fact,
and of some of its important corollaries.
For the remainder of this section let $\sigma$ be a nonempty compact subset of
$\mathbb{R}$ and let $J$ be the smallest closed interval containing $\sigma$.
Given $f:\sigma\to\mathbb{C}$, let $\iota(f):J\to\mathbb{C}$ be the extension
of $f$ formed by linearly interpolating on each of the open intervals in
$J\setminus\sigma$. It is clear that the map $\iota$ is an isometric injection
from ${BV}(\sigma)$ into ${BV}(J)$.
For a bounded set $A\subseteq\mathbb{R}$ we shall write $\mathrm{diam}(A)$ for
the diameter of $A$, that is, $\mathrm{diam}(A)=\sup A-\inf A$.
###### Proposition 2.1.
$f\in{AC}(\sigma)$ if and only if $\iota(f)\in{AC}(J)$.
###### Proof.
Since $f=\iota(f)|\sigma$, the “if” part follows immediately from the
definition.
For the “only if’ part, suppose that $f\in{AC}(\sigma)$. Fix $\epsilon>0$.
Then there exists $\delta_{0}>0$ such that if $\\{(s_{i},t_{i})\\}_{i=1}^{n}$
is a finite set of disjoint intervals with all $s_{i},t_{i}\in\sigma$ and
$\sum_{i=1}^{n}|t_{i}-s_{i}|<\delta_{0}$, then
$\sum_{i=1}^{n}|f(t_{i})-f(s_{i})|<\frac{\epsilon}{3}$.
Write $J\setminus\sigma$ as a disjoint countable union of open intervals
$O_{m}=(a_{m},b_{m})$, ordered so that
$\mathrm{diam}(O_{1})\geq\mathrm{diam}(O_{2})\geq\dots$. To avoid special
cases we allow $O_{m}=\emptyset$ for large $m$. Fix $M$ such that
$\sum_{m=M+1}^{\infty}\mathrm{diam}(O_{m})<\delta_{0}$.
Let
$g=\max_{1\leq m\leq M}\frac{|f(b_{m})-f(a_{m})|}{b_{m}-a_{m}}$
be the largest slope of the function $\iota(f)$ on the intervals
$O_{1},\dots,O_{M}$. Let $\displaystyle\delta_{1}=\frac{\epsilon}{3g}$. (If
$g=0$ then setting $\delta_{1}=1$ will suffice.) Then if
$\\{(s_{i},t_{i})\\}_{i=1}^{n}$ is a finite set of disjoint intervals with
$(s_{i},t_{i})\subseteq\bigcup_{m=1}^{M}O_{m}$ and
$\sum_{i=1}^{n}|t_{i}-s_{i}|<\delta_{1}$, then
$\sum_{i=1}^{n}|f(t_{i})-f(s_{i})|<\frac{\epsilon}{3}$.
Let $\delta=\min(\delta_{0},\delta_{1})$. Suppose that
$\\{(s_{i},t_{i})\\}_{i=1}^{n}$ is a finite set of disjoint subintervals of
$J$ such that $\sum_{i=1}^{n}|t_{i}-s_{i}|<\delta$. For each $i$, the interval
$(s_{i},t_{i})$ falls into at least one of the following cases:
1. (1)
both $s_{i}$ and $t_{i}$ are in $\sigma$.
2. (2)
$(s_{i},t_{i})\subseteq O_{m}$ for some $m$.
3. (3)
$s_{i}\not\in\sigma$ and $t_{i}\in\sigma$.
4. (4)
$s_{i}\in\sigma$ and $t_{i}\not\in\sigma$.
5. (5)
$s_{i}\not\in\sigma$ and $t_{i}\not\in\sigma$, but
$(s_{i},t_{i})\cap\sigma\neq\emptyset$.
If $(s_{i},t_{i})$ falls into Case 3 (but not Case 2) then we shall replace
the interval $(s_{i},t_{i})$ with a pair of intervals $(s_{i},u_{i})$ and
$(u_{i},t_{i})$ where $u_{i}$ is the smallest element of $\sigma$ which is
larger than $s_{i}$. Then $(s_{i},u_{i})$ falls into Case 2 and
$(u_{i},t_{i})$ falls into Case 1. The two smaller intervals have the same
total length as the original interval and will contribute at least as much to
the variation as did $(s_{i},t_{i})$.
Intervals that fall into Case 4 or Case 5 may each be split in a similar
fashion into two, or perhaps three, smaller intervals, each of which falls
into either Case 1 or Case 2. By applying this procedure then we may restrict
our attention to the situation in which every interval $(s_{i},t_{i})$ falls
into either Case 1 or Case 2.
We may therefore write $\\{1,2,\dots,n\\}$ as a union of three (possibly
nondisjoint) sets
$\displaystyle I_{1}$
$\displaystyle=\\{i\thinspace:\thinspace\hbox{$s_{i},t_{i}\in\sigma$}\\},$
$\displaystyle I_{2}$
$\displaystyle=\\{i\thinspace:\thinspace\hbox{$(s_{i},t_{i})\subseteq O_{m}$
for some $m\leq M$}\\},$ $\displaystyle I_{3}$
$\displaystyle=\\{i\thinspace:\thinspace\hbox{$(s_{i},t_{i})\subseteq O_{m}$
for some $m>M$}\\}.$
From the above construction both $\displaystyle\sum_{i\in
I_{1}}|f(t_{i})-f(s_{i})|$ and $\displaystyle\sum_{i\in
I_{2}}|f(t_{i})-f(s_{i})|$ are bounded above by
$\displaystyle\frac{\epsilon}{3}$, so it just remains to find a similar bound
for the terms with indices in $I_{3}$.
Suppose that for some $m>M$, the intervals
$(s_{i_{1}},t_{i_{1}}),\dots,(s_{i_{\ell}},t_{i_{\ell}})$ all lie in $O_{m}$.
As the subintervals are disjoint and $\iota(f)$ is linear on $O_{m}$, we have
that
$\sum_{k=1}^{\ell}|f(t_{i_{k}})-f(s_{i_{k}})|\leq|f(b_{m})-f(a_{m})|.$
Since $\sum_{m=M+1}^{\infty}|b_{m}-a_{m}|<\delta_{0}$ and
$a_{m},b_{m}\in\sigma$ for all $m$, we can now conclude that
$\sum_{i\in I_{3}}|f(t_{i})-f(s_{i})|<\frac{\epsilon}{3}$
and hence
$\sum_{i=1}^{n}|f(t_{i})-f(s_{i})|<\epsilon.$
∎
###### Corollary 2.2.
${AC}(\sigma)\subseteq{BV}(\sigma)$.
###### Proof.
Suppose that $f\in{AC}(\sigma)$. Then $\iota(f)\in{AC}(J)$ and hence
$\iota(f)\in{BV}(J)$. Using Proposition 2.2 of [1], we may deduce that
$f=\iota(f)|\sigma\in{BV}(\sigma)$. ∎
Let $\mathcal{P}$ denote the algebra of polynomials in one variable,
considered as functions on the set $\sigma$.
###### Corollary 2.3.
$\mathcal{P}\subseteq{AC}(\sigma)\subseteq\mathrm{cl}(\mathcal{P})$ (where the
closure is taken in ${BV}(\sigma)$).
###### Proof.
Note first that, since every polynomial has bounded derivative on $J$, the set
of polynomials lies in ${AC}(\sigma)$.
Suppose then that $f\in{AC}(\sigma)$ and that $\epsilon>0$. Since
${AC}(J)\cong L^{1}(J)\oplus\mathbb{C}$ and the polynomials are dense in
$L^{1}(J)$, it is easy to prove that the polynomials are dense in ${AC}(J)$.
Thus there exists a polynomial $p$ such that
$\left\lVert\iota(f)-p\right\rVert_{{BV}(J)}<\epsilon$. But in this case
$\left\lVert f-p\right\rVert_{{BV}(\sigma)}<\epsilon$ too. ∎
###### Corollary 2.4.
${AC}(\sigma)$ is a Banach subalgebra of ${BV}(\sigma)$.
###### Proof.
We first show that ${AC}(\sigma)$ is complete. Suppose then
$\\{f_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in ${AC}(\sigma)$. Then
$\\{\iota(f_{n})\\}$ is a Cauchy sequence in the Banach algebra ${AC}(J)$ and
hence it converges to some function $F\in{AC}(J)$. As noted earlier, it
follows from the definition that $F|\sigma\in AC(\sigma)$ and as
$f_{n}=\iota(f_{n})|\sigma\to F|\sigma$ in ${BV}(\sigma)$, that ${AC}(\sigma)$
is complete. That ${AC}(\sigma)$ is an algebra follows from the fact that the
polynomials form a dense subalgebra. ∎
In the classical case it is well-known that there are many continuous
functions of bounded variation which are not absolutely continuous. While this
obviously extends to the case of any $\sigma$ which contains an interval, the
situation is less clear if $\sigma$ has empty interior.
###### Question 2.5.
For which compact sets $\sigma$ is ${AC}(\sigma)=C(\sigma)\cap{BV}(\sigma)$?
###### Example 2.6.
Let $\sigma=\\{\frac{1}{n}\\}_{n=1}^{\infty}\cup\\{0\\}$. Since $\sigma$ has
only one limit point, $f\in{BV}(\sigma)$ is continuous if and only if
$f(0)=\lim_{n\to\infty}f(\frac{1}{n})$. For such a function, $\iota(f)$ is
differentiable on $J=[0,1]$ except possibly at the points of $\sigma$, and
$\iota(f)(x)=f(0)+\int_{0}^{x}\iota(f)^{\prime}(t)\,dt$. By the classical
characterization of absolutely continuous functions as indefinite integrals,
this implies that $\iota(f)\in{AC}(J)$ and hence $f\in{AC}(\sigma)$.
###### Example 2.7.
Let $\sigma$ be the standard middle third Cantor set taken inside $J=[0,1]$.
Let $F:[0,1]\to\mathbb{R}$ be the Cantor function, which is the standard
example of a function which lies in $C[0,1]\cap{BV}[0,1]$ but not in
${AC}[0,1]$. Note that $\iota(F|\sigma)=F$. This implies that
$F|\sigma\not\in{AC}(\sigma)$ — but of course $F|\sigma\in
C(\sigma)\cap{BV}(\sigma)$.
## 3\. Preliminaries
The concept of two-dimensional variation which we shall need was originally
developed in [1]. In that paper two-dimensional variation was defined in terms
of the variation along continuous parameterized curves in the plane. It was
shown in [1] that it is sufficient to work with piecewise linear curves, and
almost all proofs use this fact. In hindsight, using general continuous curves
in the definition adds an unnecessary level of complication to the theory and
we feel that it is better to work entirely with piecewise linear curves
determined by a finite ordered list of points. For the aid of the reader, we
present this simplified development below.
Suppose that $\sigma$ is a nonempty compact subset of the plane and that
$f:\sigma\to\mathbb{C}$. Suppose that
$S=\bigl{[}{\boldsymbol{x}}_{0},{\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{n}\bigr{]}$
is a finite ordered list of elements of $\sigma$. Note that the elements of
such a list do not need to be distinct. To avoid trivialities we shall assume
that $n\geq 1$. Let $\gamma_{S}$ denote the piecewise linear curve joining the
points of $S$ in order. We define the curve variation of $f$ on the set $S$ to
be
(3.1) $\operatorname{\rm cvar}(f,S)=\sum_{i=1}^{n}\left\lvert
f({\boldsymbol{x}}_{i})-f({\boldsymbol{x}}_{i-1})\right\rvert.$
Suppose that $\ell$ is a line in the plane. We say that
$\overline{\vphantom{\vbox
to5.16663pt{}}{\boldsymbol{x}}_{j}\,{\boldsymbol{x}}_{j+1}}$, the line segment
joining ${\boldsymbol{x}}_{j}$ to ${\boldsymbol{x}}_{j+1}$, is a crossing
segment of $S$ on $\ell$ if any one of the following holds:
1. (1)
${\boldsymbol{x}}_{j}$ and ${\boldsymbol{x}}_{j+1}$ lie on (strictly) opposite
sides of $\ell$.
2. (2)
$j=0$ and ${\boldsymbol{x}}_{j}\in\ell$.
3. (3)
$j>0$, ${\boldsymbol{x}}_{j}\in\ell$ and ${\boldsymbol{x}}_{j-1}\not\in\ell$.
4. (4)
$j=n-1$, ${\boldsymbol{x}}_{j}\not\in\ell$ and
${\boldsymbol{x}}_{j+1}\in\ell$.
Let $\operatorname{vf}(S,\ell)$ denote the number of crossing segments of $S$
on $\ell$. Define the variation factor of $S$ to be
$\operatorname{vf}(S)=\max_{\ell}\operatorname{vf}(S,\ell).$
Clearly $1\leq\operatorname{vf}(S)\leq n$. Informally, $\operatorname{vf}(S)$
may be thought of as the maximum number of times any line crosses
$\gamma_{S}$.
For completeness one may include the case
$S=\bigl{[}{\boldsymbol{x}}_{0}\bigr{]}$ by setting $\operatorname{\rm
cvar}(f,\bigl{[}{\boldsymbol{x}}_{0}\bigr{]})=0$ and
$\operatorname{vf}(\bigl{[}{\boldsymbol{x}}_{0}\bigr{]},\ell)=1$ whenever
${\boldsymbol{x}}_{0}\in\ell$.
The two-dimensional variation of a function $f:\sigma\rightarrow\mathbb{C}$ is
defined to be
(3.2) $\operatorname*{var}(f,\sigma)=\sup_{S}\frac{\operatorname{\rm
cvar}(f,S)}{\operatorname{vf}(S)},$
where the supremum is taken over all finite ordered lists of elements of
$\sigma$. The variation norm is
$\left\lVert f\right\rVert_{{BV}(\sigma)}=\left\lVert
f\right\rVert_{\infty}+\operatorname*{var}(f,\sigma)$
and this is used to define the set of functions of bounded variation on
$\sigma$,
${BV}(\sigma)=\\{f:\sigma\to\mathbb{C}\thinspace:\thinspace\left\lVert
f\right\rVert_{{BV}(\sigma)}<\infty\\}.$
It is shown in [1] that ${BV}(\sigma)$ is a Banach algebra under pointwise
operations.
The set $\mathcal{P}_{2}$ of polynomials in two variables in the plane may be
thought of as functions $p:\mathbb{R}^{2}\to\mathbb{C}$ of the form
$p(x,y)=\sum_{m,n=0}^{N}c_{mn}\,x^{m}y^{n}$. As is shown in section 3.6 of
[1], the simple polynomials $p_{1}(x,y)=x$ and $p_{2}(x,y)=y$ both lie in
${BV}(\sigma)$ and so the fact that ${BV}(\sigma)$ is an algebra implies that
$\mathcal{P}_{2}$ is a subalgebra of ${BV}(\sigma)$. (More formally we should
speak of the restrictions of elements of $\mathcal{P}_{2}$ to the set
$\sigma$. Here and throughout the paper we shall often omit explicit mention
of such restrictions if there is not risk of confusion.)
We define ${AC}(\sigma)$ as being the closure of $\mathcal{P}_{2}$ in
${BV}(\sigma)$ norm. We call functions in ${AC}(\sigma)$ the _absolutely
continuous functions with respect to $\sigma$_. Corollary 2.3 shows that this
definition is consistent with the natural one-dimensional definition given
earlier. Since $BV$ convergence implies uniform convergence, it is clear that
all absolutely continuous functions are in fact continuous.
It is immediately clear from the definitions that these quantities are
invariant under affine transformations of the plane. More precisely, if
$\phi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is an invertible affine transformation,
then $f\in{BV}(\sigma)$ if and only if $f\circ\phi^{-1}\in{BV}(\phi(\sigma))$,
and $\left\lVert f\circ\phi^{-1}\right\rVert_{{BV}(\phi(\sigma))}=\left\lVert
f\right\rVert_{{BV}(\sigma)}$. Importantly, invertible affine transformations
preserve absolute continuity [1, Theorem 4.1].
It is also clear that if $\sigma_{1}\subseteq\sigma$ then $\left\lVert
f\right\rVert_{{BV}(\sigma_{1})}\leq\left\lVert f\right\rVert_{{BV}(\sigma)}$
and hence if $f\in{AC}(\sigma)$ then $f|\sigma_{1}\in{AC}(\sigma_{1})$. Two
obvious, and related, questions concern implications in the reverse direction:
###### Question 3.1.
If $\sigma_{1}\subseteq\sigma$ and $f\in{AC}(\sigma_{1})$, is there an
extension $\hat{f}\in{AC}(\sigma)$ of $f$?
###### Question 3.2.
Suppose $\sigma=\sigma_{1}\cup\sigma_{2}$ and $f:\sigma\to\mathbb{C}$. If
$f|\sigma_{1}\in{AC}(\sigma_{1})$ and $f|\sigma_{2}\in{AC}(\sigma_{2})$, is
$f\in{AC}(\sigma)$?
The first of these questions is still open, although we shall discuss some
special cases (for which there is a positive answer) in section 7. As the
following example shows, it is easy to see that the answer to the second
question is negative in general, even when the sets are subsets of
$\mathbb{R}$.
###### Example 3.3.
Let $\sigma_{1}=\\{0,1,\frac{1}{3},\frac{1}{5},\dots\\}$, let
$\sigma_{2}=\\{0,\frac{1}{2},\frac{1}{4},\frac{1}{6},\dots\\}$ and let
$\sigma=\sigma_{1}\cup\sigma_{2}$. Let
$f(x)=\begin{cases}0,&x=0,\\\ (-1)^{k}x,&\hbox{$x=\frac{1}{k}$ for some
positive integer $k$.}\end{cases}$
It is easy check that $f|\sigma_{1}\in{AC}(\sigma_{1})$,
$f|\sigma_{2}\in{AC}(\sigma_{2})$, but $f$ is not even of bounded variation on
$\sigma$.
If one requires a little more of the sets $\sigma_{1}$ and $\sigma_{2}$, then
this problem can not occur. If, for example, $\sigma_{1}$ and $\sigma_{2}$ are
disjoint, then the absolute continuity of a function on each of the two
subsets implies its absolute continuity on the union. Although one can prove
this directly, we shall give this as a corollary of the Patching Lemma in
section 5. Note that the above example shows that there is no way in general
of controlling the variation of $f$ on $\sigma$ in terms of the variation of
$f$ on the two subsets.
We shall say that two nonempty compact sets $\sigma_{1}$ and $\sigma_{2}$ join
convexly if for all ${\boldsymbol{x}}\in\sigma_{1}$ and
${\boldsymbol{y}}\in\sigma_{2}$ there exists ${\boldsymbol{w}}$ on the line
joining ${\boldsymbol{x}}$ and ${\boldsymbol{y}}$ with
${\boldsymbol{w}}\in\sigma_{1}\cap\sigma_{2}$. Clearly if
$\sigma_{1}\cup\sigma_{2}$ is convex then the two sets join convexly. The
following result was stated in [3] for convex sets, although the proof only
uses the weaker property of joining convexly. This slightly more general
version will be needed in the later sections.
###### Theorem 3.4.
[3, Theorem 3.1] Suppose that $\sigma_{1},\sigma_{2}\subseteq\mathbb{C}$ are
nonempty compact sets which are disjoint except at their boundaries and that
$\sigma_{1}$ and $\sigma_{2}$ join convexly. Let
$\sigma=\sigma_{1}\cup\sigma_{2}$. If $f:\sigma\to\mathbb{C}$, then
$\max\\{\operatorname*{var}(f,\sigma_{1}),\operatorname*{var}(f,\sigma_{2})\\}\leq\operatorname*{var}(f,\sigma)\leq\operatorname*{var}(f,\sigma_{1})+\operatorname*{var}(f,\sigma_{2})$
and hence
$\max\\{\left\lVert f\right\rVert_{{BV}(\sigma_{1})},\left\lVert
f\right\rVert_{{BV}(\sigma_{2})}\\}\leq\left\lVert
f\right\rVert_{{BV}(\sigma)}\leq\left\lVert
f\right\rVert_{{BV}(\sigma_{1})}+\left\lVert
f\right\rVert_{{BV}(\sigma_{2})}.$
Thus, if $f|\sigma_{1}\in{BV}(\sigma_{1})$ and
$f|\sigma_{2}\in{BV}(\sigma_{2})$, then $f\in{BV}(\sigma)$.
We shall also need to consider some further spaces of functions. For a set
$\sigma$ containing at least 2 elements, the Banach algebra of Lipschitz
functions on $\sigma$, ${\mathrm{Lip}}(\sigma)$, is the space of all functions
$f:\sigma\to\mathbb{C}$ such that $\left\lVert
f\right\rVert_{{\mathrm{Lip}}(\sigma)}=\left\lVert
f\right\rVert_{\infty}+L_{\sigma}(f)$ is finite, where
$L_{\sigma}(f)=\sup\Bigl{\\{}\frac{|f({\boldsymbol{x}})-f({\boldsymbol{x}}^{\prime})|}{\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert}\thinspace:\thinspace{\boldsymbol{x}}\neq{\boldsymbol{x}}^{\prime}\in\sigma\Bigr{\\}}.$
As is shown in [1], ${\mathrm{Lip}}(\sigma)\subseteq{BV}(\sigma)$, but in
general ${\mathrm{Lip}}(\sigma)\not\subseteq{AC}(\sigma)$.
Let $k\in\\{1,2,3,\dots,\infty\\}$. We shall say that $f\in C^{k}(\sigma)$ if
there exists an open neighbourhood $U$ of $\sigma$ and an extension $F$ of $f$
to $U$ such that all the partial derivatives of $F$ of order less than or
equal to $k$ are continuous on $U$.
We end this section with a technical result which we shall need throughout the
paper. This simple observation, which occurs implicitly in [2], essentially
says that if one deletes elements from a list then one can only decrease the
variation factor.
###### Proposition 3.5.
Let $S$ be an ordered list of elements of $\sigma$, and let $S^{+}$ be a list
formed by adding an additional element into the list at some point. Then for
any line in the plane
$\operatorname{vf}(S,\ell)\leq\operatorname{vf}(S^{+},\ell)$ and hence
$\operatorname{vf}(S)\leq\operatorname{vf}(S^{+})$.
###### Proof.
Let $S=\bigl{[}{\boldsymbol{x}}_{0},\dots,{\boldsymbol{x}}_{n}\bigr{]}$ and
let $\ell$ be any line in the plane. Suppose first that
$S^{+}=\bigl{[}{\boldsymbol{w}},{\boldsymbol{x}}_{0},\dots,{\boldsymbol{x}}_{n}\bigr{]}$.
If ${\boldsymbol{x}}_{0}\not\in\ell$, or if ${\boldsymbol{x}}_{0}\in\ell$ and
${\boldsymbol{w}}\not\in\ell$, then each of the original line segments in $S$
retains it original status as either a crossing or non-crossing segment of
$S^{+}$ on $\ell$. If ${\boldsymbol{x}}_{0}\in\ell$ and
${\boldsymbol{w}}\in\ell$ then $\overline{\vphantom{\vbox
to5.16663pt{}}{\boldsymbol{x}}_{0}\,{\boldsymbol{x}}_{1}}$ is no longer a
crossing segment of $S^{+}$ on $\ell$, but $\overline{\vphantom{\vbox
to5.16663pt{}}{\boldsymbol{w}}\,{\boldsymbol{x}}_{0}}$ is. In either case, the
number of crossing segments is not decreased.
The proof in the case that
$S^{+}=\bigl{[}{\boldsymbol{x}}_{0},\dots,{\boldsymbol{x}}_{n},{\boldsymbol{w}}\bigr{]}$
is almost identical. The remaining case, where the additional point
${\boldsymbol{w}}$ is added between the $j$ and $(j+1)$st elements of $S$,
involves checking a slightly longer list of possibilities. More specifically,
for each of the cases 1–4 above, one needs to check that no matter where
${\boldsymbol{w}}$ lies, either $\overline{\vphantom{\vbox
to5.16663pt{}}{\boldsymbol{x}}_{j}\,{\boldsymbol{w}}}$ or
$\overline{\vphantom{\vbox
to5.16663pt{}}{\boldsymbol{w}}\,{\boldsymbol{x}}_{j+1}}$ is a crossing segment
of $S^{+}$ on $\ell$. ∎
## 4\. $C^{2}(\sigma)\subseteq{AC}(\sigma)$
Throughout this section we shall assume that $\sigma$ is a nonempty compact
subset of the plane. As was noted earlier, ${AC}(\sigma)\subseteq C(\sigma)$.
It is natural to ask what degree of smoothness is required to ensure that a
function $f$ is in ${AC}(\sigma)$. In the one-dimensional situation, it is
sufficient that $f$ be Lipschitz, but as is shown in [1, Example 4.13] this is
not longer the case for general $\sigma\subseteq\mathbb{C}$. Nonetheless,
convergence in Lipschitz norm is a useful tool in showing that a function is
absolutely continuous in this context. The following is a small rewording of
[1, Corollary 3.17].
###### Lemma 4.1.
Suppose that $\\{f_{n}\\}_{n=1}^{\infty}\subseteq{AC}(\sigma)$ and that
$f\in{\mathrm{Lip}}(\sigma)$. If $\left\lVert
f-f_{n}\right\rVert_{{\mathrm{Lip}}(\sigma)}\to 0$ then $\left\lVert
f-f_{n}\right\rVert_{{BV}(\sigma)}\to 0$ and hence $f\in{AC}(\sigma)$.
The following result appears as [1, Lemma 4.6].
###### Proposition 4.2.
Let $R=J\times K$ be a rectangle in $\mathbb{C}$. If $f\in C^{2}(R)$ then
$f\in{AC}(R)$.
The ‘proof’ in [1] suggests that this follows by approximating $f$ in
Lipschitz norm by a sequence of polynomials. While this is true, seeing how
this can be done is a little delicate.
###### Proof.
It suffices to treat the real-valued case. We may take $R=[0,1]\times[0,1]$.
The general case follows from a simple change of variables.
Suppose that $f\in C^{2}(R)$. Fix $\epsilon>0$. Then there exist polynomials
(in two variables) $g^{xx}$, $g^{xy}$ and $g^{yy}$ such that on the square
$\left\lVert\frac{\partial^{2}f}{\partial
x^{2}}-g^{xx}\right\rVert_{\infty}<\epsilon,\qquad\left\lVert\frac{\partial^{2}f}{\partial
x\partial
y}-g^{xy}\right\rVert_{\infty}<\epsilon,\qquad\left\lVert\frac{\partial^{2}f}{\partial
y^{2}}-g^{yy}\right\rVert_{\infty}<\epsilon.$
Now define $h^{x}$ for $(x,y)\in R$ by
$h^{x}(x,y)=\frac{\partial f}{\partial
x}(0,0)+\int_{0}^{x}g^{xx}(t,0)\,dt+\int_{0}^{y}g^{xy}(x,s)\,ds.$
Note that $h^{x}$ is a polynomial, and that for $(x,y)\in R$,
$\displaystyle|\frac{\partial f}{\partial x}(x,y)-h^{x}(x,y)|$
$\displaystyle=\Bigl{|}\frac{\partial f}{\partial
x}(0,0)+\int_{0}^{x}\frac{\partial^{2}f}{\partial
x^{2}}(t,0)\,dt+\int_{0}^{y}\frac{\partial^{2}f}{\partial x\partial
y}(x,s)\,ds-h^{x}(x,y)\Bigr{|}$
$\displaystyle\leq\int_{0}^{x}\bigl{|}\frac{\partial^{2}f}{\partial
x^{2}}(t,0)-g^{xx}(t,0)\bigr{|}\,dt+\int_{0}^{y}\bigl{|}\frac{\partial^{2}f}{\partial
x\partial y}(x,s)-g^{xy}(x,s)\bigr{|}\,ds$ (4.1) $\displaystyle<2\epsilon.$
Thus $\left\lVert\frac{\partial f}{\partial
x}-h^{x}\right\rVert_{\infty}<2\epsilon$. Similarly, define the polynomial
$h^{y}$ by
$h^{y}(x,y)=\frac{\partial f}{\partial
y}(0,0)+\int_{0}^{x}g^{xy}(t,y)\,dt+\int_{0}^{y}g^{yy}(0,s)\,ds.$
A similar calculation shows that $\left\lVert\frac{\partial f}{\partial
y}-h^{y}\right\rVert_{\infty}<2\epsilon$. Note also that
$\frac{\partial h^{y}}{\partial x}=g^{xy}.$
Now define a polynomial $p$ by
$p(x,y)=f(0,0)+\int_{0}^{x}h^{x}(t,0)\,dt+\int_{0}^{y}h^{y}(x,s)\,ds.$
Repeating the calculation in (4) shows that $\left\lVert
f-p\right\rVert_{\infty}<4\epsilon$. Next, for $(x,y)\in R$,
$\displaystyle\frac{\partial p}{\partial x}$
$\displaystyle=h^{x}(x,0)+\int_{0}^{y}\frac{\partial h^{y}}{\partial
x}(x,s)\,ds$ $\displaystyle=h^{x}(x,0)+\int_{0}^{y}g^{xy}(x,s)\,ds$
and
$\frac{\partial p}{\partial y}=h^{y}(x,y).$
Thus
$\displaystyle\Bigl{|}\frac{\partial f}{\partial x}(x,y)-\frac{\partial
p}{\partial x}(x,y)\Bigr{|}$ $\displaystyle=\Bigl{|}\frac{\partial f}{\partial
x}(x,0)+\int_{0}^{y}\frac{\partial^{2}f}{\partial x\partial
y}(x,s)\,ds-h^{x}(x,0)-\int_{0}^{y}g^{xy}(x,s)\,ds\Bigr{|}$
$\displaystyle\leq\bigl{|}\frac{\partial f}{\partial
x}(x,0)-h^{x}(x,0)\bigr{|}+\int_{0}^{y}\bigl{|}\frac{\partial^{2}f}{\partial
x\partial y}(x,y)-g^{xy}(x,s)\bigr{|}\,ds$ $\displaystyle<3\epsilon$
and
$\Bigl{|}\frac{\partial f}{\partial y}(x,y)-\frac{\partial p}{\partial
y}(x,y)\Bigr{|}=\Bigl{|}\frac{\partial f}{\partial
y}(x,y)-h^{y}(x,y)\Bigr{|}<2\epsilon.$
Now given ${\boldsymbol{x}}\neq{\boldsymbol{x}}^{\prime}\in R$, let
${\boldsymbol{u}}=({\boldsymbol{x}}-{\boldsymbol{x}}^{\prime})/\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert$.
It follows from the Mean Value Theorem that there exists ${\boldsymbol{\xi}}$
on the line segment joining ${\boldsymbol{x}}$ and ${\boldsymbol{x}}^{\prime}$
such that
$\frac{|(f-p)({\boldsymbol{x}})-(f-p)({\boldsymbol{x}}^{\prime})|}{\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert}=|\nabla(f-p)({\boldsymbol{\xi}})\cdot{\boldsymbol{u}}|\leq\sqrt{13}\epsilon$
and hence $\left\lVert
f-p\right\rVert_{{\mathrm{Lip}}(R)}<(4+\sqrt{13})\epsilon$.
We can therefore find a sequence $p_{n}$ of polynomials such that
$\lim_{n\to\infty}\left\lVert f-p_{n}\right\rVert_{{\mathrm{Lip}}(R)}=0$
and hence by Lemma 4.1, $f\in{AC}(R)$. ∎
It is worth noting that it is vital in this result that $f$ be differentiable
on the boundary of $R$, not just in the interior. It is easy to construct
continuous functions on $R$ which are $C^{\infty}$ on the the interior of $R$,
but which are not absolutely continuous on the whole rectangle.
###### Theorem 4.3.
Let $\sigma$ be a nonempty compact subset of the plane. Then
$C^{2}(\sigma)\subseteq{AC}(\sigma)$.
###### Proof.
Suppose that $f\in C^{2}(\sigma)$. By definition $f$ has an extension to a
$C^{2}$ function defined on some open neighbourhood of $\sigma$. Taylor’s
Theorem then implies that $f$ satisfies the hypotheses of the Whitney
Extension Theorem [9] and hence $f$ may be extended to a $C^{2}$ function $F$
on $\mathbb{R}^{2}$. (It is perhaps worth noting that this extension might be
different to the hypothesised one.)
Let $R$ denote a rectangle containing $\sigma$. By the previous proposition
$F|R$ lies in ${AC}(R)$, and hence, $f=F|\sigma$ lies in ${AC}(\sigma)$. ∎
###### Corollary 4.4.
The space $C^{2}(\sigma)$ is dense in ${AC}(\sigma)$.
We shall see in Section 6 that the hypothesis can be weakened to the
requirement that $f\in C^{1}$.
## 5\. Being absolutely continuous is a local property
In this section we shall show that being in ${AC}(\sigma)$ is determined by
the behaviour of the function in a neighbourhood of each point. We shall say
that a set $U$ is a compact neighbourhood of a point
${\boldsymbol{x}}\in\sigma$ (with respect to $\sigma$) if there exists an open
set $V$ containing ${\boldsymbol{x}}$ such that $U=\sigma\cap\overline{V}$.
###### Theorem 5.1.
[Patching Lemma] Suppose that $f:\sigma\to\mathbb{C}$. Then $f\in{AC}(\sigma)$
if and only if for every point ${\boldsymbol{x}}\in\sigma$ there exists a
compact neighbourhood $U_{\boldsymbol{x}}$ of ${\boldsymbol{x}}$ in $\sigma$
such that $f|U_{\boldsymbol{x}}\in{AC}(U_{\boldsymbol{x}})$.
The main step in proving the Patching Lemma is the following extension result.
It might be noted that this result would follow from Lemma 3.2 of [3]. The
proof of that lemma however uses the fact that ${\mathrm{CTPP}}(\sigma)$, the
space of continuous piecewise planar functions, is dense in ${AC}(\sigma)$.
Unfortunately, the proof of this given in [1] is rather inadequate. Since the
more complete proof of the density of ${\mathrm{CTPP}}(\sigma)$ which we
present in Section 6 depends on Theorem 5.1, to avoid circularity we need to
proceed here directly from the definition.
As usual we shall let $\mathop{\mathrm{int}}(R)$ denote the interior of a set
$R$, and let $\operatorname{supp}g$ denote the support of a function $g$.
###### Lemma 5.2.
Let $\emptyset\neq\sigma\subseteq\mathbb{R}^{2}$ be compact. Suppose that $R$
is a closed rectangle in $\mathbb{R}^{2}$ with
$\sigma_{1}:=R\cap\sigma\neq\emptyset$. Suppose that $g\in{AC}(\sigma_{1})$
with $\operatorname{supp}g\subset\mathop{\mathrm{int}}(R)$. Then the function
$\tilde{g}({\boldsymbol{x}})=\begin{cases}g({\boldsymbol{x}}),&\hbox{${\boldsymbol{x}}\in\sigma_{1}$,}\\\
0,&\hbox{${\boldsymbol{x}}\in\sigma\setminus\sigma_{1}$.}\end{cases}$
lies in ${AC}(\sigma)$.
$R$$\sigma$$\operatorname{supp}g$ Figure 1. The setting for Lemma 5.2.
###### Proof.
By the affine invariance of absolute continuity [1, Theorem 4.1], it suffices
to consider the case that $R=[0,1]\times[0,1]$. Choose a closed square
$R_{0}=[a,b]\times[a,b]$ (with sides parallel to those of $R$) such that
$\operatorname{supp}g\subseteq R_{0}\subseteq\mathop{\mathrm{int}}(R)$. Let
$\chi_{1}:[0,1]\to[0,1]$ be a $C^{\infty}$ bump function which is zero at and
near the endpoints, 1 on an open neighbourhood of $[a,b]$ and monotonic on the
two remaining parts of $[0,1]$. Let $\chi:R\to[0,1]$,
$\chi(x,y)=\chi_{1}(x)\chi_{1}(y)$. Then $\chi\in C^{\infty}(R)$ and, using
[1, Proposition 4.4], $\left\lVert\chi\right\rVert_{{BV}(R)}\leq 3\times 3=9$.
Fix $\epsilon>0$. Since $g\in{AC}(\sigma_{1})$, there exists a polynomial
$p\in C^{\infty}(\sigma_{1})$ such that $\left\lVert
g-p\right\rVert_{{BV}(\sigma_{1})}<\epsilon/9$. Clearly $\chi p\in
C^{\infty}(\sigma_{1})$ and
$\left\lVert g-\chi
p\right\rVert_{{BV}(\sigma_{1})}=\left\lVert\chi(g-p)\right\rVert_{{BV}(\sigma_{1})}\leq\left\lVert\chi\right\rVert_{{BV}(\sigma_{1})}\left\lVert
g-p\right\rVert_{{BV}(\sigma_{1})}<\epsilon.$
Let $\tilde{p}$ denote the extension of $\chi p$ to $\sigma$ determined by
setting
${\tilde{p}}({\boldsymbol{x}})=\begin{cases}\chi
p({\boldsymbol{x}}),&\hbox{${\boldsymbol{x}}\in\sigma_{1}$,}\\\
0,&\hbox{${\boldsymbol{x}}\in\sigma\setminus\sigma_{1}$.}\end{cases}$
Note that $\tilde{p}\in C^{\infty}(\sigma)\subseteq{AC}(\sigma)$. Let
$\delta=\tilde{g}-\tilde{p}$. We shall now show that
$\left\lVert\delta\right\rVert_{{BV}({\sigma})}\leq 5\epsilon$.
Let $S=[{\boldsymbol{x}}_{0},{\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{m}]$
be an ordered list of elements in $\sigma$. Partition the indices $1,\dots,m$
into
$\displaystyle J_{1}$
$\displaystyle=\\{j\thinspace:\thinspace\hbox{${\boldsymbol{x}}_{j-1}\in\sigma_{1}$
and ${\boldsymbol{x}}_{j}\in\sigma_{1}$}\\},$ $\displaystyle J_{2}$
$\displaystyle=\\{j\thinspace:\thinspace\hbox{${\boldsymbol{x}}_{j-1}\not\in\sigma_{1}$
and ${\boldsymbol{x}}_{j}\not\in\sigma_{1}$}\\},$ $\displaystyle J_{3}$
$\displaystyle=\\{1,\dots,m\\}\setminus(J_{1}\cup J_{2})$
so that (interpreting empty sums as being zero)
$\displaystyle\sum_{j=1}^{m}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|$
$\displaystyle=\sum_{j\in
J_{1}}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|+\sum_{j\in
J_{2}}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|$
$\displaystyle\qquad\qquad\qquad+\sum_{j\in
J_{3}}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|$ (5.1)
$\displaystyle\leq\sum_{j\in
J_{1}}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|+|J_{3}|\left\lVert\delta\right\rVert_{\infty}.$
${\boldsymbol{x}}_{0}$${\boldsymbol{x}}_{1}$${\boldsymbol{x}}_{2}$${\boldsymbol{x}}_{3}$${\boldsymbol{x}}_{4}$${\boldsymbol{x}}_{5}$${\boldsymbol{x}}_{6}$${\boldsymbol{x}}_{7}$${\boldsymbol{x}}_{8}$$R$$\sigma$
Figure 2. In this example $J_{1}=\\{1,5\\}$, $J_{2}=\\{3,7\\}$,
$J_{3}=\\{2,4,6,8\\}$ and
$S_{1}=[{\boldsymbol{x}}_{0},{\boldsymbol{x}}_{1},{\boldsymbol{x}}_{4},{\boldsymbol{x}}_{5}]$.
Suppose first that $J_{1}\neq\emptyset$. Let
$S_{1}=[{\boldsymbol{x}}_{j_{0}},\dots,{\boldsymbol{x}}_{j_{\ell}}]$ be the
sublist of $S$ consisting of all the ${\boldsymbol{x}}_{j}$ such that
${\boldsymbol{x}}_{j}\in\sigma_{1}$ and at least one of
${\boldsymbol{x}}_{j-1}$ and ${\boldsymbol{x}}_{j+1}$ also lie in
$\sigma_{1}$. Note that $S_{1}$ is a nonempty list and that, by Lemma 3.5,
$\operatorname{vf}(S_{1})\leq\operatorname{vf}(S)$. Thus
$\displaystyle\sum_{j\in
J_{1}}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|$
$\displaystyle\leq\sum_{i=1}^{\ell}|\delta({\boldsymbol{x}}_{j_{i}})-\delta({\boldsymbol{x}}_{j_{i-1}})|$
$\displaystyle\leq\left\lVert\delta\right\rVert_{{BV}(\sigma_{1})}\operatorname{vf}(S_{1})$
$\displaystyle\leq\left\lVert\delta\right\rVert_{{BV}(\sigma_{1})}\operatorname{vf}(S).$
Of course if $J_{1}=\emptyset$ then this estimate holds trivially.
We now need to deal with the edges corresponding to $J_{3}$. Note that each
element of $J_{3}$ corresponds to $\gamma_{S}$ crossing one of the four edge
lines of $R$. It follows that $\operatorname{vf}(S)\geq|J_{3}|/4$ or
$|J_{3}|\leq 4\operatorname{vf}(S)$. Substituting these into Equation 5 gives
that
$\displaystyle\frac{1}{\operatorname{vf}(S)}\sum_{j=1}^{m}|\delta({\boldsymbol{x}}_{j})-\delta({\boldsymbol{x}}_{j-1})|$
$\displaystyle\leq\left\lVert\delta\right\rVert_{{BV}(\sigma_{1})}+4\left\lVert\delta\right\rVert_{\infty}$
$\displaystyle\leq 5\left\lVert\delta\right\rVert_{{BV}(\sigma_{1})}$
$\displaystyle<5\epsilon.$
Taking the supremum over all ordered lists $S$ shows that
$\left\lVert\tilde{g}-\tilde{p}\right\rVert_{{BV}(\sigma_{1})}\leq 5\epsilon$.
It follows that $\tilde{g}\in{AC}(\sigma)$. ∎
Proof of Theorem 5.1. The forward implication is obvious.
For the reverse implication, for each ${\boldsymbol{x}}$, choose a compact
neighbourhood $U_{\boldsymbol{x}}=\mathrm{cl}(V_{\boldsymbol{x}})\cap\sigma$
such that $f|U_{\boldsymbol{x}}\in{AC}(U_{\boldsymbol{x}})$. One may, by
taking a further restriction, assume that each $U_{\boldsymbol{x}}$ is a
rectangle.
By compactness we can choose a finite open subcover $V_{1},\dots,V_{m}$ of
$\sigma$. Now choose $C^{\infty}$ functions
$\chi_{1},\dots,\chi_{m}:\mathbb{R}^{2}\to[0,1]$ such that
$\operatorname{supp}\chi_{j}\subseteq V_{j}$ for each $j$ and
$\sum_{j=1}^{m}\chi_{j}=1$ on $\sigma$.
Fix $\epsilon>0$. For $j=1,\dots,m$, $f|U_{j}\in{AC}(U_{j})$, so there exists
a polynomial $p_{j}$ such that $\left\lVert
f-p_{j}\right\rVert_{{BV}(U_{j})}\leq\epsilon/m$. Let $f_{j}=\chi_{j}f$. Then
$\operatorname{supp}f_{j}\subseteq V_{j}$ and hence, by Lemma 5.2, it has a
natural extension $\tilde{f}_{j}\in{AC}(\sigma)$. Note that
$\tilde{f}_{j}=\chi_{j}f$, hence $\sum_{j=1}^{m}\tilde{f}_{j}=f$ and so
$f\in{AC}(\sigma)$. ∎
###### Corollary 5.3.
Suppose that $\sigma_{1}$ and $\sigma_{2}$ are disjoint nonempty compact sets
in the plane, that $\sigma=\sigma_{1}\cup\sigma_{2}$ and that
$f:\sigma\to\mathbb{C}$. If $f|\sigma_{1}\in AC(\sigma_{1})$ and
$f|\sigma_{2}\in AC(\sigma_{2})$ then $f\in{AC}(\sigma)$.
## 6\. ${\mathrm{CTPP}}$
For many proofs it is easier to work with elements of a suitable dense subset
of ${AC}(\sigma)$ rather than general absolutely continuous functions. Planar
functions, that is those of the form $F(x,y)=ax+by+c$, are particularly easy
to deal with since the variation of such a function over a compact set
$\sigma$ is equal to
$\operatorname*{var}(F,\sigma)=\max_{\sigma}F-\min_{\sigma}F.$
It is clear that the right-hand-side is a lower bound for
$\operatorname*{var}(F,\sigma)$. To see that one gets equality, one can apply
a suitable affine transformation to reduce the problem to looking at functions
of the form $F(x,y)=a^{\prime}x+c^{\prime}$. The variation of such functions
is equal to the one-dimensional variation of the function $x\mapsto
a^{\prime}x+c^{\prime}$, and this gives the above expression.
For functions of one variable, continuous piecewise linear functions form a
dense subset of ${AC}[a,b]$, so one would hope that continuous piecewise
planar functions would fulfil a analogous role in the two-dimensional case. To
be more precise, we recall the appropriate definitions.
We shall say that a set $P\subseteq\mathbb{R}^{2}$ is a polygon if it is a
compact simply connected set whose boundary consists of a finite number of
line segments. It follows from the Two Ears Theorem [8] that all polygons can
be triangulated.
Suppose then that $P$ is a polygon in $\mathbb{R}^{2}$. Let
$\mathcal{A}=\\{A_{i}\\}_{i=1}^{n}$ be a triangulation of $P$. To be definite,
the triangles $A_{i}$ are assumed to be proper and closed, and have pairwise
disjoint interiors. We shall say that a function $F:P\to\mathbb{C}$ is
triangularly piecewise planar over $\mathcal{A}$ if $F|A_{i}$ is planar for
all $i$ (that is $F(x,y)=a_{i}x+b_{i}y+c_{i}$ for $(x,y)\in A_{i}$). The set
of all such functions will be denoted by ${\mathrm{CTPP}}(P,\mathcal{A})$. It
is clear that ${\mathrm{CTPP}}(P,\mathcal{A})\subseteq C(P)$. We define the
set of continuous and triangularly piecewise planar functions on $P$ as
${\mathrm{CTPP}}(P)=\bigcup_{\mathcal{A}}{\mathrm{CTPP}}(P,\mathcal{A}).$
We now extend the definition to an arbitrary nonempty compact subset $\sigma$
of the plane.
###### Definition 6.1.
A function $f:\sigma\to\mathbb{C}$ is continuous and triangularly piecewise
planar with respect to $\sigma$ if there exists a polygon $P$ which contains
$\sigma$, and $F\in{\mathrm{CTPP}}(P)$ such that $F|\sigma=f$. The set of all
such functions is denoted by ${\mathrm{CTPP}}(\sigma)$.
One can extend a function $f\in{\mathrm{CTPP}}(\sigma)$ to any polygon $P_{0}$
which contains $\sigma$.
###### Lemma 6.2.
Suppose that $f\in{\mathrm{CTPP}}(\sigma)$ and that $P_{0}$ is a polygon
containing $\sigma$. Then there exists a function
$F_{0}\in{\mathrm{CTPP}}(P_{0})$ such that $f=F_{0}|P_{0}$.
###### Proof.
By definition, there exists a polygon $P$ containing $\sigma$, a triangulation
$\mathcal{A}=\\{A_{i}\\}_{i=1}^{n}$ of $P$ and
$F\in{\mathrm{CTPP}}(P,\mathcal{A})$ such that $F|\sigma=f$. Let $R$ be any
rectangle whose interior contains both $P$ and $P_{0}$. Then the set
$R\setminus\mathop{\mathrm{int}}(P)$ can be triangulated, and so we can
produce a triangulation $\mathcal{A}^{\prime}=\\{A_{i}\\}_{i=1}^{m}$ of $R$
which contains the triangles of $\mathcal{A}$. Let
$P_{k}=\cup_{i=1}^{k}A_{i}$. The ‘ear-clipping’ triangulation algorithm allows
us to do this in a way that for $n+1\leq k\leq m$, the triangle $A_{k}$ has at
least one side adjoining $P_{k-1}$, and at least one side disjoint from
$P_{k-1}$ (except at the vertices).
One may now extend $F$, triangle by triangle. If $F$ has been defined on
$P_{k-1}$, then there exists a planar function on $A_{k}$ which agrees with
$F$ at these intersection points, and hence we can extend $F$ to $P_{k}$.
The triangulation $\mathcal{A}^{\prime}$ now generates a triangulation
$\mathcal{A}_{0}$ of $P_{0}$: for each $i$, $A_{i}\cap P_{0}$ is a union of
polygons and hence can be written as a union of triangles. Thus
$F_{0}=F|P_{0}\in{\mathrm{CTPP}}(P_{0},\mathcal{A}_{0})$. Furthermore $F_{0}$
is an extension of $f$. ∎
###### Lemma 6.3.
${\mathrm{CTPP}}(\sigma)$ is a vector space.
###### Proof.
The only point to check is that ${\mathrm{CTPP}}(\sigma)$ is closed under
addition. Suppose that $f,g\in{\mathrm{CTPP}}(\sigma)$. Let $P$ be a polygon
containing $\sigma$. By Lemma 6.2 there exist $F,G\in{\mathrm{CTPP}}(P)$ be
such that $F|\sigma=f$ and $G|\sigma=g$. The sum $F+G$ is clearly continuous
and, since we can use a common triangulation for $F$ and $G$, is planar on
polygonal regions of $R$. Hence $F+G\in{\mathrm{CTPP}}(R)$ which proves the
result. ∎
The next results show that all such functions are Lipschitz and hence of
bounded variation on $P$. In particular, on each of the spaces
${\mathrm{CTPP}}(P,\mathcal{A})$ we get an equivalence of norms. For a
triangle $A$, let $r_{A}$ denote the inradius of $A$.
###### Lemma 6.4.
Suppose that $A$ is a triangle and that $F:A\to\mathbb{C}$ is planar on $A$.
Then $L_{A}(F)$, the Lipschitz constant of $F$ on $A$, satisfies
$L_{A}(F)\leq\frac{2}{r_{A}}\left\lVert F\right\rVert_{\infty}$
and hence
$\displaystyle\left\lVert F\right\rVert_{\infty}$
$\displaystyle\leq\left\lVert
F\right\rVert_{{\mathrm{Lip}}(A)}\leq\left(1+\frac{2}{r_{A}}\right)\left\lVert
F\right\rVert_{\infty}$
###### Proof.
Since $F$ is planar, $\nabla\\!F$ is constant and, for
${\boldsymbol{x}}\neq{\boldsymbol{x}}^{\prime}\in A$,
$\frac{|F({\boldsymbol{x}})-F({\boldsymbol{x}}^{\prime})|}{\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert}=\Bigl{|}\nabla\\!F\cdot\left(\frac{{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}}{\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert}\right)\Bigr{|}.$
This quantity only depends on the direction from ${\boldsymbol{x}}^{\prime}$
to ${\boldsymbol{x}}$. Choose a unit vector ${\boldsymbol{u}}$ in the plane so
that $L_{A}(F)=|\nabla\\!F\cdot{\boldsymbol{u}}|$. The definition of $r_{A}$
ensures that there exists ${\boldsymbol{x}}\in A$ such that
${\boldsymbol{x}}^{\prime}={\boldsymbol{x}}+r_{A}{\boldsymbol{u}}$ also lies
in $A$. Thus
$L_{A}(F)=\frac{|F({\boldsymbol{x}})-F({\boldsymbol{x}}^{\prime})|}{\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert}\leq\frac{2}{r_{A}}\left\lVert
F\right\rVert_{\infty},$
which gives the bounds on $\left\lVert F\right\rVert_{{\mathrm{Lip}}(A)}$. ∎
###### Theorem 6.5.
${\mathrm{CTPP}}(\sigma)\subseteq{\mathrm{Lip}}(\sigma)\subseteq{BV}(\sigma)$.
###### Proof.
Let $R$ be any rectangle containing $\sigma$. Suppose that
$f\in{\mathrm{CTPP}}(\sigma)$. By Lemma 6.2 there exists a triangulation
$\mathcal{A}=\\{A_{i}\\}_{i=1}^{n}$ of $R$ and
$F\in{\mathrm{CTPP}}(R,\mathcal{A})$ such that $F|\sigma=f$.
Let $r=\min_{1\leq i\leq n}r_{A_{i}}$. Suppose that
${\boldsymbol{x}},{\boldsymbol{x}}^{\prime}\in R$. The line segment
$\overline{{\boldsymbol{x}}{\boldsymbol{x}}^{\prime}}$ joining
${\boldsymbol{x}}$ and ${\boldsymbol{x}}^{\prime}$ can be written as a union
of finitely many subsegments, denoted
$\overline{{\boldsymbol{x}}_{j-1}{\boldsymbol{x}}_{j}}$ ($j=1,\dots,m$), with
each subsegment entirely contained in just one triangle $T_{j}$ in
$\mathcal{A}$. Then, using Lemma 6.4,
$\displaystyle|F({\boldsymbol{x}})-F({\boldsymbol{x}}^{\prime})|$
$\displaystyle\leq\sum_{j=1}^{m}|F({\boldsymbol{x}}_{j})-F({\boldsymbol{x}}_{j-1})|$
$\displaystyle\leq\sum_{j=1}^{m}L_{T_{j}}(F)\left\lVert{\boldsymbol{x}}_{j}-{\boldsymbol{x}}_{j-1}\right\rVert$
$\displaystyle\leq\sum_{j=1}^{m}\frac{2}{r}\left\lVert
F\right\rVert_{\infty}\left\lVert{\boldsymbol{x}}_{j}-{\boldsymbol{x}}_{j-1}\right\rVert$
$\displaystyle=\frac{2}{r}\left\lVert
F\right\rVert_{\infty}\left\lVert{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rVert$
and so $L_{\sigma}(f)\leq L_{R}(F)\leq\frac{2}{r}\left\lVert
F\right\rVert_{\infty}$.
The second inclusion follows from Lemma 3.15 of [1]. Indeed
$\operatorname*{var}(F,R)\leq D_{R}L_{R}(F)$, where $D_{R}$ denotes the
diameter of $R$, and so
$\left\lVert f\right\rVert_{{BV}(\sigma)}\leq\left\lVert
F\right\rVert_{{BV}(R)}\leq\Bigl{(}1+\frac{2D_{R}}{r}\Bigr{)}\left\lVert
F\right\rVert_{\infty}.$
∎
Our aim now is to show that ${\mathrm{CTPP}}(\sigma)$ forms a dense subset of
${AC}(\sigma)$.
Suppose that $f\in{\mathrm{CTPP}}(\sigma)$ with respect to the triangulation
$\mathcal{A}$ of a rectangle $R$. We say that ${\boldsymbol{x}}\in\sigma$ is
* •
a planar point for $f$ if it lies in exactly one triangle in $\mathcal{A}$;
* •
an edge point for $f$ if it lies in exactly two triangles in $\mathcal{A}$;
* •
a vertex point for $f$ if it lies in three or more triangles in $\mathcal{A}$.
Clearly these three possibilities are exhaustive and mutually exclusive. Note
that the classification of ${\boldsymbol{x}}$ depends on the triangulation,
but we shall suppress this in the terminology. In Figure 3, the points
${\boldsymbol{x}}_{1}$ and ${\boldsymbol{x}}_{2}$ are planar points,
${\boldsymbol{x}}_{3}$ and ${\boldsymbol{x}}_{4}$ are edge points, and
${\boldsymbol{x}}_{5}$ is a vertex point for this triangulation.
$R$${\boldsymbol{x}}_{1}$${\boldsymbol{x}}_{2}$${\boldsymbol{x}}_{3}$${\boldsymbol{x}}_{4}$${\boldsymbol{x}}_{5}$
Figure 3. The classification of points with respect to a triangulation.
Our aim is to show that ${\mathrm{CTPP}}(\sigma)$ forms a dense subspace of
${AC}(\sigma)$. The first step is to show that if
$f\in{\mathrm{CTPP}}(\sigma)$ then $f\in{AC}(\sigma)$. The method of proof
will be to show that for each ${\boldsymbol{x}}\in\sigma$ there exists a
compact neighbourhood $U_{\boldsymbol{x}}$ of ${\boldsymbol{x}}$ such that
$f|U_{\boldsymbol{x}}\in{AC}(U_{\boldsymbol{x}})$, and to then apply the
Patching Lemma. If ${\boldsymbol{x}}$ is a planar point for $f$, one can
clearly take a small rectangle $R_{\boldsymbol{x}}$ around ${\boldsymbol{x}}$
such that $f$ is planar, and hence absolutely continuous, on
$R_{\boldsymbol{x}}\cap\sigma$.
###### Lemma 6.6.
Suppose that ${\boldsymbol{x}}$ is an edge point for
$f\in{\mathrm{CTPP}}(\sigma)$ (with respect to some triangulation
$\mathcal{A}$). Then there exists a compact neighbourhood $U_{\boldsymbol{x}}$
of ${\boldsymbol{x}}$ such that
$f|U_{\boldsymbol{x}}\in{AC}(U_{\boldsymbol{x}})$.
###### Proof.
Using the affine invariance, we may assume that ${\boldsymbol{x}}=0$, and that
the two triangles meet on the line $x=0$. Thus, there exists some small square
$R$ centred at the origin such that if $(x,y)\in\sigma\cap R$ then
$f(x,y)=\begin{cases}a_{1}x+b_{1}y+c_{1},&x\geq 0,\\\
a_{2}x+b_{2}y+c_{2},&x\leq 0.\end{cases}$
Since $f$ is continuous, then we have $b_{1}=b_{2}$ and $c_{1}=c_{2}$. Let
$f_{1}(x,y)=\begin{cases}a_{1}x,&x\geq 0,\\\ a_{2}x,&x\leq
0,\end{cases}\qquad\qquad f_{2}(x,y)=b_{1}y+c_{1}.$
Let $U_{\boldsymbol{x}}=\sigma\cap R$. It follows from [1, Proposition 4.4]
that $f_{1},f_{2}\in{AC}(U_{\boldsymbol{x}})$, and hence
$f\in{AC}(U_{\boldsymbol{x}})$. ∎
The most difficult case is that of vertex points. We shall use the above
results to show that on a compact neighbourhood where a piecewise planar
function has only one vertex point, we may approximate the function
arbitrarily closely by ${AC}$ functions. In order to estimate the error in
this estimation we require a number of simple lemmas.
Let $Q=Q_{t}$ is the square $[-t,t]\times[-t,t]\subseteq\mathbb{R}^{2}$. We
shall say that a function $f:Q\to\mathbb{C}$ is star-planar on $Q$ if there
exists a partitioning $\mathcal{A}=\\{A_{i}\\}_{i=1}^{n}$ of $Q$ given by
$n\geq 2$ rays starting at the origin such that $f$ is planar on each set
$A_{i}$. (We shall not need this in the proof, but one can make each set
$A_{i}$ triangular by adding the rays from the origin to the four corners of
$Q$.)
###### Lemma 6.7.
If $f$ is star-planar on $Q$ with respect to $\\{A_{i}\\}_{i=1}^{n}$ then
$f\in{BV}(Q)$ with
$\operatorname*{var}(f,Q)\leq 2n\sup_{{\boldsymbol{x}},{\boldsymbol{w}}\in
Q}|f({\boldsymbol{x}})-f({\boldsymbol{w}})|.$
###### Proof.
First note that if $f$ is star-planar on $Q$ with respect to
$\\{A_{i}\\}_{i=1}^{n}$, then it is star-planar with respect to a finer
partition $\\{A_{i}^{\prime}\\}_{i=1}^{m}$ with $m\leq 2n$ formed by extending
the rays to full lines. On each region $A_{i}^{\prime}$, $f$ is essentially a
function of just one variable, and so it has variation given by
$\sup_{{\boldsymbol{x}},{\boldsymbol{w}}\in
A_{i}^{\prime}}|f({\boldsymbol{x}})-f({\boldsymbol{w}})|$. The finer partition
has the property that one can piece the regions together to form larger and
larger convex blocks, and hence use Theorem 3.4 to obtain the result. ∎
###### Proposition 6.8.
Suppose that ${\boldsymbol{x}}$ is a vertex point for
$f\in{\mathrm{CTPP}}(\sigma)$. Then there exists a compact neighbourhood
$U_{\boldsymbol{x}}$ of ${\boldsymbol{x}}$ such that
$f|U_{\boldsymbol{x}}\in{AC}(U_{\boldsymbol{x}})$.
###### Proof.
Note that by the affine invariance property, we may assume that
${\boldsymbol{x}}=(0,0)$. Also, by replacing $f$ with $f-f(0,0)$ it suffices
to prove the result in the case that $f(0,0)=0$.
Fix a rectangle $R$ containing $\sigma$ and a triangulation $\mathcal{A}$ of
$R$ for which $(0,0)$ is a vertex point. We shall regard $f$ as being defined
on the whole of $R$.
As above, let $Q_{t}=[-t,t]\times[-t,t]$. One may choose $\delta>0$ so that
for $0<s\leq\delta$, $Q_{s}$ lies entirely inside $R$ and contains no other
vertex points for $f$. Thus, $f$ is star-planar on such $Q_{s}$ with respect
to the partitioning of that set generated by $\mathcal{A}$. Let
$U_{\boldsymbol{x}}=Q_{\delta}\cap\sigma$. Our aim now is to show that we may
approximate $f|U_{\boldsymbol{x}}$ by absolutely continuous functions. It will
suffice to show that given any $\epsilon>0$ there exists
$h\in{AC}(Q_{\delta})$ with $\left\lVert
f-h\right\rVert_{{BV}(Q_{\delta})}<\epsilon$.
Fix then $\epsilon>0$. Since $f$ is continuous, it follows from Lemma 6.7 that
there exists $s\leq\delta$ such that $\left\lVert
f\right\rVert_{{BV}(Q_{s})}<\epsilon/10$. Define the function
$g_{s}:[-\delta,\delta]\to\mathbb{R}$ to be the function with the graph given
in Figure 4.
$-\delta$$-s$$-s/2$$\delta$$s$$s/2$$g_{s}$ Figure 4. The function $g_{s}$.
Clearly $g_{s}\in{AC}[-\delta,\delta]$ with $\left\lVert
g_{s}\right\rVert_{{BV}[-\delta,\delta]}=3$. By [1, Proposition 4.4], the
functions $(x,y)\mapsto g_{s}(x)$ and $(x,y)\mapsto g_{s}(y)$ are both in
${AC}(Q_{\delta})$ with ${BV}(Q_{\delta})$ norm equal to three. It follows
that their product $\chi(x,y)=g_{s}(x)\,g_{s}(y)$ also lies in
${AC}(Q_{\delta})$ and that
$\left\lVert\chi\right\rVert_{{BV}(Q_{\delta})}\leq 10$.
Let $h=f\chi$. Let $A=Q_{\delta}\setminus(-s/3,s/3)^{2}$. Suppose that
${\boldsymbol{w}}\in A$. Then ${\boldsymbol{w}}$ is either a planar point or
an edge point for $f$, so by Lemma 6.6 and the preceding remark, there is a
compact neighbourhood $V_{\boldsymbol{w}}$ of ${\boldsymbol{w}}$ (with respect
to $Q_{\delta}$) such that $f|V_{\boldsymbol{w}}\in{AC}(V_{\boldsymbol{w}})$.
Clearly $\chi|V_{\boldsymbol{w}}\in{AC}(V_{\boldsymbol{w}})$ too and hence
$h|V_{\boldsymbol{w}}\in{AC}(V_{\boldsymbol{w}})$. On the other hand, if
${\boldsymbol{w}}\in(-s/3,s/3)^{2}$ then $h=0$ on an open neighbourhood of
${\boldsymbol{w}}$ and so again we can choose a compact neighbourhood
$V_{\boldsymbol{w}}$ of ${\boldsymbol{w}}$ such that
$h|V_{\boldsymbol{w}}\in{AC}(V_{\boldsymbol{w}})$. It follows from the
Patching Lemma (Theorem 5.1) that $h\in{AC}(Q_{\delta})$.
Now
$\left\lVert f-h\right\rVert_{{BV}(Q_{\delta})}=\left\lVert
f(1-\chi)\right\rVert_{{BV}(Q_{\delta})}\leq\left\lVert
f\right\rVert_{{BV}(Q_{\delta})}\left\lVert
1-\chi\right\rVert_{{BV}(Q_{\delta})}<\epsilon$
which completes the proof. ∎
Combining the previous results and discussion with the Patching Lemma gives
the following.
###### Corollary 6.9.
${\mathrm{CTPP}}(\sigma)\subseteq{AC}(\sigma)$.
###### Theorem 6.10.
Let $R=[0,1]\times[0,1]$. Then
$C^{1}(R)\subseteq\mathrm{cl}({\mathrm{CTPP}}(R))$ and hence
$C^{1}(R)\subseteq{AC}(R)$.
###### Proof.
Suppose that $f\in C^{1}(R)$. We shall show that $f$ can be approximated
arbitrarily closely by functions in ${\mathrm{CTPP}}(R)$.
Fix $\epsilon>0$. Choose $\delta>0$ such that for all
${\boldsymbol{x}},{\boldsymbol{y}}\in R$ with
$\left\lVert{\boldsymbol{x}}-{\boldsymbol{y}}\right\rVert<\delta$,
(6.1)
$|f({\boldsymbol{x}})-f({\boldsymbol{y}})|<\epsilon,\quad\hbox{and}\quad\left\lVert\nabla\\!f({\boldsymbol{x}})-\nabla\\!f({\boldsymbol{y}})\right\rVert<\epsilon.$
Choose an integer $n>\sqrt{2}/\delta$. Triangulate $R$ by drawing horizontal
and vertical lines at multiples of $\frac{1}{n}$, and diagonals as in Figure
5. Each triangle then has diameter less than $\delta$. Let $g$ be the element
of ${\mathrm{CTPP}}(R)$ which agrees with $f$ at all of the vertices of this
triangulation.
$\frac{1}{n}$$\frac{2}{n}$$\dots$$1$$\frac{1}{n}$$\frac{2}{n}$$\vdots$$1$
Figure 5. The triangulation $\mathcal{A}$ used in the proof of Theorem 6.10.
Fix a triangle $A$ in this triangulation. The above bounds imply that there
exist $m,M$ such that $M-m<\epsilon$ and $m\leq f\leq M$ on $A$. Since $g$ is
planar on $A$, $m\leq g\leq M$ on $A$ too and hence $|f-g|<\epsilon$ on $A$.
Thus $\left\lVert f-g\right\rVert_{\infty}<\epsilon$.
The more delicate estimate concerns the Lipschitz constant of $d=f-g$. Suppose
first that ${\boldsymbol{x}}\neq{\boldsymbol{y}}$ lie in the same triangle $A$
and let $\ell\subseteq A$ denote the line segment joining ${\boldsymbol{x}}$
and ${\boldsymbol{y}}$. By the Mean Value Theorem there exists
$\boldsymbol{q}\in\ell$ for which
$|d({\boldsymbol{x}})-d({\boldsymbol{y}})|=\left\lVert\nabla\\!d(\boldsymbol{q})\right\rVert\,\left\lVert{\boldsymbol{x}}-{\boldsymbol{y}}\right\rVert.$
Now as $g$ is planar, $\nabla\\!g=(g_{x_{1}},g_{x_{2}})$ is constant. So,
using the Mean Value Theorem on $g$ along the sides of $A$ parallel to the
coordinate axes, and the fact that $f=g$ on the vertices of $A$, one sees that
there exist ${\boldsymbol{\xi}}$ and ${\boldsymbol{\eta}}$ on the boundary of
$A$ such that
$\nabla\\!g(\boldsymbol{q})=\left(\frac{\partial f}{\partial
x_{1}}({\boldsymbol{\xi}}),\frac{\partial f}{\partial
x_{2}}({\boldsymbol{\eta}})\right).$
Thus
$\left\lVert\nabla\\!d(\boldsymbol{q})\right\rVert^{2}=\left(\frac{\partial
f}{\partial x_{1}}(\boldsymbol{q})-\frac{\partial f}{\partial
x_{1}}({\boldsymbol{\xi}})\right)^{2}+\left(\frac{\partial f}{\partial
x_{2}}(\boldsymbol{q})-\frac{\partial f}{\partial
x_{2}}({\boldsymbol{\eta}})\right)^{2}<2\epsilon^{2}$
by (6.1), and so
(6.2)
$|d({\boldsymbol{x}})-d({\boldsymbol{y}})|<\sqrt{2}\,\epsilon\,\left\lVert{\boldsymbol{x}}-{\boldsymbol{y}}\right\rVert.$
One may now extend (6.2) to general ${\boldsymbol{x}}\neq{\boldsymbol{y}}\in
R$ by splitting the line segment between ${\boldsymbol{x}}$ and
${\boldsymbol{y}}$ into segments which lie entirely in a single triangle and
then using the triangle inequality. Thus,
$\left\lVert
f-g\right\rVert_{{\mathrm{Lip}}(R)}\leq\epsilon+\sqrt{2}\epsilon.$
Since ${\mathrm{CTPP}}(\sigma)\subseteq{AC}(R)$, it now follows from Lemma 4.1
that $f\in{AC}(R)$. ∎
It now remains to show that ${\mathrm{CTPP}}(\sigma)$ is dense in
${AC}(\sigma)$.
###### Theorem 6.11.
${\mathrm{CTPP}}(\sigma)$ is dense in ${AC}(\sigma)$.
###### Proof.
Suppose that $f\in{AC}(\sigma)$. Fix a closed rectangle $R$ containing
$\sigma$. Making use of the affine invariance of the norms, it suffices assume
that $R=[0,1]\times[0,1]$. Given any $\epsilon>0$, there exists a polynomial
$p$ such that $\left\lVert f-p\right\rVert_{{BV}(\sigma)}<\epsilon/2$. Since
$p\in C^{1}(R)$, Theorem 6.10 implies that there exists
$g\in{\mathrm{CTPP}}(R)$ with $\left\lVert
p-g\right\rVert_{{BV}(R)}<\epsilon/2$. Thus $\left\lVert
f-g\right\rVert_{{BV}(\sigma)}<\epsilon$. ∎
The result of Theorem 6.10 can now be extended to general compact sets.
###### Theorem 6.12.
Suppose that $\sigma$ is a nonempty compact subset of the plane. Then
$C^{1}(\sigma)$ is a dense subset of ${AC}(\sigma)$.
###### Proof.
Suppose that $f\in C^{1}(\sigma)$, and so it admits a $C^{1}$ extension (which
we shall also denote $f$) on some open neighbourhood $V$ of $\sigma$. Suppose
that ${\boldsymbol{x}}\in\sigma$. Then there exists a closed rectangle
$R_{{\boldsymbol{x}}}$ centred at ${\boldsymbol{x}}$ which lies inside $V$. By
Theorem 6.10 $f|R_{\boldsymbol{x}}\in AC(R_{x})$. The set
$U_{\boldsymbol{x}}=R_{\boldsymbol{x}}\cap\sigma$ is a compact neighbourhood
of ${\boldsymbol{x}}$ and $f|U_{\boldsymbol{x}}\in AC(U_{x})$, so we can use
the Patching Lemma to deduce that $f\in{AC}(\sigma)$. The density of
$C^{1}(\sigma)$ follows from the fact that the polynomials lie in
$C^{1}(\sigma)$. ∎
In many situations we would like to be able to require that a
${\mathrm{CTPP}}$ approximation to an ${AC}$ function agrees with the function
at certain specified points. In the proof of Theorem 6.10 above, the function
$g\in{\mathrm{CTPP}}(R)$ agrees with the $C^{1}$ function $f$ at each vertex
of the triangulation. It is easy to remove the $C^{1}$ condition on $f$.
###### Lemma 6.13.
Let $A$ be a triangle in the plane and suppose that $f\in{AC}(A)$. For all
$\epsilon>0$ there exists $g\in{\mathrm{CTPP}}(A)$ such that $\left\lVert
f-g\right\rVert_{{BV}(A)}<\epsilon$ and such that
$f({\boldsymbol{x}})=g({\boldsymbol{x}})$ at each of the vertices of $A$.
###### Proof.
By Theorem 6.11 there exists $g_{0}\in{\mathrm{CTPP}}(A)$ such that
$\left\lVert f-g_{0}\right\rVert_{{BV}(A)}<\epsilon/4$. Let $h$ be the planar
function on $A$ which agrees with $f-g_{0}$ at each of the three vertices.
Then
$\left\lVert h\right\rVert_{{BV}(A)}=\left\lVert
h\right\rVert_{\infty}+\left(\max_{A}h-\min_{A}h\right)\leq 3\left\lVert
h\right\rVert_{\infty}\leq 3\left\lVert
f-g_{0}\right\rVert_{{BV}(A)}<\frac{3\epsilon}{4}.$
The function $g=g_{0}+h$ has the required properties. ∎
###### Theorem 6.14.
Suppose that $\sigma$ is a nonempty compact set and that $f\in{AC}(\sigma)$.
Given any finite set of points
$\\{{\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{n}\\}\subseteq\sigma$ and any
$\epsilon>0$ there exists $g\in{\mathrm{CTPP}}(\sigma)$ such that $\left\lVert
f-g\right\rVert_{{BV}(\sigma)}<\epsilon$ and such that
$f({\boldsymbol{x}}_{i})=g({\boldsymbol{x}}_{i})$ at each point $x_{i}$.
###### Proof.
By Theorem 6.11 there exists $g_{0}\in{\mathrm{CTPP}}(\sigma)$ such that
$\left\lVert f-g_{0}\right\rVert_{{BV}(\sigma)}<\frac{\epsilon}{4n+2}$. Using
Proposition 2.10 of [3] one can see that the function
$b(x,y)=\max(\min(1-|x|,1-|y|),0)$
is in ${\mathrm{CTPP}}(R_{0})$, with variation at most $4$, on any rectangle
$R_{0}$ containing $[-1,1]\times[-1,1]$. Choose $\delta>0$ so that the squares
centred at the points ${\boldsymbol{x}}_{i}$ with side length $2\delta$ are
all disjoint. Now define
$h({\boldsymbol{x}})=\sum_{i=1}^{n}(f({\boldsymbol{x}}_{i})-g_{0}({\boldsymbol{x}}_{i}))\,b\Bigl{(}\frac{{\boldsymbol{x}}-{\boldsymbol{x}}_{i}}{\delta}\Bigr{)}.$
Then $h\in{\mathrm{CTPP}}(R)$ on any suitable rectangle containing $\sigma$.
Note that
$|f({\boldsymbol{x}}_{i})-g_{0}({\boldsymbol{x}}_{i})|\leq\left\lVert
f-g\right\rVert_{{BV}(\sigma)}<\frac{\epsilon}{4n+2}$ so (using Proposition
3.7 of [1] and the invariance of norms under affine transformations)
$\operatorname*{var}(h,R)<n\ \frac{\epsilon}{4n+2}\
\operatorname*{var}(b,R_{0})<\frac{4n}{4n+2}\ \epsilon.$
Since $\left\lVert h\right\rVert_{\infty}<\frac{\epsilon}{4n+2}$ we have that
$\left\lVert h\right\rVert_{{BV}(R)}<\left(\frac{4n+1}{4n+2}\right)\epsilon$.
Let $g=g_{0}+h$. Then $g\in{\mathrm{CTPP}}(\sigma)$. Clearly
$g({\boldsymbol{x}}_{i})=f({\boldsymbol{x}}_{i})$ and
$\left\lVert f-g\right\rVert_{{BV}(\sigma)}\leq\left\lVert
f-g_{0}\right\rVert_{{BV}(\sigma)}+\left\lVert
h\right\rVert_{{BV}(\sigma)}<\epsilon$
as required. ∎
## 7\. Extension and joining results
In this final section we shall return to the considerations of Questions 3.1
and 3.2. These problems, which arise naturally in the study of ${AC}(\sigma)$
operators, are closely related.
Suppose then that $\sigma_{1}\subseteq\sigma$ and that $f\in{AC}(\sigma_{1})$.
Let $\sigma_{2}=\mathrm{cl}(\sigma\setminus\sigma_{1})$. In order to find an
extension $\hat{f}\in{AC}(\sigma)$ one typically must first define $\hat{f}$
on $\sigma_{2}$ in such a way that $\hat{f}\in{AC}(\sigma_{2})$, and then show
that the absolute continuity is preserved on the set $\sigma$.
If one has some convexity, then it is relatively easy to prove ‘extension’ and
‘joining’ results. The following joining result is a minor adaptation of Lemma
3.2 from [3].
###### Lemma 7.1.
Suppose that $\sigma$ is a nonempty compact convex set which intersects the
$x$-axis. Let
$\sigma_{1}=\\{(x,y)\in\sigma\thinspace:\thinspace y\geq
0\\},\qquad\sigma_{2}=\\{(x,y)\in\sigma\thinspace:\thinspace y\leq 0\\}.$
Suppose that $f:\sigma\to\mathbb{C}$. If $f|\sigma_{1}\in{AC}(\sigma_{1})$ and
$f|\sigma_{2}\in{AC}(\sigma_{2})$, then $f\in{AC}(\sigma)$ and
$\left\lVert f\right\rVert_{{BV}(\sigma)}\leq\left\lVert
f_{1}\right\rVert_{{BV}(\sigma_{1})}+\left\lVert
f_{2}\right\rVert_{{BV}(\sigma_{2})}.$
$\sigma_{1}$$\sigma_{2}$ Figure 6. Diagram for Lemma 7.1.
###### Proof.
Let $\sigma_{\mathbb{R}}=\sigma_{1}\cap\sigma_{2}=[a,b]\times\\{0\\}$. To
avoid trivial cases we shall assume that $(a,b)\times\\{0\\}$ lies in the
interior of $\sigma$. The convexity of $\sigma$ implies that there exists a
polygon $P$ containing $\sigma$ such that $\\{x\,:\,(x,0)\in P\\}=[a,b]$. Let
$P^{+}=\\{(x,y)\in P\,:\,y\geq 0\\}$ and let $P^{-}=\\{(x,y)\in P\,:\,y\leq
0\\}$.
Suppose first that $f$ vanishes on $\sigma_{\mathbb{R}}$. Fix $\epsilon>0$. As
$f|\sigma_{1}\in{AC}(\sigma_{1})$ there exists
$g_{1}\in{\mathrm{CTPP}}(\sigma_{1})$ with $\left\lVert
f-g_{1}\right\rVert_{{BV}(\sigma_{1})}<\epsilon/4$. Note that
$q(x)=g_{1}(x,0)$ is a piecewise linear function on $\sigma_{\mathbb{R}}$ with
$\left\lVert q\right\rVert_{{BV}(\sigma_{\mathbb{R}})}=\left\lVert
g_{1}\right\rVert_{{BV}(\sigma_{\mathbb{R}})}\leq\left\lVert
f-g_{1}\right\rVert_{{BV}(\sigma_{1})}<\epsilon/4$. Extend $g_{1}$ to
$\sigma_{2}$ by setting, for $(x,y)\in\sigma_{2}\setminus\sigma_{1}$,
$g_{1}(x,y)=\begin{cases}q(a),&\hbox{if $x<a$,}\\\ q(x),&\hbox{if $a\leq x\leq
b$,}\\\ q(b),&\hbox{if $x>b$.}\end{cases}$
We claim that $g_{1}\in{\mathrm{CTPP}}(\sigma)$. Note that by Lemma 6.2, there
is a triangulation $\mathcal{A}^{+}$ of $P^{+}$ such that $g_{1}|\sigma_{1}$
admits an extension $G^{+}\in{\mathrm{CTPP}}(P^{+},\mathcal{A}^{+})$. Since
$g_{1}|\sigma_{2}$ depends in a piecewise linear way on the $x$ variable (and
is independent of the $y$ variable),
$g_{1}|\sigma_{2}\in{\mathrm{CTPP}}(\sigma_{2})$ and so there is a
triangulation $\mathcal{A}^{-}$ of $P^{-}$ such that $g_{1}|\sigma_{2}$ admits
an extension $G^{-}\in{\mathrm{CTPP}}(P^{-},\mathcal{A}^{-})$. Since $P^{+}$
and $P^{-}$ only meet along $\sigma_{\mathbb{R}}$,
$G(x,y)=\begin{cases}G^{+}(x,y),&\hbox{if $(x,y)\in P^{+}$,}\\\
G^{-}(x,y),&\hbox{if $(x,y)\in P^{-}$}\end{cases}$
is an extension of $g_{1}$ in
${\mathrm{CTPP}}(P,\mathcal{A}^{+}\cup\mathcal{A}^{-})$ which proves the
claim.
Using Proposition 4.4 in [1] we may deduce that
$\operatorname*{var}(g_{1},\sigma_{2})=\operatorname*{var}(q,\sigma_{\mathbb{R}})$
and hence that $\left\lVert g_{1}\right\rVert_{{BV}(\sigma_{2})}<\epsilon/4$.
Similarly, we may construct $g_{2}\in{\mathrm{CTPP}}(\sigma)$ such that
$\left\lVert f-g_{2}\right\rVert_{{BV}(\sigma_{2})}<\epsilon/4$ and
$\left\lVert g_{2}|\sigma_{1}\right\rVert_{{BV}(\sigma_{1})}<\epsilon/4$. Let
$g=g_{1}+g_{2}\in{\mathrm{CTPP}}(\sigma)$. Then
$\left\lVert
f-g\right\rVert_{{BV}(\sigma_{1})}<\frac{\epsilon}{2}\quad\hbox{and}\quad\left\lVert
f-g\right\rVert_{{BV}(\sigma_{2})}<\frac{\epsilon}{2}$
so, since $\sigma$ is convex, Theorem 3.4 gives $\left\lVert
f-g\right\rVert_{{BV}(\sigma)}<\epsilon$. It follows that $f\in{AC}(\sigma)$.
For general $f$, define $h:\sigma\to\mathbb{C}$ by
$h(x,y)=\begin{cases}f(a,0),&\hbox{if $x<a$,}\\\ f(x,0),&\hbox{if $a\leq x\leq
b$,}\\\ f(b,0),&\hbox{if $x>b$.}\end{cases}$
and $f_{1}=f-h$. Since $h\in{AC}(\sigma)$ (using [1, Proposition 4.4]),
$f_{1}|\sigma_{1}\in{AC}(\sigma_{1})$, $f_{1}|\sigma_{2}\in{AC}(\sigma_{2})$
and $f_{1}$ vanishes on the real axis. It follows that $f_{1}$ and
consequently $f$ are both in ${AC}(\sigma)$.
The norm estimate is given by Theorem 3.4. ∎
###### Remark 7.2.
The algorithm for defining $g_{1}$ in the above proof might fail to produce an
element of ${\mathrm{CTPP}}(\sigma)$ in the absence of convexity. As an
example, consider the connected compact set
$\sigma=\\{(x,y)\,:\,x^{2}\leq|y|\leq 1\\}$ and let
$g_{1}(x,y)=\begin{cases}x,&\hbox{if $y\geq 0$,}\\\ 0,&\hbox{if
$y<0$.}\end{cases}$
In this case $g_{1}|\sigma_{1}\in{\mathrm{CTPP}}(\sigma_{1})$ and
$g_{1}|\sigma_{2}\in{\mathrm{CTPP}}(\sigma_{2})$, but
$g_{1}\not\in{\mathrm{CTPP}}(\sigma)$ since one cannot join the piecewise
planar parts in a continuous way on any polygon containing $\sigma$. Note
however that one can use an approximation argument to show that
$g_{1}\in{AC}(\sigma)$.
Obviously, using affine invariance, one can replace the real line in Lemma 7.1
with any line in the plane.
The hypothesis of convexity in Lemma 7.1 can be relaxed somewhat at the cost
of having less control over the norm of the joined function. Without aiming
for maximum generality, we can now extend this sort of result to more general
situations.
###### Lemma 7.3.
Let $R=J\times K$ be a rectangle in $\mathbb{R}^{2}$ and suppose that
$\sigma_{0}\subseteq R$ is the graph of a continuous convex function
$\phi:J\to K$. If $f\in{AC}(\sigma_{0})$ then there exists an extension
$g\in{AC}(R)$ of $f$ with $\left\lVert g\right\rVert_{{BV}(R)}\leq
2\left\lVert f\right\rVert_{{BV}(\sigma_{0})}$.
###### Proof.
We shall start by defining a map $\Psi:{BV}(\sigma_{0})\to{BV}(J)$. Suppose
then that $f\in{BV}(\sigma_{0})$. First consider the function
$J\to\mathbb{C}$, defined by ${\hat{f}}(x)=f(x,\phi(x))$. Suppose that
$S=[x_{0},\dots,x_{n}]$ is a finite increasing list of elements of $J$. Let
$\check{S}=[(x_{0},\phi(x_{0})),\dots,(x_{n},\phi(x_{m}))]\in\sigma_{0}$. Then
$\displaystyle\operatorname{\rm
cvar}({\hat{f}},S)=\sum_{i=1}^{n}|{\hat{f}}(x_{i})-{\hat{f}}(x_{i-1})|$
$\displaystyle=\sum_{i=1}^{n}|f(x_{i},\phi(x_{i}))-f(x_{i-1},\phi(x_{i-1}))|$
$\displaystyle\leq\frac{\operatorname{vf}(\check{S})\operatorname{\rm
cvar}(f,\check{S})}{\operatorname{vf}(\check{S})}$
$\displaystyle\leq\operatorname{vf}(\check{S})\operatorname*{var}(f,\sigma_{0}).$
Recall that as $J$ is a real interval, $\operatorname*{var}({\hat{f}},J)$ is
the supremum of $\operatorname{\rm cvar}({\hat{f}},S)$ taken over all such
increasing lists of elements $S$. Also, as $\phi$ is convex,
$\operatorname{vf}(\check{S})\leq 2$. Thus
$\operatorname*{var}({\hat{f}},J)\leq 2\operatorname*{var}(f,\sigma_{0})$. It
follows easily now that setting $\Psi(f)={\hat{f}}$ defines a bounded linear
operator from ${BV}(\sigma_{0})$ to ${BV}(J)$.
Suppose that $p(x,y)=\sum_{k,\ell}c_{k,\ell}x^{k}y^{\ell}$ is a polynomial in
two variables. Since $\phi$ is a continuous convex function on a bounded
interval, it is absolutely continuous (see [10, Theorem 14.12]). Then
$\Psi(p)(x)=\sum_{k,\ell}c_{k,\ell}x^{k}\phi(x)^{\ell}$ is also absolutely
continuous on $J$. Since $\Psi$ is continuous, it must therefore map
${AC}(\sigma_{0})$ into ${AC}(J)$. (In fact the map is clearly an
isomorphism.)
For $(x,y)\in R$, let $g(x,y)={\hat{f}}(x)$. By Proposition 4.4 of [1], $g$ is
absolutely continuous on $R$ and $\left\lVert
g\right\rVert_{{BV}(R)}=\|\hat{f}\|_{{BV}(J)}\leq 2\left\lVert
f\right\rVert_{BV(\sigma_{0})}$. ∎
###### Remark 7.4.
The factor of $2$ in the above lemma is necessary, at least for the given
construction. Let $\phi:[-1,1]\to[0,1]$, $\phi(x)=|x|$, and $f(x,|x|)=|x|$. If
$\sigma_{0}$ is the graph of $\phi$ then
$\operatorname*{var}(f,\sigma_{0})=1$, but
$\operatorname*{var}({\hat{f}},[-1,1])=2$.
One can obviously prove analogous theorems to cover situations where the
function $\phi$ is less well-behaved, at the expense of replacing the constant
$2$ with the ‘variation factor’ of the graph of $\phi$. The method of proof of
the lemma will certainly fail for a highly oscillatory curve $\phi$.
###### Lemma 7.5.
Let $R\subseteq\mathbb{R}^{2}$ be a rectangle with centre ${\boldsymbol{x}}$.
Let $S$ be any closed sector of $R$ with vertex ${\boldsymbol{x}}$ and let
$\sigma$ be the part of the boundary of $S$ consisting of the two sides of $S$
that meet at ${\boldsymbol{x}}$. If $f\in{AC}(\sigma)$ then there is an
extension $g\in{AC}(R)$ of $f$ with $\left\lVert g\right\rVert_{{BV}(R)}\leq
2\left\lVert f\right\rVert_{{BV}{\sigma}}$.
###### Proof.
By affine invariance it suffices to consider the case where ${\boldsymbol{x}}$
is the origin and $\sigma=\\{(x,y)\in R\thinspace:\thinspace y=\alpha|x|\\}$
for some $\alpha\in\mathbb{R}$. Let $R_{1}$ be the smallest rectangle with
sides parallel to the axes which contains $R$ and let $\sigma_{1}=\\{(x,y)\in
R_{1}\thinspace:\thinspace y=\alpha|x|\\}$. It is easy to extend $f$ to
$\sigma_{1}$, without increase in norm, by making it constant on
$\sigma_{1}\setminus\sigma_{2}$. By Lemma 7.1 we can now extend $f$ to
$g_{1}\in{AC}(R_{1})$ and then let $g=g_{1}|R$. ∎
###### Theorem 7.6.
Suppose that $\sigma_{1}$ and $\sigma_{2}$ are nonempty compact subsets of the
plane with polygonal boundaries, and that $\sigma=\sigma_{1}\cup\sigma_{2}$.
Suppose that $f:\sigma\to\mathbb{C}$. If $f|\sigma_{1}\in{AC}(\sigma_{1})$ and
$f|\sigma_{2}\in{AC}(\sigma_{2})$ then $f\in{AC}(\sigma)$.
###### Proof.
It suffices to consider the case where $\sigma_{1}$ and $\sigma_{2}$ have
disjoint interiors. If this were not the case, then we could replace
$\sigma_{2}$ with $\mathrm{cl}(\sigma_{2}\setminus\sigma_{1})$.
Suppose that ${\boldsymbol{x}}\in\sigma$. If ${\boldsymbol{x}}$ lies in only
one of the sets $\sigma_{1}$ or $\sigma_{2}$, or if ${\boldsymbol{x}}$ lies in
the interior of either of these sets, then Theorem 5.1 implies that there
exists a compact neighbourhood $U_{{\boldsymbol{x}}}$ of ${\boldsymbol{x}}$ in
$\sigma$ such that $f|U_{{\boldsymbol{x}}}\in{AC}(U_{{\boldsymbol{x}}})$.
The remaining case is when ${\boldsymbol{x}}$ lies in the intersection of the
boundaries of $\sigma_{1}$ and $\sigma_{2}$. Since these sets have polygonal
boundaries, we can choose a small square $R$ centred at ${\boldsymbol{x}}$ so
that $R\cap\sigma$ consists of a finite number of sectors each with vertex
${\boldsymbol{x}}$. Indeed we can choose a collection of $n$ lines through
${\boldsymbol{x}}$ which split $R$ into sectors $S_{1},\dots,S_{2n}$ in such a
way that for $1\leq i\leq 2n$, $\mathop{\mathrm{int}}(S_{i})$ is a subset of
exactly one of $\sigma_{1}$, $\sigma_{2}$ or $R\setminus\sigma$. For
convenience we shall number the sectors consecutively so that
$S_{1},\dots,S_{n}$ lie on one side of one of the lines, and the remaining
sectors lie on the other side. We can now extend $f$ to the sectors where
$\mathop{\mathrm{int}}(S_{i})\cap\sigma=\emptyset$ using Lemma 7.5. (Note that
if more than one such sector is contiguous, you should apply Lemma 7.5 to
their union to obtain the extension.) This leads to an extension $\hat{f}$ of
$f$ to all of $R$. Then $\hat{f}|S_{i}\in{AC}(S_{i})$ for all $i$.
$\sigma_{1}$$\sigma_{2}$$\sigma_{1}$${\boldsymbol{x}}$ Figure 7. Sectors
meeting at the vertex ${\boldsymbol{x}}$.
Repeated use of Lemma 7.1 (and the subsequent remark) shows that
$\hat{f}\in{AC}(S_{1}\cup\dots\cup S_{n})$ and that
$\hat{f}\in{AC}(S_{n+1}\cup\dots\cup S_{2n})$. Applying that lemma one more
time implies that $\hat{f}\in{AC}(R)$. In particular,
$f|(R\cap\sigma)\in{AC}(R\cap\sigma)$.
It now follows from the Patching Lemma that $f\in{AC}(\sigma)$. ∎
## References
* [1] B. Ashton and I. Doust, Functions of bounded variation on compact subsets of the plane, Studia Math. 169 (2005), 163–188.
* [2] B. Ashton and I. Doust, A comparison of algebras of functions of bounded variation, Proc. Edinb. Math. Soc. (2) 49 (2006), 575–591.
* [3] B. Ashton and I. Doust, ${AC}(\sigma)$ operators, J. Operator Theory 65 (2011), 255–279.
* [4] B. Ashton and I. Doust, Compact ${AC}(\sigma)$ operators, Integral Equations Operator Theory 63 (2009), 459–472.
* [5] D. Bongiorno, Absolutely continuous functions in $\mathbb{R}^{n}$, J. Math. Anal. Appl. 303 (2005) 119–134.
* [6] J. A. Clarkson and C. R. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc. 35 (1933), 824–854.
* [7] M. Csörnyei, Absolutely continuous functions of Rado, Reichelderfer, and Malý, J. Math. Anal. Appl. 252 (2000) 147–166.
* [8] G. H. Meisters, Polygons have ears, Amer. Math. Monthly 82 (1975), 648–651.
* [9] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89.
* [10] J. Yeh, Lectures on Real Analysis, World Scientific, Singapore, 2000.
|
arxiv-papers
| 2013-12-06T09:08:44 |
2024-09-04T02:49:55.098738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ian Doust, Michael Leinert and Alan Stoneham",
"submitter": "Ian Doust",
"url": "https://arxiv.org/abs/1312.1806"
}
|
1312.2075
|
Dirac particles’ tunnelling from 5-dimensional rotating black strings
influenced by the generalized uncertainty principle
Deyou Chen 111E-mail: [email protected]
Institute of Theoretical Physics, China West Normal University,
Nanchong 637009, China
Abstract: The standard Hawking formula predicts the complete evaporation of
black holes. Taking into account effects of quantum gravity, we investigate
fermions’ tunnelling from a 5-dimensional rotating black string. The
temperature is determined not only by the string, but also affected by the
quantum number of the emitted fermion and the effect of the extra spatial
dimension. The quantum correction slows down the increase of the temperature,
which naturally leads to the remnant in the evaporation.
## 1 Introduction
The semi-classical tunnelling method is an effective way to describe the
Hawking radiation [2]. Using this method, the tunnelling behavior of massless
particles across the horizon was veritably described in [3]. In the research,
the varied background spacetime was taken into account. The tunnelling rate
was related to the change of the Bekenstein-Hawking entropy and the
temperature was higher than the standard Hawking temperature. In the former
researches, the standard temperatures were derived [4, 5, 6, 7, 8, 9], which
imply the complete evaporation of black holes. Thus the varied background
spacetime accelerates the black holes’ evaporation. This result was also
demonstrated in other complicated spacetimes [10, 11, 12, 13, 14]. Extended
this work to massive particles, the tunnelling radiation of general spacetimes
was investigated in [15, 16]. The same result was derived by the relation
between the phase velocity and the group velocity.
In [17], the standard Hawking temperatures were recovered by fermions
tunnelling across the horizons. In the derivation, the action of the emitted
particle was derived by the Hamilton-Jacobi equation [18]. This derivation is
based on the complex path analysis [19]. In this method, we don’t need the
consideration of that the particle moves along the radial direction [20, 21,
22, 23]. This is a difference from the work of Parikh and Wilczek [3].
The tunnelling radiation beyond the semi-classical approximation was discussed
in [24, 25, 26]. Their ansatz is also based on the Hamilton-Jacobi method. The
key point is to expand the action in a powers of $\hbar$. Using the expansion,
one can get the quantum corrections over the semiclassical value. The
corrected temperature is lower than the standard Hawking temperature. The
higher order correction entropies were derived by the first law of black hole
thermodynamics.
Taking into account effects of quantum gravity, the semi-classical tunnelling
method was reviewed in the recent work [27, 28]. In [27], the tunnelling of
massless particles through quantum horizon of a Schwarzschild black hole was
investigated by the influence of the generalized uncertainty principle (GUP).
Through the modified commutation relation between the radial coordinate and
the conjugate momentum and the deformed Hamiltonian equation, the radiation
spectrum with the quantum correction was derived. The thermodynamic quantities
were discussed. In the fermionic fields, with the consideration of effects of
quantum gravity, the generalized Dirac equation in curved spacetime was
derived by the modified fundamental commutation relations [29], which is [28]
$\displaystyle\left[i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta
m^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{m}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right.$
$\displaystyle\left.+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right]\psi=0.$ (1)
This derivation is based on the existence of a minimum measurable length. The
length can be realized in a model of GUP
$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\left[1+\beta(\Delta
p)^{2}+\beta<p>^{2}\right],$ (2)
where $\beta=\beta_{0}\frac{l^{2}_{p}}{\hbar^{2}}$ is a small value,
$\beta_{0}<10^{34}$ is a dimensionless parameter and $l_{p}$ is the Planck
length. Eq. (2) was derived by the modified Heisenberg algebra
$\left[x_{i},p_{j}\right]=i\hbar\delta_{ij}\left[1+\beta p^{2}\right]$, where
$x_{i}$ and $p_{i}$ are position and momentum operators defined respectively
as [29, 30]
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle x_{0i},$ $\displaystyle
p_{i}$ $\displaystyle=$ $\displaystyle p_{0i}(1+\beta p_{0}^{2}),$ (3)
$p_{0}^{2}=\sum p_{0j}p_{0j}$, $x_{0i}$ and $p_{0j}$ satisfy the canonical
commutation relations $\left[x_{0i},p_{0j}\right]=i\hbar\delta_{ij}$. Thus the
minimal position uncertainty is gotten as
$\displaystyle\Delta x$ $\displaystyle=$
$\displaystyle\hbar\sqrt{\beta}\sqrt{1+\beta<p>^{2}},$ (4)
which means that the minimum measurable length is $\Delta
x_{0}=\hbar\sqrt{\beta}$ [29]. To let $\Delta x_{0}$ have a physical meaning,
the condition $\beta>0$ must be satisfied. It was showed in [29]. Based on the
GUP, the black hole’s remnant was first researched by Adler et al. [31].
Incorporate eq. (3) into the Dirac equation in curved spacetime, the modified
Dirac equation was derived[28]. Using this modified equation, fermions’
tunnelling from the Schwarzschild spacetime was investigated. The temperature
was showed to be related to the quantum number of the emitted fermion. An
interested result is that the quantum correction slows down the increase of
the temperature. It is natural to lead to the remnant.
In this paper, taking into account effects of quantum gravity, we investigate
fermions’ tunnelling from a 5-dimensional rotating black string. The key point
in this paper is to construct a tetrad and five gamma matrices. The result
shows that in the frame of quantum gravity, the temperature is affected not
only by the quantum number of the emitted fermion, but also by the effect of
the extra compact dimension. The quantum correction slows down the increase of
the temperature. The remnant is naturally observed in the evaporation.
In the next section, we perform the dragging coordinate transformation on the
metric and construct five gamma matrices, then investigate the fermion’s
tunnelling from the 5-dimensional rotating string. The remnant is observed.
Section 3 is devoted to our conclusion.
## 2 Tunnelling radiation with the influence of the generalized uncertainty
principle
The Kerr metric describes a rotating black hole solution of the Einstein
equations in four dimensions. When we add an extra compact spatial dimension
to it, the metric becomes
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta}{\rho^{2}}\left(dt-a\sin^{2}\theta
d\varphi\right)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[adt-(r^{2}+a^{2})d\varphi\right]^{2}$
(5)
$\displaystyle+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+g_{zz}dz^{2},$
where $\Delta=r^{2}-2Mr+a^{2}=(r-r_{+})(r-r_{-})$,
$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$, $g_{zz}$ is usually set to $1$. The
above metric describes a rotating uniform black string.
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ are the locations of the event (inner)
horizons. $M$ and $a$ are the mass and angular momentum unit mass of the
string, respectively. A fermion’s motion satisfies the generalized Dirac
equation (1). To investigate the tunnelling behavior of the fermion, it can
directly choose a tetrad and construct gamma matrices from the metric (5). The
metric (5) describes a rotating spacetime. The energy and mass near the
horizons are dragged by the rotating spacetime. It is not convenient to
discuss the fermion’s tunnelling behavior. For the convenience of constructing
the tetrad and gamma matrices, we perform the dragging coordinate
transformation $d\phi=d\varphi-\Omega dt$, where
$\Omega=\frac{\left(r^{2}+a^{2}-\Delta\right)a}{\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}\sin^{2}\theta},$ (6)
on the metric (5). Then the metric (5) takes on the form as
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-F(r)dt^{2}+\frac{1}{G(r)}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}+g_{zz}dz^{2}$
(7) $\displaystyle=$
$\displaystyle-\frac{\Delta\rho^{2}}{(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}{\theta}}dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+g_{zz}dz^{2}$
$\displaystyle+\rho^{2}d\theta^{2}+\frac{\sin^{2}{\theta}}{\rho^{2}}\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}{\theta}\right]d\phi^{2}.$
Now the tetrad is directly constructed from the above metric. It is
$\displaystyle
e_{\mu}^{a}=diag(\sqrt{F},1/\sqrt{G},\sqrt{g_{\theta\theta}},\sqrt{g_{\phi\phi}},\sqrt{g_{zz}}).$
(8)
Then gamma matrices are easily constructed as follows
$\displaystyle\gamma^{t}=\frac{1}{\sqrt{F}}\left(\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right),$
$\displaystyle\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right),$ (13)
$\displaystyle\gamma^{r}=\sqrt{G}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (18)
$\displaystyle\gamma^{z}=\sqrt{g^{zz}}\left(\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right).$ (21)
When measure the quantum property of a spin-1/2 fermion, we can get two
values. They correspond to two states with spin up and spin down. The wave
functions of two states of a fermion in the metric (7) spacetime take on the
form as
$\displaystyle\psi_{\left(\uparrow\right)}=\left(\begin{array}[]{c}A\\\ 0\\\
B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\uparrow}\left(t,r,\theta,\phi,z\right)\right),$
(26)
$\displaystyle\psi_{\left(\downarrow\right)}=\left(\begin{array}[]{c}0\\\ C\\\
0\\\
D\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\downarrow}\left(t,r,\theta,\phi,z\right)\right),$
(31)
where $A,B,C,D$ are functions of $(t,r,\theta,\phi,z)$, and $I$ is the action
of the fermion, $\uparrow$ and $\downarrow$ denote the spin up and spin down,
respectively. In this paper, we only investigate the state with spin up. The
analysis of the state with spin down is parallel. To use the WKB
approximation, we insert the wave function (26) and the gamma matrices into
the generalized Dirac equation (1). Dividing by the exponential term and
considering the leading terms yield four equations. They are
$\displaystyle-\frac{B}{\sqrt{F}}\partial_{t}I-B\sqrt{G}(1-\beta
m^{2})\partial_{r}I+A\sqrt{g^{zz}}(1-\beta m^{2})\partial_{z}I$
$\displaystyle-Am(1-\beta m^{2}-\beta
Q)+B\beta\sqrt{G}Q\partial_{r}I-A\beta\sqrt{g^{zz}}Q\partial_{z}I=0,$ (32)
$\displaystyle\frac{A}{\sqrt{F}}\partial_{t}I-A\sqrt{G}(1-\beta
m^{2})\partial_{r}I-B\sqrt{g^{zz}}(1-\beta m^{2})\partial_{z}I$
$\displaystyle-Bm(1-\beta m^{2}-\beta
Q)+A\beta\sqrt{G}Q\partial_{r}I+B\beta\sqrt{g^{zz}}Q\partial_{z}I=0,$ (33)
$\displaystyle-B\left(i\sqrt{g^{\theta\theta}}\partial_{\theta}I+\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)(1-\beta
m^{2}-\beta Q)=0,$ (34)
$\displaystyle-A\left(i\sqrt{g^{\theta\theta}}\partial_{\theta}I+\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)(1-\beta
m^{2}-\beta Q)=0,$ (35)
where
$Q=g^{rr}\left({\partial_{r}I}\right)^{2}+g^{\theta\theta}\left({\partial_{\theta}I}\right)^{2}+g^{\phi\phi}\left({\partial_{\phi}I}\right)^{2}+g^{zz}\left({\partial_{z}I}\right)^{2}$.
It is difficult to get the expression of the action from the above equations.
Considering the property of the spacetime, we carry out separation of
variables as
$\displaystyle I=-(\omega-j\Omega)t+W(r)+\Theta(\theta,\phi)+Jz,$ (36)
where $\omega$ is the energy of the emitted fermion, $j$ is the angular
momentum and $J$ is a conserved momentum corresponding to the compact
dimension. Eqs. (34) and (35) are irrelevant to $A,B$. Inserting Eq. (36) into
them yields
$\displaystyle
i\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+\sqrt{g^{\phi\phi}}\partial_{\phi}\Theta=0,$
(37)
which implies that $\Theta$ is a complex function other than the constant
solution. In the former research, it was found that the contribution of
$\Theta$ could be canceled in the derivation of the tunnelling rate. Using Eq.
(37), an important relation is easily gotten as
$\displaystyle
g^{\theta\theta}(\partial_{\theta}\Theta)^{2}+g^{\phi\phi}(\partial_{\phi}\Theta)^{2}=0.$
(38)
Now our interest is the first two equations. Inserting Eq. (36) into Eqs. (32)
and (33), canceling $A$ and $B$ and neglecting the higher order terms of
$\beta$, we get
$\displaystyle A(\partial_{r}W)^{4}+B(\partial_{r}W)^{2}+C=0,$ (39)
where
$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\beta G^{2}F,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle-[1-4\beta
g^{zz}\left({\partial_{z}I}\right)^{2}]GF,$ $\displaystyle C$ $\displaystyle=$
$\displaystyle[1-2\beta m^{2}-2\beta
g^{zz}\left({\partial_{z}I}\right)^{2}](m^{2}-g^{zz}\left({\partial_{z}I}\right)^{2})F+\left({\partial_{t}I}\right)^{2}.$
(40)
Solving the above equation at the event horizon yields the imaginary part of
the radial action. Based on the invariance under canonical transformations, we
adopt the method developed in [33]. The tunnelling rate is
$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp[-\frac{1}{\hbar}Im\oint p_{r}dr]=exp\left[-\frac{1}{\hbar}Im\left(\int
p_{r}^{out}dr-\int p_{r}^{in}dr\right)\right]$ (41) $\displaystyle=$
$\displaystyle exp\left[\mp\frac{2}{\hbar}Im\int p_{r}^{out,in}dr\right].$
In the above equation, $\oint p_{r}dr$ is invariant under canonical
transformations. Here let $p_{r}=\partial_{r}W$. Thus the solutions of $Im\int
p_{r}^{out,in}dr$ are determined by Eq. (39), which is
$\displaystyle Im\oint p_{r}dr$ $\displaystyle=$ $\displaystyle 2ImW^{out}$
(42) $\displaystyle=$ $\displaystyle 2Im\int
dr\sqrt{\frac{(E-j\Omega)^{2}+(1-2\beta m^{2}-2\beta
g^{zz}J^{2})(m^{2}-g^{zz}J^{2})F}{GF(1-4\beta g^{zz}J^{2})}}$
$\displaystyle\times\left[1+\beta\left(\frac{(E-j\Omega)^{2}}{F}+m^{2}-g^{zz}J^{2}\right)\right]$
$\displaystyle=$ $\displaystyle
2\pi\frac{(\omega-j\Omega_{+})(r_{+}^{2}+a^{2})}{r_{+}-r_{-}}\left[1+\beta\Xi(J,\theta,r_{+},j)\right],$
where $g^{zz}=1$, $\Omega_{+}=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular
velocity at the event horizon. $\Xi(J,\theta,r_{+},j)$ is a complicated
function of $J,\theta,r_{+},j$, therefore, we don’t write down here. It should
be that $\Xi(J,\theta,r_{+},j)>0$. If adopt Eq. (42) to calculate the
tunnelling rate, we will derive two times Hawking temperature, which was
showed in [32]. This is not in consistence with the standard temperature. With
careful observations, Akhmedova et. al. found that the contribution coming
from the temporal part of the action was ignored [33]. When they took into
account the temporal contribution, the factor of two in the temperature was
resolved.
To find the temporal contribution, we use the Kruskal coordinates $(T,R)$. The
region exterior to the string $(r>r_{+})$ is described by
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ (43)
where
$r_{*}=r+\frac{1}{2\kappa_{+}}ln\frac{r-r_{+}}{r_{+}}-\frac{1}{2\kappa_{-}}ln\frac{r-r_{-}}{r_{-}}$
is the tortoise coordinate, and
$\kappa_{\pm}=\frac{r_{+}-r_{-}}{2(r_{\pm}^{2}+a^{2})}$ denote the surface
gravity at the outer (inner) horizons. The description of the interior region
is given by
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t).$ (44)
To connect these two patches across the horizon, we need to rotate the time
$t$ as $t\rightarrow t-i\kappa_{+}\frac{\pi}{2}$. As pointed in [33], this
rotation would lead to an additional imaginary contribution coming from the
temporal part, namely, $Im(E\Delta t^{out,in})=\frac{1}{2}\pi E\kappa_{+}$,
where $E=\omega-j\Omega_{+}$. Thus the total temporal contribution is
$Im(E\Delta t)=\pi E\kappa_{+}$. Therefore, the tunnelling rate is
$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp\left[-\frac{1}{\hbar}\left(Im(E\Delta t)+Im\oint p_{r}dr\right)\right]$
(45) $\displaystyle=$
$\displaystyle-4\pi\frac{(\omega-j\Omega_{+})(r_{+}^{2}+a^{2})}{\hbar(r_{+}-r_{-})}\left[1+\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right].$
This is the Boltzman factor expression and implies the temperature
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})\left[1+\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right]}$
(46) $\displaystyle=$ $\displaystyle
T_{0}\left[1-\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right],$
where $T_{0}=\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})}$ is the standard
Hawking temperature of the Kerr string and shares the same expression of the
temperature of the 4-dimensional Kerr black hole. It shows that the corrected
temperature is determined by the mass, angular momentum and extra dimension of
the string, but also affected by the quantum number (energy, mass and angular
momentum) of the fermion. Therefore, the property of the emitted fermion
affects the temperature when the effects of quantum gravity are taken into
account.
When $a=0$, the metric (5) is reduced to the Schwarzschild string metric. Then
the imaginary part of the radial action (42) is reduced to
$\displaystyle Im\oint p_{r}dr$ $\displaystyle=$ $\displaystyle 2\pi\omega
r_{+}\left[1+\beta\left(2\omega^{2}+3m^{2}/2+J^{2}/2\right)\right].$ (47)
Adopting the same process, we get the temperature of the Schwarzschild string
as
$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{\hbar}{4\pi
r_{+}\left[1+\beta\left(\omega^{2}+3m^{2}/4+J^{2}/4\right)\right]}$ (48)
$\displaystyle=$ $\displaystyle\frac{\hbar}{8\pi
M}\left[1-\beta\left(\omega^{2}+3m^{2}/4+J^{2}/4\right)\right].$
It shows that the effect of the extra dimension and the quantum number
(energy, mass and angular momentum) of the fermion affect the temperature of
the Schwarzschild string. It is obviously that the quantum correction slows
down the crease of the temperature. Finally, the string can not evaporate
completely and there is a blanched state. At the this state, the remnant is
left. The effect of the extra dimension plays an role of impediment during the
evaporation. When $J=0$, Eq. (48) describes the temperature of the
4-dimensional Schwarzschild black hole. The remnant was derived as $\geq
M_{p}/{\beta_{0}}$, where $M_{p}$ is the Planck mass and $\beta_{0}$ is a
dimensionless parameter accounting for quantum gravity effects [28].
## 3 Conclusion
In this paper, we investigated the fermion’s tunnelling from the 5-dimensional
Kerr string spacetime. To incorporate the influence of quantum gravity, we
adopted the generalized Dirac equation derived in [28]. The corrected
temperature is not only determined by the mass, angular momentum and extra
dimension, but also affected by the quantum number of the emitted fermion. The
quantum correction slows down the increase of the temperature. Finally, the
balance state appears. At this state, the string can not evaporate completely
and the remnant is left. This can be seen as the direct consequence of the
generalized uncertainty principle.
Acknowledgements
This work is supported by the National Natural Science Foundation of China
with Grant No. 11205125.
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|
arxiv-papers
| 2013-12-07T09:18:49 |
2024-09-04T02:49:55.124620
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Deyou Chen",
"submitter": "Deyou Chen",
"url": "https://arxiv.org/abs/1312.2075"
}
|
1312.2134
|
# Substrate enhanced superconductivity in Li-decorated graphene
T. P. Kaloni1, A. V. Balatsky2,3,4, and U. Schwingenschlögl1
[email protected] 1PSE Division, KAUST, Thuwal 23955-6900,
Kingdom of Saudi Arabia
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA
3Center for Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA
4Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm Sweden
###### Abstract
We investigate the role of the substrate for the strength of the electon
phonon coupling in Li-decorated graphene. We find that the interaction with a
$h$-BN substrate leads to a significant enhancement from $\lambda_{0}=0.62$ to
$\lambda_{1}=0.67$, which corresponds to a 25% increase of the transition
temperature from $T_{c0}=10.33$ K to $T_{c1}=12.98$ K. The superconducting
gaps amount to 1.56 meV (suspended) and 1.98 meV (supported). These findings
open up a new route to enhanced superconducting transition temperatures in
graphene-based materials by substrate engineering.
###### pacs:
74.78.Db, 63.20.Dj, 81.05.Uw, 31.15.Ar
## I Introduction
Recent observations in alkali-doped graphene have opened exciting venues to
superconductivity accomplished by doping mouri . Most theoretical estimates of
the electron-phonon coupling so far have assumed suspended graphene as a base,
since this geometry makes calculations more direct and less computationally
costly. However, most of the engineered superconducting graphene samples use a
substrate. Hence, it is important to characterize the role of the substrate on
the superconductivity in atomically thin graphene. We have performed a first-
principles study of the role of the substrate on the phonon spectrum and the
electron-phonon coupling and find not only that the interaction with the
substrate is relevant but that in the case of a $h$-BN substrate the electron-
phonon coupling can be enhanced by as much as 9% so that $T_{c}$ can be
expected to reach 12.98 K, a 25% increase. This observation points to a new
direction in the search for novel superconducting materials: substrate-
engineered superconductivity, where the nascent superconducting states are
significantly enhanced by the coupling to a properly chosen substrate.
Graphite intercalated compounds are characterized by a nearly free electron
band, which upon increased doping crosses the Fermi energy ($E_{F}$).
Empirically, there are intercalated compounds that exhibit superconductivity
with a transition temperature of a few to about 10 K. An empirical correlation
between the crossing of the chemical potential and the onset of
superconductivity was first put forward in Ref. Littlewood and subsequently
was called the “Cambridge criterion” balatsky . Superconductivity in Ca-
intercalated bilayer graphene has been predicted with a sizable $T_{c}=11.5$ K
by analyzing this criterion in Refs. balatsky ; jishi . Recently, the
prediction has been verified experimentally for Ca-intercalated graphene on
either the Si or the C face of a SiC substrate, finding $T_{c}=7$ K Li .
Experimentally, it also has been observed that KC8 graphite valla and
K-intercalated few layer graphene on SiC are superconducting jacs , where
theoretical arguments for superconductivity in the latter material have been
presented in Ref. kaloni-epl . To complete the list of superconducting C
allotropes we also mention that undoped single and multiwall nanotubes exhibit
superconductivity with a sizable $T_{c}\sim 10$ to 12 K Tang ; Takesue .
Today, the highest value of $T_{c}=38$ K is observed experimentally in Cs3C60
Alexey .
It was already pointed out that not all intercalants lead to an enhanced
$T_{c}$ in graphite Littlewood . A high $T_{c}$ can be obtained when the
distance between the intercalated atom and the graphene plane is small so that
the deformation potential is large Boeri . Our observations are consistent
with this mechanism: The distance between the intercalant and the graphene
plane is 2.62 Å, 2.47 Å, and 2.26 Å for non-superconducting BaC6 Boeri1 ,
superconducting SrC6 with $T_{c}=1.65$ K Boeri1 , and superconducting CaC6
with $T_{c}=11.5$ K Emery ; Ellerby , respectively, see Table I.
compound | distance | $T_{c}$
---|---|---
BaC6 | 2.62 Å | 0 K
SrC6 | 2.47 Å | 1.65 K
CaC6 | 2.26 Å | 11.5 K
Table 1: Compound, perpendicular distance of the intercalated atom from the
center of the C hexagon, and superconducting transition temperature.
In this paper we report a first-principles study of the electron-phonon
coupling to estimated the values of $\lambda$ and $T_{c}$ for Li-decorated
suspended graphene and Li-decorated graphene on a $h$-BN substrate. We show
that the presence of the substrate enhances the electron-phonon coupling and
superconducting transition temperature, which reflects a significant impact of
the interaction of the electronic states with the substrate on the phonon
mediated superconductivity in doped graphene.
## II Results and discussion
The unit cell of Li-decorated monolayer graphene comprises 6 C atoms and 1 Li
atom in a $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ geometry, where the Li atom
lies above the center of the C hexagon in a distance of 1.76 Å, slightly
smaller than the value reported in Ref. mouri . The possible reason for the
latter is inclusion of the van der Waals interaction in our calculations,
which is expected to provide a correct interlayer spacing. The structural
arrangements of Li-decorated graphene suspended and supported by a $h$-BN
substrate are presented in Figs. 1(a) and 1(b). The electronic band structures
obtained for $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ suspended graphene
without and with Li-decoration are shown in Figs. 2(a) and 2(b). It is well
known that the C $\pi$ and $\pi^{*}$ orbitals form a Dirac cone at the Fermi
energy. Due to Brillouin zone backfolding, the Dirac cone appears at the
$\Gamma$-point and not at the K-point as in the case of the primitive unit
cell of graphene.
Figure 1: Crystal structure of Li-decorated graphene (a) suspended and (b)
supported by a $h$-BN substrate.
The electronic band structure of Li-decorated graphene is found to be modified
significantly as compared to that of pristine graphene. The nearly free
electron Li $s$ band crosses the Fermi level, due to charge transfer from Li
to C. As a consequence, the “Cambridge criterion” is satisfied and the system
should be a superconductor. We will comment later on this phonomenon by
analyzing the strength of the electron-phonon coupling. A gap of 0.38 eV opens
1.56 eV below the Fermi level, as to be expected kaloni-cpl ; Farjam . In Fig.
2(b) the partially occupied parabolic bands indicated by arrows are due to Li
$s$ states, compare Figs. 2(a) and 2(b). It has been reported that the carrier
density in Li-decorated monolayer and Li-intercalated multilayer graphene with
and without substrate can differ by a factor of 100 from that of pristine
graphene kaloni-cpl .
Figure 2: Electronic band structure of (a) suspended C6, (b) suspended C6Li,
(c) C6 on $h$-BN, and (d) C6Li on $h$-BN. Li $s$ bands crossing the Fermi
level are indicated by arrows.
For Li-decorated graphene on $h$-BN, see Fig. 1(b), the separation between
graphene and the substrate is found to be 3.39 Å, which is close to the values
for superlattices of graphene and $h$-BN as well as graphene on a $h$-BN
substrate. The perpendicular distance of the Li atom to the graphene plane is
1.77 Å. The lost sublattice symmetry (only each third C hexagon is occupied by
a Li atom) is responsible for a band gap of 90 meV. This value agrees well
with previous reports Quhe ; Naveh ; kaloni-jmc , which also applies to the
fact that the B and N states appear far away from the Fermi level. The
electronic band structure in Fig. 2(d) clearly shows that a nearly free
electron Li $s$ band crosses the Fermi level, satisfying the “Cambridge
criterion” and thus pointing to superconductivity in the system. The nature
and magnitude of the gap at the $\Gamma$-point just below the Fermi level are
similar to Fig. 2(b).
At this point we assume that the basic mechanism of the superconductivity in
the suspended and supported cases is the same as in Ca-intercalated graphene,
i. e., electron-phonon driven pairing balatsky . It has been proposed that the
dopant-induced soft phonon modes contribute substantially to the electron-
phonon coupling Mazin ; Littlewood and it is known that the motion of the
adatom is responsible for about half of the coupling, while the other half is
due to the C atoms Calandra ; Sanna ; Rosenmann . The presence of the Li $s$
states around the Fermi energy alone cannot be sufficient to give a large
electron-phonon coupling mouri , but the coupling to the out-of-plane C
vibrations plays an important role due to transitions between the C $\pi^{*}$
and Li $s$ states. The Li $s$ band enhances the coupling Boeri and, hence,
the transition temperature. For this reason, we calculate the phonon
dispersion, see Fig. 3(a), and $\alpha^{2}F(\omega)$, see Fig. 3(b), and
estimate the strength of the electron-phonon coupling $\lambda$ using Eq. (2).
For the phonon dispersion of Li-decorated suspended graphene we find that most
modes between 300 cm-1 and 500 cm-1 are due to a mixture of Li and out-of-
plane C vibrations. The pure out-of-plane modes appear from 500 cm-1 to 900
cm-1 and higher energy C-C stretching modes from 900 cm-1 to 1515 cm-1. The
modes from 300 cm-1 to 500 cm-1 are responsible for the electron-phonon
coupling. This also can be seen from $\alpha^{2}F(\omega)$ as addressed in
Fig. 3(b). Experimentally, for pristine graphene the frequency of the G-mode
is 1580 cm-1 mouri1 , which softens to 1515 cm-1 under Li decoration. The
softening can be attributed to charge transfer from Li to graphene and the
induced stronger electron-phonon coupling, in agreement with findings for the
molecular/atomic charge transfer in graphene Rao ; carbon . We obtain for the
electron-phonon coupling $\lambda=0.62$ and estimate for the superconducting
transition temperature $T_{c}=10.33$ K. This value is slightly higher than
that of Ref. mouri , since we take into account the van der Waals interaction
to achieve an accurate distance to the Li atom.
Figure 3: Electron-phonon dispersion of Li-decorated graphene (a) suspended
and (c) supported by a $h$-BN substrate. (b,c) Corresponding Eliashberg
functions.
Our central result is that an enhancement of the superconductivity in Li-
decorated graphene can be achieved by the application of a $h$-BN substrate.
The phonon modes between 100 cm-1 and 300 cm-1 at the $\Gamma$-point are
attributed to the Li vibrations, out-of-plane C vibrations, and $h$-BN
substrate, see Fig. 3(c). There are also substrate modes around 850 cm-1 as
well as higher energy modes between 900 cm-1 and 1430 cm-1, which are due to
both the substrate and C-C stretching. The softening of the modes as compared
to the suspended system is due to the interaction with the substrate, as
observed experimentally, for example, in graphite supported by Ni(111) new .
The modes in the range from 100 cm-1 to 300 cm-1 are responsible for a shift
in $\alpha^{2}F(\omega)$ see Fig. 3(d), and enhancement of $\lambda$ and
$T_{c}$. We obtain $\lambda=0.67$ (as compared to $\lambda=0.62$ in the
suspended system) and thus a higher $T_{c}=12.98$ K, which is a 25% increase
with respect to the suspended system. The clearly indicates that one can take
full advantage of the substrate to boost $T_{c}$. Similarly, it has been
observed experimentally in the FeSe0.5Te0.5 superconductor that $T_{c}$ is
enhanced by 15% if the material is supported Johnson , while the details of
the band structure are very different in the present material. The most likely
reason for the obtained enhancement of $T_{c}$ by the application of a $h$-BN
substrate are stronger spin fluctuations due to the lattice mismatch of 1.4%.
Finally, we estimate the superconducting gap $\Delta_{sc}$ by the relation
$1.75k_{B}T_{c}=\Delta_{sc}$ Parker , where $k_{B}$ is the Boltzmann constant.
We obtain for Li-decorated suspended and supported graphene, respectively,
values of 1.56 meV and 1.98 meV.
## III conclusion
In conclusion, using density functional theory we have investigated the role
of the substrate for the electron-phonon coupling in Li-decorated suspended
and supported graphene. We find that the interaction with a $h$-BN substrate
significantly enhances the electron-phonon coupling to $\lambda=0.67$ as
compared to $\lambda=0.62$ in the suspended case. The transition temperature
thus is enhanced by 25% to 12.98 K. The superconducting gap for the suspended
and supported systems is found to be 1.56 meV and 1.98 meV, respectively. Our
results show that graphene-based nanomaterials can be tailored by properly
choosing the substrate to robustly increase the superconducting transition
temperature.
###### Acknowledgements.
We thank G. Profeta for fruitful discussions. This work is supported by US
DOE, ERC-DM-321031, and VR.
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## IV Appendix
### IV.1 Computational details
All the results are obtained from density functional theory in the local
density approximation. The van der Waals interaction is taken into account via
Grimme’s scheme grime . We use the Quantum-ESPRESSO code paolo with norm-
conserving pseudopotentials and a plane wave cutoff energy of 70 Ryd. A
Monkhorst-Pack $32\times 32\times 1$ k-mesh is employed for the optimization
of the lattice parameters and the ionic relaxation and a $48\times 48\times 1$
k-mesh for refining the electronic structure. We achieve an energy convergence
of $10^{-7}$ eV and a force convergence of 0.002 eV/Å. Li-decorated monolayer
graphene is modeled by a $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ supercell
with $a=b=2.26$ to that a Li atom is added on each third hollow site. Phonon
frequencies are determined by density functional perturbation theory for
evaluating the effects of the adatoms on the phonon spectrum, using the scheme
described in Ref. Mod . The phonon dispersion is calculated with a $24\times
24\times 1$ k-mesh. We study the effect of the substrate on the strength of
the electron-phonon coupling and the transition temperature for a supercell
with Li-decorated monolayer graphene on top of $h$-BN with $a=b=4.32$ Å and
$c=15$ Å (to avoid artificial interaction due to the periodicity). Note that
graphene on $h$-BN can be synthesized due to the small lattice mismatch of
only 1.4% and interacts only weakly with the substrate Dean ; Xue ; Yang . By
construction of the supercell of the suspended system, with 6 C atoms and 1 Li
atom, there are 21 phonon modes, whereas we have 39 modes for the supported
system.
### IV.2 Superconducting transition temperature
The Allen-Dynes formula dynes ; allen , which is a modification of McMillan’s
formula mcmillan , is used to calculate
$T_{c}=\frac{<\omega>_{\log}}{1.20}\exp\Big{(}-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\Big{)}.$
(1)
The terms $<\omega>_{\log}$, $\lambda$, and $\mu^{*}$ are the logarithmic
frequency average, electron-phonon coupling constant, and effective Coulomb
repulsion, respectively. Moreover, the dimensionless parameter
$\lambda=2\int_{0}^{\infty}\frac{d\omega\alpha^{2}F(\omega)}{\omega}$ (2)
measures the strength of the Eliashberg function
$\alpha^{2}F(\omega)=N_{\uparrow}(0)\frac{\sum_{kk^{\prime}}|M_{kk^{\prime}}|^{2}\delta(\omega-\omega_{q})\delta(E_{k})\delta(E_{k^{\prime}})}{\sum_{kk^{\prime}}\delta(E_{k})\delta(E_{k^{\prime}})},$
(3)
where $k$ and $q$ represent the electron band index and phonon wave number,
respectively. In addition, $N_{\uparrow}(0)$ is the single-spin density of
states at the Fermi surface and $M_{kk^{\prime}}$ is the matrix element for
electron-phonon coupling. The effective Coulomb repulsion (also called Coulomb
pseudopotential) is given by anderson
$\frac{1}{\mu^{*}}=\frac{1}{\mu}+\ln\left(\frac{\omega_{el}}{\omega_{ph}}\right),$
(4)
where $\omega_{el}$ is the plasma frequency and $\omega_{ph}$ the frequency
cutoff in $\alpha^{2}F(\omega)$. The Coulomb coupling $\mu$ is given by the
product of the density of states at the Fermi surface and the matrix element
of the screened Coulomb interaction averaged over the Fermi surface. We use
$\mu^{*}=0.115$ in agreement with the experimental observation of the critical
temperature for bulk CaC6 Emery
|
arxiv-papers
| 2013-12-07T19:28:18 |
2024-09-04T02:49:55.132199
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. P. Kaloni, A. V. Balatsky, and U. Schwingenschl\\\"ogl",
"submitter": "Thaneshwor Prashad Kaloni",
"url": "https://arxiv.org/abs/1312.2134"
}
|
1312.2267
|
# IRCI Free Range Reconstruction for SAR Imaging with Arbitrary Length OFDM
Pulse
Tian-Xian Zhang, Xiang-Gen Xia, and Lingjiang Kong Tian-Xian Zhang and
Lingjiang Kong are with the School of Electronic Engineering, University of
Electronic Science and Technology of China, Chengdu, Sichuan, P.R. China,
611731. Fax: +86-028-61830064, Tel: +86-028-61830768, E-mail:
[email protected], [email protected]. Zhang’s research was
supported by the Fundamental Research Funds for the Central Universities under
Grant ZYGX2012YB008 and by the China Scholarship Council (CSC) and was done
when he was visiting the University of Delaware, Newark, DE 19716, USA. Xiang-
Gen Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA. Email: [email protected]. Xia’s
research was partially supported by the Air Force Office of Scientific
Research (AFOSR) under Grant FA9550-12-1-0055.
###### Abstract
Our previously proposed OFDM with sufficient cyclic prefix (CP) synthetic
aperture radar (SAR) imaging algorithm is inter-range-cell interference (IRCI)
free and achieves ideally zero range sidelobes for range reconstruction. In
this OFDM SAR imaging algorithm, the minimum required CP length is almost
equal to the number of range cells in a swath, while the number of subcarriers
of an OFDM signal needs to be more than the CP length. This makes the length
of a transmitted OFDM sequence at least almost twice of the number of range
cells in a swath and for a wide swath imaging, the transmitted OFDM pulse
length becomes long, which may cause problems in some radar applications. In
this paper, we propose a CP based OFDM SAR imaging with arbitrary pulse
length, which has IRCI free range reconstruction and its pulse length is
independent of a swath width. We then present a novel design method for our
proposed arbitrary length OFDM pulses. Simulation results are presented to
illustrate the performances of the OFDM pulse design and the arbitrary pulse
length CP based OFDM SAR imaging.
EDICS: RAS-SARI (Synthetic aperture radar/sonar and imaging), RAS-IMFR (Radar
image formation and reconstruction).
###### Index Terms:
Cyclic prefix (CP), inter-range-cell interference (IRCI), orthogonal
frequency-division multiplexing (OFDM) pulse, range reconstruction, synthetic
aperture radar (SAR) imaging.
## I Introduction
Orthogonal frequency-division multiplexing (OFDM) signals are firstly
presented for radar signal processing in [1], and recently studied and used in
radar applications, such as moving target detection [2, 3, 4], low-grazing
angle target tracking [5] and ultrawideband (UWB) radar applications [6]. The
common OFDM signals for digital communications, such as the digital audio
broadcast (DAB), digital video broadcast (DVB), Wireless Fidelity (WiFi) or
worldwide inoperability for microwave access (WiMAX) signals, are also
investigated for radar applications in [7, 8, 9, 10, 11, 12]. Using OFDM
signals for synthetic aperture radar (SAR) applications is proposed in [13,
14, 15, 16, 17, 18]. In [13, 14], an adaptive OFDM signal design is studied
for range ambiguity suppression in SAR imaging. The reconstruction of cross-
range profiles is studied in [16, 17]. However, all the existing OFDM SAR
signal processing algorithms have not considered the feature of OFDM signals
with sufficient cyclic prefix (CP) as used in communications systems. In [19],
we have proposed a sufficient CP based OFDM SAR imaging algorithm. By using a
sufficient CP, the inter-range-cell interference (IRCI) free and ideally zero
range sidelobes for range reconstruction can be obtained, which provides an
opportunity for high range resolution SAR imaging. On the other hand,
according to our analysis, the CP length, the transmitted OFDM pulse length
and the minimum radar range need to be increased with the increase of a swath
width, since the sufficient CP length is almost equal to the number of range
cells in a swath, while the number of subcarriers of the OFDM signal needs to
be more than the CP length. Then, the transmitted OFDM sequence with
sufficient CP should be at least almost twice of the number of range cells in
a swath. Meanwhile, the CP sequence needs to be removed at the receiver to
achieve the IRCI free range reconstruction. Thus, this algorithm may need a
long transmitted pulse and suffer high transmitted energy redundancy in case
of wide swath imaging, which may cause problems in some radar applications.
Although OFDM signals have been widely used in practical digital
communications and studied for radar applications, the potential high peak-to-
average power ratio (PAPR) of OFDM signals may cause problems for
communications applications [20] and radar applications [3], because the
envelope of OFDM signals is time-varying. In power amplifier of the
transmitter, a constant envelope waveform can be magnified efficiently in the
saturation region. However, the amplifier should be operated in the limited
linear region for a time-varying signal to avoid causing nonlinear distortion.
Many PAPR reduction techniques have been studied as, for example, in [21].
In this paper, we propose a sufficient CP based OFDM SAR imaging with
arbitrary pulse length that is independent of a swath width. Firstly, we
establish the arbitrary pulse length OFDM SAR imaging system model by
considering the feature of OFDM signals with sufficient CP, where the CP part
is all zero. We then derive a sufficient CP based range reconstruction
algorithm with an OFDM pulse, whose length is independent of a swath width. To
investigate the signal-to-noise ratio (SNR) degradation caused by the range
reconstruction, we also analyze the change of noise power in every step of the
range reconstruction. By considering the PAPR of the transmitted OFDM pulses
and the SNR degradation within the range reconstruction, we propose a new OFDM
pulse design method. We then present some simulations to demonstrate the
performance of the proposed OFDM pulse design method. By comparing with the
range Doppler algorithm (RDA) SAR imaging method using LFM signals, we present
some simulations to illustrate the performance of the proposed the arbitrary
pulse length OFDM SAR imaging algorithm. We find that, with a designed
arbitrary length OFDM pulse from our proposed method, this algorithm can still
maintain the advantage of IRCI free range reconstruction with insignificant
SNR degradation and completely avoid the energy redundancy.
The remainder of this paper is organized as follows. In Section II, we briefly
recall the CP based OFDM SAR algorithm proposed in [19] and describe the
problem of interest. In Section III, we propose CP based arbitrary pulse
length OFDM SAR. In Section IV, we propose a new arbitrary length OFDM
sequence design algorithm. In Section V, we show some simulation results.
Finally, in Section VI, we conclude this paper.
## II CP Based OFDM SAR and Problem Formulation
In this section, we first briefly recall the CP based OFDM SAR model proposed
in [19] and then see its required pulse length problem. Consider the
monostatic broadside stripmap SAR geometry as shown in Fig. 1. The radar
platform is moving parallelly to the $y$-axis with an instantaneous coordinate
$(0,y_{p}(\eta),H_{p})$, $H_{p}$ is the altitude of the radar platform, $\eta$
is the relative azimuth time referenced to the time of zero Doppler, $T_{a}$
is the synthetic aperture time defined by the azimuth time extent the target
stays in the antenna beam. For convenience, let us choose the azimuth time
origin $\eta=0$ to be the zero Doppler sample. Consider an OFDM signal with
$N$ subcarriers, a bandwidth of $B$ Hz, and let
${\mbox{\boldmath{$S$}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$
represent the complex weights transmitted over the subcarriers, $(\cdot)^{T}$
denotes the transpose, and $\sum_{i=0}^{N-1}\left|S_{i}\right|^{2}=1$. Note
that, although this sequence $S_{i}$ is rather general, in [19], a pseudo
random sequence $S_{i}$ with constant module is proposed to be used for
achieving the optimal SNR at the receiver. Then, a discrete time OFDM signal
is the inverse fast Fourier transform (IFFT) of the vector $S$ and the
corresponding time domain OFDM signal is
$s(t)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{j2\pi k\Delta
ft\right\\},\ t\in\left[0,T+T_{GI}\right],$ (1)
where $\Delta f=\frac{B}{N}=\frac{1}{T}$ is the subcarrier spacing.
$\left[0,T_{GI}\right)$ is the time duration of the guard interval that
corresponds to the CP in the discrete time domain as we shall see later in
more details and its length $T_{GI}$ will be specified later too, $T$ is the
length of the OFDM signal excluding CP. Due to the periodicity of the
exponential function $\textrm{exp}(\cdot)$ in (1), the tail part of $s(t)$ for
$t$ in $\left(T,T+T_{GI}\right]$ is the same as the head part of $s(t)$ for
$t$ in $\left[0,T_{GI}\right)$.
Figure 1: Monostatic stripmap SAR geometry.
After the demodulation to baseband, the complex envelope of the received
signal from all the range cells in the swath can be written in terms of fast
time $t$ and slow time $\eta$
$u(t,\eta)=\frac{1}{\sqrt{N}}\sum\limits_{m}{g_{m}}\textrm{exp}\left\\{-j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\}\sum\limits_{k=0}\limits^{N-1}S_{k}\textrm{exp}\left\\{\frac{j2\pi
k}{T}\left[t-\frac{2R_{m}(\eta)}{c}\right]\right\\}+w(t,\eta),$ (2)
where $f_{c}$ is the carrier frequency, $g_{m}$ is the radar cross section
(RCS) coefficient caused from the scatterers in the $m$th range cell within
the radar beam footprint, and $c$ is the speed of light. $w(t,\eta)$
represents the noise. $R_{m}(\eta)=\sqrt{\bar{R}_{m}^{2}+v_{p}^{2}\eta^{2}}$
is the instantaneous slant range between the radar and the $m$th range cell
with the coordinate $(x_{m},y_{m},0)$,
$\bar{R}_{m}=\sqrt{x_{m}^{2}+H_{p}^{2}}$ is the slant range when the radar
platform and the target in the $m$th range cell are the closest approach, and
$v_{p}$ is the effective velocity of the radar platform.
At the receiver, the received signal is sampled by the A/D converter with
sampling interval length $T_{s}=\frac{1}{B}$ and the range resolution is
$\rho_{r}=\frac{c}{2B}=\frac{c}{2}T_{s}$. Assume that the swath width for the
radar is $R_{w}$. Then, a range profile can be divided into
$M=\frac{R_{w}}{\rho_{r}}$ range cells that is determined by the radar system.
According to the analysis in [19], $M$ range cells correspond to $M$ paths in
communications, which include one main path (i.e., the nearest range cell) and
$M-1$ multipaths. In order to avoid the IRCI (corresponding to the intersymbol
interference (ISI) in communications) between different range cells, the CP
length should be at least equal to the number of multipaths ($M-1$). For
convenience, we set CP length as $M-1$ in [19], and then the guard interval
length in (1) is $T_{GI}=(M-1)T_{s}$. Notice that $T=NT_{s}$. Thus, the time
duration of an OFDM pulse is $T_{o}=T+T_{GI}=(N+M-1)T_{s}$. Meanwhile, to
completely avoid the IRCI between different range cells, the number, $N$, of
the OFDM signal subcarriers should satisfy $N\geq M$ as we have analyzed in
[19] and also well understood in communications applications [21]. Therefore,
the transmitted pulse duration $T_{o}$ is increased with the increase of the
swath width. For example, if we want to increase the swath width to $10$ km,
the transmitted pulse duration $T_{o}$ should be increased to about $133.3\
\mu$s. The pulse length here is much longer than the traditional radar pulse,
which might be a problem, especially, for covert/military radar applications.
Therefore, it is important to achieve OFDM SAR imaging with arbitrary pulse
length that is independent of a swath width, and in the meantime it also has
ideally zero IRCI. This is the goal of the remainder of this paper.
## III CP Based Arbitrary Pulse Length OFDM SAR
The main idea of the following study is to generate a pulse $s(t),\
t\in\left[0,T+T_{GI}\right]$, such that $s(t)=0$ for $t\in[0,T_{GI})$ and also
for $t\in\left(T,T+T_{GI}\right]$ with an arbitrary $T$ for $T>T_{GI}$, and
$s(t)$ is an OFDM signal in (1) for $t\in\left[T_{GI},T\right]$. However, if
the non-zero segment $s(t)$ for $t\in\left[T_{GI},T\right]$ is directly a
segment of an arbitrary OFDM signal in (1), the whole sampled discrete time
sequence of $s(t),\ 0\leq t\leq T+T_{GI}$: $s_{n}=s_{n}(nT_{s}),\ 0\leq n\leq
N+M-2$, that is zero at the head and tail ends from the above design idea of
$s(t)$, may not be from a sampling of any OFDM pulse in (1) for
$t\in\left[0,T+T_{GI}\right]$. Thus, such a pulse may not be used in the IRCI
free range reconstruction as in [19]. The key of this paper is to generate
such a pulse $s(t)$ with the above property of zero-valued head and tail, and
in the meantime, its sampled discrete time sequence $s_{n}$ is also a sampled
discrete time sequence of an OFDM pulse in (1) for
$t\in\left[0,T+T_{GI}\right]$. Since the non-zero pulse length is $T-T_{GI}$
and $T$ is arbitrary, the non-zero pulse length is also arbitrary and
independent of a swath width. The details is given in the following
subsections.
### III-A Received signal model
In order to better understand the IRCI free range reconstruction, let us first
see the receive signal model. Going back to (2), for the $m$th range cell,
$R_{m}(\eta)=R_{0}(\eta)+m\rho_{r}$, where $R_{0}(\eta)$ is the instantaneous
slant range between the radar and the first range cell in the swath as in
[19]. Then, the part $t-\frac{2R_{m}(\eta)}{c}$ in (2) is equivalent to
$t-\frac{2R_{m}(\eta)}{c}=t-t_{0}(\eta)-mT_{s}$, where the constant time delay
$t_{0}(\eta)=\frac{2R_{0}(\eta)}{c}$ is independent of $m$ for a given slow
time $\eta$. Let the sampling be aligned with the start of the received signal
after $t_{0}(\eta)$ seconds for the first arriving version of the transmitted
signal, $u(t,\eta)$ in (2) can be converted to the discrete time linear
convolution of the transmitted sequence with the weighting RCS coefficients
$d_{m}$, i.e.,
$\tilde{u}_{n}=\sum_{m=0}^{M-1}d_{m}s_{n-m}+\tilde{w}_{n},\
n=0,1,\ldots,N+2M-3,$ (3)
where
$d_{m}=g_{m}\textrm{exp}\left\\{-j4\pi f_{c}\frac{R_{m}(\eta)}{c}\right\\},\\\
$ (4)
in which $4\pi f_{c}\frac{R_{m}(\eta)}{c}$ in the exponential is the azimuth
phase, and $s_{n}$ is the sampled discrete time sequence, $s_{n}=s(nT_{s})$,
of the transmitted pulse $s(t)$ during $t\in\left[0,T+T_{GI}\right]$ for
$T=NT_{s}$ and $T_{GI}=(M-1)T_{s}$. Since the range reconstruction in the SAR
imaging algorithm proposed in [19] in the following is only based on the
discrete time signal model in (3), what matters in the range reconstruction is
the discrete time sequence $s_{n}=s(nT_{s})$, where $s_{n}=0$ for $n<0$. If
the sequence
${{\mbox{\boldmath{$s$}}}^{\prime}}=\left[s_{0},s_{1},\ldots,s_{N+M-2}\right]^{T}$
in (3) has the following zero head and tail property:
$\left[s_{0},\ldots,s_{M-2}\right]^{T}=\left[s_{N},\ldots,s_{N+M-2}\right]^{T}=\mathbf{0}^{(M-1)\times
1},$ (5)
then, in terms of the range reconstruction later, the transmitted pulse $s(t)$
is equivalent to that with $s(t)=0$ for $t\in[0,T_{GI})$ and
$t\in\left(T,T+T_{GI}\right]$. It is also equivalent to an OFDM pulse in (1)
such that its sampled version
$s_{n}=s(nT_{s})=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{\frac{j2\pi
kn}{N}\right\\},\ n=0,1,\ldots,N+M-2,$ (6)
has the property (5).
In summary, our proposed transmitted pulse of an arbitrary length $s(t)$ of
non-zero is that $s(t)=0$ for $t\in[0,T_{GI})$ and
$t\in\left(T,T+T_{GI}\right]$ and $s(t)$ has the OFDM form (1) for
$t\in\left[T_{GI},T\right]$ with an arbitrary $T$ of $T>T_{GI}$, where the
sampled version $s_{n}$ of the analog waveform/pulse in (1) satisfies the zero
head and tail property (5). Note that, since $T-T_{GI}$ is arbitrary and
$s(t)$ is only non-zero in the interval $\left[T_{GI},T\right]$, its non-zero
pulse length is arbitrary. Furthermore, since for the sequence
${\mbox{\boldmath{$s$}}}^{\prime}$, its both head and tail parts are the same
of all zeroes with length $M-1$, the head part is a CP of the tail part and
thus it fits to the sufficient CP based SAR imaging proposed in [19].
Based on the above analysis, in what follows, we assume that an OFDM pulse in
(1) satisfies the zero head and tail property (5) for its sampled discrete
time sequence $s_{n}$ and thus, it is equivalent to a pulse of length
$T-T_{GI}$ as described above in terms of the range reconstruction. So, for
convenience, we may use these two kinds of pulses interchangeably. Note that
the reason why these two kinds of analog waveforms are not the same is because
a non-zero OFDM signal in (1) can not be all zero for $t$ in any interval of a
non-zero length.
From (6), it is clear that the time domain OFDM sequence
${\mbox{\boldmath{$s$}}}=\left[s_{0},s_{1},\ldots,s_{N-1}\right]^{T}$ is just
the $N$-point IFFT of the vector
${\mbox{\boldmath{$S$}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$. In the
SAR imaging algorithm proposed in [19], $N\geq M$ is required, which is the
same as $T>T_{GI}$. However, from the above study, there are only $N-M+1$ non-
zero values in the sequence $s$ and $N$ can be arbitrary as long as $N\geq M$.
In this case, the transmitted sequence is just
${\mbox{\boldmath{$s$}}}_{t}=\left[s_{M-1},s_{M},\cdots,s_{N-1}\right]^{T}\in\mathbb{C}^{(N-M+1)\times
1}$. Then, the first and the last $M-1$ samples of the received signal
$\tilde{u}_{n}$ in (3) do not contain any useful signal111In [19], the first
and the last $M-1$ samples of the received signal $\tilde{u}_{n}$ in (3)
contain received target energy (or useful signal), but they are redundant and
removed at the receiver to obtain $u_{n}$ and IRCI free range reconstruction.,
$d_{m}$. Thus, we can start the sampling at $\tilde{u}_{M-1}$ as
$u_{n}=\sum_{m=0}^{M-1}d_{m}s_{n-m+M-1}+w_{n},\ n=0,1,\ldots,N-1.$ (7)
Now the question is how to design such an arbitrary length pulse, which is
studied next after the range reconstruction algorithm is introduced.
### III-B Range compression
In this subsection, we develop the range compression according to the above
OFDM received signal model. The received signal
${\mbox{\boldmath{$u$}}}=\left[u_{0},u_{1},\ldots,u_{N-1}\right]^{T}$ in (7)
is equivalent to the following representation
${\mbox{\boldmath{$u$}}}={\mbox{\boldmath{$H$}}}{\mbox{\boldmath{$s$}}}_{t}+{\mbox{\boldmath{$w$}}},$
(8)
where ${\mbox{\boldmath{$w$}}}=\left[w_{0},w_{1},\ldots,w_{N-1}\right]^{T}$ is
the noise vector and $H$ is the $N$ by $N-M+1$ matrix:
${\mbox{\boldmath{$H$}}}=\begin{bmatrix}d_{0}&0&\cdots&0\\\
d_{1}&d_{0}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&0\\\
d_{M-1}&d_{M-2}&\cdots&\vdots\\\ 0&\ddots&\ddots&\vdots\\\
\vdots&\ddots&d_{M-1}&d_{M-2}\\\ 0&\cdots&0&d_{M-1}\end{bmatrix}.$ (9)
The OFDM demodulator then performs the $N$-point fast Fourier transform (FFT)
on the vector $u$:
$\begin{array}[]{ll}U_{i}&=\frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}u_{n}\textrm{exp}\left\\{\frac{-j2\pi
in}{N}\right\\}\\\ &=D_{i}S_{i}^{\prime}+W_{i},\ i=0,1,\ldots,N-1,\end{array}$
(10)
where $\left[S_{0}^{\prime},S_{1}^{\prime},\cdots,S_{N-1}^{\prime}\right]^{T}$
is the $N$-point FFT of the sequence
$\left[{\mbox{\boldmath{$s$}}}_{t}^{T},\mathbf{0}^{1\times M-1}\right]^{T}$, a
cyclic shift of the time domain OFDM sequence $s$ of amount $M-1$, i.e.,
$S_{i}^{\prime}=S_{i}\textrm{exp}\left\\{\frac{j2\pi i(M-1)}{N}\right\\},$
(11)
${\mbox{\boldmath{$W$}}}=\left[W_{0},\ldots,W_{N-1}\right]^{T}$ is the
$N$-point FFT of the noise vector $w$, and
$D_{i}=\sum_{m=0}^{M-1}d_{m}\textrm{exp}\left\\{\frac{-j2\pi mi}{N}\right\\}.$
(12)
Then, the estimate of $D_{i}$ is
$\hat{D}_{i}=\frac{U_{i}}{S_{i}^{\prime}}=D_{i}+\frac{W_{i}}{S_{i}^{\prime}},\
i=0,1,\ldots,N-1.$ (13)
The vector
${\mbox{\boldmath{$D$}}}=\left[D_{0},D_{1},\ldots,D_{N-1}\right]^{T}$ is just
the $N$-point FFT of the vector $\sqrt{N}{\mbox{\boldmath{$\gamma$}}}$, where
$\gamma$ is
${\mbox{\boldmath{$\gamma$}}}=\left[d_{0},d_{1},\cdots,d_{M-1},\underbrace{0,\cdots,0}_{N-M}\right]^{T}.$
(14)
So, the estimate of $d_{m}$ can be achieved by the $N$-point IFFT of the
vector
$\hat{{\mbox{\boldmath{$D$}}}}=\left[\hat{D}_{0},\hat{D}_{1},\ldots,\hat{D}_{N-1}\right]^{T}$:
$\hat{d}_{m}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}\hat{D}_{i}\textrm{exp}\left\\{\frac{j2\pi
mi}{N}\right\\},\ m=0,\ldots,M-1.\\\ $ (15)
Then, we obtain the following estimates of the $M$ weighting RCS coefficients:
$\hat{d}_{m}={\sqrt{N}}d_{m}+\hat{w}_{m}^{\prime},\ m=0,\ldots,M-1,$ (16)
where $\hat{w}_{m}^{\prime}$ is from the noise. In $(\ref{hatdm2})$, $d_{m}$
can be recovered without any IRCI from other range cells.
After the range compression, combining the equations (2)-(4) and (16), we
obtain
$\hat{g}_{m}=\hat{d}_{m}\textrm{exp}\left\\{j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\},$
and the range compressed signal can be written as
$u_{ra}(t,\eta)=\sqrt{N}\sum_{m=0}^{M-1}\hat{g}_{m}\delta\left(t-\frac{2R_{m}(\eta)}{c}\right)\textrm{exp}\left\\{-j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\}+w_{ra}(t,\eta),$ (17)
where $\delta\left(t-\frac{2R_{m}(\eta)}{c}\right)$ is the delta function with
non-zero value at $t=\frac{2R_{m}(\eta)}{c}$, which indicates that, for every
$m$, the estimate $\hat{g}_{m}$ of the RCS coefficient value $g_{m}$ is not
affected by any IRCI from other range cells after the range compression. In
the delta function, the target range migration is incorporated via the azimuth
varying parameter $\frac{2R_{m}(\eta)}{c}$. Also, the azimuth phase in the
exponential is unaffected by the range compression. In summary, the above
range compression provides an IRCI free range reconstruction.
Notice that unlike the processing in [19] where the first and the last $M-1$
samples of the received signal are removed and thus cause significant
transmitted energy waste for a wide swath imaging, in the above range
reconstruction algorithm, all the transmitted energy is used for the range
compression without any waste. Since the transmitted OFDM pulse time duration
is $T-T_{GI}$, the minimum radar range is $\frac{c\left(T-T_{GI}\right)}{2}$
that is also independent of a swath width. Different from [19] where the CP
part is not zero, the pulse repetition interval $T_{PRI}$ becomes
$T_{PRI}=\frac{1}{\textrm{PRF}}>\left(\frac{2R_{w}}{c}+\left(T-T_{GI}\right)\right),$
where $R_{w}$ is the swath width and PRF is the pulse repetition frequency
(PRF). We want to emphasize here that the minimum radar range and the maximum
PRF of our proposed OFDM SAR in this paper are the same as those in the
existing SAR systems, such as LFM SAR, when the same transmitted pulse time
duration is used [22, 23].
In the above range compression, the processes of FFT in (10), estimation in
(13) and IFFT in (15) are applied. Thus, it is necessary to analyze the
changes of the noise power in each step of the range compression. Assume that
$w_{n}$ in (7) is a complex white Gaussian variable with zero-mean and
variance $\sigma^{2}$, i.e., $w_{n}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$
for all $n$. Since the FFT operation is unitary, the additive noise power does
not change after the process of (10). Thus, $W_{i}$ also obeys
$W_{i}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$ for all $i$. However, let
$\bar{W}_{i}=\frac{W_{i}}{S_{i}^{\prime}}$ in (13), then the variance of
$\bar{W}_{i}$ is changed to $\frac{\sigma^{2}}{\left|S_{i}\right|^{2}}$,
where, from (11), $\left|S_{i}^{\prime}\right|=\left|S_{i}\right|$, and thus
$\bar{W}_{i}\sim\mathcal{CN}\left(0,\frac{\sigma^{2}}{\left|S_{i}\right|^{2}}\right),\
i=0,\ldots,N-1$. Moreover, after the IFFT operation in (15) we have finished
the range compression and the noise power of $\hat{w}_{m}^{\prime}$ in (16) is
$\frac{\sigma^{2}}{N}\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}$ and in
the meantime $\hat{w}_{m}^{\prime}$, that follows the distribution
$\mathcal{CN}\left(0,\frac{\sigma^{2}}{N}\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}\right)$,
is equivalent to the noise $w_{ra}(t,\eta)$ in (17). Thus, from (16), we can
obtain the SNR of the $m$th range cell after the range compression as,
$\textrm{SNR}_{m}=\frac{N^{2}\left|d_{m}\right|^{2}}{\sigma^{2}\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}}.$
(18)
Notice that, we can obtain a larger $\textrm{SNR}_{m}$ with a smaller value of
$\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}$ by designing $S_{i}$. With
the normalized energy constraint $\sum_{i=0}^{N-1}\left|S_{i}\right|^{2}=1$,
when $S_{i}$ has constant module for all $i$, i.e.,
$\left|S_{0}\right|=\left|S_{1}\right|=\ldots=\left|S_{N-1}\right|=\frac{1}{\sqrt{N}}$,
we obtain the minimal value of
$\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}=N^{2}$. In this case, the
maximal SNR after the range compression can be obtained as
$\textrm{SNR}_{max}=\frac{\left|d_{m}\right|^{2}}{\sigma^{2}}.$ (19)
Thus, the optimal signal $S_{i}$ should have constant module for all $i$,
otherwise, the SNR after the range compression will be degraded. To evaluate
the change of SNR, we define the SNR degradation factor as
$\xi=\frac{\textrm{SNR}_{m}}{\textrm{SNR}_{max}}=\frac{N^{2}}{\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}}.$
(20)
Notice that $\textrm{SNR}_{m}$ and $\textrm{SNR}_{max}$ are related to the
$m$th range cell in a swath, however, since $\xi\in\left(0,1\right]$ is
independent of the noise power $\sigma^{2}$ and $d_{m}$, the above $\xi$ can
be used to evaluate the SNR degradation after the range compression for all
range cells. A larger $\xi$ denotes a less noise power enhancement (or a less
SNR degradation) caused by the estimation processing in (13), and the
generated signal $S_{i}$ is closer to the optimal one.
Since the length of the transmitted OFDM sequence
${\mbox{\boldmath{$s$}}}_{t}$ is $N_{t}=N-M+1$, from the normalized energy
constraint of ${\mbox{\boldmath{$s$}}}_{t}$, the mean transmitted power of
${\mbox{\boldmath{$s$}}}_{t}$ is $\frac{1}{N_{t}}$. Thus, the SNR of the
signal received from the $m$th range cell before range reconstruction is
$\overline{\textrm{SNR}}_{m}=\frac{\left|d_{m}\right|^{2}}{N_{t}\sigma^{2}}.$
(21)
We notice that the maximal SNR of the $m$th range cell after the range
compression $\textrm{SNR}_{max}$ in (19) is equal to
$N_{t}\overline{\textrm{SNR}}_{m}$, and the range reconstruction SNR gain is
the same as that using LFM pulses with the same transmitted signal parameters
[22, 23]. However, because of the sidelobes of the autocorrelation function
using LFM pulses, the IRCI will occur in the range reconstruction that
degrades the signal-to-interference-plus-noise ratio (SINR). Considering the
$M$ range cells in a swath, the interference of the $m$th range cell from
other range cells in the swath is
$\textrm{I}_{m}=\sum_{k=\textrm{max}\left\\{-m,\ -({N_{t}}-1)\right\\},\ k\neq
0}^{\textrm{min}\left\\{M-m-1,\ {N_{t}}-1\right\\}}d_{m+k}z(k),$ (22)
where $z(k)$ is the autocorrelation function of the LFM pulse, i.e.,
$z(k)=\sum_{n=0}^{{N_{t}}-1}l(n)l^{*}(n-k),\ \left|k\right|\leq{N_{t}-1},$
(23)
and $(\cdot)^{*}$ denotes the complex conjugate, $l(n),\
n=0,\ldots,{N_{t}}-1$, are the values of a transmitted LFM sequence. ${N_{t}}$
denotes the length of the LFM sequence that is equal to the length of the OFDM
sequence we use in this paper.
Thus, the SINR of the signal after the range reconstruction using an LFM pulse
is
$\textrm{SINR}_{m}=\frac{\left|d_{m}\right|^{2}}{\left|\textrm{I}_{m}\right|^{2}+\sigma^{2}}.$
(24)
To investigate the mean SINR, for convenience, we consider the mean power of
range cells as $E\left[d_{m}d_{m}^{*}\right]=\sigma_{d}^{2}$. Then, the mean
interference power, caused by the sidelobes, of each range cell in the swath
is
$E\left[\left|\textrm{I}_{m}\right|^{2}\right]=\sigma_{d}^{2}\sum\limits_{k=\textrm{max}\left\\{-m,\
-({N_{t}}-1)\right\\},\ k\neq 0}^{\textrm{min}\left\\{M-m-1,\
{N_{t}}-1\right\\}}\left|z(k)\right|^{2}.$ (25)
In this case, the mean SINR of the signal after the range reconstruction using
an LFM pulse is
$\textrm{SINR}_{\textrm{LFM}}=\frac{\sigma_{d}^{2}}{E\left[\left|\textrm{I}_{m}\right|^{2}\right]+\sigma^{2}}.$
(26)
For given $M$ and $N_{t}$, $\textrm{SINR}_{\textrm{LFM}}$ versus
$\frac{\sigma_{d}^{2}}{\sigma^{2}}$ can be calculated using (25)-(26) and will
be shown in the next section of simulations. Notice that since a random
sequence has the same level of the sidelobe magnitudes of the autocorrelation
values as an LFM signal does [19], the above SINR analysis also applies to the
range reconstruction in the random noise SAR imaging.
In contrast, for the IRCI free range reconstruction by using an OFDM pulse,
the SINR is equal to the SNR of the signal after the range reconstruction,
since for every range cell, there is no inter-range-interference from other
range cells. If the lower bound of the module of the OFDM sequence $S$ is
$S_{min}$, i.e., $\left|S_{i}\right|>S_{min}$ for all $i=0,1,\ldots,N-1$, we
can obtain
$\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}<NS_{min}^{-2}.$
Then, from (18), the SNR for the $m$th range cell signal is lower bounded by
$\textrm{SNR}_{m}=\frac{N^{2}\left|d_{m}\right|^{2}}{\sigma^{2}\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}}>\frac{N\left|d_{m}\right|^{2}}{\sigma^{2}S_{min}^{-2}}.$
(27)
Thus, the SINR for all range cells after the range reconstruction is also
lower bounded by
$\textrm{SINR}_{\textrm{OFDM}}=E\left[\textrm{SNR}_{m}\right]=\frac{N^{2}\sigma_{d}^{2}}{\sigma^{2}\sum\limits_{i=0}^{N-1}\left|S_{i}\right|^{-2}}>\frac{N\sigma_{d}^{2}}{\sigma^{2}S_{min}^{-2}}.$
(28)
A remark to the lower bound for the SINR in (28) is that it does not depend on
the swath width $M$, which is because our proposed OFDM SAR imaging algorithm
with our proposed arbitrary length OFDM pulses is IRCI free and the pulse
length does not depend on a swath width. Therefore, it is particularly
interesting in wide swath SAR imaging applications. Based on the above
analysis, the task here is to generate an OFDM sequence with a larger $\xi$
(or a less SNR degradation) by designing a sequence $S_{i}$ with larger
$S_{min}$. This motivates the following OFDM sequence design.
## IV New OFDM Sequence Design
First of all, an OFDM pulse of any segment in (1) is determined by a weight
sequence
${{\mbox{\boldmath{$S$}}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$ that
is determined by its $N$-point IFFT
${{\mbox{\boldmath{$s$}}}}=\left[s_{0},s_{1},\ldots,s_{N-1}\right]^{T}$. Thus,
an OFDM pulse design is equivalent to the design of its weight sequence $S$ or
the $N$-point IFFT, $s$, of $S$. From the studies in the preceding sections,
an arbitrary length OFDM pulse $s(t)$ supported only in
$\left[T_{GI},T\right]$ for $T>T_{GI}$ with its sampled sequence
$s_{n}=s(nT_{s})$ should be designed as follows.
1) Sequence $s$ should satisfy the zero head condition in (5). When this
condition is satisfied and the $N$-point FFT, $S$, of $s$, is used as the
weight sequence in the OFDM pulse in (1) denoted as $s_{1}(t)$, let its
segment (or truncated version) only supported on $\left[T_{GI},T\right]$ be
denoted by $s(t)$ that is $0$ for
$t\in\left[0,T_{GI}\right)\cup\left(T,T+T_{GI}\right]$ and equals $s_{1}(t)$
for $t\in\left[T_{GI},T\right]$. Then, pulse $s(t)$ is still an OFDM pulse on
its support and has length $T-T_{GI}$ of support (i.e., non-zero values) and
this length can be arbitrary and independent of a swath width. Furthermore,
$s(t)$ has the same discrete-time sequence
${{\mbox{\boldmath{$s$}}}^{\prime}}$ as the OFDM pulse $s_{1}(t)$ does, which,
thus, satisfies the zero head and tail condition (5). From the study in the
preceding section, transmitting pulse $s(t)$ leads to the IRCI free range
reconstruction in SAR imaging.
2) To avoid enhancing the noise as the estimation processing in (13) and
achieve the maximal possible SNR after the range compression, the complex
weights $S_{i}$ should be as constant module as possible for all $i$. In other
words, $S_{min}$ should be as large as possible.
3) The PAPR of the transmitted OFDM pulse $s(t)$ in (1) for
$t\in\left[T_{GI},T\right]$ should be minimized so that its transmitting and
receiving can be implemented easier. Otherwise, a delta pulse would serve 1)
and 2) above, but it has infinite bandwidth and infinite PAPR and can not be
transmitted [23].
Unfortunately, it looks like that there is no closed-form solution of an OFDM
sequence $s$ that simultaneously satisfies the above requirements 1)-3). It
would be easy to have a sequence
${{\mbox{\boldmath{$s$}}}}=\left[s_{0},s_{1},\ldots,s_{N-1}\right]^{T}$ to
satisfy the zero head condition in (5), i.e., $s_{n}=0$ for $n=0,1,\ldots,M-2$
as mentioned in the above 1). However, its FFT, $S$, may not have constant
module or may not be even close to constant module. A natural idea is to
modify this sequence $S$ to be closer to constant module and then take its
IFFT to go back to the time domain $s$ and also obtain the continuous waveform
$s(t)$. Then, this $s$ may not satisfy the zero head condition in (5) anymore.
Furthermore, the PAPR of the continuous waveform $s(t)$ for
$t\in\left[T_{GI},T\right]$ may be high. In this case, we may modify $s$ and
in the meantime add some constraint to limit the PAPR of $s(t)$ for
$t\in\left[T_{GI},T\right]$. Our OFDM sequence design idea is to do the above
process iteratively until a pre-set iteration number is reached and/or a
desired sequence $s$ is obtained.
Figure 2: Block diagram of the OFDM sequence design algorithm.
To clearly describe the design algorithm, let us better understand the PAPR
calculation for an analog waveform. For a sufficiently accurate PAPR
estimation of a transmitted OFDM pulse, we usually consider its oversampled
discrete time sequence, i.e., a time domain OFDM sequence
$\tilde{{\mbox{\boldmath{$s$}}}}=\left[\tilde{s}_{0},\ldots,\tilde{s}_{LN-1}\right]^{T}$
by $L$ times over-sampling of the continuous waveform $s(t)$ with complex
weights ${\mbox{\boldmath{$S$}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$
in (1) for a sufficiently large $L$ [20], i.e.,
$\tilde{s}_{n}=\frac{1}{\sqrt{LN}}\sum_{i=0}^{N-1}S_{i}\textrm{exp}\left\\{\frac{j2\pi
ni}{LN}\right\\},\ n=0,\ldots,LN-1,$ (29)
which can be implemented by the $LN$-point IFFT of the sequence
$\left[S_{0},S_{1},\ldots,S_{N-1},0,0,\ldots,0\right]^{T}$ of length $LN$.
Then, the PAPR of the transmitted OFDM pulse can be defined as
$\textrm{PAPR}=\frac{\underset{n=0,\ldots,LN-1}{\mathop{\textrm{max}}}\left|\tilde{s}_{n}\right|^{2}}{\frac{1}{LN}\sum_{n=0}^{LN-1}\left|\tilde{s}_{n}\right|^{2}}.$
(30)
Since $s$ and $S$ are FFT pairs, starting with $s$ and starting with $S$ are
equivalent. For the convenience to deal with the PAPR issue, our proposed
iterative algorithm starts with an initial random constant modular sequence
${{\mbox{\boldmath{$S$}}}}^{(0)}\in\mathbb{C}^{N\times 1}$ and obtains
$\tilde{{\mbox{\boldmath{$s$}}}}^{(q)}\in\mathbb{C}^{LN\times 1}$ using (29)
as shown in Fig. 2.
Since the first $M-1$ samples of our desired sequence $s$ should be equal to
zero, after the $L$ times over-sampling of the analog waveform, the first
$L(M-1)$ samples in sequence $\tilde{s}_{n}^{(q)},\ 0\leq n\leq L(M-1)-1$,
should be equal to zero. Thus, we apply the following time domain filter to
the newly obtained sequence $\tilde{{\mbox{\boldmath{$s$}}}}^{(q)}$:
$h(n)=\left\\{\begin{array}[]{ll}0,\ 0\leq n\leq L(M-1)-1\\\ 1,\ L(M-1)\leq
n\leq LN-1\end{array}\right.,$ (31)
as $\check{s}^{(q)}_{n}=\tilde{s}_{n}^{(q)}h(n),\ n=0,\ldots,LN-1$, to obtain
a new sequence
$\check{{\mbox{\boldmath{$s$}}}}^{(q)}=\left[\check{s}^{(q)}_{0},\ldots,\check{s}^{(q)}_{LN-1}\right]^{T}$.
After this truncation, we then add a PAPR constraint to the segment of the
non-zero elements of this sequence to obtain the next new sequence
$\hat{s}_{n}^{(q)}$ by clipping $\check{s}_{n}^{(q)}$ as follows. The time
domain clipping can be defined as, [24],
$\displaystyle\hat{s}^{(q)}_{n}$
$\displaystyle=\left\\{\begin{array}[]{ll}\textrm{T}_{q}\frac{\check{s}^{(q)}_{n}}{|\check{s}^{(q)}_{n}|},&\textrm{if}\
|\check{s}^{(q)}_{n}|>\textrm{T}_{q}\\\ \check{s}^{(q)}_{n},&\textrm{if}\
|\check{s}^{(q)}_{n}|\leq\textrm{T}_{q}\end{array}\right.,$ (32c)
$\displaystyle\textrm{T}_{q}$
$\displaystyle=\sqrt{\textrm{PAPR}_{d}P_{tav}^{(q)}},$ (32d)
where $L(M-1)\leq n\leq LN-1$, and $\hat{s}^{(q)}_{n}=0$ for
$n=0,\ldots,L(M-1)-1$.
$P_{tav}^{(q)}=\frac{1}{L(N-M+1)}{\sum\limits_{n=L(M-1)}^{LN-1}\left|\check{s}^{(q)}_{n}\right|^{2}}$
is the average power of the non-zero elements in sequence
$\check{{\mbox{\boldmath{$s$}}}}^{(q)}$. $\textrm{T}_{q}$ is the clipping
level in the $q$th iteration which is updated in each iteration according to
the average power $P_{tav}^{(q)}$ and a constant value $\textrm{PAPR}_{d}$
that is a lower bound for a desired PAPR.
After the $LN$-point FFT operation to $\hat{s}^{(q)}_{n}$, we obtain the
frequency domain sequence $\tilde{{\mbox{\boldmath{$S$}}}}^{(q)}$. To
constrain the out-of-band radiation caused by the time domain filtering and
clipping, we also use a filter in the frequency domain:
$H(i)=\left\\{\begin{array}[]{ll}1,\ 0\leq i\leq N-1\\\ 0,\ N\leq i\leq
LN-1\end{array}\right..$ (33)
And the output sequence $\check{S}_{i}^{(q)}$ can be obtained by
$\check{S}^{(q)}_{i}=\tilde{S}^{(q)}_{i}H(i),\ i=0,\ldots,LN-1$. To deal with
the constant module issue of the frequency domain sequence $S$, then, the
following frequency domain clipping is used:
${S}^{(q+1)}_{i}=\left\\{\begin{array}[]{ll}\sqrt{P_{fav}^{(q)}}\left(1+G_{f}\right)\frac{\check{S}^{(q)}_{i}}{|\check{S}^{(q)}_{i}|},&\textrm{if}\
|\check{S}^{(q)}_{i}|>\sqrt{P_{fav}^{(q)}}\left(1+G_{f}\right)\\\
\sqrt{P_{fav}^{(q)}}\left(1-G_{f}\right)\frac{\check{S}^{(q)}_{i}}{|\check{S}^{(q)}_{i}|},&\textrm{if}\
|\check{S}^{(q)}_{i}|<\sqrt{P_{fav}^{(q)}}\left(1-G_{f}\right)\\\
\check{S}^{(q)}_{i},&\ \textrm{otherwise}\end{array}\right..$ (34)
where $0\leq i\leq N-1$, and sequence
${{\mbox{\boldmath{$S$}}}}^{(q+1)}=\left[S^{(q+1)}_{0},S^{(q+1)}_{1},\ldots,S^{(q+1)}_{N-1}\right]^{T}$
is obtained. And
$P_{fav}^{(q)}=\frac{1}{N}\sum\limits_{i=0}^{N-1}\left|\check{S}^{(q)}_{i}\right|^{2}$
is the average power of the non-zero elements in sequence
$\check{{\mbox{\boldmath{$S$}}}}^{(q)}$. $G_{f}$ is a factor that we use to
control the upper and lower bounds for sequence ${S}^{(q+1)}_{i}$. Thus, the
module of sequence ${S}^{(q+1)}_{i}$ is constrained as
$\left|{S}^{(q+1)}_{i}\right|\in\left[\sqrt{P_{fav}^{(q)}}\left(1-G_{f}\right),\sqrt{P_{fav}^{(q)}}\left(1+G_{f}\right)\right]$.
A smaller $G_{f}$ denotes that a closer-to-constant modular sequence
${{\mbox{\boldmath{$S$}}}}^{(q+1)}$ can be obtained.
The above procedure is done for $q=0,1,\ldots$, when $q<Q$, where $Q$ is a
pre-set maximum iteration number. When $q=Q$, the iteration stops and then
$N$-point IFFT is applied to
${{\mbox{\boldmath{$S$}}}}^{(Q)}\in\mathbb{C}^{N\times 1}$ to obtain
$\tilde{{\mbox{\boldmath{$s$}}}}\in\mathbb{C}^{N\times 1}$. After that, a time
domain filter, i.e.,
$\tilde{h}(n)=\left\\{\begin{array}[]{ll}0,\ 0\leq n\leq M-2\\\ 1,\ M-1\leq
n\leq N-1\end{array}\right.,$
is applied to $\tilde{{\mbox{\boldmath{$s$}}}}$ to obtain sequence
$\check{{\mbox{\boldmath{$s$}}}}=\left[\check{s}_{0},\ldots,\check{s}_{N-1}\right]^{T}$,
where $\check{s}_{n}=\tilde{s}_{n}\tilde{h}(n),\ n=0,\ldots,N-1$. In order to
normalize the energy of the sequence $s$ to $1$, we use the normalization to
the time domain sequence $\check{{\mbox{\boldmath{$s$}}}}$ as
$s_{n}=\frac{\check{s}_{n}}{\sqrt{\sum\limits_{k=M-1}^{N-1}\left|\check{s}_{k}\right|^{2}}},\
n=0,\ldots,N-1,$
and obtain the OFDM sequence $s$ in (6) that satisfies the zero head condition
in (5). Finally, $S$ can be obtained by taking the $N$-point FFT of $s$. The
PAPR of the non-zero part of $s_{n}$ for $M-1\leq n\leq N-1$ can be calculated
using (29) and (30) and the noise power enhancement factor $\xi$ in (20) can
also be calculated from $S$.
Notice that, after the last iteration, the filtering operation in time domain
is applied to $\tilde{{\mbox{\boldmath{$s$}}}}$ to obtain $s$, which will
cause some out-of-band radiation to $S$. However, comparing to the OFDM
sequence energy, the out-of-band radiation energy is much smaller and can be
ignored as we shall see later in the simulations in the next section.
Therefore, for a given swath width and radar range resolution, we can obtain
$M$. Then, for any $N$ with $N\geq M$, by using the above OFDM pulse design
method, we can obtain an OFDM sequence $s$ with $M-1$ zeros at the head part
of $s$ and $N-M+1$ non-zero values in the remaining part of $s$, and also its
$N$-point FFT $S$. With this $S$ as the weights in (1), the OFDM pulse $s(t)$
in (1) for $t\in\left[T_{GI},T\right]$ can be obtained. Since $N$ or
correspondingly $T$ can be chosen arbitrarily, the pulse length, $T-T_{GI}$,
of $s(t)$ can be arbitrary and independent of $M$ (or the swath width).
Let us go back to the mean SINR in (28) using OFDM pulses. Note that the
constant module sequence ${S_{i}}$ is achieved when
$\left|S_{i}\right|=\frac{1}{\sqrt{N}}$ for all $i,\ i=0,1,\ldots,N-1$.
According to our numerous simulations, we find that it is not difficult to
generate an OFDM sequence $S_{i}$ with $\left|S_{i}\right|\geq
0.8\frac{1}{\sqrt{N}},\ i=0,\ldots,N-1$, using our proposed OFDM pulse design
algorithm above, which can be seen in the next section. Simulations about the
above SINR comparison are also provided in the next section.
## V Simulation Results
In this section, by using simulations we first see the performance of our
proposed OFDM sequence/pulse design of arbitrary length. We then see the
performance of the IRCI free range reconstruction in SAR imaging with our
proposed arbitrary length OFDM pulse.
### V-A Performance of the OFDM pulse design
Figure 3: The CDFs for different $Q$ with $\textrm{PAPR}_{d}$ $=1$ dB and
$G_{f}=5\%$: (a) PAPR; (b) SNR degradation factor.
Figure 4: The CDFs for different $\textrm{PAPR}_{d}$ with $Q=20$ and
$G_{f}=10\%$: (a) PAPR; (b) SNR degradation factor.
Figure 5: The CDFs for different $G_{f}$ with $Q=20$ and $\textrm{PAPR}_{d}$
$=1$ dB: (a) PAPR; (b) SNR degradation factor.
In this subsection, we first discuss the performance of the OFDM pulse design
algorithm. For simplicity, we set $M=96$ and $N=128$. To achieve a
sufficiently accurate PAPR estimate, we set the over-sampling ratio $L=4$
[20]. Then, we can generate an OFDM sequence $s$ with $M-1=95$ zeros at the
head part of $s$. We evaluate the PAPR and the SNR degradation factor $\xi$ by
using the standard Monte Carlo technique with $5\times 10^{5}$ independent
trials. In each trial, the $i$th element of initial sequence
${{\mbox{\boldmath{$S$}}}}^{(0)}$ is set as
$S_{i}^{(0)}=e^{j2\pi\varphi_{i}},\ i=0,\ldots,N-1$, where $\varphi_{i}$ is
uniformly distributed over the interval $[0,2\pi]$. In Figs. 3-5, we plot the
cumulative distribution functions (CDF) of the PAPR and the SNR degradation
factor $\xi$. The curves in Fig. 3 denote that, with the increase of the
maximum iteration number $Q$, the PAPR decreases and the $\xi$ increases to
$1$. In Fig. 3, more than $10\%$, $40\%$ and $60\%$ of the PAPRs of the OFDM
sequences are less than $3.5$ dB when $Q$ is equal to $10$, $20$ and $40$,
respectively. In Fig. 3, the probability of $\xi>-0.4\ \textrm{dB}\approx
0.91$, i.e., $\textrm{Pr}\left(\xi>0.91\right)=1-\textrm{Pr}\left(\xi\leq
0.91\right)$, is about $60\%$, $75\%$ and $78\%$ for $Q$ is equal to $10$,
$20$ and $40$, respectively. $\xi>-0.4\ \textrm{dB}\approx 0.91$ denotes that
the SNR of the received signal after the range reconstruction (using the
designed OFDM pulse) is more than $91\%$ of the maximum SNR using constant
modular weights $S_{i}$. Thus, the SNR degradation of the CP based SAR imaging
algorithm can be insignificant by using our designed arbitrary length OFDM
pulses. We also plot the CDFs for different $\textrm{PAPR}_{d}$ with $Q=20$
and $G_{f}=10\%$ in Fig. 4. The curves in Fig. 4 show that the PAPR change is
more sensitive than the $\xi$ change for different $\textrm{PAPR}_{d}$.
Specifically, the curves in Fig. 4 indicate that the PAPR of a designed $s$ is
significantly increased for the increase of $\textrm{PAPR}_{d}$. And the
curves in Fig. 4 denote that the SNR degradation becomes less when
$\textrm{PAPR}_{d}$ is higher. Similarly, the curves in Fig. 5 indicate that
the PAPR of $s$ is decreased and the SNR degradation is increased, when
$G_{f}$ is increased.
TABLE I: The numbers of Monte Carlo trials for $\xi$ and PAPR with $\textrm{PAPR}_{d}$ $=1$ dB and $G_{f}=5\%$ | $\xi\geq-0.1$ dB | $\xi\geq-0.2$ dB | $\xi\geq-0.4$ dB
---|---|---|---
PAPR $\leq 2$ dB | 4 | 5 | 7
PAPR $\leq 2.5$ dB | 145 | 1511 | 2134
PAPR $\leq 3$ dB | 615 | 35036 | 69735
Total number of trails: $5\times 10^{5}$
In practice, we want to generate an OFDM sequence $s$ with the minimal PAPR as
well as the minimal SNR degradation. However, according to the above analysis
the PAPR and $\xi$ are interacting each other. Therefore, it is necessary to
consider the constraints of both PAPR and $\xi$ at the same time. In Table I,
we count the numbers of trails under different conditions of the PAPR and
$\xi$ within the $5\times 10^{5}$ Monte Carlo independent trails for $Q=40$,
$\textrm{PAPR}_{d}$ $=1$ dB and $G_{f}=5\%$. Although only $4$ trails meet the
constraints of PAPR$\leq 2$ dB and $\xi\geq-0.1$ dB, it can also indicate that
an OFDM sequence with both low PAPR and low SNR degradation can be achieved by
using our proposed OFDM pulse design algorithm. We also count the numbers of
trails under different conditions of $S_{min}$ in Table II. The number of
trails for $S_{min}\geq 0.8\frac{1}{\sqrt{N}}$ are $14415$, especially, there
are $7$ trails with $S_{min}\geq 0.88\frac{1}{\sqrt{N}}$. These results
indicate that it is not difficult to generate an OFDM sequence $S$ with
$S_{min}\geq 0.8\frac{1}{\sqrt{N}}$. Specifically, a more excellent OFDM
sequence with lower PAPR, larger $\xi$, and larger $S_{min}$ can be obtained
by doing more Monte Carlo trails or with a larger iteration number $Q$, since
in practice, the same OFDM pulse is used for SAR imaging and can be generated
off-line. In all of the above simulations, the out-of-band radiation energy of
$S$ is less than $10^{-30}$ and thus it can be completely ignored.
TABLE II: The numbers of Monte Carlo trials for $S_{min}$ with $\textrm{PAPR}_{d}$ $=1$ dB and $G_{f}=5\%$ $S_{min}\geq 0.88\frac{1}{\sqrt{N}}$ | $S_{min}\geq 0.85\frac{1}{\sqrt{N}}$ | $S_{min}\geq 0.8\frac{1}{\sqrt{N}}$ | $S_{min}\geq 0.5\frac{1}{\sqrt{N}}$
---|---|---|---
7 | 371 | 14415 | 353782
Total number of trails: $5\times 10^{5}$
Figure 6: The SINRs after the range reconstructions using an LFM pulse and a
designed OFDM pulse: (a) SINRs of all the $M$ range cells; (b) The zoom-in
image of (a).
We also investigate the SINRs of the signals after the range reconstructions
by using an LFM pulse and a designed OFDM pulse with $N=128$ in Fig. 6. The
parameters of the LFM pulse are the same as the OFDM pulse, such as the
transmitted pulse time duration, bandwidth and transmitted signal energy. We
randomly choose a designed OFDM sequence with PAPR $=1.84$ dB, $\xi=-0.11$ dB
and $S_{min}=0.8\frac{1}{\sqrt{N}}$. The randomly generated weighting RCS
coefficients, $d_{m},\ m=0,\ldots,M-1$, are included in $M=96$ range cells in
a swath with $\frac{\sigma_{d}^{2}}{\sigma^{2}}=8$ dB. Then, the transmitted
sequence length is $N_{t}=33$ that is independent of $M$. The SINRs of all the
$M$ range cells are shown in Fig. 6. This figure indicates that the SINRs by
using a designed OFDM pulse are larger than the SINRs by using an LFM pulse.
The details from the $50$th range cell to the $80$th range cell are shown in
its zoom-in image in Fig. 6.
Figure 7: The mean SINR comparison using an LFM pulse and a designed OFDM
pulse.
In Fig. 7, we plot the SINRs when using an LFM pulse as (26) as well as the
SINRs and the lower bounds using the above designed OFDM pulse with
$S_{min}=0.8\frac{1}{\sqrt{N}}$ as (28) versus
$\frac{\sigma_{d}^{2}}{\sigma^{2}}$. The curves denote that the SINR lower
bounds using the OFDM pulse are insignificantly smaller than the SINRs using
the LFM pulse for $\frac{\sigma_{d}^{2}}{\sigma^{2}}<6$ dB. However, the SINR
lower bounds using the OFDM pulse are larger than the SINRs using the LFM
pulse for $\frac{\sigma_{d}^{2}}{\sigma^{2}}>6$ dB. Moreover, the advantage of
the SINR lower bounds by using the OFDM pulse is more obvious when
$\frac{\sigma_{d}^{2}}{\sigma^{2}}$ is larger. Furthermore, the true SINRs
using the OFDM pulse are about $1.4$ dB larger than their lower bounds, never
smaller than the SINRs using the LFM pulse for small
$\frac{\sigma_{d}^{2}}{\sigma^{2}}$, and obviously larger than the SINRs using
the LFM pulse for $\frac{\sigma_{d}^{2}}{\sigma^{2}}>0$ dB. These results
indicate that the range reconstruction SNR degradation using a designed OFDM
pulse is insignificant, and the advantage by using a designed OFDM pulse is
more significant when noise power $\sigma^{2}$ becomes smaller.
### V-B Performance of the SAR imaging
In this subsection, we present some simulations and discussions for the
proposed CP based arbitrary OFDM pulse length range reconstruction for SAR
imaging. The azimuth processing is similar to the conventional stripmap SAR
imaging [22], and a fixed value of $R_{c}$ located at the center of the range
swath is set as the reference range cell for azimuth processing as what is
commonly done in SAR image simulations. For comparison, we also consider the
range Doppler algorithm (RDA) using LFM signals222Since the performance of
random noise SAR is similar to LFM SAR, we do not present any simulation
results of random noise SAR here. For more comparisons between OFDM SAR
imaging, LFM SAR imaging, and random noise SAR imaging, we refer to [19]. as
shown in the block diagram of Fig. 8. In Fig. 8 (b), the secondary range
compression (SRC) is implemented in the range and azimuth frequency domain,
the same as the Option 2 in [22, Ch. 6.2].
Figure 8: Block diagram of SAR imaging processing: (a) CP based OFDM SAR; (b)
LFM SAR.
Figure 9: Profiles of a point spread function: (a) range profiles; (b) azimuth
profiles.
The simulation parameters are set as in a typical SAR system: PRF = $800$ Hz,
the bandwidth is $B=150$ MHz, the antenna length is $L_{a}=1$ m, the carrier
frequency $f_{c}=9$ GHz, the synthetic aperture time is $T_{a}=1$ sec, the
effective radar platform velocity is $v_{p}=150$ m/sec, the platform height of
the antenna is $H_{p}=5$ km, the slant range swath center is $R_{c}=5\sqrt{2}$
km, the sampling frequency $f_{s}=150$ MHz.
Firstly, the normalized range profiles and azimuth profiles of a point spread
function are shown in Fig. 9. It can be seen that the range sidelobes are much
lower for the OFDM signal than those of the LFM signal. And the azimuth
profiles of the point spread function are similar for these two signals.
We also consider a single range line (a cross range) with $M=10000$ range
cells in a $10$ km wide swath, and targets (non-zero RCS coefficients) are
included in $7$ range cells located from $7050$ m to $7100$ m, the amplitudes
are randomly generated and shown as the red circles in Fig. 10, and the RCS
coefficients of the other range cells are set to be zero (for a better
display, only a segment of the swath is indicated in Fig. 10). In this
simulation, we use a designed OFDM pulse with PAPR $=1.93$ dB, $\xi=-0.14$ dB
and time duration333For the algorithm in [19], by setting $N=M$, the OFDM
pulse time duration with sufficient length CP is at least
$T+T_{GI}=\frac{10000}{150}+\frac{9999}{150}\ \mu\textrm{s}\approx 133.3\
\mu$s as mentioned in Section II. $T-T_{GI}=5\ \mu$s, which is independent of
the swath width. For $T_{GI}=\frac{M-1}{f_{s}}$, $N=Tf_{s}=10749$. The
transmitted LFM pulse duration is also $5\ \mu$s. The normalized imaging
results are shown as the blue asterisks in Fig. 10. The imaging results
without noise are shown in Fig. 10 and Fig. 10. Since there is no IRCI between
different range cells, the results indicate that the OFDM SAR imaging is
precise as shown in Fig. 10. However, because of the influence of range
sidelobes of the LFM signal, some weak targets, for example, those located at
$7063$ m and $7073$ m, are submerged by the interference from the nearby
targets and thus can not be imaged correctly as shown in Fig. 10. We also give
the imaging results of LFM SAR and OFDM SAR in Fig. 10 and Fig. 10,
respectively, when the noise power of the raw radar data is $\sigma^{2}=0.05$,
and in Fig. 10 and Fig. 10, respectively, when $\sigma^{2}=0.1$. These results
can also indicate the better performance of the proposed OFDM SAR. The
performance advantage of the OFDM SAR is more obvious for a smaller noise
power, for example, when $\sigma^{2}=0.05$, which is consistent with the
results in Fig. 7. Note that, for a better display and recognizability, we
consider that only $7$ range cells in the swath contain targets. In a
practical SAR imaging, much more targets (non-zero RCS coefficients) are
included and then the IRCI of LFM (or random noise) SAR will be more serious.
Thus, the performance advantage of the OFDM SAR over LFM or random noise SAR
will be more obvious because of its IRCI free range reconstruction.
Figure 10: A range line imaging results. Red circles denote the real target
amplitudes, blue asterisks denote the imaging results. (a) LFM SAR without
noise; (b) OFDM SAR without noise; (c) LFM SAR with noise of variance
$\sigma^{2}=0.05$; (d) OFDM SAR with noise of variance $\sigma^{2}=0.05$; (e)
LFM SAR with noise of variance $\sigma^{2}=0.1$; (f) OFDM SAR with noise of
variance $\sigma^{2}=0.1$.
## VI Conclusion
In this paper, we proposed a novel sufficient CP based OFDM SAR imaging
algorithm with arbitrary pulse length that is independent of a swath width by
using our newly proposed and designed OFDM pulses. This OFDM SAR imaging
algorithm can provide the advantage of IRCI free range reconstruction and
avoid the energy redundancy. We first established the arbitrary pulse length
OFDM SAR imaging system model and then derived the range reconstruction
algorithm with free IRCI. We also analyzed the SINR after the range
reconstruction and compared it with that using LFM signals. By considering the
PAPR of a transmitted OFDM pulse and the SNR degradation of the range
reconstruction, we proposed a novel OFDM pulse design method. We finally gave
some simulations to demonstrate the performance of the proposed OFDM pulse
design method. By comparing with the RDA SAR imaging using LFM signals, we
provided some simulations to illustrate the advantage, such as higher SINR
after the range reconstruction, of the proposed arbitrary pulse length OFDM
SAR imaging algorithm. The main contributions of this paper can be summarized
as:
* •
When a sufficient CP length is at least $M-1$, where $M$ is the number of
range cells within a swath, an OFDM sequence of length $N$,
${\mbox{\boldmath{$s$}}}\in\mathbb{C}^{N\times 1}$, with at least $M-1$
consecutive zero elements in the head part is generated by an OFDM pulse
design method and thus, the transmitted OFDM sequence is
${\mbox{\boldmath{$s$}}}_{t}\in\mathbb{C}^{(N-M+1)\times 1}$ of length
$N+M-1$.
* •
With our proposed OFDM sequence/pulse design, a transmitted OFDM pulse length
can be arbitrary and independent of a swath width, which is critical in wide
swath IRCI free SAR imaging applications.
* •
With a designed OFDM pulse, no CP in the transmitted sequence needs to be
removed in the receiver. Thus, the transmitted energy redundancy can be
avoided.
* •
The proposed SAR imaging algorithm may cause some SNR degradation. However,
the degradation is insignificant according to our simulations. Comparing with
LFM SAR, the performance advantage of the OFDM SAR is more obvious for a
smaller noise power. Moreover, with our proposed OFDM pulse design method, a
better OFDM sequence with a lower PAPR can be generated by setting a larger
maximum iteration number $Q$, and the SNR degradation by using this OFDM
sequence becomes less.
## References
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* [3] ——, “Adaptive OFDM radar for target detection in multipath scenarios,” _Signal Processing, IEEE Transactions on_ , vol. 59, no. 1, pp. 78–90, 2011\.
* [4] S. Sen, “PAPR-constrained Pareto-Optimal waveform design for OFDM-STAP radar,” _Geoscience and Remote Sensing, IEEE Transactions on_ , online published, 2013: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6587082&tag=1. DOI. 10.1109/TGRS.2013.2274593.
* [5] S. Sen and A. Nehorai, “OFDM MIMO radar with mutual-information waveform design for low-grazing angle tracking,” _Signal Processing, IEEE Transactions on_ , vol. 58, no. 6, pp. 3152–3162, 2010.
* [6] D. Garmatyuk, J. Schuerger, K. Kauffman, and S. Spalding, “Wideband OFDM system for radar and communications,” in _Radar Conference, 2009 IEEE_ , Pasadena, CA, 2009, pp. 1–6.
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|
arxiv-papers
| 2013-12-08T21:36:41 |
2024-09-04T02:49:55.144726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tian-Xian Zhang, Xiang-Gen Xia, and Lingjiang Kong",
"submitter": "Tian-Xian Zhang",
"url": "https://arxiv.org/abs/1312.2267"
}
|
1312.2294
|
# Scattering theory for nonlinear Schrödinger equations with inverse-square
potential
Junyong Zhang Department of Mathematics, Beijing Institute of Technology,
Beijing 100081 China, and Department of Mathematics, Australian National
University, Canberra ACT 0200, Australia [email protected] and
Jiqiang Zheng Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France,
and Institut Universitaire de France [email protected]
###### Abstract.
We study the long-time behavior of solutions to nonlinear Schrö-dinger
equations with some critical rough potential of $a|x|^{-2}$ type. The new
ingredients are the interaction Morawetz-type inequalities and Sobolev norm
property associated with $P_{a}=-\Delta+a|x|^{-2}$. We use such properties to
obtain the scattering theory for the defocusing energy-subcritical nonlinear
Schrödinger equation with inverse square potential in energy space
$H^{1}(\mathbb{R}^{n})$.
Key Words: Nonlinear Schrödinger equation; Inverse square potential; Well-
posedness; Interaction Morawetz estimates; Scattering.
AMS Classification: 35P25, 35Q55.
## 1\. Introduction
This paper is devoted to the scattering theory for the nonlinear defocusing
Schrödinger equation
$\begin{cases}i\partial_{t}u-P_{a}u=|u|^{p-1}u\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{n}\\\
u|_{t=0}=u_{0}\in H^{1}(\mathbb{R}^{n})\end{cases}$ (1.1)
where $u:\mathbb{R}_{t}\times\mathbb{R}_{x}^{n}\to\mathbb{C}$ and
$P_{a}=-\Delta+a|x|^{-2}$ with $a>-\lambda_{n}:=-(n-2)^{2}/4$ and $n\geq 3$.
The elliptic operator $P_{a}$ is the self-adjoint extension of
$-\Delta+a|x|^{-2}$. It is well-known that in the range
$-\lambda_{n}<a<1-\lambda_{n}$, the extension is not unique; see [20, 36]. In
this case, we do make a choice among the possible extensions, such as
Friedrichs extension [20, 25].
The scale-covariance elliptic operator $P_{a}=-\Delta+a|x|^{-2}$ appearing in
(1.1) plays a key role in many problems of physics and geometry. The heat and
Schrödinger flows for the elliptic operator $-\Delta+a|x|^{-2}$ have been
studied in the theory of combustion (see [37]), and in quantum mechanics (see
[20]). The mathematical interest in these equations with $a|x|^{-2}$ however
comes mainly from the fact that the potential term is homogeneous of degree
$-2$ and therefore scales exactly the same as the Laplacian. There is
extensive literature on properties of the Schrödinger semigroup of operators
$e^{it\mathrm{H}}$ generated by $\mathrm{H}=-\Delta+V(x)$, where a potential
$V(x)$ is less singular than the inverse square potential at the origin, for
instance, when it belongs to the Kato class; see [10, 29, 30, 31]. The inverse
square potential $a|x|^{-2}$ does not belong to the Kato class and it is well
known that such singular potential belongs to a borderline case, where both
the strong maximum principle and Gaussian bound of the heat kernel for $P_{a}$
fail to hold when $a$ is negative. Because of this, it brings some
difficulties to study the heat and dispersive equations with the inverse
square potential; see [3, 37]. Fortunately, the Strichartz estimates, an
essential tool for studying the behavior of solutions to nonlinear Schrödinger
equations and wave equations, have been developed by Burq-Planchon-Stalker-
Tahvildar-Zadeh [3, 4]. In the study of Strichartz estimates for the
propagators $e^{it(\Delta+V)}$, the decay $V(x)\sim|x|^{-2}$ is borderline in
order to guarantee validity of Strichartz estimate; see Goldberg-Vega-
Visciglia [15]. And also it is known that for any potential
$V(x)\sim|x|^{-2-\epsilon}$, the Strichartz estimates are satisfied, without
any further assumption on the monotonicity or regularity of $V(x)$; see
Rodnianski-Schlag [27]. Moreover recently the Hardy type potentials have been
further studied in Fanelli-Felli-Fontelos-Primo [11]. It is well-known that
inverse square potential is in some sense critical for the spectral theory.
This is closely related to the fact that the angular momentum barrier
$k(k+1)/|x|^{2}$ is exactly same type as the inverse square potential. As a
consequence, the authors [24, 39] showed some more Strichartz estimates and
restriction estimates for wave equation with inverse square potential by
assuming additional angular regularity. In this paper, we study the scattering
theory of nonlinear Schrödinger equation (1.1) with the critical decay inverse
square potential.
The scattering theory of the nonlinear Schrödinger equation with no potential,
that is $a=0$, has been intensively studied in [1, 2, 5, 6, 9, 13, 14]. For
the energy-subcritical case: $p\in(1+\tfrac{4}{n},1+\tfrac{4}{n-2})$ when
$n\geq 3$, and $p\in(1+\tfrac{4}{n},+\infty)$ when $n\in\\{1,2\\}$, one can
obtain the global well-posedness for (1.1) with $a=0$ by using the mass and
energy conservation due to the lifespan of the local solution depending only
on the $H^{1}$-norm of the initial data. In [14], Ginibre-Velo established the
scattering theory in the energy space $H^{1}(\mathbb{R}^{n})$ by using the
classical Morawetz estimate for low spatial and almost finite propagator speed
for high spatial. The dispersive estimate is an essential tool in their
argument. However in our setting, in particular when $a$ is negative, even
though we have the Strichartz estimates, we do not know whether the dispersive
estimate holds or not. Later, Tao-Visan-Zhang [35] gave a simplified proof for
the result in [14] by making use of the following interaction Morawetz
estimate
$\big{\|}|\nabla|^{\frac{3-n}{4}}u\big{\|}_{L_{t}^{4}(I;L_{x}^{4}(\mathbb{R}^{n}))}^{2}\leq
C\|u_{0}\|_{L^{2}}\sup_{t\in I}\|u(t)\|_{\dot{H}^{\frac{1}{2}}},\quad n\geq
3.$ (1.2)
For this estimate, Visciglia [38] gave an alternative application of the
interaction Morawetz estimate.
To prove the scattering theory, we follow Tao-Visan-Zhang’s argument. Thus one
of our main task is to establish an interaction Morawetz estimate for the
nonlinear Schrödinger equation (1.1) with inverse square potential. To this
end, we have to treat an error term caused by the potential. Though we cannot
show the positivity of this error term, we can control the error term by the
quantity in the right hand side of (1.2). The method to control the error term
is, as with the classical Morawetz inequality, a ‘multiplier’ argument based
on the first order differential operator
$A=\frac{1}{2}(\partial_{r}-\partial_{r}^{*})$. The two key points are the
positivity of the commutator $[A,\Delta-a|x|^{-2}]$ and the homogeneity of the
potential $a|x|^{-2}$, which is the same as Laplacian’s scaling. Thus we
finally obtain an analogue of the interaction Morawetz-type estimate. We are
known that the Leibniz rule plays a role in proving the well-posedness.
However, we do not know whether the Leibniz rule associated with the operator
$P_{a}$ holds or not. Instead, we show the equivalence of the Sobolev norms
based on the operator $P_{a}$ and the standard Sobolev norms based on the
Laplacian by using results on the boundedness of Riesz transform Hassell-Lin
[17] and heat kernel estimate [22, 23]. Though the Sobolev norms equivalence
result partially implies Sobolev algebra property associated with $P_{a}$, it
is enough for considering the scattering theory in energy space
$H^{1}(\mathbb{R}^{n})$ to obtain our main result.
The main purpose of this paper is to prove the following result.
###### Theorem 1.1.
Let $n\geq 3$ and let $p\in\big{(}1+\tfrac{4}{n},1+\tfrac{4}{n-2}\big{)}$.
Assume that $a>-\frac{4p}{(p+1)^{2}}\lambda_{n}$ and $u_{0}\in
H^{1}(\mathbb{R}^{n})$. Then the solution $u$ to (1.1) is global. Moreover,
the solution $u$ scatters if $a\geq\frac{4}{(p+1)^{2}}-\lambda_{n}$ for $n\geq
4$, and $a\geq 0$ for $n=3$.
Here the solution $u$ to (1.1) scatters means that there exists a unique
$u_{\pm}\in H^{1}(\mathbb{R}^{n})$ such that
$\lim_{t\to\pm\infty}\|u(t)-e^{itP_{a}}u_{\pm}\|_{H^{1}_{x}}=0.$
###### Remark 1.2.
The assumption $p\in\big{(}1+\tfrac{4}{n},1+\tfrac{4}{n-2}\big{)}$ is needed
for the scattering result; the lower bound $p>1+4/n$ can be improved when one
only considers the global well-posedness result; see Remark 5.2 below. Since
we mainly focus the scattering theory, we only consider the case that $p$
belongs to $\big{(}1+\tfrac{4}{n},1+\tfrac{4}{n-2}\big{)}$ .
###### Remark 1.3.
If $a\geq\frac{4}{(p+1)^{2}}-\lambda_{n}$ for $n\geq 4$ and $a\geq 0$ for
$n=3$ , the theorem gives scattering result for NLS (1.1) with all
$p\in\big{(}1+\tfrac{4}{n},1+\tfrac{4}{n-2}\big{)}$. This result is new and
allows some negative inverse-square potential when $n\geq 4$. Indeed, the
restriction on $a$ is from
$a\geq\max\\{\frac{4}{(p+1)^{2}}-\lambda_{n},\frac{1}{4}-\lambda_{n}\\}$ where
the latter is needed in the establishment of interaction Morawetz estimate.
###### Remark 1.4.
The implicit requirement that $a>-\lambda_{n}$ not only serves for the
positivity of the operator $P_{a}$ but also needs to bound the kinetic energy.
If the solution $u$ of (1.1) has sufficient decay at infinity and smoothness,
it conserves mass
$M(u)=\int_{\mathbb{R}^{n}}|u(t,x)|^{2}dx=M(u_{0})$ (1.3)
and energy
$E(u(t))=\tfrac{1}{2}\int_{\mathbb{R}^{n}}|\nabla
u(t)|^{2}dx+\tfrac{a}{2}\int_{\mathbb{R}^{n}}\tfrac{|u(t)|^{2}}{|x|^{2}}dx+\tfrac{1}{p+1}\int_{\mathbb{R}^{n}}|u(t)|^{p+1}dx=E(u_{0}).$
(1.4)
The paper is organized as follows. In Section $2$, as a preliminaries, we give
some notations, recall the Strichartz estimate and prove a generalized Hardy
inequality. Section $3$ is devoted to proving the interaction Morawetz-type
estimates for (1.1). We show a result about the Sobolev norm equivalence in
Section $4$. In Section 5, we utilize Morawetz-type estimates and the
equivalence of Sobolev norm to prove Theorem 1.1.
Acknowledgments: The authors would like to thank Andrew Hassell and Changxing
Miao for their helpful discussions and encouragement. They also would like to
thank the referee for useful comments. This research was supported by
PFMEC(20121101120044), Beijing Natural Science Foundation(1144014), National
Natural Science Foundation of China (11401024) and Discovery Grant DP120102019
from the Australian Research Council.
## 2\. Preliminaries
In this section, we first introduce some notation, and then recall the
Strichartz estimates and also give two remarks about the inhomogeneous
Strichartz estimate at the endpoint. We conclude this section by showing a
generalized Hardy inequality.
### 2.1. Notations
First, we give some notations which will be used throughout this paper. To
simplify the expression of our inequalities, we introduce some symbols
$\lesssim,\thicksim,\ll$. If $X,Y$ are nonnegative quantities, we use
$X\lesssim Y$ or $X=O(Y)$ to denote the estimate $X\leq CY$ for some $C$, and
$X\thicksim Y$ to denote the estimate $X\lesssim Y\lesssim X$. We use $X\ll Y$
to mean $X\leq cY$ for some small constant $c$. We use $C\gg 1$ to denote
various large finite constants, and $0<c\ll 1$ to denote various small
constants. For any $r,1\leq r\leq\infty$, we denote by $\|\cdot\|_{r}$ the
norm in $L^{r}=L^{r}(\mathbb{R}^{n})$ and by $r^{\prime}$ the conjugate
exponent defined by $\frac{1}{r}+\frac{1}{r^{\prime}}=1$. We denote $a_{\pm}$
to be any quantity of the form $a\pm\epsilon$ for any $\epsilon>0$. We define
$\lambda_{n}$ by $\lambda_{n}=(n-2)^{2}/4$.
### 2.2. Strichartz estimates:
To state the Strichartz estimate, we need the following definition
###### Definition 2.1 (Admissible pairs).
A pair of exponents $(q,r)$ is called _Schrödinger admissible_ , or denote by
$(q,r)\in\Lambda_{0}$ if
$2\leq
q,r\leq\infty,~{}\tfrac{2}{q}=n\big{(}\tfrac{1}{2}-\tfrac{1}{r}\big{)},~{}\text{and}~{}(q,r,n)\neq(2,\infty,2).$
The Strichartz estimates for the solution of the linear Schrödinger equation
have been developed by Burq-Planchon- Stalker-Tahvildar-Zadeh [3].
###### Proposition 2.2 (Linear Strichartz estimate [3]).
Let $a>-\lambda_{n}$ and let $(q,r)\in\Lambda_{0}$. Then there exists a
positive constant $C$ depending on $(n,q,r,a)$, such that
$\|e^{itP_{a}}u_{0}\|_{L_{t}^{q}L_{x}^{r}(\mathbb{R}\times\mathbb{R}^{n})}\leq
C\|u_{0}\|_{L^{2}}.$ (2.1)
Furthermore, we have the estimates associated with $P_{a}$
$\big{\|}P_{a}^{1/2}\big{(}e^{itP_{a}}u_{0}\big{)}\big{\|}_{L_{t}^{q}L_{x}^{r}(\mathbb{R}\times\mathbb{R}^{n})}\leq
C\big{\|}P_{a}^{1/2}u_{0}\big{\|}_{L^{2}_{x}(\mathbb{R}^{n})}\simeq\|u_{0}\|_{\dot{H}^{1}}.$
(2.2)
By the duality argument and the Christ-Kiselev lemma [7], we obtain the
inhomogeneous Strichartz estimates except the endpoint
$(q,r)=(\tilde{q},\tilde{r})=(2,\tfrac{2n}{n-2})$.
###### Proposition 2.3 (Inhomogeneous Strichartz estimates).
Let $a>-\lambda_{n}$. Suppose $u:I\times\mathbb{R}^{n}\to\mathbb{C}$ is a
solution to $(i\partial_{t}+\Delta-\frac{a}{|x|^{2}})u=f$ with initial data
$u_{0}$. Then for any $(q,r),~{}(\tilde{q},\tilde{r})\in\Lambda_{0}$ except
$(q,r)=(\tilde{q},\tilde{r})=(2,\tfrac{2n}{n-2})$, we have
$\|u\|_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{n})}\lesssim\|u(t_{0})\|_{L^{2}(\mathbb{R}^{n})}+\|f\|_{L_{t}^{\tilde{q}^{\prime}}L_{x}^{\tilde{r}^{\prime}}(I\times\mathbb{R}^{n})},$
(2.3)
and moreover
$\big{\|}P_{a}^{1/2}u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{n})}\lesssim\|u(t_{0})\|_{\dot{H}_{x}^{1}(\mathbb{R}^{n})}+\big{\|}P_{a}^{1/2}f\big{\|}_{L_{t}^{\tilde{q}^{\prime}}L_{x}^{\tilde{r}^{\prime}}(I\times\mathbb{R}^{n})}.$
(2.4)
###### Remark 2.4.
Since the dispersive estimate possibly fails when $a<0$ (for wave equation see
[26]), thus one cannot directly follow Keel-Tao’s [18] argument to obtain the
inhomogeneous Strichartz estimates for the endpoint. However, if
$|a|\leq\epsilon$ where $\epsilon$ is a small enough constant depending on the
Strichartz estimates’ constant and the norm $\||x|^{-2}\|_{L^{n/2,2}}$, then
one can prove inhomogeneous Strichartz estimates at the endpoint. For
simplicity, we assume the initial data $u(t_{0})=0$. Indeed, we can write that
$u(t,x)=\int_{0}^{t}e^{i(t-s)P_{a}}f(s)ds=\int_{0}^{t}e^{i(t-s)\Delta}\left(-a{|x|^{-2}}u+f(s)\right)ds.$
(2.5)
By the endpoint inhomogeneous Strichartz estimates for the classical
Schrödinger equation on Lorentz space [18], we have
$\begin{split}\left\|u\right\|_{L^{2}_{t}L^{\frac{2n}{n-2}}}\leq\left\|u\right\|_{L^{2}_{t}L^{\frac{2n}{n-2},2}}&\leq
C\left(\epsilon\left\||x|^{-2}u\right\|_{L^{2}L^{\frac{2n}{n+2},2}}+\|f\|_{L^{2}L^{\frac{2n}{n+2},2}}\right)\\\
&\leq
C\left(\epsilon\||x|^{-2}\|_{L^{\frac{n}{2},2}}\left\|u\right\|_{L^{2}L^{\frac{2n}{n-2}}}+\|f\|_{L^{2}L^{\frac{2n}{n+2}}}\right).\end{split}$
(2.6)
If $\epsilon$ is small such that $C^{2}\epsilon<1$, then we obtain the
endpoint inhomogeneous Strichartz estimates. We believe that one can remove
the small assumption in the endpoint inhomogeneous Strichartz estimates by
following the argument of Hassell-Zhang [16] and considering the spectral
measure of Laplacian with inverse square potential on the metric cone.
###### Remark 2.5.
The endpoint inhomogeneous Strichartz estimate implies the uniform Sobolev
estimate
$\|(P_{a}-\alpha)^{-1}\|_{L^{r}\to L^{r^{\prime}}}\leq C,\quad
r=\frac{2n}{n+2},$ (2.7)
where $C$ is independent of $\alpha\in\mathbb{C}$. This estimate was proved by
Kenig- Ruiz-Sogge [19] for the flat Laplacian without potential, and by
Guillarmou-Hassell[12] for the Laplacian on nontrapping asymptotically conic
manifolds. To see this, we choose $w\in C_{c}^{\infty}(\mathbb{R}^{n})$ and
$\chi(t)$ equal to $1$ on $[-T,T]$ and zero for $|t|\geq T+1$, and let
$u(t,x)=\chi(t)e^{i\alpha t}w(x)$. Then
$(i\partial_{t}+P_{a})u=f(t,z),\quad f(t,z):=\chi(t)e^{i\alpha
t}(P_{a}-\alpha)w(z)+i\chi^{\prime}(t)e^{i\alpha t}w(z).$
Applying the endpoint inhomogeneous Strichartz estimate, we obtain
$\|u\|_{L^{2}_{t}L^{r^{\prime}}_{z}}\leq C\|f\|_{L^{2}_{t}L^{r}_{z}}.$
From the specific form of $u$ and $f$ we have
$\|u\|_{L^{2}_{t}L^{r^{\prime}}_{z}}=\sqrt{2T}\|w\|_{L^{r^{\prime}}}+O(1),\quad\|f\|_{L^{2}_{t}L^{r}_{z}}=\sqrt{2T}\|(P_{a}-\alpha)w\|_{L^{r}}+O(1).$
Taking the limit $T\to\infty$, we find that
$\|w\|_{L^{r^{\prime}}}\leq C\|(P_{a}-\alpha)w\|_{L^{r}}.$
This implies the uniform Sobolev estimate (2.7). Thus we have (2.7) for
$|a|\leq\epsilon$ by previous argument.
### 2.3. The generalized Hardy inequality
We need the following generalized Hardy inequality:
###### Lemma 2.6 (Hardy inequality).
Let $1<p<\infty,0\leq s<{\frac{n}{p}}$, then there exists a constant $C$ such
that for all $u\in\dot{H}^{s}_{p}(\mathbb{R}^{n})$,
$\displaystyle\int_{\mathbb{R}^{n}}\frac{|u(x)|^{p}}{|x|^{sp}}dx\leq
C\|u\|_{\dot{H}^{s}_{p}}^{p}.$ (2.8)
###### Remark 2.7.
This is an improved and extension result of Hardy inequality in Cazenave [5]
and Zhang [40]. The proof heavily relies on the boundedness of the Hardy-
Littlewood maximal operator on $L^{p}$ for $p>1$.
###### Proof.
It is obvious for $s=0$, hence we only consider $0<s<\frac{n}{p}<n$. We begin
this proof by recalling the definition of Riesz potential $I_{\alpha}f$ for
$0<\alpha<n$
$I_{\alpha}f(x)=(-\Delta)^{-\frac{\alpha}{2}}f(x)=C_{n,\alpha}\int_{\mathbb{R}^{n}}|x-y|^{-n+{\alpha}}f(y)dy,$
where $C_{n,\alpha}$ is a constant depending on $\alpha$ and $n$. The norm of
homogenous Sobolev space is given by
$\|f\|_{{\dot{H}}^{s}_{p}}=\|(-\Delta)^{\frac{s}{2}}f(x)\|_{p}=\|I_{-s}f\|_{p}.$
Let ${I_{-s}u}=f$, then $u=I_{s}f$. Thus it suffices to show that
$\displaystyle\bigg{\|}\frac{I_{s}f}{|x|^{s}}\bigg{\|}_{p}\leq C\|f\|_{p}.$
(2.9)
To do so, we write
$\displaystyle Af(x)$
$\displaystyle:=\frac{I_{s}f(x)}{|x|^{s}}=\int_{\mathbb{R}^{n}}\frac{f(y)}{|x-y|^{n-s}|x|^{s}}dy$
$\displaystyle=\int_{|x-y|\leq
100|x|}\frac{f(y)}{|x-y|^{n-s}|x|^{s}}dy+\int_{|x-y|\geq
100|x|}\frac{f(y)}{|x-y|^{n-s}|x|^{s}}dy$
$\displaystyle=:A_{1}f(x)+A_{2}f(x).$
To prove (2.9), it suffices to show that both $A_{1}$ and $A_{2}$ are strong
$(p,p)$ type.
We first consider $A_{1}f$. Notice that $s>0$, we have
$\displaystyle A_{1}f(x)$ $\displaystyle=\sum_{j\leq
0}\int_{|x-y|\sim{2^{j}}100|x|}\frac{|f(y)|}{|x-y|^{n-s}|x|^{s}}dy$
$\displaystyle\leq\sum_{j\leq
0}\int_{|x-y|\sim{2^{j}}100|x|}\frac{|f(y)|}{({2^{j}}100|x|)^{n-s}|x|^{s}}dy$
$\displaystyle\leq\sum_{j\leq
0}\frac{1}{(2^{j}|x|)^{n}}\int_{|x-y|\leq{2^{j}}100|x|}|f(y)|dy\cdot 2^{js}$
$\displaystyle\leq C{\sum_{j\leq 0}2^{js}}Mf(x)\leq C^{\prime}Mf(x),$
where $M$ is the Hardy-Littlewood maximal operator. By the boundedness of the
Hardy-Littlewood maximal operator for $p>1$, we obtain $\|A_{1}f\|_{p}\leq
C\|f\|_{p}$.
Next we consider $A_{2}f$. We note that
$\begin{split}A_{2}f(x)&=\int_{|x-y|\geq
100|x|}\frac{f(y)}{|x-y|^{n-s}|x|^{s}}dy\\\
&\leq\int_{|y|\geq{99|x|}}\frac{|f(y)|}{|x-y|^{n-s}|x|^{s}}dy\leq
C\int_{|y|\geq{99|x|}}\frac{|f(y)|}{|y|^{n-s}|x|^{s}}dy=:B_{2}f(x).\end{split}$
For any $g\in\ L^{p^{\prime}}(\mathbb{R}^{n})$, we write
$\displaystyle\langle B_{2}f(x),g(x)\rangle$
$\displaystyle=\int_{\mathbb{R}^{n}}{\int_{|y|\geq{99|x|}}\frac{|f(y)|g(x)}{|y|^{n-s}|x|^{s}}dydx}$
$\displaystyle=\int_{\mathbb{R}^{n}}{\frac{1}{|y|^{n-s}}\int_{|x|\leq\frac{|y|}{99}}\frac{g(x)}{|x|^{s}}dx\cdot|f(y)|dy}$
$\displaystyle=\langle Tg(y),|f(y)|\rangle,$
where
$Tg(y)={\frac{1}{|y|^{n-s}}\int_{|x|\leq\frac{|y|}{99}}\frac{g(x)}{|x|^{s}}dx.}$
To prove the operator $B_{2}$ is strong $(p,p)$ type, it is sufficient to show
by duality
$\displaystyle\|Tg(y)\|_{p^{\prime}}\leq C\|g\|_{p^{\prime}}.$ (2.10)
If we prove $B_{2}$ is strong $(p,p)$ type, so is $A_{2}$. Hence we need prove
(2.10). Since $0<s<n$, we can choose $q>1$ such that ${sq^{\prime}}<n$. We
have $q>\frac{n}{n-s}$. Thus we have by Hölder’s inequality
$\displaystyle{|Tg(y)|}$
$\displaystyle\leq{\frac{1}{|y|^{n-s}}\bigg{(}\int_{|x|\leq\frac{|y|}{99}}|g(x)|^{q}dx\bigg{)}^{\frac{1}{q}}\bigg{(}\int_{|x|\leq\frac{|y|}{99}}\frac{1}{|x|^{sq^{\prime}}}dx\bigg{)}^{\frac{1}{q^{\prime}}}}$
$\displaystyle\leq
C\frac{1}{|y|^{n-s}}\cdot\bigg{(}\int_{|x|\leq\frac{|y|}{99}}|g(x)|^{q}dx\bigg{)}^{\frac{1}{q}}\cdot{|y|^{{(n-{sq^{\prime}})}\frac{1}{q^{\prime}}}}$
$\displaystyle\leq C\frac{1}{|y|^{\frac{n}{q}}}\cdot\bigg{(}\int_{|x-y|\leq
2|y|}|g(x)|^{q}dx\bigg{)}^{\frac{1}{q}}$ $\displaystyle\leq
C(M(|g|^{q}))^{\frac{1}{q}}(y).$
For all $p^{\prime}>q>{\frac{n}{n-s}}$, one has $1<p<n/s$. Since
$p^{\prime}>q$, the boundedness of Hardy-Littlewood maximal operator gives
$\left\|(M(|g|^{q}))^{\frac{1}{q}}\right\|_{L^{p^{\prime}}}=\left\|M(|g|^{q})\right\|_{L^{\frac{p^{\prime}}{q}}}^{\frac{1}{q}}\leq
C\left\||g|^{q}\right\|_{L^{\frac{p^{\prime}}{q}}}^{\frac{1}{q}}=C\|g\|_{L^{p^{\prime}}}.$
Therefore we obtain $T$ is strong $(p^{\prime},p^{\prime})$ type, hence we
proves (2.10). Thus we conclude the proof of this lemma.
∎
## 3\. Morawetz-type estimates
In this section, we derive the quadratic Morawetz identity for (1.1) and then
establish the interaction Morawetz estimate. The Morawetz estimate provides us
a decay of the solution to the NLS with an inverse square potential, which
will help us study the asymptotic behavior of the solutions in the energy
space in next section. More precisely, we have
###### Proposition 3.1 (Morawetz-type estimates).
Let $u$ be an $H^{\frac{1}{2}}$-solution to (1.1) on the spacetime slab
$I\times\mathbb{R}^{n}$, the dimension $n\geq 3$ and
$a>\tfrac{1}{4}-\lambda_{n}$, then we have
$\big{\|}|\nabla|^{\frac{3-n}{2}}(|u|^{2})\big{\|}_{L^{2}(I;L^{2}(\mathbb{R}^{n}))}\leq
C\|u(t_{0})\|_{L^{2}}\sup_{t\in I}\|u(t)\|_{\dot{H}^{\frac{1}{2}}},~{}t_{0}\in
I,$ (3.1)
and hence
$\big{\|}|\nabla|^{\frac{3-n}{4}}u\big{\|}_{L_{t}^{4}(I;L_{x}^{4}(\mathbb{R}^{n}))}^{2}\leq
C\|u(t_{0})\|_{L^{2}}\sup_{t\in I}\|u(t)\|_{\dot{H}^{\frac{1}{2}}}.$ (3.2)
###### Remark 3.2.
When $n=3$, it is obvious to see that the result also holds for $a=0$; see
[8].
###### Proof.
We consider the NLS equation in the form of
$i\partial_{t}u+\Delta u=gu$ (3.3)
where $g=g(\rho,|x|)$ is a real function of $\rho=|u|^{2}=2T_{00}$ and $|x|$.
We first recall the conservation laws for free Schrödinger in Tao [34]
$\begin{split}\partial_{t}T_{00}+\partial_{j}T_{0j}=0,\\\
\partial_{t}T_{0j}+\partial_{k}T_{jk}=0,\end{split}$
where the mass density quantity $T_{00}$ is defined by
$T_{00}=\tfrac{1}{2}|u|^{2},$ the mass current and the momentum density
quantity $T_{0j}=T_{j0}$ is given by
$T_{0j}=T_{j0}=\mathrm{Im}(\bar{u}\partial_{j}u)$, and the quantity $T_{jk}$
is
$T_{jk}=2\mathrm{Re}(\partial_{j}u\partial_{k}\bar{u})-\tfrac{1}{2}\delta_{jk}\Delta(|u|^{2}),$
(3.4)
for all $j,k=1,...n,$ and $\delta_{jk}$ is the Kroncker delta. Note that the
kinetic terms are unchanged, we see that for (3.3)
$\begin{split}\partial_{t}T_{00}+\partial_{j}T_{0j}&=0,\\\
\partial_{t}T_{0j}+\partial_{k}T_{jk}&=-\rho\partial_{j}g.\end{split}$ (3.5)
By the density argument, we may assume sufficient smoothness and decay at
infinity of the solutions to the calculation and in particular to the
integrations by parts. Let $h$ be a sufficiently regular real even function
defined in $\mathbb{R}^{n}$, e.g. $h=|x|$. The starting point is the auxiliary
quantity
$J=\tfrac{1}{2}\langle|u|^{2},h\ast|u|^{2}\rangle=2\langle T_{00},h\ast
T_{00}\rangle.$
Define the quadratic Morawetz quantity $M=\tfrac{1}{4}\partial_{t}J$. Hence we
can precisely rewrite
$M=-\tfrac{1}{2}\langle\partial_{j}T_{0j},h\ast
T_{00}\rangle-\tfrac{1}{2}\langle
T_{00},h\ast\partial_{j}T_{0j}\rangle=-\langle T_{00},\partial_{j}h\ast
T_{0j}\rangle.$ (3.6)
By (3.5) and integration by parts, we have
$\begin{split}\partial_{t}M&=\langle\partial_{k}T_{0k},\partial_{j}h\ast
T_{0j}\rangle-\langle T_{00},\partial_{j}h\ast\partial_{t}T_{0j}\rangle\\\
&=-\sum_{j,k=1}^{n}\langle T_{0j},\partial_{jk}h\ast T_{0j}\rangle+\langle
T_{00},\partial_{jk}h\ast
T_{jk}\rangle+\langle\rho,\partial_{j}h\ast(\rho\partial_{j}g)\rangle.\end{split}$
For our purpose, we note that
$\begin{split}\sum_{j,k=1}^{n}\langle T_{0k},\partial_{jk}h\ast
T_{0j}\rangle&=\big{\langle}\mathrm{Im}(\bar{u}\nabla
u),\nabla^{2}h\ast\mathrm{Im}(\bar{u}\nabla u)\big{\rangle}\\\
&=\big{\langle}\bar{u}\nabla u,\nabla^{2}h\ast\bar{u}\nabla
u\rangle-\langle\mathrm{Re}(\bar{u}\nabla
u),\nabla^{2}h\ast\mathrm{Re}(\bar{u}\nabla u)\big{\rangle}.\end{split}$ (3.7)
Therefore it yields that
$\begin{split}\partial_{t}M=&\big{\langle}\mathrm{Re}(\bar{u}\nabla
u),\nabla^{2}h\ast\mathrm{Re}(\bar{u}\nabla
u)\big{\rangle}-\big{\langle}\bar{u}\nabla u,\nabla^{2}h\ast\bar{u}\nabla
u\big{\rangle}\\\
&+\Big{\langle}\bar{u}u,\partial_{jk}h\ast\big{(}\mathrm{Re}(\partial_{j}u\partial_{k}\bar{u})-\tfrac{1}{4}\delta_{jk}\Delta(|u|^{2})\big{)}\Big{\rangle}+\big{\langle}\rho,\partial_{j}h\ast(\rho\partial_{j}g)\big{\rangle}.\end{split}$
From the observation
$\begin{split}-\big{\langle}\bar{u}u,\partial_{jk}h\ast\delta_{jk}\Delta(|u|^{2})\big{\rangle}=\big{\langle}\nabla(|u|^{2}),\Delta
h\ast\nabla(|u|^{2})\big{\rangle},\end{split}$
we write
$\begin{split}\partial_{t}M=\tfrac{1}{2}\langle\nabla\rho,\Delta
h\ast\nabla\rho\rangle+R+\big{\langle}\rho,\partial_{j}h\ast(\rho\partial_{j}g)\big{\rangle},\end{split}$
(3.8)
where $R$ is given by
$\begin{split}R&=\big{\langle}\bar{u}u,\nabla^{2}h\ast(\nabla\bar{u}\nabla
u)\big{\rangle}-\big{\langle}\bar{u}\nabla u,\nabla^{2}h\ast\bar{u}\nabla
u\big{\rangle}\\\
&=\tfrac{1}{2}\int\Big{(}\bar{u}(x)\nabla\bar{u}(y)-\bar{u}(y)\nabla\bar{u}(x)\Big{)}\nabla^{2}h(x-y)\Big{(}u(x)\nabla
u(y)-u(y)\nabla u(x)\Big{)}\mathrm{d}x\mathrm{d}y.\end{split}$
Since the Hessian of $h$ is positive definite, we have $R\geq 0$. Integrating
over time in an interval $[t_{1},t_{2}]\subset I$ yields
$\begin{split}\int_{t_{1}}^{t_{2}}\Big{\\{}\frac{1}{2}\langle\nabla\rho,\Delta
h\ast\nabla\rho\rangle+\langle\rho,\partial_{j}h\ast(\rho\partial_{j}g)\rangle+R\Big{\\}}\mathrm{d}t=-\langle
T_{00},\partial_{j}h\ast T_{0j}\rangle\big{|}_{t=t_{1}}^{t=t_{2}}.\end{split}$
From now on, we choose $h(x)=|x|$. One can follow the arguments in [8] to
bound the right hand by the quantity
$\Big{|}\mathrm{Im}\int_{\mathbb{R}^{2n}}|u(x)|^{2}\frac{x-y}{|x-y|}\bar{u}(y)\nabla
u(y)dxdy\Big{|}\leq C\sup_{t\in
I}\|u(t)\|^{2}_{L^{2}}\|u(t)\|^{2}_{\dot{H}^{\frac{1}{2}}}.$
Therefore we conclude
$\int_{t_{1}}^{t_{2}}\big{\langle}\rho,\partial_{j}h\ast(\rho\partial_{j}g)\big{\rangle}dt+\big{\|}|\nabla|^{\frac{3-n}{2}}(|u|^{2})\big{\|}_{L^{2}(I;L^{2}(\mathbb{R}^{n}))}\leq
C\sup_{t\in I}\|u(t)\|_{L^{2}}\|u(t)\|_{\dot{H}^{\frac{1}{2}}}.$ (3.9)
Now we consider the term
$\begin{split}P&:=\big{\langle}\rho,\nabla h\ast(\rho\nabla
g)\big{\rangle}.\end{split}$
Consider $g(\rho,|x|)=\rho^{(p-1)/2}+V(x)$, then we can write $P=P_{1}+P_{2}$
where
$\begin{split}P_{1}=\big{\langle}\rho,\nabla
h\ast\big{(}\rho\nabla(\rho^{(p-1)/2})\big{)}\big{\rangle}=\frac{p-1}{p+1}\big{\langle}\rho,\Delta
h\ast\rho^{(p+1)/2}\big{\rangle}\geq 0\end{split}$ (3.10)
and
$\begin{split}P_{2}=\iint\rho(x)\nabla
h(x-y)\rho(y)\nabla\big{(}V(y)\big{)}\mathrm{d}x\mathrm{d}y.\end{split}$
(3.11)
Comparing (1.1) and (3.3), we see $V(x)=a|x|^{-2}$. We claim that
$\begin{split}\Big{|}\int_{t_{1}}^{t_{2}}P_{2}dt\Big{|}=&\Big{|}2a\int_{t_{1}}^{t_{2}}\iint|u(x)|^{2}\frac{(x-y)}{|x-y|}\cdot
y|y|^{-4}|u(y)|^{2}\mathrm{d}x\mathrm{d}ydt\Big{|}\\\ \lesssim&\sup_{t\in
I}\|u_{0}\|_{L^{2}}^{2}\|u(t)\|_{\dot{H}^{\frac{1}{2}}}^{2}.\end{split}$
To show this, it suffices to show
$\begin{split}\int_{t_{1}}^{t_{2}}\int|x|^{-3}|u(t,x)|^{2}\mathrm{d}xdt\lesssim\sup_{t\in
I}\|u(t)\|_{\dot{H}^{\frac{1}{2}}}^{2}.\end{split}$ (3.12)
Let $A=\partial_{r}+\frac{n-1}{2r}$ and $H=\Delta-a|x|^{-2}$, where $r=|x|$.
Now we consider the quantity $\langle Au,u\rangle$. Since $u_{t}=i(Hu-f(u))$
with $f(u)=|u|^{p-1}u$, we have
$\begin{split}\partial_{t}\langle
Au,u\rangle=i\big{\langle}[A,H]u,u\big{\rangle}+i\big{(}\langle
Au,f(u)\rangle-\langle Af(u),u\rangle\big{)}.\end{split}$
We first consider the term from the nonlinear part
$\begin{split}\Big{(}\langle Au,f(u)\rangle-\langle
Af(u),u\rangle\Big{)}=\Big{(}\langle\partial_{r}u,f(u)\rangle+\langle
f(u),\partial_{r}u\rangle+\langle
f(u),\tfrac{n-1}{|x|}u\rangle\Big{)}.\end{split}$
By assuming $u$ rapidly tends to zero as $r\rightarrow\infty$, we obtain
$\displaystyle\Big{(}\langle Au,f(u)\rangle-\langle Af(u),u\rangle\Big{)}$
$\displaystyle=\frac{2}{p+1}\int_{\mathbb{R}^{n}}\partial_{r}\big{(}|u|^{p+1}(x)\big{)}dx+\int_{\mathbb{R}^{n}}\frac{(n-1)|u|^{p+1}}{|x|}dx$
$\displaystyle=(n-1)\frac{p-1}{p+1}\int_{\mathbb{R}^{n}}\frac{|u|^{p+1}}{|x|}dx.$
(3.13)
Now we consider the term from linear part. Note that the commutator
$[A,H]=\begin{cases}-2\Delta_{\mathbb{S}^{n-1}}r^{-3}+c\delta+2ar^{-3}\qquad
n=3;\\\
-2\Delta_{\mathbb{S}^{n-1}}r^{-3}+\frac{1}{2}(n-1)(n-3)r^{-3}+2ar^{-3}\qquad\qquad
n\geq 4.\\\ \end{cases}$ (3.14)
where the constant $c>0$ and $\delta$ is the delta function. Integrating on a
finite time interval $[t_{1},t_{2}]$, we have for $n=3$
$\begin{split}i^{-1}\langle
Au,u\rangle\big{|}_{t_{1}}^{t_{2}}=&2\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|\nabla_{\theta}u|^{2}}{r^{3}}dxdt+2\int_{t_{1}}^{t_{2}}|u(t,0)|^{2}dt\\\
&+2a\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|u|^{2}}{r^{3}}dxdt\\\
&+\frac{2(p-1)}{p+1}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|u|^{p+1}}{r}dxdt\end{split}$
(3.15)
and for $n\geq 4$
$\begin{split}i^{-1}\langle
Au,u\rangle\big{|}_{t_{1}}^{t_{2}}=&2\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|\nabla_{\theta}u|^{2}}{r^{3}}dxdt\\\
&+\big{[}\frac{1}{2}(n-1)(n-3)+2a\big{]}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|u|^{2}}{r^{3}}dxdt\\\
&+(n-1)\frac{p-1}{p+1}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|u|^{p+1}}{r}dxdt.\end{split}$
(3.16)
Note $a>\tfrac{1}{4}-\lambda_{n}$, the constant before the quantity
$\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{n}}\frac{|u|^{2}}{r^{3}}dxdt$ is
strictly positive. From (3.15) and (3.16), we have by interpolation and
Hardy’s inequality
$\begin{split}\int_{t_{1}}^{t_{2}}\int\frac{|u(t,x)|^{2}}{|x|^{3}}\mathrm{d}xdt\lesssim_{a}\sup_{t\in[t_{1},t_{2}]}\bigg{(}\Big{|}\int_{\mathbb{R}^{n}}\partial_{r}u\bar{u}dx\Big{|}+\int_{\mathbb{R}^{n}}\frac{|u|^{2}}{|x|}dx\bigg{)}\lesssim_{a}\sup_{t\in[t_{1},t_{2}]}\|u(t)\|_{\dot{H}^{\frac{1}{2}}}^{2}.\end{split}$
Therefore, we conclude the proof of Proposition 3.1.
∎
###### Remark 3.3.
Our strategy can be used to prove the same estimates for the Schrödinger
equation with potential $a|x|^{\sigma}$ where $a\leq 0$ and $\sigma>1$. In
particular $\sigma=2$, we prove the interaction Morawetz estimates for
defocusing NLS with repulsive harmonic potential. This possibly is used to
remove the radial assumption in the repulsive case of [21], which studied the
scattering theory of the energy-critical defocusing NLS with harmonic
potential. In contrast to the inverse square potential, the error term brought
by the potential is positive. Indeed, we consider $V(x)=a|x|^{\sigma}$ in
(3.11). From the observation $\nabla h$ is odd, it follows that
$\begin{split}P_{2}=\int\rho(x)\rho(y)\nabla
h(y-x)\nabla\big{(}V(x)\big{)}\mathrm{d}x\mathrm{d}y=-\int\rho(x)\rho(y)\nabla
h(x-y)\nabla\big{(}V(x)\big{)}\mathrm{d}x\mathrm{d}y.\end{split}$
Thus we can write
$\begin{split}P_{2}=\frac{1}{2}\int\rho(x)\rho(y)\nabla
h(x-y)\cdot\big{[}\nabla\big{(}V(y)\big{)}-\nabla\big{(}V(x)\big{)}\big{]}\mathrm{d}x\mathrm{d}y.\end{split}$
(3.17)
By the mean value theorem, we easily see that
$\displaystyle\nabla h(x-y)\cdot\big{(}\nabla V(y)-\nabla V(x)\big{)}=$
$\displaystyle\nabla h(x-y)\cdot\Big{(}\nabla V(y)-\nabla
V\big{(}(x-y)+y\big{)}\Big{)}$ $\displaystyle=$ $\displaystyle-\nabla
h(x-y)\cdot\int_{0}^{1}(x-y)\cdot\nabla^{2}V\big{(}y+\theta(x-y)\big{)}d\theta$
$\displaystyle=$
$\displaystyle-|x-y|^{-1}\int_{0}^{1}(x-y)\otimes(x-y)\nabla^{2}V\big{(}y+\theta(x-y)\big{)}d\theta.$
We see that $\nabla^{2}V$ is negative when $V(x)=a|x|^{\sigma}$ with $a<0$ and
$\sigma>1$ by using
$\nabla^{2}(|x|^{\sigma})=\sigma|x|^{\sigma-2}\bigg{(}I_{n\times
n}+(\sigma-2)\frac{(x_{1},\cdots,x_{n})^{T}}{|x|}\cdot\frac{(x_{1},\cdots,x_{n})}{|x|}\bigg{)},$
whose the $k$-order principal minor determinant is
$\bigg{|}I_{k\times
k}+(\sigma-2)\Big{(}\frac{x_{1}}{|x|},\cdots,\frac{x_{k}}{|x|}\Big{)}^{T}\cdot\Big{(}\frac{x_{1}}{|x|},\cdots,\frac{x_{k}}{|x|}\Big{)}\bigg{|}=1+(\sigma-2)\frac{x_{1}^{2}+\cdots+x_{k}^{2}}{|x|^{2}}>0.$
Hence we can check that $P_{2}$ is nonnegative when $V(x)=a|x|^{\sigma}$ with
$a\leq 0$ and $\sigma>1$. This together with (3.9) and (3.10) concludes the
proof of the case $\sigma>1$.
## 4\. Sobolev norm equivalence
In this section, we study the equivalence of the Sobolev norms based on the
operator $P_{a}$ and the standard Sobolev norms based on the Laplacian. For
simplicity, we define
$r_{0}=\tfrac{2n}{\min\\{n+2+\sqrt{(n-2)^{2}+4a},2n\\}},\quad
r_{1}=\tfrac{2n}{\max\\{n-\sqrt{(n-2)^{2}+4a},0\\}}.$ (4.1)
The purpose of this section is to prove the following
###### Proposition 4.1 (Sobolev norm equivalence).
Let $n\geq 3,~{}a>-\lambda_{n}$. Then, there exist constants
$C_{1},~{}C_{2}>0$ depending on $(n,a,s)$ such that
$\bullet$ when $s=1$ and
$r\in(r_{0},r_{0}^{\prime})\cap(r^{\prime}_{1},r_{1})\cap(\frac{n}{n-1},n)$
$C_{1}\|f\|_{\dot{H}_{r}^{1}(\mathbb{R}^{n})}\leq\|P_{a}^{1/2}f\|_{L^{r}(\mathbb{R}^{n})}\leq
C_{2}\|f\|_{\dot{H}_{r}^{1}(\mathbb{R}^{n})},~{}~{}\forall~{}f\in\dot{H}_{r}^{1}(\mathbb{R}^{n});$
(4.2)
$\bullet$ if $a\geq 0$, for $0\leq s\leq 1$ and
$r\in(r_{0}^{\prime}/(r_{0}^{\prime}-s),r_{1}/s)\cap(1,n/s)$
$C_{1}\|f\|_{\dot{H}_{r}^{s}(\mathbb{R}^{n})}\leq\|P_{a}^{s/2}f\|_{L^{r}(\mathbb{R}^{n})}\leq
C_{2}\|f\|_{\dot{H}_{r}^{s}(\mathbb{R}^{n})},~{}~{}\forall~{}f\in\dot{H}_{r}^{s}(\mathbb{R}^{n}).$
(4.3)
###### Remark 4.2.
This result generalizes the equivalent result of the Sobolev norms in [3],
which proved
$\|f\|_{\dot{H}^{s}(\mathbb{R}^{n})}\sim\|P_{a}^{s/2}f\|_{L^{2}(\mathbb{R}^{n})}$
for $-1\leq s\leq 1$ and $a>-\lambda_{n}$.
###### Remark 4.3.
From the classical Sobolev result, we know that (4.2) with $a=0$ holds for
$1<r<\infty$. However, it is easy to check that the interval
$(r_{0},r_{1})\rightarrow(1,n)$ as $a\rightarrow 0$. It was pointed out to the
authors by Andrew Hassell that the dependence on $a$ in the boundedness of
Riesz transform is not continuous, which means that the equivalent norm result
is strongly influenced by the inverse square potential even though for
sufficiently small $|a|$.
This proposition follows from
###### Proposition 4.4.
There exist constants $c_{r}$ and $C_{r}$ satisfying the following estimates:
$\bullet$ If $a>-\lambda_{n}$, we have for $r\in(r_{0},r_{1})$
$\|\nabla f\|_{L^{r}}\leq C_{r}\|P_{a}^{\frac{1}{2}}f\|_{L^{r}};$ (4.4)
In addition, the reverse estimate holds for $r\in(r_{0},r_{1})$ and
$1<r,r^{\prime}<n$
$\|P_{a}^{\frac{1}{2}}f\|_{L^{r^{\prime}}}\leq c_{r}\|\nabla
f\|_{L^{r^{\prime}}}.$ (4.5)
$\bullet$ If $a\geq 0$, we have for $0\leq s\leq 2$ and $r\in(1,n/s)$
$\|P_{a}^{\frac{s}{2}}f\|_{L^{r}}\leq C_{r}\||\nabla|^{s}f\|_{L^{r}}$ (4.6)
In addition, the reverse estimate holds for $0\leq s\leq 1$ and
$r\in(r_{0}^{\prime}/(r_{0}^{\prime}-s),r_{1}/s)$
$\|P_{a}^{\frac{s}{2}}f\|_{L^{r}}\geq c_{r}\||\nabla|^{s}f\|_{L^{r}}$ (4.7)
For $a\geq 0$, since the heat kernel $e^{-tP_{a}}$ satisfies the Gaussian
upper bounds, we can follow D’Ancona-Fanelli-Vega-Visciglia’s [10] argument,
which are in spirt of Sikora-Wright [32] proving the boundedness of imaginary
power operators and Stein-Weiss complex interpolation theorem. When $a<0$, the
Gaussian upper bound of the heat kernel fails, we have to resort to the
boundedness of the Riesz transform, which was proved in Hassell-Lin [17]. That
is why we have to restrict $s=1$. However, this is enough for considering the
wellposedness and scattering theory in energy space $H^{1}$. We believe that
one could establish the equivalence of the Sobolev norm on metric cone and
perturbated by the inverse square potential, which is a topic we plan to
address in future articles.
###### Proof.
We first consider (4.4), which is a consequence of the boundedness of Riesz
transform $\nabla P_{a}^{-1/2}$. The theorem of Hassell-Lin [17] established
$L^{r}$-boundness of Riesz transform of Schrödinger operator with inverse
square potential on a metric cone. The result implies that if
$r\in(r_{0},r_{1})$ where $r_{0},r_{1}$ are defined in (4.1), then Riesz
transform $\nabla P_{a}^{-\frac{1}{2}}$ is bounded on $L^{r}$.
Next we use a duality argument to show (4.5). Since
$P_{a}^{\frac{1}{2}}C_{0}^{\infty}$ is dense in $L^{r}$ (see [28, Appendix]),
then
$\|P_{a}^{\frac{1}{2}}f\|_{L^{r^{\prime}}}=\sup\Big{\\{}\langle
P_{a}^{\frac{1}{2}}f,P_{a}^{\frac{1}{2}}g\rangle:g\in
C_{0}^{\infty}(\mathbb{R}^{n}),\|P_{a}^{\frac{1}{2}}g\|_{L^{r}}\leq
1\Big{\\}}.$
Therefore by the definition of the square root of $P_{a}$, we see
$\begin{split}\|P_{a}^{\frac{1}{2}}f\|_{L^{r^{\prime}}}&\leq|\langle
P_{a}^{\frac{1}{2}}f,P_{a}^{\frac{1}{2}}g\rangle|=|\langle P_{a}f,g\rangle|\\\
&\leq\|\nabla f\|_{L^{r^{\prime}}}\|\nabla
g\|_{L^{r}}+\left\|f/{|x|}\right\|_{L^{r^{\prime}}}\left\|g/{|x|}\right\|_{L^{r}}.\end{split}$
If $1<r,r^{\prime}<n$, the Hardy inequality (2.8) (see below) implies
$\begin{split}\|P_{a}^{\frac{1}{2}}f\|_{L^{r^{\prime}}}\leq C\|\nabla
f\|_{L^{r^{\prime}}}\|\nabla g\|_{L^{r}}.\end{split}$ (4.8)
By (4.4), hence we prove (4.5).
Next we prove (4.6) and (4.7). We first recall two-side estimates of the heat
kernel associated to the operator $P_{a}$, which were found independently by
Liskevich-Sobol [22, Remarks at the end of Sec. 1] and Milman-Semenov [23,
Theorem 1].
###### Lemma 4.5 (Heat kernel boundedness).
Assume $a>-\lambda_{n}$ and let $H(t,x,y)$ be the kernel of the operator
$e^{-tP_{a}}$. Then there exist positive constants $C_{1},C_{2}$ and
$c_{1},c_{2}$ such that for all $t>0$ and all
$x,y\in\mathbb{R}^{n}\setminus\\{0\\}$
$\begin{split}C_{1}&\varphi_{\sigma}(x,t)\varphi_{\sigma}(y,t)t^{-\frac{n}{2}}\exp\left(-|x-y|^{2}/c_{1}t\right)\leq
H(t,x,y)\\\ &\leq
C_{2}\varphi_{\sigma}(x,t)\varphi_{\sigma}(y,t)t^{-\frac{n}{2}}\exp\left(-|x-y|^{2}/c_{2}t\right),\end{split}$
(4.9)
where the weight function
$\varphi_{\sigma}(x,t)=\begin{cases}\left(\frac{\sqrt{t}}{|x|}\right)^{\sigma}\quad&\mathrm{if}~{}|x|\leq\sqrt{t},\\\
\quad 1\quad&\mathrm{if}~{}|x|\geq\sqrt{t}\\\ \end{cases}$ (4.10)
and $\sigma=\sigma(a)=\frac{1}{2}(n-2)-\frac{1}{2}\sqrt{(n-2)^{2}+4a}$.
###### Remark 4.6.
We notice that $\sigma(a)\leq 0$ for $a\geq 0$ and $0<\sigma(a)<(n-2)/2$ for
$a\in(-\lambda_{n},0)$.
We need a theorem of Sikora-Wright [32] that established a weak type estimate
for imaginary powers of self-adjoint operator, defined by spectral theory. The
result implies that if the heat kernel $H(t,x,y)$ associated with the operator
$\mathrm{H}$ satisfies that
$H(t,x,y)\lesssim t^{-\frac{n}{2}}\exp\left(-|x-y|^{2}/c_{2}t\right),$
then for all $y\in\mathbb{R}$ the imaginary powers $\mathrm{H}^{iy}$ is
weak-$(1,1)$. By Lemma 4.5 and the well known Gaussian upper for heat kernel
$e^{-t\Delta}$, the operators $P_{a}^{iy}$ $(a\geq 0)$ and $(-\Delta)^{-iy}$
satisfy weak-$(1,1)$ type estimate of $O(1+|y|)^{n/2}$. On the other hand, the
operators $P_{a}^{iy}$ and $(-\Delta)^{-iy}$ are obviously bounded on $L^{2}$
by the spectral theory on Hilbert space. Hence the operators $P_{a}^{iy}$ and
$(-\Delta)^{-iy}$ are bounded on $L^{r}$ for all $1<r<\infty$.
Now we define the analytic family of operators for $z\in\mathbb{C}$
$T_{z}=P_{a}^{z}(-\Delta)^{-z}$ (4.11)
where $P_{a}^{z}=\int_{0}^{\infty}\lambda^{z}dE_{\sqrt{P_{a}}}(\lambda)$ and
$(-\Delta)^{z}$ is defined by the Fourier transform. Writing $z=x+iy$ for
$x\in[0,1]$, we have
$T_{z}=P_{a}^{iy}P_{a}^{x}(-\Delta)^{-x}(-\Delta)^{-iy},\quad
y\in\mathbb{R},~{}x\in[0,1].$
When $\operatorname*{Re}z=0$, then we have for all $1<p_{0}<\infty$
$\|T_{z}\|_{L^{p_{0}}\rightarrow L^{p_{0}}}=\|T_{iy}\|_{L^{p_{0}}\rightarrow
L^{p_{0}}}\leq C(1+|y|)^{n(1-\frac{2}{p_{0}})}.$ (4.12)
Notice $P_{a}f=-\Delta f+a|x|^{-2}f$, it follows from the Hardy inequality
(2.8) that
$\left(\int_{\mathbb{R}^{n}}|P_{a}f|^{p}dx\right)^{\frac{1}{p}}\leq
C\left(\int_{\mathbb{R}^{n}}|\Delta f|^{p}dx\right)^{\frac{1}{p}},\quad
1<{p}<\frac{n}{2}.$
Therefore for $\operatorname*{Re}z=1$, we have by the $L^{p}$-boundedness of
$P_{a}^{iy}$ and $(-\Delta)^{-iy}$
$\|T_{1+iy}\|_{L^{p_{1}}\rightarrow
L^{p_{1}}}\leq\|P_{a}(-\Delta)^{-1}\|_{L^{p_{1}}\rightarrow L^{p_{1}}}\leq
C(1+|y|)^{n(1-\frac{2}{p_{1}})},1<{p_{1}}<\frac{n}{2}.$ (4.13)
Applying complex interpolation to (4.12) and (4.13), we obtain for real number
$\sigma\in[0,1]$
$\|T_{\sigma}\|_{L^{r}\rightarrow L^{r}}\leq C,$ (4.14)
where $1/r=(1-\sigma)/p_{0}+\sigma/p_{1}$ with $1<p_{0}<\infty$ and
$1<p_{1}<n/2$. This gives $r\in(1,n/(2\sigma))$, hence $1<r<n/s$ which proves
(4.6).
Finally we prove (4.7). We similarly define the analytic family of operators
for $z\in\mathbb{C}$
$\widetilde{T}_{z}=(-\Delta)^{z}P_{a}^{-z}.$ (4.15)
Writing $z=x+iy$ for $x\in[0,1/2]$, we have
$\widetilde{T}_{z}=(-\Delta)^{iy}(-\Delta)^{x}P_{a}^{-x}P_{a}^{-iy},\quad
y\in\mathbb{R},~{}x\in[0,1/2].$ (4.16)
As before the operators $P_{a}^{-iy}$ and $(-\Delta)^{iy}$ are bounded on
$L^{r}$ for all $1<r<\infty$. On the other hand, we can use the dual argument
as above and boundedness of classical Riesz transform
$\nabla(-\Delta)^{-\frac{1}{2}}$ to show
$\|(-\Delta)^{\frac{1}{2}}f\|_{L^{r}}\leq C\|\nabla f\|_{L^{r}},\quad
1<r<\infty.$
By (4.4), we hence have
$\|\widetilde{T}_{z}\|_{L^{p_{1}}\rightarrow L^{p_{1}}}\leq
C,\quad\text{for}~{}p_{1}\in(r_{0},r_{1}),\quad\operatorname*{Re}z=1/2.$
(4.17)
When $\operatorname*{Re}z=0$, then we have for all $1<p_{0}<\infty$
$\|\widetilde{T}_{z}\|_{L^{p_{0}}\rightarrow
L^{p_{0}}}=\|T_{iy}\|_{L^{p_{0}}\rightarrow L^{p_{0}}}\leq
C(1+|y|)^{n(1-\frac{2}{p_{0}})}.$ (4.18)
By the Stein-Weiss interpolation, we obtain for real number $\sigma\in[0,1/2]$
$\|\widetilde{T}_{\sigma}\|_{L^{r}\rightarrow L^{r}}\leq C,$ (4.19)
where $1/r=(1-2\sigma)/p_{0}+2\sigma/{p_{1}}$ with $1<p_{0}<\infty$, and
$p_{1}\in(r_{0},r_{1})$. This implies
$r\in(r_{0}^{\prime}/(r_{0}^{\prime}-2\sigma),r_{1}/(2\sigma))$. Note
$\sigma=s/2$, we prove (4.7).
∎
## 5\. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. The key points are the Strichartz
estimate, Leibniz rule obtained by the equivalence Sobolev norm and the
Morawetz type estimate.
### 5.1. Global well-posedness theory
By the mass and energy conservation, the global well-posedness follows from
###### Proposition 5.1 (Local well-posedness theory).
Let $n\geq 3$ and $1+\frac{2}{n-2}\leq p<1+\frac{4}{n-2}$. Assume that
$a>-\frac{4}{(p+1)^{2}}\lambda_{n}$ and $u_{0}\in H^{1}(\mathbb{R}^{n})$. Then
there exists $T=T(\|u_{0}\|_{H^{1}})>0$ such that the equation (1.1) with
initial data $u_{0}$ has a unique solution $u$ with
$u\in C(I;H^{1}(\mathbb{R}^{n}))\cap
L_{t}^{q}(I;H^{1}_{r}(\mathbb{R}^{n})),\quad I=[0,T),$ (5.1)
where the pair $(q,r)\in\Lambda_{0}$ satisfies
$(q,r)=\begin{cases}\left(\tfrac{4(p+1)}{(n-2)(p-1)},\tfrac{n(p+1)}{n+p-1}\right),\quad\text{if}\quad
a\geq 0;\\\
\left(q,p\left(\frac{1}{(2n/(n+2))_{+}}+\frac{p-1}{n}\right)^{-1}\right),\quad\text{if}\quad\min\\{1-\lambda_{n},0\\}\leq
a<0;\\\
\left(q,p\left(\frac{1}{(r_{1}^{\prime})_{+}}+\frac{p-1}{n}\right)^{-1}\right),\quad\text{if}\quad-\frac{4}{(p+1)^{2}}\lambda_{n}<a<\min\\{1-\lambda_{n},0\\}.\end{cases}$
(5.2)
Here $r_{1}$ is defined in (4.1).
###### Remark 5.2.
One can release the restriction $p\geq 1+\frac{2}{n-2}$ to $p>1$ if $a\geq 0$.
If $a<0$, one can also improve the range of $p$ which depends on $a$. We do
not give the detail since this result is enough for showing the scattering.
###### Proof.
We follow the standard Banach fixed point argument to prove this result. To
this end, we consider the map
$\Phi(u(t))=e^{itP_{a}}u_{0}-i\int_{0}^{t}e^{i(t-s)P_{a}}(|u|^{p-1}u(s))ds$
(5.3)
on the complete metric space $B$
$\displaystyle B:=\big{\\{}$ $\displaystyle u\in Y(I)\triangleq
C_{t}(I;H^{1})\cap L_{t}^{q}(I;H^{1}_{r}):\ \|u\|_{Y(I)}\leq
2CC_{1}\|u_{0}\|_{H^{1}}\big{\\}}$
with the metric
$d(u,v)=\big{\|}u-v\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{n})}$. We
need to prove that the operator $\Phi$ defined by $(\ref{inte3})$ is well-
defined on $B$ and is a contraction map under the metric $d$ for $I$.
To see this, we first consider the case $a\geq 0$. Therefore we have by
Proposition 4.1 for $n\geq 3$
$\|\nabla f\|_{L^{r}}\simeq
C_{r}\|P_{a}^{\frac{1}{2}}f\|_{L^{r}},\quad\forall~{}r\in(1,n).$ (5.4)
Let $(q,r)=\big{(}\tfrac{4(p+1)}{(n-2)(p-1)},\tfrac{n(p+1)}{n+p-1}\big{)}$. It
is easy to verify that $(q,r)\in\Lambda_{0}$ such that
$r,~{}{r}^{\prime}\in(1,n)$. Then we have by Strichartz estimate and (5.4)
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq$ $\displaystyle
C_{1}\big{\|}\langle
P_{a}^{1/2}\rangle\Phi(u)\big{\|}_{L_{t}^{q}(I;L_{x}^{r})}$
$\displaystyle\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}\big{\|}\langle
P_{a}^{1/2}\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{{q}^{\prime}}L_{x}^{{r}^{\prime}}(I\times\mathbb{R}^{n})}$
$\displaystyle\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}C_{2}\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{{q}^{\prime}}L_{x}^{{r}^{\prime}}}.$
Choose $\theta=1-\frac{(p-1)(n-2)}{4}$, then we have by Leibniz rule, Hölder’s
inequality and Sobolev inequality
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}C_{2}T^{\theta}\|u\|_{L_{t}^{q}(I;L^{\frac{nr}{n-r}})}^{p-1}\|u\|_{L_{t}^{q}(I;H_{r}^{1})}.$
Note $\theta>0$ for $p\in[1+\frac{4}{n},1+\frac{4}{n-2})$ and
$\|u\|_{Y(I)}\leq 2CC_{1}\|u_{0}\|_{H^{1}}$ if $u\in B$, we see that for $u\in
B$,
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}C_{2}T^{\theta}(2CC_{1}\|u_{0}\|_{H^{1}})^{p}.$
Taking $T$ sufficiently small such that
$C_{2}T^{\theta}(2CC_{1}\|u_{0}\|_{H^{1}})^{p}<\|u_{0}\|_{H^{1}},$
we have $\Phi(u)\in B$ for $u\in B$. On the other hand, by the same argument
as before, we have for $u,v\in B$,
$\displaystyle d\big{(}\Phi(u),\Phi(v)\big{)}\leq$ $\displaystyle
CT^{\theta}\big{(}\|u\|_{Y(I)}^{p-1}+\|v\|_{Y(I)}^{p-1}\big{)}d(u,v).$
Thus we derive by taking $T$ small enough
$d\big{(}\Phi(u),\Phi(v)\big{)}\leq\tfrac{1}{2}d(u,v).$
Next we consider the case $a\in(-\frac{4p}{(p+1)^{2}}\lambda_{n},0)$. In this
case we can choose $(q,r)\in\Lambda_{0}$ such that
$\left(\tfrac{1}{q},\tfrac{1}{r}\right)=\left(\tfrac{1}{q},\tfrac{1}{p}\left(\tfrac{1}{\widetilde{r}^{\prime}}+\tfrac{p-1}{n}\right)\right),$
where we denote $\widetilde{r}^{\prime}$ to be
$\widetilde{r}^{\prime}=\begin{cases}\left(\tfrac{2n}{n+2}\right)_{+},\quad\text{if}\quad\min\\{1-\lambda_{n},0\\}\leq
a<0;\\\
(r_{1}^{\prime})_{+},\quad\text{if}\quad-\frac{4p}{(p+1)^{2}}\lambda_{n}<a<\min\\{1-\lambda_{n},0\\}.\end{cases}$
(5.5)
Since $a<0$, one has $1/{r_{1}^{\prime}}=({n+\sqrt{(n-2)^{2}+4a}})/(2n)$. And
so we can verify $r\in(r_{0},r_{1})$ by $a>-{4p\lambda_{n}}/{(p+1)^{2}}$.
Hence we have by Proposition 4.1
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq$ $\displaystyle
C_{1}\big{\|}\langle
P_{a}^{1/2}\rangle\Phi(u)\big{\|}_{L_{t}^{\infty}(I;L_{x}^{2})\cap
L_{t}^{q}(I;L_{x}^{r})}.$
Since the operator $P_{a}^{\frac{1}{2}}$ commutates with the Schrödinger
propagator $e^{itP_{a}}$, we have by (2.4) and (4.4)
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}\big{\|}\langle
P_{a}^{1/2}\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{\widetilde{q}^{\prime}}L_{x}^{\widetilde{r}^{\prime}}(I\times\mathbb{R}^{n})},$
where $(\widetilde{q},\widetilde{r})\in\Lambda_{0}$. Note that
$\widetilde{r}^{\prime}\in(r_{1}^{\prime},r_{0}^{\prime})\cap(\frac{n}{n-1},n)\cap(\frac{2n}{n+2},2)$
when $a<0$. By Proposition 4.4 and Leibniz rule for the operator
$\langle\nabla\rangle$, we obtain that for $\theta$ as above
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{\widetilde{q}^{\prime}}L_{x}^{\widetilde{r}^{\prime}}(I\times\mathbb{R}^{n})}$
$\displaystyle\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}T^{\theta}\|u\|^{p-1}_{L^{q}_{t}(I;L^{\frac{nr}{n-r}}_{x}(\mathbb{R}^{n}))}\|u\|_{L^{q}_{t}(I;H^{1}_{r}(\mathbb{R}^{n}))}$
$\displaystyle\leq$ $\displaystyle
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}T^{\theta}\|u\|^{p}_{L^{q}_{t}(I;H^{1}_{r}(\mathbb{R}^{n}))},$
where $\frac{1}{\widetilde{q}^{\prime}}=\theta+\frac{p}{q}$ and
$\frac{1}{\widetilde{r}^{\prime}}=\frac{(n-r)(p-1)}{nr}+\frac{1}{r}$. For
$u\in B$, one must has $\|u\|_{Y(I)}\leq 2CC_{1}\|u_{0}\|_{H^{1}}$, hence we
have
$\displaystyle\big{\|}\Phi(u)\big{\|}_{Y(I)}\leq
CC_{1}\|u_{0}\|_{H^{1}}+CC_{1}C_{2}T^{\theta}(2CC_{1}\|u_{0}\|_{H^{1}})^{p}.$
We take $T$ sufficiently small such that
$C_{2}T^{\theta}(2CC_{1}\|u_{0}\|_{H^{1}})^{p}<\|u_{0}\|_{H^{1}},$
hence $\Phi(u)\in B$ for $u\in B$. Similarly, we have
$d\big{(}\Phi(u),\Phi(v)\big{)}\leq\tfrac{1}{2}d(u,v).$
The standard fixed point argument gives a unique solution $u$ of (1.1) on
$I\times\mathbb{R}^{n}$ which satisfies the bound (5.1).
∎
###### Lemma 5.3 (The boundedness of kinetic energy).
For $a>-\lambda_{n}$, there exists $c=c(n,a)>0$ such that
$\|u(t)\|_{\dot{H}^{1}}^{2}<cE(u(t)).$ (5.6)
###### Proof.
We recall the sharp Hardy’s inequality that for $n\geq 3$
$\int_{\mathbb{R}^{n}}\tfrac{|u(x)|^{2}}{|x|^{2}}dx\leq\tfrac{4}{(n-2)^{2}}\int_{\mathbb{R}^{n}}|\nabla
u|^{2}dx.$ (5.7)
Thus,
$\displaystyle E(u)=$ $\displaystyle\tfrac{1}{2}\int|\nabla
u(t)|^{2}+\tfrac{a}{2}\int\tfrac{|u(t)|^{2}}{|x|^{2}}+\tfrac{1}{p+1}\int|u(t)|^{p+1}$
$\displaystyle\geq$
$\displaystyle\tfrac{1}{2}\min\left\\{1,1+\tfrac{4a}{(n-2)^{2}}\right\\}\int|\nabla
u(t)|^{2}.$
Note that $a>-\lambda_{n}$, this implies (5.6). ∎
By using Proposition 5.1, mass and energy conservations and this lemma, we
conclude the proof of global well-posed result of Theorem 1.1.
### 5.2. Scattering theory
Now we use the global interaction Morawetz estimate (3.2)
$\big{\|}|\nabla|^{\frac{3-n}{4}}u\big{\|}_{L_{t}^{4}(\mathbb{R};L_{x}^{4}(\mathbb{R}^{n}))}^{2}\leq
C\|u_{0}\|_{L^{2}}\sup_{t\in\mathbb{R}}\|u(t)\|_{\dot{H}^{\frac{1}{2}}},$
(5.8)
to prove the scattering theory part of Theorem 1.1. Since the construction of
the wave operator is standard, we only show the asymptotic completeness.
Let $u$ be a global solution to (1.1). Using (5.6) and (5.8), we have by
interpolation
$\|u\|_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}(\mathbb{R}\times\mathbb{R}^{n})}\leq
C(E,M),$ (5.9)
where the constant $C$ depends on the energy $E$ and mass $M$. Let $\eta>0$ be
a small constant to be chosen later and split $\mathbb{R}$ into
$L=L(\|u_{0}\|_{H^{1}})$ finite subintervals $I_{j}=[t_{j},t_{j+1}]$ such that
$\|u\|_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}(I_{j}\times\mathbb{R}^{n})}\leq\eta.$
(5.10)
We first consider the case $a\geq 0$. Define
$\big{\|}\langle\nabla\rangle
u\big{\|}_{S^{0}(I)}:=\sup_{(q,r)\in\Lambda_{0}:r\in[2,\min\\{n_{-},(\frac{2n}{n-2})_{-}\\}]}\big{\|}\langle\nabla\rangle
u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{n})}.$
Using the Strichartz estimate and (5.4) , we obtain
$\displaystyle\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(I_{j})}\lesssim$
$\displaystyle\|u(t_{j})\|_{H^{1}}+\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2n}{n+2}}(I_{j}\times\mathbb{R}^{n})}.$
(5.11)
Let $\epsilon>0$ to be determined later, and
$r_{\epsilon}=\frac{2n}{n-(4/(2+\epsilon))}$. On the other hand, we use the
Leibniz rule and Hölder’s inequality to obtain
$\displaystyle\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2n}{n+2}}}\lesssim$
$\displaystyle\big{\|}\langle\nabla\rangle
u\big{\|}_{L_{t}^{2+\epsilon}(I_{j};L_{x}^{r_{\epsilon}})}\|u\|^{p-1}_{L_{t}^{\frac{2(p-1)(2+\epsilon)}{\epsilon}}L_{x}^{\frac{n(p-1)(2+\epsilon)}{4+\epsilon}}}.$
When $n\geq 4$, we can choose $\epsilon>0$ to be small enough such that
${2(p-1)(2+\epsilon)}/{\epsilon}>n+1$ and
$2\leq\frac{n(p-1)(2+\epsilon)}{4+\epsilon}<\tfrac{2n}{n-2}$ for all
$p\in(1+\frac{4}{n},1+\frac{4}{n-2})$. Therefore we use interpolation to
obtain
$\displaystyle\|u\|_{L_{t}^{\frac{2(p-1)(2+\epsilon)}{\epsilon}}L_{x}^{\frac{n(p-1)(2+\epsilon)}{4+\epsilon}}}\leq
C\|u\|^{\alpha}_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}(I_{j}\times\mathbb{R}^{n})}\|u\|^{\beta}_{L_{t}^{\infty}L_{x}^{\frac{2n}{n-2}}(I_{j}\times\mathbb{R}^{n})}\|u\|^{\gamma}_{L_{t}^{\infty}L_{x}^{2}(I_{j}\times\mathbb{R}^{n})},$
where $\alpha>0,\beta,\gamma\geq 0$ satisfy $\alpha+\beta+\gamma=1$ and
$\displaystyle\begin{cases}\frac{\epsilon}{2(p-1)(2+\epsilon)}&=\frac{\alpha}{n+1}+\frac{\beta}{\infty}+\frac{\gamma}{\infty},\\\
\frac{4+\epsilon}{n(p-1)(2+\epsilon)}&=\frac{(n-1)\alpha}{2(n+1)}+\frac{(n-2)\beta}{2n}+\frac{\gamma}{2}.\end{cases}$
It is easy to verify these requirements for
$p\in(1+\frac{4}{n},1+\frac{4}{n-2})$. Since $r_{\epsilon}\in[2,n_{-}]$ and
$r_{\epsilon}<\tfrac{2n}{n-2}$ for $\epsilon>0$ and $n\geq 4$, we have
$\displaystyle\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2n}{n+2}}}\lesssim$
$\displaystyle\big{\|}\langle\nabla\rangle
u\big{\|}_{L_{t}^{2+\epsilon}(I_{j};L_{x}^{r_{\epsilon}})}\|u\|^{\alpha(p-1)}_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}(\mathbb{R}\times\mathbb{R}^{n})}\|u\|^{(\beta+\gamma)(p-1)}_{L_{t}^{\infty}H^{1}_{x}(I_{j}\times\mathbb{R}^{n})}$
$\displaystyle\leq$ $\displaystyle
C\eta^{\alpha(p-1)}\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(I_{j})}.$
Plugging this into (5.11) and noting that $\alpha(p-1)>0$, we can choose
$\eta$ to be small enough such that
$\displaystyle\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(I_{j})}\leq
C(E,M,\eta).$
Hence we have by the finiteness of $L$
$\displaystyle\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(\mathbb{R})}\leq
C(E,M,\eta,L).$ (5.12)
When $n=3$, we choose $\epsilon=2_{+}$, and then $r_{\epsilon}=3_{-}$. If
$p\in(\frac{7}{3},4]$, then ${2(p-1)(2+\epsilon)}/{\epsilon}>4$ and
$2\leq\frac{3(p-1)(2+\epsilon)}{4+\epsilon}\leq 6$. Arguing as before, we can
estimate $\|\langle\nabla\rangle u\|_{S^{0}(I_{j})}$. If $p\in(4,5)$, we use
interpolation to show that
$\displaystyle\|u\|_{L_{t}^{\frac{2(p-1)(2+\epsilon)}{\epsilon}}L_{x}^{\frac{3(p-1)(2+\epsilon)}{4+\epsilon}}}\leq
C\|u\|^{\alpha}_{L_{t}^{4}L_{x}^{4}(I_{j}\times\mathbb{R}^{3})}\|u\|^{\beta}_{L_{t}^{\infty}L_{x}^{6}(I_{j}\times\mathbb{R}^{3})}\|u\|^{\gamma}_{L_{t}^{6}L_{x}^{18}(I_{j}\times\mathbb{R}^{3})},$
where $\alpha>0,\beta,\gamma\geq 0$ satisfy $\alpha+\beta+\gamma=1$ and
$\displaystyle\begin{cases}\frac{\epsilon}{2(p-1)(2+\epsilon)}&=\frac{\alpha}{4}+\frac{\beta}{\infty}+\frac{\gamma}{6},\\\
\frac{4+\epsilon}{3(p-1)(2+\epsilon)}&=\frac{\alpha}{4}+\frac{\beta}{6}+\frac{\gamma}{18}.\end{cases}$
It is easy to solve these equations for $p\in(4,5)$. Since
$r_{\epsilon}\in[2,3_{-}]$ for $\epsilon=2_{+}$, we have
$\displaystyle\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{6}{5}}}\lesssim$
$\displaystyle\big{\|}\langle\nabla\rangle
u\big{\|}_{L_{t}^{2+\epsilon}(I_{j};L_{x}^{r_{\epsilon}})}\|u\|^{\alpha(p-1)}_{L_{t}^{4}L_{x}^{4}(\mathbb{R}\times\mathbb{R}^{3})}\|u\|^{\beta(p-1)}_{L_{t}^{\infty}H^{1}_{x}(I_{j}\times\mathbb{R}^{3})}\|\langle\nabla\rangle
u\|^{\gamma(p-1)}_{L_{t}^{6}L_{x}^{\frac{18}{7}}(I_{j}\times\mathbb{R}^{3})}$
$\displaystyle\leq C\eta^{\alpha(p-1)}\big{\|}\langle\nabla\rangle
u\big{\|}^{1+\gamma(p-1)}_{S^{0}(I_{j})}.$
Hence arguing as above we have (5.12) for $n=3$.
We secondly consider the case $\frac{4}{(p+1)^{2}}-\lambda_{n}<a<0$ and $n\geq
4$. Let $2^{*}=\frac{2n}{n-2}$. Define
$\big{\|}\langle\nabla\rangle
u\big{\|}_{S^{0}(I)}:=\sup_{(q,r)\in\Lambda_{0}:r\in[2,(\frac{2n}{n-2})_{-}]}\big{\|}\langle\nabla\rangle
u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{n})}.$
For $1-\lambda_{n}<a<0$, we have
$[(2^{*})^{\prime},2^{*}]\subset(r_{0},r_{1})$ and
$2^{*},(2^{*})^{\prime}\in(1,n)$. Using the Strichartz estimate and (4.5) , we
obtain
$\displaystyle\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(I_{j})}\lesssim$
$\displaystyle\|u(t_{j})\|_{H^{1}}+\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2n}{n+2}}(I_{j}\times\mathbb{R}^{n})}.$
If we choose $r_{\epsilon}$ as before, we have
$r_{\epsilon}\in[2,2^{\ast}]\subset(r_{0},r_{1})$ in this case. Then we can
closely follow the previous argument to obtain (5.12). For
$\frac{4}{(p+1)^{2}}-\lambda_{n}\leq a<1-\lambda_{n}$, by using the Strichartz
estimate and (4.5), we instead obtain for some $\alpha>0,\beta\geq 1$
$\displaystyle\big{\|}\langle\nabla\rangle u\big{\|}_{S^{0}(I_{j})}\lesssim$
$\displaystyle\|u(t_{j})\|_{H^{1}}+\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{q_{1}^{\prime}}L_{x}^{(r_{1})_{-}^{\prime}}(I_{j}\times\mathbb{R}^{n})}$
$\displaystyle\lesssim$
$\displaystyle\|u(t_{j})\|_{H^{1}}+\|u\|^{\alpha(p-1)}_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}(I_{j}\times\mathbb{R}^{n})}\big{\|}\langle\nabla\rangle
u\big{\|}^{\beta}_{S^{0}(I_{j})},$
where $(q_{1},(r_{1})_{-})\in\Lambda_{0}$ and $r_{1}$ is given in (4.1). Hence
we also obtain (5.12).
Finally, we utilize (5.12) to show asymptotic completeness. We need to prove
that there exist unique $u_{\pm}$ such that
$\lim_{t\to\pm\infty}\|u(t)-e^{itP_{a}}u_{\pm}\|_{H^{1}_{x}}=0,\quad
P_{a}=-\Delta+\tfrac{a}{|x|^{2}}.$
By time reversal symmetry, it suffices to prove this for positive times. For
$t>0$, we will show that $v(t):=e^{-itP_{a}}u(t)$ converges in $H^{1}_{x}$ as
$t\to+\infty$, and denote $u_{+}$ to be the limit. In fact, we obtain by
Duhamel’s formula
$v(t)=u_{0}-i\int_{0}^{t}e^{-i\tau P_{a}}(|u|^{p-1}u)(\tau)d\tau.$ (5.13)
Hence, for $0<t_{1}<t_{2}$, we have
$v(t_{2})-v(t_{1})=-i\int_{t_{1}}^{t_{2}}e^{-i\tau
P_{a}}(|u|^{p-1}u)(\tau)d\tau.$
Arguing as before, we deduce that for some $\alpha>0,\beta\geq 1$
$\displaystyle\|v(t_{2})-v(t_{1})\|_{H^{1}(\mathbb{R}^{n})}=$
$\displaystyle\Big{\|}\int_{t_{1}}^{t_{2}}e^{-i\tau
P_{a}}(|u|^{p-1}u)(\tau)d\tau\Big{\|}_{H^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}\langle\nabla\rangle(|u|^{p-1}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2n}{n+2}}([t_{1},t_{2}]\times\mathbb{R}^{n})}$
$\displaystyle\lesssim$
$\displaystyle\|u\|_{L_{t}^{n+1}L_{x}^{\frac{2(n+1)}{n-1}}([t_{1},t_{2}]\times\mathbb{R}^{n})}^{\alpha(p-1)}\big{\|}\langle\nabla\rangle
u\big{\|}^{\beta}_{S^{0}([t_{1},t_{2}])}$ $\displaystyle\to$ $\displaystyle
0\quad\text{as}\quad t_{1},~{}t_{2}\to+\infty.$
As $t$ tends to $+\infty$, the limitation of (5.13) is well defined. In
particular, we find the asymptotic state
$u_{+}=u_{0}-i\int_{0}^{\infty}e^{-i\tau P_{a}}(|u|^{p-1}u)(\tau)d\tau.$
Therefore, we conclude the proof of Theorem 1.1.
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|
arxiv-papers
| 2013-12-09T02:27:13 |
2024-09-04T02:49:55.154535
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junyong Zhang and Jiqiang Zheng",
"submitter": "Junyong Zhang",
"url": "https://arxiv.org/abs/1312.2294"
}
|
1312.2358
|
# Exact Recovery for Sparse Signal via Weighted $\ell_{1}$ Minimization
Shenglong Zhou, Naihua Xiu, Yingnan Wang, Lingchen Kong Dec 8, 2013.
Department of Applied Mathematics, Beijing Jiaotong University, Beijing
100044, P. R. China (e-mail: [email protected], [email protected],
[email protected], [email protected]). Revised at Jan 28, 2014.
###### Abstract
Numerical experiments in literature on compressed sensing have indicated that
the reweighted $\ell_{1}$ minimization performs exceptionally well in
recovering sparse signal. In this paper, we develop exact recovery conditions
and algorithm for sparse signal via weighted $\ell_{1}$ minimization from the
insight of the classical NSP (null space property) and RIC (restricted
isometry constant) bound. We first introduce the concept of WNSP (weighted
null space property) and reveal that it is a necessary and sufficient
condition for exact recovery. We then prove that the RIC bound by weighted
$\ell_{1}$ minimization is
$\delta_{ak}<\sqrt{\frac{a-1}{a-1+\gamma^{2}}},$
where $a>1$, $0<\gamma\leq 1$ is determined by an optimization problem over
the null space. When $\gamma<1$ this bound is greater than
$\sqrt{\frac{a-1}{a}}$ from $\ell_{1}$ minimization. In addition, we also
establish the bound on $\delta_{k}$ and show that it can be larger than the
sharp one $1/3$ via $\ell_{1}$ minimization and also greater than $0.4343$ via
weighted $\ell_{1}$ minimization under some mild cases. Finally, we achieve a
modified iterative reweighted $\ell_{1}$ minimization (MIRL1) algorithm based
on our selection principle of weight, and the numerical experiments
demonstrate that our algorithm behaves much better than $\ell_{1}$
minimization and iterative reweighted $\ell_{1}$ minimization (IRL1)
algorithm.
###### Index Terms:
compressed sensing, exact recovery, weighted $\ell_{1}$ minimization, null
space property, restricted isometry constant, MIRL1 algorithm
## I Introduction
With dramatic advances in technology in recent years, various research fields,
ranging from applied mathematics, computer science to engineering, have
involved to recover some original $n$-dimensional but sparse data (e.g.,
signals and images) from linear measurement with dimension far fewer than $n$.
This essential idea in terms of signal was first formulated as compressed
sensing (CS) by Donoho [12], Cand$\grave{\textmd{e}}$s, Romberg and Tao [8]
and Cand$\grave{\textmd{e}}$s and Tao [9]. Since then myriads of researchers
have been lured to this area as a consequence of its extensive applications in
signal processing, communications, astronomy, biology, medicine, seismology
and so forth, and thus brought fruitful theoretical results, see, e.g., survey
papers [2, 24] and monographs [14, 16, 23].
To acquire a sparse presentation $x\in\mathbb{R}^{n}$ of an underdetermined
system of the form $\Phi x=b$, where $b\in\mathbb{R}^{m}$ is the available
measurement and $\Phi\in\mathbb{R}^{m\times n}$ is a known measurement matrix
(with $m<n$ ), the underlying model is the following $\ell_{0}$ _minimization_
$\displaystyle\textup{min}~{}\|x\|_{0},~{}~{}\textup{s.t.}~{}\Phi x=b,$ (1)
where $\|x\|_{0}$ is $\ell_{0}$-norm of the vector $x\in\mathbb{R}^{n}$, i.e.,
the number of nonzero entries in $x$. Model (1) is a combinatorial
optimization problem with a prohibitive complexity if solved by enumeration,
and thus does not appear tractable.
One common alternative approach is to solve (1) via its convex _$\ell_{1}$
minimization_
$\displaystyle\textup{min}~{}\|x\|_{1},~{}~{}\textup{s.t.}~{}\Phi x=b.$ (2)
The use of $\ell_{1}$ relaxation has become so widespread that it could
arguably be considered the modern least squares , see, e.g., [2, 3, 4, 5, 6,
7, 19, 22, 25, 27, 29].
Inspired by the efficiency of $\ell_{1}$ minimization, it is natural to ask,
for example, whether a different (but perhaps again convex) alternative to
$\ell_{0}$ minimization might also find the correct solution, but with a lower
measurement requirement than $\ell_{1}$ minimization.
Earlier numerical experiments indicated that the reweighted $\ell_{1}$
minimization does outperform unweighted $\ell_{1}$ minimization in many
situations [10, 11, 16, 23, 27, 28]. Therefore, reweighted $\ell_{1}$
relaxation for model (1) in decade have drawn large numbers of researchers to
pay their attention on sparse signal recover due to its numerical
computational advantage.
Because of this, there have been many researchers concentrated on studying the
theoretical aspects of the weighted $\ell_{1}$ minimization [17, 21]. In this
paper, as a sequence, we also consider the theoretical properties of the
_weighted $\ell_{1}$ minimization_
$\displaystyle\textup{min}~{}\|\omega\circ x\|_{1},~{}~{}\textup{s.t.}~{}\Phi
x=b,$ (3)
where $\circ$ denotes the Hadamard product, that is $\|w\circ
x\|_{1}=\sum\omega_{i}|x_{i}|$, and $0<\omega_{i}\leq 1,~{}i=1,\cdots,n.$ Here
if we let $\omega$ as
$\displaystyle\omega_{i}=$ $\displaystyle 1-\epsilon,~{}~{}i\in T,$
$\displaystyle\omega_{i}=$ $\displaystyle 1,~{}~{}~{}~{}~{}~{}i\in T^{C},$
where $0<\epsilon<1$, $T$ is the subset of $\left\\{1,2,\cdots,n\right\\}$ and
$T^{C}$ notates the complementary set of $T$ in
$\left\\{1,2,\cdots,n\right\\}$, then (3) can be written as
$\displaystyle\textup{min}~{}\|x\|_{1}-\epsilon\|x_{T}\|_{1},~{}~{}\textup{s.t.}~{}\Phi
x=b,$ (4)
where $x_{T}\in\mathbb{R}^{n}$ denotes the vector equals to $x$ on an index
set $T$ and zero elsewhere. It is evident that model (4) is a specific form of
the difference of two convex functions programming (DC programming, see, e.g.,
[20]).
For the sake of convenience to illustrate, we can draw a picture (see, Fig I,
where $\Phi$ and $b$ are given as Example II.5) to comprehend the advantage of
weighted $\ell_{1}$ minimization what is absent in $\ell_{1}$ minimization.
[0,c,, Some cases that $\ell_{1}$ minimization will fail to recover the sparse
signal while exact recovery can be succeeded via weighted $\ell_{1}$
minimization. (a) Sparse signal $x^{(0)}=(0,0,2)^{T}$, feasible set $\Phi
x=b$, and in $\ell_{1}$ ball there exists an
$x^{(1)}=(\frac{3}{4},\frac{3}{4},0)^{T}$ but
$\|x^{(1)}\|_{0}>\|x^{(0)}\|_{0}$. (b) In weighted $\ell_{1}$ ball, there does
not exist an $x\neq x^{(0)}$ such that $\|x\|_{0}\leq\|x^{(0)}\|_{0}$.] Now
let us recollect the theoretical properties of the standard $\ell_{1}$
minimization (2). We know that the _null space property_ (NSP) is the
necessary and sufficient condition for (2) to reconstruct the system $b=\Phi
x$ exactly [13, 19, 26]. The NSP is recalled as follows.
###### Definition I.1 (NSP).
A matrix $\Phi\in\mathbb{R}^{m\times n}$ satisfies the null space property of
order $k$ if for all subsets $S\in\mathcal{C}_{n}^{k}$ it holds
$\displaystyle\left\|h_{S}\right\|_{1}<\left\|h_{S^{C}}\right\|_{1}$ (5)
for any $h\in\mathcal{N}(\Phi)\setminus\\{0\\}$, where
$\mathcal{N}(\Phi)=\\{h\in\mathbb{R}^{n}\large|~{}\Phi h=0\\}$ and
$\mathcal{C}_{n}^{k}=\left\\{S\subset\\{1,2,\cdots,n\\}~{}\large|~{}|S|=k\right\\}$
.
Another most popular sufficient condition for exact sparse recovery is related
to the _Restricted Isometry Property_ (RIP) originated by
Cand$\grave{\textmd{e}}$s and Tao [9].
###### Definition I.2 (RIP).
For $k\in\\{1,2,\cdots,n\\}$, the restricted isometry constant is the smallest
positive number $\delta_{k}$ such that
$\displaystyle(1-\delta_{k})\|x\|_{2}^{2}\leq\|\Phi
x\|_{2}^{2}\leq(1+\delta_{k})\|x\|_{2}^{2}$ (6)
holds for all $k$-sparse vector $x\in\mathbb{R}^{n}$, i.e., $\|x\|_{0}\leq k$.
Current upper bounds on the restricted isometry constants (RICs) via
$\ell_{1}$ minimizations for exact signal recovery were emerged in many
studies [1, 3, 5, 6, 7, 22, 29], such as $\delta_{2k}<0.5746$ jointly with
$\delta_{8k}<1$ [29], an improved bound $\delta_{2k}<\frac{4}{\sqrt{41}}$ [1],
sharp ones $\delta_{2k}<\frac{\sqrt{2}}{2}$ [7] and $\delta_{k}<\frac{1}{3}$
[5]. As for the weighted $\ell_{1}$ minimization, literature [17] presented us
the upper bound on $\delta_{k}$ might be $\delta_{k}<0.4343$ under some cases.
The main contributions of this paper are four aspects:
* •
The WNSP, one necessary and sufficient condition for exact recovery via the
weighted $\ell_{1}$ minimization, has been established, and then we comprehend
its weakness compared to the standard NSP by illustrating some examples.
* •
We then prove that the RIC bound by weighted $\ell_{1}$ minimization is
$\delta_{ak}<\sqrt{\frac{a-1}{a-1+\gamma^{2}}},$
where $a>1$, $0<\gamma\leq 1$ is determined by an optimization problem over
the null space of $\Phi$. When $\gamma<1$ this bound is greater than
$\sqrt{\frac{a-1}{a}}$ from $\ell_{1}$ minimization, which signifies that the
scale of the undetermined measurement matrices, satisfying the RIP to ensure
exact recovery via weighted $\ell_{1}$ minimization, is larger than those via
$\ell_{1}$ minimization.
* •
The bound on $\delta_{k}$ has been given as well, and the result shows that it
can be larger than the sharp one $\frac{1}{3}$ via $\ell_{1}$ minimization and
also greater than $0.4343$ under some mild cases.
* •
Finally, based on the RIC theory, we achieve a modified iterative reweighted
$\ell_{1}$ minimization (MIRL1) algorithm by establishing an effective way to
add the weights. The numerical experiments demonstrate our method behaves much
better than non-weighted $\ell_{1}$ minimization and iterative reweighted
$\ell_{1}$ minimization (MIRL1) algorithm.
The organization of this paper is as follows. In Section II, we establish the
necessary and sufficient condition for exact recovery via weighted $\ell_{1}$
minimization. And then by acquiring the upper bound on RIC, we set up another
sufficient condition and give some examples to illustrate our results in
Section III. The design of modified iterative reweighted $\ell_{1}$
minimization algorithm and numerical experiments will be presented in Section
IV. We make a conclusion in Section V and give all of proofs in the last
section.
## II Weighted Null Space Property
The _Null Space Property_ (NSP) is the necessary and sufficient condition for
relaxation (2) to exactly recover problem (1). We know that
$\mathcal{N}(\Phi)$ is a convex cone, also a subspace in $\mathbb{R}^{n}$,
which means we can concentrate all information on one of its bases. Here we
define a subset $\mathcal{N}_{\varsigma}$ from $\mathcal{N}(\Phi)$ by
$\displaystyle\mathcal{N}_{\varsigma}=\\{h\in\mathbb{R}^{n}\large|~{}h\in\mathcal{N}(\Phi),\|h\|_{1}=\varsigma\\},$
(7)
where $\varsigma>0$ and any $\mathcal{N}_{\varsigma}$ is a base of
$\mathcal{N}(\Phi)$. Since the fact of
$\mathcal{N}(\Phi)\setminus\\{0\\}=\underset{\varsigma>0}{\bigcup}\mathcal{N}_{\varsigma}$,
we can cast the NSP as follows.
###### Definition II.1.
A matrix $\Phi\in\mathbb{R}^{m\times n}$ satisfies the null space property of
order $k$ if for all subsets $S\in\mathcal{C}_{n}^{k}$ it holds
$\displaystyle\left\|h_{S}\right\|_{1}<\left\|h_{S^{C}}\right\|_{1}$ (8)
for any $h\in\mathcal{N}_{1}$.
###### Lemma II.2.
Definition _I.1_ is equivalent to Definition _II.1_.
Likewise, we give a _Weighted Null Space Property_ (WNSP) for the weighted
$\ell_{1}$ minimization (3). Actually, literature [21] has already shown us
the WNSP, here we will formulate it based on our Definition II.1.
###### Definition II.3 (WNSP).
For a given weight $\omega\in\mathbb{R}^{n}$, a matrix
$\Phi\in\mathbb{R}^{m\times n}$ satisfies the weighted null space property of
order $k$ if for all subsets $S\in\mathcal{C}_{n}^{k}$ it holds
$\displaystyle\left\|\omega\circ h_{S}\right\|_{1}<\left\|\omega\circ
h_{S^{C}}\right\|_{1}$ (9)
for any $h\in\mathcal{N}_{1}.$
Similarly, by Lemma II.2, the WNSP that is built up on subset
$\mathcal{N}_{1}$ also holds for the entire space
$\mathcal{N}(\Phi)\setminus\\{0\\}$. For clearness, we will concentrate all
sequent analysis on the subset $\mathcal{N}_{1}$ instead of
$\mathcal{N}(\Phi)\setminus\\{0\\}$. Based on the WNSP we have the following
recovery result linked to the weighted $\ell_{1}$ minimization.
###### Theorem II.4.
Every $k$-sparse vector $\hat{x}\in\mathbb{R}^{n}$ is the unique solution of
the weighted minimization $(\ref{lw})$ with $b=\Phi\hat{x}$ if and only if
$\Phi$ satisfies the WNSP of order $k$.
Now let us utilize two examples, which both satisfy the WNSP we defined while
does not content the NSP, to illustrate the WNSP is a weaker exact recovery
condition than the NSP.
###### Example II.5.
Let the measurement matrix $\Phi$ and observation vector $b$ be given as
$\Phi=\left(\begin{array}[]{ccc}4/5&0&3/10\\\ 0&4/5&3/10\\\
\end{array}\right),~{}~{}b=\left(\begin{array}[]{c}3/5\\\ 3/5\\\
\end{array}\right).$
Clearly, the unique solution of $\ell_{0}$ and $\ell_{1}$ minimizations are
$x^{(0)}=(0,0,2)^{T}$ and $x^{(1)}=(\frac{3}{4},\frac{3}{4},0)^{T}$
respectively. If setting
$\omega_{2}=\omega_{1},\omega_{3}<\frac{3}{4}\omega_{1}$ and
$\omega_{1}\in(0,1]$, we can verify that $x^{(0)}$ is also the unique solution
of the weighted $\ell_{1}$ minimization (For more clearness, one can see Fig
I).
For any $h=(h_{1},h_{2},h_{3})^{T}\in\mathcal{N}_{1}$, by directly
calculating, we have $h=(\frac{3}{8}h_{3},\frac{3}{8}h_{3},-h_{3})^{T}$ with
$h_{3}=4/7$ . Then for all subset $S\in\mathcal{C}_{3}^{1}$ and the given
$\omega$ it holds
$\displaystyle\left\|\omega\circ h_{S}\right\|_{1}<\left\|\omega\circ
h_{S^{C}}\right\|_{1}.$
From Theorem II.4, the weighted $\ell_{1}$ minimization can exactly recover
the sparsest solution of $\Phi x=b$. It is worth mentioning that this $\Phi$
does not satisfy the NSP due to
$|h_{3}|\nless|\frac{3}{4}h_{3}|=|h_{1}|+|h_{2}|$ and thus the standard
$\ell_{1}$ minimization will fail to exact recovery.∎
###### Example II.6.
Let the measurement matrix $\Phi$ and observation vector $b$ be given as
$\Phi=\left(\begin{array}[]{ccccc}3/4&-1/2&3/8&1/2&-1/4\\\
3/4&-1/2&-1/8&1/2&0\\\ 0&1/4&3/8&-1/8&-3/8\\\ \end{array}\right),$
and $b=\left(1/2,1/2,-1/8\right)^{T}$.
It is easy to verify the unique solution of $\ell_{0}$ and $\ell_{1}$
minimizations are $x^{(0)}=(0,0,0,1,0)^{T}$ and
$x^{(1)}=(\frac{1}{3},-\frac{1}{2},0,0,0)^{T}$ respectively. If setting
$\omega_{2}=\frac{2}{3}\omega_{1},\omega_{4}=\frac{1}{2}\omega_{1},\omega_{3}=\omega_{5}=\omega_{1}$
and $\omega_{1}\in(0,1]$, we can verify that $x^{(0)}$ is also the optimal
solution of the weighted $\ell_{1}$ minimization.
For any $h=(h_{1},h_{2},h_{3},h_{4},h_{5})^{T}\in\mathcal{N}_{1}$, by directly
calculating, $h$ with $\|h\|_{1}=1$ has the following formation
$\displaystyle
h=\left(\frac{-8h_{2}+13h_{5}}{12},h_{2},\frac{h_{5}}{2},\frac{4h_{2}-3h_{5}}{2},h_{5}\right)^{T}.$
Then for all subset $S\in\mathcal{C}_{5}^{1}$ and the given $\omega$ it holds
$\displaystyle\left\|\omega\circ h_{S}\right\|_{1}<\left\|\omega\circ
h_{S^{C}}\right\|_{1}.$
From Theorem II.4, the weighted $\ell_{1}$ minimization can exactly recover
the sparsest solution of $\Phi x=b$. It is worth mentioning that this $h$ does
not satisfy the NSP due to $|2h_{2}|\nless|\frac{2}{3}h_{2}|+|h_{2}|$ when
$|h_{5}|=0$ and thus the standard $\ell_{1}$ minimization will fail to exact
recovery.∎
## III Restricted Isometry Property
In this section, we will study a sufficient condition, _Restricted Isometry
Property_ (RIP), for the weighted $\ell_{1}$ minimization (3) to exactly
recover model (1). The first lemma about the sparse representation of a
polytope established by Cai and Zhang [7] will be very useful to prove our
result, whose description is recalled bellow.
###### Lemma III.1.
For a positive number $\alpha$ and a positive integer $s$, define the polytope
$T(\alpha,s)\subset\mathbb{R}^{n}$ by
$T(\alpha,s)=\left\\{v\in\mathbb{R}^{n}\large~{}|~{}\|v\|_{\infty}\leq\alpha,\|v\|_{1}\leq
s\alpha\right\\}.$
For any $v\in\mathbb{R}^{n}$, define the set
$U(\alpha,s,v)\subset\mathbb{R}^{n}$ of sparse vectors by
$\displaystyle
U(\alpha,s,v)=\\{u\in\mathbb{R}^{n}\large~{}|~{}{\emph{supp}}(u)\subseteq{\emph{supp}}(v),\|u\|_{0}\leq
s,$ $\displaystyle\|u\|_{1}=\|v\|_{1},\|u\|_{\infty}\leq\alpha\\}.$
Then $v\in T(\alpha,s)$ if and only if $v$ is in the convex hull of
$U(\alpha,s,v)$. In particular, any $v\in T(\alpha,s)$ can be expressed as
$v=\sum_{i=1}^{N}\lambda_{i}u_{i},$ where $N\geq 1$ is an integer and
$0\leq\lambda_{i}\leq 1,\sum_{i=1}^{N}\lambda_{i}=1,u_{i}\in
U(\alpha,s,v),i=1,2,\cdots,N.$
In order to analyze and acquire the upper bounds on RIC, we first design a way
of weighing and introduce some notations. We will see that the way of weighing
plays a crucial role in obtaining our main results in this section. Let
$T_{0}$ and $\widehat{h}$ be the optimal solution of the following model
$\displaystyle(T_{0},\widehat{h}):=\underset{T\in\mathcal{C}_{n}^{k},h\in\mathcal{N}_{1}}{\textmd{argmax}}\|h_{T}\|_{1}.$
(10)
For a constant $0<\gamma\leq 1$, we define $\omega$ based on $T_{0}$ as
$\displaystyle\omega_{i}=$ $\displaystyle\gamma,~{}~{}~{}i\in T_{0},$ (11)
$\displaystyle\omega_{i}=$ $\displaystyle 1,~{}~{}~{}i\in T_{0}^{C},$
where $T_{0}^{C}$ is the complementary set of $T_{0}$ in
$\left\\{1,2,\cdots,n\right\\}$.
From (10) and (11) we manage to decide the locations where the entries should
be added a weight $\gamma$, which implies that the way to define the weight
$\omega$, in a sense, give us a hint to acquire a meaningful and practical
weight to pursue the sparse solution, despite we can not easily value those
weights since (10) is a combinational optimization problem.
###### Lemma III.2.
Let $T_{0}$ and $\widehat{h}$ be defined as $(\ref{t0})$. If $T_{0}$ uniquely
exists, then there exists $\omega$ defined as $(\ref{wt0})$ with $0<\gamma<1$
such that
$\displaystyle\|\omega\circ\widehat{h}_{T_{0}}\|_{1}=\underset{T\in\mathcal{C}_{n}^{k},h\in\mathcal{N}_{1}}{\max}\|\omega\circ
h_{T}\|_{1}.$ (12)
If $T_{0}$ exists but not uniquely, then $\omega$ defined as $(\ref{wt0})$
with $\gamma=1$ satisfies $(\ref{maxw})$.
###### Lemma III.3.
Let $T_{0}$ and $\widehat{h}$ be defined by $(\ref{t0})$. For the given
$\omega$ as $(\ref{wt0})$, if
$\displaystyle\|\widehat{h}_{T_{0}^{C}}\|_{1}>\gamma\|\widehat{h}_{T_{0}}\|_{1}$
(13)
holds, then the WNSP of order $k$ is followed.
Now we give our result associated with getting the upper bounds on RICs:
$\delta_{ak}$ for some $a>1$ and $\delta_{k}$.
###### Theorem III.4.
For the given $\gamma$ and $\omega$ as $(\ref{t0})$ and $(\ref{wt0})$, if
$\displaystyle\delta_{ak}<\sqrt{\frac{a-1}{a-1+\gamma^{2}}}$ (14)
holds for some $a>1$, then each $k$ sparse minimizer $\hat{x}$ of the weighted
$\ell_{1}$ minimization $(\ref{lw})$ is the solution of $(\ref{l0})$.
From (14) we list TABLE I by taking different values of $\gamma.$
TABLE I: Bounds on $\delta_{2k},\delta_{3k}$ and $\delta_{4k}$ with different cases. $\gamma$ | $\delta_{2k}$ | $\delta_{3k}$ | $\delta_{4k}$
---|---|---|---
1 | $\sqrt{2}/2$ | $~{}~{}~{}\sqrt{6}/3$ | $\sqrt{3}/2$
3/4 | 0.800 | 0.883 | 0.917
1/2 | 0.894 | 0.942 | 0.960
1/4 | 0.970 | 0.984 | 0.989
###### Theorem III.5.
For the given $\gamma$ and $\omega$ as $(\ref{t0})$ and $(\ref{wt0})$, if
$\displaystyle\delta_{k}<$ $\displaystyle\frac{1}{1+2\lceil\gamma
k\rceil/k},~{}~{}~{}\text{for even number}~{}k\geq 2,$ (15)
$\displaystyle\delta_{k}<$ $\displaystyle\frac{1}{1+\frac{2\lceil\gamma
k\rceil}{\sqrt{k^{2}-1}}},~{}~{}~{}~{}~{}\text{for odd number}~{}k\geq 3,$
(16)
holds, where $\lceil a\rceil$ denotes the smallest integer that is no less
than $a$, then each $k$ sparse minimizer $\hat{x}$ of the weighted $\ell_{1}$
minimization $(\ref{lw})$ is the solution of $(\ref{l0})$.
From (15) and (16) above, we list different RIC bounds on $\delta_{k}$ in
TABLE II by setting various $\gamma$ and $k$. From the table one cannot
difficultly find that under some mild situation, the upper bounds are greater
than $0.4343$ in [17].
TABLE II: Bounds on $\delta_{k}$ with different cases. $\gamma$ | $k\geq 2$ is even | $k\geq 3$ is odd
---|---|---
1 | $1/3$ | $0.3203$
3/4 | $3/8~{}(k\geq 4)$ | $0.3797~{}(k\geq 5)$
1/2 | $1/2~{}(k\geq 2)$ | $\sqrt{6}-2~{}(k\geq 5)$
1/4 | $2/3~{}(k\geq 4)$ | $3-\sqrt{6}~{}(k\geq 5)$
1/6 | $3/4~{}(k\geq 6)$ | $0.7101~{}(k\geq 5)$
To end this section we present two examples to illustrate Theorem III.4, which
both result in $\ell_{1}$ minimization failing to recover the sparsest
solution of $\ell_{0}$ problem while successful recovery with the help of the
weighted $\ell_{1}$ minimization .
###### Example III.6.
We consider Example II.5 again.
The optimal solution of $\ell_{0}$ is $x^{(0)}=(0,0,2)^{T}$. The unique
solution of $\ell_{1}$ minimization is $x^{(1)}=(3/4,3/4,0)^{T}$. From
$h=(\frac{3}{8}h_{3},\frac{3}{8}h_{3},-h_{3})^{T}\in\mathcal{N}_{1}$ with
$h_{3}=\frac{4}{7}$, $|h_{3}|$ is the largest entry of $h$, i.e.
$T_{0}=\\{3\\}$ uniquely exists. Therefore by setting
$\frac{3}{8}<\omega_{3}=\gamma<0.418$ and $\omega_{1}=\omega_{2}=1$, we have
$\gamma\|h_{\\{3\\}}\|_{1}<\|h_{\\{1,2\\}}\|_{1}$, which means that $x^{(0)}$
is the unique solution of weighted $\ell_{1}$ minimization from Lemma III.3
and Theorem II.4.
On the other hand, we directly calculate that $\delta_{2}=0.9224$ with
$n=3,k=2$ by the following formula (see [22, 29])
$\displaystyle\delta_{k}=\max_{S\in\mathcal{C}_{n}^{k}}\|\Phi_{S}^{T}\Phi_{S}-I_{k}\|,$
(17)
where $\|\cdot\|$ denotes the spectral norm of a matrix. Since $T_{0}$
uniquely exists and $\gamma<0.418$, it yields $\delta_{2}<0.9226$ from (14) by
taking $a=2,k=1$. Hence the $\ell_{0}$ minimization can be exactly
reconstructed by the weighted $\ell_{1}$ minimization from our Theorem III.4.∎
###### Example III.7.
We consider Example _II.6_ again.
The optimal solution of $\ell_{0}$ is $x^{(0)}=(0,0,0,1,0)^{T}$. The unique
solution of $\ell_{1}$ minimization is
$x^{(1)}=(\frac{1}{3},-\frac{1}{2},0,0,0)^{T}$. Since for any
$h\in\mathcal{N}_{1}$, $h$ with $\|h\|_{1}=1$ has the formation
$\displaystyle
h=\left(-2h_{2}/3+13h_{5}/12,h_{2},h_{5}/2,2h_{2}-3h_{5}/2,h_{5}\right)^{T}.$
Simply calculating
$(T_{0},\widehat{h})=\underset{T\in\mathcal{C}_{5}^{1},h\in\mathcal{N}_{1}}{\textmd{argmax}}\|h_{T}\|_{1}$,
it follows that
$T_{0}=\\{4\\},~{}\widehat{h}=\left(-2h_{2}/3,h_{2},0,2h_{2},0\right)^{T},~{}h_{2}=6/11,$
which manifests that $T_{0}$ uniquely exists. Therefore by setting
$\omega_{4}=\gamma=0.3$ and $\omega_{1}=\omega_{2}=\omega_{3}=\omega_{5}=1$,
we have $\gamma\|h_{\\{4\\}}\|_{1}<\|h_{\\{1,2,3,5\\}}\|_{1}$, which means
that $x^{(0)}$ is the unique solution of weighted $\ell_{1}$ minimization from
Lemma III.3 and Theorem II.4.
On the other hand, we compute $\delta_{2}=0.9572$ by (17) with $n=5,k=2$.
Since $T_{0}$ uniquely exists and $\gamma=0.3$, it yields $\delta_{2}<0.9578$
from (14) by taking $a=2,k=1$. And thus the $\ell_{0}$ minimization can be
exactly recovered via the weighted $\ell_{1}$ minimization from Theorem
III.4.∎
To end this section, we will illustrate the rationality of the extra
assumption that _$T_{0}$ defined by $(\ref{t0})$ uniquely exists_, and the
relationships between WNSP and NSP, WNSP and RIP via constructing some
instances.
###### Remark III.8.
Although $T_{0}$ defined by $(\ref{t0})$ always exists but not uniquely
sometimes. However, from Examples _III.6_ and _III.7_ , we can see the
assumption that $T_{0}$ uniquely exists is actually not a strong assumption at
least to a certain extent. Therefore our assumption is meaningful to achieve
the goal of pursuing the sparse solution exactly.
###### Remark III.9.
_i)_ WNSP is evidently an extension of NSP, and thus it is a weaker condition
than NSP for exact revoery;
_ii)_ For some measurement matrices $\Phi$, there might be lots of $\omega$
satisfying WNSP but exist relatively fewer numbers of $\omega$ contenting
$(\ref{delta1})$, which manifests the condition $(\ref{delta1})$ is stronger
than $(\ref{wnsp})$ from WNSP.
We draw a graphic to illustrate the relationship between WNSP, NSP and RIP
based on the statements above.
[0,c,,The relationship between WNSP, NSP and RIP, the dashed area denotes the
scale of matrices that satisfy the RIP via weighted $\ell_{1}$ minimization. ]
## IV Numerical Experiments
In this section, we will propose a modified iterative reweighted $\ell_{1}$
minimization (MIRL1) algorithm, where the weights are designed based on the
theoretical results on the null space of $\Phi$. Simulation tests and signal
experiments are also provided.
### IV-A Modified Iterative Reweighted $\ell_{1}$ Minimization
Considering the following formula:
$\displaystyle\textup{min}~{}\frac{1}{2}\|\Phi x-b\|_{2}^{2}+\mu\|\omega\circ
x\|_{1}:=f(x),$ (18)
where $\mu>0$ is a penalty parameter. Let $L\geq\lambda_{\max}(\Phi^{T}\Phi)$.
Then for any $x,y\in\mathbb{R}^{n}$, we have
$\displaystyle\frac{1}{2}\|\Phi x-b\|_{2}^{2}+\mu\|\omega\circ x\|_{1}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\|\Phi
y-b\|_{2}^{2}+\langle\Phi^{T}(\Phi y-b),x-y\rangle+\frac{L}{2}\|x-y\|_{2}^{2}$
$\displaystyle+\mu\|\omega\circ x\|_{1}$ $\displaystyle:=$ $\displaystyle
F(x,y)$
Evidently, for any $x,y\in\mathbb{R}^{n}$, we have
$F(x,y)\geq f(x)~{}~{}\text{and }~{}~{}F(x,x)=f(x),$
which means that $F$ is a majorization of $f$. Using this majorization
function, we start with an initial iteration $x^{0}$ and update $x^{t}$ by
solving
$\displaystyle x^{t+1}=\textup{argmin}_{x\in\mathbb{R}^{n}}~{}F(x,x^{t}),$
(19)
which is equivalent to
$\displaystyle x^{t+1}$ $\displaystyle=$
$\displaystyle\textup{argmin}_{x\in\mathbb{R}^{n}}~{}\frac{L}{2}\|x-\widetilde{x}^{t}\|_{2}^{2}+\mu\|\omega\circ
x\|_{1}$ (20) $\displaystyle=$
$\displaystyle\textmd{sign}(\widetilde{x}^{t})\circ\max\left\\{|\widetilde{x}^{t}|-\frac{\mu}{L}\omega,0\right\\}$
where
$\widetilde{x}^{t}:=x^{t}-\frac{1}{L}\Phi^{T}(\Phi x^{t}-b),$
$|x|=(|x_{1}|,|x_{2}|,\cdots,|x_{n}|)^{T}$ and $\textmd{sign}(x)$ denotes the
signum function of $x$ . Here we need to indicate how to define the weight
$\omega$. As we mentioned in Section III, since the weight $\omega$ is
depended on the null space of $\Phi$, we take $T^{\tau}$ and $\omega^{\tau}$
as
$\displaystyle T^{\tau}$ $\displaystyle=$
$\displaystyle\textup{argmax}_{T\in\mathcal{C}_{n}^{k^{\tau}}}~{}\|(h^{\tau})_{T}\|_{1},~{}~{}\tau=1,2,\cdots$
(21) $\displaystyle\omega_{i}^{\tau}=$
$\displaystyle\left[\frac{|h^{\tau}_{i}|+\epsilon}{\max_{j\in(T^{\tau})^{C}}|h^{\tau}_{j}|}\right]^{q-1}~{}~{}~{}~{}~{},i\in
T^{\tau},~{}~{}$ (22) $\displaystyle\omega_{i}^{\tau}=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},i\in(T^{\tau})^{C},$
(23)
where
$h^{\tau}=x^{\tau}-x^{\tau-1},~{}~{}~{}~{}k^{\tau}=|\textmd{supp}(x^{\tau})|$
and $0<q\leq 1,\epsilon>0$ is sufficiently small.
###### Remark IV.1.
We simply interpret the weights as $(\ref{Tk})$–$(\ref{wk2})$. Simply
verifying from $(\ref{Tk})$–$(\ref{wk2})$, we have
$|h^{\tau}_{i}|\geq|h^{\tau}_{j}|,\forall~{}i\in
T^{\tau},\forall~{}j\in(T^{\tau})^{C}$, and thus
$|h^{\tau}_{i}|+\epsilon>\max_{j\in(T^{\tau})^{C}}|h^{\tau}_{j}|,\forall~{}i\in
T^{\tau},$ which indicates
$0\leq\left[\frac{|h^{\tau}_{i}|+\epsilon}{\max_{j\in(T^{\tau})^{C}}|h^{\tau}_{j}|}\right]^{q-1}<1,~{}~{}~{}\forall~{}i\in
T^{\tau}.$
###### Remark IV.2.
To the best of our knowledge, the weights given by $(\ref{Tk})$–$(\ref{wk2})$
are different from those in the existing literature, see, e.g., [10, 11, 15,
23, 28]. By partitioning the index set into parts $T^{\tau}$ and
$(T^{\tau})^{C}$ based on $h^{\tau}$, we endow the entries in two parts with
corresponding weights. Moreover, we give weights $\omega^{\tau}$ from
$h^{\tau}$ and no longer directly utilize $x^{\tau}$ to value the weight like
$\omega_{i}^{\tau+1}=\frac{1}{|x_{i}^{\tau}|+\epsilon}$ in [10] or
$\omega_{i}^{\tau+1}=\frac{1}{(|x_{i}^{\tau}|+\epsilon)^{1-q}}$, $q\in(0,1)$
in [15], which can be uniformly written as
$\displaystyle\omega_{i}^{\tau+1}=\left[\frac{1}{|x_{i}^{\tau}|+\epsilon}\right]^{1-q},~{}~{}~{}~{}q\in[0,1).$
(24)
Recall the well-known iterative reweighted $\ell_{1}$ minimization algorithm
(IRL1) [10], we present the algorithm framework of our proposed modified
version in TABLE III.
TABLE III: The framework of MIRL1 . Modified Iterative Reweighted $\ell_{1}$
Minimization (MIRL1)
---
Initialize $x^{0},\omega^{1},M,\mu^{1}$ and
$L\geq\sigma_{\max}(\Phi^{T}\Phi)$;
For $\tau$=1: M
Initialize $x^{\tau,1}=x^{\tau-1}$;
While
$\|x^{\tau,t+1}-x^{\tau,t}\|_{2}\geq\eta^{\tau}\max\\{1,\|x^{\tau,t}\|_{2}\\}$
$\widetilde{x}^{\tau,t}=x^{\tau,t}-\frac{1}{L}\Phi^{T}(\Phi x^{\tau,t}-b)$;
$x^{\tau,t+1}=\textmd{sign}\left(\widetilde{x}^{\tau,t}\right)\circ\max\left\\{|\widetilde{x}^{\tau,t}|-\frac{\mu^{\tau}}{L}\omega^{\tau},0\right\\}$.
End
Update $x^{\tau}=x^{\tau,t+1}$;
Update $\omega^{\tau+1}$ from $x^{\tau-1},x^{\tau}$ based on (21), (22) and
(23);
End
Evidently, the framework of MIRL1 will go back to that of iterative reweighted
$\ell_{1}$ minimization (IRL1) algorithm or iterative $\ell_{1}$ minimization
(IL1) algorithm if we update $\omega^{\tau+1}$ based on (24) or
$\omega^{\tau+1}=(1,1,\cdots,1)^{T}$, respectively.
### IV-B Computational Results: Exact Recovery
We first consider the recovery without noise (exact recovery):
$y=\Phi x.$
Before proceeding to the computational results, we need to define some
notations and data sets. For convenience and clear understanding in the graph
presentations and some comments, we use the notations: $L_{1}$, $WL_{1}$,
$ML_{1}$ to represent the IL1 (namely derived from updating
$\omega^{\tau+1}=(1,1,\cdots,1)^{T}$ in the framework), the IRL1 and MIRL1
respectively. Since the weight $\omega$ in (22)–(23) and (24) is associated
with the parameter $q\in(0,1)$, we shortly write the methods as
$WL_{1}(q=\varsigma)$ and $ML_{1}(q=\varsigma)$, particularly taking
$\varsigma=0.1,0.25,0.5,0.6,0.75,0.9$ in our whole numerical experiments. For
each data set, the random matrix $\Phi$ and vector $b$ are generated by the
following matlab codes:
$\displaystyle
x_{\textmd{orig}}=\textmd{zeros}(n,1),~{}~{}~{}~{}~{}y=\textmd{randperm}(n),$
$\displaystyle x_{\textmd{orig}}(y(1:k))=\textmd{randn}(k,1),$
$\displaystyle\Phi=\textmd{randn}(m,n),~{}~{}~{}~{}~{}~{}b=\Phi*x_{\textmd{orig}}.$
The stopping criterias for the inner loops in our method are given by
$\eta^{\tau}=\mu^{\tau}10^{-4},$ where parameters
$\mu^{\tau}(\tau=1,2\cdots,8)$ are taken by
$\mu^{\tau}\in\left\\{1,1/5,(1/5)^{2},\cdots,(1/5)^{6},(1/5)^{7}\right\\}\cdot\left\|\Phi^{T}b\right\|_{\infty},$
which implies $\mu^{1}=(1/5)^{0}=1$ and $M=8$. We always initialize the start
points
$x^{0}=\textrm{ones}(n,1),~{}~{}~{}~{}~{}~{}~{}\omega^{1}=\textrm{ones}(n,1),~{}~{}~{}~{}~{}~{}~{}\epsilon=10^{-4}.$
Figure 1: Sparsity yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$.
Figure 2: Sparsity yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$.
For each fixed $q=0.1,0.25,0.50.6,0.75,0.9$, we randomly generate $20$ samples
and respectively apply IL1, IRL1 and MIRL1 algorithms to problem (18). From
Figs IV-B and IV-B, the red lines and the blue ’$+$’s stand for the sparsity
of $x_{\textmd{orig}}$ and recovered solutions, respectively. One can not be
difficult to see the comments below.
* •
For any $q$, sparsity of the optimal solutions derived from MRIL1 is closer
(almost equal) to the true sparsity than that from RIL1 and IL1.
* •
When $q=0.1,0.25$, RIL1 performs relatively bad (also see TABLE IV) while IL1
and MRIL1 still works steadily. Then with the increasing of
$q(=0.5,0.6,0.75,0.9)$, although there occasionally appears some bad cases,
under such circumstance RIL1 and MRIL1 perform moderately better than IL1.
* •
Since there is no restriction on the CPU time to run the algorithms, IL1 has
cost much more time than RIL1 and MRIL1, which contributes to obtaining the
solutions whose sparsity is equal to the true one. Hence, that is reasonable
for some $q$ the recovery effect is better for IL1 than that of RIL1.
TABLE IV: Sparsity yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.5,0.9$. Sample | $m=512,n=2048$, True Sparsity$=85$
---|---
$L_{1}$ | $q=0.1$ | $q=0.5$ | $q=0.9$
$WL_{1}$ | $ML_{1}$ | $WL_{1}$ | $ML_{1}$ | $WL_{1}$ | $ML_{1}$
1 | 85 | 85 | 85 | 85 | 85 | 85 | 85
2 | 85 | 85 | 85 | 85 | 85 | 85 | 85
3 | 85 | 84 | 85 | 85 | 85 | 85 | 85
4 | 85 | 85 | 85 | 85 | 85 | 85 | 85
5 | 85 | 83 | 85 | 85 | 85 | 85 | 85
6 | 85 | 83 | 85 | 85 | 85 | 85 | 85
7 | 85 | 85 | 85 | 84 | 85 | 85 | 85
8 | 85 | 85 | 85 | 85 | 85 | 85 | 85
9 | 85 | 84 | 85 | 85 | 85 | 85 | 85
10 | 85 | 84 | 85 | 84 | 85 | 85 | 85
| Average error $\|\Phi x-b\|_{2}$
($10^{-4}$) | 12 | 7972 | 7 | 54 | 7 | 11 | 7
| Average error $\|x-x_{\textmd{org}}\|_{2}$
($10^{-4}$) | 0.35 | 213 | 0.1 | 2 | 0.11 | 0.15 | 0.11
| Average CPU time
(second) | 5.67 | 3.00 | 3.69 | 3.14 | 3.38 | 3.17 | 3.16
Figure 3: Error yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$ . Figure
4: Error yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$.
From Figs IV-B–IV-B, several comments can be derived.
* •
For any $q$, the average errors $\|\Phi x-b\|_{2}$ (or
$\|x-x_{\textmd{orig}}\|_{\infty}$) by MRIL1 are basically smaller than those
of IL1 and RIL1; Particularly, when $q=0.1$ or $0.25$, the average errors are
much higher than others from MRIL1 and even from IL1. However with $q$ being
no less than $0.5$, the average errors $\|x-x_{\textmd{orig}}\|_{\infty}$
almost become lower than IL1 but still higher than MRIL1;
* •
For RIL1, different $q$ would lead to some fluctuations of the average errors
$\|\Phi x-b\|_{2}$ (or $\|x-x_{\textmd{orig}}\|_{\infty}$) which likely become
intense with $n$ increasing, whilst MRIL1 would generate relatively stable
errors’ fluctuations regardless of $q$;
* •
For any $q$, Figs IV-B and IV-B see upward trends of the average errors from
RIL1 with the ascend of $n$, whereas for any $q$ one can find that there are
the downward trends of the average errors $\|x-x_{\textmd{orig}}\|_{\infty}$
resulted from MRIL1 when $n$ increases.
* •
The average CPU time generated by MRIL1 and RIL1 are basically equal, which
are all much shorter than those from IL1. More specifically, all of them
increase with the rise of dimension $n$, and the time spent by MRIL1 is
slightly greater than that of RIL1 probably due to the computation of the
weight (22)–(23).
Figure 5: Time yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$.
Figure 6: Time yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$.
From Figs IV-B–IV-B and TABLE V, one can conclude the following comments.
* •
For any $q\in\\{0.1,0.25,0.50.6,0.75,0.9\\}$, the average errors $\|\Phi
x-b\|_{2}$ (or $\|x-x_{\textmd{orig}}\|_{\infty}$) by MRIL1 are quite small
(almost reach from $10^{-3}$ to $10^{-5}$), which are much lower than those
from IRL1 (most of whose values are greater than $10^{-3}$).
* •
Errors $\|\Phi x-b\|_{2}$ and $\|x-x_{\textmd{orig}}\|_{\infty}$ are basically
equal for each $q$ when $n$ is fixed; In addition, the former increase while
the latter decrease with the dimension $n$ rising;
* •
From TABLE V, it is not of difficulty to see that our approach runs very fast,
particularly when the sparsity $k=0.01n$, only 34.60 second is needed to
pursue the sparse solution.
Figure 7: Error $\|\Phi x-b\|_{2}$ and $\|x-x_{\textmd{orig}}\|_{\infty}$
yielded by MIRL1. Figure 8: Error $\|\Phi x-b\|_{2}$ and
$\|x-x_{\textmd{orig}}\|_{\infty}$ yielded by IRL1.
TABLE V: Average error and CPU time yielded by MIRL1 without noise . | $n$ | $\|\Phi x-b\|_{2}$ | $\|x-x_{\textmd{org}}\|_{2}$ | CPU time
---|---|---|---|---
$k=0.05n$ | 1280 | 4.02e-04 | 1.17e-05 | 1.40
5120 | 1.70e-03 | 1.09e-05 | 13.71
7680 | 2.10e-03 | 8.70e-06 | 29.45
10240 | 3.40e-03 | 1.09e-05 | 52.30
$k=0.01n$ | 1280 | 1.94e-04 | 5.52e-06 | 0.555
5120 | 7.99e-04 | 5.86e-06 | 8.634
7680 | 1.36e-03 | 6.85e-06 | 18.80
10240 | 1.34e-03 | 5.03e-06 | 34.60
### IV-C Computational Results: Recovery with Noise
We now consider the recovery with noise:
$y=\Phi x+\xi,$
where the noise $\xi$ obeys the normal distribution with zero expectation and
$\sigma^{2}$ variance, namely $\xi\sim N(0,\sigma^{2})$. Here we take
$\sigma=0.01$. Under the noise case, from Figs IV-C and IV-C, one can not be
difficult to see the comments below.
* •
For any $q$, sparsity derived from MRIL1 is closer to the true sparsity than
that from RIL1 and IL1.
* •
When $q=0.1,0.25,0.5,0.6$, the results from RIL1 are excessively sparse, and
then $q=0.75,0.9$, RIL1 begins to perform as well as the MRIL1 which always
performs steadily well. However IL1 always does not obtain the true sparsity.
Figure 9: Sparsity yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$.
Figure 10: Sparsity yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$.
From Figs IV-C–IV-C, several comments can be derived.
* •
For any $q$, the average errors $\|\Phi x-b\|_{2}$ (or
$\|x-x_{\textmd{orig}}\|_{\infty}$) by MRIL1 are smaller than those of IL1 and
RIL1; Particularly, when $q=0.9$, IRL1, MRIL1 and IL1 basically proceed
identically well, which indicates RIL1 method is overly dependent on the
parameter $q$;
* •
The average CPU time generated by RIL1 are smallest, and close behind is the
MRIL1 for any $q$. IL1 costs the longest time to pursue the optimal solutions.
Figure 11: Error yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$ . Figure
12: Error yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$. Figure 13:
Time yielded by IL1, IRL1 and MIRL1 when $q=0.1,0.25,0.5$.
Figure 14: Time yielded by IL1, IRL1 and MIRL1 when $q=0.6,0.75,0.9$.
## V Conclusion
In this paper, we have established weighted null space property and RIC bounds
through the weighted $\ell_{1}$ minimization for exact sparse recovery. The
upper bounds on RICs from the weighted $\ell_{1}$ minimization are better in
some cases than the current results from $\ell_{1}$ minimization, and moreover
the way presented in this paper, in a sense, gives us a hint to construct the
weight to pursue the sparse solution. As a consequence, these results
strengthen the theoretical foundation of the reweighted $\ell_{1}$
minimization approach utilized extensively in sparse signal recovery.
Moreover, the proposed method based on our new RIC theory provides an
effective access to locate the none zero entries of the original sparse
solution.
## Acknowledgement
The work was supported in part by the National Basic Research Program of China
(2010CB732501), and the National Natural Science Foundation of China
(11171018, 71271021).
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## VI Appendix
Proof of Lemma II.2
On one hand, it is obvious for Definition I.1 to get Definition II.1. On the
other hand, if $\Phi$ satisfies the null space property defined by Definition
II.1, that is, for all subsets $S\in\mathcal{C}_{n}^{k}$ it holds
$\displaystyle\left\|h_{S}\right\|_{1}<\left\|h_{S^{C}}\right\|_{1}$
for any $h\in\mathcal{N}_{1}$. For any
$h^{\prime}\in\mathcal{N}(\Phi)\setminus\\{0\\}$ with
$\|h^{\prime}\|_{1}=\varsigma>0$, we have
$\|\frac{h^{\prime}}{\varsigma}\|_{1}\in\mathcal{N}_{1}$ and thus
$\displaystyle\left\|\left(h^{\prime}/\varsigma\right)_{S}\right\|_{1}<\left\|\left(h^{\prime}/\varsigma\right)_{S^{C}}\right\|_{1}$
which is equal to
$\displaystyle\left\|h^{\prime}_{S}\right\|_{1}<\left\|h^{\prime}_{S^{C}}\right\|_{1}.$
Henceforth, $\Phi$ also satisfies the null space property defined by
Definition I.1.∎
Proof of Theorem II.4
(Sufficiency) Let us assume the WNSP of order $k$ holds. For a given $\omega$,
any $h\in\mathcal{N}(\Phi)$ with $\|h\|_{1}=\varsigma>0$ and all subsets
$S\in\mathcal{C}_{n}^{k}$, from (9) it follows that
$\left\|\omega\circ(h/\varsigma)_{S}\right\|_{1}<\left\|\omega\circ(h/\varsigma)_{S^{C}}\right\|_{1}$,
which is equivalent to
$\displaystyle\left\|\omega\circ h_{S}\right\|_{1}<\left\|\omega\circ
h_{S^{C}}\right\|_{1}.$ (25)
Hence, for any $k$-sparse vector $\hat{x}\in\mathbb{R}^{n}$,
$h\in\mathcal{N}(\Phi)$ and all subsets $S\in\mathcal{C}_{n}^{k}$, from (25)
we obtain,
$\displaystyle 0$ $\displaystyle<$ $\displaystyle\sum_{i\in
S^{C}}\omega_{i}|h_{i}|-\sum_{i\in S}\omega_{i}|h_{i}|$ $\displaystyle\leq$
$\displaystyle\sum_{i\in S^{C}}\omega_{i}|h_{i}|+\sum_{i\in
S}\omega_{i}\left(|\hat{x}_{i}+h_{i}|-|\hat{x}_{i}|\right).$
Since $\widehat{S}:=(\textmd{supp}(\hat{x})^{T},0)^{T}\in\mathcal{C}_{n}^{k}$,
together with the inequality above, we have
$\displaystyle\|\omega\circ\hat{x}\|_{1}$ $\displaystyle=$
$\displaystyle\sum_{i\in\widehat{S}}\omega_{i}|\hat{x}_{i}|$ $\displaystyle<$
$\displaystyle\sum_{i\in\widehat{S}^{C}}\omega_{i}|h_{i}|+\sum_{i\in\widehat{S}}\omega_{i}|\hat{x}_{i}+h_{i}|$
$\displaystyle=$ $\displaystyle\|\omega\circ(\hat{x}+h)\|_{1}.$
This established the required minimality of $\|\omega\circ x\|_{1}$.
(Necessity) Assume every $k$-sparse vector $\hat{x}\in\mathbb{R}^{n}$ is the
unique solution of $\|\omega\circ x\|_{1}$ subject to $\Phi x=\Phi\hat{x}$.
Then, in particular, for any $h\in\mathcal{N}_{1}$ and all subsets
$S\in\mathcal{C}_{n}^{k}$, the $k$-sparse vector $h_{S}$ is the unique
solution of $\|\omega\circ x\|_{1}$ subject to $\Phi x=\Phi h_{S}$. Since
$\Phi h=0$, we have $\Phi h_{S}=\Phi(-h_{S^{C}})$, which means that
$\displaystyle\left\|\omega\circ h_{S}\right\|_{1}<\left\|\omega\circ
h_{S^{C}}\right\|_{1}.$ (26)
The whole proof is completed immediately. ∎
Proof of Lemma III.2
If $T_{0}$ defined as $(\ref{t0})$ uniquely exists, by denoting
$T_{1}\in\mathcal{C}_{n}^{k}\setminus\\{T_{0}\\}$ as
$\displaystyle(T_{1},\widetilde{h}):=\underset{T\in\mathcal{C}_{n}^{k}\setminus\\{T_{0}\\},h\in\mathcal{N}_{1}}{\textmd{argmax}}\|h_{T}\|_{1},$
and taking
$0<\frac{\left\|\widetilde{h}_{T_{1}}\right\|_{1}}{\left\|\widehat{h}_{T_{0}}\right\|_{1}}<\gamma<1$
by the uniqueness of $T_{0}$,
$\|\omega\circ\widehat{h}_{T_{0}}\|_{1}=\gamma\|\widehat{h}_{T_{0}}\|_{1}>\|\widetilde{h}_{T_{1}}\|_{1}\geq\left\|h_{T}\right\|_{1}\geq\left\|\omega\circ
h_{T}\right\|_{1}$
holds for any $h\in\mathcal{N}_{1}$ and any
$T\in\mathcal{C}_{n}^{k}\setminus\\{T_{0}\\}$.
If $T_{0}$ exists but not uniquely, it is evident that $\omega$ defined as
$(\ref{wt0})$ with $\gamma=1$ satisfies $(\ref{maxw})$.∎
Proof of Lemma III.3
First we show the following fact based on our notations
$\displaystyle\|\omega\circ\widehat{h}\|_{1}=\underset{h\in\mathcal{N}_{1}}{\min}\|\omega\circ
h\|_{1}.$ (27)
Since $(\ref{wt0})$ with $0<\gamma\leq 1$, for any $h\in\mathcal{N}_{1}$ we
have
$\displaystyle\|\omega\circ\widehat{h}\|_{1}$ $\displaystyle=$
$\displaystyle\|\omega\circ\widehat{h}_{T_{0}}\|_{1}+\|\widehat{h}_{T_{0}^{C}}\|_{1}$
$\displaystyle=$
$\displaystyle\gamma\|\widehat{h}_{T_{0}}\|_{1}+1-\|\widehat{h}_{T_{0}}\|_{1}$
$\displaystyle\leq$ $\displaystyle(\gamma-1)\|h_{T_{0}}\|_{1}+1$
$\displaystyle=$
$\displaystyle(\gamma-1)\|h_{T_{0}}\|_{1}+\|h_{T_{0}}\|_{1}+\|h_{T_{0}^{C}}\|_{1}$
$\displaystyle=$ $\displaystyle\|\omega\circ
h_{T_{0}}\|_{1}+\|h_{T_{0}^{C}}\|_{1}$ $\displaystyle=$
$\displaystyle\|\omega\circ h\|_{1},$
where the first inequality is resulted from (10).
Then to prove WNSP, namely to show
$\displaystyle\left\|\omega\circ h_{T^{C}}\right\|_{1}>\left\|\omega\circ
h_{T}\right\|_{1}$
holds for any $h\in\mathcal{N}_{1}$ and any $T\in\mathcal{C}_{n}^{k}$. By the
definition of $T_{0}$ in (10), if (13) holds, that is
$\displaystyle\|\omega\circ\widehat{h}_{T_{0}^{C}}\|_{1}=\|\widehat{h}_{T_{0}^{C}}\|_{1}>\gamma\|\widehat{h}_{T_{0}}\|_{1}=\|\omega\circ\widehat{h}_{T_{0}}\|_{1},$
then for any $h\in\mathcal{N}_{1}$ and any $T\in\mathcal{C}_{n}^{k}$, we have
$\displaystyle\left\|\omega\circ h_{T^{C}}\right\|_{1}$ $\displaystyle=$
$\displaystyle\|\omega\circ h\|_{1}-\left\|\omega\circ h_{T}\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\|\omega\circ\widehat{h}\|_{1}-\left\|\omega\circ
h_{T}\right\|_{1}$ $\displaystyle=$
$\displaystyle\|\omega\circ\widehat{h}_{T_{0}}\|_{1}+\|\omega\circ\widehat{h}_{T_{0}^{C}}\|_{1}-\left\|\omega\circ
h_{T}\right\|_{1}$ $\displaystyle>$ $\displaystyle
2\|\omega\circ\widehat{h}_{T_{0}}\|_{1}-\left\|\omega\circ h_{T}\right\|_{1}$
$\displaystyle>$ $\displaystyle\|\omega\circ h_{T}\|_{1},$
the first and last inequalities follow from (27) and Lemma III.2
respectively.∎
Proof of Theorem III.4
From Lemma III.3 and Theorem II.4, to pursue WNSP, we only need to check
$(\ref{t0nsp})$, that is
$\displaystyle\|\widehat{h}_{T_{0}^{C}}\|_{1}>\gamma\|\widehat{h}_{T_{0}}\|_{1}.$
For simplicity we shortly denote hereafter $h=\widehat{h}$, from above
inequality we suppose on the contrary that
$\displaystyle\|h_{T_{0}^{C}}\|_{1}\leq\gamma\left\|h_{T_{0}}\right\|_{1}.$
By setting $\beta:=\left\|h_{T_{0}}\right\|_{1}/k$, then we have
$\|h_{T_{0}^{C}}\|_{1}\leq\gamma k\beta.$
We now divide $h_{T_{0}^{C}}$ into two parts, $h_{T_{0}^{C}}=h^{(1)}+h^{(2)}$,
where
$\displaystyle h^{(1)}_{i}=$
$\displaystyle(h_{T_{0}^{C}})_{i},~{}~{}~{}|(h_{T_{0}^{C}})_{i}|>\beta/t,$
$\displaystyle h^{(1)}_{i}=$ $\displaystyle
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{otherwise},$ $\displaystyle
h^{(2)}_{i}=$
$\displaystyle(h_{T_{0}^{C}})_{i},~{}~{}~{}|(h_{T_{0}^{C}})_{i}|\leq\beta/t,$
$\displaystyle h^{(2)}_{i}=$ $\displaystyle
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{otherwise},$
and $t>0$ satisfies $\gamma kt$ being an integer. Therefore $h^{(1)}$ is
$\gamma kt$-sparse as a result of facts that
$\|h^{(1)}\|_{1}\leq\|h_{T_{0}^{C}}\|_{1}\leq\gamma k\beta$ and all non-zero
entries of $h^{(1)}$ has magnitude larger than $\frac{\beta}{t}$. By letting
$\|h^{(1)}\|_{0}=m$, then it produces
$\displaystyle\|h^{(2)}\|_{1}=\|h_{\overline{}T_{0}^{C}}\|_{1}-\|h^{(1)}\|_{1}\leq\left[\gamma
kt-m\right]\beta/t,~{}~{}~{}~{}~{}~{}$ (28)
$\displaystyle\|h^{(2)}\|_{\infty}\leq\beta/t.$ (29)
Applying Lemma III.1 with $s=\gamma kt-m$, it makes $h^{(2)}$ be expressed as
a convex combination of sparse vectors, i.e.,
$h^{(2)}=\sum_{i=1}^{N}\lambda_{i}u_{i},$
where $u_{i}$ is $(\gamma kt-m)$-sparse,
$\|u_{i}\|_{1}=\|h^{(2)}\|_{1},\|u_{i}\|_{\infty}\leq\beta/t,~{}i=1,2,\cdots,N$.
Henceforth,
$\displaystyle\|u_{i}\|_{2}^{2}$ $\displaystyle\leq$ $\displaystyle(\gamma
kt-m)\|u_{i}\|_{\infty}^{2}\leq\frac{\gamma k}{t}\beta^{2}$ (30)
$\displaystyle\leq$
$\displaystyle\frac{\gamma}{t}\|h_{T_{0}}\|_{2}^{2}\leq\frac{\gamma}{t}\|h_{T_{0}}+h^{(1)}\|_{2}^{2},$
where the third and last inequalities are as the consequences of the
$\|h_{T_{0}}\|_{1}\leq\sqrt{k}\|h_{T_{0}}\|_{2}$, and disjoint supports of
$h_{T_{0}}$ and $h^{(1)}$ respectively.
For any $\mu\geq 0$, denoting $\eta_{i}=h_{T_{0}}+h^{(1)}+\mu u_{i}$, we
obtain
$\displaystyle\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\eta_{i}/2$ (31)
$\displaystyle=$ $\displaystyle h_{T_{0}}+h^{(1)}+\mu h^{(2)}-\eta_{i}/2$
$\displaystyle=$
$\displaystyle(\frac{1}{2}-\mu)\left(h_{T_{0}}+h^{(1)}\right)-\mu u_{i}/2+\mu
h,$
where $\eta_{i},\sum_{i=j}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i}-\mu h$
are all $\left(\gamma kt+k\right)$-sparse vectors thanks to the sparsity of
$\|h_{T_{0}}\|_{0}\leq k$, $\|h^{(1)}\|_{0}=m$ and $\|u_{i}\|_{0}\leq\gamma
kt-m$. Since $\Phi h=0$, together with (31), we have
$\Phi(\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i})=\Phi((\frac{1}{2}-\mu)(h_{T_{0}}+h^{(1)})-\frac{1}{2}\mu
u_{i}).$
Following the proof of Theorem 1.1 in [7], we easily elicit
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|\Phi(\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i})\|_{2}^{2}=\frac{1}{4}\sum_{i=1}^{N}\lambda_{i}\|\Phi\eta_{i}\|_{2}^{2}.$
(32)
Setting $\mu=\frac{\sqrt{(t+\gamma)t}-t}{\gamma}>0$, if it holds that
$\displaystyle\delta:=\delta_{\gamma kt+k}<\sqrt{\frac{t}{t+\gamma}},$ (33)
then combining (32) with (33), we get
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|\Phi((\frac{1}{2}-\mu)(h_{T_{0}}+h^{(1)})-\frac{1}{2}\mu
u_{i})\|_{2}^{2}$
$\displaystyle-\frac{1}{4}\sum_{i=1}^{N}\lambda_{i}\|\Phi\eta_{i}\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle(1+\delta)\sum_{i=1}^{N}\lambda_{i}[(\frac{1}{2}-\mu)^{2}\|h_{T_{0}}+h^{(1)}\|_{2}^{2}+\frac{\mu^{2}}{4}\|u_{i}\|_{2}^{2}]$
$\displaystyle-\frac{1-\delta}{4}\sum_{i=1}^{N}\lambda_{i}(\|h_{T_{0}}+h^{(1)}\|_{2}^{2}+\mu^{2}\|u_{i}\|_{2}^{2})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}[((1+\delta)(\frac{1}{2}-\mu)^{2}-\frac{1-\delta}{4})\cdot$
$\displaystyle\|h_{T_{0}}+h^{(1)}\|_{2}^{2}+\frac{1}{2}\delta\mu^{2}\|u_{i}\|_{2}^{2}]$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|h_{T_{0}}+h^{(1)}\|_{2}^{2}\cdot$
$\displaystyle\left[\mu^{2}-\mu+\delta(\frac{1}{2}-\mu+(1+\frac{\gamma}{2t})\mu^{2})\right]$
$\displaystyle=$ $\displaystyle\|h_{T_{0}}+h^{(1)}\|_{2}^{2}\cdot$
$\displaystyle\left[\mu^{2}-\mu+\delta(\frac{1}{2}-\mu+(1+\frac{\gamma}{2t})\mu^{2})\right]$
$\displaystyle=$
$\displaystyle\|h_{T_{0}}+h^{(1)}\|_{2}^{2}\left(\frac{1}{2}-\mu+(1+\frac{\gamma}{2t})\mu^{2}\right)\cdot$
$\displaystyle\left[\delta-\sqrt{\frac{t}{t+\gamma}}\right]$ $\displaystyle<$
$\displaystyle 0,$
where the second inequality is derived from (30). Obviously, this is a
contradiction.
For condition (33), setting $t=\frac{a-1}{\gamma}$, it follows that
$\displaystyle\delta_{ak}<\sqrt{\frac{a-1}{a-1+\gamma^{2}}}.$
Hence we complete the proof. ∎
In order to prove the result Theorem III.4, we need another important concept
in the RIP framework, the restricted orthogonal constants (ROC), proposed in
[9].
###### Definition VI.1.
Define the restricted orthogonal constants (ROCs) of order $k_{1},k_{2}$ for a
matrix $\Phi\in\mathbb{R}^{m\times n}$ as the smallest non-negative number
$\theta_{k_{1},k_{2}}$ such that
$\displaystyle\left|\left\langle\Phi h_{1},\Phi
h_{2}\right\rangle\right|\leq\theta_{k_{1},k_{2}}\|h_{1}\|_{2}\|h_{2}\|_{2}$
(34)
for all $k_{1}$-sparse vector $h_{1}\in\mathbb{R}^{n}$ and $k_{2}$-sparse
vector $h_{2}\in\mathbb{R}^{n}$ with disjoint supports.
The next lemma blending with Lemmas 3.1, 5.1 and 5.4 in [6] centers on several
properties of the restricted orthogonal constants (ROCs).
###### Lemma VI.2.
Let $k_{1},k_{2},k\leq n,\lambda\geq 0$ and $\mu\geq 1$ such that $\mu k_{2}$
is an integer. Suppose $h_{1},h_{2}\in\mathbb{R}^{n}$ have disjoint supports
and $h_{1}$ is $k_{1}$-sparse. If $\|h_{2}\|_{1}\leq\lambda k_{2}$ and
$\|h_{2}\|_{\infty}\leq\lambda$, then the restricted orthogonal constants
satisfy
$\displaystyle\left|\left\langle\Phi h_{1},\Phi
h_{2}\right\rangle\right|\leq\theta_{k_{1},k_{2}}\|h_{1}\|_{2}\lambda\sqrt{k_{2}},$
(35) $\displaystyle\theta_{k_{1},\mu
k_{2}}\leq\sqrt{\mu}\theta_{k_{1},k_{2}},$ (36)
and
$\displaystyle\theta_{k,k}<$
$\displaystyle~{}~{}~{}~{}2\delta_{k},~{}~{}~{}~{}~{}~{}~{}~{}\text{for any
even}~{}k\geq 2,$ (37) $\displaystyle\theta_{k,k}<$
$\displaystyle\frac{2k}{\sqrt{k^{2}-1}}\delta_{k},~{}~{}~{}~{}\text{for any
odd}~{}k\geq 3.$ (38)
Proof of Theorem III.5
Similar to the proof of Theorem II.4, we suppose on the contrary that
$\widehat{h}\in\mathcal{N}_{1}$ (also shortly denote hereafter
$h=\widehat{h}$) such that
$\|h_{T_{0}^{C}}\|_{1}\leq\gamma\|h_{T_{0}}\|_{1}.$
Setting $\beta=k^{-1}\|h_{T_{0}}\|_{1}$, then we have
$\|h_{T_{0}^{C}}\|_{1}\leq\gamma k\beta\leq\lceil\gamma k\rceil\beta$ and
$\|h_{T_{0}^{C}}\|_{\infty}\leq\beta$. In fact, if
$\|h_{T_{0}^{C}}\|_{\infty}>\beta$, then (10) will contribute to
$k\beta=\|h_{T_{0}}\|_{1}\geq k\|h_{T_{0}^{C}}\|_{\infty}>k\beta$. Thus it
follows that
$\displaystyle|\langle\Phi h_{T_{0}},\Phi h_{T_{0}^{C}}\rangle|$
$\displaystyle\leq$ $\displaystyle\theta_{k,\lceil\gamma
k\rceil}\|h_{T_{0}}\|_{2}\sqrt{\lceil\gamma k\rceil}\beta$ $\displaystyle\leq$
$\displaystyle\theta_{k,k}\sqrt{\frac{\lceil\gamma
k\rceil}{k}}\|h_{T_{0}}\|_{2}\sqrt{\lceil\gamma k\rceil}\beta$
$\displaystyle\leq$ $\displaystyle\theta_{k,k}\frac{\lceil\gamma
k\rceil}{k}\|h_{T_{0}}\|_{2}^{2},$
where the first and second inequalities hold by (35) and (36) in Lemma VI.2
respectively, and the last inequality is derived from H$\ddot{o}$lder
inequality, i.e., $\|h_{T_{0}}\|_{1}\leq\sqrt{k}\|h_{T_{0}}\|_{2}$. If it
holds
$\displaystyle\delta_{k}+\theta_{k,k}\frac{\lceil\gamma k\rceil}{k}<1,$ (39)
we have
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left|\left\langle\Phi
h_{T_{0}},\Phi h\right\rangle\right|$ $\displaystyle\geq$
$\displaystyle\left|\left\langle\Phi h_{T_{0}},\Phi
h_{T_{0}}\right\rangle\right|-|\langle\Phi h_{T_{0}},\Phi
h_{T_{0}^{C}}\rangle|$ $\displaystyle\geq$
$\displaystyle(1-\delta_{k})\|h_{T_{0}}\|_{2}^{2}-\theta_{k,k}\frac{\lceil\gamma
k\rceil}{k}\|h_{T_{0}}\|_{2}^{2}$ $\displaystyle=$
$\displaystyle\left(1-\delta_{k}-\theta_{k,k}\frac{\lceil\gamma
k\rceil}{k}\right)\|h_{T_{0}}\|_{2}^{2}$ $\displaystyle>$ $\displaystyle 0.$
Obviously, this is a contradiction. By (37) and (38) in Lemma VI.2, when
$k\geq 2$ is even, it yields
$\delta_{k}+\theta_{k,k}\frac{\lceil\gamma
k\rceil}{k}<\left(1+\frac{2\lceil\gamma k\rceil}{k}\right)\delta_{k},$
and when $k\geq 3$ is odd, it generates that
$\delta_{k}+\theta_{k,k}\frac{\lceil\gamma
k\rceil}{k}<\left(1+\frac{2\lceil\gamma
k\rceil}{\sqrt{k^{2}-1}}\right)\delta_{k}.$
Therefore the theorem is accomplished thanks to conditions (15) and (16)
enabling (39) to hold. ∎
Shenglong Zhou is a PhD student in Department of Applied Mathematics, Beijing
Jiaotong University. He received his BS degree from Beijing Jiaotong
University of information and computing science in 2011. His research field is
theory and methods for optimization. Naihua Xiu is a Professor in Department
of Applied Mathematics, Beijing Jiaotong University. He received his PhD
degree in Operations Research from Academy Mathematics and System Science of
the Chinese Academy of Science in 1997. He was a Research Fellow of City
University of Hong Kong from 2000 to 2002, and he was a Visiting Scholar of
University of Waterloo from 2006 to 2007. His research interest includes
variational analysis, mathematical optimization, mathematics of operations
research. Yingnan Wang is a research assistant in Department of Applied
Mathematics, Beijing Jiaotong University. She received her PhD degree in
Operations Research from Beijing Jiaotong University in 2011. From 2011 to
2013, she was a Post-Doctoral Fellow of Department of Combinatorics and
Optimization, Faculty of Mathematics, University of Waterloo, Canada. Her
research interests are in sparse optimization, non-smooth optimization and
analysis, robust optimization. Lingchen Kong is an associate Professor in
Department of Applied Mathematics, Beijing Jiaotong University. He received
his PhD degree in Operations Research from Beijing Jiaotong University in
2007. From 2007 to 2009, he was a Post-Doctoral Fellow of Department of
Combinatorics and Optimization, Faculty of Mathematics, University of
Waterloo, Canada. His research interests are in sparse optimization,
mathematics of operations research.
|
arxiv-papers
| 2013-12-09T09:47:03 |
2024-09-04T02:49:55.165358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shenglong Zhou, Naihua Xiu, Yingnan Wang, Lingchen Kong",
"submitter": "Shenglong Zhou",
"url": "https://arxiv.org/abs/1312.2358"
}
|
1312.2552
|
# Abstract Interpretation of Temporal Concurrent Constraint Programs 111This
paper has been accepted for publication in Theory and Practice of Logic
Programming (TPLP), Cambridge University Press.
MORENO FALASCHI
Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche
Università di Siena Italy
E-mail: [email protected] CARLOS OLARTE
Departamento de Electrónica y Ciencias de la Computación
Pontificia Universidad Javeriana-Cali Colombia
E-mail: [email protected] CATUSCIA PALAMIDESSI
INRIA and LIX
Ecole Polytechnique France
E-mail: [email protected]
(15 May 2013; 3 December 2013)
###### Abstract
Timed Concurrent Constraint Programming (tcc) is a declarative model for
concurrency offering a logic for specifying reactive systems, i.e. systems
that continuously interact with the environment. The universal tcc formalism
(utcc) is an extension of tcc with the ability to express mobility. Here
mobility is understood as communication of private names as typically done for
mobile systems and security protocols. In this paper we consider the
denotational semantics for tcc, and we extend it to a “collecting” semantics
for utcc based on closure operators over sequences of constraints. Relying on
this semantics, we formalize a general framework for data flow analyses of tcc
and utcc programs by abstract interpretation techniques. The concrete and
abstract semantics we propose are compositional, thus allowing us to reduce
the complexity of data flow analyses. We show that our method is sound and
parametric with respect to the abstract domain. Thus, different analyses can
be performed by instantiating the framework. We illustrate how it is possible
to reuse abstract domains previously defined for logic programming to perform,
for instance, a groundness analysis for tcc programs. We show the
applicability of this analysis in the context of reactive systems.
Furthermore, we make also use of the abstract semantics to exhibit a secrecy
flaw in a security protocol. We also show how it is possible to make an
analysis which may show that tcc programs are suspension free. This can be
useful for several purposes, such as for optimizing compilation or for
debugging.
###### keywords:
Timed Concurrent Constraint Programming, Process Calculi, Abstract
Interpretation, Denotational Semantics, Reactive Systems
## 1 Introduction
Concurrent Constraint Programming (ccp) [Saraswat et al. (1991), Saraswat
(1993)] has emerged as a simple but powerful paradigm for concurrency tied to
logic that extends and subsumes both concurrent logic programming [Shapiro
(1989)] and constraint logic programming [Jaffar and Lassez (1987)]. The ccp
model combines the traditional operational view of process calculi with a
_declarative_ one based upon logic. This combination allows ccp to benefit
from the large body of reasoning techniques of both process calculi and logic.
In fact, ccp-based calculi have successfully been used in the modeling and
verification of several concurrent scenarios such as biological, security,
timed, reactive and stochastic systems [Saraswat et al. (1991), Olarte and
Valencia (2008b), Nielsen et al. (2002a), Saraswat et al. (1994), Jagadeesan
et al. (2005)] (see a survey in [Olarte et al. (2013)]).
In the ccp model, agents interact by _telling_ and _asking_ pieces of
information (_constraints_) on a shared store of partial information. The type
of constraints that agents can tell and ask is parametric in an underlying
constraint system. This makes ccp a flexible model able to adapt to different
application domains.
The ccp model has been extended to consider the execution of processes along
time intervals or time-units. In tccp [de Boer et al. (2000)], the notion of
time is identified with the time needed to ask and tell information to the
store. In this model, the information in the store is carried through the
time-units. On the other hand, in Timed ccp (tcc) [Saraswat et al. (1994)],
stores are not automatically transferred between time-units. This way,
computations during a time-unit proceed monotonically but outputs of two
different time-units are not supposed to be related to each other. More
precisely, computations in tcc take place in bursts of activity at a rate
controlled by the environment. In this model, the environment provides a
stimulus (input) in the form of a constraint. Then the system, after a finite
number of internal reductions, outputs the final store (a constraint) and
waits for the next interaction with the environment. This view of _reactive
computation_ is akin to synchronous languages such as Esterel [Berry and
Gonthier (1992)] where the system reacts continuously with the environment at
a rate controlled by the environment. Hence, these languages allow to program
safety critical applications as control systems, for which it is fundamental
to provide tools aiming at helping to develop correct, secure, and efficient
programs.
Universal tcc [Olarte and Valencia (2008b)] (utcc), adds to tcc the
expressiveness needed for _mobility_. Here we understand mobility as the
ability to communicate private names (or variables) much like in the
$\pi$-calculus [Milner et al. (1992)]. Roughly, a tcc _ask_ process
$\mathbf{when}\ c\ \mathbf{do}\ P$ executes the process $P$ only if the
constraint $c$ can be entailed from the store. This idea is generalized in
utcc by a parametric ask that executes $P[\vec{t}/\vec{x}]$ when the
constraint $c[\vec{t}/\vec{x}]$ is entailed from the store. Hence the
variables in $\vec{x}$ act as formal parameters of the ask operator. This
simple change allowed to widen the spectrum of application of ccp-based
languages to scenarios such as verification of security protocols [Olarte and
Valencia (2008b)] and service oriented computing [López et al. (2009)].
Several domains and frameworks (e.g., [Cousot and Cousot (1992), Armstrong et
al. (1998), Codish et al. (1999)] ) have been proposed for the analysis of
logic programs. The particular characteristics of timed ccp programs pose
additional difficulties for the development of such tools in this language.
Namely, the concurrent, timed nature of the language, and the synchronization
mechanisms based on entailment of constraints (blocking asks). Aiming at
statically analyzing utcc as well as tcc programs, we have to consider the
additional technical issues due to the infinite internal computations
generated by parametric asks as we shall explain later.
We develop here a _compositional_ semantics for tcc and utcc that allows us to
describe the behavior of programs and collects all concrete information needed
to properly abstract the properties of interest. This semantics is based on
closure operators over sequences of constraints along the lines of [Saraswat
et al. (1994)]. We show that parametric asks in utcc of the form
$(\mathbf{abs}\ \vec{x};c)\,P$ can be neatly characterized as closure
operators. This characterization is shown to be somehow dual to the semantics
for the local operator $(\mathbf{local}\,\vec{x})\,P$ that restricts the
variables in $\vec{x}$ to be local to $P$. We prove the semantics to be fully
abstract w.r.t. the operational semantics for a significant fragment of the
calculus.
We also propose an abstract semantics which approximates the concrete one. Our
framework is formalized by abstract interpretation techniques and is
parametric w.r.t. the abstract domain. It allows us to exploit the work done
for developing abstract domains for logic programs. Moreover, we can make new
analyses for reactive and mobile systems, thus widening the reasoning
techniques available for tcc and utcc, such as type systems [Hildebrandt and
López (2009)], logical characterizations [Mendler et al. (1995), Nielsen et
al. (2002a), Olarte and Valencia (2008b)] and semantics [Saraswat et al.
(1994), de Boer et al. (1995), Nielsen et al. (2002a)].
The abstraction we propose proceeds in two-levels. First, we approximate the
constraint system leading to an abstract constraint system. We give the
sufficient conditions which have to be satisfied for ensuring the soundness of
the abstraction. Next, to obtain efficient analyses, we abstract the infinite
sequences of (abstract) constraints obtained from the previous step. Our
semantics is then computable and compositional. Thus, it allows us to master
the complexity of the data-flow analyses. Moreover, the abstraction _over-
approximates_ the concrete semantics, thus preserving safety properties.
To the best of our knowledge, this is the first attempt to propose a
compositional semantics and an abstract interpretation framework for a
language adhering to the above-mentioned characteristics of utcc. Hence we can
develop analyses for several applications of utcc or its sub-calculus tcc (see
e.g., [Olarte et al. (2013)]). In particular, we instantiate our framework in
three different scenarios. The first one presents an abstraction of a
cryptographic constraint system. We use the abstract semantics to bound the
number of messages that a spy may generate, in order to exhibit a secrecy flaw
in a security protocol written in utcc. The second one tailors an abstract
domain for groundness and type dependency analysis in logic programming to
perform a groundness analysis of a tcc program. This analysis is proven useful
to derive a property of a control system specified in tcc. Finally, we present
an analysis that may show that a tcc program is suspension free. This analysis
can be used later for optimizing compilation or for debugging purposes.
The ideas of this paper stem mainly from the works of the authors in [de Boer
et al. (1995), Falaschi et al. (1997a), Falaschi et al. (1997b), Nielsen et
al. (2002a), Olarte and Valencia (2008a)] to give semantic characterization of
ccp calculi and from the works in [Falaschi et al. (1993), Codish et al.
(1994), Falaschi et al. (1997a), Zaffanella et al. (1997), Falaschi et al.
(2007)] to provide abstract interpretation frameworks to analyze concurrent
logic-based languages. A preliminary short version of this paper without
proofs was published in [Falaschi et al. (2009)]. In this paper we give many
more examples and explanations. We also refine several technical details and
present full proofs. Furthermore, we develop a new application for analyzing
suspension-free tcc programs.
The rest of the paper is organized as follows. Section 2 recalls the notion of
constraint system and the operational semantics of tcc and utcc. In Section 3
we develop the denotational semantics based on sequences of constraints. Next,
in Section 4, we study the abstract interpretation framework for tcc and utcc
programs. The three instances and the applications of the framework are
presented in Section 5. Section 6 concludes.
## 2 Preliminaries
Process calculi based on the ccp paradigm are parametric in a _constraint
system_ specifying the basic constraints agents can tell and ask. These
constraints represent a piece of (partial) information upon which processes
may act. The constraint system hence provides a signature from which
constraints can be built. Furthermore, the constraint system provides an
_entailment_ relation ($\vdash$) specifying inter-dependencies between
constraints. Intuitively, $c\vdash d$ means that the information $d$ can be
deduced from the information represented by $c$. For example, $x>60\vdash
x>42$.
Here we consider an abstract definition of constraint systems as cylindric
algebras as in [de Boer et al. (1995)]. The notion of constraint system as
first-order formulas [Smolka (1994), Nielsen et al. (2002a), Olarte and
Valencia (2008b)] can be seen as an instance of this definition. All results
of this paper still hold, of course, when more concrete systems are
considered.
###### Definition 1 (Constraint System)
A cylindric constraint system is a structure
${\mathbf{C}}=\langle\mathcal{C},\leq,\sqcup,\operatorname{\textup{{t}}},\operatorname{\textup{{f}}},{\mathit{V}ar},\exists,D\rangle$
s.t.
\-
$\langle\mathcal{C},\leq,\sqcup,\operatorname{\textup{{t}}},\operatorname{\textup{{f}}}\rangle$
is a lattice with $\sqcup$ the $\mathit{l}ub$ operation (representing the
logical _and_), and $\operatorname{\textup{{t}}}$,
$\operatorname{\textup{{f}}}$ the least and the greatest elements in
$\mathcal{C}$ respectively (representing true and false). Elements in
$\mathcal{C}$ are called _constraints_ with typical elements
$c,c^{\prime},d,d^{\prime}...$. If $c\leq d$ and $d\leq c$ we write $c\cong
d$. If $c\leq d$ and $c\not\cong d$, we write $c<d$.
-${\mathit{V}ar}$ is a denumerable set of variables and for each $x\in{\mathit{V}ar}$ the function $\exists x:\mathcal{C}\to\mathcal{C}$ is a cylindrification operator satisfying: (1) $\exists x(c)\leq c$. (2) If $c\leq d$ then $\exists x(c)\leq\exists x(d)$. (3) $\exists x(c\sqcup\exists x(d))\cong\exists x(c)\sqcup\exists x(d)$. (4) $\exists x\exists y(c)\cong\exists y\exists x(c)$. (5) For an increasing chain $c_{1}<c_{2}<c_{3}...$, $\exists x\bigsqcup_{i}c_{i}\cong\bigsqcup_{i}\exists x(c_{i})$.
\- For each $x,y\in{\mathit{V}ar}$, the constraint $d_{xy}\in D$ is a
_diagonal element_ and it satisfies: (1)
$d_{xx}\cong\operatorname{\textup{{t}}}$. (2) If $z$ is different from $x,y$
then $d_{xy}\cong\exists z(d_{xz}\sqcup d_{zy})$. (3) If $x$ is different from
$y$ then $c\leq d_{xy}\sqcup\exists x(c\sqcup d_{xy})$.
The cylindrification operators model a sort of existential quantification,
helpful for hiding information. We shall use ${\mathit{f}v}(c)=\\{x\in Var\ |\
\exists x(c)\not\cong c\\}$ to denote the set of free variables that occur in
$c$. If $x$ occurs in $c$ and $x\not\in{\mathit{f}v}(c)$, we say that $x$ is
bound in $c$. We use ${\mathit{b}v}(c)$ to denote the set of bound variables
in $c$.
Properties (1) to (4) are standard. Property (5) is shown to be required in
[de Boer et al. (1995)] to establish the semantic adequacy of ccp languages
when infinite computations are considered. Here, the continuity of the
semantic operator in Section 3 relies on the continuity of $\exists$ (see
Proposition 3.8). Below we give some examples on the requirements to satisfy
this property in the context of different constraint systems.
The diagonal element $d_{xy}$ can be thought of as the equality $x=y$.
Properties (1) to (3) are standard and they allow us to define substitutions
of the form $[t/x]$ required, for instance, to represent the substitution of
formal and actual parameters in procedure call. We shall give a formal
definition of them in Notation 2.
Let us give some examples of constraint systems. The finite domain constraint
system (FD) [Hentenryck et al. (1998)] assumes variables to range over finite
domains and, in addition to equality, one may have predicates that restrict
the possible values of a variable to some finite set, for instance $x<42$.
The Herbrand constraint system $\mathcal{H}$ consists of a first-order
language with equality. The entailment relation is the one we expect from
equality, for instance, $f(x,y)=f(g(a),z)$ must entail $x=g(a)$ and $y=z$.
$\mathcal{H}$ may contain non-compact elements to represent the limit of
infinite chains. To see this, let $s$ be the successor constructor, $\exists
y(x=s(s^{n}(y)))$ be denoted as the constraint $\texttt{gt}(x,n)$ (i.e.,
$x>n$) and $\\{\texttt{gt}(x,n)\\}_{n}$ be the ascending chain
$\texttt{gt}(x,0)<\texttt{gt}(x,1)<\cdots$. We note that $\exists
x(\texttt{gt}(x,n))=\operatorname{\textup{{t}}}$ for any $n$ and then,
$\bigsqcup\\{\exists x(\texttt{gt}(x,n))\\}_{n}=\operatorname{\textup{{t}}}$.
Property (5) in Definition 1 dictates that $\exists
x\bigsqcup\\{\texttt{gt}(x,n)\\}_{n}$ must be equal to
$\operatorname{\textup{{t}}}$ (i.e., there exists an $x$ which is greater than
any $n$). For that, we need a constraint, e.g., $\texttt{inf}(x)$ (a non-
compact element), to be the limit $\bigsqcup\\{\texttt{gt}(x,n)\\}_{n}$. We
know that $\texttt{inf}(x)\vdash\texttt{gt}(x,n)$ for any $n$ and then,
$\bigsqcup\\{\texttt{gt}(x,n)\\}_{n}=\texttt{inf}(x)$ and $\exists
x(\texttt{inf}(x))=\operatorname{\textup{{t}}}$ as wanted. A similar
phenomenon arises in the definition of constraint system as Scott information
systems in [Saraswat et al. (1991)]. There, constraints are represented as
finite subsets of _tokens_ (elementary constraints) built from a given set
$D$. The entailment is similar to that in Definition 1 but restricted to
compact elements, i.e., a constraint can be entailed only from a finite set of
elementary constraints. Moreover, $\exists$ is extended to be a continuous
function, thus satisfying Property (5) in Definition 1. Hence, the Herbrand
constraint system in [Saraswat et al. (1991)] considers also a non-compact
element (different from $\operatorname{\textup{{f}}}$) to be the limit of the
chain $\\{\texttt{gt}(x,n)\\}_{n}$.
Now consider the Kahn constraint system underlying data-flow languages where
equality is assumed along with the constant $\operatorname{\mathit{nil}}$ (the
empty list), the predicate $\texttt{nempty}(x)$ ($x$ is not
$\operatorname{\mathit{nil}}$), and the functions $\texttt{first}(x)$ (the
first element of $x$), $\texttt{rest}(x)$ ($x$ without its first element) and
$\texttt{cons}(x,y)$ (the concatenation of $x$ and $y$). If we consider the
Kahn constraint system in [Saraswat et al. (1991)], the constraint $c$ defined
as
$\\{\texttt{first}(\texttt{tail}^{n}(x))=\texttt{first}(\texttt{tail}^{n}(y))\mid
n\geq 0\\}$ does not entail $\\{x=y\\}$ since the entailment relation is
defined only on compact elements. In Definition 1, we are free to decide if
$c$ is different or not from $x=y$. If we equate them, the constraint $x=y$ is
not longer a compact element and then, one has to be careful to only use a
compact version of “$=$” in programs (see Definition 2). A similar situation
occurs with the Rational Interval Constraint System [Saraswat et al. (1991)]
and the constraints $\\{x\in[0,1+1/n]\mid n\geq 0\\}$ and $x\in[0,1]$.
All in all many different constraint systems satisfy Definition 1.
Nevertheless, one has to be careful since the constraint systems might not be
the same as what is naively expected due to the presence of non-compact
elements.
We conclude this section by setting some notation and conventions about terms,
sequences of constraints, substitutions and diagonal elements. We first lift
the relation $\leq$ and the cylindrification operator to sequences of
constraints.
###### Notation 1 (Sequences of Constraints)
We denote by $\mathcal{C}^{\omega}$ (resp. $\mathcal{C}^{*})$ the set of
infinite (resp. finite) sequences of constraints with typical elements
$w,w^{\prime},s,s^{\prime},...$. We use $W,W^{\prime},S,S^{\prime}$ to range
over subsets of $\mathcal{C}^{\omega}$ or $\mathcal{C}^{*}$. We use
$c^{\omega}$ to denote the sequence $c.c.c...$. The length of $s$ is denoted
by $|s|$ and the empty sequence by $\epsilon$. The $i$-th element in $s$ is
denoted by $s(i)$. We write $s\leq s^{\prime}$ iff $|s|\leq|s^{\prime}|$ and
for all $i\in\\{1,\ldots,|s|\\}$, $s^{\prime}(i)\vdash s(i)$. If
$|s|=|s^{\prime}|$ and for all $i\in\\{1,...,|s|\\}$ it holds $s(i)\cong
s^{\prime}(i)$, we shall write $s\cong s^{\prime}$. Given a sequence of
variables $\vec{x}$, with $\exists\vec{x}(c)$ we mean $\exists x_{1}\exists
x_{2}...\exists x_{n}(c)$ and with $\exists\vec{x}(s)$ we mean the pointwise
application of the cylindrification operator to the constraints in $s$.
We shall assume that the diagonal element $d_{xy}$ is interpreted as the
equality $x=y$. Furthermore, following [Giacobazzi et al. (1995)], we extend
the use of $d_{xy}$ to consider terms as in $d_{xt}$. More precisely,
###### Convention 1 (Diagonal elements)
We assume that the constraint system under consideration contains an equality
theory. Then, diagonal elements $d_{xy}$ can be thought of as formulas of the
form $x=y$. We shall use indistinguishably both notations. Given a variable
$x$ and a term $t$ (i.e., a variable, constant or $n$-place function of $n$
terms symbol), we shall use $d_{xt}$ to denote the equality $x=t$. Similarly,
given a sequence of distinct variables $\vec{x}$ and a sequence of terms
$\vec{t}$, if $|\vec{x}|=|\vec{t}|=n$ then $d_{\vec{x}\vec{t}}$ denotes the
constraint $\bigsqcup\limits_{1\leq i\leq n}x_{i}=t_{i}$. If
$|\vec{x}|=|\vec{t}|=0$ then $d_{\vec{x}\vec{t}}=\operatorname{\textup{{t}}}$.
Given a set of diagonal elements $E$, we shall write $E\Vdash
d_{\vec{x}\vec{t}}$ whenever $d_{i}\vdash d_{\vec{x}\vec{t}}$ for some
$d_{i}\in E$. Otherwise, we write $E\not\Vdash d_{\vec{x}\vec{t}}$.
Finally, we set the notation for substitutions.
###### Notation 2 (Admissible substitutions)
Let $\vec{x}$ be a sequence of pairwise distinct variables and $\vec{t}$ be a
sequence of terms s.t. $|\vec{t}|=|\vec{x}|$. We denote by
$c[\vec{t}/\vec{x}]$ the constraint $\exists\vec{x}(c\sqcup
d_{\vec{x}\vec{t}})$ which represents abstractly the constraint obtained from
$c$ by replacing the variables $\vec{x}$ by $\vec{t}$. We say that $\vec{t}$
is admissible for $\vec{x}$, notation $adm(\vec{x},\vec{t})$, if the variables
in $\vec{t}$ are different from those in $\vec{x}$. If $|\vec{x}|=|\vec{t}|=0$
then trivially $adm(\vec{x},\vec{t})$. Similarly, we say that the substitution
$[\vec{t}/\vec{x}]$ is admissible iff $adm(\vec{x},\vec{t})$. Given an
admissible substitution $[\vec{t}/\vec{x}]$, from Property (3) of diagonal
elements in Definition 1, we note that $c[\vec{t}/\vec{x}]\sqcup
d_{\vec{x}\vec{t}}\vdash c$.
### 2.1 Reactive Systems and Timed CCP
Reactive systems [Berry and Gonthier (1992)] are those that react continuously
with their environment at a rate controlled by the environment. For example, a
controller or a signal-processing system, receives a stimulus (input) from the
environment. It computes an output and then, waits for the next interaction
with the environment.
In the ccp model, the shared store of constraints grows monotonically, i.e.,
agents cannot drop information (constraints) from it. Then, a system that
changes the state of a variable as in “${\mathit{s}ignal=on}$” and
“${\mathit{s}ignal=off}"$ leads to an inconsistent store.
Timed ccp (tcc) [Saraswat et al. (1994)] extends ccp for reactive systems.
Time is conceptually divided into _time intervals_(or _time-units_). In a
particular time interval, a ccp process $P$ gets an input $c$ from the
environment, it executes with this input as the initial _store_ , and when it
reaches its resting point, it _outputs_ the resulting store $d$ to the
environment. The resting point determines also a residual process $Q$ which is
then executed in the next time-unit. The resulting store $d$ is not
automatically transferred to the next time-unit. This way, computations during
a time-unit proceed monotonically but outputs of two different time-units are
not supposed to be related to each other. Therefore, the variable
${\mathit{s}ignal}$ in the example above may change its value when passing
from one time-unit to the next one.
###### Definition 2 (tcc Processes)
The set $Proc$ of tcc processes is built from the syntax
$\begin{array}[]{lll}P,Q&:=&\mathbf{skip}\ \ |\ \ \mathbf{tell}(c)\ \ |\
\mathbf{when}\ c\ \mathbf{do}\ P\ \ |\ \ P\parallel Q\ \ |\ \
(\mathbf{local}\,\vec{x})\,P\ \ |\\\ &&\mathbf{next}\,P\ \ |\ \
\mathbf{unless}\ c\ \mathbf{next}\,P\ \ |\ \ p(\vec{t})\end{array}$
where $c$ is a compact element of the underlying constraint system. Let
$\mathcal{D}$ be a set of process declarations of the form
$p(\vec{x})\operatorname{:\\!--}P$. A tcc program takes the form
$\mathcal{D}.P$. We assume $\mathcal{D}$ to have a unique process definition
for every process name, and recursive calls to be guarded by a
${\mathbf{n}ext}$ process.
The process $\mathbf{skip}$ does nothing thus representing inaction. The
process $\mathbf{tell}(c)$ adds $c$ to the store in the current time interval
making it available to the other processes. The process $\mathbf{when}\ c\
\mathbf{do}\ P$ _asks_ if $c$ can be deduced from the store. If so, it behaves
as $P$. In other case, it remains blocked until the store contains at least as
much information as $c$. The parallel composition of $P$ and $Q$ is denoted by
$P\,\parallel\,Q$. Given a set of indexes $I=\\{1,...,n\\}$, we shall use
$\prod\limits_{i\in I}P_{i}$ to denote the parallel composition
$P_{1}\parallel...\parallel P_{n}$. The process $(\mathbf{local}\,\vec{x})\,P$
_binds_ $\vec{x}$ in $P$ by declaring it private to $P$. It behaves like $P$,
except that all the information on the variables $\vec{x}$ produced by $P$ can
only be seen by $P$ and the information on the global variables in $\vec{x}$
produced by other processes cannot be seen by $P$.
The process $\mathbf{next}\,P$ is a _unit-delay_ that executes $P$ in the next
time-unit. The _time-out_ $\mathbf{unless}\ c\ \mathbf{next}\,P$ is also a
unit-delay, but $P$ is executed in the next time-unit if and only if $c$ is
not entailed by the final store at the current time interval. We use
$\mathbf{next}^{n}P$ as a shorthand for $\mathbf{next}\dots\mathbf{next}\,P$,
with $\mathbf{next}$ repeated $n$ times.
We extend the definition of free variables to processes as follows:
${\mathit{f}v}(\mathbf{skip})=\emptyset$;
${\mathit{f}v}(\mathbf{tell}(c))={\mathit{f}v}(c)$;
${\mathit{f}v}(\mathbf{when}\ c\ \mathbf{do}\
Q)={\mathit{f}v}(c)\cup{\mathit{f}v}(Q)$; ${\mathit{f}v}(\mathbf{unless}\ c\
\mathbf{next}\,Q)={\mathit{f}v}(c)\cup{\mathit{f}v}(Q)$;
${\mathit{f}v}(Q\parallel
Q^{\prime})={\mathit{f}v}(Q)\cup{\mathit{f}v}(Q^{\prime})$;
${\mathit{f}v}((\mathbf{local}\,\vec{x})\,Q)={\mathit{f}v}(Q)\setminus\vec{x}$;
${\mathit{f}v}(\mathbf{next}\,Q)={\mathit{f}v}(Q)$;
${\mathit{f}v}(p(\vec{t}))=vars(\vec{t})$ where $vars(\vec{t})$ is the set of
variables occurring in $\vec{t}$. A variable $x$ is bound in $P$ if $x$ occurs
in $P$ and $x\notin{\mathit{f}v}(P)$. We use ${\mathit{b}v}(P)$ to denote the
set of bound variables in $P$.
Assume a (recursive) process definition $\ \ p(\vec{x})\operatorname{:\\!--}P\
\ $ where ${\mathit{f}v}(P)\subseteq\vec{x}$. The call $p(\vec{t})$ reduces to
$P[\vec{t}/\vec{x}]$. Recursive calls in $P$ are assumed to be guarded by a
$\mathbf{next}\,$ process to avoid non-terminating sequences of recursive
calls during a time-unit (see [Saraswat et al. (1994), Nielsen et al.
(2002a)]).
In the forthcoming sections we shall use the idiom $!\,P$ defined as follows:
###### Notation 3 (Replication)
The replication of $P$, denoted as $!\,P$, is a short hand for a call to a
process definition
$\texttt{bang}_{P}()\operatorname{:\\!--}P\parallel\mathbf{next}\,\texttt{bang}_{P}()$.
Hence, $!\,P$ means
$P\parallel\mathbf{next}\,P\parallel\mathbf{next}\,^{2}P...$.
### 2.2 Mobile behavior and utcc
As we have shown, interaction of tcc processes is asynchronous as
communication takes place through the shared store of partial information.
Similar to other formalisms, by defining local (or private) variables, tcc
processes specify boundaries in the interface they offer to interact with each
other. Once these interfaces are established, there are few mechanisms to
modify them. This is not the case e.g., in the $\pi$-calculus [Milner et al.
(1992)] where processes can change their communication patterns by exchanging
their private names. The following example illustrates the limitation of $ask$
processes to communicate values and local variables.
###### Example 1
Let $\operatorname{\textup{{out}}}(\cdot)$ be a constraint and let
$P=\mathbf{when}\ \operatorname{\textup{{out}}}(x)\ \mathbf{do}\ R$ be a
system that must react when receiving a stimulus (i.e., an input) of the form
$\operatorname{\textup{{out}}}(n)$ for $n>0$. We notice that $P$ in a store
$\operatorname{\textup{{out}}}(42)$ does not execute $R$ since
$\operatorname{\textup{{out}}}(42)\not\vdash\operatorname{\textup{{out}}}(x)$.
The key point in the previous example is that $x$ is a free-variable and
hence, it does not act as a formal parameter (or place holder) for every term
$t$ such that $\operatorname{\textup{{out}}}(t)$ is entailed by the store.
In [Olarte and Valencia (2008b)], tcc is extended for _mobile reactive_
systems leading to _universal timed_ ccp (utcc). To model mobile behavior,
utcc replaces the ask operation $\mathbf{when}\ c\ \mathbf{do}\ P$ with a
parametric ask construction, namely $(\mathbf{abs}\ \vec{x};c)\,P$. This
process can be viewed as a $\lambda$-_abstraction_ of the process $P$ on the
variables $\vec{x}$ under the constraint (or with the _guard_) $c$.
Intuitively, for all admissible substitution $[\vec{t}/\vec{x}]$ s.t. the
current store entails $c[\vec{t}/\vec{x}]$, the process $(\mathbf{abs}\
\vec{x};c)\,P$ performs $P[\vec{t}/\vec{x}]$. For example, $(\mathbf{abs}\
x;\operatorname{\textup{{out}}}(x))\,R$ in a store entailing both
$\operatorname{\textup{{out}}}(z)$ and $\operatorname{\textup{{out}}}(42)$
executes $R[42/x]$ and $R[z/x]$.
###### Definition 3 (utcc Processes and Programs)
The utcc processes and programs result from replacing in Definition 2 the
expression $\mathbf{when}\ c\ \mathbf{do}\ P$ with $(\mathbf{abs}\
\vec{x};c)\,P$ where the variables in $\vec{x}$ are pairwise distinct.
When $|\vec{x}|=0$ we write $\mathbf{when}\ c\ \mathbf{do}\ P$ instead of
$(\mathbf{abs}\ \epsilon;c)\,P$. Furthermore, the process $(\mathbf{abs}\
\vec{x};c)\,P$ binds $\vec{x}$ in $P$ and $c$. We thus extend accordingly the
sets ${\mathit{f}v}(\cdot)$ and ${\mathit{b}v}(\cdot)$ of free and bound
variables.
From a programming point of view, we can see the variables $\vec{x}$ in the
abstraction $(\mathbf{abs}\ \vec{x};c)\,P$ as the formal parameters of $P$. In
fact, the utcc calculus was introduced in [Olarte and Valencia (2008b)] with
replication ($!\,P$) and without process definitions since replication and
abstractions are enough to encode recursion. Here we add process definitions
to properly deal with tcc programs with recursion which are more expressive
than those without it (see [Nielsen et al. (2002b)]) and we omit replication
to avoid redundancy in the set of operators (see Notation 3). We thus could
have dispensed with the next-guarded restriction in Definition 2 for utcc
programs. Nevertheless, in order to give a unified presentation of the
forthcoming results, we assume that utcc programs adhere also to that
restriction.
We conclude with an example of mobile behavior where a process $P$ sends a
local variable to $Q$. Then, both processes can communicate through the shared
variable.
###### Example 2 (Scope extrusion)
Assume two components $P$ and $Q$ of a system such that $P$ creates a local
variable that must be shared with $Q$. This system can be modeled as
$\begin{array}[]{lll l
lll}P&=&(\mathbf{local}\,x)\,(\mathbf{tell}(\operatorname{\textup{{out}}}(x))\parallel
P^{\prime})&&Q&=&(\mathbf{abs}\
z;\operatorname{\textup{{out}}}(z))\,Q^{\prime}\end{array}$
We shall show later that the parallel composition of $P$ and $Q$ evolves to a
process of the form $P^{\prime}\parallel Q^{\prime}[x/z]$ where $P^{\prime}$
and $Q^{\prime}$ share the local variable $x$ created by $P$. Then, any
information produced by $P^{\prime}$ on $x$ can be seen by $Q^{\prime}$ and
vice versa.
### 2.3 Operational Semantics (SOS)
We take inspiration on the structural operational semantics (SOS) for linear
ccp in [Fages et al. (2001), Haemmerlé et al. (2007)] to define the behavior
of processes. We consider _transitions_ between _configurations_ of the form
$\left\langle{\vec{x};P;c}\right\rangle$ where $c$ is a constraint
representing the current store, $P$ a process and $\vec{x}$ is a set of
distinct variables representing the bound (local) variables of $c$ and $P$. We
shall use $\gamma,\gamma^{\prime},\ldots$ to range over configurations.
Processes are quotiented by $\equiv$ defined as follows.
###### Definition 4 (Structural Congruence)
Let $\equiv$ be the smallest congruence satisfying: (1) $P\equiv Q$ if they
differ only by a renaming of bound variables (alpha-conversion); (2)
$P\parallel\mathbf{skip}\equiv P$; (3) $P\parallel Q\equiv Q\parallel P$; and
(4) $P\parallel(Q\parallel R)\equiv(P\parallel Q)\parallel R$.
The congruence relation $\equiv$ is extended to configurations by decreeing
that
$\left\langle{\vec{x};P;c}\right\rangle\equiv\left\langle{\vec{y};Q;d}\right\rangle$
iff $(\mathbf{local}\,\vec{x})\,P\equiv(\mathbf{local}\,\vec{y})\,Q$ and
$\exists\vec{x}(c)\cong\exists\vec{y}(d)$.
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0.0pt\vbox{\hrule width=115.10762pt}\nointerlineskip\kern 2.0pt\moveright
0.0pt\hbox{$\left\langle{\vec{x};p(\vec{t});d}\right\rangle\longrightarrow\left\langle{\vec{x};P[\vec{t}/\vec{x}];d}\right\rangle$}}}&&\hbox{
\kern 0.0pt\raise 7.04001pt\hbox{$\mathrm{R}_{UNL}$}\kern 0.0pt\kern
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2.0pt\moveright 0.0pt\vbox{\hrule width=147.73392pt}\nointerlineskip\kern
2.0pt\moveright 0.0pt\hbox{$\left\langle{\vec{x};\mathbf{unless}\ c\
\mathbf{next}\,P;d}\right\rangle\longrightarrow\left\langle{\vec{x};\mathbf{skip};d}\right\rangle$}}}\\\
\\\ \lx@intercol\mbox{ Observable Transition}\hfil\lx@intercol\\\
\lx@intercol\hfil\mathrm{R}_{OBS}\
\frac{\raisebox{2.84544pt}{$\left\langle{\emptyset;P;c}\right\rangle\longrightarrow^{*}\left\langle{\vec{x};Q;d}\right\rangle\not\longrightarrow$}}{\raisebox{-5.69046pt}{$P\stackrel{{\scriptstyle\,\,(c,\exists\vec{x}(d))\,\,}}{{\,\,===\Longrightarrow}}(\mathbf{local}\,\vec{x})\,F(Q)$}}\
\mbox{\ \ where \ \
}\par{F}(P)=\left\\{\begin{array}[]{ll}F(\mathbf{skip})=F((\mathbf{abs}\
\vec{x};c;D)\,Q)=\mathbf{skip}\\\ F(P_{1}\parallel P_{2})=F(P_{1})\parallel
F(P_{2})\\\ F(\mathbf{next}\,Q)=F(\mathbf{unless}\ c\
\mathbf{next}\,Q)=Q\end{array}\right.\hfil\lx@intercol\end{array}$
Figure 1: SOS. In $\mathrm{R}_{STR}$, $\equiv$ is given in Definition 4. In
$\mathrm{R}_{ABS}$ and $\mathrm{R}_{CALL}$, $adm(\vec{x},\vec{t})$ is defined
in Notation 2. In $\mathrm{R}_{ABS}$, $E$ is assumed to be a set of diagonal
elements and $\not\Vdash$ is defined in Convention 1. In
$\mathrm{R}_{STRVAR}$, ${\mathit{n}f}(d)$ is defined in Notation 4.
Transitions are given by the relations $\longrightarrow$ and $\Longrightarrow$
in Figure 1. The _internal_ transition
$\left\langle{\vec{x};P;c}\right\rangle\longrightarrow\left\langle{\vec{x}^{\prime};P^{\prime};c^{\prime}}\right\rangle$
should be read as “$P$ with store $c$ reduces, in one internal step, to
$P^{\prime}$ with store $c^{\prime}$ ”. We shall use $\longrightarrow^{*}$ as
the reflexive and transitive closure of $\longrightarrow$. If
$\gamma\longrightarrow\gamma^{\prime}$ and
$\gamma^{\prime}\equiv\gamma^{\prime\prime}$ we write
$\gamma\longrightarrow\equiv\gamma^{\prime\prime}$. Similarly for
$\longrightarrow^{*}$.
The _observable transition_
$P\stackrel{{\scriptstyle\,\,(c,d)\,\,}}{{\,\,===\Longrightarrow}}R$ should be
read as “$P$ on input $c$, reduces in one _time-unit_ to $R$ and outputs $d$”.
The observable transitions are obtained from finite sequences of internal
ones.
The rules in Figure 1 are easily seen to realize the operational intuitions
given in Section 2.1. As clarified below, the seemingly missing rule for a
$\mathbf{next}$ process is given by $\mathrm{R}_{OBS}$. Before explaining such
rules, let us introduce the following notation needed for
$\mathrm{R}_{STRVAR}$.
###### Notation 4 (Normal Form)
We observe that the store $c$ in a configuration takes the form
$\exists\vec{x}_{1}(d_{1})\sqcup...\sqcup\exists\vec{x}_{n}(d_{n})$ where each
$\vec{x}_{i}$ may be an empty set of variables. The normal form of $c$,
notation ${\mathit{n}f}(c)$, is the constraint obtained by renaming the
variables in $c$ such that for all $i,j\in 1..n$, if $i\neq j$ then the
variables in $\vec{x}_{i}$ do not occur neither bound nor free in $d_{j}$. It
is easy to see that $c\cong{\mathit{n}f}(c)$.
\- $\mathrm{R}_{TELL}$ says that the process $\mathbf{tell}(c)$ adds $c$ to
the current store $d$ (via the lub operator of the constraint system) and then
evolves into $\mathbf{skip}$.
\- $\mathrm{R}_{PAR}$ says that if $P$ may evolve into $P^{\prime}$, this
reduction also takes place when running in parallel with $Q$.
\- The process $(\mathbf{local}\,\vec{y})\,Q$ adds $\vec{y}$ to the local
variables of the configuration and then evolves into $Q$. The side conditions
of the rule $\mathrm{R}_{LOC}$ guarantee that $Q$ runs with a different set of
variables from those in the store and those used by other processes.
\- We extend the transition relation to consider processes of the form
$(\mathbf{abs}\ \vec{y};c;E)\,Q$ where $E$ is a set of diagonal elements. If
$E$ is empty, we write $(\mathbf{abs}\ \vec{y};c)\,Q$ instead of
$(\mathbf{abs}\ \vec{y};c;\emptyset)\,Q$. If $d$ entails $c[\vec{t}/\vec{y}]$,
then $P[\vec{t}/\vec{y}]$ is executed (Rule $\mathrm{R}_{ABS}$). Moreover, the
abstraction persists in the current time interval to allow other potential
replacements of $\vec{y}$ in $P$. Notice that $E$ is augmented with
$d_{\vec{y}\vec{t}}$ and the side condition $E\not\Vdash d_{\vec{y}\vec{t}}$
prevents executing $P[\vec{t}/\vec{y}]$ again. The process
$P[\vec{t}/\vec{y}]$ is obtained by equating $\vec{y}$ and $\vec{t}$ and then,
hiding the information about $\vec{y}$, i.e.,
$(\mathbf{local}\,\vec{y})\,(!\,\mathbf{tell}(d_{\vec{y}\vec{t}})\parallel
P)$.
\- Rule $\mathrm{R}_{STRVAR}$ allows us to _open_ the scope of existentially
quantified constraints in the store (see Example 3 below). If $\gamma$ reduces
to $\gamma^{\prime}$ using this rule then $\gamma\equiv\gamma^{\prime}$.
\- Rule $\mathrm{R}_{STR}$ says that one can use the structural congruence on
processes to continue a derivation (e.g., to do alpha conversion). It is worth
noticing that we do not allow in this rule to transform the store via the
relation $\equiv$ on configurations and then, via $\cong$ on constraints. We
shall discuss the reasons behind this choice in Example 3.
-What we observe from $p(\vec{t})$ is $P[\vec{t}/\vec{x}]$ where the formal parameters are substituted by the actual parameter (Rule $\mathrm{R}_{CALL}$).
\- Since the process $P=\mathbf{unless}\ c\ \mathbf{next}\,Q$ executes $Q$ in
the next time-unit only if the final store at the current time-unit does not
entail $c$, in the rule $\mathrm{R}_{UNL}$ $P$ evolves into $\mathbf{skip}$ if
the current store $d$ entails $c$.
For the observable transition relation, rule $\mathrm{R}_{OBS}$ says that an
observable transition from $P$ labeled with $(c,\exists\vec{x}(d))$ is
obtained from a terminating sequence of internal transitions from
$\left\langle{\emptyset;P;c}\right\rangle$ to
$\left\langle{\vec{x};Q;d}\right\rangle$. The process to be executed in the
next time interval is $(\mathbf{local}\,\vec{x})\,F(Q)$ (the “future” of $Q$).
$F(Q)$ is obtained by removing from $Q$ the ${\mathbf{a}bs}$ processes that
could not be executed and by “unfolding” the sub-terms within $\mathbf{next}$
and $\mathbf{unless}$ expressions. Notice that the output of a process hides
the local variables ($\exists\vec{x}(d)$) and those variables are also hidden
in the next time-unit ($(\mathbf{local}\,\vec{x})\,F(Q)$).
Now we are ready to show that processes in Example 2 evolve into a
configuration where a (local) variable can be communicated and shared.
###### Example 3 (Scope Extrusion and Structural Rules)
Let $P$ and $Q$ be as in Example 2. In the following we show the evolution of
the process $P\parallel Q$ starting from the store $\exists
w(\operatorname{\textup{{out}}}(w))$:
$\begin{array}[]{llll}\mbox{\tiny 1}&\left\langle{\emptyset;P\parallel
Q;\exists
w(\operatorname{\textup{{out}}}(w))}\right\rangle&\longrightarrow^{*}&\left\langle{\\{x\\};\mathbf{tell}(\operatorname{\textup{{out}}}(x))\parallel
P^{\prime}\parallel Q;\exists
w(\operatorname{\textup{{out}}}(w))}\right\rangle\\\ \mbox{\tiny
2}&&\longrightarrow^{*}&\left\langle{\\{x\\};P^{\prime}\parallel Q;\exists
w(\operatorname{\textup{{out}}}(w))\sqcup\operatorname{\textup{{out}}}(x)}\right\rangle\\\
\mbox{\tiny 3}&&\longrightarrow^{*}&\left\langle{\\{x,w\\};P^{\prime}\parallel
Q;\operatorname{\textup{{out}}}(w)\sqcup\operatorname{\textup{{out}}}(x)}\right\rangle\\\
\mbox{\tiny 4}&&\longrightarrow^{*}&\left\langle{\\{x,w\\};P^{\prime}\parallel
Q_{1}\parallel
Q^{\prime}[w/z];\operatorname{\textup{{out}}}(w)\sqcup\operatorname{\textup{{out}}}(x)}\right\rangle\\\
\mbox{\tiny 5}&&\longrightarrow^{*}&\left\langle{\\{x,w\\};P^{\prime}\parallel
Q_{2}\parallel Q^{\prime}[w/z]\parallel
Q^{\prime}[x/z];\operatorname{\textup{{out}}}(w)\sqcup\operatorname{\textup{{out}}}(x)}\right\rangle\end{array}$
where $Q_{1}=(\mathbf{abs}\
z;\operatorname{\textup{{out}}}(z);\\{d_{wz}\\})\,Q^{\prime}$ and
$Q_{2}=(\mathbf{abs}\
z;\operatorname{\textup{{out}}}(z);\\{d_{wz},d_{xz}\\})\,Q^{\prime}$. Observe
that $P^{\prime}$ and $Q^{\prime}[x/z]$ share the local variable $x$ created
by $P$. The derivation from line 2 to line 3 uses the Rule
$\mathrm{R}_{STRVAR}$ to _open_ the scope of $w$ in the store $\exists
w(\operatorname{\textup{{out}}}(w))$. Let $c_{1}=\exists
w(\operatorname{\textup{{out}}}(w))\sqcup\operatorname{\textup{{out}}}(x)$
(store in line 2) and $c_{2}=\operatorname{\textup{{out}}}(x)$. We know that
$c_{1}\cong c_{2}$. As we said before, Rule $\mathrm{R}_{STR}$ allows us to
replace structural congruent processes ($\equiv$) but it does not modify the
store via the relation $\cong$ on constraints. The reason is that if we
replace $c_{1}$ in line 2 with $c_{2}$, then we will not observe the execution
of $Q^{\prime}[w/x]$.
### 2.4 Observables and Behavior
In this section we study the input-output behavior of programs and we show
that such relation is a function. More precisely, we show that the input-
output relation is a (partial) upper closure operator. Then, we characterize
the behavior of a process by the sequences of constraints such that the
process cannot add any information to them. We shall call this behavior the
strongest postcondition. This relation is fundamental to later develop the
denotational semantics for tcc and utcc programs.
Next lemma states some fundamental properties of the internal relation. The
proof follows from simple induction on the inference
$\gamma\longrightarrow\gamma^{\prime}$.
###### Lemma 1 (Properties of $\longrightarrow$)
Assume that
$\left\langle{\vec{x};P;c}\right\rangle\longrightarrow\left\langle{\vec{x}^{\prime};Q;d}\right\rangle$.
Then, $\vec{x}\subseteq\vec{x}^{\prime}$. Furthermore:
1\. (Internal Extensiveness):
$\exists\vec{x}^{\prime}(d)\vdash\exists\vec{x}(c)$, i.e., the store can only
be augmented.
2\. (Internal Potentiality): If $e\vdash c$ and $d\vdash e$ then
$\left\langle{\vec{x};P;e}\right\rangle\longrightarrow\equiv\left\langle{\vec{x}^{\prime};Q;d}\right\rangle$,
i.e., a stronger store triggers more internal transitions.
4\. (Internal Restartability):
$\left\langle{\vec{x};P;d}\right\rangle\longrightarrow\equiv\left\langle{\vec{x}^{\prime};Q;d}\right\rangle$.
#### 2.4.1 Input-Output Behavior
Recall that tcc and utcc allows for the modeling of reactive systems where
processes react according to the stimuli (input) from the environment. We
define the behavior of a process $P$ as the relation of its outputs under the
influence of a sequence of inputs (constraints) from the environment. Before
formalizing this idea, it is worth noticing that unlike tcc, some utcc
processes may exhibit infinitely many internal reductions during a time-unit
due to the $\mathbf{abs}$ operator.
###### Example 4 (Infinite Behavior)
Consider a constant symbol “$a$”, a function symbol $f$, a unary predicate
(constraint) $c(\cdot)$ and let $Q=(\mathbf{abs}\
x;c(x))\,\mathbf{tell}(c(f(x)))$. Operationally, $Q$ in a store $c(a)$ engages
in an infinite sequence of internal transitions producing the constraints
$c(f(a))$, $c(f(f(a)))$, $c(f(f(f(a))))$ and so on.
The above behavior will arise, for instance, in applications to security as
those in Section 5.1. We shall see that the model of the attacker may generate
infinitely many messages (constraints) if we do not restrict the length of the
messages (i.e., the number of nested applications of $f$).
###### Definition 5 (Input-Output Behavior)
Let $s=c_{1}.c_{2}...c_{n}$,
$s^{\prime}=c_{1}^{\prime}.c_{2}^{\prime}...c_{n}^{\prime}$ (resp.
$w=c_{1}.c_{2}...$, $w^{\prime}=c_{1}^{\prime}.c_{2}^{\prime}...$) be finite
(resp. infinite) sequences of constraints. If
$P=P_{1}\stackrel{{\scriptstyle\,\,(c_{1},c_{1}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}P_{2}\stackrel{{\scriptstyle\,\,(c_{2},c_{2}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}...P_{n}\stackrel{{\scriptstyle\,\,(c_{n},c_{n}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}P_{n+1}$
(resp.
$P=P_{1}\stackrel{{\scriptstyle\,\,(c_{1},c_{1}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}P_{2}\stackrel{{\scriptstyle\,\,(c_{2},c_{2}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}...$
) , we write
$P\stackrel{{\scriptstyle\,\,(s,s^{\prime})\,\,}}{{\,\,===\Longrightarrow}}$
(resp.
$P\stackrel{{\scriptstyle\,\,(w,w^{\prime})\,\,}}{{\,\,===\Longrightarrow_{\omega}}}$).
We define the _input-output_ behavior of $P$ as
$\mathit{i}o{(P)}=\mathit{i}o^{\mathit{f}in}{(P)}\cup\mathit{i}o^{\mathit{i}nf}{(P)}$
where
$\begin{array}[]{lll}\mathit{i}o^{\mathit{f}in}{(P)}&=&\\{(s,s^{\prime})\ |\
P\stackrel{{\scriptstyle\,\,(s,s^{\prime})\,\,}}{{\,\,===\Longrightarrow}}\\}\mbox{
for }s,s^{\prime}\in\mathcal{C}^{*}\\\
\mathit{i}o^{\mathit{i}nf}{(P)}&=&\\{(w,w^{\prime})\ |\
P\stackrel{{\scriptstyle\,\,(w,w^{\prime})\,\,}}{{\,\,===\Longrightarrow_{\omega}}}\\}\mbox{
for }w,w^{\prime}\in\mathcal{C}^{\omega}\end{array}$
We recall that the observable transition
($\stackrel{{\scriptstyle\,\,\,\,}}{{\,\,===\Longrightarrow}}$) is defined
through a finite number of internal transitions (rule $\mathrm{R}_{OBS}$ in
Figure 1). Hence, it may be the case that for some utcc processes (e.g., $Q$
in Example 4), $\mathit{i}o^{\mathit{i}nf}=\emptyset$. For this reason, we
distinguish finite and infinite sequences in the input-output behavior
relation. We notice that if $w\in\mathit{i}o^{\mathit{i}nf}(P)$ then any
finite prefix of $w$ belongs to $\mathit{i}o^{\mathit{f}in}(P)$. We shall call
_well-terminated_ the processes which do not exhibit infinite internal
behavior.
###### Definition 6 (Well-termination)
The process $P$ is said to be _well-terminated_ w.r.t. an infinite sequence
$w$ if there exists $w^{\prime}\in\mathcal{C}^{\omega}$ s.t.
$(w,w^{\prime})\in\mathit{i}o^{\mathit{i}nf}(P).$
Note that tcc processes are well-terminated since recursive calls must be
${\mathbf{n}ext}$ guarded. The fragment of well-terminated utcc processes has
been shown to be a meaningful one. For instance, in [Olarte and Valencia
(2008a)] the authors show that such fragment is enough to encode Turing-
powerful formalisms and [López et al. (2009)] shows the use of this fragment
in the declarative interpretation of languages for structured communications.
We conclude here by showing that the utcc calculus is deterministic. The
result follows from Lemma 1 (see A).
###### Theorem 1 (Determinism)
Let $s,w$ and $w^{\prime}$ be (possibly infinite) sequences of constraints. If
both $(s,w)$, $(s,w^{\prime})\in{\mathit{i}o}(P)$ then $w\cong w^{\prime}$.
#### 2.4.2 Closure Properties and Strongest Postcondition
The $\mathbf{unless}$ operator is the only construct in the language that
exhibits non-monotonic input-output behavior in the following sense: Let
$P=\mathbf{unless}\ c\ \mathbf{next}\,Q$ and $s\leq s^{\prime}$. If
$(s,w),(s^{\prime},w^{\prime})\in{\mathit{i}o}(P)$, it may be the case that
$w\not\leq w^{\prime}$. For example, take $Q=\mathbf{tell}(d)$,
$s=\operatorname{\textup{{t}}}^{\omega}$ and
$s^{\prime}=c.\operatorname{\textup{{t}}}^{\omega}$. The reader can verify
that $w=\operatorname{\textup{{t}}}.d.\operatorname{\textup{{t}}}^{\omega}$,
$w^{\prime}=c.\operatorname{\textup{{t}}}^{\omega}$ and then, $w\not\leq
w^{\prime}$.
###### Definition 7 (Monotonic Processes)
We say that $P$ is a monotonic process if it does not have occurrences of
${\mathbf{u}nless}$ processes. Similarly, the program $\mathcal{D}.P$ is
monotonic if $P$ and all $P_{i}$ in a process definition
$p_{i}(\vec{x})\operatorname{:\\!--}P_{i}$ are monotonic.
Now we show that ${\mathit{i}o}(P)$ is a _partial upper closure operator_ ,
i.e., it is a function satisfying _extensiveness_ and _idempotence_.
Furthermore, if $P$ is _monotonic_ , ${\mathit{i}o}(P)$ is a _closure
operator_ satisfying additionally monotonicity. The proof of this result
follows from Lemma 1 (see details in A).
###### Lemma 2 (Closure Properties)
Let $P$ be a process. Then, ${\mathit{i}o}(P)$ is a function. Furthermore,
${\mathit{i}o}(P)$ is a partial upper closure operator, namely it satisfies:
Extensiveness: If $(s,s^{\prime})\in{\mathit{i}o}(P)$ then $s\leq s^{\prime}$.
Idempotence: If $(s,s^{\prime})\in{\mathit{i}o}(P)$ then
$(s^{\prime},s^{\prime})\in{\mathit{i}o}(P)$.
Moreover, if $P$ is monotonic, then:
Monotonicity: If $(s_{1},s_{1}^{\prime})\in{\mathit{i}o}(P)$,
$(s_{2},s_{2}^{\prime})\in{\mathit{i}o}(P)$ and $s_{1}\leq s_{2}$, then
$s_{1}^{\prime}\leq s_{2}^{\prime}$.
A pleasant property of closure operators is that they are uniquely determined
by their set of fixpoints, here called the _strongest postcondition_.
###### Definition 8 (Strongest Postcondition)
Given a utcc process $P$, the strongest postcondition of $P$, denoted by
${\mathit{s}p}(P)$, is defined as the set
$\\{s\in\mathcal{C}^{\omega}\cup\mathcal{C}^{*}\ |\
(s,s)\in{\mathit{i}o}(P)\\}$.
Intuitively, $s\in{\mathit{s}p}(P)$ iff $P$ under input $s$ cannot add any
information whatsoever, i.e. $s$ is a quiescent sequence for $P$. We can also
think of ${\mathit{s}p}(P)$ as the set of sequences that $P$ can output under
the influence of an arbitrary environment. Therefore, proving whether $P$
satisfies a given property $A$, in the presence of any environment, reduces to
proving whether ${\mathit{s}p}(P)$ is a subset of the set of sequences
(outputs) satisfying the property $A$. Recall that
$\mathit{i}o(P)=\mathit{i}o^{\mathit{f}in}(P)\cup\mathit{i}o^{\mathit{i}nf}(P)$.
Therefore, the sequences in ${\mathit{s}p}(P)$ can be finite or infinite.
We conclude here by showing that for the monotonic fragment, the input-output
behavior can be retrieved from the strongest postcondition. The proof of this
result follows straightforward from Lemma 2 and it can be found in A.
###### Theorem 2
Let $min$ be the minimum function w.r.t. the order induced by $\leq$ and $P$
be a monotonic process. Then, $(s,s^{\prime})\in{\mathit{i}o}(P)\mbox{\ \ iff\
\ }s^{\prime}=min({\mathit{s}p}(P)\cap\\{w\ |\ s\leq w\\})$.
## 3 A Denotational model for TCC and UTCC
As we explained before, the strongest postcondition relation fully captures
the behavior of a process considering any possible output under an arbitrary
environment. In this section we develop a denotational model for the strongest
postcondition. The semantics is compositional and it is the basis for the
abstract interpretation framework that we develop in Section 4.
Our semantics is built on the closure operator semantics for ccp and tcc in
[Saraswat et al. (1991), Saraswat et al. (1994)] and [de Boer et al. (1997),
Nielsen et al. (2002a)]. Unlike the denotational semantics for utcc in [Olarte
and Valencia (2008a)], our semantics is more appropriate for the data-flow
analysis due to its simpler domain based on sequences of constraints instead
of sequences of temporal formulas. In Section 6 we elaborate more on the
differences between both semantics.
Roughly speaking, the semantics is based on a continuous immediate consequence
operator $T_{\mathcal{D}}$, which computes in a bottom-up fashion the
_interpretation_ of each process definition $p(\vec{x})\operatorname{:\\!--}P$
in $\mathcal{D}$. Such an interpretation is given in terms of the set of the
quiescent sequences for $p(\vec{x})$.
Assume a utcc program $\mathcal{D}.P$. We shall denote the set of process
names with their formal parameters in $\mathcal{D}$ as ${\mathit{P}rocHeads}$.
We shall call _Interpretations_ the set of functions in the domain
${\mathit{P}rocHeads}\rightarrow{\mathcal{P}}(\mathcal{C}^{\omega})$. We shall
define the semantics as a function
$[\\![\cdot]\\!]_{I}:({\mathit{P}rocHeads}\rightarrow{\mathcal{P}}(\mathcal{C}^{\omega}))\rightarrow({\mathit{P}roc}\rightarrow{\mathcal{P}}(\mathcal{C}^{\omega}))$
which given an interpretation $I$, associates to each process a set of
sequences of constraints.
$\begin{array}[]{llcl}\mathrm{D}_{SKIP}&[\\![\mathbf{skip}]\\!]_{I}&=&\mathcal{C}^{\omega}\\\
\mathrm{D}_{TELL}&[\\![\mathbf{tell}(c)]\\!]_{I}&=&\uparrow\\!\\!c.\mathcal{C}^{\omega}\\\
\mathrm{D}_{ASK}&[\\![\mathbf{when}\ c\ \mathbf{do}\
P]\\!]_{I}&=&\overline{\uparrow\\!\\!c}.\mathcal{C}^{\omega}\ \cup\
(\uparrow\\!\\!c.\mathcal{C}^{\omega}\cap[\\![P]\\!]_{I})\\\
\mathrm{D}_{ABS}&[\\![(\mathbf{abs}\
\vec{x};c)\,P]\\!]_{I}&=&\operatorname{\forall\forall\
\\!}\vec{x}([\\![\mathbf{when}\ c\ \mathbf{do}\ P]\\!]_{I})\\\
\mathrm{D}_{PAR}&[\\![P\parallel
Q]\\!]_{I}&=&[\\![P]\\!]_{I}\cap[\\![Q]\\!]_{I}\\\
\mathrm{D}_{LOC}&[\\![(\mathbf{local}\,\vec{x})\,P]\\!]_{I}&=&\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]_{I})\\\
{\mathrm{D}_{NEXT}}&[\\![\mathbf{next}\,P]\\!]_{I}&=&\mathcal{C}.[\\![P]\\!]_{I}\\\
{\mathrm{D}_{UNL}}&[\\![\mathbf{unless}\ c\
\mathbf{next}\,P]\\!]_{I}&=&\overline{\uparrow\\!\\!c}.[\\![P]\\!]_{I}\ \cup\
\uparrow\\!\\!c.\mathcal{C}^{\omega}\\\
\mathrm{D}_{CALL}&[\\![{p(\vec{t})}]\\!]_{I}&=&I(p(\vec{t}))\end{array}$
Figure 2: Semantic Equations for tcc and utcc constructs. Operands “.”,
$\uparrow\\!\\!$ , $\operatorname{\forall\forall\ \\!}$ and $\
\operatorname{\exists\exists\ \\!}$ are defined in Notation 5. $\overline{A}$
denotes the set complement of $A$ in $\mathcal{C}^{\omega}$.
Before defining the semantics, we introduce the following notation.
###### Notation 5 (Closures and Operators on Sequences)
Given a constraint $c$, we shall use $\uparrow\\!\\!c$ (the upward closure) to
denote the set $\\{d\in\mathcal{C}\ |\ d\vdash c\\}$, i.e., the set of
constraints entailing $c$. Similarly, we shall use $\uparrow\\!\\!s$ to denote
the set of sequences $\\{s^{\prime}\in\mathcal{C}^{\omega}\ |\ s\leq
s^{\prime}\\}$. Given $S\subseteq\mathcal{C}^{\omega}$ and
$\mathcal{C}^{\prime}\subseteq\mathcal{C}$, we shall extend the use of the
sequences-concatenation operator “.” by declaring that $c.S=\\{c.s\ |\ s\in
S\\}$, $\mathcal{C}^{\prime}.s=\\{c.s\ |\ c\in\mathcal{C}^{\prime}\\}$ and
$\mathcal{C}^{\prime}.S=\\{c.s\ |\ c\in\mathcal{C}^{\prime}\mbox{ and }s\in
S\\}$. Furthermore, given a set of sequences of constraints
$S\subseteq\mathcal{C}^{\omega}$, we define:
$\begin{array}[]{lll}\operatorname{\exists\exists\
\\!}\vec{x}(S)&=&\\{s\in\mathcal{C}^{\omega}\ |\ \mbox{ there exists
}s^{\prime}\in S\mbox{ s.t.
}\exists\vec{x}(s)\cong\exists\vec{x}(s^{\prime})\\}\\\
\operatorname{\forall\forall\ \\!}\vec{x}(S)&=&\\{\exists\vec{y}(s)\in S\ |\
\vec{y}\subseteq{\mathit{V}ar},s\in S\mbox{ and for all
}s^{\prime}\in\mathcal{C}^{\omega},\mbox{ if
}\exists\vec{x}(s)\cong\exists\vec{x}(s^{\prime})\mbox{,}\\\
&&\qquad\qquad\quad\ \ d_{\vec{x}\vec{t}}^{\omega}\leq s^{\prime}\mbox{ and
}adm(\vec{x},\vec{t})\mbox{ then }s^{\prime}\in S\\}\end{array}$
The operators above are used to define the semantic equations in Figure 2 and
explained in the following. Recall that $[\\![P]\\!]_{I}$ aims at capturing
the strongest postcondition (or quiescent sequences) of $P$, i.e. those
sequences $s$ such that $P$ under input $s$ cannot add any information
whatsoever. The process $\mathbf{skip}$ cannot add any information to any
sequence and hence, its denotation is $\mathcal{C}^{\omega}$ (Equation
$\mathrm{D}_{SKIP}$). The sequences to which $\mathbf{tell}(c)$ cannot add
information are those whose first element entails $c$, i.e., the upward
closure of $c$ (Equation $\mathrm{D}_{TELL}$). If neither $P$ nor $Q$ can add
any information to $s$, then $s$ is quiescent for $P\parallel Q$. (Equation
$\mathrm{D}_{PAR}$).
We say that $s$ is an ${\vec{x}}$-variant of $s^{\prime}$ if
$\exists\vec{x}(s)\cong\exists\vec{x}(s^{\prime})$, i.e., $s$ and $s^{\prime}$
differ only on the information about $\vec{x}$. Let
$S=\operatorname{\exists\exists\ \\!}\vec{x}(S^{\prime})$. We note that $s\in
S$ if there is an $\vec{x}$-variant $s^{\prime}$ of $s$ in $S^{\prime}$.
Therefore, a sequence $s$ is quiescent for $Q=(\mathbf{local}\,\vec{x})\,P$ if
there exists an $\vec{x}$-variant $s^{\prime}$ of $s$ s.t. $s^{\prime}$ is
quiescent for $P$. Hence, if $P$ cannot add any information to $s^{\prime}$
then $Q$ cannot add any information to $s$ (Equation $\mathrm{D}_{LOC}$).
The process $\mathbf{next}\,P$ has no influence on the first element of a
sequence. Hence if $s$ is quiescent for $P$ then $c.s$ is quiescent for
$\mathbf{next}\,P$ for any $c\in\mathcal{C}$ (Equation $\mathrm{D}_{NEXT}$).
Recall that the process $Q=\mathbf{unless}\ c\ \mathbf{next}\,P$ executes $P$
in the next time interval if and only if the guard $c$ cannot be deduced from
the store in the current time-unit. Then, a sequence $d.s$ is quiescent for
$Q$ if either $s$ is quiescent for $P$ or $d$ entails $c$ (Equation
$\mathrm{D}_{UNL}$). This equation can be equivalently written as
$\mathcal{C}.[\\![P]\\!]_{I}\ \cup\ \uparrow\\!\\!c.\mathcal{C}^{\omega}$.
Recall that the interpretation $I$ maps process names to sequences of
constraints. Then, the meaning of $p(\vec{t})$ is directly given by the
interpretation $I$ (Rule $\mathrm{D}_{CALL}$).
Let $Q=\mathbf{when}\ c\ \mathbf{do}\ P$. A sequence $d.s$ is quiescent for
$Q$ if $d$ does not entail $c$. If $d$ entails $c$, then $d.s$ must be
quiescent for $P$ (rule $\mathrm{D}_{ASK}$). In some cases, for the sake of
presentation, we may write this equations as:
$[\\![\mathbf{when}\ c\ \mathbf{do}\ P]\\!]_{I}=\\{d.s\ |\ \mbox{ if }d\vdash
c\mbox{ then }d.s\in[\\![P]\\!]_{I}\\}$
Before explaining the Rule $\mathrm{D}_{ABS}$, let us show some properties of
$\operatorname{\forall\forall\ \\!}\vec{x}(\cdot)$. First, we note that the
$\vec{x}$-variables satisfying the condition $d_{\vec{x}\vec{t}}^{\omega}\leq
s$ in the definition of $\operatorname{\forall\forall\ \\!}$ are equivalent
(see the proof in B).
###### Observation 1 (Equality and $\vec{x}$-variants)
Let $S\subseteq\mathcal{C}^{\omega}$, $\vec{z}\subseteq{\mathit{V}ar}$ and
$s,w\in\mathcal{C}^{\omega}$ be $\vec{x}$-variants such that
$d_{\vec{x}\vec{t}}^{\omega}\leq s$, $d_{\vec{x}\vec{t}}^{\omega}\leq w$ and
$adm(\vec{x},\vec{t})$. (1) $s\cong w$. (2)
$\exists\vec{z}(s)\in\operatorname{\forall\forall\ \\!}\vec{x}(S)$ iff
$s\in\operatorname{\forall\forall\ \\!}\vec{x}(S)$.
Now we establish the correspondence between the sets
$\operatorname{\forall\forall\ \\!}\vec{x}([\\![P]\\!]_{I})$ and
$[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$ which is fundamental to understand the way
we defined the operator $\operatorname{\forall\forall\ \\!}$.
###### Proposition 1
$s\in\operatorname{\forall\forall\ \\!}\vec{x}([\\![P]\\!]_{I})$ if and only
if $s\in[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$ for all admissible substitution
$[\vec{t}/\vec{x}]$.
###### Proof 3.3.
($\Rightarrow$)Let $s\in\operatorname{\forall\forall\
\\!}\vec{x}([\\![P]\\!]_{I})$ and $s^{\prime}$ be an $\vec{x}$-variant of $s$
s.t. $d_{\vec{x}\vec{t}}^{\omega}\leq s^{\prime}$ where
$adm(\vec{x},\vec{t})$. By definition of $\operatorname{\forall\forall\ \\!}$,
we know that $s^{\prime}\in[\\![P]\\!]_{I}$. Since
$d_{\vec{x}\vec{t}}^{\omega}\leq s^{\prime}$ then
$s^{\prime}\in[\\![P]\\!]_{I}\cap\uparrow\\!\\!(d_{\vec{x}\vec{t}}^{\omega})$.
Hence $s\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]_{I}\cap\uparrow\\!\\!(d_{\vec{x}\vec{t}}^{\omega}))$
and we conclude $s\in[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$.
($\Leftarrow$) Let $[\vec{t}/\vec{x}]$ be an admissible substitution. Suppose,
to obtain a contradiction, that $s\in[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$, there
exists $s^{\prime}$ $\vec{x}$-variant of $s$ s.t.
$d_{\vec{x}\vec{t}}^{\omega}\leq s^{\prime}$ and
$s^{\prime}\notin[\\![P]\\!]_{I}$ (i.e., $s\notin\operatorname{\forall\forall\
\\!}\vec{x}([\\![P]\\!]_{I})$). Since $s\in[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$
then $s\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]_{I}\cap\uparrow\\!\\!d_{\vec{x}\vec{t}}^{\omega})$.
Therefore, there exists $s^{\prime\prime}$ $\vec{x}$-variant of $s$ s.t.
$s^{\prime\prime}\in[\\![P]\\!]_{I}$ and $d_{\vec{x}\vec{t}}^{\omega}\leq
s^{\prime\prime}$. By Observation 1, $s^{\prime}\cong s^{\prime\prime}$ and
thus, $s^{\prime}\in[\\![P]\\!]_{I}$, a contradiction.
A sequence $d.s$ is quiescent for the process $Q=(\mathbf{abs}\ x;c)\,P$ if
for all admissible substitution $[\vec{t}/\vec{x}]$, either $d\not\vdash
c[\vec{t}/\vec{x}]$ or $d.s$ is also quiescent for $P[\vec{t}/\vec{x}]$, i.e.,
$d.s\in\operatorname{\forall\forall\ \\!}\vec{x}([\\![(\mathbf{when}\ c\
\mathbf{do}\ P)]\\!]_{I})$ (rule $\mathrm{D}_{ABS}$). Notice that we can
simply write Equation $\mathrm{D}_{ABS}$ by unfolding the definition of
$\mathrm{D}_{ASK}$ as follows:
$[\\![(\mathbf{abs}\ \vec{x};c)\,P]\\!]_{I}=\operatorname{\forall\forall\
\\!}\vec{x}(\overline{\uparrow\\!\\!c}.\mathcal{C}^{\omega}\ \cup\
(\uparrow\\!\\!c.\mathcal{C}^{\omega}\cap[\\![P]\\!]_{I}))$
The reader may wonder why the operator $\operatorname{\forall\forall\ \\!}$
(resp. Rule $\mathrm{D}_{ABS}$) is not entirely dual w.r.t.
$\operatorname{\exists\exists\ \\!}$ (resp. Rule $\mathrm{D}_{LOC}$), i.e.,
why we only consider $\vec{x}$-variants entailing $d_{\vec{x}\vec{t}}$ where
$[\vec{t}/\vec{x}]$ is an admissible substitution. To explain this issue, let
$Q=(\mathbf{abs}\ x;c)\,P$ where $c=\operatorname{\textup{{out}}}{(x)}$ and
$P=\mathbf{tell}(\operatorname{\textup{{out}}}^{\prime}(x))$. We know that
$s=(\operatorname{\textup{{out}}}(a)\wedge\operatorname{\textup{{out}}}^{\prime}(a)).\operatorname{\textup{{t}}}^{\omega}\in{\mathit{s}p}(Q)$
for a given constant $a$. Suppose that we were to define:
$[\\![Q]\\!]_{I}={\ \\{s\ |\ \mbox{for all $x$-variant $s^{\prime}$ of $s$ if
}s^{\prime}(1)\vdash c\mbox{ then }s^{\prime}\in[\\![P]\\!]_{I}\\}}$
Let
$c^{\prime}=\operatorname{\textup{{out}}}(a)\wedge\operatorname{\textup{{out}}}^{\prime}(a)\wedge\operatorname{\textup{{out}}}(x)$
and $s^{\prime}=c^{\prime}.\operatorname{\textup{{t}}}^{\omega}$. Notice that
$s^{\prime}$ is an $x$-variant of $s$, $s^{\prime}(1)\vdash c$ but
$s^{\prime}\notin[\\![P]\\!]_{I}$ (since
$c^{\prime}\not\vdash\operatorname{\textup{{out}}}^{\prime}(x)$). Then
$s\notin[\\![Q]\\!]_{I}$ under this naive definition of $[\\![Q]\\!]_{I}$. We
thus consider only the $\vec{x}$-variants $s^{\prime}$ s.t. each element of
$s^{\prime}$ entails $d_{\vec{x}\vec{t}}$. Intuitively, this condition
requires that $s^{\prime}(1)\vdash c\sqcup d_{\vec{x}\vec{t}}$ in Equation
$\mathrm{D}_{ABS}$ and hence that $s^{\prime}(1)\vdash c[\vec{t}/\vec{x}]$.
Furthermore $s\in[\\![P[\vec{t}/\vec{x}]]\\!]_{I}$ realizes the operational
intuition that $P$ runs under the substitution $[\vec{t}/\vec{x}]$. The
operational rule $\mathrm{R}_{STRVAR}$ makes also echo in the design of our
semantics: the operator $\operatorname{\forall\forall\ \\!}$ considers
constraints of the form $\exists\vec{z}(s)$ where $\vec{z}$ is a (possibly
empty) set of variables, thus allowing us to open the existentially quantified
constraints as shown in the following example.
###### Example 3.4 (Scope extrusion).
Let $P=\mathbf{when}\ \operatorname{\textup{{out}}}(x)\ \mathbf{do}\
\mathbf{tell}(\operatorname{\textup{{out}}}^{\prime}(x))$, $Q=(\mathbf{abs}\
\vec{x};\operatorname{\textup{{out}}}(x))\,\mathbf{tell}(\operatorname{\textup{{out}}}^{\prime}(x))$.
We know that $[\\![Q]\\!]_{I}=\operatorname{\forall\forall\
\\!}x([\\![P]\\!]_{I})$. Assume that $d.s\in[\\![P]\\!]_{I}$. Then, $d$ must
be in the set:
$C=\\{\exists
x(\operatorname{\textup{{out}}}(x)),\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}^{\prime}(x),\exists
x(\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}^{\prime}(x)),\operatorname{\textup{{out}}}(y),\operatorname{\textup{{out}}}(y)\sqcup\operatorname{\textup{{out}}}^{\prime}(y)\cdots\\}$
where either, $d\not\vdash\operatorname{\textup{{out}}}(x)$ or
$d\vdash\operatorname{\textup{{out}}}^{\prime}(x)$. We note that: (1)
$(\exists x(\operatorname{\textup{{out}}}(x))).s\notin[\\![Q]\\!]_{I}$ since
$\operatorname{\textup{{out}}}(x)\not\in C$. Similarly, $\exists
y(\operatorname{\textup{{out}}}(y)).s\notin[\\![Q]\\!]_{I}$ since
$\operatorname{\textup{{out}}}(y)\in C$ but the $x$-variant
$\operatorname{\textup{{out}}}(x)\sqcup d_{xy}\not\in C$ (it does not entail
$\operatorname{\textup{{out}}}^{\prime}(x)$). (3)
$\operatorname{\textup{{out}}}(y).s\not\in[\\![P]\\!]_{I}$ for the same
reason. (4) Let
$e=(\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}^{\prime}(x))$.
We note that $e.s\in[\\![Q]\\!]_{I}$ since $e\in C$ and there is not an
admissible substitution $[t/x]$ s.t. $\exists x(e)\cong\exists x(e[t/x])$. (5)
Let
$e=(\operatorname{\textup{{out}}}(y)\sqcup\operatorname{\textup{{out}}}^{\prime}(y))$.
Then, $e.s\in[\\![Q]\\!]_{I}$ since $e\in C$ and the $x$-variant $e\sqcup
d_{xy}\in C$. (6) Finally, if $e=\exists
x(\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}^{\prime}(x)).s$,
then $e.s\in[\\![Q]\\!]_{I}$ as in (4) and (5).
### 3.1 Compositional Semantics
We choose as semantic domain $\mathbb{E}=(E,\sqsubseteq^{c})$ where $E=\\{X\
|\ X\in\mathcal{P}(\mathcal{C}^{\omega})\mbox{ and
}\operatorname{\textup{{f}}}^{\omega}\in X\\}$ and $X\sqsubseteq^{c}Y$ iff
$X\supseteq Y$. The bottom of $\mathbb{E}$ is then $\mathcal{C}^{\omega}$ (the
set of all the sequences) and the top element is the singleton
$\\{\operatorname{\textup{{f}}}^{\omega}\\}$ (recall that
$\operatorname{\textup{{f}}}$ is the greatest element in
($\mathcal{C},\leq$)). Given two interpretations $I_{1}$ and $I_{2}$, we write
$I_{1}\sqsubseteq^{c}I_{2}$ iff for all $p$,
$I_{1}(p)\sqsubseteq^{c}I_{2}(p)$.
###### Definition 3.5 (Concrete Semantics).
Let $[\\![\cdot]\\!]_{I}$ be defined as in Figure 2. The semantics of a
program $\mathcal{D}.P$ is the least fixpoint of the continuous operator:
$\begin{array}[]{lll}T_{\mathcal{D}}(I)(p(\vec{t}))=[\\![Q[\vec{t}/\vec{x}]]\\!]_{I}\mbox{
if }p(\vec{x})\operatorname{:\\!--}Q\in\mathcal{D}\end{array}$
We shall use $[\\![P]\\!]$ to represent
$[\\![P]\\!]_{\mathit{l}fp(T_{\mathcal{D}})}$.
In the following we prove some fundamental properties of the semantic operator
$T_{\mathcal{D}}$, namely, monotonicity and continuity. Before that, we shall
show that $\operatorname{\forall\forall\ \\!}$ is a closure operator and it is
continuous on the domain $\mathbb{E}$.
###### Lemma 3.6 (Properties of $\operatorname{\forall\forall\ \\!}$).
$\operatorname{\forall\forall\ \\!}$ is a closure operator, i.e., it satisfies
(1) Extensivity: $S\sqsubseteq^{c}\operatorname{\forall\forall\
\\!}\vec{x}(S)$; (2) Idempotency: $\operatorname{\forall\forall\
\\!}\vec{x}(\operatorname{\forall\forall\
\\!}\vec{x}(S))=\operatorname{\forall\forall\ \\!}\vec{x}(S)$; and (3)
Monotonicity: If $S\sqsubseteq^{c}S^{\prime}$ then
$\operatorname{\forall\forall\
\\!}\vec{x}(S)\sqsubseteq^{c}\operatorname{\forall\forall\
\\!}\vec{x}(S^{\prime})$. Furthermore, (4) $\operatorname{\forall\forall\
\\!}$ is continuous on $(E,\sqsubseteq^{c})$.
###### Proof 3.7.
The proofs of (1),(2) and (3) are straightforward from the definition of
$\operatorname{\forall\forall\ \\!}\vec{x}$. The proof of (4) proceeds as
follows. Assume a non-empty ascending chain
$S_{1}\sqsubseteq^{c}S_{2}\sqsubseteq^{c}S_{3}\sqsubseteq^{c}...$. Lubs in $E$
correspond to set intersection. We shall prove that
$\bigcap\operatorname{\forall\forall\
\\!}\vec{x}(S_{i})=\operatorname{\forall\forall\ \\!}\vec{x}(\bigcap S_{i})$.
The “$\subseteq$” part (i.e., $\sqsupseteq^{c}$) is trivial since
$\operatorname{\forall\forall\ \\!}$ is monotonic. As for the
$\bigcap\operatorname{\forall\forall\
\\!}\vec{x}(S_{i})\subseteq\operatorname{\forall\forall\ \\!}\vec{x}(\bigcap
S_{i})$ part, by extensiveness we know that $\operatorname{\forall\forall\
\\!}\vec{x}(S_{i})\subseteq S_{i}$ for all $S_{i}$ and then,
$\bigcap\operatorname{\forall\forall\ \\!}\vec{x}(S_{i})\subseteq\bigcap
S_{i}$. Let $s\in\bigcap\operatorname{\forall\forall\ \\!}\vec{x}(S_{i})$. By
definition we know that $s$ and all $\vec{x}$-variant $s^{\prime}$ of $s$
satisfying $d_{\vec{x}\vec{t}}^{\omega}\leq s^{\prime}$ for
$adm(\vec{x},\vec{t})$ belong to $\bigcap\operatorname{\forall\forall\
\\!}\vec{x}(S_{i})$ and then in $\bigcap S_{i}$. Hence,
$s\in\operatorname{\forall\forall\ \\!}\vec{x}(\bigcap S_{i})$ and we conclude
$\bigcap\operatorname{\forall\forall\
\\!}\vec{x}(S_{i})\subseteq\operatorname{\forall\forall\ \\!}\vec{x}(\bigcap
S_{i})$.
###### Proposition 3.8 (Monotonicity of $[\\![\cdot]\\!]$ and continuity of
$T_{\mathcal{D}}$).
Let $P$ be a process and $I_{1}\sqsubseteq^{c}I_{2}\sqsubseteq^{c}I_{3}...$ be
an ascending chain. Then,
$[\\![P]\\!]_{I_{i}}\sqsubseteq^{c}[\\![P]\\!]_{I_{i+1}}$ (Monotonicity).
Moreover,
$[\\![P]\\!]_{\bigsqcup_{I_{i}}}=\bigsqcup_{I_{i}}[\\![P]\\!]_{I_{i}}$
(Continuity).
###### Proof 3.9.
Monotonicity follows easily by induction on the structure of $P$ and it
implies the the “$\sqsupseteq^{c}$” part of continuity. As for the part
“$\sqsubseteq^{c}$” we proceed by induction on the structure of $P$. The
interesting cases are those of the local and the abstraction operator. For
$P=(\mathbf{local}\,\vec{x})\,Q$, by inductive hypothesis we know that
$[\\![Q]\\!]_{\bigsqcup_{I_{i}}}\sqsubseteq^{c}\bigsqcup_{I_{i}}[\\![Q]\\!]_{I_{i}}$.
Since $\exists$ (and therefore $\operatorname{\exists\exists\ \\!}$) is
continuous (see Property (5) in Definition 1), we conclude
$\operatorname{\exists\exists\
\\!}_{\vec{x}}([\\![Q]\\!]_{\bigsqcup_{I_{i}}})\sqsubseteq^{c}\bigsqcup_{I_{i}}\operatorname{\exists\exists\
\\!}_{\vec{x}}([\\![Q]\\!]_{I_{i}})$. The result for $P=(\mathbf{abs}\
\vec{x};c)\,Q$ follows similarly from the continuity of
$\operatorname{\forall\forall\ \\!}$ (Lemma 3.6).
$\begin{array}[]{lll}I_{1}\
:&p\to\uparrow\\!\\!\operatorname{\textup{{out}}}_{a}(x).\mathcal{C}^{\omega}\
\cap\
\mathcal{C}.\uparrow\\!\\!\operatorname{\textup{{out}}}_{a}(y).\mathcal{C}^{\omega}\mbox{
i.e.,
}p\to\uparrow\\!\\!\operatorname{\textup{{out}}}_{a}(x).\uparrow\\!\\!\operatorname{\textup{{out}}}_{a}(y).\mathcal{C}^{\omega}\\\
&q\to\operatorname{\forall\forall\ \\!}z(A.\mathcal{C}^{\omega})\cap\
\mathcal{C}.I_{\bot}(q)\mbox{ i.e., }q\to\operatorname{\forall\forall\
\\!}z(A).I_{\bot}(q)\\\
&r\to\mathcal{C}^{\omega}\cap\mathcal{C}^{\omega}=\mathcal{C}^{\omega}\\\
I_{2}\ :&p\to I_{1}(p)\\\ &q\to\operatorname{\forall\forall\
\\!}z(A.\mathcal{C}^{\omega})\cap\ \mathcal{C}.I_{1}(q)\mbox{ i.e.,
}q\to\operatorname{\forall\forall\ \\!}z(A).\operatorname{\forall\forall\
\\!}z(A.\mathcal{C}^{\omega})\cap\
\mathcal{C}.\mathcal{C}.\mathcal{C}^{\omega}\\\ &r\to I_{1}(p)\cap I_{1}(q)\\\
\dots\\\ I_{\omega}:&p\to I_{1}(p)\\\ &q\to\operatorname{\forall\forall\
\\!}z(A).\operatorname{\forall\forall\ \\!}z(A).\operatorname{\forall\forall\
\\!}z(A)...\\\ &r\to I_{\omega}(p)\cap I_{\omega}(q)\end{array}$
Figure 3: Semantics of the processes in Example 3.10.
$A_{1}=\uparrow\\!\\!(\operatorname{\textup{{out}}}_{a}(z)\sqcup\operatorname{\textup{{out}}}_{b}(z))$,
$A_{2}=\overline{\uparrow\\!\\!\operatorname{\textup{{out}}}_{a}(z)}$ and
$A=A_{1}\cup A_{2}$. We abuse of the notation and we write
$\operatorname{\forall\forall\ \\!}z(A).S$ instead of
$\operatorname{\forall\forall\
\\!}z(A.\mathcal{C}^{\omega})\cap\mathcal{C}.S$.
###### Example 3.10 (Computing the semantics).
Assume two constraints $\operatorname{\textup{{out}}}_{a}(\cdot)$ and
$\operatorname{\textup{{out}}}_{b}(\cdot)$, intuitively representing outputs
of names on two different channels $a$ and $b$. Let $\mathcal{D}$ be the
following procedure definitions
$\begin{array}[]{lll}\mathcal{D}&=&p()\operatorname{:\\!--}\
\mathbf{tell}(\operatorname{\textup{{out}}}_{a}(x))\parallel\mathbf{next}\,\mathbf{tell}(\operatorname{\textup{{out}}}_{a}(y))\\\
&&q()\operatorname{:\\!--}\ (\mathbf{abs}\
z;\operatorname{\textup{{out}}}_{a}(z))\,(\mathbf{tell}(\operatorname{\textup{{out}}}_{b}(z)))\parallel\mathbf{next}\,q()\\\
&&r()\operatorname{:\\!--}\ p()\parallel q()\end{array}$
The procedure $p()$ outputs on channel $a$ the variables $x$ and $y$ in the
first and second time-units respectively. The procedure $q()$ resends on
channel $b$ every message received on channel $a$. The computation of
$[\\![r()]\\!]$ can be found in Figure 3. Let $s\in[\\![r()]\\!]$. Then, it
must be the case that $s\in[\\![p()]\\!]$ and then,
$s(1)\vdash\operatorname{\textup{{out}}}_{a}(x)$ and
$s(2)\vdash\operatorname{\textup{{out}}}_{a}(y)$. Since $r\in[\\![q()]\\!]$,
for $i\geq 1$, if $s(i)\vdash\operatorname{\textup{{out}}}_{a}(t)$ then
$s(i)\vdash\operatorname{\textup{{out}}}_{b}(t)$ for any term $t$. Hence,
$s(1)\vdash\operatorname{\textup{{out}}}_{b}(x)$ and
$s(2)\vdash\operatorname{\textup{{out}}}_{b}(y)$.
### 3.2 Semantic Correspondence
In this section we prove the soundness and completeness of the semantics.
###### Lemma 3.11 (Soundness).
Let $[\\![\cdot]\\!]$ be as in Definition 3.5. If
$P\stackrel{{\scriptstyle\,\,(d,d^{\prime})\,\,}}{{\,\,===\Longrightarrow}}{R}$
and $d\cong d^{\prime}$, then $d.[\\![R]\\!]\subseteq[\\![P]\\!]$.
###### Proof 3.12.
Assume that
$\langle\vec{x};P;d\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};P^{\prime};d^{\prime}\rangle\not\longrightarrow$,
$\exists\vec{x}(d)\cong\exists\vec{x}^{\prime}(d^{\prime})$. We shall prove
that $\exists\vec{x}(d).\operatorname{\exists\exists\
\\!}\vec{x}^{\prime}([\\![F(P^{\prime}))]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])$. We proceed by induction on the lexicographical
order on the length of the internal derivation and the structure of $P$, where
the predominant component is the length of the derivation. We present the
interesting cases. The others can be found in B.
Case $P=Q\parallel S$. Assume a derivation for $Q=Q_{1}$ and $S=S_{1}$ of the
form
$\begin{array}[]{lll}\langle\vec{z};Q\parallel
S,d\rangle&\longrightarrow^{*}&\langle\vec{z}\cup\vec{x}_{1}\cup\vec{y}_{1};Q_{1}\parallel
S_{1},c_{1}\sqcup e_{1}\rangle\\\
&\longrightarrow^{*}&\langle\vec{z}\cup\vec{x}_{i}\cup\vec{y}_{j};Q_{i}\parallel
S_{j},c_{i}\sqcup e_{j}\rangle\\\
&\longrightarrow^{*}&\langle\vec{z}\cup\vec{x}_{m}\cup\vec{y}_{n};Q_{m}\parallel
S_{n};c_{m}\sqcup e_{n}\rangle\not\longrightarrow\end{array}$
such that for $i>0$, each $Q_{i+1}$ (resp. $S_{i+1}$) is an evolution of
$Q_{i}$ (resp. $S_{i}$); $\vec{x}_{i}$ (resp. $\vec{y}_{j}$) are the variables
added by $Q$ (resp. $S$); and $c_{i}$ (resp $e_{j}$) is the information added
by $Q$ (resp. $S$). We assume by alpha-conversion that
$\vec{x}_{m}\cap\vec{y}_{n}=\emptyset$. We know that
$\exists\vec{z}(d)\cong\exists\vec{z},\vec{x}_{m},\vec{y}_{n}(c_{m}\sqcup
e_{n})$ and from $\mathrm{R}_{PAR}$ we can derive:
$\begin{array}[]{lll}\langle\vec{z}\cup\vec{y}_{n};Q;d\sqcup
e_{n}\rangle&\longrightarrow^{*}\equiv&\langle\vec{z}\cup\vec{x}_{m}\cup\vec{y}_{n};Q_{m},c_{m}\sqcup
e_{n}\rangle\not\longrightarrow\mbox{\ \ \ and \ \ \ }\\\
\langle\vec{z}\cup\vec{x}_{m};S;d\sqcup
c_{m}\rangle&\longrightarrow^{*}\equiv&\langle\vec{z}\cup\vec{x}_{m}\cup\vec{y}_{n};S_{n},c_{m}\sqcup
e_{n}\rangle\not\longrightarrow\end{array}$
By (structural) inductive hypothesis, we know that
$\exists\vec{z},\vec{y}_{n}(d\sqcup e_{n}).\operatorname{\exists\exists\
\\!}\vec{z},\vec{x}_{m},\vec{y}_{n}[\\![F(Q_{m})]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{z},\vec{y}_{n}([\\![Q]\\!]$) and also
$\exists\vec{z},\vec{x}_{m}(d\sqcup c_{m}).\operatorname{\exists\exists\
\\!}\vec{z},\vec{y}_{n},\vec{x}_{m}[\\![F(S_{n})]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{z},\vec{x}_{m}([\\![S]\\!])$. We note that
$\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]\cap[\\![Q]\\!])=\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])\cap[\\![Q]\\!]$ if
$\vec{x}\cap{\mathit{f}v}(Q)=\emptyset$ (see Proposition D.58 in D). Hence,
from the fact that
$\vec{x}_{m}\cap{\mathit{f}v}(S_{n})=\vec{y}_{n}\cap{\mathit{f}v}(Q_{m})=\emptyset$,
we conclude:
$\exists\vec{z}(d).\operatorname{\exists\exists\
\\!}\vec{z},\vec{x}_{m},\vec{y}_{n}([\\![F(Q_{m})]\\!]\cap[\\![F(S_{n})]\\!])\subseteq\operatorname{\exists\exists\
\\!}\vec{z}([\\![Q]\\!]\cap[\\![S]\\!])$
Case $P=(\mathbf{abs}\ \vec{x};c)\,Q$. From the rule $\mathrm{R}_{ABS}$, we
can show that
$\begin{array}[]{ll}\langle\vec{y};P;d\rangle&\longrightarrow^{*}\langle\vec{y}_{1};P_{1}\parallel
Q_{1}^{1}[\vec{t_{1}}/\vec{x}];d_{1}\rangle\\\
&\longrightarrow^{*}\langle\vec{y}_{2};P_{2}\parallel
Q_{1}^{2}[\vec{t_{1}}/\vec{x}]\parallel
Q_{2}^{1}[\vec{t_{2}}/\vec{x}];d_{2}\rangle\\\
&\longrightarrow^{*}\langle\vec{y}_{3};P_{3}\parallel
Q_{1}^{3}[\vec{t_{1}}/\vec{x}]\parallel
Q_{2}^{2}[\vec{t_{2}}/\vec{x}]\parallel
Q_{3}^{1}[\vec{t_{3}}/\vec{x}];d_{3}\rangle\\\ &\longrightarrow^{*}\cdots\\\
&\longrightarrow^{*}\langle\vec{y}_{n};P_{n}\parallel
Q_{1}^{m_{1}}[\vec{t_{1}}/\vec{x}]\parallel
Q_{2}^{m_{2}}[\vec{t_{2}}/\vec{x}]\parallel
Q_{3}^{m_{3}}[\vec{t_{3}}/\vec{x}]\parallel\cdots\parallel
Q_{n}^{m_{n}}[\vec{t_{n}}/\vec{x}];d_{n}\rangle\end{array}$
where $P_{n}$ takes the form $(\mathbf{abs}\ \vec{x};c;E_{n})\,Q$,
$E_{n}=\\{d_{\vec{x}\vec{t_{1}}},...,d_{\vec{x}\vec{t_{n}}}\\}$ and
$\exists\vec{y}(d)\cong\exists\vec{y}_{n}(d_{n})$. Hence, there is a
derivation (shorter than that for $P$) for each $d_{\vec{x}\vec{t_{i}}}\in
E_{n}$:
$\langle\vec{y}_{i};Q_{i}^{1}[\vec{t_{i}}/\vec{x}];d_{i}\rangle\longrightarrow^{*}\equiv\langle\vec{y}_{i}^{\prime};Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}];d_{i}^{\prime}\rangle\not\longrightarrow$
with $Q[\vec{t_{i}}/\vec{x}]=Q^{1}_{i}[\vec{t_{i}}/\vec{x}]$ and
$\exists\vec{y}_{i}(d_{i})\cong\exists\vec{y}_{i}^{\prime}(d_{i}^{\prime})$.
Therefore, by inductive hypothesis,
$\exists\vec{y}_{i}(d_{i}).\operatorname{\exists\exists\
\\!}\vec{y}_{i}^{\prime}[\\![F(Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{y_{i}}[\\![Q[\vec{t_{i}}/\vec{x}]]\\!]$
for all $d_{\vec{x}\vec{t_{i}}}\in E_{n}$. We assume, by alpha conversion,
that the variables added for each $Q_{i}^{j}$ are distinct and then, their
intersection is empty. Furthermore, we note that
$\exists\vec{y}(d)\cong\exists\vec{y}_{1}(d_{1})$. Since
$F(P_{n})=\mathbf{skip}$, we then conclude:
$\exists\vec{y}(d).\operatorname{\exists\exists\
\\!}\vec{y}_{n}[\\![F(P_{n}\parallel\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{y}[\\![\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q[\vec{t_{i}}/\vec{x}])]\\!]$
Let $d.s\in\operatorname{\exists\exists\
\\!}\vec{y}[\\![\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q[\vec{t_{i}}/\vec{x}])]\\!]$. For an admissible $d_{\vec{x}\vec{t}}$,
either $d\not\vdash c[\vec{t}/\vec{x}]$ or $d\vdash c[\vec{t}/\vec{x}]$. In
the first case, trivially $d.s\in[\\![(\mathbf{when}\ c\ \mathbf{do}\
Q)[\vec{t}/\vec{x}]]\\!]$. In the second case, $E_{n}\Vdash
d_{\vec{x}\vec{t}}$. Hence, $d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]$ and
$d.s\in[\\![(\mathbf{when}\ c\ \mathbf{do}\ Q)[\vec{t}/\vec{x}]]\\!]$. Here we
conclude that for all admissible $[\vec{t}/\vec{x}]$,
$d.s\in[\\![(\mathbf{when}\ c\ \mathbf{do}\ Q)[\vec{t}/\vec{x}]]\\!]$ and by
Proposition 1 we derive:
$\exists\vec{y}(d).\operatorname{\exists\exists\
\\!}\vec{y}[\\![F(\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{y}\operatorname{\forall\forall\ \\!}\vec{x}[\\![(\mathbf{when}\ c\
\mathbf{do}\ Q)]\\!]$
Case $P=p(\vec{t})$. Assume that $p(\vec{x}):-Q\in\mathcal{D}$. We can verify
that
$\langle\vec{y};p(\vec{t});d\rangle\longrightarrow\langle\vec{y};Q[\vec{t}/\vec{x}];d\rangle\longrightarrow^{*}\langle\vec{y}^{\prime};Q^{\prime};d^{\prime}\rangle\not\longrightarrow$
where $\exists\vec{y}^{\prime}(d^{\prime})\cong\exists\vec{y}(d)$. By
induction $\exists\vec{y}(d).\operatorname{\exists\exists\
\\!}\vec{y}^{\prime}[\\![F(Q^{\prime})]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{y}[\\![Q[\vec{t}/\vec{x}]]\\!]$ and we conclude
$\exists\vec{y}(d).\operatorname{\exists\exists\
\\!}\vec{y}[\\![F(Q^{\prime})]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{y}[\\![p(\vec{t})]\\!]$.
The previous lemma allows us to prove the soundness of the semantics.
###### Theorem 3.13 (Soundness).
If $s\in{\mathit{s}p}(P)$ then there exists $s^{\prime}$ s.t.
$s.s^{\prime}\in[\\![P]\\!]$.
###### Proof 3.14.
If $P$ is well-terminated under input $s$, let $s^{\prime}=\epsilon$. By
repeated applications of Lemma 3.11, $s\in[\\![P]\\!]$. If $P$ is not well-
terminated, then $s$ is finite and let
$s^{\prime}=\operatorname{\textup{{f}}}^{\omega}$ (recall that
$\operatorname{\textup{{f}}}^{\omega}$ is quiescent for any process). Via
Lemma 3.11 we can show $s.s^{\prime}\in[\\![P]\\!]$.
Moreover, the semantics approximates any infinite computation.
###### Corollary 3.15 (Infinite Computations).
Assume that $d.s\in\operatorname{\exists\exists\
\\!}\vec{x}_{1}([\\![P_{1}]\\!]\cap\uparrow\\!\\!(c_{1}.\mathcal{C}^{\omega}))$
and that
$\left\langle{\vec{x}_{1};P_{1};c_{1}}\right\rangle\longrightarrow^{*}\left\langle{\vec{x}_{i};P_{i};c_{i}}\right\rangle\longrightarrow^{*}\left\langle{\vec{x}_{n};P_{n};c_{n}}\right\rangle\longrightarrow^{*}\cdots.$
Then, $\bigsqcup\exists\vec{x}_{i}(c_{i})\leq d$.
###### Proof 3.16.
Recall that procedure calls must be next guarded. Then, any infinite behavior
in $P_{1}$ is due to a process of the form $(\mathbf{abs}\ \vec{x};c)\,Q$ that
executes $Q[\vec{t}_{i}/\vec{x}]$ and adds new information of the form
$e[\vec{t}_{i}/\vec{x}]$. By an analysis similar to that of Lemma 3.11, we can
show that $d$ entails $e[\vec{t}_{i}/\vec{x}]$.
###### Example 3.17 (Infinite behavior).
Let $P=(\mathbf{abs}\
z;\operatorname{\textup{{out}}}(z))\,(\mathbf{local}\,x)\,(\mathbf{tell}(\operatorname{\textup{{out}}}(x)))$
and let $c=\operatorname{\textup{{out}}}(w)$. Starting from the store $c$, the
process $P$ engages in infinitely many internal transitions of the form
$\begin{array}[]{c}\left\langle{\emptyset;P;c}\right\rangle\longrightarrow^{*}\left\langle{\\{x_{1},\cdots,x_{i}\\};P_{i};\operatorname{\textup{{out}}}(x_{1})\sqcup\cdots\sqcup\operatorname{\textup{{out}}}(x_{i})\sqcup\operatorname{\textup{{out}}}(w)}\right\rangle\longrightarrow^{*}\\\
\left\langle{\\{x_{1},\cdots,x_{i},\cdots,x_{n}\\};P_{n};\operatorname{\textup{{out}}}(x_{1})\sqcup\cdots\sqcup\operatorname{\textup{{out}}}(x_{n})\sqcup\operatorname{\textup{{out}}}(w)}\right\rangle\longrightarrow^{*}\cdots\end{array}$
At any step of the computation, the observable store is
$\operatorname{\textup{{out}}}(w)\sqcup\bigsqcup\limits_{i\in 1..n}\exists
x_{i}\operatorname{\textup{{out}}}(x_{i})$ which is equivalent to
$\operatorname{\textup{{out}}}(w)$. Note also that
$\operatorname{\textup{{out}}}(w).\mathcal{C}^{\omega}\in[\\![P]\\!]$.
For the converse of Theorem 3.13, we have similar technical problems as in the
case of tcc, namely: the combination of the $\mathbf{local}$ operator with the
$\mathbf{unless}$ constructor. Thus, similarly to tcc, completeness is
verified only for the fragment of utcc where there are no occurrences of
$\mathbf{unless}$ processes in the body of $\mathbf{local}$ processes. The
reader may refer [de Boer et al. (1995), Nielsen et al. (2002a)] for
counterexamples showing that $[\\![P]\\!]\not\subseteq{\mathit{s}p}(P)$ when
$P$ is not locally independent.
###### Definition 3.18 (Locally Independent Fragment).
Let $\mathcal{D}.P$ be a program where $\mathcal{D}$ contains process
definitions of the form $p_{i}(\vec{x})\operatorname{:\\!--}P_{i}$. We say
that $\mathcal{D}.P$ is locally independent if for each process of the form
$(\mathbf{local}\,\vec{x};c)\,Q$ in $P$ and $P_{i}$ it holds that (1) $Q$ does
not have occurrences of $\mathbf{unless}$ processes; and (2) if $Q$ calls to
$p_{j}(\vec{x})$, then $P_{j}$ satisfies also conditions (1) and (2).
###### Lemma 3.19 (Completeness).
Let $\mathcal{D}.P$ be a locally independent program s.t. $d.s\in[\\![P]\\!]$.
If
$P\stackrel{{\scriptstyle\,\,(d,d^{\prime})\,\,}}{{\,\,===\Longrightarrow}}R$
then $d^{\prime}\cong d$ and $s\in[\\![R]\\!]$.
###### Proof 3.20.
Assume that $P$ is locally independent, $d.s\in[\\![P]\\!]$ and there is a
derivation of the form
$\langle\vec{x};P;d\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};P^{\prime};d^{\prime}\rangle\not\longrightarrow$.
We shall prove that
$\exists\vec{x}(d)\cong\exists\vec{x}^{\prime}(d^{\prime})$ and
$s\in\operatorname{\exists\exists\
\\!}\vec{x}^{\prime}[\\![F(P^{\prime})]\\!]$. We proceed by induction on the
lexicographical order on the length of the internal derivation
($\longrightarrow^{*}$) and the structure of $P$, where the predominant
component is the length of the derivation. The locally independent condition
is used for the case $P=(\mathbf{local}\,\vec{x};c)\,Q$. We only present the
interesting cases. The others can be found in B.
Case $P=Q\parallel S$. We know that $d.s\in[\\![Q]\\!]$ and
$d.s\in[\\![S]\\!]$ and by (structural) inductive hypothesis, there are
derivations
$\langle\vec{z};Q;d\rangle\longrightarrow^{*}\langle\vec{z}\cup\vec{x}^{\prime};Q^{\prime};d^{\prime}\sqcup
c\rangle\ \not\longrightarrow$ and
$\langle\vec{z};S;d\rangle\longrightarrow^{*}\langle\vec{z}\cup\vec{y}^{\prime};S^{\prime};d^{\prime\prime}\sqcup
e\rangle\ \not\longrightarrow$ s.t. $s\in\operatorname{\exists\exists\
\\!}\vec{z},\vec{x}^{\prime}[\\![F(Q^{\prime})]\\!]$,
$s\in\operatorname{\exists\exists\
\\!}\vec{z},\vec{y}^{\prime}[\\![F(S^{\prime})]\\!]$,
$\exists\vec{z}(d)\cong\exists\vec{z},\vec{x}^{\prime}(d^{\prime}\sqcup c)$
and
$\exists\vec{z}(d)\cong\exists\vec{z},\vec{y}^{\prime}(d^{\prime\prime}\sqcup
e)$. Therefore, assuming by alpha conversion that
$\vec{x}^{\prime}\cap\vec{y}^{\prime}=\emptyset$,
$\exists\vec{z}(d)\cong\exists\vec{z},\vec{x}^{\prime},\vec{y}^{\prime}(d^{\prime}\sqcup
d^{\prime\prime}\sqcup c\sqcup e)$ and by rule $\mathrm{R}_{PAR}$,
$\langle\vec{z};Q\parallel
S,d\rangle\longrightarrow^{*}\equiv\langle\vec{z}\cup\vec{x}^{\prime}\cup\vec{y}^{\prime};Q^{\prime}\parallel
S^{\prime};d^{\prime}\sqcup d^{\prime\prime}\sqcup c\sqcup
e\rangle\not\longrightarrow$
We note that $\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]\cap[\\![Q]\\!])=\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])\cap[\\![Q]\\!]$ if
$\vec{x}\cap{\mathit{f}v}(Q)=\emptyset$ (see Proposition D.58 in D). Since
$F(Q^{\prime}\parallel S^{\prime})=F(Q^{\prime})\parallel F(S^{\prime})$ and
$\vec{x}^{\prime}\cap{\mathit{f}v}(S^{\prime})=\vec{y}^{\prime}\cap{\mathit{f}v}(Q^{\prime})=\emptyset$,
we conclude $s\in\operatorname{\exists\exists\
\\!}\vec{z},\vec{x}^{\prime},\vec{y}^{\prime}([\\![F(Q^{\prime}\parallel
R^{\prime})]\\!])$.
Case $P=(\mathbf{abs}\ \vec{x};c)\,Q$. By using the rule $\mathrm{R}_{ABS}$ we
can show that:
$\begin{array}[]{ll}\langle\vec{x};P;d\rangle&\longrightarrow^{*}\langle\vec{y}_{1};P_{1}\parallel
Q_{1}^{1}[\vec{t_{1}}/\vec{x}];d_{1}^{1}\rangle\\\
&\longrightarrow^{*}\langle\vec{y}_{2};P_{2}\parallel
Q_{1}^{2}[\vec{t_{1}}/\vec{x}]\parallel
Q_{2}^{1}[\vec{t_{2}}/\vec{x}];d_{1}^{2}\sqcup d_{2}^{1}\rangle\\\
&\longrightarrow^{*}\langle\vec{y}_{3};P_{3}\parallel
Q_{1}^{3}[\vec{t_{1}}/\vec{x}]\parallel
Q_{2}^{2}[\vec{t_{2}}/\vec{x}]\parallel
Q_{3}^{1}[\vec{t_{3}}/\vec{x}];d_{1}^{3}\sqcup d_{2}^{2}\sqcup
d_{3}^{1}\rangle\\\ &\longrightarrow^{*}\cdots\\\
&\longrightarrow^{*}\langle\vec{y}_{n};P_{n}\parallel
Q_{1}^{m_{1}}[\vec{t_{1}}/\vec{x}]\parallel\cdots\parallel
Q_{n}^{m_{n}}[\vec{t_{n}}/\vec{x}];d_{1}^{m_{1}}\sqcup...\sqcup
d_{n}^{m_{n}}\rangle\end{array}$
where $P_{n}$ takes the form $(\mathbf{abs}\ \vec{x};c;E_{n})\,Q$ and
$E_{n}=\\{d_{\vec{x}\vec{t_{1}}},...,d_{\vec{x}\vec{t_{n}}}\\}$. In the
derivation above, $d_{i}^{j}$ represents the constraint added by
$Q_{i}^{j}[\vec{t_{i}}/\vec{x}]$. Note that
$Q[\vec{t_{i}}/\vec{x}]=Q_{i}^{1}[\vec{t_{i}}/\vec{x}]$. There is a derivation
(shorter than that for $P$) for each $d_{\vec{x}\vec{t_{i}}}\in E_{n}$ of the
form
$\langle\vec{y}_{i};Q_{i}^{1}[\vec{t_{i}}/\vec{x}];d_{i}\rangle\longrightarrow^{*}\equiv\langle\vec{y}_{i}^{\prime};Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}];d_{i}^{m_{i}}\rangle\not\longrightarrow$
Since $d.s\in[\\![P]\\!]$, by Proposition 1 we know that
$d.s\in[\\![Q_{i}^{1}[\vec{t_{i}}/\vec{x}]]\\!]$ and by induction,
$\exists\vec{y}_{i}(d_{i})\cong\exists\vec{y}_{i}^{\prime}(d_{i}^{m_{i}})$.
Furthermore, it must be the case that $s\in\operatorname{\exists\exists\
\\!}\vec{y}_{i}^{\prime}[\\![F(Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]$. Let
$e$ be the constraint $\exists\vec{y}_{n}(d_{1}^{m_{1}}\sqcup...\sqcup
d_{n}^{m_{n}})$. Given that
$\exists\vec{y}_{i}(d_{i})\cong\exists\vec{y}_{i}^{\prime}(d_{i}^{m_{i}})$, we
have $\exists\vec{x}(d)\cong e$. Furthermore, given that
$F(P_{n})=\mathbf{skip}$:
$(\mathbf{abs}\
\vec{x};c)\,Q\stackrel{{\scriptstyle\,\,(d,e)\,\,}}{{\,\,===\Longrightarrow}}(\mathbf{local}\,\vec{y}_{n})\,F\left(\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}]\right)$
Since $s\in\operatorname{\exists\exists\
\\!}\vec{y}_{i}^{\prime}[\\![F(Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]$ for
all $d_{\vec{x}\vec{t}_{i}}\in E_{n}$, we conclude
$s\in\operatorname{\exists\exists\
\\!}\vec{y}_{n}[\\![F(\prod\limits_{d_{\vec{x}\vec{t_{i}}}\in
E_{n}}Q_{i}^{m_{i}}[\vec{t_{i}}/\vec{x}])]\\!]$
Case $P=(\mathbf{local}\,\vec{x})\,Q$. By alpha conversion assume
$\vec{x}\not\in{\mathit{f}v}(d.s)$. We know that there exists
$d^{\prime}.s^{\prime}$ ($\vec{x}$-variant of $d.s$) s.t.
$d^{\prime}.s^{\prime}\in[\\![Q]\\!]$, $\exists\vec{x}(d.s)\cong d.s$ and
$d.s\cong\exists\vec{x}(d^{\prime}.s^{\prime})$. By (structural) inductive
hypothesis, there is a derivation
$\langle\vec{y};Q;d^{\prime}\rangle\longrightarrow^{*}\langle\vec{y}^{\prime};Q^{\prime};d^{\prime\prime}\rangle\not\longrightarrow$
and $\exists\vec{y}(d^{\prime})\cong\exists\vec{y}^{\prime}(d^{\prime\prime})$
and $s^{\prime}\in\operatorname{\exists\exists\
\\!}\vec{y}^{\prime}[\\![F(Q^{\prime})]\\!]$. We assume by alpha conversion
that $\vec{x}\cap\vec{y}=\emptyset$. Consider now the following derivation:
$\langle\vec{y};(\mathbf{local}\,\vec{x})\,Q;d\rangle\longrightarrow\langle\vec{x}\cup\vec{y};Q;d\rangle\longrightarrow^{*}\langle\vec{y}^{\prime\prime};Q^{\prime\prime},c\rangle\not\longrightarrow$
where $\vec{x}\cup\vec{y}\subseteq\vec{y}^{\prime\prime}$. We know that
$d^{\prime}\vdash d$ and by monotonicity, we have
$\exists\vec{y}^{\prime}(d^{\prime\prime})\vdash\exists\vec{y}^{\prime\prime}(c)$
and then, $d^{\prime}\vdash\exists\vec{y}^{\prime\prime}(c)$. We then conclude
$\exists\vec{y}(d)\vdash\exists\vec{y}^{\prime\prime}(c)$.
Since $s^{\prime}\in\operatorname{\exists\exists\
\\!}\vec{y}^{\prime}[\\![F(Q^{\prime})]\\!]$ then
$s\in\operatorname{\exists\exists\ \\!}\vec{x}\operatorname{\exists\exists\
\\!}\vec{y}^{\prime}[\\![F(Q^{\prime})]\\!]$. Nevertheless, notice that in the
above derivation of $(\mathbf{local}\,\vec{x})\,Q$, the final process is
$Q^{\prime\prime}$ and not $Q^{\prime}$. Since $Q$ is monotonic, there are no
$\mathbf{unless}$ processes in it. Furthermore, since $d^{\prime}\vdash d$, it
must be the case that $Q^{\prime}$ may contain sub-terms (in parallel
composition) of the form $R^{\prime}[\vec{t}/\vec{x}]$ resulting from a
process of the form $(\mathbf{abs}\ \vec{y};e)\,R$ s.t.
$d^{\prime\prime}\vdash e[\vec{t}/\vec{x}]$ and $c\not\vdash
e[\vec{t}/\vec{x}]$. Therefore, by Rule $\mathrm{D}_{PAR}$, it must be also
the case that $s^{\prime}\in[\\![F(Q^{\prime\prime})]\\!]$ and then,
$s\in\operatorname{\exists\exists\
\\!}\vec{x},\vec{y}^{\prime}[\\![{F(Q^{\prime\prime})}]\\!]$. Finally, note
that $\vec{y}^{\prime\prime}$ is not necessarily equal to $\vec{y}^{\prime}$.
With a similar analysis we can show that in $Q^{\prime}$ there are possibly
more $\mathbf{local}$ processes running in parallel than in $Q^{\prime\prime}$
and then, $s\in\operatorname{\exists\exists\
\\!}\vec{y}^{\prime\prime}[\\![{F(Q^{\prime\prime})}]\\!]$.
By repeated applications of the previous Lemma, we show the completeness of
the denotation with respect to the strongest postcondition relation.
###### Theorem 3.21 (Completeness).
Let $\mathcal{D}.P$ be a locally independent program, $w=s_{1}.s_{1}^{\prime}$
and $w\in[\\![P]\\!]$. If
$P\stackrel{{\scriptstyle\,\,(s_{1},s_{1}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}$
then $s_{1}\cong s_{1}^{\prime}$. Furthermore, if
$P\stackrel{{\scriptstyle\,\,(w,w^{\prime})\,\,}}{{\,\,===\Longrightarrow}}_{\omega}$
then $w\cong w^{\prime}$.
Notice that completeness of the semantics holds only for the locally
independent fragment, while soundness is achieved for the whole language. For
the abstract interpretation framework we develop in the next section, we
require the semantics to be a sound approximation of the operational semantics
and then, the restriction imposed for completeness does not affect the
applicability of the framework.
## 4 Abstract Interpretation Framework
In this section we develop an abstract interpretation framework [Cousot and
Cousot (1992)] for the analysis of utcc (and tcc) programs. The framework is
based on the above denotational semantics, thus allowing for a compositional
analysis. The abstraction proceeds as a composition of two different
abstractions: (1) we abstract the constraint system and then (2) we abstract
the infinite sequences of _abstract_ constraints. The abstraction in (1)
allows us to reuse the most popular abstract domains previously defined for
logic programming. Adapting those domains, it is possible to perform, e.g.,
groundness, freeness, type and suspension analyses of utcc programs. On the
other hand, the abstraction in (2) along with (1) allows for computing the
approximated output of the program in a finite number of steps.
### 4.1 Abstract Constraint Systems
Let us recall some notions from [Falaschi et al. (1997a)] and [Zaffanella et
al. (1997)].
###### Definition 4.22 (Descriptions).
A description $(\mathcal{C},\alpha,\mathcal{A})$ between two constraint
systems
$\begin{array}[]{lll}{\mathbf{C}}&=&\langle\mathcal{C},\leq\
,\sqcup,\operatorname{\textup{{t}}},\operatorname{\textup{{f}}},{\mathit{V}ar},\exists,D\rangle\\\
{\mathbf{A}}&=&\langle\mathcal{A},\leq^{\alpha},\sqcup^{\alpha},\operatorname{\textup{{t}}}^{\alpha},\operatorname{\textup{{f}}}^{\alpha},{\mathit{V}ar},\exists^{\alpha},D^{\alpha}\rangle\end{array}$
consists of an abstract domain $(\mathcal{A},\leq^{\alpha})$ and a surjective
and monotonic abstraction function $\alpha:\mathcal{C}\to\mathcal{A}$. We lift
$\alpha$ to sequences of constraints in the obvious way.
We shall use $c_{\alpha}$, $d_{\alpha}$ to range over constraints in
${\mathcal{A}}$ and
$s_{\alpha},s^{\prime}_{\alpha},w_{\alpha},w^{\prime}_{\alpha},$ to range over
sequences in $\mathcal{A}^{*}$ and $\mathcal{A}^{\omega}$ (the set of finite
and infinite sequences of constraints in $\mathcal{A}$). To simplify the
notation, we omit the subindex “$\alpha$” when no confusion arises. The
entailment $\vdash^{\alpha}$ is defined as in the concrete counterpart, i.e.
$c_{\alpha}\leq^{\alpha}d_{\alpha}$ iff $d_{\alpha}\vdash^{\alpha}c_{\alpha}$.
Similarly, $d_{\alpha}\cong_{\alpha}c_{\alpha}$ iff
$d_{\alpha}\vdash^{\alpha}c_{\alpha}$ and
$c_{\alpha}\vdash^{\alpha}d_{\alpha}$.
Following standard lines in [Giacobazzi et al. (1995), Falaschi et al.
(1997a), Zaffanella et al. (1997)] we impose the following restrictions over
$\alpha$ relating the cylindrification, diagonal and $lub$ operators of
${\mathbf{C}}$ and ${\mathbf{A}}$.
###### Definition 4.23 (Correctness).
Let $\alpha:\mathcal{C}\to\mathcal{A}$ be monotonic and surjective. We say
that ${\mathbf{A}}$ is _upper correct_ w.r.t. the constraint system
${\mathbf{C}}$ if for all $c\in\mathcal{C}$ and $x,y\in Var$:
(1)
$\alpha(\exists\vec{x}(c))\cong_{\alpha}\exists^{\alpha}\vec{x}(\alpha(c))$.
(2) $\alpha(d_{\vec{x}\vec{t}})\cong_{\alpha}d^{\alpha}_{\vec{x}\vec{t}}$.
Since $\alpha$ is monotonic, we also have $\alpha(c\sqcup
d)\vdash^{\alpha}\alpha(c)\sqcup^{\alpha}\alpha(d)$.
In the example below we illustrate an abstract domain for the groundness
analysis of tcc programs. Here we give just an intuitive description of it. We
shall elaborate more on this domain and its applications in Section 5.2.
###### Example 4.24 (Constraint System for Groundness).
Let the concrete constraint system ${\mathbf{C}}$ be the Herbrand constraint
system. As abstract constraint system A, let constraints be propositional
formulas representing groundness information as in $x\wedge(y\leftrightarrow
z)$ that means, $x$ is a ground variable and, $y$ is ground iff $z$ is ground.
In this setting, $\alpha(x=[a])=x$ (i.e., $x$ is a ground variable).
Furthermore, $\alpha(x=[a|y])=x\leftrightarrow y$ meaning $x$ is ground if and
only if $y$ is ground.
In the following definition we make precise the idea when an abstract
constraint approximates a concrete one.
###### Definition 4.25 (Approximations).
Let $(\mathcal{C},\alpha,\mathcal{A})$ be a description satisfying the
conditions in Definition 4.22. Given $d_{\alpha}=\alpha(d)$, we say that
$d_{\alpha}$ is the best approximation of $d$. Furthermore, for all
$c_{\alpha}\leq^{\alpha}d_{\alpha}$ we say that $c_{\alpha}$ approximates $d$
and we write $c_{\alpha}\propto d$. This definition is pointwise extended to
sequences of constraints in the obvious way (see Figure 4a).
(a) (b)
Figure 4: (a). $c^{\prime}_{\alpha}$ approximates $c$ (i.e.,
$c^{\prime}_{\alpha}\propto c$) and $c_{\alpha}=\alpha(c)$ is the best
approximation of $c$ (Definition 4.25). Since $\alpha$ is monotonic and $c\leq
d$, $c_{\alpha}\leq^{\alpha}d_{\alpha}$. In (b), assume that for all $d$ s.t.
$d\not\vdash c$, $d$ is not approximated by $c_{\alpha}$. Then, all constraint
$c^{\prime}$ approximated by $c_{\alpha}$ (the upper cone of $c$) entails $c$.
In this case, $c_{\alpha}\vdash_{\mathcal{A}}c$ (Definition 4.26).
### 4.2 Abstract Semantics
Now we define an abstract semantics that approximates the observable behavior
of a program and is adequate for modular data-flow analysis. The semantic
equations are given in Figure 5 and they are parametric on the abstraction
function $\alpha$ of the description $(\mathcal{C},\alpha,\mathcal{A})$. We
shall dwell a little upon the description of the rules $\mathrm{A}_{ASK}$ and
$\mathrm{A}_{UNL}$. The other cases are self-explanatory.
$\begin{array}[]{llcl}\mathrm{A}_{SKIP}&[\\![\mathbf{skip}]\\!]^{\alpha}_{X}&=&\mathcal{A}^{\omega}\\\
\mathrm{A}_{TELL}&[\\![\mathbf{tell}(c)]\\!]^{\alpha}_{X}&=&\uparrow\\!\\!(\alpha(c)).\mathcal{A}^{\omega}\\\
\mathrm{A}_{ASK}&[\\![\mathbf{when}\ c\ \mathbf{do}\
P]\\!]^{\alpha}_{X}&=&\overline{\Uparrow\\!\\!c}.\mathcal{A}^{\omega}\cup(\Uparrow\\!\\!c.\mathcal{A}^{\omega}\cap[\\![P]\\!]^{\alpha}_{X})\\\
\mathrm{A}_{ABS}&[\\![(\mathbf{abs}\
\vec{x};c)\,P]\\!]^{\alpha}_{X}&=&\operatorname{\forall\forall\
\\!}\vec{x}([\\![\mathbf{when}\ c\ \mathbf{do}\ P]\\!]^{\alpha}_{X})\\\
\mathrm{A}_{PAR}&[\\![P\parallel
Q]\\!]^{\alpha}_{X}&=&[\\![P]\\!]^{\alpha}_{X}\cap[\\![Q]\\!]^{\alpha}_{X}\\\
\mathrm{A}_{LOC}&[\\![(\mathbf{local}\,\vec{x})\,P]\\!]^{\alpha}_{X}&=&\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]^{\alpha}_{X})\\\
\mathrm{A}_{NEXT}&[\\![\mathbf{next}\,P]\\!]^{\alpha}_{X}&=&\mathcal{A}.[\\![P]\\!]^{\alpha}_{X}\\\
\mathrm{A}_{UNL}&[\\![\mathbf{unless}\ c\
\mathbf{next}\,P]\\!]^{\alpha}_{X}&=&\mathcal{A}^{\omega}\\\
\mathrm{A}_{CALL}&[\\![p(\vec{t})]\\!]^{\alpha}_{X}&=&X(p(\vec{t}))\end{array}$
Figure 5: Abstract denotational semantics for utcc. $\vdash_{\mathcal{A}}$ and
$\Uparrow\\!\\!$ are in Definition 4.26. $\overline{A}$ denotes the set
complement of $A$.
Given the right abstraction of the synchronization mechanism of blocking asks
in ccp is crucial to give a safe approximation of the behavior of programs. In
abstract interpretation, abstract elements are _weaker_ than the concrete
ones. Hence, if we approximate the behavior of $\mathbf{when}\ c\ \mathbf{do}\
P$ by replacing the guard $c$ with $\alpha(c)$, it could be the case that $P$
proceeds in the abstract semantics but it does not in the concrete one. More
precisely, let $d,c\in\mathcal{C}$. Notice that from
$\alpha(d)\vdash^{\alpha}\alpha(c)$ we cannot, in general, conclude $d\vdash
c$. Take for instance the constraint systems in Example 4.24. We know that
$\alpha(x=a)\cong^{\alpha}\alpha(x=b)$ but $x=a\not\vdash x=b$. Assume now we
were to define the abstract semantics of ask processes as:
$[\\![\mathbf{when}\ c\ \mathbf{do}\
Q]\\!]^{\alpha}_{X}=\overline{\uparrow(\alpha(c))}.\mathcal{A}^{\omega}\cup(\uparrow(\alpha(c)).\mathcal{A}^{\omega}\cap[\\![Q]\\!]^{\alpha}_{X})$
(1)
A correct analysis of the process $P=\mathbf{tell}(x=a)\parallel\mathbf{when}\
x=b\ \mathbf{do}\ \mathbf{tell}(y=b)$ should conclude that only $x$ is
definitely ground. Since $\alpha(x=a)\vdash^{\alpha}\alpha(x=b)$, if we use
Equation 1, the analysis ends with the result $(x\wedge
y).\mathcal{A}^{\omega}$, i.e., it wrongly concludes that $x$ and $y$ are
definitely ground.
We thus follow [Zaffanella et al. (1997), Falaschi et al. (1993), Falaschi et
al. (1997a)] for the abstract semantics of the ask operator. For this, we need
to define the entailment $\vdash_{\mathcal{A}}$ that relates constraints in
$\mathcal{A}$ and $\mathcal{C}$.
###### Definition 4.26 ($\vdash_{\mathcal{A}}$ relation).
Let $d_{\alpha}\in\mathcal{A}$ and $c\in\mathcal{C}$. We say that $d_{\alpha}$
entails $c$, notation $d_{\alpha}\vdash_{\mathcal{A}}c$, if for all
$c^{\prime}\in\mathcal{C}$ s.t. $d_{\alpha}\propto c^{\prime}$ it holds that
$c^{\prime}\vdash c$. We shall use $\Uparrow\\!\\!c$ to denote the set
$\\{d_{\alpha}\in\mathcal{A}\ |\ d_{\alpha}\vdash_{\mathcal{A}}c\\}$.
In words, the (abstract) constraint $d_{\alpha}$ entails the (concrete)
constraint $c$ if all constraints approximated by $d_{\alpha}$ entail $c$ (see
Figure 4b). Then, in Equation $\mathrm{A}_{ASK}$, we guarantee that if the
abstract computation proceeds (i.e., $d_{\alpha}\vdash_{\mathcal{A}}c$) then
every concrete computation it approximates proceeds too.
In Equations $\mathrm{D}_{ABS}$ and $\mathrm{D}_{LOC}$ we use the operators
$\operatorname{\forall\forall\ \\!}$ and $\operatorname{\exists\exists\ \\!}$
analogous to those in Notation 5. In this context, they are defined on
sequences of constraints in $\mathcal{A}^{\omega}$ and they use the elements
$\exists^{\alpha}$, $\sqcup^{\alpha}$ and $d^{\alpha}_{\vec{x}\vec{t}}$
instead of their concrete counterparts:
$\begin{array}[]{lll}\operatorname{\exists\exists\
\\!}\vec{x}(S_{\alpha})&=&\\{s_{\alpha}\in\mathcal{A}^{\omega}\ |\ \mbox{
there exists }s^{\prime}_{\alpha}\in S_{\alpha}\mbox{ s.t.
}\exists^{\alpha}\vec{x}(s_{\alpha})\cong_{\alpha}\exists^{\alpha}\vec{x}(s^{\prime}_{\alpha})\\}\\\
\operatorname{\forall\forall\
\\!}\vec{x}(S_{\alpha})&=&\\{\exists^{\alpha}\vec{y}(s_{\alpha})\in
S_{\alpha}\ |\ \vec{y}\subseteq{\mathit{V}ar},s_{\alpha}\in S_{\alpha}\mbox{
and for all }s^{\prime}_{\alpha}\in\mathcal{A}^{\omega},\\\ &&\quad\ \mbox{ if
}\exists^{\alpha}\vec{x}(s_{\alpha})\cong\exists^{\alpha}\vec{x}(s^{\prime}_{\alpha})\mbox{,}(d_{\vec{x}\vec{t}}^{\alpha})^{\omega}\leq
s_{\alpha}^{\prime}\mbox{ and }adm(\vec{x},\vec{t})\mbox{ then
}s^{\prime}_{\alpha}\in S_{\alpha}\\}\end{array}$
We omitted the superindex “$\alpha$” in these operators since it can be easily
inferred from the context.
The abstract semantics of the $\mathbf{unless}$ operator poses similar
difficulties as in the case of the ask operator. Moreover, even if we make use
of the entailment $\vdash_{\mathcal{A}}$ in Definition 4.26, we do not obtain
a safe approximation. Let us explain this. One could think of defining the
semantic equation for the unless process as follows:
$[\\![\mathbf{unless}\ c\
\mathbf{next}\,Q]\\!]^{\alpha}_{X}=\overline{\Uparrow\\!\\!c}.[\\![Q]\\!]^{\alpha}_{X}\cup\Uparrow\\!\\!c.\mathcal{A}^{\omega}$
(2)
The problem here is that $\alpha(d)\not\vdash_{\mathcal{A}}c$ does not imply,
in general, $d\not\vdash c$. Take for instance $\alpha$ in Example 4.24. We
know that $x\not\vdash_{\mathcal{A}}x=[a]$ and $x=[a]\vdash x=[a]$. Now let
$Q=\mathbf{unless}\ c\ \mathbf{next}\,\mathbf{tell}(e)$, $d$ be a constraint
s.t. $d\vdash c$ and $d_{\alpha}=\alpha(d)$. We know by rule
$\mathrm{D}_{UNL}$ that
$d.\operatorname{\textup{{t}}}^{\omega}\in[\\![Q]\\!]$. If
$\alpha(d)\not\vdash_{\mathcal{A}}c$, then by using the Equation (2), we
conclude that
$d_{\alpha}.(\operatorname{\textup{{t}}}^{\alpha})^{\omega}\notin[\\![Q]\\!]^{\alpha}$.
Hence, we have a sequence $s$ such that $s\in[\\![Q]\\!]$ and
$\alpha(s)\not\in[\\![Q]\\!]^{\alpha}$ and the abstract semantics cannot be
shown to be a sound approximation of the concrete semantics (see Theorem
4.31).
Notice that defining $d_{\alpha}\not\vdash_{\mathcal{A}}c$ as true iff
$c^{\prime}\not\vdash c$ for all $c^{\prime}$ approximated by $d_{\alpha}$
does not solve the problem. This is because under this definition,
$d_{\alpha}\not\vdash_{\mathcal{A}}c$ does not hold for any $d_{\alpha}$ and
$c$. To see this, notice that $\operatorname{\textup{{f}}}$ entails all the
concrete constraints and it is approximated by any abstract constraint.
Therefore, we cannot give a better (safe) approximation of the semantics of
$\mathbf{unless}\ c\ \mathbf{next}\,P$ than $\mathcal{A}^{\omega}$ (Rule
$\mathrm{A}_{UNL}$).
Now we can formally define the abstract semantics as we did in Section 3.
Given a description $(\mathcal{C},\alpha,\mathcal{A})$, we choose as abstract
domain is $\mathbb{A}=(A,\sqsubseteq^{\alpha})$ where $A=\\{X\ |\
X\in\mathcal{P}(\mathcal{A}^{\omega})\mbox{ and
}(\operatorname{\textup{{f}}}^{\alpha})^{\omega}\in X\\}$ and
$X\sqsubseteq^{\alpha}Y$ iff $X\supseteq Y$. The bottom and top of this domain
are similar to the concrete domain, i.e., $\mathcal{A}^{\omega}$ and
$\\{(\operatorname{\textup{{f}}}^{\alpha})^{\omega}\\}$ respectively.
###### Definition 4.27.
Let $[\\![\cdot]\\!]^{\alpha}_{X}$ be as in Figure 5. The abstract semantics
of a program $\mathcal{D}.P$ is defined as the least fixpoint of the
continuous semantic operator:
$T^{\alpha}_{\mathcal{D}}(X)(p(\vec{t}))={[\\![(Q[\vec{t}/\vec{x}])]\\!]^{\alpha}_{X}}\mbox{
if }p(\vec{x})\operatorname{:\\!--}Q\in\mathcal{D}$
We shall use $[\\![P]\\!]^{\alpha}$ to denote
$[\\![P]\\!]^{\alpha}_{\mathit{l}fp(T_{\mathcal{D}}^{\alpha})}$.
The following proposition shows the monotonicity of $[\\![\cdot]\\!]^{\alpha}$
and the continuity of $T_{\mathcal{D}}^{\alpha}$. The proof is analogous to
that of Proposition 3.8.
###### Proposition 4.28 (Monotonicity of $[\\![\cdot]\\!]^{\alpha}$ and
Continuity of $T_{\mathcal{D}}^{\alpha}$).
Let $P$ be a process and
$X_{1}\sqsubseteq^{\alpha}X_{2}\sqsubseteq^{\alpha}X_{3}...$ be an ascending
chain. Then,
$[\\![P]\\!]^{\alpha}_{X_{i}}\sqsubseteq^{c}[\\![P]\\!]^{\alpha}_{X_{i+1}}$
(Monotonicity). Moreover,
$[\\![P]\\!]^{\alpha}_{\bigsqcup_{X_{i}}}=\bigsqcup_{X_{i}}[\\![P]\\!]^{\alpha}_{X_{i}}$
(Continuity).
### 4.3 Soundness of the Approximation
This section proves the correctness of the abstract semantics in Definition
4.27. We first establish a Galois insertion between the concrete and the
abstract domains.
###### Proposition 4.29 (Galois Insertion).
Let $(\mathcal{C},\alpha^{\prime},\mathcal{A})$ be a description and
$\mathbb{E}$, $\mathbb{A}$ be the concrete and abstract domains. If
$\mathbf{A}$ is upper correct w.r.t. $\mathbf{C}$ then there exists an upper
Galois insertion $\mathbb{E}\galois{\alpha}{\gamma}\mathbb{A}$.
###### Proof 4.30.
Let $\mathbb{A}=(A,\sqsubseteq^{\alpha})$, $\mathbb{E}=(E,\sqsubseteq^{c})$
and $\alpha:E\to A$ and $\gamma:A\to E$ be defined as follows:
$\begin{array}[]{ll}\alpha(S)&=\\{\beta(s)\ |\ s\in S\\}\mbox{ for }S\in\\{X\
|\ X\in\mathcal{P}(\mathcal{C}^{\omega})\mbox{ and
}\operatorname{\textup{{f}}}^{\omega}\in X\\}\\\ \gamma(S_{\alpha})&=\\{s\ |\
\beta(s)\in S_{\alpha}\\}\mbox{ for }S_{\alpha}\in\\{X\ |\
X\in\mathcal{P}(\mathcal{A}^{\omega})\mbox{ and
}(\operatorname{\textup{{f}}}^{\alpha})^{\omega}\in X\\}\end{array}$
where $\beta$ is the pointwise extension of $\alpha^{\prime}$ over sequences.
Notice that $\beta$ is a monotonic and surjective function between
$\mathcal{C}^{\omega}$ and $\mathcal{A}^{\omega}$ and set intersection is the
lub in both $\mathbb{E}$ and $\mathbb{A}$. We conclude by the fact that any
additive and surjective function between complete lattices defines a Galois
insertion [Cousot and Cousot (1979)].
We lift, as standardly done in abstract interpretations [Cousot and Cousot
(1992)], the approximation induced by the above abstraction. Let
$I:ProcHeads\rightarrow E$, $X:ProcHeads\rightarrow A$, $\beta$ be as in
Proposition 4.29 and $p$ be a process definition. Then
$\begin{array}[]{lll}\alpha(I(p))=\\{\beta(s)\mid s\in
I(p)\\}&&\gamma(X(p))=\\{s\mid\beta(s)\in X(p)\\}\end{array}$
We conclude here by showing that concrete computations are safely approximated
by the abstract semantics.
###### Theorem 4.31 (Soundness of the approximation).
Let $(\mathcal{C},\alpha,\mathcal{A})$ be a description and ${\mathbf{A}}$ be
upper correct w.r.t. $\mathbf{C}$. Given a utcc program $\mathcal{D}.P$, if
$s\in[\\![P]\\!]$ then $\alpha(s)\in[\\![P]\\!]^{\alpha}$.
###### Proof 4.32.
Let $d_{\alpha}.s_{\alpha}=\alpha(d.s)$ and assume that $d.s\in[\\![P]\\!]$.
Then, $d.s\in[\\![P]\\!]_{I}$ where $I$ is the $lfp$ of $T_{\mathcal{D}}$. By
the continuity of $T_{\mathcal{D}}$, there exists $n$ s.t.
$I=T_{\mathcal{D}}^{n}(I_{\bot})$ (the $n$-th application of
$T_{\mathcal{D}}$). We proceed by induction on the lexicographical order on
the pair $n$ and the structure of $P$, where the predominant component is $n$.
We only present the interesting cases. The others can be found in C.
Case $P=(\mathbf{abs}\ \vec{x};c)\,Q$. Let $[\vec{t}/\vec{x}]$ be an
admissible substitution. We shall prove that $s\in[\\![(\mathbf{when}\ c\
\mathbf{do}\ Q)[\vec{t}/\vec{x}]]\\!]$ implies
$s_{\alpha}\in[\\![(\mathbf{when}\ c\ \mathbf{do}\
Q)[\vec{t}/\vec{x}]]\\!]^{\alpha}$. The result follows from Proposition 1 and
from the fact that $s_{\alpha}\in\operatorname{\forall\forall\
\\!}\vec{x}([\\![\mathbf{when}\ c\ \mathbf{do}\ Q]\\!]^{\alpha})$ iff
$s_{\alpha}\in[\\![(\mathbf{when}\ c\ \mathbf{do}\
Q)[\vec{t}/\vec{x}]]\\!]^{\alpha}$ for all $adm(\vec{x},\vec{t})$. The proof
of the previous statement is similar to that of Proposition 1 and it appears
in D.
Assume that $d\vdash c[\vec{t}/\vec{x}]$. Then,
$d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]$ and we distinguish two cases:
(1) $d_{\alpha}\vdash_{\mathcal{A}}c[\vec{t}/\vec{x}]$. Since
$d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]$ then $d.s\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![Q]\\!]\cap\uparrow\\!\\!(d_{\vec{x}\vec{t}}^{\omega}))$.
Therefore, there exists $d^{\prime}.s^{\prime}$, an $\vec{x}$-variant of
$d.s$, s.t. $d^{\prime}.s^{\prime}\in[\\![Q]\\!]$ and
$d^{\prime}.s^{\prime}\in\uparrow\\!\\!(d_{\vec{x}\vec{t}}^{\omega})$. By
(structural) inductive hypothesis,
$\alpha(d^{\prime}.s^{\prime})\in[\\![Q]\\!]^{\alpha}$. Furthermore, by
monotonicity of $\alpha$ and Property (2) in Definition 4.23, we derive
$\alpha(d^{\prime}.s^{\prime})\in\uparrow(d^{\alpha}_{\vec{x}\vec{t}})^{\omega}$.
Hence
$\alpha(d^{\prime}.s^{\prime})\in([\\![Q]\\!]^{\alpha}\cap\uparrow\\!\\!((d^{\alpha}_{\vec{x}\vec{t}})^{\omega})$.
Since $\exists\vec{x}(d.s)=\exists\vec{x}(d^{\prime}.s^{\prime})$, by Property
(1) in Definition 4.23, we have
$\exists^{\alpha}\vec{x}(\alpha(d.s))=\exists^{\alpha}\vec{x}(\alpha(d^{\prime}.s^{\prime}))$
(i.e., $\alpha(d^{\prime}.s^{\prime})$ is an $\vec{x}$-variant of
$d_{\alpha}.s_{\alpha}$). Then,
$d_{\alpha}.s_{\alpha}\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![Q]\\!]^{\alpha}\cap\uparrow\\!\\!((d^{\alpha}_{\vec{x}\vec{t}})^{\omega}))$
and we conclude
$d_{\alpha}.s_{\alpha}\in[\\![Q[\vec{t}/\vec{x}]]\\!]^{\alpha}$.
(2) $d_{\alpha}\not\vdash_{\mathcal{A}}c[\vec{t}/\vec{x}]$. Hence trivially
$d_{\alpha}.s_{\alpha}\in[\\![(\mathbf{when}\ c\ \mathbf{do}\
Q)[\vec{t}/\vec{x}]]\\!]^{\alpha}$.
We conclude by noticing that if $d\not\vdash c[\vec{t}/\vec{x}]$ then
$d_{\alpha}\not\vdash_{\mathcal{A}}c[\vec{t}/\vec{x}]$ and therefore
$d_{\alpha}.s_{\alpha}\in[\\![(\mathbf{when}\ c\ \mathbf{do}\
Q)[\vec{t}/\vec{x}]]\\!]^{\alpha}$.
Case $P\operatorname{:\\!--}p(\vec{t})$. Let
$p(\vec{x})\operatorname{:\\!--}Q$ in $\mathcal{D}$ be a process definition.
If $d.s\in[\\![p(\vec{t})]\\!]$ then $d.s\in I(p(\vec{t}))$ (recall that
$I=lfp(T_{\mathcal{D}})$). We know that $d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]$
and then, $d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]_{I^{\prime}}$ where
$I^{\prime}=T_{\mathcal{D}}^{m}(I_{\bot})$ with $m<n$. By induction, and
continuity of $T_{\mathcal{D}}^{\alpha}$, we know that
$d_{\alpha}.s_{\alpha}\in[\\![Q[\vec{t}/\vec{x}]]\\!]^{\alpha}$ and then
$d_{\alpha}.s_{\alpha}\in[\\![p(\vec{t})]\\!]^{\alpha}$.
### 4.4 Obtaining a finite analysis
As standard in Abstract Interpretation, it is possible to obtain an analysis
which terminates, by imposing several alternative conditions (see for instance
Chapter 9 in [Cousot and Cousot (1992)]). So, one possibility is to impose
that the abstract domain is noetherian (also called finite ascending chain
condition). Another possibility is to use widening operators, or to find an
abstract domain that guarantees termination after a finite number of steps.
So, our framework allows to use all this classical methodologies. In the
examples that we have developed we shall focus our attention on a special
class of abstract interpretations obtained by defining what we call a
_sequence abstraction_ mapping possibly infinite sequences of (abstract)
constraints into finite ones. Actually we can define these abstractions as
Galois connections.
###### Definition 4.33 ($k$-sequence Abstraction).
A $k$-sequence abstraction is given by the following pair of functions
$(\alpha_{k},\gamma_{k})$, with
$\alpha_{k}:(\mathcal{A}^{\omega},\leq^{\alpha})\rightarrow(\mathcal{A}_{k}^{*},\leq^{\alpha})$,
and
$\gamma_{k}:(\mathcal{A}_{k}^{*},\leq^{\alpha})\rightarrow(\mathcal{A}^{\omega},\leq^{\alpha})$.
As for the function $\alpha_{k}$, we set $\alpha_{k}(s)=s^{\prime}$ where
$s^{\prime}$ has length $k$ and $s^{\prime}(i)=s(i)$ for $i\leq k$. Similarly,
$\gamma_{k}(s^{\prime})=s$ where $s^{\prime}(i)=s(i)$ for $i\leq k$ and
$s^{\prime}(i)=\operatorname{\textup{{t}}}$ for $i>k$.
It is easy to see that, for any $k$, $(\alpha_{k},\gamma_{k})$ defines a
Galois connection between $(\mathcal{A}^{\omega},\leq^{\alpha})$ and
$(\mathcal{A}^{*}_{k},\leq^{\alpha})$. Thus it is possible to use compositions
of Galois connections for obtaining a new abstraction [Cousot and Cousot
(1992)].
If $\mathcal{A}$ in $(\mathcal{C},\alpha,\mathcal{A})$ leads to a Noetherian
abstract domain $\mathbb{A}$, then the abstraction obtained from the
composition of $\alpha$ and any $\alpha_{k}$ above guarantees that the
fixpoint of the abstract semantics can be reached in a finite number of
iterations. Actually the domain that we obtain in this way is given by
sequences cut at length $k$. The number $k$ determines the length of the cut
and hence the precision of the approximation. The bigger $k$ the better the
approximation.
## 5 Applications
This section is devoted to show some applications of the abstract semantics
developed here. We shall describe three specific abstract domains as instances
of our framework: (1) we abstract a constraint system representing
cryptographic primitives. Then we use the abstract semantics to exhibit a
secrecy flaw in a security protocol modeled in utcc. Next, (2) we tailor two
abstract domains from logic programming to perform a groundness and a type
analysis of a tcc program. We then apply this analysis in the verification of
a reactive system in tcc. Finally, (3) we propose an abstract constraint
system for the suspension analysis of tcc programs.
### 5.1 Verification of Security Protocols
The ability of utcc to express mobile behavior, as in Example 2, allows for
the modeling of security protocols. Here we describe an abstraction of a
cryptographic constraint system in order to bound the length of the messages
to be considered in a secrecy analysis. We start by recalling the constraint
system in [Olarte and Valencia (2008b)] whose terms represent the messages
generated by the protocol and cryptographic primitives are represented as
functions over such terms.
###### Definition 5.34 (Cryptographic Constraint System).
Let $\Sigma$ be a signature with constant symbols in
$\mathcal{P}\cup\mathcal{K}$, function symbols $\operatorname{\mathit{enc}}$,
$\operatorname{\mathit{pair}}$, $\operatorname{\mathit{priv}}$ and
$\operatorname{\mathit{pub}}$ and predicates
$\operatorname{\textup{{out}}}(\cdot)$ and
$\operatorname{\textup{{secret}}}(\cdot)$. Constraint in $\mathcal{C}$ are
formulas built from predicates in $\Sigma$, conjunction ($\sqcup$) and
$\exists$.
Intuitively, $\mathcal{P}$ and $\mathcal{K}$ represent respectively the
principal identifiers, e.g. $A,B,\ldots$ and keys $k,k^{\prime}$. We use
$\\{m\\}_{k}$ and $(m_{1},m_{2})$ respectively, for
$\operatorname{\mathit{enc}}(m,k)$ (encryption) and
$\operatorname{\mathit{pair}}(m_{1},m_{2})$ (composition). For the generation
of keys, $\operatorname{\mathit{priv}}(k)$ stands for the private key
associated to the value $k$ and $\operatorname{\mathit{pub}}(k)$ for its
public key.
As standardly done in the verification of security protocols, a Dolev-Yao
attacker [Dolev and Yao (1983)] is presupposed, able to eavesdrop,
disassemble, compose, encrypt and decrypt messages with available keys. The
ability to eavesdrop all the messages in transit in the network is implicit in
our model due to the shared store of constraints. The other abilities are
modeled by the following utcc processes:
$\begin{array}[]{lll}{Disam}()&\operatorname{:\\!--}&(\mathbf{abs}\
x,y;\operatorname{\textup{{out}}}(\ (x,y)\
))\,\mathbf{tell}(\operatorname{\textup{{out}}}{(x)}\sqcup\operatorname{\textup{{out}}}{(y)})\\\
{Comp}()&\operatorname{:\\!--}&(\mathbf{abs}\
x,y;\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}(y))\,\mathbf{tell}(\operatorname{\textup{{out}}}{(\
(x,y)\ )})\\\ {Enc}()&\operatorname{:\\!--}&(\mathbf{abs}\
x,y;\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}(y))\,\mathbf{tell}(\operatorname{\textup{{out}}}{(\\{x\\}_{\operatorname{\mathit{pub}}(y)})})\\\
{Dec}()&\operatorname{:\\!--}&(\mathbf{abs}\
x,y;\operatorname{\textup{{out}}}(\operatorname{\mathit{priv}}(y))\sqcup\operatorname{\textup{{out}}}(\\{x\\}_{\operatorname{\mathit{pub}}(y)}))\,\mathbf{tell}(\operatorname{\textup{{out}}}{(x)})\\\
{Pers}()&\operatorname{:\\!--}&(\mathbf{abs}\
x;\operatorname{\textup{{out}}}(x))\,\mathbf{next}\,\mathbf{tell}(\operatorname{\textup{{out}}}(x))\\\
{Spy}()&\operatorname{:\\!--}&{Disam}()\parallel{Comp}()\parallel{Enc}()\parallel{Dec}()\parallel{Pers}()\parallel\mathbf{next}\,{Spy}()\end{array}$
Since the final store is not automatically transferred to the next time-unit,
the process $Pers$ above models the ability to remember all messages posted so
far.
It is easy to see that the process ${Spy}()$ in a store
$\operatorname{\textup{{out}}}(m)$ may add messages of unbounded length. Take
for example the process ${Comp}()$ that will add the constraints
$\operatorname{\textup{{out}}}(m)$,
$\operatorname{\textup{{out}}}((m,(m,m)))$,
$\operatorname{\textup{{out}}}(((m,m),m))$ and so on.
To deal with the inherent state explosion problem in the model of the
attacker, symbolic (compact) representations of the behavior of the attacker
have been proposed, for instance in [Boreale (2001), Fiore and Abadi (2001),
Olarte and Valencia (2008b), Bodei et al. (2010)]. Here we follow the approach
of restricting the number of states to be considered in the verification of
the protocol, as for instance in [Escobar et al. (2011), Song et al. (2001),
Armando and Compagna (2008)]. Roughly, we shall cut the messages generated of
length greater than a given $\kappa$, thus allowing us to model a bounded
version of the attacker.
Before defining the abstraction, we notice that the constraint system we are
considering includes existentially quantified syntactic equations. For this
kind of equations it is necessary to refer to a solved form of them in order
to have a uniform way to compute an approximation of the constraint system. We
then consider constraints of the shape
$\exists\vec{y}(x_{1}=t_{1}(\vec{y})\sqcup...\sqcup x_{n}=t_{n}(\vec{y}))$
where $\vec{x}=x_{1},...x_{n}$ are pairwise distinct and
$\vec{x}\cap\vec{y}=\emptyset$. Here, $t(\vec{y})$ refers to a term where
${\mathit{f}v}(t(\vec{y}))\subseteq\vec{y}$. Given a constraint, its normal
form can be obtained by applying the algorithm proposed in [Maher (1988)]
where: quantifiers are moved to the outermost position and equations of the
form $f(t_{1},...,t_{n})=f(t_{1}^{\prime},...,t_{n}^{\prime})$ are replaced by
$t_{1}=t_{1}^{\prime}\sqcup...\sqcup t_{n}=t_{n}^{\prime}$; equations such as
$x=x$ are deleted; equation of the form $t=x$ are replaced by $x=t$; and given
$x=t$, if $x$ does not occur in $t$, $x$ is replaced by $t$ in $t^{\prime}$ in
all equation of the form $x^{\prime}=t^{\prime}$. For instance, the solved
form of $\exists z,y(x=f(y)\sqcup y=g(z))$ is the constraint $\exists
z(x=f(g(z)))$.
###### Definition 5.35 (Abstract secure constraint system).
Let $\mathcal{M}$ be the set of terms (messages) generated from the signature
$\Sigma$ in Definition 5.34. Let ${\mathit{l}g}:\mathcal{M}\to\mathbb{N}$ be
defined as ${\mathit{l}g}(m)=0$ if $m\in\mathcal{P}\cup\mathcal{K}\cup Var$;
${\mathit{l}g}(\\{m_{1}\\}_{m_{2}})={\mathit{l}g}(\ (m_{1},m_{2})\
)=1+{\mathit{l}g}(m_{1})+{\mathit{l}g}(m_{2})$. Let $cut_{\kappa}(m)=m$ if
${\mathit{l}g}(m)\leq\kappa$. Otherwise, $cut_{\kappa}(m)=m_{\top}$ where
$m_{\top}\notin\mathcal{M}$ represents all the messages whose length is
greater than $\kappa$. We define $\alpha(c)$ as $\alpha_{\kappa}(NF(c))$ where
$\begin{array}[]{llll l
llll}\alpha_{\kappa}(c(m))&=&c(cut_{\kappa}(m))&&\alpha_{\kappa}(d_{xt})&=&d_{xt^{\prime}}\mbox{
where }t^{\prime}=cut_{\kappa}(t)\\\ \alpha_{\kappa}(c\sqcup
c^{\prime})&=&\alpha_{\kappa}(c)\sqcup\alpha_{\kappa}(c^{\prime})&&\alpha_{\kappa}(\exists\vec{x}c)&=&\exists\vec{x}\alpha_{\kappa}(c)\\\
\end{array}$
and $NF(c)$ is a solved form of the constraint $c$. We omit the superscript
$\alpha$ in the abstract operators $\sqcup^{\alpha}$, $\exists^{\alpha}$ and
$d_{\vec{x}\vec{t}}^{\alpha}$ to simplify the notation.
We note that the previous abstraction reminds of the $depth\mbox{-}\kappa$
abstractions typically done in the analysis of logic programs (see e.g., [Sato
and Tamaki (1984)]).
We shall illustrate the use of the abstract constraint system above by
performing a secrecy analysis on the Needham-Schröder (NS) protocol [Lowe
(1996)]. This protocol aims at distributing two _nonces_ in a secure way.
Figure 6(a) shows the steps of NS where $m$ and $n$ represent the nonces
generated, respectively, by the principals $A$ and $B$. The protocol initiates
when $A$ sends to $B$ a new nonce $m$ together with her own agent name $A$,
both encrypted with $B$’s public key. When $B$ receives the message, he
decrypts it with his secret private key. Once decrypted, $B$ prepares an
encrypted message for $A$ that contains a new nonce $n$ together with the
nonce $m$ and his name $B$. $A$ then recovers the clear text using her private
key. $A$ convinces herself that this message really comes from B by checking
whether she got back the same nonce sent out in the first message. If that is
the case, she acknowledges B by returning his nonce. $B$ does a similar test.
$\begin{array}[]{llll}{\mathsf{M_{1}}}&A\to
B&:&\\{(m,A)\\}_{\operatorname{\mathit{pub}}(B)}\\\ {\mathsf{M_{2}}}&B\to
A&:&\\{(m,n,B)\\}_{\operatorname{\mathit{pub}}(A)}\\\ {\mathsf{M_{3}}}&A\to
B&:&\\{n\\}_{\operatorname{\mathit{pub}}(B)}\\\ \end{array}$
(a)
$\begin{array}[]{llll}{\mathsf{M_{1}}}&A\to
C&:&\\{(m,A)\\}_{\operatorname{\mathit{pub}}{(C)}}\\\
{\mathsf{M_{1}^{\prime}}}&C\to
B&:&\\{(m,A)\\}_{\operatorname{\mathit{pub}}{(B)}}\\\ {\mathsf{M_{2}}}&B\to
A&:&\\{(m,n,B)\\}_{\operatorname{\mathit{pub}}{(A)}}\\\ {\mathsf{M_{3}}}&A\to
C&:&\\{n\\}_{\operatorname{\mathit{pub}}{(C)}}\\\ \end{array}$
(b)
Figure 6: Steps of the Needham-Schroeder Protocol
Assume the execution of the protocol in Figure 6(b). Here $C$ is an intruder,
i.e. a malicious agent playing the role of a principal in the protocol. As it
was shown in [Lowe (1996)], this execution leads to a secrecy flaw where the
attacker $C$ can reveal $n$ which is meant to be known only by $A$ and $B$. In
this execution, the attacker replies to $B$ the message sent by $A$ and $B$
believes that he is establishing a session key with $A$. Since the attacker
knows the private key $\operatorname{\mathit{priv}}(C)$, she can decrypt the
message $\\{n\\}_{\operatorname{\mathit{pub}}{(C)}}$ and $n$ is no longer a
secret between $B$ and $A$ as intended.
We model the behavior of the principals of the NS protocol with the process
definitions in Figure 7.
$\begin{array}[]{lll}{Init}(i,r)&\operatorname{:\\!--}&(\mathbf{local}\,m)\,\mathbf{tell}(\operatorname{\textup{{out}}}(\\{(m,i)\\}_{pub(r))})\parallel\\\
&&\qquad\qquad\qquad\mathbf{next}\,(\mathbf{abs}\
x;\operatorname{\textup{{out}}}(\\{(m,x,r)\\}_{\operatorname{\mathit{pub}}(i)}))\,\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x\\}_{\operatorname{\mathit{pub}}(r))})\\\
&&\parallel\mathbf{next}\,Init(i,r)\\\
{Resp(r)}&\operatorname{:\\!--}&(\mathbf{abs}\
x,u;\operatorname{\textup{{out}}}(\\{(x,u)\\}_{\operatorname{\mathit{pub}}(r)}))\,\mathbf{next}\\\
&&\ \ \ \ \ \ \ \ \
(\mathbf{local}\,n)\,{(Secrete}(n)\parallel\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,n,r\\}_{\operatorname{\mathit{pub}}{(u)})}))\\\
&&\parallel\mathbf{next}\,Resp(r)\\\
{Secrete}(x)&\operatorname{:\\!--}&\mathbf{tell}(\operatorname{\textup{{secret}}}(x))\parallel\mathbf{next}\,{Secrete}(x)\\\
{SpKn}()&\operatorname{:\\!--}&\parallel_{A\in\mathcal{P}}\
\mathbf{tell}(\operatorname{\textup{{out}}}(A)\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{pub}}(A)))\\\
&&\parallel_{A\in Bad}\
\mathbf{tell}(\operatorname{\textup{{out}}}(\operatorname{\mathit{priv}}(A)))\\\
&&\parallel\mathbf{next}\,SpKn()\end{array}$
Figure 7: utcc model of the Needham-Schröder Protocol
Nonce generation is modeled by $\mathbf{local}$ constructs and the process
$\mathbf{tell}(\operatorname{\textup{{out}}}(m))$ models the broadcast of the
message $m$. Inputs (message reception) are modeled by $\mathbf{abs}$
processes as in Example 3. In ${Resp}$, we use the process ${Secrete}(n)$ to
state that the nonce $n$ cannot be revealed. Finally, the process ${SpKn}$
corresponds to the initial knowledge of the attacker: the names of the
principals, their public keys and the leaked keys in the set $Bad$ (e.g., the
private key of $C$ in the configuration of Figure 6 (b)).
Consider the following process:
${NS}:-\ \ {Spy}\parallel{SpKn}\parallel{Init}(A,C)\parallel{Resp}(B)$ (3)
By using the composition of $\alpha_{3}$ (as in Definition 5.35) and the
sequence abstraction $2$-$sequence$, we obtain the abstract semantics of
${NS}$ as showed in Figure 8. This allows us to exhibit the secrecy flaw of
the NS protocol pointed out in [Lowe (1996)]: Let $s=c_{1}.c_{2}$ s.t.
$s\in[\\![\mathbf{NS}]\\!]^{\alpha}$. Then, there exist a
$m_{1}$-$n_{1}$-variant $s^{\prime}=c_{1}^{\prime}.c_{2}^{\prime}$ of $s$ s.t.
$\begin{array}[]{l}c_{1}^{\prime}\vdash\operatorname{\textup{{out}}}(\\{m_{1},A\\}_{\operatorname{\mathit{pub}}(C)})\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{priv}}(C))\sqcup\operatorname{\textup{{out}}}(\\{m_{1},A\\}_{\operatorname{\mathit{pub}}(B)})\\\
c_{2}^{\prime}\vdash\operatorname{\textup{{out}}}(\\{m_{1},n_{1},A\\}_{\operatorname{\mathit{pub}}(A)})\sqcup\operatorname{\textup{{out}}}(\\{n_{1}\\}_{\operatorname{\mathit{pub}}(C)})\sqcup\operatorname{\textup{{out}}}(\operatorname{\textup{{secret}}}(n_{1}))\sqcup\operatorname{\textup{{out}}}(\operatorname{\textup{{out}}}(n_{1}))\end{array}$
This means that the nonce $n_{1}$ appears as plain text in the network and it
is no longer a secret between $A$ and $B$ as intended.
$\begin{array}[]{lcl}[\\![{Init}(A,C)]\\!]^{\alpha}&=&\operatorname{\exists\exists\
\\!}\ m_{1}\ \operatorname{\exists\exists\ \\!}\ m_{2}\ \ (\\{c_{1}.c_{2}\ |\
c_{1}\vdash^{\alpha}\operatorname{\textup{{out}}}(\\{m_{1},A\\}_{\operatorname{\mathit{pub}}{(C)}}),\
c_{2}\vdash^{\alpha}\operatorname{\textup{{out}}}(\\{A,m_{2}\\}_{\operatorname{\mathit{pub}}{(C)}})\\}\cap\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \mathcal{A}.\operatorname{\forall\forall\
\\!}x(\\{c_{2}\mid\mbox{if
}c_{2}\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(\\{m_{1},x,C\\}_{\operatorname{\mathit{pub}}(A)})\mbox{
then
}c_{2}\vdash^{\alpha}\operatorname{\textup{{out}}}(\\{x\\}_{\operatorname{\mathit{pub}}(C)})\\}))\\\
\\\ [\\![{resp}(B)]\\!]^{\alpha}&=&\operatorname{\forall\forall\ \\!}x,u(\
\operatorname{\exists\exists\ \\!}\ n_{1}\ \\{c_{1}.c_{2}\ |\ \mbox{if
}c_{1}\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(\\{x,u\\}_{\operatorname{\mathit{pub}}(B)})\mbox{
then }\\\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
c_{2}\vdash^{\alpha}\operatorname{\textup{{secret}}}(n_{1})\sqcup\operatorname{\textup{{out}}}(\\{x,n_{1},B\\}_{\operatorname{\mathit{pub}}(u)})\\})\\\
\\\ [\\![{Spy}]\\!]^{\alpha}&=&\operatorname{\forall\forall\ \\!}\ x\
(\\{c_{1}.c_{2}\ |\ \mbox{ if
}c_{1}\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(x)\mbox{ then
}c_{2}\vdash^{\alpha}\operatorname{\textup{{out}}}(x)\\})\cap S.S\mbox{ where
}\\\ \\\ S&=&\operatorname{\forall\forall\ \\!}x,y(\\{c\ |\ \mbox{ if
}c\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}(y)\mbox{
then
}c\vdash^{\alpha}\operatorname{\textup{{out}}}(\\{x,y\\})\sqcup\operatorname{\textup{{out}}}(\\{x\\}_{\operatorname{\mathit{pub}}(y)})\\}\cap\\\
&&\ \ \ \ \ \ \ \ \ \ \ \\{c\ |\ \mbox{ if
}c\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(\\{x,y\\})\mbox{ then
}c\vdash^{\alpha}\operatorname{\textup{{out}}}(x)\sqcup\operatorname{\textup{{out}}}(y)\\}\cap\\\
&&\ \ \ \ \ \ \ \ \ \ \ \\{c\ |\ \mbox{ if
}c\vdash_{\mathcal{A}}\operatorname{\textup{{out}}}(\\{x\\}_{\operatorname{\mathit{pub}}(y)})\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{priv}}(y))\mbox{
then }c\vdash^{\alpha}\operatorname{\textup{{out}}}(x)\\})\\\ \\\
[\\![{SpKn}]\\!]^{\alpha}&=&\\{c_{1}.c_{2}\ |\
c_{i}\vdash^{\alpha}\operatorname{\textup{{out}}}(\operatorname{\mathit{pub}}(A))\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{pub}}(B))\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{pub}}(C))\\}\cap\\\
&&\\{c_{1}.c_{2}\ |\
c_{i}\vdash^{\alpha}\operatorname{\textup{{out}}}(A)\sqcup\operatorname{\textup{{out}}}(B)\sqcup\operatorname{\textup{{out}}}(C)\sqcup\operatorname{\textup{{out}}}(\operatorname{\mathit{priv}}(C))\\}\\\
\\\ [\\![{NS}]\\!]^{\alpha}&=&\ \
[\\![{Spy}]\\!]^{\alpha}\cap[\\![{SpKn}]\\!]^{\alpha}\cap[\\![\mathbf{init}(A,C)]\\!]^{\alpha}\cap[\\![\mathbf{resp}(B)]\\!]^{\alpha}\\\
\end{array}$
Figure 8: Abstract semantics of the process ${NS}$ in Equation 3
### 5.2 Groundness Analysis
In logic programming one useful analysis is groundness. It aims at determining
if a variable will always be bound to a ground term. This information can be
used, e.g., for optimization in the compiler or as base for other data flow
analyses such as independence analysis, suspension analysis, etc. Here we
present a groundness analysis for a tcc program. To this end, we shall use as
concrete domain the Herbrand Constraint System and the following running
example.
$\begin{array}[]{lll}{\mathit{g}en}_{a}(x)&:-&(\mathbf{local}\,x^{\prime})\,({\mathit{a}ssign}(x,[a|x^{\prime}])\parallel\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{when}\ \mathit{g}o_{a}=[]\ \mathbf{do}\
\mathbf{next}\,{\mathit{g}en}_{a}(x^{\prime})\parallel\mathbf{when}\
\mathit{s}top_{a}=[]\ \mathbf{do}\ {\mathit{a}ssign}(x^{\prime},[]))\\\ \\\
{\mathit{a}ssign}(x,y)&:-&\mathbf{tell}(x=y)\parallel\mathbf{next}\,{\mathit{a}ssign}(x,y)\\\
\\\ {\mathit{a}ppend}(x,y,z)&:-&\mathbf{when}\ x=[]\ \mathbf{do}\
{\mathit{a}ssign}(y,z)\parallel\\\ &&\mathbf{when}\
\exists_{x^{\prime},x^{\prime\prime}}(x=[x^{\prime}\ |x^{\prime\prime}])\
\mathbf{do}\\\ &&\ \ \ \ \
(\mathbf{local}\,x^{\prime},x^{\prime\prime},z^{\prime})\,({\mathit{a}ssign}(x,[x^{\prime}|x^{\prime\prime}])\parallel{{\mathit{a}ssign}(z,[x^{\prime}|z^{\prime}])}\parallel\mathbf{next}\,{\mathit{a}ppend}(x^{\prime\prime},y,z^{\prime}))\end{array}$
Figure 9: Appending streams (Example 5.36). The process definition $gen_{b}$
is similar to $gen_{a}$ but replacing the constant $a$ with $b$.
###### Example 5.36 (Append).
Assume the process definitions in Figure 9. The process
${\mathit{g}en}_{a}(x)$ adds an “$a$” to the stream $x$ when the environment
provides $go_{a}=[]$ as input. Under input $stop_{a}=[]$,
${\mathit{g}en}_{a}(x)$ terminates the stream binding its tail to the empty
list. The process ${\mathit{g}en}_{b}$ can be explained similarly. The process
${\mathit{a}ssign}(x,y)$ persistently equates $x$ and $y$. Finally,
${\mathit{a}ppend}(x,y,z)$ binds $z$ to the concatenation of $x$ and $y$.
We shall use $Pos$ [Armstrong et al. (1998)] as abstract domain for the
groundness analysis. In $Pos$, positive propositional formulas represent
groundness dependencies among variables. For instance, $\alpha_{G}(x=[a|b])=x$
meaning that $x$ is a ground variable and
$\alpha_{G}(x=[y|z])=x\leftrightarrow(y\wedge z)$ meaning that $x$ is ground
if and only if both $y$ and $z$ are ground. Elements in this domain are
ordered by logical implication, e.g., $x\sqcup(x\leftrightarrow(y\wedge
z))\vdash_{\alpha_{G}}y$.
###### Observation 2 (Precision of Pos with respect to Synchronization)
Notice that $Pos$ does not distinguish between the empty list and a list of
ground terms: $d_{\kappa}=\alpha_{G}(x=[])=\alpha_{G}(x=[a])=x$ and then,
$d_{\kappa}\not\vdash_{\mathcal{A}}x=[]$ (see Definition 4.26). This affects
the precision of the analysis. For instance, let $P=\mathbf{tell}(x=[])$ and
$Q=\mathbf{when}\ x=[]\ \mathbf{do}\ \mathbf{tell}(y=[])$. One would expect
that the groundness analysis of $P\parallel Q$ determines that $x$ and $y$ are
ground variables. Nevertheless, it is easy to see that
$x.true^{\omega}\in[\\![P]\\!]^{\alpha_{G}}$ and then, the information added
by $\mathbf{tell}(y=[])$ is lost.
We improve the accuracy of the analysis by using the abstract domain defined
in [Codish and Demoen (1994)] to derive information about type dependencies on
terms. The abstraction is defined as follows:
$\alpha_{T}(x=t)=\left\\{\begin{array}[]{lll}\operatorname{\mathit{list}}(x,x_{s})&\mbox{if}&t=[y\
|\ x_{s}]\mbox{ for some $y$}\\\
\operatorname{\mathit{nil}}(x)&\mbox{if}&t=[]\end{array}\right.$
Informally, $list(x,x_{s})$ means $x$ is a list iff $x_{s}$ is a list and
$nil(x)$ means $x$ is the empty list. If $x$ is a list we write $list(x)$ and
$nil(x)\vdash^{\alpha_{T}}list(x)$. Elements in the domain are ordered by
logical implication.
The following constraint systems result from the reduced product [Cousot and
Cousot (1992)] of the previous abstract domains, thus allowing us to capture
groundness and type dependency information.
###### Definition 5.37 (Groundness-type Constraint System).
Let
${\mathbf{A}_{GT}}=\langle\mathcal{A},\leq^{\alpha_{GT}}\,\sqcup^{\alpha_{GT}},\operatorname{\textup{{t}}}^{\alpha_{GT}},\operatorname{\textup{{f}}}^{\alpha_{GT}},{\mathit{V}ar},\exists^{\alpha_{GT}},d^{\alpha_{GT}}\rangle$.
Given $c\in\mathcal{C}$,
$\alpha_{GT}(c)=\langle\alpha_{G}(c),\alpha_{T}(c)\rangle$. The operations
$\sqcup^{\alpha_{GT}}$ and $\exists^{\alpha_{GT}}$ correspond to logical
conjunction and existential quantification on the components of the tuple and
$d_{\vec{x}\vec{t}}^{\alpha_{GT}}$ is defined as
$\langle\alpha_{G}(\vec{x}=\vec{t}),\alpha_{T}(\vec{x}=\vec{t})\rangle$.
Finally, $\langle c_{\kappa},d_{\kappa}\rangle\leq^{\alpha_{GT}}\langle
c_{\kappa}^{\prime},d_{\kappa}^{\prime}\rangle$ iff
$c_{\kappa}^{\prime}\vdash_{\alpha_{G}}c_{\kappa}$ and
$d_{\kappa}^{\prime}\vdash_{\alpha_{T}}d_{\kappa}$.
Consider the Example 5.36 and the abstraction $\alpha$ resulting from the
composition of $\alpha_{GT}$ above and $sequence_{\kappa}$. Note that the
program makes use of guards of the form $\exists
x^{\prime},x^{\prime\prime}(x=[x^{\prime}|x^{\prime\prime}])$ and $x=[]$. Note
also that $list(x,x^{\prime})\vdash_{\mathcal{A}}\exists
x^{\prime},x^{\prime\prime}(x=[x^{\prime}|x^{\prime\prime}])$ and
$nil(x)\vdash_{\mathcal{A}}x=[]$. Roughly speaking, this guarantees that the
chosen domain is accurate w.r.t. the ask processes in the program.
The semantics of the process $P=gen_{a}(x)\parallel gen_{b}(y)\parallel
append(x,y,z)$ is depicted in Figure 10. Assume that
$s=c_{1}.c_{2}...c_{\kappa}\in[\\![P]\\!]^{\alpha}$. Let $n\leq\kappa$ and
assume that for $i<n$, $c_{i}\vdash_{\mathcal{A}}go_{a}=[]$ and
$c_{n}\vdash_{\mathcal{A}}stop_{a}=[]$. Since $s\in[\\![P]\\!]^{\alpha}$, we
know that $s\in[\\![gen_{a}(x)]\\!]^{\alpha}$ and then, we can verify that
$c_{n}\vdash^{\alpha}\langle x,list(x)\rangle$. Similarly, take $m\leq\kappa$
and assume that for $j<m$, $c_{j}\vdash_{\mathcal{A}}go_{b}=[]$ and
$c_{m}\vdash_{\mathcal{A}}stop_{b}=[]$. We can verify that
$c_{m}\vdash^{\alpha}\langle y,list(y)\rangle$. Finally, since $s\in
append(x,y,z)$, we can show that $c_{max(n,m)}\vdash^{\alpha}\langle
z,list(z)\rangle$. In words, the process $P$ binds $x$, $y$ and $z$ to ground
lists whenever the environment provides as input a series of constraints
$go_{a}=[]$ (resp. $go_{b}=[]$) followed by an input $stop_{a}=[]$ (resp.
$stop_{b}=[]$).
$\begin{array}[]{lll}\lx@intercol[\\![gen_{a}(x)\parallel gen_{b}(y)\parallel
append(x,y,z)]\\!]^{\alpha}=\operatorname{\exists\exists\
\\!}x_{1}(GA_{1})\cap\operatorname{\exists\exists\ \\!}y_{1}(GB_{1})\cap
A_{1}\hfil\lx@intercol\mbox{ where }\\\ \\\ GA_{1}&=&\uparrow\\!\\!\langle
x\leftrightarrow x_{1},list(x,x_{1})\rangle.\mathcal{A}\ \cap\\\ &&\\{c.s\
|\mbox{ if }c\vdash_{\mathcal{A}}\ go_{a}=[]\mbox{ then
}s\in\operatorname{\exists\exists\ \\!}x_{2}(GA_{2})\\}\cap\\\ &&\\{c.s\
|\mbox{ if }c\vdash_{\mathcal{A}}\ stop_{a}=[]\mbox{ then }\langle
x_{1},nil(x_{1})\rangle^{\omega}\leq^{\alpha}c.s\\}\\\ \cdots\\\
GA_{\kappa}&=&\uparrow\\!\\!\langle x_{\kappa-1}\leftrightarrow
x_{\kappa},list(x_{\kappa-1},x_{\kappa})\rangle.\epsilon\ \cap\\\
&&\\{c.\epsilon\ |\mbox{ if }c\vdash_{\mathcal{A}}\ stop_{a}=[]\mbox{ then
}c\vdash^{\alpha}\langle x_{\kappa},nil(x_{\kappa})\rangle\\}\\\ \\\
A_{1}&=&\\{c.s\ |\ \mbox{if }c\vdash_{\mathcal{A}}x=[]\mbox{ then
}(d_{yz}^{\alpha})^{\omega}\leq^{\alpha}c.s\\}\ \cap\\\ &&\\{c.s\ |\ \mbox{if
}c\vdash_{\mathcal{A}}\exists x^{\prime},x_{2}(x=[x^{\prime}|x_{2}])\mbox{
then }\\\ &&\ \ \ \ \ \ \ \ \ c.s\in\operatorname{\exists\exists\
\\!}x^{\prime}\operatorname{\exists\exists\
\\!}x_{2}\operatorname{\exists\exists\ \\!}z_{2}(\uparrow\\!\\!(\langle
x\leftrightarrow x_{2},list(x,x_{2})\rangle^{\omega})\ \cap\\\ &&\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \uparrow\\!\\!(\langle
z\leftrightarrow
z_{2},list(z,z_{2})\rangle^{\omega})\cap\mathcal{A}.A_{2})\\}\\\ \cdots\\\
A_{\kappa}&=&\\{c.\epsilon\ |\ \mbox{if
}c\vdash_{\mathcal{A}}x_{\kappa}=[]\mbox{ then
}d_{y_{\kappa}z_{\kappa}}^{\alpha}\leq^{\alpha}c\\}\ \cap\\\ &&\\{c.\epsilon\
|\ \mbox{if }c\vdash_{\mathcal{A}}\exists
x^{\prime},x_{\kappa^{\prime}}(x=[x^{\prime}|x_{\kappa^{\prime}}])\mbox{ then
}\\\ &&\ \ \ \ \ \ \ \ \ c.\epsilon\in\operatorname{\exists\exists\
\\!}x^{\prime}\operatorname{\exists\exists\
\\!}x_{\kappa^{\prime}}\operatorname{\exists\exists\
\\!}z_{\kappa^{\prime}}(\uparrow\\!\\!(\langle x_{\kappa}\leftrightarrow
x_{\kappa^{\prime}},list(x_{\kappa},x_{\kappa^{\prime}})\rangle).\epsilon\
\cap\\\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \uparrow\\!\\!(\langle z_{\kappa}\leftrightarrow
z_{\kappa^{\prime}},list(z_{\kappa},z_{\kappa^{\prime}})\rangle).\epsilon)\\}\end{array}$
Figure 10: Abstract semantics of the process $P=gen_{a}(x)\parallel
gen_{b}(y)\parallel append(x,y,z)$. Definitions of $gen_{a}(x),gen_{b}(y)$ and
$append(x,y,z)$ are given in Example 5.36. Sets $GB_{1},..,GB_{\kappa}$ are
similar to $GA_{1},..,GA_{\kappa}$ and omitted here.
#### 5.2.1 Reactive Systems
Synchronous data flow languages [Berry and Gonthier (1992)] such as Esterel
and Lustre can be encoded as tcc processes [Saraswat et al. (1994), Tini
(1999)]. This makes tcc an expressive declarative framework for the modeling
and verification of reactive systems. Take for instance the program in Figure
11, taken and slightly modified from [Falaschi and Villanueva (2006)], that
models a control system for a microwave checking that the door must be closed
when it is turned on. Otherwise, it must emit an error signal. In this model,
on, off, closed and open represent the constraints
$on=[],{\mathit{o}ff}=[],close=[]$ and $open=[]$ and the symbols $yes$, $no$,
$stop$ denote constant symbols.
The analyses developed here can provide additional reasoning techniques in tcc
for the verification of such systems. For instance, by using the groundness
analysis in the previous section, we can show that if
$c_{1}.c_{2}....c_{\kappa}\in[\\![micCtrl(Error,Button)]\\!]^{\alpha}$ and
there exists $1\leq i\leq\kappa$ s.t. $c_{i}\vdash_{\mathcal{A}}(open=[]\sqcup
on=[])$, then, it must be the case that $c_{1}\vdash^{\alpha}\langle
Error,\operatorname{\mathit{list}}(Error)\rangle$, i.e., $Error$ is a ground
variable. This means, that the system correctly binds the list $Error$ to a
ground term whenever the system reaches an inconsistent state.
$\begin{array}[]{ll}{\mathit{m}icCtrl}(Error,Signal)\operatorname{:\\!--}\\\ \
\ \ \ (\mathbf{local}\,Error^{\prime},Signal^{\prime},er,sl)\,(\\\ \ \ \ \ \ \
\ \ !\,\mathbf{tell}(Error=[er\ |\ Error^{\prime}]\sqcup Signal=[sl\ |\
Signal^{\prime}])\\\ \ \ \ \ \ \ \ \ \parallel\mathbf{when}\
\texttt{on}\sqcup\texttt{open}\ \mathbf{do}\ !\,\mathbf{tell}(er=yes\sqcup
Error^{\prime}=[]\sqcup sl={stop})\\\ \ \ \ \ \ \ \ \ \parallel\mathbf{when}\
\texttt{off}\ \mathbf{do}\
(!\,\mathbf{tell}(er=no)\parallel\mathbf{next}\,\mathit{m}icCtrl(Error^{\prime},Signal^{\prime}))\\\
\ \ \ \ \ \ \ \ \parallel\mathbf{when}\ \texttt{closed}\ \mathbf{do}\
(!\,\mathbf{tell}(er=no)\parallel\mathbf{next}\,\mathit{m}icCtrl(Error^{\prime},Signal^{\prime})))\end{array}$
Figure 11: Model for a microwave controller (see Notation 3 for the definition
of $!\,$).
###### Observation 3 (Synchronization constraints)
In several applications of tcc and utcc the environment interact with the
system by adding as input some constraints that only appear in the guard of
ask processes as $\texttt{on},\texttt{off},\texttt{open},\texttt{close}$ in
Figure 11 and $go_{a}$, $stop_{a}$ in the Figure 9. These constraints can be
thought of as “synchronization constraints” [Fages et al. (2001)].
Furthermore, since these constraints are inputs from the environment, they are
not expected to be produced by the program, i.e., they do not appear in the
scope of a tell process. In these situations, in order to improve the accuracy
of the analyses, one can orthogonally add those constraints in the abstract
domain. This can be done, for instance, with a reduced product as we did in
Definition 5.37 to give a finer approximation of the inputs $go_{a}$ and
$stop_{a}$ by adding type dependency information.
### 5.3 Suspension Analysis
In a concurrent setting it is important to know whether a given system reaches
a state where no further evolution is possible. Reaching a deadlocked
situation is something to be avoided. There are many studies on this problem
and several works developing analyses in (logic) concurrent languages (e.g.
[Codish et al. (1994), Codish et al. (1997)]). However, we are not aware of
studies available for ccp and its temporal extensions. A suspended state in
the context of ccp may happen when the guard of the ask processes are not
carefully chosen and then, none of them can be entailed. In this section we
develop an analysis that aims at determining the constraints that a program
needs as input from the environment to proceed. This can be used to derive
information about the suspension of the system. We start by extending the
concrete semantics to a collecting semantics that keeps information about the
suspension of processes. For this, we define the following constraint system.
###### Definition 5.38 (Suspension Constraint System).
Let $\mathcal{S}=\\{\bot,\texttt{ns}\\}$ s.t. $\bot\leq\texttt{ns}$. Given a
constraint system
${\mathbf{C}}=\langle\mathcal{C},\leq,\sqcup,\operatorname{\textup{{t}}},\operatorname{\textup{{f}}},{\mathit{V}ar},\exists,d\rangle$,
the suspension-constraint system $S({\mathbf{C}})$ is defined as
${\mathbf{S}}=\langle\mathcal{C}\times\mathcal{S},\leq^{s},\sqcup^{s},\langle\operatorname{\textup{{t}}},\bot\rangle,\langle\operatorname{\textup{{f}}},\texttt{ns}\rangle,Var,\exists^{s},d^{s}\rangle$
where $\leq^{s},\sqcup^{s}$ are pointwise defined,
$\exists^{s}_{\vec{x}}(\langle
c,c^{\prime}\rangle)=\langle\exists_{x}c,c^{\prime}\rangle$ and
$d^{s}_{\vec{x}\vec{t}}=\langle d_{\vec{x}\vec{t}},\bot\rangle$. Given a
constraint $c\in\mathcal{C}$, we shall use $\widehat{c}$ to denote the
constraint $\langle c,\bot\rangle$.
Let us illustrate how $S({\mathbf{C}})$ allows us to derive information about
suspension.
###### Example 5.39 (Collecting Semantics).
Let
$\mathcal{C}=\\{\operatorname{\textup{{t}}},a,b,c,d,\operatorname{\textup{{f}}}\\}$
be a complete lattice where $b\vdash a$ and $d\vdash c$, $P=\mathbf{when}\ a\
\mathbf{do}\ \mathbf{tell}(b)$ and $Q=\mathbf{when}\ c\ \mathbf{do}\
\mathbf{tell}(d)$. We know that
$[\\![P]\\!]=\\{\operatorname{\textup{{t}}},b,c,d,\operatorname{\textup{{f}}}\\}.\mathcal{C}^{\omega}$
(note that $P$ does not suspend on $b$ and $\operatorname{\textup{{f}}}$). Let
$\widehat{P}$ and $\widehat{Q}$ be defined over $S({\mathbf{C}})$ as:
$\begin{array}[]{lll}\widehat{P}=\mathbf{when}\ \widehat{a}\ \mathbf{do}\
(\mathbf{tell}(\widehat{b})\parallel\mathbf{tell}(\langle
a,\texttt{ns}\rangle))&\\!\\!&\widehat{Q}=\mathbf{when}\ \widehat{c}\
\mathbf{do}\ (\mathbf{tell}(\widehat{d})\parallel\mathbf{tell}(\langle
c,\texttt{ns}\rangle))\\\ \end{array}$
We then have:
$\begin{array}[]{rll}[\\![\widehat{P}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
b,\texttt{ns}\rangle,\langle c,\uparrow\\!\\!\bot\rangle,\langle
d,\uparrow\\!\\!\bot\rangle,\langle\operatorname{\textup{{f}}},\texttt{ns}\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\\\
[\\![\widehat{Q}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
a,\uparrow\\!\\!\bot\rangle,\langle b,\uparrow\\!\\!\bot\rangle,\langle
d,\texttt{ns}\rangle,\langle\operatorname{\textup{{f}}},\texttt{ns}\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\\\
[\\![\widehat{P}\parallel\widehat{Q}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
b,\texttt{ns}\rangle,\langle
d,\texttt{ns}\rangle,\langle\operatorname{\textup{{f}}},\texttt{ns}\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\end{array}$
where $\langle c,\uparrow\\!\\!\bot\rangle$ is a shorthand for the couple of
tuples $\langle c,\bot\rangle,\langle c,\texttt{ns}\rangle$. The process $P$
suspends on input $c$ (since $c\not\vdash a$) while $Q$ under input $c$
outputs $d$ and it does not suspend. Notice that the system $P\parallel Q$
does not block on input $b,d$ or $\operatorname{\textup{{f}}}$ and it does on
input $\operatorname{\textup{{t}}}$. Notice also that $\langle
c,\bot\rangle.s\not\in[\\![\widehat{P}\parallel\widehat{Q}]\\!]$. This means
that in a store $c$, at least one the ask processes in
$\widehat{P}\parallel\widehat{Q}$ is able to proceed. The key idea is that the
process $\mathbf{tell}(\langle c,\texttt{ns}\rangle)$ in $\widehat{Q}$ ensures
that if $\langle e,e^{\prime}\rangle\in[\\![\widehat{Q}]\\!]$ and $e\vdash c$,
then it must be the case that $e^{\prime}=\texttt{ns}$. This corresponds to
the intuition that if an ask process can evolve on a store $c$, it can evolve
under any store greater than $c$ (Lemma 1).
Next we define a program transformation that allows us to scatter suspension
information when we want to verify that none of the ask processes suspend.
###### Example 5.40.
Let $P$ and $Q$ be as in Example 5.39. Let also $\widehat{P}=\mathbf{when}\
\widehat{a}\ \mathbf{do}\ (\mathbf{tell}(\widehat{b}))$,
$\widehat{Q}=\mathbf{when}\ \widehat{c}\ \mathbf{do}\
(\mathbf{tell}(\widehat{d}))$ and
$\widehat{R}=\widehat{P}\parallel\widehat{Q}\parallel\mathbf{when}\
\widehat{a}\sqcup\widehat{c}\ \mathbf{do}\ (\mathbf{tell}(a\sqcup
c,\texttt{ns}))$. Therefore,
$\begin{array}[]{rll}[\\![\widehat{P}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
b,\uparrow\\!\\!\bot\rangle,\langle c,\uparrow\\!\\!\bot\rangle,\langle
d,\uparrow\\!\\!\bot\rangle,\langle\operatorname{\textup{{f}}},\uparrow\\!\\!\bot\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\\\
[\\![\widehat{Q}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
a,\uparrow\\!\\!\bot\rangle,\langle b,\uparrow\\!\\!\bot\rangle,\langle
d,\uparrow\\!\\!\bot\rangle,\langle\operatorname{\textup{{f}}},\uparrow\\!\\!\bot\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\\\
[\\![\widehat{R}]\\!]&=&\\{\langle\operatorname{\textup{{t}}},\uparrow\\!\\!\bot\rangle,\langle
b,\uparrow\\!\\!\bot\rangle,\langle
d,\uparrow\\!\\!\bot\rangle,\langle\operatorname{\textup{{f}}},\texttt{ns}\rangle\\}.(\mathcal{C}\times\mathcal{S})^{\omega}\end{array}$
Hence we can conclude that only under input $\operatorname{\textup{{f}}}$
neither $P$ nor $Q$ suspend.
The previous program transformation can be arbitrarily applied to subterms of
the form $P=\prod\limits_{i\in I}\mathbf{when}\ c_{i}\ \mathbf{do}\ P_{i}$.
Similarly, for verification purposes, a subterm of the form $P=(\mathbf{abs}\
\vec{x_{1}};c_{1})\,P_{1}\parallel...\parallel(\mathbf{abs}\
\vec{x_{n}};c_{n})\,P_{n}$ can be replaced by
$P^{\prime}=\widehat{P}\parallel\mathbf{when}\
(\exists{\vec{x_{1}}}\widehat{c_{1}}\sqcup...\sqcup\exists{\vec{x_{n}}}\widehat{c_{n}})\
\mathbf{do}\ \mathbf{tell}(\langle c_{1}\sqcup...\sqcup
c_{n},\texttt{ns}\rangle)$
We conclude with an example showing how an abstraction of the previous
collecting semantics allows us to analyze a protocol programmed in utcc. For
this we shall use the abstraction in Definition 5.35 to cut the terms up to a
given length.
###### Example 5.41.
Assume a protocol where agent $A$ has to send a message to $B$ through a proxy
server $S$. This situation can be modeled as follows:
$\begin{array}[]{lll}A(x,y)&\\!\\!\operatorname{:\\!--}\\!\\!&(\mathbf{local}\,m)\,(\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,y,m\\}_{pub(srv)})))\\\
S&\\!\\!\operatorname{:\\!--}\\!\\!&(\mathbf{abs}\
x,y,m;\operatorname{\textup{{out}}}(\\{x,y,m\\}_{pub(srv)}))\,{\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}))}\\!\parallel\\!\mathbf{next}\,S()\\\
B(y)&\\!\\!\operatorname{:\\!--}\\!\\!&(\mathbf{abs}\
x,m;\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}))\,B_{c}\\\
Protocol&\\!\\!\operatorname{:\\!--}\\!\\!&A(x,y)\parallel S()\parallel
B(y)\end{array}$
where $B_{c}=\mathbf{skip}$ is the continuation of the protocol that we left
unspecified.
This code is correct if the message can flow from $A$ to $B$ without any input
from the environment. This holds if the ask process in $B(y)$ does not block.
We shall then analyze the program above by replacing all $c$ with
$\widehat{c}$ and $B(y)$ with
$B^{\prime}(y)\operatorname{:\\!--}(\mathbf{abs}\
x,m;\operatorname{\widehat{\textup{{out}}}}(\\{x,m\\}_{pub(y)}))\,(\mathbf{tell}(\langle\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}),\texttt{ns}\rangle))$
Let $\alpha_{\kappa}$ be as in Definition 5.35. We choose as abstract domain
$\mathcal{A}=S(\alpha_{\kappa}(\mathcal{C}))$ and we consider sequences of
length one. In Figure 12 we show the abstract semantics. We notice that
$\langle c,\texttt{ns}\rangle$ where $c=\exists
m(\operatorname{\textup{{out}}}(\\{x,y,m\\}_{pub(srv)})\sqcup\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}))$
is in the semantics $[\\![Protocol]\\!]^{\alpha}$ and $\langle
c,\bot\rangle\notin[\\![Protocol]\\!]^{\alpha}$. We then conclude that the
protocol is able to correctly deliver the message to $B$.
Assume now that the code for the server is (wrongly) written as
$S^{\prime}\operatorname{:\\!--}(\mathbf{abs}\
x,y,m;\operatorname{\textup{{out}}}(\\{x,y,m\\}_{pub(srv)}))\,{\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(x)}))}\parallel\mathbf{next}\,S^{\prime}()$
where we changed
$\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}))$ to
$\mathbf{tell}(\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(x)}))$. We can
verify that $\langle c,\bot\rangle\in[\\![Protocol^{\prime}]\\!]^{\alpha}$
where $c=\exists
m(\operatorname{\textup{{out}}}(\\{x,y,m\\}_{pub(srv)})\sqcup\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(x)}))$.
This can warn the programmer that there is a mistake in the code.
$\begin{array}[]{rll}[\\![Protocol]\\!]^{\alpha}&=&A.\epsilon\cap
S.\epsilon\cap B.\epsilon\mbox{ where }\\\ A&=&\operatorname{\exists\exists\
\\!}m(\uparrow\\!\\!(\operatorname{\widehat{\textup{{out}}}}(\\{x,y,m\\}_{pub(srv)})))\\\
S&=&\operatorname{\forall\forall\ \\!}x,y,m(\\{\langle d,c\rangle\ |\ \mbox{
if }\langle
d,c\rangle\vdash_{\mathcal{A}}\operatorname{\widehat{\textup{{out}}}}(\\{x,y,m\\}_{pub(srv)})\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{then }\langle
d,c\rangle\leq^{\alpha}\operatorname{\widehat{\textup{{out}}}}(\\{x,m\\}_{pub(y)})\\})\\\
B&=&\operatorname{\forall\forall\ \\!}x,m(\\{\langle d,c\rangle\ |\ \mbox{ if
}\langle
d,c\rangle\vdash_{\mathcal{A}}\operatorname{\widehat{\textup{{out}}}}(\\{x,m\\}_{pub(y)})\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{then }\langle
d,c\rangle\leq^{\alpha}\langle\operatorname{\textup{{out}}}(\\{x,m\\}_{pub(y)}),\texttt{ns}\rangle\\})\
\\}\end{array}$
Figure 12: Semantics of the protocol in Example 5.41.
## 6 Concluding Remarks
Several frameworks and abstract domains for the analysis of logic programs
have been defined (see e.g. [Cousot and Cousot (1992), Codish et al. (1999),
Armstrong et al. (1998)]). Those works differ from ours since they do not deal
with the temporal behavior and synchronization mechanisms present in tcc-based
languages. On the contrary, since our framework is parametric w.r.t. the
abstract domain, it can benefit from those works.
We defined in [Falaschi et al. (2007)] a framework for the declarative
debugging of ntcc [Nielsen et al. (2002a)] programs (a non-deterministic
extension of tcc). The framework presented here is more general since it was
designed for the static analysis of tcc and utcc programs and not only for
debugging. Furthermore, as mentioned above, it is parametric w.r.t an abstract
domain. In [Falaschi et al. (2007)] we also dealt with infinite sequences of
constraints and a similar finite cut over sequences was proposed there.
In [Olarte and Valencia (2008b)] a symbolic semantics for utcc was proposed to
deal with the infinite internal reductions of non well-terminated processes.
This semantics, by means of temporal formulas, represents finitely the
infinitely many constraints (and substitutions) the SOS may produce. The work
in [Olarte and Valencia (2008a)] introduces a denotational semantics for utcc
based on (partial) closure operators over sequences of _temporal logic
formulas_. This semantics captures compositionally the _symbolic strongest
postcondition_ and it was shown to be fully abstract w.r.t. the symbolic
semantics for the fragment of locally-independent (see Definition 3.18) and
abstracted-unless free processes (i.e., processes not containing occurrences
of unless processes in the scope of abstractions). The semantics here
presented turns out to be more appropriate to develop the abstract
interpretation framework in Section 4. Firstly, the inclusion relation between
the strongest postcondition and the semantics is verified for the whole
language (Theorem 3.13) – in [Olarte and Valencia (2008a)] this inclusion is
verified only for the abstracted-unless free fragment–. Secondly, this
semantics makes use of the entailment relation over constraints rather than
the more involved entailment over first-order linear-time temporal formulas as
in [Olarte and Valencia (2008a)]. Finally, our semantics allows us to capture
the behavior of tcc programs with recursion. This is not possible with the
semantics in [Olarte and Valencia (2008a)] which was thought only for utcc
programs where recursion can be encoded. This work then provides the
theoretical basis for building tools for the data-flow analyses of utcc and
tcc programs.
For the kind of applications that stimulated the development of utcc, it was
defined entirely deterministic. The semantics here presented could smoothly be
extended to deal with some forms of non-determinism like those in [Falaschi et
al. (1997a)], thus widening the spectrum of applications of our framework.
A framework for the abstract diagnosis of timed-concurrent constraint programs
has been defined in [Comini et al. (2011)] where the authors consider a
denotational semantics similar to ours, although with several technical
differences. The language studied in [Comini et al. (2011)] corresponds to
tccp [de Boer et al. (2000)], a temporal ccp language where the stores are
monotonically accumulated along the time-units and whose operational semantics
relies on the notion of true parallelism. We note that the framework developed
in [Comini et al. (2011)] is used for abstract diagnosis rather than for
general analyses.
Our results should foster the development of analyzers for different systems
modeled in utcc and its sub-calculi such as security protocols, reactive and
timed systems, biological systems, etc (see [Olarte et al. (2013)] for a
survey of applications of ccp-based languages). We plan also to perform
freeness, suspension, type and independence analyses among others. It is well
known that this kind of analyses have many applications, e.g. for code
optimization in compilers, for improving run-time execution, and for
approximated verification. We also plan to use abstract model checking
techniques based on the proposed semantics to automatically analyze utcc and
tcc code.
Acknowledgments. We thank Frank D. Valencia, François Fages and Rémy Haemmerlé
for insightful discussions on different subjects related to this work. We also
thank the anonymous reviewers for their detailed comments. Special thanks to
Emanuele D’Osualdo for his careful remarks and suggestions for improving the
paper. This work has been partially supported by grant 1251-521-28471 from
Colciencias, and by Digiteo and DGAR funds for visitors.
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## Appendix A Detailed proofs Section 2.4
Before presenting the proof that utcc is deterministic, we shall prove the
following auxiliary result.
###### Lemma A.42 (Confluence).
Suppose that $\gamma_{0}\longrightarrow\gamma_{1}$,
$\gamma_{0}\longrightarrow\gamma_{2}$ and $\gamma_{1}\not\equiv\gamma_{2}$.
Then, there exists $\gamma_{3}$ such that
$\gamma_{1}\longrightarrow\gamma_{3}$ and
$\gamma_{2}\longrightarrow\gamma_{3}$.
###### Proof A.43.
Let $\gamma_{0}=\left\langle{\vec{x};P;c}\right\rangle$. The proof proceed by
structural induction on $P$. In each case where $\gamma_{0}$ has two different
transitions (up to $\equiv$) $\gamma_{0}\longrightarrow\gamma_{1}$ and
$\gamma_{0}\longrightarrow\gamma_{2}$, one shows the existence of $\gamma_{3}$
s.t. $\gamma_{1}\longrightarrow\gamma_{3}$ and
$\gamma_{2}\longrightarrow\gamma_{3}$.
Given a configuration $\gamma=\left\langle{\vec{x};P;c}\right\rangle$ let us
define the size of $\gamma$ as the size of $P$ as follows:
$M(\mathbf{skip})=0$, $M(\mathbf{tell}(c))=M(p(\vec{t}))=1$, $M((\mathbf{abs}\
\vec{x};c;D)\,P^{\prime})=M((\mathbf{local}\,\vec{x})\,P^{\prime})=M(\mathbf{next}\,P^{\prime})=M(\mathbf{unless}\
c\ \mathbf{next}\,P^{\prime})=1+M(P^{\prime})$ and $M(Q\parallel
R)=M(Q)+M(R)$. Suppose that
$\gamma_{0}\equiv\left\langle{\vec{x};P;c_{0}}\right\rangle$,
$\gamma_{0}\longrightarrow\gamma_{1}$, $\gamma_{0}\longrightarrow\gamma_{2}$
and $\gamma_{1}\not\equiv\gamma_{2}$. The proof proceeds by induction on the
size of $\gamma_{0}$. From the assumption $\gamma_{1}\not\equiv\gamma_{2}$, it
must be the case that the transition $\longrightarrow$ is not an instance of
the rule $\mathrm{R}_{STRVAR}$; moreover, $P$ is neither a process of the form
$\mathbf{tell}(c)$, $(\mathbf{local}\,\vec{x})\,P$, $p(\vec{t})$ or
$\mathbf{unless}\ c\ \mathbf{next}\,P^{\prime}$ (since those processes have a
unique possible transition modulo structural congruence) nor
$\mathbf{next}\,P$ or $\mathbf{skip}$ (since they do not exhibit any internal
derivation).
For the case $P=Q\parallel R$, we have to consider three cases. Assume that
$\gamma_{1}\equiv\langle\vec{x}_{1};Q_{1}\parallel R,c_{1}\rangle$ and
$\gamma_{2}\equiv\langle\vec{x}_{2};Q_{2}\parallel R,c_{2}\rangle$. Let
$\gamma^{\prime}_{0}\equiv\langle\vec{x};Q;c_{0}\rangle$,
$\gamma^{\prime}_{1}\equiv\langle\vec{x}_{1};Q_{1};c_{1}\rangle$ and
$\gamma^{\prime}_{2}\equiv\langle\vec{x}_{2};Q_{2};c_{2}\rangle$. We know by
induction that if $\gamma^{\prime}_{0}\longrightarrow\gamma^{\prime}_{1}$ and
$\gamma_{0}^{\prime}\longrightarrow\gamma^{\prime}_{2}$ then there exists
$\gamma_{3}^{\prime}\equiv\langle\vec{x}_{3};Q_{3};c_{3}\rangle$ such that
$\gamma_{1}^{\prime}\longrightarrow\gamma_{3}^{\prime}$ and
$\gamma_{2}^{\prime}\longrightarrow\gamma_{3}^{\prime}$. We conclude by
noticing that $\gamma_{1}\longrightarrow\gamma_{3}$ and
$\gamma_{2}\longrightarrow\gamma_{3}$ where
$\gamma_{3}\equiv\langle\vec{x}_{3};Q_{3}\parallel R;c_{3}\rangle$. The
remaining cases when (1) $R$ has two possible transitions and (2) when $Q$
moves to $Q^{\prime}$ and then $R$ moves to $R^{\prime}$ are similar.
Let $\gamma_{0}\equiv\langle\vec{x};P;c_{0}\rangle$ with $P=(\mathbf{abs}\
\vec{y};c;D)\,Q$. One can verify that
$\gamma_{1}\equiv\langle\vec{x}\cup\vec{x}_{1};P_{1};c_{0}\rangle$ where
$P_{1}$ takes the form $(\mathbf{abs}\
\vec{y};c;D\cup\\{d_{\vec{y}\vec{t_{1}}}\\})\,Q\parallel
Q[\vec{t_{1}}/\vec{y}]$ and
$\gamma_{2}\equiv\langle\vec{x}\cup\vec{x}_{2};P_{2};c_{0}\rangle$ where
$P_{2}$ takes the form $(\mathbf{abs}\
\vec{z};c;D\cup\\{d_{\vec{y}\vec{t_{2}}}\\})\,Q\parallel
Q[\vec{t_{2}}/\vec{y}]$. From the assumption $\gamma_{1}\not\equiv\gamma_{2}$,
it must be the case that $d_{\vec{y}\vec{t_{1}}}\not\cong
d_{\vec{y}\vec{t_{2}}}$. By alpha conversion we assume that
$\vec{x}_{1}\cap\vec{x}_{2}=\emptyset$. Let
$\gamma_{3}\equiv\langle\vec{x}\cup\vec{x}_{1}\cup\vec{x}_{2};P_{3};c_{0}\rangle$
where $P_{3}=(\mathbf{abs}\
\vec{y};c;D\cup\\{d_{\vec{y}\vec{t_{1}}},d_{\vec{y}\vec{t_{2}}}\\})\,Q\parallel
Q[\vec{t_{1}}/\vec{y}]\parallel Q[\vec{t_{2}}/\vec{y}]$. Clearly
$\gamma_{1}\longrightarrow\gamma_{3}$ and
$\gamma_{2}\longrightarrow\gamma_{3}$ as wanted.
###### Observation 4 (Finite Traces)
Let
$\gamma_{1}\longrightarrow\cdots\longrightarrow\gamma_{n}\not\longrightarrow$
by a finite internal derivation. The number of possible internal transitions
(up to $\equiv$) in any
$\gamma_{i}=\left\langle{\vec{x}_{i};P_{i};c_{i}}\right\rangle$ in the above
derivation is finite.
###### Proof A.44.
We proceed by structural induction on $P_{i}$. The interesting case is the abs
process. Let $Q=(\mathbf{abs}\ \vec{x};c)\,P$. Suppose, to obtain a
contradiction, that $c_{i}\vdash c[\vec{t}/\vec{x}]$ for infinitely many
$\vec{t}$ (to have infinitely many possible internal transitions). In that
case, it is easy to see that we must have infinitely many internal derivation,
thus contradicting the assumption that $\gamma_{n}\not\longrightarrow$.
###### Lemma A.45 (Finite Traces).
If there is a finite internal derivation of the form
$\gamma_{1}\longrightarrow\gamma_{2}\longrightarrow\cdots\longrightarrow\gamma_{n}\not\longrightarrow$
then, any derivation starting from $\gamma_{1}$ is finite.
###### Proof A.46.
We observe that recursive calls must be guarded by a next processes. Then, any
infinite behavior inside a time-unit is due to an abs process. From
Observation 4 and Lemma A.42, it follows that any derivation starting from
$\gamma_{1}$ is finite.
_Theorem 1_ (_Determinism_)
Let $s,w$ and $w^{\prime}$ be (possibly infinite) sequences of constraints. If
both $(s,w)$, $(s,w^{\prime})\in{\mathit{i}o}(P)$ then $w\cong w^{\prime}$.
###### Proof A.47.
Assume that
$P\stackrel{{\scriptstyle\,\,(c,\exists\vec{x}(d))\,\,}}{{\,\,===\Longrightarrow}}(\mathbf{local}\,\vec{x})\,F(Q)$,
$P\stackrel{{\scriptstyle\,\,(c,\exists\vec{x}^{\prime}(d^{\prime}))\,\,}}{{\,\,===\Longrightarrow}}(\mathbf{local}\,\vec{x}^{\prime})\,F(Q^{\prime})$
and let $\gamma_{1}\equiv\langle\emptyset;P;c\rangle$,
$\gamma_{2}\equiv\langle\emptyset;P;c\rangle$. If
$\gamma_{1}\not\longrightarrow$ then trivially
$\gamma_{2}\not\longrightarrow$, $d\cong d^{\prime}$ and $Q\equiv Q^{\prime}$.
Now assume that
$\gamma_{1}\longrightarrow^{*}\gamma_{1}^{\prime}\not\longrightarrow$ and
$\gamma_{2}\longrightarrow^{*}\gamma_{2}^{\prime}\not\longrightarrow$ where
$\gamma_{1}^{\prime}\equiv\langle\vec{x};Q;d\rangle$ and
$\gamma_{2}^{\prime}\equiv\langle\vec{x}^{\prime};Q^{\prime};d^{\prime}\rangle$.
By repeated applications of Lemma A.42 we conclude
$\gamma_{1}^{\prime}\equiv\gamma_{2}^{\prime}$ and then, $d\cong d^{\prime}$
and $Q\equiv Q^{\prime}$.
_Lemma 2_ (_Closure Properties_)
Let $P$ be a process. Then,
1. (1)
${\mathit{i}o}(P)$ is a function.
2. (2)
${\mathit{i}o}(P)$ is a partial closure operator, namely it satisfies:
Extensiveness: If $(s,s^{\prime})\in{\mathit{i}o}(P)$ then $s\leq s^{\prime}$.
Idempotence: If $(s,s^{\prime})\in{\mathit{i}o}(P)$ then
$(s^{\prime},s^{\prime})\in{\mathit{i}o}(P)$.
Monotonicity: Let $P$ be a monotonic process such that
$(s_{1},s_{1}^{\prime})\in{\mathit{i}o}(P)$. If
$(s_{2},s_{2}^{\prime})\in{\mathit{i}o}(P)$ and $s_{1}\leq s_{2}$, then
$s_{1}^{\prime}\leq s_{2}^{\prime}$.
###### Proof A.48.
We shall assume here that the input and output sequences are infinite. The
proof for the case when the sequences are finite is analogous. The proof of
(1) is immediate from Theorem 1. For (2), assume that $s=c_{1}.c_{2}...$,
$s^{\prime}=c_{1}^{\prime}.c_{2}^{\prime}...$ and that
$(s,s^{\prime})\in{\mathit{i}o}(P)$. We then have a derivation of the form:
$P\equiv
P_{1}\stackrel{{\scriptstyle\,\,(c_{1},c_{1}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}P_{2}\stackrel{{\scriptstyle\,\,(c_{2},c_{2}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}...P_{i}\stackrel{{\scriptstyle\,\,(c_{i},c_{i}^{\prime})\,\,}}{{\,\,===\Longrightarrow}}P_{i+1}...$
For $i\geq 1$, we also know that there is an internal derivation of the form
$\langle\emptyset;P_{i};c_{i}\rangle\longrightarrow^{*}\langle\vec{x};P_{i}^{\prime};c_{i}^{\prime}\rangle\not\longrightarrow$
where $P_{i+1}=(\mathbf{local}\,\vec{x})\,F(P_{i}^{\prime})$.
Extensiveness follows from (1) in Lemma 1.
Idempotence is proved by repeated applications of (3) in Lemma 1.
As for Monotonicity, we proceed as in [Nielsen et al. (2002a)]. Let $\preceq$
be the minimal ordering relation on processes satisfying: (1)
$\mathbf{skip}\preceq P$. (2) If $P\preceq Q$ and $P\equiv P^{\prime}$ and
$Q\equiv Q^{\prime}$ then $P^{\prime}\preceq Q^{\prime}$. (3) If $P\preceq Q$,
for every context $C[\cdot]$, $C[P]\preceq C[Q]$. Intuitively, $P\preceq Q$
represents the fact that $Q$ contains “at least as much code” as $P$. We have
to show that for every $P$, $P^{\prime}$, $c$, $c^{\prime}$ and
$\vec{x},\vec{x}^{\prime}$ if
$\langle\vec{x};P;c\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};P^{\prime};c^{\prime}\rangle\not\longrightarrow$
then for every $d\vdash c$ and $Q$ s.t. $P\preceq Q$ there
$\langle\vec{x};Q;d\rangle\longrightarrow^{*}\langle\vec{y};Q^{\prime};d^{\prime}\rangle\not\longrightarrow$
for some $\vec{y}$ and $Q^{\prime}$ with
$(\mathbf{local}\,\vec{x}^{\prime})\,F(P^{\prime})\preceq(\mathbf{local}\,\vec{y})\,F(Q^{\prime})$
and $\exists\vec{y}(d^{\prime})\vdash\exists\vec{x}^{\prime}(c^{\prime})$.
This can be proved by induction on the length of the derivation using the
following two properties:
(a) $\longrightarrow$ is monotonic w.r.t. the store, in the sense that, if
$\langle\vec{x};P;c\rangle\longrightarrow\langle\vec{x}^{\prime};P^{\prime};c^{\prime}\rangle$
then for every $d\vdash c$ and $Q$ s.t. $P\preceq Q$,
$\langle\vec{x};Q;d\rangle\longrightarrow\langle\vec{y};Q^{\prime};d^{\prime}\rangle$
where $\exists\vec{y}(d^{\prime})\vdash\exists\vec{x}^{\prime}(c^{\prime})$
and
$(\mathbf{local}\,\vec{x}^{\prime})\,P^{\prime}\preceq(\mathbf{local}\,\vec{y})\,Q^{\prime}$.
(b) For every monotonic process $P$, if
$\langle\vec{x};P;c\rangle\not\longrightarrow$ then for every $d\vdash c$ and
$Q$ such that $P\preceq Q$ we have either
$\langle\vec{x};Q;d\rangle\not\longrightarrow$ or
$\langle\vec{x};Q;d\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};Q^{\prime};d^{\prime}\rangle\not\longrightarrow$
where $\exists\vec{x}^{\prime}(d^{\prime})\vdash\exists\vec{x}(d)$ and
$(\mathbf{local}\,\vec{x})\,F(P)\preceq(\mathbf{local}\,\vec{x}^{\prime})\,F(Q^{\prime})$.
The restriction to programs which do not contain unless constructs is
essential here.
_Theorem 2_
Let $min$ be the minimum function w.r.t. the order induced by $\leq$ and $P$
be a monotonic process. Then, $(s,s^{\prime})\in{\mathit{i}o}(P)\mbox{\ \ iff\
\ }s^{\prime}=min({\mathit{s}p}(P)\cap\\{w\ |\ s\leq w\\})$
###### Proof A.49.
Let $P$ be a monotonic process and $(s,s^{\prime})\in{\mathit{i}o}(P)$. By
extensiveness $s\leq s^{\prime}$ and by idempotence,
$(s^{\prime},s^{\prime})\in{\mathit{i}o}(P)$. Let
$s^{\prime\prime}=min({\mathit{s}p}(P)\cap\\{w\ |\ s\leq w\\})$. Since
$s^{\prime}\in{\mathit{s}p}(P)$ and $s\leq s^{\prime}$, it must be the case
that $s\leq s^{\prime\prime}\leq s^{\prime}$. If
$(s^{\prime\prime},s^{\prime\prime\prime})\in{\mathit{i}o}(P)$, by
monotonicity $s^{\prime}\leq s^{\prime\prime\prime}$. Since
$s^{\prime\prime}\in{\mathit{s}p}(P)$, $s^{\prime\prime}\cong
s^{\prime\prime\prime}$ and then, $s^{\prime}\leq s^{\prime\prime}$. We
conclude $s^{\prime}\cong s^{\prime\prime}$.
## Appendix B Detailed Proofs Section 3
_Observation 1_ (_Equality and $\vec{x}$-variants_)
Let $S\subseteq\mathcal{C}^{\omega}$, $\vec{z}\subseteq{\mathit{V}ar}$ and
$s,w$ be $\vec{x}$-variants such that $d_{\vec{x}\vec{t}}^{\omega}\leq s$,
$d_{\vec{x}\vec{t}}^{\omega}\leq w$ and $adm(\vec{x},\vec{t})$. (1) $s\cong
w$. (2) $\exists\vec{z}(s)\in\operatorname{\forall\forall\ \\!}\vec{x}(S)$ iff
$s\in\operatorname{\forall\forall\ \\!}\vec{x}(S)$.
###### Proof B.50.
(1) Let $i\geq 1$, $c=s(i)$ and $d=w(i)$. We prove that $c\vdash d$ and
$d\vdash c$. We know that $c\sqcup d_{\vec{x}\vec{t}}\cong c$, $d\sqcup
d_{\vec{x}\vec{t}}\cong d$ and $\exists\vec{x}(c\sqcup
d_{\vec{x}\vec{t}})\cong\exists\vec{x}(d\sqcup d_{\vec{x}\vec{t}})$. Hence,
$c[\vec{t}/\vec{x}]\cong d[\vec{t}/\vec{x}]$. Since
$c\vdash\exists\vec{x}(c)$, we can show that $c\vdash\exists\vec{x}(d\sqcup
d_{\vec{x}\vec{t}})$ and then, $c\vdash d[\vec{t}/\vec{x}]$. Since
$d[\vec{t}/\vec{x}]\sqcup d_{\vec{x}\vec{t}}\vdash d$ (Notation 2) we conclude
$c\vdash d$. The “$d\vdash c$” side is analogous and we conclude $c\cong d$.
Property (2) follows directly from the definition of
$\operatorname{\forall\forall\ \\!}(\cdot)$.
_Lemma 3.11_
Let $[\\![\cdot]\\!]$ be as in Definition 3.5. If
$P\stackrel{{\scriptstyle\,\,(d,d^{\prime})\,\,}}{{\,\,===\Longrightarrow}}{R}$
and $d\cong d^{\prime}$, then $d.[\\![R]\\!]\subseteq[\\![P]\\!]$.
###### Proof B.51.
Assume that
$\langle\vec{x};P;d\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};P^{\prime};d^{\prime}\rangle\not\longrightarrow$,
$\exists\vec{x}(d)\cong\exists\vec{x}^{\prime}(d^{\prime})$. We shall prove
that $\exists\vec{x}(d).\operatorname{\exists\exists\
\\!}\vec{x}^{\prime}([\\![F(P^{\prime}))]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])$. We proceed by induction on the lexicographical
order on the length of the internal derivation and the structure of $P$, where
the predominant component is the length of the derivation. Here we present the
missing cases in the body of the paper.
Case $P=\mathbf{skip}$. This case is trivial.
Case $P=\mathbf{tell}(c)$. If
$\langle\vec{x};\mathbf{tell}(c);d\rangle\longrightarrow\langle\vec{x};\mathbf{skip},d\rangle$
then it must be the case that $d\cong d\sqcup c$ and $d\vdash c$. We conclude
$\exists\vec{x}(d).[\\![\mathbf{skip}]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x}([\\![\mathbf{tell}(c)]\\!])$.
Case $P=(\mathbf{local}\,\vec{x};c)\,Q$. Consider the following derivation
$\left\langle{\vec{y};(\mathbf{local}\,\vec{x})\,Q;d}\right\rangle\longrightarrow\left\langle{\vec{y}\cup\vec{x};Q;d}\right\rangle\longrightarrow^{*}\left\langle{\vec{y}\cup\vec{x}^{\prime};Q^{\prime};d^{\prime}}\right\rangle\not\longrightarrow$
where, by alpha-conversion, $\vec{x}\cap\vec{y}=\emptyset$ and
$\vec{x}\cap{\mathit{f}v}(d)=\emptyset$. Assume that
$\exists\vec{y}(d)\cong\exists\vec{y}\exists\vec{x}^{\prime}(d^{\prime})$.
Since the derivation starting from $Q$ is shorter than that starting from $P$,
we conclude $\exists\vec{y}(d).\operatorname{\exists\exists\
\\!}\vec{y},\vec{x}^{\prime}[\\![F(Q^{\prime})]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x},\vec{y}[\\![Q]\\!]$.
Case $P=\mathbf{next}\,Q$. This case is trivial since
$d.[\\![Q]\\!]\subseteq[\\![P]\\!]$ for any $d$.
Case $P=\mathbf{unless}\ c\ \mathbf{next}\,Q$. We distinguish two cases: (1)
If $d\vdash c$, then we have $\langle\vec{x};\mathbf{unless}\ c\
\mathbf{next}\,Q;d\rangle\longrightarrow\langle\vec{x};\mathbf{skip};d\rangle\not\longrightarrow$
and we conclude $\operatorname{\exists\exists\
\\!}\vec{x}(d).[\\![\mathbf{skip}]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x}[\\![\mathbf{unless}\ c\ \mathbf{next}\,P]\\!]$. (2), the case when
$d\not\vdash c$ is similar to the case of $P=\mathbf{next}\,Q$.
_Lemma 3.19_ (_Completeness_)
Let $\mathcal{D}.P$ be a locally independent program s.t. $d.s\in[\\![P]\\!]$.
If
$P\stackrel{{\scriptstyle\,\,(d,d^{\prime})\,\,}}{{\,\,===\Longrightarrow}}R$
then $d^{\prime}\cong d$ and $s\in[\\![R]\\!]$.
###### Proof B.52.
Assume that $P$ is locally independent, $d.s\in[\\![P]\\!]$ and there is a
derivation of the form
$\langle\vec{x};P;d\rangle\longrightarrow^{*}\langle\vec{x}^{\prime};P^{\prime};d^{\prime}\rangle\not\longrightarrow$.
We shall prove that $\exists x(d)\cong\exists\vec{x}^{\prime}(d^{\prime})$ and
$s\in\operatorname{\exists\exists\
\\!}\vec{x}^{\prime}[\\![F(P^{\prime})]\\!]$. We proceed by induction on the
lexicographical order on the length of the internal derivation
($\longrightarrow^{*}$) and the structure of $P$, where the predominant
component is the length of the derivation. The locally independent condition
is used for the case $P=(\mathbf{local}\,\vec{x};c)\,Q$. We present here the
missing cases in the body of the paper.
Case $\mathbf{skip}$. This case is trivial
Case $P=\mathbf{tell}(c)$. This case is trivial since it must be the case that
$d\vdash c$ and hence $d\sqcup c\cong d$.
Case $P=\mathbf{next}\,Q$. This case is trivial since
$\langle\vec{x};P;d\rangle\not\longrightarrow$ for any $d$ and $\vec{x}$ and
$F(P)=Q$.
Case $P=\mathbf{unless}\ c\ \mathbf{next}\,Q$.If $d\vdash c$ the case is
trivial. If $d\not\vdash c$ the case is similar to that of
$P=\mathbf{next}\,Q$.
Case $P=p(\vec{t})$. Assume that $p(\vec{x}):-Q\in\mathcal{D}$. If
$d.s\in[\\![p(\vec{t})]\\!]$ then $d.s\in[\\![Q[\vec{t}/\vec{x}]]\\!]$. By
using the rule $\mathrm{R}_{CALL}$ we can show that there is a derivation
$\langle\vec{y};p(\vec{x});d\rangle\longrightarrow\langle\vec{y};Q[\vec{t}/\vec{x}];d\rangle\longrightarrow^{*}\langle\vec{y}^{\prime};Q^{\prime};d^{\prime}\rangle\not\longrightarrow$
By inductive hypothesis we know that $\exists
y^{\prime}(d^{\prime})\cong\exists\vec{y}(d)$ and
$s\in\operatorname{\exists\exists\
\\!}\vec{y}^{\prime}[\\![F(Q^{\prime})]\\!]$.
## Appendix C Detailed Proofs Section 4
_Theorem 4.31_ (_Soundness of the approximation_)
Let $(\mathcal{C},\alpha,\mathcal{A})$ be a description and ${\mathbf{A}}$ be
upper correct w.r.t. $\mathbf{C}$. Given a utcc program $\mathcal{D}.P$, if
$s\in[\\![P]\\!]$ then $\alpha(s)\in[\\![P]\\!]^{\alpha}$.
###### Proof C.53.
Let $d_{\alpha}.s_{\alpha}=\alpha(d.s)$ and assume that $d.s\in[\\![P]\\!]$.
Then, $d.s\in[\\![P]\\!]_{I}$ where $I$ is the lfp of $T_{\mathcal{D}}$. By
the continuity of $T_{\mathcal{D}}$, there exists $n$ s.t.
$I=T_{\mathcal{D}}^{n}(I_{\bot})$ (the $n$-th application of
$T_{\mathcal{D}}$). We proceed by induction on the lexicographical order on
the pair $n$ and the structure of $P$, where the predominant component is the
length $n$. We present here the missing cases in the body of the paper.
Case $P=\mathbf{skip}$. This case is trivial.
Case $P=\mathbf{tell}(c)$. We must have $d\vdash c$ and by monotonicity of
$\alpha$, $d_{\alpha}\vdash^{\alpha}\alpha(c)$. We conclude
$d_{\alpha}.s_{\alpha}\in[\\![\mathbf{tell}(c)]\\!]^{\alpha}$.
Case $P=Q\parallel R$. We must have that $s\in[\\![Q]\\!]$ and
$s\in[\\![R]\\!]$. By inductive hypothesis we know that
$s_{\alpha}\in[\\![Q]\\!]^{\alpha}$ and $s_{\alpha}\in[\\![R]\\!]^{\alpha}$
and then, $s_{\alpha}\in[\\![Q\parallel R]\\!]^{\alpha}$.
Case $P=(\mathbf{local}\,\vec{x})\,Q$. It must be the case that there exists
$d^{\prime}.s^{\prime}$ $\vec{x}$-variant of $d.s$ s.t.
$d^{\prime}.s^{\prime}\in[\\![Q]\\!]$. Then, by (structural) inductive
hypothesis $\alpha(d^{\prime}.s^{\prime})\in[\\![Q]\\!]^{\alpha}$. We conclude
by using the properties of $\alpha$ in Definition 4.23 to show that
$\exists^{\alpha}\vec{x}(\alpha(d.s))=\exists^{\alpha}\vec{x}(\alpha(d^{\prime}.s^{\prime}))$,
i.e., $\alpha(d.s)$ and $\alpha(d^{\prime}.s^{\prime})$ are
$\vec{x}$-variants, and then,
$d_{\alpha}.s_{\alpha}\in[\\![(\mathbf{local}\,\vec{x})\,Q]\\!]^{\alpha}$.
Case $P=\mathbf{next}\,Q$. We know that $s\in[\\![Q]\\!]$ and by inductive
hypothesis $\alpha(s)\in[\\![Q]\\!]^{\alpha}$. We then conclude
$d_{\alpha}.s_{\alpha}\in[\\![P]\\!]^{\alpha}$.
Case $P=\mathbf{unless}\ c\ \mathbf{next}\,Q$. This case is trivial since
$\mathcal{A}$ approximates every possible concrete computation.
## Appendix D Auxiliary results
###### Proposition D.54.
Let $P$ be a process such that $\vec{x}\cap{\mathit{f}v}(P)=\emptyset$ and let
$d.s\in[\\![P]\\!]$. If $d^{\prime}.s^{\prime}$ is an $\vec{x}$-variant of
$d.s$ then $d^{\prime}.s^{\prime}\in[\\![P]\\!]$.
###### Proof D.55.
The proof proceeds by induction on the structure of $P$. We shall use the
notation $c(\vec{y})$ and $P(\vec{y})$ to denote constraints and processes
where the free variables are exactly $\vec{y}$ and we shall assume that
$\vec{y}\cap\vec{x}=\emptyset$. We assume that $d.s\in[\\![P(\vec{y})]\\!]$
and $d^{\prime}.s^{\prime}$ is an $\vec{x}$-variant of $d.s$. We consider the
following cases. The others are easy.
Case $P=\mathbf{when}\ c(\vec{y})\ \mathbf{do}\ Q(\vec{y})$. If
$d^{\prime}\vdash c(\vec{y})$ then, by monotonicity,
$\exists\vec{x}(d^{\prime})\vdash\exists\vec{x}(c(\vec{y}))$ and then
$\exists\vec{x}(d)\vdash c(\vec{y})$. Hence, it must be the case that $d\vdash
c(\vec{y})$ and $d.s\in[\\![Q(\vec{y})]\\!]$. By induction we conclude
$d^{\prime}.s^{\prime}\in[\\![Q(\vec{y})]\\!]$. If $d^{\prime}\not\vdash
c(\vec{y})$, then $\exists\vec{x}(d^{\prime})\not\vdash c(\vec{y})$ (since
$\exists\vec{x}(d^{\prime})\leq d^{\prime}$). Hence, $d\not\vdash c(\vec{y})$
and trivially, $d.s\in[\\![P]\\!]$ and so
$d^{\prime}.s^{\prime}\in[\\![P]\\!]$.
Case $P=(\mathbf{abs}\ \vec{z};c(\vec{z},\vec{y}))\,Q(\vec{z},\vec{y})$. We
know that $d.s\in\operatorname{\forall\forall\ \\!}\vec{z}[\\![\mathbf{when}\
c(\vec{z},\vec{y})\ \mathbf{do}\ Q(\vec{z},\vec{y})]\\!]$. By definition of
the operator $\operatorname{\forall\forall\ \\!}(\cdot)$,
$\exists\vec{x}(d.s)\in[\\![P]\\!]$. Since
$\exists\vec{x}(d^{\prime}.s^{\prime})\cong\exists\vec{x}(d.s)$ we conclude
$d^{\prime}.s^{\prime}\in[\\![P]\\!]$.
###### Proposition D.56.
If $\vec{x}\cap{\mathit{f}v}(P)=\emptyset$ then
$[\\![P]\\!]=\operatorname{\exists\exists\ \\!}\vec{x}[\\![P]\\!]$.
###### Proof D.57.
The case $[\\![P]\\!]\subseteq\operatorname{\exists\exists\
\\!}\vec{x}[\\![P]\\!]$ is trivial by the definition of
$\operatorname{\exists\exists\ \\!}(\cdot)$. The case
$\operatorname{\exists\exists\ \\!}\vec{x}[\\![P]\\!]\subseteq[\\![P]\\!]$,
follows directly from Proposition D.54.
###### Proposition D.58.
If $\vec{x}\not\in{\mathit{f}v}(Q)$ then $\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]\cap[\\![Q]\\!])=\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])\cap[\\![Q]\\!]$.
###### Proof D.59.
($\subseteq$): Let $d.s\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!]\cap[\\![Q]\\!])$. Then, there exists an
$\vec{x}$-variant $d^{\prime}.s^{\prime}$ s.t.
$d^{\prime}.s^{\prime}\in[\\![P]\\!]\cap[\\![Q]\\!]$. Then,
$d.s\in\operatorname{\exists\exists\ \\!}\vec{x}([\\![P]\\!])$ (by definition)
and $d.s\in[\\![Q]\\!]$ by Proposition D.54.
($\supseteq$): Let $d.s\in\operatorname{\exists\exists\
\\!}\vec{x}([\\![P]\\!])\cap[\\![Q]\\!]$. Then, there exists
$d^{\prime}.s^{\prime}$ $\vec{x}$-variant of $d.s$ s.t.
$d^{\prime}.s^{\prime}\in[\\![P]\\!]$. By Proposition D.54,
$d^{\prime}.s^{\prime}\in[\\![Q]\\!]$ and therefore,
$d.s\in\operatorname{\exists\exists\ \\!}\vec{x}([\\![P]\\!]\cap[\\![Q]\\!])$.
In Theorem 4.31, the proof of the ${\mathbf{a}bs}$ case requires the following
auxiliary results (similar to those in the concrete semantics).
###### Observation 5 (Equality and $\vec{x}$-variants)
Let $s_{\alpha}$ and $w_{\alpha}$ be $\vec{x}$-variants such that
$({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}\leq^{\alpha}s_{\alpha}$,
$({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}\leq^{\alpha}w_{\alpha}$ and
$adm(\vec{x},\vec{t})$. Then $s_{\alpha}\cong^{\alpha}w_{\alpha}$.
###### Proof D.60.
Let $c_{\alpha}=s_{\alpha}(i)$ and $d_{\alpha}=w_{\alpha}(i)$ with $i\geq 1$.
We shall prove that $c_{\alpha}\vdash^{\alpha}d_{\alpha}$ and
$d_{\alpha}\vdash^{\alpha}c_{\alpha}$. We know that
$c_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}}\cong^{\alpha}c_{a}$ and
$d_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}}\cong^{\alpha}d_{\alpha}$.
We also know that
$\exists^{\alpha}\vec{x}(c_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}})\cong^{\alpha}\exists^{\alpha}\vec{x}(d_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}})$.
Since $c_{\alpha}\vdash^{\alpha}\exists^{\alpha}\vec{x}(c_{\alpha})$, we can
show that
$c_{\alpha}\vdash^{\alpha}\exists^{\alpha}\vec{x}(d_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}})$.
Furthermore,
$\exists^{\alpha}\vec{x}(d_{\alpha}\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}})\sqcup^{\alpha}d^{\alpha}_{\vec{x}\vec{t}}\vdash^{\alpha}d_{\alpha}$
(see Notation 2). Hence, we conclude $c_{\alpha}\vdash^{\alpha}d_{\alpha}$.
The proof of $d_{\alpha}\vdash^{\alpha}c_{\alpha}$ is analogous.
###### Proposition D.61.
$s_{\alpha}\in\operatorname{\forall\forall\
\\!}\vec{x}([\\![P]\\!]^{\alpha}_{X})$ if and only if
$s\in[\\![P[\vec{t}/\vec{x}]]\\!]^{\alpha}_{X}$ for all admissible
substitution $[\vec{t}/\vec{x}]$.
###### Proof D.62.
($\Rightarrow$)Let $s_{\alpha}\in\operatorname{\forall\forall\
\\!}\vec{x}([\\![P]\\!]^{\alpha}_{X})$ and $s_{\alpha}^{\prime}$ be an
$\vec{x}$-variant of $s_{\alpha}$ s.t.
$({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}\leq^{\alpha}s_{\alpha}^{\prime}$
where $adm(\vec{x},\vec{t})$. By definition of $\operatorname{\forall\forall\
\\!}$, we know that $s^{\prime}_{\alpha}\in[\\![P]\\!]^{\alpha}_{X}$. Since
$({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}\leq^{\alpha}s_{\alpha}^{\prime}$
then
$s_{\alpha}^{\prime}\in[\\![P]\\!]^{\alpha}_{X}\cap\uparrow\\!\\!(({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega})$.
Hence, $s_{\alpha}\in\operatorname{\exists\exists\
\\!}^{\alpha}\vec{x}([\\![P]\\!]^{\alpha}_{X}\cap\uparrow\\!\\!(({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}))$
and we conclude $s_{\alpha}\in[\\![P[\vec{t}/\vec{x}]]\\!]^{\alpha}_{X}$.
($\Leftarrow$) Let $[\vec{t}/\vec{x}]$ be an admissible substitution. Suppose,
to obtain a contradiction, that
$s_{\alpha}\in[\\![P[\vec{t}/\vec{x}]]\\!]^{\alpha}_{X}$, there exists
$s^{\prime}_{\alpha}$ $\vec{x}$-variant of $s_{\alpha}$ s.t.
$({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega}\leq^{\alpha}s^{\prime}_{\alpha}$ and
$s^{\prime}_{\alpha}\notin[\\![P]\\!]^{\alpha}_{X}$ (i.e.,
$s_{\alpha}\notin\operatorname{\forall\forall\
\\!}\vec{x}([\\![P]\\!]^{\alpha}_{X})$). Since
$s_{\alpha}\in[\\![P[\vec{t}/\vec{x}]]\\!]^{\alpha}_{X}$ then
$s_{\alpha}\in\operatorname{\exists\exists\
\\!}^{\alpha}\vec{x}([\\![P]\\!]^{\alpha}_{X}\cap\uparrow\\!\\!({d^{\alpha}_{\vec{x}\vec{t}}})^{\omega})$.
Therefore, there exists $s^{\prime\prime}_{\alpha}$ $\vec{x}$-variant of
$s_{\alpha}$ s.t. $s^{\prime\prime}_{\alpha}\in[\\![P]\\!]^{\alpha}_{X}$ and
${d^{\alpha}_{\vec{x}\vec{t}}}^{\omega}\leq^{\alpha}s^{\prime\prime}_{\alpha}$.
By Observation 5, $s^{\prime}_{\alpha}\cong^{\alpha}s^{\prime\prime}_{\alpha}$
and thus, $s^{\prime}_{\alpha}\in[\\![P]\\!]^{\alpha}_{X}$, a contradiction.
|
arxiv-papers
| 2013-12-09T19:28:24 |
2024-09-04T02:49:55.180739
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moreno Falaschi, Carlos Olarte, Catuscia Palamidessi",
"submitter": "Carlos Olarte",
"url": "https://arxiv.org/abs/1312.2552"
}
|
1312.2616
|
# A HIGH STATISTICS STUDY OF THE BETA-FUNCTION IN THE SU(2) LATTICE
THERMODYNAMICS
S. S. Antropov, V. V. Skalozub
Oles Honchar Dnipropetrovsk National University,
Dnipropetrovsk, Ukraine
O. A. Mogilevsky
Bogolyubov Institute for Theoretical Physics of the National Academy
of Sciences of Ukraine, Kiev, Ukraine E-mail:[email protected]
mail:[email protected]:[email protected]
###### Abstract
The beta-function is investigated on the lattice in $SU(2)$ gluodynamics. It
is determined within a scaling hypothesis while a lattice size fixed to be
taken into account. The functions calculated are compared with the ones
obtained in the continuum limit. Graphics processing units (GPU) are used as a
computing platform that allows gathering a huge amount of statistical data.
Numerous beta-functions are analyzed for various lattices. The coincidence of
the lattice beta-function and the analytical expression in the region of the
phase transition is shown. New method for estimating a critical coupling value
is proposed.
## 1 Introduction
The beta-function is one of the main objects in quantum field theory. It
defines scaling properties of the theory in different regions of dynamic
variables. It is defined as
$\displaystyle\beta_{f}(g_{\mu})=\mu^{2}\frac{\partial\overline{g}(\mu^{2})}{\partial(\mu^{2})},$
(1)
where $\beta_{f}(g_{\mu})$ is the beta-function,
$g_{\mu}\equiv\overline{g}(\mu^{2})$ – the effective coupling constant, $\mu$
– the normalizing momentum.
For the case of the Monte-Carlo (MC) calculations in $SU(N)$ lattice
gluodynamics the beta-function has the form
$\displaystyle\beta_{f}(g)=-a\frac{dg}{da},$ (2)
where $a$ replaces the parameter $\mu^{2}$, $a$ \- is the lattice spacing.
Lattice spacing is a free parameter of the theory. In particular, the
calculation of $\beta_{f}(g)$ is one of the ways to define $a$.
In analytical approach, the beta-function is well described by an expansion as
power series of coupling constant. In the cases of quantum chromodynamics or
$SU(N)$ lattice gluodynamics, a non-perturbative beta-function attracts the
most interest.
In ref. [1] a new special method was developed. Namely, the effects connected
with the final sizes of a lattice were taken into account, and scaling near
the critical point of $SU(N)$ lattice gauge theories has been considered
without attempt to reach a continuum limit.
The goal of the present paper is the detailed investigation and development of
this approach. In $SU(2)$ gluodinamics, we calculate the beta-functions on
different lattices and compare their values with those obtained in a continuum
limit.
## 2 Analytical expression
The beta-function describes the dependence of the lattice spacing $a$ on a
coupling constant $g$
$\displaystyle\beta_{f}(g)=-a\frac{dg}{da}.$ (3)
Our calculations are based on the special form of the definition of the beta-
function [1]. Let us consider a transformation
$\displaystyle a\rightarrow a^{\prime}=ba=(1+\Delta b)a.$ (4)
Under this transformation the definition (3) becomes
$\displaystyle-a\frac{dg}{da}=-\lim_{b\to
1}\left(a\frac{g(ba)-g(a)}{ba-a}\right)=-\lim_{b\to
1}\frac{dg}{db}=\beta_{f}(g).$ (5)
The singular part of the free energy density can be described by the universal
finite-size scaling function [2]
$\displaystyle
f(t,h,N_{\sigma},N_{\tau})=\left(\frac{N_{\sigma}}{N_{\tau}}\right)^{-3}Q_{f}\left(g_{t}\left(\frac{N_{\sigma}}{N_{\tau}}\right)^{1/\nu},g_{h}\left(\frac{N_{\sigma}}{N_{\tau}}\right)^{\frac{\beta+\gamma}{\nu}}\right),$
(6)
where $\beta,\gamma,\nu$ are the critical indexes of the theory. Due to the
finite size scaling hypothesis, these indexes coincide with the critical
indexes of 3-d Ising model. The scaling function $Q_{f}$ depends on the
reduced temperature $t=\frac{T-T_{c}}{T_{c}}$ and on the external field
strength $h$ through the thermal and magnetic scaling fields
$\displaystyle g_{t}$ $\displaystyle=$ $\displaystyle c_{t}t(1+b_{t}t),$ (7)
$\displaystyle g_{h}$ $\displaystyle=$ $\displaystyle c_{h}h(1+b_{h}t)$
with non-universal coefficients $c_{t},c_{h},b_{t},b_{h}$ which are still
carrying a possible $N_{\tau}$ dependence.
The existence of the scaling function $Q$ [3, 4] allows developing a procedure
to renormalize the coupling constant $g^{-2}$ by using two different lattice
sizes $N_{\sigma},N_{\tau}$ and $N^{\prime}_{\sigma},N^{\prime}_{\tau}$
($N_{\sigma}$ is the number of lattice nods in spatial directions, $N_{\tau}$
– the number of lattice nods in time direction). Let us fix
$\frac{N^{\prime}_{\tau}}{N_{\tau}}=\frac{N^{\prime}_{\sigma}}{N_{\sigma}}=b$
and perform a scale transformation
$\displaystyle a$ $\displaystyle\rightarrow$ $\displaystyle a^{\prime}=ba,$
(8) $\displaystyle N_{\sigma}$ $\displaystyle\rightarrow$ $\displaystyle
N^{\prime}_{\sigma}=\frac{N_{\sigma}}{b},$ $\displaystyle N_{\tau}$
$\displaystyle\rightarrow$ $\displaystyle
N^{\prime}_{\tau}=\frac{N_{\tau}}{b}.$
Then the phenomenological renormalization is defined by the following equation
$\displaystyle
Q(g^{-2},N_{\sigma},N_{\tau})=Q\left((g^{\prime})^{-2},\frac{N_{\sigma}}{b},\frac{N_{\tau}}{b}\right).$
(9)
It means that the scaling function $Q$ remains unchanged if the lattice size
is rescaled by a factor $b$ and the inverse coupling $g^{-2}$ is shifted to
$(g^{\prime})^{-2}$ simultaneously. Taking the derivative with respect to the
scale parameter $b$ of the both sides of (9) and using (5) we obtain the
expression
$\displaystyle a\frac{dg^{-2}}{da}=\frac{\frac{\partial
Q(g^{-2},N_{\sigma},N_{\tau})}{\partial lnN_{\sigma}}+\frac{\partial
Q(g^{-2},N_{\sigma},N_{\tau})}{\partial lnN_{\tau}}}{\frac{\partial
Q(g^{-2},N_{\sigma},N_{\tau})}{\partial g^{-2}}}.$ (10)
Fourth derivative of $f$ in $h$ taken at $h=0$ and divided by
$\chi^{2}(\frac{N_{\sigma}}{N_{\tau}})^{3}$ is called the Binder cumulant [5]
$\displaystyle g_{4}=\frac{\frac{\partial^{4}f}{\partial
h^{4}}}{\chi^{2}(\frac{N_{\sigma}}{N_{\tau}})^{3}}\Biggm{|}_{h=0}.$ (11)
It identically coincides with the scale function [5]
$\displaystyle
g_{4}=Q_{g_{4}}\left(g_{t}\left(\frac{N_{\sigma}}{N_{\tau}}\right)^{\frac{1}{\nu}}\right).$
(12)
Binder cumulant $g_{4}$ is calculated through the Polyakov loops on a lattice
[5]
$\displaystyle g_{4}=\frac{\langle P^{4}\rangle}{\langle P^{2}\rangle^{2}}-3.$
(13)
We get the expression for the beta-function
$\displaystyle a\frac{dg^{-2}}{da}=\frac{\frac{\partial g_{4}}{\partial
lnN_{\sigma}}+\frac{\partial g_{4}}{\partial lnN_{\tau}}}{\frac{\partial
g_{4}}{\partial g^{-2}}}=\frac{1}{4}\frac{\frac{\partial g_{4}}{\partial
lnN_{\sigma}}+\frac{\partial g_{4}}{\partial lnN_{\tau}}}{\frac{\partial
g_{4}}{\partial\beta}}.$ (14)
## 3 Lattice observables
Let us calculate the beta-function using (14). As the lattice size is
discrete, it is necessary to replace the derivatives in (14) by the finite
differences which are calculated on lattices with the closest
$N_{\sigma},N_{\tau}$ (and corresponding $g_{4}(N_{\sigma},N_{\tau})$):
$\displaystyle\frac{\partial g_{4}(\beta,N_{\sigma},N_{\tau})}{\partial
lnN_{\sigma}}\rightarrow\frac{g_{4}(\beta,N^{\prime}_{\sigma},N_{\tau})-g_{4}(\beta,N_{\sigma},N_{\tau})}{ln(\beta,N^{\prime}_{\sigma}/N_{\sigma})},$
(15) $\displaystyle\frac{\partial g_{4}(\beta,N_{\sigma},N_{\tau})}{\partial
lnN_{\tau}}\rightarrow\frac{g_{4}(\beta,N_{\sigma},N^{\prime}_{\tau})-g_{4}(\beta,N_{\sigma},N_{\tau})}{ln(\beta,N^{\prime}_{\tau}/N_{\tau})}.$
Such replacement,
$\displaystyle\frac{\partial g_{4}}{\partial\beta}\rightarrow\frac{\Delta
g_{4}}{\Delta\beta},$ (16)
leads to huge computing errors. Near the phase transition area, the dispersion
is increased and the substitution (16) becomes not reasonable. For different
lattices investigated, the amount of data near the critical region varies from
$120$ up to $600$ points, but the error for (16) still remains large.
Table 1. Tested fitting curves
$Function$ | $Parameters$
---|---
$A_{1}+\frac{A_{2}-A_{1}}{1+10^{(\beta_{0}-\beta)*p}}$ | $A_{1},A_{2},\beta_{0},p$
$\frac{A_{1}-A_{2}}{1+(\frac{\beta}{\beta_{0}})^{p}}+A_{2}$ | $A_{1},A_{2},\beta_{0},p$
$\frac{A_{1}-A_{2}}{1+e^{(\beta-\beta_{0})/p}}+A_{2}$ | $A_{1},A_{2},\beta_{0},p$
Our the best fits (see Fig. 1, Tab. 2) are reached for the function
$\displaystyle g_{4}=A1+(A2-A1)/(1+10^{(\beta_{0}-\beta)*p}),$ (17)
where $A1,A2,\beta_{0},p$ are the fitting parameters.
Figure 1. Binder cumulants. Cumulants are received on lattices with
$N_{\tau}=4$, and $N_{\sigma}=8$, $12$, $16$, $24$, $28$, $32$. The higher
number of nods in the lattice corresponds with the sharper step. All curves
intersect each other in a local area and as it comes from the theory these
curves should intersect in one point (the critical point).
If one knows $g_{4}$ in an analytical form, it is possible to calculate
$\frac{\partial g_{4}}{\partial\beta}$ straightforwardly. However, the result
of $g_{4}$ calculations is a set of points. To reveal a functional dependence
on this sequence, it is necessary to apply some fitting procedure. For this
procedure we chose the step functions, since the critical area of $g_{4}$ is a
steplike (see Tab. 1).
In Tab. 2 the best fits for number of lattices are represented. We have
analyzed up to 600 points for some lattices and have reached small values
(down to $10^{-3}$) of $\chi^{2}$ function.
Table 2. Fitting of Binder cumulants by
$A_{1}+\frac{A_{2}-A_{1}}{1+10^{(\beta_{0}-\beta)*p}}$
| Parameters | | Fitting range
---|---|---|---
Lattice | $\chi^{2}$ | $A_{1}$ | $A_{2}$ | $\beta_{0}$ | $p$ | Number of points | $\beta_{min}$ | $\beta_{max}$
$N_{\tau}=4,N_{\sigma}=8$ | $0.009$ | $-1.953$ | $-0.0523$ | $2.2705$ | $-12$ | $126$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=8$ | $0.012$ | $-1.957$ | $-0.0507$ | $2.2747$ | $-11$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=12$ | $0.025$ | $-1.98$ | $-0.1$ | $2,286$ | $-24$ | $253$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=12$ | $0.011$ | $-2$ | $-0.04$ | $2,289$ | $-16$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=16$ | $0.029$ | $-2.01$ | $-0.066$ | $2.287$ | $-30.1$ | $236$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=16$ | $0.013$ | $-1.99$ | $-0.05$ | $2.292$ | $-30.9$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=20$ | $0.055$ | $-2$ | $-0.065$ | $2.291$ | $-48$ | $246$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=24$ | $0.1$ | $-2.0098$ | $0.044$ | $2.296$ | $-68$ | $126$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=24$ | $0.006$ | $-2.001$ | $0.061$ | $2.291$ | $-27$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=28$ | $0.089$ | $-2.05$ | $-0.13$ | $2.29$ | $-62$ | $626$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=28$ | $0.012$ | $-1.99$ | $-8\cdot 10^{-5}$ | $2.28$ | $-21$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=32$ | $0.12$ | $-1.984$ | $-0.2$ | $2.3$ | $-84$ | $626$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=32$ | $0.01$ | $-1.988$ | $0.014$ | $2.27$ | $-28$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=4,N_{\sigma}=36$ | $0.19$ | $-2$ | $-0.27$ | $2.3$ | $-105$ | $600$ | $2.28$ | $2.31$
$N_{\tau}=16,N_{\sigma}=20$ | $0.094$ | $-1.17$ | $-0.017$ | $2.68$ | $-7$ | $126$ | $1.7$ | $2.95$
$N_{\tau}=16,N_{\sigma}=24$ | $0.054$ | $-1.7$ | $0.04$ | $2.75$ | $-6$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=16,N_{\sigma}=28$ | $0.021$ | $-1.6$ | $-0.017$ | $2.67$ | $-17$ | $26$ | $1.7$ | $2.95$
$N_{\tau}=16,N_{\sigma}=32$ | $0.021$ | $-1.7$ | $0.03$ | $2.69$ | $-23$ | $126$ | $1.7$ | $2.95$
Now we turn to an interesting feature of these fits. Parameters of the curve,
which based on 600 data points, are nearly the same as parameters (especially
$\beta_{0}$) of the curve, which based on 25 data points. The parameter
$\beta_{0}$ coincides (to within 2 up to 3 digits) with an inverse critical
coupling constant for a corresponding lattice (see Tab. 3, ref. [2], [6]).
Table 3. Values of the inverse coupling constant
$N_{\tau}$ | $2$ | $4$ | $6$ | $8$
---|---|---|---|---
$\beta_{c}$ | $1.875$ | $2.301$ | $2.422$ | $2.508$
It is common to use the linear fits for critical point findings. Because of
the dispersion in critical region these fits need a lot of data to be
performed. Using both listed above properties one can estimate the inverse
critical coupling using just few points. For more precise calculations one can
use the function (17) with data, which are from above and below critical
region. The dispersion for these data is much less than for data, which are
near critical area, so one need much less statistics than usually.
The expression for the beta-function in lattice variables reads:
$\displaystyle\beta_{f}(\beta)=\frac{1}{\beta^{3/2}}\cdot\frac{\frac{g_{4}(\beta,N^{\prime}_{\sigma},N_{\tau})-g_{4}(\beta,N_{\sigma},N_{\tau})}{ln(N^{\prime}_{\sigma}/N_{\sigma})}+\frac{g_{4}(\beta,N_{\sigma},N^{\prime}_{\tau})-g_{4}(\beta,N_{\sigma},N_{\tau})}{ln(N^{\prime}_{\tau}/N_{\tau})}}{\frac{\partial
g_{4}(\beta,N_{\sigma},N_{\tau})}{\partial\beta}}.$ (18)
It will be used below.
## 4 Calculation of the beta-function
We chose the heat-bath as working algorithm in MC procedure. We use standard
form of Wilson action of the $SU(2)$ lattice gauge theory. In the MC
simulations, we use the hypercubic lattice $L_{t}\times L_{s}^{3}$ with
hypertorus geometry.
We use the General Purpose computation on Graphics Processing Units (GPGPU)
technology allowing studying large lattices on personal computers. Performance
analysis indicates that the GPU-based MC simulation program shows better
speed-up factors for big lattices in comparing with the CPU-based one. For the
majority lattice geometries the GPU vs. CPU (single-thread CPU execution)
speed-up factor is above 50 and for some lattice sizes could overcome the
factor 100.
The plots of dependencies of the beta-function on the inverse coupling
constant are shown below.
Figure 2. The solid line represents the beta-function in asymptotic expansion.
Dashed lines with a point - the beta-functions (18), $N_{\tau}=2$,
$N_{\sigma}=8,16,20$, $\Delta N_{\tau}=N^{\prime}_{\tau}-N_{\tau}=2$, $\Delta
N_{\sigma}=N^{\prime}_{\sigma}-N_{\sigma}=4$.
Figure 3. Same as above. Dashed lines with a point - the beta-functions (18),
$N_{\tau}=4$, $N_{\sigma}=12,20$, $\Delta
N_{\tau}=N^{\prime}_{\tau}-N_{\tau}=2$, $\Delta
N_{\sigma}=N^{\prime}_{\sigma}-N_{\sigma}=4$. The Dashed line with two points
is the beta-function received in ref. [7].
The standard deviation of the function (18) is the smallest one near the
critical point. It comes from analysis of Binder cumulants. Cumulants decrease
linearly in the critical area and change little above and belove that area.
Therefore $\frac{\partial g_{4}(\beta,N_{\sigma},N_{\tau})}{\partial\beta}$ in
the bottom of (18) comes to $0$ and leads (18) to infinity. Beta-function
values which are calculated near critical point are in good agreement with
known results [7].
## 5 Conclusions
We have performed high-statistics calculations of the beta-function in $SU(2)$
lattice gluodynamics. These calculations became possible due to technology of
GPU calculations.
The key point for our investigations is definition (5) [1]. It gives a
possibility to analyze a finite size of the lattice.
We have constructed and analyzed the lattice beta-functions for a wide range
of different lattices.
Values of all beta-functions in critical region are the same for different
functions. In particular, the values of the beta-functions (18) in critical
region are almost the same as the values obtained in ref. [7]. The fast method
of determination of the inverse critical constant on a lattice based on the
formula (17) is proposed.
## References
* [1] O. Mogilevsky, Ukr.J.Phys. Vol.51 8, 820-823 (2006).
* [2] J. Fingberg, U. M. Heller and F. Karsch, Nucl. Phys. B 392, 493 (1993) [hep-lat/9208012].
* [3] M. N. Barber, Phase Transitions and Critical Phenomena Vol. 8, ed. C. Domb and J. Lebowitz, Academic Press (1981).
* [4] V. Privman, Finite-Size Scaling and Numerical Simulations of Statistical Systems, World Scientific Publishing Co. (1990).
* [5] K. Binder, Phys. Rev. Lett. 47, 693 (1981).
* [6] A. Velytsky, Int. J. Mod. Phys. C 19, 1079 (2008) [arXiv:0711.0748 [hep-lat]].
* [7] J. Engels, F. Karsch and K. Redlich, Nucl. Phys. B 435, 295 (1995) [hep-lat/9408009].
|
arxiv-papers
| 2013-12-09T22:15:36 |
2024-09-04T02:49:55.201193
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. S. Antropov, O. A. Mogilevsky, V. V. Skalozub",
"submitter": "Serge Antropov",
"url": "https://arxiv.org/abs/1312.2616"
}
|
1312.2629
|
# Sense, Model and Identify the Load Signatures of HVAC Systems in Metro
Stations
Yongcai Wang, _Member IEEE_ Institute for Interdisciplinary Information
Sciences (IIIS)
Tsinghua University, Beijing, P. R. China, 100084
[email protected] Haoran Feng Software School, Peking University,
Beijing, P. R. China, 100084
[email protected] Yongcai Wang, _Member IEEE_ Institute for
Interdisciplinary Information
Sciences (IIIS) Tsinghua University,
Beijing, P. R. China, 100084
[email protected] Haoran Feng National Engineering Research Center
of Software Engineering, Peking University,
Beijing, P. R. China, 100084
[email protected] Xiangyu Xi Department of Automation
Tsinghua University
Beijing, P. R. China, 100084
[email protected]
###### Abstract
The HVAC systems in subway stations are energy consuming giants, each of which
may consume over 10, 000 Kilowatts per day for cooling and ventilation. To
save energy for the HVAC systems, it is critically important to firstly know
the “load signatures” of the HVAC system, i.e., the quantity of heat imported
from the outdoor environments and by the passengers respectively in different
periods of a day, which will significantly benefit the design of control
policies. In this paper, we present a novel sensing and learning approach to
identify the load signature of the HVAC system in the subway stations. In
particular, sensors and smart meters were deployed to monitor the indoor,
outdoor temperatures, and the energy consumptions of the HVAC system in real-
time. The number of passengers was counted by the ticket checking system. At
the same time, the cooling supply provided by the HVAC system was inferred via
the energy consumption logs of the HVAC system. Since the indoor temperature
variations are driven by the difference of the _loads_ and the _cooling
supply_ , linear regression model was proposed for the load signature, whose
coefficients are derived via a proposed algorithm . We collected real sensing
data and energy log data from HaiDianHuangZhuang Subway station, which is in
line 4 of Beijing from the duration of July 2012 to Sept. 2012. The data was
used to evaluate the coefficients of the regression model. The experiment
results show typical variation signatures of the loads from the passengers and
from the outdoor environments respectively, which provide important contexts
for smart control policies.
## I Introduction
Being backbone of transportation network, the subways are also major energy
consumers. As stated in a site survey conducted in [2][10], a subway line (for
example the line 1 in Beijing) can consume nearly 500 thousands $kW\cdot H$
power per day in the summer season, among which, more than 40% of energy was
consumed by the Heating Ventilation and Air Conditioning (HVAC) subsystems for
cooling and ventilation. If it is possible to save the energy consumption of
the HVAC system a few percents, for example 10%, dramatical energy (nearly 20
thousands $kW\cdot H$ per line, per day) can be saved.
A major way to save energy for the HVAC systems in established subways is to
design optimal control strategies to minimize the overall energy consumption
or operating cost of the HVAC systems while still maintaining the satisfied
indoor thermal comfort and healthy environment[14]. To optimize the design and
operation of HVAC, understanding the signature of the heating or cooling load
is the critically first step, which is to estimate the the quantity of heat
(or cold) imported from environments or passengers into the subway station in
each time unit. By learning the knowledge of the load signature, the HVAC
system can be optimally controlled to supply only necessary cooling (or
heating) efforts to meet the predicted demands, which on one hand maintains
the indoor comfort, on the other hand optimizes energy consumption.
However, because the outdoor environments and the passenger flows entering or
leaving a subway station are highly dynamic, the heating (cooling) loads of a
subway station are diverse and are hard to estimate. Some existing load
estimation methods for buildings use the construction details and material
features to estimate the heating (cooling) loads according to different
outdoor temperature, using empirical models of heat conduction, radiation and
convection [15]. However these models cannot capture the special features of
the subway station: i) the impacts of passenger flows; ii) the piston wind
pushed by trains in tunnels ; iii) the complex materials and underground
construction structures. Lacking effective methods to predict the load in the
subway station, current HVAC systems generally controlled by simple time-
driven rules, or in passive responding mode. As a result, the mismatching of
load and supply is the main reason for energy waste in current subway HVAC
systems.
To characterize the load signature of HVAC system in metro stations, in this
paper, we exploit the advantages of sensing and learning technologies. In a
subway station, i.e., HaiDianHuangZhuang station of line 4 of Beijing subway,
we deployed different kinds of environment sensors to monitor the
indoor/outdoor temperatures, humidity and CO2 moisture in realtime. The
passenger flow is recorded by the ticket checking system, and the energy
consumptions of the refrigerators, ventilators and cooling towers of the HVAC
system are monitored by the deployed smart meters. By thermal principles, we
model the load of the subway station by a regression model of the sensor
readings. On the other hand, the cooling supply generated by the HVAC system
is inferred by the working states and energy consumptions of the HVAC system.
Since the indoor temperature variations are driven by the difference of load
and cooling supply, linear equations are set up, and we proposed a search
algorithm to minimize the difference between integrated load and supply for
dealing with the noises of sensor measurement.
We show by the identified load signatures from the real data of
HaiDianHuangZhuang station, that the load of HVAC systems in the subway
stations have significant characteristics, which is jointly impacted by the
outdoor temperature and the passenger flow, both of which can be predicted in
working days and the weekends. Therefore, the derived load signature will be
useful context for further design of optimal control strategies.
The remainder of this paper is organized as following. Related works are
introduced in Section 2. Sensor deployments and the field study in
HaiDianHuangZhuang station are introduced in Section 3. We proposed load and
supply models in Section 4. Solution method and experimental results and
verification of load signatures are introduced in Section 5. Conclusion and
further works are discussed in Section 6.
## II Related Work
The autonomous, optimal control for HVAC systems has attracted great research
attentions in the studies of smart and sustainable buildings [9], which is to
determine the optimal solutions (operation mode and setpoints) that minimize
overall energy consumption or operating cost while still maintaining the
satisfied indoor thermal comfort and healthy environment [14].
This goal is the same in the subway HVAC control systems. Because the HVAC
systems contain different types of subsystems, such as gas-side and water-side
subsystems, the optimal control problems of HVAC are extremely difficult. One
of the difficulties is the lack of an exact model to describe the internal
relationships among different components. A dynamic model of an HVAC system
for control analysis was presented in [13]. The authors proposed to use
Ziegler-Nichols rule to tune the parameters to optimize PID controlle. A
metaheuristic simulation–EP (evolutionary programming) coupling approach was
developed in [5], which proposed evolutionary programming to handle the
discrete, non-linear and highly constrained optimization problems. Multi
agent-based simulation models were studied in [3] to investigate the
performance of HVAC system when occupants are participating. In [16], swarm
intelligence was utilized to determine the control policy of each equipment in
the HVAC system.
One of the most closely related work is the SEAM4US (Sustainable Energy
mAnageMent for Underground Stations) project established in 2011 in Europe[1].
It studies the metro station energy saving mainly from the modeling and
controlling aspect. Multi-agent and hybrid models were proposed to model the
complex interactions of energy consumption in the underground subways[12, 11].
Adaptive and predictive control schemes were also proposed for controlling
ventilation subsystems to save energy [7].
Another related work reported the factors affecting the range of heat transfer
in subways [8]. They show by numerical analysis that how the heat is
transferred in tunnels and stations. Reference [4] studied the environmental
characters in the subway metro stations in Cairo, Egypt, which showed the
different environment characters in the tunnel and on the surface. The most
related work is [10], which surveyed the energy consumption of Beijing subway
lines in 2008.
Different from these existing work, we deployed sensors and presented models
to study the the load signatures and distinct features of energy consumptionof
subway HVAC systems.
## III Monitor the Thermal Dynamics in Subway Station
### III-A Notations
Before introducing the deployment of sensors, we firstly define notations
which will be used in this paper, which are listed in Table I.
TABLE I: Notations defined for the load and supply models Notations | Definitions
---|---
$L(t)$ | the quantity of thermal imported from outside to inside at $t$.
$T(t)$ | the indoor temperature at $t$.
$T_{o}(t)$ | the outdoor temperature at $t$
$R_{eq}$ | heat transferring resistance from outside to inside.
$M_{air}$ | the volume of outdoor air input into the subway station
$c$ | the heat capacity of per cube air.
$T_{p}$ | the body temperature of people.
$n(t)$ | the the number of passengers at time $t$.
$M_{mix}$ | volume of mixed air
$M_{new}$ | volume of new air
$M_{ac}$ | volume of cooling air
$\alpha$ | the proportion of new air in the mixed air.
$T_{ac}$ | temperature of cooling air at the outlet of refrigerator.
$T_{mix}$ | temperature of the mixed air.
$e_{ac}$ | efficiency of of the cooling air transportation.
$M_{z}$ | the volume of air inside the subway station.
### III-B Sensor Deployment
A way to capture the thermal and environment dynamics in the subway station is
to deploy sensors to measure the indoor, outdoor temperatures, passenger flows
and power consumptions of the HVAC systems in real-time. In HaiDianHuangZhuang
subway station, which is a transferring station between line 10 and line 4 in
Beijing subway, we deployed different kinds of sensors and smart meters to
measure above information. The sensors were mainly deployed in the section of
line 4, which is operated by Hongkong MTR.
We installed temperature sensors at four points inside the subway station and
two points outside the subway to monitor the indoor and outdoor temperatures
$T(t)$ and $T_{o}(t)$ respectively. Note that $T(t)$ is calculated by the
average of indoor temperature sensors, so as $T_{o}(t)$. CO2 sensors are
installed inside the subway to measure the indoor air quality. The passenger
flow is recorded by the ticket checking system, which is denoted by $n(t)$.
Note that $n(t)$ is calculated by the sum of the checked-in and checked-out
passengers from $t-1$ to $t$.
To monitor the working state of the HVAC system, temperature sensors were
installed at the inlets and the outlets of the refrigerators to measure the
temperature of the return air $T(t)$ and the cooling air $T_{ac}(t)$.
Temperature sensors are also installed at the new air pipes and mixed air
pipes of the ventilator to measure the temperatures of new air $T_{o}(t)$ and
mixed air $T_{mix}(t)$. Note that the mixed air is the mixing of return air
and new air. The energy consumptions of different components of the HVAC
system, i.e, refrigerator, ventilator, water tower, pumps, fans etc are
measured in real-time by the embedded power meters of the HVAC system.
### III-C Observed Passenger Flow Pattern
From the data of ticket checking system, Fig. 1 shows the variation of
passenger flow as a function of time during a week from Sep. 15 to Sep. 21.
The passenger flow shows different structure in working days and weekends. In
working days there are two obvious peaks in the rush hours in the morning and
in the evening. In week ends, the passenger flow was almost uniformly
distributed from 8:00 AM to 8:00 PM.
Figure 1: Pattern of passenger flow over a week.
### III-D Observed Load Signatures
Figure 2: How the indoor temperature was affected by the outdoor temperature
and passenger flow when the HVAC system was running.
The indoor thermal condition is mainly affected by three factors: i) the
outdoor temperature; ii) the passenger flow; iii) the working state of the
HVAC system. To investigate how these factors affect the indoor temperature,
for a particular day, Sept. 4, February, a sunny day in 2012, we monitored the
variations of outdoor temperature, indoor CO2 density and indoor temperature
and plotted the results in Fig. 3. It intuitively shows how the outdoor
temperatures and passenger flows affect the variation of indoor temperature.
Note that during the monitoring, the HVAC system was working.
Fig. 3(a) shows the outdoor temperature in that day. Fig. 3(b) shows the
traffic flow variation which was recorded by the ticket checking system. Fig.
3(c ) shows the variation process of indoor temperature. By comparing these
three figures, we can see that: i) the indoor temperature curve varied between
22 centigrade and 27 centigrade, which was jointly impacted by the outdoor
environments, the passenger flows and the HVAC system; ii) there are four
peaks in the temperature curve, which are according to following reasons:
* •
The first peak is at 4:00 AM, which is because the HVAC system was off in the
morning, so the indoor temperature increases slowly.
* •
The second peak is at 8:00 AM, the rush hour in the morning. It is because the
quantity of thermal brought in by the passenger flow was more than the cooling
effects of the HVAC system.
* •
The third peak is at 2:00 PM, which is the hottest time in the day. This peak
is not obvious, because the outdoor temperature increases slowly, the HVAC
system had enough time to cool down the indoor temperature.
* •
The last peak is at 18:00 PM, the rush hour in the evening, because the
cooling effects of HVAC is less than the thermal brought in by the passengers.
These measurements show intuitively the impacts of environments and passengers
to the indoor temperature. However a quantitative model to more accurately
characterize these impacts is still lacked. We call it the load signature,
which will be modeled and learned in the next section.
## IV Model and Identify the Load Signatures
From the sensor readings, we see the typical features of the outdoor
temperature and passenger flows, but it is still unclear whose influence is
more significant to the indoor temperature. In this section, we present linear
regression model to identify the load signature.
### IV-A Load Model
###### Definition 1 (load model)
We define the quantity of heat imported from outdoor environments and the
passengers into the subway station in a time unit as the _load_ of the HVAC
system in the subway station.
$L\left(t\right)=\frac{{{T_{o}}(t)\\!\\!-\\!\\!T(t)}}{{{R_{eq}}}}+n(t)\left({{T_{p}}\\!\\!-\\!\\!T\left(t\right)}\right)+c{M_{air}}\left({{T_{o}}(t)\\!\\!-\\!\\!T(t)}\right)$
(1)
$L(t)$ contains three parts: 1) the heat imported from outdoor environments by
heat conduction through walls, roofs etc, i.e.,
$\frac{T_{o}(t)-T(t)}{R_{eq}}$; 2) the heat imported by passengers, i.e,
$n(t)\left(T_{p}-T(t)\right)$; 3) the heat imported via outdoor air, i.e.,
$cM_{air}\left(T_{o}(t)-T(t)\right)$ which is due to piston wind or wind
entered through doors. We can rewrite the equation (1) as:
$\begin{array}[]{l}L\left(t\right)=c_{p}n(t)\left({{T_{p}}-T\left(t\right)}\right)+\left({c{M_{air}}+\frac{1}{{{R_{eq}}}}}\right)\left({{T_{o}}(t)-T(t)}\right)\\\
={L_{p}}(t)+{L_{a}}(t)\end{array}$ (2)
where $L_{p}(t)=c_{p}n(t)\left({{T_{p}}-T\left(t\right)}\right)$ is only
related to the passengers, called the _passenger introduced load (PIL)_ ;
$L_{e}(t)=\left({c{M_{air}}+\frac{1}{{{R_{eq}}}}}\right)\left({{T_{o}}(t)-T(t)}\right)$
is caused by the indoor-outdoor temperature difference, which is called
_Environment Introduced Loads (EIL)_. Note that in (2), $T_{o}(t),T(t),n(t)$
are measured in realtime; $T_{p}$, $c$ are known constants; only
$\\{c_{p},M_{air},R_{eq}\\}$ are unknown variables.
### IV-B Supply Model
The HVAC system runs adaptively to response the dynamics of the loads to
control the indoor temperature at desired temperature. By assuming the indoor
air is fully mixed, the variation of indoor temperature is mainly caused by
the thermal difference of the load and the supply:
$L(t)-S(t)=cM_{z}\Delta(t)$ (3)
where $M_{z}$ is the volume of air in the subway station, which can be
calculated by the geometrical information of the station, such as the length,
width, height of the station and the tunnels.
$\Delta(t)=\left(T(t+1)-T(t)\right)$ is the temperature difference changed
from time $t$ to time $t+1$.
Since the working states of the HVAC system were fully monitored, the cooling
supply can be inferred by the sensors readings of the HVAC system. The HVAC
system in subway station has three working modes:
1. 1.
_New air mode:_ , which is used when the outdoor temperature is lower than the
indoor temperature. In this mode, the refrigerator is off; The new air is the
source to cool the indoor air.
2. 2.
_Refrigerator mode:_ is used when the outdoor temperature is higher than the
indoor temperature, during which the new air intaking is closed and the
refrigerators are working to cool the indoor air.
3. 3.
_Mixed mode:_ is used when the new air’s capacity is not enough to cool the
indoor temperature, so both the new air ventilator and a part of the
refrigerator are working.
###### Definition 2 (supply model)
We define the quantity of heat cooled down by the HVAC system in a unit time
as the _supply_ of the HVAC system, which is defined based on different
working modes of the HVAC system:
$\begin{gathered}S(t)=\hfill\\\
\left\\{{\begin{array}[]{*{20}{l}}{c{M_{new}}\left({T(t)-{T_{o}}(t)}\right),{\textrm{
New air mode}}}\\\
{\left({T_{in}^{w}(t)-T_{out}^{w}(t)}\right)V_{cool}^{w}\beta_{ac}{\textrm{,
Refrigerator mode}}}\\\
{c{M_{new}}\left({T(t)\\!\\!-\\!\\!{T_{o}}(t)}\right)\\!\\!+\\!\\!\left({T_{in}^{w}(t)\\!\\!-\\!\\!T_{out}^{w}(t)}\right)V_{cool}^{w}\beta_{ac}{\textrm{,
Mixed}}}\end{array}}\right.\hfill\\\ \end{gathered}$ (4)
Note that in (4), $M_{new}$ is the volume of new air blowed into the subway
station by the new air ventilator. $T_{in}^{w}(t)-T_{in}^{w}(t)$ is the
temperature difference of input and output water at the refrigerator;
$V_{cool}^{w}$ is the volume of the cooling water;
$\beta_{ac}=c_{cool}^{w}e_{ac}$, where $c_{cool}^{w}$ is the heat capacity of
the cooling water and $e_{ac}$ is the heat transportation efficiency of the
refrigerator. So that
$\left({T_{in}^{w}(t)-T_{out}^{w}(t)}\right)V_{cool}^{w}\beta_{ac}$ measures
the cooling supply provided by the refrigerator and $cM_{new}(T(t)-T_{o}(t))$
measures the cooling supply of the new air.
Note that $T_{o}(t),T(t),T_{in}^{w}(t),T_{out}^{w}(t)$, and $V_{cool}^{w}$ are
measured in real time by the deployed sensors. $c$ is a known constant. Only
$M_{new}$ and $\beta_{ac}$ are unknown. But the volume of air blowed by the
ventilator in a time unit can be further inferred by the power meter readings
of the ventilators. From the fan affinity laws[6], ventilators operates under
a predictable law that the air volume delivered by a ventilator is in the one-
third order of its operating power.
$M_{v}=\beta_{v}E_{v}^{\frac{1}{3}}$ (5)
So that, the supply model of the HVAC system in the subway station can be
rewritten into:
$\begin{gathered}S(t)=\hfill\\\
\left\\{{\begin{array}[]{*{20}{l}}{cE_{v}^{\frac{1}{3}}\beta_{v}\left({T(t)-{T_{o}}(t)}\right),{\textrm{
New air mode}}}\\\
{\left({T_{in}^{w}(t)-T_{out}^{w}(t)}\right)V_{cool}^{w}\beta_{ac}{\textrm{,
Refrigerator mode}}}\\\
{cE_{v}^{\frac{1}{3}}\beta_{v}\left({T(t)\\!\\!-\\!\\!{T_{o}}(t)}\right)\\!\\!+\\!\\!\left({T_{in}^{w}(t)\\!\\!-\\!\\!T_{out}^{w}(t)}\right)V_{cool}^{w}\beta_{ac}{\textrm{,
Mixed}}}\end{array}}\right.\hfill\\\ \end{gathered}$ (6)
By substituting (6) and (2), we can set up linear equations to identify the
unknown parameters in the load and supply functions.
Figure 3: Derived load signatures Vs. Variation Signatures of Supply Vs. the
relative error between integrated load and integrated supply
### IV-C Identify Load Signature by Linear Regression
Let’s consider the joint impacts of load and supply to the indoor temperature.
Without losing of generality, let’s consider the case when the HVAC is working
in the refrigerator mode, by substituting (2) and (6) into (3), we have:
$\left[{\begin{array}[]{*{20}{c}}{n(t)({T_{p}}-T(t))}\\\ {{T_{o}}(t)-T(t)}\\\
{V_{cool}^{w}(T_{in}^{w}(t)-T_{out}^{w}(t))}\end{array}}\right]^{T}\left[{\begin{array}[]{*{20}{c}}{{c_{p}}}\\\
{{\alpha}}\\\ {{-\beta_{ac}}}\end{array}}\right]=c{M_{z}}\Delta(t)$ (7)
where $\alpha=cM_{air}+\frac{1}{R_{eq}}$ is the coefficients of
$T_{o}(t)-T(t)$ in the load model, which is modeled as one unknown
coefficient. We can rewrite (7) as
$\mathbf{A}(t)\mathbf{\theta}=\mathbf{B}(t)$. Then by sensor measurements and
HVAC states from 1 to t, we can set up an overdetermined observation matrix
$\mathbf{A}_{1:t}=[\mathbf{A}(1),\mathbf{A}(2),\cdots,\mathbf{A}(t)]^{T}$, and
an observation vector
$\mathbf{B}_{1:t}=[\mathbf{B}(1),\mathbf{B}(2),\cdots,\mathbf{B}(t)]^{T}$.
Then the problem of identifying the load signature is to identify the vector
$\mathbf{\theta}$ by solving $\mathbf{A}_{1:t}\theta=\mathbf{B}_{1:t}$, with
the constraints that $c_{p},\alpha,\beta_{ac}$ are nonnegative.
## V Techniques to Solve the Regression Model by Real Data
We used real data collected from HaiDianHuangZhuang Station to calculate the
model parameters in (7) and to investigate the signatures of the loads.
### V-A Calculate Coefficients by Real Data
Data collected from HaiDianHuangZhuang station from a timespan of Aug 21th,
2013 to Aug 23th, 2013 was selected to solve the linear regression model. The
dataset provides real-time $T(t)$, $T_{ac}(t)$, $T_{in}^{w}(t)$,
$T_{out}^{w}(t)$, $V_{cool}^{w}$, and $E_{v}$, which are in one-minute
resolution. In addition, passenger flows are acquired by the ticket checking
system in per-hour resolution. We estimated the per-minute resolution
passenger amount by linear interpolations. Based on these data, the
observation matrix $\mathbf{A}_{1:t}$ is constructed and the vector
$\mathbf{B}_{1:t}$ are constructed. Note that the volume of air $M_{z}$ in the
subway station is inferred by the geometrical data of the station.
Since the coefficients are required to be nonnegative, directly applying the
least square estimation is inefficient. We propose a search algorithm to solve
this constrained optimization problem:
$\begin{gathered}\theta=\mathop{\arg\min}\limits_{\left[{{c_{p}},\alpha,{\beta_{ac}}}\right]}\frac{{\sum\limits_{i=1}^{t}{\left|{{\mathbf{A}_{i,1}}{c_{p}}+{\mathbf{A}_{i,2}}\alpha-{\mathbf{A}_{i,3}}{\beta_{ac}}-{\mathbf{B}_{i}}}\right|}}}{{\sum\limits_{i=1}^{t}{\left({{\mathbf{A}_{i,1}}{c_{p}}+{\mathbf{A}_{i,2}}\alpha}\right)}}}\hfill\\\
{\text{subject to: }}{c_{p}}>0,\alpha>0,{\beta_{ac}}\geq 0\hfill\\\
\end{gathered}$ (8)
$\mathbf{A}_{i,j}$ is the item in $i$th column and $j$th row in the matrix
$\mathbf{A}_{1:t}$. Note that we divide the accumulated absolute difference of
the loads and the supplies by the accumulated loads, which is to find the
coefficient vector that can provide the minimum relative difference between
the load vector and the supply vector. Otherwise, smaller parameters providing
smaller absolute error tend to be voted for the lacking of normalization. The
search algorithm searches all combinations of $[c_{p},\alpha]$ for
$c_{p}<1000$ and $\alpha<10000$. For each combination of $c_{p}$ and $\alpha$,
$\beta_{ac}$ that provides the minimum relative error is calculated. The
parameter set $[c_{p}^{*},\alpha^{*},-\beta_{ac}^{*}]$ which provides the
overall minimum relative error is chosen as the optimal solution of problem
(8). For the number of coefficients is limited, the computing complexity of
the algorithm is tolerable.
### V-B The Load Signatures
Another difficulty to solve (8) is that we found the vector $B_{1:t}$ is
highly zigzagging over time, which is due to the noises of the measurements of
the temperature sensors, i.e., the difference of indoor temperature of
successive time cannot be accurately measured because of the accuracy
limitation of sensors. To overcome this noise issue, we proposed to further
calculate the coefficients by minimizing the differences of the _integrated
loads_ and the _integrated supply_. We define the difference between the
integrated load and the integrated supply by
$\mathbf{C}(T)=\sum_{t=1}^{T}\mathbf{A}(t)$; the integrated indoor thermal
variation is defined by $\mathbf{D}(T)=\sum_{t=1}^{T}\mathbf{B}(t)$. Then we
solve (8) by replacing $\mathbf{A}_{i,j},\mathbf{B}_{i}$ by
$\mathbf{C}_{i,j},\mathbf{D}_{i}$. This method gives us robust estimation of
the coefficients which can tolerates the sensor noises. For the particular
dataset of August 23, 2013 of HaidianHuangZhuang station, we calculated the
optimal parameter set $\theta$ as $[83,53703,-1290071]^{T}$. When varying the
scope of the data, we found the solution vary within tolerable range of
errors.
By substituting the calculated coefficients into the load model, the derived
load signature was plotted in Fig.3a). It shows that _the loads from the
outdoor temperature take the major portion, while the thermal loads introduced
by the passengers take a small portion_. The real-time supplies calculated by
the supply model are plotted in Fig.3b). We can see the variation of supply
has similar pattern as the load. The relative error between the integrated
load and integrated supply is plotted in Fig.3c), which it is relative small
by calculating using the optimally derived parameters. It indicates that the
searching algorithm has provided a rather confident estimation to the load
signatures.
## VI Conclusion and Discussion
This paper investigated the load signatures of HVAC system in subway station
based on real data collected from subway station. By extensive sensor data
collected from environments and the HVAC system, we proposed a linear
regression to model to describe the impacts of loads and the cooling supply to
the indoor temperature. We then present a search algorithm to identify the
model coefficients by minimizing the integrated differences between load and
supply, which can tolerate the noises of sensor measurements. Experiment
results on real dataset show the proposed method can provide rather confident
load signature which highly coincides with the real-time supply measurements.
Since the load signature provide important knowledge for the energy efficient
control, we will study the optimal control strategies in our future work.
## References
* [1] Sustainable energy management for underground stations, 2011.
* [2] Beijing subway, Oct. 2013. Page Version ID: 576457338.
* [3] C. Andrews, D. Yi, U. Krogmann, J. Senick, and R. Wener. Designing buildings for real occupants: An agent-based approach. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 41(6):1077–1091, 2011.
* [4] A. H. A. Awad. Environmental study in subway metro stations in cairo, egypt. Journal of Occupational Health, 44(2):112–118, 2002.
* [5] K. Fong, V. Hanby, and T. Chow. HVAC system optimization for energy management by evolutionary programming. Energy and Buildings, 38(3):220–231, Mar. 2006.
* [6] R. W. Ford. Affinity laws. ASHRAE JOURNAL, 53(3):42–43, 2011.
* [7] A. Giretti, A. Carbonari, and M. Vaccarini. Energy saving through adaptive control of ventilation systems. Gerontechnology, 11(2), June 2012.
* [8] Z. Hu, X. Li, X. Zhao, L. Xiao, and W. Wu. Numerical analysis of factors affecting the range of heat transfer in earth surrounding three subways. Journal of China University of Mining and Technology, 18(1):67–71, Mar. 2008.
* [9] A. Kelman, Y. Ma, and F. Borrelli. Analysis of local optima in predictive control for energy efficient buildings. In 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pages 5125–5130, 2011.
* [10] M. Lu, T. He, X. Pei, and Z. Chen. Analysis of the electricity consumption and the water consumption of beijing subway. Journal of Beijing JIaotong University, 35(1):136–139, Feb. 2011\.
* [11] R. L. Roberta Ansuini. Hybrid modeling for energy saving in subway stations. 2012\.
* [12] R. Serban, H. Guo, and A. Salden. Common hybrid agent platform – sustaining the collective. In 2012 13th ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel Distributed Computing (SNPD), pages 420–427, 2012.
* [13] B. Tashtoush, M. Molhim, and M. Al-Rousan. Dynamic model of an HVAC system for control analysis. Energy, 30(10):1729–1745, July 2005.
* [14] S. Wang and Z. Ma. Supervisory and optimal control of building HVAC systems: A review. HVAC&R Research, 14(1):3–32, 2008.
* [15] D. Yan, J. Xia, W. Tang, F. Song, X. Zhang, and Y. Jiang. Dest—an integrated building simulation toolkit part i: Fundamentals. In Building Simulation, volume 1, pages 95–110. Springer, 2008\.
* [16] R. Yang and L. Wang. Optimal control strategy for HVAC system in building energy management. In Transmission and Distribution Conference and Exposition (T D), 2012 IEEE PES, pages 1–8, 2012.
|
arxiv-papers
| 2013-12-10T00:11:50 |
2024-09-04T02:49:55.207006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongcai Wang, Haoran Feng, Xiangyu Xi",
"submitter": "Yongcai Wang",
"url": "https://arxiv.org/abs/1312.2629"
}
|
1312.2632
|
# SEED: Public Energy and Environment Dataset for Optimizing HVAC Operation in
Subway Stations
Yongcai Wang, _Member IEEE_ Institute for Interdisciplinary
Information Sciences (IIIS)
Tsinghua University,
Beijing, P. R. China, 100084
[email protected] Haoran Feng National Engineering Research
Center of Software Engineering,
Peking University,
Beijing, P. R. China, 100084
[email protected] Xiao Qi Institute for Interdisciplinary
Information Sciences (IIIS)
Tsinghua University,
Beijing, P. R. China, 100084
[email protected]
###### Abstract
For sustainability and energy saving, the problem to optimize the control of
heating, ventilating, and air-conditioning (HVAC) systems has attracted great
attentions, but analyzing the signatures of thermal environments and HVAC
systems and the evaluation of the optimization policies has encountered
inefficiency and inconvenient problems due to the lack of public dataset. In
this paper, we present the Subway station Energy and Environment Dataset
(SEED), which was collected from a line of Beijing subway stations, providing
minute-resolution data regarding the environment dynamics (temperature,
humidity, CO2, etc.) working states and energy consumptions of the HVAC
systems (ventilators, refrigerators, pumps), and hour-resolution data of
passenger flows. We describe the sensor deployments and the HVAC systems for
data collection and for environment control, and also present initial
investigation for the energy disaggregation of HVAC system, the signatures of
the thermal load, cooling supply, and the passenger flow using the dataset.
## I Introduction
For low-carbon, sustainability and environment friendly living, reducing the
energy consumptions of electrical appliances has attracted great attentions,
among which, optimizing the operations of Heating Ventilation and Air
Conditioning (HVAC) systems plays a major role, because the HVAC systems are
energy consuming giants in our living environments. For example, the HVAC
systems in a commercial building may consume nearly 50% of overall energy [15]
and the HVAC system in a subway station can consume more than 40% of the total
power [14]. If we can decrease the energy consumption of the HVAC system a few
percents, for example 10%, dramatical energy can be saved.
A major way to save energy for the HVAC systems is to design optimal control
strategies to minimize the overall energy consumption while still maintaining
the satisfied indoor thermal comfort and healthy environment [20]. This
process generally needs three procedures: 1) identifying the load signatures
of the buildings and the cooling-energy patterns of the HVAC systems; 2)
designing the optimal control policies; 3) evaluating the control policies.
Current approaches generally used simulation, or model-based methods to tackle
the diversity and complexity of thermal exchanging in different kinds of
buildings. Because the buildings’ surfaces and structures are diverse and the
inner states of the HVAC system are complex to monitor, it is generally
expensive to build reasonable models, and at least in some extend lacks
fidelity in design and evaluation.
On the other hand, although it is highly relevant to use data mining or
machine learning techniques to identify the signatures of the thermal
environments and the HVAC systems, which are also powerful tools for
optimizing the control policies, very few work has been seen in this area. It
is at least partially due to lack of publicly available dataset in this
domain, which is mainly because of the difficulty for monitoring the dynamics
of the thermal environments, user states, and the generally non-accessing of
the HVAC working states. Also in the interdisciplinary areas of power and
computing, in a very closed domain, some recent published data set: REDD[13],
BLUED[2], Smart*[5] have dramatically benefited the studies in energy
disaggregation in smart homes. However, there are still few dataset regarding
the real, long-term, fine-grained working states, thermal environment
conditions and user states in HVAC systems.
In this paper, we present the Subway station Energy and Environment Dataset
(SEED), which was collected in August and September in the summer of 2013,
over multiple stations from a line of Beijing subway. It provides minute-
resolution, comprehensive data regarding the environment dynamics
(temperature, humidity, CO2, etc.), working states and energy consumptions of
the HVAC systems (ventilators, refrigerators, pumps), and hour-resolution data
of passenger flows. These data was a part of the data measured and recorded
during our projects for developing the autonomous HVAC control systems for
Beijing metro stations. For the sake of protecting the privacy of the subway
stations, the name of the stations and the lines are hidden in the public
dataset, which will not affect its usage. We describe the deployment of
sensors, the HVAC systems, the hardware and the software platform for
environment, HVAC states, and the passenger flow monitoring. We also present
initial investigation for the energy disaggregation, environment and passenger
load signature analysis and HVAC cooling supply signature analysis utilizing
the dataset. The entire dataset and the notes to explain it are available
online at: http://iiis.tsinghua.edu.cn/~yongcai/SEED/.
The remainder of this paper is organized as following. Background and related
works are introduced in Section II. Sensor deployments and the system
architecture of the subway HVAC system are introduced in Section III. The
overview of the dataset and some attributes are highlighted in Section IV. We
present basic investigations on the power disaggregation, signatures on the
loads and cooling supply in Section V. Conclusion and further works are
presented in Section VI.
## II Related Work and Background
### II-A Optimization of HVAC systems
HVAC system optimization generally includes three steps:
#### II-A1 Signature Identification
which is to identify the signatures of thermal environments and the HVAC
systems. This step generally need extensive in-field survey, measurements
under controlled HVAC operations, and post data processing. In case the in-
field measurements are infeasible because of lacking the real systems or
resources to conduct measurements, theoretical models or simulation based
models are used instead. The most widely used simulators include DeST[21] and
EnergyPlus[7], which provide detailed models to simulate the thermal
exchanging patterns in different kinds of buildings. The encoded parameters of
buildings include the size, surface styles, thickness, materials of walls,
roofs, windows, doors, and a lot of other parameters, so that it is generally
time consuming to setup an acceptable simulation model. In theoretical model
aspect, dynamic model of an HVAC system for control analysis was presented in
[18]. The authors proposed to use Ziegler-Nichols rule to tune the parameters
to optimize PID controller. Multi agent-based simulation models were studied
in [3] to investigate the performance of HVAC system when occupants are
participating. More simulation models and theoretical models can be referred
to survey in [19].
#### II-A2 Designing Optimal Control Policy
is to design adaptive control strategies or the optimal setpoints based on
rough theoretical or simulation-based models to minimize the overall energy
consumption of the HVAC system while still maintaining the required indoor
thermal comfort. Tremendous research efforts have been devoted in this area,
especially for sustainable buildings [12][10][22]. Various optimization
techniques have been exploited in existing studies, including evolutionary
computing[8], genetic algorithm and neutral networks etc [6]. A survey of the
optimization methods was conducted by [20].
#### II-A3 Evaluate the Control Policy
Since the HVAC systems are running in practical environments, it is generally
infeasible to directly test the immature control policies in the HVAC systems.
Therefore, most of the control policies are evaluated via simulations in their
design phase, which in some extend lacks the fidelity of system dynamics. By
providing public, fine-grained dataset regarding thermal environments and HVAC
system energy and state logs, all the above three steps can be benefited.
### II-B Optimizing HVAC Systems in Subway Stations
As a branch of HVAC systems for large buildings, the HVAC systems in subway
stations have also attracted great attentions. One of the most closely related
work is the SEAM4US (Sustainable Energy mAnageMent for Underground Stations)
project established in 2011 in Europe[1]. It studies the metro station energy
saving mainly from the modeling and controlling aspect. Multi-agent and hybrid
models were proposed in[17, 16], and adaptive and predictive control schemes
were proposed for controlling ventilation subsystems to save energy [9].
Another related work reported the factors affecting the range of heat transfer
in subways [11]. They showed by numerical analysis that how the heat was
transferred in tunnels and stations. Reference [4] studied the environmental
characters in the subway metro stations in Cairo, Egypt, which showed the
different environment characters in the tunnel and on the surface.
### II-C Related Datasets
This paper focuses on providing public dataset for efficiency and convenience
in studying the HVAC optimization problems. Although few datasets are
available in HVAC studies, a series of public datasets were published recently
in the area of energy disaggregation in smart homes, including REDD[13],
BLUED[2], Smart*[5] etc. The prevalence of these datasets has strongly
benefited the application of machine learning methods into energy
disaggregation area. For the related data analysis works in HVAC systems, the
most related one is [14], which surveyed the energy consumption of Beijing
subway lines in 2008, but without providing a dataset.
## III Sensing and HVAC Control Systems
The SEED dataset was constructed during our development of the autonomous HVAC
energy conservation systems for Beijing subway stations. We firstly report the
deployment of sensors by using a subway station as an example.
### III-A Sensor Deployment
Our way to capture the thermal and the environment dynamics in the subway
station is to deploy sensors to measure the indoor, outdoor temperatures,
passenger flows and power consumptions of the HVAC systems in real-time. In
subway station A (we hide the name for the sake of privacy protection), which
is a transferring station between two lines in Beijing subway, we deployed
different kinds of sensors and smart meters to measure above information. The
architecture of the station and the deployment of sensors are shown in Fig. 1,
which is from a snapshot of our subway station environment monitoring
interface.
Figure 1: The structure of subway station A and the deployment of sensors for
environment monitoring
#### III-A1 Environment Sensors
We deployed temperature, humidity and CO2 sensors at four points inside the
subway station and two points outside the subway station to monitor the indoor
and outdoor temperatures, humidity and CO2 density respectively. The sensors
are connected to a data collection server. Each sensor reports data once per
minute, so the time resolution of the environment data is one-minute. The
deployed positions of the sensors in the Station A are shown in Fig.1. Similar
sensor deployment and data collection strategy are also used in other stations
in the same line to collect the environment data in real-time.
#### III-A2 Passenger Flow
Since the thermal brought in by the passengers is also an important source of
heat, we acquired the passenger flow data from the operating company of the
subway. The passenger flow was recorded by the ticket checking system. In SEED
data set, passenger flows over multiple days in multiple stations are
provided. We will compare the different temporary patterns of the passenger
flows in the working days and in the weekends in the next section.
#### III-A3 Run-time Parameters and States of the HVAC System
By deploying power meters, sensors, and by readings from the internal sensors
of the HVAC system, the run-time parameters and working states of the HVAC
system, including the data of the refrigerators, ventilators, cooling towers,
pumps and the valves of the HVAC system are monitored. The HVAC systems in
different subway stations have the same architecture. Each HVAC system in a
station contains 3 refrigerators, 2 supply fans, 2 return fans, 2 exhaust
fans, 4 cooling pumps, 4 chilling puns, a set of valves. The sensor readings
of these devices are listed in Table I. These data is reported to the central
data collection server in one-minute time resolution.
TABLE I: List of data types provided in SEED dataset Environment Info | Type of Sensors | Type of Values
---|---|---
6 Temperature sensors | Temperature at $i$th outdoor sensor | oC
| Temperature at $i$th indoor sensor | oC
6 Humidity sensors | Humidity at $i$th indoor sensor | %
| Humidity at $i$th outdoor sensor | %
6 CO2 sensors | CO2 at $i$th outdoor sensor | mg/kg
| CO2 at $i$th indoor sensor | mg/kg
Devices of HVAC | Parameters or States | Types of Value
3 Refrigerators | Power of $i$th Refrigerator | Watt
| Current of $i$th Refrigerator | Ampere
| State of $i$th Refrigerator | 0/1
| Cool Water Temperature | oC
| Return Water Temperature | oC
2 Supply fans | Power of $i$th fan | Watt
2 Return fan | Current of $i$th fan | Ampere
2 Exhaust fan | Working State of of $i$th fan | 0/1
| Supply air temperature | oC
| Return air temperature | oC
| Exaust air temperature | oC
4 Cooling pumps | Power of $i$th pump | Watt
4 Chilling pumps | Current of $i$th pump | Ampere
| Working State of $i$th pump | 0/1
Valves | States of $i$th valve | %
Events | Operating logs | time + event
Frequency changers | Logs of frequency changers | time+ event
Passenger Flow | Data type | Types of Value
| number of checked in passengers | $n$/hour
| number of checked out passengers | $n$/hour
### III-B Hardware and Software of HVAC Monitoring and Control
#### III-B1 Hardware
The SEED dataset contains data types in above list collected from three
stations over multiple days. The HVAC systems in different stations shares the
same structure, which is illustrated in Fig.2. The devices in the HVAC systems
are all from Carrier http://www.carrier.com.cn. We added the new air
temperature sensors, return air temperature sensors and deployed Profibus DP
network to connected the sensors into the information collection server. We
have also developed the control cabinet and autonomous control logics for the
HVAC system. All data are collected by the information collection server to be
reported to the central control console in real-time.
Figure 2: Architecture of HVAC system in a subway station
#### III-B2 Software
Based on the data collected in real-time by the deployed sensors, both the
environment monitoring system and the HVAC working state monitoring systems
were developed. Fig.1 shows the snapshot of environment monitoring interface.
The overall sensing and control systems were established in the spring of 2013
and they have run during the whole summer of 2013. In the SEED data set, we
chose data from August and September, including data from both the very hot
days and data for the days when outdoor temperatures are lower in indoor
temperatures.
Figure 3: Hardware of data collection server and control cabinet
Figure 4: Interface of HVAC working state monitoring software
## IV Basic Investigation to SEED Dataset
We conducted basic research on the SEED dataset to investigate basic features
of the thermal environments in the subway stations and the features of the
HVAC system shown by the data.
1. 1.
Energy disaggregation in the HVAC system.
2. 2.
Temperature difference Vs. States of Refrigerator.
3. 3.
Signatures of the passenger flow.
4. 4.
Correlation features of CO2 and passenger flow.
5. 5.
Responding speed to cooling supply;
### IV-A Energy Disaggregation in HVAC
Because the HVAC system is an energy consuming giant, to understand how the
energy was consumed in the HVAC system is of the primary interests to many
researchers. We select data from 8.21 - 8.23, 8.29 - 8.31, and 9.1 -9.30 three
periods from a subway station to investigate the disaggregated energy
consumption in the HVAC systems (other stations have similar features), when
the outdoor temperatures are different. The average peak temperature of these
three periods are $35^{o}C$, $31^{o}$C and $27^{o}$C respectively.
Fig. 5 shows the daily average energy consumptions of the refrigerator,
chilled pump, cooling pump and fans in three periods. Some interesting
phenomena can be seen: 1) The energy consumptions of the HVAC are highly
relevant to the outdoor temperatures. _The higher is the daily average outdoor
temperature, the higher is the daily energy consumption._ 2) The fans consume
similar amount of energy in all three periods, so _the energy differences over
different periods are mainly dominated by the energy consumptions differences
of the refrigerator, chilled pump and the cooling pump._ 3) Since the pumps
work only if the refrigerator is working, so their energy consumptions are
strongly correlated. _We can basically disaggregate the consumptions of HVAC
into the consumption of ventilating (fans), which is rather stable and the
consumptions of the cooling utilities (refrigerators and pumps), which are
dynamic according to the outdoor weathers_. Reducing the consumptions of the
cooling utilities should be the major way for reducing consumptions of HVAC.
8.21.2013-8.23.2013
8.29.2013-8.31.2013
9.1.2013-9.30.2013
Figure 5: Comparing of average daily disaggregated energy consumptions over
different time periods
a) Indoor, outdoor temperature variations
b) States and currents of the refrigerators
Figure 6: Indoor outdoor temperature differences VS. the states and the
consumptions of the refrigerators.
### IV-B Temperature Differences VS. States of Refrigerators
To control the indoor temperature at the desired temperature point, the
refrigerators and the pumps work adaptively to response to the temperature
variations. We investigated via SEED dataset how the working states and energy
consumptions of the refrigerators change over a day with the variations of the
outdoor temperatures. Fig.6a) shows the temperature variations and indoor-
outdoor temperature differences over a day. Fig.6b) shows the concurrent
working states and energy consumptions of the two refrigerators in that day.
We can basically see the working loads of the refrigerators are closely
responding to the indoor-outdoor temperature differences.
### IV-C Signature of the Passenger Flow
Another observation is on the passenger flow signatures. Fig.7 shows the
patterns of passenger flows in working days and weekends of two subway
stations. One station is close to CBD and the other is close to the town
center. The two figures show that the signatures of the passenger flow are
related not only to time but also to the locations of the stations.
In time dimension, they show different patterns of passenger flow between the
working days and the weekends. In the working days, sharp peaks of passenger
flow appear at the rush hours, while in the weekends, the passenger flow
curves are different. The flow increases and decreases smoothly with peaks
generally appearing at 15:00 to 16:00 pm.
From the location dimension, for the stations close to the working places
(e.g. CBD has many office towers), the peaks in rush hours in the working days
are very sharp, while for locations close to leisure places (e.g. town
center), the peaks in the rush hours are not very sharp. In weekends, the much
less passengers go to the working places but more passengers go to the leisure
places. These signatures provides hints for the smart control of HVAC system
with consideration of time and location differences.
Figure 7: Patterns of traffic flow are related not only to time but also to
the locations of the subway stations.
### IV-D Correlated feature of CO2 density and the passenger flow
We also observed the correlated feature of CO2 density and the passenger flow
over working days and weekends. It is interesting to see that the variations
of CO2 density are highly relevant to the variations of passenger flows. The
curves of CO2 density and passenger flow for a station in Aug. 30 (a working
day) and Aug.31 (a weekend) are plotted in Fig.8. The curves of the CO2
variations and the passenger flows show similar trends at corresponding time.
This result indicates that we may infer the number of passengers by the CO2
density data in case the passenger flow is not available.
Figure 8: Comparison of CO2 density and traffic flow in working days and
weekends.
### IV-E Responding Speed of Indoor Temperature to the Cooling Supply of HVAC
Figure 9: How the indoor temperature response to the cooling supply of the
HVAC system
In the last aspect, we evaluated how does the indoor temperature respond to
the cooling supply of the HVAC system. We measured the cooling supply of the
HVAC system by the temperature of the cooling air blowed by the cooling fans.
Fig.9 shows the variation of the indoor temperature in a subway station over a
day following the temperature variations of the cooling air. We can see when
the temperature of the cooling air changes, the indoor temperature changes
quickly, which shows that the indoor temperature has short responding time to
the cooling supply from the HVAC. It indicates that in the subway stations,
the control latency is slow in the particular settings of the HVAC systems.
## V Conclusion and Discussion
The paper has introduced SEED, a publicly available dataset regarding the
environment, energy and working states of HVAC systems collected from multiple
stations of Beijing subway over multiple days from August to September 2013.
We make it publicly available for the convenience and efficiency for design
and evaluation of the optimal control policies for the HVAC systems. We
described the sensing and HVAC systems for data collection and environment
control, and also presented our basic investigation to the energy
disaggregation of HVAC, working features of the refrigerator, signatures of
the passenger flow, correlation features of CO2 and the passenger flow, and
the responding speed of indoor temperature to the cooling supplies of HVAC. In
future work, the dataset can be further investigated from different ways, such
as identifying the load signatures of the subway stations, designing and
evaluating the optimized control policies.
## Acknowledgment
This work was supported by the National Basic Research Program of China Grant
2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China
Grant 61202360, 61033001, 61061130540, 61073174.
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|
arxiv-papers
| 2013-12-10T00:29:04 |
2024-09-04T02:49:55.213415
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongcai Wang, Haoran Feng, Xiao Qi",
"submitter": "Yongcai Wang",
"url": "https://arxiv.org/abs/1312.2632"
}
|
1312.2659
|
Synchronization of Coupled Stochastic Systems Driven by Non-Gaussian Lévy
Noises 111This work has been partially supported by NSFC Grants 11071165 and
11071199, NSF of Guangxi Grants 2013GXNSFBA019008 and Guangxi Provincial
Department of Research Project Grants 2013YB102.
∗Corresponding author: A. Gu ([email protected]).
Anhui Gu, Yangrong Li
School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
China
Abstract: We consider the synchronization of the solutions to coupled
stochastic systems of $N$-stochastic ordinary differential equations (SODEs)
driven by Non-Gaussian Lévy noises ($N\in\mathbb{N})$. We discuss the
synchronization between two solutions and among different components of
solutions under certain dissipative and integrability conditions. Our results
generalize the present work obtained in Liu et al (2010) and Shen et al
(2010).
MSC: 60H10, 34F05, 37H10
Keywords: Synchronization; Lévy noise; Skorohod metric; random attractor;
càdlàg random dynamical system.
## 1 Introduction
The synchronization of coupled systems is a well-known phenomenon in both
biology and physics. Description of its diversity of occurrence can be founded
in [5], [6], [7], [8], [16], [17], [18]. Synchronization of deterministic
coupled systems has been investigated mathematically in [8], [19], [21] for
autonomous cases and in [12] for non-autonomous systems. For the stochastic
cases, we can refer to the coupled system of Itô SODEs with additive noise
[9], [11] and multiplicative noise [10], [15]. Recently, Shen et al. [15]
generalized the multiplicative case to $N$-Stratonovich SODEs. These
dissipative dynamical systems discussed above are focused on the Gaussian
noises (in terms of Brownian motion). However, complex systems in engineering
and science are often subjected to non-Gaussian fluctuations or uncertainties.
The coupled dynamical systems under non-Gaussian Lévy noises are considered in
[13], [14] and [23].
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, where
$\Omega=D(\mathbb{R},\mathbb{R}^{d})$ of càdlàg functions with the Skorohod
metric as the canonical sample space and denote by
$\mathcal{F}:=\mathcal{B}(D(\mathbb{R},\mathbb{R}^{d}))$ the Borel
$\sigma$-algebra on $\Omega$. Let $\mu_{L}$ be the (Lévy) probability measure
on $\mathcal{F}$ which is given by the distribution of a two-sided Lévy
process with paths in $\Omega$, i.e. $\omega(t)=L_{t}(\omega)$.
Define $\theta=(\theta_{t},t\in\mathbb{R})$ on $\Omega$ the shift by
$(\theta_{t}\omega)(s):=\omega(t+s)-\omega(t).$
Then the mapping $(t,\omega)\rightarrow\theta_{t}\omega$ is continuous and
measurable [1], and the (Lévy) probability measure is $\theta$-invariant, i.e.
$\mu_{L}(\theta_{t}^{-1}(A))=\mu_{L}(A),$
for all $A\in\mathcal{F}$, see [2] for more details. Consider the following
SODEs system driven by non-Gaussian Lévy noises in $\mathbb{R}^{Nd}$,
$dX_{t}^{(j)}=f^{(j)}(X_{t}^{(j)})dt+c_{j}dL_{t}^{(j)},\ \ j=1,\cdots,N,$
(1.1)
where $c_{j}\in\mathbb{R}^{d}$, are constants vectors with no components equal
to zero, $L_{t}^{(j)}$ are independent two-sided scalar Lévy processes on
$(\Omega,\mathcal{F},\mathbb{P})$ satisfying proper conditions which will be
specified later, and $f^{(j)},j=1,\cdots,N,$ are regular enough to ensure the
existence and uniqueness of solutions and satisfy the one-sided dissipative
Lipschitz conditions
$\langle
x_{1}-x_{2},f^{(j)}(x_{1})-f^{(j)}(x_{2})\rangle\leq-l\|x_{1}-x_{2}\|^{2},\ \
j=1,\cdots,N$ (1.2)
on $\mathbb{R}^{d}$ for some $l>4$. In addition to (1.2), we further assume
the following integrability condition: There exists $m_{0}>0$ such that for
any $m\in(0,m_{0}]$, and any càdlàg function
$X:\mathbb{R}\rightarrow\mathbb{R}^{d}$ with sub-exponential growth it follows
$\int^{t}_{-\infty}e^{ms}|f^{(j)}(X(s))|^{2}ds<\infty,\ \ j=1,\cdots,N.$ (1.3)
Without lose of generality, we also assume the Lipschitz constant $l\leq
m_{0}$.
Set
$x^{(j)}(t,\omega)=X_{t}^{(j)}-\bar{X}_{t}^{(j)},\ \
t\in\mathbb{R},\omega\in\Omega,j=1,\cdots,N,$
where
$\bar{X}_{t}^{(j)}=c_{j}e^{-t}\int^{t}_{-\infty}e^{s}dL_{s}^{(j)},\ \
j=1,\cdots,N,$
are the stationary solutions of the Langevin equations
$dX_{t}^{(j)}=-X_{t}^{(j)}dt+c_{j}dL_{t}^{(j)},\ \ j=1,\cdots,N.$
Then system (1.1) can be translated into the following random ordinary
differential equations (RODEs), with right-hand derivative in time
$\displaystyle\frac{dx^{(j)}}{dt_{+}}$ $\displaystyle=$ $\displaystyle
F^{(j)}(x^{(j)},\bar{X}_{t}^{(j)})$ (1.4) $\displaystyle:=$ $\displaystyle
f^{(j)}(x^{(j)}+\bar{X}_{t}^{(j)})+x^{(j)}+\bar{X}_{t}^{(j)},\ j=1,\cdots,N.$
Now we consider the linear coupled RODEs of (1.4)
$\frac{dx^{(j)}}{dt_{+}}=F^{(j)}(x^{(j)},\bar{X}_{t}^{(j)})+\lambda(x^{(j-1)}-2x^{(j)}+x^{(j+1)}),\
j=1,\cdots,N,$ (1.5)
with the coupled coefficient $\lambda>0$, where $x^{(0)}=x^{(N)}$ and
$x^{(N+1)}=x^{(1)}$. Hence (1.5) can be written as the following equivalent
SODEs
$\displaystyle dX_{t}^{(j)}$ $\displaystyle=$ $\displaystyle
f^{(j)}(X_{t}^{(j)})+\lambda(X_{t}^{(j-1)}-2X_{t}^{(j)}+X_{t}^{(j+1)})-\lambda(\bar{X}_{t}^{(j-1)}-2\bar{X}_{t}^{(j)}+\bar{X}_{t}^{(j+1)})$
(1.6) $\displaystyle+c_{j}dL_{t}^{(j)},\ j=1,\cdots,N,$
where $X_{t}^{(0)}=X_{t}^{(N)}$ and $X_{t}^{(N+1)}=X_{t}^{(1)}$. For
synchronization of solutions to RODEs system (1.5), there are two cases: one
for any two solutions and the other for components of solutions. When $N=2$,
Liu et al. [13] consider both types of synchronization. Under the one-sided
dissipative Lipschitz condition (1.2) and the integrability condition (1.3),
they firstly proved that synchronization of any two solutions occurs and the
random dynamical system generated by the solution of (1.5)N=2 has a singleton
sets random attractor, then they obtained that the synchronization between any
two components of solutions occurs as the coupled coefficient $\lambda$ tends
to infinity. The synchronization result implies that coupled dynamical system
share a dynamical feature in some asymptotic sense. Based on the work of [13]
and [15], we consider the synchronization of solutions of (1.5) in the case of
$N\geq 3$ and obtain the similar results. We show that the random dynamical
system (RDS) generated by the solution of the coupled RODEs system (1.5) has a
singleton sets random attractor which implies the synchronization of any two
solutions of (1.5). Moreover, the singleton set random attractor determines a
stationary stochastic solution of the equivalently coupled SODEs system (1.6).
We also show that any two solutions of RODEs system (1.5) converge to a
solution $Z(t,\omega)$ of the averaged RODE
$\frac{dZ}{dt_{+}}=\frac{1}{N}\sum_{j=1}^{N}f^{(j)}(\bar{X}_{t}^{(j)}+Z)+\frac{1}{N}\sum_{j=1}^{N}(\bar{X}_{t}^{(j)}+Z),$
(1.7)
as the coupling coefficient $\lambda\rightarrow\infty$. It is worth mentioning
that the generalization is not trivial because new techniques similar to [15]
are needed.
## 2 Auxiliary Lemmas
We will frequently use the following auxiliary results.
###### Lemma 2.1.
[13] (Pathwise boundedness and convergence.) Let $L_{t}$ be a two-sided Lévy
motion on $\mathbb{R}^{d}$ for which $\mathbb{E}|L_{1}|<\infty$ and
$\mathbb{E}|L_{1}|=\gamma$. Then we have
(A) $\lim_{t\rightarrow\pm\infty}\frac{1}{t}L_{t}=\gamma$, a.s.
(B) the integrals $\int_{-\infty}^{t}e^{-\delta(t-s)}dL_{s}(\omega)$ are
pathwisely uniformly bounded in $\delta>0$ on finite time intervals
$[T_{1},T_{2}]$ in $\mathbb{R}$;
(C) the integrals $\int_{T_{1}}^{t}e^{-\delta(t-s)}dL_{s}(\omega)\rightarrow
0$ as $\delta\rightarrow\infty$, pathwise on finite time intervals
$[T_{1},T_{2}]$ in $\mathbb{R}$.
###### Lemma 2.2.
(Gronwall type inequality.) Suppose that $D(t)$ is a $n\times n$ matrix and
$\Phi(t),\Psi(t)$ are $n$-dimensional vectors on $[T_{0},T]\ (T\geq T_{0},\
T,T_{0}\in\mathbb{R})$ which are sufficiently regular. If the following
inequality holds in the componentwise sense
$\frac{d}{dt_{+}}\Phi(t)\leq D(t)\Phi(t)+\Psi(t),\ t\geq T_{0},$ (2.1)
where $\frac{d}{dt_{+}}\Phi(t):=\lim_{h\downarrow
0^{+}}\frac{\Phi(t+h)-\Phi(t)}{h}$ is right-hand derivative of $\Phi(t)$. Then
$\Phi(t)\leq\mathop{\hbox{exp}}(\int_{T_{0}}^{t}D(s)ds)\Phi(T_{0})+\int_{T_{0}}^{t}\mathop{\hbox{exp}}(\int_{\tau}^{t}D(s)ds)\Psi(\tau)d\tau,\
t\geq T_{0}.$ (2.2)
###### Proof.
See Lemma 2.8 in [22] and the proof of Lemma 2.2 in [15]. ∎
###### Lemma 2.3.
[13] (Random attractor for càdlàg RDS.) Let $(\theta,\phi)$ be an RDS on
$\Omega\times\mathbb{R}^{d}$ and let $\phi$ be continuous in space, but càdlàg
in time. If there exists a family $B=\\{B(\omega),\omega\in\Omega\\}$ of non-
empty measurable compact subsets $B(\omega)$ of $\mathbb{R}^{d}$ and a
$T_{D,\omega}\geq 0$ such that
$\phi(t,\theta_{-t}\omega,D(\theta_{-t}\omega))\subset B(\omega),\ \forall
t\geq T_{D,\omega},$
for all families $D=\\{D(\omega),\omega\in\Omega\\}$ in a given attracting
universe, then the RDS $(\theta,\phi)$ has a random attractor
$\mathcal{A}=\\{\mathcal{A}(\omega),\omega\in\Omega\\}$ with the component
subsets defined for each $\omega\in\Omega$ by
$\mathcal{A}(\omega)=\bigcap_{s>0}\overline{\bigcup_{t\geq
s}\phi(t,\theta_{-t}\omega,B(\theta_{-t}\omega))}.$
Furthermore, if the random attractor consist of singleton sets, i.e.
$\mathcal{A}(\omega)=\\{X^{*}(\omega)\\}$ for some random variable $X^{*}$,
then $X^{*}_{t}(\omega)=X^{*}_{t}(\theta_{t}\omega)$ is a stationary
stochastic process.
## 3 Synchronization of Two Solutions
Consider the coupled RODEs system (1.5)
$\frac{dx^{(j)}}{dt_{+}}=F^{(j)}(x^{(j)},\bar{X}_{t}^{(j)})+\lambda(x^{(j-1)}-2x^{(j)}+x^{(j+1)}),\
j=1,\cdots,N,$ (3.1)
with initial data
$x^{(j)}(0,\omega)=x^{(j)}_{0}(\omega)\in\mathbb{R}^{d},\ \omega\in\Omega,\
j=1,\cdots,N,$ (3.2)
where $\lambda>0$, and
$F^{(j)}(x^{(j)},\bar{X}_{t}^{(j)}):=f^{(j)}(x^{(j)}+\bar{X}_{t}^{(j)})+x^{(j)}+\bar{X}_{t}^{(j)},\
j=1,\cdots,N.$ (3.3)
Here $f^{(j)}$ are regular enough to ensure the existence and uniqueness of
global solutions on $\mathbb{R}$ and satisfy the one-sided dissipative
Lipschitz condition (1.2) and integrability condition (1.3) for
$j=1,\cdots,N$.
First, we have the result of existence of stationary solutions.
###### Lemma 3.1.
Supposed the assumptions (1.2) and (1.3) be satisfied. Then the coupled RODEs
system (3.1) with initial condition (3.2) has a unique stationary solution.
###### Proof.
For any two solutions
$(x_{1}^{(1)}(t),x_{1}^{(2)}(t),\cdots,x_{1}^{(N)}(t))^{\mathbf{T}}$ and
$(x_{2}^{(1)}(t),x_{2}^{(2)}(t),\\\ \cdots,x_{2}^{(N)}(t))^{\mathbf{T}}$ of
RODEs system (3.1)-(3.2). By the dissipative Lipschitz condition (1.2), for
$j=1,\cdots,N$, we have
$\displaystyle\frac{d}{dt_{+}}\|x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\|^{2}$
$\displaystyle=$ $\displaystyle 2\langle
x_{1}^{(j)}(t)-x_{2}^{(j)}(t),\frac{d}{dt_{+}}x_{1}^{(j)}(t)-\frac{d}{dt_{+}}x_{2}^{(j)}(t)\rangle$
(3.4) $\displaystyle=$ $\displaystyle 2\langle
f^{(j)}(x_{1}^{(j)}+\bar{X}_{t}^{(j)})-f^{(j)}(x_{2}^{(j)}+\bar{X}_{t}^{(j)}),x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\rangle$
$\displaystyle+(2-4\lambda)\|x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\|^{2}$
$\displaystyle+2\lambda\langle
x_{1}^{(j-1)}(t)-x_{2}^{(j-1)}(t),x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\rangle$
$\displaystyle+2\lambda\langle
x_{1}^{(j+1)}(t)-x_{2}^{(j+1)}(t),x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\rangle$
$\displaystyle\leq$
$\displaystyle(2-2l-2\lambda)\|x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{1}^{(j-1)}(t)-x_{2}^{(j-1)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{1}^{(j+1)}(t)-x_{2}^{(j+1)}(t)\|^{2}.$
Define for $t\in\mathbb{R}$,
$\mathbf{x}(t)=(\|x_{1}^{(1)}(t)-x_{2}^{(1)}(t)\|^{2},\|x_{1}^{(2)}(t)-x_{2}^{(2)}(t)\|^{2},\cdots,\|x_{1}^{(N)}(t)-x_{2}^{(N)}(t)\|^{2})^{\mathbf{T}},$
and
$D_{\lambda}=\left(\begin{array}[]{cccccc}2-2l-2\lambda&\lambda&0&\cdots&0&\lambda\\\
\lambda&2-2l-2\lambda&\lambda&0&\cdots&0\\\
0&\lambda&2-2l-2\lambda&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\lambda&0\\\
0&\cdots&0&\lambda&2-2l-2\lambda&\lambda\\\
\lambda&0&\cdots&0&\lambda&2-2l-2\lambda\end{array}\right)_{N\times N}.$
Thus, the differential inequalities can be written as a simple form
$\mathbf{\dot{x}}(t)\leq D_{\lambda}\mathbf{x}(t),\ \mbox{-componentwise}.$
(3.5)
By Lemma 2.2, it yields from (3.5) that
$\mathbf{x}(t)\leq\mathop{\hbox{exp}}(\int_{0}^{t}D_{\lambda}ds)\mathbf{x}(0),\
\mbox{-componentwise}.$ (3.6)
Now, we firstly to estimate the upper bound of eigenvalues of the real
symmetric matrix $\int_{0}^{t}D_{\lambda}ds$. The quadratic from satisfies
$\displaystyle f(\zeta_{1},\zeta_{2},\cdots,\zeta_{N})$ $\displaystyle=$
$\displaystyle\zeta^{\mathbf{T}}(\int_{0}^{t}D_{\lambda}ds)\zeta$
$\displaystyle=$
$\displaystyle(2-2l-2\lambda)t\sum_{j=1}^{N}\zeta_{j}^{2}+2\lambda
t\sum_{j=1}^{N}\zeta_{j}\zeta_{j-1}$ $\displaystyle\leq$
$\displaystyle(2-l)t\sum_{j=1}^{N}\zeta_{j}^{2}-lt\sum_{j=1}^{N}\zeta_{j}^{2},$
where
$\zeta=(\zeta_{1},\zeta_{2},\cdots,\zeta_{N})^{\mathbf{T}}\in\mathbb{R}^{N}$
and $\zeta_{0}=\zeta_{N}$. Due to the Lipschitz constant $l>4$, we have
$f(\zeta_{1},\zeta_{2},\cdots,\zeta_{N})\leq-lt\sum_{j=1}^{N}\zeta_{j}^{2},$
which implies that the quadratic form is negative definite and eigenvalues of
$\int_{0}^{t}D_{\lambda}ds$ satisfy
$\max\\{\mu^{(1)}_{\lambda},\mu^{(2)}_{\lambda},\cdots,\mu^{(N)}_{\lambda}\\}\leq-
lt.$ (3.7)
Because of the real and symmetric properties of matrix
$\int_{0}^{t}D_{\lambda}ds$, for $j=1,\cdots,N$, we obtain
$\displaystyle\|\mathop{\hbox{exp}}(\int_{0}^{t}D_{\lambda}ds)\mathbf{x}(0)\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\mathbf{x}(0)\|^{2}\mathop{\hbox{exp}}(2\max\\{\mu^{(1)}_{\lambda},\mu^{(2)}_{\lambda},\cdots,\mu^{(N)}_{\lambda}\\})$
(3.8) $\displaystyle\leq$
$\displaystyle\|\mathbf{x}(0)\|^{2}\mathop{\hbox{exp}}(-2lt),$
which leads to
$\lim_{t\rightarrow\infty}\|x_{1}^{(j)}(t)-x_{2}^{(j)}(t)\|=0,\ j=1,\cdots,N,$
that is, all solutions of the coupled RODEs system (3.1)-(3.2) converge
pathwise to each other as time $t$ tends to infinity. The proof is finished. ∎
Now, we use the theory of random dynamical systems which generated by SDEs
driven by Lévy motion to find what the solutions of (3.1)-(3.2) will converge
to. It is easy to see from [13] that the solution
$\phi(t,\omega)=(x^{(1)}(t,\omega),x^{(2)}(t,\omega),\cdots,x^{(N)}(t,\omega))^{\mathbf{T}},\
\omega\in\Omega$
of system (3.1)-(3.2) generates a càdlàg RDS over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ with state
space $\Omega\times\mathbb{R}^{Nd}$. The RDS $(\theta,\phi)$ is continuous in
space but càdlàg in time. Recall that a stationary solution $X^{*}$ is a
stationary solution of a stochastic differential equation system may be
characterized as a stationary orbit of the corresponding RDS $(\theta,\phi)$
generated by the stochastic differential equation system, namely,
$\phi(t,\omega)X^{*}(\omega)=X^{*}(\theta_{t}\omega)$.
Then, we have the result for this RDS.
###### Theorem 3.2.
Under the conditions (1.2) and (1.3), the RDS
$\phi(t,\omega),t\in\mathbb{R},\omega\in\Omega$, has a singleton sets random
attractor given by
$\mathcal{A}_{\lambda}(\omega)=\\{(\bar{x}_{\lambda}^{(1)}(\omega),\bar{x}_{\lambda}^{(2)}(\omega),\cdots,\bar{x}_{\lambda}^{(N)}(\omega))^{\mathbf{T}}\\},$
which implies the synchronization of any two solutions of system (3.1)-(3.2).
Furthermore,
$(\bar{x}_{\lambda}^{(1)}(\theta_{t}\omega)+\bar{X}_{t}^{(1)},\bar{x}_{\lambda}^{(2)}(\theta_{t}\omega)+\bar{X}_{t}^{(2)},\cdots,\bar{x}_{\lambda}^{(N)}(\theta_{t}\omega)+\bar{X}_{t}^{(N)})^{\mathbf{T}}$
is the stationary stochastic solution of the equivalent coupled SODEs (1.6).
###### Proof.
For $j=1,\cdots,N,$ we have
$\displaystyle\frac{d}{dt_{+}}\|x^{(j)}(t)\|^{2}$ $\displaystyle=$
$\displaystyle 2\langle x^{(j)}(t),\frac{d}{dt_{+}}x^{(j)}(t)\rangle$
$\displaystyle=$ $\displaystyle 2\langle
f^{(j)}(x^{(j)}(t)+\bar{X}_{t}^{(j)}),x^{(j)}(t)\rangle+2\langle
x^{(j)}(t)+\bar{X}_{t}^{(j)},x^{(j)}(t)\rangle$
$\displaystyle-4\lambda\|x^{(j)}(t)\|^{2}+2\lambda\langle
x^{(j)}(t),x^{(j-1)}(t)\rangle+2\lambda\langle x^{(j)}(t),x^{(j+1)}(t)\rangle$
$\displaystyle\leq$ $\displaystyle 2\langle
f^{(j)}(x^{(j)}(t)+\bar{X}_{t}^{(j)})-f^{(j)}(\bar{X}_{t}^{(j)}),x^{(j)}(t)\rangle+2\langle
f^{(j)}(\bar{X}_{t}^{(j)}),x^{(j)}(t)\rangle$
$\displaystyle+(2-4\lambda)\|x^{(j)}(t)\|^{2}+2\langle\bar{X}_{t}^{(j)},x^{(j)}(t)\rangle$
$\displaystyle+2\lambda\langle x^{(j)}(t),x^{(j-1)}(t)\rangle+2\lambda\langle
x^{(j)}(t),x^{(j+1)}(t)\rangle$ $\displaystyle\leq$
$\displaystyle\|\bar{X}_{t}^{(j)}\|^{2}+|f^{(j)}(\bar{X}_{t}^{(j)})|^{2}+(4-2l-2\lambda)\|x^{(j)}(t)\|^{2}$
$\displaystyle+\lambda\|x^{(j-1)}(t)\|^{2}+\lambda\|x^{(j+1)}(t)\|^{2}.$
Analogous to (3.5), we get
$\mathbf{\dot{y}}(t)\leq\tilde{D}_{\lambda}\mathbf{y}(t)+\mathbf{g}(t),$
where
$\mathbf{y}(t)=(\|x^{(1)}(t)\|^{2},\|x^{(2)}(t)\|^{2},\cdots,\|x^{(N)}(t)\|^{2})^{\mathbf{T}},\
t\in\mathbb{R},$ $\displaystyle\mathbf{g}(t)$ $\displaystyle=$
$\displaystyle(|f^{(1)}(\bar{X}_{t}^{(1)})|^{2}+\|\bar{X}_{t}^{(1)}\|^{2},|f^{(2)}(\bar{X}_{t}^{(2)})|^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\|\bar{X}_{t}^{(2)}\|^{2},\cdots,|f^{(N)}(\bar{X}_{t}^{(N)})|^{2}+\|\bar{X}_{t}^{(N)}\|^{2},)^{\mathbf{T}},\
t\in\mathbb{R},$
and
$\tilde{D}_{\lambda}=\left(\begin{array}[]{cccccc}4-2l-2\lambda&\lambda&0&\cdots&0&\lambda\\\
\lambda&4-2l-2\lambda&\lambda&0&\cdots&0\\\
0&\lambda&4-2l-2\lambda&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\lambda&0\\\
0&\cdots&0&\lambda&4-2l-2\lambda&\lambda\\\
\lambda&0&\cdots&0&\lambda&4-2l-2\lambda\end{array}\right)_{N\times N}.$
Then by Lemma 2.2,
$\mathbf{y}(t)\leq\mathop{\hbox{exp}}(\int_{t_{0}}^{t}\tilde{D}_{\lambda}ds)\mathbf{y}(t_{0})+\int_{t_{0}}^{t}\mathop{\hbox{exp}}(\int_{\tau}^{t}\tilde{D}_{\lambda}ds)\mathbf{g}(\tau)d\tau,\
t\geq t_{0}.$
Similar to Lemma 3.1, we have
$\|\mathop{\hbox{exp}}(\int_{t_{0}}^{t}\tilde{D}_{\lambda}ds)\mathbf{y}(t_{0})\|\leq\|\mathbf{y}(t_{0})\|\mathop{\hbox{exp}}(-l(t-t_{0})),\
t\geq t_{0}.$
Define
$\rho_{\lambda}(\omega):=\int_{-\infty}^{0}\mathop{\hbox{exp}}(\int_{\tau}^{0}\tilde{D}_{\lambda}ds)\mathbf{g}(\tau)d\tau,$
(3.9)
and
$R_{\lambda}^{2}(\omega)=1+\|\rho_{\lambda}(\omega)\|^{2},$ (3.10)
and let $\mathbb{B}_{\lambda}$ be a random ball in $\mathbb{R}^{Nd}$ centered
at the origin with radius $R_{\lambda}(\omega)$. Obviously, the infinite
integral on the right-hand side of (3.9) is well-defined by Lemma 2.1 and the
integrability condition (1.3). Hence by Lemma 2.3, the coupled system has a
random attractor
$\mathcal{A}_{\lambda}=\\{\mathcal{A}_{\lambda}(\omega),\omega\in\Omega\\}$
with $\mathcal{A}_{\lambda}(\omega)\subset\mathbb{B}_{\lambda}$. By Lemma 3.1,
all solutions of (3.1)-(3.2) converge pathwise to each other, therefore,
$\mathcal{A}_{\lambda}(\omega)$ consists of singleton sets, that is
$\mathcal{A}_{\lambda}(\omega)=\\{(\bar{x}_{\lambda}^{(1)}(\omega),\bar{x}_{\lambda}^{(2)}(\omega),\cdots,\bar{x}_{\lambda}^{(N)}(\omega))^{\mathbf{T}}\\}.$
We transform the coupled RODEs (3.1) back to the coupled SODEs (1.6), the
corresponding pathwise singleton sets attractor is then equal to
$(\bar{x}_{\lambda}^{(1)}(\theta_{t}\omega)+\bar{X}_{t}^{(1)},\bar{x}_{\lambda}^{(2)}(\theta_{t}\omega)+\bar{X}_{t}^{(2)},\cdots,\bar{x}_{\lambda}^{(N)}(\theta_{t}\omega)+\bar{X}_{t}^{(N)})^{\mathbf{T}},$
which is exactly a stationary stochastic solution of the coupled SODEs (1.6)
because the Ornstein-Uhlenbeck process is stationary. ∎
## 4 Synchronization of Components of Solutions
It is known in Section 3 that all solutions of the coupled RODEs system
(3.1)-(3.2) converge pathwise to each other in the future for a fixed positive
coupling coefficient $\lambda$. Here, we would like to discuss what will
happen to solutions of the coupled RODEs system (3.1)-(3.2) as
$\lambda\rightarrow\infty$. First, we will give some lemmas which play an
important role in this section.
We need the following estimations. Suppose that
$(x_{\lambda}^{(1)}(t),x_{\lambda}^{(2)}(t),\cdots,x_{\lambda}^{(N)}(t))^{\mathbf{T}}$
is a solution of the coupled RODEs system (3.1)-(3.2). For any two different
components $x_{\lambda}^{(j)}(t),x_{\lambda}^{(k)}(t)$ of the solution for
$\forall j,k\in\\{1,2,\ldots,N\\}$,
$\displaystyle d^{k,j}_{\lambda}(t)$ $\displaystyle=$ $\displaystyle 2\langle
x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t),F^{(j)}(x_{\lambda}^{(j)},\bar{X}_{t}^{(j)})-F^{(k)}(x_{\lambda}^{(k)},\bar{X}_{t}^{(k)})\rangle$
$\displaystyle=$ $\displaystyle 2\langle
x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t),f^{(j)}(x_{\lambda}^{(j)}+\bar{X}_{t}^{(j)})-f^{(k)}(x_{\lambda}^{(k)}+\bar{X}_{t}^{(k)})\rangle$
$\displaystyle+2\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|^{2}+2\langle
x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t),\bar{X}_{t}^{(j)}-\bar{X}_{t}^{(k)}\rangle$
$\displaystyle\leq$
$\displaystyle-2l(\|x_{\lambda}^{(j)}(t)\|^{2}-\|x_{\lambda}^{(k)}(t)\|^{2})+2\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|^{2}$
$\displaystyle+2\langle
f^{(j)}(\bar{X}_{t}^{(j)})-f^{(k)}(\bar{X}_{t}^{(k)}),x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\rangle$
$\displaystyle+2\langle
x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t),\bar{X}_{t}^{(j)}-\bar{X}_{t}^{(k)}\rangle$
$\displaystyle\leq$ $\displaystyle
2\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|(\|f^{(j)}(\bar{X}_{t}^{(j)})\|+|\bar{X}_{t}^{(j)}|)$
$\displaystyle+2\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|(\|f^{(k)}(\bar{X}_{t}^{(j)})\|+|\bar{X}_{t}^{(k)}|),$
thus, for fixed $\alpha>0$, we have
$\displaystyle-\alpha\lambda\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|^{2}+d^{k,j}_{\lambda}(t)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda}(\frac{4}{\alpha}\|f^{(j)}(\bar{X}_{t}^{(j)})\|^{2})+\frac{4}{\alpha}|\bar{X}_{t}^{(j)}|^{2})+\frac{1}{\lambda}(\frac{4}{\alpha}\|f^{(k)}(\bar{X}_{t}^{(k)})\|^{2})+\frac{4}{\alpha}|\bar{X}_{t}^{(k)}|^{2}).$
Let
$C^{j,k,\alpha}_{T_{1},T_{2}}(\lambda,\omega)=\frac{4}{\alpha}\sup_{t\in[T_{1},T_{2}]}[(\|f^{(j)}(\bar{X}_{t}^{(j)})\|^{2}+|\bar{X}_{t}^{(j)}|^{2})+(\|f^{(k)}(\bar{X}_{t}^{(k)})\|^{2}+|\bar{X}_{t}^{(k)}|^{2})]$
in any bounded interval $[T_{1},T_{2}]$. Note that $\rho_{\lambda}(\omega)$ in
(3.9) satisfies
$\frac{d}{d\lambda}\|\rho_{\lambda}(\omega)\|^{2}=2\langle\rho_{\lambda}(\omega),\frac{d}{d\lambda}\rho_{\lambda}(\omega)\rangle\leq
0,$
and consequently, $\rho_{\lambda}(\omega)\leq\rho_{1}(\omega)$ for
$\lambda\geq 1$. Hence, $C^{j,k,\alpha}_{T_{1},T_{2}}(\lambda,\omega)$ is
uniformly bounded in $\lambda$ and
$-\alpha\lambda\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|^{2}+d^{k,j}_{\lambda}(t)\leq\frac{1}{\lambda}C^{j,k,\alpha}_{T_{1},T_{2}}(\lambda,\omega)$
(4.1)
uniformly for $t\in[T_{1},T_{2}]$ with
$C^{j,k,\alpha}_{T_{1},T_{2}}(\omega)=\sup_{\lambda\geq
1}C^{j,k,\alpha}_{T_{1},T_{2}}(\lambda,\omega).$
Now let us estimate the difference between any two components of a solution of
the coupled RODEs system (3.1)-(3.2) as $\lambda\rightarrow\infty$.
###### Lemma 4.1.
Provided conditions (1.2) and (1.3) are satisfied, then any two components of
a solution
$(x_{\lambda}^{(1)}(t),x_{\lambda}^{(2)}(t),\cdots,x_{\lambda}^{(N)}(t))^{\mathbf{T}}$
of the coupled RODEs system (3.1)-(3.2) uniformly vanish in any bounded time
interval when the coupling coefficient $\lambda\rightarrow\infty$, that is,
for any bounded interval $[T_{1},T_{2}]$ and $\forall t\in[T_{1},T_{2}]$, it
yields
$\lim_{\lambda\rightarrow\infty}\|x_{\lambda}^{(j)}(t)-x_{\lambda}^{(k)}(t)\|=0,\
\ \forall j,k\in\\{1,2,\ldots,N\\}.$
###### Proof.
To prove the result, we can equivalently estimate the difference between any
two adjacent components only because the first and the last components of the
solution are considered to be adjacent. We will notice that only one new term
appears in each step which continuous the process, except the last step that
ends the process.
For the difference of the first part of the solution
$(x_{\lambda}^{(1)}(t),x_{\lambda}^{(2)}(t),\cdots,x_{\lambda}^{(N)}(t))^{\mathbf{T}}$,
$\displaystyle\frac{d}{dt_{+}}\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}$
$\displaystyle=$ $\displaystyle 2\langle
x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t),F^{(1)}(x^{(1)},\bar{X}_{t}^{(1)})-F^{(2)}(x^{(2)},\bar{X}_{t}^{(2)})\rangle$
(4.2)
$\displaystyle-6\lambda\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}$
$\displaystyle+2\lambda\langle
x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t),x_{\lambda}^{(N)}(t)-x_{\lambda}^{(3)}(t)\rangle$
$\displaystyle\leq$
$\displaystyle-5\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}+\lambda\|x_{\lambda}^{(N)}(t)-x_{\lambda}^{(3)}(t)\|^{2}+d^{1,2}_{\lambda}(t)$
$\displaystyle\leq$
$\displaystyle-\beta\lambda\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}+\lambda\|x_{\lambda}^{(N)}(t)-x_{\lambda}^{(3)}(t)\|^{2}$
$\displaystyle+\frac{1}{\lambda}C^{1,2,5-\beta}_{T_{1},T_{2}}(\omega)$
uniformly for $t\in[T_{1},T_{2}]$ by (4.1). Here, we can take
$\beta=\begin{array}[]{l}\begin{cases}1-\cos\frac{N\pi}{N+2},&\mbox{N is
even},\\\ 1-\cos\frac{(N-1)\pi}{N+1},&\mbox{N is odd}.\end{cases}\end{array}$
In fact, from Lemma 4.1 in [15], we can take any
$\beta\in(-2\cos\frac{N\pi}{N+2},2)$ when $N$ is even and any
$\beta\in(-2\cos\frac{(N-1)\pi}{N+1},2)$ when $N$ is odd.
We have seen that the estimations in (4.2) generate
$x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)$. Now, we have
$\displaystyle\frac{d}{dt_{+}}\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2}$
$\displaystyle=$ $\displaystyle 2\langle
x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t),F^{(3)}(x^{(3)},\bar{X}_{t}^{(3)})-F^{(N)}(x^{(N)},\bar{X}_{t}^{(N)})\rangle$
$\displaystyle-4\lambda\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2}$
$\displaystyle+2\lambda\langle
x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t),x_{\lambda}^{(2)}(t)-x_{\lambda}^{(1)}(t)\rangle$
$\displaystyle+2\lambda\langle
x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t),x_{\lambda}^{(4)}(t)-x_{\lambda}^{(N-1)}(t)\rangle$
$\displaystyle\leq$
$\displaystyle-\beta\lambda\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2}+\lambda\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{\lambda}^{(4)}(t)-x_{\lambda}^{(N-1)}(t)\|^{2}+\frac{1}{\lambda}C^{3,N,2-\beta}_{T_{1},T_{2}}(\omega)$
uniformly for $t\in[T_{1},T_{2}]$.
Note that $x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)$ has been fixed and
$x_{\lambda}^{(4)}(t)-x_{\lambda}^{(N-1)}(t)$ is generated. Similarly, it
yields
$\displaystyle\frac{d}{dt_{+}}\|x_{\lambda}^{(4)}(t)-x_{\lambda}^{(N-1)}(t)\|^{2}$
$\displaystyle\leq$
$\displaystyle-\beta\lambda\|x_{\lambda}^{(4)}(t)-x_{\lambda}^{(N-1)}(t)\|^{2}+\lambda\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{\lambda}^{(5)}(t)-x_{\lambda}^{(N-2)}(t)\|^{2}+\frac{1}{\lambda}C^{4,N-1,2-\beta}_{T_{1},T_{2}}(\omega)$
uniformly for $t\in[T_{1},T_{2}]$.
Continue such estimations, for $j=2,3,\ldots$, we get
$\displaystyle\frac{d}{dt_{+}}\|x_{\lambda}^{(j+3)}(t)-x_{\lambda}^{(N-j)}(t)\|^{2}$
$\displaystyle\leq$
$\displaystyle-\beta\lambda\|x_{\lambda}^{(j+3)}(t)-x_{\lambda}^{(N-j)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{\lambda}^{(j+2)}(t)-x_{\lambda}^{(N-j+1)}(t)\|^{2}$
$\displaystyle+\lambda\|x_{\lambda}^{(j+4)}(t)-x_{\lambda}^{(N-j-1)}(t)\|^{2}+\frac{1}{\lambda}C^{j+3,N-j,2-\beta}_{T_{1},T_{2}}(\omega)$
uniformly for $t\in[T_{1},T_{2}]$.
We can divide the situation into two cases: $N$ is even and $N$ is odd, which
just as same as [15] did. When $N$ is even, we can rewrite the inequalities in
the matrix form
$\mathbf{\dot{u}}(t)\leq\mathbf{H}_{\lambda}\mathbf{u}(t)+\frac{1}{\lambda}\mathbf{C},$
(4.3)
which uniformly for $t\in[T_{1},T_{2}]$, where for $t\in\mathbb{R}$,
$\mathbf{u}(t)=(\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2},\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2},\cdots,\|x_{\lambda}^{(\frac{N}{2}+1)}(t)-x_{\lambda}^{(\frac{N}{2}+2)}(t)\|^{2})^{\mathbf{T}},$
$\mathbf{C}=(C^{1,2,5-\beta}_{T_{1},T_{2}}(\omega),C^{3,N,2-\beta}_{T_{1},T_{2}}(\omega),\cdots,C^{\frac{N}{2},\frac{N}{2}+3,2-\beta}_{T_{1},T_{2}}(\omega),C^{\frac{N}{2}+1,\frac{N}{2}+2,5-\beta}_{T_{1},T_{2}}(\omega))^{\mathbf{T}},$
are $\frac{N}{2}$-dimensional vectors, and
$\mathbf{H}_{\lambda}=\left(\begin{array}[]{cccccc}-\beta\lambda&\lambda&0&\cdots&0\\\
\lambda&-\beta\lambda&\lambda&\ddots&\vdots\\\ 0&\lambda&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&-\beta\lambda&\lambda\\\
0&\cdots&0&\lambda&-\beta\lambda\end{array}\right)_{\frac{N}{2}\times\frac{N}{2}}.$
By Lemma 2.2, it follows from (4.3) that
$\mathbf{u}(t)\leq
e^{(t-t_{0})\mathbf{H}_{\lambda}}\mathbf{u}(t_{0})+\frac{1}{\lambda}\int_{t_{0}}^{t}e^{(t-s)\mathbf{H}_{\lambda}}\mathbf{C}ds.$
(4.4)
By Lemma 4.1 in [15] again, $\frac{1}{\lambda}\mathbf{H}_{\lambda}$ is
negative definite, then we have
$\|e^{(t-t_{0})\mathbf{H}_{\lambda}}\mathbf{u}(t_{0})\|\leq
e^{(t-t_{0})\mu_{\max}}\|\mathbf{u}(t_{0})\|,$
where $\mu_{\max}=-\beta-2\cos\frac{N\pi}{N+2}<0$ is the maximal eigenvalue of
$\frac{1}{\lambda}\mathbf{H}_{\lambda}$. Thus (4.4) implies that
$\mathbf{u}(t)\rightarrow\mathbf{0}\ \ \mbox{as}\ \lambda\rightarrow\infty,$
and
$\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}\rightarrow 0\ \ \mbox{and}\
\
\|x_{\lambda}^{(\frac{N}{2}+1)}(t)-x_{\lambda}^{(\frac{N}{2}+2)}(t)\|^{2}\rightarrow
0,$
uniformly for $t\in[T_{1},T_{2}]$ as $\lambda\rightarrow\infty$.
Similarly, when $N$ is odd, we can rewrite the inequalities in the matrix form
$\mathbf{\dot{v}}(t)\leq\mathbf{\tilde{H}}_{\lambda}\mathbf{v}(t)+\frac{1}{\lambda}\mathbf{\tilde{C}},$
(4.5)
which uniformly for $t\in[T_{1},T_{2}]$, where for $t\in\mathbb{R}$,
$\mathbf{v}(t)=(\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2},\|x_{\lambda}^{(3)}(t)-x_{\lambda}^{(N)}(t)\|^{2},\cdots,\|x_{\lambda}^{(\frac{N+1}{2})}(t)-x_{\lambda}^{(\frac{N+1}{2}+2)}(t)\|^{2})^{\mathbf{T}},$
$\mathbf{\tilde{C}}=(C^{1,2,5-\beta}_{T_{1},T_{2}}(\omega),C^{3,N,2-\beta}_{T_{1},T_{2}}(\omega),\cdots,C^{\frac{N-1}{2},\frac{N+1}{2}+3,2-\beta}_{T_{1},T_{2}}(\omega),C^{\frac{N+1}{2},\frac{N+1}{2}+2,5-\beta}_{T_{1},T_{2}}(\omega))^{\mathbf{T}},$
are $\frac{N-1}{2}$-dimensional vectors, and
$\mathbf{\tilde{H}}_{\lambda}=\left(\begin{array}[]{cccccc}-\beta\lambda&\lambda&0&\cdots&0\\\
\lambda&-\beta\lambda&\lambda&\ddots&\vdots\\\ 0&\lambda&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&-\beta\lambda&\lambda\\\
0&\cdots&0&\lambda&-\beta\lambda\end{array}\right)_{\frac{N-1}{2}\times\frac{N-1}{2}}.$
By Lemma 2.2, it follows from (4.5) that
$\mathbf{v}(t)\leq
e^{(t-t_{0})\mathbf{\tilde{H}}_{\lambda}}\mathbf{v}(t_{0})+\frac{1}{\lambda}\int_{t_{0}}^{t}e^{(t-s)\mathbf{\tilde{H}}_{\lambda}}\mathbf{\tilde{C}}ds.$
(4.6)
Just like the even case, for uniform $t\in[T_{1},T_{2}]$, we have
$\|x_{\lambda}^{(1)}(t)-x_{\lambda}^{(2)}(t)\|^{2}\rightarrow 0,\ \ \mbox{as}\
\lambda\rightarrow\infty.$
For other adjacent components, the process above can be repeated. Hence, we
can draw a conclusion that the difference between any adjacent components of a
solution of the coupled RODEs system (3.1)-(3.2) tends to zero uniformly for
$t\in[T_{1},T_{2}]$ as the coupling coefficient goes to infinity which
completes the proof.
∎
We know that all components of a solution of system (3.1)-(3.2) have the same
limit uniformly for $t\in[T_{1},T_{2}]$ as $\lambda\rightarrow\infty$. Now, we
are in the position to find what they converge to.
###### Lemma 4.2.
If the assumptions (1.2) and (1.3) hold, then the random dynamical system
$\phi(t,\omega)$ generated by the solution of the averaged RODE system
$\frac{dZ}{dt_{+}}=\frac{1}{N}\sum_{j=1}^{N}f^{(j)}(\bar{X}_{t}^{(j)}+Z)+\frac{1}{N}\sum_{j=1}^{N}(\bar{X}_{t}^{(j)}+Z)$
(4.7)
has a singleton sets random attractor denoted by $\\{\bar{Z}(\omega)\\}$.
Furthermore,
$\bar{Z}(\theta_{t}\omega)+\frac{1}{N}\sum_{j=1}^{N}\bar{X}_{t}^{(j)}$
is the stationary stochastic solution of the equivalently averaged SODE system
$dz=\frac{1}{N}\sum_{j=1}^{N}f^{(j)}(z)dt+\frac{1}{N}\sum_{j=1}^{N}c_{j}dL^{(j)}_{t}.$
(4.8)
###### Proof.
Assume that $Z_{1}(t)$ and $Z_{2}(t)$ are two solutions of (4.7), we have
$\frac{d}{dt_{+}}\|Z_{1}(t)-Z_{2}(t)\|^{2}\leq(2-2l)\|Z_{1}(t)-Z_{2}(t)\|^{2}.$
It follows from Gronwall’s lemma that
$\|Z_{1}(t)-Z_{2}(t)\|^{2}\leq e^{(2-2l)t}\|Z_{1}(0)-Z_{2}(0)\|^{2},$
which implies
$\lim_{t\rightarrow\infty}\|Z_{1}(t)-Z_{2}(t)\|^{2}=0,$
because of the Lipschitz coefficient $l>4$. Then all solutions of (4.7)
converge pathwise to each other.
Now, we have to give what they converge to based on the theory of càdlàg
random dynamical systems. Let $Z(t)$ be a solution of (4.7), we get
$\frac{d}{dt_{+}}\|Z(t)\|^{2}\leq(4-2l)\|Z(t)\|^{2}+\frac{1}{N}\sum_{j=1}^{N}\|f^{(j)}(\bar{X}_{t}^{(j)})\|^{2}+\frac{1}{N}\sum_{j=1}^{N}|\bar{X}_{t}^{(j)}|^{2}.$
From Gronwall’s lemma, it yields for $t>t_{0}$,
$\displaystyle\|Z(t)\|^{2}$ $\displaystyle\leq$ $\displaystyle
e^{(4-2l)(t-t_{0})}\|Z(t_{0})\|^{2}$ $\displaystyle\ \ \ \ \ \
+\frac{1}{N}\sum_{j=1}^{N}\int_{t_{0}}^{t}e^{(4-2l)(t-\tau)}(\|f^{(j)}(\bar{X}_{\tau}^{(j)})\|^{2}+|\bar{X}_{\tau}^{(j)}|^{2})d\tau.$
By pathwise pullback convergence with $t_{0}\rightarrow-\infty$, the random
closed ball centered as the origin with random radius $\tilde{R}(\omega)$ is a
pullback absorbing set of $\phi(t,\omega)$, where
$\tilde{R}^{2}(\omega)=1+\frac{1}{N}\sum_{j=1}^{N}\int_{-\infty}^{0}e^{(2l-4)\tau}(\|f^{(j)}(\bar{X}_{\tau}^{(j)})\|^{2}+|\bar{X}_{\tau}^{(j)}|^{2})d\tau.$
Obviously, by Lemma 2.1 and condition (1.3), the integral defined in the
right-hand side is well-defined.
By Lemma 2.3, there exists a random attractor $\\{\bar{Z}(\omega)\\}$ for
$\phi(t,\omega)$. Since all solutions of (4.7) converge pathwise to each
other, the random attractor $\\{\bar{Z}(\omega)\\}$ are composed of singleton
sets.
Note that the averaged RODE (4.7) is transformed from the averaged SODE (4.8)
by the transformation
$Z(t,\omega)=z-\frac{1}{N}\sum_{j=1}^{N}\bar{X}_{t}^{(j)},$
so the pathwise singleton sets attractor
$\bar{Z}(\theta_{t}\omega)+\frac{1}{N}\sum_{j=1}^{N}\bar{X}_{t}^{(j)}$ is a
stationary solution of the averaged SODE (4.8) since the Ornstein-Uhlenbeck
process is stationary. ∎
Now, we will present another main result of this work.
###### Theorem 4.3.
(Synchronization under non-Gaussian Lévy noise.) Let
$(\bar{x}^{(1)}_{\lambda_{n}}(t,\omega),\bar{x}^{(2)}_{\lambda_{n}}(t,\omega),\cdots,\bar{x}^{(N)}_{\lambda_{n}}(t,\omega))^{\mathbf{T}}=(\bar{x}^{(1)}_{\lambda_{n}}(\theta_{t}\omega),\bar{x}^{(2)}_{\lambda_{n}}(\theta_{t}\omega),\cdots,\bar{x}^{(N)}_{\lambda_{n}}(\theta_{t}\omega))^{\mathbf{T}}$
be the singleton sets random attractor of the càdlàg random dynamical system
$\phi(t,\omega)$ generated by the solution of RODEs system (3.1)-(3.2), then
$((\bar{x}^{(1)}_{\lambda_{n}}(t,\omega),\bar{x}^{(2)}_{\lambda_{n}}(t,\omega),\cdots,\bar{x}^{(N)}_{\lambda_{n}}(t,\omega))^{\mathbf{T}})\rightarrow(\bar{Z}(t,\omega),\bar{Z}(t,\omega),\cdots,\bar{Z}(t,\omega))^{\mathbf{T}}$
in Skorohod metric pathwise uniformly for $t$ belongs to any bounded time-
interval $[T_{1},T_{2}]$ for any sequence $\lambda_{n}\rightarrow\infty$,
where $\bar{Z}(t,\omega)=\bar{Z}(\theta_{t}\omega)$ is the solution of the
averaged RODE (4.7) and $\bar{Z}(\omega)$ is the singleton sets random
attractor of the càdlàg random dynamical system $\phi(t,\omega)$ which
generated by the solution of averaged RODE (4.7).
###### Proof.
Define
$\bar{Z}_{\lambda}(\omega)=\frac{1}{N}\sum_{j=1}^{N}\bar{x}^{(j)}_{\lambda}(\omega),$
(4.9)
where
$\\{\bar{x}^{(1)}_{\lambda}(\omega),\bar{x}^{(2)}_{\lambda}(\omega),\cdots,\bar{x}^{(N)}_{\lambda}(\omega)\\}$
is the singleton sets random attractor of the càdlàg RDS generated by RODEs
system (3.1)-(3.2). Thus,
$\bar{Z}_{\lambda}(t,\omega)=\bar{Z}_{\lambda}(\theta_{t}\omega)$ satisfies
$\frac{d\bar{Z}_{\lambda}(t,\omega)}{dt_{+}}=\frac{1}{N}\sum_{j=1}^{N}f^{(j)}(\bar{X}_{t}^{(j)}+\bar{x}^{(j)}_{\lambda}(t,\omega))+\frac{1}{N}\sum_{j=1}^{N}(\bar{X}_{t}^{(j)}+\bar{x}^{(j)}_{\lambda}(t,\omega)),$
(4.10)
Then, we get
$\displaystyle\|\frac{d\bar{Z}_{\lambda}(t,\omega)}{dt_{+}}\|^{2}\leq\frac{2}{N}\sum_{j=1}^{N}(\|f^{(j)}(\bar{X}_{t}^{(j)}+\bar{x}^{(j)}_{\lambda}(t,\omega))\|^{2}+|\bar{X}_{t}^{(j)}+\bar{x}^{(j)}_{\lambda}(t,\omega)|^{2}),$
by the càdlàg property of the solutions in [2] and the fact that these
solutions belong to the compact ball $\mathbb{B}_{1}(\omega)$, it follows that
$\sup_{t\in[T_{1},T_{2}]}\|\frac{d\bar{Z}_{\lambda}(t,\omega)}{dt_{+}}\|\leq(\frac{2}{N}\sum_{j=1}^{N}\frac{\alpha}{4}\mathbf{C}_{T_{1},T_{2}}^{j,\bullet,\alpha}(\omega))^{\frac{1}{2}}<\infty.$
By the Ascoli-Arzel$\grave{a}$ theorem in $D([T_{1},T_{2}],\mathbb{R}^{d})$ in
[3], there exists a subsequence $\lambda_{n_{k}}\rightarrow\infty$ such that
$\bar{Z}_{\lambda_{n_{k}}}(t,\omega)$ converges to $\bar{Z}(t,\omega)$ in
Skorohod metric as $n_{k}\rightarrow\infty$.
Since difference between any two components of a solution of the coupled RODEs
system (3.1)-(3.2) tends to zero uniformly for $t\in[T_{1},T_{2}]$ as
$\lambda\rightarrow\infty$, from (4.9), we have
$\displaystyle\bar{x}^{(j)}_{\lambda_{n_{k}}}(t,\omega)=\bar{Z}_{\lambda_{n_{k}}}(t,\omega)+\frac{1}{N}\sum_{j^{\prime}\neq
j}\sum_{j^{\prime\prime}\neq
j^{\prime}}(\bar{x}^{(j^{\prime\prime})}_{\lambda_{n_{k}}}(t,\omega)-\bar{x}^{(j^{\prime})}_{\lambda_{n_{k}}}(t,\omega))\rightarrow\bar{Z}(t,\omega)$
uniformly for $t\in[T_{1},T_{2}]$ as $\lambda_{n_{k}}\rightarrow\infty$ for
$j=1,\cdots,N$. Furthermore, it follows from (4.10) that for $t\geq T_{1}$,
$\bar{Z}_{\lambda}(t,\omega)=\bar{Z}_{\lambda}(T_{1},\omega)+\frac{1}{N}\sum_{j=1}^{N}\int_{T_{1}}^{t}(f^{(j)}(\bar{X}_{s}^{(j)}+\bar{x}^{(j)}_{\lambda}(s,\omega))+(\bar{X}_{s}^{(j)}+\bar{x}^{(j)}_{\lambda}(s,\omega)))ds.$
Thus,
$\displaystyle\bar{Z}(t,\omega)=\bar{Z}(T_{1},\omega)+\frac{1}{N}\sum_{j=1}^{N}\int_{T_{1}}^{t}(f^{(j)}(\bar{X}_{s}^{(j)}+\bar{Z}(s,\omega))+(\bar{X}_{s}^{(j)}+\bar{Z}(s,\omega)))ds,$
uniformly for $t\in[T_{1},T_{2}]$ as $\lambda_{n_{k}}\rightarrow\infty$, which
implies that $\bar{Z}_{\lambda}(s,\omega)$ solves RODE (4.7). Then, we note
that all possible sequences of $\bar{Z}_{\lambda_{n_{k}}}(t,\omega)$ converges
to the same limit $\bar{Z}(t,\omega)$ uniformly for $t\in[T_{1},T_{2}]$ as
$\lambda_{n}\rightarrow\infty$. Since the RDS generated by the solutions of
RODE (4.7) has a singleton sets random attractor $\\{\bar{Z}(\omega)\\}$, the
stationary stochastic process $\bar{Z}(\theta_{t}\omega)$ must be equal to
$\bar{Z}(t,\omega)$, i.e. $\bar{Z}(t,\omega)=\bar{Z}(\theta_{t}\omega)$, which
completes the proof. ∎
As a obvious result of Theorem 3.2, we get
###### Corollary 4.4.
$((\bar{x}^{(1)}_{\lambda}(t,\omega),\bar{x}^{(2)}_{\lambda}(t,\omega),\cdots,\bar{x}^{(N)}_{\lambda}(t,\omega))^{\mathbf{T}})\rightarrow(\bar{Z}(t,\omega),\bar{Z}(t,\omega),\cdots,\bar{Z}(t,\omega))^{\mathbf{T}}$
in Skorohod metric pathwise uniformly for $t\in[T_{1},T_{2}]$ as
$\lambda\rightarrow\infty$.
###### Remark 4.5.
The results in this paper hold just in almost everywhere sense. In the
equation (1.1) we should replace the $X_{t}^{(j)}$ with $X_{t_{-}}^{(j)}$
because we must take the left limit to make sure that càdlàg solution process
$X_{t}^{(j)}$ is predictable and unique [21]. For the typographical
convenience, however, we will use $X_{t}^{(j)}$ instead of $X_{t_{-}}^{(j)}$
for the rest of the paper. Moreover, in the case of additive noise, the
distinction for left limit or not is not necessary because if we have to
consider the integral form of equation (1.1), $f^{(j)}(X_{t}^{(j)})$ has only
countable discontinuous points and is still Riemann and Legesgue integrable,
where $j=1,\cdots,N$.
## 5 Conflict of Interests
The authors declare that there is no conflict of interests regarding the
publication of this article.
## References
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|
arxiv-papers
| 2013-12-10T04:00:16 |
2024-09-04T02:49:55.219811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anhui Gu and Yangrong Li",
"submitter": "Anhui Gu Dr.",
"url": "https://arxiv.org/abs/1312.2659"
}
|
1312.2661
|
Random Attractor For Stochastic Lattice FitzHugh-Nagumo System Driven By
$\alpha$-stable Lévy Noises 111This work has been partially supported by NSFC
Grants 11071199, GXNSF Grants 2013GXNSFBA019008 and GXPDRP Grants 2013YB102.
∗Corresponding author: A. Gu ([email protected]).
Anhui Gu, Yangrong Li and Jia Li
School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
China
Abstract: The present paper is devoted to the existence of a random attractor
for stochastic lattice FitzHugh-Nagumo system driven by $\alpha$-stable Lévy
noises under some dissipative conditions.
Keywords: Synchronization; Lévy noise; Skorohod metric; random attractor;
càdlàg random dynamical system.
## 1 Introduction
We consider the following stochastic lattice FitzHugh-Nagumo system (SLFNS)
$\left\\{\begin{array}[]{l}\frac{du_{i}}{dt_{+}}=u_{i-1}-2u_{i}+u_{i+1}-\lambda
u_{i}+f_{i}(u_{i})-v_{i}\\\
\quad\quad+h_{i}+\sum_{j=1}^{N}\varepsilon_{j}u_{i}\diamond\frac{dL_{t}^{j}}{dt},\\\
\frac{dv_{i}}{dt_{+}}=\varrho u_{i}-\varpi
v_{i}+g_{i}+\sum_{j=1}^{N}\varepsilon_{j}v_{i}\diamond\frac{dL_{t}^{j}}{dt},\\\
u(0)=u_{0}=(u_{i0})_{i\in\mathbb{Z}},v(0)=v_{0}=(v_{i0})_{i\in\mathbb{Z}}\end{array}\right.$
(1.1)
where $\mathbb{Z}$ denotes the integer set, $u_{i}\in\mathbb{R}$,
$\lambda,\varrho$ and $\varpi$ are positive constants, $h_{i},\
g_{i}\in\mathbb{R}$, $f_{i}$ are smooth functions satisfying some dissipative
conditions, $\varepsilon_{j}\in\mathbb{R}$ for $j=1,...,N$, $L_{t}^{j}$ are
mutually independent $\alpha$-stable Lévy motions ($1<\alpha<2$), and
$\diamond$ denotes the Marcus sense in the stochastic term, ,
$\frac{d\cdot}{dt_{+}}$ is right-hand derivative of $\cdot(t)$ at $t$,
$\ell^{2}=(\ell^{2},(\cdot,\cdot),\|\cdot\|)$ denotes the regular space of
infinite sequences.
As we all known, noises involved in realistic systems will play an important
role as intrinsic phenomena rather than just compensation of defects in
deterministic models. Stochastic lattice dynamical systems (SLDS) arise
naturally in a wide variety of applications where the spatial structure has a
discrete character and random influences or uncertainties are taken into
account. For the recent research of SLDS, we can see e.g. [Bates et al.(2006),
Huang(2007), Caraballo & Lu (2008), Zhao & Zhou(2009), Han et al.(2011)] for
the first- or second-order lattice dynamical systems with white noises in
regular (or weight) space of infinite sequences, see e.g. [Gu (2013), Gu & Li
(2013)] for the first-order lattice dynamical systems driven by fractional
Brownian motions, see [Gu & Ai (2014)] for the first-order lattice dynamical
systems with non-Gaussian noises.
When there are no noises terms, form similar to (1.1) is the discrete of the
FitzHugh-Nagumo system which arose as modeling the signal transmission across
axons in neurobiology (see [Jones (1984)]). Lattice FitzHugh-Nagumo system was
used to stimulate the propagation of action potentials in myelinated nerve
axons (see [Elmer & Van Vleck (2005)]). Gaussian processes like Brownian
motion have been widely used to model fluctuations in engineering and science.
When lattice FitzHugh-Nagumo system perturbed by additive or multiplicative
white noises, the existence of random attractors has been proved in
[Huang(2007), Gu et al. (2012)]. To the best of our knowledge, there are no
results on the system when it is perturbed by a non-Gaussian noise (in terms
of Lévy noise).
In fact, some complex phenomena involve non-Gaussian fluctuations with
peculiar properties such as anomalous diffusion (mean square displacement is a
nonlinear power law of time) [Bouchaud & Georges (1990)] and heavy tail
distribution (non-exponential relaxation) [Yonezawa (1996)]. For this topic,
we can refer to [Shlesinger et al. (1995), Scher et al. (1991), Herrchen
(2001), Ditlevsen (1999)] for more details. A Lévy motion $L_{t}$ is a non-
Gaussian process with independent and stationary increments, i.e,. increments
$\Delta L_{t}=L_{t+\Delta t}-L_{t}$ are stationary and independent for any non
overlapping time lags $\Delta t$. Moreover, its sample paths are only
continuous in probability, namely,
$\mathbb{P}(|L_{t}-L_{t_{0}}|\geq\epsilon)\rightarrow 0$ as $t\rightarrow
t_{0}$ for any positive $\epsilon$. With a suitable modification, these path
may be taken as càdlàg, i.e., paths are continuous on the right and have
limits on the left. This continuity is weaker than the usual continuity in
time. Indeed, a càdlàg function has at most countably many discontinuities on
any time interval, which generalizes the Brownian motion to some extent (see
e.g. [Applebaum (2004)]). As a special case of Lévy processes, the symmetric
$\alpha$-stable Lévy motion plays an important role among stable processes
just like Brownian motion among Gaussian processes. A stochastic process
$\\{L_{t},t\geq 0\\}$ is called the $\alpha$-stable Lévy motions if (i)
$L_{0}=0$ a.e., (ii) $L$ has independent increments, and (iii)
$L_{t}-L_{s}\sim\mathbf{S}_{\alpha}((t-s)^{\frac{1}{\alpha}},\beta,0)$ for
$0\leq s<t<\infty$ and for some $0<\alpha\leq 2,-1\leq\beta\leq 1$, where
$\mathbf{S}_{\alpha}(\sigma,\beta,\nu)$ denotes the $\alpha$-stable
distribution with index of stability $\alpha$, scale parameter $\sigma$,
skewness parameter $\beta$ and shift parameter $\nu$; in particular,
$\mathbf{S}_{2}(\sigma,0,\mu)=N(\mu,2\sigma^{2})$ denotes the Gaussian
distribution. For more details on $\alpha$-stable distributions, we can refer
to [Sato (1999)]. It is worth mentioning that when $\alpha=2$, we have the
standard Brownian motion, which the Marcus sense stochastic terms (see e.g.
[Marcus (1981)]) reduce to the Stratonovich stochastic terms and the existence
of a random attractor for system (1.1) has been considered in [Gu et al.
(2012)]. For the further development on Lévy motions, we can refer to the
recent monographs [Applebaum (2004), Peszat & Zabczyk (2007)].
The goal of this article is to establish the existence of a random attractor
for SLFNS with the nonlinearity $f_{i}$ under some dissipative conditions and
driven by $\alpha$-stable Lévy noises with $\alpha\in(1,2)$. By virtue of an
Ornstein-Uhlenbeck process with a stationary solution, we transform system
(1.1) into a conjugated random integral equation (with a solution in the sense
of Carathéodory). Here, we assume that $1<\alpha<2$ since this is the only
case where the solutions of the Ornstein-Uhlenbeck equations for
$\alpha$-stable Lévy noises are stationary, which is vital to our purpose. Fot
the case of $0<\alpha<1$, there will be a new challenges for us for future
research.
The paper is organized as follows. In Sec. 2, we recall some basic concepts in
random dynamical systems. In Sec. 3, we give a unique solution to system (1.1)
and make sure that the solution generates a random dynamical system. We
establish the main result, that is, the existence of a random attractor
generated by system (1.1) in Sec. 4.
## 2 Random dynamical systems and random attractors
For the reader’s convenience, we introduce some basic concepts related to
random dynamical systems and random attractors, which are taken from
[Arnold(1998), Chueshov(2002), Han et al.(2011)]. Let
$(\mathbb{E},\|\cdot\|_{\mathbb{E}})$ be a separable Hilbert space and
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.
###### Definition 2.1.
A stochastic process $\\{\varphi(t,\omega)\\}_{t\geq 0,\omega\in\Omega}$ is a
continuous random dynamical system (RDS) over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ if $\varphi$
is
$(\mathcal{B}[0,\infty)\times\mathcal{F}\times\mathcal{B}(\mathbb{E}),\mathcal{B}(\mathbb{E}))$-measurable,
and for all $\omega\in\Omega$,
(i) the mapping $\varphi(t,\omega):\mathbb{E}\mapsto\mathbb{E}$,
$x\mapsto\varphi(t,\omega)x$ is continuous for every $t\geq 0$,
(ii) $\varphi(0,\omega)$ is the identity on $\mathbb{E}$,
(iii) (cocycle property)
$\varphi(s+t,\omega)=\varphi(t,\theta_{s}\omega)\varphi(s,\omega)$ for all
$s,t\geq 0$.
###### Definition 2.2.
(i) A set-valued mapping $\omega\mapsto B(\omega)\subset\mathbb{E}$ (we may
write it as $B(\omega)$ for short) is said to be a random set if the mapping
$\omega\mapsto$ dist${}_{\mathbb{E}}(x,B(\omega))$ is measurable for any
$x\in\mathbb{E}$, where dist${}_{\mathbb{E}}(x,D)$ is the distance in
$\mathbb{E}$ between the element $x$ and the set $D\subset\mathbb{E}$.
(ii) A random set $B(\omega)$ is said to be bounded if there exist
$x_{0}\in\mathbb{E}$ and a random variable $r(\omega)>0$ such that
$B(\omega)\subset\\{x\in\mathbb{E}:\|x-x_{0}\|_{\mathbb{E}}\leq
r(\omega),x_{0}\in\mathbb{E}\\}$ for all $\omega\in\Omega$.
(iii) A random set $B(\omega)$ is called a compact random set if $B(\omega)$
is compact for all $\omega\in\Omega$.
(iv) A random bounded set $B(\omega)\subset\mathbb{E}$ is called tempered with
respect to $(\theta_{t})_{t\in\mathbb{R}}$ if for a.e. $\omega\in\Omega$,
$\lim_{t\rightarrow+\infty}e^{-\gamma t}d(B(\theta_{-t}\omega))=0\ \ \mbox{for
all}\ \ \gamma>0$, where $d(B)=\sup_{x\in B}\|x\|_{\mathbb{E}}$. A random
variable $\omega\mapsto r(\omega)\in\mathbb{R}$ is said to be tempered with
respect to $(\theta_{t})_{t\in\mathbb{R}}$ if for a.e. $\omega\in\Omega$,
$\lim_{t\rightarrow+\infty}\sup_{t\in\mathbb{R}}e^{-\gamma
t}r(\theta_{-t}\omega)=0\ \ \mbox{for all}\ \ \gamma>0$.
We consider an RDS $\\{\varphi(t,\omega)\\}_{t\geq 0,\omega\in\Omega}$ over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ and
$\mathcal{D}(\mathbb{E})$ the set of all tempered random sets of $\mathbb{E}$.
###### Definition 2.3.
A random set $\mathcal{K}$ is called an absorbing set in
$\mathcal{D}(\mathbb{E})$ if for all $B\in\mathcal{D}(\mathbb{E})$ and a.e.
$\omega\in\Omega$ there exists $t_{B}(\omega)>0$ such that
$\varphi(t,\theta_{-t}\omega)B(\theta_{-t}\omega)\subset\mathcal{K}(\omega)\ \
\mbox{for all}\ \ t\geq t_{B}(\omega).$
###### Definition 2.4.
A random set $\mathcal{A}$ is called a global random $\mathcal{D}(\mathbb{E})$
attractor (pullback $\mathcal{D}(\mathbb{E})$ attractor) for
$\\{\varphi(t,\omega)\\}_{t\geq 0,\omega\in\Omega}$ if the following hold:
(i) $\mathcal{A}$ is a random compact set, i.e. $\omega\mapsto
d(x,\mathcal{A}(\omega))$ is measurable for every $x\in\mathbb{E}$ and
$\mathcal{A}(\omega)$ is compact for a.e. $\omega\in\Omega$;
(ii) $\mathcal{A}$ is strictly invariant, i.e. for $\omega\in\Omega$ and all
$t\geq 0$,
$\varphi(t,\omega)\mathcal{A}(\omega)=\mathcal{A}(\theta_{t}\omega)$;
(iii) $\mathcal{A}$ attracts all sets in $\mathcal{D}(\mathbb{E})$, i.e. for
all $B\in\mathcal{D}(\mathbb{E})$ and a.e. $\omega\in\Omega$, we have
$\lim_{t\rightarrow+\infty}d(\varphi(t,\theta_{-t}\omega)B(\theta_{-t}\omega),\mathcal{A}(\omega))=0,$
where $d(X,Y)=\sup_{x\in X}\inf_{y\in Y}\|x-y\|_{\mathbb{E}}$ is the Hausdorff
semi-metric ($X\subseteq\mathbb{E},Y\subseteq\mathbb{E}$).
###### Proposition 2.5.
(See [Han et al.(2011)] .) Suppose that
(a) there exists a random bounded absorbing set
$K(\omega)\in\mathcal{D}(\ell^{2})$, $\omega\in\Omega$, such that for any
$B(\omega)\in\mathcal{D}(\ell^{2})$ and all $\omega\in\Omega$, there exists
$T(\omega,B)>0$ yielding
$\varphi(t,\theta_{-t}\omega,B(\theta_{-t}\omega))\subset K(\omega)$ for all
$t\geq T(\omega,B)$;
(b) the RDS $\\{\varphi(t,\omega)\\}_{t\geq 0,\omega\in\Omega}$ is random
asymptotically null on $K(\omega)$, i.e., for any $\epsilon>0$, there exist
$T(\epsilon,\omega,K)>0$ and $I_{0}(\epsilon,\omega,K)\in\mathbb{N}$ such that
$\begin{split}\sup_{u\in
K(\omega)}&\sum_{|i|>I_{0}(\epsilon,\omega,K(\omega))}|\varphi_{i}(t,\theta_{-t}\omega,u(\theta_{-t}\omega))|^{2}\\\
&\leq\epsilon^{2},\quad\quad\quad\forall t\geq
T(\epsilon,\omega,K(\omega)).\end{split}$ (2.1)
Then the RDS $\\{\varphi(t,\omega,\cdot)\\}_{t\geq 0,\omega\in\Omega}$
possesses a unique global random $\mathcal{D}(\ell^{2})$ attractor given by
$\displaystyle\mathcal{\tilde{A}}(\omega)=\bigcap_{\tau\geq
T(\omega,K)}\overline{\bigcup_{t\geq\tau}\varphi(t,\theta_{-t}\omega,K(\theta_{-t}\omega))}.$
(2.2)
## 3 SLFNS driven by $\alpha$-stable Lévy noises
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, where
$\Omega=\mathcal{S}(\mathbb{R},\ell^{2})$ with Skorokhod metric as the
canonical sample space of càdlàg functions defined on $\mathbb{R}$ and taking
values in $\ell^{2}$,
$\mathcal{F}:=\mathcal{B}(\mathcal{S}(\mathbb{R},\ell^{2}))$ the associated
Borel $\sigma$-field and $\mathbb{P}$ is the corresponding (Lévy) probability
measure on $\mathcal{F}$ which is given by the distribution of a two-sided
Lévy process with paths in $\mathcal{S}(\mathbb{R},\ell^{2})$, i.e.
$\omega(t)=L_{t}(\omega)$. Let
$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\ t\in\mathbb{R},$ then the
mapping $(t,\omega)\rightarrow\theta_{t}\omega$ is continuous and measurable
(see [Arnold(1998)]), and the (Lévy) probability measure is
$\theta$-invariant, i.e.
$\mathbb{P}(\theta_{t}^{-1}(\tilde{A}))=\mathbb{P}(\tilde{A})$ for all
$\tilde{A}\in\mathcal{F}$ (see [Applebaum (2004)]).
For convenience, we now formulate system (1.1) as a stochastic differential
equation in $\ell^{2}\times\ell^{2}$. For
$u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, define
$\mathbb{A},\mathbb{B},\mathbb{B}^{*}$ to be linear operators from $\ell^{2}$
to $\ell^{2}$ as follows:
$\displaystyle(\mathbb{A}u)_{i}$ $\displaystyle=$ $\displaystyle-
u_{i-1}+2u_{i}-u_{i+1},$ $\displaystyle(\mathbb{B}u)_{i}$ $\displaystyle=$
$\displaystyle u_{i+1}-u_{i},\ \ (\mathbb{B}^{*}u)_{i}=u_{i-1}-u_{i},\ \
i\in\mathbb{Z}.$
It is easy to show that
$\mathbb{A}=\mathbb{B}\mathbb{B}^{*}=\mathbb{B}^{*}\mathbb{B}$,
$(\mathbb{B}^{*}u,u^{\prime})=(u,\mathbb{B}u^{\prime})$ for all
$u,u^{\prime}\in\ell^{2}$, which implies that $(\mathbb{A}u,u)\geq 0$.
Let $f_{i}\in\mathcal{C}(\mathbb{R})$ satisfy the conditions that
$\sup_{i\in\mathbb{Z}}|f^{\prime}(u)|$ is bounded for $u$ in bounded sets and
$f_{i}(x)x\geq 0$ for all $x\in\mathbb{R}$. Let $\tilde{f}$ be the Nemytski
operator associated with $f_{i}$, for $u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{2}$,
then $\tilde{f}(u)\in\ell^{2}$ and $\tilde{f}$ is locally Lipschitz from
$\ell^{2}$ to $\ell^{2}$ (see [Bates et al.(2006), Caraballo & Lu (2008)]). In
the sequel, when no confusion arises, we identify $\tilde{f}$ with $f$.
Let $\mathbb{E}=\ell^{2}\times\ell^{2}$, for $\Psi=(u,v)\in\mathbb{E}$, denote
the norm $\|\Psi\|^{2}:=\|\Psi\|^{2}_{\mathbb{E}}=\|u\|^{2}+\|v\|^{2}$. Then
system (1.1) can be interpreted as a system of integral equations in
$\mathbb{E}$ for $t\in\mathbb{R}$ and $\omega\in\Omega$,
$\left\\{\begin{array}[]{l}u(t)=u(0)+\int_{0}^{t}(-\mathbb{A}u(s)-\lambda
u(s)\\\ \quad\quad+f(u(s))-v(s)+h)ds\\\
\quad\quad\quad+\sum_{j=1}^{N}\int_{0}^{t}\varepsilon_{j}u(s)\diamond
dL_{t}^{j},\\\ v(t)=v(0)+\int_{0}^{t}(\varrho u(s)-\varpi v(s)+g)ds\\\
\quad\quad+\sum_{j=1}^{N}\int_{0}^{t}\varepsilon_{j}v(s)\diamond
dL_{t}^{j},\end{array}\right.$ (3.1)
where the stochastic integral is understood to be in the Marcus sense.
To prove that this stochastic equation (3.1) generates a random dynamical
system, we will transform it into a random differential equation in
$\mathbb{E}$. Now, we introduce the Ornstein-Uhlenbeck processes in $\ell^{2}$
on the metric dynamical system
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ given by the
random variable
$z(\theta_{t}\omega)=-\int^{0}_{-\infty}e^{s}\theta_{t}\omega(s)ds,\ \
t\in\mathbb{R},\omega\in\Omega.$ (3.2)
The above integrals exist in the sense of any path with a subexponential
growth, and $z$ solves the following Ornstein-Uhlenbeck equation
$dz+zdt=dL_{t},\ \ t\in\mathbb{R}.$ (3.3)
In fact, we have the following properties (see Lemma 3.1 in [Gu & Ai (2014)]):
(i) There exists a $\\{\theta_{t}\\}_{t\in\mathbb{R}}$-invariant subset
$\bar{\Omega}\in\mathcal{F}$ of full measure for a.e. $\omega\in\bar{\Omega}$,
the random variable
$z(\omega)=-\int^{0}_{-\infty}e^{s}\omega(s)ds,$
is well defined and the unique stationary solutions of (3.3) is given by
(3.2). Moreover, the mapping $t\rightarrow z(\theta_{t}\omega)$ is càdlàg;
(ii) For $\omega\in\bar{\Omega}$, the sample paths $\omega(t)$ of $L_{t}$
satisfy
$\lim_{t\rightarrow\pm\infty}\frac{\omega(t)}{t}=0,\ t\in\mathbb{R}$
and
$\lim_{t\rightarrow\pm\infty}\frac{|z(\theta_{t}\omega)|}{|t|}=\lim_{t\rightarrow\pm\infty}\frac{1}{t}\int_{0}^{t}z(\theta_{t}\omega(s))ds=0.$
Now, let $z_{j}$ be the associated Ornstein-Uhlenbeck process corresponding to
(3.3) with $L_{t}^{j}$ instead of $L_{t}$ and denote
$\Lambda(\omega)=e^{\sum_{j=1}^{N}\varepsilon_{j}z_{j}(\omega)}\mathbf{Id}_{\mathbb{E}}$,
then $\Lambda(\omega)$ is clearly a homeomorphism in $\mathbb{E}$ and the
inverse operator is well defined by
$\Lambda^{-1}(\omega)=e^{-\sum_{j=1}^{N}\varepsilon_{j}z_{j}(\omega)}\mathbf{Id}_{\mathbb{E}}$.
It is easy to verify that $\|\Lambda^{-1}(\theta_{t}\omega)\|$ has sub-
exponential growth as $t\rightarrow\pm\infty$ for $\omega\in\Omega$. Hence
$\|\Lambda^{-1}\|$ is tempered. Since the mapping of $\theta$ on
$\bar{\Omega}$ has the same properties as the original one if we choose the
trace $\sigma$-algebra with respect to $\bar{\Omega}$ to be denoted also by
$\mathcal{F}$, we can change our metric dynamical system with respect to
$\bar{\Omega}$, and still denoted the symbols by
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$.
Denote
$\xi(\theta_{t}\omega)=\sum_{j=1}^{N}\varepsilon_{j}z_{j}(\theta_{t}\omega)$,
and consider the change in variables
$\begin{split}(U(t),V(t))=&\Lambda^{-1}(\theta_{t}\omega)(u(t),v(t))\\\
\quad\quad&=e^{-\xi(\theta_{t}\omega)}(u(t),v(t)),\end{split}$
where $(u,v)$ is the solution of (3.1), then we get the evolution equations
with random coefficients but without white noise
$\left\\{\begin{array}[]{l}\frac{dU}{dt_{+}}=-\mathbb{A}U-(\lambda-\xi(\theta_{t}\omega))U\\\
\quad\quad+e^{-\xi(\theta_{t}\omega)}f(e^{\xi(\theta_{t}\omega)}U)-V+e^{-\xi(\theta_{t}\omega)}h,\\\
\frac{dV}{dt_{+}}=\varrho
U-(\varpi-\xi(\theta_{t}\omega))V+e^{-\xi(\theta_{t}\omega)}g\\\
\end{array}\right.$ (3.4)
and initial condition $(U(0),V(0))=(U_{0},V_{0})\in\mathbb{E}$.
Now, we have the following result:
###### Theorem 3.1.
Let $T>0$ and $\Psi_{0}=(U_{0},V_{0})\in\mathbb{E}$ be fixed, then the
following statements hold:
(i) For every $\omega\in\Omega$, system (3.4) has a unique solution
$\Psi(\cdot,\omega,\Psi_{0})=(U(\cdot,\omega,U_{0}),V(\cdot,\omega,V_{0}))\in\mathcal{C}([0,T),\mathbb{E})$
in the sense of Carathéodory.
(ii) For each $\omega\in\Omega$, the mapping
$\Psi_{0}\in\mathbb{E}\mapsto\Psi(\cdot,\omega,\Psi_{0})\in\mathcal{C}([0,T),\mathbb{E})$
is continuous, which implies the solution $\Psi$ of (3.4) continuously depends
on the initial data $\Psi_{0}$.
(iii) Equation (3.4) generates a continuous RDS $(\varphi(t))_{t\geq 0}$ over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$, where
$\varphi(t,\omega,\Psi_{0})=\Psi(t,\omega,\Psi_{0})$ for
$\Psi_{0}\in\mathbb{E}$, $t\geq 0$ and for all $\omega\in\Omega$. Moreover,
$\psi(t,\omega,\Psi_{0})=\Lambda(\theta_{t}\omega)\psi(t,\omega,\Lambda^{-1}(\omega)\Psi_{0})$
for $\Psi_{0}\in\mathbb{E}$, $t\geq 0$ and for all $\omega\in\Omega$, then
$\psi$ is another RDS for which the process
$(\omega,t)\rightarrow(\psi(t,\omega,\Psi_{0}))$ solves (3.1) for any initial
condition $\Psi_{0}\in\mathbb{E}$.
###### Proof.
(i) Let $F(t,U)=e^{-\xi(\theta_{t}\omega)}f(e^{\xi(\theta_{t}\omega)}U)$, for
any fixed $T>0$ and $\Psi_{0}\in\mathbb{E}$ and let $U_{1},U_{2}\in Y$, where
$Y$ is a bounded set in $\mathbb{E}$, we have
$\begin{split}\|F(t,&U_{1})-F(t,U_{2})\|\\\
&=\|e^{-\xi(\theta_{t}\omega)}f(e^{\xi(\theta_{t}\omega)}U_{1})-e^{-\xi(\theta_{t}\omega)}f(e^{\xi(\theta_{t}\omega)}U_{2})\|\\\
&\quad\quad\quad\quad\leq C_{Y}\|v_{1}-v_{2}\|,\end{split}$
where $C_{Y}$ is a constant only depending on $Y$. This implies that the
mapping $F(t,U)$ is locally Lipschitz with respect to $U$ and the Lipschitz
constant is uniformly bounded in $[0,T]$. By the standard arguments, we know
that (3.4) possesses a local solution
$\Psi(\cdot,\omega,\Psi_{0})\in\mathcal{C}([0,T_{\max}),\mathbb{E})$, where
$[0,T_{\max})$ is the maximal interval of existence of the solution of (3.4).
Next, we need to show that the local solution is a global one. By taking the
inner products of $U$ and $V$ respectively in $\ell^{2}$ with the two
equations in system (3.4), we have
$\begin{split}\frac{d}{dt_{+}}(&\|U\|^{2}+\frac{1}{\varrho}\|V\|^{2})\\\
&\leq-(\delta-2\xi(\theta_{t}\omega))(\|U\|^{2}+\frac{1}{\varrho}\|V\|^{2})\\\
&\quad\quad+\frac{1}{\delta}(\|h\|^{2}+\frac{1}{\varrho}\|g\|^{2})e^{-2\xi(\theta_{t}\omega)},\end{split}$
(3.5)
where $\delta=\min\\{\lambda,\varpi\\}$. By virtue of the special Gronwall
lemma (see Lemma 2.8 in [Robinson (2001)]), it yields that
$\begin{split}\|\Psi(t)&\|^{2}\leq e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\cdot e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\int_{0}^{t}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{0}^{s}\xi(\theta_{r}\omega)dr}ds,\end{split}$
where
$c_{1}=\frac{\max\\{1,\frac{1}{\varrho}\\}}{\delta\min\\{1,\frac{1}{\varrho}\\}}$.
Denote
$a(\omega)=2\int_{0}^{T}|\xi(\theta_{s}\omega)|ds$
and
$\begin{split}b(\omega)&=c_{1}\max_{t\in[0,T]}\\{(\|h\|^{2}+\|g\|^{2})\\\
&\cdot e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\int_{0}^{t}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{0}^{s}\xi(\theta_{r}\omega)dr}ds\\}.\end{split}$
Due to the properties of the Ornstein-Uhlenbeck process, we know that
$a(\omega),b(\omega)$ are well-defined. Then we have
$\|\Psi(t)\|^{2}\leq\|\Psi_{0}\|^{2}e^{a(\omega)}+b(\omega),$
which implies that the solution $\Psi$ is defined in any interval $[0,T]$.
(ii) Let $\Phi_{0}=(\bar{U}_{0},\bar{V}_{0})$,
$\Psi_{0}=(U_{0},V_{0})\in\mathbb{E}$, and
$\Phi(t):=(\bar{U}(t,\omega,\bar{U}_{0}),\bar{V}(t,\omega,\bar{V}_{0})),\Psi(t):=(U(t,\omega,U_{0}),V(t,\omega,V_{0}))$
be two solutions of (3.4). By denoting $Z(t)=\Phi(t)-\Psi(t)$, we have
$\begin{split}\frac{d}{dt_{+}}&\|Z(t)\|^{2}\\\ &\leq
2e^{-\xi(\theta_{t}\omega)}\|f(e^{\xi(\theta_{t}\omega)}\Phi(t))-f(e^{\xi(\theta_{t}\omega)}\Psi(t))\|\|Z\|\\\
&\quad\quad+2\xi(\theta_{t}\omega)\|Z\|^{2}\\\ &\quad\quad\quad\quad\leq
2(L_{Y^{\prime}}+\xi(\theta_{t}\omega))\|Z\|^{2}\leq\kappa\|Z\|^{2},\end{split}$
where $\kappa=2(L_{Y^{\prime}}+\max_{t\in[0,T]}|\xi(\theta_{t}\omega)|)$ is
well-defined, and $L_{Y^{\prime}}$ denotes the Lipschitz constant of $f$
corresponding to a bounded set $Y^{\prime}\in\mathbb{E}$ where $\Phi$ and
$\Psi$ belong to. By the Gronwall lemma again, we obtain
$\|Z(t)\|^{2}\leq e^{\kappa t}\|Z(0)\|^{2},$
and consequently
$\sup_{t\in[0,T]}\|\Phi(t)-\Psi(t)\|^{2}\leq e^{\kappa
T}\|\Phi_{0}-\Psi_{0}\|^{2}.$
If $\Phi_{0}=\Psi_{0}$, then the above inequality indicates that the
uniqueness and continuous dependence on the initial data of the solutions of
(3.4).
(iii) The continuity of $\varphi$ is due to (i) and (ii). The measurability of
$\psi$ follows from the properties of $\Lambda$. Here, we only remain to prove
the conjugacy between $\varphi$ and $\psi$. The verification by chain rule is
routine and thus be omitted. The proof is complete.
∎
## 4 Existence of a global random attractor
In this section, we will prove the existence of a global random attractor for
system (1.1). Since the random dynamical systems $\varphi$ and $\psi$ are
conjugated, we only have to consider the RDS $\varphi$. Firstly, we have our
main result
###### Theorem 4.1.
The SLDS $\varphi$ generated by system (3.4) has a unique global random
attractor.
In order to prove Theorem 4.1, we will use Proposition 2.5. We first need to
prove there exists an absorbing set for $\varphi$ in
$\mathcal{D}(\mathbb{E})$. Next, we will show the RDS $\varphi$ is random
asymptotically null in the sense of (2.1).
###### Lemma 4.2.
There exists a closed random tempered set $\mathcal{K}(\omega)\in$
$\mathcal{D}(\mathbb{E})$ such that for all $B\in\mathcal{D}(\mathbb{E})$ and
a.e. $\omega\in\Omega$ there exists $t_{B}(\omega)>0$ such that
$\varphi(t,\theta_{-t}\omega)B(\theta_{-t}\omega)\subset\mathcal{K}(\omega)\ \
\mbox{for all}\ \ t\geq t_{B}(\omega).$
###### Proof.
Let us start with $\Psi(t)=\varphi(t,\omega,\Psi_{0})$. Then by (3.5), we have
$\begin{split}\|\varphi(t)&\|^{2}\leq e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\cdot e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\int_{0}^{t}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{0}^{s}\xi(\theta_{r}\omega)dr}ds.\end{split}$
Now, by replacing $\omega$ with $\theta_{-t}\omega$ and $\Psi_{0}$ with
$e^{-\xi(\theta_{-t}\omega)}\Psi_{0}$, respectively, in the expression
$\varphi$, we obtain
$\begin{split}&\|\varphi(t,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0})\|^{2}\\\
&\leq e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s-t}\omega)ds}\|e^{-\xi(\theta_{-t}\omega)}\Psi_{0}\|^{2}\\\
&\quad\quad+c_{1}(\|h\|^{2}+\|g\|^{2})e^{-\delta
t+2\int_{0}^{t}\xi(\theta_{s-t}\omega)ds}\\\
&\quad\cdot\int_{0}^{t}e^{-2\xi(\theta_{s-t}\omega)+\delta
s-2\int_{0}^{s}\xi(\theta_{r-t}\omega)dr}ds\\\ &\leq e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{0}^{t}\xi(\theta_{s-t}\omega)ds}\|\Psi_{0}\|^{2}\\\
&\quad\quad+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\quad\cdot\int_{0}^{t}e^{-2\xi(\theta_{s-t}\omega)+\delta(s-t)-2\int_{s}^{t}\xi(\theta_{r-t}\omega)dr}ds\\\
&\leq e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{-t}^{0}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}\\\
&\quad\quad+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\quad\cdot\int_{-t}^{0}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds\\\ &\leq e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{-t}^{0}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}\\\
&\quad\quad+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\quad\cdot\int_{-\infty}^{0}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds.\end{split}$ (4.1)
By the properties of the Ornstein-Uhlenbeck process, we know that
$\int_{-\infty}^{0}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds<+\infty.$
Consider for any $\Psi_{0}\in B(\theta_{-t}\omega)$, we have
$\displaystyle\|\varphi(t,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0})\|^{2}$
$\displaystyle\leq$ $\displaystyle e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{-t}^{0}\xi(\theta_{s}\omega)ds}d(B(\theta_{-t}\omega))^{2}$
$\displaystyle+c_{1}(\|h\|^{2}+\|g\|^{2})\int_{-\infty}^{0}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds.$
Note that
$\displaystyle\lim_{t\rightarrow+\infty}e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{-t}^{0}\xi(\theta_{s}\omega)ds}d(B(\theta_{-t}\omega))^{2}=0,$
and denote
$\displaystyle\begin{split}R^{2}(\omega)=&1+c_{1}(\|h\|^{2}+\|g\|^{2})\\\
&\quad\cdot\int_{-\infty}^{0}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds,\end{split}$
we conclude that
$\displaystyle\mathcal{K}(\omega)=\overline{B_{\mathbb{E}}(0,R(\omega))}$
(4.2)
is an absorbing closed random set. It remains to show that
$\mathcal{K}(\omega)\in\mathcal{D}(\mathbb{E})$. Indeed, from Definition 2.2
(iv), for all $\gamma>0$, we get
$\displaystyle\begin{split}e^{-\gamma t}&R^{2}(\theta_{-t}\omega)=e^{-\gamma
t}+c_{1}e^{-\gamma t}(\|h\|^{2}+\|g\|^{2})\\\
&\cdot\int_{-\infty}^{0}e^{-2\xi(\theta_{s-t}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r-t}\omega)dr}ds\\\ &\quad\quad\quad=e^{-\gamma
t}+c_{1}e^{-\gamma t}(\|h\|^{2}+\|g\|^{2})\\\
&\cdot\int_{-\infty}^{-t}e^{-2\xi(\theta_{s}\omega)+\delta(s+t)-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds\rightarrow
0\\\ &\quad\quad\quad\mbox{as}\ \ t\rightarrow\infty,\end{split}$
which completes the proof. ∎
###### Lemma 4.3.
Let $\Psi_{0}(\omega)\in\mathcal{K}(\omega)$ be the absorbing set given by
(4.2). Then for every $\epsilon>0$, there exist
$\tilde{T}(\epsilon,\omega,\mathcal{K}(\omega))>0$ and
$\tilde{N}(\epsilon,\omega,\mathcal{K}(\omega))>0$, such that the solution
$\varphi$ of problem (3.4) is random asymptotically null, that is, for all
$t\geq\tilde{T}(\epsilon,\omega,\mathcal{K}(\omega))$,
$\displaystyle\begin{split}\sup_{\Psi\in\mathcal{K}(\omega)}\sum_{|i|>\tilde{N}(\epsilon,\omega,\mathcal{K}(\omega))}|&\varphi_{i}(t,\theta_{-t}\omega,\Psi(\theta_{-t}\omega)|^{2}\leq\epsilon^{2}.\end{split}$
###### Proof.
Choose a smooth cut-off function satisfying $0\leq\rho(s)\leq 1$ for
$s\in\mathbb{R^{+}}$ and $\rho(s)=0$ for $0\leq s\leq 1$, $\rho(s)=1$ for
$s\geq 2$. Suppose there exists a positive constant $c_{0}$ such that
$|\rho^{\prime}(s)|\leq c_{0}$ for $s\in\mathbb{R}^{+}$.
Let $N$ be a fixed integer which will be specified later, set
$x=(\rho(\frac{|i|}{N})U_{i})_{i\in\mathbb{Z}}$ and
$y=(\rho(\frac{|i|}{N})V_{i})_{i\in\mathbb{Z}}$. Then take the inner product
of the two equations in system (3.4) with $x$ and $y$ in $\ell^{2}$,
respectively, and combine the following two inequalities
$\displaystyle(AU,x)=(\tilde{B}U,\tilde{B}x)\geq-\frac{2c_{0}}{N}\|U\|^{2}\geq-\frac{2c_{0}}{N}\|\varphi\|^{2},$
and
$\displaystyle-\infty<-2e^{-\xi(\theta_{t}\omega)}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})f_{i}(e^{\xi(\theta_{t}\omega)}U_{i})U_{i}\leq
0,$
we have
$\displaystyle\frac{d}{dt_{+}}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}|^{2}+(\delta-2\xi(\theta_{t}\omega))\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}|^{2}$
$\displaystyle\quad\quad\leq\frac{c_{2}}{N}\|\varphi(t,\omega,e^{-\xi(\omega)}\Psi_{0})\|^{2}$
$\displaystyle\quad\quad\quad\quad+c_{1}e^{-\xi(\theta_{t}\omega)}\sum_{|i|\geq
N}(|h_{i}|^{2}+|g_{i}|^{2}),$
where $c_{2}=\frac{4c_{0}}{\min\\{1,\frac{1}{\varrho}\\}}$. By using the
Gronwall lemma, for $t\geq T_{\mathcal{K}}=T_{\mathcal{K}}(\omega)$, it
follows that
$\displaystyle\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}(t,\omega,e^{-\xi(\omega)}\Psi_{0}(\omega))|^{2}$
(4.5) $\displaystyle\leq$ $\displaystyle
e^{-\delta(t-T_{\mathcal{K}})+2\int_{T_{\mathcal{K}}}^{t}\xi(\theta_{s}\omega)ds}$
$\displaystyle\quad\quad\cdot\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}(t,\omega,e^{-\xi(\omega)}\Psi_{0}(\omega))|^{2}$
$\displaystyle+\frac{c_{2}}{N}\int_{T_{\mathcal{K}}}^{t}e^{-\delta(t-\tau)+2\int_{\tau}^{t}\xi(\theta_{s}\omega)ds}$
$\displaystyle\quad\quad\quad\quad\cdot\|\varphi(\tau,\omega,e^{-\xi(\omega)}\Psi_{0}(\omega))\|^{2}d\tau$
$\displaystyle+c_{1}\sum_{|i|\geq N}(|h_{i}|^{2}+|g_{i}|^{2})$
$\displaystyle\quad\quad\cdot\int_{T_{\mathcal{K}}}^{t}e^{-\delta(t-\tau)+2\int_{\tau}^{t}\xi(\theta_{s}\omega)ds-\xi(\theta_{t}\omega)}d\tau.$
Now, substitute $\theta_{-t}\omega$ for $\omega$ and estimate each term from
(4.5) to (4.5). In (4.1), with $t$ replaced with $T_{\mathcal{K}}$ and
$\omega$ with $\theta_{-t}\omega$, respectively, it follows from (4.5) that
$\displaystyle
e^{-\delta(t-T_{\mathcal{K}})+2\int_{T_{\mathcal{K}}}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle\quad\cdot\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}(T_{\mathcal{K}},\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))|^{2}$
$\displaystyle\leq$ $\displaystyle e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}$
$\displaystyle~{}~{}+c_{1}\int_{0}^{T_{\mathcal{K}}}e^{-2\xi(\theta_{s-t}\omega)+\delta(s-t)+2\int_{s}^{t}\xi(\theta_{r-t}\omega)dr}ds$
$\displaystyle\leq$ $\displaystyle e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{0}^{t}\xi(\theta_{s}\omega)ds}\|\Psi_{0}\|^{2}$
$\displaystyle~{}~{}+c_{1}\int_{-t}^{T_{\mathcal{K}}-t}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}ds.$
Due to the properties of the Ornstein-Uhlenbeck process, there exists a
$T_{1}(\epsilon,\omega,\mathcal{K}(\omega))>T_{\mathcal{K}}(\omega)$, such
that if $t>T_{1}(\epsilon,\omega,\mathcal{K}(\omega))$, then
$\displaystyle
e^{-\delta(t-T_{\mathcal{K}})+2\int_{T_{\mathcal{K}}}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle\quad\cdot\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}(T_{\mathcal{K}},\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))|^{2}$
$\displaystyle\quad\quad\quad\quad\leq\frac{\epsilon^{2}}{3}.$ (4.6)
Next, from (4.1) and (4.5), it follows that
$\displaystyle\frac{c_{2}}{N}\int_{T_{\mathcal{K}}}^{t}e^{-\delta(t-\tau)+2\int_{\tau}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle\quad\quad\cdot\|\varphi(\tau,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))\|^{2}d\tau$
$\displaystyle\leq$
$\displaystyle\frac{c_{2}}{N}\|\Psi_{0}\|^{2}(t-T_{\mathcal{K}})e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{0}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle+\frac{c_{1}c_{2}}{N}(\|h\|^{2}+\|g\|^{2})$
$\displaystyle\quad\cdot\int_{T_{\mathcal{K}}}^{t}\int_{0}^{\tau}e^{-2\xi(\theta_{s-t}\omega)+\delta(s-t)+2\int_{s}^{t}\xi(\theta_{r-t}\omega)dr}dsd\tau$
$\displaystyle\leq$
$\displaystyle\frac{c_{2}}{N}\|\Psi_{0}\|^{2}(t-T_{\mathcal{K}})e^{-\delta
t-2\xi(\theta_{-t}\omega)+2\int_{0}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle+\frac{c_{1}c_{2}}{N}(\|h\|^{2}+\|g\|^{2})$
$\displaystyle\quad\cdot\int_{T_{\mathcal{K}}}^{t}\int_{-t}^{\tau-t}e^{-2\xi(\theta_{s}\omega)+\delta
s-2\int_{s}^{0}\xi(\theta_{r}\omega)dr}dsd\tau.$
Thanks to the properties of the Ornstein-Uhlenbeck process, there exist
$T_{2}(\epsilon,\omega,\mathcal{K}(\omega))>T_{\mathcal{K}}(\omega)$ and
$N_{1}(\epsilon,\omega,\mathcal{K}(\omega))>0$ such that if
$t>T_{2}(\epsilon,\omega,\mathcal{K}(\omega))$ and
$N>N_{1}(\epsilon,\omega,\mathcal{K}(\omega))$, then
$\displaystyle\frac{c_{2}}{N}\int_{T_{\mathcal{K}}}^{t}e^{-\delta(t-\tau)+2\int_{\tau}^{t}\xi(\theta_{s-t}\omega)ds}$
$\displaystyle\cdot\|\varphi(\tau,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))\|^{2}d\tau\leq\frac{\epsilon^{2}}{3}.$
(4.7)
Since $h,g\in\ell^{2}$, by the properties of the Ornstein-Uhlenbeck process
again, we find that there exists
$N_{2}(\epsilon,\omega,\mathcal{K}(\omega))>0$ such that if
$N>N_{2}(\epsilon,\omega,\mathcal{K}(\omega))$, then from (4.5),
$\displaystyle c_{1}\sum_{|i|\geq N}(|h_{i}|^{2}+|g_{i}|^{2})$
$\displaystyle\quad\cdot\int_{T_{\mathcal{K}}}^{t}e^{-\delta(t-\tau)+2\int_{\tau}^{t}\xi(\theta_{s}\omega)ds-\xi(\theta_{t}\omega)}d\tau\leq\frac{\epsilon^{2}}{3}.$
(4.8)
Let
$\displaystyle\tilde{T}(\epsilon,\omega,\mathcal{K}(\omega))=\max\\{T_{1}(\epsilon,\omega,\mathcal{K}(\omega)),T_{2}(\epsilon,\omega,\mathcal{K}(\omega))\\},$
$\displaystyle\tilde{N}(\epsilon,\omega,\mathcal{K}(\omega))=\max\\{N_{1}(\epsilon,\omega,\mathcal{K}(\omega)),N_{2}(\epsilon,\omega,\mathcal{K}(\omega))\\}.$
Then from (4.6), (4.7) and (4.8), for
$t>\tilde{T}(\epsilon,\omega,\mathcal{K}(\omega))$ and
$N>\tilde{N}(\epsilon,\omega,\mathcal{K}(\omega))$, we get
$\displaystyle\sum_{|i|\geq
2N}|\varphi_{i}(t,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))|^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|\varphi_{i}(t,\theta_{-t}\omega,e^{-\xi(\theta_{-t}\omega)}\Psi_{0}(\theta_{-t}\omega))|^{2}\leq\epsilon^{2},$
which implies the conclusion. ∎
We are now in a position to prove our main result.
Proof of Theorem 4.1. The desired result follows directly from Lemmas 4.2 and
4.3 and Proposition 2.5. $\blacksquare$
###### Remark 4.4.
The result may have generalized the existing results (see e.g. [Huang(2007),
Gu et al. (2012)]) to some extent. First, càdlàg functions in a more wider
sense than continues ones as indicated in Introduction section; Second, here
we restrict to $1<\alpha<2$, when $\alpha=2$, the $\alpha$-stable process
actually reduces to the standard Brownian motion.
###### Remark 4.5.
Recently, some sufficient conditions for the upper-semicontinuity of
attractors for random lattice systems perturbed by small white noises have
been given in [Zhou (2012)]. Here, it is worth mentioning that all the results
on this topic are focus on the SLDS perturbed by the white noises. It will be
an interesting question left to future research.
## References
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* [Chueshov(2002)] Chueshov, I. [2002] Monotone Random Systems Theory and Applications, (Springer-Verlag, New York).
* [Ditlevsen (1999)] Ditlevsen, P. “Observation of $\alpha$-stable noise induced millennial climate changes from an ice record,” Geophys. Res. Lett. 26, pp. 1441–1444.
* [Elmer & Van Vleck (2005)] Elmer, C. & Van Vleck, E. [2005] “Spatially discrete FitzHugh-Nagumo equations,” SIAM J. Appl. Math. 96, pp. 1153-1174.
* [Gu (2013)] Gu, A. [2013] “Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions,” Int. J. Bifurcation Chaos 23, pp. 1–9.
* [Gu & Ai (2014)] Gu, A. & Ai, W. [2014] “Random attractor for stochastic lattice dynamical systems with $\alpha$-stable Lévy noises,” Commun. Nonlinear Sci. Numer. Simulat. 19, pp. 1433–1441.
* [Gu & Li (2013)] Gu, A. & Li, Y. [2013] “Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions,” arXiv:1310.7113v1.
* [Gu et al. (2012)] Gu, A., Zhou, S. & Jin, Q. [2012] “Random attractor for partly dissipative stochastic lattice dynamical systems with multiplicative white noises,” Acta. Math. Appl. Sin-E. Article in press.
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* [Huang(2007)] Huang, J. [2007] “The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises,” Physica D 233, pp. 83–94.
* [Jones (1984)] Jones, C. [1984] “Stability of the traveling wave solution of the FitzHugh-Nagumo System,” Trans. Amer. Math. Soc. 286, pp. 431–469.
* [Marcus (1981)] Marcus, S. “Modelling and approximation of stochastic differential equations driven by semimaringales,” Stochastics 4, pp. 223–245.
* [Peszat & Zabczyk (2007)] Peszat, S. & Zabczyk, J. Stochastic Partial Differential Equations with Lévy Processes, (Cambridge University Press, Cambridge).
* [Robinson (2001)] Robinson, J. Infinite-dimensional dynamical systems, (Cambrdge Unversity Press, Cambridge).
* [Sato (1999)] Sato, K. Lévy Processes and Infinitely Divisible Distributions, (Cambridge University Press, Cambridge).
* [Scher et al. (1991)] Scher, H., Shlesinger, M.& Bendler J. “Time-scale invariance in transport and relaxation,” Phys. Today, pp. 26–34.
* [Shlesinger et al. (1995)] Shlesinger, M., Zaslavsky, G., & Frisch U. Lévy flights and related topics in physics, in: Lecture Notes in Physics, (Springer-Verlag, Berlin).
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* [Zhao & Zhou(2009)] Zhao, C. & Zhou, S. [2009] “Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications,” J. Math. Anal. Appl. 354, pp. 78–95.
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|
arxiv-papers
| 2013-12-10T04:12:26 |
2024-09-04T02:49:55.226866
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anhui Gu, Yangrong Li and Jia Li",
"submitter": "Anhui Gu Dr.",
"url": "https://arxiv.org/abs/1312.2661"
}
|
1312.2724
|
# Maximal surfaces in anti-de Sitter 3-manifolds with particles
Jérémy Toulisse Department of Mathematics
Mathematics Research Unit BLG
University of Southern Califonia
3620 S. Vermont Avenue, KAP 104
Los Angeles, CA 90089-2532 [email protected]
###### Abstract.
We prove the existence of a unique maximal surface in each anti-de Sitter
(AdS) Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with
conical singularities along time-like lines) for cone angles less than $\pi$.
We interpret this result in terms of Teichmüller theory, and prove the
existence of a unique minimal Lagrangian diffeomorphism isotopic to the
identity between two hyperbolic surfaces with cone singularities when the cone
angles are the same for both surfaces and are less than $\pi$.
###### Contents
1. 1 Introduction
2. 2 AdS GHM 3-manifolds
1. 2.1 Mess parametrization
2. 2.2 Surfaces embedded in an AdS GHM 3-manifold
3. 3 AdS convex GHM 3-manifolds with particles
1. 3.1 Extension of Mess’ parametrization
2. 3.2 Maximal surface
4. 4 Existence of a maximal surface
1. 4.1 First step
2. 4.2 Second step
3. 4.3 Third step
4. 4.4 Fourth step
5. 5 Uniqueness
6. 6 Consequences
1. 6.1 Minimal Lagrangian diffeomorphisms
2. 6.2 Middle point in $\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
## 1\. Introduction
For $\theta\in(0,2\pi)$, consider the space obtained by gluing with a rotation
the boundary of an angular sector of angle $\theta$ between two half-lines in
the hyperbolic disk. We denote this singular Riemannian manifold by
$\mathbb{H}^{2}_{\theta}$. The induced metric is called local model for
hyperbolic metric with conical singularity of angle $\theta$. This metric is
hyperbolic outside the singular point.
Let $\Sigma_{\mathfrak{p}}$ be a closed oriented surface of genus $g$ with $n$
marked points $\mathfrak{p}:=(p_{1},...,p_{n})\subset\Sigma$ and
$\theta:=(\theta_{1},...,\theta_{n})\in(0,2\pi)^{n}$.
###### Definition 1.1.
A hyperbolic metric $g$ with conical singularities of angle $\theta_{i}$ at
the $p_{i}\in\mathfrak{p}$ is a (singular) metric on $\Sigma_{\mathfrak{p}}$
such that each $p_{i}\in\mathfrak{p}$ has a neighborhood isometric to a
neighborhood of the singular point in $\mathbb{H}^{2}_{\theta_{i}}$ and
$(\Sigma_{\mathfrak{p}},g)$ has constant curvature $-1$ outside the marked
points.
It has been proved by M. Troyanov [Tro91] and M.C. McOwen [McO88] that each
conformal class of metric on a surface $\Sigma_{\mathfrak{p}}$ with marked
points admits a unique hyperbolic metric with cone singularities of angle
$\theta_{i}$ at the $p_{i}$ as soon as
$\chi(\Sigma_{\mathfrak{p}})+\sum_{i=1}^{n}\left(\frac{\theta_{i}}{2\pi}-1\right)<0,$
where $\chi(\Sigma_{\mathfrak{p}})$ is the Euler characteristic of
$\Sigma_{\mathfrak{p}}$.
We denote by $\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ the space of
isotopy classes of hyperbolic metrics with cone singularities of angle
$\theta$ (where the isotopies fix each marked point). Note that, from the
theorem of Troyanov and McOwen, this space is canonically identified with the
space of marked conformal structures on $\Sigma_{\mathfrak{p}}$. As in
dimension 2, a conformal structure is equivalent to a complex structure,
$\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ is also identified with the
space of marked complex structures on $\Sigma_{\mathfrak{p}}$.
When $\mathfrak{p}=\emptyset$, $\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
corresponds to the classical Teichmüller space $\mathscr{T}(\Sigma)$ of
$\Sigma$, that is, the space of equivalence classes of marked hyperbolic
structures on $\Sigma$.
Minimal Lagrangian diffeomorphism.
###### Definition 1.2.
Let $g_{1},g_{2}\in\mathscr{T}(\Sigma)$, a minimal Lagrangian diffeomorphism
$\Psi:(\Sigma,g_{1})\longrightarrow(\Sigma,g_{2})$ is an area preserving
diffeomorphism such that its graph is a minimal surface in
$(\Sigma\times\Sigma,g_{1}\oplus g_{2})$.
In [Sch93], R. Schoen proved the existence of a unique minimal Lagrangian
diffeomorphism isotopic to the identity between two hyperbolic surfaces
$(\Sigma,g_{1})$ and $(\Sigma,g_{2})$ (see also [Lab92]).
Minimal Lagrangian diffeomorphisms are related to harmonic diffeomorphisms
(that is to diffeomorphisms whose differential minimizes the $L^{2}$ norm).
For a conformal structure $\mathfrak{c}$ on $\Sigma$ and
$g\in\mathscr{T}(\Sigma)$, the work of J.J. Eells and J.H. Sampson [ES64]
implies the existence of a unique harmonic diffeomorphism
$u:(\Sigma,\mathfrak{c})\to(\Sigma,g)$ isotopic to the identity. Given a
harmonic diffeomorphism $u$ we define its Hopf differential by
$\Phi(u):=u^{*}g^{2,0}$ (that is the $(2,0)$ part with respect to the complex
structure associated to $\mathfrak{c}$ of $u^{*}g$). The work of R. Schoen
implies that, given $g_{1},g_{2}\in\mathscr{T}(\Sigma)$, there exists a unique
conformal structure $\mathfrak{c}$ on $\Sigma$ such that
$\Phi(u_{1})+\Phi(u_{2})=0$, where
$u_{i}:(\Sigma,\mathfrak{c})\to(\Sigma,g_{i})$ is the unique harmonic
diffeomorphism isotopic to the identity. Moreover, $u_{2}\circ u_{1}^{-1}$ is
the unique minimal Lagrangian diffeomorphism isotopic to the identity between
$(\Sigma,g_{1})$ and $(\Sigma,g_{2})$.
In his thesis, J. Gell-Redman [GR15] proved the existence of a unique harmonic
diffeomorphism isotopic to the identity from a closed surface with $n$ marked
points equipped with a conformal structure to a negatively curved surface with
$n$ conical singularities of angles less than $\pi$ at the marked points
(where the isotopy fixes each marked point). In this paper, we prove the
existence of a unique minimal Lagrangian diffeomorphism isotopic to the
identity between hyperbolic surfaces with conical singularities of angles less
than $\pi$ and so we give a positive answer to [BBD+12, Question 6.3].
###### Theorem 1.3.
Given two hyperbolic metrics
$g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ with cone
singularities of angles $\theta=(\theta_{1},...,\theta_{n})\in(0,\pi)^{n}$,
there exists a unique minimal Lagrangian diffeomorphism
$\Psi:(\Sigma,g_{1})\longrightarrow(\Sigma,g_{2})$ isotopic to the identity.
In particular, this result extends the result of R. Schoen to the case of
surfaces with conical singularities of angles less than $\pi$. The proof of
this statement uses the deep connections between hyperbolic surfaces and three
dimensional anti-de Sitter (AdS) geometry.
AdS geometry. An anti-de Sitter (AdS) manifold $M$ is a Lorentz manifold of
constant sectional curvature $-1$. It is Globally Hyperbolic Maximal (GHM)
when it contains a closed Cauchy surface, that is a space-like surface
intersecting every inextensible time-like curve exactly once, and which is
maximal in a certain sense (precised in Section 2). The global hyperbolicity
condition implies in particular that $M$ is homeomorphic to
$\Sigma\times\mathbb{R}$ (where $\Sigma$ has the same topology as the Cauchy
surface). In his groundbreaking work, G. Mess [Mes07, Section 7] considered
the moduli space $\mathscr{M}(\Sigma)$ of AdS GHM structure on
$\Sigma\times\mathbb{R}$. He proved that $\mathscr{M}(\Sigma)$ is naturally
parametrized by two copies of the Teichmüller space $\mathscr{T}(\Sigma)$.
This result can be thought as an AdS analogue of the famous Bers’ simultaneous
uniformization Theorem [Ber60]. In fact, Bers’ Theorem provides a
parametrization of the moduli space $\mathscr{QF}(\Sigma)$ of quasi-Fuchsian
structures on $\Sigma\times\mathbb{R}$ by two copies of the Teichmüller space
$\mathscr{T}(\Sigma)$.
In [BBZ07], the authors proved the existence of a unique maximal space-like
surface (that is an area-maximizing surface whose induced metric is
Riemannian) in each AdS GHM metric on $\Sigma\times\mathbb{R}$. Note that
maximal surfaces are the Lorentzian analogue of minimal surfaces in Riemannian
geometry: they are characterized by the vanishing of the mean curvature field.
This result is actually equivalent to the result of R. Schoen of existence of
a unique minimal Lagrangian diffeomorphism (see [AAW00]).
A particle in an AdS GHM manifold $M$ is a conical singularity along a time-
like line. In this paper, we only consider particles with cone angles less
than $\pi$. In [BS09], F. Bonsante and J.-M. Schlenker extended Mess’
parametrization to the case of AdS GHM manifolds with particles: they gave a
parametrization of the moduli space of AdS convex GHM manifolds with particles
by two copies of the Teichmüller space
$\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$. In this paper, we study the
existence and uniqueness of a maximal surface in AdS GHM manifolds with
particles, and give a positive answer to [BBD+12, Question 6.2]. Namely, we
prove
###### Theorem 1.4.
For each AdS convex GHM 3-manifold $(M,g)$ with particles of angles less than
$\pi$, there exists a unique maximal space-like surface
$S\hookrightarrow(M,g)$.
Moreover, we prove that the existence of a unique maximal surface provides the
existence of a unique minimal Lagrangian diffeomorphism isotopic to the
identity
$\Psi:(\Sigma_{\mathfrak{p}},g_{1})\longrightarrow(\Sigma_{\mathfrak{p}},g_{2}),$
where $g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ parametrize
the AdS convex GHM metric with particles $g$.
It follows from Theorem 1.4 that one can associate to each pair of hyperbolic
metrics with conical singularities
$g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ the first and
second fundamental form of the unique maximal surface in $(M,g)$ where $g$ is
parametrized by $g_{1}$ and $g_{2}$. It gives a map
$\varphi:\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\times\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\longrightarrow
T^{*}\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}}).$
From Theorem 1.4 and using the Fundamental Theorem of surfaces in AdS
manifolds with particles (see Section 3), we prove that this map is one-to-
one. In Theorem 6.4, we give a nice geometric interpretation of $\varphi$:
given a pair of points
$g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$, there exists a
unique conformal structure $\mathfrak{c}$ on $\Sigma_{\mathfrak{p}}$ such that
$\Phi(u_{1})+\Phi(u_{2})=0$ and $u_{2}\circ u_{1}^{-1}$, where $\Phi(u_{i})$
is the Hopf differential of the harmonic map
$u_{i}:(\Sigma_{\mathfrak{p}},\mathfrak{c})\to(\Sigma_{\mathfrak{p}},g_{i})$.
We then have $\varphi(g_{1},g_{2})=\big{(}\mathfrak{c},i\Phi(u_{1})\big{)}$.
This picture extends the connections between minimal Lagrangian
diffeomorphisms and harmonic maps to the case with conical singularities.
Finally, in [Tou14], we prove the existence of a minimal map between
hyperbolic surfaces with conical singularities when the two surfaces have
different cone angles. In that case, uniqueness only holds when the cone
angles of one surface are strictly smaller than the ones of the other surface.
Acknowledgement. It is a pleasure to thank Jean-Marc Schlenker for its
patience while discussing about the paper. I would also thank Francesco
Bonsante and Thierry Barbot for helpful and interesting conversations about
this subject. I am grateful to the referee who helped to improve the paper.
## 2\. AdS GHM 3-manifolds
### 2.1. Mess parametrization
The AdS 3-space. Let $\mathbb{R}^{2,2}$ be the usual real 4-space with the
quadratic form:
$q(x):=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}.$
The anti-de Sitter (AdS) 3-space is defined by:
$\text{AdS}_{3}=\\{x\in\mathbb{R}^{2,2}\text{ such that }q(x)=-1\\}.$
With the induced metric, $\text{AdS}_{3}$ is a Lorentzian symmetric space of
dimension 3 with constant curvature $-1$ diffeomorphic to $\mathbb{D}\times
S^{1}$ (where $\mathbb{D}$ is a disk of dimension 2). In particular,
$\text{AdS}_{3}$ is not simply connected.
The Klein model of the AdS 3-space is given by the image of $\text{AdS}_{3}$
under the canonical projection
$\pi:\mathbb{R}^{2,2}\setminus\\{0\\}\longrightarrow\mathbb{RP}^{3}.$
Denote by $\text{AdS}^{3}:=\pi(\text{AdS}_{3})$. In the affine chart
$x_{4}\neq 0$ of $\mathbb{RP}^{3}$, $\text{AdS}^{3}$ is the interior of the
hyperboloid of one sheet given by the equation
$\\{x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=1\\}$, and this hyperboloid identifies with
the boundary $\partial\text{AdS}^{3}$ of $\text{AdS}^{3}$ in this chart. In
this model, geodesics are given by straight lines: space-like geodesics are
the ones which intersect the boundary $\partial\text{AdS}^{3}$ in two points,
time-like geodesics are the ones which do not have any intersection and light-
like geodesics are tangent to $\partial\text{AdS}^{3}$.
###### Remark 2.1.
This model is called Klein model by analogy with the Klein model of the
hyperbolic space. In fact, in both models, geodesics are given by straight
lines.
The isometry group. As $\partial\text{AdS}^{3}$ is a hyperboloid of one sheet,
it is foliated by two families of straight lines. We call one family the right
one and the other, the left one. The group $\text{Isom}_{+}(\text{AdS}^{3})$
of space and time-orientation preserving isometries of $\text{AdS}^{3}$
preserves each family of the foliation. Fix a space-like plane $P_{0}$ in
$\text{AdS}^{3}$, its boundary is a space-like circle in
$\partial\text{AdS}^{3}$ which intersects each line of the right (respectively
the left) family exactly once. Then $P_{0}$ provides an identification of each
family with $\mathbb{RP}^{1}$ (when changing $P_{0}$ to another space-like
plane, the identification changes by a conjugation by an element of
$\text{PSL}_{2}(\mathbb{R})$). It is proved in [Mes07, Section 7] that each
element of $\text{Isom}_{+}(\text{AdS}^{3})$ acts by projective
transformations on each $\mathbb{RP}^{1}$ and so extend to a pair of elements
in $\text{PSL}_{2}(\mathbb{R})$. So
$\text{Isom}_{+}(\text{AdS}^{3})\cong\text{PSL}_{2}(\mathbb{R})\times\text{PSL}_{2}(\mathbb{R})$.
###### Remark 2.2.
Fixing a space-like plane $P_{0}$ also provides an identification between
$\partial\text{AdS}^{3}$ and $\mathbb{RP}^{1}\times\mathbb{RP}^{1}$. In fact,
given a point $x\in\partial\text{AdS}^{3}$, there exists a unique line in the
right family a unique line the left one which pass through $x$. It follows
that $x\in\partial\text{AdS}^{3}$ gives a point in
$\mathbb{RP}^{1}\times\mathbb{RP}^{1}$. This application is bijective.
AdS GHM 3-manifold. An AdS 3-manifold is a manifold $M$ endowed with a
$(G,X)$-structure, where $G=\text{Isom}_{+}(\text{AdS}^{3})$,
$X=\text{AdS}^{3}$. That is, $M$ is endowed with an atlas of charts taking
values in $\text{AdS}^{3}$ so that the transition functions are restriction of
elements in $\text{Isom}_{+}(\text{AdS}^{3})$. An AdS 3-manifold $M$ is
Globally Hyperbolic Maximal (GHM) if it satisfies the following two
conditions:
1. (1)
Global Hyperbolicity: $M$ contains a space-like Cauchy surface, that is a
closed oriented surface which intersects every inextensible time-like curve
exactly once.
2. (2)
Maximality: $M$ cannot be strictly embedded in an AdS manifold satisfying the
same properties.
Note that the Global Hyperbolicity condition implies strong restrictions on
the topology of $M$. In particular, $M$ has to be homeomorphic to
$\Sigma\times\mathbb{R}$ where $\Sigma$ is an oriented closed surface of genus
$g>0$ (homeomorphic to the Cauchy surface). We restrict ourselves to the case
$g>1$. We denote by $\mathscr{M}(\Sigma)$ the space of AdS GHM structure on
$\Sigma\times\mathbb{R}$ considered up to isotopy, and by
$\mathscr{T}(\Sigma)$ the Teichmüller space of $\Sigma$.
We have a fundamental result due to G. Mess [Mes07, Proposition 20]:
###### Theorem 2.1 (Mess).
There is a parametrization
$\mathfrak{M}:\mathscr{M}(\Sigma)\longrightarrow\mathscr{T}(\Sigma)\times\mathscr{T}(\Sigma)$.
###### Construction of the parametrization.
To an AdS GHM structure on $M$ is associated its holonomy representation
$\rho:\pi_{1}(M)\to\text{Isom}_{+}(\text{AdS}^{3})$ (well defined up to
conjugation). As
$\text{Isom}_{+}(\text{AdS}^{3})\cong\text{PSL}_{2}(\mathbb{R})\times\text{PSL}_{2}(\mathbb{R})$
and as $\pi_{1}(M)=\pi_{1}(\Sigma)$, one can split the representation $\rho$
into two morphisms
$\rho_{1},\rho_{2}:\pi_{1}(\Sigma)\to\text{PSL}_{2}(\mathbb{R}).$
G. Mess proved [Mes07, Proposition 19] that these holonomies have maximal
Euler class $e$ (that is $|e(\rho_{l})|=|e(\rho_{r})|=2g-2$). Using Goldman’s
criterion [Gol88], he proved that these morphisms are Fuchsian holonomies and
so define a pair of points in $\mathscr{T}(\Sigma)$.
Reciprocally, as two Fuchsian holonomies $\rho_{1},\rho_{2}$ are conjugated by
an orientation preserving homeomorphism
$\phi:\mathbb{RP}^{1}\to\mathbb{RP}^{1}$ and as $\partial\text{AdS}^{3}$
identifies with $\mathbb{RP}^{1}\times\mathbb{RP}^{1}$ (fixing a totally
geodesic space-like plane $P_{0}$, see Remark 2.2), one can see the graph of
$\phi$ as a closed curve in $\partial\text{AdS}^{3}$. G. Mess proved that this
curve is nowhere time-like and is contained in an affine chart. In particular,
one can construct the convex hull of the graph of $\phi$. The holonomy
$(\rho_{1},\rho_{2}):\pi_{1}(\Sigma)\to\text{Isom}_{+}(\text{AdS}^{3})$ acts
properly discontinuously on this convex hull and the quotient is a piece of
globally hyperbolic AdS manifold. It follows from a Theorem of Y. Choquet-
Bruhat and R. Geroch [CBG69] that this piece of AdS globally hyperbolic
manifold uniquely embeds in a maximal one. So the map $\mathfrak{M}$ is a one-
to-one. ∎
### 2.2. Surfaces embedded in an AdS GHM 3-manifold
K. Krasnov and J.-M. Schlenker [KS07, Section 3] proved results about surfaces
embedded in an AdS GHM manifold. Here we state some of these results. Recall
that a space-like surface embedded in a Lorentzian manifold is maximal if its
mean curvature vanishes everywhere. The following result was proved by T.
Barbot, F. Béguin and A. Zeghib in [BBZ07]:
###### Theorem 2.2 (Barbot, Béguin, Zeghib).
Every AdS GHM 3-manifold contains a unique maximal space-like surface.
In [KS07], the authors give an explicit formula for the Mess parametrization
$\mathfrak{M}$:
###### Theorem 2.3 (Krasnov, Schlenker).
Let $S$ be a space-like surface embedded in an AdS GHM manifold $M$ whose
principal curvatures are in $(-1,1)$. We denote by E the identity map, $J$ the
complex structure on S (associated to the induced metric), $B$ its shape
operator and I its first fundamental form. We have:
$\mathfrak{M}(M)=(g_{1},g_{2}),$
where $g_{1,2}(x,y)=\textrm{I}((E\pm JB)x,(E\pm JB)y)$.
###### Remark 2.3.
In particular, they proved that the metrics $g_{1}$ and $g_{2}$ are hyperbolic
and do not depend of the choice of the surface $S$ (up to isotopy).
If we denote by $\mathscr{H}(\Sigma)$ the space of maximal space-like surfaces
in germs of AdS manifold, it is proved in [KS07] (using the Fundamental
Theorem of surfaces embedded in AdS manifolds) that this space is canonically
identified with the space of couples $(g,h)$ where $g$ is a smooth metric on
$\Sigma$ and $h$ is a symmetric bilinear form on $TS$ so that:
1. (1)
$tr_{g}(h)=0$.
2. (2)
$\delta_{g}h=0$ (where $\delta_{g}$ is the divergence operator associated to
the Levi-Civita connection of $g$).
3. (3)
$K_{g}=-1-\det_{g}(h)$ (where $K_{g}$ is the Gauss curvature). We call this
equation modified Gauss’ equation.
We recall a theorem of Hopf [Hop51]:
###### Theorem 2.4 (Hopf).
Let $g$ be a Riemannian metric on $\Sigma$ and $h$ a bilinear symmetric form
on $T\Sigma$, then:
* i.
$tr_{g}(h)=0$ if and only if $h$ is the real part of a quadratic differential
$q$ on $(\Sigma,g)$
* ii.
If i. holds, then $\delta_{g}h=0$ if and only if $q$ is holomorphic with
respect to the complex structure associated to $g$.
* iii.
if i. and ii. hold, then $g$ (respectively $h$) is the first (respectively
second) fundamental form of a maximal surface if and only if
$K_{g}=-1-\det_{g}(h)$.
Moreover, it is proved in [KS07, Lemma 3.6.] that for every conformal class
$\mathfrak{c}$ on $\Sigma$ and every $h$ real part of a holomorphic quadratic
differential $q$ on $(\Sigma,J_{\mathfrak{c}})$ (where $J_{\mathfrak{c}}$ is
the complex structure associated to $\mathfrak{c}$), there exists a unique
metric $\mathrm{g}_{0}\in\mathfrak{c}$ such that modified Gauss’ equation is
satisfied.
This result provides a canonical parametrization of $\mathscr{H}(\Sigma)$ by
$T^{*}\mathscr{T}(\Sigma)$. In this parametrization, $h$ is the real part of a
holomorphic quadratic differential, and $\mathrm{g}_{0}\in\mathfrak{c}$ is the
unique metric verifying $K_{\mathrm{g}_{0}}=-1-\det_{\mathrm{g}_{0}}(h)$. In
addition, such a surface has principal curvatures in $(-1,1)$ [KS07, Lemma
3.11.].
As every AdS GHM manifold contains a unique maximal surface, there is a
parametrization $\phi:\mathscr{M}(\Sigma)\longrightarrow
T^{*}\mathscr{T}(\Sigma)$ [KS07, Theorem 3.8]. Hence, we get an application
associated to the Mess parametrization:
$\varphi:=\phi\circ\mathfrak{M}^{-1}:T^{*}\mathscr{T}(\Sigma)\to\mathscr{T}(\Sigma)\times\mathscr{T}(\Sigma).$
## 3\. AdS convex GHM 3-manifolds with particles
In this section we define the AdS convex GHM manifolds with particles and
recall the parametrization of the moduli space of such structures. The proofs
of these results can be found in [KS07] and [BS09].
### 3.1. Extension of Mess’ parametrization
First, we are going to define the singular AdS space of dimension 3 in order
to define the AdS convex GHM manifolds with particles.
###### Definition 3.1.
Let $\theta>0$, we define
$\text{AdS}^{3}_{\theta}:=\\{(t,\rho,\varphi)\in\mathbb{R}\times\mathbb{R}_{\geq
0}\times[0,\theta)\\}$ with the metric:
$g_{\theta}=-\cosh^{2}\rho dt^{2}+d\rho^{2}+\sinh^{2}\rho d\varphi^{2}.$
###### Remark 3.1.
* •
$\text{AdS}^{3}_{\theta}$ can be obtained by cutting the universal cover of
$\text{AdS}^{3}$ along two time-like planes intersecting along the line
$l:=\\{\rho=0\\}$, making an angle $\theta$, and gluing the two sides of the
angular sector of angle $\theta$ by a rotation fixing $l$. A simple
computation shows that, outside of the singular line,
$\text{AdS}^{3}_{\theta}$ is a Lorentz manifold of constant curvature -1, and
$\text{AdS}^{3}_{\theta}$ carries a conical singularity of angle $\theta$
along $l$.
* •
In the neighborhood of the totally geodesic plane $P_{0}:=\\{t=0\\}$ given by
the points at a causal distance less than $\pi/2$ from $P_{0}$, the metric
$g_{\theta}$ also expresses
$g_{\theta}=-dt^{2}+\cos^{2}t(d\rho^{2}+\sinh^{2}\rho d\varphi^{2}).$
###### Definition 3.2.
An AdS cone-manifold is a (singular) Lorentzian 3-manifold $(M,g)$ in which
any point $x$ has a neighborhood isometric to an open subset of
$\text{AdS}^{3}_{\theta}$ for some $\theta>0$. If $\theta$ can be taken equal
to $2\pi$, $x$ is a smooth point, otherwise $\theta$ is uniquely determined.
To define the global hyperbolicity in the singular case, we need to define the
orthogonality to the singular locus:
###### Definition 3.3.
Let $S\subset\text{AdS}^{3}_{\theta}$ be a space-like surface which intersect
the singular line $l$ at a point $x$. $S$ is said to be orthogonal to $l$ at
$x$ if the causal distance (that is the “distance” along a time-like line) to
the totally geodesic plane $P$ orthogonal to the singular line at $x$ is such
that:
$\lim\limits_{y\to x,y\in S}\frac{d(y,P)}{d_{S}(x,y)}=0$
where $d_{S}(x,y)$ is the distance between $x$ and $y$ along $S$.
Now, a space-like surface $S$ in an AdS cone-manifold $(M,g)$ which intersects
a singular line $d$ at a point $y$ is said to be orthogonal to $d$ if there
exists a neighborhood $U\subset M$ of $y$ isometric to a neighborhood of a
singular point in $\text{AdS}^{3}_{\theta}$ such that the isometry sends
$S\cap U$ to a surface orthogonal to $l$ in $\text{AdS}^{3}_{\theta}$.
Now we are able to define the AdS convex GHM manifolds with particles.
###### Definition 3.4.
An AdS convex GHM manifold with particles is an AdS cone-manifold $(M,g)$
which is homeomorphic to $\Sigma_{\mathfrak{p}}\times\mathbb{R}$ (where
$\Sigma_{\mathfrak{p}}$ is a closed oriented surface with $n$ marked points),
such that the singularities are along time-like lines $d_{1},...,d_{n}$ and
have fixed cone angles $\theta_{1},..,\theta_{n}$ with $\theta_{i}<\pi$.
Moreover, we impose two conditions:
1. (1)
Convex Global Hyperbolicity $M$ contains a space-like future-convex Cauchy
surface orthogonal to the singular locus.
2. (2)
Maximality $M$ cannot be strictly embedded in another manifold satisfying the
same conditions.
###### Remark 3.2.
The condition of convexity in the definition will allow us to use a convex
core. As pointed out by the authors in [BS09], we do not know if every AdS GHM
manifold with particles is convex GHM.
###### Definition 3.5.
For $\theta:=(\theta_{1},...,\theta_{n})\in(0,\pi)^{n}$, let
$\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$ be the space of isotopy classes
of AdS convex GHM metrics on $M=\Sigma_{\mathfrak{p}}\times\mathbb{R}$ with
particles of cone angles $\theta_{i}$ along $d_{i}$.
Many results known in the non-singular case extend to the singular case (that
is with particles of angles less than $\pi$). We recall some of them here (see
[BS09], [KS07]):
1. (1)
The parametrization $\mathfrak{M}$ defined above extends to the singular case.
Namely, we have a parametrization
$\mathfrak{M}_{\theta}:\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})\longrightarrow\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\times\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
which corresponds to Mess’ parametrization when there is no particle.
2. (2)
Each AdS convex GHM 3-manifold with particles $(M,g)$ contains a minimal non-
empty convex subset called its ”convex core” whose boundary is a disjoint
union of two pleated space-like surfaces orthogonal to the singular locus
(except in the Fuchsian case which corresponds to the case where the two
metrics of the parametrization are equal. In this case, the convex core is a
totally geodesic space-like surface).
###### Remark 3.3.
The analogy between AdS GHM geometry and quasi-Fuchsian geometry explained in
the introduction extends to the case with particles. Namely, it is proved in
[LS14] and [MS09] that there exists a parametrization of the moduli space of
quasi-Fuchsian manifolds with particles which extends Bers’ parametrization.
### 3.2. Maximal surface
Let $g\in\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$ be an AdS convex GHM
metric with particles on $M=\Sigma_{\mathfrak{p}}\times\mathbb{R}$.
###### Definition 3.6.
A maximal surface in $(M,g)$ is a locally area-maximizing space-like Cauchy
surface $S\hookrightarrow(M,g)$ which is orthogonal to the singular lines.
In particular, such a maximal surface $S\hookrightarrow(M,g)$ has everywhere
vanishing mean curvature. Note that our definition differs from [KS07,
Definition 5.6] where the authors impose the boundedness of the principal
curvatures of $S$. The following Proposition shows that a maximal surface in
our sense has bounded principal curvatures:
###### Proposition 3.7.
For a maximal surface $S\hookrightarrow(M,g)$ with shape operator $B$ and
induced metric $g_{S}$, $\det_{g_{S}}(B)$ tends to zero at the intersections
with the particles. In particular, $B$ is the real part of a meromorphic
quadratic differential with at most simple poles at the singularities.
###### Proof.
Let $d$ be a particle of angle $\theta$ and set $0:=d\cap S$. We see locally
$S$ as the graph of a function $u:P_{0}\longrightarrow\mathbb{R}$ where
$P_{0}$ is the (piece of) totally geodesic plane orthogonal to $d$ at $0$. We
will show that, the induced metric $g_{S}$ on $S$ carries a conical
singularity of angle $\theta$.
Recall that a metric $g$ on a surface carries a conical singularity of angle
$\theta$ if there exists complex coordinates $z$ centered at the singularity
so that
$g=e^{2u}|z|^{2\left(\theta/2\pi-1\right)}|dz|^{2},$
where $u$ is a bounded function. We need the following lemma:
###### Lemma 3.8.
The gradient of $u$ tends to zero at the intersections with the particles.
###### Proof.
To prove this lemma, we will use Schauder estimates for solutions of uniformly
elliptic PDE’s. For the convenience of the reader, we recall these estimates.
The main reference for the theory is [GT01].
A second order linear operator $L$ on a domain $\Omega\subset\mathbb{R}^{n}$
is a differential operator of the form
$Lu=a^{ij}(x)D_{ij}u+b^{k}(x)D_{k}u+c(x)u,~{}u\in\mathscr{C}^{2}(\Omega),~{}x\in\Omega,$
where we sum over all repeated indices. We say that $L$ is uniformly elliptic
if the smallest eigenvalue of the matrix $\big{(}a_{ij}(x)\big{)}$ is bounded
from below by a strictly positive constant.
We finally define the following norms for a function $u$ on $\Omega$:
* •
$|u|_{k}:=\|u\|_{\mathscr{C}^{k}(\Omega)}.$
* •
$|u|^{(i)}_{0}:=\underset{x\in\Omega}{\sup}~{}d_{x}^{i}|u(x)|,~{}\text{where
}d_{x}=\text{dist}(x,\partial\Omega).$
* •
$|u|^{*}_{k}=\sum_{i=0}^{k}\underset{x\in\Omega,~{}|\alpha|=i}{\sup}d_{x}^{i}|D^{\alpha}u|$.
The following theorem can be found in [GT01, Theorem 6.2]
###### Theorem 3.9.
(Schauder interior estimates) Let $\Omega\subset\mathbb{R}^{n}$ be a domain
with $\mathscr{C}^{2}$ boundary and $u\in\mathscr{C}^{2}(\Omega)$ be solution
of the equation
$Lu=0$
where $L$ is uniformly elliptic so that
$\left|a^{ij}\right|_{0}^{(0)},~{}\left|b^{k}\right|^{(1)}_{0},~{}|c|^{(2)}_{0}<\Lambda.$
Then there exists a positive constant $C$ depending only on $\Omega$ and $L$
so that
$|u|^{*}_{2}\leq C|u|_{0}.$
For every domain $\Omega\subset P_{0}$ which does not contain the singular
point, $u$ satisfies the maximal surface equation (see for example [Ger83])
which is given by:
$\mathscr{L}(u):=\text{div}_{g_{S}}\big{(}v(-1,\pi^{*}\nabla u)\big{)}=0.$
Here, $\pi:S\longrightarrow P_{0}$ is the orthogonal projection,
$v=\big{(}1-\|\pi^{*}\nabla u\|^{2}\big{)}^{-1/2}$ and so $v(-1,\pi^{*}\nabla
u)$ is the unit future pointing normal vector field to $S$. Also, one easily
checks that this equation can be written
(1) $\text{div}_{g_{S}}(v\pi^{*}\nabla u)+a(x,u,\nabla u)=0,\text{ for some
function }a.$
The proof of Proposition 4.11 applies in this case and implies the $S$ is
uniformly space-like. It follows that $\pi$ is uniformly bi-Lipschitz and so
$v$ is uniformly bounded.
It follows that Equation (1) is a quasi-linear elliptic equation in the
divergence form. Moreover, if we write it in the following way:
$a^{ij}(x,u,Du)D_{ij}u+b^{k}(x,u,Du)D_{k}u+c(x,Du,u)u=0,$
it is easy to see that the equation is uniformly elliptic (in fact
$a^{ij}(x,u,Du)\geq 1$) and the coefficients satisfy conditions of Theorem 3.9
(as they are uniformly bounded on $\Omega$). Hence, we are in the good
framework to apply the Schauder estimates.
Let $x_{0}\in P_{0}\setminus\\{0\\}$ and let $2r:=\text{dist}_{S}(x_{0},0)$.
Consider the disk $D_{r}$ of radius $r$ centered at $x_{0}$. It follows from
the previous discussion that $u:D_{r}\longrightarrow\mathbb{R}$ satisfies
$\mathscr{L}u=0$. By a homothety of ratio $1/r$, send the disk $D_{r}$ to the
unit disk $(D,h_{r})$ where $h_{r}$ is the metric of constant curvature
$-r^{2}$. The function $u$ is sent to a new function
$u_{r}:(D,h_{r})\longrightarrow\mathbb{R},$
and satisfies the equation
$\mathscr{L}_{r}u_{r}=0.$
Here, the operator $\mathscr{L}_{r}$ is the maximal surface operator for the
rescaled metric $g_{r}:=-dt^{2}+\cos^{2}t.h_{r}$. In particular,
$\mathscr{L}_{r}$ is a quasi-linear uniformly elliptic operator whose
coefficients applied to $u_{r}$ satisfy the condition of Theorem 3.9.
In a polar coordinates system $(\rho,\varphi)$, the metric $h_{r}$ expresses
$h_{r}=d\rho^{2}+r^{-2}\sinh^{2}(r.\rho)d\varphi^{2}.$
As $r$ tends to zero, the metric $h_{r}$ converges $\mathscr{C}^{\infty}$ on
$D$ to the flat metric $h_{0}=d\rho^{2}+\rho^{2}d\varphi^{2}$. It follows that
the coefficients of the family of operators $(\mathscr{L}_{r})_{r\in(0,1)}$
applied to $u_{r}$ converge to the ones of the operator $\mathscr{L}_{0}$
applied to $u_{0}=\underset{r\to 0}{\lim}u_{r}$ where $\mathscr{L}_{0}$ is the
maximal surface operator associated to the metric
$g_{0}=-dt^{2}+\cos^{2}th_{0}$.
As a consequence, the family of constants $\\{C_{r}\\}$ associated to the
Schauder interior estimates applied to $\mathscr{L}_{r}(u_{r})$ are uniformly
bounded by some $C>0$.
Now, to obtain a bound on the norm of the gradient $\|\nabla u\|$ at a point
$x_{0}$ at a distance $2r$ from the singularity, we apply the Schauder
interior estimates to $\mathscr{L}_{r}(u_{r})$, where
$u_{r}:(D,h_{r})\longrightarrow\mathbb{R}$. We get
$|u_{r}|^{*}_{2}\leq C_{r}|u_{r}|_{0}\leq C|u_{r}|_{0}.$
As $\|\nabla u_{r}\|(x_{0})\leq|u|^{*}_{2}$, and as $u_{r}(x_{0})=o(2r)$
(because $S$ is orthogonal to $d$), we obtain
$\|\nabla u_{r}\|(x_{0})\leq C.o(r).$
But as $u_{r}$ is obtained by rescaling $u$ with a factor $r$, so $\|\nabla
u\|=r^{-1}\|\nabla u_{r}\|$ and we finally get:
$\|\nabla u\|=o(1).$
∎
###### Lemma 3.10.
The induced metric $g_{S}$ on $S$ carries a conical singularity of angle
$\theta$ at its intersection with the particle $d$.
###### Proof.
Recall that (see [MRS15, Section 2.2] and [JMR11, Section 2.1]) a metric $h$
carries a conical singularity of angle $\theta$ if and only if there exists
normal polar coordinates $(\rho,\varphi)\in\mathbb{R}_{>0}\times[0,2\pi)$
around the singularity so that
$g=d\rho^{2}+f^{2}(\rho,\varphi)d\varphi^{2},~{}\frac{f(\rho,\varphi)}{\rho}\underset{\rho\to
0}{\longrightarrow}\theta/2\pi.$
That is, if $g$ can be written by the matrix
$g=\left(\begin{array}[]{ll}1&0\\\
0&\left(\frac{\theta}{2\pi}\right)^{2}\rho^{2}+o(\rho^{2})\end{array}\right).$
The metric of $(M,g)$ can be locally written around the intersection of $S$
and the particle $d$ by
$g=-dt^{2}+\cos^{2}th_{\theta},$
where $h_{\theta}=d\rho^{2}+\left(\frac{\theta}{2\pi}\right)^{2}\sinh^{2}\rho
d\varphi^{2}$ is the metric of $\mathbb{H}^{2}_{\theta}$.
Setting $t=u(\rho,\varphi)$, with $u(\rho,\varphi)=o(\rho)$ and $\|\nabla
u\|=o(1)$, we get
$dt^{2}=(\partial_{\rho}u)^{2}d\rho^{2}+2\partial_{\rho}u\partial_{\varphi}ud\rho
d\varphi+(\partial_{\varphi}u)^{2}d\varphi^{2}.$
Note that, as $\|\nabla u\|=o(1)$, $\partial_{\rho}u=o(1)$ and
$\partial_{\varphi}u=o(\rho)$.
Finally, using $\cos^{2}(u)=1+o(\rho^{2})$, we get the following expression
for the induced metric on $S$:
$g_{S}=\left(\begin{array}[]{ll}1+o(1)&o(\rho)\\\
o(\rho)&\left(\frac{\theta}{2\pi}\right)^{2}\rho^{2}+o(\rho^{2})\end{array}\right).$
One easily checks that, with a change of variable, the induced metric carries
a conical singularity of angle $\theta$ at the intersection with $d$. ∎
Now the proof of Proposition 3.7 follows: suppose the second fundamental form
$\textrm{II}=g_{S}(B.,.)$ is the real part of a meromorphic quadratic
differential $q$ with a pole of order $n$. In complex coordinates, write
$q=f(z)dz^{2}$ and $g_{S}=e^{2u}|z|^{2\left(\theta/2\pi-1\right)}|dz|^{2}$
where $u$ is bounded. Then $B$ is the real part of the harmonic Beltrami
differential
$\mu:=\frac{\overline{q}}{g_{S}}=e^{-2u}|z|^{-2(\theta/2\pi-1)}\overline{f}(z)d\overline{z}\partial_{z}.$
Using the real coordinates
$z=x+iy,~{}dz=dx+idy,~{}\partial_{z}=\frac{1}{2}(\partial_{x}-i\partial_{y})$
we get
$\displaystyle B$ $\displaystyle=$
$\displaystyle\Re\left(\frac{1}{2}e^{-2u}|z|^{-2(\theta/2\pi-1)}\big{(}\Re(f)-i\Im(f)\big{)}(dx-
idy)(\partial_{x}-i\partial_{y})\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2u}|z|^{-2(\theta/2\pi-1)}\big{(}\Re(f)(dx\partial_{x}-dy\partial_{y})-\Im(f)(dx\partial_{y}-dy\partial_{x})\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2u}|z|^{-2(\theta/2\pi-1)}\left(\begin{array}[]{ll}\Re(f)&-\Im(f)\\\
-\Im(f)&-\Re(f)\end{array}\right).$
It follows that
$\text{det}_{g_{S}}(B)=-\frac{1}{4}e^{-4u}|z|^{-4(\theta/2\pi-1)}|f|^{2}=-e^{v}|z|^{-2(\theta/\pi-2+n)},$
for some bounded $v$. By (modified) Gauss equation, the curvature $K_{S}$ of
$S$ is given by
$K_{S}=-1-\text{det}_{g_{S}}(B).$
By Gauss-Bonnet formula for surface with cone singularities (see for example
[Tro91]), $K_{S}$ has to be locally integrable. But we have:
$K_{s}dvol_{S}=\big{(}-1+e^{w}|z|^{-2(\theta/2\pi-1+n)}\big{)}d\lambda,$
where $d\lambda$ is the Lebesgue measure on $\mathbb{R}^{2}$. It follows that
$K_{s}dvol_{S}$ is integrable if and only if $\theta/2\pi-1+n<1$, that is
$n\leq 1$. Note also that, for $n\leq 1$,
$\det_{g_{S}}(B)=O\left(|z|^{2(1-\theta/\pi)}\right)$ and so tends to zero at
the singularity. ∎
It is proved in [KS07] that, as in the non-singular case, we can define the
space $\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$ of maximal surfaces in a
germ of AdS convex GHM with $n$ particles of angles
$\theta=(\theta_{1},...,\theta_{n})\in(0,\pi)^{n}$. This space is still
parametrized by $T^{*}\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$. Recall
that the cotangent space
$T^{*}_{g}\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ to
$\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ at a metric
$g\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ is given by the space of
meromorphic quadratic differentials on $(\Sigma_{\mathfrak{p}},J_{g})$ (where
$J_{g}$ is the complex structure associated to $g$) with at most simple poles
at the marked points.
Moreover, given $(g,h)\in\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$, using
the Fundamental Theorem of surfaces in AdS convex GHM manifolds with
particles, one can locally reconstruct a piece of AdS globally hyperbolic
manifold with particles which uniquely embeds in a maximal one. It provides a
map from $\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$ to
$\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$. This map is bijective if and
only if each AdS convex GHM manifold $(M,g)$ contains a unique maximal
surface.
## 4\. Existence of a maximal surface
In this section, we prove the existence part of Theorem 1.4. Note that in the
Fuchsian case (that is when the two metrics of the parametrization
$\mathfrak{M}_{\theta}$ are equal), the convex core is reduced to a totally
geodesic plane orthogonal to the singular locus which is thus maximal (its
second fundamental form vanishes).
Hence, from now on, we consider an AdS convex GHM manifold with particles
$(M,g)$, where $g\in\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$ is such that
$\mathfrak{M}_{\theta}(g)\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\times\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
are two distinct points (that is $(M,g)$ is not Fuchsian). It follows from
[BS09, Section 5] that $(M,g)$ contains a convex core with non-empty interior.
The boundary of this convex core is given by two pleated surfaces: a future-
convex one and a past-convex one.
###### Proposition 4.1.
The AdS convex GHM manifold with particles $(M,g)$ contains a maximal surface
$S\hookrightarrow(M,g)$.
The proof is done in four steps:
1. Step 1
Approximate the singular metric $g$ by a sequence of smooth metrics
$(g_{n})_{n\in\mathbb{N}}$ which converges to the metric $g$, and prove the
existence for each $n\in\mathbb{N}$ of a maximal surface
$S_{n}\hookrightarrow(M,g_{n})$.
2. Step 2
Prove that the sequence $(S_{n})_{n\in\mathbb{N}}$ converges outside the
singular lines to a smooth nowhere time-like surface $S$ with vanishing mean
curvature.
3. Step 3
Prove that the limit surface $S$ is space-like.
4. Step 4
Prove that the limit surface $S$ is orthogonal to the singular lines.
### 4.1. First step
Approximation of singular metrics. Take $\theta\in(0,2\pi)$ and let
$\mathscr{C}_{\theta}\subset\mathbb{R}^{3}$ be the cone given by the
parametrization:
$\mathscr{C}_{\theta}:=\left\\{\left(u.\cos v,u.\sin
v,\text{cotan}(\theta/2).u\right),~{}(u,v)\in\mathbb{R}_{+}\times[0,2\pi)\right\\}.$
Now, consider the intersection of this cone with the Klein model of the
hyperbolic 3-space, and denote by $h_{\theta}$ the induced metric on
$\mathscr{C}_{\theta}$. Outside the apex, $\mathscr{C}_{\theta}$ is a convex
ruled surface in $\mathbb{H}^{3}$, and so has constant curvature $-1$.
Moreover, one easily checks that $h_{\theta}$ carries a conical singularity of
angle $\theta$ at the apex of $\mathscr{C}_{\theta}$. Consider the orthogonal
projection $p$ from $\mathscr{C}_{\theta}$ to the disk of equation
$\mathbb{D}:=\\{z=0\\}\subset\mathbb{H}^{3}$. We have that
$(\mathbb{D},(p^{-1})^{*}h_{\theta})$ is isometric to the local model of
hyperbolic metric with cone singularity $\mathbb{H}^{2}_{\theta}$ as defined
in the introduction.
###### Remark 4.1.
The angle of the singularity is given by $\displaystyle{\underset{\rho\to
0}{\lim}\frac{l(C_{\rho})}{\rho}}$ where $l(C_{\rho})$ is the length of the
circle of radius $\rho$ centered at the singularity.
Now, to approximate this metric, take
$(\epsilon_{n})_{n\in\mathbb{N}}\subset(0,1)$, a sequence decreasing to zero
and define a sequence of even functions
$f_{n}:\mathbb{R}\longrightarrow\mathbb{R}$ so that for each $n\in\mathbb{N}$,
$\left\\{\begin{array}[]{l}f_{n}(0)=-\epsilon_{n}^{2}.\text{cotan}(\theta/2)\\\
f_{n}^{{}^{\prime\prime}}(x)<0\ \ \forall x\in(-\epsilon_{n},\epsilon_{n})\\\
f_{n}(x)=-\text{cotan}(\theta/2).x\text{ if
}x\geqslant\epsilon_{n}.\par\end{array}\right.$
Figure 1. Graph of $f_{n}$
Consider the surface $\mathscr{C}_{\theta,n}$ obtained by making a rotation of
the graph of $f_{n}$ around the axis $(0z)$ and consider its intersection with
the Klein model of hyperbolic 3-space. Denote by $h_{\theta,n}$ the induced
metric on $\mathscr{C}_{\theta,n}$, and define
$\mathbb{H}^{2}_{\theta,n}:=(\mathbb{D},(p^{-1})^{*}h_{\theta,n})$ (where $p$
is still the orthogonal projection to the disk
$\mathbb{D}=\\{z=0\\}\subset\mathbb{H}^{3}$). By an abuse of notations, we
write $\mathbb{H}^{2}_{\theta,n}=(\mathbb{D},h_{\theta,n})$. Denote by
$B_{i}\subset\mathbb{D}$ the smallest set where the metric $h_{\theta,n}$ does
not have of constant curvature $-1$, by construction,
$B_{n}\underset{n\to\infty}{\longrightarrow}\\{0\\}$, where $\\{0\\}$ is the
center of $\mathbb{D}$. We have
###### Proposition 4.2.
For all compact $K\subset\mathbb{D}\setminus\\{0\\}$, there exists
$i_{K}\in\mathbb{N}$ such that for all $n>n_{K}$,
$h_{\theta_{|K}}=h_{\theta,n_{|K}}$.
We define the AdS 3-space with regularized singularity:
###### Definition 4.3.
For $\theta>0$, $n\in\mathbb{N}$, set
$\text{AdS}^{3}_{\theta,n}:=\\{(t,\rho,\varphi)\in(-\pi/2,\pi/2)\times\mathbb{D}\\}$
endowed with the metric:
$g_{\theta,n}=-dt^{2}+\cos^{2}th_{\theta,n}.$
By construction, there exists a smallest tubular neighborhood $V^{n}_{\theta}$
of $l=\\{0\\}\times(-\pi/2,\pi/2)$ such that
$\text{AdS}^{3}_{\theta,n}\setminus V^{n}_{\theta}$ is a Lorentzian manifold
of constant curvature $-1$.
In this way, we are going to define the regularized AdS convex GHM manifold
with particles.
For all $j\in\\{1,...,n\\}$ and $x\in d_{j}$ where $d_{j}$ is a singular line
in $(M,g)$, there exists a neighborhood of $x$ in $(M,g)$ isometric to a
neighborhood of a point on the singular line in $\text{AdS}^{3}_{\theta_{j}}$.
For $n\in\mathbb{N}$, we define the regularized metric $g_{n}$ on $M$ so that
the neighborhoods of points of $d_{j}$ are isometric to neighborhoods of
points on the central axis in $\text{AdS}^{3}_{\theta_{j},n}$. Clearly, the
metric $g_{n}$ is obtained taking locally the metric of $V^{n}_{\theta_{j}}$
in a tubular neighborhood $U^{n}_{j}$ of the singular lines $d_{j}$ for all
$j\in\\{1,...,n\\}$. In particular, outside these $U^{n}_{j}$, $(M,g_{n})$ is
a regular AdS manifold.
###### Proposition 4.4.
Let $K\subset M$ be a compact set which does not intersect the singular lines.
There exists $n_{K}\in\mathbb{N}$ such that, for all $n>n_{K}$,
$g_{n_{|K}}=g_{|K}$.
Existence of a maximal surface in each $(M,g_{n})$
We are going to prove Proposition 4.1 by convergence of maximal surfaces in
each $(M,g_{n})$. A result of Gerhardt [Ger83, Theorem 6.2] provides the
existence of a maximal surface in $(M,g_{n})$ given the existence of two
smooth barriers, that is, a strictly future-convex smooth (at least
$\mathscr{C}^{2}$) space-like surface and a strictly past-convex one. This
result has been improved in [ABBZ12, Theorem 4.3] reducing the regularity
conditions to $\mathscr{C}^{0}$ barriers.
The natural candidates for these barriers are equidistant surfaces from the
boundary of the convex core of $(M,g)$. It is proved in [BS09, Section 5] that
the future (respectively past) boundary component $\partial_{+}$ (respectively
$\partial_{-}$) of the convex core is a future-convex (respectively past-
convex) space-like pleated surface orthogonal to the particles. Moreover, each
point of the boundary components is either contained in the interior of a
geodesic segment (a pleating locus) or of a totally geodesic disk contained in
the boundary components.
For $\epsilon>0$ fixed, consider the $2\epsilon$-surface in the future of
$\partial_{+}$ and denote by $\partial_{+,\epsilon}$ the $\epsilon$-surface in
the past of the previous one. As pointed out in [BS09, Proof of Lemma 4.2],
this surface differs from the $\epsilon$-surface in the future of
$\partial_{+}$ (at the pleating locus).
###### Proposition 4.5.
For $n$ big enough, $\partial_{+,\epsilon}\hookrightarrow(M,g_{n})$ is a
strictly future-convex space-like $\mathscr{C}^{1,1}$ surface.
###### Proof.
Outside the open set $\displaystyle{U^{n}:=\bigcup_{j=1}^{n}U^{n}_{j}}$ (where
the $U_{j}^{n}$ are tubular neighborhoods of $d_{j}$ so that the curvature is
different from $-1$), $(M,g_{n})$ is isometric to $(M,g)$, and moreover,
$U^{n}_{j}\underset{n\to\infty}{\longrightarrow}d_{j}$ for each $j$. As proved
in [BS09, Lemma 5.2], each intersection of $\partial_{+}$ with a particle lies
in the interior of a totally geodesic disk contained in $\partial_{+}$. So,
there exists $n_{0}\in\mathbb{N}$ such that, for $n>n_{0}$,
$U_{i}^{j}\cap\partial_{+}$ is totally geodesic.
The fact that $\partial_{+,\epsilon}$ is $\mathscr{C}^{1,1}$ is proved in
[BS09, Proof of Lemma 4.2].
For the strict convexity outside $U^{n}$, the result is proved in [BBZ07,
Proposition 6.28]. So it remains to prove that $\partial_{+,\epsilon}\cap
U_{n}$ is strictly future-convex.
Let $d=d_{j}$ be a singular line which intersects $\partial_{+}$ at a point
$x$. As $U:=U_{n}^{j}\cap\partial_{+}$ is totally geodesic, we claim that
$U_{\epsilon}:=U_{n}^{j}\cap\partial_{+,\epsilon}$ is the $\epsilon$-surface
of $U$ with respect to the metric $g_{n}$. In fact, the space-like surface
$\mathscr{P}_{0}\subset\text{AdS}^{3}_{\theta,i}$ given by the equation
$\\{t=0\\}$ is totally geodesic and the one given by
$\mathscr{P}_{\epsilon}:=\\{t=\epsilon\\}$ is the $\epsilon$-surface of
$\mathscr{P}_{0}$ and corresponds to the $\epsilon$-surface in the past of
$\mathscr{P}_{2\epsilon}$. It follows that $U_{\epsilon}$ is obtained by
taking the $\epsilon$-time flow of $U$ along the unit future-pointing vector
field $N$ normal to $\partial_{+}$ (extended to an open neighborhood of $U$ by
the condition $\nabla_{N}^{n}N=0$, where $\nabla^{n}$ is the Levi-Civita
connection of $g_{n}$). We are going to prove that the second fundamental form
on $U_{\epsilon}$ is positive definite.
Note that in $\text{AdS}^{3}_{\theta_{j},n}$, the surfaces
$\mathscr{P}_{t_{0}}:=\\{t=t_{0}\\}$ are equidistant from the totally geodesic
space-like surface $\mathscr{P}_{0}$. Moreover, the induced metric on
$\mathscr{P}_{t_{0}}$ is $\textrm{I}_{t_{0}}=\cos^{2}(t_{0})h_{n,\theta}$ and
so, the variation of $\textrm{I}_{t_{0}}$ along the flow of $N$ is given by
$\frac{d}{dt}_{|t=t_{0}}\textrm{I}_{t}(u_{t},u_{t})=-2\cos(t_{0})\sin(t_{0}),$
for $u_{t}$ a unit vector field tangent to $\mathscr{P}_{t}$. On the other
hand, this variation is given by
$\frac{d}{dt}_{|t=t_{0}}\textrm{I}_{t}(u_{t},u_{t})=\mathscr{L}_{N}\textrm{I}_{t_{0}}(u_{t_{0}},u_{t_{0}})=2\textrm{I}_{t_{0}}(\nabla^{i}_{u_{t_{0}}}N,u_{t_{0}})=-2\textrm{II}_{t_{0}}(u_{t_{0}},u_{t_{0}}),$
where $\mathscr{L}$ is the Lie derivative and $Bu:=-\nabla_{u}N$ is the shape
operator.
It follows that $\textrm{II}_{t_{0}}$ is positive-definite for $t_{0}>0$ small
enough. So $\partial_{+,\epsilon}\hookrightarrow(M,g_{n})$ is strictly future-
convex. ∎
So we get a $\mathscr{C}^{1,1}$ barrier. The existence of a
$\mathscr{C}^{1,1}$ strictly past-convex surface is analogous. So, by [ABBZ12,
Theorem 4.3], we get that for all $n>n_{0}$, there exists a maximal space-like
Cauchy surface $S_{n}$ in $(M,g_{n})$. By re-indexing, we finally have proved
###### Proposition 4.6.
There exists a sequence $(S_{n})_{n\in\mathbb{N}}$ of space-like surfaces
where each $S_{n}\hookrightarrow(M,g_{n})$ is a maximal space-like surface.
### 4.2. Second step
###### Proposition 4.7.
There exists a subsequence of $(S_{n})_{n\in\mathbb{N}}$ converging uniformly
on each compact which does not intersect the singular lines to a surface
$S\hookrightarrow(M,g)$.
###### Proof.
For some fixed $n_{0}\in\mathbb{N}$, $(M,g_{n_{0}})$ is a smooth globally
hyperbolic manifold and so admits some smooth time function
$f:(M,g_{n_{0}})\longrightarrow\mathbb{R}$. This time function allows us to
see the sequence of maximal surfaces $(S_{n})_{n\in\mathbb{N}}$ as a sequence
of graphs on functions over $f^{-1}(\\{0\\})$ (where we suppose $0\in f(M)$).
Let $K\subset f^{-1}(\\{0\\})$ be a compact set which does not intersect the
singular lines and see locally the surfaces $S_{n}$ as graphs of functions
$u_{n}:K\longrightarrow\mathbb{R}$.
For $n$ big enough, the graphs of $u_{n}$ are pieces of space-like surfaces
contained in the convex core of $(M,g)$, so the sequence
$(u_{n})_{n\in\mathbb{N}}$ is a sequence of uniformly bounded Lipschitz
functions with uniformly bounded Lipschitz constant. By Arzelà-Ascoli’s
Theorem, this sequence admits a subsequence (still denoted by
$(u_{n})_{n\in\mathbb{N}}$) converging uniformly to a function
$u:K\longrightarrow\mathbb{R}$. Applying this to each compact set of
$f^{-1}(\\{0\\})$ which does not intersect the singular line, we get that the
sequence $(S_{n})_{n\in\mathbb{N}}$ converges uniformly outside the singular
lines to a surface $S$. ∎
Note that, as the surface $S$ is a limit of space-like surfaces, it is nowhere
time-like. However, $S$ may contains some light-like locus. We recall a
theorem of C. Gerhardt [Ger83, Theorem 3.1]:
###### Theorem 4.8.
(C. Gerhardt) Let $S$ be a limit on compact subsets of a sequence of space-
like surfaces in a globally hyperbolic space-time. Then if $S$ contains a
segment of a null geodesic, this segment has to be maximal, that is it extends
to the boundary of $M$.
So, if $S$ contains a light-like segment, either this segment extends to the
boundary of $M$, or it intersects two singular lines. The first is impossible
as it would imply that $S$ is not contained in the convex core. Thus, the
light-like locus of $S$ lies in the set of light-like rays between two
singular lines.
We now prove the following:
###### Proposition 4.9.
The sequence of space-like surfaces $(S_{n})_{n\in\mathbb{N}}$ of Proposition
4.6 converges $\mathscr{C}^{1,1}$ on each compact which does not intersect the
singular lines and light-like locus. Moreover, outside these loci, the surface
$S$ has everywhere vanishing mean curvature.
###### Proof.
For a point $x\in S$ which neither lies on a singular line nor on a light-like
locus, see a neighborhood $K\subset S$ of $x$ as the graph of a function $u$
over a piece of totally geodesic space-like plane $\Omega$. With an isometry
$\Psi$, send $\Omega$ to the totally geodesic plane
$P_{0}\subset\text{AdS}^{3}$ given by the equation
$P_{0}:=\\{(t,\rho,\varphi)\in\text{AdS}^{3},~{}t=0\\}$. We still denote by
$S_{n}$ (respectively $S$, $u$ and $\Omega$) the image by $\Psi$ of $S_{n}$
(respectively $S$, $u$ and $\Omega$). Note that, for $n\in\mathbb{N}$ big
enough, the metric $g_{n}$ coincides with the metric $g$ in a neighborhood of
$K$ in $M$. So locally around $x$, the surfaces $S_{n}$ have vanishing mean
curvature in $(M,g)$, hence their images in $\text{AdS}^{3}$ have vanishing
mean curvature.
Let $u_{n}:\Omega\longrightarrow\mathbb{R}$ be such that
$S_{n}=\text{graph}(u_{n})$. The unit future pointing normal vector to $S_{n}$
at $(x,u_{n}(x))$ is given by
$N_{n}=v_{n}.\pi^{*}(1,\nabla u_{n}),$
where $(1,\nabla u_{n})\in T_{x}\text{AdS}^{3}$ is the vector
$(\partial_{t},\nabla_{\rho}u_{n},\nabla_{\varphi}u_{n})$,
$\pi:S_{n}\longrightarrow\Omega$ is the orthogonal projection on $P_{0}$ and
$v_{n}=\big{(}1-\|\pi^{*}\nabla u_{n}\|^{2})^{-1/2}$. The vanishing of the
mean curvature of $S_{n}$ is equivalent to
$-\delta_{g}N_{n}=0,$
where $\delta_{g}$ is the divergence operator. In coordinates, this equation
reads (see also [Ger83, Equation 1.14]):
(3) $\frac{1}{\sqrt{\det g}}\partial_{i}(\sqrt{\det
g}v_{n}g^{ij}\nabla_{j}u_{n})+\frac{1}{2}v_{n}\partial_{t}g^{ij}\nabla_{i}u_{n}\nabla_{j}u_{n}-\frac{1}{2}v_{n}^{-1}g^{ij}\partial_{t}g_{ij}=0.$
Here, we wrote the metric
$g=-dt^{2}+g_{ij}(x,t)dx^{i}dx^{j},$
applying the convention of Einstein for the summation (with indices
$i,j=1,2$). The metric $g$ is taken at the points $(u_{n}(x),x)$ and $\det g$
is the determinant of the metric.
We have the following
###### Lemma 4.10.
The solutions $u_{n}$ of equation (3) are in $\mathscr{C}^{\infty}(\Omega)$.
###### Proof.
This is a bootstrap argument. From [Ger83, Theorem 5.1], we have
$u_{n}\in\mathscr{W}^{2,p}(\Omega)$ for all $p\in[1,+\infty)$ (where
$\mathscr{W}^{k,p}(\Omega)$ is the Sobolev space of functions over $\Omega$
admitting weak $L^{p}$ derivatives up to order $k$).
As $v_{n}$ is uniformly bounded from above and from below (because the surface
$S_{n}$ is space-like), and as $u_{n}\in\mathscr{W}^{2,p}(\Omega)$, the third
term of equation (3) is in $\mathscr{W}^{1,p}(\Omega)$.
For the second term, we recall the multiplication law for Sobolev space: if
$\frac{k}{2}-\frac{1}{p}>0$, then the product of functions in
$\mathscr{W}^{k,p}(\Omega)$ is still in $\mathscr{W}^{k,p}(\Omega)$. So, as
the second term of equation (3) is a product of three terms in
$\mathscr{W}^{1,p}(\Omega)$, it is in $\mathscr{W}^{1,p}(\Omega)$ (by taking
$p>2$).
Hence the first term is in $\mathscr{W}^{1,p}(\Omega)$, and so $\sqrt{\det
g}v_{n}g^{ij}\nabla_{j}u_{n}\in\mathscr{W}^{2,p}(\Omega)$. Moreover, as we can
write the metric $g$ to that $g_{ij}=0$ whenever $i\neq j$ and as $\sqrt{\det
g}g^{ii}$ are $\mathscr{W}^{2,p}(\Omega)$ and bounded from above and from
below, $v_{n}\nabla_{i}u_{n}\in\mathscr{W}^{2,p}(\Omega)$. We claim that it
implies $u_{n}\in\mathscr{W}^{3,p}$. It fact, for $f$ a never vanishing smooth
function, consider the map
$\begin{array}[]{llll}\varphi:&D\subset\mathbb{R}^{2}&\longrightarrow&\mathbb{R}^{2}\\\
&p&\longmapsto&(1-f^{2}(p)|p|^{2})^{-1/2}p,\end{array}$
where $D$ is a domain such that $f^{2}(p)|p|^{2}<1-\epsilon$ and $p\neq 0$.
The map $\varphi$ is a $\mathscr{C}^{\infty}$ diffeomorphism on its image, and
we have $\big{(}\varphi(\nabla u_{n})\big{)}_{i}\in\mathscr{W}^{2,p}(\Omega)$
for $i=1,2$ (in fact, as it is a local argument, we can always perturb
$\Omega$ so that $\nabla u_{n}\neq 0$). Applying $\varphi^{-1}$, we get
$\nabla_{i}u_{n}\in\mathscr{W}^{2,p}(\Omega)$ and so
$u\in\mathscr{W}^{3,p}(\Omega)$.
Iterating the process, we obtain that $u_{n}\in\mathscr{W}^{k,p}(\Omega)$ for
all $k\in\mathbb{N}$ and $p>1$ big enough. Using the Sobolev embedding Theorem
$\mathscr{W}^{j+k,p}(\Omega)\subset\mathscr{C}^{j,\alpha}(\Omega)\text{ for
}0<\alpha<k-\frac{2}{p},$
we get the result. ∎
Now, from Proposition 4.7,
$u_{n}\overset{\mathscr{C}^{0,1}}{\longrightarrow}u$, that is
$u_{n}\overset{\mathscr{W}^{1,p}}{\longrightarrow}u$ for all
$p\in[1,+\infty)$.
Moreover, as the sequence of graphs of $u_{n}$ converges uniformly to a space-
like graph, the sequence $(\nabla u_{n})_{n\in\mathbb{N}}$ is uniformly
bounded. From equation (3), we get that there exists a constant $C>0$ such
that for each $n\in\mathbb{N}$,
$|\partial_{i}(\sqrt{\det h}v_{n}g^{ii}\nabla_{i}u_{n})|<C.$
As $(\nabla u_{n})_{n\in\mathbb{N}}$ is uniformly bounded, the terms
$\partial_{i}v_{n}$ are also uniformly bounded and we obtain
$|\partial_{i}(\nabla_{i}u_{n})|<C^{\prime},$
for some constant $C^{\prime}$.
Thus $(\nabla_{i}u_{n})_{n\in\mathbb{N}}$ is a sequence of bounded Lipschitz
functions with uniformly bounded Lipschitz constant so admits a convergent
subsequence by Arzelà-Ascoli. It follows that
$u_{n}\overset{\mathscr{W}^{2,p}}{\longrightarrow}u,$
for all $p\in[1,+\infty)$. Thus $u$ is a solution of equation (3), and so
$u\in\mathscr{C}^{\infty}(\Omega)$. Moreover, as $u$ satisfies equation (3),
$S$ has locally vanishing mean curvature. ∎
### 4.3. Third step
###### Proposition 4.11.
The surface $S$ of Proposition 4.7 is space-like.
We are going to prove that, at its intersections with the singular lines, $S$
does not contain any light-like direction. To prove this, we are going to
consider the link of $S$ at its intersection $p$ with a particle $d$. The link
is essentially the set of rays from $p$ that are tangent to the surface.
Denote by $\alpha$ the cone angle of the singular line. We see locally the
surface as the graph of a function $u$ over a small disk
$D_{\alpha}=D_{\alpha}(0,r)=((0,r)\times[0,\alpha))\cup\\{0\\}$ contained in
the totally geodesic plane orthogonal to $d$ passing through $p$ (in
particular, $u(0)=0$).
First, we describe the link at a regular point of an AdS convex GHM manifold,
then the link at a singular point. The link of a surface at a smooth point is
a circle in a sphere with an angular metric (called HS-surface in [Sch98]). As
the surface $S$ is a priory not smooth, we will define the link of $S$ as the
domain contained between the two curves given by the limsup and liminf at zero
of $\displaystyle{\frac{u(\rho,\theta)}{\rho}}$.
The link of a point. Consider $p\in(M,g)$ not lying of a singular line. The
tangent space $T_{p}M$ identifies with the Minkowski 3-space
$\mathbb{R}^{2,1}$. We define the link of $M$ at $p$, that we denote by
$\mathscr{L}_{p}$, as the set of rays from $p$, that is the set of half-lines
from $0$ in $T_{p}M$. Geometrically, $\mathscr{L}_{p}$ is a 2-sphere, and the
metric is given by the angle ”distance”. So one can see that $\mathscr{L}_{p}$
is divided into five subsets (depending on the type of the rays and on the
causality):
* •
The set of future and past pointing time-like rays that carries a hyperbolic
metric.
* •
The set of light-like rays defines two circles called past and future light-
like circles.
* •
The set of space-like rays which carries a de Sitter metric.
To obtain the link of a point lying on a singular line of angle $\alpha\leq
2\pi$, we cut $\mathscr{L}_{p}$ along two meridian separated by an angle
$\alpha$ and glue by a rotation. We get a surface denoted
$\mathscr{L}_{p,\alpha}$ (see Figure 2).
Figure 2. Link at a singular point
The link of a surface. Let $\Sigma$ be a smooth surface in $(M,g)$ and
$p\in\Sigma$ not lying on a singular line. The space of rays from $p$ tangent
to $\Sigma$ is just the projection of the tangent plane to $\Sigma$ on
$\mathscr{L}_{p}$ and so describe a circle in $\mathscr{L}_{p}$. Denote this
circle by $\mathscr{C}_{\Sigma,p}$. Obviously, if $\Sigma$ is a space-like
surface, $\mathscr{C}_{\Sigma,p}$ is a space-like geodesic in the de Sitter
domain of $\mathscr{L}_{p}$ and if $\Sigma$ is time-like or light-like,
$\mathscr{C}_{\Sigma,p}$ intersects one of the time-like circle in
$\mathscr{L}_{p}$.
Now, if $p\in\Sigma$ belongs to a singular line of angle $\alpha$ and is not
smooth, we define the link of $\Sigma$ at $p$ as the domain
$\mathscr{C}_{\Sigma,p}$ delimited by the limsup and the liminf of
$\displaystyle{\frac{u(\rho,\theta)}{\rho}}$.
We have an important result:
###### Proposition 4.12.
Let $\Sigma$ be a nowhere time-like surface which intersects a singular line
of angle $\alpha<\pi$ at a point $p$. If $\mathscr{C}_{\Sigma,p}$ intersects a
light-like circle in $\mathscr{L}_{p,\alpha}$, then $\mathscr{C}_{\Sigma,p}$
does not cross $\mathscr{C}_{0,p}$. That is, $\mathscr{C}_{\Sigma,p}$ remains
strictly in one hemisphere (where a hemisphere is a connected component of
$\mathscr{L}_{p,\alpha}\setminus\mathscr{C}_{0,p}$).
###### Proof.
Fix a non-zero vector $u\in T_{p}(\Sigma)$ and for $\theta\in[0,\alpha)$,
denote by $v_{\theta}$ the unit vector making an angle $\theta$ with $u$.
Suppose that $v_{\theta_{0}}$ corresponds to the direction where
$\mathscr{C}_{\Sigma,p}$ intersects a light-like circle, for example, the
future light-like circle. As the surface is nowhere time-like, $\Sigma$
remains in the future of the light-like plane containing $v_{\theta_{0}}$. But
the link of a light-like plane at a non singular point $p$ is a great circle
in $\mathscr{L}_{p}$ which intersects the two different light-like circles at
the directions given by $v_{\theta_{0}}$ and $v_{\theta_{0}+\pi}$. So it
intersects $\mathscr{C}_{0,p}$ at the directions $v_{\theta_{0}\pm\pi/2}$.
Now, if $p$ belongs to a singular line of angle $\alpha<\pi$, the link of the
light-like plane which contains $v_{\theta_{0}}$ is obtained by cutting the
link of $p$ along the directions of $v_{\theta_{0}\pm\alpha/2}$ and gluing the
two wedges by a rotation (see the Figure 2). So, the link of our light-like
plane remains in the upper hemisphere, which implies the result. ∎
###### Remark 4.2.
Equivalently, we get that if $\mathscr{C}_{\Sigma,p}$ intersects
$\mathscr{C}_{0,p}$, it does not intersect a light-like circle.
In particular, if link of $\Sigma$ at $p$, $\mathscr{C}_{\Sigma,p}$ is
continuous, there exists $\eta>0$ (depending of $\alpha$) such that:
* •
If $\mathscr{C}_{\Sigma,p}$ $\theta_{0}$ intersects the future light-like
circle, then
(4) $u(\rho,\theta)\geq\eta.\rho~{}\forall\theta\in[0,\alpha),~{}\rho\ll 1.$
* •
If $\mathscr{C}_{\Sigma,p}$ $\theta_{0}$ intersects $\mathscr{C}_{0,p}$, then
(5) $u(\rho,\theta)\leq(1-\eta).\rho~{}\forall\theta\in[0,\alpha)~{}\rho\ll
1.$
Figure 3. The link remains in the upper hemisphere
These two results will be used in the next part.
Link of $S$ and orthogonality. Let $S$ be the limit surface of Proposition 4.7
and let $p\in S$ be an intersection with a singular line $d$ of angle
$\alpha<\pi$. As previously, we consider locally $S$ as the graph of a
function
$u:D_{\alpha}\rightarrow\mathbb{R}$
in a neighborhood of $p$. Let $\mathscr{C}_{S,p}\subset\mathscr{L}_{p,\alpha}$
be the “augmented” link of $S$ at $p$, that is, the connected domain contained
between the curves $\mathscr{C}_{\pm}$, where $\mathscr{C}_{+}$ is the curve
corresponding to $\displaystyle{\underset{\rho\to
0}{\limsup}\frac{u(\rho,\theta)}{\rho}}$, and $\mathscr{C}_{-}$ corresponding
to the liminf.
###### Lemma 4.13.
The curves $\mathscr{C}_{+}$ and $\mathscr{C}_{-}$ are $\mathscr{C}^{0,1}$.
###### Proof.
We give the proof for $\mathscr{C}_{-}$ (the one for $\mathscr{C}_{+}$ is
analogue). For $\theta\in[0,\alpha)$, denote by
$k(\theta):=\underset{\rho\to 0}{\liminf}\frac{u(\rho,\theta)}{\rho}.$
Fix $\theta_{0},\theta\in[0,\alpha)$. By definition, there exists a decreasing
sequence $(\rho_{k})_{k\in\mathbb{N}}\subset\mathbb{R}_{>0}$ such that
$\underset{k\to\infty}{\lim\rho_{k}}=0$ and
$\underset{k\to\infty}{\lim}\frac{u(\rho_{k},\theta_{0})}{\rho_{k}}=k(\theta_{0}).$
As $S$ is nowhere time-like, for each $k\in\mathbb{N}$, $S$ remains in the
cone of space-like and light-like geodesic from
$((\rho_{k},\theta_{0}),u(\rho_{k},\theta_{0}))\in S$. That is,
$|u(\rho_{k},\theta)-u(\rho_{k},\theta_{0})|\leq
d_{a}(\theta,\theta_{0})\rho_{k},$
where $d_{a}$ is the angular distance between two directions. So we get
$\underset{k\to\infty}{\lim}\frac{u(\rho_{k},\theta)}{\rho_{k}}\leq
k(\theta_{0})+d_{a}(\theta,\theta_{0}),$
and so
$k(\theta)\leq k(\theta_{0})+d_{a}(\theta,\theta_{0}).$
On the other hand, for all $\epsilon>0$ small enough, there exists $R>0$ such
that, for all $\rho\in(0,R)$ we have:
$u(\rho,\theta_{0})>(k(\theta_{0})-\epsilon)\rho.$
By the same argument as before, because $S$ is nowhere time-like, we get
$|u(\rho,\theta)-u(\rho,\theta_{0})|\leq d_{a}(\theta,\theta_{0})\rho,$
that is
$u(\rho,\theta)\geq u(\rho,\theta_{0})-d_{a}(\theta,\theta_{0})\rho.$
So
$u(\rho,\theta)>(k(\theta_{0})-\epsilon)\rho-d_{a}(\theta,\theta_{0})\rho,$
taking $\epsilon\to 0$, we obtain
$k(\theta)\geq k(\theta_{0})-d_{a}(\theta,\theta_{0}).$
So the function $k$ is 1-Lipschitz ∎
Now we can prove Proposition 4.11. Suppose that $S$ is not space-like, that
is, $S$ contains a light-like direction at an intersection with a singular
line. For example, suppose that $\mathscr{C}_{+}$ intersects the upper light-
like circle (the proof is analogue if $\mathscr{C}_{-}$ intersects the lower
light-like circle). The proof will follow from the following lemma:
###### Lemma 4.14.
If the curve $\mathscr{C}_{+}$ intersects the future light-like circle, then
$\displaystyle{\underset{\rho\to
0}{\liminf}\frac{u(\rho,\theta)}{\rho}\geq\eta}$ for all
$\theta\in[0,\alpha)$.
###### Proof.
As $\mathscr{C}_{+}$ intersects the upper time-like circle, there exist
$\theta_{0}\in[0,\alpha)$, and
$(\rho_{k})_{k\in\mathbb{N}}\subset\mathbb{R}_{>0}$ a strictly decreasing
sequence, converging to zero, such that
$\underset{k\to\infty}{\lim}\frac{u(\rho_{k},\theta_{0})}{\rho_{k}}=1.$
From (4), for a fixed $\eta>\widetilde{\eta}$, there exist
$k_{0}\in\mathbb{N}$ such that:
$\forall
k>k_{0},~{}u(\rho_{k},\theta)\geq\widetilde{\eta}\rho_{k}~{}\forall\theta\in[0,\alpha).$
As $S$ has vanishing mean curvature outside its intersections with the
singular locus, we can use a maximum principle. Namely, if a strictly future-
convex surface $\Sigma$ intersects $S$ at a point $x$ outside the singular
locus, then $S$ lies locally in the future of $\Sigma$ (the case is analogue
for past-convex surfaces). It follows that on an open set $V\subset
D_{\alpha}$, $\underset{x\in V}{\sup}u(x)=\underset{x\in\partial V}{\sup}u(x)$
and $\underset{x\in V}{\inf}u(x)=\underset{x\in\partial V}{\inf}u(x)$
Now, consider the open annulus $A_{k}:=D_{k}\setminus\overline{D}_{k+1}\subset
D_{\alpha}$ where $D_{k}$ is the open disk of center 0 and radius $\rho_{k}$.
As $S$ is a maximal surface, we can apply the maximum principle to $u$ on
$A_{k}$, we get:
$\inf_{A_{k}}u=\min_{\partial A_{k}}u\geq\widetilde{\eta}\rho_{k+1}.$
So, for all $\rho\in[0,r)$, there exists $k\in\mathbb{N}$ such that
$\rho\in[\rho_{k+1},\rho_{k}]$ and
(6) $u(\rho,\theta)\geq\widetilde{\eta}\rho_{k+1}.$
We obtain that, $\forall\theta\in[0,\alpha),~{}u(\rho,\theta)>0$ and so
$\displaystyle{\underset{\rho\to 0}{\liminf}\frac{u(\rho,\theta)}{\rho}\geq
0}$.
Now, suppose that
$\exists\theta_{1}\in[0,\alpha)\text{ such that
}\displaystyle{\underset{\rho\to
0}{\liminf}\frac{u(\rho,\theta_{1})}{\rho}=0},$
then there exists $(r_{k})_{k\in\mathbb{N}}\subset\mathbb{R}_{>0}$ a strictly
decreasing sequence converging to zero with
$\displaystyle{\underset{k\to\infty}{\lim}\frac{u(r_{k},\theta_{1})}{r_{k}}=0}.$
Moreover, we can choose a subsequence of $(\rho_{k})_{k\in\mathbb{N}}$ and
$(r_{k}){k\in\mathbb{N}}$ such that $r_{k}\in[\rho_{k+1},\rho_{k}[~{}\forall
k\in\mathbb{N}$.
This implies, by (5) that there exist $k_{1}\in\mathbb{N}$ such that
$\forall
k>k_{1},~{}u(r_{k},\theta)\leq(1-\widetilde{\eta})r_{k}~{}\forall\theta\in[0,\alpha).$
Now, applying the maximum principle to the open annulus
$B_{k}:=\mathscr{D}^{\prime}_{k}\setminus\overline{\mathscr{D}^{\prime}}_{k+1}\subset
D_{\alpha}$ where $\mathscr{D}^{\prime}_{k}$ is the open disk of center 0 and
radius $r_{k}$, we get:
$\underset{B_{k}}{\sup u}=\underset{\partial B_{k}}{\max
u}\leq(1-\widetilde{\eta})r_{k}.$
And so we get that for all $\rho\in[0,r)$ there exists $k\in\mathbb{N}$ with
$\rho\in[r_{k+1},r_{k}]$ and we have:
(7)
$u(\rho,\theta)\leq(1-\widetilde{\eta})r_{k}\leq(1-\widetilde{\eta})\rho_{k}.$
Now we are able to prove the lemma:
Take $\epsilon<1$, as
$\displaystyle{\lim\frac{u(\rho_{k},\theta_{0})}{\rho_{k}}=1}$, there exist
$k_{3}\in\mathbb{N}$ such that:
$\forall
k>k_{3},~{}u(\rho_{k},\theta_{0})\geq(1-\epsilon\widetilde{\eta})\rho_{k}.$
Using (7) we get:
(8) $(1-\epsilon.\widetilde{\eta}).\rho_{k}\leq
u(\rho_{k},\theta_{0})\leq(1-\widetilde{\eta})\rho_{k+1},~{}\text{and so
}~{}\frac{\rho_{k+1}}{\rho_{k}}\leq\frac{1-\epsilon.\widetilde{\eta}}{1-\widetilde{\eta}}.$
Now, as $\displaystyle{\lim\frac{u(r_{k},\theta_{0})}{r_{k}}=0}$, there exist
$N^{\prime}\in\mathbb{N}$ such that, for all $k$ bigger than $N^{\prime}$ we
have:
$u(r_{k},\theta_{1})\leq\epsilon.\widetilde{\eta}.r_{k}\leq\epsilon.\widetilde{\eta}.\rho_{k}.$
Using (6), we get:
(9) $\widetilde{\eta}.\rho_{k+1}\leq
u(r_{k},\theta_{0})\leq\epsilon.\widetilde{\eta}.\rho_{k},~{}\text{and so
}~{}\frac{\rho_{k+1}}{\rho_{k}}\leq\epsilon.$
But as $\epsilon<1$, the conditions (8) and (9) are incompatibles. ∎
Now, as the curve $\mathscr{C}_{-}$ does not cross $\mathscr{C}_{0,p}$ and is
contained in the de Sitter domain, we obtain
$l(\mathscr{C}_{-})<l(\mathscr{C}_{0,p})$ (where $l$ is the length). For
$D_{r}\subset D_{\alpha}$ the disk of radius $r$ and center 0 and
$A_{g}(u(D_{r}))$ the area of the graph of $u_{|D_{r}}$, we get:
$\displaystyle A_{g}(u(D_{r}))$ $\displaystyle\leq$
$\displaystyle\int_{0}^{r}l(\mathscr{C}_{-})\rho d\rho$ $\displaystyle<$
$\displaystyle\int_{0}^{r}l(\mathscr{C}_{0,p})\rho d\rho.$
The first inequality comes from the fact that
$\displaystyle{\int_{0}^{r}l(\mathscr{C}_{-})\rho d\rho}$ corresponds to the
area of a flat piece of surface with link $\mathscr{C}_{-}$ which is bigger
than the area of a curved surface (because we are in a Lorentzian manifold).
So, the local deformation of $S$ sending a neighborhood of $S\cap d$ to a
piece of totally geodesic disk orthogonal to the singular line would strictly
increase the area of $S$. However, as $S$ is a limit of a sequence of maximal
surfaces, such a deformation does not exist. So $\mathscr{C}_{S,p}$ cannot
cross the light-like circles.
### 4.4. Fourth step
###### Proposition 4.15.
The surface $S\hookrightarrow(M,g)$ of Proposition 4.7 is orthogonal to the
singular lines.
The proof uses a “zooming” argument: by a limit of a sequence of homotheties
and rescaling, we send a neighborhood of an intersection of the surface $S$
with a singular line to a piece of surface $V^{*}$ in the Minkowski space-time
with cone singularity (defined below). Then we prove, using the Gauss map,
that $V^{*}$ is orthogonal to the singular line and we show that it implies
the result.
###### Proof.
For $\tau>0$, define $\text{AdS}^{3}_{\theta,\tau}$ as the space
$(t,\rho,\varphi)\in(-\pi/2,\pi/2)\times\mathbb{R}_{\geq 0}\times[0,\theta)$
with the metric
$g_{\theta,\tau}=-dt^{2}+\cos^{2}(t/\tau)(d\rho^{2}+\tau^{2}\sinh^{2}(\rho/\tau)d\theta^{2}).$
Define the “zoom” map
$\begin{array}[]{lllll}\mathscr{Z}_{\tau}&:&\text{AdS}^{3}_{\theta}&\longrightarrow&\text{AdS}^{3}_{\theta,\tau}\\\
&&(\rho,\theta,t)&\longmapsto&(\tau\rho,\tau\theta,\tau t)\\\ \end{array}$
and the set
$K_{\tau}:=(K,g_{\theta,\tau}),$
where
$K:=\big{\\{}(t,\rho,\varphi)\in(-\pi/2,\pi/2)\times[0,1)\times[0,\theta)\big{\\}}$.
Let $p$ be the intersection of the surface $S\hookrightarrow(M,g)$ of
Proposition 4.7 with a singular line of angle $\theta$. By definition, there
exists an isometry $\Psi$ sending a neighborhood of $p$ in $M$ to a
neighborhood of $0:=(0,0,0)\in\text{AdS}^{3}_{\theta}$. Denote by $U$ the
image by $\Psi$ of the neighborhood of $p$ in $S$ and set
$U_{n}:=\mathscr{Z}_{n}(U)\cap K\subset\text{AdS}^{3}_{\theta,n}$ for
$n\in\mathbb{N}$. Note that the $U_{n}$ are pieces of space-like surface in
$\text{AdS}^{3}_{\theta,n}$ with vanishing mean curvature.
For all $n\in\mathbb{N}$, let
$f_{n}:[0,1]\times\mathbb{R}/\theta\mathbb{Z}\longrightarrow[-1,1]$ so that
$U_{n}=\text{graph}(f_{n})$. With respect to the metric
$d\rho^{2}+\sinh^{2}\rho d\varphi^{2}$ on
$[0,1]\times\mathbb{R}/\theta\mathbb{Z}$, the sequence
$(f_{n})_{n\in\mathbb{N}}$ is a sequence of uniformly bounded Lipschitz
functions with uniformly bounded Lipschitz constant and so converges
$\mathscr{C}^{0,1}$ to a function $f$.
###### Lemma 4.16.
Outside its intersection with the singular line, the surface
$V:=\text{graph}(f)\subset K$ is space-like and has everywhere vanishing mean
curvature with respect to the metric
$\textbf{g}_{\theta}:=-dt^{2}+d\rho^{2}+\rho^{2}d\theta^{2}.$
###### Proof.
As the surfaces $U_{n}\subset(K,g_{\theta,n})$ are space-like with everywhere
vanishing mean curvature (outside the intersection with the singular line),
they satisfy on $K\setminus\\{0\\}$ the following equation (see equation (3),
using the fact that $g_{ij}=0$ for $i\neq j$):
$\frac{1}{\sqrt{\det g_{\theta,n}}}\partial_{i}(\sqrt{\det
g_{\theta,n}}v_{n}g^{ii}_{\theta,n}\nabla_{i}f_{n})+\frac{1}{2}v_{n}\partial_{t}g^{ii}_{\theta,n}|\nabla_{i}f_{n}|^{2}-\frac{1}{2}v_{n}^{-1}g^{ii}_{\theta,n}\partial_{t}(g_{n})_{ii}=0.$
Recall that here, $\det g_{\theta,n}$ is the determinant of the induced metric
on $U_{n}\hookrightarrow(K,g_{\theta,n})$, $\nabla f_{n}$ is the gradient of
$f_{n}$ and $v_{n}:=\big{(}1-\|\pi^{*}\nabla f_{n}\|^{2}\big{)}^{-1/2}$ for
$\pi$ the orthogonal projection on $\\{(t,\rho,\varphi)\in K,~{}t=0\\}$.
As each $f_{n}$ satisfies the vanishing mean curvature equation, the same
argument as in the proof of Proposition 4.9 implies a uniform bound on the
norm of the covariant derivative of the gradient of $f_{n}$. It follows that
$f_{n}\overset{\mathscr{C}^{1,1}}{\longrightarrow}f.$
Moreover, one easily checks that on $K$,
$g_{\theta,n}\overset{\mathscr{C}^{\infty}}{\longrightarrow}\textit{{g}}_{\theta}$.
In particular $\det g_{\theta,n}$ and $v_{n}$ converge $\mathscr{C}^{1,1}$ to
$\det\textit{{g}}_{\theta}$ and $v$ (respectively). It follows that $f$ is a
weak solution of the vanishing mean curvature equation for the metric
$\textit{{g}}_{\theta}$, and so, a bootstrap argument shows it is a strong
solution. In particular, $V=\text{graph}(f)$ is a space-like surface in
$(K,\textit{{g}}_{\theta})$ with everywhere vanishing mean curvature outside
its intersection with the singular line. ∎
Consider on $\mathbb{R}^{2,1}$ the coordinates
$(t,\rho,\varphi)\in\mathbb{R}\times\mathbb{R}_{>0}\times[0,2\pi)$ so that the
metric $g$ of $\mathbb{R}^{2,1}$ writes
$g=-dt^{2}+d\rho^{2}+\rho^{2}d\varphi^{2}.$
The universal cover $\widetilde{\mathbb{R}^{2,1}\setminus d}$ of
$\mathbb{R}^{2,1}\setminus d$ (where
$d:=\\{(t,\rho,\varphi)\in\mathbb{R}^{2,1},~{}\rho=0\\}$ is the central axis)
admits natural coordinates
$(\widetilde{t},\widetilde{\rho},\widetilde{\varphi})\in\mathbb{R}\times\mathbb{R}_{>0}\times\mathbb{R}$.
In these coordinates, the projection
$\pi:\widetilde{\mathbb{R}^{2,1}\setminus
d}\longrightarrow\mathbb{R}^{2,1}\setminus d$
maps $\widetilde{\varphi}$ to the unique $\varphi\in[0,2\pi)$ with
$\widetilde{\varphi}\in\varphi+2\pi\mathbb{Z}$.
Let’s define
$r_{\theta}:\widetilde{\mathbb{R}^{2,1}\setminus
d}\longrightarrow\widetilde{\mathbb{R}^{2,1}\setminus d}$
by
$r_{\theta}(\widetilde{t},\widetilde{\rho},\widetilde{\varphi})=(\widetilde{t},\widetilde{\rho},\widetilde{\varphi}+\theta)$.
The quotient manifold
$\mathbb{R}^{2,1}_{\theta}:=\widetilde{\mathbb{R}^{2,1}\setminus d}/\langle
r_{\theta}\rangle$ inherits a singular Lorentz metric $g_{\theta}$ by pushing
forward the metric $\pi^{*}g$ of $\widetilde{\mathbb{R}^{2,1}\setminus d}$. We
call $\mathbb{R}^{2,1}_{\theta}$ with its metric the Minkowski space with cone
singularity of angle $\theta$.
The manifold $(K,\textit{{g}}_{\theta})$ of Lemma 4.16 with the central axis
$l$ removed is canonically isometric to an open subset of
$\mathbb{R}^{2,1}_{\theta}$. It follows that we can see the surface
$V^{*}:=V\setminus\\{l\cap V\\}$ as a space-like surface embedded in
$\mathbb{R}^{2,1}_{\theta}$.
Recall that the Gauss map of $V^{*}$ is the map associating to each point $x$
the unit future pointing vector normal to $V^{*}$.
###### Lemma 4.17.
The Gauss map is naturally identified with a map
$\mathscr{N}:V^{*}\longrightarrow\mathbb{H}^{2}_{\theta}$.
###### Proof.
Consider $\widetilde{V^{*}}\subset\widetilde{\mathbb{R}^{2,1}_{\theta}}$ the
lifting of $V^{*}\subset\mathbb{R}^{2,1}_{\theta}$. As $\widetilde{V^{*}}$ is
space-like, for each point $p\in\widetilde{V^{*}}$, the geodesic orthogonal to
$\widetilde{V^{*}}$ passing through $p$ either intersects the space-like
surface
$\widetilde{\mathbb{H}^{2*}}:=\\{(\widetilde{t},\widetilde{\rho},\widetilde{\varphi})\in\widetilde{\mathbb{R}^{2,1}_{\theta}},~{}\widetilde{t}=\sqrt{\widetilde{\rho}^{2}+1}\\}$
(which is a lifting of the hyperboloid $\mathbb{H}^{2}\subset\mathbb{R}^{2,1}$
with the point $(1,0,0)$ removed) or is not complete (namely, the geodesic
hits the boundary curve $\\{\widetilde{\rho}=0\\}$).
Denote by $\widetilde{\mathfrak{p}}\subset\widetilde{V^{*}}$ the set of points
so that the orthogonal geodesic is not complete. The Gauss map is thus
canonically identified with a map
$\widetilde{\mathscr{N}}:\widetilde{V*}\setminus\widetilde{\mathfrak{p}}\longrightarrow\widetilde{\mathbb{H}^{2*}}.$
It is clear that $\widetilde{\mathscr{N}}$ is equivariant with respect to the
action of $r_{\theta}$ so descends to a map
$\mathscr{N}:V^{*}\setminus\mathfrak{p}\longrightarrow\mathbb{H}^{2*}_{\theta}:=\widetilde{\mathbb{H}^{2*}}/\langle
r_{\theta}\rangle,$
where $\mathfrak{p}=\widetilde{\mathfrak{p}}/\langle r_{\theta}\rangle$. Note
that $\mathbb{H}^{2*}_{\theta}$ is isometric to the hyperbolic disk with cone
singularity (defined in the Introduction) with the center $0_{\theta}$
removed.
As $V^{*}$ is smooth, setting $\mathscr{N}(\mathfrak{p})=0_{\theta}$ gives a
smooth extension of $\mathscr{N}$ to a map
$\mathscr{N}:V^{*}\longrightarrow\mathbb{H}^{2}_{\theta}.$ ∎
###### Lemma 4.18.
The Gauss map $\mathscr{N}:V^{*}\longrightarrow\mathbb{H}^{2}_{\theta}$ is
holomorphic with respect to the complex structure associated to the reverse
orientation of $\mathbb{H}^{2}_{\theta}$.
###### Proof.
As $V^{*}$ has everywhere vanishing mean curvature, we can choose an
orthonormal framing of $TV^{*}$ such that the shape operator $B$ of $V^{*}$
expresses
$B=\begin{pmatrix}k&0\\\ 0&-k\end{pmatrix}.$
Denoting $h_{\theta}$ the metric of $\mathbb{H}^{2}_{\theta}$, we obtain that
$\mathscr{N}^{*}h_{\theta}=\textrm{I}(B.,B.)=k^{2}\textrm{I}(.,.),$
where I is the first fundamental form of $V^{*}$. That is $\mathscr{N}$ is
conformal and reverses the orientation and so is holomorphic with respect to
the holomorphic structure defined by the opposite orientation of
$\mathbb{H}^{2}_{\theta}$. ∎
###### Lemma 4.19.
The piece of surface $V^{*}\hookrightarrow\mathbb{R}^{2,1}_{\theta}$ is
orthogonal to the singular line.
###### Proof.
Fix complex coordinates $z:V^{*}\longrightarrow\mathbb{D}^{*}$ and
$w:\mathbb{H}^{2}_{\theta}\longrightarrow\mathbb{D}^{*}$. In these coordinates
systems, the metric $g_{V}$ and $h_{\theta}$ of $V^{*}$ and
$\mathbb{H}^{2}_{\theta}$ respectively express:
$g_{V}=\rho^{2}(z)|dz|^{2},~{}~{}h_{\theta}=\sigma^{2}(w)|dw|^{2}.$
Note that, as $\mathbb{H}^{2}_{\theta}$ carries a conical singularity of angle
$\theta$ at the center, $\sigma^{2}(w)=e^{2u}|w|^{2(\theta/2\pi-1)}$, where
$u$ is a bounded function.
Assuming $\mathscr{N}$ does not have an essential singularity at $0$, the
expression of $\mathscr{N}$ in the complex charts has the form:
$\mathscr{N}(z)=\frac{\lambda}{z^{n}}+f(z),\text{ where
}z^{n}f(z)\underset{z\to 0}{\longrightarrow}0$
for some $n\in\mathbb{Z}$ and non-zero $\lambda$.
Denote by $e(\mathscr{N})=\frac{1}{2}\|d\mathscr{N}\|^{2}$ the energy density
of $\mathscr{N}$. The third fundamental form of $V^{*}$ is thus given by
$\mathscr{N}^{*}h_{\theta}=e(\mathscr{N})g_{U}.$
Moreover, we have:
$e(\mathscr{N})=\rho^{-2}(z)\sigma^{2}(\mathscr{N}(z))|\partial_{z}\mathscr{N}|^{2}.$
If $n\neq 0$, we have
$|\partial_{z}\mathscr{N}|^{2}=C|z|^{2(n-1)}+o\left(|z|^{2(n-1)}\right),\text{
for some }C>0,$
and
$\sigma^{2}(\mathscr{N}(z))=e^{2v}|z|^{2n(\theta/2\pi-1)},\text{ for some
bounded }v.$
So we finally get,
$\mathscr{N}^{*}h_{\theta}=e^{2\varphi}|z|^{2(n\theta/2\pi-1)}|dz|^{2},\text{
where }\varphi\text{ is bounded.}$
For $n=0$, the same computation gives
$\mathscr{N}^{*}h_{\theta}=e^{2\varphi}|dz|^{2},\text{ where }\varphi\text{ is
bounded from above.}$
For $\mathscr{N}$ having an essential singularity, we get that for all
$n<0,~{}|z|^{n}=o\left(\rho^{2}(z)e(\mathscr{N})\right)$ and so
$\mathscr{N}^{*}h_{\theta}$ cannot have a conical singularity.
However, as the third fundamental form is the pull-back by the Gauss map of
$h_{\theta}$, it has to carry a conical singularity of angle $\theta$, that is
have the expression
$\textrm{III}=e^{2\Psi}|z|^{2(\theta/2\pi-1)}|dz|^{2},$
for some bounded $\Psi$. It implies in particular that $n=1$, and so
$\mathscr{N}(z)\underset{z\to 0}{\longrightarrow}0$, which means that $V^{*}$
is orthogonal to the singular line.
∎
The proof of Proposition 4.15 follows. For $\tau\in\mathbb{R}_{>0}$, let
$u_{\tau}\in T_{0}\text{AdS}^{3}_{\theta,\tau}$ be the unit future pointing
vector tangent to $d$ at $0=U_{\tau}\cap d$. For $x\in U_{\tau}$ close enough
to $0$, let $u_{\tau}(x)$ be the parallel transport of $u_{\tau}$ along the
unique geodesic in $U_{\tau}$ joining $0$ to $x$. Denoting by
$\mathscr{N}_{\tau}$ the Gauss map of $U_{\tau}$, we define a map:
$\psi_{\tau}(x):=g_{\theta,\tau}(u_{\tau}(x),\mathscr{N}_{\tau}(x)),$
where $g_{\theta,\tau}$ is the metric of $\text{AdS}^{3}_{\theta,\tau}$. Note
that, by construction, the value of $\psi_{\tau}(0)$ is constant for all
$\tau\in\mathbb{R}_{>0}$. As $U_{\infty}$ is orthogonal to $d$,
$\underset{\tau\to\infty}{\lim}\psi_{\tau}(0)=-1$ so in particular
$\psi_{1}(0)=-1$, that is the surface $S$ is orthogonal to the singular lines.
∎
## 5\. Uniqueness
In this section, we prove the uniqueness part of Theorem 1.4:
###### Proposition 5.1.
The maximal surface $S\hookrightarrow(M,g)$ of Proposition 4.1 is unique.
###### Proof.
Given a causal curve intersecting two space-like surfaces $S_{1}$ and $S_{2}$,
we denote by $l_{\gamma}(S_{1},S_{2})$ the causal length of $\gamma$ between
$S_{1}$ and $S_{2}$. Namely,
$l_{\gamma}(S_{1},S_{2})=\int_{t_{1}}^{t_{2}}\big{(}-g(\gamma^{\prime}(t),\gamma^{\prime}(t))\big{)}^{1/2}dt,$
where $\gamma(t_{i})\in S_{i},~{}i=1,2$.
Suppose that there exist two different maximal surfaces $S_{1}\text{ and
}S_{2}$ in $(M,g)$ where $S_{1}$ is the one of Proposition 4.1. Denote by
$\Gamma$ the set of time-like geodesics in $M$ and set
$C:=\sup_{\gamma\in\Gamma}\\{l_{\gamma}(S_{1},S_{2})\\}>0.$
Note that, from [BS09, Lemma 5.7], as $S_{1}\hookrightarrow(M,g)$ is contained
in the convex core, $C<\pi/2$. Consider
$(\gamma_{n})_{n\in\mathbb{N}}\subset\Gamma$ such that
$\underset{n\to\infty}{\lim}l(\gamma_{n})=C.$
###### Lemma 5.2.
The sequence of geodesic segments $(\gamma_{n})_{n\in\mathbb{N}}$ converges to
$\gamma\in\Gamma$.
###### Proof.
For $i=1,2$, denote by $x_{in}$ the intersection of $\gamma_{n}$ and $S_{i}$.
For $n\in\mathbb{N}$, choose a lifting $\widetilde{x_{1}}_{n}$ of $x_{1n}$ in
the universal cover $\widetilde{M}$ of $M$. This choice fixes a lifting of the
whole sequence $(x_{1n})_{n\in\mathbb{N}}$ and of
$(\gamma_{n})_{n\in\mathbb{N}}$, so of $(x_{2n})_{n\in\mathbb{N}}$. Note that
the sequence $(\widetilde{x_{1}}_{n})_{n\in\mathbb{N}}$ converges to
$\widetilde{x_{1}}\in\widetilde{S_{1}}\subset\widetilde{M}$ and, as the causal
cone of $\widetilde{x_{1}}$ intersects $\widetilde{S}_{2}$ in a compact set
containing infinitely many $\widetilde{x_{2}}_{n}$, the sequence
$(\widetilde{x_{2}}_{n})_{n\in\mathbb{N}}$ converges to $\widetilde{x}_{2}$
(up to a subsequence).
It follows that $\widetilde{x_{2}}$ projects to $x_{2}\in S_{2}$ and
$C=l_{\gamma}(S_{1},S_{2})$ where $\gamma$ is the projection of the unique
time-like geodesic joining $\widetilde{x}_{1}$ to $\widetilde{x}_{2}$. ∎
It is clear that $S_{1}$ and $S_{2}$ are orthogonal to $\gamma$ (if not, there
would exist some deformation of $\gamma$ increasing the causal length at the
first order).
For $i=1,2$, denote by $x_{i}$ the intersection of $\gamma$ and $S_{i}$, by
$P_{i}$ the (locally defined) totally geodesic plane tangent to $S_{i}$ at
$x_{i}$ and by $\pm k_{i}$ the principal curvatures of $S_{i}$ at $x_{i}$. We
can assume moreover that $k_{1}\geq k_{2}\geq 0$ and that $x_{2}$ is in the
future of $x_{1}$.
Let $u\in\mathscr{U}_{x_{1}}S_{1}$ (where $\mathscr{U}_{x}S_{1}$ is the unit
tangent bundle to $S_{1}$) be a principal direction associated to $k_{1}$. For
$\epsilon>0$, denote by $J$ the Jacobi field along $\gamma$ so that
$\left\\{\begin{array}[]{lll}J(0)&=&\epsilon u\\\
J^{\prime}(0)&=&0\end{array}\right.$
and set $\gamma_{\epsilon}:=\exp(J)\in\Gamma$ the deformation of $\gamma$
along $J$. It is clear that $\gamma_{\epsilon}$ is orthogonal to the piece of
totally geodesic plane $P_{1}$.
By definition of the curvature, we have
$l_{\gamma_{\epsilon}}(S_{1},S_{2})=l_{\gamma_{\epsilon}}(P_{1},P_{2})+(k_{1}-\kappa_{2})\epsilon^{2}+o(\epsilon^{2}).$
Here $\kappa_{2}$ is the curvature of $S_{2}$ at $x_{2}$ is the direction
$p_{\gamma}(u)$ where $p_{\gamma}:T_{x_{1}}S_{1}\longrightarrow
T_{x_{2}}S_{2}$ is the parallel transport along $\gamma$. In particular we
have $-\kappa_{2}\geq-k_{2}$ and so
$l_{\gamma_{\epsilon}}(S_{1},S_{2})\geq l_{\gamma_{\epsilon}}(P_{1},P_{2}).$
Moreover, in the proof of Proposition 4.5 we proved that the equidistant
surface in the future of a totally geodesic space-like plane in
$\text{AdS}^{3}_{\theta}$ is strictly future-convex (when the distance is less
than $\pi/2$). It follows that
$l_{\gamma_{\epsilon}}(P_{1},P_{2})>C,$
and we get a contradiction. ∎
## 6\. Consequences
### 6.1. Minimal Lagrangian diffeomorphisms
In this paragraph, we prove Theorem 1.3. Let $\Sigma$ be a closed oriented
surface endowed with a Riemannian metric $g$ and let $\nabla$ be the
associated Levi-Civita connection.
###### Definition 6.1.
A bundle morphism $b:T\Sigma\longrightarrow T\Sigma$ is Codazzi if
$d^{\nabla}b=0$, where $d^{\nabla}$ is the covariant derivative of vector
valued form associated to the connection $\nabla$.
We recall a result of [Lab92]:
###### Theorem 6.2 (Labourie).
Let $b:T\Sigma\longrightarrow T\Sigma$ be a everywhere invertible Codazzi
bundle morphism, and let $h$ be the symmetric $2$-tensor defined by
$h=g(b.,b.)$. The Levi-Civita connection $\nabla^{h}$ of $h$ satisfies
$\nabla^{h}_{u}v=b^{-1}\nabla_{u}(bv),$
and its curvature is given by:
$K_{h}=\frac{K_{g}}{\det(b)}.$
Given $g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ and
$\Psi:(\Sigma_{\mathfrak{p}},g_{1})\longrightarrow(\Sigma_{\mathfrak{p}},g_{2})$
a diffeomorphism isotopic to the identity, there exists a unique bundle
morphism $b:T\Sigma_{\mathfrak{p}}\longrightarrow T\Sigma_{\mathfrak{p}}$ so
that $g_{2}=g_{1}(b.,b.)$. We have the following characterization:
###### Proposition 6.3.
The diffeomorphism $\Psi$ is minimal Lagrangian if and only if
* i.
$b$ is Codazzi with respect to $g_{1}$,
* ii.
$b$ is self-adjoint for $g_{1}$ with positive eigenvalues.
* iii.
$\det(b)=1$.
We now prove Theorem 1.3:
Existence: Let $g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$. It
follows from Theorem 1.4 and the extension of Mess’ parametrization that we
can uniquely realize $g_{1}$ and $g_{2}$ as
$\left\\{\begin{array}[]{l}g_{1}=\textrm{I}((E+JB).,(E+JB).)\\\
g_{2}=\textrm{I}((E-JB).,(E-JB).),\end{array}\right.$
where $\textrm{I},~{}B,~{}J,~{}E$ are respectively the first fundamental form,
the shape operator, the complex structure and the identity morphism of the
unique maximal surface $S\hookrightarrow(M,g)$ and $(M,g)$ is an AdS convex
GHM space-time with particles parametrized by $(g_{1},g_{2})$.
Define the bundle morphism $b:T\Sigma_{\mathfrak{p}}\longrightarrow
T\Sigma_{\mathfrak{p}}$:
$b=(E+JB)^{-1}(E-JB).$
Note that, as the eigenvalues of $B$ are in $(-1,1)$, (from [KS07, Lemma
5.15]) the morphism $b$ is well defined. Moreover, we have
$g_{2}=g_{1}(b.,b.)$. We are going to prove that $b$ satisfies the conditions
of Proposition 6.3:
1. -
Codazzi: Denote by $D$ the Levi-Civita connection associated to I, and
consider the bundle morphism $A=(E+JB)$. From Codazzi’s equation for surfaces,
$d^{D}A=0$. From Proposition 6.2, the Levi-Civita connection $\nabla_{1}$ of
$\textrm{I}(A.,A.)$ satisfies:
$\nabla_{1u}v=A^{-1}D_{u}(Av).$
We get that $d^{\nabla_{1}}b=A^{-1}d^{D}(E-JB)=0$.
2. -
Self-adjoint:
$\displaystyle g_{1}(bx,y)$ $\displaystyle=$
$\displaystyle\textrm{I}\big{(}(E-JB)x,(E+JB)y\big{)}$ $\displaystyle=$
$\displaystyle\textrm{I}\big{(}(E+JB)(E-JB)x,y\big{)}$ $\displaystyle=$
$\displaystyle\textrm{I}\big{(}(E-JB)(E+JB)x,y\big{)}$ $\displaystyle=$
$\displaystyle\textrm{I}\big{(}(E+JB)x,(E-JB)y\big{)}$ $\displaystyle=$
$\displaystyle g_{1}(x,by).$
3. -
Positive eigenvalues: From [KS07, Lemma 5.15], the eigenvalues of $B$ are in
$(-1,1)$. So $(E\pm JB)$ has strictly positives eigenvalues and the same hold
for $b$.
4. -
Determinant 1:
$\displaystyle{\det(b)=\frac{\det(E-JB)}{\det(E+JB)}=\frac{1+\det(JB)}{1+\det(JB)}=1}$,
(because $\text{tr}(JB)=0$).
Uniqueness: Suppose that there exist
$\Psi_{1},\Psi_{2}:(\Sigma_{\mathfrak{p}},g_{1})\longrightarrow(\Sigma_{\mathfrak{p}},g_{2})$
two minimal Lagrangian diffeomorphisms. It follows from Proposition 6.3 that
there exists $b_{1},b_{2}:T\Sigma_{\mathfrak{p}}\longrightarrow
T\Sigma_{\mathfrak{p}}$ Codazzi self-adjoint with respect to $g_{1}$ with
positive eigenvalues and determinant 1 so that $g_{1}(b_{1}.,b_{1}.)$ and
$g_{2}(b_{2}.,b_{2}.)$ are in the same isotopy class.
For $i=1,2$, define
$\left\\{\begin{array}[]{ll}\textrm{I}_{i}(.,.)&=\frac{1}{4}g_{1}\big{(}(E+b_{i}).,(E+b_{i}).\big{)}\\\
B_{i}&=-J_{i}(E+b_{i})^{-1}(E-b_{i}),\\\ \end{array}\right.$
where $J_{i}$ is the complex structure associated to $\textrm{I}_{i}$.
One easily checks that $B_{i}$ is well defined and self-adjoint with respect
to $\textrm{I}_{i}$ with eigenvalues in $(-1,1)$. Moreover, we have
$b_{i}=(E+J_{i}B_{i})^{-1}(E-J_{i}B_{i}).$
Writing the Levi-Civita connection of $g_{1}$ by $\nabla$ and the one of
$\textrm{I}_{i}$ by $D^{i}$, Proposition 6.2 implies
$D^{i}_{x}y=(E+b_{i})^{-1}\nabla_{x}((E+b_{i})y).$
So we get:
$\displaystyle D^{i}B_{i}(x,y)$
$\displaystyle=(E+b_{i})^{-1}\nabla_{y}\big{(}(E+b_{i})By\big{)}-(E+b_{i})^{-1}\nabla_{y}\big{(}(E+b_{i})x\big{)}-B_{i}[x,y]$
$\displaystyle=(E+b_{i})^{-1}(\nabla(E+b_{i}))(x,y)$ $\displaystyle=0.$
And the curvature of $\textrm{I}_{i}$ satisfies
$K_{\textrm{I}_{i}}=-\det(E+JB_{i})=-1-\det(B_{i}).$
It follows that $B_{i}$ is traceless, self-adjoint and satisfies the Codazzi
and Gauss equation. Setting $\textrm{II}_{i}:=\textrm{I}_{i}(B_{i}.,.)$, we
get that $\textrm{I}_{i}$ and $\textrm{II}_{i}$ are respectively the first and
second fundamental form of a maximal surface in an AdS convex GHM manifold
with particles (that is,
$(\textrm{I}_{i},\textrm{II}_{i})\in\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$
where $\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$ is defined in Section 3).
Moreover, one easily checks that, for $i=1,2$
$\left\\{\begin{array}[]{l}g_{1}=\textrm{I}_{i}\left((E+J_{i}B_{i}).,(E+J_{i}B_{i}).\right)\\\
g_{2}=\textrm{I}_{i}\left((E-J_{i}B_{i}).,(E-J_{i}B_{i}).\right)\\\
\end{array}\right.$
It means that $(\textrm{I}_{i},\textrm{II}_{i})$ is the first and second
fundamental form of a maximal surface in $(M,g)$ (for $i=1,2$) and so, by
uniqueness,
$(\textrm{I}_{1},\textrm{II}_{1})=(\textrm{I}_{2},\textrm{II}_{2})$. In
particular, $b_{1}=b_{2}$ and $\Psi_{1}=\Psi_{2}$.
### 6.2. Middle point in $\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
Theorem 1.3 provides a canonical identification between the moduli space
$\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$ of singular AdS convex GHM
structure on $\Sigma_{\mathfrak{p}}\times\mathbb{R}$ with the space
$\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$ of maximal surfaces in germ of
singular AdS convex GHM structure (as defined in Section 3). By the extension
of Mess’ parametrization, the moduli space
$\mathscr{M}_{\theta}(\Sigma_{\mathfrak{p}})$ is parametrized by
$\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\times\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$
and by [KS07, Theorem 5.11], the space
$\mathscr{H}_{\theta}(\Sigma_{\mathfrak{p}})$ is parametrized by
$T^{*}\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$.
It follows that we get a map
$\varphi:\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\times\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})\longrightarrow
T^{*}\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}}).$
We show that this map gives a “middle point” in
$\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$:
###### Theorem 6.4.
Let $g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$ be two
hyperbolic metrics with cone singularities. There exists a unique conformal
structure $\mathfrak{c}$ on $\Sigma_{\mathfrak{p}}$ so that
$\Phi(u_{1})=-\Phi(u_{2})$
and $u_{2}\circ u_{1}^{-1}$ is minimal Lagrangian. Here
$u_{i}:(\Sigma_{\mathfrak{p}},\mathfrak{c})\longrightarrow(\Sigma_{\mathfrak{p}},g_{i})$
is the unique harmonic map isotopic to the identity and $\Phi(u_{i})$ is its
Hopf differential. Moreover,
$(g_{1},g_{2})=\varphi(\mathfrak{c},i\Phi(u_{1})).$
###### Proof.
Existence: For $g_{1},g_{2}\in\mathscr{T}_{\theta}(\Sigma_{\mathfrak{p}})$,
let
$\Psi:(\Sigma_{\mathfrak{p}},g_{1})\longrightarrow(\Sigma_{\mathfrak{p}},g_{2})$
be the unique minimal Lagrangian diffeomorphism isotopic to the identity. By
definition, the embedding
$\iota:~{}\Gamma:=\text{graph}(\Psi)\hookrightarrow(\Sigma_{\mathfrak{p}}\times\Sigma_{\mathfrak{p}},g_{1}\oplus
g_{2})$ is minimal so $\iota$ is a conformal harmonic map (see [ES64,
Proposition 4.B]).
It follows that the $i$-th projection
$\pi_{i}:(\Sigma_{\mathfrak{p}}\times\Sigma_{\mathfrak{p}},g_{1}\oplus
g_{2})\longrightarrow(\Sigma_{\mathfrak{p}},g_{i})$ restricts to a harmonic
map $u_{i}$ on $\iota(\Gamma)$. Moreover, as $\iota$ is conformal, we get
$\iota^{*}(g_{1}\oplus
g_{2})^{2,0}=0=u_{1}^{*}(g_{1})^{2,0}+u_{2}^{*}(g_{2})^{2,0}=\Phi(u_{1})+\Phi(u_{2}).$
Uniqueness: It is a direct consequence of the uniqueness part of Theorem 1.3.
Expression of $\varphi$: It follows from the extension of Mess’
parametrization (section 3.1) and Theorem 1.4 that the metrics $g_{1}$ and
$g_{2}$ can be uniquely expressed as
$\left\\{\begin{array}[]{l}g_{1}=\textrm{I}((E+JB).,(E+JB).)\\\
g_{2}=\textrm{I}((E-JB).,(E-JB).),\end{array}\right.$
where $\textrm{I},~{}B,~{}J,~{}E$ are respectively the first fundamental form,
shape operator, complex structure and identity morphism of the unique maximal
surface $S\hookrightarrow(M,g)$, and $(M,g)$ is an AdS convex GHM space-time
with particles parametrized by $(g_{1},g_{2})$.
An easy computation shows that
$g_{1}+g_{2}=2(\textrm{I}+\textrm{III}).$
As $S$ is maximal, the third fundamental form III of $S$ is conformal to I, so
$\mathfrak{c}:=[g_{1}+g_{2}]=[I],$
where $[h]$ denotes the conformal class of a metric $h$.
On the other hand, the embedding
$\iota:\Gamma=\text{graph}(\Psi)\hookrightarrow(\Sigma_{\mathfrak{p}}\times\Sigma_{\mathfrak{p}},g_{1}\oplus
g_{2})$ is conformal, so
$[\iota^{*}(g_{1}\oplus g_{2})]=[g_{1}+g_{2}]=[I]=\mathfrak{c}.$
By definition of the Hopf differential, there exists some strictly positive
function $\lambda$ so that with respect to the complex structure associated to
$\mathfrak{c}$, we have the following decomposition
$u_{1}^{*}g_{1}=\Phi(u_{1})+\lambda\iota^{*}(g_{1}\oplus
g_{2})+\overline{\Phi(u_{1})}.$
Note that in our case, the metrics $g_{1}$ and $g_{2}$ are normalized so that
$u_{i}=\text{id}$. Moreover, as $[\iota^{*}g_{1}\oplus g_{2}]=[\textrm{I}]$,
we have (for a different function $\lambda^{\prime}$)
$g_{1}=\Phi(u_{1})+\lambda^{\prime}\textrm{I}+\overline{\Phi(u_{1})}.$
Now we get
$g_{1}=\textrm{I}\big{(}(E+JB).,(E+JB).\big{)}=2\textrm{I}(JB.,.)+\textrm{I}+\textrm{III},$
in particular,
$\textrm{I}(JB.,.)=\Re\big{(}\Phi(u_{1})\big{)}.$
Let $\partial_{x},\partial_{y}\in\Gamma(TS)$ be an orthonormal framing of
principal directions of $S$. We have the following expressions in this
framing:
$B=\left(\begin{array}[]{ll}k&0\\\
0&-k\end{array}\right),~{}~{}J=\left(\begin{array}[]{ll}0&-1\\\
1&0\end{array}\right).$
Setting $dz\in\Gamma(T^{*}S\otimes\mathbb{C}),~{}dz:=dx+idy$ where $(dx,dy)$
is the framing dual to $(\partial_{x},\partial_{y})$, we obtain
$\Phi(u_{1})=-ikdz^{2}=k(dxdy+dydx)-ik(dx^{2}-dy^{2}).$
So
$\textrm{II}=\textrm{I}(B.,.)=\Re(i\Phi(u_{1})),$
and $(\mathfrak{c},i\Phi(u_{1}))$ is the maximal AdS germ associated to
$(M,g)$. ∎
## References
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* [ABBZ12] L. Andersson, T. Barbot, F. Béguin, and A. Zeghib. Cosmological time versus CMC time in spacetimes of constant curvature. Asian J. Math., 16(1):37–87, 2012.
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* [BBZ07] T. Barbot, F. Béguin, and A. Zeghib. Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on ${AdS}_{3}$. Geometriae Dedicata, 126(1):71–129, 2007.
* [Ber60] L. Bers. Simultaneous uniformization. Bull. Amer. Math. Soc., 66:94–97, 1960.
* [BS09] F. Bonsante and J.-M. Schlenker. AdS manifolds with particles and earthquakes on singular surfaces. Geom. Funct. Anal., 19(1):41–82, 2009.
* [CBG69] Y. Choquet-Bruhat and R. Geroch. Global aspects of the cauchy problem in general relativity. Communications in Mathematical Physics, 14(4):329–335, 1969.
* [ES64] J. J. Eells and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109–160, 1964.
* [Ger83] C. Gerhardt. $H$-surfaces in Lorentzian manifolds. Comm. Math. Phys., 89(4):523–553, 1983.
* [Gol88] W. Goldman. Topological components of spaces of representations. Invent. Math., 93(3):557–607, 1988.
* [GR15] J. Gell-Redman. Harmonic maps of conic surfaces with cone angles less than $2\pi$. Comm. Anal. Geom., 23(4):717–796, 2015.
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* [Hop51] H. Hopf. Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr., 4:232–249, 1951.
* [JMR11] T. D Jeffres, R. Mazzeo, and Y. A Rubinstein. Kähler-Einstein metrics with edge singularities. arXiv:1105.5216, 2011.
* [KS07] K. Krasnov and J.-M. Schlenker. Minimal surfaces and particles in 3-manifolds. Geom. Dedicata, 126:187–254, 2007.
* [Lab92] F. Labourie. Surfaces convexes dans l’espace hyperbolique et ${\mathbb{CP}}^{1}$-structures. J. London Math. Soc. (2), 45(3):549–565, 1992.
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* [Mes07] G. Mess. Lorentz spacetimes of constant curvature. Geom. Dedicata, 126:3–45, 2007.
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* [Sch93] R. M. Schoen. The role of harmonic mappings in rigidity and deformation problems. In Complex geometry (Osaka, 1990), volume 143 of Lecture Notes in Pure and Appl. Math., pages 179–200. Dekker, New York, 1993.
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|
arxiv-papers
| 2013-12-10T09:28:51 |
2024-09-04T02:49:55.243177
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "J\\'er\\'emy Toulisse",
"submitter": "Toulisse J\\'er\\'emy",
"url": "https://arxiv.org/abs/1312.2724"
}
|
1312.2737
|
# On the Evolution of BM Orionis
R. Priyatikanto [email protected] Prodi Astronomi Institut
Teknologi Bandung, Jl. Ganesha no. 10, Bandung,
Jawa Barat, Indonesia 40132
###### Abstract
BM Orionis, eclipsing binary system that located in the center of Orion Nebula
Cluster posses several enigmatic problems. Its intrinsic nature and nebular
environment make it harder to measure the physical parameters of the system,
but it is believed as Algol type binary where secondary component is pre-main
sequence star with larger radius. To assure this, several stellar models
($M_{1}=5.9$ M⊙ and $M_{2}=2.0$ M⊙) are created and simulated using MESA.
Models with rigid rotation of $\omega=10^{-5}$ rad/s exhibit considerable
similar properties during pre-main sequence stage, but 2.0 M⊙ at assumed age
of $\sim 10^{6}$ is $6.46$ times dimmer than observed secondary star. There
must be an external mechanism to fill this luminosity gap. Then, simulated
post-main sequence binary evolution of BM Ori that involves mass transfer
shows that primary star will reach helium sequence with the mass of $\sim 0.8$
M⊙ before second stage mass transfer. _Keywords : binary star, stellar and
binary evolution_
††journal: Jurnal Matematika dan Sains
## 1 Introduction
Orion (known as _Waluku_ in Javanese) is special constellation for people
around equator and becomes an icon for equatorial heaven. In the heart of this
constellation, there is a young embedded Orion Nebula Cluster (ONC) with its
exotic trapezium stars ($\theta^{1}$ Ori) as an evident of mass segregation in
young cluster [11]. Aggregates of relatively more massive stars in the center
form hierarchical multiple stellar system as already observed clearly [5].
Among 5 prominence systems ($\theta^{1}$ Ori A – E), $\theta^{1}$ Ori B is
interesting with 5 members and Algol type eclipsing binary as its
parent/central, known as BM Orionis.
BM Ori (HD37021) which located at
$\alpha_{2000}=05^{\text{h}}35^{\text{m}}16.117^{\text{d}}$ and
$\delta_{2000}=-05^{\circ}23^{\prime}6.86"$ is eclipsing binary with B V star
as primary and larger but less massive star as secondary component [19, 24].
Eventhough study about this system converges toward one conclusion about the
primary component, physical parameters of the secondary are still uncertain.
Light curve with shallow secondary minima and nearly unresolved spectra keeps
BM Ori in mystery. Several models have been proposed to explain the observed
phenomena, such as secondary star with flatten disk [10] and disk shell [12]
or spherical shell [22] surrounding the primary.
Based on its light curve, it’s obvious that BM Ori is detached system [2], but
the interaction between components will influence future evolution of the
binary and also the stability of multiple system as a whole. Primary component
will soon leave main sequence open the mass transfer channel toward its
couple. Stellar evolution in this close binary system will be studied.
This article is devided into 5 sections. The conducted observations toward BM
Ori and its environment are reviewed in Section 2. Section 3 explains the
orbital and physical properties of both primary and secondary component.
Section 4 describes the main tool and the results of the simulation. Section 5
gives the closing remarks.
Figure 1: Left panel shows radial velocity curve for primary (blue) and
secondary component (red) compiled from literatures [7, 19, 13]. Right panel
shows light curve from Hall & Garrison (1969) and Antonkhina (1989).
## 2 Observations of BM Ori
Photometric observation of BM Ori and its vicinities have been done
intensively since mid-1900s using Johnson UBV filter [9, 3]. Variability of
this eclipsing binary has period of $\sim 6.5$ days and amplitude of $\sim
0,7$ mag. These early photometric study also concluded that secondary
component is 3 times larger that the primary.
Spectroscopic observation of this system has its own challanges which come
from intrinsic properties of the binary and the nebular environment of ONC.
Although successfully measured radial velocity of primary component, Johnson
(1965) and Doremus (1970) haven’t identify the spectra of secondary component.
Several years later, Popper & Plavec (1976) used D-line ($\lambda\sim
4050\aa$) to measure secondary star’s kinematics. Next decade, Ismailov (1988)
analysed He I lines that represents primary component and metal lines (Fe I,
Ca II and Mg II) that represents the secondary. Emission profile that was
found among the metal lines may indicate the presence of a shell.
Spectroscopic observations during eclipse have also been carried out to gain
microturbulence speed and abundances of some important metals [24, 23]. On the
other hand, the abundace of the primary star just yet determined
qualitatively. The only conclusion was that primary star has Helium abundance
comparable to the Sun.
In addition to photometry and spectroscopy, direct imaging and astrometry
observations to clarify multiplicity of trapezium stars have been conducted
[5]. From those observations, $\theta^{1}$ Ori B is confirmed as multiple
stellar system with (at least) 5 member, BM Ori ($\theta^{1}$ Ori B1) becomes
the central body, surrounded by B2, B3 and less massive B4.
## 3 Properties of BM Ori
### 3.1 Orbital Properties
BM Ori located in the heart of ONC, 418 pc away from the Sun [16]. This
eclipsing binary has nearly circular orbit with period of 6.470525 days [9]
and orbital separation of $\sim 30$ R⊙. Here is the ephemeris of BM Ori:
$T_{\text{min}}=JD2440265.343+6.470525-E$ (1)
This eclipsing binary with nearly $90^{\circ}$ inclination is believed as
detached system which experiences partial eclipse. The _Roche lobe_ filling
factor of this system is 0.16 and 0.90 for primary and secondary component
respectively [2]. Table 1 summarizes orbital parameters from literatures.
Table 1: Orbital parameters of BM Ori from literatures that consist of orbital separation ($a$), eccentricity ($e$), inclination ($i$) and mass ratio ($q$). Reference | $a$(R⊙) | $e$ | $i$ (∘) | $q$ | metode
---|---|---|---|---|---
Struve & Titus (1994) | … | $0.14$ | … | … | spectroscopy
Parenago (1957) | $50$ | $0.14$ | $87.7$ | $0.25$ | spectroscopy
Hall & Garrison (1969) | $32\pm 3$ | … | $83.8\pm 2.1$ | $0.52$ | phtometry
Popper & Plavec (1976) | $29.0\pm 1.5$ | … | $83\pm 4$ | $0.30$ | spectroscopy
AlNaimy & AlSikab (1983) | $29$ | … | … | $0.55$ | phtometry
Ismailov (1988) | $29$ | $0.15$ | … | $0.37$ | spectroscopy
Table 2: Physical parameters of BM Ori’s primary component from literatures, which are Hall & Garrison (1969) [HG69], Popper & Plavec (1976) [PP76], Antonkhina et al. (1989) [An89]. Parameter | HG69 | PP76 | An89 | Adopted
---|---|---|---|---
MK class | B2–B3 | B3V | B2V | B3V
$(B-V)_{0}$ | $0.08$ | $-0.21\pm 0.02$ | … | $-0.21$
$M_{V}$ | $0.7$ | $-0.8\pm 0.3$ | … | $-0.70$
$T_{\text{eff}}$ | $18700-22000$ | $18700$ | $22000$ | $18700$
$R_{\text{eq}}/$R⊙ | $2.5$ | $3.0\pm 0.4$ | $2.1$ | $3.0$
oblateness | … | $0.87$ | $1.00$ | $1.0$
$M/$M⊙ | $5.4$ | $5.9\pm 0.8$ | $5.9\pm 0.9$ | $5.9$
abundance | … | … | … | $Z=0.02$
$v_{\text{rot}}[km/s]$ | … | $300$ | … | $300$
Table 3: Physical parameters of the secondary compiled from literatures such as Hall & Garrison (1969) [HG69], Popper & Plavec (1976) [PP76], Antonkhina et al. (1989) [An89], Vitrichenko & Plachinda (2000) [VP00], Vitrichenko & Klochkova (2001) [VK01]. Parameter | HG69 | PP76 | An89 | VP00 | Adopted
---|---|---|---|---|---
MK class | A1 | A5–F0 | A3–A4 | G2III | A5-A6
$(B-V)_{0}$ | $0.07$ | $0.17\pm 0.10$ | … | … | $0.17$
$M_{V}$ | $-1.1$ | $0.2\pm 0.4$ | … | … | $-0.55$
$T_{\text{eff}}$ | $9400$ | $7200-8200$ | $9020$ | 5740 | $8000$
$R_{\text{eq}}/$R⊙ | $8.5$ | $7.0\pm 0.1$ | $8.0$ | $2.5$ | $7.0$
oblateness | … | $0.57$ | $0.74$ | … | $1.0$
$M/$M⊙ | $2.8$ | $1.8\pm 0.4$ | $2.15\pm 0.4$ | $2.5$ | $2.0$
abundance | … | … | … | $[M/H]=-0.5^{\text{dex}}$ | $Z=0.02$
$v_{\text{rot}}[km/s]$ | … | $50-100$ | … | $60$ | $60$
### 3.2 Primary Component
Primary component has visual magnitude of $V=8.37$ and intrinsic color of
$(B-V)_{0}=-0.21$, $(U-B)_{0}=-0.80$ after correction using $E_{B-V}=0.30$.
Assuming distance modulus of 8.2, Popper & Plavec (1976) derived absolute
magnitude of $M_{V}=-0.80$ and confirmed that the primary is B2-3 type main
sequence star. More accurate distance ($d=418$ pc) from parallax measurement
[16] and assuming $A_{V}=3.0E_{B-V}$, the absolute magnitude of this star can
be recalculated.
$M_{V}=m_{V}+5-5\log(d)-A_{V}$ (2)
Then, using $T_{\text{eff}}=18700$ K from spectroscopic observations, it has
bolometric correction of $BC=1.94$ [schmidt82]. Implying:
$\displaystyle M_{\text{bol}}$ $\displaystyle=M_{V}-BC=-2.64$
$\displaystyle\log(L_{1}/L_{\odot})$ $\displaystyle=2.94$ $\displaystyle
R_{1}/R_{\odot}$ $\displaystyle=2.85$
This derived value is in agreement with derived values from light curve
analysis which range from 2.1R⊙ [2] and 3.0R⊙ or $3.4\pm 0.6$ which is derived
from observed survace gravity [19].
As expected, the primary component is a fast rotating star with velocity of
$250-300$ km/s [19], but still below its critical velocity ($\sim 1000$ km/s).
This B star also blows stellar wind, responsibles with observed He I and metal
emission lines [13]. X-ray source that coincides with BM Ori (COUP 778) can be
explained by shock wind mechanism related to that stellar wind [20]. Tabel 2
summarize physical parameters of the primary.
### 3.3 Secondary Component
Secondary component of BM Ori is so hard to be observed that its physical
parameters is not well determined. This star is believed to be pre-main
sequence star with $V=8.52$ that experiences gravitational contraction. Popper
& Plavec (1976) got $M_{V}=0.2\pm 0.4$ and $(B-V)_{0}=0.17_{-0.03}^{+0.10}$,
while Hall & Garrison (1969) got $(U-B)_{0}=-0.02$ which indicates ultraviolet
excess of $\delta_{U-B}=0.7$ as observed in another pre-main sequence star.
Radio observation and detection of non-thermal emission from BM Ori may
related to flare activity of the secondary [8].
Color index of $(B-V)_{0}=0.17$ corresponds to main sequence star with
temperature of $7200-8200$ K (A5-F0), but Antonkhina et al. (1989) derived
$T_{\text{eff}}=9020$ K according the light curve while Vitrichenko &
Plachinda (2000) got $T_{\text{eff}}=5740$ K based on the surface gravity. For
secondary, determination of surface temperature and spectral class is not easy
since the spectrum is not well-resolved from the primary.
Star’s magnitude and luminosity can also be recalculated assuming
$T_{\text{eff}}\approx 8000$ K (average value from literatures) and
$BC=-0.14$. It yieds:
$\displaystyle M_{V}$ $\displaystyle=-0.55$ $\displaystyle M_{\text{bol}}$
$\displaystyle=-0.68$ $\displaystyle\log(L_{2}/L_{\odot})$
$\displaystyle=2.17$ $\displaystyle R_{2}/R_{\odot}$ $\displaystyle=6.35$
From the orbit, the secondary component has mass of $\sim 2$ M⊙ and radius of
$\sim 7$ R⊙. It rotates with velocity of $50-100$ km/s, much slower compared
to its pair [19, 24]. Tidal attraction from the primary causes higher
oblateness of this star.
Figure 2: Evolutionary track of primary component with mass of $M_{1}=5.9$ M⊙
(blue) and secondary with mass of $M_{2}=2.0$ M⊙ (red). Thick and thin lines
represent pre and post main sequence stages respectively, while dotted lines
belong to rotating models. Filled circles mark the position of assumed model
or the best fit toward observed parameters (circles).
## 4 Structure and Evolution of BM Ori
In this study, the structure and evolution of the stars are simulated using
Module for Experiments in Stellar Astrophysics (MESA, [18]). This program is
developed according Eggleton’s code, adopting updated physical data. This code
is constructed using Fortran95 which can be compiled in multi-processor
device. Various case of stellar evolution, ranging from pre-main sequence
evolution to final collapse of a star can be simulated using this code. Binary
evolution that involves mass transfer can also be treated.
Three different models are generated and evolved with MESA. Each model
consists of 5.9 M⊙ primary and 2.0 M⊙ secondary component with metalicity of
$Z=0.02$ (solar metalicity). The first two models start from Hayashi track
with enormous size and luminosity, but with different rotation nature: one
without rotation and the other rotates with $\omega_{i}=1.5\times 10^{-8}$
rad/s. This value of angular velocity is choosen in order to make rotating
main sequence star with observed rotation velocity ($v_{1}\approx 300$ km/s
and $v_{2}\approx 50$ km/s). In these two models, no binary interaction
calculated. The last model is the binary evolution model starts from Zero Age
Main Sequence (ZMAS) through advanced evolution including mass transfer.
Table 4: Global parameter of the models without mass transfer at $\log(t)\approx 6.20$. Parameter | primary | secondary
---|---|---
| rotating | non-rotating | rotating | non-rotating
$\log(t)$ | $6.200$ | $6.200$ | $6.230$ | $6.209$
$M/$M⊙ | $5.900$ | $5.900$ | $2.000$ | $2.000$
$\log(L/L_{\astrosun})$ | $2.970$ | $3.021$ | $1.366$ | $1.366$
$R/$R⊙ | $3.083$ | $2.978$ | $2.547$ | $2.547$
$T_{\text{eff}}$ [K] | $18174$ | $19045$ | $7943$ | $7962$
$\log(g)$ | $4.230$ | $4.260$ | $3.927$ | $3.930$
### 4.1 Evolution Toward Main Sequence
Departing from Hayashi track, both primary and secondary star contracts to
attain new hydrodynamic equilibrium as main sequence star, but with different
time scale (less massive star spends more time). During this evolutionary
stage, there is no significant difference between non-rotating and rotating
models. This result has similar trend compared to rotating model of Martin &
Claret (1996), though their model has smaller mass and faster rotation.
Rotating stars have different gravity potential which may influence their
internal structure. This is more clearly demonstrated in the post-main
sequence evolution.
### 4.2 Comparison with Observed Properties
To make comparison between the model and the observed propertis is not
straightforward process since the age of both stars are not precisely
determined. Previous study gave a possible range of $10^{5}-10^{6}$ years.
Primary component has already reached main sequence at age of $8\times 10^{5}$
years. Its structure does not change much during main sequence stage that
lasts until the age of $\sim 10^{7}$ years.
The present age of secondary component is harder to approximate because of its
pre-main sequence nature. But, as binary component with nearly circular orbit,
it is more likely that secondary star formed almost in the same epoch,
together with its pair. Although theory of binary star formation doesn’t
demand simultaneous formation, observation bring evidents toward coevality [4,
17]. Then age range of $10^{6}-10^{7}$ years for both components is reasonable
in order to compare model and observation.
As plotted in HR diagram (Figure 2), main sequence of 5.9 M⊙ model is rather
fit to measured temperature and luminosity of the primary component. Both
rotating and non-rotating model show almost similar properties, but rotating
model has a bit smaller effective temperature (see Table 4).
On the other hand, evolutionary track of 2 M⊙ models are located below the
observed properties of secondary star. At the age of $1.6\times 10^{6}$ or
$\log(t)=6.2$ both rotating and non-rotating model have the highest
luminosity, but still 6.46 dimmer. Difference between these two luminosity
demands external processes to occure and add the stellar luminosity.
Accumulative reflection from primary component and heating by stellar wind may
be sufficiently cover gap [12]. Later process is expected to give more
contribution.
Figure 3: HR Diagram (left) and mass-radius plot (left) of the donor (blue)
and accretor (red). Dashed line in the left panel marks ZAMS while dashed lin
in the left marks Roche lobe radius of each star. Letter A–H marks
evolutionary stages as described in the text.
### 4.3 Post-Main Sequence Evolution
As relatively close binary system, BM Ori experiences complicated evolution
involved mass transfer. After leaving main sequence at age of $\sim 10^{7}$
years, primary component starts to expand, fills its Roche lobe and initiates
mass transfer. This post-main sequence mass transfer is an example of case B
of Kippenhahn & Wiegert (1967).
In this model, primary ($M_{1}=5.9$ M⊙) and secondary star ($M_{2}=2.0$ M⊙)
are in the main sequence with similar age. Both stars orbit the center of mass
in circular orbit with orbital period of $P=6.47$ days. And here are
evolutionary stages experienced by the system:
1. 1.
Primary component leaves main sequence when hydrogen fuel in the center is
exhausted. The core shrinks while the envelope expands makes the star fills
its Roche lobe (stage B in Fig 3).
2. 2.
Non-conservative mass transfer occurs, sometimes accreting star (secondary)
gains same amount of mass transfer from the donor (primary). In this model,
mass transfer rate is kept to be constant at $\dot{M}=10^{-5}$ M⊙/tahun,
almost similar to the model of De Greve & de Loore (1976) for intermediate
mass system.
3. 3.
Expansion rate of the primary is overwhelmed such that the envelope has much
larger radius compared to the orbital separation, common envelope is
established. At the same time, star ignites helium burning (stage D).
4. 4.
Donor star reaches a new equilibrium as helium star with smaller size and mass
($M_{1}^{\prime}=0.8$ M⊙). On the other hand, accretor becomes more massive
($M_{2}^{\prime}=4.4$ M⊙) while the orbit becomes larger ($a^{\prime}=40.3$ R⊙
and $P=13.00$ days).
5. 5.
After 10 Gyr, helium star leave its stable condition and expands again. Second
stage of mass transfer is initiated (stage G) sets aside smaller mass (stage
H) when the simulation is terminated.
## 5 Closing Remarks
In this study, previous observations and studies about BM Ori as an
interesting eclipsing binary in the heart of Orion Nebula Cluster are
reviewed. However, physical parameters of secondary component are note well-
determined. Standard model with assumed parameters of $M_{1}=5.9$ M⊙ and
$M_{2}=2.0$ M⊙ does fit with primary component but not for secondary. There
must be an external mechanism occurs around the secondary to fill the
luminoasity gap.
Simulated binary evolution after main sequence stage shows that mass transfer
will transform the donor star to become helium star with stripped envelope.
During this stable stage, total mass of the system is around 5.2 M⊙, much
lower that initial total mass. Beside that, orbital parameters are change
toward larger separation of $a^{\prime}=40$ R⊙ and shorter period of
$P^{\prime}=13$ days. This condition undoubtfully influence the stability of
$\theta^{1}$ Ori B multiple system. Further dynamical analysis need to be done
to assess this.
## References
## References
* [1] AlNaimy & Alsikab. 1984, Ap&SS.103,115A
* [2] Antonkhina et al. 1989, SvAL, 15, 362A
* [3] Arnold, C.N. & Hall, D.S. 1976, AcA, 26, 91A
* [4] Brandner, W. & Zinnecker, H. 1997, A&A, 321, 220
* [5] Close, L.M. et al. 2012, ApJ, 749, 180C
* [6] De Greve, J.P. & De Loore, C. 1976, Ap&SS, 43, 35D
* [7] Doremus, C. 1970, PASP, 82, 745D
* [8] Felli, M., Churchwell, E., Wilson, T.L. and Taylor, G.B. 1993, A&AS, 98, 137F
* [9] Hall & Garrison. 1969, PASP, 81, 771H auschildt, P. H., Allard, F., Ferguson, J., Baron, E., & Alexander, D. 1999b, ApJ, 525, 871
* [10] Hall, D.S. 1971, IAU Colloquium No. 15, Bamberg Variable Star Colloquium, p. 217
* [11] Hillendbrand, L.A. 1997, AJ, 113, 1733H
* [12] Huang, S. 1975, APJ, 195, 127
* [13] Ismailov. 1988, SvAL, 14, 138I
* [14] Kippenhahn, R., Kohl, K. & Weigert, A. 1967, Z, Astrophys, 66, 58
* [15] Martin, E. L. & Claret, A. 1996, 306, 408
* [16] Menten, K. M., Reid, M. J., Forbrich, J. & Brunthaler, A. 2007, A&A, 474, 515M
* [17] Palla, F. & Stahler, S. W. 2001, ApJ, 553, 299
* [18] Paxton, B., Bildsten, L., Dotter, A., Herwig, F., Lasaffre & P., Timmes, F. 2011, AJSS, 192, 3
* [19] Popper, D. & Plavec, M. 1976, ApJ, 205, 462P
* [20] Stelzer, B., Flaccomio, E., Montmerle, T., Micela, G., Sciortino, S., Favata, F., Preibisch, T. & Feigelson, E. D. 2005, ApJS, 160, 557S
* [21] Struve, O. & Titus, J. 1944, ApJ, 99, 84S
* [22] Vitrichenko, E.A. 1998, Pisma Astron. Zh., 24, 708
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* [24] Vitrichenko & Plachinda, 2000, AstL, 26, 290V
|
arxiv-papers
| 2013-12-10T10:04:12 |
2024-09-04T02:49:55.255322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rhorom Priyatikanto",
"submitter": "Rhorom Priyatikanto",
"url": "https://arxiv.org/abs/1312.2737"
}
|
1312.2742
|
# Suryakala-Nusantara: Documenting Indonesian Sundials
R. Priyatikanto [email protected] Prodi Astronomi Institut
Teknologi Bandung, Jl. Ganesha no. 10, Bandung,
Jawa Barat, Indonesia 40132
###### Abstract
Sundial is the ancient or classic timekeeper device, especially prior to the
invention of mechanical clock. In the classical Islamic civilization, the
daily movement of the Sun becomes main indicator of praying time, which can be
deduced using sundial. This kind of device probably permeated to Indonesia
during the Islamic acculturation. Since then, the development of astronomical
knowledge, technology, art and architectural in classical Indonesia are
partially reflected into sundial. These historical attractions of sundial
demand comprehensive documentation and investigation of Indonesian sundial
which are rarely found in the current literatures. The required spatial and
temporal information regarding Indonesian sundial can be collected by general
public through citizen science scheme. This concept may answer scientific
curiosity of a research and also educate the people, expose them with science.
In this article, general scheme of citizen science are discussed, its
application for sundial study in Indonesia is proposed as Suryakala-Nusantara
program.
Keyword: sundial – astronomy outreach – citizen science
††journal: Proceeding of the Indonesian Astronomy Ascossiation (HAI) Seminar
2013
## 1 Introduction
A sundial, in its broadest sense, is any device that uses the motion of the
apparent sun to cause a shadow or a spot of light to fall on a reference scale
indicating the passage of time. Although the oldest sundial was created by
Egyptians in 1500 BC [7], the existence of this device in Islamic civilization
came later. Muslim inherited sundial system from the Greeks, who have strong
tradition of sundial, during the conquering era around the seventh century.
Sundials are then developed and utilized for religious purpose, to indicate
the time of midday (_zuhr_) and afternoon (_ashr_) prayer. A number Islamic
scientist, such as _Habash al-Hasib_ and _Thabit ibn Qurra_ , were born and
continued the development of sundial knowledge and design in medieval ages
[1].
Figure 1: Infographic of a sundial that lays in the front rear of Kauman
Mosque in Ungaran, Central Java. It is severly damaged with no dial/gnomon to
cast a shadow. From the existing features, it can be deduced that this
equatorial sundial is a special sundial for praying-timekeeping only.
(Photographed by the author)
Islamic concepts and its following civilization were then come to Indonesia in
fourteenth century mainly through trading activities. Since then, the Islamic
ideology and knowledge were widely spread in the Indonesian archipelago. In
line with this process, sundial as timekeeper was brought to the archipelago.
A large number of mosques are built with different architectural identitiy;
some of them are accompanied by sundial.
Unfortunately, the study of the sundial establishment and utilization in
Indonesian Islamic civilization has not been precisely conducted and even
overshadowed by the studies of the mosque architecture which are more common
in literatures. Adequate documentation regarding this timekeeper is hard to
find. We do not really know which one is the oldest sundial in Indonesia or
even where to find them. Because of its role as timekeeper has been altered by
the presence of mechanical and digital clock, sundial gains less attention
these days. There are not a few numbers of sundials that have been weathered
by time, one of which is a sundial or bancet in Masjid Kauman Ungaran, Jawa
Tengah (Figure 1).
In contrary, similar documentation and research are conducted overseas. Among
them are Kim et al. [5] and Kim & Lee [6] who documented numbers of Korean
sundials which are established in 14th century. The main points of these
studies are to understand how the sundials work and to restore some of the
defective sundials which they considered as valuable relics. Ferrari [4], who
registers himself as _North America Sundial Society_ (NASS), wrote about
medieval sundials constructed by Ottoman in Northern Italy. This kind of
research provides an outlook regarding astronomy knowledge, technology, and
architectural development of the people at that time. Beside that, the
spreading pattern of sundial designs become an important issue to be
addressed.
The urgency of sundial documentation and investigation can be responded by
utilization of citizen science approach. General public can be actively
involved in the documentation of sundials which are distributed on a vast area
of Indonesian archipelago and a broad range of time.
This article discusses about the implementation of citizen science scheme in
the form of Suryakala-Nusantara. The discussion is limited to the realm of
Islamic sundial, though the scope of Suryakala-Nusantara program can be
expanded, involving more ancient sundials. By involving as many as people and
exploiting the flexibility of the information sharing through the internet,
Suryakala-Nusantara aims to record and document the existence of a sundial in
Indonesia. The resulting data/documentation can be used for further studies in
many fields such as history, art, architecture, and ethno astronomy.
General explanation about citizen science approach will be given in Section 2,
followed by the implementation of the concept into Suryakala-Nusantara program
in Section 3, while its on-line database scheme is mentioned in Section 4.
Several issues and opportunities will be discussed in the last section (5).
## 2 Leaning on Citizen Science
Citizen science can broadly be defined as the involvement of general public or
volunteers or non-expert in scientific activities [3]. This kind of
collaboration in astronomy has an equally impressive history. In 1874 the
British government funded the Transit of Venus project to measure the Earth’s
distance to the sun engaging the existing amateur astronomers to support data
collection all over the globe. In 1932, British Trust for Ornithology was
founded in order to harness the efforts of amateur birdwatchers for the
benefit of science and nature conservation [9].
However, a new era of citizen science is just rising in the expansion of
today’s information systems [9]. Public involvement through citizen science
can be categorized into two leading branches, namely data collection and data
processing. Numbers of biodiversity researches rely upon citizen science for
data collection, for example e-Bird111www.ebird.com which is endorsed by _The
Cornell Lab of Ornithology_ (CLO) in 2002. On the other hand, _Galaxy Zoo_
222www.galaxyzoo.org,startedin2007 becomes an example of citizen science
project with data processing modus, public are encouraged to access Sloan
Digital Sky Survey (SDSS) image of galaxies and do some classifications. In
Indonesia, a birdwatcher community called _Indonesian Ornithological Union_
333kukila2004.wordpress.com becomes the one example of established citizen
science project with several reports/publications produced.
Research project may gain benefits by adopting citizen science scheme since it
encompasses a broad scope of space and time. Moreover, citizen science is not
only serves positive impact on the scientific research and discovery, but also
on the science literacy of the general public [2]. Engagement of the public in
citizen-science-based program increases the awareness of science issues and
development or the scientific processes that shape the whole human knowledge.
However, this special approach requires a slightly different design when
compared to regular research plan. Bonney et al. [2] proposed a basic model
for the development of citizen science programs, especially for massive data
collection. The development of this model is mainly based on the CLO
activities, converges to the following steps:
* 1.
Choose scientific questions for which data collection relies on basic skills
of the common participants. More training or supporting materials are required
to prepare the participants for higher level involvement.
* 2.
Form a team of scientist, educator, engineer, or evaluator team. The whole
chain-process of receiving, archiving, analysing, visualizing, and
disseminating project data request a team of multi-discipline people.
* 3.
Develop, test, and refine protocols, data forms, and educational support
materials. These factors are needed to ensure the quality of the publicly
collected data. Right destination is reached through the right way.
* 4.
Recruit participants through publication via various media such as printed
media, e-mail, social media, or workshop at conferences of potential
participants or their leaders.
* 5.
Train participants to provide them with sufficient knowledge and skills for
data collection or analysis.
* 6.
Accept, edit, and display data. The achieved data need to be available for
further analysis, not only for professional scientists, but also the general
public.
* 7.
Analyse and interpret data. Raw data from the public need to be analysed to
get conclusive points as planned earlier. Here, the professional scientist
takes the leading role of the research.
* 8.
Disseminate results to the scientific community and the public. Each segment
demands its proper media, e.g. scientific journals or online article for the
public.
* 9.
Measure outcomes in order to evaluate the achievement of the project.
These steps become inspiring model for the Suryakala-Nusantara establishment.
## 3 Composing Suryakala-Nusantara
Suryakala-Nusantara will be formulated according to the model of Bonney et al.
[2]. Additional variables (such as funding) are need to taken into
consideration.
As previously mentioned, the main objective of Suryakala-Nusantara program is
to map the Indonesian sundials, especially those are related to Islamic
culture and civilization. The main question to be answered in this research is
how the distribution of archipelagic sundials in the dimension of space and
time. For this purpose, scientific protocol should includes: (1) take picture
clearly, (2) record the location, and (3) find the establishment time of the
sundial. Then, the contributor uploads the acquired data to the database of
Suryakala-Nusantara.
Sundial picture can be captured using various devices from cell phones to SLR
(single-lens reflex) camera which are widely distributed. There are several
things to be considered during the photo shoot in order to provide all
possible information. Size, orientation, shape, scale and marking detail are 5
informations that should be included in the picture.
The contributor should aware the location of the sundial, at least the name of
the mosque where it lies, the area/city and the province. Higher spatial
accuracy, e.g. precise geographic coordinate of the sundial, will be required
when the case studies are conducted upon particular sundials.
The last protocol is the most challenging step since the date of mosque or
sundial establishment is not always present explicitly. Interviewing the
elders or scholars becomes an alternative way to obtain the appropriate
temporal information.
To support these protocols, public or contributors need to get educational
supports which contain basic theory sundial, types, and how the sundial works.
Nontechnical aspects, such as art, architecture, and history of general
sundial also become subject of the supporting materials.
## 4 Compiling Suryakala-Nusantara
This citizen science program will be enriched by the digital data exchange and
done via internet. The free domain website, for instance the wordpress domain,
can be employed as the basis homepage in this project. Although it can
accommodate every basic information about the Suryakala Nusantara project, in
this case the fundamental understanding, scientific protocol, and also
education materials. But, the storage capacity of the free domain website is
not so large in quantity.
Picture sharing site like flickr.com provide the reliable alternative. One
account, both individual or in group, registered in this site are allowed to
upload to 1 TB without any charge. The picture description form and discussion
forum are also available to simplify the data validation process. Beside,
flickr.com is also well-recognized for its scientific discoveries, e.g. newly
found species[8].
Readers can visit www.flickr.com/groups/suryakala-nusantara/ for contribution
or further information.
## 5 Perspectives
The Indonesian classical sundial is a valuable cultural heritage that offers
indicator of architectural, technology, art and astronomical knowledge
development in the archipelago. It is our task to know, investigate and
conserve that heritage. Nevertheless, there are many things to do because of
the documentation of Indonesian sundial is rarely found in the literatures.
Citizen science scheme provides a new opportunities in order to transform the
physical existence of the sundial into (at least) more secure document.
Because of its broad range in space and time domain, it is suitable to be
applied in Indonesia through Suryakala-Nusantara program. After being launched
in HAI seminar, this program will be promoted widely through the blooming
social media.
However, there are several emerging issues related to the data-collection-
based citizen science [9, 10]:
* 1.
Coordination between the participants need to be conducted in order to ensure
the achievement of the main research objectives.
* 2.
Personal and social motivation need to be triggered and maintained from the
strong and publicly attractive research backgrounds.
* 3.
Data validation or quality controls become a crucial process to consider for
significant scientific product.
* 4.
Retention of the core team and participants. Most of the citizen-science-based
researches deal with a long period of time. It is usual that citizen science
program brings forth a new community, motivated by discoveries.
* 5.
More funding is needed to accomplish deeper research, especially for in-situ
case study of particular sundial.
Suryakala-Nusantara that relies heavily on citizen science needs to consider
and overcomes these issues.
## Acknoledgements
The author would like to thank Prof. Dr. Suhardja D. Wiramihadja, Dr. Premana
W. Premadi, Dr. Taufiq Hidayat, Dr. Mahasena Putra for the discussions and
valuable insights.
## References
## References
* Berggren [2001] J.L. Berggren. Sundials in medieval Islamic science and civilization. _The Compendium_ , 8:8, 2001.
* Bonney et al. [2009] R. Bonney, Cooper, J. C.B., Dickinson, S. Kelling, T. Phillips, K.V. Rosenberg, and J. Shirk. Citizen science: A developing tool for expanding science knowledge and scientific literacy. _BioScience_ , 59:977, 2009.
* Dickinson et al. [2010] J.L. Dickinson, B. Zuckerberg, and D.N. Bonter. Citizen science as an ecological research tool: Challenges and benefits. _Annual Review of Ecology, Evolution, and Systematics_ , 41:149, 2010.
* Ferrari [2010] G. Ferrari. The Ottoman sundials in Aiello del Friuli. _The Compendium_ , 17:31, 2010.
* Kim et al. [2010] S.H. Kim, K. Lee, and Y.S. Lee. A study on the sundials of the kang family of _Jinju_. _J. Astron. Space Sci_ , 27:161, 2010.
* Lee and Kim [2011] Y.S. Lee and S.H. Kim. A study for the restoration of the sundials in King Sejong era. _J. Astron. Space Sci_ , 28:143, 2011.
* Mayall and Mayall [1938] R.N. Mayall and M.L. Mayall. _Sundials: How to Know, Use, and Make Them_. Hale, Cushman & Flint, 1938.
* Perkins [2012] S. Perkins. Scienceshot: New species discovered, thanks to flickr. news.sciencemag.org, August 2012.
* Silvertown [2009] J. Silvertown. A new dawn for citizen science. _Trends in Ecology & Evolution_, 24:467, 2009.
* Webb et al. [2010] C.O. Webb, J.W.F. Silk, and T. Triono. Biodiversity inventory and informatics in Southeast Asia. _Biodiversity Conservation_ , 19:955, 2010.
|
arxiv-papers
| 2013-12-10T10:18:31 |
2024-09-04T02:49:55.261410
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rhorom Priyatikanto",
"submitter": "Rhorom Priyatikanto",
"url": "https://arxiv.org/abs/1312.2742"
}
|
1312.2797
|
050005 2013 J. J. Niemela V. Lakshminarayanan, Waterloo University, Canada
050005
We report a series of experiments on laser pulsed photoacoustic excitation in
turbid polymer samples addressed to evaluate the sound speed in the samples
and the presence of inhomogeneities in the bulk. We describe a system which
allows the direct measurement of the speed of the detected waves by engraving
the surface of the piece under study with a fiduciary pattern of black lines.
We also describe how this pattern helps to enhance the sensitivity for the
detection of an inhomogeneity in the bulk. These two facts are useful for
studies in soft matter systems including, perhaps, biological samples. We have
performed an experimental analysis on Grilon®samples in different situations
and we show the limitations of the method.
# Enhancement of photoacoustic detection of inhomogeneities in polymers
P. Grondona [inst1] H. O. Di Rocco [inst2] D. I. Iriarte [inst2] J. A.
Pomarico [inst2]
H. F. Ranea-Sandoval [inst2] G. M. Bilmes[inst3] E-mail:
[email protected]
(7 December 2012; 19 June 2013)
††volume: 5
99 inst1 Universidad Nacional de Rosario. Facultad de Ciencias Bioquímicas y
Farmacéuticas. Rosario (Santa Fe) Argentina. inst2 Instituto de Física “Arroyo
Seco”, Universidad Nacional del Centro de la Provincia de Buenos Aires. Calle
Pinto 399, B7000GHG, Tandil (Buenos Aires) Argentina. inst3 Centro de
Investigaciones Ópticas (CONICET-CIC) and Facultad de Ingeniería Universidad
Nacional de La Plata, La Plata. Argentina.
## 1 Introduction
In highly light-scattering materials, such as certain types of polymers,
turbid liquids, glassy structures, and body organs, inspection and monitoring
of internal features were made possible by means of X-Ray irradiation until
the development of ultrasound imaging. The former has the well-known
disadvantage that in biological tissues it may trigger degenerative processes
in the cells, and in non-biological samples, X-Ray inspection is not always
simple to perform directly in the production line. Ultrasound imaging is very
helpful in these situations.
On the other hand, visible light optical tomography and optical topography is
nowadays reaching the status of clinical resource in the detection and
monitoring of several types of tumors and for non-invasive evaluation of
oxygenation of tissues in biological samples. In non-clinical applications it
can be used for the detection of abnormal bodies within materials, which is of
great importance in quality control in several areas of technology. These
techniques were derived from the study of light propagating in turbid media,
and applied afterward to biological samples and medical imaging of different
parameters often using polymers as phantoms of biological tissues [1, 2, 3, 4,
5, 6, 7].
The photoacustic effect (PA) provides a method of analysis that has been used
in clear fluids and has sufficiently proven its capability for detecting very
low concentrations of absorbing species in a mixture or solution; it has also
been used for the monitoring of molecular processes in different environments
as shown in references [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. This paper
intends to make a contribution on the application of the PA in soft matter,
namely the detection of inclusions in polymer samples and the direct
determination of the speed of sound in the material used for the samples.
The PA technique has the advantage that acoustic waves do not scatter as light
does in the characteristic lengths of many experimental situations. Even if
the excitation light undergoes scattering, the location of an inhomogeneity
within the bulk of the sample can be achieved by detecting the remnant of the
shock wave generated at an absorbing region or at an interface at which the
speed of the sound waves changes. Repeating this inspection at other relative
positions of the laser and the acoustic detector and with the aid of a
suitable algorithm, a sufficiently precise location of a single inhomogeneity
of simple geometry can thus, in principle, be resolved, together with some
information about its composition (using at least two wavelength for the
excitation), provided the speed of sound is known (see, for example, Ref.
[20]). An example of this is presented in Ref. [21] in a rear-detection scheme
used for detecting inhomogeneities in subsurface inhomogeneities in metals.
The PA detection of bodies included in a turbid medium may provide
complementary information to diffuse light propagation studies in that medium.
Namely, it could bring an independent value for the absorption coefficient,
and it thus may help in the solution of the inverse problem in optical
tomography of samples.
The speed of sound determination relies on the fact that the acoustic signal
picked by the transducer arrives at times proportional to the distance from
the laser beam that generates the shock wave to that transducer. A drawback
with the photoacustic method applied to turbid materials is that the light
scattered by the bulk generates a pressure pulse on the detector if it is in
contact with a free surface of the sample explored. Consequently, the time of
arrival of the pressure signal at the detector is insensitive to the relative
position of the laser and the sensor. Hence, the speed of the waves involved
in the PA signal is difficult to determine and requires an adequate procedure
to evaluate it. This is one of the motivations of this contribution.
For this paper, we used a laser pulsed photoacustic system equipped with a PZT
in contact with the sample made of polymer Grilon®which is representative of a
turbid medium, to show how the presence of controlled, fiduciary absorbing
regions in the surface of a sample are used as local wave generators that
allow the determination of the speed of waves in materials despite of the
light scattering described. We have engraved in the surface of the samples a
pattern of stripes of absorbing material. In this way, we have a greater
signal whose contribution may be discriminated from the signal generated by
the light scattered by the sample material. We demonstrate that this fiduciary
pattern is useful also to enhance the photoacustic signal, and that from that
signal the presence of inhomogeneities in a medium may be inferred.
Other successful recent approaches to the problem of detection of tumoral
tissues in biological samples can be found in Refs. [18, 19].
## 2 Experimental
A scheme of the pulsed photoacustic system used in all the experiments is
shown in Fig. 1, which is essentially the same that can be used to determine
the speed of sound in liquids and in clear samples.
Figure 1: Experimental setup of the experiment. The parts are: the Nd+3:YAG
laser (NdL), the positioning device (X-TS), the Pinhole (PH) to clip the laser
beam (LB), and the oscilloscope (DO), the amplifier with a DC power supply
(APS). The DO is synchronized with the laser via the laser pulse
synchronization (LPS) cable.
We used a pulsed Nd+3:YAG laser emitting at $1.06\,\mu$m with a pulse duration
of approximately $10$ ns, at energies between $0.5$ mJ to 50 mJ. The laser
beam was clipped by means of a pinhole in order to reduce the original laser
beam size and to use a uniform spot thus reducing the power impinging on the
samples. This pinhole was held at the far end of a beam dump for security
reasons. In the results we present here, we have used two pinholes of $1$ mm
and $1.5$ mm in diameter which shall be specified in each experiment. This is
the diameter of the laser impinging on the sample, as inferred from sensitive
photographic paper. For acoustic detection, a ceramic $4\times 4$ mm2 PZT
transducer was strongly pressed against one of the free surfaces of the
sample, namely the one normal to that facing the laser. The photoacustic
signals were amplified and processed by means of a Tektronix TDS 3032B, $300$
MHz digital oscilloscope, averaging at least 64 signals before displaying the
photoacustic signal.
Samples used were square-section parallelepipeds, $10$ mm width, and $39$ mm
high, all made from the same polymer Grilon®piece. The samples were placed in
a C-clamp, with the PZT cage in one of its arms. The fiduciary pattern
engraved on one of the faces of some samples consists of five grooves of
approximately $1$ mm width and $0.2$ mm deep, filled with thick black paint,
separated by stripes of material which retain the natural turbid white color
of the polymer (which we call “clear” for short) of $1$ mm, whose lengths are
approximately 70% the length of the face. A second type of sample prepared in
a similar fashion, but with a centered cylindrical hole of 3 mm diameter
drilled in it parallel to the surfaces of the sample in all cases mentioned,
was also used in the experiments in order to compare the signals with the
former. This cavity was alternatively emptied or filled with deionized water.
We call “sample 1” the one drilled with the cylindrical cavity, and “sample 2”
the one without the hole. Figure 2 is a sketch of sample 1 with a schematic
representation of the fiduciary pattern used. The PZT and the laser beam
relative positions are displayed, together with the approximate position of
the cavity.
Figure 2: The Grilon®sample prepared for the surface absorption experiments.
The shadowed region (PZT) is the location of the transducer with respect to
the impinging laser beam direction (LBD). The cavity is a $3$ mm diameter
hole, whose position (HP) is shown for the samples that have drilled cavities.
The cavity may be empty or filled with water. The height of the samples is
$39$ mm and has a $10$ mm square base. It has five grooves (FP) in its front
face, painted in black to enhance absorption.
We obtained two types of signals, those from samples without holes and those
from samples with centered holes. Each type was subdivided into signals taken
with the laser impinging on the blank surface, and those taken with the laser
impinging on the patterned surface. Besides, there are signals obtained from
the samples with cavities, either empty or filled with water.
In each sample, the laser point of impact was moved from the farthest possible
position to the nearest with respect to the PZT. This was accomplished by
means of a $1\,\mu$m precision, step motor movable stage, Zaber Model T-LA60A,
controlled by a PC interface.
## 3 Results
In order to properly analyze the results, we calibrate the response of the
system to increasing laser pulse energy. To this end we irradiate a blank
surface of sample 2 at a point near the center of the face, and we plotted the
amplitude of the first maximum of the acoustic signal as a function of the
laser pulse energy. The result is displayed in Fig. 3 and shows linearity in
the energy range used.
Figure 3: PAS vs. laser pulse energy. The linearity of the PA response
(squares) to laser excitation is evident in the plots. The black dots
represent the PA signal when the laser impinges on a black stripe nearly at
the center of the front face of a striped sample.
In the same plot, we display three points (including the origin) which are the
maxima of the signal at the same location of a striped sample face, but
impinging on a black groove. As it can be seen, the signal nearly trebles its
maximum peak for the same excitation energy. In both experiments, the pinhole
used was $1.5$ mm in diameter.
After ascertaining the linearity of the response, and the fact that there is
an evident dependence of the PAS on the absorbance of the surface, we obtain a
profile of three of the grooves of sample 2 by plotting the value of the
amplitude of the first peak of the PA signal versus the relative distance
between the PZT and the excited region using the same pinhole as before. The
result of this is shown in Fig. 4. It can be seen that 1) the groove profile
is neatly resolved, and 2) there is an improvement of the signal generated in
the black stripes which decreases as the distance increases. Since the stripes
and the laser beam have approximately the same transverse size of the grooves,
the resulting profile is somehow rounded off, but this is not important in
what we aim to prove here.
Figure 4: The centers of clear bands and black grooves are clearly resolved by
scanning the surface with the laser. The signals were taken at $100\,\mu$m
displacement from each other. The energy of PAS decreases with distance of the
laser beam to PZT. The increment in signal due to the grooves is more than
6-fold with respect to the signal due to the bulk polymer.
We could determine the speed of sound in the sample from a plot of the time
position of the beginning of the first peak of the acoustic signal (arrival
time), as a function of the distance between the impinging point on the sample
and the PZT detector. But when we try to do that, experiments demonstrate that
the time elapsed since a laser pulse triggers a digital oscilloscope and the
appearance of the PA signal is the same regardless of the distance between the
impinging laser beam and the detector, due mainly to the light scattered by
the bulk of the polymer that hits the PZT. This poses a problem in the
evaluation of the speed of the waves.
To avoid that difficulty, we use the signal produced if the laser hits in the
black grooves, generated only by the absorption at the grooves. We obtain this
by subtracting from the PAS signal measured when the laser impinges in a black
groove, the PAS signal obtained in a “clear” region nearby. To this end, we
moved the sample slightly away from the previous black stripe, so the first
PAS signal was obtained with the beam impinging in a black groove and the
second PAS signal was obtained with the beam impinging in a white stripe.
Figure 5 shows the determination of the speed of sound in sample 2 by this
method. Since the plot uses as input the maxima of the amplitude of the
signals, the straight line would not cross the origin. The extrapolated value
for zero-crossing corresponds approximately to the amplitude of the first
maximum of the signal in clear samples.
Figure 5: Time of the first maximum of the processed signals vs. the position
of the impinging point of the laser presents a linear correlation
($R\,\approx\,0.994$) that yields a value of $v=(2333\,\pm\,133)$ ms-1 for the
speed of sound. All these signals were taken with a pinhole of $1$ mm
diameter.
Evaluation of the slope of the resulting line allows the calculation of the
value of speed, as $v=(2333\pm 133)$ ms-1, which compares well with calculated
data determined by using the properties of the polymer [22]. Since the
vibration of the whole sample has distinctive frequencies, the FFT of the PA
signal provides another estimation of the speed of waves, once the frequency
sequence is properly found. The Fast Fourier Transform (FFT) analysis was used
both, to estimate the sound speed in the Grilon®sample via the frequencies
identified in the spectrum and their spacing, knowing that the piece is a
parallelepiped of known dimensions, and to define a scale for the energy of
the pulse. The details of this procedure are straightforward calculations
[23]. We found it useful to use the power spectrum for defining the energy
instead of the integral of the temporal pulse and that is the parameter we use
in the presentation of the results.
Figures 6 and 7 show an FFT treatment of the signals obtained from the
following three cases: solid Grilon®sample, sample with empty cavity and
sample with the hole filled with deionized water. The PAS energy used in this
figure is a measure of the energy content of the acoustic pulse, as evaluated
from the power spectrum of the signal.
In Fig. 6, we display the results obtained impinging with the laser on clear
faces, and in Fig. 7 we show the results impinging with the laser on the
patterned faces. It is clear that a distinctive feature arises near the center
of the sample in the patterned faces where a black stripe is located, which is
not visible in the clear-face analysis.
Figure 6: The acoustic energy (FFT power integral of the PAS) vs. the relative
distance between the laser beam and the PZT in Grilon®samples in a face with
no fiduciary pattern (clear sample). The energy diminishes as the distance to
the detector increases. References in the insert: Triangles represent the
cavity empty in a clear sample. Circles are for cavity filled with water in
clear samples. Rhombi are for clear samples without the cavity. Figure 7: The
energy deposited by the laser on a white stripe and on the black grooves for
the three cases analyzed as seen in the insert. The circles represent a
grooved sample with the cavity filled with water. Triangles and rhombi are for
grooved samples and the empty cavity, and no cavity, respectively.
The acoustic energy deposited in the patterned faces is more than one order of
magnitude higher than that obtained in the clear faces when the inclusion is
present (compare Fig. 6 with Fig. 7). The water-filled hole and the empty hole
are also clearly distinguishable from the solid Grilon®response.
## 4 Analysis and Conclusions
We have performed an experimental analysis of the photoacustic signal in
Grilon®polymer but it can be extended to other materials as epoxy resins used
also as biological phantoms. We have shown that such applications are viable
for quantitative determinations. The above results can be used to determine
the presence and some optical characteristics of an inhomogeneity embedded in
this type of materials.
All the acoustic signals detected by the PZT in this soft turbid solid
material begin at approximately the same time after the laser trigger fires,
regardless of the relative distance from the impact point of the excitation to
the PZT acoustic detector, due mainly to scattering, making this method
useless to evaluate the speed of sound by the scanning standard procedure.
The differences in those PA signals are difficult to analyze. Therefore, to
evaluate the speed of the acoustic waves and to gather information about the
presence of a cavity or inhomogeneity in the polymer, we have developed a
method that used a regular pattern on the surface of the sample, consisting on
parallel clear stripes of the base material and grooves filled with highly
absorbent black paint. In the black grooves, the localized absorption provides
a strong shock wave at the surface. By comparing the time of appearance of the
signal in different positions of the surface, it is possible to estimate the
speed of those waves in the polymer.
For each of the zones in the pattern, the PA curves undergo a change of shape
and amplitude from signals in the white zones to the signals obtained in the
black stripes, being this strong evidence of the effect of the inhomogeneity
in the signal. A Power FFT was performed on each signal in order to provide
another means to determine whether a black or a white stripe is excited, and
the integral of the FFT provides a measure of the energy absorbed by the
sample in each case. Please note that the energy of the laser was fixed to a
value that avoids bleaching of the paint, being in all cases below $500\,\mu$J
per pulse.
The signals generated at these inhomogeneities provide a well defined point of
absorption and thus a definite path for the sound generated by the light-
absorption mechanism which is very distinctive from other mechanisms of
excitation of the PZT. It also reveals the presence of a surface inhomogeneity
once the contributions of other sources of acoustic waves are identified
making suitable use of reference signals. All the conclusions of this work
must be under the proviso that the PZT has a limited frequency band.
Although the above results were obtained for a soft polymer with a fiduciary
painted pattern, they can be extended to other type of resins with charge of
dyes or other absorbent particles. We are confident that with minor
modifications it can be used for the determination of properties of materials
of biological interest as well.
The results shown in Figs. 6 and 7 confirm that employing one of the surfaces
of the sample conveniently patterned, and scanning it for detection of
ultrasound signals, can be used to determine the presence of an inhomogeneity,
albeit its precise location and size is not well defined by this procedure and
it should be complemented by similar determinations at other relative
positions of the laser and the PZT. The increase in the signal with respect to
the background material is at least one order of magnitude or better.
When using this technique in phantoms used in medical applications, one should
take care of the fact that there are limitations in several aspects, such as
the power involved in each pulse avoiding any kind of damage, and that using
other wavelengths would be better suited for biological tissues which involve
blood. Other type of samples are being currently inspected by modifications of
the procedure reported here so to adapt it to gelled phantoms.
The conclusions are, in short, that the system is sensitive to the presence of
the inhomogeneity, and that the higher absorbance of the painted stripes in
the surface allows not only to evaluate the speed of sound (which is essential
to any tomographic technique) but also improves the detectivity by enhancing
the energy released as mechanical waves. This is a non-trivial result since in
the modeling of the propagation of the laser light in the turbid substance,
scattering is predominant, but still is sufficient for the detection of
inhomogeneities through changes in the absorption. The technique based on the
PA is simple and has the advantage that it can be adapted to be used in larger
samples or in samples of biological interest. The procedure of using a single
acoustic detector for the signals produced by the laser scanning of the
surface under study, has an advantage over multiple detector arrangements in
the sense that with a suitable fiduciary pattern the method can provide
information about the speed of the waves involved in the signal. This is
interesting because the data processing would not depend on generic
information about its value.
###### Acknowledgements.
PG wants to thank the Red Nacional de Laboratorios de Óptica for financial
help and partial funding during the experiments and to InterU System for
providing a grant for the completion of the experiments. This work partially
funded by Universidad Nacional del Centro de la Provincia de Buenos Aires,
Agencia Nacional de Promoción Científica y Tecnológica (PICT 0570) and CONICET
(PIP 384). HODR, DII, JAP and HFRS are members of Carrera del Investigador
Científico, Consejo Nacional de Investigaciones Científicas y Técnicas
(Argentina). GMB is member of Carrera del Investigador Científico, Comisión de
Investigaciones Científicas de la Provincia de Buenos Aires (Argentina).
Authors wish to thank Nicolás A. Carbone for help in the final preparation of
the manuscript.
## References
* [1] A Ishimaru, Wave propagation and scattering in random media, Academic Press, New York (1978).
* [2] J Ripoll Lorenzo, Light diffusion in turbid media with biomedical applications, Ph. D. Thesis, Universidad Autónoma de Madrid, Spain (2000).
* [3] A K Dunn, H Bolay, M A Moskowitz, D A Boas, Dynamic imaging of cerebral blood flow using laser speckle, J. Cerebr. Blood F. Met. 21, 195 (2001).
* [4] P N den Outer, Th M Nieuwenhuizen, Ad Lagendijk, Location of objects in multiple-scattering media, J. Opt. Soc. Am. A 10, 1209 (1993).
* [5] D A Boas, M A O’Leary, B Chance, A G Yodh, Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal to noise analysis, Appl. Opt. 36, 75 (1997).
* [6] D Contini, H Liszka, A Sassaroli, G Zaccanti, Imaging of highly turbid media by absorption method, Appl. Opt. 35, 2315 (1996).
* [7] A C Tam, Applications of photoacustic sensing techniques, Rev. Mod. Phys. 58, 381 (1986).
* [8] C K N Patel, A C Tam, Pulsed optoacoustic spectroscopy of condensed matter, Rev. Mod. Phys. 53, 517 (1981).
* [9] A A Oraevsky, A A Karabutov, Optoacoustic tomography in Biomedical Photonics, Ed. Tuan Vo-Dinh, CRC Press, Chapter 17 (2002).
* [10] R O Esenaliev, A A Karabutov, A A Oraevsky, IEEE J. Sel. Topics in Quantum. Electron. 5, 981 (1999).
* [11] L Nicolaides, A Mandelis, M Munidasa, Experimental and image-inversion optimization aspects of thermal wave diffraction tomography microscopy, AIP Conf. Proc. 463, 8 (1998).
* [12] P C Beard, Photoacustic imaging of blood vessel equivalent phantoms, Proc. SPIE 4618, 54 (2002).
* [13] E Zhang, J Laufer, P Beard, Backward-mode multiwavelength photoacustic scanner using a planar Fabry–Perot polymer film ultrasound sensor for high-resolution three-dimensional imaging of biological tissues, Appl. Opt. 47, 561 (2008).
* [14] S Fantini, M A Franceschini, E Gratton, Quantitative determination of the absorption spectra of chromophores in strongly scattered media. A light-emitting-diode based technique, Appl. Opt. 33, 5204 (1994).
* [15] G M Bilmes, O E Martínez, P Seré, D J Orzi, A Pignotti, On line photoacustic measurement of residual dirt on steel plates, AIP Conf. Proc. 557, 1944 (2001).
* [16] K H Song, E W Stein, J A Margenthaler, L V Wang, Noninvasive photoacoustic identification of sentinel lymph nodes containing methylene blue in vivo in a rat model, J. Biomed. Opt. 13, 054033 (2008).
* [17] Z Xu, Ch Li, L V Wang, Photoacustic tomography of water in phantoms and tissue, J. Biomed. Opt. 15, 036019 (2010).
* [18] L Xi, X Li, L Yao, Design and evaluation of a hybrid photoacoustic tomography and diffuse optical tomography system for breast cancer detection, Med. Phys. 39, 2584 (2012).
* [19] B Wang, Q Zhao, Photoacoustic tomography and fluorescence molecular tomography: A comparative study based on indocyanine green, Med. Phys. 39, 2512 (2012).
* [20] M Xu, L V Wang, Photoacoustic imaging in biomedicine, Rev. Sci. Instrum. 77, 041101 (2006).
* [21] R Takaue, H Tobimatsu, M Matsunaga, K Hosokawa, Detection of surface grooves and subsurface inhomogeneities in metals by transmission correlation photoacoustics, J. Appl. Phys. 59, 3975 (1986).
* [22] Data on mechanical properties of Grilon can be found in http://engr.bd.psu.edu/rxm61/METBD470/
Lectures/PolymerProperties%20from%20
CES.pdf and in http://www.inoxidable.com/
propiedades1.htm.
* [23] P Grondona, Caracterización de un sistema fotoacústico en el IR cercano para estudios en medios turbios. Algunas aplicaciones al estudio en fantomas de polímeros con inclusiones, Master Degree Thesis, Universidad Nacional de Rosario, Argentina (2009).
|
arxiv-papers
| 2013-12-10T13:47:13 |
2024-09-04T02:49:55.267939
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Grondona, H. O. Di Rocco, D. I. Iriarte, J. A. Pomarico, H\\'ector\n F. Ranea-Sandoval, G. M. Bilmes",
"submitter": "Hector F. Ranea-Sandoval",
"url": "https://arxiv.org/abs/1312.2797"
}
|
1312.2804
|
# A Novel Software Tool for Analysing NT® File System Permissions
Simon Parkinson and Andrew Crampton School of Informatics
University of Huddersfield
HD1 3DH, UK
Email: [email protected]
###### Abstract
Administrating and monitoring New Technology File System (NTFS) permissions
can be a cumbersome and convoluted task. In today’s data rich world there has
never been a more important time to ensure that data is secured against
unwanted access. This paper identifies the essential and fundamental
requirements of access control, highlighting the main causes of their
misconfiguration within the NTFS. In response, a number of features are
identified and an efficient, informative and intuitive software-based solution
is proposed for examining file system permissions. In the first year that the
software has been made freely available it has been downloaded and installed
by over four thousand users111Available at: http://eprints.hud.ac.uk9743
and http://download.cnet.comNTFSPermissionsExplorerSnapIn30002094_475325639.
## I Introduction
Controlling access permissions to a given file system is an important aspect
of data security. Having a secure and flexible way of viewing and managing
access control should be a standard requirement of all modern file systems.
This should certainly be true of the New Technology File System (NTFS), since
NTFS is currently the most common file system in use. This is mainly due to
Microsoft’s dominance of computing operating systems. Surprisingly, however,
no such flexibility exists for the NTFS and the process for determining access
controls is cumbersome at best.
The NTFS implements access control with the use of Access Control Lists
(ACLs). Each file system object (folder or file) will have an associated ACL
for controlling access. An ACL contains a list of ACEs (Access Control
Entities). Each ACE contains information regarding the interacting user or
group, and the level of access that they will be granted.
It is well reported that from observing an ACE that the following information
can be established [1, 2, 3]:
1. 1.
The user or group that the ACE applies to.
2. 2.
The level of granted permission for a user or group.
3. 3.
Information regarding the prorogation of the permission down the directory
hierarchy
The way in which users are required to interact with ACEs and ACLs in the NTFS
results in the following peculiarities:
1. 1.
Permissions are interacted with on a per object level, rather than per user
[4]. This does not allow for the administrator to evaluate user permission
across a whole directory structure.
2. 2.
Interacting with a single ACL using Windows Explorer as seen in Figure 1
requires the traversal of four different interfaces. Interacting with multiple
ACLs soon becomes a cumbersome task, which could ultimately result in
permissions being overlooked.
3. 3.
Not only is the administrator required to examine users or groups within the
ACL, they have to remember, or explore, group association to evaluate the
inheritance of permissions from different groups.
It is well reported that these time-consuming peculiarities result in the
potential for errors to occur, which could ultimately result in users being
denied access, or in the worst case, the possibility for unwanted access to
occur [5, 6, 7, 4, 3].
Previous efforts to provide a solution to the identified problems [4] have
been mostly successful, however, since their production the NTFS has evolved
to allow for the specification of fine- and -coarse grained file system
permissions [8]. This brings additional complexity as not only can the
standard six permission levels be granted, there is the possibility to create
‘special permissions’ which are constructed from any combination of the
possible fourteen permission attributes.
Microsoft provide a variety of command line utilities [9, 10, 11] and third-
party solutions are also available [12] to examine permission allocation.
However, the shortcomings of these utilities make none of them serve as a
single solution. These shortcomings can be summarised as the inabilities to:
1. 1.
Show both fine- and coarse-grained permissions.
2. 2.
Examine permissions on multiple folders at once.
3. 3.
Evaluate permissions per user rather than per object.
There is insufficient literature available to suggest that freely available
tools have been developed to significantly aid with the administration and
reporting of NTFS permissions [1, 2, 3], as well as providing detailed
information regarding the low-level implementation NTFS access control [13].
There are few research papers aimed at understanding NTFS access control [14,
15] and how it can be improved through better administration [8]. One author
has provided a formal model of NTFS access control, describing fundamentals of
rigours implementation [16], but there is no indication of the production of
any tools that make this available for system administrators.
One paper provides the results for an alternative management interface for
NTFS permissions [7]. Through careful consideration to human and computer
interaction, an application was designed where they could performed
administration tasks significantly faster, whilst reducing potential errors.
However, the work is restricted to only viewing file system permissions for a
single directory at any one time. Since the work was been published, there is
no evidence that the tool has been made available in the public domain. Other
work includes using novel ways to represent security policies [17]. This work
is also concerned with temporal aspects of managing file system permissions,
whereas the work in this paper is also concerned with providing useful
features to aid the quality of the analysis and help to reduce
misconfiguration.
This paper starts by giving a detailed description of how NTFS implements file
system permissions, highlighting complexities that result in misconfiguration.
A design is then provided, detailing how a software tool can be used to help
overcome the complexities, reducing misconfiguration. The next section
discusses the functionality of the produced piece of software. This section
describes how the functionality can be used to overcome the highlighted
complexities by using real-world examples where possible. Finally, we conclude
by discussing the beneficial impacts that the solution can bring, and suggest
future developments.
Figure 1: Analysing NTFS file system permissions using Windows Explorer
## II NTFS Access Control
In this section we describe the inner-workings of the NTFS as regards to
permission management. It is necessary to investigate the following aspects to
motivate the designed solution.
### II-A Access control structure
The NTFS follows in the footsteps of Microsoft’s object-oriented approach to
implementation. This means that the file system is made up of multiple file
and folder objects, and any subject within the operating system (user or
process) can request operations on the objects.
To control access to file system objects, the NTFS implements Access Control
Lists (ACLs) by applying an ACL to each object within the file system. Each
ACL will contain a Security Identifier (SID) which is a unique key that
identifies the owner of the object and the primary associated group. The
structure of the ACL is a sequential storage mechanism which contains access
control entries (ACEs). An ACE is an element within an ACL which dictates the
level of access given to the interacting subject. The ACE contains a SID that
identifies the particular subject, an access mask which contains information
regarding the level of permissions and the inheritance flags. Figure 2
illustrates the logical structure of an ACL and associated ACEs.
Figure 2: Access Control List illustration
### II-B Access Mask
An ACE within the NTFS is made up of a combination of fourteen individual
permission attributes. The NTFS provides six levels of standard coarse-grained
permission that consist of a combination of predefined attributes. It is also
the case that NTFS allows for the creation of special coarse-grained
permissions which consist of any combination of the fourteen individual
attributes[3].
TABLE I: Bit mask Bit / Bit range | Description | Example
---|---|---
0-15 | Object specific access rights | Read Data, Execute, Append Data
16-22 | Standard security access rights | Delete ACE, Write ACL, Write owner
23 | Access to ACL | Access System Security
24-27 | Reserved | n/a
28 | Generic all | $29\cup 30\cup 21$
29 | Generic Execute | All needed to execute
30 | Generic Write | All needed to write to a file
31 | Generic Read | All needed to read a file
The access mask is represented by a thirty-two-bit vector. Table I identifies
the use of each bit within the vector. It is evident from the table that the
standard coarse-grained permissions are represented as follows;
TABLE II: Standard coarse grained permission bits Coarse-grained level | Set bit(s)
---|---
Read | bit31
Write | bit30
List folder contents | bit31 $\cup$ bit29
Read and execute | bit31 $\cup$ bit29
Modify | bit31 $\cup$ bit29 $\cup$ bit30
Full control | bit28
Fine-grained special permissions are represented by using the bits within the
range of zero to fifteen. Creating a special permission for most is a very
useful feature; however, it can often be a source of confusion as it requires
the complete understanding of the authority that each attribute holds [18].
A good example of having to use special permissions is when you wish to assign
a group of users the standard privilege elevation of modify for all the
contents of a shared folder. However, creating an ACE with the modify
permission on the folder explicitly will result in the user being able to
delete the folder itself rather than the child objects (Table I). To get
around this problem we would simply assign the group or user the default
permission level of Modify, and then go and modify the permissions’ attributes
turning it into a special permission so that only subfolders and files can be
deleted.
### II-C Propagation and Inheritance
It is necessary to discuss the different mechanisms behind the way that NTFS
permissions can propagate throughout the directory structure. Within the ACL
there are two types of ACE; (1) Explicit and (2) Inherited. Explicit entries
are those that are applied directly to the objects’ ACL, whereas inherited are
those that are propagated from their parent object. The type of ACE allows to
determine whether the permission was assigned directly to the directory in
question (explicit) or if it was inherited from the directory that it resides
within (inherited).
This mechanism is controlled by the bit-flag within each ACE as seen in Figure
2. Table III shows the standard three coarse-grained levels of propagation and
explains their use.
TABLE III: Propagation and inheritance Bit | Name | Use
---|---|---
1 | container inherit ace | Applies the ACE to all the children objects
2 | no propagate inherit ace | Propagates the ACE to the child object without bit 1 being set, therefore, stopping propagation at the first level.
3 | inherit only ace | The ACE only applies to children objects. (i.e. does not apply to container)
Furthermore, the creation of fine-grained special file system permissions also
allows for the creation of custom fine-grained inheritance rules. Special
inherited permissions can be different depending on whether the ACE has the
container inherit ace bit flag set which controls whether the ACE is applied
to all the children objects or not. The creation of fine- grained propagation
rules can easily be overlooked and can ultimately result in the unintended
propagation of access.
One of the main difficulties with access propagation with the NTFS is
correctly evaluating the effective propagation rules. For a user to view the
propagation rules the same situation as viewing the effective permission
applies, where the user is required to traverse through the several Windows
interface to retrieve the required information as seen in Figure 4.
### II-D Accumulation
Accumulation is the possibility for the subject to receive the effective
permission of multiple different policies. This feature is prominent within
the NTFS resulting in the possibility for a subject to receive permissions
from multiple different ACEs within the same ACL. Furthermore, any subject
that interacts with the NTFS can be assigned to any number of groups, which
can be entered into the ACE. This means that the user does not have to be
directly entered into the ACE, they could simply be a member of the group that
is entered.
The policy combination is handled within the operating system by the Local
Security Authority Subsystem Service (LSASS). This service combines the
permissions together to effectively create the union of all the policies.
There are few complexities within permission accumulation due to the
structured way in which ACEs are processed. These are:
1. 1.
Explicit permissions take precedence over inherited permissions.
2. 2.
Explicit deny permissions always take precedence over apply permissions.
3. 3.
Permissions inherited from closer relatives take precedence over relatives.
further away.
It might expect that deny permissions always take precedence over apply
permissions to ensure that during the policy combination stage the user always
operates as the least possible privilege elevation. However, the first point
regarding explicit permissions taking precedence over inherited permissions
can result in a situation where an inherited deny permission is never reached.
Considering the folder structure in Figure 3, where the folder Accounting has
an explicit deny permission for the Everyone group, which is set to propagate
to all its children. This means that all the subfolders to the Accounting
folder will receive an inherited deny Everyone ACE. If the case was to arise,
like in this example, where a single user now requires access to the Plan
folder, an explicit ACE to allow access could be entered. Now when the user
visits the Plan folder, the LSASS would process the explicit allow permission
first and allow for it to take precedence over any other permission. This goes
against a fundamental aspect of policy combination to ensure that a deny
permission is never ignored. If the case where a user is able to ignore a deny
permission to receive access was to either intentionally or unintentionally
arise, the system administrator needs to be made aware of this situation.
Figure 3: Explicit beford inherited demonstration
To summarise, the precedence hierarchy for policy accumulation is as follows:
1. 1.
Explicit deny.
2. 2.
Explicit allow.
3. 3.
Inherited deny.
4. 4.
Inherited allow.
In addition to the explicit permissions taking precedence over inherited
permissions, inherited permissions that of closer distance to the invoked
object will take precedence over more distant relatives. For example, a
folder’s inherited permissions will take precedence over those from their
grandparent.
Accounting for permission accumulation has currently been made possible by
using the standard Windows Explorer feature of displaying the effective
permission. This feature allows for the user to enter a specified user or
group and the effective permission that they hold on that specific directory
will be displayed. Unfortunately, performing this evaluation on several
folders soon becomes infeasible.
### II-E Group Membership
A fundamental aspect of access control within the NTFS is that of group
membership. A subject (group, user or process) that interacts with the file
system can be a member of any group. This means that permissions can be
inherited from any of the associated groups if they are entered within any
ACL. Subjects, in this case users, will often be grouped together by
(separation of duty) to make management easier, and as Hanner, 1999 [4]
identifies, understanding effective file permissions can become significantly
more complex by group association. To correctly evaluate a user’s effective
permissions you would have to know which groups they are a member of. We
should note that this is not directly related to the mechanism of how NTFS
implements access control, it is an unavoidable component of how Microsoft
allows for users, groups and processes to be managed by group association.
## III Novel Solution
This section describes the design of a solution based on the NTFS’s inner-
workings which can cause the identified administrative complexities as seen in
Section II.
### III-A Coarse- and Fine-Grained Permissions
As previously described, the NTFS allows for the standard set of coarse
permissions, but also allows for the creation of special fine-grained
permissions.
An alternative method of display, special permissions could be displayed by a
character-to-attribute representation. This way a string can be constructed to
display the full granularity of the permission by only using little space. For
example, if a special permission was constructed to have the attributes
enabled:
1. 1.
Read (R).
2. 2.
Write (W).
3. 3.
Delete subfolders and files (Dc).
4. 4.
Read permissions (Rp).
5. 5.
Change permissions (Cp).
Using the character-to-attribute would results in the production of the string
‘R-W-Dc-Rp-Cp’. After some time the user would become accustom to this
relationship and the key would no longer be required.
### III-B Multiple Folders
Input: Initial directory $d$
Input: Set of ACEs to be filtered out $F=(f_{1},f_{2},f_{3},\ldots,f_{n})$
Output: Set of ordered directories and ACEs
$P=(d_{1},(p_{1},p_{2},p_{3},\ldots,p_{n}))$ where $d_{n}$ is the directory
and $p_{n}$ are the permission entries for that directory.
1 Algorithm _algo(__)_
3 2 $P\leftarrow$ proc(_d_)
5 4 return
6
7
1 Procedure _proc(_directory d_)_
2 $pACL\leftarrow d(ACL)$
3 foreach _subdirectory $c$ of $d$_ do
4 $cACL\leftarrow c(ACL)$
5 if _$cACL\;!=pACL$_ then
6 foreach _ACE $a$ in $cACL$_ do
7 if _$a\not\in F$_ then
8 if _$isSpecial(a)$_ then
9 $p\leftarrow compress(p)$
10
11 else
12 $p\leftarrow a$
13 end if
14 $P\leftarrow(c,p)$
15 proc(_c_)
16 end if
17
18
19 end if
20
21
22
Algorithm 1 Depth-first recursive directory search, analysing and filtering
security permissions.
It has previously been identified that Windows Explorer allows for the
examination of an objects’ ACL, however, it is often the case that evaluating
multiple ACLs is necessary. A useful way to view multiple ACLs would be to
allow the examination of a whole directory structure simultaneously. This
would provide the means to also examine how the propagation and inheritance
aspects of the ACLs are interacting. Algorithm 1 describes the recursive
depth-first examination search technique that has been implemented for
analysing the permissions of multiple folders. This algorithm traverses the
directory structure, analysing each directories permissions. In each analysis,
the algorithm evaluates whether:
1. 1.
It is necessary to display the current ACL to the user based on whether it is
different from the parent’s ACL.
2. 2.
Each ACE in the ACL contains a special permission.
3. 3.
Report the ACE to the user, displaying the level of permission.
### III-C Compression
As seen on line 9 of Algorithm 1, a compress function is called if a special
permission is identified. This compress function performs the character-to-
attribute mapping as described in Section III-A. In this method, an enumerated
type is used for changing the permission attributes to the associated
character.
### III-D Filtering
Filtering of groups is easily performed as shown on line 7 of Algorithm 1
where a check is made to ensure that the current ACE $a$ is not present in the
set of groups to filer $F$. This provides the facility to filter for multiple
user or group objects, therefore removing excess information.
### III-E Per User View
When performing a per user search of the file system, Algorithm 1 is used,
however, line 7 is substituted with a condition to check that the ACE in
question is the one that is being searched for ($a\in F$). This means that all
groups and user objects are excluded if they are not represent in the filter
list. When viewing per user, the filer list contains the user or group that
the user wants to analyse.
### III-F Accumulation
Algorithm 1 identifies provides a search strategy that can report the file
system permissions for an entire directory structure, whilst considering
compression and filtering. Although the returned permission information is
what is visible in the ACE, it might not be the user’s effective permission as
no consideration to permission accumulation as described in Section II-D is
taken. Algorithm 2 provides an alternative method where the search
concentrates on calculating the effective permission that the user and or
group hold. Algorithm 2 shows an algorithm that can be used to store the
explicit $ex$ and inherited $in$ permissions based on the inheritance and
propagation. This algorithm considers both the inheritance and deny
hierarchies. For speed purposes the algorithm can identify deny permissions
and stop the algorithm from continuing the examine the ACL. Line 16 shows that
once the explicit and inherited permissions have been identified a function is
then called to calculate the effective permission. In this algorithm
$calculatedEffective(explicit,inherited)$ represents a native Microsoft .NET
command that is able to return the effective permission. Using this native
method ensures that the correct effective permission is reported.
Input: Initial directory $d$
Input: Initial group or user $u$
Output: Set of ordered directories and ACEs
$P=(d_{1},(p_{1},p_{2},p_{3},\ldots,p_{n}))$ where $d_{n}$ is the directory
and $p_{n}$ are the permission entries for that directory.
1 Algorithm _algo(__)_
3 2 $P\leftarrow$ proc(_d_)
5 4 return
6
7
1 Procedure _proc(_directory d_)_
2 $pACL\leftarrow d(ACL)$
3 foreach _subdirectory $c$ of $d$_ do
4 $cACL\leftarrow c(ACL)$
5 if _$cACL\;!=pACL$_ then
6 $ex=\emptyset$, $in=\emptyset$
7 foreach _ACE $a$ in $cACL$_ do
8 if _$isExplicitDeny(a)$_ then
9 $P\leftarrow(c,a)$
10 break
11
12 else
13 else if _$isExplicitAllow(a)$_ then
14 $ex\leftarrow a$
15
16 else if _$isInherited(a)$_ then
17 $in\leftarrow a$
18
19 $P\leftarrow(c,calculatedEffective(ex,in))$
20
21 end if
22
23
24 end if
25
26
27
Algorithm 2 Depth-first recursive directory search, returning the effective
permission of a specified user or group.
### III-G Group Membership
User and group membership is fundamental mechanism that allows users to
inherit file system permissions from group objects. A simple recursive method
can be used to examine a user or groups membership. There are two possible
directions in which the group membership can be analysed. The first is to
examine which groups an object is a member of. This is where a search is
performed to recursively report which groups a user or group is a member of.
The second method is the members of displaying a user or groups members. This
is where a recursive search is performed to reporting on a groups members.
## IV Developed Solution
The developed software-based tool is programmed in C# .NET 3.5 with the use of
the Microsoft Management Console (MMC) System Development Kit (SDK) to produce
a MMC SnapIn application. The motivation behind making the application run in
the MMC was to bring consistency with other Microsoft management tool,
therefore, making the software self-intuitive for the users.
The software runs under the credentials of the executing user, therefore, only
receiving access to view file system permissions that they have been assigned
to. The software runs in real-time, processing the desired ACLs upon request.
This means that the software requires only a minimal amount of installation,
and does not require an additional database to store permission entries. The
overheads caused by the application on both the host machine and any
interacting file servers are very small and do not affect normal performance
at all.
In this remaining of this section, the provided functionality is discussed,
using examples where possible.
### IV-A Application Layout
Figure 4: Developed MMC Application
As seen in Figure 4, the interface has three main sections. Firstly on the
left is the control pane. The control pane is where the user can see all the
physical and remote mounted NTFS volumes. The user is able to browse the
folder structure of all local and remote drives in a Windows standard
hierarchical tree view. In addition, any effective permission searches that
the user performs will be listed here. The middle pane is where the associated
results from the item selected within the control pane are displayed. On the
right is the action pane. This pane contains functionality associated with
each of the items selected within the control pane that can affect the
contents of the results pane.
The results pane shows the ACL for the specified local or remote drive,
providing that the executing user has permission to view the ACL. This pane
contains the same ACL information as present in the Windows Explorer
interface. The ACEs are classified into the standard NTFS sets although List
Folders is not classed as a set because the permission is the same as Read &
Execute, just the propagation is different, which is correctly displayed.
### IV-B Coarse- and Fine-Grained Permissions
As described in the design, the application does have a different way of
representing special permissions. To allow the user to easily and correctly
see the fine-grained permissions the special permissions are displayed as a
hyphen separated character string, where each character is associated with a
different special permission attribute.
As shown in Figure 5 the group ‘BUILTIN\Users’ has a special permission entry
that is displayed by the hyphenated character string. On further inspection of
this permission it is possible to view the character-to-attribute
relationship, which is also displayed in Figure 5. After using the application
we might start to remember the character-to-attribute relationship, meaning
that we do not need to inspect the special permission, therefore, further
speeding up the process of reporting fine-grained special permissions. The
results pane also shows information regarding whether each permission (ACE) is
an allow or deny permission, and also the propagation level of each of the ACE
entries.
Figure 5: Developed MMC Application
### IV-C Traversal View and Custom Filter
Another highlighted problem was difficulties within trying to view the ACL for
multiple folders at any one time. The developed application avoids this issue
by firstly allowing a user to simply traverse the file system in the control
pane to view the ACL for a single folder, and secondly, allowing the user to
view the ACLs for a whole directory in one traversal view. To reduce the
quantity of displayed information and help display what is useful to the user,
by default the traversal view will only show the ACL for a folder that is not
the same as its parents’. A custom filter has also been implemented so that
the user can select groups and users that they do not wish to include in the
traversal view.
Figure 6 shows the results pane when the traversal function is applied to the
local folder C:\Users. The illustration also shows the filter interface where
the user can select groups that they wish to remove from view. The traversal
view also displays both fine- and coarse-grained permissions in the same way
as the individual view where the permissions are classified as the standard or
special sets.
Figure 6: Traversal view with custom filter
### IV-D Permission Accumulation
Policy combination can be one of the most time consuming aspects of the NTFS
when trying to evaluate the permission that a subject holds on any given
location. As described earlier, accumulation of deny and access permissions,
group membership as well as consideration to the ACE processing hierarchy
results in several complication factors to the evaluation. The developed
application has a built-in search feature to show the exact effective
permissions for a given subject on the selected location. Figure 7 shows the
interface after performing a custom search for the user ‘simon-PC\simon’ on
the directory ‘C:\User’. The same logic applies when performing a search where
only permissions that differ from their parent object are displayed by
default, and special permissions are displayed using the hyphenated character
representation.
Figure 7: Permissions accumulation search results
## V Conclusions
We began by examining in detail the workings of access control within the NTFS
to highlight the potential causes of complexity, which could ultimately lead
to unintended access. Next, we discussed the common usability problems that
can be experienced when examining NTFS permissions. Following this, we
developed a Microsoft Management Console SnapIn application to provide a new
way of examining NTFS permissions that can help overcome the identified
complexities. We believe that our study and software solution helps to improve
file system security by providing an intuitive, efficient and thorough method
for permission examination.
This paper provides a contribution to system administrators by aiding them
with permission examination and allocation. The requirement to provide a
software-based tool to overcome the identified complexities can be established
from the in excess of four thousand downloads the tool has received since
production. This shows that NTFS administrators are actively seeking support
for their duties. In addition to the number of downloads, the tool has also
received promotion through a rated software site[19] and a useful list of
system administration tools[20]. This emphasises how requirement for such
tool.
## VI Future Scope
Future work involves allowing for the user to modify file system permissions
once a problem has been identified. Another possibility is a software tool
that can automatically identify configuration problems and suggest intelligent
solutions.
## VII Acknowledgement
The authors would like to express great thanks to Michele Puri of the European
University Institution for passing on vast amounts of knowledge regarding the
implementation and administration of NTFS permissions within a large
organisation. Thanks should also be expressed to Alan Radley and Malcolm
Merrington of the University of Huddersfield for providing additional insight
to the problems and for testing the developed software.
## References
* [1] C. Russel, S. Crawford, and J. Gerend, _Microsoft windows server 2003 administrator’s companion_. Microsoft Press, 2003.
* [2] D. A. Solomon, “Microsoft windows internals: Microsoft windows server 2003, windows xp, and windows 2000.”
* [3] _Microsoft Windows Server 2003, Administrator’s Companion_ , 2nd ed. Microsoft Press, 2006.
* [4] K. Hanner and R. Hörmanseder, “Managing windows nt file system permissions— a security tool to master the complexity of microsoft windows nt file system permissions,” _Journal of Network and Computer Applications_ , vol. 22, no. 2, pp. 119 – 131, 1999. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1084804599900863
* [5] K. Beznosov, P. Inglesant, J. Lobo, R. Reeder, and M. E. Zurko, “Usability meets access control: challenges and research opportunities,” in _Proceedings of the 14th ACM symposium on Access control models and technologies_ , ser. SACMAT ’09. New York, NY, USA: ACM, 2009, pp. 73–74. [Online]. Available: http://doi.acm.org/10.1145/1542207.1542220
* [6] X. Cao and L. Iverson, “Intentional access management: making access control usable for end-users,” in _Proceedings of the second symposium on Usable privacy and security_ , ser. SOUPS ’06. New York, NY, USA: ACM, 2006, pp. 20–31. [Online]. Available: http://doi.acm.org/10.1145/1143120.1143124
* [7] R. A. Maxion and R. W. Reeder, “Improving user-interface dependability through mitigation of human error,” _International Journal of Human-Computer Studies_ , vol. 63, no. 1–2, pp. 25 – 50, 2005, ¡ce:title¿HCI research in privacy and security¡/ce:title¿. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1071581905000601
* [8] S. De Capitani di Vimercati, S. Paraboschi, and P. Samarati, “Access control: principles and solutions,” _Software: Practice and Experience_ , vol. 33, no. 5, pp. 397–421, 2003. [Online]. Available: http://dx.doi.org/10.1002/spe.513
* [9] Microsoft, “How to use xcalcs.vbs to modify ntfs permissions,” 2006. [Online]. Available: http://support.microsoft.com/kb/825751
* [10] “Accesschk v5.01,” 2010. [Online]. Available: http://technet.microsoft.com/en-gb/sysinternals/bb664922
* [11] Microsoft, “Accessenum v1.32,” 2006. [Online]. Available: http://technet.microsoft.com/en-us/sysinternals/bb897332
* [12] “Security explorer v7.5.0,,” 2010. [Online]. Available: http://www.scriptlogic.com/products/security-explorer/
* [13] B. Carrier, _File system forensic analysis_. Addison-Wesley Boston, 2005, vol. 3.
* [14] L.-y. WANG and J.-w. JU, “Analysis of ntfs file system structure,” _Computer Engineering and Design_ , vol. 3, p. 018, 2006.
* [15] L. J. Z. Yue, “The main data structure of ntfs file system,” _Computer Engineering and Applications_ , vol. 8, p. 038, 2003.
* [16] J. Crampton, G. Loizou, and G. O’Shea, “A logic of access control,” _The Computer Journal_ , vol. 44, no. 2, pp. 137–149, 2001.
* [17] R. W. Reeder, L. Bauer, L. F. Cranor, M. K. Reiter, K. Bacon, K. How, and H. Strong, “Expandable grids for visualizing and authoring computer security policies,” in _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ , ser. CHI ’08. New York, NY, USA: ACM, 2008, pp. 1473–1482. [Online]. Available: http://doi.acm.org/10.1145/1357054.1357285
* [18] O. Thomas, “Are ntfs and share permissions a bit too complicated,” _Windows IT Pro_ , vol. 16, p. 78, 2010.
* [19] SoftSea.com, “Ntfs permissions explorer snapin,” http://www.softsea.com/review/NTFS-Permissions-Explorer-SnapIn.html, accessed: 2013-08-17.
* [20] C. Goggi, “101 free admin tools,” http://www.gfi.com/blog/101-free-admin-tools, accessed: 2013-08-17.
|
arxiv-papers
| 2013-12-10T14:05:42 |
2024-09-04T02:49:55.274487
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Simon Parkinson and Andrew Crampton",
"submitter": "Simon Parkinson Mr",
"url": "https://arxiv.org/abs/1312.2804"
}
|
1312.2844
|
# mARC: Memory by Association and Reinforcement of Contexts
Norbert Rimoux and Patrice Descourt
Marvinbot S.A.S, France
###### Abstract
This paper introduces the memory by Association and Reinforcement of Contexts
(mARC). mARC is a novel data modeling technology rooted in the second
quantification formulation of quantum mechanics. It is an all-purpose
incremental and unsupervised data storage and retrieval system which can be
applied to all types of signal or data, structured or unstructured, textual or
not. mARC can be applied to a wide range of information classification and
retrieval problems like e-Discovery or contextual navigation. It can also
formulated in the artificial life framework a.k.a Conway ” Game Of Life”
Theory. In contrast to Conway approach, the objects evolve in a massively
multidimensional space. In order to start evaluating the potential of mARC we
have built a mARC-based Internet search engine demonstrator with contextual
functionality. We compare the behavior of the mARC demonstrator with Google
search both in terms of performance and relevance. In the study we find that
the mARC search engine demonstrator outperforms Google search by an order of
magnitude in response time while providing more relevant results for some
classes of queries.
## 1 Introduction
At the onset of 20th century, it was generally believed that a complex system
was the sum of its constituents. Furthermore, each constituent could be
analyzed independently of the others and reassembled together to bring the
whole system back.
Since the advent of quantum physics with Dirac, Heinsenberg, Schr dinger,
Wigner, etc. and the debate about the incompleteness of the probabilistic
formulation of quantum mechanics which arose between Einstein and the
Copenhagen interpretation of quantum physics led by Niels Bohr in 1935, the
paradigm enounced in the beginning of this paragraph has been seriously
questioned.
The EPR thought experiment $[$Einstein1935$]$ at the heart of this debate
opened the path for Bell inequalities which concern measurements made by
observers on pairs of particles that have interacted and then separated. In
quantum theory, such particles are still strongly entangled irrespective of
the distance between them. According to Einstein’s local reality principle
(due to the finiteness of the speed of light), there is a limit to the
correlation of subsequent measurements of the particles.
This experiment opened the path to Aspect’s experiments between 1980 and 1982
which showed a violation of Bell inequalities and proved the non-local and
non-separable orthodox formulation of quantum theory without hidden variables
$[$Aspect1981, Aspect1982$]$.
After this holistic shift in the former Newtonian and Cartesian paradigms,
Roger Penrose and others have argued that quantum mechanics may play an
essential role in cognitive processes $[$Penrose1989, Penrose1997$]$. This
contrasts with most current mainstream biophysics research on cognitive
processes where the brain is modeled as a neural network obeying classical
physics. We may so wonder if, for any artificial intelligence system seen as a
complex adaptive system (CAS), quantum entanglement should not be an inner
feature such as emergence, self-organization and co-evolution.
$[$Rijsbergen2004$]$ further demonstrates that several models of information
retrieval (IR) can be expressed in the same framework used to formulate the
general principles of quantum mechanics.
Building on this principle, we have designed and implemented a complex
adaptive system, the memory by Association and Reinforcement of Contexts
(mARC) that can efficiently tackle the most complex information retrieval
tasks.
The remainder of this paper is organized as follows: Section 2 describes the
current approaches to machine learning and describes related work. Section 3
describes mARC. Section 4 compares the performance and relevance of results of
a mARC-based search demonstrator and Google search. Finally, section 5 draws
some conclusions and describes future work.
## 2 Current Approaches
### 2.1 Text Mining
Text mining covers a broad range of related topics and algorithms for text
analysis. It spans many different communities among which: natural language
processing, named entity recognition, information retrieval, text
summarization, dimensionality reduction, information extraction, data mining,
machine learning (supervised, unsupervised and semi-supervised) and many
applications domains such as the World Wide Web, biomedical science, finance
and media industries.
The most important characteristic of textual data is that it is sparse and
high dimensional. A corpus can be drawn from a lexicon of about one hundred
thousand words, but a given text document from this corpus may contain only a
few hundred words. This characteristic is even more prominent when the
documents are very short (tweets, emails, messages on a Facebook wall, etc.).
While the lexicon of a given corpus of documents may be large, the words are
typically correlated with one another. This means that the number of concepts
(or principal components) in the data is much smaller than the feature space.
This advocates for the careful design of algorithms which can account for word
correlations.
Mathematically speaking, a corpus of text documents can be represented as a
huge, massively high-dimensional, sparse term/document matrix. Each entry in
this matrix is the normalized frequency of a given term in the lexicon in a
given document. Term frequency-inverse document frequency (TF-IDF) is
currently the most accurate and fastest normalization statistic that can take
into account the proper normalization between the local and global importance
of a given word inside a document with respect to the corpus. Note, however,
that it has been shown recently that binary weights give more stable
indicators of sentence importance than word probability and TF-IDF in topic
representation for text summarization $[$Gupta2007$]$.
Because of the huge size and the sparsity of the text/document matrix, all
correlation techniques suffer from the curse of dimensionality. Moreover, the
variability in word frequencies and document lengths also creates a number of
issues with respect to document representation and normalization. These are
critical to the relevance, efficiency and scalability of state of the art
classification, information extraction, or statistical machine learning
algorithms.
Textual data can be analyzed at different representation levels. The primary
and most widely investigated representation in practical applications is the
bag of words model. However, for most applications, being able to represent
text information semantically enables a more meaningful analysis and text
mining. This requires a major shift in the canonical representation of textual
information to a representation in terms of named entities such as people,
organizations, locations and their respective relations $[$Etzioni2011$]$.
Only the proper representation of explicit and implicit contextual
relationships (instead of a bag of words) can enable the discovery of more
interesting patterns. $[$Etzioni2011$]$ underscores the urgent need to go
beyond the keyword approximation paradigm. Looking at the fast expanding body
scientific literature from which people struggle to make sense, gaining
insight into the semantics of the encapsulated information is urgently needed
$[$Lok2010$]$.
Unfortunately, state of the art methods in natural language processing are
still not robust enough to work well in unrestricted heterogeneous text
domains and generate accurate semantic representations of text. Thus, most
text mining approaches currently rely on the word-based representations,
especially the bag of words model. This model, despite losing the positioning
and relational information in the words, is generally much simpler to deal
with from an algorithmic point of view $[$Aggarwal2012$]$.
Although statistical learning and language have so far been assumed to be
intertwined, this theoretical presupposition has rarely been tested
empirically $[$Misyak2012$]$. As emphasized by Clark in $[$Clark1973$]$,
current investigators of words, sentences, and others language materials
almost never provide statistical evidence that their findings generalize
beyond the specific sample of language materials they have chosen. Perhaps the
most frustrating aspect of statistical language modeling is the contrast
between our intuition as speakers of natural languages and the over-simplistic
nature of our most successful models $[$Rosenfeld2000$]$.
Supervised learning methods exploit training data which is manually created,
annotated, tagged and classified by human beings in order to train a
classifier or regression function that can be used to compute predictions on
new data. This learning paradigm is largely in use in commercial machine
language processing tools to extract information and relations about facts,
people and organizations. This requires large training data sets and numerous
human annotators and linguists for each language that needs to be processed.
The current methods comprise rules-based classifiers, decision trees, nearest
neighbors classifiers, neural networks classifiers, maximal margins
classifiers (like support vector machines) and probabilistic classifiers like
conditional random fields (CRF) for name entity recognition, Bayesian networks
(BN) and Markov processes such as Hidden Markov Models (HMMs) (currently used
in part-of-speech tagging and speech recognition), maximum-entropy Markov
models (MEMMs), and Markov Random Fields. CRF has been applied to a wide
variety of problems in natural language processing, including POS tagging
$[$Lafferty2001$]$, shallow parsing $[$Sha2003$]$, and named entity
recognition $[$McCallum2003$]$ as an alternative to the related HMMs.
Many statistical learning algorithms treat the learning task as a sequence
labeling problem. Sequence labeling is a general machine learning technique.
It has been used to model many natural language processing tasks including
part-of-speech tagging, chunking and named entity recognition. It assumes we
are given a sequence of observations. Usually each observation is represented
as feature vectors which interact through feature functions to compute
conditional probabilities.
As a simple example, let us consider $x_{1:N}$ be a set of observations (e.g.
words in a document), and $z_{1:N}$ the hidden labels (e.g. tags). Let us also
assume that each observation can be expressed in terms of F features. A linear
chain conditional random field de?nes the conditional probability that a given
tag is associated with a document knowing that a given word has been observed
as:
$p(z_{1:N}x_{1:N})=\frac{1}{Z}e^{\sum_{n=1}^{N}{\sum_{i=1}^{F}{\lambda_{i}}f_{i}(z_{n-1},z_{n},x_{1}:N,n)}}$
Z is just there to ensure that all the probabilities sum to one, i.e. it is a
normalization factor. For example, we can de?ne a simple feature function
which produces binary values: it is 1 if the current word is ” John”, and if
the current state $z_{n}$ is ” PERSON”:
$f_{1}(z_{n-1},z_{n},x_{1:N},n)=\left\\{\begin{array}[]{lr}1&ifz_{n}="PERSON"andx_{n}="John"\\\
0&otherwise\\\ \end{array}\right.$
How this feature is used depends on its corresponding weight ?1 . If ?1 $>$ 0,
whenever f1 is active (i.e. we see the word John in the sentence and we assign
it the tag PERSON), it increases the probability of the tag sequence z1:N.
This is another way of saying ” the CRF model should prefer the tag PERSON for
the word John”.
A common way to assign a label to each observation is to model the joint
probability as a Markov process where the generation of a label or an
observation is dependent only on one or a few previous labels and/or
observations. This technique is currently extensively used in the industry.
Although Markov chains are ef?cient at encoding local word interactions, the
n-gram model clearly ignores the rich syntactic and semantic structures that
constrain natural languages $[$Ming2012$]$. Attempting to increase the order
of an n-gram to capture longer range dependencies in natural language
immediately runs into the dimensionality curse $[$Bengio2003$]$.
Unfortunately, from a computational point of view, even if we restrict the
process to be linear (depending only on one predecessor) the task is highly
demanding in computational resources. The major di?erence between CRFs and
MEMMs is that in CRFs the label of the current observation can depend not only
on previous labels but also on future labels.
In mathematical graph theory terms, CRFs are undirected graph models while
both HMMs and MEMMs are directed graph models. Usually, linear-chain CRFs are
used for sequence labeling problems in natural language processing where the
current label depends on the previous label and the next label as well as the
observations. In linear-chain CRFs long-range features cannot be de?ned.
General CRFs allow long-range features but are too expensive to perform exact
inference. Sarawagi and Cohen have proposed semi-Markov conditional random
?elds as a compromise $[$Saragawi2005$]$. In semi-Markov CRFs, labels are
assigned to segments of the observation sequence and features can measure
properties of these segments. Exact learning and inference on semi-Markov CRFs
is thus computationally feasible and consequently achieves better performance
than standard CRFs because they take into account long-range features.
HMM models have been applied to a wide variety of problems in information
extraction and natural language processing, especially POS tagging
$[$Kupiec1992$]$ and named entity recognition $[$Bikel1999$]$. Taking POS
tagging as an example, each word is labeled with a tag indicating its
appropriate part of speech, resulting in annotated text, such as: ” $[$VB
heat$]$ $[$NN water$]$ $[$IN in$]$ $[$DT a$]$ $[$JJ large$]$ $[$NN vessel$]$”.
Given a sequence of words, e.g. ” heat water in a large vessel”, the task is
to assign a sequence of labels e.g. ” VB NN IN DT JJ NN”, for the words. HMM
models determine the sequence of labels by maximizing a joint probability
distribution computed from the manually annotated training data. In practice,
Markov processes like HMM require independence assumptions among the random
variables in order to ensure tractable inference.
The primary advantage of CRFs over HMMs is their conditional nature resulting
in the relaxation of the independence assumption. However, the problem of
exact inference in CRFs is nevertheless intractable. Similarly to HMMs, the
parameters are typically learned by maximizing the likelihood of training data
and need rely on iterative techniques such as iterative scaling
$[$Lafferty2001$]$ and gradient-descent methods $[$Sha2003$]$.
All these models depend on multiple parameters to define the underlying prior
probabilistic distributions used to generate the posterior distributions which
describe the observed labeled data in order to infer classification on
unlabeled data. Canonical well know and well-studied probability distributions
like Gaussian, multinomial, Poisson, or Dirichlet are primarily used in these
models. The paradigmatic mathematical formulation of these models in terms of
”cost”, ”score” or ”energy” functions rely on the maximization of the latter.
Unfortunately, these models are embedded in huge multi-dimensional spaces.
Finding the set of parameters which actually minimize these functions is a
combinatorial optimization problem and is known to be NP-hard. Heuristic
algorithms to compute the parameters are fairly complex and difficult to
implement $[$Teyssier2012$]$. Moreover, parameter estimation for the prior
distribution functions is essentially based on conditional counting with
various normalization and regularization smoothing schemes to correct for
sparseness of a given occurrence in the observed and training data. These
parameterization schemes greatly vary in the literature and there is no
canonical or natural heuristic to determine them for each application domain.
The learning algorithms for these probabilistic models try to ?nd maximum-
likelihood estimation (MLE) and maximum a posteriori probability (MAP)
estimators for the parameters in these models. Most of the time, no closed
form solutions can be provided.
In order to be able to make predictions from these models, canonical learning
schemes such as Expectation-Maximization (EM) $[$Blei2003$]$ $[$Borman2004$]$,
Gibbs sampling and Markov Chain Monte Carlo are used extensively
$[$Andrieu2003$]$. In recent years, the main research trend in this field has
been in the context of two classes of text data:
* •
Dynamic Applications
The large amount of text data being generated by dynamic applications such as
social networks or online chat applications has created a tremendous need for
clustering streaming text. Such streaming applications must be applicable to
text which is not very clean, as is often the case for social networks.
* •
Heterogeneous Applications
Text applications increasingly arise in heterogeneous applications in which
the text is available in the context of links, and other heterogeneous
multimedia data. For example, in social networks such as Flickr, clustering
often needs to be applied. Therefore, it is critical to effectively adapt
text-based algorithms to heterogeneous multimedia scenarios.
Unsupervised learning techniques do not require any training data and
therefore no manual effort. The two main applications are clustering and topic
modeling. The basic idea behind topic modeling is to create a probabilistic
generative model for the text documents in the corpus. The main approach is to
represent a corpus as a function of hidden random variables, the parameters of
which are estimated using a particular document collection.
There are two basic methods for topic modeling: Probabilistic Latent Semantic
Indexing (PLSI) $[$Hofmann1999$]$ and Latent Dirichlet Allocation (LDA)
$[$Blei2004$]$. Supervised information extraction comprises Hidden Markov
models, Conditional Random Fields or Support Vector Machines. These
techniques are currently heavily in use in the machine learning industry. All
these techniques require the preprocessing of documents through manual
annotation. For domain-speci?c information extraction systems, the annotated
documents have to come from the target domain. For example, in order to
evaluate gene and protein name extraction, biomedical documents such as PubMed
abstracts are used. If the purpose is to evaluate general information
extraction techniques, standard benchmark data sets can be used. Commonly used
evaluation data sets for named entity recognition include MUC
$[$Grishman1996$]$, CoNLL-2003 $[$Tjong2003$]$ and ACE $[$ACE$]$. For relation
extraction, ACE data sets are usually used. Currently, state-of-the-art named
entity recognition methods can achieve around 90% of F-1 scores (geometric
mean of precision and recall) when trained and tested on the same domain
$[$Tjong2003$]$.
For relation extraction, state-of-the-art performance is lower than that of
named entity recognition. On the ACE 2004 benchmark dataset, for example, the
best F-1 score is around 77% for the seven major relation types
$[$LongHua2008$]$.
It is generally observed that person entities are easier to extract, followed
by locations and then organizations. It is important to note that when there
is a domain change, named entity recognition performance can drop
substantially. There have been several studies addressing the domain
adaptation problem for named entity recognition $[$Jiang2006$]$.
Another new direction is open information extraction, where the system is
expected to extract all useful entity relations from a large, diverse corpus
such as the World Wide Web. The output of such systems includes not only the
arguments involved in a relation but also a description of the relation
extracted from the text.
In $[$Banko2008$]$, Banko and Etzioni have introduced an un-lexicalized CRF-
based method for open information extraction. This method is based on the
observation that although di?erent relation types have very di?erent semantic
meanings, there exists a small set of syntactic patterns that covers the
majority of the semantic relation mentions. The method categorizes binary
relationships using a compact set of lexico-syntactic patterns. The heuristics
are designed to capture dependencies typically obtained via syntactic parsing
and semantic role labeling. For example, a heuristic used to identify positive
examples is the extraction of noun phrases participating in a subject verb-
object relationship e.g. ” $<$Einstein$>$ received $<$the Nobel Prize$>$ in
1921.” An example of a heuristic that locates negative examples is the
extraction of objects that cross the boundary of an adverbial clause, e.g. ”
He studied $<$Einstein’s work$>$ when visiting $<$Germany$>$”.
The set of features used by CRF is largely similar to those used by state-of-
the-art relation extraction systems. They include part-of-speech tags
(predicted using a separately trained maximum-entropy model), regular
expressions (e.g. detecting capitalization, punctuation, etc.), context words,
and conjunctions of features occurring in adjacent positions within six words
to the left and six words to the right of the current word. The Open IE system
extracts different relationships with a precision of 88.3% and a recall of
45.2%. However, the CRF-based IE system (O-CRF) has a number of limitations,
most of which are shared with other systems that perform extraction from
natural language text. First, O-CRF only extracts relations that are
explicitly mentioned in the text; implicit relationships that could inferred
from the text would need to be inferred from O-CRF extractions. Second, O-CRF
focuses on relationships that are primarily word-based, and not indicated
solely from punctuation or document-level features. Finally, relations must
occur between entity names within the same sentence.
With the fast growth of textual data on the Web, we expect that future work on
information extraction will need to deal with even more diverse and noisy
text. Weakly supervised and unsupervised methods will play a larger role in
information extraction. The various user-generated content on the Web such as
Wikipedia articles will also become important resources to provide some kind
of supervision for $[$Aggarwal2012$]$.
In some applications, prior knowledge may be available about the kinds of
clusters available in the underlying data. This prior knowledge may take on
the form of labels attached with the document which indicate its underlying
topic.
Such knowledge can be very useful in creating signi?cantly more coherent
clusters, especially when the total number of clusters is large. The process
of using such labels to guide the clustering process is referred to as semi-
supervised clustering. This form of learning is a bridge between the
clustering and classi?cation problem, because it uses the underlying class
structure, but is not completely tied down by the speci?c structure. As a
result, this approach is applicable to both the clustering and classi?cation
scenarios. The most natural way of incorporating supervision into the
clustering process is partitional clustering methods such as k-means. This is
because supervision can be easily incorporated by changing the seeds in the
clustering process $[$Aggarwal2004, Basu2002$]$. A number of probabilistic
frameworks have also been designed for semi-supervised clustering
$[$Nigam1998, Basu2004$]$.
However real world applications in these fields currently lack scalable and
robust methods for natural language understanding and modeling
$[$Aggarwal2012$]$. For example, current information extraction algorithms
mostly rely on costly, non-incremental, and time consuming supervised learning
and generally only work well when sufficient structured and homogeneous
training data is available. This requirement drastically restricts the
practical application domains of these techniques $[$Aggarwal2012$]$.
All the models described above are computationally intensive. The e?ciency of
the learning algorithms is always an issue, especially for large scale data
sets which are quite common for text data. In order to deal with such large
datasets, algorithms with linear or even sub-linear time complexity are
required, for which parallelism can be used to speed up computation.
MapReduce $[$Dean2004$]$ is a programming model for processing large data
sets, and the name of an implementation of the model by Google. MapReduce is
typically used to do distribute computations on clusters of computers. Apache
Hadoop (http://hadoop.apache.org) is an open-source implementation of
MapReduce. It supports data-intensive distributed applications and running
these applications on large clusters of commodity hardware. The major
algorithmic challenges in map-reduce computations involve balancing a
multitude of factors such as the number of machines available for
mappers/reducers, their memory requirements, and communication cost (total
amount of data sent from mappers to reducers) $[$Foto2012$]$.
Figure 1 (taken from $[$Hockenmaier$]$) presents the training time for
syntactic translation models using Hadoop. On the right, the benefit of
distributed computation quickly outweighs the overhead of a MapReduce
implementation on a 3-node cluster. However, on the left, we see that
exporting the data to the distributed file system incurs cost nearly equal to
that of the computation itself.
Existing tools do not lend themselves to sophisticated data analysis at the
scale many users would like $[$Maden2012$]$. Tools such as SAS, R, and Matlab
support relatively sophisticated analysis, but are not designed to scale to
datasets that exceed even the memory of a single computer. Tools that are
designed to scale, such as relational DBMSs and Hadoop, do not support these
algorithms out of the box. Additionally, neither DBMSs nor MapReduce are
particularly efficient at handling high incoming data rates and provide little
out-of-the-box support for techniques such as approximation, single-pass/sub
linear algorithms, or sampling that might help ingest massive volumes of data.
Figure 1: MapReduce/Hadoop comparison on training data
Several research projects are trying to bridge the gap between large-scale
data processing platforms such as DBMSs and MapReduce, and analysis packages
such as SAS, R, and Matlab. These typically take one of three approaches:
extend the relational model, extend the MapReduce/Hadoop model, or build
something entirely different. In the relational camp are traditional vendors
such as Oracle, with products like its Data Mining extensions, as well as
upstarts such as Greenplum with its Mad Skills project. However, machine
learning algorithms often require considerably sophisticated users, especially
with regard to selecting features for training and choosing model structure
(for instance, for regression or in statistical graphical models).
In the past two decades $[$DU2012$]$, most work in speech and language
processing has used ” shallow” models which lack multiple layers of adaptive
nonlinear features. Current speech recognition systems, for example, typically
use Gaussian mixture models (GMMs), to estimate the observation (or emission)
probabilities of hidden Markov models (HMMs) $[$Singh2012$]$. GMMs are
generative models that have only one layer of latent variables. Instead of
developing more powerful models, most of the research has focused on finding
better ways of estimating the GMM parameters so that error rates are decreased
or the margin between different classes is increased. The same observation
holds for natural language processing (NLP) in which maximum entropy (MaxEnt)
models and conditional random fields (CRFs) have been popular for the last
decade. Both of these approaches use shallow models whose success largely
depends on the use of carefully handcrafted features.
Shallow models have been effective in solving many simple or well-constrained
problems, but their limited modeling power can cause difficulties when dealing
with more complex real-world applications. For example, a state-of-the-art
GMM-HMM based speech recognition system that achieves less than 5% word error
rate (WER) on read English may exceed 15% WER on spontaneous speech collected
under real usage scenarios due to variations in environment, accent, speed,
co-articulation, and channel.
Existing deep models like hierarchical HMMs or Higher Order Conditional Random
Fields (HRCRFs) and multi-level detection-based systems are quite limited in
exploiting the full potential that deep learning techniques can bring to
advance the state of the art in speech and language processing.
### 2.2 Recommender Systems
Recommender systems apply data mining techniques and prediction algorithms to
the prediction of users’ interest on information, products and services among
vast amounts of available items (e.g. Amazon, Netflix, movieLens, and
VERSIFY). The growth of information on the Internet as well as the number of
website visitors add key challenges to recommender systems $[$Almazro2010$]$.
Two recommendation techniques are currently extensively used in the industry
$[$Zhou2012$]$: content based filtering (CBF) and collaborative filtering
(CF). The content-based approach recommends items whose content is similar to
content that the user has previously viewed or selected. The CBF systems
relies on an extremely variable specific representation of items features,
e.g. for a movie CBF, each film is featured by genre, actors, director, etc.
Knowledge-based recommendation attempts to suggest objects based on inferences
about user needs and preferences. In some sense, all recommendation techniques
could be described as doing some kind of inference. Knowledge-based approaches
are particular in that they have functional knowledge: they have knowledge
about how a particular item meets a particular user need and can therefore
reason about the relationship between a need and a possible recommendation.
The user profile can be any knowledge structure that supports this inference.
In the simplest case, as in Google, it may simply be the query that the user
has formulated. In others, it may be a more detailed representation of the
user needs $[$Burke2002$]$.
The features retained to feed recommendation systems are generally created by
human beings. Building the set of retained features is of course very time
consuming, expensive and highly subjective. This subjectivity may impair the
classification and recommendation efficiency of the system.
Collaborative filtering (CF) systems collect information about users by asking
them to rate items and make recommendations based on the highly rated items by
users with similar taste. CF approaches make recommendations based on the
ratings of items by a set of users (neighbors) whose rating profiles are most
similar to that of the target user. In contrast to CBF systems, CF systems
rely on the availability of user profiles which capture past ratings and do
not require any human intervention for tagging content because item knowledge
is not required. CF is the most widely used approach for building recommender
systems. It is currently used by Amazon to recommend books, CDs and many other
products. Some systems combine CBF and CF techniques to improve and enlarge
the capabilities of both approaches.
The quality and availability of user profiles is critical to the accuracy of
recommender system. This information can be implicitly gathered by software
agents that monitor user activities such as real time click streams and
navigation patterns. Other agents collect explicit information about user
interest from the ratings and items selected. Both the explicit and implicit
methods have strengths and weaknesses. On the one hand, explicit interactions
are more accurate because they come directly from the user but require a much
greater user involvement. On the other hand, implicit monitoring requires
little or no burden on the user but inferences drawn from the user interaction
do not faithfully measure user interests. Hence, user profiles are often
difficult to obtain and their quality is also both hard to ensure and assess.
Current existing user profiling for recommender systems is mainly using user
rating data. Hundreds of thousands of items and users are simultaneously
involved in a recommender system, while only a few items are viewed, rated or
selected by users. Sarwar et al. $[$Sarwar2001$]$ have reported that the
density of the available ratings in commercial recommender systems is often
less than 1%. Moreover, new users start with a blank profile without selecting
or rating any items at all. These situations are commonly referred to as data
sparseness and cold start problem. The current recommender algorithms are
impeded by the sparseness and cold start problems.
With the increased importance of recommender systems in e-commerce and social
networks, the deliberate injection of false user rating data has also
intensified. A simple, yet effective attack on recommender systems is to
deliberately create a large number of fake users with pseudo ratings to favor
or disfavor a particular product. With such fake information, user profile
data can become unreliable.
In summary, without sufficient knowledge about users, even the most
sophisticated recommendation strategies are not be able to make satisfactory
recommendations. The cold start, data sparseness and malicious ratings are
outstanding problems for user profiling. These make user profiles the weakest
link in the whole recommendation process.
To tackle these issues, social recommender systems use user-generated
(created) contents which comprise various forms of media and creative works as
written, audio, visual and combined created by users explicitly and pro-
actively $[$Pu2012$]$. Another path to improve performance, combine the above
techniques in so-called hybrid recommenders $[$Burke2002$]$.
## 3 mARC
### 3.1 Principle of Operation
The Memory by Association and Reinforcement of Contexts (mARC) is an
incremental, unsupervised and adaptive learning and pattern recognition
system. Its ground principles allow the automatic detection and recognition of
different types of patterns which are contextually linked.
mARC is built upon the premises introduced in $[$USP2004$]$. Companies such as
IBM, Seagate Technology, and Nuance Communications have referenced this work
in their patents and products.
Unlike systems such as feed-forward or recurrent neural networks and guided
propagation networks (GPN), mARC does not require a large memory space to run
and has a fast response time. Furthermore, artificial neural network systems
require the weights to be known before the network can be deployed and their
capability to recognize patterns in known systems are limited
$[$Papert1969$]$.
The core of mARC is a fractal self-organized network whose basic element is
called a cell. A cell is an abstract structure used to encode any pattern from
the incoming signal or any pattern from feedback signal inside the network.
The fractal structure naturally emerges as a consequence of the building and
learning processes taking place inside the whole network.
A mARC server consists of the following elements:
* •
A networking socket.
* •
A reading head or sensorial layer.
* •
A highly-optimized integrated binary database for fast storage and indexing of
the input signal.
* •
A core referred to as knowledge.
* •
An application programming interface (API) which allows interaction with the
core.
The network is initially empty, i.e. it does not contain any cells. At the top
of the network is a reading head which reads a causal one-dimensional numeric
input signal.
In the input signal flow, the relative event time (causal appearance)
describes the position of an event relative to another event. This can be seen
as a relative time quantification between two event occurrences. As an
example, if the incoming signal flow is 838578, sampled as 83—85—78 coding for
the word SUN in extended ASCII, the event U appears after event S and prior to
event N. This is the relative time quantification of the event U in the
context of the pattern, the word SUN in this example. In general, events are
handled at the cell level and relative event times are handled at the global
network level.
The mARC implementation described in this paper is calibrated to sample the
input signal byte-wise. In other words, it interprets the input signal as
extended ASCII. As the ASCII input signal is presented to the network, it is
transcoded into cells in the network. The network grows according to the input
signal pattern. The input signal is composed of basic components or events in
some order of occurrence linked by unknown causal patterns.
If a cell matching the basic component is found, that cell is reinforced
(reinforcement learning and recognition) in the network. If a cell does not
exist, a new cell is created to hold the basic component. As the cells are
propagated in the network, a path encoding the pattern is automatically
inserted in the structure of the network.
The learning and building processes are deeply intertwined. At any given time,
the network contains a plurality of cell structures enabled to be linked to
parent cells, cousin cells, and children cells in what we refer to as a
”tricel” physical structure. Each cell controls its own behavioral functions
and transfers control to the next linked cells (self-signal forward and
backward internal and external propagation).
A cell may have an attribute type of termination or glue. A termination
attribute marks the end of a learned and recognized segment in a pattern. A
glue attribute indicates that a cell is an embedded event in a pattern. That
is, a termination attribute typically marks an end of a significant recognized
pattern. The termination cell may also include a link to another sub-network
where related patterns are stored. These networks further aid in identifying
an input pattern.
In other words, the network itself is the resultant of deeply inter-related
and interacting layers of cells which draw a huge and massively multi-
dimensional knowledge non-directed graph in the mathematical sense.
### 3.2 The mARC Programming Model
Interacting with mARC is performed via an application programming interface
(API). The purpose of the API is to translate the internal structures of the
mARC knowledge into object collections which are easier to handle
procedurally.
For a text signal-oriented mARC, objects are typically words, compound
expressions or phrases. The API automatically translates the inner contextual
information of the mARC knowledge into weighted values for each object in a
set according to their generality and activity with respect to the whole
knowledge.
We distinguish two kinds of sets. We call genuine or canonical context, a set
of patterns which are genuinely correlated by the core. We call generic
context a context which is manually created using the API.
For example, let us assume that we want to probe the knowledge about the
pattern bee. The API contains a specific command for this. We instruct the API
to build an empty context and put the pattern bee in it. For now, the context
has no genuine meaning with respect to the knowledge. The resulting context is
generic. The API allows us to retrieve the genuine contexts from this generic
context.
The genuine contexts are learned by the knowledge automatically from the
corpus which has been submitted to it. The API allows the manipulation of the
genuine contexts to perform true contextual analysis from the knowledge
extracted from a corpus. Each element of a context (generic or genuine) is
associated with two numerical values or weights internally computed from the
knowledge: the generality and the activity. The activities of each element in
a generic context have no meaning; they are arbitrarily fixed by the user. The
activities are reevaluated by the knowledge once the genuine contexts issued
from the generic context are retrieved from the knowledge.
The generality of an element inside a genuine context is a numerical estimate
of the corresponding human notion with respect to the corpus which has been
learned. The activity of an element inside a genuine context is an algebraic
measure of the intensity of the coupling of each constituent of this context
with respect to all the connections in the knowledge. The strength of this
coupling is proportional to the number of connections between an element and
its corresponding linked elements in the knowledge network.
## 4 Key Differentiators
mARC presents a number of key differentiators compared to other data
processing and querying technologies:
1. 1.
Independence from the data
mARC is independent from the nature of the input signal. For example, mARC
extracts contexts from textual data independently of the language the text is
written in. mARC handles any textual data as a numerical signal. In essence,
it is therefore a general numerical signal analysis processing unit. Right
now, it is restricted to handle byte-wise sampled signal i.e. Latin 9 or
extended ASCII.
1. 2.
Access time
Access to contextual data is at least one order of magnitude faster than
access to data using classical SQL-based language.
1. 3.
Noise filtering and error correction
Assuming enough contextual information is available, useful data can be
filtered from noise. Data can also be reconstructed by mARC if it has been
fragmented or altered.
1. 4.
Storage efficiency
mARC auto-regulates the amount of storage allocated to index the contextual
information. The size of the context information depends on the density of the
relationships in the data set but is bounded by O (log n) of the data set
size.
For plain text data, the context space typically evolves from O (n) for a
small data set to O (log n).
1. 5.
Ease of programming
The mARC APIs provide an easy programmatic access to the context information.
This allows developers to efficiently develop context-aware data management
applications.
## 5 Applications
mARC has a broad potential for applications. It is particularly well suited to
big data applications.
* •
Keyword-oriented search engines.
* •
Context-oriented search engines. Contextual search is to be understood as the
intuitive meaning of contexts in free form texts. E.g. the terms of a request
or of an article, are not to be present in the result of a user request, or in
a similarity process. Contextual text or request processing is able to solve
ambiguities, and to extract the discriminant or low frequency significant
information.
* •
Contextual meta search engine, to enhance existing search facilities
* •
Contextual indexation algorithms to enhance existing search facilities
* •
User request profiling (solving ambiguous requests by user context)
* •
User profiling (indexing each user by its requests or other criterions)
* •
Contextual document routing inside a global information system
* •
Contextual document matching with a given static ontology
* •
Contextual survey of documents flows
* •
Contextual similarity matching between documents
## 6 Experimental Results
In order to demonstrate some of the benefits of mARC, we have built a basic
World Wide Web search engine demonstrator using the mARC APIs. We use it to
study the performance of mARC-based search engines with that of a high-
performance procedural search engine: Google search.
## 7 mARC Search Engine Demonstrator
The mARC search engine demonstrator provides search features similar to Google
search: keyword-based queries and auto-completion of search queries.
The demonstrator provides additional functionality not currently accessible to
procedural search engines:
* •
Search for contextually-related articles, called similar article function in
the remainder of the paper.
* •
Query auto-completion based on pattern association (noisy recognition of
misspelled queries).
* •
Meta-search engine for image retrieval.
For the purpose of this study, the demonstrator has been restricted in order
to be comparable with a keyword-based or N-gram based search engine like
Google. The full contextual search engine cannot be used in this study because
it would not easily allow a side by side comparison with a procedural search
engine like Google, mainly because it does not handle keywords in the Google
sense.
Data Corpus
The study is performed on both the English and French Wikipedia corpuses. For
the comparison, the mARC demonstrator indexes 3.5 million English articles and
1 million French articles and Google indexes 3.9 million English articles and
1.4 million articles.
The demonstrator index is built from local snapshots of the Wikipedia French
and English corpuses taken previously. On the other hand, the Google index is
kept up to date quasi-real-time. This explains the difference in the number of
articles indexed. We do believe, however, that the difference in the size of
the corpuses does not significantly affect the conclusions of this study.
## 8 Validity of the Study
The Google architecture is distributed on a very large scale $[$GSA$]$. The
demonstrator is hosted on an Intel CoreI5-based server running Windows 7. This
can make performance comparison claims difficult to back due to the difference
in architectures, raw computing power, size of the indices, network latencies,
etc. In the following, we provide elements to justify the validity of the
comparison.
Google sells search appliances which allow deploying the Google search engine
within an enterprise. The physical servers sold by Google are equivalent in
specifications to the one used to run the mARC demonstrator. More details
about the Google Search Appliance can be found at $[$GSA$]$.
Google advertises a minimum 50 ms. response time and an average response time
of less than one second for a corpus of 300000 to 1000000 documents for the
Google Search Appliance. Pareto’s rule gives an approximate 250 ms. average
response time per request.
The user forums for the Google Search Appliance report a lower performance of
the Google Search Appliance compared to the Internet search engine. Google
advertises a 250 ms. average response time for its Internet search engine.
The choice of Wikipedia for the analysis is also relevant. Google search
largely favors Wikipedia when returning search results and Wikipedia
consistently appears in the top five results returned by Google search
$[$IPS2012$]$. Google search is highly optimized for Wikipedia. Therefore, we
believe that restricting the comparison between the mARC demonstrator and
Google search to the Wikipedia corpus does not put Google search at a
disadvantage.
Another potential objection to the results presented this study is
scalability. We are comparing the performance a dedicated demonstrator to a
search engine which handles three to four billion requests per day and indexes
30 billion documents.
Given the structure of the World Wide Web and the redundancy rate in
documents, Google implements a binary tree for the data. With each server
managing 108 primary documents, the binary tree is 10 levels deep for 30.109
documents. Therefore, each request involves a cluster of at most 10 servers,
11 with an http front-end server.
In addition, Google optimizes requests by dispatching the request to several
clusters in parallel. The cluster which has cached the request has the
shortest response time. We estimate that the number of concurrent cluster
varies between 1 and 25 depending on the load. This gives us an average number
of 120 servers participating simultaneously to the resolution of a request.
Google advocates 250 servers involved in the resolution of each request
$[$Google2012$]$.
The Google search infrastructure is dimensioned to sustain 4.109 requests per
day, which is 46300 requests per second. The number of servers to ensure a 1
second response time is 46300 x 11 = 509300. The number of servers operated by
Google is estimated to be around 1.7 million so the load of a Google search
server is therefore comparable to the observed load on the mARC demonstrator
server.
These considerations lead us to believe that the response time comparison
between Google search and the mARC demonstrator is valid.
In the following sections, we analyze some of the results gathered with the
mARC search engine demonstrator to evaluate how the mARC claims stand up to
experimentation.
## 9 Independence from the Data Set
In the demonstrator, indexation and search are identical for the English and
French corpuses. There is no language-specific customization. We can easily
demonstrate the same independence from the data set on the Wikipedia corpus in
other languages.
However, we have made two simplifying assumptions in the implementation of the
demonstrator:
* •
The input signal is segmented into 8 bits packets.
* •
The space character is implicitly used to segment the input signal.
As a consequence of this simplification, the demonstrator does not currently
allow the validation of the claim of universal independence from the data.
Nevertheless, it proves a minima the independence from the language.
## 10 Storage Efficiency
The following table presents the size in MB of various data elements for the
mARC search engine demonstrator: size of the mARC contextual RAM, size of the
index and the inverse resolution database, as well as the stored data set size
corresponding to the whole English and French Wikipedia corpuses.
Corpus | mARC | Index | Inverse resolution | Total | Data | Ratio %
---|---|---|---|---|---|---
Fr | 500 | 900 | 731 | 2100 | 4000 | 52.5
En | 600 | 1600 | 1500 | 3700 | 11000 | 33.6
From this data, we observe the following:
* •
The size of the mARC does not grow linearly with the size of the data set.
Rather, it grows in log (data size).
* •
The size of the index is at most around 50% of the size of the data set. The
index contains all the information necessary to implement the search
functionality.
It should be noted that comparable full text search functionality provided by
relational database vendors or search engines such as Indri or Sphinx requires
indices which are 100% to 300% of the data set size $[$Turtle2012$]$ depending
on the settings of the underlying indexation API. The size of the Google index
was not available at the time of writing.
Furthermore, mARC is at a relative disadvantage when doing keyword-based
search (which is needed for this comparison). A mARC-based search engine using
the context information more directly (as exemplified by the similar article
feature of the demonstrator) would leverage more of the power of mARC. This
approach would reduce the overall memory footprint of the mARC search engine
metadata by one order of magnitude.
## 11 Response Time
We have measured the response time for the two search engines over two classes
of requests:
* •
Popular queries. A set of a hundred requests among the most popular for
English and French Wikipedia at the time of the study $[$techxav2009$]$.
* •
Complex queries. For this measurement, we use the title of a Wikipedia article
returned by the search engine in response to a query as the query (i.e.
copy/paste). This allows us to take into account the trend towards larger
requests which has been observed in recent years $[$WIKI2001$]$.
In order to account for any caching effects, each query is run four times in
the experiments. The first time to measure the response time for a non-cached
request and the subsequent times to average the response time after the
request has been cached.
We measure the response time for each query run. The response time reported in
the results only accounts for the wall clock time taken by the search engine
to resolve the requests. We exclude all network, protocol, and response
formatting overheads from the analysis.
The average response time is extrapolated using Pareto’s 80/20 rule:
First request (non-cached) * 0.2 + average (next 3 requests) * 0.8
In addition, the true recall rate is also measured. It should be noted that
for the Google search engine, the recall rate returned by the server in the
query result is potential. E.g.:
About 158,000,000 results (0.19 seconds)
We measure the true recall rate by navigating to the last page of results
returned by Google. In order to limit the results to the most relevant
articles, Google prunes out articles with similar contents. Including the
pruned-out articles in the query does not significantly affect the recall
rate. In our measurements, the real recall rate never exceeded 800 results.
For the mARC demonstrator, the real recall rate is displayed. All articles are
directly accessible from the results page.
The experimental results are summarized in the following table:
| Avg. response time (ms.) | Real recall rate (articles)
---|---|---
| mARC | Google | Ratio | mARC | Google | Ratio
Popular queries (EN) | 12.3 | 132.3 | 16.4 | 808 | 621 | 1.30
Popular queries (FR) | 11.6 | 119.1 | 20.5 | 647 | 415 | 1.56
Complex queries (EN) | 19.3 | 261.3 | 15 | 887 | 302 | 2.94
Complex queries (FR) | 13.3 | 279.2 | 24.1 | 778 | 299 | 2.60
The results show significantly better response times for the mARC
demonstrator. The detailed results are presented in appendix 1.
In terms of computing resources, the mARC CPU utilization on the demonstrator
is measured to less than 10% of the response time. The remainder of the
response time is disk access, formatting, API and communication overhead.
Similar results are not available for Google.
For the popular requests, the average response time for Google when restricted
to the domains en.wikipedia.org and fr.wikipedia.org are respectively 119 and
132 ms. The response time for the same requests without the domain restriction
is around 320 ms. This measurement is consistent with Google’s advertised
average response time of 250ms. From this we can deduce that:
* •
Google optimizes the response time for popular domains, such as Wikipedia.
* •
The Google servers are lightly loaded, as indicated by the small variance of
response times.
This gives us reasonable confidence that the results reported in this paper
are meaningful. mARC shows response times over an order of magnitude better
than Google (see Appendix for numerical details).
It should be noted that with mARC once the initial results page has been
access, all the results have been cached. As a consequence, the average access
time to a page containing the next 20 results is in the order of 5 ms. With
Google, displaying the next results is equivalent to issuing a new (non-
cached) request between 70 and 300 ms. for each page.
In addition, it should be noted that the mARC demonstrator does not perform
any optimization on the request itself. Each result page change causes the
request to be completely re-evaluated, as in Google search. A trivial
optimization would be to keep the results in a session variable to optimize
the scanning of the cached results on the mARC server. This would reduce the
response time to about 0.5 ms. per results page, independently of the
complexity of the query.
Given that in practice the average request generates navigation to 2.5 pages,
we can interpolate the average response time of a mARC-based search engine to
be less than 5 ms. which is 25 times faster than Google search.
## 12 Search Relevance
Even though search relevance is a largely subjective notion and cannot be
accurately measured, it is nevertheless very real and important. The only
valid measurement technique would be some form of double blind testing rating
user satisfaction. Nevertheless, we attempt in the following to provide some
insights into the differences in relevance between a mARC-based search engine
and a procedural search engine like Google.
The Google search algorithm is well documented in the literature. In the
following, we focus on describing the search strategies implemented in the
mARC demonstrator centered around:
1. 1.
Keyword-based queries and more pattern-sensitive detection.
2. 2.
Context similarity-based queries and associative-sensitive detection.
The mARC search demonstrator resolves queries using both search strategies in
parallel. The results are displayed in two columns on the results page to
allow for easy comparison as shown in figure 2 below.
Figure 2: Figure 2: mARC demonstrator search results
## 13 Keyword-based Search
The first column (Zone 1) displays the results for the keyword-based search.
The current implementation of mARC does not feature customizable indexation
strategies. Therefore, keyword-based search is only approximated through the
API.
In this mode, there is no contextual evaluation of the request. This
restriction allows the demonstrator to emulate as closely as possible the
operation of a procedural search engine like Google. To this effect, the query
routine favors elementary word contexts over associative contexts. In
practice, the behavior is essentially similar to a pure keyword-based
approach, nevertheless with a touch of implicit associativity.
We observe the following trends:
* •
For generic and well know query terms, the shape is preponderant. Both the
titles and the bodies of the Wikipedia articles returned as results contain
the keywords. Compared to a pure keyword-based request which only returns
normalized relevance with respect to the matched keywords, the contextual
activation greater than 100% indicates that the returned article is
contextually over-activated and thus contextually plainly meaningful. In other
words, the non-normalized (as in this demonstrator) confidence rate over 100%
means that the resulting documents contain not only the keywords, as a
significant pattern, but also a part of the contexts associated with this
keyword inside the knowledge.
Example query: programming.
* •
For more qualified queries, associativity becomes preponderant. This means
that the articles ranked as most relevant by mARC may not contain the terms of
the request.
Example query: Ferdinand de Saussure
Out of the 41 first results returned (all evaluated to be relevant by a panel
of human observers), 9 results have been retrieved through associative
contexts and do not contain the terms Ferdinand de Saussure.
There are more differences in behavior compared to a purely procedural
approach.
Adding terms to a query is equivalent to adding shape contexts. The contexts
interact with each other. Since the search is focused on shape, this is
equivalent to an intersection of keywords for the most activated articles. The
search results are ordered by the analogy with the shape of the request (in
the title or in the body of the article). Then, as the activation decreases,
sub-contexts start appearing up to a point of disjoint shape sub-contexts with
low activation. In this configuration, activation $<$=100 implies a shape and
association more and more disjoint from the request.
Article titles are not privileged. But since titles are small contexts, the
relative importance of each term is more contrasted.
Example queries: thallium, skirt, oxygen, oxygen + nitrogen, octane rating
In summary, for shape-based search we observe the following when comparing the
results returned by the mARC demonstrator and Google:
* •
One term queries
Example queries: history, orange, metempsychosis
For generic requests like history, the results are very similar; only
differing in the order of the results. The mARC demonstrator has a slight
tendency to order the results in categories: history and geographical context
(history of various countries), then history of well know countries (U.S.A.,
France, etc.), then the broad general categories such as history of sciences,
history of literature, economy, military, etc.
For generic and ambiguous requests like orange, the behavior is roughly the
same for both search engines. However, we observe a slightly better tendency
for mARC to vary the semantic contexts on the first few result pages.
For targeted requests (i.e. request that do not yield a lot of results), we
observe that the mARC demonstrator returns significantly more relevant results
than Google. The reason is that semantic contexts are weighted more heavily
and return matches for both form and substance. The return rate is higher.
Overall, we observe that the more targeted the request, the more relevant the
results returned by the mARC demonstrator are.
* •
Two or three terms queries
We observe that the two search engine can return very different results for
these request:
* •
If the terms have little relationship between them (e.g. vertebrate
politics), Google returns a list of articles containing all terms but without
real semantic connection. To the contrary, the mARC demonstrator tries to
consolidate the two contexts and varies the results on the first few result
pages. Articles containing all terms are generally not activated enough to be
presented.
* •
If the terms are connected with equivalent generalities, both search engines
return comparable results, e.g. for Roddenberry and Spock.
* •
If the terms are connected with disparate generalities, e.g. wine and quantum
Google returns more relevant results. The mARC demonstrator tends to return
only one to five seemingly relevant articles.
* •
If the terms are precise, the mARC context associativity kicks in. More
results are returned and are more relevant than Google’s (e.g. Stegastes
fuscus, Tantalum 180m, Niobe daughter Tantalus, Amyclas, hemoprotein, cyanide
intoxication, organophosphate intoxication, chrome cancer (professional
disease), anaerobic respiration).
It should be noted that Google is not very sensitive to the ordering of terms
within the query. The mARC demonstrator can be if the order carries a semantic
change. E.g. red green and green red are treated as equal by the mARC while
Paris Hilton and Hilton Paris are not.
Overall, we find that Google search provides slightly more relevant results in
the case of keywords-based search. This can be easily understood. On one hand
Google search relies intensively rely on user requests to improve the search
results. On the other hand we, as humans, use Google search on a daily basis.
In a way we are self-trained to know what results to expect. It is in this
query range that the trio intersection/return rate/relevance is the least
random.
However, the real-time article similarity matching provided by the mARC
demonstrator offers dynamic query disambiguation capabilities which are out of
the reach of Google search.
* •
Long queries
These requests are either article titles (four or more terms) or copied/pasted
from article text.
For these queries, the mARC demonstrator provides indisputably more relevant,
better categorized results than Google search. The request contains enough
contextual information for the mARC to evaluate and classify the articles in a
more relevant way than Google search.
On a number of categorical articles from Wikipedia, we observe that the mARC
demonstrator and Google search return very similar results for the first two
or three results pages.
Example: list of IATA airport codes
Surprisingly, Google search returns the correct article as result, even though
the article does not contain all of the terms of the query. The reason for
this behavior is that the Wikipedia article files often contain links (added
by the authors) which point to articles relevant to the topic. Google uses
these links to improve its search relevance. The mARC does not use this
metadata and only considers the text of the articles.
It is interesting to note that both search engines return the same results in
this case. This emphasizes the ability of the mARC to detect semantic
relationships. mARC does a comparable job in finding semantic relationships
between articles as the Wikipedia authors.
## 14 Similar Article Search
The similar article search results are displayed on the right column on the
results page (Zone 2). A similar capability is not available for Google
search.
Example: Orange and SA (”Similar Articles” search button) in the different
articles returned in Zone 1.
We find that the similar article search feature enhances the keyword-based
search results in a very interesting and significant way. The conjunct use of
the pattern-based search and similarity-based search allows a semantic-driven
navigation from the initial query with low risk of ambiguity. It gives access
to different, yet relevant, results which are not accessible through keyword-
based search.
In addition, similarity-based search helps categorize the results of the
keyword-based query.
This is a novel and unique feature of mARC.
## 15 Ease of Programming
The PHP code which implements the similar article functionality in the
demonstrator is shown below. The context detection and selection logic is
entirely provided in a generic manner by mARC.
public function connexearticles ($rowid) {
// similar article
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.CLEAR’);
$this-$>$s-$>$Execute($this-$>$session, ’RESULTS.CLEAR’);
$this-$>$s-$>$Execute($this-$>$session,
’CONTEXTS.SET’,’KNOWLEDGE’,$this-$>$knw );
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.NEW’);
$this-$>$s-$>$Execute($this-$>$session, ’TABLE:wikimaster2.TOCONTEXT’,$rowid);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.DUP’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.EVALUATE’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.FILTERACT’,’25’,’true’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.NEWFROMSEM’,’1’,’-1’,’-1’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.SWAP’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.DROP’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.SWAP’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.DUP’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.ROLLDOWN’,’3’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.UNION’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.EVALUATE’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.INTERSECTION’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.NORMALIZE’);
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.FILTERACT’,’25’,’true’ );
$this-$>$s-$>$Execute($this-$>$session, ’CONTEXTS.TORESULTS’,’false’,’25’);
$this-$>$s-$>$Execute($this-$>$session, ’RESULTS.SelectBy’,’Act’,’$>$’,’95’);
$this-$>$s-$>$Execute($this-$>$session, ’RESULTS.SortBy’,’Act’,’false’);
$this-$>$s-$>$Execute($this-$>$session, ’RESULTS.GET’,’ResultCount’);
$count = $this-$>$s-$>$KMResults;
}
## 16 Conclusion and Future Work
This paper has presented the basic principles of mARC and studied its
application to Internet search. The results indicate that a mARC-based search
engine has the potential to be an order of magnitude faster yet more relevant
than current commercial search engines.
In the current mARC implementation, sampling of the incoming signal is limited
to eight bits. We are currently working on improving the sensorial layer
(reading head) to sample UTF-8 signals. This will enable the mARC search
engine to read and learn complex scripted languages such a Chinese,
Vietnamese, Hindi or Arabic and all other languages.
In a later stage, we will investigate non-sampled incoming signals to enable
mARC to process any kind of noisy, weakly-correlated signal.
Finally, we are also working on other application domains of the mARC besides
text mining.
## Acknowledgements
The authors would like to thank Prof. Claude Berrou, member of the French
Academy of sciences and IEEE Fellow for his comments on earlier revisions of
this document.
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## 17 Appendix: Experimental Results
Wikipedia En | mARC | Ratio | Google
---|---|---|---
| | | | | | | | | | | | |
| Returned Results (real) | Req1 (ms.) out of cache | Req 2 (ms.) | Req3 (ms.) | Req 3 (ms.) | average (ms.) | ratio | Returned Results (real) | Req1 (ms.) out of cache | Req 2 (ms.) | Req3 (ms.) | Req 3 (ms.) | average (ms.)
2012 | 800 | 17 | 5.2 | 5.3 | 5.26 | 7.6 | 20.5 | 800 | 220 | 160 | 130 | 130 | 156.0
2009 Swine Flu outbreak | 899 | 21 | 5.1 | 5.1 | 4.8 | 8.2 | 18.0 | 531 | 300 | 100 | 100 | 130 | 148.0
Abraham Lincoln | 1071 | 23.3 | 5.3 | 5.3 | 5.4 | 8.9 | 14.6 | 667 | 200 | 100 | 130 | 110 | 130.7
Adolf Hitler | 1015 | 14.8 | 4.7 | 4.7 | 4.7 | 6.7 | 20.1 | 629 | 250 | 90 | 130 | 100 | 135.3
America’s Next Top Model | 667 | 13.5 | 3.5 | 3.5 | 3.5 | 5.5 | 23.4 | 643 | 230 | 110 | 100 | 100 | 128.7
American Idol | 800 | 19.6 | 5.5 | 5.4 | 5.4 | 8.3 | 17.6 | 659 | 300 | 90 | 110 | 120 | 145.3
Anal sex | 962 | 123 | 5.8 | 7.6 | 5.6 | 29.7 | 4.8 | 626 | 260 | 110 | 130 | 100 | 142.7
Australia | 651 | 12.3 | 3.7 | 3 | 3.1 | 5.1 | 28.6 | 600 | 300 | 110 | 100 | 110 | 145.3
Barack Obama | 1131 | 17.3 | 6.9 | 6.9 | 6.9 | 9.0 | 15.4 | 645 | 250 | 110 | 120 | 100 | 138.0
Batman | 683 | 38.6 | 3.5 | 3.5 | 3.4 | 10.5 | 12.7 | 660 | 200 | 140 | 100 | 110 | 133.3
Bleach manga | 804 | 30.2 | 6 | 6 | 5.9 | 10.8 | 14.1 | 598 | 280 | 110 | 140 | 110 | 152.0
Canada | 800 | 14.2 | 5 | 5 | 5 | 6.8 | 22.4 | 632 | 260 | 120 | 120 | 140 | 153.3
China | 989 | 19.3 | 5.9 | 6.3 | 5.9 | 8.7 | 19.4 | 600 | 270 | 140 | 160 | 130 | 168.7
Current events portal | 897 | 254 | 4.4 | 4 | 4 | 54.1 | 2.3 | 665 | 230 | 90 | 100 | 110 | 126.0
Deadpool comics | 281 | 12 | 1.5 | 1.5 | 1.4 | 3.6 | 42.5 | 596 | 320 | 100 | 110 | 120 | 152.0
Deaths in 2009 | 800 | 18.71 | 8.6 | 5 | 5 | 8.7 | 16.2 | 700 | 250 | 110 | 100 | 130 | 140.7
Facebook | 800 | 12.9 | 5.3 | 5.4 | 5.3 | 6.8 | 19.3 | 641 | 220 | 100 | 120 | 110 | 132.0
Family Guy | 800 | 11.9 | 5.5 | 5.5 | 5.5 | 6.8 | 24.5 | 676 | 190 | 220 | 110 | 150 | 166.0
Farrah Fawcett | 875 | 11 | 5.4 | 5.1 | 5.2 | 6.4 | 23.5 | 502 | 310 | 110 | 110 | 110 | 150.0
Favicon.ico | 144 | 29.12 | 0.9 | 0.9 | 0.9 | 6.5 | 20.2 | 295 | 260 | 100 | 110 | 90 | 132.0
Featured content portal | 1220 | 8.9 | 7.9 | 7.9 | 7.9 | 8.1 | 14.8 | 690 | 200 | 100 | 110 | 90 | 120.0
France | 601 | 22.2 | 2.7 | 2.7 | 2.6 | 6.6 | 23.7 | 700 | 260 | 120 | 120 | 150 | 156.0
George W. Bush | 1084 | 107.89 | 6.9 | 6.4 | 6.4 | 26.8 | 4.8 | 646 | 210 | 110 | 110 | 110 | 130.0
Germany | 642 | 24.9 | 2.9 | 2.9 | 3 | 7.3 | 20.1 | 700 | 230 | 120 | 140 | 120 | 147.3
Global warming | 938 | 7.9 | 4.8 | 4.9 | 4.9 | 5.5 | 27.3 | 632 | 240 | 150 | 110 | 120 | 149.3
Google | 800 | 7.9 | 5.9 | 5.5 | 5.5 | 6.1 | 20.0 | 663 | 210 | 90 | 110 | 100 | 122.0
Henry VIII of England | 1232 | 70.88 | 10 | 9.5 | 9.6 | 21.9 | 6.4 | 662 | 260 | 100 | 110 | 120 | 140.0
Heroes TV series | 800 | 18.5 | 5.6 | 5.7 | 5.6 | 8.2 | 18.3 | 654 | 230 | 120 | 130 | 140 | 150.0
Hotmail | 136 | 21.9 | 0.7 | 0.7 | 0.7 | 4.9 | 27.3 | 451 | 180 | 120 | 110 | 140 | 134.7
House TV series | 800 | 15.4 | 5.2 | 5.3 | 5.3 | 7.3 | 23.6 | 670 | 340 | 160 | 110 | 120 | 172.0
Human penis size | 671 | 24.4 | 3.5 | 3.3 | 3.3 | 7.6 | 18.4 | 499 | 270 | 100 | 110 | 110 | 139.3
India | 653 | 13.3 | 3 | 2.9 | 3 | 5.0 | 25.7 | 582 | 220 | 100 | 110 | 110 | 129.3
Internet Movie Database | 1126 | 60.66 | 6 | 5.9 | 5.8 | 16.9 | 8.9 | 669 | 200 | 160 | 130 | 120 | 149.3
Jade Goody | 1317 | 388.7 | 6.6 | 6.2 | 6.1 | 82.8 | 1.5 | 518 | 250 | 100 | 100 | 90 | 127.3
Japan | 800 | 15.8 | 5.2 | 5.1 | 5.1 | 7.3 | 20.1 | 700 | 250 | 120 | 110 | 130 | 146.0
Jonas Brothers | 917 | 28.3 | 4.7 | 5.2 | 4.5 | 9.5 | 13.3 | 648 | 190 | 110 | 110 | 110 | 126.0
Kim Kardashian | 478 | 55.67 | 2.7 | 2.4 | 2.5 | 13.2 | 8.9 | 529 | 250 | 80 | 80 | 90 | 116.7
Kristen Stewart | 1283 | 297 | 6.3 | 6.1 | 6.2 | 64.4 | 2.1 | 593 | 260 | 90 | 90 | 120 | 132.0
Lady Gaga | 887 | 22.3 | 4.4 | 4.5 | 4.3 | 8.0 | 14.7 | 640 | 240 | 80 | 90 | 90 | 117.3
Lil Wayne | 989 | 41.4 | 4.2 | 4.2 | 4.2 | 11.6 | 9.8 | 644 | 210 | 90 | 90 | 90 | 114.0
List of Family Guy episodes | 638 | 14.5 | 3.3 | 3.1 | 3.1 | 5.4 | 21.7 | 595 | 230 | 90 | 90 | 90 | 118.0
List of Heroes episodes | 804 | 96.45 | 3.3 | 3 | 3 | 21.8 | 5.2 | 589 | 210 | 80 | 80 | 110 | 114.0
List of House episodes | 638 | 13.3 | 3.3 | 3.1 | 3.5 | 5.3 | 21.3 | 626 | 230 | 90 | 80 | 80 | 112.7
List of Presidents of the United States | 1152 | 33.2 | 10 | 9.9 | 9.9 | 14.6 | 9.3 | 700 | 280 | 90 | 110 | 100 | 136.0
List of sex positions | 1185 | 105.2 | 5 | 4.9 | 5.3 | 25.1 | 5.1 | 588 | 270 | 100 | 100 | 80 | 128.7
Lost season 5 | 800 | 5.8 | 5.7 | 5.7 | 5.8 | 5.7 | 21.7 | 654 | 250 | 90 | 100 | 90 | 124.7
Martin Luther King Jr | 1293 | 40.3 | 7.9 | 7.9 | 7.8 | 14.4 | 11.8 | 653 | 380 | 130 | 120 | 100 | 169.3
Masturbation | 466 | 7.8 | 2.7 | 2.6 | 2.6 | 3.7 | 30.7 | 619 | 190 | 90 | 80 | 110 | 112.7
Megan Fox | 1022 | 21.4 | 8.6 | 8.7 | 8.7 | 11.2 | 14.6 | 572 | 270 | 80 | 120 | 210 | 163.3
Metallica | 604 | 35.28 | 3.2 | 3.2 | 3.2 | 9.6 | 13.3 | 675 | 240 | 100 | 90 | 110 | 128.0
Mexico | 629 | 16.6 | 3 | 3 | 2.9 | 5.7 | 25.2 | 626 | 250 | 120 | 130 | 100 | 143.3
Michael Jackson | 943 | 28.57 | 4.4 | 4.3 | 4.3 | 9.2 | 12.9 | 651 | 220 | 80 | 100 | 100 | 118.7
Mickey Rourke | 697 | 30.6 | 3.5 | 3.5 | 3.4 | 8.9 | 14.5 | 587 | 260 | 100 | 90 | 100 | 129.3
Miley Cyrus | 802 | 26.75 | 4.1 | 4 | 4 | 8.6 | 15.3 | 642 | 230 | 110 | 100 | 110 | 131.3
MySpace | 800 | 25.2 | 5.5 | 5.5 | 5.4 | 9.4 | 13.5 | 681 | 210 | 140 | 90 | 90 | 127.3
Naruto | 726 | 18.5 | 4.6 | 4.5 | 4.6 | 7.4 | 16.0 | 618 | 240 | 90 | 80 | 90 | 117.3
Natasha Richardson | 1449 | 485.6 | 6.9 | 7.1 | 6.8 | 102.7 | 1.3 | 579 | 190 | 120 | 110 | 120 | 131.3
New York City | 623 | 29.6 | 3.4 | 3.2 | 3.2 | 8.5 | 22.4 | 700 | 370 | 150 | 160 | 130 | 191.3
Penis | 800 | 5.6 | 5.2 | 5.2 | 5.2 | 5.3 | 21.6 | 671 | 210 | 90 | 90 | 90 | 114.0
Pornography | 800 | 5.3 | 5.3 | 5.3 | 5.3 | 5.3 | 22.6 | 730 | 240 | 100 | 80 | 90 | 120.0
Relapse album | 774 | 20.54 | 5.1 | 5.2 | 5.3 | 8.3 | 13.6 | 601 | 230 | 80 | 90 | 80 | 112.7
Rhianna | 375 | 129.8 | 1.4 | 1.3 | 1.3 | 27.0 | 6.0 | 247 | 300 | 140 | 110 | 130 | 161.3
Robert Pattinson | 251 | 16.22 | 1.5 | 1.4 | 1.3 | 4.4 | 33.8 | 566 | 230 | 100 | 100 | 180 | 147.3
Russia | 642 | 38.85 | 3.1 | 3.1 | 3.1 | 10.3 | 12.6 | 700 | 220 | 110 | 90 | 120 | 129.3
Scrubs TV series | 800 | 19.34 | 4 | 3.9 | 4 | 7.0 | 19.7 | 95 | 360 | 80 | 90 | 80 | 138.7
Selena Gomez | 648 | 29.43 | 3.1 | 3 | 3 | 8.3 | 14.4 | 618 | 210 | 90 | 110 | 90 | 119.3
Sex | 800 | 5.7 | 54 | 5.8 | 5.4 | 18.5 | 7.0 | 636 | 210 | 100 | 140 | 90 | 130.0
Sexual intercourse | 961 | 20.22 | 4.8 | 4.8 | 4.8 | 7.9 | 15.8 | 646 | 250 | 100 | 90 | 90 | 124.7
Slumdog Millionaire | 550 | 21 | 2.7 | 2.7 | 2.8 | 6.4 | 17.4 | 609 | 210 | 80 | 90 | 90 | 111.3
Israel | 663 | 22.34 | 3.6 | 3.3 | 3.7 | 7.3 | 17.3 | 700 | 230 | 90 | 110 | 100 | 126.0
Star Trek film | 852 | 14.1 | 4.1 | 4.3 | 4.1 | 6.2 | 21.8 | 651 | 270 | 100 | 90 | 110 | 134.0
Swine flu | 589 | 12.72 | 2.6 | 2.5 | 2.7 | 4.6 | 25.4 | 591 | 240 | 90 | 90 | 80 | 117.3
Taylor Swift | 791 | 48.64 | 3.8 | 3.5 | 3.6 | 12.6 | 9.2 | 635 | 180 | 90 | 120 | 90 | 116.0
Terminator Salvation | 520 | 56.91 | 2.3 | 2.3 | 2.3 | 13.2 | 9.3 | 552 | 240 | 80 | 110 | 90 | 122.7
The Beatles | 800 | 18.68 | 5.5 | 5.5 | 5.5 | 8.1 | 13.8 | 700 | 230 | 80 | 80 | 90 | 112.7
The Dark Knight film | 984 | 25.3 | 5.2 | 5.1 | 5 | 9.1 | 15.2 | 621 | 240 | 100 | 140 | 100 | 138.7
The Notorious B.I.G. | 598 | 16.3 | 3.1 | 3.1 | 3 | 5.7 | 21.1 | 642 | 190 | 100 | 110 | 100 | 120.7
Transformers 2 | 800 | 48.22 | 5.2 | 5.1 | 5.3 | 13.8 | 9.0 | 644 | 210 | 100 | 110 | 100 | 124.7
Transformers: Revenge of the Fallen | 802 | 18.81 | 5 | 4.9 | 4.9 | 7.7 | 14.7 | 624 | 220 | 80 | 90 | 90 | 113.3
Tupac Shakur | 635 | 20.67 | 3.1 | 3.2 | 3.1 | 6.6 | 19.3 | 622 | 200 | 150 | 80 | 100 | 128.0
Twilight | 800 | 22.73 | 5.5 | 5.4 | 5.5 | 8.9 | 11.7 | 700 | 190 | 80 | 90 | 80 | 104.7
Twilight 2008 film | 800 | 7.3 | 5.6 | 5.4 | 5.5 | 5.9 | 20.9 | 610 | 200 | 110 | 90 | 110 | 122.7
Twitter | 800 | 18.95 | 5.4 | 5.4 | 5.5 | 8.1 | 15.2 | 634 | 260 | 90 | 90 | 90 | 124.0
United Kingdom | 899 | 27.5 | 4.3 | 4.1 | 4.4 | 8.9 | 14.9 | 600 | 250 | 100 | 110 | 100 | 132.7
Vagina | 728 | 14.7 | 5.4 | 4.6 | 4.7 | 6.9 | 15.5 | 640 | 160 | 90 | 100 | 90 | 106.7
Valentine’s Day | 902 | 52.48 | 5.3 | 5.2 | 5.2 | 14.7 | 8.6 | 667 | 260 | 100 | 90 | 90 | 126.7
Vietnam War | 1026 | 33.87 | 4.6 | 4.7 | 4.7 | 10.5 | 13.1 | 700 | 250 | 120 | 120 | 90 | 138.0
Watchmen film | 494 | 13.6 | 2.8 | 2.8 | 3 | 5.0 | 23.3 | 531 | 210 | 110 | 90 | 80 | 116.7
William Shakespeare | 800 | 15.14 | 5.5 | 5.6 | 8.2 | 8.2 | 14.2 | 660 | 180 | 100 | 100 | 100 | 116.0
Windows 7 | 800 | 5.74 | 5.6 | 6.3 | 5.4 | 5.8 | 20.8 | 650 | 240 | 90 | 90 | 90 | 120.0
Wolverine comics | 905 | 16.6 | 5.7 | 5.7 | 57 | 21.6 | 6.2 | 700 | 240 | 120 | 90 | 110 | 133.3
World War I | 640 | 13.19 | 3.1 | 3.5 | 3.3 | 5.3 | 24.8 | 664 | 240 | 120 | 90 | 100 | 130.7
World War II | 1040 | 18.03 | 4.7 | 4.6 | 4.7 | 7.3 | 15.5 | 700 | 210 | 90 | 90 | 90 | 114.0
X-Men Origins: Wolverine | 1399 | 52.69 | 10.6 | 10.3 | 10.3 | 18.9 | 6.2 | 624 | 220 | 90 | 90 | 90 | 116.0
YouTube | 800 | 5.6 | 5.5 | 5.4 | 5.4 | 5.5 | 19.6 | 641 | 190 | 80 | 90 | 90 | 107.3
| | | | | | | | | | | | |
| | 41.4 | 5.2 | 4.7 | 5.2 | 12.3 | 16.4 | | 239.4 | 105.4 | 104.9 | 106.1 | 132.3
2010 request sample | mARC | Ratio | Google
---|---|---|---
wikipedia fr | | | | | | | | | | | | |
| Results (real) | Req1 (ms.) out of cache | Req 2 (ms.) | Req3 (ms.) | Req 3 (ms.) | average (ms.) | ratio | Results (real) | Req1 (ms.) out of cache | Req 2 (ms.) | Req3 (ms.) | Req 3 (ms.) | average (ms.)
Facebook | 299 | 7.29 | 2.2 | 1.7 | 1.6 | 2.9 | 38.1 | 620 | 170 | 110 | 90 | 90 | 111.3
youtube | 482 | 11.7 | 2.8 | 2.8 | 2.8 | 4.6 | 23.6 | 681 | 180 | 90 | 90 | 90 | 108.0
jeux | 800 | 6.04 | 5.9 | 5.9 | 5.9 | 5.9 | 20.6 | 700 | 210 | 110 | 100 | 90 | 122.0
you | 840 | 15.28 | 5.3 | 5.4 | 5.3 | 7.3 | 15.1 | 695 | 180 | 90 | 100 | 90 | 110.7
yahoo | 639 | 4.29 | 4 | 4 | 4 | 4.1 | 27.1 | 642 | 190 | 90 | 90 | 90 | 110.0
tv | 800 | 6.5 | 5.7 | 5.7 | 5.7 | 5.9 | 21.2 | 679 | 220 | 100 | 90 | 110 | 124.0
orange | 837 | 11.3 | 5.2 | 5.3 | 5.3 | 6.5 | 19.9 | 689 | 230 | 110 | 100 | 100 | 128.7
meteo | 183 | 23.63 | 1.4 | 1.2 | 1.3 | 9.2 | 13.6 | 678 | 180 | 110 | 90 | 100 | 125.3
le bon coin | 858 | 21.3 | 6.3 | 6.2 | 6.2 | 9.2 | 13.6 | 498 | 240 | 90 | 110 | 90 | 125.3
hotmail | 215 | 17.98 | 1.3 | 1.7 | 1.3 | 4.7 | 26.4 | 570 | 200 | 90 | 120 | 110 | 125.3
yahoo mail | 802 | 7.36 | 6.9 | 6.5 | 6.5 | 6.8 | 15.9 | 396 | 180 | 80 | 90 | 100 | 108.0
web mail | 828 | 12.93 | 8.8 | 9 | 8.6 | 9.6 | 11.3 | 545 | 170 | 100 | 90 | 90 | 108.7
iphone | 200 | 8.9 | 1.3 | 1.2 | 1 | 2.7 | 37.3 | 600 | 160 | 90 | 80 | 90 | 101.3
jeux.fr | 1060 | 14.3 | 10.4 | 10.8 | 11 | 11.4 | 9.3 | 587 | 160 | 100 | 90 | 90 | 106.7
roland garros | 899 | 23.55 | 5 | 4.6 | 4.7 | 8.5 | 12.9 | 591 | 190 | 90 | 90 | 90 | 110.0
robert pattinson | 639 | 157.31 | 22 | 21 | 23 | 49.1 | 1.6 | 124 | 150 | 60 | 70 | 60 | 80.7
mappy michelin | 658 | 13.1 | 5.2 | 5.3 | 5.1 | 6.8 | 17.2 | 27 | 210 | 90 | 110 | 80 | 116.7
le monde | 1143 | 21.68 | 6.8 | 6.8 | 6.8 | 9.8 | 12.1 | 671 | 180 | 110 | 90 | 110 | 118.7
figaro | 617 | 15.7 | 4 | 3.9 | 3.8 | 6.3 | 20.3 | 674 | 250 | 120 | 90 | 80 | 127.3
tf1 | 385 | 9.54 | 2.5 | 2.6 | 2.5 | 3.9 | 33.0 | 680 | 290 | 110 | 80 | 80 | 130.0
le parisien | 605 | 7.04 | 4.3 | 4.4 | 4.3 | 4.9 | 20.7 | 643 | 170 | 90 | 80 | 80 | 100.7
liberation | 308 | 6.1 | 1.9 | 1.9 | 1.8 | 2.7 | 36.6 | 684 | 150 | 80 | 80 | 100 | 99.3
sarkozy | 800 | 7.08 | 5.7 | 6.1 | 5.9 | 6.1 | 16.8 | 642 | 170 | 90 | 80 | 90 | 103.3
20 minutes | 577 | 75.63 | 5.8 | 5.5 | 5.6 | 19.6 | 6.0 | 661 | 180 | 130 | 100 | 80 | 118.7
obama | 267 | 6.1 | 1.5 | 1.7 | 1.6 | 2.5 | 45.3 | 634 | 180 | 90 | 110 | 90 | 113.3
news | 800 | 9.71 | 6.7 | 6.5 | 6.5 | 7.2 | 15.3 | 657 | 150 | 100 | 100 | 100 | 110.0
les echos | 1509 | 82.03 | 8.5 | 9.3 | 8.3 | 23.4 | 4.6 | 645 | 180 | 90 | 80 | 100 | 108.0
pub orange | 1053 | 19.46 | 10 | 10.3 | 10.4 | 12.1 | 11.1 | 383 | 260 | 100 | 90 | 120 | 134.7
pub vittel | 581 | 33.65 | 4.2 | 4.3 | 4.6 | 10.2 | 11.6 | 36 | 180 | 100 | 100 | 110 | 118.7
pub tf1 | 782 | 7.2 | 7.3 | 7.2 | 7.2 | 7.2 | 17.3 | 453 | 210 | 90 | 110 | 110 | 124.7
pub sfr | 558 | 10.5 | 5.2 | 4.6 | 4.6 | 5.9 | 19.0 | 85 | 190 | 90 | 90 | 100 | 112.7
pub renault | 801 | 13.1 | 9.6 | 9.6 | 9.6 | 10.3 | 7.8 | 251 | 270 | 80 | 10 | 10 | 80.7
pub oasis | 800 | 8.08 | 7.9 | 8 | 7.9 | 8.0 | 16.0 | 114 | 250 | 90 | 110 | 90 | 127.3
pub nike | 635 | 6.9 | 5.7 | 5.6 | 5.6 | 5.9 | 21.9 | 105 | 190 | 90 | 110 | 140 | 128.7
pub iphone | 600 | 6 | 5 | 5.1 | 4.8 | 5.2 | 22.3 | 125 | 190 | 90 | 110 | 90 | 115.3
pub free | 755 | 24.6 | 4.1 | 4.1 | 3.9 | 8.1 | 14.8 | 504 | 230 | 100 | 90 | 90 | 120.7
pub evian | 486 | 13.9 | 3.8 | 3.9 | 3.9 | 5.9 | 21.6 | 67 | 220 | 90 | 90 | 130 | 126.7
twilight | 701 | 17.2 | 5.1 | 5.4 | 5.1 | 7.6 | 15.1 | 614 | 160 | 120 | 90 | 100 | 114.7
michael jackson | 1090 | 20.5 | 7.6 | 7.1 | 7 | 9.9 | 13.2 | 670 | 200 | 110 | 100 | 130 | 130.7
wat | 167 | 7.9 | 0.89 | 0.92 | 0.9 | 2.3 | 53.6 | 605 | 230 | 100 | 90 | 100 | 123.3
programme tnt | 692 | 20.94 | 8.2 | 8.2 | 12.72 | 12.0 | 10.2 | 491 | 220 | 100 | 100 | 90 | 121.3
naruto shippuden | 311 | 110.3 | 2.1 | 1.9 | 2.1 | 23.7 | 4.6 | 344 | 180 | 90 | 90 | 90 | 108.0
streaming | 250 | 5.6 | 1.4 | 1.3 | 1.3 | 2.2 | 46.6 | 422 | 150 | 90 | 90 | 90 | 102.0
m6 replay | 286 | 7.8 | 1.9 | 1.8 | 2 | 3.1 | 35.1 | 72 | 180 | 80 | 100 | 90 | 108.0
one piece | 658 | 11.2 | 4.7 | 4.8 | 4.6 | 6.0 | 21.6 | 618 | 220 | 110 | 90 | 120 | 129.3
Twitter | 191 | 2.9 | 1 | 1 | 0.9 | 1.4 | 87.7 | 618 | 220 | 100 | 90 | 90 | 118.7
Swine Flu | 190 | 10.94 | 1.1 | 1.2 | 1 | 3.1 | 32.4 | 69 | 190 | 80 | 80 | 70 | 99.3
Stock Market | 807 | 11.2 | 8.8 | 8.5 | 8.5 | 9.1 | 16.0 | 251 | 210 | 120 | 130 | 140 | 146.0
Farrah Fawcett | 251 | 27.5 | 1.5 | 1.5 | 1.4 | 6.7 | 16.4 | 83 | 200 | 80 | 90 | 90 | 109.3
Patrick Swayze | 1258 | 19.35 | 9.1 | 9.2 | 9.1 | 11.2 | 9.9 | 146 | 220 | 80 | 70 | 100 | 110.7
Cash for Clunkers | 400 | 18.43 | 2.8 | 2.8 | 2.8 | 5.9 | 14.2 | 2 | 140 | 70 | 70 | 70 | 84.0
Jon and Kate Gosselin | 1040 | 25.11 | 11.2 | 12 | 11.3 | 14.2 | 5.6 | 4 | 120 | 70 | 70 | 70 | 80.0
Billy Mays | 1153 | 142.7 | 7.5 | 7.2 | 7.3 | 34.4 | 3.2 | 83 | 190 | 90 | 100 | 80 | 110.0
Jaycee Dugard | 26 | 16.9 | 0.4 | 0.5 | 0.4 | 3.7 | 23.6 | 15 | 240 | 50 | 50 | 50 | 88.0
Jean Sarkozy | 1047 | 168.271 | 94 | 97 | 95 | 109.9 | 1.2 | 608 | 210 | 100 | 120 | 100 | 127.3
Rihanna | 123 | 2.8 | 0.7 | 0.7 | 0.6 | 1.1 | 120.1 | 587 | 270 | 110 | 90 | 90 | 131.3
Zohra Dhati | 53 | 2.8 | 0.5 | 0.4 | 0.4 | 0.9 | 119.1 | 5 | 140 | 90 | 100 | 110 | 108.0
Salma Hayek et Fran ois Pinault | 469 | 26.421 | 3.3 | 3.2 | 3.2 | 7.9 | 9.9 | 8 | 190 | 50 | 50 | 50 | 78.0
Fr d ric Mitterrand | 1013 | 113.09 | 9 | 9.4 | 9.2 | 30.0 | 4.4 | 586 | 200 | 130 | 110 | 100 | 130.7
Roman Polanski | 468 | 16.02 | 2.9 | 2.9 | 2.8 | 5.5 | 26.9 | 591 | 260 | 100 | 100 | 160 | 148.0
Loana | 77 | 69.68 | 0.9 | 0.8 | 0.8 | 14.6 | 6.9 | 160 | 131 | 90 | 90 | 100 | 100.9
Caster Semenya | 106 | 74.58 | 1 | 0.8 | 0.9 | 15.6 | 5.2 | 44 | 170 | 60 | 60 | 60 | 82.0
Jacques S gu la | 1144 | 27.86 | 20 | 20 | 19.9 | 21.5 | 7.1 | 140 | 310 | 130 | 100 | 110 | 152.7
Yann Barthes | 749 | 158.229 | 4.7 | 4.4 | 4.4 | 35.2 | 5.4 | 113 | 250 | 290 | 120 | 120 | 191.3
Le miracle de l’Hudson | 1400 | 125.08 | 17 | 19 | 17.6 | 39.3 | 2.8 | 167 | 430 | 130 | 110 | 120 | 110.0
Barack Obama | 459 | 14.23 | 2.4 | 2.6 | 2.4 | 4.8 | 24.8 | 615 | 170 | 90 | 120 | 110 | 119.3
La crise conomique | 1076 | 15.6 | 6.3 | 6.9 | 6.3 | 8.3 | 16.8 | 700 | 220 | 130 | 120 | 110 | 140.0
Greve aux Antilles | 952 | 30.54 | 7.5 | 7.5 | 8.6 | 12.4 | 13.9 | 248 | 300 | 150 | 120 | 150 | 172.0
S isme en Italie | 1000 | 18.75 | 8.6 | 9.1 | 8.6 | 10.8 | 11.5 | 549 | 180 | 90 | 90 | 150 | 124.0
La grippe A | 631 | 46.93 | 5 | 4 | 4 | 12.9 | 8.9 | 612 | 200 | 100 | 90 | 90 | 114.7
Le malaise pr sidentiel | 799 | 17.9 | 8.3 | 8.1 | 8.5 | 10.2 | 13.4 | 204 | 230 | 110 | 120 | 110 | 136.7
Hadopi | 225 | 27.05 | 1.1 | 1 | 1.1 | 6.3 | 18.9 | 438 | 220 | 90 | 90 | 100 | 118.7
Le proces Clearstream | 517 | 31.5 | 5.6 | 5.9 | 5.6 | 10.9 | 12.0 | 109 | 240 | 110 | 100 | 100 | 130.7
Madonna | 800 | 20.25 | 6.5 | 6.3 | 6.3 | 9.1 | 13.1 | 641 | 200 | 100 | 100 | 100 | 120.0
U2 | | 11.35 | 2.9 | 3.1 | 2.8 | 4.6 | 22.7 | 602 | 150 | 90 | 90 | 100 | 104.7
Diam’s | 142 | 16.58 | 0.73 | 0.78 | 0.74 | 3.9 | 28.9 | 378 | 220 | 80 | 90 | 90 | 113.3
Mylene Farmer | 540 | 24.45 | 3 | 3.1 | 2.9 | 7.3 | 14.3 | 586 | 160 | 90 | 90 | 90 | 104.0
Les Beatles remast ris s | 520 | 19.56 | 3.7 | 4.1 | 3.7 | 7.0 | 23.7 | 110 | 320 | 140 | 120 | 120 | 165.3
Johnny Hallyday | 822 | 8.8 | 5.6 | 6 | 5.5 | 6.3 | 23.7 | 535 | 270 | 110 | 130 | 120 | 150.0
Lady Gaga | 489 | 74.68 | 3.8 | 3.7 | 3.7 | 17.9 | 6.6 | 579 | 220 | 90 | 90 | 100 | 118.7
La s paration d’Oasis | 1360 | 28.62 | 15.7 | 15.7 | 15.7 | 18.3 | 10.1 | 207 | 280 | 260 | 100 | 120 | 184.0
Prince a Paris | 800 | 9.9 | 7.7 | 7.7 | 7.7 | 8.1 | 20.8 | 679 | 260 | 180 | 120 | 140 | 169.3
David Guetta | 736 | 109.37 | 27 | 26 | 24 | 42.4 | 2.8 | 529 | 200 | 90 | 100 | 100 | 117.3
| | | | | | | | | | | | |
| | 30.5 | 6.8 | 6.8 | 6.8 | 11.6 | 20.5 | | 207.0 | 101.8 | 94.3 | 98.2 | 119.1
Wikipedia En | mARC Search Engine demonstrator | Ratio | Google
---|---|---|---
| Results (real) | Req1 (ms) out of cache | Req2 (ms) | Req3 (ms) | Req3 (ms) | average (ms) | | Results(real) | Req1(ms) out of cache | Req2 (ms) | Req3 (ms) | Req 3 (ms) | average (ms)
Mathematical formulations of quantum mechanics | 1360 | 55.1 | 8.6 | 8.7 | 8.7 | 18.0 | 10.2 | 581 | 280 | 160 | 160 | 160 | 184.0
Philosophical interpretation of classical physics | 1284 | 51.1 | 8.9 | 8.6 | 8.9 | 17.3 | 11.5 | 536 | 350 | 160 | 160 | 160 | 198.0
Governor General’s Award for English language non fiction | 893 | 77.4 | 5.4 | 5.7 | 5.3 | 19.9 | 5.8 | 545 | 220 | 90 | 90 | 90 | 116.0
John Breckinridge (Attorney General) | 1145 | 42.97 | 7.5 | 7.5 | 7.5 | 14.6 | 12.2 | 540 | 300 | 130 | 150 | 160 | 177.3
Popular Front for the Liberation of Palestine General Command | 959 | 73.2 | 5.9 | 6.4 | 5.9 | 19.5 | 6.7 | 484 | 230 | 120 | 110 | 90 | 131.3
The Six Wives of Henry VIII (TV series) | 891 | 18.93 | 6.4 | 6 | 5.9 | 8.7 | 20.2 | 449 | 370 | 140 | 110 | 130 | 175.3
List of Chancellors of the University of Cambridge | 1060 | 17.35 | 9.2 | 8.6 | 8.6 | 10.5 | 15.9 | 592 | 300 | 140 | 150 | 110 | 166.7
International Council of Unitarians and Universalists | 1045 | 48.44 | 8 | 7.8 | 7.8 | 16.0 | 8.7 | 573 | 270 | 110 | 110 | 100 | 139.3
International Council of Unitarians and Universalists | 739 | 232.4 | 4.6 | 4.7 | 4.3 | 50.1 | 2.6 | 362 | 280 | 90 | 90 | 90 | 128.0
Finitely generated abelian group | 655 | 29.3 | 3.3 | 3.4 | 3.3 | 8.5 | 21.6 | 288 | 240 | 170 | 170 | 170 | 184.0
Structure theorem for finitely generated modules over a principal ideal domain | 495 | 102.11 | 2.4 | 2.4 | 2.4 | 22.3 | 10.7 | 69 | 310 | 210 | 220 | 230 | 238.0
Asimov’s Biographical Encyclopedia of Science and Technology | 1401 | 93.15 | 8.4 | 8.8 | 8.4 | 25.5 | 7.2 | 271 | 260 | 160 | 160 | 170 | 182.7
Aalto University School of Science and Technology | 974 | 19.59 | 6 | 5.9 | 5.9 | 8.7 | 13.2 | 180 | 250 | 80 | 80 | 80 | 114.0
List of historical sites associated with Ludwig van Beethoven | 604 | 63.97 | 3.4 | 3 | 3.1 | 15.3 | 11.0 | 104 | 270 | 140 | 150 | 140 | 168.7
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The cinema of the Soviet Union , not to be confused with Cinema of Russia despite Russian language films being predominant in both genres, includes several film contributions of the constituent republics of the Soviet Union reflecting elements of | 600 | 141.53 | 5.4 | 5.7 | 4.7 | 32.5 | 8.4 | 23 | 760 | 140 | 170 | 150 | 274.7
Niger is home to a number of national parks and protected areas , including two UNESCO-MAB Biosphere Reserves. The protected areas of Niger normally have a designation and status determined by the Government of Niger. | 715 | 75.6 | 5.4 | 4.7 | 4.7 | 19.1 | 32.9 | 4 | 860 | | 860 | 850 | 628.0
The term hamburger or burger can also be applied to the patty meat patty on its own, especially in the UK. There are several accounts of the invention of the hamburger | 672 | 71.32 | 4.6 | 4.1 | 4.5 | 17.8 | 12.3 | 132 | 590 | 130 | 120 | 130 | 219.3
Rockwell International was a major American manufacturing conglomerate (company) conglomerate in the latter half of the 20th century, involved in aircraft, the space industry, both defense-oriented and commercial electronics, automotive and truck | 656 | 63.81 | 5 | 4.7 | 4.9 | 16.7 | 38.2 | 1 | 850 | 810 | 130 | 810 | 636.7
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Les Monstres du fond des mers | 794 | 10.853 | 4.7 | 4.7 | | 5.7 | 22.7 | 482 | 220 | 100 | 110 | 110 | 129.3
Liste des lacs et mers int rieures de la Terre du Milieu | 504 | 31.17 | 3.8 | 3.9 | 3.7 | 9.3 | 22.6 | 419 | 420 | 130 | 170 | 170 | 209.3
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Liste d’arch ologues par ordre alphab tique | 1474 | 30.53 | 9.4 | 8.8 | 9.4 | 13.5 | 11.1 | 556 | 270 | 150 | 100 | 110 | 150.0
Liste des noms de famille les plus courants au Qu bec, par ordre alphab tique H | 1399 | 39.9 | 10.6 | 12.3 | 10.5 | 16.9 | 7.2 | 584 | 250 | 100 | 90 | 80 | 122.0
Moulin a vent de l’ le Saint Bernard de Ch teauguay | 802 | 49.37 | 5.7 | 5.7 | 5.7 | 14.4 | 16.0 | 200 | 330 | 210 | 200 | 210 | 231.3
C sar de la meilleure actrice dans un second r le | 1227 | 68.334 | 11.7 | 11.2 | 11.8 | 22.9 | 7.4 | 546 | 280 | 110 | 210 | 110 | 170.7
La litt rature fran aise comprend l’ensemble des oeuvres crites par des auteurs de nationalit fran aise ou de langue fran aise . Son histoire commence en ancien fran ais au Moyen Age et se perp tue aujourd’hui. Chanson de geste La Litt r | 556 | 86.47 | 6.9 | 6.1 | 6.1 | 22.4 | 20.6 | 7 | 970 | 540 | 300 | 160 | 460.7
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La r sonance est un ph nomene selon lequel certains systemes physiques ( lectriques, m caniques…) sont sensibles a certaines fr quences. Un systeme r sonant peut accumuler une nergie, si celle-ci est appliqu e sous forme p riodique, et proche d’une fr | 653 | 20.66 | 6.2 | 6.2 | 6.2 | 9.1 | 67.0 | 3 | 1060 | 930 | 290 | 270 | 609.3
Brevet de technicien sup rieur Techniques physiques pour l’industrie et le laboratoire | 450 | 47.6 | 3.7 | 3.8 | 3.8 | 12.5 | 13.5 | 115 | 310 | 140 | 130 | 130 | 168.7
Dipl me universitaire de technologie Mesures physiques | 519 | 47.32 | 3.7 | 3.6 | 3.7 | 12.4 | 14.0 | 340 | 310 | 160 | 140 | 120 | 174.0
| | | | | | | | | | | | |
| | 42.4 | 6.1 | 6.1 | 6.2 | 13.3 | 24.1 | | 455.0 | 285.8 | 224.2 | 195.8 | 279.2
|
arxiv-papers
| 2013-12-10T15:56:53 |
2024-09-04T02:49:55.283416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Norbert Rimoux, Patrice Descourt",
"submitter": "Patrice Descourt",
"url": "https://arxiv.org/abs/1312.2844"
}
|
1312.2950
|
050004 2013 G. Mindlin 050004
Realistic mathematical modeling of voice production has been recently boosted
by applications to different fields like bioprosthetics, quality speech
synthesis and pathological diagnosis. In this work, we revisit a two-mass
model of the vocal folds that includes accurate fluid mechanics for the air
passage through the folds and nonlinear properties of the tissue. We present
the bifurcation diagram for such a system, focusing on the dynamical
properties of two regimes of interest: the onset of oscillations and the
normal phonation regime. We also show theoretical support to the nonlinear
nature of the elastic properties of the folds tissue by comparing theoretical
isofrequency curves with reported experimental data.
# Revisiting the two-mass model of the vocal folds
M. F. Assaneo [inst1] M. A. Trevisan[inst1] E-mail: [email protected]
mail: [email protected]
(2 March 2013; 5 June 2013)
††volume: 5
99 inst1 Laboratorio de Sistemas Dinámicos, Depto. de Física, FCEN,
Universidad de Buenos Aires. Pabellón I, Ciudad Universitaria, 1428EGA Buenos
Aires, Argentina.
## 1 Introduction
In the last decades, a lot of effort was devoted to develop a mathematical
model for voice production. The first steps were made by Ishizaka and Flanagan
[1], approximating each vocal fold by two coupled oscillators, which provide
the basis of the well known two-mass model. This simple model reproduces many
essential features of the voice production, like the onset of self sustained
oscillation of the folds and the shape of the glottal pulses.
Early analytical treatments were restricted to small amplitude oscillations,
allowing a dimensional reduction of the problem. In particular, a two
dimensional approximation known as the flapping model was widely adopted by
the scientific community, based on the assumption of a transversal wave
propagating along the vocal folds [2, 3]. Moreover, this model was also used
to successfully explain most of the features present in birdsong [4, 5].
Faithful modeling of the vocal folds has recently found new challenges:
realistic articulatory speech synthesis [6, 7, 8], diagnosis of pathological
behavior of the folds [9, 10] and bioprosthetic applications [11]. Within this
framework, the 4-dimensional two-mass model was revisited and modified. Two
main improvements are worth noting: a realistic description of the vocal fold
collision [13, 14] and an accurate fluid mechanical description of the glottal
flow, allowing a proper treatment of the hydrodynamical force acting on the
folds [15, 8].
In this work, we revisit the two-mass model developed by Lucero and Koenig
[7]. This choice represents a good compromise between mathematical simplicity
and diversity of physical phenomena acting on the vocal folds, including the
main mechanical and fluid effects that are partially found in other models
[15, 13]. It was also successfully used to reproduce experimental temporal
patterns of glottal airflow. Here, we extend the analytical study of this
system: we present a bifurcation diagram, explore the dynamical aspects of the
oscillations at the onset and normal phonation and study the isofrequency
curves of the model.
This work is organized as follows: in the second section, we describe the
model. In the third section, we present the bifurcation diagram, compare our
solutions with those of the flapping model approximation and analyze the
isofrecuency curves. In the fourth and last section, we discuss our results.
## 2 The model
Each vocal fold is modeled as two coupled damped oscillators, as sketched in
Fig. 1.
Figure 1: Sketch of the two-mass model of the vocal folds. Each fold is
represented by masses $m_{1}$ and $m_{2}$ coupled to each other by a
restitution force $k_{c}$ and to the laryngeal walls by $K_{1}$ and $K_{2}$
(and dampings $B_{1}$ and $B_{2}$), respectively. The displacement of each
mass from the resting position $x_{0}$ is represented by $x_{1}$ and $x_{2}$.
The different aerodynamic pressures $P$ acting on the folds are described in
the text.
Assuming symmetry with respect to the saggital plane, the left and right mass
systems are identical (Fig. 1) and the equation of motion for each mass reads
$\displaystyle\dot{x_{i}}$ $\displaystyle=y_{i}$ (1)
$\displaystyle\dot{y_{i}}$
$\displaystyle=\frac{1}{m_{i}}\left[f_{i}-K_{i}(x_{i})-B_{i}(x_{i},y_{i})-k_{c}(x_{i}-x_{j})\right],$
for $i,j=1$ or 2 for lower and upper masses, respectively. $K$ and $B$
represent the restitution and damping of the folds tissue, $f$ the
hydrodynamic force, $m$ is the mass and $k_{c}$ the coupling stiffness. The
horizontal displacement from the rest position $x_{0}$ is represented by $x$.
We use a cubic polynomial for the restitution term [Eq. (2)], adapted from [1,
7]. The term with a derivable step-like function $\Theta$ [Eq. (5)] accounts
for the increase in the stiffness introduced by the collision of the folds.
The restitution force reads
$\displaystyle K_{i}$ $\displaystyle(x_{i})=k_{i}x_{i}(1+100{x_{i}}^{2})$ (2)
$\displaystyle+\Theta\left(\frac{x_{i}+x_{0}}{x_{0}}\right)3k_{i}(x_{i}+x_{0})[1+500{(x_{i}+x_{0})}^{2}],$
with
$\displaystyle\Theta(x)=\left\\{\begin{array}[]{rl}0&\text{if }x\leq 0\\\
\frac{x^{2}}{8\text{ }10^{-4}+x^{2}}&\text{if }x>0\end{array}\right.,$ (5)
where $x_{0}$ is the rest position of the folds.
For the damping force, we have adapted the expression proposed in [7], making
it derivable, arriving at the following equation:
$\displaystyle B_{i}(x_{i})=$ (6)
$\displaystyle\left[1+\Theta\left(\frac{x_{i}+x_{0}}{x_{0}}\right)\frac{1}{\epsilon_{i}}\right]r_{i}(1+850{x_{i}}^{2})y_{i},$
where $r_{i}=2\epsilon_{i}\sqrt{k_{i}m_{i}}$, and $\epsilon_{i}$ is the
damping ratio.
In order to describe the hydrodynamic force that the airflow exerts on the
vocal folds, we have adopted the standard assumption of small inertia of the
glottal air column and the model of the boundary layer developed in [7, 11,
15]. This model assumes a one-dimensional, quasi-steady incompressible airflow
from the trachea to a separation point. At this point, the flow separates from
the tissue surface to form a free jet where the turbulence dissipates the
airflow energy. It has been experimentally shown that the position of this
point depends on the glottal profile. As described in [15], the separation
point located at the glottal exit shifts down to the boundary between masses
$m_{1}$ and $m_{2}$ when the folds profile becomes more divergent than a
threshold [Eq. (11)].
Viscous losses are modeled according to a bi-dimensional Poiseuille flow [Eqs.
(8) and (11)]. The equations for the pressure inside the glottis are
$\displaystyle P_{in}$ $\displaystyle=P_{s}+\frac{\rho
u_{g}^{2}}{2a_{1}^{2}},$ (7) $\displaystyle P_{12}$
$\displaystyle=P_{in}-\frac{12\mu u_{g}d_{1}l_{g}^{2}}{a_{1}^{3}},$ (8)
$\displaystyle P_{21}$ $\displaystyle=\left\\{\begin{array}[]{rl}\frac{12\mu
u_{g}d_{2}l_{g}^{2}}{a_{2}^{3}}+P_{out}&\text{if }a_{2}>k_{s}a_{1}\\\
0&\text{if }a_{2}\leq k_{s}a_{1}\end{array},\right.$ (11) $\displaystyle
P_{out}$ $\displaystyle=0.$ (12)
As sketched in Fig. 1, the pressures exerted by the airflow are: $P_{in}$ at
the entrance of the glottis, $P_{12}$ at the upper edge of $m_{1}$, $P_{21}$
at the lower edge of $m_{2}$, $P_{out}$ at the entrance of the vocal tract and
$P_{s}$ the subglottal pressure.
The width of the folds (in the plane normal to Fig. 1) is $l_{g}$; $d_{1}$ and
$d_{2}$ are the lengths of the lower and upper masses, respectively. $a_{i}$
are the cross-sections of the glottis, $a_{i}=2l_{g}(x_{i}+x_{0})$; $\mu$ and
$\rho$ are the viscosity and density coefficient of the air; $u_{g}$ is the
airflow inside the glottis, and $k_{s}=1.2$ is an experimental coefficient. We
also assume no losses at the glottal entrance [Eq. (7)], and zero pressure at
the entrance of the vocal tract [Eq. (12)].
The hydrodynamic force acting on each mass reads:
$\displaystyle f_{1}=\left\\{\begin{array}[]{rl}d_{1}l_{g}P_{s}&\text{if
}x_{1}\leq-x_{0}\text{ or }x_{2}\leq-x_{0}\\\ \frac{P_{in}+P_{12}}{2}&\text{in
other case}\end{array}\right.$ (15)
$\displaystyle f_{2}=\left\\{\begin{array}[]{rl}d_{2}l_{g}P_{s}&\text{if
}x_{1}>-x_{0}\text{ and }x_{2}\leq-x_{0}\\\ 0&\text{if }x_{1}\leq-x_{0}\\\
\frac{P_{21}+P_{out}}{2}&\text{in other case}\end{array}\right.$ (19)
Following [1, 7, 10], these functions represent opening, partial closure and
total closure of the glottis. Throughout this work, piecewise functions
$P_{21}$, $f_{1}$ and $f_{2}$ are modeled using the derivable step-like
function $\Theta$ defined in Eq. (5).
## 3 Analysis of the model
### 3.1 Bifurcation diagram
The main anatomical parameters that can be actively controlled during the
vocalizations are the subglottal pressure $P_{s}$ and the folds tension
controlled by the laryngeal muscles. In particular, the action of the
thyroarytenoid and the cricothyroid muscles control the thickness and the
stiffness of folds. Following [1], this effect is modeled by a parameter $Q$
that scales the mechanic properties of the folds by a cord-tension parameter:
$k_{c}=Qk_{c0}$, $k_{i}=Qk_{i0}$ and $m_{i}=\frac{m_{i0}}{Q}$. We therefore
performed a bifurcation diagram using these two standard control parameters
$P_{s}$ and $Q$.
Five main regions of different dynamic solutions are shown in Fig. 2. At low
pressure values (region I), the system presents a stable fixed point. Reaching
region II, the fixed point becomes unstable and there appears an attracting
limit cycle. At the interface between regions I and II, three bifurcations
occur in a narrow range of subglottal pressure (Fig. 3, left panel), all along
the $Q$ axis. The right panel of Fig. 3 shows the oscillation amplitude of
$x_{2}$. At point A, oscillations are born in a supercritical Hopf
bifurcation. The amplitude grows continuously for increasing $P_{s}$ until
point B, where it jumps to the upper branch. If the pressure is then
decreased, the oscillations persist even for lower pressure values than the
onset in A. When point C is reached, the oscillations suddenly stop and the
system returns to the rest position. This onset-offset oscillation hysteresis
was already reported experimentally in [12].
Figure 2: Bifurcation diagram in the plane of subglottal pressure and fold
tension ($Q$,$P_{s}$). The insets are two-dimensional projections of the flow
on the ($v_{1}$,$x_{1}$) plane, the red crosses represent unstable fixed
points and the dotted lines unstable limit cycles. Normal voice occurs at
$(Q,P_{s})\sim(1,800)$. The color code represents the linear correlation
between $(x_{1}-x_{2})$ and $(y_{1}+y_{2})$: from dark red for $R=1$ to dark
blue for $R=0.6$. This diagram was developed with the help of AUTO
continuation software [20]. The rest of the parameters were fixed at
$m_{1}=0.125$ g, $m_{2}=0.025$ g, $k_{10}=80$ N/m, $k_{20}=8$ N/m, $k_{c}=25$
N/m, $\epsilon_{1}=0.1$, $\epsilon_{2}=0.6$, $l_{g}=1.4$ cm, $d_{1}=0.25$ cm,
$d_{2}=0.05$ cm and $x_{0}=0.02$ cm.
The branch AB depends on the viscosity. Decreasing $\mu$, points A and B
approach to each other until they collide at $\mu=0$, recovering the result
reported in [3, 10, 14], where the oscillations occur as the combination of a
subcritical Hopf bifurcation and a cyclic fold bifurcation.
On the other hand, the branch BC depends on the separation point of the jet
formation. In particular, for increasing $k_{s}$, the folds become stiffer and
the separation point moves upwards toward the output of the glottis. From a
dynamical point of view, points C and B approach to each other until they
collapse. In this case, the oscillations are born at a supercritical Hopf
bifurcation and the system presents no hysteresis, as in the standard flapping
model [17].
Figure 3: Hysteresis at the oscillation onset-offset. Left panel: zoom of the
interface between regions I and II. The blue and green lines represent folds
of cycles (saddle-node bifurcations in the map). The red line is a
supercritical Hopf bifurcation. Right panel: the oscillation amplitude of
$x_{2}$ as a function of the subglottal pressure $P_{s}$, at $Q=1.71$. The
continuation of periodic solutions was realized with the AUTO software package
[20].
Regions II and III of Fig. 2 are separated by a saddle-repulsor bifurcation.
Although this bifurcation does not represent a qualitative dynamical change
for the oscillating folds, its effects are relevant when the complete
mechanism of voiced sound production is considered. Voiced sounds are
generated as the airflow disturbance produced by the oscillation of the vocal
folds is injected into the series of cavities extending from the laryngeal
exit to the mouth, a non-uniform tube known as the vocal tract. The
disturbance travels back and forth along the vocal tract, that acts as a
filter for the original signal, enhancing the frequencies of the source that
fall near the vocal tract resonances. Voiced sounds are in fact perceived and
classified according to these resonances, as in the case of vowels [18].
Consequently, one central aspect in the generation of voiced sounds is the
production of a spectrally rich signal at the sound source level.
Interestingly, normal phonation occurs in the region near the appearance of
the saddle-repulsor bifurcation. Although this bifurcation does not alter the
dynamical regime of the system or its time scales, we have observed that part
of the limit cycle approaches the stable manifold of the new fixed point (as
displayed in Fig. 4), therefore changing its shape. This deformation is not
restricted to the appearance of the new fixed point but rather occurs in a
coarse region around the boundary between II and III, as the flux changes
smoothly in a vicinity of the bifurcation. In order to illustrate this effect,
we use the spectral content index SCI [21], an indicator of the spectral
richness of a signal: $SCI=\sum_{k}A_{k}f_{k}/(\sum_{k}A_{k}f_{0})$, where
$A_{k}$ is the Fourier amplitude of the frequency $f_{k}$ and $f_{0}$ is the
fundamental frequency. As the pressure is increased, the SCI of $x_{1}(t)$
increases (upper right panel of Fig. 4), observing a boost in the vicinity of
the saddle-repulsor bifurcation that stabilizes after the saddle point is
generated.
Thus, the appearance of this bifurcation near the region of normal phonation
could indicate a possible mechanism to further enhance the spectral richness
of the sound source, on which the production of voiced sounds ultimately
relies.
Figure 4: A projection of the limit cycle for $x_{1}$ and the stable manifold
of the saddle point, for parameters consistent with normal phonatory
conditions, $(Q,P_{s})=(1,850)$ (region III). Left inset: projection in the
3-dimensional space ($y_{1}$, $x_{1}$, $x_{2}$). Right inset: Spectral content
index of $x_{1}(t)$ as a function of $P_{s}$ for a fixed value of $Q=0.95$. In
green, the value at which the saddle-repulsor bifurcation takes place.
In the boundary between regions III and IV, one of the unstable points created
in the saddle-repulsor bifurcation undergoes a subcritical Hopf bifurcation,
changing stability as an unstable limit cycle is created [19]. Finally,
entering region V, the stable and the unstable cycles collide and disappear in
a fold of cycles where no oscillatory regimes exist.
In Fig. 2, we also display a color map that quantifies the difference between
the solutions of the model and the flapping approximation. The flapping model
is a two dimensional model that, instead of two masses per fold, assumes a
wave propagating along a linear profile of the folds, i.e., the displacement
of the upper edge of the folds is delayed $2\tau$ with respect to the lower.
The cross sectional areas at glottal entry and exit ($a_{1}$ and $a_{2}$) are
approximated, in terms of the position of the midpoint of the folds, by
$\displaystyle\left\\{\begin{array}[]{rl}a_{1}=2l_{g}(x_{0}+x+\tau\dot{x})\\\
a_{2}=2l_{g}(x_{0}+x-\tau\dot{x})\end{array}\right.,$ (22)
where $x$ is the midpoint displacement from equilibrium $x_{0}$, and $\tau$ is
the time that the surface wave takes to travel half the way from bottom to
top. Equation (22) can be rewritten as $(x_{1}-x_{2})=\tau(y_{1}+y_{2})$. We
use this expression to quantify the difference between the oscillations
obtained with the two-mass model solutions and the ones generated with the
flapping approximation, computing the linear correlation coefficient between
$(x_{1}-x_{2})$ and $(y_{1}+y_{2})$. As expected, the correlation coefficient
$R$ decreases for increasing $P_{s}$ or decreasing $Q$. In the region near
normal phonation, the approximation is still relatively good, with $R\sim
0.8$. As expected, the approximation is better for increasing $x_{0}$, since
the effect of colliding folds is not included in the flapping model.
### 3.2 Isofrequency curves
One basic perceptual property of the voice is the pitch, identified with the
fundamental frequency $f_{0}$ of the vocal folds oscillation. The production
of different pitch contours is central to language, as they affect the
semantic content of speech, carrying accent and intonation information.
Although experimental data on pitch control is scarce, it was reported that it
is actively controlled by the laryngeal muscles and the subglottal pressure.
In particular, when the vocalis or interarytenoid muscle activity is inactive,
a raise of the subglottal pressure produces an upraising of the pitch [16].
Figure 5: Relationship between pitch and restitution forces. Left panels:
isofrequency curves in the plane ($Q$,$P_{s}$). Right panels: Curves
$f_{0}(P_{s})$ for $Q$=0.9, $Q$=0.925 and $Q$=0.95. In the upper panels, we
used the model with the cubic nonlinear restitution of Eq. (2). In the lower
panels, we show the curves obtained with a linear restitution,
$K_{i}(x_{i})=k_{i}x_{i}+\Theta(\frac{x_{i}+x_{0}}{x_{0}})3k_{i}(x_{i}+x_{0})$.
Compatible with these experimental results, we performed a theoretical
analysis using $P_{s}$ as a single control parameter for pitch. In the upper
panels of Fig. 5, we show isofrequency curves in the range of normal speech
for our model of Eqs. (1) to (19). Following the ideas developed in [22] for
the avian case, we compare the behavior of the fundamental frequency with
respect to pressure $P_{s}$ in the two most usual cases presented in the
literature: the cubic [1, 7] and the linear [10, 14] restitutions. In the
lower panels of Fig. 5, we show the isofrequency curves that result from
replacing the cubic restitution by a linear restitution
$K_{i}(x_{i})=k_{i}x_{i}+\Theta(\frac{x_{i}+x_{0}}{x_{0}})3k_{i}(x_{i}+x_{0})$.
Although the curves $f_{0}(P_{s})$ are not affected by the type of restitution
at the very beginning of oscillations, the changes become evident for higher
values of $P_{s}$, with positive slopes for the cubic case and negative for
the linear case. This result suggests that a nonlinear cubic restitution force
is a good model for the elastic properties of the oscillating tissue.
## 4 Conclusions
In this paper, we have analyzed a complete two-mass model of the vocal folds
integrating collisions, nonlinear restitution and dissipative forces for the
tissue and jets and viscous losses of the air-stream. In a framework of
growing interest for detailed modeling of voice production, the aspects
studied here contribute to understanding the role of the different physical
terms in different dynamical behaviors.
We calculated the bifurcation diagram, focusing in two regimes: the
oscillation onset and normal phonation. Near the parameters of normal
phonation, a saddle repulsor bifurcation takes place that modifies the shape
of the limit cycle, contributing to the spectral richness of the glottal flow,
which is central to the production of voiced sounds. With respect to the
oscillation onset, we showed how jets and viscous losses intervene in the
hysteresis phenomenon.
Many different models for the restitution properties of the tissue have been
used across the literature, including linear and cubic functional forms. Yet,
its specific role was not reported. Here we showed that the experimental
relationship between subglottal pressure and pitch is fulfilled by a cubic
term.
###### Acknowledgements.
This work was partially funded by UBA and CONICET.
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|
arxiv-papers
| 2013-12-10T13:33:18 |
2024-09-04T02:49:55.302833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mar\\'ia Florencia Assaneo, Marcos A. Trevisan",
"submitter": "Marcos A. Trevisan",
"url": "https://arxiv.org/abs/1312.2950"
}
|
1312.3079
|
# Molecular Clouds in the North American and Pelican Nebulae: Structures
Shaobo Zhang11affiliation: Purple Mountain Observatory, & Key Laboratory for
Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, China;
[email protected] 22affiliation: Graduate University of the Chinese Academy
of Sciences, 19A Yuquan Road, Shijingshan District, Beijing 100049, China , Ye
Xu11affiliation: Purple Mountain Observatory, & Key Laboratory for Radio
Astronomy, Chinese Academy of Sciences, Nanjing 210008, China;
[email protected] , Ji Yang11affiliation: Purple Mountain Observatory, & Key
Laboratory for Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008,
China; [email protected]
###### Abstract
We present observations of 4.25 square degree area toward the North American
and Pelican Nebulae in the $J=1-0$ transitions of 12CO, 13CO, and C18O. Three
molecules show different emission area with their own distinct structures.
These different density tracers reveal several dense clouds with surface
density over 500 $M_{\sun}$ pc-2 and a mean H2 column density of 5.8, 3.4, and
11.9$\times 10^{21}$ cm-2 for 12CO, 13CO, and C18O, respectively. We obtain a
total mass of $5.4\times 10^{4}~{}M_{\odot}$ (12CO), $2.0\times
10^{4}~{}M_{\odot}$ (13CO), and $6.1\times 10^{3}~{}M_{\odot}$ (C18O) in the
complex. The distribution of excitation temperature shows two phase of gas:
cold gas ($\sim$10 K) spreads across the whole cloud; warm gas ($>$20 K)
outlines the edge of cloud heated by the W80 H II region. The kinetic
structure of the cloud indicates an expanding shell surrounding the ionized
gas produced by the H II region. There are six discernible regions in the
cloud including the Gulf of Mexico, Caribbean Islands and Sea, Pelican’s Beak,
Hat, and Neck. The areas of 13CO emission range within 2-10 pc2 with mass of
(1-5)$\times 10^{3}~{}M_{\odot}$ and line width of a few km s-1. The different
line properties and signs of star forming activity indicate they are in
different evolutionary stages. Four filamentary structures with complicated
velocity features are detected along the dark lane in LDN 935. Furthermore, a
total of 611 molecular clumps within the 13CO tracing cloud are identified
using the ClumpFind algorithm. The properties of the clumps suggest most of
the clumps are gravitationally bound and at an early stage of evolution with
cold and dense molecular gas.
###### Subject headings:
stars: formation – ISM: molecules – ISM: kinematics and dynamics
## 1\. Introduction
The study of massive star formation is limited. The molecular clouds within a
few hundred parsecs of the sun provide an ideal environment for improving our
knowledge of star forming process. Among these clouds, low-mass star-forming
regions constitute the majority of the population, while regions with massive
clumps and dense clusters like the Orion nebula are infrequent. The North
American (NGC 7000) and Pelican (IC 5070) Nebulae (referred to as the “NAN
complex” hereafter) are together one of the nearby ($\sim$600 pc, Laugalys &
Straižys 2002) star forming regions with large numbers of massive stars. This
is the next closest region showing signs of massive star formation after
Orion, but has been rarely studied to-date.
The studies of molecules (Bally & Scoville, 1980; Dobashi, Bernard, Yonekura
et al., 1994) and near-infrared extinction (Cambrésy, Beichman, Jarrett et
al., 2002) all confirm substantial quantities of molecular gas along the Lynds
Dark Nebula (LDN) 935 (Lynds, 1962) which lies between the North American and
Pelican nebulae. All three objects (NGC 7000, IC 5070, and LDN 935) are
thought to be a part of W80, a large H II region mainly in the background.
Comerón & Pasquali (2005) identified an O5V star, 2MASS J205551.25+435224.6,
hidden behind the LDN 935 cloud to be the ionizing star of the H II region.
Mid-infrared observations as Mid-course Space Experiment (MSX, Egan, Shipman,
Price et al. 1998) have found several Infrared dark clouds (IRDCs) in LDN 935
which indicates the existence of a cold, dense environment in the molecular
cloud. Other signposts of on-going star formation, such as HH objects, and
H$\alpha$ emission-line stars (e.g., Bally & Reipurth 2003; Comerón & Pasquali
2005; Armond, Reipurth, Bally et al. 2011, etc.), are also found in the NAN
complex. However, studies of molecules in the NAN complex, which can reveal
both the spatial and velocity structures, have only been conducted in a few
small regions or are limited by resolution.
In this work, we use molecular data tracing different environments to study
the properties of the individual regions, filamentary structures, and clumps
in the NAN complex. There is a divergence in the distance estimation of the
complex as discussed by Wendker, Baars, & Benz (1983); Straizys, Kazlauskas,
Vansevicius et al. (1993); Cersosimo, Muller, Figueroa Vélez et al. (2007),
etc and reviewed by Reipurth & Schneider (2008). In our calculation, we adopt
a commonly used distance of 600 pc based on multi-color photometric results
for hundreds stars (Laugalys & Straižys, 2002; Laugalys, Straižys, Vrba et
al., 2007).
## 2\. Observations and Data Reduction
We observed the NAN complex in 12CO (1$-$0), 13CO (1$-$0), and C18O (1$-$0)
with the Purple Mountain Observatory Delingha (PMODLH) 13.7 m telescope as one
of the scientific demonstration regions for Milky Way Imaging Scroll Painting
(MWISP) project111http://www.radioast.nsdc.cn/yhhjindex.php from May 27 to
June 3, 2011. The three CO lines were observed simultaneously with the 9-beam
superconducting array receiver (SSAR) working in sideband separation mode and
with the fast Fourier transform spectrometer (FFTS) employed (Shan, Yang, Shi
et al., 2012). The typical receiver noise temperature ($T_{\rm rx}$) is about
30 K as given by status
report222http://www.radioast.nsdc.cn/zhuangtaibaogao.php of PMODLH.
Our observations were made in 17 cells of dimension 30′$\times$30′ and covered
an area of total 4.25 deg2 (466 pc2 at the distance of 600 pc) as shown in
Figure 1. The cells were mapped using the on-the-fly (OTF) observation mode,
with the standard chopper wheel method for calibration (Penzias & Burrus,
1973). In this mode, the telescope beam is scanned along lines in RA and Dec
directions on the sky at a constant rate of 50″/sec, and receiver records
spectra every 0.3 sec. Each cell was scanned in both RA and Dec direction to
reduce the fluctuation of noise perpendicular to the scanning direction.
Further observations were made toward the regions with C18O detection to
improve their signal to noise ratios. The typical system temperature during
observations was $\sim$280 K for 12CO and $\sim$185 K for 13CO and C18O.
After removing the bad channels in the spectra, we calibrated the antenna
temperature ($T_{a}^{*}$) to the main beam temperature ($T_{\rm mb}$) with a
main beam efficiency of 44% for 12CO and 48% for 13CO and C18O. The calibrated
OTF data were then re-gridded to 30″pixels and mosaicked to a FITS cube using
the GILDAS software package (Guilloteau & Lucas, 2000). A first order baseline
was applied for the spectra. The resulting rms noise is 0.46 K for 12CO at the
resolution of 0.16 km s-1, 0.31 K for 13CO and 0.22 K for C18O at 0.17 km s-1.
Such noise level corresponds to a typical integration time of $\sim$30 sec in
each resolution element. A summary of the observation parameters is provided
in Table 1
Table 1Observation Parameters
Line | $\nu_{0}$ | HPBW | $T_{\rm sys}$ | $\eta_{\rm mb}$ | $\delta v$ | $T_{\rm mb}$ rms noise
---|---|---|---|---|---|---
($J=1-0$) | (GHz) | (″) | (K) | | (km s-1) | (K)
12CO | 115.271204 | 52$\pm$3 | 220-500 | 43.6% | 0.160 | 0.46
13CO | 110.201353 | 52$\pm$3 | 150-310 | 48.0% | 0.168 | 0.31
C18O | 109.782183 | 52$\pm$3 | 150-310 | 48.0% | 0.168 | 0.22
Note. — The columns show the line observed, the rest frequency of the line,
the half-power beam width of the telescope, the system temperature, main beam
efficiency, velocity resolution and rms noise of main beam temperature. The
beam width and main beam efficiency are given by status report of the
telescope.
## 3\. Result
### 3.1. General Distribution of Molecular Cloud
Figures 2-5 show the distributions of 12CO, 13CO and C18O emissions. The
distributions are elongated in the southeast-northwest direction along the
dark lane. 12CO presents bright, complex, extended emission throughout the
mosaic, while 13CO presents several condensations, and C18O only appears at
those brightest parts. From the distribution of molecules, we distinguish by
eye six regions and designated their names following Rebull, Guieu, Stauffer
et al. (2011). Positions of these regions are indicated on the composed image
in Figure 2. The brightest portions in all three lines are the Gulf of Mexico
to the southeast, and the Pelican to the northwest. Between these, there are
filamentary structures (the Caribbean Islands) and extended feature to the
south (the Caribbean Sea) with few pixels of C18O detection. The Caribbean
Islands and Sea regions are spatially coincident along the line of sight but
are separate in the velocity dimension.
Figure 1.— The location of observation coverage, superimposed on the second
Palomar Observatory Sky Survey (POSS II) red image. Green crosses mark the
T-Tauri type stars identified by Herbig (1958) Figure 2.— A composite color
image of the NAN complex made from the integrated intensity map, with 12CO in
blue, 13CO in green, and C18O in red, respectively. The spectra are integrated
over $-20\sim 20$ km s-1 for 12CO and 13CO, and $-10\sim 10$ km s-1 for C18O.
We also overlay outlines of the six regions with their names on the plot.
Figure 3.— Integrated intensity contours and gray-scale map of 12CO. The
spectra are integrated over $-20\sim 20$ km s-1. The contours are from 10 K km
s-1($\sim 9\sigma$) at intervals of 15 K km s-1, and the gray-scale colors
correspond to a linear stretch of integrated intensity. Figure 4.— Integrated
intensity contours and gray-scale map of 13CO. The spectra are integrated over
$-20\sim 20$ km s-1. The contours are from 4 K km s-1($\sim 5\sigma$) at
intervals of 6 K km s-1, and the gray-scale colors correspond to a linear
stretch of integrated intensity. Figure 5.— Integrated intensity contours and
gray-scale map of C18O. The spectra are integrated over $-10\sim 10$ km s-1.
The contours are from 2 K km s-1($\sim 5\sigma$) at intervals of 1.2 K km s-1,
and the gray-scale colors correspond to a linear stretch of integrated
intensity.
The channel map in Figure 6 illustrates the velocity structure of the
molecules in the NAN complex. Three 13CO filaments are clearly presented in
the velocity ranges of $-$7 to $-$4, $-$3 to $-$2, and $-$1 to 0 km s-1. The
latter two filaments connect the Gulf of Mexico and the Pelican’s Hat regions.
The emissions in the Gulf of Mexico indicate an arc feature from 0 to 2 km
s-1. Along with the Caribbean Sea, they show complicated structures in the
following positive velocity panels. There is another filamentary structure
near the Pelican’s Beak, in the velocity range 3 to 4 km s-1.
Figure 6.— Channel map of 13CO in the NAN complex. The central velocity of
each channel, in km s-1, is marked on the top left corner of each map.
The velocity-coded image shown in Figure 7 indicates the velocity distribution
of the emission peak of 13CO. Near the center of the whole complex, there are
several velocity components with high peak separation, and three filamentary
structures showing with different color overlapping each other. The velocity
components of the Pelican region in the northwest are relatively simple, while
the peak velocities in the southeast show a component around 0 km s-1, which
outlines the Gulf of Mexico region, and another separated extended components
at 3-4 km s-1. Such velocity structure could also be seen on the position-
velocity map in Figure 8 along the axis through the full length of the complex
in Figure 7. In the center region of the whole complex, the molecular emission
near $-$1 km s-1 is lacking and forms a cavity structure. Bally & Scoville
(1980) pointed out that the molecular gas in the northwest part of the NAN
complex belongs to an expanding shell surrounding the ionized gas produced by
the W80 H II region.
Figure 7.— Velocity-coded 13CO map. The color image shows the velocity
distribution of the emission peak of 13CO. The black long arrow indicates the
axis of the position-velocity map in Figure 8. The black dot on the axis
indicate the position of the ionizing star. A azimuthally averaged, around the
ionizing star within the sector region, position-velocity map is given in
Figure 9. The positions of four filamentary structures are shown with white
lines. Figure 8.— Position-velocity map along the axis shown in Figure 7. The
spectrum on each position is averaged along a 1° width line perpendicular to
the axis. The gray-scale background indicates 12CO, black contours indicate
13CO, and red contours indicate C18O. The lowest contour is 10$\sigma$ and the
contour interval is 10$\sigma$ (0.28 K) for 13CO and 5$\sigma$ (0.1 K) for
C18O. Projected positions of six regions are marked.
In Figure 9, we illustrate the kinematic of molecular shell near the Pelican
region in detail. We could derive a expansion velocity of $\sim$5 km s-1. The
Pelican’s Hat at the far end is $\sim$14 pc away from the center of the H II
region. The cloud near the ionizing star at $\sim$0 km s-1 connects to the
Gulf of Mexico region. Its velocity is close to the rest velocity of the whole
complex, which is probably because the molecular gas in these region has not
been penetrated by the shock of H II region (Bally & Scoville, 1980). In
Figure 8, we could further find there is a velocity gradient of $\sim$0.2 km
s-1 pc-1 within the complex along the axis of the position-velocity map.
Figure 9.— Azimuthally average position-velocity map around the ionizing star
within the sector region shown in Figure 7. The gray-scale background
indicates 12CO, and black contours indicate 13CO. The lowest contour is
5$\sigma$ (0.4 K) and the contour interval is 5$\sigma$.
Our mapping region contains total areas of 403 pc2 with 12CO detection, 225
pc2 with 13CO detection, and 18 pc2 with C18O detection over 3$\sigma$ at the
distance of 600 pc. Under the assumption of local thermodynamic equilibrium
(LTE), we derive the excitation temperature from the radiation temperature of
12CO. The distribution of excitation temperature shown in Figure 10 indicates
gases of two different temperatures within the NAN complex: localized warm gas
($>$20 K) in the Caribbean Islands, Pelican’s Neck and Beak, and in some small
clouds to the southeast; and extended cold gas ($\sim$10 K) distributed
throughout the whole of the dark nebula. The warm gas clearly matches the edge
of the whole cloud, suggesting the warm clouds are heated by the background H
II regions.
Figure 10.— A gray-scale map of excitation temperature in the NAN complex. The
excitation temperature is derived from the radiation temperature of 12CO
within the velocity range of $-20\sim 20$ km s-1.
We further calculate the column density and LTE mass with the 13CO data
following the process given by Nagahama, Mizuno, Ogawa et al. (1998) and
adopting a 13CO abundance of $N({\rm H_{2}})/N({\rm{}^{13}CO})=7\times
10^{5}$. We obtain a total mass of $2.0\times 10^{4}~{}M_{\odot}$ in the NAN
complex. Using the abundance of $N({\rm H_{2}})/N({\rm C^{18}O})=7\times
10^{6}$ (Castets & Langer, 1995), a LTE mass based on C18O data can also be
derived as $6.1\times 10^{3}~{}M_{\odot}$. If we simply use the CO-to-H2 mass
conversion factor $X$ of $1.8\times 10^{20}~{}{\rm
cm^{-2}(K~{}km~{}s^{-1})^{-1}}$ given in the CO survey of Dame, Hartmann, &
Thaddeus (2001), a mass of $5.4\times 10^{4}~{}M_{\odot}$ can be derived for
the complex. The mass of inner denser gas traced by 13CO accounts for 36% of
the mass in a larger area traced by 12CO, while the mass in a few small dense
cloud traced by C18O accounts for 11% of the total mass.
In our calculation, we obtained a mean H2 column density of $5.8\times
10^{21}$ cm-2 based on 12CO emission by averaging all pixels with line
detection. Similar method produces a mean column density of 3.4, and
11.9$\times 10^{21}$ cm-2 traced by 13CO and C18O, respectively. We show the
surface density map for the three molecular species in Figure 11. All three
tracers show a maximum surface density over 500 $M_{\sun}$ pc-2 in the
Pelican’s Neck region, while the Gulf of Mexico region is optically thick with
high surface density only in the C18O map. The $3\sigma$ noise at $T_{\rm
ex}=10$ K within velocity width of 40 km s-1 correspond to 14, 19, and 146
$M_{\odot}$ pc-2 in 12CO, 13CO, and C18O map, respectively. Therefore the mass
hidden under our detection limit of 13CO is lower than $5.4\times
10^{2}~{}M_{\odot}$, which indicates that the discrepancy in the obtained mass
between 12CO and 13CO is mainly the result of the different emission area
tracing by them. The hidden mass for C18O is $1.4\times 10^{3}~{}M_{\odot}$ at
most, significantly lower than the total mass traced by 13CO. This means that
we have detected over 80% of mass in our C18O observation area.
Bally & Scoville (1980) observed a similar field in the NAN complex and
estimated the LTE mass as $(3-6)\times 10^{4}~{}M_{\odot}$ for a distance of 1
kpc and 13CO abundance of $N({\rm H_{2}})/N({\rm{}^{13}CO})=1\times 10^{6}$.
For the same parameters as we used, it would correspond to (1-2)$\times
10^{4}~{}M_{\odot}$. Cambrésy, Beichman, Jarrett et al. (2002) obtained a mass
of $4.5\times 10^{4}~{}M_{\odot}$ for a distance of 580 pc in their near-
infrared extinction study covering an area of 6.25 deg2 in the NAN complex.
These discrepancy of mass might be due to the dust-to-gas ratio or the $X$
factor.
Figure 11.— The surface density of H2 traced by 12CO, 13CO, and C18O with the
same dynamical range. The abundance $N({\rm H_{2}})/N({\rm{}^{13}CO})=7\times
10^{5}$ (Nagahama, Mizuno, Ogawa et al., 1998) and $N({\rm H_{2}})/N({\rm
C^{18}O})=7\times 10^{6}$ (Castets & Langer, 1995) are adopted in the surface
density calculation.
### 3.2. Features in Individual Regions
Several discernible regions and filamentary structures can be identified in
our observations. The spectra observed toward the regions are shown in Figure
12. The variety of line profile and intensity ratios indicates distinct
kinematic and chemistry environments. Their properties probed by different
tracers are summarized in Table 2 and details for each region are listed
below.
Figure 12.— Typical spectra in the selected regions with names in the upper
left corner. These spectra are averaged within a 2′$\times$2′ box centered at
the positions (RA, Dec) marked at the lower right corner.
Table 2Properties of regions
| | | 12CO | | 13CO | | C18O | |
---|---|---|---|---|---|---|---|---|---
Region | $T_{\rm ex}$ | | area | $N_{\rm H_{2}}$ | $M$ | | area | $N_{\rm H_{2}}$ | $M$ | $\Delta v$ | | area | $N_{\rm H_{2}}$ | $M$ | | $S({\rm C^{18}O})\over S({\rm{}^{13}CO})$
| (K) | | (pc2) | ($10^{22}{\rm cm}^{-2}$) | ($10^{3}M_{\odot}$) | | (pc2) | ($10^{22}{\rm cm}^{-2}$) | ($10^{3}M_{\odot}$) | (km s-1) | | (pc2) | ($10^{22}{\rm cm}^{-2}$) | ($10^{2}M_{\odot}$) | |
Gulf of Mexico | 14.2 | | 18.2 | 1.2 | 11.8 | | 5.7 | 1.3 | 5.4 | 5.93 | | 2.0 | 2.5 | 32.0 | | 0.13
Pelican’s Hat | 12.2 | | 16.4 | 0.7 | 4.1 | | 4.6 | 0.6 | 1.7 | 4.37 | | 1.3 | 1.3 | 8.4 | | 0.14
Pelican’s Neck | 22.8 | | 4.4 | 1.6 | 6.0 | | 3.4 | 1.6 | 3.1 | 2.99 | | 0.6 | 2.4 | 9.9 | | 0.07
Pelican’s Beak | 15.9 | | 7.5 | 1.1 | 4.8 | | 2.2 | 0.8 | 1.3 | 2.24 | | 0.3 | 1.1 | 1.3 | | 0.08
Caribbean Islands | 18.7 | | 19.2 | 1.3 | 12.3 | | 2.2 | 1.3 | 5.0 | 2.98 | | 0.3 | 3.6 | 8.5 | | 0.11
Caribbean Sea | 12.1 | | 41.6 | 0.7 | 8.1 | | 10.2 | 0.4 | 2.5 | 3.86 | | - | - | - | | -
Note. — The properties of the regions in the NAN complex, including excitation
temperature, area within the half maximum contour line, mean column density of
H2, and mass of 12CO, 13CO, and C18O, line width of averaged spectra for 13CO,
and integrated intensity ratio of C18O to 13CO. The column density and mass
for 12CO are derived with a constant CO-to-H2 mass conversion factor, and
those for 13CO, and C18O are derived under the LTE assumption. The C18O
properties in Caribbean Sea is missing because of the low C18O detection rate
in this region.
The “Gulf of Mexico” (GoM) is the largest and the most massive region with all
three line detections in the southeast of the NAN complex. Two major clumps
can be found in this region: one in the north (GoM N) with weak C18O emission,
and one in the south (GoM S), with strong C18O emission indicating a pair of
parallel arcs, which closely matches the morphology of the filamentary dark
cloud in Spitzer mid-infrared image (Guieu, Rebull, Stauffer et al., 2009;
Rebull, Guieu, Stauffer et al., 2011). The 12CO spectra are flat-topped,
indicating high opacity at these locations. Under the LTE assumption, we find
a low excitation temperature ($\sim$14 K), large line width, and high column
density in this region. The intensity ratio in Figure 13, which also indicates
the relative abundance of C18O to 13CO, shows a higher ratio in the GoM S.
Rebull, Guieu, Stauffer et al. (2011) reported a young stellar objects (YSOs)
cluster is associated with the GoM region. A high concentration of T-Tauri
type stars (Herbig, 1958) and an association of H2O maser (Toujima, Nagayama,
Omodaka et al., 2011) are found in the GoM N. No IRAS point sources are
associated with either clump. These evidence suggest active star formation in
the GoM region, and the GoM N region is relatively more evolved than GoM S.
Figure 13.— Integrated intensity ratio of C18O to 13CO. The two panels show
the same region as that in Figure 5. Names of the regions are marked on the
map.
The “Pelican”s Hat” locates to the north of Pelican’s head, and is similar to
but smaller than the GoM S. The excitation temperature in this region is low
($\sim$12 K), and the relative abundance of C18O to 13CO is higher than those
in other regions in Pelican nebula. The line emission resembles the morphology
of mid-infrared extinction in Spitzer image (Guieu, Rebull, Stauffer et al.,
2009; Rebull, Guieu, Stauffer et al., 2011).
The “Pelican”s Neck” region to the west of the NAN complex shows the most
intense 12CO emission in our survey. It presents a high excitation temperature
of $\sim$23 K, narrow line width, and weak C18O emission. The molecular
emission shows a bright feature oriented in the north-south direction, with a
sharp cut-off towards the east. Several IRAS sources are associated with the
peaks on the 13CO map. A position-velocity slice along the east edge of the
Pelican’s Neck (as in Figure 14) reveals a weak component at $\sim$3 km s-1
that is separate from the molecular clump and forms a cavity near IRAS
20489+4410 and IRAS 20490+4413 in the velocity dimension. Such a structure
could be the result of an embedded H II region.
The molecular emission in Pelican’s Neck matches the morphology of the
brightest surface brightness region in Spitzer mid-infrared image, and it is
at the west edge of the Pelican Cluster, an active star forming cluster of
YSOs identified by Rebull, Guieu, Stauffer et al. (2011). A clustering of
T-Tauri type stars (Herbig, 1958) was found around the molecular clump. These
all indicate that the Pelican’s Neck is a warm region with active star
formation.
Figure 14.— Left: the 13CO integrated intensity map of Pelican’s Neck. Red
stars indicate the position of IRAS point sources. The names of three massive
sources are indicated. The blue arrow indicates the axis of the position-
velocity map. Right: position-velocity map along the axis shown in the left
panel. The gray-scale background indicates 12CO, dashed contours indicate
13CO, and red solid contours indicate C18O. The lowest contour is 10$\sigma$
and the contour interval is 15$\sigma$ for 13CO and 10$\sigma$ for C18O. The
vertical green line indicates the rest velocity averaged over the whole
region. Projected positions of three IRAS sources are marked.
The “Pelican”s Beak” region is a small elongated region to the southeast of
Pelican’s Neck. The excitation temperature is intermediate ($\sim$15 K) with
weak C18O emission. The molecules protrude along a filament to the south at
the velocity of 3 km s-1. Its properties may suggest an intermediate stage
between the cold dense regions (e.g. GoM, Pelican’s Hat) and the warm active
regions (e.g. Pelican’s Neck, Caribbean Islands).
The “Caribbean Islands” are several bright clumps extending from the west of
GoM and to the east of Pelican’s head. The southern half of Caribbean Islands
is spatially coincident with the Caribbean Sea. The channel map indicates
these “Islands” are part of a filamentary structure (see 3.3). They associate
with several highly localized nebulous bright blobs in Spitzers mid-infrared
image (Rebull, Guieu, Stauffer et al., 2011). These clumps show a high
excitation temperature, narrow line width, and low relative abundance of C18O.
These properties indicate a similar situation to that in Pelican’s Neck.
Together with Pelican’s Neck, the north part of Caribbean Islands forms a
cavity structure at the position of the Pelican Cluster which can be seen on
the 13CO integrated intensity map.
The molecular cloud in the northernmost part of this region is associated with
an H II regions, G085.051$-$0.182 at $-0.2$ km s-1, identified by Lockman,
Pisano, & Howard (1996). Figure 15 shows that the H II region is not
associated with any dense molecular clumps at its rest velocity. Dense and
heated gas with temperature $\sim$27 K appears within the velocity from $-$6
to $-3$ km s-1 near the position of the H II region, while diffuse clumps are
shown in the panels with positive velocities. The position-velocity map shows
an incomplete asymmetric molecular shell around the H II region. It is notable
that the densest part of the heated clumps tracing by C18O presents a slightly
higher velocity, which is closer to the rest velocity of the H II region than
those tracing by 12CO and 13CO. These indicate that the H II region undergoes
an asymmetric expansion within the parent molecular cloud.
Figure 15.— Top: channel map of 13CO near H II region in the Caribbean Islands
region. The cross in each panel marks the position of the H II region,
G085.051$-$0.182, reported by Lockman, Pisano, & Howard (1996). The central
velocity of each channel, in km s-1, is marked on the top left corner of each
map. The blue arrow is the axis of the position-velocity map. Bottom:
position-velocity map along the axis shown in the top panel. The gray-scale
background indicates 12CO, dashed contours indicate 13CO, and red solid
contours indicate C18O. The lowest contour is 5$\sigma$ for 13CO and
10$\sigma$ for C18O, with the contour interval of 5$\sigma$. The cross
indicates the rest velocity and the position of the H II region.
The “Caribbean Sea” is a diffuse extended cloud to the west of GoM at the
velocity of 3 km s-1 with low excitation temperature and optical depth. 12CO
are detected in a large area, and weak C18O emission can only be detected at a
few positions. This region shows the lowest column density among all the
regions but its total mass is relatively high.
### 3.3. Filamentary Structures
In our observations with velocity dimensions, we resolve three separate
filamentary structures (designated as F-1, F-2, F-3 in ascending velocity
order) nearly parallel to each other along the dark lane in the NAN complex.
Another filament (F-4) is also resolved near Pelican’s Beak region. Figure 7
shows the positions of the filaments with different color representing their
different velocity. Figure 16 and 17 shows the morphology and velocity
structure of these filaments. We found elongated molecular clumps along these
filaments. F-1, which contains the bright clumps in Caribbean Islands,
presents a complex twisted spatial and velocity structure, with a ring-like
structure near $\rm 20^{h}54^{m}.5,+44^{\circ}19^{\prime}$. F-2 and F-3 are
discontinuous, and together with the Pelican’s Hat region, they form a hub-
filament structure (Myers, 2009). The northwest and the southeast section of
F-2 show opposite velocity gradient directions. The northwest section of F-3
bends to the east with higher velocity and surrounds the Pelican Cluster. Both
F-2 and F-4 show clear velocity gradient along their axes in the position-
velocity map.
Figure 16.— 13CO moment maps of the filaments showing the integrated intensity
(left), rest velocity (middle), and line width (right). On each panel, the
zeroth moment contour lines are overlaid from 7$\sigma$ with 10$\sigma$
intervals. The name and the velocity range of each filament is marked in the
top left corner in each panel.
Figure 17.— 13CO position-velocity map of the four filamentary structures
along the axes shown in the integrated intensity maps of Figure 16. The lowest
contour is 5$\sigma$ and the contour interval is 5$\sigma$ for each panel. Red
dashed lines indicate the velocity range of each filaments. Figure 18.—
Position averaged 13CO spectra of the filaments. Spectra are moved upwards for
clarity. The name of each filament is marked on the left of each spectrum.
We show the averaged spectra in Figure 18 and some physical properties of the
filaments are listed in Table 3. A typical 13CO line width of 3.3 km s-1 is
shown. These filamentary structures show similar optical depths, while F-1 and
F-4 have a higher excitation temperature. We could estimate the mass per unit
length by dividing the mass of filaments by their spatial dimension. F-1 shows
a higher mass per unit length than that in the other filaments. The twisted
structure in F-1 may cause an overestimation of this measurement. A maximum,
critical linear mass density needed to stabilize a cylinder structure can be
calculated with $(M/l)_{\rm max}=84(\Delta v)^{2}M_{\odot}{\rm pc}^{-1}$ in
the turbulent support case, where $\Delta v$ is the line width in unit of km
s-1 (Jackson, Finn, Chambers et al., 2010). This means our filaments are
gravitationally stable on the assumption of the 13CO abundance we adopted.
Table 3Properties of filaments
Filament | $T_{\rm ex}$ | $\Delta v$(13CO) | $\tau$(13CO) | $M$ | $M/l$
---|---|---|---|---|---
| (K) | (km s-1) | | ($M_{\odot}$) | ($M_{\odot}$ pc-1)
F-1 | 16 | 3.20 | 0.33 | 1401 | 107
F-2 | 12 | 3.77 | 0.34 | 416 | 30
F-3 | 12 | 3.52 | 0.36 | 487 | 32
F-4 | 17 | 2.75 | 0.26 | 196 | 38
Note. — The properties of the filaments in the NAN complex, including
excitation temperature, line width of averaged spectra, optical depth of 13CO,
mass, and mass per unit length. These typical values are the results averaged
within the 10$\sigma$ contour line of each filament.
### 3.4. Clump Identification
We use the FINDCLUMPS tool in the CUPID package (a library of Starlink
package) to identify molecular clumps in the obtained 13CO FITS cube. The
ClumpFind algorithm is applied in the process of identification. The algorithm
first contours the data and searches for peaks to locate the clumps, and then
follows them down to lower intensities. We set the parameters
TLOW=5$\times$RMS and DELTAT=3$\times$RMS, where TLOW determines the lowest
level to contour a clump, and DELTAT represents the gap between contour levels
which determines the lowest level at which to resolve merged clumps (Williams,
de Geus, & Blitz, 1994). The parameters of each clump, such as the position,
velocity, size in RA and Dec directions, and one-dimensional velocity
dispersion, are directly obtained in this process. The clump size has removed
the effect of beam width, and velocity dispersion is also de-convolved from
the velocity resolution. The morphology of the clumps are checked by eye
within the three-dimension RA-Dec-velocity space to pick out clumps with
meaningful structures. We then mark every clump on their velocity channel in
the 13CO cube to confirm the morphology and emission intensity of the
molecular gas within the clumps. In addition, clumps with pixels that touch
the edge of the data cube are removed. 22 clumps are removed in these checking
steps. Eventually, a total of 611 clumps are identified, and the position,
velocity, and size of the clumps as illustrated in Figure 19 are consistent
with the spatial and velocity distribution of the molecular gas.
Figure 19.— Clumps identified in 13CO data cube. The circles indicate the
clump positions on the integrated intensity map of 13CO. The colors of the
circles represent the velocities the clumps, while the circles are scaled
according to the sizes of clumps.
We extract the excitation temperature for each clump from their 12CO datacube
under LTE assumption and can then derive the LTE mass. The parameters of the
clumps are listed in Table 4. The clump size are derived from the geometric
mean of the clump size in two directions. Figure 20 shows the distributions of
clump size, excitation temperature, and volume density, which yield typical
properties of $\sim$0.3 pc, 13 K, and 8$\times 10^{3}$ cm-3, respectively. The
left panel in Figure 21 shows the distribution of three-dimensional velocity
dispersion estimated as $\sigma_{v\rm 3D}=\sqrt{3}\sigma_{v\rm 1D}$. The
thermal portion in the velocity dispersion is $\sigma_{\rm
Thermal}=\sqrt{kT_{\rm kin}/m}$, where $k$ is the Boltzmann constant, $m$ is
the mean molecular mass, and $T_{\rm kin}$ is the kinetic temperature equal to
the excitation temperature, while the non-thermal portion is $\sigma_{\rm Non-
thermal}=\sqrt{\sigma_{v\rm 1D}^{2}-\sigma_{\rm Thermal}^{2}}$. The
distribution of the thermal and non-thermal velocity dispersion are shown in
Figure 21. There are 568 (93%) clumps with $\sigma_{\rm Non-thermal}$ larger
than $\sigma_{\rm Thermal}$. The mean ratio of $\sigma_{\rm Non-thermal}$ and
$\sigma_{\rm Thermal}$ is 1.57. This suggests that non-thermal broadening
mechanisms (e.g., rotation, turbulence, etc) play a dominant role in the
clumps.
Table 4Properties of clumps
Clump | R.A. | Dec. | Velocity | $\Delta{\rm R.A.}$ | $\Delta{\rm Dec.}$ | $R$ | $\delta_{v\rm 1D}$ | $T_{\rm peak}$ | $T_{\rm ex}$ | $\Sigma$ | $n_{\rm H_{2}}$ | $M_{\rm LTE}$ | $\alpha_{\rm Vir}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) | (km s-1) | (″) | (″) | (pc) | (km s-1) | (K) | (K) | ($M_{\sun}\rm pc^{-2}$) | $10^{3}\rm cm^{-3}$ | ($M_{\odot}$) |
1 | 20 48 01.2 | +43 42 59.4 | +0.01 | 115.9 | 120.7 | 0.17 | 0.35 | 4.40 | 16.67 | 23.9 | 11.2 | 15.6 | 1.52
2 | 20 48 01.6 | +43 34 43.8 | +1.13 | 139.2 | 178.4 | 0.23 | 0.35 | 4.57 | 17.73 | 31.9 | 10.1 | 33.2 | 0.98
3 | 20 48 15.2 | +43 40 59.4 | +1.50 | 128.9 | 160.3 | 0.21 | 0.24 | 3.62 | 15.32 | 17.0 | 5.2 | 13.1 | 1.07
4 | 20 48 20.6 | +43 31 04.7 | +0.99 | 56.5 | 94.2 | 0.10 | 0.28 | 3.27 | 14.66 | 6.9 | 5.6 | 1.8 | 5.00
5 | 20 48 37.6 | +43 48 11.8 | +1.01 | 194.6 | 319.4 | 0.36 | 0.56 | 3.73 | 18.72 | 40.1 | 5.7 | 74.4 | 1.74
6 | 20 48 41.2 | +43 39 38.6 | +1.67 | 52.0 | 31.2 | 0.06 | 0.21 | 7.21 | 14.41 | 9.8 | 29.7 | 1.6 | 1.80
7 | 20 48 42.3 | +44 21 38.6 | $-$4.26 | 47.5 | 65.4 | 0.08 | 0.36 | 5.19 | 17.79 | 15.3 | 26.4 | 3.9 | 3.09
8 | 20 48 44.2 | +43 40 15.1 | +2.25 | 32.2 | 79.2 | 0.07 | 0.19 | 6.69 | 13.16 | 7.2 | 10.6 | 1.2 | 2.58
9 | 20 48 44.8 | +43 52 52.4 | +1.47 | 169.5 | 193.1 | 0.26 | 0.27 | 3.61 | 16.00 | 17.9 | 3.0 | 14.8 | 1.46
10 | 20 48 46.3 | +44 15 17.7 | +1.95 | 45.3 | 80.5 | 0.09 | 0.40 | 6.49 | 24.47 | 38.2 | 53.3 | 9.9 | 1.63
Note. — The properties of the clumps in the NAN complex. Columns are clump
number, clump position (R.A. and Dec.), rest velocity, clump size in R.A. and
Dec. direction, clump radius, one-dimensional velocity dispersion, temperature
of emission peak, excitation temperature, surface density, volume density, LTE
mass, and virial parameter. The entire table is published in its entirety in
the electronic edition. A portion is shown here for guidance regarding its
form and content.
Figure 20.— Distribution of clump size (left), excitation temperature
(middle), and volume density (right). The size is the geometric mean of the
size in the R.A. and Dec. direction, and density is derive under the spherical
assumption. The range and typical value of each property are marked on each
plot. Figure 21.— Histogram of three-dimensional velocity dispersion (left),
and the thermal (middle) and non-thermal (right) one dimensional velocity
dispersions. The range and typical value of each velocity dispersion are
marked on each panel.
## 4\. Discussion
### 4.1. Comparison with Other Star Formation Regions
In a typical low-mass star-forming region, the Taurus region, Goldsmith,
Heyer, Narayanan et al. (2008) gives a LTE column density of $\leq 10^{22}$
cm-2 for the most dense region based on the data from the Five College Radio
Astronomy Observatory (FCRAO) survey (Narayanan, Heyer, Brunt et al., 2008).
This column density is lower than the averaged column density we derived in
several dense regions of the NAN complex. Qian, Li, & Goldsmith (2012)
searched for clumps in the Taurus survey data and derived a typical mean H2
density of $\sim$2000 cm-3, lower than the clump density in NAN complex of
8000 cm-3. In addition, we found a number of clumps with densities over 104
cm-3, which is hardly seen in the Taurus region. The 13CO line width (0.4-2.2
km s-1) in NAN complex is also slightly higher than that in Taurus (0.5-1.7 km
s-1). These may indicate potential massive stars are forming in some of the
dense clumps in the NAN complex.
In an active high-mass star-forming region, such as the Orion Nebula, a survey
of the Orion A region by Nagahama, Mizuno, Ogawa et al. (1998) yielded an
averaged column density similar to our results. Their survey found regions
with excitation temperature $\geq$ 20 K in most areas, and an even higher
temperature of $\geq$ 60 K in the Orion KL region. Meanwhile, the high
temperature regions in the NAN complex are limited to those around the Pelican
Cluster. In fact, the statistical properties of the clumps we identified in
the NAN complex are similar to those of the Planck cold dense core (Planck
Collaboration, Ade, Aghanim et al., 2011) of the Orion complex as studied by
Liu, Wu, & Zhang (2012). These results suggest most of the clumps, especially
the cold ones, in the NAN complex are in an early evolutionary stage of star
formation dominated by a non-thermal environment.
### 4.2. Gravitational Stability of the Clumps
The gravitational stability of clump determines whether the molecular clump
could further collapse and form a star cluster. We firstly calculate the
escape velocity ($v_{\rm escape}=\sqrt{2GM_{\rm LTE}/R}$) for each clump and
compare with its three-dimensional velocity dispersion. The escape velocities
range from 0.21 to 2.84 km s-1 with a typical value of 0.64 km s-1. About 493
(72%) clumps have velocity dispersion smaller than escape velocity, and only 8
(1%) clumps have velocity dispersion larger than twice the escape velocity. We
note that the clumps with high $\sigma_{v\rm 3D}$ to $v_{\rm escape}$ ratios
are faint with low emission peaks.
By simply assuming the clumps have a density profile of $\rho(r)=r^{-k}$ with
power-law index $k=1$, we could further derive the virial mass using the
standard equation (e.g. Solomon, Rivolo, Barrett et al. 1987; Evans 1999):
$M_{\rm Vir}=1164R\sigma_{v\rm 1D}^{2}[M_{\odot}]$, where the clump size $R$
is in pc, and three-dimensional velocity dispersion $\sigma_{v\rm 1D}$ is in
km s-1. A steeper power-law index of $k$ would result in a lower estimation of
virial mass. The virial parameter, defined as the ratio of virial mass to LTE
mass: $\alpha_{\rm Vir}=M_{\rm Vir}/M_{\rm LTE}$, describes the competition of
internal supporting energy against the gravitational energy. We find a typical
virial parameter of 2.5 in our clump sample. The virial masses are comparable
to the LTE masses. There are 588 (96%) clumps with virial mass larger than LTE
mass, and 221 (36%) clumps with virial parameter larger than 3. The clumps
with high virial parameter ($\alpha_{\rm Vir}>10$) are all faint ones with
emission peak lower than 3.3 K. The clumps with $\alpha_{\rm Vir}<1$ are
virialized and could be collapsing, while the clumps with higher $\alpha_{\rm
Vir}$ could be in a stable or expanding state unless they are external
pressure confined. Alternatively, it is possible that the faint clumps may be
transient entities (Ballesteros-Paredes, 2006).
Figure 22 shows the spacial distribution of clumps with virial parameter
coded. It is notable that the clumps close to virial equilibrium associate
with dense gas mainly around the Pelican region. The clumps in the Gulf of
Mexico region present slightly higher virial parameters than those in the
other dense regions, and most of the clumps with weak molecular emissions
especially those in the Caribbean Sea region are far from equilibrium state.
We compare $M_{\rm Vir}$ and $M_{\rm LTE}$ in the left panel of Figure 23. The
massive clumps tend to have a lower virial parameter. The mass relationship
can be fitted with a power-law of $M_{\rm Vir}/(M_{\odot})=(4.26\pm
0.16)[M_{\rm LTE}/(M_{\odot})]^{(0.75\pm 0.02)}$. The power index we obtained
is slightly higher than the value in Orion B (0.67) reported by Ikeda &
Kitamura (2009) and Planck cold clumps (0.61) reported by Liu, Wu, & Zhang
(2012), moreover, significantly higher than the index of pressure-confined
clumps ($\alpha_{\rm Vir}\propto M_{\rm LTE}^{-2/3}$) as given by Bertoldi &
McKee (1992). Although our molecular observations reveal several clumps in the
NAN complex could be exposed to strong external pressure from ionising
radiation and winds of massive stars, the comparability and the high power
index of virial and LTE mass suggest that most molecular clumps in the NAN
complex are gravitationally bound rather than pressure confined.
We could also derived the Jeans mass with $M_{\rm Jeans}=17.3{T_{\rm
kin}}^{1.5}n^{-0.5}M_{\odot}$ (Gibson, Plume, Bergin et al., 2009) and plot
their relationship with LTE mass in the right panel of Figure 23. Such
relationship could be described with a power-law of $M_{\rm
Jeans}/(M_{\odot})=(7.82\pm 0.27)[M_{\rm LTE}/(M_{\odot})]^{(0.12\pm 0.01)}$.
The flat power index indicates that the LTE masses of most massive clumps are
substantially larger than their Jeans mass, suggesting that these clumps will
further fragment and may not form individual proto-stars but proto-clusters.
Figure 22.— Distribution of clumps with virial parameter coded. The dots
represent the clumps overlaid on the integrated intensity map of 13CO. The
colors of the dots indicate the virial parameter of the clumps.
Figure 23.— Top: Virial mass-LTE mass relation of the clumps. Bottom: Jeans
mass-LTE mass relation of the clumps. The dot-dashed green line indicates a
mass ratio of 1. The solid red line shows a power-law fit to the relationship.
The dashed blue line in the left panel indicates the median mass ratio.
### 4.3. Larson Relationship and Mass Function of Clumps
Larson (1981) presents a correlation between the velocity dispersion and the
region size (range from 0.1 to 100 pc), known as the Larson relationship. The
Larson relationship was suggested to exist by several work (Leung, Kutner, &
Mead, 1982; Myers, Linke, & Benson, 1983), but some recent molecular surveys
suggest weak or no correlation between line width and size of molecular clouds
(Onishi, Mizuno, Kawamura et al., 2002; Liu, Wu, & Zhang, 2012). Figure 24
shows the relationship between size and three dimensional velocity dispersion
for our clumps. A fitting to the data gives a correlation of $\sigma_{v\rm
3D}/({\rm km~{}s^{-1}})=(1.00\pm 0.03)\times[{\rm Size}/({\rm pc})]^{(0.43\pm
0.02)}$, with a correlation coefficient of 0.63. The power index is slightly
larger than 0.39 given by Larson (1981). The correlation is not strong, which
might be the result of small dynamic range, and of the scattering of velocity
dispersion and clump size (0.06-1.26 pc) we found. The dynamic range is
limited by the sensitivity of observations. A uniform survey with sufficient
high sensitivity will improve the completeness of less intense clumps with low
column density and small size. On the other hand, Liu, Wu, & Zhang (2012)
pointed out that turbulence plays a dominant role in shaping the clump
structures and density distribution at a large scale, while the small-scale
clumps are easily affected by the fluctuations of density and temperature.
This will cause a large scattering of line width broadening induced by other
factor other than turbulence at small scales. Such scattering of the velocity
dispersion may result in a weak or even absent relationship.
Figure 24.— Larson relationship for the clumps. The red line indicates a
linear fitting to the clump size and velocity dispersion relation.
We then study the clump mass function (CMF) in Figure 25 based on the clump
mass sample we derived. A power-law distribution of d$N$/d log$M\propto
M^{-\gamma}$ is fitted with our data. Our power index (0.95) is lower than the
stellar initial mass function (IMF) of 1.35 give by Salpeter (1955). Several
(sub)millimeter continuum studies (Testi & Sargent, 1998; Johnstone & Bally,
2006; Reid & Wilson, 2006) and molecular observations (Ikeda & Kitamura, 2009)
obtained CMFs which are consistent with the Salpter IMF, while Kramer,
Stutzki, Rohrig et al. (1998) reported a flatter power index of 0.6-0.8 in
their CO isotopes study of seven molecular clouds. The similarity between CMF
and IMF power indices could simply be explained by a constant star formation
efficiency unrelated to the mass and self-similar cloud structure, based on a
scenario of one-to-one transformation from cores to stars (Lada, Muench,
Rathborne et al., 2008). However, such scenario is oversimplified, and ignores
the fragmentation in cores whose masses exceed the Jeans mass. Fragmentation
in prestellar cores has been observed and discussed by several work (Goodwin,
Kroupa, Goodman et al., 2007; Chen & Arce, 2010; Maury, André, Hennebelle et
al., 2010). In addition, a simulation by Swift & Williams (2008) suggested
that the obtained IMF is similar to the input CMF even when different
fragmentation modes are considered.
Figure 25.— Clump mass function (CMF) for the clumps. Red line fit the power-
law distribution from 15 to 300 M⊙.
### 4.4. YSOs in Molecular Cloud
Cambrésy, Beichman, Jarrett et al. (2002) identified nine young stellar
clusters in the NAN complex, and Guieu, Rebull, Stauffer et al. (2009)
provided a list of more than 1600 YSOs in their four Infrared Array Camera
(IRAC) bands study with the Spitzer Space Telescope. Lately, Rebull, Guieu,
Stauffer et al. (2011) incorporated their Multiband Imaging Photometer for
Spitzer (MIPS) observations with earlier archival data, and identified a list
of 1286 YSOs in the NAN complex. We compare the distribution of the YSOs from
Rebull, Guieu, Stauffer et al. (2011) with our molecular observations in
Figure 26. The Class I and flat sources are concentrated in cold and dense
molecular clouds, especially in the Gulf of Mexico and the Pelican’s Hat
region, while the Class II sources are spread across the cloud with low
molecular opacity, and only a few YSOs are associated with the diffuse
Caribbean Sea region. The molecular properties associated with different
classes of YSOs are extracted and studied in Figure 27. The histograms
indicate that the Class I and flat sources match the distribution of molecular
clouds and prefer a cold dense environment with excitation temperature of
$\sim$14 K and column density of $\sim$1022 cm-2.
Three main YSO clusters are identified from the sample of Rebull, Guieu,
Stauffer et al. (2011). Two of these with a great fraction of Class I and flat
objects are associated with the molecular cloud of the Gulf of Mexico and the
Pelican’s Hat region which shows low temperature and high C18O abundance. The
third cluster, the Pelican Cluster, is surrounded by the Pelican’s Neck, the
Pelican’s Beak, and the Caribbean Islands. Although the Class II sources
constitute a higher fraction in the Pelican Cluster, most of the Class I and
flat objects appear on the east and west edges of the cluster. This
distribution is consistent with the molecular distribution in which the
molecular gas in the central area is dispersed and surrounded by clouds with
higher molecular temperature and low C18O abundance. The YSO proportion in the
clusters suggests a younger stage of evolution in the most south-eastern and
north-western parts of the NAN complex, and an older stage in the center of
the Pelican Cluster. If the complex velocity structures in surrounding regions
of the Pelican Cluster are indeed the results of feedbacks from the massive
cluster members, the cluster may be triggering the star formation in the
molecular cloud across a span of over 5 pc and $\sim$10 km s-1.
Figure 26.— YSOs in the NAN complex identified by Rebull, Guieu, Stauffer et
al. (2011). The gray-scale image is the integrated intensity map of 13CO. Red
dots are Class I, green are flat, blue are Class II, and purple are Class III.
Figure 27.— The molecular properties (left: excitation temperature; right:
column density of H2) associated with YSOs, with the same color code as Figure
26.
We then compare our clump results with the distribution of YSOs, by separating
the clumps spatially associated with YSOs from those containing no YSO. The
Class III YSOs are not considered, as the Class III catalogue is not complete
and their distribution is not associate with molecular cloud. A total of 143
clumps are found to be associated with YSOs. The discrepancies in their
physical properties are shown in Figure 28. The clumps associated with YSOs
present a higher velocity dispersion, clump size, and excitation temperature,
while the discrepancy of the CMF indices is not significant. Further
observations with higher signal-to-noise ratio and resolution are needed to
extend the limit of mass completeness in CMF comparison.
Figure 28.— The histogram of three-dimensional velocity dispersion (upper
left), clump size (upper right), excitation temperature (lower left) and CMF
(lower right). Red lines represent the clumps associated with YSOs, while
black lines represent the clumps without YSOs. The median values and CMF
indices are marked on the plot.
## 5\. Summary
We have presented the PMODLH mapping observations for an area of 4.25 deg2
toward the North American and Pelican Nebulae molecular cloud complex in 12CO,
13CO, and C18O lines. The main results are listed below:
The molecules distribution is along the dark lane in the southeast-northwest
direction. 12CO emission is bright, extended, while 13CO and C18O emissions
are compact. The channel map shows intricate structures within the complex,
and filamentary structures are revealed. Position-velocity slice along the
full length of the cloud reveals a molecular shell surrounding the W80 H II
region. Gases of two different temperatures are seen in the distribution of
excitation temperature.
The surface density map shows several dense clouds with surface density over
500 $M_{\sun}$ pc-2 in the complex. We have derived a total mass of $2.0\times
10^{4}~{}M_{\odot}$ (13CO) and $6.1\times 10^{3}~{}M_{\odot}$ (C18O) under the
LTE assumption with uniform molecular abundance, and $5.4\times
10^{4}~{}M_{\odot}$ with the constant CO-to-H2 factor in the NAN complex. Such
a discrepancy in mass may be due to the different extent which the molecules
are tracing.
Six regions are discerned in the molecular maps, each with different emission
characteristics. Their sizes, column densities, and masses vary with different
density tracers. The properties of low temperature, high column density, and
high C18O abundance found in the Gulf of Mexico, and Pelican’s Hat regions
indicate a young stage of massive star formation, while the properties of the
Pelican’s Neck, Pelican’s Beak, and Caribbean Islands regions represent a hot,
dense, and more evolved environment probably affected by the Pelican Cluster.
Only the Caribbean Sea region shows little sign of star formation.
Four filamentary structures are found in the NAN complex. They show complex
structures such as a twisted spatial distribution or opposite velocity
gradient directions, but these filaments all seem in a gravitationally stable
state.
We have identified 611 clumps using the ClumpFind algorithm in the NAN
complex, and yield a typical size, excitation temperature, and density of
$\sim$0.3 pc, 13 K, and 8$\times 10^{3}$ cm-3, respectively. Most of the
clumps are non-thermal dominated and in an early evolutionary stage of star
formation. The comparison of virial and LTE mass of the clumps indicates that
most clumps are gravitationally bound. We obtain a clump mass function index
$\gamma=0.95$. The clumps associate with YSOs present more evolved features
comparing with those having no association.
This work is based on observations made with the Delingha 13.7-m telescope of
the Purple Mountain Observatory. The authors appreciate all the staff members
of the observatory for their help during the observations. We acknowledge John
Bally, the referee of this paper for his valuable comments which helped to
considerably improve the quality of the manuscript. This work is supported by
the Chinese NSF through grants NSF 11133008, NSF 11073054, NSF 11233007, and
the Key Laboratory for Radio Astronomy, CAS.
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|
arxiv-papers
| 2013-12-11T08:49:00 |
2024-09-04T02:49:55.311249
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaobo Zhang, Ye Xu, Ji Yang",
"submitter": "Shaobo Zhang",
"url": "https://arxiv.org/abs/1312.3079"
}
|
1312.3089
|
# Quasiperiodic graphs at the onset of chaos
B. Luque1, M. Cordero-Gracia1, M. Gómez1, and A. Robledo2 1 Dept. Matemática
Aplicada y Estadística. ETSI Aeronáuticos, Universidad Politécnica de Madrid,
Spain.
2 Instituto de Física y Centro de Ciencias de la Complejidad, Universidad
Nacional Autónoma de México, Mexico.
###### Abstract
We examine the connectivity fluctuations across networks obtained when the
horizontal visibility (HV) algorithm is used on trajectories generated by
nonlinear circle maps at the quasiperiodic transition to chaos. The resultant
HV graph is highly anomalous as the degrees fluctuate at all scales with
amplitude that increases with the size of the network. We determine families
of Pesin-like identities between entropy growth rates and generalized graph-
theoretical Lyapunov exponents. An irrational winding number with pure
periodic continued fraction characterizes each family. We illustrate our
results for the so-called golden, silver and bronze numbers.
###### pacs:
05.45.Ac, 05.90.+m, 05.10.Cc
## I Introduction
The onset of chaos is a prime dynamical phenomenon that has attracted
continued attention motivated by the aim to both expand its understanding and
to explore its manifestations in many fields of study strogatz1 . From a
theoretical viewpoint, chaotic attractors generated by low-dimensional
dissipative maps have ergodic and mixing properties and, not surprisingly,
they can be described by a thermodynamic formalism compatible with Boltzmann-
Gibbs (BG) statistics dorfman1 . But at the transition to chaos, the infinite-
period accumulation point of periodic attractors, these two properties are
lost and this suggests the possibility of exploring the limit of validity of
the BG structure in a precise but simple enough setting. The horizontal
visibility (HV) algorithm luque1 ; luque2 that transforms time series into
networks has offered luque3 ; luque4 ; luque5 ; luque6 ; luque7 a view of
chaos and its genesis in low-dimensional maps from an unusual perspective
favorable for the appreciation and understanding of basic features. Here we
present the scaling and entropic properties associated with the connectivity
of HV networks obtained from trajectories at the quasiperiodic onset of chaos
of circle maps hilborn1 and show that this is an unusual but effective
setting to observe the universal properties of this phenomenon.
The three well-known routes to chaos in low-dimensional dissipative systems,
period-doubling, intermittency and quasiperiodicity, have been analyzed
recently luque3 ; luque4 ; luque5 ; luque6 ; luque7 via the HV formalism, and
complete sets of graphs, that encode the dynamics of all trajectories within
the attractors along these routes, have been determined. These graphs display
structural and entropic properties through which a distinct characterization
of the families of time series spawned by these deterministic systems is
obtained. The quantitative basis for these results is provided by the
corresponding analytical expressions for the degree distributions. The graph
at the transition to chaos has been studied only for the period-doubling route
for which connectivity expansion an entropy growth rates have been determined
and found to be linked by Pesin-like identities luque5 . Here we present
results for the transition to chaos for the quasiperiodic route that expand on
this findings and suggest that structural and entropic properties of such
networks are linked by Pesin-like equalities that use generalizations of the
ordinary Lyapunov and BG entropy expressions.
We refer to Pesin-like identities as those that were first found to occur at
the period-doubling transitions to chaos that link generalized Lyapunov
exponents to entropy growth rates at finite, but all, iteration times
baldovin1 ; mayoral1 . Recently luque5 these identities were retrieved in a
network context via the HV method. Pesin-like identities differ from the
genuine Pesin identity, the single positive Lyapunov exponent version of the
Pesin theorem pesin1 , for chaotic attractors in one-dimensional iterated
maps. The Pesin identity links asymptotic quantities that are invariant under
coordinate transformations, whereas the finite-time Pesin-like identities,
that appear for vanishing ordinary Lyapunov exponent are coordinate dependent.
However, in the case of period doubling it has been seen that the identities
remain valid when different coordinate systems are used to determine them, as
in Refs. luque5 and baldovin1 .
The rest of this paper is as follows: We first recall the HV algorithm luque1
; luque2 that converts a time series into a network and focus on the
quasiperiodic graphs luque6 as the specific family of HV graphs generated by
the standard circle map. We then expose the universal scale-invariant
structure of the graphs that arise at the infinite period accumulation points
by focusing on the golden ratio route. We describe the diagonal structure of
these graphs when represented by the exponential of the connectivity, and
introduce a generalized graph-theoretical Lyapunov exponent appropriate for
the subexponential growth of connectivity fluctuations. Subsequently, we show
how the collapse of the diagonal structure into a single one represents the
scale-invariant property that governs the degree fluctuations. Following this,
we analyze the network expression for the entropy rate of growth and find a
spectrum of Pesin-like identities. Finally, we show that all the previous
results can be generalized by considering winding numbers given by any
quadratic irrational. We discuss our results.
Figure 1: Six levels of the Farey tree and the periodic motifs of the graphs
associated with the corresponding rational fractions $p/q$ taken as dressed
winding numbers $\omega$ in the circle map (for space reasons only two of
these are shown at the sixth level). (a) In order to show how graph
inflationary process works, we have highlighted an example using different
grey tones on the left side. See Ref. luque6 for details. (b) First five
steps along the Golden ratio route, $b=1$ (thick solid line); (c) First three
steps along the Silver ratio route, $b=2$ (thick dashed line).
## II Quasiperiodic graphs at the golden ratio onset of chaos
The idea of extracting graphs from time series is hardly new and over the past
years several approaches have been proposed and are currently developed
Crutchfield ; zhang06 ; kyriakopoulos07 ; xu08 ; donner10 ; donner11 ;
donner11-2 ; campanharo11 . The HV approach is chosen here because of both its
simplicity of implementation and its capability to produce analytical results
in closed form for quantities that are generally difficult to determine. As we
see below this is corroborated for the present enterprise. For the circle map
it has been possible to determine previously the relevant dynamical quantities
at the transition to chaos only for the golden route robledo1 . In contrast,
in the present study it has been possible to generalize this result
effortlessly for an infinite number of routes to chaos associated with all the
quadratic irrational numbers.
The horizontal visibility (HV) algorithm is a general method to convert time
series data into a graph luque1 ; luque2 and is minimally stated as follows:
assign a node $i$ to each datum $\theta_{i}$ of the time series
$\\{\theta_{i}\\}_{i=1,2,...}$ of real data, and then connect any pair of
nodes $i$, $j$ if their associated data fulfill the criterion $\theta_{i}$ ,
$\theta_{j}$ $>$ $\theta_{n}$ for all $n$ such that $i<n<j$. We note that the
HV algorithm is related to the permutation entropy scheme Bandt in which the
problem of the partition of symbols of a time series is sorted out by simple
comparison of nearest-neighbor values within the series. The HV method
addresses in a similar way this problem, but in addition it makes use of
comparisons of values between neighbors that can be separated by long
distances, and consequently it stores additional information of the series in
the structure of the resulting HV graph.
The HV method has been applied luque6 to trajectories generated by the
standard circle map Landau ; Ruelle ; Shenker ; Kadanoff ; Rand ; Rand2 ;
hilborn1 given by
$\theta_{t+1}=f_{\Omega,K}(\theta_{t})=\theta_{t}+\Omega-\frac{K}{2\pi}\sin(2\pi\theta_{t}),\;\textrm{mod}\;1,$
(1)
representative of the general class of nonlinear circle maps:
$\theta_{t+1}=f_{\Omega,K}(\theta_{t})=\theta_{t}+\Omega+K\cdot
g(\theta_{t}),\;\textrm{mod}\;1$, where $g(\theta)$ is a periodic function
that fulfills $g(\theta+1)=g(\theta)$. This family of maps exhibit universal
properties that are preserved by the HV algorithm luque6 so that without loss
of generality we explain below our findings in terms of the standard circle
map, where $\theta_{t}$, $0\leq\theta_{t}<1$, is the dynamical variable, the
control parameter $\Omega$ is called _bare winding number_ , and $K$ is a
measure of the strength of the nonlinearity. The _dressed winding number_ for
the map is defined as the limit of the ratio:
$\omega\equiv\lim_{t\rightarrow\infty}(\theta_{t}-\theta_{0})/t$. For $K\leq
1$ trajectories are periodic (locked motion) when the corresponding dressed
winding number $\omega(\Omega)$ is a rational number $p/q$ and quasiperiodic
when it is irrational. For $K=1$ (critical circle map) locked motion covers
the entire interval of $\Omega$ leaving only a multifractal subset of $\Omega$
unlocked.
The periodic time series of period $q$ that constitutes the trajectory within
an attractor with $\omega(\Omega)=p/q$ is represented in the HV graph by the
repeated concatenation of a motif, a number of which are shown in Fig. 1. The
display of these motifs in the Farey tree in Fig. 1 helps visualize the
inflationary process that takes place when the HV network grows at the onset
of chaos luque6 . For illustrative purposes in Fig. 1 we show the periodic
motifs of the HV graphs that are associated with the irreducible rational
numbers $p/q\in[0,1]$, and we place them on the Farey tree hilborn1 along
which routes to chaos take place. A well-studied case is the sequence of
rational approximations of $\omega_{\infty}=\phi^{-1}=(\surd
5-1)/2=0.618034...$ , the reciprocal of the golden ratio, which yields winding
numbers $\\{\omega_{n}=F_{n-1}/F_{n}\\}_{n=1,2,3...}$ where $F_{n}$ is the
Fibonacci number generated by the recurrence $F_{n}=F_{n-1}+F_{n-2}$ with
$F_{0}=1$ and $F_{1}=1$. The first few steps of this route can be seen in Fig.
1(b).
The trajectories generated by the map with initial condition $\theta_{0}=1$ at
the golden ratio onset of chaos define a multifractal attractor that forms a
striped pattern of positions when plotted in logarithmic scales, i.e.
$\ln\theta_{t}$ vs $\ln t$. See Fig. 3 in Ref. robledo1 . This attractor
corresponds to the accumulation point
$\Omega_{\infty}=\lim_{n\rightarrow\infty}\Omega_{n}$ of bare winding numbers
$\Omega_{n}$ that characterize superstable trajectories of periods $F_{n}$,
$n=1,2,3,...$, $\Omega_{\infty}=0.606661...$ robledo1 . A sample of this time
series is shown in the top panel of Fig. 2. In the bottom panel of the same
figure we plot, in logarithmic scales, the outcome of the HV method with use
of the variable $\exp k(N)$, where $k(N)$ is the degree of node $N$ in the
graph generated by the time series $\theta_{t}$ (that is, $N\equiv
t=1,2,3,...$). Notice that the distinctive striped pattern of the attractor
robledo1 is present in the figure, although in a simplified manner where the
fine structure is replaced by single lines of constant degree. The HV
algorithm transforms the multifractal attractor into a discrete set of
connectivities.
## III Diagonal structure of the connectivity fluctuations
It is clear from the bottom panel of Fig. 2 that the degree $k(N)$, and also
$\exp k(N)$, fluctuates when $N$ is increased step by step via a deterministic
pattern of ever increasing amplitude. Notice also in the same panel the
diagonal lines that are drawn to connect sequences of node-connectivity
($N,k$) values; there is a main diagonal followed by two other diagonals close
to each other. These ($N,k$) sequences fall asymptotically along parallel
straight lines, that begin after the initial steps from the lowest values of
the degree, $k=2$ or $k=3$, skip the absent $k=4$, and reach the values $k=5$
or $k=6$, and therefore the sequences obey a power law with the same exponent.
There are many more sequences along same-slope diagonals, not highlighted in
the figure, arranged in close groups and that trace all other possible
connectivities $k(N)$. See also Fig. 3 in Ref. robledo1 . It is by examining
the dependence of $k(N)$ along each member of this family of diagonals that
the scaling and entropic properties of the network are determined.
Figure 2: Top: Positions $\theta_{t}$ as a function of ${\small t}$ for the
first ${\small 55}$ data for the orbit with initial condition
${\small\theta}_{0}{\small=1}$ at the golden ratio onset of chaos (see text)
of the critical circle map ${\small K=1}$. The data highlighted are associated
with specific subsequences of nodes (see text). Bottom: Log-log plot of
${\small\exp k(N)}$ as a function of the node $N$ for the HV graph generated
from same time series as as for the upper panel but for $3\times 10^{2}$
iterations, where ${\small N=t}$. The distinctive band pattern of the
attractor manifests through a pattern of single lines of constant degree. The
node positions of some node subsequences along diagonals is highlighted as
guide lines to the eye. The inset shows the collapse of all nodes in the graph
into a single diagonal (see text).
Thus, the ($N,k$) pairs in the graph define a structure in diagonals
$d=1,2,3,...$, and on each diagonal $d$ we label the particular nodes that lie
on it as $n=0,1,2,...$ Thus, $N(n;d)$ indicates the node/time for the $n$-th
position on diagonal $d$. For example, in the first and main diagonal $d=1$ in
Fig. 2 we have $N(0;1)=1=F_{1}$, $N(1;1)=3=F_{3}$, $N(2;1)=8=F_{5}$,
$N(3;1)=21=F_{7}$,… As it can be seen in the top panel of Fig. 2, the matching
positions $\theta_{t}$, $t=F_{2n+1}$ (highlighted) grow monotonically when
removed from the rest of the time series, and according to the HV algorithm
this implies increasing values for the degrees of their corresponding nodes.
For $d=2$ (the second diagonal in Fig. 2) $N(0;2)=2$, $N(1;2)=6$, $N(2;2)=16$,
$N(3;2)=42$,… All the nodes $N(n;d)$ can be expressed via the recurrence
formula
$\displaystyle N(0;d)$ $\displaystyle=$
$\displaystyle\textrm{mex}\\{N(n;i):1\leq i<d,n\geq 0\\},$ $\displaystyle
N(1;d)$ $\displaystyle=$ $\displaystyle 2N(0;d)+d,$ $\displaystyle N(n;d)$
$\displaystyle=$ $\displaystyle 3N(n-1;d)-N(n-2;d),$ (2)
with $d=1,2,...$ and $n=0,1,2,..$., where the term mex stands for
MinimumEXclude value conway1 that in this case it means the smallest value of
$N$ that has not appeared in the previous diagonals. In fraenkel1 it is
demonstrated that every integer $N$ appears only once under the above
recurrence and this exotic enumeration occurs in a natural way in the golden
ratio route. In fact, all the time labels $n$ along the diagonals $d=1,2,...$
can be expressed as Fibonacci numbers
$F_{n}^{(d)}=F_{n-1}^{(d)}+F_{n-2}^{(d)}$ with different initial conditions
for each one of them,
$\displaystyle F_{0}^{(d)}$ $\displaystyle=$ $\displaystyle d,\
F_{1}^{(d)}=N(0;d),$ $\displaystyle N(n;d)$ $\displaystyle=$ $\displaystyle
F_{2n+1}^{(d)}.$ (3)
This recurrence is the consequence of the inflationary process that takes
place in the generation of graphs via the golden ratio route luque6 . Notice
that this route goes through successive approximants of the continued fraction
$[1,1,1,...]$ (see Fig. 1b). These approximants permanently alternate from
larger to smaller to larger values around the golden number, such that an
approximant graph is generated by concatenation of the two preceding
approximant graphs alternating the order of concatenation at each stage. This
can be seen explicitly in Fig. 3.
Figure 3: First substructures of the quasiperiodic graph associated with the
golden ratio route to chaos. The resulting patterns follow from the universal
order with which an orbit visits the positions of the attractor. The
quasiperiodic graph associated with the time series generated at the onset of
chaos ($n\rightarrow\infty$) is the result of an infinite application of the
inflationary process by which a graph at period $F_{2n+1}$ is generated out of
graphs at periods $F_{2n}$ and $F_{2n-1}$ luque6 . The first few node/time
steps along the first diagonal ($d=1$) are highlighted.
The recurrence formula in Eq. (2) can be solved leading to an explicit
expression convenient for our purposes. First, it can be demonstrated
fraenkel1 that
$\displaystyle N(0;d)$ $\displaystyle=$
$\displaystyle\lfloor(d-1)\phi\rfloor+1,$ $\displaystyle N(n;d)$
$\displaystyle=$ $\displaystyle\lfloor N(n-1;d)\phi^{2}\rfloor+1.$ (4)
Then, use of the approximation $N(n;d)\approx N(n-1;d)\phi^{2}$ and of the
definition $C_{d}\equiv
N(1;d)=\big{\lfloor}(\lfloor(d-1)\phi\rfloor+1)\phi^{2}\big{\rfloor}+1$ yields
the solution
$N(n;d)=C_{d}\phi^{2n-2},\ n\geq 1.$ (5)
This equation captures the values $N(n;d)$ along the diagonals starting always
from $n=1$, that, as we can observe in the bottom panel of Fig. 2, are the
nodes with connectivities $k=5$ or $k=6$. Furthermore, all the (parallel
straight- line) diagonals can be collapsed into a single one by first
redefining the connectivities in each of them such that the degree is zero in
the initial position $n=1$. To do this it is only necessary to subtract $5$ or
$6$ according to the given diagonal, with the outcome that
$\widetilde{k}=2n-2$ with $n=1,2,...$ To get the collapse it is sufficient to
introduce the change of variable $\widetilde{N}(n;d)=N(n;d)/C_{d}$ so that
$\widetilde{N}(n;d)=\phi^{2n-2}$. We can see the result in the inset in the
bottom panel of Fig. 2. To keep notation simple we make use of this variable
and write $k$ instead of $\widetilde{k}$ from now on.
Figure 4: Log-log plot of the distance between two nearby trajectories
$l_{t}=|\theta_{t}-\theta_{t}^{\prime}|$ close to
${\small\theta}_{0}{\small=1}$, where ${\small l}_{0}{\small=10}^{-4}$,
measured at times ${\small t=N(n;d)}$, ${\small n=0,1,2,...}$, along the main
diagonal ${\small d=1}$ at the transition to chaos for the golden, silver and
bronze routes (see text).
## IV Generalized Lyapunov exponents at the accumulation point of the golden
ratio route to chaos
We define now a connectivity expansion rate for the graph under study. The
formal network analog of the sensitivity to initial conditions in the map is
luque5
$\xi(N(n;d))\equiv\frac{\exp(k(n))}{\exp(k(1))}=\exp(k(n)),$ (6)
since $k(1)=k(N(1;d))=0$. That is, we compare the expansion $\exp(k(n))$ with
the minimal $\exp(k(1))=1$ occurring always at nodes at positions $N(1;d)$.
From Eq. (5) we have
$k(N(n;d))=2n-2=\ln\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}},$ (7)
or
$\xi(N(n;d))=\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}}.$ (8)
The standard network Lyapunov exponent is defined as
$\lambda\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\ln\xi(N),$ (9)
but since Eq. (8) indicates that the bounds of the fluctuations of $\xi(N)$
grow with $N$ slower than $\exp N$ we have $\lambda=0$, in agreement to the
ordinary Lyapunov exponent at the onset of chaos.
To get a suitable expansion rate that grows linearly with the size of the
network, we deform the ordinary logarithm in $\ln{\xi(N)}=k(N)$ into
$\ln_{q}\xi(N)$ by an amount $q>1$ such that $\ln_{q}\xi(N)$ depends linearly
in $N$, where $\ln_{q}x\equiv(x^{1-q}-1)/(1-q)$ and $\ln x$ is restored in the
limit $q\rightarrow 1$ Koelink ; robledo2 . And through this deformation we
define the generalized graph-theoretical Lyapunov exponent as
$\lambda_{q}\equiv\frac{1}{\Delta N}\ln_{q}\xi(N),$ (10)
where $\Delta N=N(n;d)-C_{d}$ is the node distance or iteration time duration
between an initial node $N(1;d)$ where $d$ is fixed and $N(n;d)$ is the final
node position. From Eq. (8) we obtain
$\lambda_{q}(d)=\frac{1}{N-C_{d}}\frac{\left(\frac{N}{C_{d}}\right)^{\frac{1-q}{\ln\phi}}-1}{1-q}=\frac{1}{C_{d}\ln\phi},$
(11)
where the degree of deformation $q$ is found to be $q=1-\ln\phi$. This way we
have determined a spectrum of generalized Lyapunov exponents $\lambda_{q}(d)$,
one for each diagonal $d=1,2,...$ in Fig.2. The largest value is for the main
diagonal, $\lambda_{q}(1)=(C_{1}\ln\phi)^{-1}$, and the others gradually
decrease as $d\rightarrow\infty$.
## V $q$-deformed entropy expression and Pesin-like identities
Having obtained the family of generalized Lyapunov exponents $\lambda_{q}(d)$
from a suitable expansion rate $\ln_{q}\xi(N)$, we proceed to analyze the
entropic properties of the network. At the transition to chaos for the golden
ratio the HV method creates a single network that represents many different
trajectories. Trajectories initiated at different positions of the attractor
produce networks related to each other by a node translation equal to the
number of iterations needed from one initial position $\theta_{0}^{(1)}$ to
reach the second $\theta_{0}^{(2)}$. The two positions appear in the
trajectory initiated at $\theta_{0}=0$ at times $t_{1}$ and $t_{2}$,
$\theta_{0}^{(1)}=\theta_{t_{1}}$ and $\theta_{0}^{(2)}=\theta_{t_{2}}$, and
the node translation is $\delta N=t_{2}-t_{1}>0$. This shift property can be
visualized in Fig. 2, and is implicated in the derivation of Eq. (10) for
$\lambda_{q}(d)$. But also, trajectories initiated at positions off the
attractor, but sufficiently close to a position of this set generate the same
network, as the HV method distinguishes differences in trajectory positions
only when they surpass threshold values. There is a basic property of
trajectories at the onset of chaos that combines with the previous remark and
that can be used to describe the rate of entropy growth of the network with
its size. This property is that for a small interval of length $l_{0}$ with
$\mathcal{N}$ uniformly-distributed initial conditions around, say,
$\theta_{0}=0$, all trajectories behave similarly, remain uniformly-
distributed at later times and follow the concerted pattern shown in Fig. 3 in
Ref. robledo1 . Studies of entropy growth associated with an initial
distribution of positions with iteration time $t$ of several chaotic maps
latora1 have established that a linear growth occurs during an intermediate
stage in the evolution of the entropy, after an initial transient dependent on
the initial distribution and before an asymptotic approach to a constant
equilibrium value. In relation to this it was found, both at the period-
doubling baldovin1 ; mayoral1 and at the quasiperiodic golden ratio robledo1
transitions to chaos, that (i) there is no initial transient if the initial
distribution is uniform and defined around a small interval of an attractor
position, and (ii) the distribution remains uniform for an extended period of
time due to the subexponential dynamics. In Fig. 4 we demonstrate this
property by presenting the time evolution of the distance between to nearby
trajectories, say the endpoints of the interval of length $l_{t}$ containing
the $\mathcal{N}$ uniformly-distributed positions at time $t$, for the golden
ratio transition to chaos, and also for other quasiperiodic transitions to
chaos along other routes discussed below. But the time evolution of the
trajectory distances in Fig. 4 can also be that between any pair of adjacent
positions in the initial uniform distribution and therefore the trajectories
distribution remains uniform after continued iterations.
We denote the above-referred distribution by $\pi(t)=1/W(t)$ where
$W(0)=l_{0}/\mathcal{N}$ is the number of cells that cover the initial
interval $l_{0}$. As stated, all such trajectories give rise to the same HV
graph, and at iteration times, say, of the form $t=N(n;d)$, $n=1,2,3,\dots$,
the HV criterion assigns $k=2n-2$ links to the common node $N(n;d)$. The
distribution $\pi$ is defined in the map but we can look at its
$n$-dependence, $\pi(N(n;d))$, if the scaling properties of the network retain
the scaling property of $\pi$ in the map. We can corroborate this and also
that the entropic properties derived from this distribution are connected to
the network Lyapunov exponents described in the previous section. The scaling
property of the network that yields the collapse of the diagonals in Fig. 2
described above implies that the uniform distributions $\pi$ for the
consecutive node-connectivity pairs ($N(n;d),2n-2$) and ($N(n+1;d),2(n+1)-2$)
along the same diagonal $d$ scale with the same factors and this leads us to
conclude that the $n$-dependence for these distributions is
$\pi(N)=W_{n}^{-1}=\exp(-2n+2).$ (12)
But since
$W_{n}=\exp(2n-2)=\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}},$ (13)
the ordinary entropy associated with $\pi$ grows logarithmically with the
number of nodes $N$, $S_{1}\left[\pi(N)\right]=\ln W_{j}\sim\ln N$. However,
the $q$-deformed entropy
$S_{q}\left[\pi(N)\right]\equiv\ln_{q}W_{n}=\frac{1}{1-q}\left[W_{n}^{1-q}-1\right],$
(14)
where the amount of deformation $q$ of the logarithm has the same value as
before, grows linearly with $N$, as $W_{n}$ can be rewritten as
$W_{n}=\exp_{q}[\lambda_{q}\Delta N],$ (15)
with $q=1-\ln\phi$ and $\lambda_{q}(d)=(C_{d}\ln\phi)^{-1}$. Therefore, if we
define the entropy growth rate
$h_{q}\left[\pi(N)\right]\equiv\frac{1}{\Delta N}S_{q}\left[\pi(N)\right]$
(16)
we obtain
$h_{q}\left[\pi(N)\right]=\lambda_{q}(d),$ (17)
a Pesin-like identity at the onset of chaos (effectively one identity for each
subsequence of node numbers , $n=1,2,3,\dots$, given each by a value of
$d=1,2,3,...$).
Figure 5: Top: Positions $\theta_{t}$ as a function of ${\small t}$ for the
first ${\small 70}$ data for the orbit with initial condition
${\small\theta}_{0}{\small=1}$ at the silver number onset of chaos (see text)
of the critical circle map ${\small K=1}$. The data highlighted are associated
with specific subsequences of nodes (see text). Bottom: Log-log plot of
${\small\exp k(N)}$ as a function of the node ${\small N}$ for the HV graph
generated from same time series as as for the upper panel but for $3\times
10^{2}$ iterations, where ${\small N=t}$. The distinctive band pattern of the
attractor manifests through a pattern of single lines of constant degree. The
node positions of some node subsequences along diagonals is highlighted as
guide lines to the eye.
## VI Quasiperiodic graphs at the onset of chaos for quadratic irrationals
We can generalize the above results for every quadratic irrational in $[0,1]$
with pure periodic continued fraction representation:
$\phi_{b}^{-1}=[b,b,b,...]=[\overline{b}]$ ($b=1$ , $2$, $3$, correspond to
the golden, silver and bronze routes, respectively). These irrationals are the
solutions of the equation $x^{2}-bx-1=0$, where $b$ is a natural number. The
dressed winding number is now
$\omega_{\infty}=\lim_{n\rightarrow\infty}[1-(F_{n-1}/F_{n})]=\phi_{b}^{-1}$with
$F_{n}=bF_{n-1}+F_{n-2}$, $F_{0}=0$, $F_{1}=1$ and the route to chaos is the
infinite sequence of attractors with periods $F_{n}$, $n=1,2,3,...$(Notice now
$F_{n}$ is only a Fibonacci number when $b=1$). The first few steps of the
silver route $b=2$ can be seen in Fig. 1(c), whereas Fig. 5 shows results for
the attractor at the onset of chaos via this route. Similarly to Fig. 2 for
$b=1$, in the top panel of Fig. 5 is the time series for the first $70$
iteration times, while in the bottom panel of the same figure we plot, in
logarithmic scales, the outcome of the HV method with use of the variable
$\exp k(N)$. As it can be observed the networks for the two cases are
qualitatively similar, although there are differences, mainly the absence of
even connectivities when $k>5$.
This absence can be verified by inspection of the degree distribution
$P_{\infty}(k)$ for the graphs at the $\omega_{\infty}=\phi_{b}^{-1}$
accumulation points luque6
$P_{\infty}(k)=\left\\{\begin{array}[]{ll}\phi_{b}^{-1}&k=2\\\
1-2\phi_{b}^{-1}&k=3\\\
(1-\phi_{b}^{-1})\phi_{b}^{(3-k)/b}&k=bn+3,\;n\in\mathbb{N}\\\
0&\mathrm{otherwise,}\end{array}\right.$ (18)
where we can see explicitly which values of $k$ are not present for a given
value of $b$. This and other connectivity properties can be worked out from
the inflation process of the graphs. See Fig. 6.
Figure 6: First substructures of the quasiperiodic graph associated to (a)
the silver number ${\small b=2}$ and (b) the bronze number ${\small b=3}$
routes to chaos. The resulting patterns follow from the universal order with
which an orbit visits the positions of the attractors. The quasiperiodic graph
associated with the time series generated at the onset of chaos
($n\rightarrow\infty$) is the result of an infinite application of the
inflationary process by which a graph at period $F_{2n}$ is generated out of
graphs at periods $F_{2n-2}$ and $F_{2n-1}$ luque6 .
We will center our attention on the first diagonal $d=1$. For every $b$, the
node positions on the first diagonal, $n=1,2,3,...$, are
$N(n;1)=F_{2n},$ (19)
that with the use of the generalized Binet formula
$F_{n}=\frac{1}{\sqrt{b^{2}+4}}\left[\phi_{b}^{n}-\left(\frac{-1}{\phi_{b}}\right)^{n}\right]\approx\frac{\phi_{b}^{n}}{\sqrt{b^{2}+4}},$
can be written as
$N(n;1)\approx\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2n}=\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2}\phi_{b}^{2n-2}=C_{b}\phi_{b}^{2n-4},$
(20)
where the position $n=1$ is
$N(1;1)=F_{2}\approx\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2}\equiv C_{b}.$ (21)
We note that the connectivity of the first node is $k(n=1)=b+3$ and in general
$k(n)=b+3+2b(n-1)$, $n\geq 2$. As before we redefine the connectivities such
that the degree is zero at the initial position $n=1$, $k(n)=2b(n-1)$,
$n=1,2,3,...$ Following the same procedure as in Section 4, from Eq. (20) we
have
$k(N(n;1))=2b(n-1)=\ln\left(\frac{N}{C_{b}}\right)^{\frac{1}{\ln\phi_{b}}},$
(22)
and use of it in the sensitivity $\xi(N(n;1))\equiv\exp(k(n))$ yields
$\xi(N(n;1))=\left(\frac{N}{C_{b}}\right)^{\frac{1}{\ln\phi_{b}}}.$ (23)
Since all the features required for the $q$-deformation described in Section 4
are present for general $b$, we obtain for the generalized Lyapunov exponent
the expression
$\lambda_{q}^{(b)}(1)=\frac{1}{N-C_{b}}\frac{\left(\frac{N}{C_{b}}\right)^{\frac{1-q}{\ln\phi_{b}}}-1}{1-q}=\frac{1}{C_{b}\ln\phi_{b}},$
(24)
where $q=1-\ln\phi_{b}$. Likewise, the contents of Section 5 can also be
reproduced for general $b$ with the result that
$h_{q}\left[\pi(N)\right]=\lambda_{q}^{(b)}(1).$ (25)
## VII Summary and discussion
At the quasiperiodic onset of chaos the HV method leads to a self-similar
network with a structure illustrated by the related periodic networks obtained
from the sequence of attractors of finite periods along the route to chaos
luque6 . Under the HV algorithm many nearby trajectory positions lead to the
same network, since only when the values of trajectory positions cross a
threshold the corresponding node increases its degree with new links. (See the
succinct definition of the algorithm and the top panel in Fig. 2). Therefore
trajectories off the attractor but close to it transform into the same network
structure. As we have seen the fluctuations of the degree capture the
anomalous but basic behavior of the fluctuations of the sensitivity to initial
conditions at the transition to chaos robledo1 . The graph-theoretical
analogue of the sensitivity was identified as $\exp(k)$ while the amplitude of
the variations of $k$ grows logarithmically with the number of nodes $N$.
These deterministic fluctuations are described by a discrete spectrum of
generalized graph-theoretical Lyapunov exponents that are shown to relate to
an equivalent spectrum of generalized entropy growth rates, yielding a set of
Pesin-like identities. This behavior is similar to what was observed for the
case of the more straightforward period-doubling accumulation point luque5 .
The definitions of these quantities involve a scalar deformation of the
ordinary logarithmic function that ensures their linear growth with the number
of nodes. Therefore the entropy expression involved is extensive and of the
Tsallis type with a precisely fixed value of the deformation index $q$,
$q=1-\ln\phi_{b}$, where $\phi_{b}$ is the inverse of the irrational (dressed)
winding number.
We have considered special families of time series and converted each into a
network, each family consists of the trajectories associated with an attractor
at the quasiperiodic transition to chaos of circle maps. The attractors
studied are defined by a winding number given by a quadratic irrational or,
equivalently, by a pure periodic continued fraction. Each winding number
singles out a specific route to chaos. Amongst these we described in some
detail the so-called golden route, but also we have shown results for those
known as the silver and bronze routes luque6 . See Figs. 1 and 4. The HV
algorithm proved to be capable of generating a single network that contains
the scaling and entropic properties of the trajectories associated with each
attractor. The results presented here are of the same kind as those obtained
for the period-doubling route to chaos luque5 suggesting that the HV networks
associated with the onset of chaos are useful for describing the universal
properties at these special systems. The Pesin identity is a reflection of a
basic connection between BG statistical mechanics and chaos so that our
results provide elements for an analogous connection for the case of
nonergodic and nonmixing dynamics at vanishing ordinary Lyapunov exponent.
Acknowledgements. We acknowledge financial support by the Comunidad de Madrid
(Spain) through Project No. S2009ESP-1691 (B.L.), support from CONACyT & DGAPA
(PAPIIT IN100311)-UNAM (Mexican agencies) (A.R.).
## References
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|
arxiv-papers
| 2013-12-11T09:32:44 |
2024-09-04T02:49:55.320453
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bartolo Luque, Marta Cordero-Gracia, Mariola G\\'omez, and Alberto\n Robledo",
"submitter": "Bartolo Luque",
"url": "https://arxiv.org/abs/1312.3089"
}
|
1312.3388
|
# Online Bayesian Passive-Aggressive Learning
Tianlin Shi Institute for Interdisciplinary Information Sciences, Tsinghua,
Beijing Jun Zhu Department of Computer Science and Technology, Tsinghua,
Beijing
###### Abstract
Online Passive-Aggressive (PA) learning is an effective framework for
performing max-margin online learning. But the deterministic formulation and
estimated single large-margin model could limit its capability in discovering
descriptive structures underlying complex data. This paper presents online
Bayesian Passive-Aggressive (BayesPA) learning, which subsumes the online PA
and extends naturally to incorporate latent variables and perform
nonparametric Bayesian inference, thus providing great flexibility for
explorative analysis. We apply BayesPA to topic modeling and derive efficient
online learning algorithms for max-margin topic models. We further develop
nonparametric methods to resolve the number of topics. Experimental results on
real datasets show that our approaches significantly improve time efficiency
while maintaining comparable results with the batch counterparts.
bayesian, passive, aggressive, medlda, medhdp, machine learning, ICML
## 1 Introduction
Online learning is an effective way to deal with large-scale applications,
especially applications with streaming data. Among the popular algorithms,
online Passive-Aggressive (PA) learning (Crammer et al., 2006) provides a
generic framework for online large-margin learning, with many applications
(McDonald et al., 2005; Chiang et al., 2008). Though enjoying strong
discriminative ability suitable for predictive tasks, existing online PA
methods are formulated as a point estimate problem by optimizing some
deterministic objective function. This may lead to some inconvenience. For
example, a single large-margin model is often less than sufficient in
describing complex data, such as those with rich underlying structures.
On the other hand, Bayesian methods enjoy great flexibility in describing the
possible underlying structures of complex data. Moreover, the recent progress
on nonparametric Bayesian methods (Hjort, 2010; Teh et al., 2006a) further
provides an increasingly important framework that allows the Bayesian models
to have an unbounded model complexity, e.g., an infinite number of components
in a mixture model (Hjort, 2010) or an infinite number of units in a latent
feature model (Ghahramani & Griffiths, 2005), and to adapt when the learning
environment changes. For Bayesian models, one challenging problem is posterior
inference, for which both variational and Monte Carlo methods can be too
expensive to be applied to large-scale applications. To scale up Bayesian
inference, much progress has been made on developing online variational Bayes
(Hoffman et al., 2010; Mimno et al., 2012) and online Monte Carlo (Ahn et al.,
2012) methods. However, due to the generative nature, Bayesian models are lack
of the discriminative ability of large-margin methods and usually less than
sufficient in performing discriminative tasks.
Successful attempts have been made to bring large-margin learning and Bayesian
methods together. For example, maximum entropy discrimination (MED) (Jaakkola
et al., 1999) made a significant advance in conjoining max-margin learning and
Bayesian generative models, mainly in the context of supervised learning and
structured output prediction (Zhu & Xing, 2009). Recently, much attention has
been focused on generalizing MED to incorporate latent variables and perform
nonparametric Bayesian inference, in many contexts including topic modeling
(Zhu et al., 2012), matrix factorization (Xu et al., 2012), and multi-task
learning (Jebara, 2011; Zhu et al., 2011). However, posterior inference in
such models remain a big challenge. It is desirable to develop efficient
online algorithms for these Bayesian max-margin models.
To address the above problems of both the existing online PA algorithms and
Bayesian max-margin models, this paper presents online Bayesian Passive-
Aggressive (BayesPA) learning, a general framework of performing online
learning for Bayesian max-margin models. We show that online BayesPA subsumes
the standard online PA when the underlying model is linear and the parameter
prior is Gaussian. We further show that another major significance of BayesPA
is its natural generalization to incorporate latent variables and to perform
nonparametric Bayesian inference, thus allowing online BayesPA to have the
great flexibility of (nonparametric) Bayesian methods for explorative analysis
as well as the strong discriminative ability of large-margin learning for
predictive tasks. As concrete examples, we apply the theory of online BayesPA
to topic modeling and derive efficient online learning algorithms for max-
margin supervised topic models (Zhu et al., 2012). We further develop
efficient online learning algorithms for the nonparametric max-margin topic
models, an extension of the nonparametric topic models (Teh et al., 2006a;
Wang et al., 2011) for predictive tasks. Extensive empirical results on real
data sets show significant improvements on time efficiency and maintenance of
comparable results with the batch counterparts.
## 2 Bayesian Passive-Aggressive Learning
In this section, we present a general perspective on online max-margin
Bayesian inference.
### 2.1 Online PA Learning
The goal of online supervised learning is to minimize the cumulative loss for
a certain prediction task from the sequentially arriving training samples.
Online Passive-Aggressive (PA) algorithms (Crammer et al., 2006) achieve this
goal by updating some parameterized model $\bm{w}$ (e.g., the weights of a
linear SVM) in an online manner with the instantaneous losses from arriving
data $\\{\bm{x}_{t}\\}_{t\geq 0}$ and corresponding responses
$\\{y_{t}\\}_{t\geq 0}$. The losses
$\ell_{\epsilon}(\bm{w};\bm{x}_{t},y_{t})$, as they consider, could be the
hinge loss $(\epsilon-y_{t}\bm{w}^{\top}\bm{x}_{t})_{+}$ for binary
classification or the $\epsilon$-insensitive loss
$(|y_{t}-\bm{w}^{\top}\bm{x}_{t}|-\epsilon)_{+}$ for regression, where
$\epsilon$ is a hyper-parameter and $(x)_{+}=\max(0,x)$. The Passive-
Aggressive update rule is then derived by defining the new weight
$\bm{w}_{t+1}$ as the solution to the following optimization problem:
$\min_{\bm{w}}{\frac{1}{2}||\bm{w}-\bm{w_{t}}||^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{s.t.:}\leavevmode\nobreak\
\ell_{\epsilon}(\bm{w};\bm{x}_{t},y_{t})=0.$ (1)
Intuitively, if $\bm{w_{t}}$ suffers no loss from the new data, i.e.,
$\ell_{\epsilon}(\bm{w}_{t};\bm{x}_{t},y_{t})=0$, the algorithm _passively_
assigns $\bm{w}_{t+1}=\bm{w}_{t}$; otherwise, it aggressively projects
$\bm{w_{t}}$ to the feasible zone of parameter vectors that attain zero loss.
With provable bounds, (Crammer et al., 2006) shows that online PA algorithms
could achieve comparable results to the optimal classifier $\bm{w}^{*}$. In
practice, in order to account for inseparable training samples, soft margin
constraints are often adopted and the resulting learning problem is
$\min_{\bm{w}}{\frac{1}{2}||\bm{w}-\bm{w_{t}}||^{2}}+2c\ell_{\epsilon}(\bm{w};\bm{x}_{t},y_{t}),$
(2)
where $c$ is a positive regularization parameter. For problems (1) and (2)
with samples arriving one at a time, closed-form solutions can be derived
(Crammer et al., 2006).
### 2.2 Online BayesPA Learning
Instead of updating a point estimate of $\bm{w}$, online Bayesian PA (BayesPA)
sequentially infers a new posterior distribution $q_{t+1}(\bm{w})$, either
parametric or nonparametric, on the arrival of new data $(\bm{x}_{t},y_{t})$
by solving the following optimization problem:
$\begin{array}[]{rl}\underset{q(\bm{w})\in\mathcal{F}_{t}}{\operatorname{min}}&\leavevmode\nobreak\
\text{KL}[q(\bm{w})||q_{t}(\bm{w})]-\mathbb{E}_{q(\bm{w})}[\log
p(\bm{x}_{t}|\bm{w})]\\\ \text{s.t.:}&\leavevmode\nobreak\
\leavevmode\nobreak\
\ell_{\epsilon}[q(\bm{w});\bm{x}_{t},y_{t}]=0,\end{array}$ (3)
where $\mathcal{F}_{t}$ is some distribution family, e.g., the probability
simplex $\mathcal{P}$. In other words, we find a posterior distribution
$q_{t+1}(\bm{w})$ in the feasible zone that is not only close to
$q_{t}(\bm{w})$ by the commonly used KL-divergence, but also has a high
likelihood of explaining new data. As a result, if Bayes’ rule already gives
the posterior distribution $q_{t+1}(\bm{w})\propto
q_{t}(\bm{w})p(\bm{x}_{t}|\bm{w})$ that suffers no loss (i.e.,
$\ell_{\epsilon}=0$), BayesPA _passively_ updates the posterior following just
Bayes’ rule; otherwise, BayesPA _aggressively_ projects the new posterior to
the feasible zone of posteriors that attain zero loss. We should note that
when no likelihood is defined (e.g., $p(\bm{x}_{t}|\bm{w})$ is independent of
$\bm{w}$), BayesPA will passively set $q_{t+1}(\bm{w})=q_{t}(\bm{w})$ if
$q_{t}(\bm{w})$ suffers no loss. We call it non-likelihood BayesPA.
In practical problems, the constraints in (3) could be unrealizable. To deal
with such cases, we introduce the soft-margin version of BayesPA learning,
which is equivalent to minimizing the objective function
$\mathcal{L}(q(\bm{w}))$ in problem (3) with a regularization term (Cortes &
Vapnik, 1995):
$\displaystyle
q_{t+1}(\bm{w})=\underset{q(\bm{w})\in\mathcal{F}_{t}}{\operatorname{argmin}}\leavevmode\nobreak\
\mathcal{L}(q(\bm{w}))+2c\ell_{\epsilon}(q(\bm{w});\bm{x}_{t},y_{t}).$ (4)
For the max-margin classifiers that we focus on in this paper, two loss
functions $\ell_{\epsilon}(q(\bm{w});\bm{x}_{t},y_{t})$ are common — the hinge
loss of an _averaging classifier_ that makes predictions using the rule
$\hat{y}_{t}=\textrm{sign}\leavevmode\nobreak\
\mathbb{E}_{q(\bm{w})}[\bm{w}^{\top}\bm{x}_{t}]$:
$\ell_{\epsilon}^{Avg}[q(\bm{w});\bm{x}_{t},y_{t}]=(\epsilon-
y_{t}\mathbb{E}_{q(\bm{w})}[\bm{w}^{\top}\bm{x}_{t}])_{+}$
and the expected hinge loss of a _Gibbs classifier_ that randomly draws a
classifier $\bm{w}\sim q(\bm{w})$ to make predictions using the rule
$\hat{y}_{t}=\textrm{sign}\leavevmode\nobreak\ \bm{w}^{\top}\bm{x}_{t}$:
$\ell_{\epsilon}^{Gibbs}[q(\bm{w});\bm{x}_{t},y_{t}]=\mathbb{E}_{q(\bm{w})}[(\epsilon-
y_{t}\bm{w}^{\top}\bm{x}_{t})_{+}].$
They are closely connected via the following lemma due to the convexity of the
function $(x)_{+}$.
###### Lemma 2.1.
The expected hinge loss $\ell_{\epsilon}^{\text{Gibbs}}$ is an upper bound of
the hinge loss $\ell_{\epsilon}^{\text{Avg}}$, that is,
$\ell_{\epsilon}^{\text{Gibbs}}\geq\ell_{\epsilon}^{\text{Avg}}$.
Before developing BayesPA learning for practical problems, we make several
observations.
###### Lemma 2.2.
If $q_{0}(\bm{w})=\mathcal{N}(0,I)$, $\mathcal{F}_{t}=\mathcal{P}$ and we use
$\ell_{\epsilon}^{Avg}$, the non-likelihood BayesPA subsumes the online PA.
This can be proved by induction. First, we can show that
$q_{t}(\bm{w})=\mathcal{N}(\bm{\mu}_{t},I)$ is a normal distribution with an
identity covariance matrix. Second, we can show that the posterior mean
$\bm{\mu}_{t}$ is updated in the same way as in the online PA. We defer the
detailed proof to Appendix A.
###### Lemma 2.3.
If $\mathcal{F}_{t}=\mathcal{P}$ and we use $\ell_{\epsilon}^{Gibbs}$, the
update rule of online BayesPA is
$q_{t+1}(\bm{w})=\frac{q_{t}(\bm{w})p(\bm{x}_{t}|\bm{w})e^{-2c(\epsilon-
y_{t}\bm{w}^{\top}\bm{x}_{t})_{+}}}{\Gamma(\bm{x}_{t},y_{t})},$ (5)
where $\Gamma(\bm{x}_{t},y_{t})$ is the normalization constant.
Therefore, the posterior $q_{t}(\bm{w})$ in the previous round $t$ becomes a
prior, while the newly observed data and its loss function provide a
likelihood and an unnormalized pseudo-likelihood respectively.
Mini-Batches. A useful technique to reduce the noise in data is the use of
mini-batches. Suppose that we have a mini-batch of data points at time $t$
with an index set $B_{t}$, denoted as $\bm{X}_{t}=\\{\bm{x}_{d}\\}_{d\in
B_{t}},\bm{Y}_{t}=\\{y_{d}\\}_{d\in B_{t}}$. The Bayesian PA update equation
for this mini-batch is simply,
$q_{t+1}(\bm{w})=\underset{q\in\mathcal{F}_{t}}{\operatorname{argmin}}{\leavevmode\nobreak\
\mathcal{L}(q(\bm{w}))+2c\ell_{\epsilon}(q(\bm{w});\bm{X}_{t},\bm{Y}_{t})},$
where $\ell_{\epsilon}(q(\bm{w});\bm{X}_{t},\bm{Y}_{t})=\sum_{d\in
B_{t}}{\ell_{\epsilon}(q(\bm{w});\bm{x}_{d},y_{d})}$.
### 2.3 Learning with Latent Structures
To expressively explain complex real-word data, Bayesian models with latent
structures have been extensively developed. The latent structures could
typically be characterized by two kinds of latent variables — _local latent
variables_ $\bm{h}_{d}$ ($d\geq 0$) that characterize the hidden structures of
each observed data $\bm{x}_{d}$ and _global variables_ $\bm{\mathcal{M}}$ that
capture the common properties shared by all data.
The goal of Bayesian PA learning with latent structures is therefore to update
the distribution of $\bm{\mathcal{M}}$ as well as weights $\bm{w}$ based on
each incoming mini-batch $(\bm{X}_{t},\bm{Y}_{t})$ and their corresponding
latent variables $\bm{H}_{t}=\\{\bm{h}_{d}\\}_{d\in B_{t}}$. Because of the
uncertainty in $\bm{H}_{t}$, we extend BayesPA to infer the joint posterior
distribution, $q_{t+1}(\bm{w},\bm{\mathcal{M}},\bm{H}_{t})$, as solving
$\displaystyle\underset{q\in\mathcal{F}_{t}}{\operatorname{min}}{\leavevmode\nobreak\
\mathcal{L}(q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t}))+2c\ell_{\epsilon}(q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t});\bm{X}_{t},\bm{Y}_{t})},$
(6)
where
$\mathcal{L}(q)\\!=\\!\text{KL}[q||q_{t}(\bm{w},\bm{\mathcal{M}})p_{0}(\bm{H}_{t})]-\mathbb{E}_{q}[\log
p(\bm{X}_{t}|\bm{w},\\\ \bm{\mathcal{M}},\bm{H}_{t})]$ and
$\ell_{\epsilon}(q;\bm{X}_{t},\bm{Y}_{t})$ is some cumulative margin-loss on
the min-batch data induced from some classifiers defined on the latent
variables $\bm{H}_{t}$ and/or global variables $\bm{\mathcal{M}}$. Both the
averaging classifiers and Gibbs classifiers can be used as in the case without
latent variables. We will present concrete examples in the next section.
Before diving into the details, we should note that in real online setting,
only global variables are maintained in the bookkeeping, while the local
information in the streaming data is forgotten. However, (6) gives us a
distribution of $(\bm{w},\bm{\mathcal{M}})$ that is coupled with the local
variables $\bm{H}_{t}$. Although in some cases we can marginalize out the
local variables $\bm{H}_{t}$, in general we would not obtain a closed-form
posterior distribution $q_{t+1}(\bm{w},\bm{\mathcal{M}})$ for the next
optimization round, especially in dealing with some involved models like
MedLDA (Zhu et al., 2012). Therefore, we resort to approximation methods,
e.g., by posing additional assumptions about
$q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t})$ such as the mean-field assumption,
$q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t})=q(\bm{w})q(\bm{\mathcal{M}})q(\bm{H}_{t})$.
Then, we can solve the problem via an iterative procedure and use the optimal
distribution $q^{*}(\bm{w})q^{*}(\bm{\mathcal{M}})$ as
$q_{t+1}(\bm{w},\bm{\mathcal{M}})$. More details will be provided in next
sections.
## 3 Online Max-Margin Topic Models
We apply the theory of online BayesPA to topic modeling and develop online
learning algorithms for max-margin topic models. We also present a
nonparametric generalization to resolve the number of topics in the next
section.
### 3.1 Batch MedLDA
A max-margin topic model consists of a latent Dirichlet allocation (LDA) (Blei
et al., 2003) model for describing the underlying topic representations and a
max-margin classifier for predicting responses. Specifically, LDA is a
hierarchical Bayesian model that treats each document as an admixture of
topics, $\bm{\Phi}=\\{\bm{\phi}_{k}\\}_{k=1}^{K}$, where each topic
$\bm{\phi}_{k}$ is a multinomial distribution over a $W$-word vocabulary. Let
$\bm{\theta}$ denote the mixing proportions. The generative process of
document $d$ is described as
$\displaystyle\bm{\theta}_{d}\sim$ $\displaystyle\text{Dir}(\bm{\alpha}),$
$\displaystyle z_{di}\sim\text{Mult}(\bm{\theta}_{d}),\leavevmode\nobreak\
x_{di}$ $\displaystyle\sim\text{Mult}(\bm{\phi}_{z_{di}}),\leavevmode\nobreak\
\forall i\in[n_{d}]$
where $z_{di}$ is a topic assignment variable and $\text{Mult}(\cdot)$ is a
multinomial distribution. For Bayesian LDA, the topics are drawn from a
Dirichlet distribution, i.e., $\bm{\phi}_{k}\sim\text{Dir}(\bm{\gamma})$.
Given a document set $\bm{X}=\\{\bm{x}_{d}\\}_{d=1}^{D}$. Let
$\bm{Z}=\\{\bm{z}_{d}\\}_{d=1}^{D}$ and
$\bm{\Theta}=\\{\bm{\theta}_{d}\\}_{d=1}^{D}$. LDA infers the posterior
distribution $p(\bm{\Phi},\bm{\Theta},\bm{Z}|\bm{X})\propto
p_{0}(\bm{\Phi},\bm{\Theta},\bm{Z})p(\bm{X}|\bm{Z},\bm{\Phi})$ via Bayes’
rule. From a variational point of view, the Bayes posterior is equivalent to
the solution of the optimization problem:
$\min\limits_{q\in\mathcal{P}}\leavevmode\nobreak\
\mathrm{KL}[q(\bm{\Phi},\bm{\Theta},\bm{Z})||p(\bm{\Phi},\bm{\Theta},\bm{Z}|\bm{X})].$
The advantage of the variational formulation of Bayesian inference lies in the
convenience of posing restrictions on the post-data distribution with a
regularization term. For supervised topic models (Blei & McAuliffe, 2010; Zhu
et al., 2012), such a regularization term could be a loss function of a
prediction model $\bm{w}$ on the data $\bm{X}=\\{\bm{x}_{d}\\}_{d=1}^{D}$ and
response signals $\bm{Y}=\\{y_{d}\\}_{d=1}^{D}$. As a regularized Bayesian
(RegBayes) model (Jiang et al., 2012), MedLDA infers a distribution of the
latent variables $\bm{Z}$ as well as classification weights $\bm{w}$ by
solving the problem:
$\displaystyle\min\limits_{q\in\mathcal{P}}\leavevmode\nobreak\
\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{\Theta},\bm{Z}))+2c\sum\limits_{d=1}^{D}{\ell_{\epsilon}(q(\bm{w},\bm{z}_{d});\bm{x}_{d},y_{d})},$
where
$\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{\Theta},\bm{Z}))=\mathrm{KL}[q(\bm{w},\bm{\Phi},\bm{\Theta},\bm{Z})||p(\bm{w},\bm{\Phi},\bm{\Theta},\\\
\bm{Z}|\bm{X})]$ . To specify the loss function, a linear discriminant
function needs to be defined with respect to $\bm{w}$ and $\bm{z}_{d}$
$f(\bm{w},\bm{z}_{d})=\bm{w}^{\top}\bar{\bm{z}}_{d},$ (7)
where $\bar{\bm{z}}_{dk}=\frac{1}{n_{d}}\sum_{i}{\mathbb{I}[z_{di}=k]}$ is the
average topic assignments of the words in document $d$. Based on the
discriminant function, both averaging classifiers with the hinge loss
$\ell^{Avg}_{\epsilon}(q(\bm{w},\bm{z}_{d});\bm{x}_{d},y_{d})=(\epsilon-
y_{d}\mathbb{E}_{q}[f(\bm{w},\bm{z}_{d})])_{+},$ (8)
and Gibbs classifiers with the expected hinge loss
$\ell^{Gibbs}_{\epsilon}(q(\bm{w},\bm{z}_{d});\bm{x}_{d},y_{d})=\mathbb{E}_{q}[(\epsilon-
y_{d}f(\bm{w},\bm{z}_{d}))_{+}],$ (9)
have been proposed, with extensive comparisons reported in (Zhu et al., 2013a)
using batch learning algorithms.
### 3.2 Online MedLDA
To apply the online BayesPA, we have the global variables
$\bm{\mathcal{M}}=\bm{\Phi}$ and local variables
$\bm{H}_{t}=(\bm{\Theta}_{t},\bm{Z}_{t})$. We consider Gibbs MedLDA because as
shown in (Zhu et al., 2013a) it admits efficient inference algorithms by
exploring data augmentation. Specifically, let $\zeta_{d}=\epsilon-
y_{d}f(\bm{w},\bm{z}_{d})$ and
$\psi(y_{d}|\bm{z}_{d},\bm{w})=e^{-2c(\zeta_{d})_{+}}$. Then in light of Lemma
2.3, the optimal solution to problem (6),
$q_{t+1}(\bm{w},\bm{\mathcal{M}},\bm{H}_{t})$, is
$\displaystyle\frac{q_{t}(\bm{w},\bm{\mathcal{M}})p_{0}(\bm{H}_{t})p(\bm{X}_{t}|\bm{H}_{t},\bm{\mathcal{M}})\psi(\bm{Y}_{t}|\bm{H}_{t},\bm{w})}{\Gamma(\bm{X}_{t},\bm{Y}_{t})},$
where $\psi(\bm{Y}_{t}|\bm{H}_{t},\bm{w})=\prod_{d\in
B_{t}}\psi(y_{d}|\bm{h}_{d},\bm{w})$ and $\Gamma(\bm{X}_{t},\bm{Y}_{t})$ is a
normalization constant. To potentially improve the inference accuracy, we
first integrate out the local variables $\bm{\Theta}_{t}$ by the conjugacy
between a Dirichlet prior and a multinomial likelihood (Griffiths & Steyvers,
2004; Teh et al., 2006b). Then we have the local variables
$\bm{H}_{t}=\bm{Z}_{t}$. By the equality (Zhu et al., 2013a):
$\psi(y_{d}|\bm{z}_{d},\bm{w})=\int_{0}^{\infty}{\psi(y_{d},\lambda_{d}|\bm{z}_{d},\bm{w})d\lambda_{d}},$
(10)
where
$\psi(y_{d},\lambda_{d}|\bm{z}_{d},\bm{w})=(2\pi\lambda_{d})^{-1/2}\exp(-\frac{(\lambda_{d}+c\zeta_{d})^{2}}{2\lambda_{d}})$,
the collapsed posterior $q_{t+1}(\bm{w},\bm{\Phi},\bm{Z}_{t})$ is a marginal
distribution of $q_{t+1}(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t})$, which
equals to
$\displaystyle\frac{p_{0}(\bm{Z}_{t})q_{t}(\bm{w},\bm{\Phi})p(\bm{X}_{t}|\bm{Z}_{t},\bm{\Phi})\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})}{\Gamma(\bm{X}_{t},\bm{Y}_{t})},$
where $\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})\\!=\\!\prod_{d\in
B_{t}}\psi(y_{d},\lambda_{d}|\bm{z}_{d},\bm{w})$ and
$\bm{\lambda}_{t}=\\{\lambda_{d}\\}_{d\in B_{t}}$ are augmented variables,
which are also locally associated with individual documents. In fact, the
augmented distribution $q_{t+1}(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t})$
is the solution to the problem:
$\displaystyle\underset{q\in\mathcal{P}}{\operatorname{min}}{\leavevmode\nobreak\
\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t}))-\mathbb{E}_{q}[\log\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})]},$
(11)
where
$\mathcal{L}(q)=\mathrm{KL}[q(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t})\|q_{t}(\bm{w},\bm{\Phi})p_{0}(\bm{Z}_{t})]-\\\
\mathbb{E}_{q}[\log p(\bm{X}_{t}|\bm{Z}_{t},\bm{\Phi})]$. We can show that
this objective is an upper bound of that in the original problem (6). See
Appendix B for details.
With the mild mean-field assumption that
$q(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t})\\\
=q(\bm{w})q(\bm{\Phi})q(\bm{Z}_{t},\bm{\lambda}_{t})$, we can solve (11) via
an iterative procedure that alternately updates each factor distribution
(Jordan et al., 1998), as detailed below.
Global Update: By fixing the distribution of local variables,
$q(\bm{Z}_{t},\bm{\lambda}_{t})$, and ignoring irrelevant variables, we have
the mean-field update equations:
$\displaystyle q(\bm{\Phi}_{k})\propto
q_{t}(\bm{\Phi}_{k})\exp(\mathbb{E}_{q(\bm{Z}_{t})}[\log
p_{0}(\bm{Z}_{t})p(\bm{X}|\bm{Z}_{t},\bm{\Phi})]),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k$ $\displaystyle q(\bm{w})\propto
q_{t}(\bm{w})\exp(\mathbb{E}_{q(\bm{Z}_{t},\bm{\lambda}_{t})}[\log
p_{0}(\bm{Z}_{t})\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})]).$
If initially
$q_{0}(\bm{\Phi}_{k})=\text{Dir}(\Delta_{k1}^{0},...,\Delta_{kW}^{0})$ and
$q_{0}(\bm{w})=\mathcal{N}(\bm{w};\bm{\mu}^{0},\bm{\Sigma}^{0})$, by induction
we can show that the inferred distributions in each round has a closed form,
namely, $q_{t}(\bm{\Phi}_{k})=\text{Dir}(\Delta_{k1}^{t},...,\Delta_{kW}^{t})$
and $q_{t}(\bm{w})=\mathcal{N}(\bm{w};\bm{\mu}^{t},\bm{\Sigma}^{t})$. For the
above update equations, we have
$q(\bm{\Phi}_{k})=\text{Dir}(\Delta_{k1}^{*},...,\Delta_{kW}^{*}),$ (12)
where $\Delta_{kw}^{*}=\Delta_{kw}^{t}+\sum_{d\in
B_{t}}\sum_{i\in[n_{d}]}{\gamma_{di}^{k}\cdot\mathbb{I}[x_{di}=w]}$ for all
words $w$ and $\gamma_{di}^{k}=\mathbb{E}_{q(\bm{z}_{d})}\mathbb{I}[z_{di}=k]$
is the probability of assigning word $x_{di}$ to topic $k$, and
$q(\bm{w})=\mathcal{N}(\bm{w};\bm{\mu}^{*},\bm{\Sigma}^{*}),$ (13)
where the posterior paramters are computed as
$(\bm{\Sigma}^{*})^{-1}=(\bm{\Sigma}^{t})^{-1}+c^{2}\sum_{d\in
B_{t}}\mathbb{E}_{q(\bm{z}_{d},\lambda_{d})}[\lambda_{d}^{-1}\bar{\bm{z}}_{d}\bar{\bm{z}}_{d}^{\top}]$
and
$\bm{\mu}^{*}=\bm{\Sigma}^{*}(\bm{\Sigma}^{t})^{-1}\bm{\mu}^{t}+\bm{\Sigma}^{*}\cdot
c\sum_{d\in
B_{t}}\mathbb{E}_{q(\bm{z}_{d},\lambda_{d})}[y_{d}(1+c\epsilon\lambda_{d}^{-1})\bar{\bm{z}}_{d}]$.
Local Update: Given the distribution of global variables,
$q(\bm{\Phi},\bm{w})$, the mean-field update equation for
$(\bm{Z}_{t},\bm{\lambda}_{t})$ is
$\displaystyle q(\bm{Z}_{t},\bm{\lambda}_{t})\propto$ $\displaystyle
p_{0}(\bm{Z}_{t})\prod\limits_{d\in
B_{t}}\frac{1}{\sqrt{2\pi\lambda_{d}}}\exp\Big{(}\sum\limits_{i\in[n_{d}]}\Lambda_{z_{di},x_{di}}$
(14) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\mathbb{E}_{q(\bm{\Phi},\bm{w})}[\frac{(\lambda_{d}+c\zeta_{d})^{2}}{2\lambda_{d}}]\Big{)},$
where
$\Lambda_{z_{di},x_{di}}=\mathbb{E}_{q(\Phi)}[\log(\Phi_{z_{di},x_{di}})]=\Psi(\Delta_{z_{di},x_{di}}^{*})-\Psi(\sum_{w}{\Delta_{z_{di},w}^{*}})$
and $\Psi(\cdot)$ is the digamma function, due to the distribution in (12).
But it is impossible to evaluate the expectation in the global update using
(14) because of the huge number of configurations for
$(\bm{Z}_{t},\bm{\lambda}_{t})$. As a result, we turn to Gibbs sampling and
estimate the required expectations using multiple empirical samples. This
hybrid strategy has shown promising performance for LDA (Mimno et al., 2012).
Specifically, the conditional distributions used in the Gibbs sampling are as
follows:
For $\bm{Z}_{t}$: By canceling out common factors, the conditional
distribution of one variable $z_{di}$ given $\bm{Z}_{t}^{\neg di}$ and
$\bm{\lambda}_{t}$ is
$\begin{array}[]{rl}&\\!q(z_{di}\\!=\\!k|\bm{Z}_{t}^{\neg
di},\bm{\lambda}_{t})\\!\propto\\!(\alpha\\!+\\!C_{dk}^{\neg
di})\\!\exp\\!\Big{(}\frac{cy_{d}(c\epsilon+\lambda_{d})\mu_{k}^{*}}{n_{d}\lambda_{d}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\Lambda_{k,x_{di}}-\frac{c^{2}(\mu_{k}^{*2}+\Sigma_{kk}^{*}+2(\mu_{k}^{*}\bm{\mu}^{*}+\bm{\Sigma}_{\cdot,k}^{*})^{\top}\bm{C}_{d}^{\neg
di})}{2n_{d}^{2}\lambda_{d}}\Big{)},\end{array}$ (15)
where $\bm{\Sigma}_{\cdot,k}^{*}$ is the $k$-th column of $\bm{\Sigma}^{*}$,
$\bm{C}_{d}^{\neg di}$ is a vector with the $k$-th entry being the number of
words in document $d$ (except the $i$-th word) that are assigned to topic $k$.
For $\bm{\lambda}_{t}$: Let $\bar{\zeta}_{d}=\epsilon-
y_{d}\bar{\bm{z}}_{d}^{\top}\bm{\mu}^{*}$. The conditional distribution of
each variable $\lambda_{d}$ given $\bm{Z}_{t}$ is
$\begin{array}[]{rl}q(\lambda_{d}|\bm{Z}_{t})\propto&\frac{1}{\sqrt{2\pi\lambda_{d}}}\exp\left(-\frac{c^{2}\bar{\bm{z}}_{d}^{\top}\bm{\Sigma}^{*}\bar{\bm{z}}_{d}+(\lambda_{d}+c\bar{\zeta}_{d})^{2}}{2\lambda_{d}}\right)\\\
\\\
=&\mathcal{GIG}\left(\lambda_{d};\frac{1}{2},1,c^{2}(\bar{\zeta}_{d}^{2}+\bar{\bm{z}}_{d}^{\top}\bm{\Sigma}^{*}\bar{\bm{z}}_{d})\right),\end{array}$
(16)
a generalized inverse gaussian distribution (Devroye, 1986). Therefore,
$\lambda_{d}^{-1}$ follows an inverse gaussian distribution
$\mathcal{IG}(\lambda_{d}^{-1};\frac{1}{c\sqrt{\bar{\zeta}_{d}^{2}+\bar{\bm{z}}_{d}^{\top}\bm{\Sigma}^{*}\bar{\bm{z}}_{d}}},1)$,
from which we can draw a sample in constant time (Michael et al., 1976).
For training, we run the global and local updates alternately until
convergence at each round of PA optimization, as outlined in Alg. 1. To make
predictions on testing data, we then draw one sample of $\hat{\bm{w}}$ as the
classification weight and apply the prediction rule. The inference of
$\bm{\bar{z}}$ for testing documents is the same as in (Zhu et al., 2013a).
Algorithm 1 Online MedLDA
1: Let
$q_{0}(\bm{w})=\mathcal{N}(0;v^{2}I),q_{0}(\bm{\phi}_{k})=\text{Dir}(\gamma),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k$.
2: for $t=0\to\infty$ do
3: Set $q(\bm{\Phi},\bm{w})=q_{t}(\bm{\Phi},\bm{w})$. Initialize $\bm{Z}_{t}$.
4: for $i=1\to\mathcal{I}$ do
5: Draw samples
$\\{\bm{Z}_{t}^{(j)},\bm{\lambda}_{t}^{(j)}\\}_{j=1}^{\mathcal{J}}$ from (15,
16).
6: Discard the first $\beta$ burn-in samples ($\beta<\mathcal{J}$).
7: Use the rest $\mathcal{J}-\beta$ samples to update $q(\bm{\Phi},\bm{w})$
following (12, 13).
8: end for
9: Set $q_{t+1}(\bm{\Phi},\bm{w})=q(\bm{\Phi},\bm{w})$.
10: end for
## 4 Online Nonparametric MedLDA
We present online nonparametric MedLDA for resolving the unknown number of
topics, based on the theory of hierarchical Dirichlet process (HDP) (Teh et
al., 2006a).
### 4.1 Batch MedHDP
HDP provides an extension to LDA that allows for a nonparametric inference of
the unknown topic numbers. The generative process of HDP can be summarized
using a stick-breaking construction (Wang & Blei, 2012), where the stick
lengths $\bm{\pi}=\\{\pi_{k}\\}_{k=1}^{\infty}$ are generated as:
$\begin{array}[]{l}\pi_{k}=\bar{\pi}_{k}\prod\limits_{i<k}(1-\bar{\pi}_{i}),\leavevmode\nobreak\
\bar{\pi}_{k}\sim\text{Beta}(1,\gamma),\leavevmode\nobreak\
\textrm{for}\leavevmode\nobreak\ k=1,...,\infty,\end{array}$
and the topic mixing proportions are generated as
$\bm{\theta}_{d}\sim\text{Dir}(\alpha\bm{\pi}),\leavevmode\nobreak\
\textrm{for}\leavevmode\nobreak\ d=1,...,D$. Each topic $\bm{\phi}_{k}$ is a
sample from a Dirichlet base distribution, i.e.,
$\bm{\phi}_{k}\sim\text{Dir}(\bm{\eta})$. After we get the topic mixing
proportions $\bm{\theta}_{d}$, the generation of words is the same as in the
standard LDA.
To augment the HDP topic model for predictive tasks, we introduce a classifier
$\bm{w}$ and define the linear discriminant function in the same form as (7),
where we should note that since the number of words in a document is finite,
the average topic assignment vector $\bar{\bm{z}}_{d}$ has only a finite
number of non-zero elements. Therefore, the dot product in (7) is in fact
finite. Let $\bm{\bar{\pi}}=\\{\bar{\pi}_{k}\\}_{k=1}^{\infty}$. We define
MedHDP as solving the following problem to infer the joint posterior
$q(\bm{w},\bm{\bar{\pi}},\bm{\Phi},\bm{\Theta},\bm{Z})$111Given
$\bm{\bar{\pi}}$, $\bm{\pi}$ can be computed via the stick breaking process.:
$\displaystyle\min\limits_{q\in\mathcal{P}}{\mathcal{L}(q(\bm{w},\bm{\bar{\pi}},\bm{\Phi},\bm{\Theta},\bm{Z}))}+2c\sum\limits_{d=1}^{D}{\ell_{\epsilon}(q(\bm{w},\bm{z}_{d});\bm{x}_{d},\bm{y}_{d})},$
where
$\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{\bar{\pi}},\bm{\Theta},\bm{Z}))=\mathrm{KL}[q(\bm{w},\bm{\Phi},\bm{\bar{\pi}},\bm{\Theta},\bm{Z})||p(\bm{w},\\\
\bm{\bar{\pi}},\bm{\Phi},\bm{\Theta},\bm{Z}|\bm{X})]$, and the loss function
could be either (8) or (9), leading to the MedHDP topic models with either
averaging or Gibbs classifiers.
### 4.2 Online MedHDP
To apply the online BayesPA, we have the global variables
$\bm{\mathcal{M}}=(\bm{\bar{\pi}},\bm{\Phi})$, and the local variables
$\bm{H}_{t}=(\bm{\Theta}_{t},\bm{Z}_{t})$. We again focus on the expected
hinge loss (9) in this paper. As in online MedLDA, we marginalize out
$\bm{\Theta}_{t}$ and adopt the same data augmentation technique with the
augmented variables $\bm{\lambda}_{t}$. Furthermore, to simplify the sampling
scheme, we introduce auxiliary latent variables
$\bm{S}_{t}=\\{\bm{s}_{d}\\}_{d\in B_{t}}$, where $s_{dk}$ represents the
number of occupied tables serving dish $k$ in a Chinese Restaurant Process
(Teh et al., 2006a; Wang & Blei, 2012). By definition, we have
$p(\bm{Z}_{t},\bm{S}_{t}|\bm{\bar{\pi}})=\prod_{d\in
B_{t}}p(\bm{s}_{d},\bm{z}_{d}|\bm{\bar{\pi}})$ and
$p(\bm{s}_{d},\bm{z}_{d}|\bm{\bar{\pi}})\propto\prod_{k=1}^{\infty}{S(n_{d}\bar{z}_{dk},s_{dk})(\alpha\pi_{k})^{s_{dk}}},$
(17)
where $S(a,b)$ are unsigned Stirling numbers of the first kind (Antoniak,
1974). It is not hard to verify that
$p(\bm{z}_{d}|\bm{\bar{\pi}})=\sum_{\bm{s}_{d}}{p(\bm{s}_{d},\bm{z}_{d}|\bm{\bar{\pi}}}$).
Therefore, we have local variables
$\bm{H}_{t}=(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})$, and the target
collapsed posterior
$q_{t+1}(\bm{w},\bm{\bar{\pi}},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t})$ is the
marginal distribution of
$q_{t+1}(\bm{w},\bm{\bar{\pi}},\bm{\Phi},\bm{H}_{t})$, which is the solution
of the problem:
$\displaystyle\underset{q\in\mathcal{F}_{t}}{\operatorname{min}}{\leavevmode\nobreak\
\mathcal{L}(q(\bm{w},\bm{\bar{\pi}},\bm{\Phi},\bm{H}_{t})\\!)\\!-\\!\mathbb{E}_{q}[\log\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})]},$
(18)
where
$\mathcal{L}(q)=\mathrm{KL}[q||q_{t}(\bm{w},\bm{\bar{\pi}},\bm{\Phi})p(\bm{Z}_{t},\bm{S}_{t}|\bm{\bar{\pi}})p(\bm{X}_{t}|\bm{Z}_{t},\bm{\Phi})]$
. As in online MedLDA, we solve (18) via an iterative procedure detailed
below.
Global Update: By fixing the distribution of local variables,
$q(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})$, and ignoring the irrelevant
terms, we have the mean-field update equations for $\bm{\Phi}$ and $\bm{w}$,
the same as in (12) and (13), while for $\bar{\bm{\pi}}$, we have
$q(\bar{\pi}_{k})\propto q_{t}(\bar{\pi}_{k})\prod_{d\in
B_{t}}\exp(\mathbb{E}_{q(\bm{h}_{d})}[\log
p(\bm{s}_{d},\bm{z}_{d}|\bm{\bar{\pi}})]).$ (19)
By induction, we can show that
$q_{t}(\bar{\pi}_{k})=\text{Beta}(u_{k}^{t},v_{k}^{t})$, a Beta distribution
at each step, and the update equation is
$q(\bar{\pi}_{k})=\text{Beta}(u_{k}^{*},v_{k}^{*}),$ (20)
where $u_{k}^{*}=u_{k}^{t}+\sum_{d\in
B_{t}}\mathbb{E}_{q(\bm{s}_{d})}{[s_{dk}]}$ and
$v_{k}^{*}=v_{k}^{t}+\sum_{d\in
B_{t}}\mathbb{E}_{q(\bm{s}_{d})}{[\sum_{j>k}{s_{dj}}]}$ for $k=\\{1,2,...\\}$.
Since $\bm{Z}_{t}$ contains only finite number of discrete variables, we only
need to maintain and update the global distribution for a finite number of
topics.
Local Update: Fixing the global distribution
$q(\bm{w},\bm{\bar{\pi}},\bm{\Phi})$, we get the mean-field update equation
for $(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})$:
$q(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})\propto\tilde{q}(\bm{Z}_{t},\bm{S}_{t})\tilde{q}(\bm{Z}_{t},\bm{\lambda}_{t})$
(21)
where
$\tilde{q}(\bm{Z}_{t},\bm{S}_{t})=\exp(\mathbb{E}_{q(\bm{\Phi)}q(\bm{\bar{\pi}})}[\log
p(\bm{X}_{t}|\bm{\Phi},\bm{Z}_{t})+\log
p(\bm{Z}_{t},\bm{S}_{t}|\bm{\bar{\pi}})])$ and
$\tilde{q}(\bm{Z}_{t},\bm{\lambda}_{t})=\exp(\mathbb{E}_{q(\bm{w})}[\log\psi(\bm{Y}_{t},\\\
\bm{\lambda}_{t}|\bm{w},\bm{Z}_{t})])$. To overcome the the potentially
unbounded latent space, we take the ideas from (Wang & Blei, 2012) and adopt
an approximation for $\tilde{q}(\bm{Z}_{t},\bm{S}_{t})$:
$\tilde{q}(\bm{Z}_{t},\bm{S}_{t})\approx\mathbb{E}_{q(\bm{\Phi)}q(\bm{\bar{\pi}})}[p(\bm{X}|\bm{\Phi},\bm{Z}_{t})p(\bm{Z}_{t},\bm{S}_{t}|\bm{\bar{\pi}})].$
(22)
Instead of marginalizing out $\bm{\bar{\pi}}$ in (22), which is analytically
difficult, we sample $\bm{\bar{\pi}}$ jointly with
$(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})$. This leads to the following Gibbs
sampling scheme:
For $\bm{Z}_{t}$: Let $K$ be the current inferred number of topics. The
conditional distribution of one variable $z_{di}$ given $\bm{Z}_{t}^{\neg
di}$, $\bm{\lambda}_{t}$ and $\bm{\bar{\pi}}$ can be derived from (21) with
$\bm{s}_{d}$ marginalized out for convenience:
$\begin{array}[]{ll}&q(z_{di}=k|\bm{Z}_{t}^{\neg
di},\bm{\lambda_{t}},\bm{\bar{\pi}})\propto\frac{(\alpha\pi_{k}+C_{dk}^{\neg
di})(C_{kx_{di}}^{\neg di}+\Delta_{kx_{di}}^{*})}{\sum_{w}{(C_{kw}^{\neg
di}+\Delta_{kw}^{*})}}\\\
&\exp\\!\Big{(}\\!\frac{cy_{d}(c\epsilon+\lambda_{d})\mu_{k}^{*}}{n_{d}\lambda_{d}}\\!-\\!\frac{c^{2}(\mu_{k}^{*2}+\Sigma_{kk}^{*}+2(\mu_{k}^{*}\bm{\mu}^{*}+\bm{\Sigma}_{\cdot,k}^{*})^{\top}\bm{C}_{d}^{\neg
di})}{2n_{d}^{2}\lambda_{d}}\Big{)}.\end{array}$
Besides, for $k>K$ and symmetric Dirichlet prior $\bm{\eta}$, this becomes
$q(z_{di}=k|\bm{Z}_{t}^{\neg
di},\bm{\lambda_{t}},\bm{\bar{\pi}})\propto\alpha\pi_{k}/W$, and therefore the
total probability of assigning a new topic is
$q(z_{di}>K|\bm{Z}_{t}^{\neg
di},\bm{\lambda}_{t},\bm{\bar{\pi}})\propto\alpha\left(1-\sum\limits_{k=1}^{K}{\pi_{k}}\right)/W.$
For $\bm{\lambda}_{t}$: The conditional distribution
$q(\lambda_{d}|\bm{Z}_{t},\bm{S}_{t},\bm{\bar{\pi}})$ is the same as (16).
For $\bm{S}_{t}$: The conditional distribution of $s_{dk}$ given
$\bm{Z}_{t},\bm{\bar{\pi}},\\\ \bm{\lambda}_{t}$ can be derived from the joint
distribution (17):
$q(s_{dk}|\bm{Z}_{t},\bm{\lambda}_{t},\bm{\bar{\pi}})\propto{S(n_{d}\bar{z}_{dk},s_{dk})(\alpha\pi_{k})^{s_{dk}}}$
(23)
For $\bm{\bar{\pi}}$: It can be derived from (21) that given
$(\bm{Z}_{t},\bm{S}_{t},\bm{\lambda}_{t})$, each $\bar{\pi}_{k}$ follows the
beta distribution, $\bar{\pi}_{k}\sim\mathrm{Beta}(a_{k},b_{k})$, where
$a_{k}=u_{k}^{*}+\sum_{d\in B_{t}}s_{dk}$ and $b_{k}=v_{k}^{*}+\sum_{d\in
B_{t}}\sum_{j>k}s_{dj}$.
Similar to online MedLDA, we iterate the above steps till convergence for
training.
## 5 Experiments
We demonstrate the efficiency and prediction accuracy of online MedLDA and
MedHDP, denoted as _paMedLDA_ and _paMedHDP_ , on the 20Newsgroup (20NG) and a
large Wikipedia dataset. A sensitivity analysis of the key parameters is also
provided. Following the same setting in (Zhu et al., 2012), we remove a
standard list of stop words. All of the experiments are done on a normal
computer with single-core clock rate up to 2.4 GHz.
### 5.1 Classification on 20Newsgroup
We perform multi-class classification on the entire 20NG dataset with all the
20 categories. The training set contains 11,269 documents, with the smallest
category having 376 documents and the biggest category having 599 documents.
The test set contains 7,505 documents, with the smallest and biggest
categories having 259 and 399 documents respectively. We adopt the ”one-vs-
all” strategy (Rifkin & Klautau, 2004) to combine binary classifiers for
multi-class prediction tasks.
Figure 1: Test errors with different number of passes through the 20NG
training dataset. Left: LDA-based models. Right: HDP-based models.
We compare paMedLDA and paMedHDP with their batch counterparts, denoted as
_bMedLDA_ and _bMedHD_ P, which are obtained by letting the batch size $|B|$
be equal to the dataset size $D$, and Gibbs MedLDA, denoted as _gMedLDA_ ,
(Zhu et al., 2013a), which performs Gibbs sampling in the batch manner. We
also consider online unsupervised topic models as baselines, including sparse
inference for LDA (_spLDA_) (Mimno et al., 2012), which has been demonstrated
to be superior than online variational LDA (Hoffman et al., 2010) in
performance, and truncation-free online variational HDP (_tfHDP_) (Wang &
Blei, 2012), which has been shown to be promising in nonparametric topic
modeling. For both of them, we learn a linear SVM with the topic
representations using LIBSVM (Chang & Lin, 2011). The performances of other
batch supervised topic models, such as sLDA (Blei & McAuliffe, 2010) and
DiscLDA (Lacoste-Julien et al., 2008), are reported in (Zhu et al., 2012). For
all LDA-based topic models, we use symmetric Dirichlet priors
$\bm{\alpha}=1/K\cdot\bm{1},\bm{\gamma}=0.5\cdot\bm{1}$; for all HDP-based
topic models, we use $\alpha=5,\gamma=1,\bm{\eta}=0.45\cdot\bm{1}$; for all
MED topic models, we use $\epsilon=164,c=1,v=1$, the choice of which is not
crucial to the models’ performance as shown in (Zhu et al., 2013a).
We first analyze how many processed documents are sufficient for each model to
converge. Figure 1 shows the prediction accuracy with the number of passes
through the entire 20NG dataset, where $K=80$ for parametric models and
$(\mathcal{I},\mathcal{J},\beta)=(1,2,0)$ for BayesPA. As we could observe, by
solving a series of latent BayesPA learning problems, paMedLDA and paMedHDP
fully explore the redundancy of documents and converge in one pass, while
their batch counterparts need many passes as burn-in steps. Besides, compared
with the online unsupervised learning algorithms, BayesPA topic models utilize
supervising-side information from each mini-batch, and therefore exhibit a
faster convergence rate in discrimination ability.
Next, we study each model’s best performance possible and the corresponding
training time. To allow for a fair comparison, we train each model until the
relative change of its objective is less than $10^{-4}$. Figure 2 shows the
accuracy and training time of LDA-based models on the whole dataset with
varying numbers of topics. Similarly, Figure 3 shows the accuracy and training
time of HDP-based models, where the dots stand for the mean inferred numbers
of topics, and the lengths of the horizontal bars represent their standard
deviations. As we can see, BayesPA topic models, at the power of online
learning, are about 1 order of magnitude faster than their batch counterparts
in training time. Furthermore, thanks to the merits of Gibbs sampling, which
does not pose strict mean-field assumptions about the independence of latent
variables, BayesPA topic models parallel their batch alternatives in accuracy.
Figure 2: Classification accuracy and running time of paMedLDA and comparison
models on the 20NG dataset. Figure 3: Classification accuracy and running time
of paMedHDP and comparison models on the 20NG dataset.
### 5.2 Further Discussions
We provide further discussions on BayesPA learning for topic models. First, we
analyze the models’ sensitivity to some key parameters. Second, we illustrate
an application with a large Wikipedia dataset containing 1.1 million
documents, where class labels are not exclusive.
#### 5.2.1 Sensitivity Analysis
Batch Size $|B|$: Figure 4 presents the test errors of BayesPA topic models
as a function of training time on the entire 20NG dataset with various batch
sizes, where $K=40$. We can see that the convergence speeds of different
algorithms vary. First of all, the batch algorithms suffer from multiple
passes through the dataset and therefore are much slower than the online
alternatives. Second, we could observe that algorithms with medium batch sizes
($|B|=64,256$) converge faster. If we choose a batch size too small, for
example, $|B|=1$, each iteration would not provide sufficient evidence for the
update of global variables; if the batch size is too large, each mini-batch
becomes redundant and the convergence rate reduces.
Figure 4: Test errors of paMedLDA (left) and paMedHDP (right) with different
batch sizes on the 20NG dataset.
Number of iterations $\mathcal{I}$ and samples $\mathcal{J}$: Since the time
complexity of Algorithm 1 is linear in both $\mathcal{I}$ and $\mathcal{J}$,
we would like to know how these parameters influence the quality of the
trained model. First, notice that the first $\beta$ samples are discarded as
burn-in steps. To understand how large $\beta$ is sufficient, we consider the
settings of the pairs $(\mathcal{J},\beta)$ and check the prediction accuracy
of Algorithm 1 for $K=40,|B|=512$, as shown in Table 1.
Table 1: Effect of the number of samples and burn-in steps. $\mathcal{J}$ $\beta$ | 0 | 2 | 4 | 6 | 8
---|---|---|---|---|---
1 | 0.783 | | | |
3 | 0.803 | 0.799 | | |
5 | 0.808 | 0.803 | 0.792 | |
9 | 0.806 | 0.806 | 0.806 | 0.804 | 0.796
We can see that accuracies closer to the diagonal of the table are relatively
lower, while settings with the same number of kept samples, e.g.
$(\mathcal{J},\beta)=(3,0),(5,2),(9,6)$, yield similar results. The number of
kept samples exhibits a more significant role in the performance of BayesPA
topic models than the burn-in steps.
Next, we analyze which setting of $(\mathcal{I},\mathcal{J})$ guarantees good
performance. Figure 5 presents the results. As we can see, for
$\mathcal{J}=1$, the algorithms suffer from the noisy approximation and
therefore sacrifices prediction accuracy. But for larger $\mathcal{J}$, simply
$\mathcal{I}=1$ is promising, possibly due to the redundancy among mini-
batches.
Figure 5: Classification accuracies and training time of (a): paMedLDA, (b):
paMedHDP, with different combinations of $(\mathcal{I},\mathcal{J})$ on the
20NG dataset.
#### 5.2.2 Multi-Task Classification
For multi-task classification, a set of binary classifiers are trained, each
of which identifies whether a document $\bm{x}_{d}$ belongs to a specific
task/category $\bm{y}_{d}^{\tau}\in\\{+1,-1\\}$. These binary classifiers are
allowed to share common latent representations and therefore could be attained
via a modified BayesPA update equation:
$\underset{q\in\mathcal{F}_{t}}{\operatorname{min}}{\leavevmode\nobreak\
\mathcal{L}(q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t}))+2c\sum\limits_{\tau=1}^{\mathcal{T}}\ell_{\epsilon}(q(\bm{w},\bm{\mathcal{M}},\bm{H}_{t});\bm{X}_{t},\bm{Y}_{t}^{\tau})}$
where $\mathcal{T}$ is the total number of tasks. We can then derive the
multi-task version of Passive-Aggressive topic models, denoted by _paMedLDA-
mt_ and _paMedHDP-mt_ , in a way similar to Section 3.2 and 4.2.
We test paMedLDA-mt and paMedHDP-mt as well as comparison models, including
bMedLDA-mt, bMedHDP-mt and gMedLDA-mt (Zhu et al., 2013b) on a large Wiki
dataset built from the Wikipedia set used in PASCAL LSHC challenge 2012 222See
http://lshtc.iit.demokritos.gr/.. The Wiki dataset is a collection of
documents with labels up to 20 different kinds, while the data distribution
among the labels is balanced. The training/testing split is 1.1 million / 5
thousand. To measure performance, we use F1 score, the harmonic mean of
precision and recall.
Figure 6 shows the F1 scores of various models as a function of training time.
We can see that BayesPA topic models are again about 1 order of magnitude
faster than the batch alternatives and yet produce comparable results.
Therefore, BayesPA topic models are potentially extendable to large-scale
multi-class settings.
Figure 6: F1 scores of various models on the 1.1M wikipedia dataset.
## 6 Conclusions and Future Work
We present online Bayesian Passive-Aggressive (BayesPA) learning as a new
framework for max-margin Bayesian inference of online streaming data. We show
that BayesPA subsumes the online PA, and more significantly, generalizes
naturally to incorporate latent variables and to perform nonparametric
Bayesian inference, therefore providing great flexibility for explorative
analysis. Based on the ideas of BayesPA, we develop efficient online learning
algorithms for max-margin topic models as well as their nonparametric
extensions. Empirical experiments on several real datasets demonstrate
significant improvements on time efficiency, while maintaining comparable
results.
As future work, we are interested in showing provable bounds in its
convergence and limitations. Furthermore, better understanding its
mathematical structure would allow one to design more involved BayesPA
algorithms for various models. We are also interested in developing highly
scalable, distributed (Broderick et al., 2013) BayesPA learning paradigms,
which will better meet the demand of processing massive real data available
today.
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## Appendix A: Proof of Lemma 2.2
In this section, we prove Lemma 2.2. We should note that our deviations below
also provide insights for the developments of online BayesPA algorithms with
the averaging classifiers.
###### Proof.
We prove for the more generalized soft-margin version of BayesPA learning,
which can be reformulated using a slack variable $\xi$:
$\begin{array}[]{ccc}&q_{t+1}(\bm{w})=&\underset{q(\bm{w})\in\mathcal{P}}{\operatorname{argmin}}\leavevmode\nobreak\
\text{KL}[q(\bm{w})||q_{t}(\bm{w})]+c\xi\\\
&&\text{s.t.}:y_{t}\mathbb{E}_{q}[{\bm{w}}^{\top}\bm{x}_{t}]\geq\epsilon-\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \xi\geq 0.\end{array}$ (24)
Similar to Corollary 5 in (Zhu et al., 2012), the optimal solution
$q^{*}(\bm{w})$ of the above problem can be derived from its functional
Lagrangian and has the following form:
$q^{*}(\bm{w})=\frac{1}{\Gamma(\tau)}q_{t}(\bm{w})\exp(\tau
y_{t}\bm{w}^{\top}\bm{x}_{t})$ (25)
where $\Gamma(\tau)$ is a normalization term and $\tau$ is the optimal
solution to the dual problem:
$\begin{array}[]{rl}\max\limits_{\tau}&{\leavevmode\nobreak\
\tau\epsilon-\log\Gamma(\tau)}\\\ \text{s.t. }&0\leq\tau\leq c\end{array}$
(26)
Using this primal-dual interpretation, we first prove that for prior
$p_{0}(\bm{w})=\mathcal{N}(\bm{w}_{0},I)$,
$q_{t}(\bm{w})=\mathcal{N}(\bm{\mu}_{t},I)$ for some $\bm{\mu}_{t}$ in each
round $t=0,1,2,...$. This can be shown by induction. Assume for round $t$, the
distribution $q_{t}(\bm{w})=\mathcal{N}(\bm{\mu}_{t},I)$. Then for round
$t+1$, the distribution by (25) is
$q_{t+1}(\bm{w})=\frac{1}{\mathcal{C}\cdot\Gamma(\tau)}\exp\Big{(}-\frac{1}{2}||\bm{w}-(\bm{\mu}_{t}+\tau
y_{t}\bm{x}_{t})||^{2}\Big{)}$ (27)
where $\mathcal{C}$ is some constant. Therefore, the distribution
$q_{t+1}(\bm{w})=\mathcal{N}(\mu_{t}+\tau\bm{x}_{t},I)$. As a by-product, the
normalization term $\Gamma(\tau)=\sqrt{2\pi}\exp(\tau
y_{t}\bm{x}_{t}^{\top}\bm{\mu}_{t}+\frac{1}{2}\tau^{2}\bm{x}_{t}^{\top}\bm{x}_{t})$.
Next, we show that $\bm{\mu}_{t+1}=\bm{\mu}_{t}+\tau y_{t}\bm{x}_{t}$ is the
optimal solution of the online Passive-Aggressive update rule (Crammer et al.,
2006). To see this, we plug the derived $\Gamma(\tau)$ into (26), and obtain
$\begin{array}[]{rl}\max\limits_{\tau}&{\leavevmode\nobreak\
\tau-\frac{1}{2}\tau^{2}\bm{x}_{t}^{\top}\bm{x}_{t}-\tau
y_{t}\bm{\mu}_{t}^{\top}\bm{x}_{t}}\\\ \text{s.t. }&0\leq\tau\leq
c\end{array}$ (28)
which is exactly the dual form of the online Passive-Aggressive update rule:
$\begin{array}[]{rl}\bm{\mu}_{t+1}^{*}=&\arg\min{\leavevmode\nobreak\
||\bm{\mu}-\bm{\mu}_{t}||^{2}+c\xi}\\\ \text{s.t.
}&y_{t}\bm{\mu}^{\top}\bm{x}_{t}\geq\epsilon-\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \xi\geq 0,\end{array}$ (29)
the optimal solution to which is $\bm{\mu}_{t+1}^{*}=\bm{\mu}_{t}+\tau
y_{t}\bm{x}_{t}$. It is then clear that $\mu_{t+1}=\mu_{t+1}^{*}$. ∎
## Appendix B:
We show the objective in (11) is an upper bound of that in (6), that is,
$\begin{array}[]{l}\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t}))-\mathbb{E}_{q}[\log(\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w}))]\\\
\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\geq\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{Z}_{t}))+2c\sum\limits_{d\in
B_{t}}{\mathbb{E}_{q}[(\xi_{d})_{+}]}\end{array}$ (30)
where
$\mathcal{L}(q)=\mathrm{KL}[q||q_{t}(\bm{w},\bm{\Phi})q_{0}(\bm{Z}_{t})]$.
###### Proof.
We first have
$\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{Z}_{t},\bm{\lambda}_{t}))=\mathbb{E}_{q}[\log\frac{q(\bm{\lambda}_{t}\leavevmode\nobreak\
|\leavevmode\nobreak\
\bm{w},\bm{\Phi},\bm{Z}_{t})q(\bm{w},\bm{\Phi},\bm{Z}_{t})}{q_{t}(\bm{w},\bm{\Phi},\bm{Z}_{t})}],$
and
$\mathcal{L}(q(\bm{w},\bm{\Phi},\bm{Z}_{t}))=\mathbb{E}_{q}[\log\frac{q(\bm{w},\bm{\Phi},\bm{Z}_{t})}{q_{t}(\bm{w},\bm{\Phi},\bm{Z}_{t})}]$
Comparing these two equations and canceling out common factors, we know that
in order for (30) to make sense, it suffices to prove
$\mathbb{H}[q^{\prime}]-\mathbb{E}_{q^{\prime}}[\log(\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})]\geq
2c\sum\limits_{d\in B_{t}}{\mathbb{E}_{q^{\prime}}[(\xi_{d})_{+}]}$ (31)
is uniformly true for any given $(\bm{w},\bm{\Phi},\bm{Z}_{t})$, where
$\mathbb{H}(\cdot)$ is the entropy operator and
$q^{\prime}=q(\bm{\lambda}_{t}\leavevmode\nobreak\ |\leavevmode\nobreak\
\bm{w},\bm{\Phi},\bm{Z}_{t})$. The inequality (31) can be reformulated as
$\mathbb{E}_{q^{\prime}}[\log\frac{q^{\prime}}{\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})}]\geq
2c\sum\limits_{d\in B_{t}}{\mathbb{E}_{q^{\prime}}[(\xi_{d})_{+}]}$ (32)
Exploiting the convexity of the function $\log(\cdot)$, i.e.
$-\mathbb{E}_{q^{\prime}}[\log\frac{\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})}{q^{\prime}}]\geq-\log\int_{\bm{\lambda}_{t}}{\psi(\bm{Y}_{t},\bm{\lambda}_{t}|\bm{Z}_{t},\bm{w})\leavevmode\nobreak\
d\bm{\lambda}_{t}},$
and utilizing the equality (10), we then have (32) and therefore prove (30). ∎
|
arxiv-papers
| 2013-12-12T02:46:07 |
2024-09-04T02:49:55.340275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tianlin Shi and Jun Zhu",
"submitter": "Tianlin Shi",
"url": "https://arxiv.org/abs/1312.3388"
}
|
1312.3522
|
# Sparse Matrix-based Random Projection for Classification
Weizhi Lu, Weiyu Li, Kidiyo Kpalma and Joseph Ronsin
###### Abstract
As a typical dimensionality reduction technique, random projection can be
simply implemented with linear projection, while maintaining the pairwise
distances of high-dimensional data with high probability. Considering this
technique is mainly exploited for the task of classification, this paper is
developed to study the construction of random matrix from the viewpoint of
feature selection, rather than of traditional distance preservation. This
yields a somewhat surprising theoretical result, that is, the sparse random
matrix with exactly one nonzero element per column, can present better feature
selection performance than other more dense matrices, if the projection
dimension is sufficiently large (namely, not much smaller than the number of
feature elements); otherwise, it will perform comparably to others. For random
projection, this theoretical result implies considerable improvement on both
complexity and performance, which is widely confirmed with the classification
experiments on both synthetic data and real data.
###### Index Terms:
Random Projection, Sparse Matrix, Classification, Feature Selection, Distance
Preservation, High-dimensional data
## I Introduction
Random projection attempts to project a set of high-dimensional data into a
low-dimensional subspace without distortion on pairwise distance. This brings
attractive computational advantages on the collection and processing of high-
dimensional signals. In practice, it has been successfully applied in numerous
fields concerning categorization, as shown in [1] and the references therein.
Currently the theoretical study of this technique mainly falls into one of the
following two topics. One topic is concerned with the construction of random
matrix in terms of distance preservation. In fact, this problem has been
sufficiently addressed along with the emergence of Johnson-Lindenstrauss (JL)
lemma [2]. The other popular one is to estimate the performance of traditional
classifiers combined with random projection, as detailed in [3] and the
references therein. Specifically, it may be worth mentioning that, recently
the performance consistency of SVM on random projection is proved by
exploiting the underlying connection between JL lemma and compressed sensing
[4] [5].
Based on the principle of distance preservation, Gaussian random matrices [6]
and a few sparse $\\{0,\pm 1\\}$ random matrices [7, 8, 9] have been
sequentially proposed for random projection. In terms of implementation
complexity, it is clear that the sparse random matrix is more attractive.
Unfortunately, as it will be proved in the following section II-B, the sparser
matrix tends to yield weaker distance preservation. This fact largely weakens
our interests in the pursuit of sparser random matrix. However, it is
necessary to mention a problem ignored for a long time, that is, random
projection is mainly exploited for various tasks of classification, which
prefer to maximize the distances between different classes, rather than
preserve the pairwise distances. In this sense, we are motivated to study
random projection from the viewpoint of feature selection, rather than of
traditional distance preservation as required by JL lemma. During this study,
however, the property of satisfying JL lemma should not be ignored, because it
promises the stability of data structure during random projection, which
enables the possibility of conducting classification in the projection space.
Thus throughout the paper, all evaluated random matrices are previously
ensured to satisfy JL lemma to a certain degree.
In this paper, we indeed propose the desired $\\{0,\pm 1\\}$ random projection
matrix with the best feature selection performance, by theoretically analyzing
the change trend of feature selection performance over the varying sparsity of
random matrices. The proposed matrix presents currently the sparsest
structure, which holds only one random nonzero position per column. In theory,
it is expected to provide better classification performance over other more
dense matrices, if the projection dimension is not much smaller than the
number of feature elements. This conjecture is confirmed with extensive
classification experiments on both synthetic and real data.
The rest of the paper is organized as follows. In the next section, the JL
lemma is first introduced, and then the distance preservation property of
sparse random matrix over varying sparsity is evaluated. In section III, a
theoretical frame is proposed to predict feature selection performance of
random matrices over varying sparsity. According to the theoretical
conjecture, the currently known sparsest matrix with better performance over
other more dense matrices is proposed and analyzed in section IV. In section
V, the performance advantage of the proposed sparse matrix is verified by
performing binary classification on both synthetic data and real data. The
real data incudes three representative datasets in dimension reduction: face
image, DNA microarray and text document. Finally, this paper is concluded in
section VI.
## II Preliminaries
This section first briefly reviews JL lemma, and then evaluates the distance
preservation of sparse random matrix over varying sparsity.
For easy reading, we begin by introducing some basic notations for this paper.
A random matrix is denoted by $\mathbf{R}\in\mathbb{R}^{k\times d}$, $k<d$.
$r_{ij}$ is used to represent the element of $\mathbf{R}$ at the $i$-th row
and the $j$-th column, and $\mathbf{r}\in\mathbb{R}^{1\times d}$ indicates the
row vector of $\mathbf{R}$. Considering the paper is concerned with binary
classification, in the following study we tend to define two samples
$\mathbf{v}\in\mathbb{R}^{1\times d}$ and $\mathbf{w}\in\mathbb{R}^{1\times
d}$, randomly drawn from two different patterns of high-dimensional datasets
$\mathcal{V}\subset\mathbb{R}^{d}$ and $\mathcal{W}\subset\mathbb{R}^{d}$,
respectively. The inner product between two vectors is typically written as
$\langle\mathbf{v},\mathbf{w}\rangle$. To distinguish from variable, the
vector is written in bold. In the proofs of the following lemmas, we typically
use $\Phi(*)$ to denote the cumulative distribution function of $N(0,1)$. The
minimal integer not less than $*$, and the the maximum integer not larger than
$*$ are denoted with $\lceil*\rceil$ and $\lfloor*\rfloor$ .
### II-A Johnson-Lindenstrauss (JL) lemma
The distance preservation of random projection is supported by JL lemma. In
the past decades, several variants of JL lemma have been proposed in [10, 11,
12]. For the convenience of the proof of the following Corollary 2, here we
recall the version of [12] in the following Lemma 1. According to Lemma 1, it
can be observed that a random matrix satisfying JL lemma should have
$\mathds{E}(r_{ij})=0$ and $\mathds{E}(r_{ij}^{2})=1$.
###### Lemma 1.
[12] Consider random matrix $\mathbf{R}\in\mathbb{R}^{k\times d}$, with each
entry $r_{ij}$ chosen independently from a distribution that is symmetric
about the origin with $\mathds{E}(r_{ij}^{2})=1$. For any fixed vector
$\mathbf{v}\in\mathbb{R}^{d}$, let
$\mathbf{v}^{\prime}=\frac{1}{\sqrt{k}}\mathbf{R}\mathbf{v}^{T}$.
* •
Suppose $B=\mathds{E}(r_{ij}^{4})<\infty$. Then for any $\epsilon>0$,
$\displaystyle\text{\emph{Pr}}(\|\mathbf{v}^{\prime}\|^{2}\leq(1-\epsilon)\|\mathbf{v}\|^{2})\leq
e^{-\frac{(\epsilon^{2}-\epsilon^{3})k}{2(B+1)}}$ (1)
* •
Suppose $\exists L>0$ such that for any integer $m>0$,
$\mathds{E}(r_{ij}^{2m})\leq\frac{(2m)!}{2^{m}m!}L^{2m}$. Then for any
$\epsilon>0$,
$\displaystyle\text{
\emph{Pr}}(\|\mathbf{v}^{\prime}\|^{2}\geq(1+\epsilon)L^{2}\|\mathbf{v}\|^{2})$
$\displaystyle\leq((1+\epsilon)e^{-\epsilon})^{k/2}$ (2) $\displaystyle\leq
e^{-(\epsilon^{2}-\epsilon^{3})\frac{k}{4}}$
### II-B Sparse random projection matrices
Up to now, only a few random matrices are theoretically proposed for random
projection. They can be roughly classified into two typical classes. One is
the Gaussian random matrix with entries i.i.d dawn from $N(0,1)$ , and the
other is the sparse random matrix with elements satisfying the distribution
below:
$r_{ij}=\sqrt{q}\times\left\\{\begin{array}[]{cl}1&\text{with
probability}~{}1/2q\\\ 0&\text{with probability}~{}1-1/q\\\ -1&\text{with
probability}~{}1/2q\end{array}\right.$ (3)
where $q$ is allowed to be 2, 3 [7] or $\sqrt{d}$ [8]. Apparently the larger
$q$ indicates the higher sparsity.
Naturally, an interesting question arises: can we continue improving the
sparsity of random projection? Unfortunately, as illustrated in Lemma 2, the
concentration of JL lemma will decrease as the sparsity increases. In other
words, the higher sparsity leads to weaker performance on distance
preservation. However, as it will be disclosed in the following part, the
classification tasks involving random projection are more sensitive to feature
selection rather than to distance preservation.
###### Lemma 2.
Suppose one class of random matrices $R\in\mathbb{R}^{k\times d}$, with each
entry $r_{ij}$ of the distribution as in formula (3), where $q=k/s$ and $1\leq
s\leq k$ is an integer. Then these matrices satisfy JL lemma with different
levels: the sparser matrix implies the worse property on distance
preservation.
###### Proof.
With formula (3), it is easy to derive that the proposed matrices satisfy the
distribution defined in Lemma 1. In this sense, they also obey JL lemma if the
two constraints corresponding to formulas (1) and (2) could be further proved.
For the first constraint corresponding to formula (1):
$\displaystyle B$ $\displaystyle=\mathds{E}(r_{ij}^{4})$ (4)
$\displaystyle=(\sqrt{k/s})^{4}\times(s/2k)+(-\sqrt{k/s})^{4}\times(s/2k)$
$\displaystyle=k/s<\infty$
then it is approved.
For the second constraint corresponding to formula (2):
for any integer $m>0$, derive $\mathds{E}(r^{2m})=(k/s)^{m-1}$, and
$\frac{\mathds{E}(r_{ij}^{2m})}{(2m)!L^{2m}/(2^{m}m!)}=\frac{2^{m}m!k^{m-1}}{s^{m-1}(2m)!L^{2m}}.$
Since $(2m)!\geq m!m^{m}$,
$\frac{\mathds{E}(r_{ij}^{2m})}{(2m)!L^{2m}/(2^{m}m!)}\leq\frac{2^{m}k^{m-1}}{s^{m-1}m^{m}L^{2m}},$
let $L=(2k/s)^{1/2}\geq\sqrt{2}(k/s)^{(m-1)/2m}/\sqrt{m}$, further derive
$\frac{\mathds{E}(r_{ij}^{2m})}{(2m)!L^{2m}/(2^{m}m!)}\leq 1.$
Thus $\exists L=(2k/s)^{1/2}>0$ such that
$\mathds{E}(r_{ij}^{2m})\leq\frac{(2m)!}{2^{m}m!}L^{2m}$
for any integer $m>0$. Then the second constraint is also proved.
Consequently, it is deduced that, as $s$ decreases, $B$ in formula (4) will
increase, and subsequently the boundary error in formula (4) will get larger.
And this implies that the sparser the matrix is, the worse the JL property. ∎
## III Theoretical Framework
In this section, a theoretical framework is proposed to evaluate the feature
selection performance of random matrices with varying sparsity. As it will be
shown latter, the feature selection performance would be simply observed, if
the product between the difference between two distinct high-dimensional
vectors and the sampling/row vectors of random matrix, could be easily
derived. In this case, we have to previously know the distribution of the
difference between two distinct high-dimensional vectors. For the possibility
of analysis, the distribution should be characterized with a unified model.
Unfortunately, this goal seems hard to be perfectly achieved due to the
diversity and complexity of natural data. Therefore, without loss of
generality, we typically assume the i.i.d Gaussian distribution for the
elements of difference between two distinct high-dimensional vectors, as
detailed in the following section III-A. According to the law of large
numbers, it can be inferred that the Gaussian distribution is reasonable to be
applied to characterize the distribution of high-dimensional vectors in
magnitude. Similarly to most theoretical work attempting to model the real
world, our assumption also suffers from an obvious limitation. Empirically,
some of the real data elements, in particular the redundant (indiscriminative)
elements, tend to be coherent to some extent, rather than being absolutely
independent as we assume above. This imperfection probably limits the accuracy
and applicability of our theoretical model. However, as will be detailed
later, this problem can be ignored in our analysis where the difference
between pairwise redundant elements is assume to be zero. This also explains
why our theoretical proposal can be widely verified in the final experiments
involving a great amount of real data. With the aforementioned assumption, in
section III-B, the product between high-dimensional vector difference and row
vectors of random matrices is calculated and analyzed with respect to the
varying sparsity of random matrix, as detailed in Lemmas 3-5 and related
remarks. Note that to make the paper more readable, the proofs of Lemmas 3-5
are included in the Appendices.
### III-A Distribution of the difference between two distinct high-
dimensional vectors
From the viewpoint of feature selection, the random projection is expected to
maximize the difference between arbitrary two samples $\mathbf{v}$ and
$\mathbf{w}$ from two different datasets $\mathcal{V}$ and $\mathcal{W}$,
respectively. Usually the difference is measured with the Euclidean distance
denoted by $\lVert\mathbf{R}\mathbf{z}^{T}\rVert_{2}$,
$\mathbf{z}=\mathbf{v}-\mathbf{w}$. Then in terms of the mutual independence
of $\mathbf{R}$, the search for good random projection is equivalent to
seeking the row vector $\hat{\mathbf{r}}$ such that
$\hat{\mathbf{r}}=\operatorname*{arg\,max}_{\mathbf{r}}\\{|\langle\mathbf{r},\mathbf{z}\rangle|\\}.$
(5)
Thus in the following part we only need to evaluate the row vectors of
$\mathbf{R}$. For the convenience of analysis, the two classes of high-
dimensional data are further ideally divided into two parts,
$\mathbf{v}=[\mathbf{v}^{f}~{}\mathbf{v}^{r}]$ and
$\mathbf{w}=[\mathbf{w}^{f}~{}\mathbf{w}^{r}]$, where $\mathbf{v}^{f}$ and
$\mathbf{w}^{f}$ denote the feature elements containing the discriminative
information between $\mathbf{v}$ and $\mathbf{w}$ such that
$\mathds{E}(v^{f}_{i}-w^{f}_{i})\neq 0$, while $\mathbf{v}^{r}$ and
$\mathbf{w}^{r}$ represent the redundant elements such that
$\mathds{E}(v^{r}_{i}-w^{r}_{i})=0$ with a tiny variance. Subsequently,
$\mathbf{r}=[\mathbf{r}^{f}~{}\mathbf{r}^{r}]$ and
$\mathbf{z}=[\mathbf{z}^{f}~{}\mathbf{z}^{r}]$ are also seperated into two
parts corresponding to the coordinates of feature elements and redundant
elements, respectively. Then the task of random projection can be reduced to
maximizing $|\langle\mathbf{r}^{f},\mathbf{z}^{f}\rangle|$, which implies that
the redundant elements have no impact on the feature selection. Therefore, for
simpler expression, in the following part the high-dimensional data is assumed
to have only feature elements except for specific explanation, and the
superscript $f$ is simply dropped. As for the intra-class samples, we can
simply assume that their elements are all redundant elements, and then the
expected value of their difference is equal to 0, as derived before. This
means that the problem of minimizing the intra-class distance needs not to be
further studied. So in the following part, we only consider the case of
maximizing inter-class distance, as described in formula (5).
To explore the desired $\hat{\mathbf{r}}_{i}$ in formula (5), it is necessary
to know the distribution of $\mathbf{z}$. However, in practice the
distribution is hard to be characterized since the locations of feature
elements are usually unknown. As a result, we have to make a relaxed
assumption on the distribution of $\mathbf{z}$. For a given real dataset, the
values of $v_{i}$ and $w_{i}$ should be limited. This allows us to assume that
their difference $z_{i}$ is also bounded in amplitude, and acts as some
unknown distribution. For the sake of generality, in this paper $z_{i}$ is
regarded as approximately satisfying the Gaussian distribution in magnitude
and randomly takes a binary sign. Then the distribution of $z_{i}$ can be
formulated as
$z_{i}=\left\\{\begin{array}[]{cl}x&\text{with probability}~{}1/2\\\
-x&\text{with probability}~{}1/2\end{array}\right.$ (6)
where $x\in N(\mu,\sigma^{2})$, $\mu$ is a positive number, and
Pr$(x>0)=1-\epsilon$, $\epsilon=\Phi(-\frac{\mu}{\sigma})$ is a small positive
number.
### III-B Product between high-dimensional vector and random sampling vector
with varying sparsity
This subsection mainly tests the feature selection performance of random row
vector with varying sparsity. For the sake of comparison, Gaussian random
vectors are also evaluated. Recall that under the basic requirement of JL
lemma, that is $\mathds{E}(r_{ij})=0$ and $\mathds{E}(r_{ij}^{2})=1$, the
Gaussian matrix has elements i.i.d drawn from $N(0,1)$, and the sparse random
matrix has elements distributed as in formula (3) with $q\in\\{d/s:1\leq s\leq
d,s\in\mathds{N}\\}$.
Then from the following Lemmas 3-5, we present two crucial random projection
results for the high-dimensional data with the feature difference element
$|z_{i}|$ distributed as in formula (6):
* •
Random matrices will achieve the best feature selection performance as only
one feature element is sampled by each row vector; in other words, the
solution to the formula (5) is obtained when $\mathbf{r}$ randomly has $s=1$
nonzero elements;
* •
The desired sparse random matrix mentioned above can also obtain better
feature selection performance than Gaussian random matrices.
Note that, for better understanding, we first prove a relatively simple case
of $z_{i}\in\\{\pm\mu\\}$ in Lemma 3, and then in Lemma 4 generalize to a more
complicated case of $z_{i}$ distributed as in formula (6). The performance of
Gaussian matrices on $z_{i}\in\\{\pm\mu\\}$ is obtained in Lemma 5.
###### Lemma 3.
Let $\mathbf{r}=[r_{1},...,r_{d}]$ randomly have $1\leq s\leq d$ nonzero
elements taking values $\pm\sqrt{d/s}$ with equal probability, and
$\mathbf{z}=[z_{1},...,z_{d}]$ with elements being $\pm\mu$ equiprobably,
where $\mu$ is a positive constant. Given
$f(\mathbf{r},\mathbf{z})=|\langle\mathbf{r},\mathbf{z}\rangle|$, there are
three results regarding the expected value of $f(r_{i},z)$:
$\mathds{E}(f)=2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\lceil\frac{s}{2}\rceil
C_{s}^{\lceil\frac{s}{2}\rceil}$;
$\mathds{E}(f)|_{s=1}=\mu\sqrt{d}>\mathds{E}(f)|_{s>1}$;
$\mathop{\lim}\limits_{s\rightarrow\infty}\frac{1}{\sqrt{d}}\mathds{E}(f)\rightarrow\mu\sqrt{\frac{2}{\pi}}$.
###### Proof.
Please see Appendix A. ∎
Remark on Lemma 3: This lemma discloses that the best feature selection
performance is obtained, when only one feature element is sampled by each row
vector. In contrast, the performance tends to converge to a lower level as the
number of sampled feature elements increases. However, in practice the desired
sampling process is hard to be implemented due to the few knowledge of feature
location. As it will be detailed in the next section, what we can really
implement is to sample only one feature element with high probability. Note
that with the proof of this lemma, it can also be proved that if $s$ is odd,
$\mathds{E}(f)$ fast decreases to $\mu\sqrt{2d/\pi}$ with increasing $s$; in
contrast, if $s$ is even, $\mathds{E}(f)$ quickly increases towards
$\mu\sqrt{2d/\pi}$ as $s$ increases. But for arbitrary two adjacent $s$ larger
than 1, their average value on $\mathds{E}(f)$, namely
$(\mathds{E}(f)|_{s}+\mathds{E}(f)|_{s+1})/2$, is very close to
$\mu\sqrt{2d/\pi}$. For clarity, the values of $\mathds{E}(f)$ over varying
$s$ are calculated and shown in Figure 1, where instead of $\mathds{E}(f)$,
$\frac{1}{\mu\sqrt{d}}\mathds{E}(f)$ is described since only the varying $s$
is concerned. The specific character of $\mathds{E}(f)$ ensures that one can
still achieve better performance over others by sampling $s=1$ element with a
relative high probability, along with the occurrence of a sequence of $s$
taking consecutive values slightly larger than 1.
|
---|---
(a) | (b)
Figure 1: The process of $\frac{1}{\mu\sqrt{d}}\mathds{E}(f)$ converging to
$\sqrt{2/\pi}~{}(\approx 0.7979)$ with increasing $s$ is described in (a); and
in (b) the average value of two $\frac{1}{\mu\sqrt{d}}\mathds{E}(f)$ with
adjacent $s~{}(>1)$, namely
$\frac{1}{2\mu\sqrt{d}}(\mathds{E}(f)|_{s}+\mathds{E}(f)|_{s+1})$, is approved
very close to $\sqrt{2/\pi}$. Note that $\mathds{E}(f)$ is calculated with the
formula provided in Lemma 3.
###### Lemma 4.
Let $\mathbf{r}=[r_{1},...,r_{d}]$ randomly have $1\leq s\leq d$ nonzero
elements taking values $\pm\sqrt{d/s}$ with equal probability, and
$\mathbf{z}=[z_{1},...,z_{d}]$ with elements distributed as in formula (6).
Given $f(\mathbf{r},\mathbf{z})=|\langle\mathbf{r},\mathbf{z}\rangle|$, it is
derived that:
$\mathds{E}(f)|_{s=1}>\mathds{E}(f)|_{s>1}$
if
$(\frac{9}{8})^{\frac{3}{2}}[\sqrt{\frac{2}{\pi}}+(1+\frac{\sqrt{3}}{4})\frac{2}{\pi}(\frac{\mu}{\sigma})^{-1}]+2\Phi(-\frac{\mu}{\sigma})\leq
1$.
###### Proof.
Please see Appendix B. ∎
Remark on Lemma 4: This lemma expands Lemma 3 to a more general case where
$|z_{i}|$ is allowed to vary in some range. In other words, there is an upper
bound on $\frac{\sigma}{\mu}$ for $\mathds{E}(f)|_{s=1}>\mathds{E}(f)|_{s>1}$,
since $\Phi(-\frac{\mu}{\sigma})$ decreases monotonically with respect to
$\frac{\mu}{\sigma}$. Clearly the larger upper bound for $\frac{\sigma}{\mu}$
allows more variation of $|z_{i}|$. In practice the real upper bound should be
larger than that we have derived as a sufficient condition in this lemma.
###### Lemma 5.
Let $\mathbf{r}=[r_{1},...,r_{d}]$ have elements i.i.d drawn from $N(0,1)$,
and $\mathbf{z}=[z_{1},...,z_{d}]$ with elements being $\pm\mu$ equiprobably,
where $\mu$ is a positive constant. Given
$f(\mathbf{r},\mathbf{z})=|\langle\mathbf{r},\mathbf{z}\rangle|$, its expected
value $\mathds{E}(f)=\mu\sqrt{\frac{2d}{\pi}}$.
###### Proof.
Please see Appendix C. ∎
Remark on Lemma 5: Comparing this lemma with Lemma 3, clearly the row vector
with Gaussian distribution shares the same feature selection level with sparse
row vector with a relatively large $s$. This explains why in practice the
sparse random matrices usually can present comparable classification
performance with Gaussian matrix. More importantly, it implies that the
sparsest sampling process provided in Lemma 3 should outperform Gaussian
matrix on feature selection.
## IV Proposed sparse random matrix
The lemmas of the former section have proved that the best feature selection
performance can be obtained, if only one feature element is sampled by each
row vector of random matrix. It is now interesting to know if the condition
above can be satisfied in the practical setting, where the high-dimensional
data consists of both feature elements and redundant elements, namely
$\mathbf{v}=[\mathbf{v}^{f}~{}\mathbf{v}^{r}]$ and
$\mathbf{w}=[\mathbf{w}^{f}~{}\mathbf{w}^{r}]$. According to the theoretical
condition mentioned above, it is known that the row vector
$\mathbf{r}=[\mathbf{r}^{f}~{}\mathbf{r}^{r}]$ can obtain the best feature
selection, only when $||\mathbf{r}^{f}||_{0}=1$, where the quasi-norm
$\ell_{0}$ counts the number of nonzero elements in $\mathbf{r}^{f}$. Let
$\mathbf{r}^{f}\in\mathds{R}^{d_{f}}$, and
$\mathbf{r}^{r}\in\mathds{R}^{d_{r}}$, where $d=d_{f}+d_{r}$. Then the desired
row vector should have $d/d_{f}$ uniformly distributed nonzero elements such
that $\mathds{E}(||\mathbf{r}^{f}||_{0})=1$. However, in practice the desired
distribution for row vectors is often hard to be determined, since for a real
dataset the number of feature elements is usually unknown.
In this sense, we are motivated to propose a general distribution for the
matrix elements, such that $||\mathbf{r}^{f}||_{0}=1$ holds with high
probability in the setting where the feature distribution is unknown. In other
words, the random matrix should hold the distribution maximizing the ratio
$\text{Pr}(||\mathbf{r}^{f}||_{0}=1)/\text{Pr}(||\mathbf{r}^{f}||_{0}\in\\{2,3,...,d_{f}\\})$.
In practice, the desired distribution implies that the random matrix has
exactly one nonzero position per column, which can be simply derived as below.
Assume a random matrix $\mathbf{R}\in\mathbb{R}^{k\times d}$ randomly holding
$1\leq s^{\prime}\leq k$ nonzero elements per _column_ , equivalently
$s^{\prime}d/k$ nonzero elements per _row_ , then one can derive that
$\displaystyle\text{Pr}(||\mathbf{r}^{f}||_{0}=1)/\text{Pr}(||\mathbf{r}^{f}||_{0}\in\\{2,3,...,d_{f}\\})$
(7)
$\displaystyle=\frac{\text{Pr}(||\mathbf{r}^{f}||_{0}=1)}{1-\text{Pr}(||\mathbf{r}^{f}||_{0}=0)-\text{Pr}(||\mathbf{r}^{f}||_{0}=1)}$
$\displaystyle=\frac{C_{d_{f}}^{1}C_{d_{r}}^{s^{\prime}d/k-1}}{C_{d}^{s^{\prime}d/k}-C_{d_{r}}^{s^{\prime}d/k}-C_{d_{f}}^{1}C_{d_{r}}^{s^{\prime}d/k-1}}$
$\displaystyle=\frac{d_{f}d_{r}!}{\frac{d!(d_{r}-s^{\prime}d/k+1)!}{s^{\prime}d/k(d-s^{\prime}d/k)!}-\frac{d_{r}!(d_{r}-s^{\prime}d/k+1)}{s^{\prime}d/k}-d_{f}d_{r}!}$
From the last equation in formula (7), it can be observed that the increasing
$s^{\prime}d/k$ will reduce the value of formula (7). In order to maximize the
value, we have to set $s^{\prime}=1$. This indicates that the desired random
matrix has only one nonzero element per column.
The proposed random matrix with exactly one nonzero element per column
presents two obvious advantages, as detailed below.
* •
In complexity, the proposed matrix clearly presents much higher sparsity than
existing random projection matrices. Note that, theoretically the very sparse
random matrix with $q=\sqrt{d}$ [8] has higher sparsity than the proposed
matrix when $k<\sqrt{d}$. However, in practice the case $k<\sqrt{d}$ is
usually not of practical interest, due to the weak performance caused by large
compression rate $d/k$ ($>\sqrt{d}$).
* •
In performance, it can be derived that the proposed matrix outperforms other
more dense matrices, if the projection dimension $k$ is not much smaller than
the number $d_{f}$ of feature elements included in the high-dimensional
vector. To be specific, from Figure 1, it can be observed that the dense
matrices with column weight $s^{\prime}>1$ share comparable feature selection
performance, because as $s^{\prime}$ increases they tend to sample more than
one feature element (namely $||\mathbf{r}^{f}||_{0}>1$) with higher
probability. Then the proposed matrix with $s^{\prime}=1$ will present better
performance than them, if $k$ ensures $||\mathbf{r}^{f}||_{0}=1$ with high
probability, or equivalently the ratio
$\text{Pr}(||\mathbf{r}^{f}||_{0}=1)/\text{Pr}(||\mathbf{r}^{f}||_{0}\in\\{2,3,...,d_{f}\\})$
being relatively large. As shown in formula (7), the condition above can be
better satisfied, as $k$ increases. Inversely, as $k$ decreases, the feature
selection advantage of the proposed matrix will degrade. Recall that the
proposed matrix is weaker than other more dense matrices on distance
preservation, as demonstrated in section II-B. This means that the proposed
matrix will perform worse than others when its feature selection advantage is
not obvious. In other words, there should exist a lower bound for $k$ to
ensure the performance advantage of the proposed matrix, which is also
verified in the following experiments. It can be roughly estimated that the
lower bound of $k$ should be on the order of $d_{f}$, since for the proposed
matrix with column weight $s^{\prime}=1$, the $k=d_{f}$ leads to
$\mathds{E}(||\mathbf{r}^{f}||_{0})=d/k\times d_{f}/d=1$. In practice, the
performance advantage seemingly can be maintained for a relatively small
$k(<d_{f})$. For instance, in the following experiments on synthetic data, the
lower bound of $k$ is as small as $d_{f}/20$. This phenomenon can be explained
by the fact that to obtain performance advantage, the probability
$\text{Pr}(||\mathbf{r}^{f}||_{0}=1)$ is only required to be relatively large
rather than to be equal to 1, as demonstrated in the remark on Lemma 3.
## V Experiments
### V-A Setup
This section verifies the feature selection advantage of the proposed
currently sparest matrix (StM) over other popular matrices, by conducting
binary classification on both synthetic data and real data. Here the synthetic
data with labeled feature elements is provided to specially observe the
relation between the projection dimension and feature number, as well as the
impact of redundant elements. The real data involves three typical datasets in
the area of dimensionality reduction: face image, DNA microarray and text
document. As for the binary classifier, the classical support vector machine
(SVM) based on Euclidean distance is adopted. For comparison, we test three
popular random matrices: Gaussian random matrix (GM), sparse random matrix
(SM) as in formula (3) with $q=3$ [7] and very sparse random matrix (VSM) with
$q=\sqrt{d}$ [8].
The simulation parameters are introduced as follows. It is known that the
repeated random projection tends to improve the feature selection, so here
each classification decision is voted by performing 5 times the random
projection [13]. The classification accuracy at each projection dimension $k$
is derived by taking the average of 100000 simulation runs. In each
simulation, four matrices are tested with the same samples. The projection
dimension $k$ decreases uniformly from the high dimension $d$. Moreover, it is
necessary to note that, for some datasets containing more than two classes of
samples, the SVM classifier randomly selects two classes to conduct binary
classification in each simulation. For each class of data, one half of samples
are randomly selected for training, and the rest for testing.
TABLE I: Classification accuracies on the synthetic data which have $d=2000$ and redundant elements suffering from three different varying levels $\sigma_{r}$. The best performance is highlighted in bold. The lower bound of projection dimension $k$ that ensures the proposal outperforming others in all datasets is highlighted in bold as well. Recall that the acronyms GM, SM, VSM and StM represent Gaussian random matrix, sparse random matrix with $q=3$, very sparse rand matrix with $q=\sqrt{d}$, and the proposed sparsest random matrix, respectively. | $k$ | 50 | 100 | 200 | 400 | 600 | 800 | 1000 | 1500 | 2000
---|---|---|---|---|---|---|---|---|---|---
$\sigma_{r}=8$ | GM | 70.44 | 67.93 | 84.23 | 93.31 | 95.93 | 97.17 | 97.71 | 98.35 | 98.74
SM | 70.65 | 67.90 | 84.43 | 93.03 | 95.97 | 96.86 | 97.78 | 98.36 | 98.80
VSM | 70.55 | 68.05 | 84.46 | 93.19 | 96.00 | 96.99 | 97.68 | 98.38 | 98.76
StM | 70.27 | 68.09 | 84.66 | 94.22 | 97.11 | 98.03 | 98.67 | 99.37 | 99.57
$\sigma_{r}=12$ | GM | 64.89 | 63.06 | 76.08 | 85.04 | 88.46 | 90.21 | 91.16 | 92.68 | 93.32
SM | 64.67 | 62.66 | 75.85 | 85.03 | 88.30 | 90.09 | 91.21 | 92.70 | 93.30
VSM | 65.17 | 62.95 | 76.12 | 85.14 | 88.80 | 90.46 | 91.37 | 92.88 | 93.64
StM | 64.85 | 63.00 | 76.82 | 88.41 | 93.51 | 96.12 | 97.59 | 99.13 | 99.68
$\sigma_{r}=16$ | GM | 60.90 | 59.42 | 70.13 | 78.26 | 81.70 | 83.82 | 84.74 | 86.50 | 87.49
SM | 60.86 | 59.58 | 69.93 | 78.04 | 81.66 | 83.85 | 84.79 | 86.55 | 87.39
VSM | 60.98 | 59.87 | 70.27 | 78.49 | 81.98 | 84.36 | 85.27 | 86.98 | 87.81
StM | 61.09 | 59.29 | 71.58 | 84.56 | 91.65 | 95.50 | 97.24 | 98.91 | 99.30
### V-B Synthetic data experiments
#### V-B1 Data generation
The synthetic data is developed to evaluate the two factors as follows:
* •
the relation between the lower bound of projection dimension $k$ and the
feature dimension $d_{f}$;
* •
the negative impact of redundant elements, which are ideally assumed to be
zero in the previous theoretical proofs.
To this end, two classes of synthetic data with $d_{f}$ feature elements and
$d-d_{f}$ redundant elements are generated in two steps:
* •
randomly build a vector $\tilde{\mathbf{v}}\in\\{\pm 1\\}^{d}$, then define a
vector $\tilde{\mathbf{w}}$ distributed as $\tilde{w}_{i}=-\tilde{v}_{i}$, if
$1\leq i\leq d_{f}$, and $\tilde{w}_{i}=\tilde{v}_{i}$, if $d_{f}<i\leq d$;
* •
generate two classes of datasets $\mathcal{V}$ and $\mathcal{W}$ by i.i.d
sampling $v^{f}_{i}\in N(\tilde{v}_{i},\sigma_{f}^{2})$ and $w^{f}_{i}\in
N(\tilde{w}_{i},\sigma_{f}^{2})$, if $1\leq i\leq d_{f}$; and $v^{r}_{i}\in
N(\tilde{v}_{i},\sigma_{r}^{2})$ and $w^{r}_{i}\in
N(\tilde{w}_{i},\sigma_{r}^{2})$, if $d_{f}<i\leq d$.
Subsequently, the distributions on pointwise distance can be approximately
derived as $|v_{i}^{f}-w_{i}^{f}|\in N(2,2\sigma_{f}^{2})$ for feature
elements and $(v_{i}^{r}-w_{i}^{r})\in N(0,2\sigma_{r}^{2})$ for redundant
elements, respectively. To be close to reality, we introduce some
unreliability for feature elements and redundant elements by adopting
relatively large variances. Precisely, in the simulation $\sigma_{f}$ is fixed
to 8 and $\sigma_{r}$ varies in the set $\\{8,12,16\\}$. Note that, the
probability of $(v_{i}^{r}-w_{i}^{r})$ converging to zero will decrease as
$\sigma_{r}$ increases. Thus the increasing $\sigma_{r}$ will be a challenge
for our previous theoretical conjecture derived on the assumption of
$(v_{i}^{r}-w_{i}^{r})=0$. As for the size of the dataset, the data dimension
$d$ is set to 2000, and the feature dimension $d_{f}=1000$. Each dataset
consists of 100 randomly generated samples.
#### V-B2 Results
Table I shows the classification performance of four types of matrices over
evenly varying projection dimension $k$. It is clear that the proposal always
outperforms others, as $k>200$ (equivalently, the compression ratio
$k/d>0.1$). This result exposes two positive clues. First, the proposed matrix
preserves obvious advantage over others, even when $k$ is relatively small,
for instance, $k/d_{f}$ is allowed to be as small as 1/20 when $\sigma_{r}=8$.
Second, with the interference of redundant elements, the proposed matrix still
outperforms others, which implies that the previous theoretical result is also
applicable to the real case where the redundant elements cannot be simply
neglected.
### V-C Real data experiments
Three types of representative high-dimensional datasets are tested for random
projection over evenly varying projection dimension $k$. The datasets are
first briefly introduced, and then the results are illustrated and analyzed.
Note that, the simulation is developed to compare the feature selection
performance of different random projections, rather than to obtain the best
performance. So to reduce the simulation load, the original high-dimensional
data is uniformly downsampled to a relatively low dimension. Precisely, the
face image, DNA, and text are reduced to the dimensions 1200, 2000 and 3000,
respectively. Note that, in terms of JL lemma, the original high dimension
allows to be reduced to arbitrary values (not limited to 1200, 2000 or 3000),
since theoretically the distance preservation of random projection is
independent of the size of high-dimensional data [7].
#### V-C1 Datasets
* •
Face image
* –
AR [14] : As in [15], a subset of 2600 frontal faces from 50 males and 50
females are examined. For some persons, the faces were taken at different
times, varying the lighting, facial expressions (open/closed eyes, smiling/not
smiling) and facial details (glasses/no glasses). There are 6 faces with dark
glasses and 6 faces partially disguised by scarfs among 26 faces per person.
* –
Extended Yale B [16, 17]: This dataset includes about 2414 frontal faces of 38
persons, which suffer varying illumination changes.
* –
FERET [18]: This dataset consists of more than 10000 faces from more than 1000
persons taken in largely varying circumstances. The database is further
divided into several sets which are formed for different evaluations. Here we
evaluate the 1984 _frontal_ faces of 992 persons each with 2 faces separately
extracted from sets _fa_ and _fb_.
* –
GTF [19]: In this dataset, 750 images from 50 persons were captured at
different scales and orientations under variations in illumination and
expression. So the cropped faces suffer from serious pose variation.
* –
ORL [20]: It contains 40 persons each with 10 faces. Besides slightly varying
lighting and expressions, the faces also undergo slight changes on pose.
TABLE II: Classification accuracies on five face datasets with dimension $d=1200$. For each projection dimension $k$, the best performance is highlighted in bold. The lower bound of projection dimension $k$ that ensures the proposal outperforming others in all datasets is highlighted in bold as well. Recall that the acronyms GM, SM, VSM and StM represent Gaussian random matrix, sparse random matrix with $q=3$, very sparse random matrix with $q=\sqrt{d}$, and the proposed sparsest random matrix, respectively. | $k$ | 30 | 60 | 120 | 240 | 360 | 480 | 600
---|---|---|---|---|---|---|---|---
AR | GM | 98.67 | 99.04 | 99.19 | 99.24 | 99.30 | 99.28 | 99.33
SM | 98.58 | 99.04 | 99.21 | 99.25 | 99.31 | 99.30 | 99.32
VSM | 98.62 | 99.07 | 99.20 | 99.27 | 99.30 | 99.31 | 99.34
StM | 98.64 | 99.10 | 99.24 | 99.35 | 99.48 | 99.50 | 99.58
Ext-YaleB | GM | 97.10 | 98.06 | 98.39 | 98.49 | 98.48 | 98.45 | 98.47
SM | 97.00 | 98.05 | 98.37 | 98.49 | 98.48 | 98.45 | 98.47
VSM | 97.12 | 98.05 | 98.36 | 98.50 | 98.48 | 98.45 | 98.48
StM | 97.15 | 98.06 | 98.40 | 98.54 | 98.54 | 98.57 | 98.59
FERET | GM | 86.06 | 86.42 | 86.31 | 86.50 | 86.46 | 86.66 | 86.57
SM | 86.51 | 86.66 | 87.26 | 88.01 | 88.57 | 89.59 | 90.13
VSM | 87.21 | 87.61 | 89.34 | 91.14 | 92.31 | 93.75 | 93.81
StM | 87.11 | 88.74 | 92.04 | 95.38 | 96.90 | 97.47 | 97.47
GTF | GM | 96.67 | 97.48 | 97.84 | 98.06 | 98.09 | 98.10 | 98.16
SM | 96.63 | 97.52 | 97.85 | 98.06 | 98.09 | 98.13 | 98.16
VSM | 96.69 | 97.57 | 97.87 | 98.10 | 98.13 | 98.14 | 98.16
StM | 96.65 | 97.51 | 97.94 | 98.25 | 98.40 | 98.43 | 98.53
ORL | GM | 94.58 | 95.69 | 96.31 | 96.40 | 96.54 | 96.51 | 96.49
SM | 94.50 | 95.63 | 96.36 | 96.38 | 96.48 | 96.47 | 96.48
VSM | 94.60 | 95.77 | 96.33 | 96.35 | 96.53 | 96.55 | 96.46
StM | 94.64 | 95.75 | 96.43 | 96.68 | 96.90 | 97.04 | 97.05
* •
DNA microarray
* –
Colon [21]: This is a dataset consisting of 40 colon tumors and 22 normal
colon tissue samples. 2000 genes with highest intensity across the samples are
considered.
* –
ALML [22]: This dataset contains 25 samples taken from patients suffering from
acute myeloid leukemia (AML) and 47 samples from patients suffering from acute
lymphoblastic leukemia (ALL). Each sample is expressed with 7129 genes.
* –
Lung [23] : This dataset contains 86 lung tumor and 10 normal lung samples.
Each sample holds 7129 genes.
* •
Text document [24]111Publicly available at
http://www.cad.zju.edu.cn/home/dengcai/Data/TextData.html
* –
TDT2: The recently modified dataset includes 96 categories of total 10212
documents/samples. Each document is represented with vector of length 36771.
This paper adopts the first 19 categories each with more than 100 documents,
such that each category is tested with 100 randomly selected documents.
* –
20Newsgroups (version 1): There are 20 categories of 18774 documents in this
dataset. Each document has vector dimension 61188. Since the documents are not
equally distributed in the 20 categories, we randomly select 600 documents for
each category, which is nearly the maximum number we can assign to all
categories.
* –
RCV1: The original dataset contains 9625 documents each with 29992 distinct
words, corresponding to 4 categories with 2022, 2064, 2901, and 2638 documents
respectively. To reduce computation, this paper randomly selects only 1000
documents for each category.
#### V-C2 Results
TABLE III: Classification accuracies on three DNA datasets with dimension $d=2000$. For each projection dimension $k$, the best performance is highlighted in bold. The lower bound of projection dimension $k$ that ensures the proposal outperforming others in all datasets is highlighted in bold as well. Recall that the acronyms GM, SM, VSM and StM represent Gaussian random matrix, sparse random matrix with $q=3$, very sparse random matrix with $q=\sqrt{d}$, and the proposed sparsest random matrix, respectively. | $k$ | 50 | 100 | 200 | 400 | 600 | 800 | 1000 | 1500
---|---|---|---|---|---|---|---|---|---
Colon | GM | 77.16 | 77.15 | 77.29 | 77.28 | 77.46 | 77.40 | 77.35 | 77.55
SM | 77.23 | 77.18 | 77.16 | 77.36 | 77.42 | 77.42 | 77.39 | 77.54
VSM | 76.86 | 77.19 | 77.34 | 77.52 | 77.64 | 77.61 | 77.61 | 77.82
StM | 76.93 | 77.34 | 77.73 | 78.22 | 78.51 | 78.67 | 78.65 | 78.84
ALML | GM | 65.11 | 66.22 | 66.96 | 67.21 | 67.23 | 67.24 | 67.28 | 67.37
SM | 65.09 | 66.16 | 66.93 | 67.25 | 67.22 | 67.31 | 67.31 | 67.36
VSM | 64.93 | 67.32 | 68.52 | 69.01 | 69.15 | 69.16 | 69.25 | 69.33
StM | 65.07 | 68.38 | 70.43 | 71.39 | 71.75 | 71.87 | 72.00 | 72.11
Lung | GM | 98.74 | 98.80 | 98.91 | 98.96 | 98.95 | 98.96 | 98.95 | 98.97
SM | 98.71 | 98.80 | 98.92 | 98.97 | 98.96 | 98.98 | 98.97 | 98.97
VSM | 98.81 | 99.21 | 99.48 | 99.57 | 99.58 | 99.61 | 99.61 | 99.61
StM | 98.70 | 99.48 | 99.69 | 99.70 | 99.69 | 99.72 | 99.68 | 99.65
TABLE IV: Classification accuracies on three Text datasets with dimension $d=3000$. For each projection dimension $k$, the best performance is highlighted in bold. The lower bound of projection dimension $k$ that ensures the proposal outperforming others in all datasets is highlighted in bold as well. Recall that the acronyms GM, SM, VSM and StM represent Gaussian random matrix, sparse random matrix with $q=3$, very sparse random matrix with $q=\sqrt{d}$, and the proposed sparsest random matrix, respectively. | k | 150 | 300 | 600 | 900 | 1200 | 1500 | 2000
---|---|---|---|---|---|---|---|---
TDT2 | GM | 83.64 | 83.10 | 82.84 | 82.29 | 81.94 | 81.67 | 81.72
SM | 83.61 | 82.93 | 83.10 | 82.28 | 81.92 | 81.55 | 81.76
VSM | 82.59 | 82.55 | 82.72 | 82.20 | 81.74 | 81.47 | 81.78
StM | 82.52 | 83.15 | 84.06 | 83.58 | 83.42 | 82.95 | 83.35
Newsgroup | GM | 75.35 | 74.46 | 72.27 | 71.52 | 71.34 | 70.63 | 69.95
SM | 75.21 | 74.43 | 72.29 | 71.30 | 71.07 | 70.34 | 69.58
VSM | 74.84 | 73.47 | 70.22 | 69.21 | 69.28 | 68.28 | 68.04
StM | 74.94 | 74.20 | 72.34 | 71.54 | 71.53 | 70.46 | 70.00
RCV1 | GM | 85.85 | 86.20 | 81.65 | 78.98 | 78.22 | 78.21 | 78.21
SM | 86.05 | 86.19 | 81.53 | 79.08 | 78.23 | 78.14 | 78.19
VSM | 86.04 | 86.14 | 81.54 | 78.57 | 78.12 | 78.05 | 78.04
StM | 85.75 | 86.33 | 85.09 | 83.38 | 82.30 | 81.39 | 80.69
Tables II-IV illustrate the classification performance of four classes of
matrices on three typical high-dimensional data: face image, DNA microarray
and text document. It can be observed that, all results are consistent with
the theoretical conjecture stated in section IV. Precisely, the proposed
matrix will always perform better than others, if $k$ is larger than some
thresholds, i.e. $k>120$ (equivalently, the compression ratio $k/d>1/10$) for
all face image data, $k>100$ ($k/d>1/20$) for all DNA data, and $k>600$
($k/d>1/5$) for all text data. Note that, for some individual datasets, in
fact we can obtain smaller thresholds than the uniform thresholds described
above, which means that for these datasets, our performance advantage can be
ensured in lower projection dimension. It is worth noting that our performance
gain usually varies across the types of data. For most data, the gain is on
the level of around $1\%$, except for some special cases, for which the gain
can achieve as large as around $5\%$. Moreover, it should be noted that the
proposed matrix can still present comparable performance with others (usually
inferior to the best results not more than $1\%$), even as $k$ is smaller than
the lower threshold described above. This implies that regardless of the value
of $k$, the proposed matrix is always valuable due to its lower complexity and
competitive performance. In short, the extensive experiments on real data
sufficiently verifies the performance advantage of the theoretically proposed
random matrix, as well as the conjecture that the performance advantage holds
only when the projection dimension $k$ is large enough.
## VI Conclusion and Discussion
This paper has proved that random projection can achieve its best feature
selection performance, when only one feature element of high-dimensional data
is considered at each sampling. In practice, however, the number of feature
elements is usually unknown, and so the aforementioned best sampling process
is hard to be implemented. Based on the principle of achieving the best
sampling process with high probability, we practically propose a class of
sparse random matrices with exactly one nonzero element per column, which is
expected to outperform other more dense random projection matrices, if the
projection dimension is not much smaller than the number of feature elements.
Recall that for the possibility of theoretical analysis, we have typically
assumed that the elements of high-dimensional data are mutually independent,
which obviously cannot be well satisfied by the real data, especially the
redundant elements. Although the impact of redundant elements is reasonably
avoided in our analysis, we cannot ensure that all analyzed feature elements
are exactly independent in practice. This defect might affect the
applicability of our theoretical proposal to some extent, whereas empirically
the negative impact seems to be negligible, as proved by the experiments on
synthetic data. In order to validate the feasibility of the theoretical
proposal, extensive classification experiments are conducted on various real
data, including face image, DNA microarray and text document. As it is
expected, the proposed random matrix shows better performance than other more
dense matrices, as the projection dimension is sufficiently large; otherwise,
it presents comparable performance with others. This result suggests that for
random projection applied to the task of classification, the proposed
currently sparsest random matrix is much more attractive than other more dense
random matrices in terms of both complexity and performance.
## Appendix A.
Proof of Lemma 3
###### Proof.
Due to the sparsity of $\mathbf{r}$ and the symmetric property of both $r_{j}$
and $z_{j}$, the function $f(\mathbf{r},\mathbf{z})$ can be equivalently
transformed to a simpler form, that is
$f(x)=\mu\sqrt{\frac{d}{s}}|\sum_{i=1}^{i=s}x_{i}|$ with $x_{i}$ being $\pm 1$
equiprobably. With the simplified form, three results of this lemma are
sequentially proved below.
* 1)
First, it can be easily derived that
$\mathds{E}(f(x))=\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=1}^{s}(C_{s}^{i}|s-2i|)$
then the solution to $\mathds{E}(f(x))$ turns to calculating
$\sum_{i=0}^{s}(C_{s}^{i}|s-2i|)$, which can be deduced as
$\sum_{i=0}^{s}(C_{s}^{i}|s-2i|)=\left\\{\begin{array}[]{cl}2sC_{s-1}^{\frac{s}{2}-1}&if~{}s~{}is~{}even\\\\[5.0pt]
2sC_{s-1}^{\frac{s-1}{2}}&if~{}s~{}is~{}odd\\\ \end{array}\right.$
by summing the piecewise function
$C_{s}^{i}|s-2i|=\left\\{\begin{array}[]{ll}sC_{s-1}^{0}&if~{}i=0\\\\[6.0pt]
sC_{s-1}^{s-i-1}-sC_{s-1}^{i-1}&if~{}1\leq i\leq\frac{s}{2}\\\\[6.0pt]
sC_{s-1}^{i-1}-sC_{s-1}^{s-i-1}&if~{}\frac{s}{2}<i<s\\\\[6.0pt]
sC_{s-1}^{s-1}&if~{}i=s\\\ \end{array}\right.$
Further, with $C_{s-1}^{i-1}=\frac{i}{s}C_{s}^{i}$, it can be deduced that
$\sum_{i=0}^{s}(C_{s}^{i}|s-2i|)=2\lceil\frac{s}{2}\rceil
C_{s}^{\lceil\frac{s}{2}\rceil}$
Then the fist result is obtained as
$\mathds{E}(f)=2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\lceil\frac{s}{2}\rceil
C_{s}^{\lceil\frac{s}{2}\rceil}$
* 2)
Following the proof above, it is clear that
$\mathds{E}(f(x))|_{s=1}=f(x)|_{s=1}=\mu\sqrt{d}$. As for
$\mathds{E}(f(x))|_{s>1}$, it is evaluated under two cases:
* –
if $s$ is odd,
$\frac{\mathds{E}(f(x))|_{s}}{\mathds{E}(f(x))|_{s-2}}=\frac{\frac{2}{\sqrt{s}}\frac{1}{2^{s}}\frac{s+1}{2}C_{s}^{\frac{s+1}{2}}}{\frac{2}{\sqrt{s-2}}\frac{1}{2^{s-2}}\frac{s-1}{2}C_{s-2}^{\frac{s-1}{2}}}=\frac{\sqrt{s(s-2)}}{s-1}<1$
namely, $\mathds{E}(f(x))$ decreases monotonically with respect to $s$.
Clearly, in this case $\mathds{E}(f(x))|_{s=1}>\mathds{E}(f(x))|_{s>1}$;
* –
if $s$ is even,
$\frac{\mathds{E}(f(x))|_{s}}{\mathds{E}(f(x))|_{s-1}}=\frac{\frac{2}{\sqrt{s}}\frac{1}{2^{s}}\frac{s}{2}C_{s}^{\frac{s}{2}}}{\frac{2}{\sqrt{s-1}}\frac{1}{2^{s-1}}\frac{s}{2}C_{s-1}^{\frac{s}{2}}}=\sqrt{\frac{s-1}{s}}<1$
which means $\mathds{E}(f(x))|_{s=1}>\mathds{E}(f(x))|_{s>1}$, since $s-1$ is
odd number for which $\mathds{E}(f(x))$ monotonically decreases.
Therefore the proof of the second result is completed.
* 3)
The proof of the third result is developed by employing Stirling’s
approximation [25]
$s!=\sqrt{2\pi
s}(\frac{s}{e})^{s}e^{\lambda_{s}},~{}~{}~{}1/(12s+1)<\lambda_{s}<1/(12s).$
Precisely, with the formula of $\mathds{E}(f(x))$, it can be deduced that
* –
if $s$ is even,
$\mathds{E}(f(x))=\mu\sqrt{ds}\frac{1}{2^{s}}\frac{s!}{\frac{s}{2}!\frac{s}{2}!}=\mu\sqrt{\frac{2d}{\pi}}e^{\lambda_{s}-2\lambda_{\frac{s}{2}}}$
* –
if $s$ is odd,
$\mathds{E}(f(x))=\mu\sqrt{d}\frac{s+1}{\sqrt{s}}\frac{1}{2^{s}}\frac{s!}{\frac{s+1}{2}!\frac{s-1}{2}!}=\mu\sqrt{\frac{2d}{\pi}}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}$
Clearly
$\mathop{\lim}\limits_{s\rightarrow\infty}\frac{1}{\sqrt{d}}\mathds{E}(f(x))\rightarrow\mu\sqrt{\frac{2}{\pi}}$
holds, whenever $s$ is even or odd.
∎
## Appendix B.
Proof of Lemma 4
###### Proof.
Due to the sparsity of $\mathbf{r}$ and the symmetric property of both $r_{j}$
and $z_{j}$, it is easy to derive that
$f(\mathbf{r},\mathbf{z})=|\langle\mathbf{r},\mathbf{z}\rangle|=\sqrt{\frac{d}{s}}|\sum_{j=1}^{s}z_{j}|$.
This simplified formula will be studied in the following proof. To present a
readable proof, we first review the distribution shown in formula (6)
$z_{j}\sim\left\\{\begin{array}[]{lr}N(\mu,\sigma)&\text{with
probability}~{}1/2\\\ N(-\mu,\sigma)&\text{ with probability}~{}1/2\\\
\end{array}\right.$
where for $x\in N(\mu,\sigma)$, $\text{Pr}(x>0)=1-\epsilon$,
$\epsilon=\Phi(-\frac{\mu}{\sigma})$ is a tiny positive number. For notational
simplicity, the subscript of random variable $z_{j}$ is dropped in the
following proof. To ease the proof of the lemma, we first need to derive the
expected value of $|x|$ with $x\sim N(\mu,\sigma^{2})$:
$\displaystyle\mathds{E}(|x|)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{|x|}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx$
$\displaystyle=\int_{-\infty}^{0}\frac{-x}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx+\int_{0}^{\infty}\frac{x}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx$
$\displaystyle=-\int_{-\infty}^{0}\frac{x-\mu}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx+\int_{0}^{\infty}\frac{x-\mu}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx$
$\displaystyle+\mu\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx-\mu\int_{-\infty}^{0}\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}dx$
$\displaystyle=\frac{\sigma}{\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}|_{-\infty}^{0}-\frac{\sigma}{\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}|^{\infty}_{0}+\mu\text{Pr}(x>0)-\mu\text{Pr}(x<0)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\sigma
e^{-\frac{\mu^{2}}{2\sigma^{2}}}+\mu(1-2\text{Pr}(x<0))$
$\displaystyle=\sqrt{\frac{2}{\pi}}\sigma
e^{-\frac{\mu^{2}}{2\sigma^{2}}}+\mu(1-2\Phi(-\frac{\mu}{\sigma}))$
which will be used many a time in the following proof. Then the proof of this
lemma is separated into two parts as follows.
* 1)
This part presents the expected value of $f(r_{i},z)$ for the cases $s=1$ and
$s>1$.
* –
if $s=1$, $f(\mathbf{r},\mathbf{z})=\sqrt{d}|z|$; with the the probability
density function of $z$:
$p(z)=\frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(z-\mu)^{2}}{2\sigma^{2}}}+\frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(z+\mu)^{2}}{2\sigma^{2}}}$
one can derive that
$\displaystyle\mathds{E}(|z|)$
$\displaystyle=\int_{-\infty}^{\infty}|z|p(z)d_{z}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\frac{|z|}{\sqrt{2\pi}\sigma}e^{\frac{-(z-\mu)^{2}}{2\sigma^{2}}}dz+\frac{1}{2}\int_{-\infty}^{\infty}\frac{|z|}{\sqrt{2\pi}\sigma}e^{\frac{-(z+\mu)^{2}}{2\sigma^{2}}}dz$
with the previous result on $\mathds{E}(|x|)$, it is further deduced that
$\mathds{E}(|z|)=\sqrt{\frac{2}{\pi}}\sigma
e^{-\frac{\mu^{2}}{2\sigma^{2}}}+\mu(1-2\Phi(-\frac{\mu}{\sigma}))$
Recall that $\Phi(-\frac{\mu}{\sigma})=\epsilon$, so
$\mathds{E}(f)=\sqrt{d}\mathds{E}(|z|)=\sqrt{\frac{2d}{\pi}}\sigma_{\mu}e^{-\frac{\mu^{2}}{2\sigma^{2}}}+\mu\sqrt{d}(1-2\Phi(-\frac{\mu}{\sigma}))\approx\mu\sqrt{d}$
if $\epsilon$ is tiny enough as illustrated in formula (6).
* –
if $s>1$, $f(\mathbf{r},\mathbf{z})=\sqrt{\frac{d}{s}}|\sum_{j=1}^{s}z|$; let
$t=\sum_{j=1}^{s}z$, then according to the symmetric distribution of $z$, $t$
holds $s+1$ different distributions:
$t\sim N((s-2i)\mu,s\sigma^{2})~{}\text{with
probability}~{}\frac{1}{2^{s}}C_{s}^{i}$
where $0\leq i\leq s$ denotes the number of $z$ drawn from
$N(-\mu,\sigma^{2})$. Then the PDF of $t$ can be described as
$p(t)=\frac{1}{2^{s}}\sum_{i=0}^{s}C_{s}^{i}\frac{1}{\sqrt{2\pi
s}\sigma}e^{\frac{-(t-(s-2i)\mu)^{2}}{2s\sigma^{2}}}$
then,
$\displaystyle\mathds{E}(|t|)$
$\displaystyle=\int_{-\infty}^{\infty}|t|p(t)dt$
$\displaystyle=\frac{1}{2^{s}}\sum_{i=0}^{s}C_{s}^{i}\int_{-\infty}^{\infty}|t|\frac{1}{\sqrt{2\pi
s}\sigma}e^{\frac{-(t-(s-2i)\mu)^{2}}{2s\sigma^{2}}}dt$
$\displaystyle=\frac{1}{2^{s}}\sum_{i=0}^{s}C_{s}^{i}\\{\sqrt{\frac{2s}{\pi}}\sigma
e^{\frac{-(s-2i)^{2}\mu^{2}}{2s\sigma^{2}}}+\mu|s-2i|[1-2\Phi(\frac{-|s-2i|\mu}{\sqrt{s}\sigma})]\\}$
subsequently, the expected value of $f(r_{i},z)$ can be expressed as
$\displaystyle\mathds{E}(f)$
$\displaystyle=\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=0}^{s}(C_{s}^{i}|s-2i|)+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=0}^{s}C_{s}^{i}e^{\frac{-(s-2i)^{2}\mu^{2}}{2s\sigma^{2}}}$
$\displaystyle-2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=0}^{s}[C_{s}^{i}|s-2i|\Phi(\frac{-|s-2i|\mu}{\sqrt{s}\sigma})]$
* 2)
This part derives the upper bound of the aforementioned
$\mathds{E}(f)|_{s>1}$. For simpler expression, the three factors of above
expression for $\mathds{E}(f)|_{s>1}$ are sequentially represented by $f_{1}$,
$f_{2}$ and $f_{3}$, and then are analyzed, respectively.
* –
for
$f_{1}=\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=0}^{s}(C_{s}^{i}|s-2i|)$,
it can be rewritten as
$f_{1}=2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}C_{s}^{\lceil\frac{s}{2}\rceil}\lceil\frac{s}{2}\rceil$
* –
for
$f_{2}=\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=0}^{s}C_{s}^{i}e^{\frac{-(s-2i)^{2}\mu^{2}}{2s\sigma^{2}}}$,
first, we can bound
$\left\\{\begin{array}[]{ll}e^{\frac{-(s-2i)^{2}\mu^{2}}{2s\sigma^{2}}}<\text{exp}(-\frac{\mu^{2}}{\sigma^{2}})&\text{if}~{}i<\alpha~{}\text{or}~{}i>\alpha\\\\[6.0pt]
e^{\frac{-(s-2i)^{2}\mu^{2}}{2s\sigma^{2}}}\leq 1&\text{if}~{}\alpha\leq i\leq
s-\alpha\\\ \end{array}\right.$
where $\alpha=\lceil\frac{s-\sqrt{s}}{2}\rceil$. Take it into $f_{2}$,
$\displaystyle f_{2}$
$\displaystyle<\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=0}^{\alpha-1}C_{s}^{i}e^{\frac{-\mu^{2}}{\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=s-\alpha+1}^{s}C_{s}^{i}e^{\frac{-\mu^{2}}{2\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=\alpha}^{s-\alpha}C_{s}^{i}$
$\displaystyle<\sigma\sqrt{\frac{2d}{\pi}}e^{\frac{-\mu^{2}}{\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sum_{i=\alpha}^{s-\alpha}C_{s}^{i}$
Since $C_{s}^{i}\leq C_{s}^{\lceil s/2\rceil}$,
$\displaystyle f_{2}$
$\displaystyle<\sigma\sqrt{\frac{2d}{\pi}}e^{\frac{-\mu^{2}}{\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}(\lfloor\sqrt{s}\rfloor+1)C_{s}^{\lceil
s/2\rceil}$
$\displaystyle\leq\sigma\sqrt{\frac{2d}{\pi}}e^{\frac{-\mu^{2}}{\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\sqrt{s}C_{s}^{\lceil
s/2\rceil}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}C_{s}^{\lceil s/2\rceil}$
$\displaystyle\leq\sigma\sqrt{\frac{2d}{\pi}}e^{\frac{-\mu^{2}}{\sigma^{2}}}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}\frac{2}{\sqrt{s}}C_{s}^{\lceil
s/2\rceil}{\lceil\frac{s}{2}\rceil}+\sigma\sqrt{\frac{2d}{\pi}}\frac{1}{2^{s}}C_{s}^{\lceil
s/2\rceil}$
with Stirling’s approximation,
$f_{2}<\left\\{\begin{array}[]{ll}\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}+\sqrt{d}\frac{2}{\pi}\sigma
e^{\lambda_{s}-2\lambda_{s/2}}+\sqrt{\frac{d}{s}}\frac{2}{\pi}\sigma
e^{\lambda_{s}-2\lambda_{s/2}}&\text{if $s$ is even}\\\\[10.0pt]
\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}+\sqrt{{d}}\frac{2\sigma}{\pi}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}\\\
+\sqrt{{d}}\frac{2\sigma}{\pi}\frac{\sqrt{s}}{s+1}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}&\text{if
$s$ is odd}\end{array}\right.$
* –
for
$f_{3}=-2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=0}^{s}[C_{s}^{i}|s-2i|\Phi(\frac{-|s-2i|\mu}{\sqrt{s}\sigma})]$,
with the previous defined $\alpha$,
$\displaystyle f_{3}$
$\displaystyle\leq-2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=\alpha}^{s-\alpha}[C_{s}^{i}|s-2i|\Phi(\frac{-|s-2i|\mu}{\sqrt{s}\sigma})]$
$\displaystyle\leq-2\mu\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=\alpha}^{s-\alpha}[C_{s}^{i}|s-2i|\Phi(\frac{-\mu}{\sigma})]$
$\displaystyle=-2\mu\epsilon\sqrt{\frac{d}{s}}\frac{1}{2^{s}}\sum_{i=\alpha}^{s-\alpha}[C_{s}^{i}|s-2i|]$
$\displaystyle=-2\mu\epsilon\sqrt{\frac{d}{s}}\frac{1}{2^{s}}(2sC_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-2sC_{s-1}^{\alpha-1})$
$\displaystyle=-4\mu\epsilon\sqrt{ds}\frac{1}{2^{s}}(C_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-C_{s-1}^{\alpha-1})$
$\displaystyle\leq 0$
finally, we can further deduce that
$\displaystyle\mathds{E}(f)|_{s>1}=f_{1}+f_{2}+f_{3}$
$\displaystyle<\left\\{\begin{array}[]{ll}2\mu\frac{1}{2^{s}}\sqrt{\frac{d}{s}}C_{s}^{\lceil\frac{s}{2}\rceil}+\frac{2\sigma}{\pi}\sqrt{d}e^{\lambda_{s}-2\lambda_{\frac{s}{2}}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}+\sqrt{\frac{d}{s}}\frac{2}{\pi}\sigma
e^{\lambda_{s}-2\lambda_{s/2}}\\\
-4\mu\epsilon\sqrt{ds}\frac{1}{2^{s}}(C_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-C_{s-1}^{\alpha-1})&\text{if
$s$ is even}\\\\[12.0pt]
2\mu\frac{1}{2^{s}}\sqrt{\frac{d}{s}}C_{s}^{\lceil\frac{s}{2}\rceil}+\frac{2\sigma}{\pi}\sqrt{d}\frac{s^{2}}{s^{2}-1}^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}\\\
+\sqrt{{d}}\frac{2\sigma}{\pi}\frac{\sqrt{s}}{s+1}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}-4\mu\epsilon\sqrt{ds}\frac{1}{2^{s}}(C_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-C_{s-1}^{\alpha-1})&\text{if
$s$ is odd}\end{array}\right.$
$\displaystyle=\left\\{\begin{array}[]{ll}(\sqrt{\frac{2d}{\pi}}\mu+\frac{4\sigma}{\pi}\sqrt{d})e^{\lambda_{s}-2\lambda_{\frac{s}{2}}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}+\sqrt{\frac{d}{s}}\frac{2}{\pi}\sigma
e^{\lambda_{s}-2\lambda_{s/2}}\\\
-4\mu\epsilon\sqrt{ds}\frac{1}{2^{s}}(C_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-C_{s-1}^{\alpha-1})&\text{if
$s$ is even}\\\\[12.0pt]
(\sqrt{\frac{2d}{\pi}}\mu+\frac{4\sigma}{\pi}\sqrt{d})(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}\\\
+\sqrt{{d}}\frac{2\sigma}{\pi}\frac{\sqrt{s}}{s+1}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}e^{\lambda_{s}-\lambda_{\frac{s+1}{2}}-\lambda_{\frac{s-1}{2}}}-4\mu\epsilon\sqrt{ds}\frac{1}{2^{s}}(C_{s-1}^{\lceil{\frac{s}{2}-1}\rceil}-C_{s-1}^{\alpha-1})&\text{if
$s$ is odd}\end{array}\right.$
* 3)
This part discusses the condition for
$\mathds{E}(f)|_{s>1}<\mathds{E}(f)|_{s=1}=\sqrt{\frac{2d}{\pi}}\sigma
e^{-\frac{\mu^{2}}{2\sigma^{2}}}+\mu\sqrt{d}(1-2\Phi(-\frac{\mu}{\sigma}))$
by further relaxing the upper bound of $\mathds{E}(f)|_{s>1}$.
* –
if $s$ is even, since $f_{3}\leq 0$,
$\displaystyle\mathds{E}(f)|_{s>1}$
$\displaystyle<(\sqrt{\frac{2d}{\pi}}\mu+\frac{2\sigma}{\pi}\sqrt{d})e^{\lambda_{s}-2\lambda_{\frac{s}{2}}}+\sqrt{\frac{d}{s}}\frac{2}{\pi}\sigma
e^{\lambda_{s}-2\lambda_{s/2}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}$
$\displaystyle\leq(\sqrt{\frac{2d}{\pi}}\mu+\frac{2\sigma}{\pi}\sqrt{d})+\sqrt{\frac{d}{s}}\frac{2}{\pi}\sigma+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}$
$\displaystyle=\mu\sqrt{d}(\sqrt{\frac{2}{\pi}}+(1+\frac{1}{\sqrt{s}})\frac{2\sigma}{\pi\mu})+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}$
Clearly $\mathds{E}(f)|_{s>1}<\mathds{E}(f)|_{s=1}$, if
$\sqrt{\frac{2}{\pi}}+(1+\frac{1}{\sqrt{2}})\frac{2\sigma}{\pi\mu}\leq
1-2\Phi(-\frac{\mu}{\sigma})$. This condition is well satisfied when
$\mu>>\sigma$, since $\Phi(-\frac{\mu}{\sigma})$ decreases monotonically with
increasing $\mu/\sigma$.
* –
if $s$ is odd, with $f_{3}\leq 0$,
$\displaystyle\mathds{E}(f)|_{s>1}$
$\displaystyle<(\sqrt{\frac{2d}{\pi}}\mu+\frac{2\sigma}{\pi}\sqrt{d})(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}+\sqrt{{d}}\frac{2\sigma}{\pi}\frac{\sqrt{s}}{s+1}(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}$
It can be proved that $(\frac{s^{2}}{s^{2}-1})^{\frac{s}{2}}$ decreases
monotonically with respect to $s$. This yields that
$\displaystyle\mathds{E}(f)|_{s>1}<(\sqrt{\frac{2d}{\pi}}\mu+(1+\frac{\sqrt{3}}{4})\frac{2\sigma}{\pi}\sqrt{d})(\frac{3^{2}}{3^{2}-1})^{\frac{3}{2}}+\sqrt{\frac{2d}{\pi}}\sigma
e^{\frac{-\mu^{2}}{2\sigma^{2}}}$
in this case $\mathds{E}(f)|_{s>1}<\mathds{E}(f)|_{s=1}$, if
$(\frac{9}{8})^{\frac{3}{2}}(\sqrt{\frac{2}{\pi}}+(1+\frac{\sqrt{3}}{4})\frac{2\sigma}{\pi\mu})\leq
1-2\Phi(-\frac{\mu}{\sigma})$.
Summarizing above two cases for $s$ , finally
$\mathds{E}(f)|_{s>1}<\mathds{E}(f)|_{s=1},~{}\text{if}~{}(\frac{9}{8})^{\frac{3}{2}}[\sqrt{\frac{2}{\pi}}+(1+\frac{\sqrt{3}}{4})\frac{2}{\pi}(\frac{\mu}{\sigma})^{-1}]+2\Phi(-\frac{\mu}{\sigma})\leq
1$
∎
## Appendix C.
Proof of Lemma 5
First, one can rewrite
$f(\mathbf{r},\mathbf{z})=|\Sigma_{j=1}^{j=d}(r_{j}z_{j})|=\mu|x|$, where
$x\in N(0,d)$, since i.i.d $r_{j}\in N(0,1)$ and $z_{j}\in\\{\pm\mu\\}$ with
equal probability. Then one can prove that
$\displaystyle\mathds{E}(|x|)$
$\displaystyle=\int_{-\infty}^{0}\frac{-x}{\sqrt{2\pi
d}}e^{-\frac{x^{2}}{2d}}dx+\int_{0}^{\infty}\frac{x}{\sqrt{2\pi
d}}e^{-\frac{x^{2}}{2d}}dx$
$\displaystyle=2\int_{0}^{\infty}\frac{\sqrt{d}}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2d}}d\frac{x^{2}}{2d}$
$\displaystyle=2\sqrt{d}\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\alpha}d\alpha$
$\displaystyle=\sqrt{\frac{2d}{\pi}}$
Finally, it is derived that
$\mathds{E}(f)=\mu\mathds{E}(|x|)=\mu\sqrt{\frac{2d}{\pi}}$.
## References
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* [4] R. Calderbank, S. Jafarpour, and R. Schapire, “Compressed learning: Universal sparse dimensionality reduction and learning in the measurement domain,” _Technical Report_ , 2009.
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* [6] P. Indyk and R. Motwani, “Approximate nearest neighbors: Towards removing the curse of dimensionality,” _in Proceedings of the 30th Annual ACM Symposium on Theory of Computing_ , pp. 604–613, 1998.
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|
arxiv-papers
| 2013-12-12T15:26:57 |
2024-09-04T02:49:55.353510
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weizhi Lu and Weiyu Li and Kidiyo Kpalma and Joseph Ronsin",
"submitter": "Weizhi Lu",
"url": "https://arxiv.org/abs/1312.3522"
}
|
1312.3593
|
Hadron Molecules Revisted
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA
###### Abstract
Hadron Molecules are particles made out of hadrons that are held together by
self interactions. In this report we discuss seven such molecules and their
self interactions. The $f_{0}(980)$, $a_{0}(980)$, $f_{1}(1400)$, $\Delta
N(2150)$ and $\pi_{1}(1400)$ molecular structure is given. We predict that two
more states the $K\overline{K}K(1500)$ and $a_{1}(1400)$ should be found.
## 1 Introduction
The first two molecular states $f_{0}(980)$ and $a_{0}(980)$ are the
isosinglet and the isotriplet states of the $K$$\overline{K}$ bound system[1].
This binding requires a quark-spin hyperfine interaction in the over all $q$
$q$ $\overline{q}$ $\overline{q}$ system. We will see that this binding is
different from the particle exchange mechanisms that bind the rest of
molecules of this report. The exchanges of Ref.[1] that bind the
$K$$\overline{K}$ system are quark exchanges where the quark-spin hyperfine
interaction leads to an attractive potential. This attractive potential can
only make states if the mesons of the fall apart mode $q$ $\overline{q}$ \-
$q$ $\overline{q}$ are below threshold. Thus we have states of
$K$$\overline{K}$ lying just below threshold in scalar channel($0^{++}$). This
is somewhat like the deuteron in the $p$ $n$ system.
The work of Ref.[1] has one flaw with regards to long-range color van der
Waals-type forces[2]. It has been pointed out that confining potentials of the
type used in Ref.[1] leads to a long-range power-law residual potential
between color singlets of $r^{-2}$ between two mesons. Ref.[1] calculates the
potential between mesons to be given by
$V_{vdW}=-{{20MeV}\over{r^{2}}}.$ (1)
This is to be compared with the coulomb force between charged mesons
$V_{coulomb}=-{{1.5MeV}\over{r}}.$ (2)
The r in equation 1 and equation 2 is given in fm. Let us compare the
potential strength between them by setting them equal
$V_{vdW}=V_{coulomb}=-{{20MeV}\over{r^{2}}}=-{{1.5MeV}\over{r}}.$ (3)
This occurs at the distance of 13.3 fm which is a very large distance. The
size of the scalar bound states are much smaller 2 fm at best. This long range
van der Waals force implies that gluons are massless(like the photon) and can
travel to the edge of the universe in a virtual state(like photons in EM-
fields). Gluons can only try to travel to the edge if they are in a color
singlet state (glueballs or glueloops). This would lead to an exponential
cutoff
$V_{r}=-{{e^{-\mu r}}\over{r}},$ (4)
where $\mu$ is the glueball mass. Sine mesons are much lighter than glueballs,
and the pion is the lightest, it is the longest carrier of the strong force.
Meson exchanges will be the binding force of the other hadron molecules of
this report.
The report is organized in the following manner:
Sec. 1 is the introduction to $f_{0}(980)$ and $a_{0}(980)$. Sec. 2 looks at
particle exchange calculations and the generation of $f_{1}(1400)$. Sec. 3
consider a dibaryon state $\Delta N(2150)$ and the similarity to
$f_{1}(1400)$. This similarity predicts the $a_{1}(1400)$ state. Sec. 4 an
exotic meson state $1^{-+}$ which is seen in $\eta$ $\pi$ p-wave scattering
$\pi_{1}(1400)$ is explained. Sec. 5 predicts another molecular state
$K\overline{K}K(1500)$ which should be found. Sec. 6 is the summary and
discussion.
## 2 $K$$\overline{K}$$\pi$ as an interacting system
In order to bind $K$$\overline{K}$$\pi$ together, we need to develop a
dynamical theory that uses particle exchange mechanism not the quark exchange
that led to the van der Walls forces[2]. It is standard to break up the
scattering of a three particle system into a sum of isobar spectator
scatterings[3]. To complete this task we develop a unitary isobar model which
has long-range particle exchange forces. We will assume that the only
interaction among the three particles are isobars decaying into two particles
where one of these particle exchange with the spectator forming another
isobar. This one-particle-exchange(OPE) occurs in the $K^{*}$$\overline{K}$,
$\overline{K^{*}}$$K$ and $a_{0}$$\pi$ isobar systems. See Figure 1(a) for the
OPE mechanism(note $a_{0}$ is the $\delta$ isobar which is its older name). We
choose as our dynamical framework the Blankenbecler-Sugar formalism[4] which
yields set of coupled integral equations for amplitudes $X(K^{*}\rightarrow
K^{*}$), $X(K^{*}\rightarrow\overline{K^{*}}$), $X(K^{*}\rightarrow a_{0}$),
$X(\overline{K^{*}}\rightarrow K^{*}$),
$X(\overline{K^{*}}\rightarrow\overline{K^{*}}$),
$X(\overline{K^{*}}\rightarrow a_{0}$), $X(a_{0}\rightarrow K^{*}$),
$X(a_{0}\rightarrow\overline{K^{*}}$), and $X(a_{0}\rightarrow a_{0}$). These
amplitudes describe the isobar quasi-two-body processes
$K^{*}\overline{K}\rightarrow K^{*}\overline{K}$,
$K^{*}\overline{K}\rightarrow\overline{K^{*}}K$, $K^{*}\overline{K}\rightarrow
a_{0}\pi$, etc., whose solution are Lorentz invariant, and satisfy two- and
three-body unitarity, and cluster properties. In operator formalism these
equations have the structure(a schematic representation is show in Figure
1(b)),
$X_{ba}=B_{ba}(W_{E})+B_{bc}(W_{E})G_{c}(W_{E})X_{ca}(W_{E});a,b,c=K^{*},\overline{K^{*}},a_{0}.$
(5)
In equation 5, $W_{E}$ is the overall c.m. energy of the three-particle
system. The index $c$ which is summed over represent each isobar. Integration
is over solid angles where each total angular momentum is projected out as an
individual set of coupled equations. The c.m. momentum of the spectator which
has the same magnitude as the isobar is integrated from zero to $\infty$. Thus
all effective masses of isobars are probed starting at the kinematic limit
down to negative $\infty$. All details are given in Ref.[5]. $G_{c}(W_{E})$ is
the propagator of isobar $c$ and $W_{E}$ determines the kinematic limit where
c.m. momentum of the spectator and the isobar is zero. In Figure 2 we show the
Breit-Wigner shape of the $K^{*}$ propagator $G_{K^{*}}(2.0)$ as a function of
isobar mass($K\pi$). In Figure 3 we show the Breit-Wigner shape of the $a_{0}$
propagator $G_{a_{0}}(2.04)$ as a function of isobar mass($K\overline{K}$).
Figure 1: (a) Long-range one-particle-exchange(OPE) mechanism. Isobars
$K^{*}$, $\overline{K^{*}}$, or $\delta$($a_{0}$ newer name) plus
$\overline{K}$, $K$ or $\pi$ which absorbs the exchange particle which has
decayed from the isobar forming another isobar. (b) Unitary sum of OPE
diagrams in terms of coupled integral equations.
Figure 2: The absolute value squared of the imaginary part divided by the
propagator for $K\pi$ propagation of the $K^{*}$ which is an $I={{1}\over{2}}$
and $J=1$ mode. This is equal to the square of the T-matrix scattering of this
$K\pi$ mode.
Figure 3: The absolute value squared of the imaginary part divided by the
propagator for $K\overline{K}$ propagation of the $a_{0}$ which is an $I=1$
and $J=0$ mode. This is equal to the square of the T-matrix scattering of this
$K\overline{K}$ mode.
Equation 5 can be rewriting as
$\sum_{k}(\delta_{ik}-M_{ik})X_{kj}=B_{ij};i,j,k=K^{*},\overline{K^{*}},a_{0}.$
(6)
This Fredholm integral equation leads to a Fredholm determinant as a function
of $W_{E}$ for each partial wave or total $J$ projection. We have solved this
Fredholm determinant for two $J^{PC}$ states $0^{-+}$ and $1^{++}$. The
results of this analysis is shown in Figure 4. We see no binding effect in the
$0^{-+}$ determinant, while in the $1^{++}$ channel there is a large effect
around $1.40$ GeV. At the energy of $1.40$ GeV the $K^{*}$ and
$\overline{K^{*}}$ the peaks of Figure 2 are just coming into play. Since for
$1^{++}$ they are in a s-wave they have maximum effect. In the $0^{-+}$ these
peak are suppressed by a p-wave barrier. Around the mass $1.40$ GeV one can
form a picture of the system being $K$$\overline{K}$($a_{0}$) molecule at the
center of gravity with a light pion revolving in a p-wave orbit. The momentum
of the pion is such that at each half-revolution a $K^{*}$ or a
$\overline{K^{*}}$ is formed(see Figure 5). The phase shift and production
cross sections of this molecular state is explored in detail in Ref.[5].
Figure 4: The value of 1 over the Fredholm determinant squared for $J^{PC}$ =
$1^{++}$ and $J^{PC}$ = $0^{-+}$ as a function of $K$$\overline{K}$$\pi$
mass(smooth curves).
Figure 5: The meson system mainly resonates in the s-wave
$K^{*}$$\overline{K}$ and $K$$\overline{K^{*}}$ mode with a pion rotating in a
p-wave about a $K$ $\overline{K}$ system which forms a isospin triplet. The
pion moves back and forth forming $K^{*}$ and $\overline{K^{*}}$ states with
one $K$ or $\overline{K}$.
Figure 6: The dibaryon system mainly resonates in the s-wave $\Delta$ $N$ mode
with a pion rotating in a p-wave about a spin aligned $N$ $N$ system which
forms a isospin singlet. The pion moves back and forth forming $\Delta$ states
with one nucleon and then the other.
## 3 Two more molecular states
### 3.1 Dibaryon state $\Delta N(2150)$ as a molecule.
The dibaryon state interacts in three two-body scattering channels. Its mass
is 2.15 GeV and has a strong interaction resonance decay width of 100 MeV. It
interacts in the $N$$N$ d-wave spin anti-aligned[6], $d$$\pi$ p-wave spin
aligned[7], and $\Delta$$N$ s-wave spin aligned[8]. The dibaryon system mainly
resonates in the s-wave $\Delta$$N$ mode with a pion rotating in a p-wave
about a spin aligned $N$$N$ system which forms a isospin singlet. The pion
moves back and forth forming $\Delta$ states with one nucleon and then the
other(see Figure 6). All three isospin states of the pion can be achieved in
this resonance. Thus we can have $\pi^{+}$$d$, $\pi^{0}$$d$, and $\pi^{-}$$d$
states. If the pion is absorbed by any of the nucleons it under goes a spin
flip producing a d-wave $N$$N$ system. The resonance decays into $N$$N$,
$\pi$$d$, or $\pi$$N$$N$. In the last section we saw a meson system that had
an analogous orbiting pion in a p-wave mode about a $K\overline{K}$ in a
s-wave[5]. Both systems have a similar lifetime or width of $\sim$ .100
GeV[9].
### 3.2 $a_{1}(1400)$ state is predicted
Unlike the $f_{1}(1400)$ the $\Delta N(2150)$ has an isosinglet at the center
of motion. The $K$$\overline{K}$ isosinglet state of Sec. 1 could form the
center of motion for an isotriplet molecular state $a_{1}(1400)$. The set of
integral equation would be the same as in the $f_{1}(1400)$ case making a
similar Fredholm determinant. Like for $\Delta N(2150)$ which had a $d\pi$
decay mode, one would expect that there would be a $f_{0}(980)$ $\pi$ decay
mode. We can calculate the branching ration of $f_{1}(1400)$ to $a_{0}$$\pi$
from the Dalitz plot calculated using equation 20 of Ref.[5]. The ratio in the
plot going into $a_{0}$$\pi$ is 22%. The reason this mode is so small is
because $\sqrt{Imag(D_{a_{0}})}\over{|D_{a_{0}}|}$ is much smaller than
$\sqrt{Imag(D_{K^{*}})}\over{|D_{K^{*}}|}$[5]. Where as the ratios of
$Imag(D_{a_{0}})\over{|D_{a_{0}}|}$ and $Imag(D_{K^{*}})\over{|D_{K^{*}}|}$
are one at resonance(see Figure 2 and 3). For the $\Delta N(2150)$ the $d\pi$
branching ratio is 25%[9]. We should expect that the branching of
$a_{1}(1400)\rightarrow f_{0}\pi$ should be the same as
$f_{1}(1400)\rightarrow a_{0}\pi$. Dr. Suh-Urk Chung has claimed such a state
has been observed[10].
## 4 Exotic state $J^{PC}$ = $1^{-+}$ $\pi_{1}(1400)$ as a molecule.
In Sec. 2 we explained the $f_{1}$(1420) seen in $\overline{K}K\pi$[5].
Following the same approach we can demonstrate the possibility that the
$\pi_{1}$(1400) is a $\overline{K}K\pi\pi$ molecule, where the
$\overline{K}K\pi$ in a relative s-wave with the other $\pi$ orbiting them in
a p-wave. Since the $\overline{K}K\pi$ is resonating as the $\eta$(1295), it
is possible that the offshell $\overline{K}K\pi(\eta)$ would couple to the
ground state $\eta$, thus creating a $\eta\pi$ p-wave decay mode.
As was done in Ref.[5], we need to arrange a set of Born terms connecting all
of the possible intermediate isobar states of the $\overline{K}K\pi\pi$ system
($\eta$(1295)$\pi$, $a_{0}$(980)$\rho$(770), $K_{1}$(1270)$\overline{K}$ or
$\overline{K_{1}}(1270)$$K$). We assume that the only interaction among the
particles occurs through one-particle exchange (OPE), thus connecting the
above isobar states (Figure 7). In order to completely derive the dynamics one
would have to develop a true four-body scattering mechanism with OPE Born
terms connecting two- and three-body isobar states. We can take a short cut
and use the three-body formalism developed in Ref. [5], if we note that the
set of diagrams (Figure 8) could be summed using a true four-body formalism,
and be replaced by the Born term of Figure 9. Here the $a_{0}$(980) is treated
as a stable particle and the $\pi\pi$ p-wave phase shift ($\rho_{med}$) is
assumed to be modified by the sum of terms in Figure 8. With this assumption,
then binding can occur if we use the $N/D$ propagators for the
$\eta$(1295)(see Figure 10) and $\rho_{med}$(see Figure 11). In Figure 11 we
also show the unaltered p-wave phase shift ($\rho$). Figure 12 shows the final
state enhancement times the $\eta$(1295) $\pi$ p-wave kinematics. The bump is
driven by the collision on the Dalitz plot of the $\eta$(1295) Breit-Wigner
(Figure 10) and the rapid increase of the $\pi\pi$ p-wave phase shift (Figure
11).
We have suggested the possibility that the $\pi_{1}$(1400) is a final state
interaction for the $K\bar{K}\pi$ system in a s-wave orbiting by a $\pi$ in a
p-wave. The $\eta\pi$ decay mode is generated by the off shell appearance of
the $\eta$ from the $K\bar{K}\pi$ system ($0^{-+}$). Our model thus predicts
that a strong $J^{PC}=1^{-+}$ should be seen in the $K\bar{K}\pi\pi$ system at
around 1.4 GeV/c2. If the $\pi_{1}$(1400) is only seen in the $\eta\pi$
channel then its hard to understand three facts about its production. First,
that the force between the $\eta$ and $\pi$ in a p-wave should be repulsive
(QCD) [11]. This is not a problem if the $\eta\pi$ is a minor decay mode.
Second, why should the production be so small compared to the $a_{2}$ which
has only a 14% branching to $\eta\pi$? One would think it should be produced
in unnatural parity exchange not natural. Again this is not a problem if minor
decay mode.
Figure 7: One-particle-exchange (OPE) Born terms for $\overline{K}K\pi\pi$
system.
Figure 8: The set of infinite terms where all $K$ and $\overline{K}$ exchanges
are summed.
Figure 9: The Born that is used in the three-body effective analysis, where
the $\pi\pi$ p-wave is altered by the sum of terms in Figure 8.
Figure 10: The absolute value squared of the imaginary part of the $\eta$
(1290) propagator divided by the complete propagator, thus forming the square
of the T-matrix scattering amplitude.
Figure 11: The absolute value squared of the imaginary part of the $\pi\pi$
p-wave phase shift: the solid line is the modified phase shift; the dashed
line is the original vacuum phase shift which is the $\rho$ meson.
Figure 12: The value of 1 over the Fredholm determinate squared times the
kinematics of p-wave $\pi\eta$(1290).
Finally, it is reasonable to think that the largest decay amplitude would be
the modes that have an $a_{0}$(980) in the final state. However in Ref.[5] the
same conclusion was initially drawn, except when one puts in all the numerical
factors the $a_{0}$ modes become suppressed. The explanation comes from the
very powerful attraction of the kaons in the $a_{0}$ mode. The isobar decay
amplitude is proportional to $\sqrt{N}/D$ both $N$ and $D$ are large numbers
while the ratio is near one at the threshold(see last section and Ref.[5]).
Thus the decay amplitude becomes proportional to $1/\sqrt{N}$ . We predict
that the major mode could be $\pi\pi$ p-wave having no $\rho$ peak (work
above) forming a $K\pi\pi$ or a $\overline{K}\pi\pi~{}J^{p}=1^{+}$ plus a
$\overline{K}$ or $K$ with overall $G$-parity minus. The $K\pi\pi$ should more
or less be a phase space distribution.
## 5 $K\overline{K}K(1500)$ state is predicted
We saw in Sec.1 and Sec. 2 that the $K$$\overline{K}$ system had attraction
through the $a_{0}(980)$ resonance. It seems only natural to investigate the
possibility that a three-K molecule might exist. This is only worthwhile if we
consider only exotic quantum numbers. The only exotic quantum number which can
be be obtained is the isotopic spin. Thus a set of coupled equations for the
$K$$\overline{K}$$K$ system in a overall s-wave with isotopic spin of
$3\over{2}$ is created[5]. The Fredholm determinate squared times of the
equations is shown in Figure 13.
Isopin spin $3\over{2}$ implies there are four states
$K^{+}$$\overline{K^{0}}$$K^{+}$,$K^{+}$$K^{-}$$K^{+}$,
$K^{0}$$\overline{K^{0}}$$K^{0}$, and $K^{0}$$K^{-}$$K^{0}$. The
$K^{+}$$\overline{K^{0}}$$K^{+}$ is double charged. There would also be a
$K^{-}$$K^{0}$$K^{-}$ which is the anti-matter state of the
$K^{+}$$\overline{K^{0}}$$K^{+}$. These states are unique to this type of
binding mechanism.
## 6 Summary and Discussion
In this report we have discussed seven possible hadron molecular states. These
states are particles made out of hadrons that are held together by self
interactions. The seven molecules and their self interactions are explored.
The $f_{0}(980)$, $a_{0}(980)$ relied quark exchange forces which made states
of $K$$\overline{K}$ lying just below threshold in scalar channel($0^{++}$).
This is somewhat like the deuteron in the $p$ $n$ system. The $f_{1}(1400)$,
$\Delta N(2150)$ and $\pi_{1}(1400)$ molecular structure are held together by
long range particle exchange mechanisms not the quark exchange that led to the
van der Walls forces[2]. These exchange mechanisms also predicts that two more
states the $K\overline{K}K(1500)$ and $a_{1}(1400)$ should be found. For the
$a_{1}(1400)$ the set of integral equation would be the same as in the
$f_{1}(1400)$ case making a similar Fredholm determinant. Like for $\Delta
N(2150)$ which had a $d\pi$ decay mode, one would expect that there would be a
$f_{0}(980)$ $\pi$ decay mode. Dr. Suh-Urk Chung has claimed such a state has
been observed[10].
Figure 13: The value of 1 over the Fredholm determinant squared for $J^{P}$ =
$0^{-}$ as a function of $K$$\overline{K}$$K$ mass(smooth curves).
## 7 Acknowledgments
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.
## References
* [1] J. Weinstein and N. Isgur, Phys. Rev.Lett. 48 (1982) 659; Phys. Rev. D 27 (1983) 588; Phys. Rev. D 41 (1990) 2236.
* [2] O.W. Greenburg and H.J. Lipkin, Nucl. Phys. A370 (1981) 349.
* [3] D.J. Herndon, Phys. Rev. D 11 (1975) 3165.
* [4] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051.
* [5] R. Longacre, Phys. Rev. D 42 (1990) 874.
* [6] R.A. Arndt et al., Phys. Rev. C 76 (2007) 025209.
* [7] C.H. Oh et al., Phys. Rev. C 56 (1997) 635.
* [8] D. Schiff and J. Tran Thanh Van, Nucl. Phys. B5 (1968) 529.
* [9] R. Longacre,arXiv:1311.3609[hep-ph].
* [10] Suh-Urk Chung(private communication).
* [11] T. Barnes(private communication).
|
arxiv-papers
| 2013-12-12T19:06:06 |
2024-09-04T02:49:55.363451
|
{
"license": "Public Domain",
"authors": "Ron Longacre",
"submitter": "Ron S. Longacre",
"url": "https://arxiv.org/abs/1312.3593"
}
|
1312.3606
|
# Strong squeezing and robust entanglement in cavity electromechanics
Eyob A. Sete1 and Hichem Eleuch2 1Department of Electrical Engineering,
University of California, Riverside, California 92521, USA
2Department of Physics, McGill University, Montreal, Canada H3A 2T8
###### Abstract
We investigate nonlinear effects in an electromechanical system consisting of
a superconducting charge qubit coupled to transmission line resonator and a
nanomechanical oscillator, which in turn is coupled to another transmission
line resonator. The nonlinearities induced by the superconducting qubit and
the optomechanical coupling play an important role in creating optomechanical
entanglement as well as the squeezing of the transmitted microwave field. We
show that strong squeezing of the microwave field and robust optomechanical
entanglement can be achieved in the presence of moderate thermal decoherence
of the mechanical mode. We also discuss the effect of the coupling of the
superconducting qubit to the nanomechanical oscillator on the bistability
behaviour of the mean photon number.
###### pacs:
42.50.Wk 85.85.+j 42.50.Lc 42.65.Pc
## I Introduction
Cavity optomechanics, where the electromagnetic mode of the cavity is coupled
to the mechanical motion via radiation pressure force, has attracted a great
deal of renewed interest in recent years Nori13 . Such coupling of macroscopic
objects with the cavity field can be used to directly investigate the
limitation of the quantum-based measurements and quantum information protocols
Bra92 ; Man97 ; Bos97 . Furthermore, optomechanical coupling is a promising
approach to create and manipulate quantum states of macroscopic systems. Many
quantum and nonlinear effects have been theoretically investigated. Examples
include, squeezing of the transmitted field Fab94 ; Woo08 ; Set12 ,
entanglement between the cavity mode and the mechanical oscillator Zha03 ;
Pin05 ; Vit08 , optical bistability Tre96 ; Dor83 ; Mey85 ; Jia12 ; Set12 ,
side band ground state cooling Oco10 ; Teu11 among others. In particular, the
squeezing of the transmitted field and the optomechanical entanglement
strongly rely on the nonlinearity induced by the optomechanical interaction
which couples the position of the oscillator to the intensity of the cavity
mode. Recently, relatively strong optomechanical squeezing has been realized
experimentally by exploiting the quantum nature of the mechanical interaction
between the cavity mode and a membrane mechanical oscillator embedded in an
optical cavity Pur13 .
On the other hand, demonstrations of ground state cooling, manipulation, and
detection of mechanical states at the quantum level require strong coupling,
where the rate of energy exchange between the mechanical oscillator and the
cavity field exceeds the rates of dissipation of energy from either system.
Although the control and measurement of a single microwave phonon has already
been demonstrated Oco10 , the phonon states appeared to be short-lived.
However, for practical applications mechanical states should survive longer
than the operation time. This unwanted property is due to the fact that
mechanical resonators performance degrades as the fundamental frequency
increases Eki05 .
In order to observe the quantum mechanical effects in cavity optomechanics,
one needs to reach the strong coupling regime and overcome the thermal
decoherence. This has been exceedingly difficult to experimentally demonstrate
in cavity optomechanics schemes. An alternative approach to realize strong
coupling is to use electromechanical systems, where the mechanical motion is
coupled to superconducting circuits embedded in transmission line resonators
Rab04 ; Gel04 ; Gel05 ; Zho06 ; Wen08 ; Wen082 ; Teu11 ; Nic12 ; Yin12 .
Teuful et al. Teu11 have recently realized strong coupling and quantum
enabled regimes using electromechanical systems composed of low-loss
superconducting circuits. These systems fulfil the requirements for
experimentally observing and controlling the theoretically predicted quantum
effects Zha03 ; Pin05 ; Vit08 ; Tre96 ; Dor83 ; Mey85 ; Jia12 ; Set12 . In
this regard, much attention has been paid in exploiting experimentally
accessible electromechanical systems Rab04 ; Gel04 ; Gel05 ; Zho06 ; Wen08 ;
Wen082 ; Nic12 ; Yin12 .
In this work, we investigate the squeezing and the optomechanical
entanglement, in an electromechanical system in which a superconducting charge
qubit is coupled to a transmission line microwave resonator and a movable
membrane, simulating the mechanical motion. The membrane is also capacitively
coupled to a second transmission line resonator (see Fig. 1). In the strong
dispersive limit, the coupling of the superconducting qubit with the resonator
and the nanomechanical oscillator gives rise to an effective nonlinear
coupling between the resonator and the nanomechanical oscillator. In effect,
there are two types of nonlinearities in our system: the nonlinear interaction
between the first resonator and the nanomechanical oscillator mediated by the
superconducting qubit and the nonlinear interaction induced by the
optomechanical coupling between the nanomechanical oscillator and the second
microwave resonator. We find that presence of the superconducting qubit-
induced nonlinearity increases the pump power required to observe the bistable
behaviour of the mean photon number in the second resonator. We show that the
combined effect of these nonlinearities leads to strong squeezing of the
transmitted field in the presence of thermal fluctuations. The squeezing is
controllable by changing the microwave drive pump power. Using logarithmic
negatively as entanglement measure, we also show that the mechanical motion is
entangled with the second resonator mode in the steady state. The generated
entanglement is shown to be robust against thermal decoherence.
Figure 1: Schematics of our model. A Cooper pair box, consists of two
Josephson junctions, is coupled to a superconducting transmission line
resonator ($\text{TLR}_{1}$) and a nanomechanical oscillator. In general, the
interaction between the qubit (the Cooper pair box) and the nanomechanical
oscillator is nonlinear, which depends on the variable capacitor, $C_{q}$. A
second superconducting transmission line resonator ($\text{TLR}_{2}$) is
capacitively coupled to the nanomechanical oscillator. The radio frequency
(rf) source produces a microwave field, which populates the second resonator
$\text{TLR}_{2}$ via a small capacitance.
## II Model and Hamiltonian
The electromechanical system considered here is schematically depicted in Fig.
1. A superconducting transmission line resonator ($\text{TLR}_{1}$) is placed
close to the Cooper-pair box, which is coupled to a large superconducting
reservoir via two identical Josephson junctions of capacitance $C_{J}$ and
Josephson energy $E_{J}$. This effectively forms a superconducting quantum
interface device (SQUID) and is also a basic configuration for superconducting
charge qubit Wen08 . The state of the qubit can be controlled by the gate
voltage $V_{\textit{g}}$ through a gate capacitance $C_{\textit{g}}$. The
qubit is further coupled to a nanomechanical oscillator via capacitance
$C_{\rm q}$ that depends on the position x of the membrane (the green line in
Fig. 1) from the equilibrium position. Since the amplitude is close to the
zero point fluctuation $\text{x}_{\rm zpf}$, the first order correction to the
displacement is enough to describe the capacitance. We introduce a
dimensionless position operator as $x=\text{x}/\text{x}_{\rm zpf}$, which can
be expressed in terms of the annihilation and creation operators as
$x=b+b^{{\dagger}}$. Thus, the Hamiltonian of the nanomechanical oscillator of
frequency $\hbar\omega_{\rm m}$ is given by $\hbar\omega_{\rm
m}(b^{{\dagger}}b+1/2)$ (in our analysis we drop the constant term
$\hbar\omega_{\rm m}/2$). If the distance between the membrane and the other
arm of the capacitor is $d$ at $\text{x}=0$, then the corresponding
capacitance is $C^{(0)}_{q}=\epsilon_{\rm m}S/d$, where $S$ is the surface
area of the electrode and $\epsilon_{\rm m}$ is the permittivity of free
space. At the displacement $d-\text{x}$ the capacitance reads
$C_{q}(\text{x})=C^{(0)}_{q}/(1-\text{x}/d)\simeq C^{(0)}_{q}+C^{(1)}x$, where
$C^{(1)}_{q}=x_{\rm zpf}C^{(0)}_{q}/d$. To create a tunable coupling between
the microwave resonator and the circuit elements, a gate voltage $V_{q}$ is
applied.
The Hamiltonian that describes the interaction of the qubit with the resonator
$\text{TLR}_{1}$ and the nanomechanical oscillator, in the rotating wave
approximation, is given by Wen08 (we take $\hbar=1$)
$H_{1}=-\frac{1}{2}\omega_{q}\sigma_{z}+\textit{g}_{c}(c^{{\dagger}}\sigma_{-}+c\sigma_{+})+\textit{g}_{b}(b^{\dagger
2}\sigma_{-}+b^{2}\sigma_{+}),$ (1)
where $\omega_{\rm q}$ is the transition frequency of the qubit,
$\textit{g}_{b}$ and $\textit{g}_{c}$ are the microwave resonator-qubit and
nanomechanical oscillator-qubit couplings, respectively. The qubit operators
are defined by $\sigma_{z}=\left|e\right\rangle\left\langle
e\right|-\left|g\right\rangle\left\langle
g\right|,\sigma_{+}=(\sigma_{-})^{\dagger}=\left|e\right\rangle\left\langle
g\right|$ with $\left|g\right\rangle$ and $\left|e\right\rangle$ representing
the ground and the excited states of the qubit; $b$ and $c$ are the
annihilation operators of the mechanical mode and the first resonator
$\text{TLR}_{1}$ mode.
Furthermore, the nanomechanical oscillator is coupled to the second
transmission line resonator ($\text{TLR}_{2}$), which is externally driven by
a microwave field of frequency $\omega_{\rm d}$. This coupling is described by
the Hamiltonian
$H_{2}=\textit{g}_{a}a^{{\dagger}}a(b^{{\dagger}}+b)+i\varepsilon(a^{{\dagger}}e^{-i\omega_{\rm
d}t}-ae^{i\omega_{\rm d}t}),$ (2)
where $a$ the annihilation operator for the resonator $\text{TLR}_{2}$ mode;
$\textit{g}_{a}$ is the resonator-mechanical mode coupling constant,
$\varepsilon=\sqrt{2\kappa_{a}P/\hbar\omega_{a}}$ is the amplitude of the
microwave drive of $\text{TLR}_{2}$ with $P$ being the corresponding power,
$\kappa_{a}$ the resonator damping rate, and $\omega_{a}$ the resonator
frequency. The free energies of the mechanical oscillator and the two
resonators read
$H_{0}=\omega_{\rm m}b^{{\dagger}}b+\omega_{a}a^{{\dagger}}a+\omega_{\rm
c}c^{{\dagger}}c,$ (3)
where $\omega_{\rm m}$ is the mechanical oscillator frequency and $\omega_{\rm
c}$ is the frequency of $\text{TLR}_{1}$.
Next, we apply the unitary transformation that effectively eliminates the
degrees of freedom of the qubit [in fact the transformation diagonalizes the
interaction part of the Hamiltonian (1)]. This can be achieved by applying a
unitary transformation defined by
$H=U(H_{0}+H_{1}+H_{2})U^{{\dagger}},$
where
$U=\exp\left[\frac{g_{c}}{\Delta_{qc}}(c\sigma_{+}-c^{{\dagger}}\sigma_{-})+\frac{g_{b}}{\Delta_{qm}}(b^{2}\sigma_{+}-b^{\dagger
2}\sigma_{-})\right],$
in which $\Delta_{qc}=\omega_{\rm q}-\omega_{c}$ and $\Delta_{qm}=\omega_{\rm
q}-2\omega_{\rm m}$. In the dispersive limit,
$\Delta_{qm},\Delta_{qc}\gg\sqrt{g_{b}^{2}+g_{c}^{2}}$, the transformation
yields an approximate Hamiltonian
$\displaystyle H$ $\displaystyle\approx\omega_{a}a^{{\dagger}}a+\omega_{\rm
m}b^{{\dagger}}b+\omega_{c}c^{{\dagger}}c+\alpha(b^{{\dagger}2}c+c^{{\dagger}}b^{2})\sigma_{z}$
$\displaystyle+\textit{g}_{a}a^{{\dagger}}a(b^{{\dagger}}+b)+i\varepsilon(a^{{\dagger}}e^{-i\omega_{\rm
d}t}-ae^{i\omega_{\rm d}t}),$ (4)
where
$\alpha=\textit{g}_{b}\textit{g}_{c}\left(\Delta_{qc}+\Delta_{qm}\right)/2\Delta_{qm}\Delta_{qc}$
is an effective nonlinear coupling between the nanomechanical oscillator and
the resonator $\text{TLR}_{1}$. If the qubit is adiabatically kept in the
ground state, the effective Hamiltonian reduces to
$\displaystyle H$ $\displaystyle\approx\omega_{a}a^{{\dagger}}a+\omega_{\rm
m}b^{{\dagger}}b+\omega_{c}c^{{\dagger}}c-\alpha(b^{{\dagger}2}c+c^{{\dagger}}b^{2})$
$\displaystyle+\textit{g}_{a}a^{{\dagger}}a(b^{{\dagger}}+b)+i\varepsilon(a^{{\dagger}}e^{-i\omega_{\rm
d}t}-ae^{i\omega_{\rm d}t}).$ (5)
Note that if there is strong thermal excitation which promotes the qubit to
the excited state, then as follows from (II) the sign of the coupling strength
obviously change from $-\alpha$ to $\alpha$. The effective nonlinear coupling
between the resonator $\text{TLR}_{1}$ and the mechanical mode does not have
the same form as the usual optomechanical coupling (e.g., the coupling between
$\text{TLR}_{2}$ and the mechanical mode). This is because the former is
mediated by a qubit, while the latter is a direct intensity-dependent
coupling.
### II.1 Quantum Langevin equations
The dynamics of our system can be described by the quantum Langevin equations
that take into account the loss of microwave photons from each resonator and
the damping of the mechanical motion due to the membrane’s thermal bath. In a
frame rotating with the microwave drive frequency $\omega_{\rm d}$, the
nonlinear quantum Langevin equations read
$\dot{a}=-\left(i\Delta_{a}+\frac{\kappa_{a}}{2}\right)a-i\textit{g}_{a}a(b^{{\dagger}}+b)+\varepsilon+\sqrt{\kappa_{a}}a_{\text{in}},$
(6) $\dot{b}=-(i\omega_{\rm
m}+\frac{\gamma_{b}}{2})b-i\text{g}_{a}a^{{\dagger}}a-2i\alpha
cb^{{\dagger}}+\sqrt{\gamma_{m}}b_{\text{in}},$ (7)
$\dot{c}=-\left(i\omega_{c}+\frac{\kappa_{c}}{2}\right)c+i\alpha
b^{2}+\sqrt{\kappa_{c}}c_{\text{in}},$ (8)
where $\Delta_{a}=\omega_{a}-\omega_{\rm d}$, and $\kappa_{c}$ and
$\gamma_{m}$ are, respectively, the damping rates for the first resonator
$\text{TLR}_{1}$ and mechanical oscillator. We assume that the resonators
thermal baths and that of the mechanical bath are Markovian and hence the
noise operators $a_{\text{in}},b_{\text{in}}$, and $c_{\text{in}}$ satisfy the
following correlation functions:
$\langle
A_{\text{in}}^{{\dagger}}(\omega)A_{\text{in}}(\omega^{\prime})\rangle=2\pi
n_{A}\delta(\omega+\omega^{\prime}),$ (9) $\langle
A_{\text{in}}(\omega)A_{\text{in}}^{{\dagger}}(\omega^{\prime})\rangle=2\pi(n_{A}+1)\delta(\omega+\omega^{\prime}),$
(10)
with $n_{A}^{-1}=\exp(\hbar\omega_{A}/k_{B}T_{A})-1$, where $k_{B}$ is the
Boltzmann constant and $A=a,b,c$, and the noise operators have zero-mean
values, $\langle a_{\rm in}\rangle=\langle b_{\rm in}\rangle=\langle c_{\rm
in}\rangle=0$.
### II.2 Optical bistability in resonator photon number
It is well-known that for strong enough pump power and in the red-detuned
($\omega_{d}-\omega_{a}<0$) regime, an optomechanical coupling gives rise to
optical bistability. Here we investigate the effect of the nonlinearity
induced by the superconducting qubit on the bistable behaviour. Solving the
expectation values of Eqs. (6)-(8) in the steady state we obtain
$\langle a\rangle=\frac{\varepsilon}{i\Delta_{\rm f}+\kappa_{a}/2},$ (11)
$\langle b\rangle=\frac{-ig_{a}|\langle a\rangle|^{2}}{i\omega_{\rm
m}+\gamma_{\rm m}/2}-i\frac{2\alpha\langle c\rangle\langle
b^{{\dagger}}\rangle}{i\omega_{\rm m}+\gamma_{\rm m}/2},$ (12) $\langle
c\rangle=i\frac{\alpha\langle b\rangle^{2}}{i\omega_{c}+\kappa_{c}/2},$ (13)
where $\Delta_{\rm f}=\Delta_{a}+g_{a}(\langle b\rangle+\langle
b^{{\dagger}}\rangle)$ is an effective detuning for second resonator.
Combining these equations, we obtain the coupled equations for the mean photon
number $I_{a}=|\langle a\rangle|^{2}$ in the second resonator and the mean
phonon number $I_{b}=|\langle b\rangle|^{2}$ as
Figure 2: Bistability behaviour for mean photon number in the second resonator
$I_{a}$ (a) in the presence of the nonlinear coupling $\alpha\neq 0$
[$F(I_{b})<1$] (b) in the absence of the nonlinear coupling
$\alpha=0$[$F(I_{b})=1$]. The parameters used are: frequencies $\omega_{\rm
m}/2\pi=10~{}\text{MHz}$, $\omega_{a}/2\pi=7.5~{}\text{GHz}$,
$\omega_{c}/2\pi=2.5~{}\text{GHz}$, $\omega_{q}/2\pi=3~{}\text{GHz}$,
$\omega_{d}/2\pi=7~{}\text{GHz}$, couplings
$g_{a}/\pi=460\text{Hz}$,$g_{b}/2\pi=2\text{MHz}$,
$g_{c}/2\pi=30~{}\text{MHz}$, and damping rates
$\kappa_{a}/2\pi=10^{5}~{}\text{Hz}$, $\gamma_{\rm m}/2\pi=50~{}\text{Hz}$,
and $\kappa_{c}/2\pi=10^{5}~{}\text{Hz}$.
$I_{a}\left[\left(\Delta_{a}-F(I_{b})\frac{2\textit{g}_{a}^{2}\omega_{\rm
m}I_{a}}{\omega_{\rm m}^{2}+(\gamma_{\rm
m}/2)^{2}}\right)^{2}+\left(\frac{\kappa_{a}}{2}\right)^{2}\right]=|\varepsilon|^{2},$
(14) $\displaystyle I_{a}^{2}=$
$\displaystyle\frac{I_{b}[(1+I_{b}\beta_{1})^{2}+I_{b}^{2}\beta_{2}^{2}]^{2}[\omega_{\rm
m}^{2}+(\gamma_{\rm m}/2)^{2}]^{2}/g_{a}^{2}}{[\omega_{\rm
m}(1+I_{b}\beta_{1})+\frac{\gamma_{\rm
m}}{2}I_{b}\beta_{2}]^{2}+[\frac{\gamma_{\rm
m}}{2}(1+I_{b}\beta_{1})-\omega_{\rm m}I_{b}\beta_{2}]^{2}}$ (15)
where
$F(I_{b})=\frac{1+I_{b}\beta_{1}+\frac{\gamma_{\rm m}}{2\omega_{\rm
m}}I_{b}\beta_{2}}{(1+I_{b}\beta_{1})^{2}+I_{b}^{2}\beta_{2}^{2}},$ (16)
$\beta_{1}=\frac{2\alpha^{2}(\omega_{\rm m}\omega_{c}-\gamma_{\rm
m}\kappa_{c}/4)}{[\omega_{\rm m}^{2}+(\gamma_{\rm
m}/2)^{2}][\omega_{c}^{2}+(\kappa_{c}/2)^{2}]},$ (17)
$\beta_{2}=\frac{\alpha^{2}(\omega_{\rm m}\kappa_{c}+\omega_{c}\gamma_{\rm
m})}{[\omega_{\rm m}^{2}+(\gamma_{\rm
m}/2)^{2}][\omega_{c}^{2}+(\kappa_{c}/2)^{2}]}.$ (18)
Figure 3: Bistability behaviour for mean photon number in the second resonator
$I_{a}$ (blue solid curve) and mean photon number $I_{b}$ (red dashed curve)
as a function of the pump power in the presence of the superconducing qubit
($\alpha\neq 0,F(I_{b}<1)$). All parameters are the same as in Fig. 2.
We immediately see from Eq. (14) that in the absence of the superconducting
circuit, which amounts to setting $\alpha=0$ in (17) and (18), the factor $F$
that appears in (14) becomes, $F(I_{b})=1$. The resulting equation reproduces
the cubic equation for the mean photon number $I_{a}$ as in the standard
optomechanical coupling Set12 , which is known to exhibit bistable behaviour.
In general, for electromechanical system considered here, $F(I_{b})<1$ (for
typical experimental parameters Teu11 ), thus yielding the same form of cubic
equation for $I_{a}$. In Fig. 2, we plot the mean photon number $I_{a}$ as
function of the pump power in the presence and absence of the superconducting
qubit. Figure 2a shows, in the presence of the qubit ($\alpha\neq 0$), the
bistability behaviour only appears when the microwave resonator is pumped at
nW range. For example, for the parameters used in Fig. (2)a, the lower tuning
point is obtained at $P\approx 28\text{nW}$. The hysteresis then follows the
arrow and jumps to the upper branch. Then scanning the pump power towards
zero, one obtains the other turning point at very low pump power
$P=0.02\text{pW}$. On the other hand, in the absence of the superconducting
qubit (see Fig. 2b), the pump power required to achieve the bistable behaviour
reduces to the pW range, with the lower turning point appearing at
$P=0.26\text{pW}$. Therefore, the bistable behaviour in the mean photon number
in the second resonator can be observed at relatively high pump power when the
nanomechanical oscillator coupled to the superconducting qubit.Therefore, when
the nanomechanical oscillator is coupled to the superconducting qubit, a
relatively high power is needed to observe a bistable behavior.
Furthermore, according to Eq. (II.2), since $\alpha/\omega_{\rm m}\ll
1(\beta_{i}\approx 0)$, the mean photon number $I_{a}$ is related to the
phonon number via $I_{a}^{2}=I_{b}[\omega_{\rm m}^{2}+(\gamma_{\rm
m}/2)^{2}]/g_{a}^{2}$, indicating that the phonon number also exhibits
bistability. Figure 3 compares the bistable behavior for both $I_{a}$ and
$I_{b}$. As can be seen from this figure, the bistability occurs at the same
power range; however, their corresponding photon and phonon numbers are
different by four orders of magnitude. Note that, as expected, all the
bistable behaviours are observed in the red detuned regime,
$\Delta_{a}=\omega_{a}-\omega_{d}>0$. From application viewpoint, the bistable
behaviour can used as a fast optical switching.
### II.3 Fluctuations about the classical mean value
The quantum Langevin equations [Eqs. (6)-(8)] can be solved analytically by
adopting a linearization scheme Set10 ; Set11 in which the operators are
expressed as the sum of their mean values plus fluctuations, that is,
$a=\langle a\rangle+\delta a$, $b=\langle b\rangle+\delta b$, and $c=\langle
c\rangle+\delta c$. The equations for fluctuation operators then read
$\displaystyle\delta\dot{a}=-\left(i\Delta_{\rm
f}+\frac{\kappa_{a}}{2}\right)\delta a-i\textit{g}_{a}\langle a\rangle(\delta
b+\delta b^{{\dagger}})+\sqrt{\kappa_{a}}a_{\text{in}},$ (19)
$\displaystyle\delta\dot{b}=$ $\displaystyle-\left(i\omega_{\rm
m}+\frac{\gamma_{\rm m}}{2}\right)\delta b-i\textit{g}_{a}(\langle
a^{{\dagger}}\rangle\delta a+\langle a\rangle\delta a^{{\dagger}})$
$\displaystyle-2i\alpha[\langle c\rangle\delta b^{{\dagger}}+\langle
b^{{\dagger}}\rangle\delta c]+\sqrt{\gamma_{\rm m}}b_{\text{in}},$ (20)
$\displaystyle\delta\dot{c}=-\left(i\omega_{c}+\frac{\kappa_{c}}{2}\right)\delta
c+2i\alpha\langle b\rangle\delta b+\sqrt{\kappa_{c}}c_{\text{in}}.$ (21)
The solutions to these equations can easily be obtained in frequency domain.
To this end, writing the Fourier transform of Eqs. (19)-(21) and their complex
conjugates, we get
$\mathcal{A}\mathcal{U}=\mathcal{N},$ (22)
where
$\mathcal{A}=\left(\begin{array}[]{cccccc}\eta_{+}&0&G&G&0&0\\\
0&\eta_{-}&G^{*}&G^{*}&0&0\\\ -G^{*}&G&v_{+}&\mathcal{C}&\mathcal{B}^{*}&0\\\
G^{*}&-G&\mathcal{C}^{*}&v_{-}&0&\mathcal{B}\\\ 0&0&\mathcal{B}&0&u_{+}&0\\\
0&0&0&\mathcal{B}^{*}&0&u_{-}\\\ \end{array}\right),$ (23)
$\mathcal{U}=(\delta a,\delta a^{{\dagger}},\delta b,\delta
b^{{\dagger}},\delta c,\delta c^{{\dagger}})^{T}$ and
$\mathcal{N}=(\sqrt{\kappa_{a}}a_{\rm in},\sqrt{\kappa_{a}}a_{\rm
in}^{{\dagger}},\sqrt{\gamma_{\rm m}}b_{\rm in},\sqrt{\gamma_{\rm m}}b_{\rm
in}^{{\dagger}},\sqrt{\kappa_{c}}c_{\rm in},\sqrt{\kappa_{c}}c_{\rm
in}^{{\dagger}})^{T}$ with $\eta_{\pm}=\kappa_{a}/2+i(\omega\pm\Delta_{\rm
f})$, $v_{\pm}=\gamma_{\rm m}/2+i(\omega\pm\omega_{\rm m})$, and
$u_{\pm}=\kappa_{c}/2+i(\omega\pm\omega_{c})$, $G=i\textit{g}_{a}\langle
a\rangle,\mathcal{B}=-2i\alpha\langle b\rangle,\mathcal{C}=2i\alpha\langle
c\rangle.$
The solution for the fluctuation operator $\delta a$ of the second resonator
field has the form
$\delta
a(\omega)=\xi_{1}a_{\text{in}}+\xi_{2}a_{\text{in}}^{{\dagger}}+\xi_{3}b_{\text{in}}+\xi_{4}b_{\text{in}}^{{\dagger}}+\xi_{5}c_{\text{in}}+\xi_{6}c_{\text{in}}^{{\dagger}}.$
(24)
The explicit expression for the coefficients $\xi_{i}$ are given in the
Appendix. Similarly, the expressions for $\delta b(\omega)$ and $\delta
c(\omega)$ can be obtained from (23). In the following, we use (24) to analyze
the squeezing of the transmitted microwave field from the second resonator.
## III Squeezing spectrum
It was shown that the optomechanical coupling can lead to squeezing of the
nanomechanical motion, which can be inferred by measuring the squeezing of the
transmitted microwave field Fab94 ; Woo08 ; Set11 . Here we investigate the
squeezing properties of the transmitted microwave field in the presence of the
nonlinearity induced by superconducting qubit [represented by the effective
coupling $\alpha$ in Eq. (II)] as well as the nonlinearity due to the
optomechanical coupling [represented by coupling $\textit{g}_{a}$ in Eq.
(II)]. The stationary squeezing spectrum of the transmitted field is given by
$\displaystyle S(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\langle\delta
X_{\phi}^{\text{out}}(t+\tau)\delta
X_{\phi}^{\text{out}}(t)\rangle_{\text{ss}}e^{i\omega\tau}$
$\displaystyle=\langle\delta X_{\phi}^{\text{out}}(\omega)\delta
X_{\phi}^{\text{out}}(\omega)\rangle$ (25)
where $\delta X_{\phi}^{\text{out}}=e^{i\phi}\delta
a_{\text{out}}+e^{-i\phi}\delta a^{{\dagger}}_{\text{out}}$ with
$a_{\text{out}}=\sqrt{\kappa_{a}}\delta a-a_{\text{in}}$ being the input-
output relation Mil94 and $\phi$ the measurement phase angle determined by
the local oscillator. The squeezing spectrum can be put in the form
$S(\omega)=1+C_{a^{{\dagger}}a}^{\text{out}}+e^{-2i\phi}C_{aa}^{\text{out}}+e^{2i\phi}C_{a^{{\dagger}}a^{{\dagger}}}^{\text{out}},$
(26)
where $\langle\delta a_{\text{out}}(\omega)\delta
a_{\text{out}}(\omega^{\prime})\rangle=2\pi
C_{aa}^{\text{out}}(\omega)\delta(\omega+\omega^{\prime})$ and $\langle\delta
a_{\text{out}}(\omega)^{{\dagger}}\delta
a_{\text{out}}(\omega^{\prime})\rangle=2\pi
C_{a^{{\dagger}}a}^{\text{out}}(\omega)\delta(\omega+\omega^{\prime})$. The
degree of squeezing depends on the direction of the measurement of the
quadratures, thus can be optimized over the phase angle $\phi$. Using the
angle which optimizes the squeezing Set111 , we obtain
$S_{\text{opt}}^{(\pm)}(\omega)=1+2C_{a^{{\dagger}}a}^{\text{out}}(\omega)\pm
2|C_{aa}^{\text{out}}(\omega)|.$ (27)
$S_{\text{opt}}^{(-)}$ corresponds to the spectrum of the squeezed quadrature,
while $S_{\text{opt}}^{(+)}$ represents the spectrum of the unsqueezed
quadrature. Using the solution (24) and the correlation properties of the
noise forces (9) and (10), the spectrum of the squeezed quadrature takes the
form
$\displaystyle
S_{\text{opt}}^{(-)}(\omega)=1+2C_{a^{{\dagger}}a}^{\text{out}}(\omega)-2|C_{aa}^{\text{out}}(\omega)|,$
(28)
where
$\displaystyle C_{a^{{\dagger}}a}^{\text{out}}(\omega)=$
$\displaystyle\kappa_{a}\big{[}n_{a}\xi_{1}(\omega)\xi_{1}^{*}(-\omega)+(n_{a}+1)\xi_{2}(\omega)\xi_{2}^{*}(-\omega)$
$\displaystyle+n_{b}\xi_{3}(\omega)\xi_{3}^{*}(-\omega)+(n_{b}+1)\xi_{4}(\omega)\xi_{4}^{*}(-\omega)$
$\displaystyle+n_{c}\xi_{5}(\omega)\xi_{5}^{*}(-\omega)+(n_{c}+1)\xi_{6}(\omega)\xi_{6}^{*}(-\omega)\big{]}$
$\displaystyle-2\sqrt{\kappa_{a}}n_{a}[\xi_{1}(\omega)+\xi_{1}^{*}(-\omega)]+n_{a},$
(29) $\displaystyle C_{aa}^{\text{out}}(\omega)=$
$\displaystyle\kappa_{a}\big{[}n_{a}\xi_{1}(\omega)\xi_{2}^{*}(-\omega)+(n_{a}+1)\xi_{1}^{*}(-\omega)\xi_{2}(\omega)$
$\displaystyle+n_{b}\xi_{3}(\omega)\xi_{4}^{*}(-\omega)+(n_{b}+1)\xi_{3}^{*}(-\omega)\xi_{4}(\omega)$
$\displaystyle+n_{c}\xi_{5}(\omega)\xi_{6}^{*}(-\omega)+(n_{c}+1)\xi_{5}^{*}(-\omega)\xi_{6}(\omega)\big{]}$
$\displaystyle-\sqrt{\kappa_{a}}[n_{a}\xi_{2}^{*}(-\omega)+(n_{a}+1)\xi_{2}(\omega)].$
(30)
Based on the definition of the quadrature $\delta X_{\varphi}^{\text{out}}$,
the microwave field is squeezed when the value of the squeezing spectrum is
below the standard quantum limit, $S_{\text{opt}}^{(-)}(\omega)=1$.
Figure 4: Plots of the squeezing spectrum of the transmitted microwave field
[Eq. (28)] for drive pump power $P=8~{}\mu\text{W}$, for drive frequency
$\omega_{d}/2\pi=7.4999~{}\text{GHz}$, for membrane’s bath temperature
$T_{b}=50~{}\text{mK}$, for bath temperature of the first resonator,
$T_{c}=2~{}\text{K}$, and for various bath temperatures of the second
resonator: (a) $T_{a}=150~{}\text{mK}$ (solid green curve), (b)
$T_{b}=250~{}\text{mK}$ (dashed red curve), and (c) $T_{b}=350~{}\text{mK}$
(dot-dashed black curve). The horizontal solid line represents the standard
quantum limit [$S_{\text{opt}}^{(-)}(\omega)=1$], below which indicates
squeezing. All other parameters are the same as in Fig. 2.
In Fig. 4, we plot the squeezing spectrum of the microwave field as a function
of the temperature $T_{a}$ of the second resonator thermal bath. As can be
seen from this figure, the microwave field exhibits squeezing with the degree
of squeezing strongly relying on the thermal bath temperature, $T_{a}$.
Obviously, the amount of squeezing degrades as the thermal temperature
increases and it ultimately disappears when the bath temperature reaches at
$T_{a}\approx 600~{}\text{mK}$ for the parameters used in Fig. 4. We also
found that the degree of squeezing is less sensitive to the first resonator
thermal bath temperature $T_{c}$. This is because the second resonator is not
directly coupled to the first resonator thermal bath, though it is indirectly
coupled via the nanomechanical oscillator through a low-loss capacitor. The
other interesting aspect is that the spectrum shows double dips for strong
enough pump power indicating that the optomechanical interaction reached the
strong coupling regime, a requirement to observe quantum mechanical effects.
It is worth mentioning that to make sure that the squeezing is determined in
the stable regime, the microwave drive frequency $\omega_{d}$ is deliberately
chosen close to resonance frequency of the second resonator $\omega_{a}$.
Figure 5: Plots of the squeezing spectrum vs the microwave drive pump power P
($\mu$W) for the bath temperature of the first resonator $T_{c}=2~{}\text{K}$,
the membrane’s bath temperature, $T_{b}=10~{}\text{mK}$, and for different
values of the bath temperature $T_{a}$ of the second resonator: (a)
$T_{a}=250~{}\text{mK}$ (dot-dashed black curve), (b) $T_{a}=150~{}\text{mK}$
(dashed red curve), and (c) $T_{a}=50~{}\text{mK}$ (solid green curve). All
other parameters are the same as in Fig. 2. Figure 6: Plots of the squeezing
spectrum (in logarithmic scale) vs the bath temperature of the second
resonator $T_{a}$ for a pump power $P=10~{}\mu\text{W}$, for the bath
temperature $T_{c}=2~{}\text{K}$ of the first resonator, and for different
values of the membrane’s bath temperature, $T_{b}=1~{}\text{K}$ (dotted blue
curve), $T_{b}=0.25~{}\text{K}$ (dot-dashed black curve),
$T_{b}=0.05~{}\text{K}$ (dashed red curve), $T_{b}=0.01~{}\text{K}$ (solid
green curve). All other parameters are the same as in Fig. 2.
The other important external parameter that can be used to control the degree
of the squeezing is the strength of the microwave drive. The dependence of the
squeezing on the drive pump power is illustrated in Fig. 5. When the microwave
drive frequency is close to the resonator frequency, that is, when
$\Delta_{a}/2\pi=0.1\text{MHz}$, the squeezing gradually develops as the pump
power is increased to the range of few $\mu\text{W}$. Further increase in the
pump power strength leads to an optimum squeezing that can possibly be
achieved for a given value of temperature of the thermal baths. For example,
for $T_{a}=10\text{mK},T_{b}=10\text{mK}$, and $T_{c}=2\text{K}$, the maximum
squeezing is $\approx 97\%$ below the standard quantum limit at a pump power
$P\approx 10\mu\text{W}$. However, when the pump power is increased beyond
$P\approx 10\text{mW}$, the degree of squeezing sharply decrease and becomes
strongly dependent on $T_{a}$. The other interesting aspect is that although
the bath temperature $T_{a}$ is increased to $250\text{mK}$, there exists an
optimum power for which the squeezing is still the maximum achievable. Even
though the overall squeezing is due to both nonlinearities induced by the
effective coupling between the first resonator and the nanomechanical
oscillator and the optomechanical coupling, the enhancement of the squeezing
with pump power is mainly due to the optomechanical coupling. This is because
the pump power directly affects the intensity in the second resonator
($\text{TLR}_{2}$), which in turn increases the strength of the optomechanical
coupling.
Fixing the power ($P=10\text{mW}$) at which the squeezing is maximum, it is
important to understand the interplay between the bath temperatures $T_{a}$
and $T_{b}$ in determining the degree of squeezing of the microwave field.
Figure 6 shows that the squeezing persists up to $T_{a}\approx\text{2K}$.
While the degree of squeezing is weakly dependent on the thermal bath
temperature $T_{b}$ of the nanomechanical oscillator when $T_{a}>0.1\text{K}$,
the squeezing decreases with increasing $T_{b}$ for $T_{a}<0.1\text{K}$.
Therefore, a strong and robust squeezing can be achieved by tuning the pump
power close to $P=10\mu\text{W}$ while keeping the bath temperatures
$T_{a},T_{b}$ within $\lesssim 1$K range.
## IV Optomechanical entanglement
It has been shown that the optomechanical coupling gives rise to entanglement
between the resonator field and mechanical motion Zha03 ; Pin05 ; Vit08 . Here
we analyze the robustness of the optomechanical entanglement against thermal
decoherence in the presence of the two different nonlinearities discussed
earlier. We also analyze how the degree entanglement depends on the drive pump
power and the detuning $\Delta_{a}$. In order to investigate the
optomechanical entanglement, it is more convenient to use the quadrature
operators defined by
$\displaystyle X_{a}=\frac{1}{\sqrt{2}}(\delta a+\delta
a^{{\dagger}}),Y_{a}=\frac{1}{\sqrt{2}i}(\delta a-\delta a^{{\dagger}}),$ (31)
$\displaystyle X_{b}=\frac{1}{\sqrt{2}}(\delta b+\delta
b^{{\dagger}}),Y_{b}=\frac{1}{\sqrt{2}i}(\delta b-\delta b^{{\dagger}}),$ (32)
$\displaystyle X_{c}=\frac{1}{\sqrt{2}}(\delta c+\delta
c^{{\dagger}}),Y_{c}=\frac{1}{\sqrt{2}i}(\delta c-\delta c^{{\dagger}}).$ (33)
The equations of motion for these quadrature operators can be put in a matrix
form
$\dot{u}(t)=Mu(t)+f(t),$ (34)
where
$R=\left(\begin{array}[]{cccccc}-\kappa_{a}/2&\Delta_{f\rm}&-2\text{g}_{a}\eta_{b}&0&0&0\\\
\Delta_{f\rm}&-\kappa_{a}/2&-2\text{g}_{a}\mu_{a}&0&0&0\\\ 0&0&-\gamma_{\rm
m}/2+2\alpha\mu_{c}&\omega_{\rm
m}-2\alpha\eta_{c}&-2\alpha\mu_{b}&2\alpha\eta_{b}\\\
-2\text{g}_{a}\eta_{a}&-2\text{g}_{a}\mu_{a}&-(\omega_{\rm
m}+2\alpha\eta_{c})&-(\gamma_{\rm
m}/2+2\alpha\mu_{c})&-2\alpha\eta_{b}&-2\alpha\mu_{b}\\\
0&0&-2\alpha\mu_{b}&-2\alpha\eta_{b}&-\kappa_{c}/2&\omega_{c}\\\
0&0&2\alpha\eta_{b}&-2\alpha\mu_{b}&-\omega_{c}&-\kappa_{c}/2\\\
\end{array}\right),u=\left(\begin{array}[]{c}\delta X_{a}\\\ \delta Y_{a}\\\
\delta X_{b}\\\ \delta Y_{b}\\\ \delta X_{c}\\\ \delta
Y_{c}\end{array}\right),f=\left(\begin{array}[]{c}\sqrt{\kappa_{a}}X_{a}^{\text{in}}\\\
\sqrt{\kappa_{a}}Y_{a}^{\text{in}}\\\ \sqrt{\gamma_{\rm
m}}X_{b}^{\text{in}}\\\ \sqrt{\gamma_{\rm m}}Y_{b}^{\text{in}}\\\
\sqrt{\kappa_{c}}X_{c}^{\text{in}}\\\
\sqrt{\kappa_{c}}Y_{c}^{\text{in}}\end{array}\right),$ (35)
where $\eta_{L}=\frac{1}{2}(\langle L\rangle+\langle L^{{\dagger}}\rangle)$,
$\mu_{L}=\frac{1}{2i}(\langle L\rangle-\langle L^{{\dagger}}\rangle)$ and
$X_{L}^{\text{in}}=(\delta L_{\text{in}}+\delta
L^{{\dagger}}_{\text{in}})/\sqrt{2}$, $Y_{L}^{\text{in}}=i(\delta
L^{{\dagger}}_{\text{in}}-\delta L_{\text{in}})/\sqrt{2}$, where $L=a,b,c$.
In this work, we are interested in the steady state optomechanical
entanglement. It is then sufficient to focus on the subspace spanned by the
second resonator and mechanical mode (the upper left $4\times 4$ matrix in
$R$). To study the stationary optomechanical entanglement, one needs to find a
stable solution for Eq. (34), so that it reaches a unique steady state
independent of the initial condition. Since we have assumed the quantum noises
$a_{\rm in},b_{\rm in}$ and $c_{\rm in}$ to be zero-mean Gaussian noises and
the corresponding equations for fluctuations $(\delta a,\delta b$, and $\delta
c$) are linearized, the quantum steady state for fluctuations is simply a
zero-mean Gaussian state, which is fully characterized by $4\times 4$
correlation matrix $V_{ij}=[\langle
u_{i}(\infty)u_{j}(\infty)+u_{j}(\infty)u_{i}(\infty)\rangle]/2$. The solution
to Eq. (34),
$u(t)=M(t)u(0)+\int_{0}^{t}dt^{\prime}M(t^{\prime})f(t-t^{\prime})$, where
$M(t)=\exp(Rt)$, is stable and reaches steady state when all of the
eigenvalues of $R$ have negative real parts so that $M(\infty)=0$. The
stability condition can be derived by applying the Routh-Hurwitz criterion
DeJ87 . For all results presented in this paper, the stability conditions are
satisfied. When the system is stable one easily get
$\mathcal{V}_{ij}=\sum_{lm}\int_{0}^{\infty}dt^{\prime}\int_{0}^{\infty}dt^{\prime\prime}M_{il}(t^{\prime})M_{jm}\Pi_{lm}(t^{\prime}-t^{\prime\prime}),$
(36)
where the stationary noise correlation matrix is give by $\Pi_{lm}=[\langle
f_{l}(t)f_{m}(t^{\prime\prime})+f_{m}(t^{\prime\prime})f_{l}(t)\rangle]/2$,
where $f_{i}$ is the $i$th element of the column vector $f$. Since all noise
correlations are assumed to be Markovian (delta-correlated) and all components
of $f(t)$ are uncorrelated, the noise correlation matrix takes a simple form
$\Pi_{lm}(t^{\prime}-t^{\prime\prime})=D_{lm}\delta(t^{\prime}-t^{\prime\prime})$,
where
$\displaystyle D=$
$\displaystyle\text{Diag}[\kappa_{a}(2n_{a}+1)/2,\kappa_{a}(2n_{b}+1)/2,\gamma_{\rm
m}(2n_{b}+1)/2,$ $\displaystyle\gamma_{\rm
m}(2n_{b}+1)/2,\kappa_{c}(2n_{c}+1)/2,\kappa_{c}(2n_{c}+1)/2]$ (37)
is the diagonal matrix. As a result, Eq. (36) becomes
$V=\int_{0}^{\infty}dt^{\prime}M(t^{\prime})DM(t^{\prime})^{\rm T}$. When the
stability conditions are satisfied, i.e., $M(\infty)=0$, one readily obtain an
equation for steady state correlation matrix
$R\mathcal{V}+\mathcal{V}R^{\rm T}=-D.$ (38)
Equation (38) is a linear equation (also known as Lyapunov equation) for
$\mathcal{V}$ and can be solved in straight-forward manner. However, the
solution for our system is rather lengthy and will not be presented here. We
instead solve (38) numerically to analyze the optomechanical entanglement.
In order to analyze the optomechanical entanglement, we employ the logarithmic
negativity $E_{N}$, a quantity which has been proposed as a measure of
bipartite entanglement Vid02 . For continuous variables, $E_{N}$ is defined as
$E_{N}=\max[0,-\ln 2\chi],$ (39)
where
$\chi=2^{-1/2}\left[\sigma-\sqrt{\sigma^{2}-4\text{det}\mathcal{V}}\right]^{1/2}$
is the lowest simplistic eigenvalue of the partial transpose of the $4\times
4$ correlation matrix $\mathcal{V}$ with
$\sigma=\det\mathcal{V}_{A}+\det\mathcal{V}_{B}-2\det\mathcal{V}_{AB}$. Here
$\mathcal{V}_{A}$ and $\mathcal{V}_{B}$ represent the second resonator field
and mechanical mode, respectively, while $\mathcal{V}_{AB}$ describes the
optomechanical correlation. These matrices are elements of the $2\times 2$
block form of the correlation matrix
$\mathcal{V}\equiv\left(\begin{array}[]{cc}\mathcal{V}_{A}&\mathcal{V}_{AB}\\\
\mathcal{V}_{AB}^{T}&\mathcal{V}_{B}\\\ \end{array}\right).$ (40)
Any two modes are said to be entangled when the logarithmic negativity $E_{N}$
is positive.
Figure 7: Plots of the logarithmic negativity $E_{N}$ vs the temperature of
the first resonator thermal bath, $T_{c}$ for the drive pump power
$P=1~{}\mu\text{W}$, $\Delta_{a}/2\pi=0.1\text{MHz}$ and for different values
of the second resonator thermal bath temperature $T_{a}$= $50~{}\text{mK}$
(dotted blue curve), $100~{}\text{mK}$ (dotdashed black curve),
$150~{}\text{mK}$ (dashed red curve), and $200~{}\text{mK}$ (solid green
curve). All other parameters the same as in Fig. 2. Figure 8: Plots of the
logarithmic negativity $E_{N}$ vs the detuning $\Delta_{a}$ for the thermal
bath temperature of the first resonator $T_{c}=50~{}\text{mK}$ and for the
thermal bath temperature of the second resonator $T_{a}$= $100~{}\text{mK}$
and for different values of the microwave drive pump power
$P=0.5~{}\mu\text{W}$ (dotted blue curve), $1.0~{}\mu\text{W}$ (dashed red
curve), and $2.0~{}\mu\text{W}$ (solid green curve). All other parameters as
the same as in Fig. 2.
In Fig. 7, we plot the logarithmic negativity $E_{N}$ as a function the
thermal bath temperature $T_{c}$ of the first resonator while varying the
thermal bath temperature $T_{a}$ of the second resonator at a fixed drive pump
power, $P=1\mu\text{W}$. This figure shows that the mechanical mode is
entangled with the resonator mode of the second resonator in the steady state.
The entanglement strongly relies on the bath temperatures $T_{a}$ and $T_{c}$
of the first and second resonators, respectively. In general, the
optomechanical entanglement degrades as the thermal bath temperatures
increases. For instance, when the temperature of the second resonator fixed at
$50\text{mK}$, the entanglement survives until the bath temperature $T_{c}$ of
the first resonator reaches about $100\text{mK}$. If the temperature $T_{a}$
is further increased, the critical temperature $T_{c}$ above which the
entanglement disappears decreases. Therefore, at constant pump power, the
entanglement can be controlled by tuning the bath temperatures of the two
resonators.
Another system parameter that can be used as an external knob to control the
degree of entanglement is the detuning $\Delta_{a}$. Figure 8 illustrates the
logarithmic negativity versus the detuning $\Delta_{a}$ for different values
of the pump power. Close to resonance ($\Delta_{a}=0$) and for the pump power
$P\gtrsim 1.2\mu\text{W}$, there is no optomechanical entanglement; however,
the entanglement between the nanomechanical oscillator and the resonator field
arises when the detuning is further increased, and reaches stationary values
for $\Delta_{a}/2\pi\simeq\omega_{\rm m}/2\pi=10\text{MHz}$, which is
consistent with the results in the literature Vit08 . The interesting aspect
of our result is that the entanglement persists for wide range of detuning
$\Delta_{a}$, opposed to the results reported for systems which only involve
the optomechanical coupling Vit08 .
## V Conclusion
We analyzed the squeezing and optomechanical entanglement in electromechanical
system in which a superconducting charge qubit is coupled to a transmission
line resonator and a movable membrane, which in turn is coupled to a second
transmission line resonator. We show that due the nonlinearities induced by
the optomechanical coupling and the superconducting qubit, the transmitted
microwave field exhibits strong squeezing. Besides, we showed that robust
optomechanical entanglement can be achieved by tuning the bath temperature of
the two resonators. We also showed that the generated entanglement can be
controlled for appropriate choice of the input drive pump power and the
detuning of the drive frequency from the resonator frequency. Merging of
optomechanics with electrical circuits opens new avenue for an alternative way
to explore creation and manipulation of quantum states of microscopic systems.
###### Acknowledgements.
The authors thank Konstantin Dorfman for useful discussions. This research was
funded by the Office of the Director of National Intelligence (ODNI),
Intelligence Advanced Research Projects Activity (IARPA), through the Army
Research Office Grant No. W911NF-10-1-0334. All statements of fact, opinion or
conclusions contained herein are those of the authors and should not be
construed as representing the official views or policies of IARPA, the ODNI,
or the U.S. Government. We also acknowledge support from the ARO MURI Grant
No. W911NF-11-1-0268.
## Appendix A The terms that appear in Eq. (24)
The coefficients that appear in Eq. (24) are given by
$\displaystyle\xi_{1}$
$\displaystyle=\frac{\sqrt{\kappa_{a}}}{d(\omega)}\eta_{-}[(u_{-}v_{-}-|\mathcal{B}|^{2})(u_{+}v_{+}-|\mathcal{B}|^{2})-u_{-}u_{+}|\mathcal{C}|^{2}]$
$\displaystyle-\frac{\sqrt{\kappa_{a}}|G|^{2}}{d(\omega)}\big{\\{}[(u_{-}-u_{+})|\mathcal{B}|^{2}+(|\mathcal{B}|^{2}$
$\displaystyle+u_{-}u_{+}[v_{-}-v_{+}+2i\text{Im}(\mathcal{C})]\big{\\}},$
(41) $\displaystyle\xi_{2}=$
$\displaystyle\frac{\sqrt{\kappa_{a}}G^{2}}{d(\omega)}\Big{\\{}[(u_{-}-u_{+})|\mathcal{B}|^{2}$
$\displaystyle+u_{-}u_{+}[v_{-}-v_{+}+2i\text{Im}(\mathcal{C})]\Big{\\}},$
(42) $\xi_{3}=\frac{\sqrt{\gamma_{\rm
m}}G\eta_{-}}{d(\omega)}u_{+}\left(|\mathcal{B}|^{2}+u_{-}\mathcal{C}^{*}-u_{-}v_{-}\right),$
(43) $\xi_{4}=\frac{\sqrt{\gamma_{\rm
m}}G}{d(\omega)}\eta_{-}u_{-}\left(|\mathcal{B}|^{2}+u_{+}\mathcal{C}^{*}-u_{+}v_{+}\right),$
(44)
$\xi_{5}=-\frac{\sqrt{\kappa_{c}}G}{d(\omega)}\eta_{-}\mathcal{B}^{*}\left(|\mathcal{B}|^{2}+u_{-}\mathcal{C}^{*}-u_{-}v_{-}\right),$
(45)
$\xi_{6}=-\frac{\sqrt{\kappa_{c}}G}{d(\omega)}\eta_{-}\mathcal{B}\left(|\mathcal{B}|^{2}+u_{-}\mathcal{C}-u_{+}v_{+}\right),$
(46)
where
$\displaystyle d(\omega)=$
$\displaystyle[(u_{-}v_{-}-|\mathcal{B}|^{2})(v_{+}u_{+}-|\mathcal{B}|^{2})-u_{-}u_{-}|\mathcal{C}|^{2}]\eta_{-}\eta_{+}$
$\displaystyle+|G|^{2}\Big{\\{}u_{-}[|\mathcal{B}|^{2}+u_{+}(v_{-}-v_{+}+2i\text{Im}(\mathcal{C}))]$
$\displaystyle-u_{+}|\mathcal{B}|^{2}\Big{\\}}(\eta_{-}-\eta_{+}).$ (47)
## References
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* (17) T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, Phys. Rev. X 3, 031012 (2013)
* (18) K.L. Ekinci and M.L. Roukes, Rev. Sci. Instrum. 76, 061101 (2005).
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* (21) P. Rabl, A. Shnirman, and P. Zoller, Phys. Rev. B 70, 205304 (2004).
* (22) X. Zhou and A. Mizel, Phys. Rev. Lett. 97, 267201 (2006).
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* (24) W. Y. Huo and G.L. Long, Appl. Phys. Lett. 10, 133102(2008).
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* (28) E. A. Sete, S. Das, and H. Eleuch, Phys. Rev. A 83, 023822 (2011).
* (29) D.F. Walls and G. J. Milburn, Quantum Optics (Spinger-Verlag, Berlin, 2008).
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|
arxiv-papers
| 2013-12-12T20:03:01 |
2024-09-04T02:49:55.369491
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eyob A. Sete and H. Eleuch",
"submitter": "Eyob A. Sete",
"url": "https://arxiv.org/abs/1312.3606"
}
|
1312.3677
|
# Foil Diffuser Investigation with GEANT4
Joseph M. Fabritius II, Konstantin Borozdin, Peter Walstrom
###### Abstract
An investigation into the appropriate materials for use as a diffuser foil in
electron radiography was undertaken in GEANT4. Simulations were run using
various refractory materials to determine a material of appropriate Z number
such that energy loss is minimal. The plotted results of angular spread and
energy spread are shown. It is concluded that higher Z number materials such
as tungsten, tantalum, platinum or uranium could be used as diffuser
materials. Also, an investigation into the handling of bremsstrahlung,
multiple coulomb scattering, and ionization in GEANT4 was performed.
## 1 Motivation
In deciding on the best material for a diffuser foil for increasing the
angular spread of the accelerator beam in electron radiography, one must take
into account four physical phenomena:
1. 1.
Multiple Coulomb scattering (the desired effect, which increases the angular
spread of the beam). This is mostly due to elastic scattering from nuclei and
is approximately described by the Particle Data Group (PDG) formula 27.14.
2. 2.
Ionization energy loss and straggling (the part of the energy loss not due to
bremsstrahlung)
3. 3.
Energy loss due to bremsstrahlung
4. 4.
Melting temperature
Choosing a diffuser foil for electron radiography experiments requires the
material must be both refractory, that is resistant to melting or deformation
under high temperatures, and to also have a low energy spread so that the
effects of chromatic blur are lessened. Chromatic blurring effects can be
directly seen in the energy loss of the beam through the material. By choosing
a material with less energy loss an appropriate diffuser material can be
found. To accomplish this task several elements were chosen from the Particle
Data Group table on atomic properties of materials. Those materials that were
found to have a melting point of over 1400 K were chosen for investigation:
carbon (graphite), silicon, iron, tantalum, tungsten, platinum, and uranium.
The optimal material Z was found by first choosing the foil thickness of each
refractory material so that a specified angular spread, $\theta_{rms}$, was
achieved. A desired angular spread of 0.2 mRad was chosen as the reference of
comparison between diffuser materials. The diffuser thickness was calculated
using the multiple scattering distribution equation:
$\theta_{rms}=\frac{13.6\textnormal{MeV}}{\beta
cp}z\sqrt{\frac{x}{X_{0}}}\left[1+0.038\frac{x}{X_{0}}\right]$ (1)
where $\beta c$ is the electron speed, $z$ is the particle charge number, $x$
is the thickness of the material, and $X_{0}$ is the radiation length of the
material. The main material dependence is in the factor
$\sqrt{\frac{x}{X_{0}}}$, where $x$ is the thickness in g/cm2 and $X_{0}$ is
the radiation length, also in g/cm2. The above equation was taken from the PDG
journal(2010 pg 290, 27.14) on Multiple scattering through small angles.
The simulations for investigating the foil materials were run in GEANT4 using
the same code for previous electron radiography investigations. A pencil beam
of 12 GeV electrons was fired at a slab of material with a detector situated
just beyond the object for capturing deflected electrons. Secondary particles
were ignored in the detector for these simulations. The geometry is shown in
the figure below.
Figure 1: The test simulation geometry. Figure is not to scale.
Using Equation (1), the initial material thicknesses were used for preliminary
simulations. With these initial simulations the angular distribution of the
beam through the diffuser was plotted in ROOT to verify the spread was 0.2
mRad. The thickness of the material slab was then incrementally adjusted until
the resulting angular spread was 0.2 $\pm$ 0.09 mRad. The calculated and
adjusted thicknesses are presented in the table below.
Material | Z | Melting | Calculated | Adjusted
---|---|---|---|---
| number | Point (K) | Thickness (m) | Thickness (m)
Graphite | 6 | 3600** | 0.007736 | 0.006900
Silicon | 14 | 1687 | 0.003847 | 0.003100
Titanium | 22 | 1941 | 0.001470 | 0.001150
Iron | 26 | 1811 | 0.000722 | 0.000600
Tantalum | 73 | 3293.15 | 0.000168 | 0.000120
Tungsten | 74 | 3695 | 0.000144 | 0.000110
Platinum | 78 | 4098 | 0.000125 | 0.000090
Uranium | 92 | 1408 | 0.000129 | 0.000095
| | **Sublimation | |
| | Temperature | |
## 2 Plots and Analysis
For each diffuser material, a histogram of the energy loss of the electron
beam through the material was created. When compared along the same energy
range it can be seen that for the lower Z materials there is a much larger
energy loss, evident in the RMS values shown in the plots below. The peak of
the histogram can be seen to decrease as Z number decreases.
Figure 2: Energy loss histograms plotted for the lower Z materials. The energy
scale was focused on the range of 11994 MeV to 12000 MeV, with 500 bins.
Figure 3: Energy loss histograms plotted for the higher Z materials. The
energy scale was focused on the range of 11994 MeV to 12000 MeV, with 500
bins.
Higher Z materials evidently have less energy loss, and will thus make for
better diffusers in electron radiography as the chromatic blur effects will be
lessened than with lower Z. To examine a better comparison of the higher Z
materials the energy histograms were replotted in a smaller interval to focus
on the peak area. From the newer energy plots it is apparent that the peaks
and RMS values are close enough that there is no appreciable difference and
the choice of diffuser material will depend on other criteria, such as
availability of material in foil form, or secondary particle creation. Further
investigative studies will be required.
Figure 4: Energy loss histograms plotted for the higher Z materials. The
energy scale was focused on the range of 11999 MeV to 12000 MeV, with 500
bins.
## 3 GEANT4 Physics Investigation
Apart from the refractory nature of the material, the three other processes
are important in our investigation. The Multiple Coulomb scattering effect is
described by Equation (1) above.
The deterministic part of the ionization energy loss
$\frac{dE}{dx}_{\textit{ion.}}$ (in units of MeV-cm2/g) has a somewhat more
complicated material dependence, including some dependence on the mean
ionization potential of the material, but the main factor is trend is that
$\frac{dE}{dx}_{\textit{ion.}}$ increases as $\frac{Z}{A}$ increases. This
dependence is illustrated by Fig. 5, which is a plot of the minimum
$\frac{dE}{dx}$ for various elements $vs.$ $\frac{Z}{A}$. Random ionization
energy straggling, which is added to the deterministic energy loss, giving a
Landau distribution for thin objects, also increases as $\frac{Z}{A}$
increases.
Bremsstrahlung energy loss is small compared to ionization energy loss for
thin foils. In the thick limit, where an electron emits a substantial number
of “hard” photons, total bremsstrahlung energy loss is proportional to beam
energy, i.e. $\frac{dE}{dx}\approx\frac{E_{0}}{X_{0}}$, where $E_{0}$ is the
incident energy. However, for typical diffuser foils, we are in the “thin”
limit, where the probability of a “hard” bremsstrahlung event is low (the
definition of a hard event is somewhat arbitrary, but it can be taken to be
emission of a photon with an energy of 0.1% of the incident electron energy).
Figure 5: Miniumum ionization $\frac{dE}{dx}$ in MeV-cm2/g vs. $\frac{Z}{A}$
for various materials. The materials in order of increasing $\frac{Z}{A}$ are
U, W, Be, Cu, Al, and C. The outlier is Be.
Using the MCS mean-angle formula and ignoring the log factor, we can write for
the foil thicknesss $x_{\textit{foil}}$ required to get a certain mean MCS
angle $\theta_{0}$, $x_{\textit{foil}}=C(E)X_{0}\theta_{0}^{2}$, where $C(E)$
is approximately material-independent and contains the dependence on the
electron energy. On the other hand, the ionization energy loss distribution
for a particular foil thickness $x$ scales roughly as $\frac{Z}{A}$, so the
ionization energy loss in a foil of a particular material with a thickness
that gives a specified mean scattering angle $\theta_{0}$ is $\Delta
E_{\textit{ion}}\sim\frac{Z}{A}\hskip 3.61371ptX_{0}\theta_{0}^{2}$. Since
both $\frac{Z}{A}$ and $X_{0}$ decrease with increasing atomic weight, this
favors high-Z diffuser materials, provided that their melting temperature is
high.
To investigate the dominant effect in electron deflection within the GEANT4
code a simple scheme was developed. Using the same simulation set-up as the
diffuser investigation, a simplified Physics List was written that only
included the processes G4eBremsstrahlung and G4eMultipleScattering.
Simulations consisted of firing 1 million electrons at a 168 $\mu$m slab of
tantalum. Three separate simulations were run with only bremsstrahlung, only
multiple Coulomb Scattering, and both processes active. Histograms of the
angular spread were plotted and are presented below. It is obvious from the
plots that angular deflection is dominated by multiple Coulomb scattering,
with electron bremsstrahlung only contributing a small amount to the
deflection of the electron as it travels through the tantalum.
Figure 6: Histograms of angular distribution. TOP: Both the G4eBremsstrahlung
and G4eMultipleScattering processes were active for this simulation. BOTTOM
LEFT: Only the G4eMultipleScattering process was active for this simulation.
BOTTOM RIGHT: Only the G4eBremsstrahlung process was active for this
simulation.
After investigating the angular spread effects of the physical processes in
the GEANT4 code we also wanted to confirm the energy loss effects of those
processes. The prior manufactured physics list was modified to include the
G4eIonisation process and simulations were run with all three processes, and
with only G4eIonisation active. The results, shown in the figure below,
confirm that the energy loss of the electrons through the tantalum sample is
dominated by the ionization process.
Figure 7: Histograms of energy loss. TOP: Only G4eBremsstrahlung active, the
majority of electrons( 67.5$\%$) did not lose energy and passed right through
the foil. BOTTOM: Only G4eIonisation active. This is the dominant effect on
energy loss, as seen when compared to the energy loss diagrams using a full
physics list.
We were also curious about the angular dependence of the energy loss from
bremsstrahlung in GEANT4. Histograms were created by plotting logarithmic
angle versus logarithm of total energy and subtracting the electrons final
energy at the detector. The final plot shows there is a correlation between
energy loss and angle, so another simulation was run using 12 MeV electrons
instead of 12 GeV electrons to see how the correlation would change or if the
relation was a static product of a random distribution. Both plots are
presented below, and it can be seen that the relation becomes steeper for
higher energy particles.
Figure 8: Histogram of angle distribution versus energy loss for 12 GeV
electron beam incident on 168 $\mu$m tantalum slab. Figure 9: Histogram of
angle distribution versus energy loss for 12 MeV electron beam incident on 168
$\mu$m tantalum slab.
|
arxiv-papers
| 2013-12-12T23:47:11 |
2024-09-04T02:49:55.377978
|
{
"license": "Public Domain",
"authors": "Joseph M. Fabritius II, Konstantin Borozdin, Peter Walstrom",
"submitter": "Joseph Fabritius II",
"url": "https://arxiv.org/abs/1312.3677"
}
|
1312.3781
|
aainstitutetext: Institute of Theoretical Physics,
China West Normal University,
Nanchong, 637009, Chinabbinstitutetext: Center for Theoretical Physics,
College of Physical Science and Technology,
Sichuan University,
Chengdu, 610051, China
# Remnants, fermions’ tunnelling and effects of quantum gravity
D.Y. Chen a Q.Q. Jiang b P. Wang b and H. Yang [email protected]
[email protected] [email protected] [email protected]
###### Abstract
The remnants are investigated by fermions’ tunnelling from a 4-dimensional
charged dilatonic black hole and a 5-dimensional black string. Based on the
generalized uncertainty principle, effects of quantum gravity are taken into
account. The quantum numbers of the emitted fermions affects the Hawking
temperatures. For the black hole, the quantum gravity correction slows down
the increase of the temperature, which leads to the remnant left in the
evaporation. For the black string, the existence of the remnant is determined
by the quantum gravity correction and effects of the extra compact dimension.
## 1 Introduction
The standard Hawking formula predicts the complete evaporation of black holes.
In the original research SWH , the formula was gotten in the frame of quantum
field theory on curved spacetime. It is based on the Heisenberg uncertainty
principle (HUP). Therefore, it is natural to find that the complete
evaporation is a direct consequence of the HUP. The semi-classical tunnelling
method put forward by Parikh and Wilczek is an effective way to research on
Hawking radiation PW . With the consideration of the variable background
spacetime, the tunnelling behavior of photons across the horizons was
described veritably. The corrected temperatures were gotten and higher than
the standard Hawking temperatures SWH . This result indicates that the
variable spacetimes speed up the increases of the temperatures and the black
holes evaporate completely. The extension of this method to the tunnelling
radiation of massive scalar particles was found in the subsequent work ZZ ;
JWC . The Hamilton-Jacobi ansatz is another version of the tunnelling method
ANVZ ; KM1 . Adopting this ansatz, the standard Hawking temperatures were
recovered by fermions’ tunnelling across the horizons of the black holes KM2 .
All of these results lead to that black holes evaporate completely and there
are no remnants left AAS .
On the other hand, various theories of quantum gravity, such as string theory,
loop quantum gravity and quantum geometry, predict the existence of the
minimal observable length PKT ; ACV ; KPP ; LJG ; GAC ; NIC2 . This view is
supported by the Gedanken experiment FS . An effective model to realize this
minimal length is the generalized uncertainty principle (GUP),
$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\left[1+\beta(\Delta
p)^{2}\right],$ (1)
which is derived by the modified fundamental commutation relations.
$\beta=\beta_{0}\frac{l^{2}_{p}}{\hbar^{2}}$ is a small value,
$\beta_{0}<10^{34}$ is a dimensionless parameter and $l_{p}$ is the Planck
length. Kempf et. al. first modified the commutation relations and got
$\left[x_{i},p_{j}\right]=i\hbar\delta_{ij}\left[1+\beta p^{2}\right]$, where
$x_{i}$ and $p_{i}$ are position and momentum operators defined respectively
as KMM
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle x_{0i},$ $\displaystyle
p_{i}$ $\displaystyle=$ $\displaystyle p_{0i}(1+\beta p^{2}),$ (2)
$x_{0i}$ and $p_{0j}$ satisfy the canonical commutation relations
$\left[x_{0i},p_{0j}\right]=i\hbar\delta_{ij}$. The modification is not
unique. Other modifications are referred to AK ; FB ; ADV ; DV ; NIC3 .
These modifications play an important role on the black hole physics. Based on
the GUP, it was found that there is no existence of black holes at LHC in AFA
. The black hole thermodynamics was discussed in XW ; BJM ; KP ; ZDM ,
respectively. The relation between the area and entropy and the corrected
Hawking temperatures were gotten. An interested result is that the remnants
exist in black holes’ evaporation ACS ; SGC ; BG ; LX ; NS ; NIC1 .
Incorporating the GUP into the tunnelling radiation in scalar fields, the
corrected Hawking temperatures in the Schwarzschild and the noncommutative
spacetimes were obtained NS ; NM . Using the modified commutation relation
between the radial coordinate and the conjugate momentum and considering the
natural cutoffs as minimal and maximal momentum, the tunnelling rates were
derived in NS . The interesting result is that the minimal mass and the
maximum temperature in the scalar field were found.
In this paper, taking into account effects of quantum gravity, we investigate
the tunnelling radiation of fermions from a 4-dimensional charged dilatonic
black hole and a 5-dimensional black string. The remnants are discussed by the
corrected Hawking temperatures. The temperatures are affected by the quantum
numbers (mass, charge and energy) of the emitted fermions. For the dilatonic
black hole, the quantum gravity correction slows down the increase of the
Hawking temperature. It is natural to lead to the remnant existed in the
evaporation. In the black string spacetime, the quantum gravity correction and
the effect of the extra compact dimension affect the evaporation.
The rest is outlined as follows. In the next section, based on the modified
commutation relations put forward in KMM , we modify the Dirac equation in
curved spacetime. In section 3, with the consideration of effects of quantum
gravity, the fermion’s tunnelling from the charged dilatonic black hole is
investigated and the remnant is derived. In section 4, we investigate the
fermion’s tunnelling from the black string. The evaporation of the string is
discussed. Section 5 is devoted to our conclusion.
## 2 Generalized Dirac equation
In this section, we adopt the modified operators of position and momentum in
eqn. (2) to modify the Dirac equation in curved spacetime. To achieve this
purpose, we first introduce the GUP into Dirac equation. We then generalize
Dirac equation to curved background by standard process. Respecting covariance
is certainly of importance during the derivations. Under this constraint, the
modification of Dirac equation in flat spacetime based on GUP can be uniquely
determined NK ; HBH ; OS ; MK . The square of momentum operators is
$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle
p_{i}p^{i}=-\hbar^{2}\left[{1-\beta\hbar^{2}\left({\partial_{j}\partial^{j}}\right)}\right]\partial_{i}\cdot\left[{1-\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)}\right]\partial^{i}$
(3) $\displaystyle\simeq$
$\displaystyle-\hbar^{2}\left[{\partial_{i}\partial^{i}-2\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)\left({\partial^{i}\partial_{i}}\right)}\right].$
In the last step, the higher order terms of $\beta$ are neglected. In the
theory of quantum gravity, the generalized frequency takes on the form as WG
$\displaystyle\tilde{\omega}=E(1-\beta E^{2}),$ (4)
with the definition of energy operator $E=i\partial_{t}$. Using the energy
mass shell condition $p^{2}+m^{2}=E^{2}$, we get the generalized expression of
energy NS ; WG ; NK ; HBH
$\displaystyle\tilde{E}=E[1-\beta(p^{2}+m^{2})].$ (5)
Then, in flat background, the modified Dirac equation based on GUP follows
straightforwardly as in NK . In curved spacetime, the Dirac equation with an
electromagnetic field takes on the form as
$\displaystyle
i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}+\frac{i}{\hbar}eA_{\mu}\right)\psi+\frac{m}{\hbar}\psi=0,$
(6)
where $\Omega_{\mu}\equiv\frac{i}{2}\omega_{\mu}\,^{ab}\Sigma_{ab}$,
$\Sigma_{ab}=\frac{i}{4}\left[{\gamma^{a},\gamma^{b}}\right]$,
$\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}$, $\omega_{\mu}\,^{ab}$ is the spin
connection defined by
$\omega_{\mu}\,^{a}\,{}_{b}=e_{\nu}\,^{a}e^{\lambda}\,_{b}\Gamma^{\nu}_{\mu\lambda}-e^{\lambda}\,_{b}\partial_{\mu}e_{\lambda}\,^{a}$,
$\Gamma^{\nu}_{\mu\lambda}$ is the ordinary connection and $e^{\lambda}\,_{b}$
is the tetrad. The Greek indices are raised and lowered by the curved metric
$g_{\mu\nu}$, while the Latin indices are governed by the flat metric
$\eta_{ab}$. The construction of a tetrad satisfies the following relations
$g_{\mu\nu}=e_{\mu}\,^{a}e_{\nu}\,^{b}\eta_{ab},\hskip
14.22636pt\eta_{ab}=g_{\mu\nu}e^{\mu}\,_{a}e^{\nu}\,_{b},\hskip
14.22636pte^{\mu}\,_{a}e_{\nu}\,^{a}=\delta^{\mu}_{\nu},\hskip
14.22636pte^{\mu}\,_{a}e_{\mu}\,^{b}=\delta_{a}^{b}.$ (7)
Therefore, it is readily to construct the $\gamma^{\mu}$’s in curved spacetime
as
$\gamma^{\mu}=e^{\mu}\,_{a}\gamma^{a},\hskip
19.91692pt\left\\{{\gamma^{\mu},\gamma^{\nu}}\right\\}=2g^{\mu\nu}.$ (8)
To modify the Dirac equation, we rewrite eqn. (6) as
$\displaystyle-i\gamma^{0}\partial_{0}\psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\psi,$
(9)
namely,
$\displaystyle
i\partial_{0}\psi=-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\psi.$
(10)
The left hand side of the above equation is related to the energy. Using the
generalized expression of energy (5), we get the modified Dirac equation as
follows NK ; HBH
$\displaystyle i\partial_{0}\Psi$ $\displaystyle=$
$\displaystyle-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1-\beta
p^{2}-\beta m^{2}\right)\Psi$ (11) $\displaystyle=$
$\displaystyle-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\Psi.$
The last equal sign was derived by the expression of the square of momentum
operators in eqn. (3) and the neglect of the higher order terms of $\beta$. In
this equation, $\Psi$ is the generalized Dirac field. Thus the modified Dirac
equation in curved spacetime is
$\displaystyle-i\gamma^{0}\partial_{0}\Psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\Psi,$ (12)
which is rewritten as
$\displaystyle\left[i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta
m^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{m}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right.$
$\displaystyle\left.+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right]\Psi=0.$ (13)
When $eA_{\mu}=0$, it describes the Dirac equation without the electromagnetic
field. In the following sections, we adopt eqn. (13) to investigate fermions’
tunnelling across the horizons of the 4-dimensional and the 5-dimensional
spacetimes. If one only considers the modification of momenta in the Dirac
equation. It is not covariant. From covariance, therefore, we modified the
momenta and energy in the Dirac equation.
## 3 The remnant in the 4-dimensional dilatonic black hole
The general solution of dilatonic black holes GHSHH was derived from the
action
$\displaystyle
S=\int{dx^{4}\sqrt{-g}\left[-R+2(\Delta\Phi)^{2}+e^{-2\alpha\Phi}F^{2}\right]},$
(14)
which describes the standard matter, gravity coupled to a Maxwell field and a
dilaton. $\alpha$ is a parameter expressed the strength of coupling of the
dilation field $\Phi$ to the Maxwell field $F$. It reduces to the usual
Einstein-Maxwell scalar theory when $\alpha=0$, while it is part of the low
energy action of string theory when $\alpha=1$. The metric of the spherically
symmetric charged dilatonic black hole is given by
$\displaystyle
ds^{2}=-f\left(r\right)dt^{2}+\frac{1}{f\left(r\right)}dr^{2}+R^{2}(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (15)
with the electromagnetic potential
$A_{\mu}=\left(A_{t},0,0,0\right)=\left(\frac{Q}{r},0,0,0\right)$, where
$\displaystyle R\left(r\right)$ $\displaystyle=$ $\displaystyle
r\left(1-\frac{r_{-}}{r}\right)^{\frac{\alpha^{2}}{1+\alpha^{2}}},$
$\displaystyle f\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}.$
(16)
The event horizon is located at $r=r_{+}$ for all $\alpha$ and $r_{+}>r_{-}$.
The mass and charge of the black hole are represented by
$M=\frac{r_{+}}{2}+\frac{r_{-}}{2}\left(\frac{1-\alpha^{2}}{1+\alpha^{2}}\right)$
and $Q=\sqrt{\frac{r_{+}r_{-}}{1+\alpha^{2}}}$, respectively.
For a spin-1/2 fermion, there are two states corresponding to spin up and spin
down. In this paper, we only consider the state with spin up without loss of
generality. The investigation of the state with spin down is parallel. The
motion of a fermion in the dilaton black hole obeys the generalized Dirac
equation (13). To describe the motion, we first suppose that the wave function
of the fermion with spin up state takes on the form as
$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\theta,\phi\right)\right),$
(21)
where $I$ is the action of the fermion with spin up state and $A$ and $B$ are
functions of $(t,r,\theta,\phi)$. To solve the equation (13), one should
construct gamma matrices. The construction of gamma matrices is relied on a
tetrad. It is straightforward to get the tetrad from the metric (15) as
$\displaystyle
e_{\mu}^{a}=\rm{diag}\left(\sqrt{f},1/\sqrt{f},R,R\sin\theta\right).$ (22)
Then the gamma matrices is constructed as
$\displaystyle\gamma^{t}=\frac{1}{\sqrt{f\left(r\right)}}\left(\begin{array}[]{cc}0&\rm
I\\\ -\rm I&0\end{array}\right),$
$\displaystyle\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (27)
$\displaystyle\gamma^{r}=\sqrt{f\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right).$ (32)
In the above equations, $\sqrt{g^{\theta\theta}}=R^{-1}$,
$\sqrt{g^{\phi\phi}}=(R\sin\theta)^{-1}$, $\rm I$ is the unit matrix,
$\sigma^{i}$ are the Pauli matrices,
$\displaystyle\sigma^{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma^{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma^{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (39)
We insert the gamma matrices and the wave function into the equation (13) and
divide by the exponential term. Applying the WKB approximation, we get the
resulting equations to leading order in $\hbar$. They are decoupled into four
equations
$\displaystyle-B\frac{1}{\sqrt{f}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}I-Am\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]$
$\displaystyle+B\beta\sqrt{f}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]+Am\left(1-\beta
m^{2}\right)$ $\displaystyle-B\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\beta\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right)\right]=0,$
(40)
$\displaystyle A\frac{1}{\sqrt{f}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}I-Bm\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]$
$\displaystyle+A\beta\sqrt{f}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]+Bm\left(1-\beta
m^{2}\right)$ $\displaystyle+A\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\beta\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right)\right]=0,$
(41)
$\displaystyle A\left\\{-(1-\beta
m^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right.$
$\displaystyle\left.-i(1-\beta
m^{2})\sqrt{g^{\phi\phi}}\partial_{\phi}I+i\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right\\}=0,$
(42)
$\displaystyle B\left\\{-(1-\beta
m^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right.$
$\displaystyle\left.-i(1-\beta
m^{2})\sqrt{g^{\phi\phi}}\partial_{\phi}I+i\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right\\}=0.$
(43)
It is difficult to solve the action from the above equations. Considering
properties of the dilatonic spacetime and the above equations, we carry out
separation of variables on the action and get
$I=-\omega t+W(r)+\Xi(\theta,\phi),$ (44)
where $\omega$ is the energy of the emitted fermion.
From eqns. (40)-(43), it is found that eqn. (42) and eqn. (43) are irrelevant
to $A$ and $B$ and can be reduced into the same equation. Then inserting eqn.
(44) into eqn. (42) and eqn. (43) and rewriting the equation yield
$\displaystyle\left(\sqrt{g^{\theta\theta}}\partial_{\theta}\Xi+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Xi\right)\times$
$\displaystyle\left[\beta g^{rr}(\partial_{r}W)^{2}+\beta
g^{\theta\theta}(\partial_{\theta}\Xi)^{2}+\beta
g^{\phi\phi}(\partial_{\phi}\Xi)^{2}+\beta m^{2}-1\right]=0,$ (45)
which implies
$\displaystyle\sqrt{g^{\theta\theta}}\partial_{\theta}\Xi+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Xi=0.$
(46)
Thus the solution of $\Xi$ can be gotten. It is a complex function other than
the trivial solution of constant. This complex function produces a
contribution on the action. However, it has no contribution on the tunelling
rate since the contributions of the outgoing and ingoing solutions are
canceled in the calculation. From the above equation, it is easily to derive
the relation
$\displaystyle
g^{\theta\theta}\left(\partial_{\theta}\Xi\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}\Xi\right)^{2}=0.$
(47)
Now we focus our attention on the radial action. Inserting eqn. (44) into
eqns. (40) and (41) and using eqn. (47), we get
$\displaystyle B\frac{\omega}{\sqrt{f}}-B\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}W-Am\beta
g^{rr}\left(\partial_{r}W\right)^{2}+B\beta\sqrt{f}g^{rr}\left(\partial_{r}W\right)^{3}$
$\displaystyle-B\frac{eA_{t}}{\sqrt{f}}\left(1-\beta m^{2}-\beta
g^{rr}\left(\partial_{r}W\right)^{2}\right)+Am\left(1-\beta m^{2}\right)=0,$
(48)
$\displaystyle-A\frac{\omega}{\sqrt{f}}-A\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}W-Bm\beta
g^{rr}\left(\partial_{r}W\right)^{2}+B\beta\sqrt{f}g^{rr}\left(\partial_{r}W\right)^{3}$
$\displaystyle+A\frac{eA_{t}}{\sqrt{f}}\left(1-\beta m^{2}-\beta
g^{rr}\left(\partial_{r}W\right)^{2}\right)+Bm\left(1-\beta m^{2}\right)=0.$
(49)
In the above equations, $A$ and $B$ are irrelevant to the result. Eliminating
them yields
$C_{6}\left(\partial_{r}W\right)^{6}+C_{4}\left(\partial_{r}W\right)^{4}+C_{2}\left(\partial_{r}W\right)^{2}+C_{0}=0,$
(50)
where
$\displaystyle C_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}f^{4},$
$\displaystyle C_{4}$ $\displaystyle=$ $\displaystyle\beta
f^{3}\left(m^{2}\beta-2\right)-\beta^{2}f^{2}e^{2}A_{t}^{2},$ $\displaystyle
C_{2}$ $\displaystyle=$ $\displaystyle
f^{2}\left(1-\beta^{2}m^{4}\right)+2\beta feA_{t}\left[-\omega+eA_{t}(1-\beta
m^{2})\right],$ $\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle-m^{2}f\left(1-\beta m^{2}\right)^{2}-\left[\omega-
eA_{t}\left(1-\beta m^{2}\right)\right]^{2}.$ (51)
Keeping the leading order terms of $\beta$ and solving $W$ at the event
horizon, we derive the imaginary part of the radial action
$\displaystyle ImW_{\pm}$ $\displaystyle=$ $\displaystyle\pm\int
dr\frac{1}{f}\sqrt{m^{2}f+\left[\omega-eA_{t}(1-\beta
m^{2})\right]^{2}}\left(1+\beta
m^{2}+\beta\frac{\tilde{\omega}^{2}}{f}-\frac{\beta
eA_{t}\tilde{\omega}}{f}\right)$ (52) $\displaystyle=$
$\displaystyle\pm\pi\frac{r_{+}}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}(\omega-
eA_{t+})\times\left(1+\beta\Pi\right),$
where
$\displaystyle\tilde{\omega}$ $\displaystyle=$ $\displaystyle\omega-eA_{t},$
$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle
m^{2}+\frac{m^{2}}{\omega_{0}}eA_{t+}+\frac{1}{2}\frac{m^{2}\left(\omega_{0}-eA_{t+}\right)}{\omega-
eA_{t+}(1-\beta
m^{2})}-\frac{eA_{t+}\left(\omega_{0}-eA_{t+}\right)}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}$
(53)
$\displaystyle+\frac{\omega_{0}}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}\left[2\omega_{0}+eA_{t+}-\frac{r_{-}(1-\alpha^{2})\left(2\omega_{0}-eA_{t+}\right)}{(r_{+}-r_{-})(1+\alpha^{2})}-\frac{2e^{2}Q^{2}}{\omega_{0}r_{+}}\right],$
$+(-)$ denotes the outgoing(ingoing) solutions, $\omega_{0}=\omega-eA_{t+}$,
$A_{t}=\frac{Q}{r}$, $A_{t+}=\frac{Q}{r_{+}}$ is the electromagnetic potential
at the event horizon. Using the relations between $M$, $Q$ and $r_{\pm}$, it
is found that $\Pi>0$. Thus the tunnelling rate of the fermion with spin up
state at the event horizon is
$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{P_{(emission)}}{P_{(absorption)}}=\frac{\exp{\left(-\frac{2}{\hbar}ImI_{+}\right)}}{\exp{\left(-\frac{2}{\hbar}ImI_{-}\right)}}=\frac{exp\left(-\frac{2}{\hbar}ImW_{+}-\frac{2}{\hbar}Im{\Xi}\right)}{exp\left(-\frac{2}{\hbar}ImW_{-}-\frac{2}{\hbar}Im{\Xi}\right)}$
(54) $\displaystyle=$ $\displaystyle
exp\left[-4\pi\frac{r_{+}}{\hbar\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}(\omega-
eA_{t+})\times\left(1+\beta\Pi\right)\right].$
This is the Boltzmann factor with the Hawking temperature at the event horizon
of the dilatonic black hole taking
$\displaystyle
T=\frac{\hbar\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}{4\pi
r_{+}\left(1+\beta\Pi\right)}=T_{0}\left(1-\beta\Pi\right),$ (55)
where $T_{0}=\frac{\hbar}{4\pi
r_{+}}\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}$
is the standard Hawking temperature. It is evidently that the corrected
Hawking temperature appears and is lower than the standard one. This
temperature is also lower than that derived by the semi-classical tunnelling
method PW . The correction value is determined not only by the mass and charge
of the black hole but also by the quantum number (energy, mass and charge) of
the emitted fermion. Due to the radiation, the temperature increases. Eqn.
(55) shows that the quantum gravity correction slows down the increase of the
temperature during the radiation. This correction therefore causes the
radiation ceased at some particular temperature, leaving the remnant mass.
Eqn. (55) describes the temperature of the Reissner-Nordstrom black hole when
$\alpha=0$ and that of the Schwarzschild black hole when $\alpha=0$ and $Q=0$.
Using an assumption that the emitted particle is massless, we estimate the
remnant of the Schwarzschild black hole. The evaporation stops when
$\left(M-dM\right)\left(1+\beta\omega^{2}\right)\simeq M$. With the
observation $dM=\omega$ and $\beta=\frac{\beta_{0}}{M_{p}^{2}}$, we get the
the remnant as
$M_{R}\simeq\frac{M_{p}^{2}}{\beta_{0}\omega}\geq\frac{M_{p}}{\beta_{0}}$,
where we assumed that the maximal energy of the emitted particle is the Planck
mass $M_{p}$ .
## 4 The remnant in the 5-dimensional black string
In this section, we investigate the remnant by the fermion’s tunnelling from a
5-dimensional spacetime. The emitted fermion is supposed to be uncharged.
Therefore, the electromagnetic effect in eqn. (13) is not taken into account
here. The 4-dimensional Schwarzschild metric is a static spherically symmetric
solution to the vacuum Einstein equations. When an extra compact spatial
dimension $z$ is added, the metric becomes
$\displaystyle
ds^{2}=-F\left(r\right)dt^{2}+\frac{1}{F\left(r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)+dz^{2},$ (56)
where $F=1-\frac{r_{h}}{r}$, $r_{h}=2M$ is the location of the event horizon
and $M$ is proportional to the black hole mass. The metric (56) describes a
neutral uniform black string. Here we investigate a fermion tunnelling from
this string.
We still only consider the spin up state. The wave function of the fermion
with spin up state is now assumed as
$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\theta,\phi,z\right)\right),$
(61)
$A$ and $B$ are functions of $(t,r,\theta,\phi,z)$. The fermion’s motion
satisfies the generalized Dirac equation. Now the tetrad is different from
that in the above section. It is
$e_{\mu}^{a}=\rm{diag}\left(\sqrt{F},1/\sqrt{F},r,r\sin\theta,1\right)$. Then
gamma matrices are constructed as follows
$\displaystyle\gamma^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{F\left(r\right)}}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),\hskip
19.91692pt\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (66) $\displaystyle\gamma^{r}$
$\displaystyle=$
$\displaystyle\sqrt{F\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),\hskip
19.91692pt\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right),$ (71) $\displaystyle\gamma^{z}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right).$ (74)
To apply the WKB approximation, we insert the wave function and the gamma
matrices into the Dirac equation and divide by the exponential term.
Multiplying by $\hbar$, the equations to leading order in $\hbar$ are obtained
and decoupled into four equations
$\displaystyle-iA\frac{1}{\sqrt{F}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{F}\partial_{r}I+Am\left(1-\beta m^{2}\right)+A\left(1-\beta
m^{2}\right)\partial_{z}I$
$\displaystyle+\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+\left(\partial_{z}I\right)^{2}\right]\left(B\sqrt{F}\partial_{r}I-A\partial_{z}I-Am\right)=0,$
(75)
$\displaystyle iB\frac{1}{\sqrt{F}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{F}\partial_{r}I+Bm\left(1-\beta m^{2}\right)-B\left(1-\beta
m^{2}\right)\partial_{z}I$
$\displaystyle+\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+\left(\partial_{z}I\right)^{2}\right]\left(A\sqrt{F}\partial_{r}I+B\partial_{z}I-Bm\right)=0,$
(76)
$\displaystyle
A\left(\sqrt{g^{\theta\theta}}\partial_{\theta}I+i\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)\times$
$\displaystyle\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}+(\partial_{z}I)^{2}+\beta
m^{2}-1\right]=0.$ (77)
$\displaystyle
B\left(\sqrt{g^{\theta\theta}}\partial_{\theta}I+i\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)\times$
$\displaystyle\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}+(\partial_{z}I)^{2}+\beta
m^{2}-1\right]=0.$ (78)
It is also difficult to solve the action from the above equations. Considering
the property of the black string spacetime, we carry out separation of
variables as
$\displaystyle I=-\omega t+W(r)+\Theta(\theta,\phi)+Jz,$ (79)
where $\omega$ is the energy, $J$ is a conserved momentum and describes a
constant of motion corresponding to the compact dimension $z$.
We first focus our attention on the last two equations. They are irrelevant to
$A$ and $B$ and can be reduced to the same equation. Inserting eqn. (79) into
them, we get
$\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Theta=0$
since the summation of factors in the square brackets in eqn. (77) and (78)
can not be zero. Thus the solution of $\Theta$ is a complex function (other
than the constant solution). The following relation,
$\displaystyle
g^{\theta\theta}\left(\partial_{\theta}\Theta\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}\Theta\right)^{2}=0,$
(80)
is easily obtained. Return to eqns. (76) and (77). Inserting eqns. (79) and
(80) into them and eliminating $A$ and $B$, we get the equation of the radial
action
$\displaystyle
D_{6}\left(\partial_{r}W\right)^{6}+D_{4}\left(\partial_{r}W\right)^{4}+D_{2}\left(\partial_{r}W\right)^{2}+D_{0}=0$
(81)
where
$\displaystyle D_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}F^{4},$
$\displaystyle D_{4}$ $\displaystyle=$ $\displaystyle-2\beta
xF^{3}-\left(m^{2}-J\right)\beta^{2}F^{3},$ $\displaystyle D_{2}$
$\displaystyle=$ $\displaystyle x^{2}F^{2}+2\beta
xF^{2}\left(m^{2}-J\right)-i2\beta J\omega F^{3/2},$ $\displaystyle D_{0}$
$\displaystyle=$
$\displaystyle-\left(m^{2}-J\right)x^{2}F-\omega^{2}+i2J\omega x\sqrt{F},$
$\displaystyle x$ $\displaystyle=$ $\displaystyle 1-\beta m^{2}-\beta J^{2}.$
(82)
Neglect higher order terms of $\beta$ and solve the equation (81) at the event
horizon. We only interest the imaginary part of the action because the
tunnelling rate is determined by it. The imaginary part is
$\displaystyle ImW_{\pm}$ $\displaystyle=$ $\displaystyle\pm
Im\int{dr\frac{\sqrt{\omega^{2}+\left(m^{2}-J^{2}\right)F-i2\sqrt{F}J\omega}}{F}\left[1+\beta\left(m^{2}-J^{2}+\frac{\omega^{2}-2iJ\omega\sqrt{F}}{F}\right)\right]}$
(83) $\displaystyle=$ $\displaystyle\pm\pi\omega
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right],$
where $+(-)$ are the outgoing(ingoing) solutions. Thus the tunnelling rate of
the uncharged fermion across the event horizon of the 5-dimensional black
string is
$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\exp{\left(-\frac{2}{\hbar}ImI_{+}\right)}}{\exp{\left(-\frac{2}{\hbar}ImI_{-}\right)}}=\frac{exp\left(-\frac{2}{\hbar}ImW_{+}-\frac{2}{\hbar}Im{\Theta}\right)}{exp\left(-\frac{2}{\hbar}ImW_{-}-\frac{2}{\hbar}Im{\Theta}\right)}$
(84) $\displaystyle=$ $\displaystyle exp\left\\{-\frac{1}{\hbar}4\pi\omega
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right]\right\\},$
which shows that the Hawking temperature is
$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{\hbar}{4\pi
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right]}$
(85) $\displaystyle=$ $\displaystyle
T_{0}\left[1-\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right].$
In the above equation, $T_{0}=\frac{\hbar}{4\pi r_{h}}$ is the standard
Hawking temperature. It shows that the corrected Hawking temperature is not
only determined by the quantum number (energy and mass) of the emitted fermion
but also affected by the effect of the extra compact dimension.
It is of interest to discuss the value of $3m^{2}+4\omega^{2}-2J^{2}$. When
$3m^{2}+4\omega^{2}>2J^{2}$, it is easily found that the corrected temperature
is lower than the standard Hawking temperature. This implies that the
combination of the quantum gravity correction and the effect of the extra
compact dimension slows down the increase of the temperature caused by the
radiation. Finally, the black string should be in a stable balanced state. At
this state, the remnant is left. The special case is $J=0$. In this case, the
fermion’s motion is limited in the 4-dimensional spacetime. Thus eqn. (85)
reduces to the temperature of the 4-dimensional Schwarzschild black hole.
When $3m^{2}+4\omega^{2}<2J^{2}$, the corrected temperature is higher than the
standard one. It shows that the black string accelerates the evaporation and
there is no remnant left. If $3m^{2}+4\omega^{2}=2J^{2}$, the effect of the
quantum gravity correction and that of the extra dimension are canceled. Then
the standard Hawking temperature appears and results in the complete
evaporation. Therefore, the evaporation of the black string is affected by the
quantum gravity correction and the effect of the extra compact dimension.
## 5 Conclusion
In this paper, based on the modified fundamental commutation relation, we
modified the HUP and investigated the fermions’ tunnelling across the horizons
of the 4-dimensional charged dilatonic black hole and the 5-dimensional
neutral black string. The corrected Hawking temperatures were gotten. The
remnants were discussed by the temperatures. For the dilatonic black hole, the
correction is determined not only by the mass and charge of the black hole but
also by the quantum number (mass, charge and energy) of the emitted fermion.
The interesting point is that the quantum gravity correction slows down the
increase of the Hawking temperature. It is natural to lead to the remnant left
in the evaporation. For the black string, the temperature is affected by the
quantum number (mass and energy) of the emitted fermion and the effect of
extra compact dimension. The existence of the remnant is determined by the
combined effect of the quantum gravity correction and the compact dimension.
In NIC4 , noncommutative black holes was discussed. In RMS ; SVZR , the
quantum tunnelling radiation were researched beyond the semiclassical
approximation. The corrected Hawking temperatures were derived and also lower
than the standard semiclassical Hawking temperatures.
###### Acknowledgements.
This work is supported in part by the NSFC (Grant Nos. 11205125, 11005016,
11005086, 11175039 and 11375121), the Innovative Research Team in College of
Sichuan Province (Grant No. 13TD0003) and SiChuan Province Science Foundation
for Youths (Grant No. 2012JQ0039).
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|
arxiv-papers
| 2013-12-13T11:49:06 |
2024-09-04T02:49:55.384880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Deyou Chen, Qingquan Jiang, Peng Wang and Haitang Yang",
"submitter": "Deyou Chen",
"url": "https://arxiv.org/abs/1312.3781"
}
|
1312.3838
|
050003 2013 G. Mindlin 050003
Since its first formulations almost a century ago, mathematical models for
disease spreading contributed to understand, evaluate and control the epidemic
processes. They promoted a dramatic change in how epidemiologists thought of
the propagation of infectious diseases. In the last decade, when the
traditional epidemiological models seemed to be exhausted, new types of models
were developed. These new models incorporated concepts from graph theory to
describe and model the underlying social structure. Many of these works merely
produced a more detailed extension of the previous results, but some others
triggered a completely new paradigm in the mathematical study of epidemic
processes. In this review, we will introduce the basic concepts of
epidemiology, epidemic modeling and networks, to finally provide a brief
description of the most relevant results in the field.
# Invited review: Epidemics on social networks
M. N. Kuperman[inst1, inst2] E-mail: [email protected]
(6 April 2013; 3 June 2013)
††volume: 5
99 inst1 Consejo Nacional de Investigaciones Científicas y Técnicas,
Argentina. inst2 Centro Atómico Bariloche and Instituto Balseiro, 8400 S. C.
de Bariloche, Argentina
## 1 Introduction
With the development of more precise and powerful tools, the mathematical
modeling of infectious diseases has become a crucial tool for making decisions
associated to policies on public health. The scenario was completely different
at the beginning of the last century, when the first mathematical models
started to be formulated. The rather myopic comprehension of the
epidemiological processes was evidenced during the most dramatic epidemiologic
events of the last century, the pandemic 1918 flu. The lack of a mathematical
understanding of the evolution of epidemics gave place to an inaccurate
analysis of the epidemiological situation and subsequent failed assertion of
the success of the immunization strategy. During the influenza pandemic of
1892, a viral disease, Richard Pfeiffer isolated bacteria from the lungs and
sputum of patients. He installed, among the medical community, the idea that
these bacteria were the cause of influenza. At that moment, the bacteria was
called Pfeiffer’s bacillus or Bacillus influenzae, while its present name
keeps a reminiscence of Pfeiffer’s wrong hypothesis: Haemophilus influenzae.
Though there were some dissenters, the hypothesis of linking influenza with
this pathogen was widely accepted from then on. Among the supporters of
Pfeiffer hypothesis was William Park, at the New York City Health Department,
who in view of the fast progression of the flu in USA, developed a vaccine and
antiserum against Haemophilus influenzae on October 1918. Shortly afterwards
the Philadelphia municipal laboratory released thousands of doses of the
vaccine that was constituted by a mix of killed streptococcal, pneumococcal,
and H. influenzae bacteria. Several other attempts to develop similar vaccines
followed this initiative. However, none of these vaccines prevented viral
influenza infection. The present consensus is that they were even not
protective against the secondary bacterial infections associated to influenza
because the vaccine developers at that time could not identify, isolate, and
produce all the disease-causing strains of bacteria. Nevertheless, a wrong
evaluation of the evolution of the disease and a lack of epidemiological
knowledge led to the conclusion that the vaccine was effective. If we look at
Fig. 1 corresponding to the weekly influenza death rates in a couple of U.S.
cities taken from Ref. [1], we observe a remarkable decay after vaccination,
in week 43. This decay was inaccurately attributed to the effect of
vaccination as it corresponds actually to a normal and expected development of
an epidemics without immunization.
Figure 1: Weekly “Spanish influenza” death rates in Baltimore (circles) and
San Francisco (squares) from 1918 to 1919. Data taken from Ref. [1].
The inaccurate association between H. influenzae and influenza persisted until
1933, when the viral etiology of the flu was established. But Pfeiffer’s
influenza bacillus, finally named Haemophilus influenzae, accounts in its
denomination for this persistent mistake.
The formulation of mathematical models in epidemiology has a tradition of more
than one century. One of the first successful examples of the mathematical
explanation of epidemiological situations is associated with the study of
Malaria. Ronald Ross was working at the Indian Medical Service during the last
years of the 19th century when he discovered and described the life-cycle of
the malaria parasite in mosquitoes and developed a mathematical model to
analyze the dynamics of the transmission of the disease [2, 3, 4]. His model
linked the density of mosquitoes and the incidence of malaria among the human
population. Once he had identified the anopheles mosquitoes as the vector for
malaria transmission, Ross conjectured that malaria could be eradicated if the
ratio between the number of mosquitoes and the size of the human population
was carried below a threshold value. He based his analysis on a simple
mathematical model.
Ross’ model was based on a set of deterministic coupled differential
equations. He divided the human population into two groups, the susceptible,
with proportion $S_{h}$ and the infected, with proportion $I_{h}$. After
recovery, any formerly infected individual returned to the susceptible class.
This is called a SIS model. The mosquito population was also divided into two
groups (with proportions $S_{m}$ and $I_{m}$), with no recovery from
infection. Considering equations for the fraction of the population in each
state, we have $S+I=1$ for both humans and mosquitoes and the model is reduced
to a set of two coupled equations
$\displaystyle\frac{dI_{h}}{dt}$ $\displaystyle=$ $\displaystyle
abfI_{m}(1-I_{h})-rI_{h}$ (1) $\displaystyle\frac{dI_{m}}{dt}$
$\displaystyle=$ $\displaystyle acI_{h}(1-I_{m})-\mu_{m}I_{m},$
where $a$ is the man biting rate, $b$ is the proportion of bites that produce
infection in humans, $c$ is the proportion of bites by which one susceptible
mosquito becomes infected, $f$ is the ratio between the number of female
mosquitoes and humans, $r$ is the average recovery rate of human and $\mu_{m}$
is the rate of mosquito mortality.
One of the parameters to quantify the intensity of the epidemics propagation
is the basic reproductive rate $R_{0}$, that measures the average number of
cases produced by an initial case throughout its infectious period. $R_{0}$
depends on several factors. Among them, we can mention the survival time of an
infected individual, the necessary dose for infection, the duration of
infectiousness in the host, etc. $R_{0}$ allows to determine whether or not an
infectious disease can spread through a population: an infection can spread in
a population only if $R_{0}>1$ and can be maintained in an endemic state when
$R_{0}=1$ [5]. In the case of malaria, $R_{0}$ is defined as the number of
secondary cases of malaria arising from a single case in an susceptible
population. For the model described by Eq. (1)
$R_{0}=\frac{ma^{2}bc}{r\mu_{m}}.$ (2)
It is clear that the choice of the parameters affects $R_{0}$. The main result
is that it is possible to reduce $R_{0}$ by increasing the mosquito mortality
and reducing the biting rate. For his work on malaria, Ross was awarded the
Nobel Prize in 1902.
Ross’ pioneering work was later extended to include other ingredients and
enhance the predictability power of the original epidemiological model [5, 6,
7, 8, 9, 10, 11].
Some years after Ross had proposed his model, a couple of seminal works
established the basis of the current trends in mathematical epidemiology. Both
models consider the population divided into three epidemiological groups or
compartments: susceptible (S), infected (I) and recovered (R).
On the one hand, Kermack and McKendrick [12] proposed a SIR model that
expanded Ross’ set of differential equations. The model did not consider the
existence of a vector, but a direct transmission from an infected individual
to a susceptible one. A particular case of the original model, in which there
is no age dependency of the transmission and recovery rate, is the classical
SIR model that will be explained later.
On the other hand, Reed and Frost [13] developed a SIR discrete and stochastic
epidemic model to describe the relationship between susceptible, infected and
recovered immune individuals in a population. It is a chain binomial model of
epidemic spread that was intended mainly for teaching purposes, but that is
the starting point of many modern epidemiological studies. The model can be
mapped into a recurrence equation that defines what will happen at a given
moment depending on what has happened in the previous one,
$I_{t+1}=S_{t}(1-(1-\rho)^{I_{t}}),$ (3)
where $I_{t}$ is the number of cases at time $t$, $S_{t}$ is the number of
susceptible individuals at time $t$ and $\rho$ is the probability of
contagion.
The basic assumption of these SIR models, which is present in almost any
epidemiological work, is that the infection is spread directly from infectious
individuals to susceptible ones after a certain type of interaction between
them. In turn, these newly infected individuals will develop the infection to
become infectious. After a defined period of time, the infected individuals
heal and remain permanently immune. The interaction between any two
individuals of the population is considered as a stochastic process with a
defined probability of occurrence that most of the deterministic model
translates into a contact rate.
Given a closed population and the number of individuals in each state, the
calculation of the evolution of the epidemics is straightforward. The epidemic
event is over when no infective individuals remain.
While many classic deterministic epidemiological models were having success at
describing the dynamics of an infectious disease in a population, it was noted
that many involved processes could be better described by stochastic
considerations and thus a new family of stochastic models was developed [14,
15, 16, 17, 18, 19]. Sometimes, deterministic models introduce some colateral
mistakes due to the continuous character of the involved quantities.An example
of such a case is discussed in Ref. [20]. In Ref. [21], the authors proposed a
deterministic model to describe the prevalence of rabies among foxes in
England. They predicted a sharp decaying prevalence of the rabies up to
negligible levels, followed by an unexpected new outbreak of infected foxes.
The spontaneous outbreak after the apparent disappearing of the rabies is due
to a fictitious very low endemic level of infected foxes, as explained in Ref.
[20]. The former one is one among several examples of how stochastic models
contributed to a better understanding and explanation of some observed
phenomena but, as their predecessors, they considered a mean field scheme in
the set of differential equations.
Traditional epidemiological models have successfully describe the generalities
of the time evolution of epidemics, the differential effect on each age group,
and some other relevant aspects of an epidemiological event. But all of them
are based on a fully-mixing approximation, proposing that each individual has
the same probability of getting in touch with any other individual in the
population. The real underlying pattern of social contacts shows that each
individual has a finite set of acquaintances that serve as channels to promote
the contagion. While the fully mixed approximation allows for writing down a
set of differential equations and a further exploitation of a powerful
analytic set of tools, a better description of the structure of the social
network provides the models with the capacity to compute the epidemic dynamics
at the population scale from the individual-level behavior of infections, with
a more accurate representation of the actual contact pattern. This, in turn,
reflects some emergent behavior that cannot be reproduced with a system based
on a set of differential equation under the fully mixing assumption. One of
the most representative examples of this behavior is the so called herd
immunity, a form of immunity that occurs when the vaccination of a significant
portion of the population is enough to block the advance of the infection on
other non vaccinated individuals. Additionally, some network models allow also
for an analytic study of the described process. It is not surprising then that
during the last decade, a new tendency in epidemiological modeling emerged
together with the inclusion of complex networks as the underlying social
topology in any epidemic event. This new approach proves to contribute with a
further understanding of the dynamics of an epidemics and unveils the crucial
effect of the social architecture in the propagation of any infectious
disease.
In the following section, we will introduce some generalities about
traditional epidemiological models. In section III, we will present the most
commonly used complex networks when formulating an epidemiological model. In
section IV, we will describe the most relevant results obtained by modeling
epidemiological processes using complex networks to describe the social
topology. Next, we will introduce the concept of herd protection or immunity
and a discussion of some of the works that treat this phenomenon.
## 2 Basic Epidemiological Models
Two main groups can be singled out among the deterministic models for the
spread of infectious diseases which are transmitted through person-to-person
contact: the SIR and the SIS. The names of these models are related to the
different groups considered as components of the population or epidemiological
compartments: S corresponds to susceptible, I to infected and R to removed.
The S group represents the portion of the population that has not been
affected by the disease but may be infected in case of contact with a sick
person. The I group corresponds to those individuals already infected and who
are also responsible for the transmission of the disease to the susceptible
group. The removed group R includes those individuals recovered from the
disease who have temporary or permanent immunity or, eventually, those who
have died from the illness and not from other causes. These models may or may
not include the vital dynamics, associated with birth and death processes. Its
inclusion depends on the length of time over which the spread of the disease
is studied.
### 2.1 The SIR Model
As mentioned before, in 1927, Kermack and McKendrick [12] developed a
mathematical model in which they considered a constant population divided into
three epidemiological groups : susceptible, infected and recovered. The
equations of a SIR model are
$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI$
$\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma I$
(4) $\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma I,$
where the involved quantities are the proportion of individuals in each group.
As the population is constant,
$S(t)+I(t)+R(t)=1.$ (5)
The SIR model is used when the disease under study confers permanent immunity
to infected individuals after recovery or, in extreme cases, it kills them.
After the contagious period, the infected individual recovers and is included
in the R group. These models are suitable to describe the behavior of
epidemics produced by virus agent diseases (measles, chickenpox, mumps, HIV,
poliomyelitis) [22].
The model formulated through Eq. (2.1) assumes that all the individuals in the
population have the same probability of contracting the disease with a rate of
$\beta$, the contact rate. The number of infected increases proportionally to
both the number of infected and susceptible. The rate of recovery or removal
is proportional to the number of infected only. $\gamma$ represents the mean
recovery rate, ( $1/\gamma$ is the mean infective period). It is assumed that
the incubation time is negligible and that the rates of infection and recovery
are much faster than the characteristic times associated to births and deaths.
Usually, the initial conditions are set as
$S(0)>0,\,\,\,I(0)>0\,\,\mbox{and }R(0)=0.$ (6)
It is straightforward to show that
$\left.\frac{dI}{dt}\right|_{t=0}=I(0)(\beta S(0)-\gamma),$ (7)
and that the sign of the derivative depends on the value of
$S_{c}=\frac{\gamma}{\beta}$. When $S(t)>S_{c}$, the derivative is positive
and the number of infected individuals increases. When $S(t)$ goes below this
threshold, the epidemic starts to fade out.
A rather non intuitive result can be obtained from Eq. 2.1. We can write
$\displaystyle\frac{dS}{dR}$ $\displaystyle=$ $\displaystyle-\frac{S}{\rho}$
(8) $\displaystyle\Rightarrow$ $\displaystyle S=S_{0}\exp[-R/\rho]\geq
S_{0}\exp[-N/\rho]>0$ $\displaystyle\Rightarrow$ $\displaystyle
0<S(\infty)\leq N.$
The epidemics stops when $I(t)=0$, so we can set $I(\infty)=0$, so
$R(\infty)=N-S(\infty)$. From (8),
$\displaystyle S(\infty)$
$\displaystyle=S_{0}\exp\left[-\frac{R(\infty)}{\rho}\right]$
$\displaystyle=S_{0}\exp\left[-\frac{N-S(\infty)}{\rho}\right].$ (9)
The last equation is a transcendent expression with a positive root
$S(\infty)$.
Taking (9), we can calculate the total number of susceptible individuals
throughout the whole epidemic process
$I_{\mbox{\small total}}=I_{0}+S_{0}-S(\infty).$ (10)
As $I(t)\to 0$ and $S(t)\to S(\infty)>0$, we conclude that when the epidemics
end, there is a portion of the population that has not been affected
The previous model can be extended to include vital dynamics [23], delays
equations [24], age structured population, migration [25], and diffusion. In
any case, all these generalizations only introduce some slight changes on the
steady states of the system, or in the case of spatially extended models,
travelling waves [26].
Figure 2 displays the typical behavior of the density of individuals in each
of the epidemiological compartments described by Eq. (2.1). Compare this with
the pattern shown in Fig. 1.
Figure 2: Temporal behavior of the proportion of individuals in each of the
three compartments of the SIR model.
### 2.2 The SIS Model
The SIS model assumes that the disease does not confer immunity to infected
individuals after recovery. Thus, after the infective period, the infected
individual recovers and is again included in the S group. Therefore, the model
presents only two epidemiological compartments, S and I. This model is
suitable to describe the behavior of epidemics produced by bacterial agent
diseases (meningitis, plague, venereal diseases) and by protozoan agent
diseases (malaria) [22]. We can write the equations for a general SIS model
assuming again that the population is constant,
$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI+\gamma
I$ $\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma
I.$ (11)
As the relation $S+I=1$ holds, Eq. (2.2) can be reduced to a single equation,
$\frac{dI}{dt}=(\beta-\gamma)I-\beta I^{2}.$ (12)
The solution of this equation is
$I(t)=(1-\frac{\gamma}{\beta})\frac{C\exp[(\gamma-\beta)t]}{1+C\exp[(\gamma-\beta)t]},$
(13)
where $C$ is defined by the initial conditions as
$C=\frac{\beta i_{0}}{\beta(1-i_{0})-\gamma}.$ (14)
If $I_{0}$ is small and $\beta>\gamma$, the solution is a logistic growth that
saturates before the whole population is infected, the stationary value is
$I_{s}=\frac{\beta-\gamma}{\beta}$. It can be shown that $R_{0}=\beta/\gamma$.
This sets the condition for the epidemic to persist.
### 2.3 Other models
The literature on epidemiological models includes several generalizations
about the previous ones to adapt the description to the particularities of a
specific infectious disease [27]. One possibility is to increase the number of
compartments to describe different stages of the state of an individual during
the epidemic spread. Among these models, we can mention the SIRS, a simple
extension of the SIR that does not confer a permanent immunity to recovered
individuals and after some time they rejoin the susceptible group,
$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI++\lambda
R$ $\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma
I$ (15) $\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma
I-\lambda R.$
Other models include more epidemiological groups or compartments, such as the
SEIS and SEIR model, that take into consideration the exposed or latent period
of the disease, by defining an additional compartment E.
There are several diseases in which there is a vertical transient immunity
transmission from a mother to her newborn. Then, each individual is born with
a passive immunity acquired from the mother. To indicate this, an additional
group P is added.
The range of possibilities is rather extended, and this is reflected in the
title of Ref. [27]: “A thousand and one epidemiological models”. There are a
lot of possibilities to define the compartment structure. Usually, this
structure is represented as a transfer chart indicating the flow between the
compartments and the external contributions. Figure 3 shows an example of a
diagram for a SEIRS model, taken from Ref. [27].
Figure 3: Transfer diagram for a SEIRS models. Taken from Ref. [27].
Horizontal incidence refers to a contagion due to a contact between a
susceptible and infectious individual, vertical incidence account for the
possibility for the offspring of infected parents to be born infected, such as
with AIDS, hepatitis B, Chlamydia, etc.
Many of the previous models have been expanded, including stochastic terms.
One of the most relevant differences between the deterministic and stochastic
models is their asymptotic behavior. A stochastic model can show a solution
converging to the disease-free state when the deterministic counterpart
predicts an endemic equilibrium. The results obtained from the stochastic
models are generally expressed in terms of the probability of an outbreak and
of its size and duration distribution [14, 15, 16, 17, 18, 19].
## 3 Complex Networks
A graph or network is a mathematical representation of a set of objects that
may be connected between them through links. The interconnected objects are
represented by the nodes (or vertices) of the graph while the connecting links
are associated to the edges of the graph. Networks can be characterized by
several topological properties, some of which will be introduced later. Social
links are preponderantly non directional (symmetric), though there are some
cases of social directed networks. The set of nodes attached to a given node
through these links is called its neighborhood. The size of the neighborhood
is the degree of the node.
While the study of graph theory dates back to the pioneering works of Erd s
and Renyi in the 1950s [28], their gradual colonization of the modern
epidemiological models has only started a decade ago. The attention of
modelers was drawn to graph theory when some authors started to point out that
the social structure could be mimicked by networks constructed under very
simple premises [30, 34]. Since then, a huge collection of computer-generated
networks have been studied in the context of disease transmission. The
underlying rationale for the use of networks is that they can represent how
individuals are distributed in social and geographical space and how the
contacts between them are promoted, reinforced or inhibited, according to the
rules of social dynamics. When the population is fully mixed, each individual
has the same probability of coming into contact with any other individual.
This assumption makes it possible to calculate the effective contact rates
$\beta$ as the product of the transmission rate of the disease, the effective
number of contacts per unit time and the proportion of these contacts that
propagate the infection. The formulation of a mean field model is then
straightforward. However, in real systems, the acquaintances of each
individual are reduced to a portion of the whole population. Each person has a
set of contacts that shapes the local topology of the neighborhood. The whole
social architecture, the network of contacts, can be represented with a graph.
In the limiting case when the mean degree of the nodes in a network is close
to the total number of nodes, the difference between a structured population
and a fully mixed one fades out. The differences are noticeable when the
network is diluted, i.e., the mean degree of the node is small compared with
the size of the network. This will be a necessary condition for all the
networks used to model disease propagation. In the following paragraphs, we
will introduce the most common families of networks used for epidemiological
modeling.
Lattices. When incorporating a network to a model, the simplest case is
considering a grid or a lattice. In a squared $d$ dimensional lattice, each
node is connected to $2d$ neighbors. Individuals are regularly located and
connected with adjacent neighbors; therefore, contacts are localized in space.
Figure 4 shows, among others, an example of a two dimensional square lattice
Figure 4: Scheme of four kinds of networks: (a) Lattice, (b)scale free, (c)
Exponential, (d) Small World.
Small-world networks. The concept of Small World was introduced by Milgram in
1967 in order to describe the topological properties of social communities and
relationships [29]. Some years ago, Watts and Strogatz introduced a model for
constructing networks displaying topological features that mimic the social
architecture revealed by Milgram. In this model of Small World (SW) networks a
single parameter $p$, running from 0 to 1, characterizes the degree of
disorder of the network, ranging from a regular lattice to a completely random
graph [30]. The construction of these networks starts from a regular, one-
dimensional, periodic lattice of N elements and coordination number $2K$. Each
of the sites is visited, rewiring K of its links with probability $p$. Values
of $p$ within the interval [0,1] produce a continuous spectrum of small world
networks. Note that $p$ is the fraction of modified regular links. A schematic
representation of this family of networks is shown in Fig. 5.
Figure 5: Representation of several Small World Networks constructed according
the algorithm presented in Ref. [30]. As the disorder degree increases, there
number of shortcuts grow replacing some of the original (ordered network)
links.
To characterize the topological properties of the SW networks, two magnitudes
are calculated. The first one, $L(p)$, measures the mean topological distance
between any pair of elements in the network, that is, the shortest path
between two vertices, averaged over all pairs of vertices. Thus, an ordered
lattice has $L(0)\sim N/K$, while, for a random network, $L(1)\sim
ln(N)/ln(K)$. The second one, $C(p)$, measures the mean clustering of an
element’s neighborhood. $C(p)$ is defined in the following way: Let us
consider the element $i$, having $k_{i}$ neighbors connected to it. We denote
by $c_{i}(p)$ the number of neighbors of element $i$ that are neighbors among
themselves, normalized to the value that this would have if all of them were
connected to one another; namely, $k_{i}(k_{i}-1)/2$. Now, $C(p)$ is the
average, over the system, of the local clusterization $c_{i}(p)$. Ordered
lattices are highly clustered, with $C(0)\sim 3/4$, and random lattices are
characterized by $C(1)\sim K/N$. Between these extremes, small worlds are
characterized by a short length between elements, like random networks, and
high clusterization, like ordered ones.
Figure 6: In this figure, we show the mean values of the clustering
coefficient $C$ and the path length $L$ as a function of the disorder
parameter $p$. Note the fast decay of $L$ and the presence of a region where
the value adopted by $L$ is similar to the one corresponding to total
disorder, while the value adopted by $C$ is close to the one corresponding to
the ordered case.
Other procedures for developing similar social networks have been proposed in
Ref. [31] where instead of rewiring existing links to create shortcuts, the
procedure add links connecting two randomly chosen nodes with probability $p$.
In Fig. 7, we show an example, analogous to the one shown in Fig. 5.
Figure 7: Representation of several Small World Networks constructed according
the algorithm presented in Ref. [31]. As the disorder degree increases, three
number of shortcuts as well as the number of total links grow.
Random networks. There are different families of networks with random genesis
but displaying a wide spectra of complex topologies. In random networks, the
spatial position of individuals is irrelevant and the links are randomly
distributed. The iconic Erdös-Rényi (ER) random graphs are built from a set of
nodes that are randomly connected with probability $p$, independently of any
other existing connection. The degree distribution, i.e., the number of links
associated to each node, is binomial and when the number of nodes is large, it
can be approximated by a Poisson distribution [32].
Figure 8: This figure shows examples of (a) ER and (b) BA networks. The figure
also displays the connectivity distribution $P(k)$, that follows a binomial
distribution for the ER networks and a power law for BA networks.
In Ref. [33], the authors propose a formalism based on the generating function
that permits to construct random networks with arbitrary degree distribution.
The mechanism of construction also allows for further analytic studies on
these networks. In particular, networks can be chosen to have a power law
degree distribution. This case will be presented in the next paragraphs.
Scale-free network. As mentioned before, one of the most revealing measures of
a network is its degree of distribution, i.e., the distribution of the number
of connections of the nodes. In most real networks, it is far from being
homogeneous, with highly connected individuals on one extreme and almost
isolated nodes on the other. Scale-free networks provide a means of achieving
such extreme levels of heterogeneity.
Scale-free networks are constructed by adding new individuals to a core, with
a connection mechanism that imitates the underlying process that rules the
choice of social contacts. The Barabási - Albert (BA) model algorithm, one of
the triggers of the present huge interest on scale-free networks, uses a
preferential attachment mechanism [34]. The algorithm starts from a small
nucleus of connected nodes. At each step, a new node is added to the network
and connected to $m$ existing nodes. The probability of choosing a node
$p_{i}$ is proportional to the number of links that the existing node already
has
$p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},$
where $k_{i}$ is the degree of node $i$. That means that the new nodes have a
preference to attach themselves to the most “popular” nodes. One salient
feature of these networks is that their degree distribution is scale-free,
following a power law of the form
$P\left(k\right)\sim k^{-3}.$
A sketch of the typical topology of the last two networks is shown in Fig. 8.
While the degree distribution of the ER network has a clear peak and is close
to homogeneous, the topology of the BA network is dominated by the presence of
hub, highly connected nodes. The figure also displays the typical degree
distribution $P(k)$ for each case.
Over the last years, many other attachment mechanisms have been proposed to
obtain scale-free networks with other adjusted properties such as the
clustering coefficient, higher moments of the degree distribution [35, 36, 37,
38].
Coevolutive or adaptive topology. When one of the former examples of networks
is chosen as a model for the social woven, there is an implicit assumption:
the underlying social topology is frozen. However, this situation does not
reflect the observed fact that in real populations, social and migratory
phenomena, sanitary isolation or other processes can lead to a dynamic
configuration of contacts, with some links being eliminated, other being
created. If the time span of the epidemics is long enough, the social network
will change and these changes will not be reflected if the topology remains
fixed. This is particularly important in small groups. The social dynamics,
including the epidemic process, can shape the topology of the network,
creating a feedback mechanism that can favor or attempt against the
propagation of an infectious disease. For this reason, some models consider a
coevolving network, with dynamic links that change the aspect of the networks
while the epidemics occur.
## 4 Epidemiological Models on Networks
In this section, we will discuss several models based on the use of complex
networks to mimic the social architecture. The discussion will be organized
according to the topology of these underlying networks.
Lattices. Lattices were the first attempt to represent the underlying topology
of the social contacts and thus to analyze the possible effect of interactions
at the individual level. These models took distance from the paradigmatic
fully mixed assumption and focused on looking for those phenomena that a mean
field model could not explain. Still, the lattices cannot fully capture the
role of inhomogeneities. As the individuals are located on a regular grid,
mostly two dimensional, the neighborhood of each node is reduced to the
adjacent nodes, inducing only short range or localized interactions. A typical
model considers that the nodes can be in any of the epidemiological states or
compartments. The dynamic of the epidemics evolves through a contact process
[39] and the evolutive rules do not differ too much from traditional cellular
automata models [40]. Disease transmission is modeled as a stochastic process.
Each infected node has a probability $p_{i}$ of infecting a neighboring
susceptible node. Once infected, the individuals may recover from infection
with a probability $p_{r}$; i.e., the infective stage lasts typically
$1/p_{r}$. From the infective phase, the individuals can move back to the
susceptible compartment or the recovered phase, depending on whether the
models are SIS or SIR. Usually, a localized infectious focus is introduced
among the population. The transient shows a local and slow development of the
disease that at the initial stage involves the growing of a cluster, with the
infection propagating at its boundary, like a traveling wave. After the
initial transient, SIS, SIR and SIRS models behave in different ways. The
initially local dynamics that can or cannot propagate to the whole system is
what introduces a completely new behavior in this spatially extended model. In
Ref. [41], the author argued the infective clusters behave as the clusters in
the directed percolation model. Figure 9 shows an example of the behavior of
the asymptotic value of infected individuals under SIS dynamics in a two
dimensional square lattice. The figure reflects the results found in Ref.
[42]. The parameter $f$ is associated to the infectivity of infectious
individuals, closely related to the contact rate. We observe the inset
displaying the scaling of the data with a power-like curve
$A|f-f_{c}|^{\alpha}$, with $\alpha\approx 0.5$ [42].
Figure 9: SIS model. Asymptotic value of infected individuals as a function
of the infectivity of infectious individuals. The inset displays the scaling
of the data with a power-like curve $A|f-f_{c}|^{\alpha}$, with $\alpha\approx
0.5$. Adapted from Ref. [42].
As mentioned before, Kermack and Mckendrick [12] proved the existence of a
propagation threshold for the disease invading a susceptible population. The
lattice based SIR models introduce a different threshold. The simulations show
that epidemics can just remain localized around the initial focus or turn into
a pandemic, affecting the entire population. The most dramatic examples of
real pandemic are the Black Plague between the 1300 and 1500 and the Spanish
Flu, in 1917-1918. Both left a wake of death and terror while crossing the
European continent. The predicted new threshold established a limit below in
which the pandemic behaviour is not achieved.
Figure 10: SIR model. Asymptotic value of susceptible individuals as a
function of the infectivity of infectious individuals. The inset displays the
scaling of the data with a power-like curve $A|f-f_{c}|^{\alpha}$, with
$\alpha\approx 0.5$. Adapted from Ref. [42].
Some works about epidemic propagation on lattices are analogous to forest fire
models [43], with the characteristic feature that the frequency distributions
of the epidemic sizes and duration obey a power-law. In Ref. [44, 45], the
authors exploit these analogies to explain the observed behavior of measles,
whooping cough and mumps in the Faroe Islands. The observed data display a
power-like behavior.
Random networks. Most of the models based on random graphs were previous to
the renewed interest on complex networks. A simple but effective idea for the
study of the dynamics of diseases on random networks is the contact process
proposed in Ref. [46] that produces a branching phenomena while the infection
propagates. In Ref. [47], the authors use a E-R network with an approximately
Poisson degree distribution. A common feature to all these models is that the
rate of the initial transient growth is smaller than the corresponding to
similar models in fully-mixed populations. This effect can be easily
understood noting that, on the one hand, the degree of a given initially
infected node is typically small, thus having a limited number of susceptible
contacts. On the other hand, there is a self limiting process due to the fact
that the same infection propagation predates the local availability of
susceptible targets.
A different analytical approach to random networks is presented in Ref. [48].
The author shows that a family of variants of the SIR model can be solved
exactly on random networks built by a generating function method and appealing
to the formalism of percolation models. The author analyzes the propagation of
a disease in networks with arbitrary degree distributions and heterogeneous
infectiveness times and transmission probabilities. The results include the
particular case of scale-free networks, that will be discussed later.
Small-world networks. As mentioned above, regular networks can exhibit high
clustering but long path lengths. On the other extreme, random networks have a
lot of shortcuts between two distant individuals, but a negligible clustering.
Both features affect the propagative behavior on any modeled disease. The
spread of infectious diseases on SW networks has been analyzed in several
works. The interested was triggered by the fact that even a small number of
random connections added to a regular lattice, following for example the
algorithm described in Ref. [30], produces unexpected macroscopic effects. By
sharing topological properties from random and ordered networks, SW networks
can display complex propagative patterns. On the one hand, the high level of
clustering means that most infection occurs locally. On the other hand,
shortcuts are vehicles for the fast spread of the epidemic to the entire
population.
In Ref. [51], the authors study a SI model and show that shortcuts can
dramatically increase the possibility of an epidemic event. The analysis is
based on bond percolation concepts. While the result could be easily
anticipated due to the long range propagative properties of shortcuts, the
authors find an important analytic result. It was a study of a SIRS models
that showed for the first time the evidence of a dramatic change in the
behavior of an epidemic due to changes in the underlying social topology [52].
By specifically analyzing the effect of clustering on the dynamics of an
epidemics, the authors show that a SIRS model on a SW network presents two
distinct types of behavior. As the rewiring parameter $p$ increases, the
system transits from an endemic state, with a low level of infection to
periodic oscillations in the number of infected individuals, reflecting an
underlying synchronization phenomena. The transition from one regime to the
other is sharp and occurs at a finite value of $p$. The reason behind this
phenomenon is still unknown. Figure 11 shows the temporal behavior of the
number of infected individuals for three values of the rewiring parameter $p$,
as found in Ref. [52].
Figure 11: Asymptotic behavior of the number of infected individuals in three
SW networks with different degrees of disorder $p$. The emergence of a
synchronized pattern is evident in the bottom graph.
It would not be responsible to affirm that SW networks reflect all the real
social structures. However, they capture essential aspects of such
organization that play central roles in the propagation of a diseases, namely,
the clustering coefficient and the short social distance between individuals.
Understanding that there are certain limitations, SW networks help to mimic
different social organizations that range from rural population to big cities.
There are more sophisticated models of networks with topologies that are more
closely related to real social organizations at large scale. These networks
are characterized by a truncated power law distribution of the degree of the
nodes and by values of clustering and mean distance corresponding to the small
world regime.
Scale-free networks. Scale-free networks captured the attention of
epidemiologists due to the close resemblance between their extreme degree
distribution and the pattern of social contacts in real populations. A power
law degree distribution presents individuals with many contacts and who play
the role of super-spreaders. A higher number of contacts implies a greater
risk of infection and correspondingly, a higher “success” as an infectious
agent. Some scale-free networks present positive assortativity. That
translates into the fact that highly connected nodes are connected among them.
This local structures can be used to model the existence of core groups of
high-risk individuals, that help to maintain sexually transmitted diseases in
a population dominated by long-term monogamous relationships [53]. Models of
disease spread through scale-free networks showed that the infection is
concentrated among the individuals with highest degree [54, 48]. One of the
most surprising results is the one found in Ref. [54]. There, the authors show
that no matter the values taken by the relevant epidemiological parameter,
there is no epidemic threshold. Once installed in a scale-free network, the
disease will always propagate, independently of $R_{0}$. Remember that when
analyzed under the fully mixed assumption, the studied SIS model has a
threshold. The authors perform analytic and numerical calculations of the
propagation of the disease, to show the lack of thresholds. Later, in Ref.
[55], it was pointed out that networks with divergent second moments in the
degree distribution will show no epidemic threshold. The B A network fulfills
this condition. In Ref. [56, 57], the authors analyze the structure of
different networks of sexual encounters, to find that it has a pattern of
contact closely related to a power law. They also discuss the implications of
such structure on the propagation of venereal diseases
Co-evolutionary networks. Co-evolutionary or adaptive networks take into
account the own dynamics of the social links. In some occasions, the
characteristic times associated to changes in social connections are
comparable with the time scales of an epidemic process. Some other times, the
presence of n infectious core induces changes in social links. Consider for
example a case when the population of susceptible individuals after learning
about the existence of infectious individuals try to avoid them, or another
case when the health policies promote the isolation of infectious individuals
[58]. The behavior of models based on adaptive network is determined by the
interplay of two different dynamics that sometimes have competitive effects.
On the one hand, we have the dynamics of the disease propagation. On the other
hand, the network dynamics that operates to block the advance of the
infection. The later is dominated by the rewiring rate of the network, which
affects the fraction of susceptible individuals connected to infective ones.
The most obvious choice is to eliminate the infectious contacts of a
susceptible individual by deleting or replacing them with noninfectious ones.
The net effect is an effective reduction of the infection rate. While static
networks typically predict either a single attracting endemic or disease-free
state, the adaptive networks show a new phenomenon, a bistable situation
shared by both states. The bistability appears for small rewiring rates [58,
59, 60, 61]. In Ref. [61], the authors consider a contact switching dynamics.
All links connecting a susceptible agents with an infective one is broken with
a rate $r$. The susceptible node is then connected to a new neighbor, randomly
chosen among the entire population. The authors show that reconnection can
completely prevent an epidemics, eliminating the disease. The main conclusion
is that the mechanism that they propose, contact switching, is a robust and
effective control strategy. Figure 12 displays the results found in Ref. [61],
where two completely different types of behavior can be distinguish as the
rewiring parameter $r$ changes. The crossover from one regime to the other is
a second order phase transition.
$\lambda$
Figure 12: These two panels show the equilibrium fraction number of infected
individuals, as a function of the infectivity of the disease, $\lambda$. Lines
are analytic results, symbols are numerical simulations. Adapted from Ref.
[61].
## 5 Immunization in networks
Any epidemiological model can reproduce the fact that the number of
individuals in a population who are effectively immune to a given infection
depends on the proportion of previously infected individuals and the
proportion who have been efficiently vaccinated.
For some time, the epidemiologists knew about an emerging effect called herd
protection (or herd immunity). They discovered the occurrence of a global
immunizing effect verified when the vaccination of a significant portion of a
population provides protection for individuals who have not or cannot
developed immunity. Herd protection is particularly important for diseases
transmitted from person to person. As the infection progresses through the
social links, its advance can be disrupted when many individuals are immune
and their links to non immune subjects are no longer valid channels of
propagation. The net effect is that the greater the proportion of immune
individuals is, the smaller the probability that a susceptible individual will
come into contact with an infectious one. The vaccinated individuals will not
contract neither transmit the disease, thus establishing a firewall between
infected and susceptible individuals.
While taking profit from the herd protection is far from being an optimal
public health policy, it is still taken into consideration when individuals
cannot be vaccinated due, for example, to immune disorders or allergies. The
herd protection effect is equivalent to reduce the $R_{0}$ of a disease. There
is a threshold value for the proportion of necessary immune individuals in a
population for the disease not to persist or propagate. Its value depends on
the efficacy of the vaccine but also on the virulence of the disease and the
contact rate. If the herd effect reduces the risk of infection among the
uninfected enough, then the infection may no longer be sustainable within the
population and the infection may be eliminated. In a real population, the
emergence of herd immunity is closely related to the social architecture.
While many fully mixed models can describe the phenomenon, the real effect is
much more accurately reproduced by models based on Social Networks. One of the
most expected result is to quantify how the shape of a social network can
affect the level of vaccination required for herd immunity. There is a related
phenomenon, not discussed here, that consists in the propagation of real
immunity from a vaccinated individual to a non vaccinated one. This is called
contact immunity and has been verified for several vaccines, such as the OPV
[62].
The models to quantify the success of immunization of the population propose a
targeted immunization of the populations.
It is well established that immunization of randomly selected individuals
requires immunizing a very large fraction of the population, in order to
arrest epidemics that spread upon contact between infected individuals.
In Ref. [63], the authors studied the effects of immunization on an SIR
epidemiological model evolving on a SW network. In the absence of
immunization, the model exhibits a transition from a regime where the disease
remains localized to a regime where it spreads over a portion of the system.
The effect of immunization reveals through two different phenomena. First,
there is an overall decrease in the fraction of the population affected by the
disease. Second, there is a shift of the transition point towards higher
values of the disorder. This can be easily understood as the effective average
number of susceptible neighbors per individual decreases. Targeted
immunization that is applied by vaccinating those individuals with the highest
degree, produces a substantial improvement in disease control. It is
interesting to point out that this improvement occurs even when the degree
distribution over small-world networks is relatively uniform, so that the best
connected sites do not monopolize a disproportionately high number of links.
Figure 13 shows an example of the results found in Ref. [63], where the author
compare the amount of non vaccinated individuals that are infected for
different levels of vaccination, $\rho$, and different degrees of disorder of
the SW network $p$, as defined in Ref. [30].
Figure 13: Fraction $r$ of the non-vaccinated population that becomes
infected during the disease propagation, as a function of the disorder
parameter $p$, for various levels of random immunization (upper) and targeted
immunization (bottom). Adapted from Ref. [63].
In a scale-free network, the existence of individuals of an arbitrarily large
degree implies that there is no level of uniform random vaccination that can
prevent an epidemic propagation, even extremely high densities of randomly
immunized individuals can prevent a major epidemic outbreak. The discussed
susceptibility of these networks to epidemic hinders the implementation of a
prevention strategy different from the trivial immunization of all the
population [54, 55, 66].
Taking into account the inhomogeneous connectivity properties of scale-free
networks can help to develop successful immunization strategies. The obvious
choice is to vaccinate individuals according to their connectivity. A
selective vaccination can be very efficient, as targeting some of the super-
spreaders can be sufficient to prevent an epidemic [67, 55].
The vaccination of a small fraction of these individuals increases quite
dramatically the global tolerance to infections of the network.
When comparing the uniform and the targeted immunization procedures [67], the
results indicate that while uniform immunization does not produce any
observable reduction of the infection prevalence, the targeted immunization
inhibits the propagation of the infection even at very low immunization
levels. These conclusions are particularly relevant when dealing with sexually
transmitted diseases, as the number of sexual partners of the individuals
follows a distribution pattern close to a power law.
Targeted immunization of the most highly connected individuals [64, 65, 67]
proves to be effective, but requires global information about the architecture
of network that could be unavailable in many cases. In Ref. [68], the authors
proposed a different immunization strategy that does not use information about
the degree of the nodes or other global properties of the network but achieves
the desired pattern of immunization. The authors called it acquaintance
immunization as the targeted individuals are the acquaintances of randomly
selected nodes. The procedure consists of choosing a random fraction $p_{i}$
of the nodes, selecting a random acquaintance per node with whom they are in
contact and vaccinating them. The strategy operates at the local level. The
fraction $p_{i}$ may be larger than 1, for a node might be chosen more than
once, but the fraction of immunized nodes is always less than 1. This strategy
allows for a low vaccination level to achieve the immunization threshold. The
procedure is able to indirectly detect the most connected individuals, as they
are acquaintances of many nodes so the probability of being chosen for
vaccination is higher.
## 6 Final remarks
The mathematical modeling of the propagation of infectious diseases transcends
the academic interest. Any action pointing to prevent a possible pandemic
situation or to optimize the vaccination strategies to achieve critical
coverage are the core of any public health policy. The understanding of the
behavior of epidemics showed a sharp improvement during the last century,
boosted by the formulation of mathematical models. However, for a long time,
many important aspects regarding the epidemic processes remained unexplained
or out of the scope of the traditional models. Perhaps, the most important one
is the feedback mechanism that develops between the social topology and the
advance of an infectious disease. The new types of models developed during the
last decade made an important contribution to the field by incorporating a
mean of describing the effect of the social pattern. While a quantitative
analysis of a real situation still demands huge computational resources, the
mathematical foundations to develop it are already laid. The is too much to do
yet, but the breakthrough produced by these new models based on complex
networks is already undeniable.
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|
arxiv-papers
| 2013-12-12T18:44:12 |
2024-09-04T02:49:55.397663
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marcelo N. Kuperman",
"submitter": "Marcelo N. Kuperman",
"url": "https://arxiv.org/abs/1312.3838"
}
|
1312.3993
|
# Identities of symmetry for $(h,q)-$extension of higher-order Euler
polynomials
Dae San Kim, Taekyun Kim, Jong Jin Seo 1
Department of Mathematics
Sogang University
Seoul 121-742, Republic of Korea [email protected] 2
Department of Mathematics
Kwangwoon University
Seoul 139-701, Republic of Korea [email protected] 3
Department of Applied Mathematics
Pukyong National University
Busan 608-737, Republic of Korea [email protected]
###### Abstract.
In this paper, we study some symmetric properties of the multiple $q-$Euler
zeta function. From these properties, we derive several identities of symmetry
for the $(h,q)-$extension of higher-order Euler polynomials, which is an
answer to a part of open question in $[7]$.
###### Key words and phrases:
multiple $q-$Euler zeta function, $(h,q)-$extension of higher-order Euler
polynomials
## 1\. Introduction
Let $\mathbb{C}$ be the complex number field. We assume that $q\in\mathbb{C}$
with $|q|<1$ and the $q-$number is defined by $[x]_{q}=\frac{1-q^{x}}{1-q}$.
Note that $\lim_{q\rightarrow 1}{[x]_{q}=x}.$ As is well known, the higher-
order Euler polynomials $E_{n}^{(r)}(x)$ are defined by the generating
function to be
$F^{(r)}(x,t)=\left(\frac{2}{e^{t}+1}\right)^{r}e^{xt}=\sum_{n=0}^{\infty}E_{n}^{(r)}(x)\frac{t^{n}}{n!},\
\ \textnormal{(see [4], [16]),}$ (1.1)
where $|t|<\pi$.
When $x=0,E_{n}^{(r)}=E_{n}^{(r)}(0)$ are called the Euler numbers of order
$r.$ Recently, the second author defined the $(h,q)-$extension of higher-order
Euler polynomials, which is given by the generating function to be
$\begin{split}F_{q}^{(h,r)}(x,t)=&[2]_{q}^{r}\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}q^{\sum_{j=1}^{r}(h-j+1)m_{j}}(-1)^{\sum_{j=1}^{r}m_{j}}e^{[m_{1}+\cdot\cdot\cdot+m_{r}+x]_{q}t}\\\
=&\sum_{n=0}^{\infty}E_{n,q}^{(h,r)}(x)\frac{t^{n}}{n!},\ \ \textnormal{(see
[6], [8]),}\end{split}$ (1.2)
where $h\in{\mathbb{Z}}$ and $r\in{\mathbb{Z}}_{\geq 0}.$
Note that $\lim_{q\rightarrow
1}F_{q}^{(h,r)}(x,t)=(\frac{2}{e^{t}+1})^{r}e^{xt}=\sum_{n=0}^{\infty}E_{n}^{(r)}(x){\frac{t^{n}}{n!}}.$
By $(1.2)$, we get
$\
F_{q}^{(h,r)}(t,x)=[2]_{q}^{r}\sum_{m=0}^{\infty}{{m+r-1}\choose{m}}_{q}(-q^{h-r+1})^{m}e^{[m+x]_{q}t},\
\ \textnormal{(see [6], [8]),}$ (1.3)
where
${{x}\choose{m}}_{q}={\frac{[x]_{q}[x-1]_{q}[x-2]_{q}\cdot\cdot\cdot[x-m+1]_{q}}{[m]_{q}!}}.$
From $(1.3)$, we can derive the following equation:
$\begin{split}E_{n,q}^{(h,r)}(x)=&{\frac{[2]_{q}^{r}}{(1-q)^{n}}}\sum_{l=0}^{n}{n\choose
l}{\frac{(-q^{x})^{l}}{(-q^{h-r+l+1}:q)}_{r}}\\\
=&[2]_{q}^{r}\sum_{m=0}^{\infty}{m+r-1\choose
m}_{q}(-q^{h-r+1})^{m}[m+x]_{q}^{n},\ \ \textnormal{(see [6]),}\end{split}$
(1.4)
where $(x:q)_{n}=(1-x)(1-xq)\cdot\cdot\cdot(1-xq^{n-1}).$
In $[6]$ and $[8]$, the second author constructed the multiple $q-$Euler zeta
function which interpolates the $(h,q)-$ extension of higher-order Euler
polynomials at negative integers as follows :
$\begin{split}\zeta_{q,r}^{(h)}(s,x)=&{\frac{1}{\Gamma(s)}}\int_{0}^{\infty}F_{q}^{(h,r)}(x,t)t^{s-1}dt\\\
=&[2]_{q}^{r}\sum_{m_{1},\cdot\cdot\cdot\
,m_{r}=0}^{\infty}\frac{(-1)^{m_{1}+\cdot\cdot\cdot+m_{r}}q^{\sum_{j=1}^{r}(h-j+1)m_{j}}}{[m_{1}+\cdot\cdot\cdot+m_{r}+x]_{q}^{s}},\
\ \textnormal{(see [6]),}\end{split}$ (1.5)
where ${h,s}\in{\mathbb{C}},x\in{\mathbb{R}}$ with
$x\neq{0,-1,-2,\cdot\cdot\cdot}.$
From $(1.5)$, we have
$\begin{split}\zeta_{q,r}^{(h)}(s,x)=[2]_{q}^{r}\sum_{m=0}^{\infty}{{m+r-1}\choose{m}}_{q}{(-q^{h-j+1})^{m}}\frac{1}{[m+x]_{q}^{s}}.\end{split}$
(1.6)
Using the Cauchy residue theorem and Laureut series in $(1.5)$, we obtain the
following lemma.
###### Lemma 1.1.
For $n\in{\mathbb{Z}}_{\geq 0}$ and $h\in{\mathbb{Z}},$ we have
$\zeta_{q,r}^{(h)}(-n,x)=E_{n,q}^{(h,r)}(x),\ \ \textnormal{(see [6], [8])}.$
In $[7]$, the second author introduced many identities of symmetry for Euler
and Bernoulli polynomials which are derived from the $p$-adic integral
expression of the generating function and suggested an open problem about
finding identities of symmetry for the Carlitz’s type $q$-Euler numbers and
polynomials.
When $x=0$, $E_{n,q}^{(h,r)}=E_{n,q}^{(h,r)}(0)$ are called the $(h,q)$-Euler
numbers of order $r$.
From $(1.3)$ and $(1.4)$, we can derive the following equation :
$\begin{split}E_{n,q}^{(h,r)}(x)=(q^{x}E_{q}^{(h,r)}+[x]_{q})^{n}=\sum_{l=0}^{n}{{n}\choose{l}}q^{lx}E_{l,q}^{(h,r)}[x]_{q}^{n-l},\end{split}$
(1.7)
with the usual convention about replacing $(E_{q}^{(h,r)})^{n}$ by
$E_{n,q}^{(h,r)}.$
Recently, Y. Simsek introduced recurrence symmetric identities for
$(h,q)$-Euler polynomials and alternating sums of powers of consecutive
$(h,q)$\- integers (see $[16]$).
In this paper, we investigate some symmetric properties of the multiple
$q$-Euler zeta function. From our investigation, we give some new identities
of symmetry for the $(h,q)$-extension of higher-order Euler polynomials, which
is an answer to a part of open question in $[7]$.
## 2\. Identities for $(h,q)-$extension of higher-order Euler Polynomials
In this section, we assume that $h\in{\mathbb{Z}}$ and $a,b\in{\mathbb{N}}$
with $a\equiv 1(mod\ 2)$ and $b\equiv 1(mod\ 2).$ Now, we observe that
$\begin{split}&\frac{1}{[2]_{q^{a}}^{r}}\zeta_{q^{a},r}^{(h)}\left(s,bx+\frac{b({j_{1}+\cdot\cdot\cdot+j_{r}})}{a}\right)\\\
&\ \ \ \ \
=\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}\frac{(-1)^{m_{1}+\cdot\cdot\cdot+m_{r}}q^{a\sum_{j=1}^{r}(h-j+1)m_{j}}}{[m_{1}+\cdot\cdot\cdot+m_{r}+bx+\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}]_{q^{a}}^{s}}\\\
&\ \ \ \ \
=[a]_{q}^{s}\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}\frac{q^{a\sum_{j=1}^{r}(h-j+1)m_{j}}{(-1)^{m_{1}+\cdot\cdot\cdot+m_{r}}}}{[b\sum_{l=1}^{r}j_{l}+{abx}+{a}\sum_{l=1}^{r}{m_{l}}]_{q}^{s}}\\\
&\ \ \ \ \
=[a]_{q}^{s}\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}\sum_{i_{1},\cdot\cdot\cdot,i_{r}=0}^{b-1}\frac{(-1)^{\sum_{l=1}^{r}(i_{l}+bm_{l})}q^{a\sum_{j=1}^{r}(h-j+1)(i_{j}+m_{j}b)}}{[ab(x+\sum_{l=1}^{r}m_{l})+b\sum_{l=1}^{r}j_{l}+a\sum_{l=1}^{r}i_{l}]_{q}^{s}}.\\\
\end{split}$ (2.1)
Thus, by $(2.1)$, we get
$\begin{split}&\frac{[b]_{q}^{s}}{[2]_{q^{a}}^{r}}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{\sum_{l=1}^{r}j_{l}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}\zeta_{{q^{a}},r}^{(h)}\left(s,bx+\frac{b(j_{1}+\cdot\cdot\cdot\,+j_{r})}{a}\right)\\\
=&[a]_{q}^{s}[b]_{q}^{s}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}\sum_{i_{1},\cdot\cdot\cdot,i_{r}=0}^{b-1}\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}\frac{(-1)^{\sum_{l=1}^{r}(i_{l}+j_{l}+m_{l})}q^{a\sum_{l=1}^{r}(h-l+1)i_{l}+b\sum_{l=1}^{r}(h-l+1)j_{l}}}{[ab(x+\sum_{l=1}^{r}m_{l})+\sum_{l=1}^{r}(bj_{l}+ai_{l})]_{q}^{s}}\\\
&\times q^{ab\sum_{l=1}^{r}m_{l}}.\end{split}$ (2.2)
By the same method as $(2.2)$, we see that
$\begin{split}&\frac{[a]_{q}^{s}}{[2]_{q^{b}}^{r}}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{b-1}(-1)^{\sum_{l=1}^{r}j_{l}}q^{a\sum_{l=1}^{r}(h-l+1)j_{l}}\zeta_{{q^{b}},r}^{(h)}\left(s,ax+\frac{a(j_{1}+\cdot\cdot\cdot\,+j_{r})}{b}\right)\\\
=&[b]_{q}^{s}[a]_{q}^{s}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{b-1}\sum_{i_{1},\cdot\cdot\cdot,i_{r}=0}^{a-1}\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}\frac{(-1)^{\sum_{l=1}^{r}(i_{l}+j_{l}+m_{l})}q^{a\sum_{l=1}^{r}(h-l+1)j_{l}+b\sum_{l=1}^{r}(h-l+1)i_{l}}}{[ab(x+\sum_{l=1}^{r}m_{l})+\sum_{l=1}^{r}(bi_{l}+aj_{l})]_{q}^{s}}\\\
&\times q^{ab\sum_{l=1}^{r}m_{l}}.\end{split}$ (2.3)
Therefore, by$(2.2)$ and $(2.3)$, we obtain the following theorem.
###### Theorem 2.1.
For $a,b\in{\mathbb{N}}$, with $a\equiv 1(mod\ 2)$ and $b\equiv 1(mod\ 2)$, we
have
$\begin{split}&[2]_{q^{b}}^{r}[b]_{q}^{s}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}\zeta_{{q^{a}},r}^{(h)}\left(s,bx+\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right)\\\
&=[2]_{q^{a}}^{r}[a]_{q}^{s}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{b-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{a\sum_{l=1}^{r}(h-l+1)j_{l}}\zeta_{{q^{b}},r}^{(h)}\left(s,ax+\frac{a(j_{1}+\cdot\cdot\cdot+j_{r})}{b}\right).\\\
\end{split}$
From Lemma $1.1$ and Theorem $2.1$, we can derive the following theorem.
###### Theorem 2.2.
For $n\in{\mathbb{Z}_{\geq 0}}$ and $a,b\in{\mathbb{N}}$, with $a\equiv 1(mod\
2)$ and $b\equiv 1(mod\ 2)$, we have
$\begin{split}&[2]_{q^{b}}^{r}[a]_{q}^{n}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}E_{n,{q^{a}}}^{(h,r)}\left(bx+\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right)\\\
&=[2]_{q^{a}}^{r}[b]_{q}^{n}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{b-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{a\sum_{l=1}^{r}(h-l+1)j_{l}}E_{n,{q^{b}}}^{(h,r)}\left(ax+\frac{a(j_{1}+\cdot\cdot\cdot+j_{r})}{b}\right).\\\
\end{split}$
By $(1.4)$, we easily see that
$\begin{split}E_{n,q}^{(h,k)}(x+y)&=(q^{x+y}E_{q}^{(h,k)}+[x+y]_{q})^{n}=(q^{x+y}E_{q}^{(h,k)}+q^{x}[y]_{q}+[x]_{q})^{n}\\\
&=\left(q^{x}(q^{y}E_{q}^{(h,k)}+[y]_{q})+[x]_{q}\right)^{n}=\sum_{i=0}^{n}{{n}\choose{i}}q^{ix}E_{i,q}^{(h,k)}(y)[x]_{q}^{n-i}.\end{split}$
(2.4)
Therefore, by $(2.4)$, we obtain the following proposition.
###### Proposition 2.3.
For $n\geq 0$, we have
$\begin{split}E_{n,q}^{(h,k)}(x+y)&=\sum_{i=0}^{n}{n\choose
i}q^{ix}E_{i,q}^{(h,k)}(y)[x]_{q}^{n-i}=\sum_{i=0}^{n}{{n}\choose{i}}q^{(n-i)x}E_{n-i,q}^{(h,k)}(y)[x]_{q}^{i}.\\\
\end{split}$
From Proposition$2.3$, we note that
$\begin{split}&\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}E_{n,{q^{a}}}^{(h,r)}\left(bx+\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right)\\\
&=\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}\sum_{i=0}^{n}{n\choose
i}q^{ia\left(\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right)}E_{i,{q^{a}}}^{(h,r)}(bx)\\\
&\times\left[\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right]_{q^{a}}^{n-i}\\\
&=\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}\sum_{i=0}^{n}{n\choose
i}q^{(n-i){b(j_{1}+\cdot\cdot\cdot+j_{r})}}E_{n-i,{q^{a}}}^{(h,r)}(bx)\\\
&\times\left[\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right]_{q^{a}}^{i}\\\
&=\sum_{i=0}^{n}{n\choose
i}\left(\frac{[b]_{q}}{[a]_{q}}\right)^{i}E_{n-i,{q^{a}}}^{(h,r)}(bx)\sum_{j_{1},\cdot\cdot\cdot\,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h+n-l-i+1)j_{l}}\left[j_{1}+\cdot\cdot\cdot+j_{r}\right]_{q^{b}}^{i}\\\
&=\sum_{i=0}^{n}{n\choose
i}\left(\frac{[b]_{q}}{[a]_{q}}\right)^{i}E_{n-i,{q^{a}}}^{(h,r)}(bx)S_{n,i,{q^{b}}}^{(h,r)}(a),\\\
\end{split}$ (2.5) $\begin{split}where\ \
S_{n,i,q}^{(h,r)}(a)=\sum_{j_{1},\cdot\cdot\cdot\,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{\sum_{l=1}^{r}(h+n-l-i+1)j_{l}}\left[j_{1}+\cdot\cdot\cdot+j_{r}\right]_{q}^{i}.\\\
\end{split}$ (2.6)
By $(2.5)$, we get
$\begin{split}&[2]_{q^{b}}^{r}[a]_{q}^{n}\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{b\sum_{l=1}^{r}(h-l+1)j_{l}}E_{n,{q^{a}}}^{(h.r)}\left(bx+\frac{b(j_{1}+\cdot\cdot\cdot+j_{r})}{a}\right)\\\
&=[2]_{q^{b}}^{r}\sum_{i=0}^{n}{n\choose
i}[a]_{q}^{n-i}[b]_{q}^{i}E_{n-i,{q^{a}}}^{(h,r)}(bx)S_{n,i,{q^{b}}}^{(h,r)}(a).\\\
\end{split}$ (2.7)
By the same method as $(2.7)$, we see that
$\begin{split}&[2]_{q^{a}}^{r}[b]_{q}^{n}\sum_{j_{1},\cdot\cdot\cdot\,j_{r}=0}^{b-1}(-1)^{j_{1}+\cdot\cdot\cdot+j_{l}}q^{a\sum_{l=1}^{r}(h-l+1)j_{l}}E_{n,{q^{b}}}^{(h,r)}\left(ax+\frac{a(j_{1}+\cdot\cdot\cdot+j_{r})}{b}\right)\\\
&=[2]_{q^{a}}^{r}\sum_{i=0}^{n}{n\choose
i}[b]_{q}^{n-i}[a]_{q}^{i}E_{n-i,{q^{b}}}^{(h,r)}(ax)S_{n,i,{q^{a}}}^{(h,r)}(b).\\\
\end{split}$ (2.8)
Therefore, by $(2.7)$ and $(2.8)$, we obtain the following theorem.
###### Theorem 2.4.
For ${a,b}\in\mathbb{N}$ with $a\equiv 1(mod\ 2)$ and $b\equiv 1(mod\
2),n\in{\mathbb{Z}}_{\geq 0}$,
let
$S_{n,i,q}^{(h,r)}(a)={\sum_{j_{1},\cdot\cdot\cdot,j_{r}=0}^{a-1}}(-1)^{j_{1}+\cdot\cdot\cdot+j_{r}}q^{{\sum_{l=1}^{r}}(h+n-l-i+1)j_{l}}[j_{1}+\cdot\cdot\cdot+j_{r}]_{q}^{i}.$
Then we have
$[2]_{q^{b}}^{r}\sum_{i=0}^{n}{n\choose
i}[a]_{q}^{n-i}[b]_{q}^{i}E_{n-i,q^{a}}^{(h,r)}(bx)S_{n,i,q^{b}}^{(h,r)}(a)=[2]_{q^{a}}^{r}\sum_{i=0}^{n}{n\choose
i}[b]_{q}^{n-i}[a]_{q}^{i}E_{n-i,q^{b}}^{(h,r)}(ax)S_{n,i,q^{a}}^{(h,r)}(b).$
It is not difficult to show that
$[x+y+m]_{q}(u+v)-[x]_{q}v=[x]_{q}u+q^{x}[y+m]_{q}(u+v).$ (2.9)
From $(2.9)$, we note that
$\begin{split}&e^{[x]_{q}u}{\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}}q^{{\sum_{j=1}^{r}}(h-j+1)m_{j}}(-1)^{{\sum_{j=1}^{r}}m_{j}}e^{{[m_{1}+\cdot\cdot\cdot+m_{r}+y]_{q}}q^{x}(u+v)}\\\
&=e^{-[x]_{q}v}{\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}}q^{{\sum_{j=1}^{r}}(h-j+1)m_{j}}(-1)^{{\sum_{j=1}^{r}}m_{j}}e^{{[x+y+m_{1}+\cdot\cdot\cdot+m_{r}]_{q}}(u+v)}.\end{split}$
(2.10)
The left hand side of $(2.10)$ multiplied by $[2]_{q}^{r}$ is given by
$\begin{split}&[2]_{q}^{r}e^{[x]_{q}u}{\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}}q^{{\sum_{j=1}^{r}}(h-j+1)m_{j}}(-1)^{{\sum_{j=1}^{r}}m_{j}}e^{{[m_{1}+\cdot\cdot\cdot+m_{r}+y]_{q}}q^{x}(u+v)}\\\
&\ \ \ \
=e^{[x]_{q}u}{\sum_{n=0}^{\infty}}q^{nx}E_{n,q}^{(h,r)}(y)\frac{1}{n!}(u+v)^{n}\\\
&\ \ \ \
=\left({\sum_{l=0}^{\infty}}[x]_{q}^{l}\frac{u^{l}}{l!}\right)\left({\sum_{n=0}^{\infty}}q^{nx}E_{n,q}^{(h,r)}(y){\sum_{k=0}^{n}}\frac{u^{k}}{k!(n-k)!}v^{n-k}\right)\\\
&\ \ \ \
=\left({\sum_{l=0}^{\infty}}[x]_{q}^{l}\frac{u^{l}}{l!}\right)\left({\sum_{k=0}^{\infty}}{\sum_{n=0}^{\infty}}q^{(n+k)x}E_{n+k,q}^{(h,r)}(y)\frac{u^{k}}{k!}\frac{v^{n}}{n!}\right)\\\
&\ \ \ \
=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\left({\sum_{k=0}^{m}}{m\choose
k}q^{(n+k)x}E_{n+k,q}^{(h,r)}(y)[x]_{q}^{m-k}\right)\frac{u^{m}}{m!}\frac{v^{n}}{n!}\end{split}$
(2.11)
The right hand side of $(2.10)$ multiplied by $[2]_{q}^{r}$ is given by
$\begin{split}&[2]_{q}^{r}e^{-[x]_{q}v}{\sum_{m_{1},\cdot\cdot\cdot,m_{r}=0}^{\infty}}q^{{\sum_{j=1}^{r}}(h-j+1)m_{j}}(-1)^{{\sum_{j=1}^{r}}m_{j}}e^{{[x+y+m_{1}+\cdot\cdot\cdot+m_{r}]_{q}}(u+v)}\\\
&\ \ \ \
=e^{-[x]_{q}v}{\sum_{n=0}^{\infty}}E_{n,q}^{(h,r)}(x+y)\frac{1}{n!}(u+v)^{n}\\\
&\ \ \ \
=\left({\sum_{l=0}^{\infty}}\frac{(-[x]_{q})^{l}}{l!}v^{l}\right)\left({\sum_{m=0}^{\infty}}{\sum_{k=0}^{\infty}}E_{m+k,q}^{(h,r)}(x+y)\frac{u^{m}}{m!}\frac{v^{k}}{k!}\right)\\\
&\ \ \ \
=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\left({\sum_{k=0}^{n}}{n\choose
k}E_{m+k,q}^{(h,r)}(x+y)(-[x]_{q})^{n-k}\right)\frac{u^{m}}{m!}\frac{v^{n}}{n!}\\\
&\ \ \ \
=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\left({\sum_{k=0}^{n}}{n\choose
k}E_{m+k,q}^{(h,r)}(x+y)q^{(n-k)x}[-x]_{q}^{n-k}\right)\frac{u^{m}}{m!}\frac{v^{n}}{n!}.\end{split}$
(2.12)
Therefore, by $(2.10)$, $(2.11)$ and $(2.12)$, we obtain the following
theorem.
###### Theorem 2.5.
For $m,n\geq 0$, we have
$\sum_{k=0}^{m}{m\choose
k}q^{(n+k)x}E_{n+k,q}^{(h,r)}(y)[x]_{q}^{m-k}=\sum_{k=0}^{n}{n\choose
k}E_{m+k,q}^{(h,r)}(x+y)q^{(n-k)x}[-x]_{q}^{n-k}.$
Remark. Recently, several authors have studied $(h,q)-$extension of Bernoulli
and Euler polynomials $\textnormal{(see[1]-[5],\ [9]-[17])}.$
## References
* 1.
* 2.
* 3. S. Araci, J. Seo, D. Erdal, New construction weighted $(h,q)-$Genocchi numbers and polynomials related to zeta type functions, Discrete Dyn. Nat. Soc. (2011), Art. ID 487490, 7 pp.
* 4. I. N. Cangul, H. Ozden, Y. Simsek, Generating functions of the $(h,q)-$ extension of twisted Euler polynomials and numbers. Acta Math. Hungar. 120 (2008), no. 3, 281-299.
* 5. M. Cenkci, The $p-$adic generalized twisted $(h,q)-$Euler-l-function and its applications, Adv. Stud. Contemp. Math. 15 (2007), no. 1, 37-47.
* 6. D. V. Dolgy, D. J. Kang, T. Kim, B. Lee, Some new identities on the twisted $(h,q)-$Euler numbers and $q-$Bernstein polynomials, J. Comput. Anal. Appl. 14(2012), no. 5, 974-984.
* 7. D. S. Kim, N. Lee, J. Na, K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (II), J. Math. Anal. Appl. 379 (2011), no. 1, 388-400.
* 8. T. Kim, New approach to $q-$Euler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), no. 2, 218-225.
* 9. T. Kim, Symmetry $p-$adic invariant integral on $\mathbb{Z}_{p}$ for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267-1277.
* 10. T. Kim, q-Euler numbers and polynomials associated with $p-$adic $q-$integrals, J. Nonlinear Math. Phys. 14 (2007), no. 1, 15-27.
* 11. T. Kim, A family of $(h,q)-$zeta function associated with $(h,q)-$Bernoulli numbers and polynomials, J. Comput. Anal. Appl. 14 (2012), no. 3, 402-409.
* 12. T. Mansour, A.Sh. Shabani, Generalization of some inequalities for the $(q_{1},\cdot\cdot\cdot,q_{s})-$gamma function, Matematiche (Catania) 67 (2012), no. 2, 119-130.
* 13. B. Kurt, Some formulas for the multiple twisted $(h,q)-$Euler polynomials and numbers, Appl. Math. Sci. (Ruse) 5 (2011), no. 25-28, 1263-1270.
* 14. H. Ozden, Y. Simsek, Interpolation function of the $(h,q)-$extension of twisted Euler numbers, Comput. Math. Appl. 56 (2008), no. 4, 898-908.
* 15. H. Ozden, I. N. Cangul, Y. Simsek, Remarks on sum of procucts of $(h,q)-$ twisted Euler polynomials and numbers, J. Inequal. Appl.(2008). Art. ID 816129, 8 pp.
* 16. K. H. Park, On interpolation functions of the generalized twisted $(h,q)-$ Euler polynomials, J.Inequal. Appl. (2009), Art. ID 946569,17 pp.
* 17. S.-H. Rim, S.-J. Lee, Some identities on the twisted $(h.q)-$ Genocchi numbers and polynomials associated with $q-$Bernstein polynomials, Int. J. Math. Math. Sci. (2011), Art. ID 482840, 8 pp.
* 18. Y. Simsek, Complete sum of products of $(h.q)-$extension of Euler polynomials and numbers, J. Difference Equ. Appl. 16 (2010), no. 11, 1331-1348.
* 19. Y. Simsek, Twisted $(h.q)-$Bernoulli numbers and polynomials related to twisted $(h.q)-$zeta function and L-function, J. Math. Anal. Appl. 324 (2006), no.2, 790-804.
* 20.
|
arxiv-papers
| 2013-12-14T01:09:05 |
2024-09-04T02:49:55.416226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dae San Kim and Taekyun Kim",
"submitter": "Taekyun Kim",
"url": "https://arxiv.org/abs/1312.3993"
}
|
1312.4105
|
# Focus Point from Direct Gauge Mediation
Sibo Zheng Department of Physics, Chongqing University, Chongqing 401331, P.
R. China
###### Abstract
This paper is devoted to reconcile the tension between theoretic expectation
from naturalness and the present LHC limits on superpartner mass bounds. We
argue that in SUSY models of direct gauge mediation the focusing phenomenon
appears, which dramatically reduces the fine tuning associated to 126 GeV
Higgs boson. This type of model is highly predictive in mass spectrum, with
multi-TeV third generation, $A_{t}$ term of order 1 TeV, gluino mass above LHC
mass bound, and light neutralinos and charginos beneath 1 TeV.
## I I. Introduction
As the LHC keeps running, the searches of supersymmetry (SUSY) signals such as
stop/gluino, sbottom and Higgs mass discovered at 126 GeV 125 continue to
push their mass bounds towards to multi-TeV range ATLAS2013 ; CMS2013 . On the
other hand, the argument of naturalness requires the masses of third
generation scalars, the Higgsinos and gluinos should be $\sim$ 1 TeV. This is
the present status of SUSY.
To reconcile the experimental limits and expectation of naturalness, either of
them needs subtle reconsiderations. In this paper, we consider relaxing the
upper bounds from argument of naturalness. The upper bounds on above soft
breaking parameters arise from the significant contribution to renormalization
group (RG) running for up-type Higgs mass squared $m^{2}_{H_{\mu}}$, which
connects to the electroweak (EW) scale through electroweak symmetry breaking
(EWSB) condition (for $\tan\beta>10$ in the context of the minimal
supersymmetric model (MSSM)),
$\displaystyle{}m^{2}_{Z}\simeq-2\mu^{2}-2m^{2}_{H_{\mu}},$ (1)
Naively, low fine tuning implies the value of $\mu$ and $\mid m_{H_{\mu}}\mid$
at EW scale should be both near EW scale. But there exists an exception. In
some cases, there is significantly cancellation among the RGE corrections
arising from soft breaking parameters to $m^{2}_{H_{\mu}}$, although their
input values are far beyond 1 TeV. This is known as focusing phenomenon
FPSUSY1 ; FPSUSY2 .
The early attempts in FPSUSY1 ; FPSUSY2 ; 1201.4338 ; 1303.1622 were mainly
restricted to SUSY models near grand unification scale (GUT). One recent work
related to focus point SUSY deals with gaugino mediation gaugino . In this
text, we consider gauge mediated (GM) SUSY models with intermediate or low
messenger scale $M$ (for a review see, e.g., Giudice ). Since the focusing
phenomenon can be analytically estimated only if the gaugino masses dominate
over all other soft breaking masses, or they are small in compared with the
third-generation scalar masses (with FPSUSY3 or without FPSUSY1 ; FPSUSY2
$A$ terms ), following this observation, in this paper we study direct GM
model, in which the gaugino masses are naturally small due to the fact that
gaugino masses of order $\mathcal{O}(F)$ vanishes Yanagida .
Another rational for employing direct GM models is that focusing phenomenon
can be understood as a result of hidden symmetry. Because without directly
gauging global symmetries of the model, there would be larger symmetries
maintained in the hidden theory. Otherwise, without the protection of symmetry
tiny deviation for model parameters from their focus point values leads to
significant fine tuning again, and the model is actually unnatural.
As we will see, there are three free input parameters in our model. Two of
them are fixed so as to induce focusing phenomenon, leaving an overall mass
parameter $m_{0}$. The fit to 126 GeV Higgs boson discovered at the LHC then
determines the magnitude of this parameter, with $m_{0}\sim 4-7$ TeV. Thus,
our model is highly predictive in mass spectrum.
In section IIA, we introduce the model in detail. In section IIB, we discuss
the focusing phenomenon, the boundary conditions for such structure and the
mass spectrum at EW scale. In section IIC, we discuss the possibility of
uplifting the gluino mass above LHC lower bound while keeping the focusing.
Finally we conclude in section III.
## II II. The Model
### II.1 A. Setup
In contrast to Agashe , in which non-minimal GM model was employed to discuss
focusing phenomenon, we study SUSY models that don’t spoil the grand
unification of SM gauge couplings and restrict to the context of direct GM.
The messenger fields include chiral quark superfields $q+q^{\prime}$ and their
bi-fundamental fields $\bar{q}+\bar{q^{\prime}}$, lepton superfields
$l+l^{\prime}$ and their bi-fundamental fields $\bar{l}+\bar{l^{\prime}}$, and
singlet $S$ and its bi-fundamental field $\bar{S}$. They transform under
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ as, respectively,
$\displaystyle{}q,~{}q^{\prime}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{3},\mathbf{1},-\frac{1}{3}\right),$
$\displaystyle\bar{q},~{}\bar{q^{\prime}}$ $\displaystyle\sim$
$\displaystyle\left(\bar{\mathbf{3}},\mathbf{1},\frac{1}{3}\right),$
$\displaystyle l,l^{\prime}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\mathbf{2},\frac{1}{2}\right),$ (2)
$\displaystyle\bar{l},~{}\bar{l^{\prime}}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\bar{\mathbf{2}},-\frac{1}{2}\right)$
$\displaystyle S,~{}\bar{S}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\mathbf{1},0\right)$
So, these messenger multiplets complete a $\mathbf{5}+\bar{\mathbf{5}}$
representation of SM gauge group. The renormalizable superpotential consistent
with SM gauge symmetry is given by 111It belongs to general Wess-Zumino model,
which can be completed as effective theory of strong dynamics at low energy
ISS . The direct gauge mediation arises after gauging the global symmetries in
the weak theory and identifying them as SM gauge groups.,
$\displaystyle{}W=fX+Xq\bar{q}+Xl\bar{l}+m(q^{\prime}\bar{q}+q\bar{q^{\prime}})+m(l^{\prime}\bar{l}+l\bar{l^{\prime}}).$
where $X=M+F\theta^{2}$, denotes the SUSY-breaking sector with nonzero $F$
term. In what follows, we will consider $N$ copies of such messengers
multiplets, with $N<6$ so as to maintain the grand unification of SM gauge
couplings.
For the purpose of focusing we add a deformation to superpotential Eq.(II.1),
$\displaystyle{}W=\lambda H_{u}S\bar{l}.$ (4)
This superpotential can be argued to be natural by either imposing a hidden
$U(1)_{X}$ symmetry soft2 or matter parity 1007.3323 . For example, we can
impose $U(1)_{X}$ charges of fields as,
$\displaystyle{}q_{X}(X,~{}\phi_{i},~{}\bar{\phi}_{i},~{}H_{u},H_{d})=(1,-1/2,-1/2,1,-1)$
(5)
where $\phi_{i}=\\{q,q^{\prime},l,l^{\prime},S\\}$. In addition, this hidden
symmetry forbids some operators such as $H_{d}Sl$.
In Eq.(II.1) we have assumed unified mass parameter $m$ and ignored the Yukawa
coefficients for simplicity. For $m<M$ which we adopt in this paper the soft
scalar mass spectrum induced by superpotential Eq.(II.1) is the same as that
of minimal GM at the leading order. Since the minimal GM can not induce
focusing phenomenon, the deformation to the scalar mass spectrum due to Eq.(4)
is thus crucial for our purpose. In particular, Eq.(4) gives rise to a
negative one-loop contribution to $m^{2}_{H_{u}}$ with suppression factor
$F/M^{2}$. Unless we take $\sqrt{F}<<M$, the sign of $m^{2}_{H_{u}}$ would be
negative, it will not lead to focusing (see explanation around Eq.(51)).
Therefore, we are restricted to choose
$\displaystyle{}m<M,~{}~{}\text{and}~{}~{}\sqrt{F}<<M.$ (6)
For detailed calculation of the deviation to scalar mass spectrum given by
Eq.(4), We refer the reader to soft1 ; soft2 . With small SUSY breaking given
by Eq.(6), $m^{2}_{H_{\mu}}$ will be uplifted as required for focusing.
One can verify that gaugino masses at one loop of order $\mathcal{O}(F)$
vanish due to the fact the mass matrix of messengers
$\displaystyle{}\mathcal{M}=\left(\begin{array}[]{ll}X&m\\\
m&0\end{array}\right)$ (9)
satisfies $\det\mathcal{M}=\text{const}$ as long as $m$ doesn’t vanish,
although $m$ is small in comparison with scale $M$. So we expect that the RGE
for $m^{2}_{H_{\mu}}$ is dominated by stop mass squared $m^{2}_{Q_{3}}$,
$m^{2}_{u_{3}}$, and Eq.(4) induced A term.
### II.2 B. Focusing And Mass Spectrum
Following the observation FPSUSY1 ; FPSUSY2 ; FPSUSY3 that the REGs for
$A_{t}$ and scalar masses such as $m^{2}_{H_{\mu}}$ are affected by both
themselves and gluino masses, while the RGE for gluino mass is only affected
by itself, we can solve the RGEs for soft scalar masses,
$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{H_{\mu}}(Q)\\\
m^{2}_{u_{3}}(Q)\\\ m^{2}_{Q_{3}}(Q)\\\ A^{2}_{t}(Q)\\\ \end{array}\right)$
$\displaystyle=$
$\displaystyle\kappa_{12}I^{2}(Q)\left(\begin{array}[]{ccccc}3\\\ 2\\\ 1\\\
6\\\ \end{array}\right)+\kappa_{6}I(Q)\left(\begin{array}[]{ccccc}3\\\ 2\\\
1\\\ 0\\\ \end{array}\right)$ (22) $\displaystyle+$
$\displaystyle\kappa_{0}\left(\begin{array}[]{ccccc}1\\\ 0\\\ -1\\\ 0\\\
\end{array}\right)+\kappa^{\prime}_{0}\left(\begin{array}[]{ccccc}0\\\ 1\\\
-1\\\ 0\\\ \end{array}\right).$ (31)
for small gluino masses (in compared with above scalar soft masses). Here,
$\displaystyle{}I(Q)=\exp\left(\int^{\ln Q}_{\ln
M}\frac{6y^{2}_{t}(Q^{\prime})}{8\pi^{2}}d\ln Q^{\prime}\right)$ (32)
which depends on $M$ and RGE for top Yukawa. In Fig. 1 we show the numerical
value of $I$ as function of $M$, with the context of MSSM below scale $M$.
Figure 1: $I$ as function of $M$ for the context of MSSM below scale $M$.
In particular, $I(1\text{TeV})\simeq 0.527$ for $M=10^{8}$ GeV.
The condition for focusing phenomenon can be derived from Eq.(22) by imposing
$m^{2}_{H_{\mu}}(1\text{ TeV})\simeq 0$. Define
$m^{2}_{H_{\mu}}(M)=+m^{2}_{0}$. Similar to FPSUSY3 we choose $x$ to
parameterize the splitting between $m^{2}_{Q_{3}}(M)$ and $m^{2}_{u_{3}}(M)$,
and $y$ to be directly related to $A_{t}(M)$. In the case of small SUSY
breaking the mass spectrum which induces focusing at scale $\mu=1\text{TeV}$
reads as,
$\displaystyle{}m^{2}_{0}\left(\begin{array}[]{c}1\\\ 1.41+x-1.58y\\\
1.82-x-3.16y\\\ 9y\\\ \end{array}\right)_{M}\rightarrow
m^{2}_{0}\left(\begin{array}[]{c}0\\\ 0.74+x-1.58y\\\ 1.48-x-3.16y\\\ 1.66y\\\
\end{array}\right)_{\mu}$ (41)
Alternatively we rescale parameter $x$ as in FPSUSY3 such that
$m^{2}_{Q_{3}}$ only depends on $x$. For $m^{2}_{H_{\mu}}(M)=-m^{2}_{0}$,
Eq.(41) is instead of,
$\displaystyle{}m^{2}_{0}\left(\begin{array}[]{c}-1\\\ -1.41+x-1.58y\\\
-1.82-x-3.16y\\\ 9y\\\ \end{array}\right)_{M}\rightarrow
m^{2}_{0}\left(\begin{array}[]{c}0\\\ -0.74+x-1.58y\\\ -1.48-x-3.16y\\\
1.66y\\\ \end{array}\right)$ (51)
This parameterization appears when $F/M^{2}\rightarrow 1$. In this limit,
$m^{2}_{H_{\mu}}$ is dominated by the one-loop negative contribution
proportional to Yukawa coupling $\lambda$. From Eq.(51), there is no
consistent solution to $x$ and $y$ in this case.
Soft masses in Eq.(22) at scale $\mu=1\text{TeV}$ are functions of Yukawa
coupling $\lambda$, number of messenger pairs $N$, ratio $F/M^{2}$ and SUSY-
breaking mediated scale $M$. From Eq. (41) one connects the variables $(x,y)$
and the model parameters $\lambda$ and $N$. For the three input parameters
$m_{0}$, $x$ and $y$ (with $M$ fixed) for focusing in the model, two of them
can be fixed by the choices of $\lambda$ and $N$. We choose $x$ and $y$ for
analysis. Fig.2 shows the plots of $x$ (dotted) and $y$ (solid) as function of
$\alpha_{\lambda}$ and $N$. For each $N$ the focus point values of $x$ and $y$
are read from the crossing points between vertical line and solid curve
(dotted curve ) for $y$ ($x$) . Therefore, there is only one free parameter
left in the model by imposing the focusing condition, which is very predictive
in the mass spectrum and signal analysis.
Since we perform our analysis in perturbative theory, in order to avoid Landau
pole up to GUT scale, the Yukawa coupling $\alpha_{\lambda}$ is upper bounded,
$\sim$0.1 for our choice of messenger scale. The dotted and solid horizontal
lines in fig. 2 refer to allowed ranges for $x$ and $y$, respectively. These
ranges are derived from the requirement that the stop soft masses aren’t
tachyon-like and the $A_{t}$ squared is positive. Following these we obtain,
$\displaystyle{}0<y<0.40,~{}-0.74<x<1.48,$ $\displaystyle
1.58y-0.74<x<1.48-3.16y,$ (52) $\displaystyle 1.58y-1.41<x<1.82-3.16y.$
It is easy to verify that for each $N$ the crossing points satisfy the
constraints above.
Figure 2: Plots of $x$ (dotted) and $y$ (solid) as function of
$\alpha_{\lambda}$ for $N=\\{1,2,3,4\\}$. The red, blue, purple and black
curves correspond to $N=1,2,3,4$, respectively. For each $N$ the focus point
value are read from the crossing points between vertical line and solid curve
for $y$ and dottoed curve for $x$, respectively. The dotted (solid) horizontal
lines refer to range allowed for $x$ ($y$).
With focusing phenomenon we have single free parameter, namely $m_{0}$ at
hand. It can be uniquely determined in terms of the mass of Higgs boson
observed at the LHC. Fig. 2 shows how $m_{h}$ changes as parameter $m_{0}$ for
different $N$s. The two-loop level Higgs boson mass in the MSSM is given by
Carena ,
$\displaystyle{}m_{h}^{2}$ $\displaystyle=$ $\displaystyle
m_{Z}^{2}\cos^{2}2\beta+\frac{3m^{4}_{t}}{4\pi^{2}\upsilon^{2}}\left\\{\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)+\frac{1}{2}\tilde{A}_{t}+\frac{1}{16\pi^{2}}\left(\frac{3}{2}\frac{m^{2}_{t}}{\upsilon^{2}}-32\pi\alpha_{3}\right)\left[\tilde{A}_{t}+\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)\right]\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)\right\\}$
(53)
Here $\upsilon=174$ GeV and
$\tilde{A}_{t}=\frac{2X^{2}_{t}}{M^{2}_{S}}\left(1-\frac{X^{2}_{t}}{12M^{2}_{S}}\right)$,
with $X_{t}=A_{t}-\mu\cot\beta$. We focus on large $\tan\beta$ region. For
$\tan\beta\geq 20$, the fit to Higgs boson mass doesn’t change much. From
fig.3 one observes that $m_{0}\sim 4.0-7.0$ due to the fit to 126 GeV Higgs
boson.
Figure 3: $m_{h}$ vs $m_{0}$ for different $N$s, with $N=1,2,3,4$ from bottom
to top, respectively. Multi-TeV $m_{0}$ is required by the 126 GeV Higgs
boson.
Substituting the values of $m_{0}$ from fig.3 and $x$, $y$ from fig.2 into
Eq.(41) we find the mass spectrum, which is shown in table 1.
The choice on large $\tan\beta$ might be forbidden by possibly large B$\mu$
term induced by Eq.(4). As noted in 1007.3323 , B$\mu\sim\mu A_{t}$. In terms
of electroweak symmetry breaking condition, we have
$\sin(2\beta)\simeq\text{B}\mu/m^{2}_{0}\sim(A_{t}/m_{0})^{2}\cdot(\mu/A_{t})$.
With a small $\mu$ term of order $\sim 300-500$ GeV (as shown in table 1) at
messenger scale $M$, one does not have to worry about $\mu$ being made very
large by radiative correction involving heavy soft scalar masses (see e.g.,
soft1 ). So, one obtains $\sin(2\beta)$ of order $\sim(1/4)^{2}\cdot(1/4)$
from table 1, and the choice on large value of $\tan\beta$ is not violated by
B$\mu$ term.
### II.3 C. Gaugino Mass
As mentioned above due to $\det\mathcal{M}=\text{const}$ gaugino masses vanish
at one-loop level of order $\mathcal{O}(F)$ and at the two-loop level of order
$\mathcal{O}(F)$. Their leading contributions appear at one-loop level of
order $\mathcal{O}(F^{3}/M^{5})$ Yanagida . Under small SUSY-breaking limit
the magnitude of gaugino mass relative to $m_{Q_{3}}$ at input scale is given
by 222We thank the referee for pointing out a critical error in estimation of
gaugino mass in the previous version of this manuscript.,
$\displaystyle{}\frac{m_{\tilde{g}_{i}}}{m_{Q_{3}}}\sim\left(\frac{F}{M^{2}}\right)^{2}\cdot\frac{\sqrt{N}\alpha_{i}}{\sqrt{2\times\left(\frac{4}{3}\alpha^{2}_{3}(M)+\frac{3}{4}\alpha^{2}_{2}(M)+\frac{1}{60}\alpha^{2}_{1}(M)\right)}}$
Using one-loop RGEs for gluino masses, we find their values at the
renormalization scale $\mu=1$ TeV. One observes from Eq.(II.3) that the gluino
mass is far below the 2013 LHC bound $\simeq 1.3$ TeV due to the suppression
by factor $F^{2}/M^{4}$.
Without extra significant modifications to the gaugino mass spectrum, LHC
bound would exclude this simple model, despite it provides a natural
explanation of Higgs boson mass and is consistent with present experimental
limits. Here, we propose a recipe 0612139 in terms of imposing small
modification to superpotential $\delta
W=m^{\prime}\left(\bar{l^{\prime}}l^{\prime}+\bar{q^{\prime}}q^{\prime}\right)$,
with small mass $m^{\prime}<m$. These mass terms are consistent with gauge
symmetries and matter parity of messenger sector.
If so, Eq.(9) will be instead of
$\displaystyle{}\mathcal{M}=\left(\begin{array}[]{ll}X&m\\\
m&m^{\prime}\end{array}\right)$ (57)
The correction to soft scalar mass spectrum is of order $\mathcal{O}(m^{\prime
4}/m^{4})$ and very weak. However, the correction to gaugino mass, which is of
order,
$\displaystyle{}m_{\tilde{g}_{i}}\simeq
N\cdot\frac{\alpha_{i}}{4\pi}\cdot\frac{F}{m}\cdot\frac{m^{\prime}}{m}$ (58)
can be large enough to reconcile with the LHC bound when $m^{\prime}/m$ is
larger than $F^{2}/M^{4}$. For example, we choose $N=1$, $M=10^{8}$ GeV and
$m=0.1M$. Then $m_{0}\sim 7$ TeV and $\sqrt{F}\sim 8.2\cdot 10^{6}$ GeV, and
further $m_{\tilde{g}_{3}}\sim 7\cdot 10^{-3}\cdot m^{\prime}$ from Eq.(58).
LHC gluino mass bound requires $m^{\prime}\geq 2\cdot 10^{5}$ GeV, which is
consistent with the constraint $m^{\prime}<m<M$. The bino and wino masses are
both near 1 TeV. So they are the main target of 14-TeV LHC.
| $N=1$ | $N=2$ | $N=3$ | $N=4$
---|---|---|---|---
$m_{0}$ | $7.0$ | $5.9$ | $4.0$ | $3.5$
$m_{\tilde{t}_{1}}$ | $3.12$ | $3.62$ | $4.54$ | 4.83
$m_{\tilde{t}_{2}}$ | $7.65$ | $4.98$ | $4.80$ | 6.0
$A_{t}$ | $1.64$ | $1.48$ | $1.50$ | 1.50
$\mu$ | $0.50$ | $0.42$ | $0.28$ | 0.24
Table 1: Given a focus point, input mass parameter $m_{0}$ (in unit of TeV)
required for $m_{h}=126$ GeV and corresponding soft mass spectrum (in unit of
TeV) at renormalization scale $\mu=1$ TeV in the context of MSSM, for
different values of messenger number $N$.
## III III. Discussion
From mass spectrum of table 1, the main source for fine tuning arises from
$\mu$ term. The fine tuning parameter c, which is defined as
$c=\max\\{c_{i}\\}$, with
$\displaystyle c_{i}=\mid\partial\ln m^{2}_{Z}/\partial\ln a_{i}\mid$
where $a_{i}$ are the soft mass parameters involved, has been reduced from
$\sim 2000$ to $\sim 20$ due to the focusing phenomenon.
As for other indirect experimental limits such as flavor changing neutral
violation, the model feels comfortable. Because the masses of the three-
generation sleptons and first two-generation squarks are all of order $\sim$
multi-TeV, with highly degeneracy in each sector.
What about the sensitivity of our results to the messenger scale ? At first,
assuming that there exists a completion of strong dynamics at high energy
indicates that $M$ should be smaller than the GUT scale. Typically, we have
$M<10^{10}$ GeV in the context of direct gauge mediation. For the case of low-
scale mediation, i.e, $M<10^{8}$ GeV, the gluino mass is already close to the
2013 LHC mass bound. In other words, $M=10^{8}$ GeV as we studied in detail is
a reference value for intermediate scale SUSY model. The promising signals for
this simple and natural model include searching gluino, neutralinos and
charginos at the LHC.
Along this line it is of interest to extend the model-independent focusing
condition to the whole energy range below GUT scale Zheng , and construct
natural SUSY models in the context of either direct or non-direct GM.
$\mathbf{Acknowledgement}$ The work is supported in part by Natural Science
Foundation of China under grant No.11247031 and 11405015.
## References
* (1) $\mathbf{ATLAS}$ Collaboration, Phys. Lett. B710, 49 (2012), arXiv:1202.1408[hep-ex]; The $\mathbf{CMS}$ Collaboration, Phys. Lett. B710, 26 (2012), arXiv:1202.1488[hep-ex].
* (2) $\mathbf{ATLAS}$ Collaboration, ATLAS SUSY 2013 Stop Summary, https://twiki.cern.ch/twiki/pub/AtlasPublic/CombinedSummaryPlots
/ATLAS_directstop _all_SUSY2013.pdf.
* (3) $\mathbf{CMS}$ Collaboration,CMS SUSY 2013 Stop Summary, https://twiki.cern.ch/ twiki/pub/CMSPublic/
SUSYSMSSummaryPlots8TeV/SUSY2013T2ttT6.pdf.
* (4) J. L. Feng, K. T. Matchev, and T. Moroi, Phys. Rev. Lett. 84, 2322 (2000),
* (5) J. L. Feng, K. T. Matchev, and T. Moroi, Phys. Rev. D61, 075005 (2000),
* (6) F. Brummer and W. Buchmuller, JHEP 1205, 006 (2012), arXiv:1201.4338 [hep-ph].
* (7) F. Brümmer, M. Ibe and T. T. Yanagida, Phys. Lett. B 726, 364 (2013), arXiv:1303.1622 [hep-ph].
* (8) T. Yanagida and N. Yokozaki, Phys. Letts. B 722 (2013) 355.
* (9) J. L. Feng and D. Sanford, Phys. Rev. D86, 055015 (2012),
* (10) G. F. Giudice and R. Rattazzi, Phys. Repts. 322 (1999) 419.
* (11) M. Carena, J. R. Espinosa, M. Quiros, C. E. M. Wagner, Phys. Letts. B 355 (1995) 209.
* (12) K.-I. Izawa, Y. Nomura, K. Tobe and T. Yanagida, Phys. Rev. D56 (1997) 2886.
* (13) K. Agashe, Phys. Rev. D61 (2000) 115006.
* (14) K. A. Intriligator, N. Seiberg and D. Shih, JHEP 04 (2006) 021.
* (15) S. Zheng, Eur. Phys. J. C 74, 2724 (2014), arXiv:1308.5377 [hep-ph].
* (16) N. Craig, S. Knapen, D. Shih and Y. Zhao, JHEP 1303, 154 (2013), arXiv:1206.4086 [hep-ph].
* (17) S. Zheng et al, to appear (2014).
* (18) K. Hamaguchi and N. Yokozaki, Phys. Lett. B 694, 398 (2011), arXiv:1007.3323 [hep-ph].
* (19) R. Kitano, H. Ooguri and Y. Ookouchi, Phys. Rev. D 75, 045022 (2007), [hep-ph/0612139].
|
arxiv-papers
| 2013-12-15T03:05:43 |
2024-09-04T02:49:55.424932
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1312.4105"
}
|
1312.4117
|
# Precision measurement of transverse velocity distribution of a strontium
atomic beam
F. Gao1,2 H. Liu1,2 P. Xu1 X. Tian1,2 Y. Wang1 J. Ren1 Haibin Wu3
[email protected] Hong Chang1,3 [email protected] 1 CAS Key Laboratory
of Time and Frequency Primary Standards, National Time Service Center, Xi’an
710600, China 2 University of Chinese Academy of Sciences, Beijing 100049,
China 3 State Key Laboratory of Precision Spectroscopy, Department of
Physics, East China Normal University, Shanghai 200062, China
###### Abstract
We measure the transverse velocity distribution in a thermal Sr atomic beam
precisely by velocity-selective saturated fluorescence spectroscopy. The use
of an ultrastable laser system and the narrow intercombination transition line
of Sr atoms mean that the resolution of the measured velocity can reach 0.13
m/s, corresponding to 90$\mu K$ in energy units. The experimental results are
in very good agreement with the results of theoretical calculations. Based on
the spectroscopic techniques used here, the absolute frequency of the
intercombination transition of 88Sr is measured using an optical-frequency
comb generator referenced to the SI second through an H maser, and is given as
434 829 121 318(10) kHz.
Atomic (or molecular) beams, which have now become standard laboratory tools,
play very important roles in the field of atomic, molecular and optical
physics. These beams have been widely used in the determination of atomic
structures atomstructure , measurement of physical constants physicsconstant ,
studies of chemical reactions reaction and atomic frequency standards
frequencystandard . In all these applications, measurement of the velocity
distribution of the atomic beams is both necessary and highly important. Many
spectroscopic techniques have been developed to perform these measurements
beam1 ; beam2 ; beam3 ; beam4 ; beam5 ; beam6 ; beam7 ; beam8 . However, the
precise determination of the velocity distribution of an atomic beam, and
especially that of a well collimated atomic beam, remains challenging because
of the low signal-to-noise ratio and relatively low spectroscopy resolution of
these beams.
In this paper, we report the precise measurement of the transverse velocity
distribution of a well-collimated thermal strontium beam by velocity-selective
saturated fluorescence spectroscopy. The measurement accuracy of the
transverse velocity of the atomic beam is greatly increased by the combination
of an ultrastable laser system and high-resolution spectroscopy of the
intercombination transition of 88Sr. The measurement resolution for the
velocity can reach 0.13 m/s, corresponding to 90 $\mu K$ in energy units.
Because of the confinement of the cylinder nozzle of the atomic beam, the
transverse velocity distribution is no longer the well-known Maxwell-Boltzmann
distribution. The measured results show good agreement with the theoretically
predicted results theory . Using the spectroscopic techniques developed here,
the absolute frequencies of the isotopes of strontium atoms are measured using
an optical frequency comb generator referenced to the SI second through an H
maser. Particular attention is paid to the intercombination transition of
88Sr. The frequency is measured as 434 829 121 318(10) kHz.
Figure 1: Experimental setup. (a) The narrowed linewidth of the laser system
(689 nm). PBS: polarized beam splitter, ULE: ultra-low expansion optical
cavity, PDH: Pound-Drever-Hall frequency locking system. (b) Experimental
setup used for velocity-selective saturated fluorescence spectroscopy. The
frequencies of the two counterpropagating laser beams are scanned using a saw-
tooth waveform at 0.1 Hz. PMF: single-mode polarization-maintaining fiber,
AOM: acousto-optic modulator, VCO: voltage-controlled oscillator, PSC: power
stabilization servo controller, M-S: Earth magnetic shield. (c) Optical
frequency measurement using a fiber frequency comb (Menlo FC1500), with
respect to an H maser at a radio frequency signal of 10 MHz.
Velocity-selective saturation (fluorescence) spectroscopy is a powerful
technique that is widely used for high resolution measurements. We use it here
to measure the velocity distribution of the atomic beam precisely. The
measurement principle is as follows. Consider two-level atoms interacting with
the counterpropagating probe beam and pump beam; the probability of the atoms
being in excited states is proportional to the velocity of the atoms, the
light intensities and the scattering rate of the excited state $\gamma$. When
the probe and pump beams have the same frequency, the group of atoms with zero
velocity see the light beams at resonance and show a Lamb dip at the atomic
resonant frequency with a large Doppler background. The peak amplitude can be
used to determine the number of atoms. When the frequencies of the pump and
probe lasers are different, only the atoms with velocity $v=\Delta\omega/(2k)$
see the light beams at resonance, where $\Delta\omega$ is the frequency
difference between the beams and $k$ is the wave vector of the laser light.
The spectroscopic resolution of this method is limited by the natural
linewidth of the excited state, and therefore the velocity resolution is
$v=(\gamma/f)c$, where $f$ and $c$ are the resonant frequency and the light
speed in a vacuum, respectively. The atom used here is strontium, which has
four stable natural isotopes, comprising bosonic 88Sr (82.58%), 86Sr (9.86%),
84Sr (0.56%), with nuclear spin I=0, and fermionic 87Sr (7.0%), with I=9/2.
The intercombination line $5^{1}S_{0}-5^{3}P_{1}$, because of its high
frequency fraction ($f/\gamma$), has been widely studied in optical frequency
standards frequency1 ; frequency3 ; frequency4 . In this paper, we use this
transition to precisely measure the velocity distribution of the atomic beam.
The experimental setup is shown in Fig. 1. A commercial extended cavity diode
laser (ECDL) (Topical-110) is used as the spectroscopy laser, and can
typically deliver 16 mW at 689 nm. The laser linewidth is reduced by locking
the laser to an ultrastable optical cavity via the Pound-Drever-Hall scheme;
the phase modulation is produced by a resonant electro-optic modulator (EOM),
which is driven at 5 MHz. The cavity is made of an ultralow expansion glass
with a finesse of 12000. To avoid a residual standing wave in the EOM, which
induces spurious amplitude modulation (AM) on the locking signal, a 60 dB
optical isolator is placed between the EOM and the cavity. The power of the
locking beam is maintained at as low a level as possible to minimize the
frequency shift caused by light heating and optical feedback. The linewidth
and the fractional frequency drift are about 200 Hz and $2.8\times 10^{-13}$
at 1 s, respectively.
The strontium atomic beam is obtained from strontium metal heated to 823 K
(550 oC) in an oven. A bundle of stainless steel capillaries (Monel-400) are
used to collimate the beam. The length and the internal diameter of the
capillary are 8 mm and 200 $\mu m$, respectively. The residual atomic beam
divergence is 25 mrad, and the typical atomic flux is estimated to be
$10^{12}/s$. The vacuum for the atomic beam is maintained at $1\times
10^{-8}$Torr with a 40 l/s ion pump.
Velocity-selective saturation (fluorescence) spectroscopy is obtained by the
use of two counterpropagating laser beams (i.e. the probe and pump beams)
perpendicular to the atomic beam. These beams are generated by two acousto-
optical modulators (AOMs) with the same double-pass configurations. The AOMs
are controlled by the same oscillator but with different voltage-controlled
oscillators (VCOs). A homemade circuit is used to prevent the oscillator
frequencies from disturbing each other. The frequency resolution is
approximately 0.1 Hz. The power is stabilized by a servo, which is better than
$10^{-4}$. Both the pump and probe beams have been expanded to have 1.2 cm
waists (1/$e^{2}$ diameter). The fluorescence signal is collected by a large
diameter lens in a direction that is orthogonal to the atomic beam and the
laser light beams on a high-sensitivity detector. A magnetic shield was placed
in the interaction region to prevent the Zeeman effect being caused by the
Earth’s magnetic field.
Figure 2: Velocity-selective saturated fluorescence spectra: (a)
$\Delta=0MHz$, corresponding to the condition where a group of atoms with
velocity $v=0$ is measured; (b) $\Delta=5MHz$, corresponding to the condition
where a group of atoms with velocity $v=3.45m/s$ is measured.
By scanning the pump and probe light beams at different frequencies, we
observe the velocity-selective saturated fluorescence spectra. Fig. 2 shows
two typical spectra for $\Delta=0$ and $\Delta=5\,MHz$, where $\Delta$ is the
frequency difference between the probe and pump light beams. The linewidth of
the central peak in Fig. 2(a) is approximately 180 kHz, which is larger than
its natural linewidth of 7.5 kHz. This effect mainly stems from power
broadening. The beam intensity of 800 $\mu W/cm^{2}$ (the saturation intensity
is 3 $\mu W/cm^{2}$) is chosen to produce a large signal-to-noise ratio for
the spectroscopy signal. The contributions of second-order Doppler broadening
(0.55 kHz) and transit time broadening (1.06 kHz) are small and can be
neglected. With this linewidth for the spectroscopy, the transverse velocity
of the magnitude at 0.13 $m/s$ (corresponding to 90 $\mu K$ in energy units)
can be measured.
Figure 3: The transverse velocity distribution of the atomic beam. The black
dots represent the data for the experimental results. The error bars denote
statistical fluctuations that arise from measurement uncertainty for the
amplitudes of the signals. The red solid curve shows the results of the
theoretical calculations. All parameters are taken from the experimental
measurements. There are no free parameters.
For different frequency detunings of the pump and probe light beams, groups of
atoms with different velocities see the light beams at resonance. The number
of atoms can be measured from the amplitude of the saturated fluorescence
spectroscopy signal, and therefore the velocity distribution can be obtained,
and is shown in Fig. 3. The number of atoms has been normalized with respect
to the number of atoms when $\Delta=0$. The maximum frequency scanning range
for our AOMs is approximately $20\,MHz$, corresponding to $13.78\,m/s$ for the
maximum measurable transverse velocity of the atoms. Fig. 3 clearly shows that
the transverse velocity distribution of the thermal atomic beam is no longer a
Maxwell-Boltzmann-like distribution. The confinement of the capillaries leads
to this new distribution, which was predicted in Ref. theory as
$P(v,a)=\frac{|v|\exp(-v^{2}/v_{0}^{2})\Gamma(-1/2,v^{2}L^{2}/v_{0}^{2}a^{2})}{\sqrt{8\pi
v_{0}^{2}(1+a^{2}/L^{2})^{1/2}}},$ (1)
where $v_{0}\equiv\sqrt{2k_{B}T/m}$ is the most probable velocity for the
atoms, $k_{B}$ is the Boltzmann constant, T is the oven temperature, $m$ is
the mass of the atoms, $a$ is the internal diameter of the collimator and $L$
is the length of the collimator. $\Gamma(-1/2,v^{2}L^{2}/v_{0}^{2}a^{2})$ is
the incomplete Gamma function. Based on the experimental parameters, the
results of the theoretical calculation of the velocity distribution are
plotted as the solid curve (red curve) in Fig. 3. There are no free
parameters. The theoretical curve clearly shows very good agreement with our
measurement results. The influence of the three other isotopes, 86Sr, 87Sr and
84Sr, on the measurement results is small because of large frequency
differences and the low natural abundance of these isotopes.
Figure 4: (a) Frequency measurement of the ${}^{88}Sr$ $5^{1}S_{0}\to
5^{3}P_{1}$ transition. The error bars correspond to the standard deviation
for each data set. (b) Comparison of the optical frequency measurement of the
intercombination line of ${}^{88}Sr$ with the results of previous measurements
Refs mea1 ; mea2 . The black triangle represents the data from Ref mea1 , the
red square represents the data from Ref mea2 , and the blue dot represents the
data from our measurement. Table 1: Optical frequency measurement results
and uncertainties for all four natural isotopes of strontium atoms.
Isotopes | $5^{1}S_{0}\to 5^{3}P_{1}$ | Frequency (kHz)
---|---|---
${}^{88}Sr$ | $J=0$$\to J^{\prime}=1$ | 434 829 121 318 (10)
${}^{87}Sr$ | $F=9/2$$\to F^{\prime}=7/2$ | 434 830 473 227 (45)
$F=9/2\to F^{\prime}=9/2$ | 434 829 342 995 (55)
$F=9/2\to F^{\prime}=11/2$ | 434 827 879 835 (50)
${}^{86}Sr$ | $J=0$$\to J^{\prime}=1$ | 434 828 957 500 (15)
${}^{84}Sr$ | $J=0$$\to J^{\prime}=1$ | 434 828 769 730 (100)
We reduce the power of the pump and probe light beams and use our narrowed
linewidth laser (200 Hz), and the saturation fluorescence spectroscopy
linewidth is approximately 55 kHz. The optical frequency of the Sr atoms is
measured using a commercial optical-frequency comb (Menlo FC1500). The
repetition rate and the carrier offset envelope frequency are locked to an H
maser. Fig. 4(a) shows the results of measurements of the 88Sr transition
frequency taken over a period of several days. The error bars correspond to
the standard deviation. The absolute frequency of the intercombination
transition of 88Sr is 434 829 121 318(10) kHz. A comparison with the results
of previous measurements in Ref mea1 ; mea2 is presented in Fig. 4(b), and
shows reasonable agreement. The optical frequencies of 86Sr, 87Sr and 84Sr are
measured using high laser intensities (2.1 $mW/cm^{2}$, 4.3 $mW/cm^{2}$, and
12 $mW/cm^{2}$, respectively); because of the low natural abundance of these
isotopes, the high power is required to obtain a good signal-to-noise ratio.
The measurement results for the optical frequencies of all four natural
isotopes of atomic strontium are summarized in Table I.
In conclusion, we report on the precise measurement of the transverse velocity
distribution of a well-collimated thermal Sr atomic beam using the velocity-
selective saturation fluorescence spectroscopy technique. The combination of
the ultrastable laser system and the narrow linewidth of the intercombination
transition means that the velocity measurement resolution is greatly improved.
The detectable minimum of the velocity is approximately $0.13\,m/s$
(corresponding to $90\,\mu K$ in energy units). The velocity distribution is
no longer likely to be a normal Maxwell-Boltzmann distribution. Instead, it
shows a counter-intuitive umbrella shape. The measured data are in very good
agreement with the results of the theoretical calculations. We also measured
the optical frequency of the $5^{1}S_{0}\to 5^{3}P_{1}$ transition through an
optical-frequency comb generator referenced to the SI second via an H maser.
The measured absolute frequency of the intercombination transition of 88Sr is
comparable to the results of previous measurements. Accurate optical frequency
values for all other isotopes are also presented, and these results may
provide a benchmark for subsequent measurements.
This research is supported by the National Natural Science Foundation of China
(Grant Nos. 11074252 and 61127901), and the Key Projects of the Chinese
Academy of Sciences (Grant No. KJZD-EW-W02).
## References
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* (8) U. Brinkmann, J. Kluge and K. Pippert, J. Appl. Phys. 51, 4612 (1980).
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* (10) G. Di Domenico, G. Mileti, and P. Thomann, Phys. Rev. A 64, 043408 (2001).
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* (14) N. Poli, F. Y. Wang, M. G. Tarallo, A. Alberti, M. Prevedelli, and G. M. Tino, Phys. Rev. Lett. 1069, 038501 (2011).
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|
arxiv-papers
| 2013-12-15T07:08:24 |
2024-09-04T02:49:55.431401
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Gao, H. Liu, P. Xu, X. Tian, Y. Wang, J. Ren, Haibin Wu, Hong Chang",
"submitter": "Haibin Wu",
"url": "https://arxiv.org/abs/1312.4117"
}
|
1312.4195
|
# On an interpolative Schrödinger equation and an alternative classical limit
111Author’s note: Archival version of an old preprint restored from obsolete
electronic media. This was first submitted to Phys. Rev. D back in 1992 but
rejected as being of “insufficient interest”. It describes a generalized
dynamical system which contains an exact embedding of the classical
Hamiltonian point mechanics alongside the entire non-relativistic quantum
theory. The two are joined by a one-parameter family of deformed dynamics in a
dimensionless parameter with $\hbar$ kept constant throughout. Originally, the
work was presented as a toy model called $\lambda$–dynamics for the purpose of
illustrating the absurdity of the Copenhagen Interpretation conception of a
“classical domain” sitting alongside a “quantum domain”. The relevance of the
work today is primarily mathematical. This preprint will be superceded by a
more contemporary study of this system in relation to the Renormalization
Group and the connection between classical and quantum dynamics. This work
posted under Creative Commons 3.0 - Attribution License.
K. R. W. Jones Physics Department, University of Queensland,
St Lucia 4072, Brisbane, Australia.
(Revised version 13/10/92)
###### Abstract
We introduce a simple deformed quantization prescription that interpolates the
classical and quantum sectors of Weinberg’s nonlinear quantum theory. The
result is a novel classical limit where $\hbar$ is kept fixed while a
dimensionless mesoscopic parameter, $\lambda\in[0,1]$, goes to zero. Unlike
the standard classical limit, which holds good up to a certain timescale, ours
is a precise limit incorporating true dynamical chaos, no dispersion, an
absence of macroscopic superpositions and a complete recovery of the
symplectic geometry of classical phase space. We develop the formalism, and
discover that energy levels suffer a generic perturbation. Exactly, they
become $E(\lambda^{2}\hbar)$, where $\lambda=1$ gives the standard prediction.
Exact interpolative eigenstates can be similarly constructed. Unlike the
linear case, these need no longer be orthogonal. A formal solution for the
interpolative dynamics is given, and we exhibit the free particle as one
exactly soluble case. Dispersion is reduced, to vanish at $\lambda=0$. We
conclude by discussing some possible empirical signatures, and explore the
obstructions to a satisfactory physical interpretation.
###### pacs:
03.65.Bz, 02.30.$+$g, 03.20.$+$i, 0.3.65.Db
††preprint: UQ Theory: October1992
## I Introduction
It is generally thought that classical dynamics is a limiting case of quantum
dynamics. Certainly, the subclass of coherent states admit a rigorous
reduction of quantum dynamics to classical dynamics as $\hbar\rightarrow 0$lim
. However, as many authors have notedyaf , this result does not hold for all
quantum states. To see this most clearly imagine that we are in the deep
semiclassical regime. I may choose two coherent states centered about
different points in phase space. These follow the classical trajectories over
some finite time interval with an error, and dispersion, that can be made as
small as one pleases. Any linear combination of these is also a solution of
the Schrödinger evolution, but it need not follow any classical trajectorysup
.
Ordinarily we solve this problem by prohibiting the appearance of such states
at the classical levelcat . This can be partially justified using measurement
as a means to remove coherencesdec . For practical purposes the dilemma is of
no consequencecon . We are not forbidden to use the old theory, when
appropriatebor . The problem is thus one of consistency (for example, Ford et
al.for argue that quantum suppression of dynamical chaoscha spells trouble
for the correspondence principle). If quantum theory is universal then why
does it not give us a clean, simple, and chaotic reductionred ?
In this paper we outline such a reduction. To do this we must pay a heavy
price and forsake the assumption of universality. Keeping that which is good,
we require a generalized theory which contains both classical and quantum
dynamics. The only candidate we know of is Weinberg’s nonlinear quantum
theorysw1 ; sw2 . Elsewhere we used this to recast exact Hamiltonian classical
mechanicsjon . Here we develop a way to pass smoothly between both regimes.
Our motivation is curiosity; to find a nice way to do this, irrespective of
what it might mean. However, where possible we have attempted to interpret the
formalism as physical theory. This is fraught with interpretative difficulty,
but some generic empirical signatures can be extracted.
To set the scene, quantum theory is superbly successful. In looking around to
find trouble’s mark, we can think of no place but the classical regime
(gravitation, the most classical theory, remains the hardest uncraked nut). It
is at the interface between the microworld and the macroworld that aesthetic
dissaffection arises, for it is here that quantum stochasticity and
measurement prove necessary. Most “resolutions”, “new interpretations”,
whatever…, depart little from the orthodox theory. Here our philosophy is to
first enlarge quantum dynamics and then seek a natural way to blend the
classical and quantum components together. The interpolative dynamics is then
put forward as a candidate to describe a regime that borders the cut we
customarily make in everyday calculations. We make a guess at some kind of
general theoretical structure in which to think around the questions. Without
evidence that quantum theory fails we can do no more. Why do it then? Because
when no alternative exists we are unlikely to find any failure.
## II Classical mechanics in Weinberg’s theory
Unlike regular classical mechanics, the carbon copy within Weinberg’s theory
employs wavefunctions, $\hbar$ and the commutation relation
$[\hat{q},\hat{p}]=i\hbar$. To form it we take any classical function, say
$H(q,p)$, and turn it into a Weinberg observablesw3 via the ansatz
$h_{0}(\psi,\psi^{*})\equiv\langle\psi|H(\langle\hat{q}\rangle,\langle\hat{p}\rangle)|\psi\rangle,$
(1)
where $\langle\hat{q}\rangle\equiv\langle\psi|\hat{q}|\psi\rangle/n$,
$\langle\hat{p}\rangle\equiv\langle\psi|\hat{p}|\psi\rangle/n$, with
$n=\langle\psi|\psi\rangle$. Commutators are then replaced by the Weinberg
bracket,
$[g,h]_{\rm W}\equiv g\star h-h\star g,$ (2)
where $g\star h=\delta_{\psi}g\delta_{\psi^{*}}h$ and $\delta_{\psi}$, and
$\delta_{\psi^{*}}$ are shorthand for functional derivativesfun . Canonical
commutators then translate to:
$[\langle\hat{q}\rangle,\langle\hat{q}\rangle]_{\rm W}=0$,
$[\langle\hat{p}\rangle,\langle\hat{p}\rangle]_{\rm W}=0$, and
$[\langle\hat{q}\rangle,\langle\hat{p}\rangle]_{\rm W}=i\hbar/n$. The equation
of motion now reads,
$i\hbar\frac{dg}{dt}=[g,h]_{\rm W}.$ (3)
Taking the special functionals (1) one showsjon that $[g,h]_{\rm W}=i\hbar
n\\{G,H\\}_{\rm PB}$. Then, since the dynamics is norm preserving, we have
$dn/dt=0$ and (3) reduces to
$\frac{dG}{dt}=\\{G,H\\}_{\rm
PB}\equiv\partial_{\langle\hat{q}\rangle}G\partial_{\langle\hat{p}\rangle}H-\partial_{\langle\hat{p}\rangle}G\partial_{\langle\hat{q}\rangle}H.$
(4)
Hitherto, noncommutativity was thought to embody the essential difference
between the classical and quantum theories. Now we see things differently, (3)
reduces to (4) for any value of $\hbar$.
What, then, is the fundamental difference? To see this, we simply compare the
classical functional ansatz (1) to the Weinberg analogue of canonical
quantization,
$h_{1}(\psi,\psi^{*})\equiv\langle\psi|\hat{H}(\hat{q},\hat{p})|\psi\rangle.$
(5)
Now equation (3) reduces to the familiar result
$i\hbar\frac{d}{dt}\langle\psi|\hat{G}|\psi\rangle=\langle\psi|[\hat{G},\hat{H}]|\psi\rangle.$
(6)
Clearly, Weinberg’s theory is general enough to embrace both standard quantum
theory and a novel wave version of Hamiltonian classical mechanics.
Our point of departue for an alternative classical limit is the recovery of a
familiar result. Comparing (1) and (5), we write
$h_{1}=h_{0}\left(1+\frac{h_{1}-h_{0}}{h_{0}}\right).$ (7)
At any $\hbar$ the classical approximation is good for those $\psi$ such that
$(h_{1}-h_{0})/h_{0}\ll 1$. Two features deserve explicit note: the smaller is
$\hbar$ the better is the approximation for a given $\psi$; and, for all
non–zero $\hbar$, there exist states such that the criterion fails.
## III An interpolative domain?
Some functionals $h(\psi,\psi^{*})$ are classical, of form (1), others are
quantal, of form (5), while most are neither. Since both sectors are disjoint
for all $\hbar$ we seek an interpolation which joins them. In physical terms,
we imagine that the correspondence principle is to be taken literally. Thus we
speculate that, some objects, composed of many quantum particles, act as a
collective mesoparticlemes , with a center of mass dynamics that is neither
strictly quantum nor strictly classical, but some curious blend of both. For
simplicity, we assume that a one–particle equation can do this many–particle
job.
## IV Deformed quantization
### IV.1 The mathematical notion
To formulate this concept we generalize the central idea of canonical
quantization and postulate a map which sends any classical phase space
function $H(q,p)$ into a one–parameter family of interpolative Weinberg
observables $h_{\lambda}(\psi,\psi^{*})$. Symbolically, we write
${\cal Q}^{\lambda}_{\psi}\vdash
H(q,p)\stackrel{{\scriptstyle\lambda}}{{\mapsto}}h_{\lambda}(\psi,\psi^{*}),$
(8)
and call ${\cal Q}^{\lambda}_{\psi}$ a deformed quantization. Imposing (1) and
(5) as known boundary conditions, we interpret $\lambda\in[0,1]$ as a
dimensionless index of mesoscopic effects.
Since $\lambda$ is to govern emergence of classical behaviour we expect it to
depend upon some function of particle size, mass, number, or mixture thereof.
There is no way to guess this. Some authors suggest that gravity could have
something to do with itgra . Here we pick $\lambda(m)\equiv
1/(1+(m/m_{P})^{\alpha})$, for some $\alpha>0$, where $m_{P}=2.177\times
10^{-5}{\rm g}$ is the Planck mass to illustrate how the proposal might
workpla . However, we emphasize that $\lambda$ is an adjustable parameter
which cannot be fixed within this framework.
### IV.2 The specific proposal
With only the boundary conditions known we cannot fix (8) uniquely. However,
since the ansatz (1) contains only expectations, and (5) only operators, it is
suggestive to deform the particle coordinates via the simple convex
combinationlie :
$\displaystyle\hat{q}_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\lambda\hat{q}+(1-\lambda)\langle\hat{q}\rangle,$ (9)
$\displaystyle\hat{p}_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\lambda\hat{p}+(1-\lambda)\langle\hat{p}\rangle.$ (10)
This prescription is unique among linear combinations once we impose the
physical constraints:
$q_{\lambda}\equiv\langle\hat{q}_{\lambda}\rangle=\langle\hat{q}\rangle$, and
$p_{\lambda}\equiv\langle\hat{p}_{\lambda}\rangle=\langle\hat{p}\rangle$.
These enforce invariance of both the center of mass coordinates, and the
canonical Weinberg bracket relations under deformation.
Having chosen the deformed operators we now select the obvious generalization
of canonical quantization:
${\cal Q}^{\lambda}_{\psi}\vdash
H(q,p)\stackrel{{\scriptstyle\lambda}}{{\mapsto}}h_{\lambda}(\psi,\psi^{*})=\langle\psi|\hat{H}^{\lambda}|\psi\rangle,$
(11)
where $\hat{H}^{\lambda}\equiv\hat{H}(\hat{q}_{\lambda},\hat{p}_{\lambda})$,
and, for definiteness, we assume that $\hat{q}_{\lambda}$ and
$\hat{p}_{\lambda}$ are Weyl–orderedwey . As we now show, (11) gives an
interpolative dynamical system with some interesting properties. For
inessential simplicity we treat only systems with one classical degree of
freedom. The generalization is straightforward.
## V The reduced Weinberg bracket
### V.1 A general reduction lemma
Of fundamental importance is the effect of the ansatz (11) upon the bracket
(2). We begin with a computation for the more general class of functionals
$h(\psi,\psi^{*})\equiv\langle\psi|\hat{H^{\prime}}(\hat{q},\langle\hat{q}\rangle;\hat{p},\langle\hat{p}\rangle)|\psi\rangle,$
(12)
with $H^{\prime}(q_{1},q_{2};p_{1},p_{2})$ an auxilliary $c$–number function.
Applying the chain rule first, the functional derivative of this expands to
$\displaystyle\delta_{\psi}h=\langle\psi|\hat{H^{\prime}}+$ (13)
$\displaystyle\hskip
19.91684pt\langle\psi|\partial_{\langle\hat{q}\rangle}\hat{H^{\prime}}|\psi\rangle\delta_{\psi}\langle\hat{q}\rangle+\langle\psi|\partial_{\langle\hat{p}\rangle}\hat{H^{\prime}}|\psi\rangle\delta_{\psi}\langle\hat{p}\rangle.$
Evaluating $\delta_{\psi}\langle\hat{q}\rangle$ and
$\delta_{\psi}\langle\hat{p}\rangle$ gives the bra–like pair:
$\displaystyle\delta_{\psi}\langle\hat{q}\rangle$ $\displaystyle=$
$\displaystyle\langle\psi|(\hat{q}-\langle\hat{q}\rangle)/n,$ (14)
$\displaystyle\delta_{\psi}\langle\hat{p}\rangle$ $\displaystyle=$
$\displaystyle\langle\psi|(\hat{p}-\langle\hat{p}\rangle)/n.$ (15)
Taking hermitian adjoints of (13), (14) and (15) gives the ket–like quantities
$\delta_{\psi^{*}}h$, $\delta_{\psi^{*}}\langle\hat{q}\rangle$, and
$\delta_{\psi^{*}}\langle\hat{p}\rangle$. Using these rules it becomes a
simple matter to expand
$[g,h]_{\rm
W}=\delta_{\psi}g\delta_{\psi^{*}}h-\delta_{\psi^{*}}g\delta_{\psi}h.$
In reducing the expansion it is helpful to identify like terms and to make
frequent use of (14), (15) and their adjoints. Of special utility is a family
of results like
$\delta_{\psi}\langle\hat{q}\rangle\hat{H^{\prime}}|\psi\rangle-\langle\psi|\hat{H^{\prime}}\delta_{\psi^{*}}\langle\hat{q}\rangle=i\hbar\langle\psi|\partial_{\;\hat{p}\;}\hat{H}^{\prime}|\psi\rangle/n,$
where $\partial_{\hat{q}}\equiv[\bullet,\hat{p}]/i\hbar$, and
$\partial_{\hat{p}}\equiv[\hat{q},\bullet]/i\hbar$. Then, after some
cancellation using canonical bracket relations, and some rearrangement, we
find that
$\displaystyle[g,h]_{\rm
W}=\langle\psi|[\hat{G^{\prime}},\hat{H^{\prime}}]|\psi\rangle$ (16)
$\displaystyle\mbox{}+i\hbar\left\\{\langle\psi|\partial_{\langle\hat{q}\rangle}\hat{G^{\prime}}|\psi\rangle\langle\psi|\;\partial_{\hat{p}}\;\hat{H^{\prime}}|\psi\rangle-\langle\psi|\;\partial_{\hat{p}}\;\hat{G^{\prime}}|\psi\rangle\langle\psi|\partial_{\langle\hat{q}\rangle}\hat{H^{\prime}}|\psi\rangle\right\\}/n$
$\displaystyle\mbox{}+i\hbar\left\\{\langle\psi|\;\partial_{\hat{q}}\;\hat{G^{\prime}}|\psi\rangle\langle\psi|\partial_{\langle\hat{p}\rangle}\hat{H^{\prime}}|\psi\rangle-\langle\psi|\partial_{\langle\hat{p}\rangle}\hat{G^{\prime}}|\psi\rangle\langle\psi|\;\partial_{\hat{q}}\;\hat{H^{\prime}}|\psi\rangle\right\\}/n$
$\displaystyle\mbox{}+i\hbar\left\\{\langle\psi|\partial_{\langle\hat{q}\rangle}\hat{G^{\prime}}|\psi\rangle\langle\psi|\partial_{\langle\hat{p}\rangle}\hat{H^{\prime}}|\psi\rangle-\langle\psi|\partial_{\langle\hat{p}\rangle}\hat{G^{\prime}}|\psi\rangle\langle\psi|\partial_{\langle\hat{q}\rangle}\hat{H^{\prime}}|\psi\rangle\right\\}/n.$
This expression is rather more general than is required, but displays the
essential origin of our next result.
### V.2 Reduction for interpolative observables
To treat the interpolative case (11) we choose
$H^{\prime}=H(\lambda q_{1}+(1-\lambda)q_{2},\lambda p_{1}+(1-\lambda)p_{2}).$
Then, $\partial_{\hat{q}}\hat{H^{\prime}}=\lambda\hat{H}^{\lambda}_{q}$,
$\partial_{\langle\hat{q}\rangle}\hat{H^{\prime}}=(1-\lambda)\hat{H}^{\lambda}_{q}$,
$\partial_{\hat{p}}\hat{H^{\prime}}=\lambda\hat{H}^{\lambda}_{p}$, and
$\partial_{\langle\hat{p}\rangle}\hat{H^{\prime}}=(1-\lambda)\hat{H}^{\lambda}_{p}$,
where $\hat{H}^{\lambda}_{q}$ and $\hat{H}^{\lambda}_{p}$ denote the quantized
classical partials of $H(q,p)$. Thus (16) becomes
$\displaystyle[g_{\lambda},h_{\lambda}]_{\rm
W}=\langle\psi|[\hat{G}^{\lambda},\hat{H}^{\lambda}]|\psi\rangle$ (17)
$\displaystyle\mbox{}+i\hbar(1-\lambda^{2})\left\\{\langle\psi|\hat{G}_{q}^{\lambda}|\psi\rangle\langle\psi|\hat{H}_{p}^{\lambda}|\psi\rangle-\langle\psi|\hat{G}_{p}^{\lambda}|\psi\rangle\langle\psi|\hat{H}_{q}^{\lambda}|\psi\rangle\right\\}/n.$
Thus the ansatz (11) collects the three residual terms of (16) into a
“mean–field” Poisson bracketmft . The scale factor $(1-\lambda^{2})$ now
controls the mixture of quantum and classical effectsmoy .
## VI An Interpolative Schrödinger equation
### VI.1 The equation of motion for expectation values
Substituting (17) into (3), and using the property that $dn/dt=0$, now gives
$\displaystyle\frac{d\langle
G^{\lambda}\rangle}{dt}\equiv\langle[\hat{G}^{\lambda},\hat{H}^{\lambda}]\rangle/i\hbar+$
(18) $\displaystyle\hskip
28.45274pt(1-\lambda^{2})\left\\{\langle\hat{G}_{q}^{\lambda}\rangle\langle\hat{H}_{p}^{\lambda}\rangle-\langle\hat{G}_{p}^{\lambda}\rangle\langle\hat{H}_{q}^{\lambda}\rangle\right\\},$
where $\langle\bullet\rangle\equiv\langle\psi|\bullet|\psi\rangle/n$. This
provides an interpolative analogue of the standard Schrödinger picture
equation of motion for expectation values.
Of course, at $\lambda=0$ all deformed operators commute and the first term
vanishes. We are thus left with the second term alone and (4) drops out
directly. The other limit $\lambda=1$ kills the second term, operators revert
to their standard canonical quantizations and (6) results. So the commutator
term is certainly “quantum” and the bracket term is certainly “classical”.
### VI.2 An interpolative Ehrenfest theorem
Applying (18) to the coordinate operators now gives an interpolative
Ehrenfest–type theoremehr :
$\displaystyle\frac{d\langle\hat{q}_{\lambda}\rangle}{dt}$ $\displaystyle=$
$\displaystyle+\langle\hat{H}^{\lambda}_{p}\rangle$ (19)
$\displaystyle\frac{d\langle\hat{p}_{\lambda}\rangle}{dt}$ $\displaystyle=$
$\displaystyle-\langle\hat{H}^{\lambda}_{q}\rangle.$ (20)
Using this we obtain valuable insight about how wave propagation is affected
by $\lambda$. For instance, choosing $H(q,p)=p^{2}/2m+V(q)$, we find that the
only change appears in the force term. After some rearrangement, this reads
$\displaystyle\frac{d\langle\hat{p}_{\lambda}\rangle}{dt}$ $\displaystyle=$
$\displaystyle-\langle
V_{q}(\langle\hat{q}\rangle+\lambda[\hat{q}-\langle\hat{q}\rangle])\rangle$
(21) $\displaystyle=$
$\displaystyle-\sum_{k=0}^{\infty}\frac{\lambda^{k}}{k!}\langle[\hat{q}-\langle\hat{q}\rangle]^{k}\rangle\partial_{q}^{k+1}V(\langle\hat{q}\rangle).$
Looking at this we see that $\lambda$ controls the range at which the
wavefunction $\psi$ probes the potential $V(q)$. At the classical extreme,
wavepackets feel only the classical force at their centre, whereas, in the
quantum extreme, this is averaged over spacemes .
### VI.3 The interpolative wave equation
Consider now Weinberg’s generalized Schrödinger equationsw4
$i\hbar\frac{d\psi}{dt}=\delta_{\psi^{*}}h.$
Although nonlinear, standard Hilbert space methods are easily adapted by using
the definition (11), along with the hermitian adjoints of (13), (14) and (15),
to introduce an effective Hamiltonian operator,
$\displaystyle\hat{H}_{\rm
eff}^{\lambda}(\psi,\psi^{*})\equiv\hat{H}^{\lambda}+$ (22)
$\displaystyle(1-\lambda)\left\\{\langle
H^{\lambda}_{q}\rangle(\hat{q}-\langle\hat{q}\rangle)+\langle
H^{\lambda}_{p}\rangle(\hat{p}-\langle\hat{p}\rangle)\right\\},$
such that $\delta_{\psi^{*}}h=\hat{H}_{\rm
eff}^{\lambda}(\psi,\psi^{*})|\psi\rangle$. This operator defines the
interpolative Schrödinger equationkib ,
$i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}_{\rm
eff}^{\lambda}(\psi,\psi^{*})|\psi\rangle.$ (23)
One verifies easily that $\lambda=1$ returns the ordinary linear Schrödinger
equation. However, for $\lambda\neq 1$ the operator is generally
$\psi$–dependent.
This property is responsible for the failure of many standard results, such as
the superposition principle, preservation of the global inner product between
distant states, and the orthogonality of eigenvectors for self–adjoint
operators. Proofs of these assume that $\hat{H}$ is the same for any quantum
state.
Choosing $H(q,p)=p^{2}/2m+V(q)$, we set $\hat{q}=q$ and
$\hat{p}=-i\hbar\partial_{q}$. Then defining,
$\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle
n^{-1}\int_{-\infty}^{\infty}\psi(q,t)^{*}q\psi(q,t)\,dq$ (24) $\displaystyle
P(t)$ $\displaystyle=$ $\displaystyle
n^{-1}\int_{-\infty}^{\infty}\psi(q,t)^{*}\\{-i\hbar\partial_{q}\psi(q,t)\\}\,dq,$
(25) $\displaystyle F(t)$ $\displaystyle=$ $\displaystyle
n^{-1}\int_{-\infty}^{\infty}\psi(q,t)^{*}V_{q}(\lambda
q+(1-\lambda)Q(t))\psi(q,t)\,dq,$ (26)
equations (22) and (23) yield the explicit nonlinear integrodifferential wave
equation,
$\displaystyle i\hbar\frac{\partial\psi(q,t)}{\partial
t}=\frac{1}{2m}\left\\{-\lambda^{2}\hbar^{2}\partial_{q}^{2}-2i\hbar(1-\lambda^{2})P(t)\partial_{q}-(1-\lambda^{2})P^{2}(t)\right\\}\psi(q,t)$
$\displaystyle+\left\\{V(\lambda
q+(1-\lambda)Q(t))+(1-\lambda)F(t)(q-Q(t))\right\\}\psi(q,t).$
The nonlinearity of (23) lies in those terms carrying state dependent
parameters (24), (25) and (26). Given the complexity of this form, abstract
operator techniques are preferable. Calculations with the explicit equation
are hideous.
Of particular interest is the case $\lambda=0$. From (22) we compute the
effective classical Hamiltonian
$\displaystyle\hat{H}_{\rm eff}^{0}(\psi,\psi^{*})\equiv
H(\langle\hat{q}\rangle,\langle\hat{p}\rangle)+$ (27) $\displaystyle
H_{q}(\langle\hat{q}\rangle,\langle\hat{p}\rangle)(\hat{q}-\langle\hat{q}\rangle)+H_{p}(\langle\hat{q}\rangle,\langle\hat{p}\rangle)(\hat{p}-\langle\hat{p}\rangle).$
Combining (23) and (27) now gives us a classical Schrödinger equation.
## VII A classical Schrödinger equation
From (4), we know that all solutions $\psi(t)$ must have expectations,
$Q(t)\equiv\langle\hat{q}\rangle$ and $P(t)\equiv\langle\hat{p}\rangle$, that
precisely follow the classical trajectories of any chosen $H(q,p)$, for all
time, and for all values of $\hbar$. We now construct the explicit solution.
### VII.1 An heuristic overview
For short time intervals, $\Delta t$, we can assume that (27) is constant. In
the simplest approximation, we let $\psi_{t_{0}}$ be the initial wavefunction,
and construct the infinitesimal unitary propagator
$\hat{U}_{\Delta t}\approx\exp\left\\{-\frac{i\Delta t}{\hbar}\hat{H}_{\rm
eff}^{0}(\psi_{t_{0}},\psi^{*}_{t_{0}})\right\\}.$ (28)
Then, since (27) is linear in $\hat{q}$ and $\hat{p}$, it follows that (28) is
a member of the Heisenberg–Weyl grouphwg . Operators of this type assume the
general form,
$\hat{U}[Q,P;S]\equiv\exp\left\\{\frac{i}{\hbar}\left[S\hat{1}+P\hat{q}-Q\hat{p}\right]\right\\},$
(29)
and obey the operator relations:
$\displaystyle\hat{U}^{\dagger}[Q,P;S]\hat{q}\hat{U}[Q,P;S]$ $\displaystyle=$
$\displaystyle\hat{q}+Q\hat{1},$ (30)
$\displaystyle\hat{U}^{\dagger}[Q,P;S]\hat{p}\hat{U}[Q,P;S]$ $\displaystyle=$
$\displaystyle\hat{p}+P\hat{1}.$ (31)
Comparing (28) and (29), and rewriting the definition (27) in the form,
$\hat{H}_{\rm
eff}^{0}=-\left\\{QH_{q}+PH_{p}-H\right\\}\hat{1}+H_{q}\hat{q}+H_{p}\hat{p},$
(32)
now gives the approximate result
$\displaystyle|\psi_{t_{0}+\Delta t}\rangle$ $\displaystyle\approx$
$\displaystyle\hat{U}_{\Delta t}|\psi_{t_{0}}\rangle$ $\displaystyle=$
$\displaystyle\hat{U}[+H_{p}\Delta t,-H_{q}\Delta t;\Delta
S]|\psi_{t_{0}}\rangle,$
with $\Delta S=\\{Q(t_{0})H_{q}+P(t_{0})H_{p}-H\\}\Delta t$. Invoking (30) and
(31), it follows that:
$\displaystyle Q(t_{0}+\Delta t)$ $\displaystyle\approx$ $\displaystyle
Q(t_{0})+H_{p}(Q(t_{0}),P(t_{0}))\Delta t,$ $\displaystyle P(t_{0}+\Delta t)$
$\displaystyle\approx$ $\displaystyle P(t_{0})-H_{q}(Q(t_{0}),P(t_{0}))\Delta
t.$
These considerations show how the effective Hamiltonian (27) propagates any
wave $\psi$ along classical trajectories, as expected from equation (4).
### VII.2 The exact treatment
Suppose we construct the operator $U[Q(t),P(t);S(t)]$ using parameters $Q(t)$
and $P(t)$ that are obtained from solving Hamilton’s equations for the initial
conditions, $Q(t_{0})$, and $P(t_{0})$. Specifically, we demand that,
$\displaystyle\dot{P}(t)$ $\displaystyle=$ $\displaystyle-\partial_{q}H(Q,P),$
(33) $\displaystyle\dot{Q}(t)$ $\displaystyle=$
$\displaystyle+\partial_{p}H(Q,P),$ (34)
for all $t\geq t_{0}$. Then, choosing $\psi_{0}$ to be an arbitrary state with
both coordinate expectation values equal to zero, we construct the trial
solution
$|\psi_{t}\rangle=U[Q(t),P(t);S(t)]|\psi_{0}\rangle,\;\;t\geq t_{0}.$
Equation (4) is now trivially satisified. To verify (23) note that
$\hat{U}[t]$ determines,
$\hat{H}(t)\equiv
i\hbar\left\\{\frac{d}{dt}\hat{U}[t]\right\\}\hat{U}^{\dagger}[t].$ (35)
Then, using the Weyl multiplication rulehwg ,
$\displaystyle\hat{U}[Q_{2},P_{2};S_{2}]\hat{U}^{\dagger}[Q_{1},P_{1};S_{1}]=e^{i/2\hbar\\{P_{1}Q_{2}-Q_{1}P_{2}\\}}\times$
(36) $\displaystyle\hskip
28.45274pt\hat{U}[Q_{2}-Q_{1},P_{2}-P_{1};S_{2}-S_{1}],$
and (35), we compute:
$\displaystyle\hat{H}_{\rm eff}^{0}$ $\displaystyle=$ $\displaystyle
i\hbar\lim_{\delta t\rightarrow 0}\frac{\hat{U}[Q(t+\delta t),P(t+\delta
t);S(t+\delta t)]\hat{U}^{\dagger}[Q(t),P(t);S(t)]-\hat{1}}{\delta t}$
$\displaystyle=$ $\displaystyle i\hbar\lim_{\delta t\rightarrow
0}\frac{e^{i\delta
t/2\hbar\\{P\dot{Q}-Q\dot{P}\\}}\hat{U}^{\dagger}[\dot{Q}\delta
t,\dot{P}\delta t;\dot{S}\delta t]-\hat{1}}{\delta t}$ $\displaystyle=$
$\displaystyle-\\{(P\dot{Q}-Q\dot{P})/2+\dot{S}\\}-\dot{P}\hat{q}+\dot{Q}\hat{p}.$
Comparing this to (32), we first pick out (33) and (34) as necessary
conditions. Then, looking at the constant term, we solve for $\dot{S}$ to
obtain $\dot{S}=1/2(P\dot{Q}-Q\dot{P})-H$. Integrating $\dot{S}$ now gives the
exact classical propagator,
$\hat{U}[t]=\exp\left\\{\frac{i}{\hbar}\left[\phi(t)\hat{1}+P(t)\hat{q}-Q(t)\hat{p}\right]\right\\},$
(37)
where $Q(t)$ and $P(t)$ obey (33), and the phase factor $\phi(t)$ reads
$\phi(t)=\int_{t_{0}}^{t}\left(\frac{P\dot{Q}-Q\dot{P}}{2}\right)-H(Q,P)\,d\tau.$
(38)
Unlike ordinary classical mechanics, our wave version has an extra degree of
freedom; a phase factor. As one might have expectedact , this phase records
the classical action. However, unlike linear theory, the phase–to–action
correspondence is now exact.
### VII.3 Phase anholonomy effects
Interestingly, (38) contains a simple Berry phaseber . To isolate this we
employ the Aharanov–Anandanaap formula,
$\dot{\gamma(t)}=i\langle\tilde{\psi}|\\{d/dt|\tilde{\psi}\rangle\\}$, where
$\tilde{\psi}$ is a ray–space trajectory. If $|\tilde{\psi}(0)\rangle$ is any
state with vanishing coordinate expectations, then a ray path can be
parametrized as
$|\tilde{\psi}(t)\rangle=\hat{U}[Q(t),P(t);0]|\tilde{\psi}(0)\rangle,$ to
give,
$\displaystyle\dot{\gamma(t)}$ $\displaystyle=$ $\displaystyle
i\langle\tilde{\psi}(0)|\hat{U}^{\dagger}[t]\left\\{\frac{d}{dt}\hat{U}[t]\right\\}|\tilde{\psi}(0)\rangle$
$\displaystyle=$
$\displaystyle\langle\tilde{\psi}(0)|(P\dot{Q}-Q\dot{P})/2-\dot{P}\hat{q}+\dot{Q}\hat{p}|\tilde{\psi}(0)\rangle/\hbar$
$\displaystyle=$ $\displaystyle(P\dot{Q}-Q\dot{P})/2\hbar.$
On a closed loop $\Gamma$, we find
$\int_{0}^{T}P\dot{Q}\,dt=+\oint_{\Gamma}P\,dQ$, and
$\int_{0}^{T}Q\dot{P}\,dt=-\oint_{\Gamma}P\,dQ$, where $T$ is the circuit time
and signs are fixed by the sense of traversal. Thus,
$\gamma(\Gamma)=+\frac{1}{\hbar}\oint_{\Gamma}P\,dQ.$ (39)
This explicit relationship suggests that geometric phases upon closed loops
might well be interpreted as the natural action variables of quantum
mechanics.
### VII.4 Explicit wavefunction solutions
Returning to (37), we now seek explicit wavefunction solutions. Passing to the
Schrödinger representation, $\hat{q}\mapsto q$, and
$\hat{p}\mapsto-i\hbar\partial_{q}$, we note the standard resulthwg ,
$U[Q,P;0]\psi(q)=e^{-iPQ/2\hbar}e^{iPq/\hbar}\psi(q-Q).$ (40)
Then, given any state $\psi_{0}(q)$ with both expectation values equal to
zero, equation (37) yields
$\psi(q,t)=e^{i\phi(t)/\hbar}e^{-iP(t)Q(t)/2\hbar}e^{iP(t)q/\hbar}\psi_{0}(q-Q(t)).$
Looking at this we see directly that all waves propagate without dispersion.
The arbitrary wave envelope $\psi_{0}(q)$ preserves its shape while being
moved around in Hilbert space via its expectation value parameters $Q(t)$ and
$P(t)$. Therefore, no interference or tunnelling is possible in this limit. A
wave–packet must reflect or pass a barrier with certainty, just as a point
particle does in ordinary classical mechanics. Suppose we fire a packet at a
double slit. Then it must go through either one or the other slit, or it must
strike the slit screen and return. Hence it is possible to view interference
and diffraction phenomena as products of linear dynamics. Pick the right kind
of nonlinear propagation, and they evaporate altogethereva .
### VII.5 The recovery of classical phase space
Since the wave aspects are frozen out, we can now build a faithful analogue of
classical phase space. To define this, we introduce the coordinate map,
$\Pi\vdash{\cal H}\mapsto{\bf R}^{2}\mbox{ where
}\Pi[\psi]=(\langle\hat{q}\rangle,\langle\hat{p}\rangle).$
The appropriate mathematical object involves a partition of Hilbert space into
disjoint sets of wavefunctions which share identical coordinate expectations.
These sets are defined as the $\Pi$–induced equivalence classes,
$\tilde{\psi}(Q,P)=\\{\psi\in{\cal H}\vdash\Pi[\psi]=(Q,P)\in{\bf R}^{2}\\}.$
One can now treat the labels $(Q,P)$ as points, just like in ordinary
classical phase space. Each emblazons a bag of $\Pi$–equivalent wavefunctions.
We think of the classical limit as a dynamical regime where $\psi$ does not
matter, only its parameters $(Q,P)$. The original classical Hamiltonian
$H(q,p)$ now determines, via the ansatz (1), and equations, (23), and (27), a
symplectomorphism of this phase spacesym ; der .
## VIII The interpolative propagator
### VIII.1 The Liouville equation
Introducing a Liouville operator ${\cal L}_{h}\equiv[\bullet,h]_{\rm W}$, such
that ${\cal L}_{h}\circ g\equiv[g,h]_{\rm W}$ with iterated “powers”: ${\cal
L}_{h}^{k+1}\circ g=[{\cal L}_{h}^{k}\circ g,h]_{\rm W}$, we can obtain a
formal solution to (3) via exponentiation of the “tangent vector” identity
$\frac{d}{dt}\equiv{\cal L}_{h}/i\hbar$. Thus,
$g_{t}=\exp\left\\{-i(t-t_{0}){\cal L}_{h}/\hbar\right\\}\circ g_{t_{0}},$
(41)
where $L_{\Delta t}\equiv e^{-i(t-t_{0}){\cal L}_{h}/\hbar}$ is the Liouville
propagator. Now, ${\cal L}_{h}\circ(f+g)={\cal L}_{h}\circ f+{\cal L}_{h}\circ
g$, so $L_{\Delta t}$ is a linear operator on the vector space of Weinberg
observables. However, because ${\cal L}_{h}$ depends, via $h$, upon $\psi$,
the object $L_{\Delta t}$ is usually a nonlinear operator when acting on
wavefunctions. Therefore, one must be exceedingly careful to distinguish the
trivial pseudo–superposition
$(f+g)_{t}(\psi,\psi^{*})=f_{t}(\psi,\psi^{*})+g_{t}(\psi,\psi^{*}),$
which is always valid, from the special trajectorial superposition property
$(\psi+\phi)(t)=\psi(t)+\phi(t).$
This is valid when $h(\psi,\psi^{*})$ is a linear functional in both slotslin
, but fails in general (one sees this easily from (23), if $\hat{H}$ depends
upon $\psi$ then we cannot add operators for different states).
### VIII.2 The classical propagator
Using the identity $[g_{0},h_{0}]_{\rm W}=i\hbar n\\{G,H\\}_{\rm PB}$, valid
for functionals of type (1), and the fact that $dn/dt=0$, we recover the
well–known classical result:
$G_{t}=G_{t_{0}}+\\{G_{t_{0}},H_{t_{0}}\\}_{\rm
PB}(t-t_{0})+\frac{1}{2!}\\{\\{G_{t_{0}},H_{t_{0}}\\}_{\rm
PB},H_{t_{0}}\\}_{\rm PB}(t-t_{0})^{2}+\ldots.$
Similarly, one can use (41) to expand a formal solution for the classical
Schrödinger equation. Here there is no need given the exact solution (37).
### VIII.3 The quantum propagator
For quantum functionals, as defined by (4), we invoke the identity
$[g_{1},h_{1}]_{\rm W}=\langle\psi|[\hat{G},\hat{H}]|\psi\rangle$, and (41)
becomes:
$\langle\hat{G}\rangle_{t}=\langle\hat{G}\rangle_{t_{0}}+\langle[\hat{G},\hat{H}]\rangle_{t_{0}}(t-t_{0})/i\hbar+\frac{1}{2!}\langle[[\hat{G},\hat{H}],\hat{H}]\rangle_{t_{0}}(t-t_{0})^{2}/(i\hbar)^{2}+\ldots.$
Similarly, using $[\psi,h_{1}]_{\rm W}=\hat{H}|\psi\rangle$, one gets
$|\psi_{t}\rangle=e^{-i(t-t_{0})\hat{H}/\hbar}|\psi_{t_{0}}\rangle$. In this
special case the propagator does not depend upon $|\psi_{t_{0}}\rangle$.
This property encodes the superposition principle. All complexity lies in the
propagator, which happens to be independent of the initial condition for
linear theory. More generally this is not the case. Treating function–valued
curves $\psi(t)$ as “trajectories”, the overlap:
${\cal
D}(\psi,\psi^{\prime})=1-|\langle\psi|\psi^{\prime}\rangle|^{2}\;\;\mbox{where}\;\;{\cal
D}\in[0,1],$ (42)
need not be constant in time. Divergence, and the possibility of strong
divergence (i.e. “exponential”, in some sense), is thus permitted in the
nonlinear sector. To formalize this notion one can look to extend the
KS–entropy, or the classical Lyapunov exponent to Weinberg’s theoryrei via
use of the metric (42) (see jp , for its properties).
### VIII.4 Dynamical chaos in the interpolative regime?
In the interpolative case, an explicit computation of the iterated bracket
(17) is prohibitive. Nevertheless, the existence of a formal solution permits
direct study of the formal computability properties of both the classical and
quantal dynamics. Ford et al.’s algorithmic information theory approachfor to
the study of “quantum chaos” might extend in this direction.
On the numerical front, one needs to ascertain when, and how, exactly, quantum
suppression of chaos is switched off. Certainly, it must happen at some
$\lambda\in[0,1]$. Since (17) has a Poisson bracket contribution for every
$\lambda\neq 1$, this is the candidate chaos factoryfei .
## IX Interpolative eigenstates
### IX.1 The fundamental variational principle
Weinberg has generalized the eigenstates of linear quantum theory as
stationary points of the normalized observables via the simple variational
principlesw5 ,
$\delta\left(\frac{h(\psi,\psi^{*})}{n(\psi,\psi^{*})}\right)=0,$ (43)
which is equivalent tosw1 :
$\displaystyle\delta_{\psi^{*}}\left(\frac{h}{n}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\delta_{\psi^{*}}h-\frac{h}{n^{2}}\delta_{\psi^{*}}n=0,$
(44) $\displaystyle\delta_{\psi}\;\,\left(\frac{h}{n}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\delta_{\psi}\;\,h-\frac{h}{n^{2}}\delta_{\psi}\;\,n=0.$
(45)
In the linear case this reduces to the familiar result
$\hat{H}|\psi\rangle=E|\psi\rangle$. Given the form of (22), we expect a
similar result for the special interpolative observables (11).
### IX.2 Some preliminary observations
Suppose, first of all, that $\psi$ is a stationary point of the Weinberg
observable $h(\psi,\psi^{*})$. Then, if $a(\psi,\psi^{*})$ is any other
Weinberg observable, we can use the definitions (2), (44) and (45) to compute
$\displaystyle[a,h]_{\rm W}$ $\displaystyle=$
$\displaystyle\delta_{\psi}a\delta_{\psi^{*}}h-\delta_{\psi}h\delta_{\psi^{*}}a$
(46) $\displaystyle=$
$\displaystyle\frac{h}{n}\left(\delta_{\psi}a\delta_{\psi^{*}}n-\delta_{\psi}n\delta_{\psi^{*}}a\right)$
$\displaystyle=$ $\displaystyle\frac{h}{n}[a,n]_{\rm W}=0.$
This property generalizes the obvious fact that an eigenstate $\psi$ of the
linear operator $\hat{H}$, must return
$\langle\psi|[\hat{A},\hat{H}]|\psi\rangle=0$, for all $\hat{A}$.
As an immediate consequence of (46) we deduce, via the expressions (19) and
(20), that
$\langle\hat{H}^{\lambda}_{q}\rangle=0,\mbox{ and
}\langle\hat{H}^{\lambda}_{p}\rangle=0,$ (47)
of necessity.
### IX.3 The interpolative eigenvalue equation
To construct the stationarity conditions for (22), we substitute
$\delta_{\psi^{*}}h=\hat{H}_{\rm eff}^{\lambda}|\psi\rangle$ into (44),
identify $\delta_{\psi^{*}}n=|\psi\rangle$, and obtain the eigenvalue equation
$\hat{H}_{\rm eff}^{\lambda}|\psi\rangle=\langle\hat{H}_{\rm
eff}^{\lambda}\rangle|\psi\rangle.$ (48)
Combining (47) with (22) we see that
$(1-\lambda)\left\\{\langle
H^{\lambda}_{q}\rangle(\hat{q}-\langle\hat{q}\rangle)+\langle
H^{\lambda}_{p}\rangle(\hat{p}-\langle\hat{p}\rangle)\right\\}\equiv 0,$ (49)
which reduces (48) to
$\hat{H}^{\lambda}|\psi\rangle=E^{\lambda}|\psi\rangle,$ (50)
with the deformed eigenvalue,
$E^{\lambda}\equiv\langle\hat{H}_{\rm
eff}^{\lambda}\rangle=\langle\hat{H}^{\lambda}\rangle.$
So (48) implies (50). Passing in the other direction, we assume that
$\lambda\neq 0$, and notice that:
$\lambda\langle\hat{H}^{\lambda}_{q}\rangle=\langle[\hat{H}^{\lambda},\hat{p}]\rangle/i\hbar,\;\mbox{
and
}\lambda\langle\hat{H}^{\lambda}_{p}\rangle=\langle[\hat{q},\hat{H}^{\lambda}]\rangle/i\hbar,$
whence (50) implies (47), (49), and thus (48).
To treat the exceptional point $\lambda=0$, we invoke (47) alone, and deduce
that the classical stationary states of the deformed dynamical system comprise
all $\psi$ such that $\langle\hat{q}\rangle$ and $\langle\hat{p}\rangle$ lie
at a fixed point of the classical Hamiltonian flow (as one might have
guessed). Clearly, such states have infinite degeneracy, with a deformed
eigenvalue that is precisely the classical energy at the fixed point.
### IX.4 The general solution via linear quantum theory
Equation (50) is simpler than (48), but there remains a bothersome difficulty
in that
$\hat{H}^{\lambda}=\hat{H}(\lambda\hat{q}+(1-\lambda)\langle\hat{q}\rangle,\lambda\hat{p}+(1-\lambda)\langle\hat{p}\rangle).$
(51)
Although the expectation values are stationary, we have to solve (50)
self–consistently.
To fix this trouble, we bootstrap from solutions of the simpler, linear,
eigenvalue problem,
$\hat{H}(\lambda\hat{q},\lambda\hat{p})|\psi\rangle=E^{\lambda}|\psi\rangle.$
(52)
Defining the new operators: $\hat{q}^{\prime}\equiv\lambda\hat{q}$, and
$\hat{p}^{\prime}\equiv\lambda\hat{p}$, we observe that
$[\hat{q}^{\prime},\hat{p}^{\prime}]=i\hbar^{\prime}$ with
$\hbar^{\prime}=\lambda^{2}\hbar$. Equation (52) is, therefore, just the
standard eigenvalue problem with a rescaled value of $\hbar$.
Given a parametric family of $\hbar$–dependent eigenstates $\psi(q;\hbar)$,
eigenvalues $E(\hbar)$, and eigenstate expectations, $Q(\hbar)$, and
$P(\hbar)$, for the ordinary Schrödinger problem, we identify:
$\displaystyle\hbar^{\prime}$ $\displaystyle\mapsto$
$\displaystyle\hbar^{\prime}=\lambda^{2}\hbar$ $\displaystyle\hat{q}^{\prime}$
$\displaystyle\mapsto$ $\displaystyle q^{\prime}=\lambda q$
$\displaystyle\hat{p}^{\prime}$ $\displaystyle\mapsto$
$\displaystyle-i\hbar^{\prime}\partial_{q^{\prime}}=-i(\lambda^{2}\hbar)\partial_{(\lambda
q)}=\lambda(-i\hbar\partial_{q}).$
Thus the solution to (52) is obtained by applying the rescalings
$q\mapsto\lambda q$ and $\hbar\mapsto\lambda^{2}\hbar$ to the known solutions
for the $\lambda=1$ problem. Imposing the constraint,
$\int_{-\infty}^{\infty}\psi(\lambda q)\psi^{*}(\lambda q)\,dq=1$, now fixes
the renormalized quantities:
$\displaystyle\langle q|\psi_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\lambda^{1/2}\psi(\lambda q;\lambda^{2}\hbar),$ (53)
$\displaystyle E^{\lambda}$ $\displaystyle=$ $\displaystyle
E(\lambda^{2}\hbar),$ (54) $\displaystyle Q^{\lambda}$ $\displaystyle=$
$\displaystyle\langle\psi_{\lambda}|\hat{q}|\psi_{\lambda}\rangle=Q(\lambda^{2}\hbar)/\lambda,$
(55) $\displaystyle P^{\lambda}$ $\displaystyle=$
$\displaystyle\langle\psi_{\lambda}|\hat{p}|\psi_{\lambda}\rangle=P(\lambda^{2}\hbar)/\lambda.$
(56)
Using these expressions we can construct a solution to the general problem
(50).
First we form, after (29), and using (55) and (56), the Weyl operator,
$\hat{V}\equiv\hat{U}[(1-\lambda)Q^{\lambda},(1-\lambda)P^{\lambda}].$ (57)
Applying this to both sides of (52) gives,
$\hat{V}^{\dagger}\hat{H}(\lambda\hat{q},\lambda\hat{p})\hat{V}\hat{V}^{\dagger}|\psi_{\lambda}\rangle=E^{\lambda}\hat{V}^{\dagger}|\psi_{\lambda}\rangle.$
Thus we can identify,
$|\psi^{\prime}_{\lambda}\rangle=\hat{V}^{\dagger}|\psi_{\lambda}\rangle$ (58)
as an eigenstate of the new operator,
$\hat{V}^{\dagger}\hat{H}(\lambda\hat{q},\lambda\hat{p})\hat{V}$ with the
eigenvalue $E^{\lambda}$ unchanged.
Using (30), (31) and (58) we compute:
$\displaystyle\langle\psi^{\prime}_{\lambda}|\hat{q}|\psi^{\prime}_{\lambda}\rangle/n$
$\displaystyle=$
$\displaystyle\langle\psi_{\lambda}|\hat{q}-(1-\lambda)Q^{\lambda}|\psi_{\lambda}\rangle/n=\lambda
Q^{\lambda},$ (59)
$\displaystyle\langle\psi^{\prime}_{\lambda}|\hat{p}|\psi^{\prime}_{\lambda}\rangle/n$
$\displaystyle=$
$\displaystyle\langle\psi_{\lambda}|\hat{p}-(1-\lambda)P^{\lambda}|\psi_{\lambda}\rangle/n=\lambda
P^{\lambda}.$ (60)
Similarly,
$\displaystyle\hat{V}^{\dagger}\hat{H}(\lambda\hat{q},\lambda\hat{p})\hat{V}=$
$\displaystyle\hat{H}(\lambda[\hat{q}+(1-\lambda)Q^{\lambda}],\lambda[\hat{p}+(1-\lambda)P^{\lambda}]).$
Combining these relations, and comparing to (51), we verify that solves (50)
self–consistently. To pass in the other direction, we start with a solution to
(50), pick $\hat{V}$ as the inverse of (57), with $Q^{\lambda}$ and
$P^{\lambda}$ determined from (59) and (60), and obtain, via (58), a solution
of (52).
Making use of (53), (55), (56) and the disentanglement relation (40),
$\displaystyle\psi_{\lambda}(q)=\lambda^{1/2}e^{-i(1-\lambda)P(\lambda^{2}\hbar)(\lambda
q)/(\lambda^{2}\hbar)}$ (61) $\displaystyle\hskip 56.9055pt\times\psi(\lambda
q+(1-\lambda)Q(\lambda^{2}\hbar);\lambda^{2}\hbar),$
where all indicated functions are obtained as solutions to the standard
Schrödinger problem ($\lambda=1$).
Recall the harmonic oscillator wavefunctionsmor ,
$\psi_{n}(q;\hbar)=(2^{n}n!)^{-1/2}(\beta/\pi)^{1/4}e^{-\beta
q^{2}/2}H_{n}(q\beta^{1/2}),$ (62)
where $H_{n}(z)=(-1)^{n}e^{z^{2}}(d^{n}/dz^{n})e^{-z^{2}}$, with
$\beta(\hbar)=m\omega/\hbar$. Since the position and momentum expectations of
these vanish, it is easy to verify that (62) are invariant under the
transformation (61).
Although (61) looks singular at $\lambda=0$, this need not always be the case.
As a matter of curiosity, we wonder which class of Hamiltonians have
eigenstates that are fixed points of this abstract mapping.
### IX.5 Degeneracies and the failure of orthogonality
Some minor trouble arises if (52) is degenerate. Then (57) must be applied, in
turn, to each member of the invariant subspace associated with $E^{\lambda}$,
so as to generate a corresponding interpolative eigensubspace. Thus one can
think of the solutions to (50) as being constructed by applying the nonlinear
mapping (57) to the entire Hilbert space. Evidently, the usual linear
eigenvector orthogonality relations are preserved, if, and only if, all
eigenvectors of (52) happen to share identical coordinate expectations.
Although the form of $E^{\lambda}$ suggests, on first sight, that we are
merely taking $\hbar\rightarrow 0$ via a circuitous route, the failure of
orthogonality shows that the two approaches are, in fact, fundamentally
different. One distinguishes this limit from the standard classical limit via
the modification to eigenfunctions (examine (61)).
Another clear distinguishing feature is that we cannot superpose the nonlinear
eigensolutions
$|\psi^{\prime}_{\lambda}(t)\rangle=e^{-i(t-t_{0})E^{\lambda}/\hbar}|\psi^{\prime}_{\lambda}(t_{0})\rangle,$
to get a solution of (23).
### IX.6 A connection between quantum eigenstates and
fixed points of the classical Hamiltonian flow?
Given that $\lambda=0$ eigenstates lie at fixed points of the classical
Hamiltonian flow, we conjecture that:
$\displaystyle\lim_{\lambda\rightarrow 0}E(\lambda^{2}\hbar)$ $\displaystyle=$
$\displaystyle E^{0}_{\rm f.p.},$ (63) $\displaystyle\lim_{\lambda\rightarrow
0}Q(\lambda^{2}\hbar)$ $\displaystyle=$ $\displaystyle Q^{0}_{\rm f.p.},$ (64)
$\displaystyle\lim_{\lambda\rightarrow 0}P(\lambda^{2}\hbar)$ $\displaystyle=$
$\displaystyle P^{0}_{\rm f.p.}.$ (65)
Two problems confound a proof. Firstly, continuity of the defining variational
problem, (43), is essential, but the infinite degeneracy of solutions at
$\lambda=0$ contradicts this. Secondly, at this same point the auxilliary
problem, (52), is obviously singular. So the known $\lambda=0$ behaviour need
not always connect with the above limits.
For example, parity arguments applied to the quartic double well potential,
$V(q)=(q^{2}-1)(q^{2}+1)$, show that (64) fails. Eigenstates have vanishing
expectation so the two stable fixed points are missed out. Either conditions
of broken symmetry must obtain, or the correct statement is more subtle.
For exact single fixed point problems, the limit (63) is easily verifiedver .
The harmonic oscillator obeys it,
$E^{\lambda}=\lambda^{2}\hbar\omega(n+1/2)\rightarrow E^{0}=0,$
as does the hydrogen atom,
$E^{\lambda}_{n}=-\frac{Z^{2}e^{4}m_{e}}{2n^{2}\lambda^{4}\hbar^{2}}\rightarrow
E^{0}=-\infty,$
(if we treat the origin as a fixed point). A soluble example with two fixed
points is Calogero’s problemcal ,
$\left\\{-\alpha\frac{\partial^{2}}{\partial q^{2}}+\beta q^{2}+\gamma
q^{-2}\right\\}\psi(q)=E\psi(q),$
with the eigenfunctionsabs ,
$\psi(q)=(\kappa
q)^{a+1/2}e^{-\kappa^{2}q^{2}/2}L^{a}_{n}(\kappa^{2}q^{2}),\;\;n=0,1,2,\ldots.$
where $\kappa=(\beta/\alpha)^{1/4}$, $a=1/2(1+4\gamma/\alpha)^{1/2}$, and
$4\gamma/\alpha>-1$. The classical fixed points lie at
$q=\pm(\gamma/\beta)^{1/4}$, with energy $2(\gamma/\beta)^{1/2}$. Taking
Calogero’s eigenvalue formula
$E_{n}=(\alpha\beta)^{1/2}(2+2a+4n),$
we let $\alpha\rightarrow 0$ and verify (63).
Thus the energy result seems quite general. Indeed one can take the EBK
semiclassical quantization ruleebk ,
$\oint_{\Gamma}p\,dq=2\pi\hbar(n+\alpha/4)$ and deduce that, as
$\hbar\rightarrow 0$, the symplectic area enclosed by the classical periodic
orbits $\Gamma_{n}(\hbar)$ must vanish. Now we assume that a continuously
parametrized family of periodic orbits with this property must converge upon
some classical fixed point. Then EBK connects a quantized energy level with
the action parameter labelling the “disappearing torus”. It appears that
integrable Hamiltonians must respect (63).
## X Uncertainty products and dispersion
### X.1 Generalized dispersion
To develop a generalized uncertainty relation we recall the usual definition,
$\Delta^{2}_{a}\equiv\langle\psi|(\hat{A}-\langle\hat{A}\rangle)^{2}|\psi\rangle$,
where $\hat{A}$ is a linear operator. Then for
$a\equiv\langle\psi|\hat{A}|\psi\rangle$, we observe that
$\Delta^{2}_{a}=a\star a-a^{2}/n,$ (66)
where $a\star a\equiv\delta_{\psi}a\delta_{\psi^{*}}a$. If we assume that $a$
commutes with all its $\star$–product powers, then, Weinberg arguessw6 , the
usual probability interpretation is retained. Thus $a\star a$ is the average
of the square, $a^{2}$ the average squared, and (66) is a generalized
dispersion observable.
### X.2 A generalized uncertainty principle?
Given a second observable $b$, whose $\star$–powers again commute, we treat
$\delta_{\psi^{*}}a$ and $\delta_{\psi^{*}}b$ as kets, their adjoints as bras,
and set
$|\alpha\rangle=\delta_{\psi^{*}}a-a/n\delta_{\psi^{*}}n,\mbox{ and
}|\beta\rangle=\delta_{\psi^{*}}b-b/n\delta_{\psi^{*}}n.$
Substituting these into the Schwartz inequalitysch ,
$\langle\alpha|\alpha\rangle\langle\beta|\beta\rangle\geq|\langle\alpha|\beta\rangle|^{2}$,
we collect $\star$–products to obtain the inequality
$(a\star a\\!-\\!a^{2}/n)(b\star b\\!-\\!b^{2}/n)\\!\geq\\!\left|(a\star
b\\!-\\!ab/n)\right|^{2}.$ (67)
Working on the right hand side, we have
$a\star b-ab/n=\frac{1}{2}[a,b]_{\rm W}+\frac{1}{2}[a,b]^{+}_{\rm W}-ab/n,$
with $[a,b]^{+}_{\rm W}\equiv a\star b+b\star a$. Taking the square norm, we
observe that $1/2[a,b]_{\rm W}$ is pure imaginary, while $1/2[a,b]^{+}_{\rm
W}-ab/n$, is pure real. Given that the real term vanishes on the minimum
uncertainty states, (67) permits the simpler, weakened, form
$\Delta^{2}_{a}\Delta^{2}_{b}\geq\frac{1}{4}\left|[a,b]_{\rm W}\right|^{2}.$
(68)
Although this inequality bears a striking resemblance to the standard
Heisenberg–Robertson relationrob , it is only properly motivated if $a$ and
$b$ are observables whose $\star$–powers commute. Caution is advisable since
the right and left members of (67) need not be invariant under general
nonlinear canonical transformations.
Although (67) has the formal properties of dispersion, its physical
interpretation is unclear. If dispersion depends upon the coordinate system,
we can make little of it, except perhaps to distinguish the value zero as
being special.
### X.3 A simple example: coordinate functionals
For a simple example, we take the deformed coordinate functionals
$q_{\lambda}$ and $p_{\lambda}$. Since these commute with their
$\star$–powers, we have
$\Delta^{2}_{q_{\lambda}}\Delta^{2}_{p_{\lambda}}\geq\frac{1}{4}\left|[q_{\lambda},p_{\lambda}]_{\rm
W}\right|^{2}=\frac{\hbar^{2}}{4}.$
Thus deformation preserves the generalized uncertainty principle (68), and
coordinate dispersions are seen to obey the usual interpretative rules.
### X.4 Wider validity?: classical observables
Interestingly, the general stationarity conditions (44) and (45) imply, via
(66), that dispersion must vanish for generalized stationary states. This is
the most cogent physical reason for believing that (67) may be of general
significance.
For example, using (13) we compute,
$\Delta_{h_{0}}^{2}=(\partial_{q}H)^{2}\Delta_{q}^{2}+2(\partial_{q}H)(\partial_{p}H)\Delta_{qp}^{2}+(\partial_{p}H)^{2}\Delta_{p}^{2},$
where,
$\Delta_{qp}^{2}\equiv\frac{1}{2}\langle\psi|(\hat{p}-\langle\hat{p}\rangle)(\hat{q}-\langle\hat{q}\rangle)+(\hat{q}-\langle\hat{q}\rangle)(\hat{p}-\langle\hat{p}\rangle)|\psi\rangle.$
Thus classical dispersion is just a “quantized” version of gaussian quadrature
error analysis. Dispersion vanishes at classical fixed points, as does the
right hand member of (68) for quantities in involution (i.e. with zero Poisson
bracket).
More generally the interpolative dispersion does not seem to have any ready
interpretation. We therefore doubt that the concept is useful, except as a
means to study the spreading of quantum states under evolution.
### X.5 Interpolative dynamics of dispersion
Since the generalized dispersions formed via rule (66) are again homogeneous
of degree one, we can use the evolution equation (3). For instance, from (9),
(10), and the formula (17) we compute
$[\Delta^{2}_{q_{\lambda}},h_{\lambda}]_{\rm W}$ and
$[\Delta^{2}_{p_{\lambda}},h_{\lambda}]_{\rm W}$, to obtain:
$\displaystyle\frac{d\Delta^{2}_{q_{\lambda}}}{dt}$ $\displaystyle=$
$\displaystyle+\lambda
n\left\\{\langle[\hat{q},\hat{H}^{\lambda}_{p}]^{+}\rangle-2\langle\hat{q}\rangle\langle\hat{H}^{\lambda}_{p}\rangle\right\\},$
(69) $\displaystyle\frac{d\Delta^{2}_{p_{\lambda}}}{dt}$ $\displaystyle=$
$\displaystyle-\lambda
n\left\\{\langle[\hat{p},\hat{H}^{\lambda}_{q}]^{+}\rangle-2\langle\hat{p}\rangle\langle\hat{H}^{\lambda}_{q}\rangle\right\\}.$
(70)
No matter what the chosen state $\psi$, or Hamiltonian $H$, dispersion is
smoothly switched off as $\lambda\rightarrow 0$.
## XI The interpolative free particle
To illustrate the preceding formal material we consider the interpolative free
particle Hamiltonian:
$\hat{H}^{\lambda}_{\rm
eff}\equiv\frac{\hat{p}_{\lambda}^{2}}{2m}+\frac{\langle\hat{p}\rangle}{m}(\hat{p}-\langle\hat{p}\rangle).$
(71)
Using either the propagator formula (41), or the fact that the momentum
$P_{0}=\langle\hat{p}\rangle$ is a constant of the motion (via equations (19)
and (20)), we see that the free particle propagator is just
$\hat{U}_{\Delta t}=\exp\left\\{\frac{-i\Delta
t}{\hbar}\left(a\hat{p}^{2}+b\hat{p}+c\hat{1}\right)\right\\},$ (72)
where, from (71), the constants $a$,$b$ and $c$ read:
$a=\frac{\lambda^{2}}{2m},\;b=\frac{(1-\lambda^{2})P_{0}}{m},\;\mbox{and}\;c=\frac{(\lambda^{2}-1)P^{2}_{0}}{m}.$
(73)
The problem is now easily solved using the deformed free particle Green’s
function,
$\displaystyle K_{\lambda}(q^{\prime},q;\Delta t)\equiv$ (74)
$\displaystyle\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}e^{-i\Delta
t\left(ap^{2}+bp+c\hat{1}\right)/\hbar}e^{+i(q^{\prime}-q)p/\hbar}\,dp,$
such that,
$\psi(q^{\prime},t_{0}+\Delta
t)=\int_{-\infty}^{\infty}K_{\lambda}(q^{\prime},q;\Delta
t)\psi(q,t_{0})\,dq.$ (75)
Evaluating (76) we get,
$K_{\lambda}(q^{\prime},q;\Delta
t)=(\pi/i\gamma)^{-1/2}e^{-i\kappa}e^{i\gamma\left[q-(q^{\prime}-\delta)\right]^{2}},$
(76)
where,
$\gamma=1/4a\hbar\Delta t,\;\delta=b\Delta t,\;\mbox{and}\;\kappa=c\Delta
t/\hbar.$ (77)
Choosing an initial gaussian at the origin,
$\psi(q,t_{0})=(\pi/2\alpha)^{-1/4}e^{-\alpha q^{2}+i\beta q},$ (78)
with appropriate width and momentum parameters,
$\alpha=\frac{1}{4\sigma_{q}^{2}},\;\mbox{and}\;\beta=\frac{P_{0}}{\hbar},$
(79)
we substitute (76) and (78) into (75), and compute the evolved gaussian state,
$\psi(q,t_{0}+\Delta
t)=(\pi/2\alpha)^{-1/4}[(\alpha-i\gamma)/i\gamma]^{-1/2}e^{-i[\kappa+\gamma(q-\delta)^{2}]}\exp\left\\{-\frac{\gamma^{2}[q-(\delta+\beta/2\gamma)]^{2}}{(\alpha-i\gamma)}\right\\},$
(80)
where primes are now dropped. Next we form,
$|\psi(q,t_{0}+\Delta
t)|^{2}=\left(\frac{\pi(\alpha^{2}+\gamma^{2})}{2\alpha\gamma^{2}}\right)^{-1/2}\exp\left\\{-\frac{2\alpha\gamma^{2}[q-(\delta+\beta/2\gamma)]^{2}}{(\alpha^{2}+\gamma^{2})}\right\\},$
(81)
and use (73), (77) and (79), to pick out the evolved packet centre and
dispersion formulæ:
$\displaystyle q_{0}(t_{0}+\Delta t)$ $\displaystyle=$
$\displaystyle\frac{P_{0}\Delta t}{m},$ (82)
$\displaystyle\sigma_{q}^{2}(t_{0}+\Delta t)$ $\displaystyle=$
$\displaystyle\sigma_{q}^{2}(t_{0})\left\\{1+\frac{\lambda^{4}\hbar^{2}(\Delta
t)^{2}}{4m^{2}\sigma_{q}^{4}(t_{0})}\right\\}.$ (83)
We check that interpolative particles propagate at the desired classical
velocity $P_{0}/m$. Moreover, as with the energies $E^{\lambda}$, the formula
(83) is identical to the standard linear one, except that $\hbar$ is replaced
by $\lambda^{2}\hbar$. Compare the $\lambda=0$ behaviour with standard quantum
theory. For any mass $m$, there exists some time interval $\Delta t_{c}$, such
that a particle will eventually disperse so as to fill the entire known
universe. Ordinarily, we dispense with this difficulty by stating that the
interval is far too long to matter, and that particles are, in any case,
localized by measurements long before the situation gets out of hand. In
contrast, the limit (83) offers greater descriptive (not prescriptive) power
in that we can hang the value $\lambda=0$ upon this circumstance.
## XII Prospects for empirical test
### XII.1 Where does linearity apply, for sure?
There have been numerous stringent tests of quantum linearity performed upon
microscopic systems. Each of these has yielded a null resultexp . Bollinger et
alexp , have bounded the Weinberg nonlinearity in Beryllium nuclei
spin–precession experiments at less than $4$ parts in $10^{-27}$. Other
indirect tests, such as the atomic version of Young’s double slit
experimentyou , and inversion tunnelling in small molecules like Ammonia
provide strong evidence against nonlinearity in atomic scale systems.
### XII.2 How might nonlinearity emerge?
The quantum dynamics of isolated systems observed in today’s laboratory must
therefore be linear to a very high degree of precision. If nonlinearity lies
somewhere, then it seems that one must look for its effects in a new place.
Either that, or one argues that this exact version of Hamiltonian classical
dynamics, formulated as a wave theory for any value of $\hbar$, is just a
bizarre mathematical accident, put there expressly to tease us.
A clear question emerges. Is quantum theory always linear with an approximate
classical limit; or is there a more general nonlinear theory which is linear
for small systems and progressively nonlinear until we recover an exact
classical limit?
Two distinct physical interpretations appear possible. Either the
$\psi$–dependent operators express a statistical result that should then be
traced to environment–induced fluctations (decoherencedec ); or, since (3) is
deterministic, the nonlinearity might reflect a purely causal coupling to the
environment (a back–reaction or self–energy effect). In either case, it seems
plausible that nonlinearity should become larger the less isolated, and more
entangled, a quantum system becomes.
### XII.3 In search of a mesoscopic “elementary particle”
Most elementary particles have internal structure. However, if empirical
energy scale is decoupled from the internal degrees of freedom, then we can
exploit a structureless one–particle approximation.
In particle physics one reveals internal structure by building a higher energy
accelerator. To test any one–particle wave equation one needs an inverted
version of this program. The goal is to screen the known internal degrees of
freedom and get the detector energies low enough (or sideband them on a more
accessible frequency).
To make a mesoscopic “elementary particle” we could take a spherical
macromolecule, or perhaps a microspheremsp . Then we charge it, or magnetize
it, and find an ingenious way to measure this and weigh itwei . Then we give
the particle a moment of some kind, put it in a well and couple it to coherent
radiation in an accessible range (probably microwaves). Then it is feasible,
in principle, to resolve the quantized energy levels. Nobody does this now
because it seems impossible to get the thermal background cool enough, or the
characteristic frequencies high enough, to be able to resolve the levels of a
particle in, say, the microgram range. The lighter our particle the easier the
experiment, but the further we are likely to be from the classical regime.
### XII.4 A possible empirical signature
Suppose we can do this at some mass (or size) scale. Given the standard
prediction for energy levels $E(\hbar)$, one needs to use the spectroscopic
data, along with the known particle mass etc., to measure Planck’s constant
(assuming the radiation law $\Delta E=\hbar\nu$). If this were to exhibit a
monotonic decrease as one passes to more classical systems, then has evidence
for a perturbative energy level shift, like the $E(\lambda^{2}\hbar)$ effect.
Because the classical and quantal Weinberg energy functionals differ, one
might expect something similar for any interpolative scheme. Our investigation
is thus helpful, if only to show that any observed discrepancy of this kind
deserves careful attention.
## XIII Theoretical difficulties
Given that experimental tests of the validity of exact linear quantum theory
in the classical domain are so very difficult; we now highlight some of the
severe problems the nonlinear theory generates. It may be that strong
exclusions can be found via this route.
### XIII.1 The free nature of $\lambda$
This is the most obvious problem. Without positive empirical evidence one
cannot fix $\lambda$. The only thing we learn is what kind of effects one
might need to look for. There does not seem to be any way around this problem.
Remember also that (11) is just a postulate. Canonical quantization is not the
only route to generalization.
### XIII.2 Lack of manifest algebraic closure
From (16), we see that the interpolative observables (11), do not manifestly
comprise a subalgebra, except at $\lambda=0,1$. This ugly mathematical feature
strongly suggests that the interpolation is unphysical. A subalgebra may show
up using coordinate free methods (i.e. write (11) in terms of $\star$–products
and ordinary products). However, this fact, and the general complexity of the
interpolative domain, leads us to conclude that (11) has no fundamental
physical content, other than as a guide to formulating empirical questions.
At a deeper level we obtain a sieve: “What existence and uniqueness
constraints apply to a one–parameter family of Weinberg subalgebras which
joins the classical and quantum regimes?”.
### XIII.3 Problems with measurement: a provisional
probabilistic interpretation
Weinberg has emphasizedsw7 that generalization of the probability
interpretation to nonlinear observables is defeated by non–associativity of
the functional $\star$–product. Nor can we use the Hilbert space inner
product, since this is not a canonical invariant in the nonlinear sector of
the theory.
How else might we get a probability interpretation? Since problems arise due
to nonlinearity, the natural place to look for the “right” idea is in this
sector. It is much easier to specialize a working result; than to generalize
from a special one.
Classical statistical physics employs densities $\rho(q,p)$ on phase space.
Liouville’s theorem preserves normalization and Hamilton’s equations determine
evolution of the ensemble. One can then discuss classical measurement as a
stochastic diffusive process superimposed upon the dynamics, and justify
statistical mechanics via the ergodic hypothesislif .
Classical expectations are phase space averages
$\bar{f}=\int\rho(q,p)f(q,p)\,dpdq,$ (84)
where $\rho(q,p)$ is stationary.
Since Weinberg’s theory specializes the Hamiltonian formalism (homogeneity is
a constraint upon the hamiltonian) we can try and carry this over directly.
The key is to find an invariant measure upon quantum states.
Canonical invariance of the symplectic form $dp\wedge dq$ (and thus its
exterior powers), implies Liouville’s theoremarn . In Weinberg’s theory we
identify the corresponding canonically invariant symplectic form
$\sum_{k=1}^{D}d\psi^{*}_{j}\wedge d\psi_{j}$. Taking exterior powers of this
we get Liouville’s theorem, and an induced invariant measure on the projective
Hilbert space of normalized states.
Exploiting canonical invariance of the norm $n$, we focus on functionals that
are homogeneous of degree $p$, and define the measurekj1 :
$\int
F(\psi,\psi^{*})\,d\hat{\Omega}_{\tilde{\psi}}\equiv\frac{\Gamma(D)}{\Gamma(D+p)}\int
F(\psi,\psi^{*})e^{-n}\,\prod_{j=1}^{D}\pi^{-1}d{\rm Re}[\psi_{j}]d{\rm
Im}[\psi_{j}],$ (85)
where $d\hat{\Omega}_{\tilde{\psi}}$ emphasizes the analogy with solid angle.
To get a good $D\rightarrow\infty$ limit, we set $p=0$ on the right hand side
(divide $F$ by $n^{p}$ when taking the average).
The formula (85) is immediately recognized as the standard functional measure
of path integralssch , or the theory of gaussian random fieldsgud .
Now let $\rho(\psi,\psi^{*})$ be any positive Weinberg observable satisfying,
$\int\rho(\psi,\psi^{*})\,d\hat{\Omega}=1.$
The uniform density becomes $n(\psi,\psi^{*})$, the norm functional. A
nontrivial example is,
$\rho^{\phi}_{N}(\psi,\psi^{*})=n^{1-N}\frac{\Gamma(D+N)}{\Gamma(N)\Gamma(D)}|\langle\psi|\phi\rangle|^{2N},$
(86)
where the factor $n^{1-N}$ makes this homogeneous of degree one, so that (3)
applies. On averaging we set this to $n^{-N}$. As $N\rightarrow\infty$, (86)
peaks strongly about $\phi$. This density plays the role of a delta function
on states.
To see this, we use the formulakj2 ,
$\displaystyle\int|\langle\phi|\psi\rangle|^{2}f(|\langle\omega|\psi\rangle|^{2})\,d\hat{\Omega}_{\tilde{\psi}}$
$\displaystyle=$
$\displaystyle\frac{1}{D-1}(1-|\langle\phi|\omega\rangle|^{2})\int
f(|\langle\psi|\omega\rangle|^{2})\,d\hat{\Omega}_{\tilde{\psi}}$ (87)
$\displaystyle+\frac{1}{D-1}(D|\langle\phi|\omega\rangle|^{2}-1)\int|\langle\psi|\omega\rangle|^{2}f(|\langle\psi|\omega\rangle|^{2})\,d\hat{\Omega}_{\tilde{\psi}}.$
Defining generalized quantum averages in the classical fashion,
$\bar{a}=\int\rho(\psi,\psi^{*})a(\psi,\psi^{*})\,d\hat{\Omega}_{\tilde{\psi}},$
(88)
we choose the bilinear functional,
$a_{1}(\psi,\psi^{*})=\langle\psi|\hat{A}|\psi\rangle=\sum_{j=1}^{D}a_{j}|\langle\psi|\omega_{j}\rangle|^{2},$
(89)
where $A_{j}$ are the eigenvalues of $\hat{A}$, and $|\omega_{j}\rangle$ its
eigenvectors. Substituting (89) and (86) into (88), we use (87) and the
definition (85) to verify that,
$\displaystyle\int\rho^{\phi}_{N}(\psi,\psi^{*})a_{1}(\psi,\psi^{*})\,d\hat{\Omega}_{\tilde{\psi}}=$
(91)
$\displaystyle\sum_{j=1}^{D}\left\\{\frac{D}{(D-1)(D+N)}+\frac{N}{D+N}\left(1-\frac{D}{N(D-1)}\right)A_{j}|\langle\phi|\omega_{j}\rangle|^{2}\right\\}.$
Keeping $D$ fixed, and taking $N\rightarrow\infty$, we recover the desired
result
$\bar{a}_{1}=\int\rho^{\phi}_{\infty}(\psi,\psi^{*})a_{1}(\psi,\psi^{*})\,d\hat{\Omega}_{\tilde{\psi}}=\langle\phi|\hat{A}|\phi\rangle.$
(92)
Thus quantal expectation values can be reinterpreted as phase space averages
with respect to the delta distribution $\rho^{\phi}_{\infty}$.
If we let $D\rightarrow\infty$, in heuristic fashion, and choose classical
Weinberg functionals, then (85) induces the standard Liouville measure over
the phase space of coordinate expectations, and we recover the classical
result (84).
More generally, one consider $\rho(\psi,\psi^{*})$ as defining the density
matrix,
$\hat{\rho}=\int\rho(\psi,\psi^{*})|\psi\rangle\langle\psi|\,d\hat{\Omega}_{\tilde{\psi}}.$
(93)
Linearity of the trace operation and positivity of the probability density
ensures that, $\hat{\rho}>0$ and ${\rm Tr}[\hat{\rho}]=1$. This connection is
many–to–one, so that $\rho(\psi,\psi^{*})$ is a “hidden”, or “indeterminable”,
representation of $\hat{\rho}$. Nevertheless,
$\bar{a}_{1}=\int\rho(\psi,\psi^{*})a_{1}(\psi,\psi^{*})\,d\hat{\Omega}_{\tilde{\psi}}={\rm
Tr}[\hat{\rho}\hat{A}].$
Pure states become delta function ensembles, whereas smeared densities
generate the mixed states.
Rephrasing quantum averages in this fashion, we acquire a common statistical
language for both classical and quantum physics. Next we need to incorporate a
generalized theory of measurement. Thusfar we can only do this by appeal to
the known result. Nevertheless, our hope is that a suitably generalized
perspective might reveal (94) as a special case, whose inner product nature is
accidental to the linear sector, but somehow necessary.
For a quantum system in state $\phi$ subjected to a complete measurement with
the operator $\hat{A}$, we have the jump process $\phi\mapsto\omega_{j}$
occuring with conditional probability,
$p(\omega_{j}|\phi)=|\langle\phi|\omega_{j}\rangle|^{2}.$ (94)
Post measurement, we have a probability density peaked as delta function
spikes on each of the eigenvectors, with weight given by rule (94):
$\rho(\psi,\psi^{*})=\sum_{k=1}^{D}|\langle\phi|\omega_{j}\rangle|^{2}\rho^{\omega_{j}}_{\infty}(\psi,\psi^{*}).$
(95)
Substituting this into the rule (88), we verify that
$\bar{a}=\langle\phi|\hat{A}|\phi\rangle$. This is the same as the result
(92), but the underlying distribution over states is different.
How can we understand this? Although (86), with $N\rightarrow\infty$, and (95)
generate exactly the same statistics, their dynamical properties under (3) are
different. If $\hat{A}$ were the Hamiltonian then (95) is a stationary
probability density. In contrast, the density (86) must be time–dependent,
unless $\phi$ happens to be an eigenstate of $\hat{A}$.
Thus the stationary states of the hamiltonian flow appear rather special to
the quantum case, they are associated with stationary probability densities
which are sums of delta functions upon these. This suggests that we should
incorporate (94) as a dynamical resultbel , albeit via stochastic dynamicsnot
.
Master equations, either classical, or quantal, are the canonical examples of
this paradigmfok . They encapsulate stochastic evolution of individual
ensemble members via a deterministic equation of Fokker–Planck type. Adopting
this formal route, we postulate the generalized nonlinear quantum master
equationmil :
$\frac{d\rho}{dt}=\frac{1}{i\hbar}[\rho,h]_{\rm
W}+\frac{\Gamma}{(i\hbar)^{2}}[[\rho,a]_{\rm W},a]_{\rm W},$ (96)
where $h$ is the “free evolution” and $a$ is the “measurement functional”,
while $\Gamma$ is a phenomenological parameter (zero if measurement is
switched off).
Given (96) we must solve for the stationary probability density
$\rho_{\infty}$ (defined as the limit $t\rightarrow\infty$) generated from a
chosen initial condition $\rho_{0}$. Assuming existence of $\rho_{\infty}$,
the averaging rule (88) provides the statistical prediction.
From (96) the stationarity condition reads $\dot{\rho}=0$. To recast this we
introduce Liouville operators: ${\cal L}_{h}\equiv[\bullet,h]_{\rm W}$, and
${\cal L}_{a}\equiv[\bullet,a]_{\rm W}$, to get:
$({\cal L}_{h}-{\cal L}_{a}\circ{\cal L}_{a})\circ\rho=0.$ (97)
This specifies the kernel of a linear operator,
${\cal L}_{\rm M}\equiv{\cal L}_{h}-{\cal L}_{a}\circ{\cal L}_{a},$ (98)
on the space of Weinberg functionals (the operator acts in the adjoint
representation of this Lie algebra). Thus we identify ${\cal L}_{\rm M}$ as
the formal “measurement operator” which is to describe an $a$–measurement
perfomed upon a system undergoing $h$–evolution.
Conveniently, linearity of (98) implies a spectral theory. Intuitively, we
expect the spectrum of (98) to determine the decay rate of $\rho_{0}$ to
$\rho_{\infty}$, and also the class of initial conditions $\rho_{0}$, upon
which a given measurement will be good (in the sense that we get to a
stationary density, or arbitrarily close to it, in a finite interaction time).
If $\rho^{k}_{\infty}$ is a finite set $k\in[1,M]$ of stationary densities,
then so too is the linear combination,
$\rho_{\infty}=\sum_{k=1}^{M}w_{k}\rho^{k}_{\infty},$ (99)
provided only that the weights $w_{k}$ sum to unity. Thus a measurement theory
somewhat analagous to that of linear quantum theory exists, even when
orthogonality is relaxed (recovery of this, on the space of density
functionals, would require (98) to be self–adjoint).
The generalization is certainly suggestive. Significantly, the equation (96)
is not a hamiltonian flow, but it has the desired physical property of being
expressed purely via canonically invariant Weinberg brackets.
Unsolved problems aside, a consistent, and inclusive, statistical
interpretation of nonlinear quantum theory is conceivable via appeal to
stochastic dynamics.
### XIII.4 Thermodynamic constraints
Although $\hbar$ is fixed, harmonic oscillator energy levels have the reduced
spacing $\lambda^{2}\hbar\omega$ and the $n$th stationary solution now reads:
$|n_{t}\rangle=e^{-i\lambda^{2}\omega(n+1/2)(t-t_{0})}|n_{t_{0}}\rangle.$
(100)
However, from (19) and (20) one verifies that a gaussian wave packet
oscillates at the classical frequency $\omega$, for all $\lambda$. Thus we
encounter the bizzare circumstance that a transition between two stationary
states suggests the photon frequency $\nu=\lambda^{2}\omega$, while the
classical radiation frequency remains $\omega$.
We could try and fix this by letting $\nu=\omega$ so that the photon energies
become $\lambda^{2}\hbar\nu$. However, that leads to the horrible consequence
that photons must either, be confined to a single $\lambda$–sector, or, change
their frequency at each interaction with matter. Worse still, the deformed
Planck black body factor (excluding degeneracy),
$\frac{e^{-\lambda^{2}\hbar\nu/kT}}{1-e^{-\lambda^{2}\hbar\nu/kT}},$ (101)
detonates at $\lambda=0$. Presto, an ultraviolet catastrophe! The only way to
salvage this disaster is to postulate that photons are always $\lambda=1$
particles. To defend that, superposition of light waves is regularly observed
at the classical level, whereas that of matter waves is not.
Thus a deformed harmonic oscillator must have two characteristic frequencies.
This curious property offends cherished physical intuition. However, as shown
in great depth by Weinbergsw8 , such behaviour is common. Moreover, because
the state preparations differ, it would be impossible to observe both
frequencies in a single experiment. Evidently, there is no ambiguity or
contradiction, a situation not unlike wave–particle duality. How one could
ever detect this is a problem, unless perhaps thermodynamics can do it for us
via some modification of specific heats. Certainly, the deformed black body
rule would predict this; but given the historical importance of that problem
it is hard to believe that there is any discrepancy lurking in the data.
Currently it is assumed that material and radiative oscillators must be
quantized in the same way. Certainly radiation must be consistently quantized,
because it mediates interaction between material particles. That leaves us in
some doubt as to whether material oscillators must obey the same rule of
consistency. Clarification of this issue is probably the most powerful
constraint upon any modified quantum theory. Nobody would reject
thermodynamics.
## XIV Conclusion
In summary, we have embedded both Hamiltonian classical mechanics and linear
quantum theory as two disjoint dynamical sectors of Weinberg’s generalized
nonlinear theory. To explore the idea of a mesoscopic regime we then studied
one technique for interpolation. Although not fully constrained, our method is
simple, general, and has some desirable physical features. The result is an
alternative classical limit whereby quantal evolution is smoothly transformed
into classical evolution as we vary a single dimensionless control parameter
$\lambda$. Significantly, this works for any value of $\hbar$.
At the level of mathematical physics, we have a new tool for comparing
classical and quantal dynamics. This can be put to immediate use in studies of
“quantum chaos”. The ability to turn dynamical chaos on and off via $\lambda$,
whatever the magnitude of $\hbar$, provides a new probe of the origin of
dynamical chaos suppression, and the potential for exposing some interesting
phenomena in the transition regime. We will return to study this later, with a
parting comment that the interpretative problems have no bearing upon this
pursuit.
Concerning the working hypothesis that nonlinearity emerges at the classical
level, we stress that the evident success of linear quantum theory for
microscopic systems is not in dispute. Rather we imagine that a complex of
atomic systems, a whole molecule, a block of solid, glass of beer, cat, flea
on cat, or ribbon of its DNA, has gotten complicated enough that the dynamics
for the $\psi$ of its centre of mass is described by a nonlinear theory.
This attempt at a physical interpretation is imprecisely formulated. The
mathematics is unwieldy, and devoid of predictive power. Given its complexity,
we do not believe that the interpolative technique has any fundamental
physical content. Nevertheless, the one–particle assumption at least enables
us to compute deformed energies $E(\lambda^{2}\hbar)$, and show that the free
particle has uniformly suppressed dispersion as $\lambda\rightarrow 0$. Thus
we settle upon the view that the proper role of interpolative dynamical
studies is to guide tests of the universality of linear quantum theory.
Fundamental questions of this nature demand careful scrutiny. Indeed, the idea
of emergent nonlinearity, bizarre as it may be, is consistent with both the
observed linearity of isolated atomic scale systems and the fact that
classical mechanics describes the familiar world of our senses so well. In the
current climate one is led to reject a complete recovery of classical theory,
because it implies that there is a nonlinear regime, and so linear quantum
theory could not be considered universal. We suggest that if our prejudices
demand that we invent reasons to ignore simple mathematical facts, then
physics is in very serious trouble.
## XV Acknowledgments
Portions of this work were carried out at the University of Melbourne,
Australia; University of Houston, Texas; University of Texas at Austin;
Institute of Advanced Study Princeton; and my current address. I am grateful
to: B.H.J. McKellar, A.G. Klein, S. Adler, S. Weinberg, S.C. Moss, and M.
Eisner for their hospitality, and for useful discussions. Conversation or
correspondence with: A.J. Davies, S. Dyrting, O. Bonfim, N.E. Frankel, Z.
Ficek, G.J. Milburn, V. Kowalenko, H. Wiseman, R. Volkas and I.C. Percival
sharpened the ideas, and provided encouragement. Support from the University
of Melbourne, A.G. Klein, B.H.J. McKellar and N.E. Frankel, via a Visiting
Research Fellowship, and a Special Studies Travel Grant are further
acknowledged, along with an A.R.C. postdoctoral fellowship.
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* (3) If $\psi_{1}(t)$ and $\psi_{2}(t)$ are solutions then so too is $\psi(t)=\alpha\psi_{1}(t)+\beta\psi_{2}(t)$, even if these states are non–orthogonal. The normalisation is time independent; absorb it into $\alpha$ and $\beta$. Then, using results from Ref. lim , one can choose states such that $\langle\hat{A}\rangle_{\psi_{1}}(t)$ and $\langle\hat{A}\rangle_{\psi_{2}}(t)$ follow the classical trajectories. But now $\langle\hat{A}\rangle_{\psi}(t)=|\alpha|^{2}\langle\hat{A}\rangle_{\psi_{1}}(t)+|\beta|^{2}\langle\hat{A}\rangle_{\psi_{2}}(t)+2{\rm Re}[\alpha^{*}\beta\langle\psi_{1}(t)|\hat{A}|\psi_{2}(t)\rangle],$ which need not follow any classical trajectory.
* (4) The paradox of Schrödinger’s cat is thus avoided by the practical assertion that non–local, alive and dead or correlated singlet felines are unpreparable.
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* (14) This class consists of all real–valued functionals of $\psi=(\psi_{1},\ldots,\psi_{d})$ such that $h(\lambda\psi,\psi^{*})=\lambda h(\psi,\psi^{*})=h(\psi,\lambda\psi^{*})$, for all complex $\lambda$, or, equivalently (Ref.sw1 ), $\frac{\partial h}{\partial\psi_{k}}\psi_{k}=h=\frac{\partial h}{\partial\psi^{*}_{k}}\psi^{*}_{k},$ with summation over $k$ implicit. Although the norm $n=\psi^{*}_{k}\psi_{k}$ is invariant, there is no invariant meaning for the global inner product. To motivate Hilbert space methods we observe that homogeneity implies, $h=\psi^{*}_{k}\frac{\partial^{2}h}{\partial\psi^{*}_{k}\partial\psi_{l}}\psi_{l},$ whatever the chosen coordinate system. Thus, at each $\psi$, $h$ fixes an Hermitian form, a local inner product, a local orthonormal basis and, consequently, a tangent Hilbert space ${\cal H}$, its dual, and a space of linear operators ${\cal L}({\cal H})$ acting on these. The local inner product is not invariant under general canonical transformations. It seems that demanding such invariance characterizes the usual linear theory [see the analysis by R. Cirelli, A. Mania and L. Pizzocchero, Int. J. Mod. Phys. A6, 2133 (1991)]. This has important interpretational consequences (we can’t use the projection postulate). However, all of our computations can still be carried out in Hilbert space in a representation independent fashion (this is like fixing a system of Euclidean coordinates in classical mechanics for the purpose of displaying the motion). (N.B. locality means the mathematical kind in the space of all $\psi$; physically these objects are non–local.)
* (15) All computations follow from the bilinear result $\delta_{\psi}\langle\psi|\hat{A}|\psi\rangle=\langle\psi|\hat{A}$, where $\hat{A}$ is any linear operator. This device, from Ref.jon , permits direct comparison with standard theory. As per Ref.sw3 inner products apply between quantities defined at the same $\psi$.
* (16) This hypothesis was posed in the context of a nonlinear wave theory by I. Bialynicki-Birula and J. Mycielski, Ann. Phys. N.Y. 100, (1976). Null results for their log–nonlinear wave equation include: C.G. Shull, D.K. Attwood, J. Arthur and M.A. Horne, Phys. Rev. Lett. 44, 765 (1980); and R. Gähler, A.G. Klein, and A. Zeilinger, Phys. Rev. 23, 1611 (1981). Because our wave equation recovers exact classical theory, which is known to be empirically accurate in a certain domain, the theory posed here is much harder to exclude outright.
* (17) R. Penrose, in Quantum Concepts in Space and Time, edited by C.J. Isham and R. Penrose (Oxford, Oxford, 1986); R. Penrose, The Emperor’s New Mind, (Oxford, Oxford, 1989) pp367–370; N. Rosen, Found. Phys. 16, 687 (1986); and A. Peres, Nucl. Phys. 48, 622 (1963).
* (18) For instance, if we take a “typical” $A\approx 50$ atom, we get $m/m_{P}=0(10^{-17})$. For $\alpha=1$ we find the spectral perturbation is sub–Lamb shift ($\lambda$ is almost unity so the order of $\hbar$ does not matter). For $\alpha=2$, it is $O(10^{-34})$, (cf. Bollinger et al. Ref.exp ). This mischief continues ad infinitum. Pick any $f$ such that $f(0)=1$ and $f(\infty)=0$ with $\lambda(m)=f(m/m_{P})$. The free nature of $\lambda$ is an very serious defect. However, the Copenhagen interpretation shares a similar inability to pin down Heisenberg’s cut. Bell called this situation the shifty split [J.S. Bell, Phys. World, 3, 33 (1990)].
* (19) Expectations generate “particle”–like evolution and operators “wave”–like evolution. Varying the mix effectively controls wave–particle duality. The uncertainty principle stands, so precise measurability is not implied. More generally we can take an arbitrary Lie algebra of operators $\hat{A}_{k}$, and replace all commutators $[\hat{A}_{j},\hat{A}_{k}]=iC_{jk}^{l}\hat{A}_{l}$ by their Weinberg bracket equivalents. The $C_{jk}^{l}$ are preserved by $\lambda$–deformation. Taking $\lambda\rightarrow 0$ we get a “classical limit” for any quantum system, even those with no classical analogue.
* (20) Deformed Weyl–ordered quantization is defined via the obvious generalization, $\displaystyle{\cal Q}_{\psi}^{\lambda}\circ H(q,p)\equiv$ $\displaystyle(2\pi\hbar)^{-2}\int_{R^{4}}e^{i[\sigma(\hat{p}_{\lambda}-p)+\tau(\hat{q}_{\lambda}-q)]/\hbar}H(q,p)\,d\sigma d\tau dqdp,$ of the standard Weyl operator fourier transform. One checks easily that: $\partial_{\hat{q}}{\cal Q}_{\psi}^{\lambda}=\lambda{\cal Q}_{\psi}^{\lambda}\partial_{q}$ and $\partial_{\hat{q}}{\cal Q}_{\psi}^{\lambda}=\lambda{\cal Q}_{\psi}^{\lambda}\partial_{q}$. For the standard theory ($\lambda=1$) see: H. Weyl, Z. Physik. 46, 1 (1927); H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950) pp272–280; N.H. McCoy, Proc. Nat. Acad. Sci. 18, 674 (1932); K.E. Cahill and R.J. Glauber, Phys. Rev. 177, 1857, 1882 (1969); G.S. Agarwal and E. Wolf, Phys. Rev. D2, 2161, 2187, 2206 (1970); F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. (N.Y.) 111, 61, 111 (1978); and M. Hillery, R.F. O’Connell, M.O. Sculley and E.P. Wigner, Phys. Rep. 106, 121 (1984).
* (21) Connections between nonlinearity, mean–field theory, and/or dynamical chaos have been examined in many places. For example, Primas [H. Primas, in Sixty–Two Years of Uncertainty edited by A.I. Miller (Plenum, New York, 1990)], discusses this in connection with early work by Onsager [L. Onsager, J. Amer. Chem. Soc. 58, 1486 (1936)]. More recently, see P. Boná, Comenius University Report, Faculty of Mathematics and Physics Report No. Ph10-91, 1991 (unpublished), and references therein. In connection with chaos, see D. David, D.D. Holm, and M.V. Tratnik, Phys. Lett. 138A, 29 (1989); W.M. Zhang, D.H. Feng, J.M. Yuan and S.H. Wang, Phys. Rev. A40, 438 (1989); and W.M. Zhang, D.H. Feng and J.M. Yuan, Phys. Rev. A42, 7125 (1990). The factorization algorithm common to much of this work, and given detailed study in Ref. yaf often generates dynamical chaos in what began as a non–chaotic quantum model. This is because the $O(1/N)$ error contol of large $N$ limits is rapidly overcome in any chaotic regime of the classical system. This can have important, sometimes dire, consequences for studies that seek to match theory to experiment via this approximation.
* (22) To place both terms on equal footing in $\lambda$ and $\hbar$ one adapts Moyal’s calculus [J.E. Moyal, Proc. Camb. Phil. Soc. 45, 99 (1949)] to prove that, $\langle[\hat{G}^{\lambda},\hat{H}^{\lambda}]\rangle/i\hbar=\lambda^{2}\langle\\{G,H\hat{\\}}^{\lambda}_{\rm M}\rangle,$ where $\\{G,H\hat{\\}}^{\lambda}_{\rm M}$ denotes the deformed Weyl–quantization of the Moyal bracket, $\\{\bullet,\bullet\\}_{\rm M}\equiv\frac{2}{\hbar}\sin\left(\frac{\hbar}{2}\left[\frac{\partial}{\partial Q}\frac{\partial}{\partial P}-\frac{\partial}{\partial P}\frac{\partial}{\partial Q}\right]\right),$ of the classical functions $G$ and $H$. See also, T. F. Jordan and E.C.G Sudarshan, Rev. Mod. Phys. 33, 515 (1961); and G.A. Baker Jr., Phys. Rev. 109, 2198 (1958).
* (23) Compare, Messiah Ref.lim
* (24) Ref.sw1 equation 2.12, we use $\hbar\neq 1$
* (25) A Schrödinger equation of this type, with a $\psi$–dependent Hermitian operator, appears in T. Kibble, Commun. Math. Phys. 64, 73 (1978). We stumbled across it in: K.R.W. Jones, University of Melbourne Report No. UM-P-91/47 (unpublished) 1991.
* (26) W.M. Zhang, D.H. Feng and R. Gilmore, Rev. Mod. Phys. 62 (1990), 867; A. Perelemov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986). J.R. Klauder and B.S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).
* (27) P.A.M. Dirac, Phys. Zeit. der Sowjet. 3 64 (1933); R.P. Feynmann, Rev. Mod. Phys. 20 267 (1948). The reprints appear in: Selected Papers on Quantum Electrodynamics edited by J. Schwinger (Dover, New York, 1958).
* (28) M.V. Berry, Proc. Roy. Soc. Lond. A392, 45 (1984); M.V. Berry, in Geometric Phases in Physics edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989).
* (29) Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); see also, J. Anandan and L. Stodolsky, Phys. Rev. D35, 2597 (1987).
* (30) This is why we adopt the interpretation noted at Ref.lie . The $Q(t)$ and $P(t)$ are not precisely measurable, but they can “guide” $\psi$ along a classical path.
* (31) V.I. Arnol’d, Mathematical Methods of Classical Mechanics 2nd edn. (Springer–Verlag, Berlin, 1989) chap 8.
* (32) Elsewhere, we have derived the classical Schrödinger equation [K.R.W. Jones, University of Melbourne Report No. UM-P-91/45, 1991 (unpublished)]. In 1927 Weyl proved there is but one projective representation of the Abelian group of translations on the plane [H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950) pp272–280]. Exploiting this fact we can rewrite Hamilton’s equations in the operator form, $i\hbar\frac{d}{dt}\hat{U}[Q,P]=\hat{H}(Q,P)\hat{U}[Q,P],$ where $\hat{U}[Q,P]$ is a member of the Heisenberg–Weyl group and the Hamiltonian reads, $\hat{H}(Q,P)=H+H_{Q}(\hat{q}-Q)+H_{P}(\hat{p}-P),$ with $H(Q,P)$ the classical Hamiltonian. The solution is the operator–valued trajectory, $\tilde{U}[Q,P]=e^{\frac{i}{\hbar}\int L\,dt}\hat{U}[Q,P],$ where, $\int L\,dt=\int(P\dot{Q}-Q\dot{P})/2-H(Q,P)\,dt$ and $Q(t)$, and $P(t)$ solve Hamilton’s equations. This is verified by differentiation. Then we invoke the Stone–von Neumann theorem [M.H. Stone, Proc. Nat. Acad. Sci. 16, 172 (1932); J. von Neumann, Math. Ann. 104 570 (1931)], and note that any Hilbert space which carries an irrep. of the Heisenberg–Weyl group is unitarily equivalent to the standard Schrödinger representation. We then place a ket on the right to get the wave evolution. Thus the projective revision of classical theory automatically gives us: some constant $\hbar$, wavefunctions, canonical commutation relations, and the classical Schödinger equation. P. Boná (private communication, see Ref.mft ) informs me that he has obtained a similar result.
* (33) It is a folk prejudice of the quantum chaos community that the linearity of quantum theory has nothing to do with chaos suppression because the classical Liouville equation has this trivial linearity property. Wider study of nonlinear quantum theory, via numerical simulations, should help decide the matter.
* (34) Given that strobe maps are so useful as test examples, it might be interesting to study one–parameter families of nonlinear Floquet maps defined by, $|\psi_{n+1}\rangle={\cal U}^{\mu}_{\Delta t}(|\psi_{n}\rangle)\equiv e^{-\frac{i\Delta t}{\hbar}\hat{H}_{\rm eff}^{1-\mu}(\psi_{n},\psi_{n}^{*})}|\psi_{n}\rangle,$ where $t=n\Delta t$, $n=0,1,2\ldots$ and $\mu=1-\lambda$ is the nonlinearity parameter [compare: M.J. Feigenbaum, in Universality in Chaos edited by P. Cvitanović (Adam Hilger, Bristol, 1986)]. Of interest is the fact that quantum systems with suppressed chaos must be perturbed via $\mu$ towards their nonintegrable classical counterparts.
* (35) L.E. Reichl, The transition to chaos in conservative classical systems: quantum manifestations (Springer–Verlag, New York, 1992).
* (36) J.P. Provost and G. Vallee, Commun. Math. Phys. 76, 289 (1980).
* (37) Ref. 11 §3
* (38) P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953).
* (39) We can find no single–fixed point exactly soluble problem which does not, but these are terribly un–representative examples.
* (40) F. Calogero, J. Math. Phys. 10, 2197 (1969); and F. Calogero, J. Math. Phys. 12, 419 (1971).
* (41) M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, 1964) eq. 22.6.18 p781.
* (42) See Eckhardt Ref.cha pp.224-235; V.P. Maslov and M.V. Fedoriuk, Semi–Classical approximations in quantum mechanics (Reidel, Holland, 1981); J.B. Keller, Ann. Phys. (N.Y.) 4, 180 (1958); and references therein.
* (43) Ref. 11 §5
* (44) L.I. Schiff, Quantum Mechanics 3rd edn. (McGraw-Hill, New York, 1968).
* (45) W. Heisenberg, The Physical Principles of the Quantum Theory (University of Chicago, Chicago, 1930); and H.P. Robertson, Phys. Rev. 34, (1929).
* (46) Numerous high precision tests of linearity have been performed using the Weinberg theory (see Ref.sw2 for the proposal). Early examples include: J.J. Bollinger, D.J. Heinzen, W.M. Itano, S.L. Gilbert and D.J. Wineland, Phys. Rev. Lett. 63, 1031 (1989); T.E. Chupp and R.J. Hoare, Phys. Rev. Lett. 64, 2261 (1990); R.L. Walsworth, I.F. Silvera, E.M. Mattison and R.F.C. Vessot, Phys. Rev. Lett. 64, 2599 (1990).
* (47) O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991); D.W. Keith, C.R. Ekstrom, Q.A. Turchette, and D.E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
* (48) Microspherules have been developed for use in guided drug delivery, P. Guiot and P. Couvreur, Polymeric Nanoparticles and Microspheres (CRC Press, Florida, 1986). They can be manufactured down to $1-100\mu$m. At a notional specific gravity of unity (they are prepared in suspension), this corresponds to $m\approx 10^{-9}$–$10^{-15}$ kg. For a natural frequency of $10^{9}$ Hz (microwaves), we need a “spring constant” $k=\omega^{2}m\approx 10^{3}$–$10^{6}$ N${\rm m}^{-1}$. Over one particle radius (a simple measure of stress) this is $10^{-3}$–$10^{2}$ N. This may be feasible at the lower end. Single–atom trapping technology might scale for this purpose [see: H. Dehmelt, Rev. Mod. Phys. 62, 525 (1990); W. Paul, Rev. Mod. Phys. 62, 531 (1990); and N.F. Ramsey, Rev. Mod. Phys. 62, 541 (1990)].
* (49) For true elementary particles the requisite parameters can be measured in different experiments, for a composite particle it becomes very much harder.
* (50) Ref. 11 §5
* (51) L.D. Landau and E.M. Lifschitz, Statistical Physics (Pergamon, Oxford, 1989). One must keep the ideas put forward here distinct from quantum statistical mechanics. Measurement probabilities differ from thermodynamic ones.
* (52) See Ref. sym pp206–207.
* (53) K.R.W Jones, Ann. Phys. (N.Y.) 207, 140 (1991).
* (54) L. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
* (55) S.P. Gudder, Stochastic Methods in Quantum Mechanics (North–Holland, Amsterdam, 1979).
* (56) K.R.W. Jones, J. Phys. A 24, 121 (1991); K.R.W. Jones, J. Phys. A. 24, 1237 (1991).
* (57) This idea is the basis of much current work on measurement modelling. The approach is particularly useful in quantum optics. See, for example: Gisin and Percival Ref. bor ; and H.M. Wiseman and G.J. Milburn, University of Queensland, Preprint (1992). An early example, is Bohm and Bub, Ref. bor . They sought to interpret their model as a non–local hidden variables theory.
* (58) H. Risken, The Fokker–Planck Equation (Springer–Verlag, Berlin, 1989).
* (59) S. Dyrting (private communication) 1992, told me of this possibility. The idea of elevating such formal double–commutator type equations to fundamental status is briefly explored in G.J. Milburn, Phys. Rev. A44, 5401 (1991); and references therein. Milburn shows how such behaviour can be made to emerge from a “shortest tick of the universal clock” postulate.
* (60) This may involve non–local hidden variables traced to the unknown wavefunction of the environment. J. Polchinski, Phys. Rev. Lett. 66, 397 (1991) has pointed out some subtle difficulties of nonlinear theories in relation to EPR–type experiments. However, if the quantum statistics are correctly recovered, then singlet states cannot be used for superluminal communication. See the discussion in R.J. Glauber, Ann. N.Y. Acad. Sci. 480, 336 (1986).
* (61) Many characteristic effects of this kind are developed in Ref. 11 §6. Although our interpolation does not lie in the class considered by Weinberg, the stringent exclusions made there appear to extend to this work also.
|
arxiv-papers
| 2013-12-15T21:46:37 |
2024-09-04T02:49:55.442660
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K.R.W. Jones",
"submitter": "Kingsley Jones",
"url": "https://arxiv.org/abs/1312.4195"
}
|
1312.4241
|
# The index of Dirac operators on incomplete edge spaces
Pierre Albin University of Illinois, Urbana-Champaign [email protected]
and Jesse Gell-Redman Department of Mathematics, University of Toronto
[email protected]
###### Abstract.
We derive a formula for the index of a Dirac operator on an incomplete edge
space satisfying a “geometric Witt condition.” We accomplish this by cutting
off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer
index theorem, and taking a limit. We deduce corollaries related to the
existence of positive scalar curvature metrics on incomplete edge spaces.
## Introduction
Ever since Cheeger’s celebrated study of the spectral invariants of singular
spaces [20, 21, 22] there has been a great deal of research to extend our
understanding of geometric analysis from smooth spaces. Index theory in
particular has been extended to spaces with isolated conic singularities quite
successfully (beyond the papers of Cheeger see, e.g., [19, 37, 28, 29]) and
was used by Bismut and Cheeger to establish their families index theorem on
manifolds with boundary [12, 13, 14].
The fact that Bismut and Cheeger used, [12, Theorem 1.5], is that for a Dirac
operator (though not any Dirac-type operator) on a space with a conic
singularity, the null space of $L^{2}$ sections naturally corresponds to the
null space of the Dirac operator on the manifold with boundary obtained by
excising the singularity and imposing the ‘Atiyah-Patodi-Singer boundary
condition’ [7], provided an induced Dirac operator on the link has no kernel.
Indeed Cheeger points out in [20] that one can recover the Atiyah-Patodi-
Singer index theorem from the index formula for spaces with conic
singularities.
In this paper we consider a Dirac operator on a space with non-isolated conic
singularities, also known as an ‘incomplete edge space’, and the Dirac
operator on the manifold with boundary obtained by excising a tubular
neighborhood of the singularity and imposing the Atiyah-Patodi-Singer boundary
condition. Although the relation between the domains of these two Dirac
operators is much more complicated than in the case of isolated conic
singularities, we show that under a “geometric Witt assumption” analogous to
that used by Bismut-Cheeger, the index of these operators coincide. Thus we
obtain a formula for the index of the Dirac operator on the singular space as
the ‘adiabatic limit’ of the index of the Dirac operator with Atiyah-Patodi-
Singer boundary conditions.
$\varepsilon$$X$$M_{\varepsilon}$$B\simeq Y$ Figure 1. The singular space $X$
obtained by collapsing the fibers of the boundary fibration of $M.$ The spaces
$M_{\varepsilon}$ play a central role in our proofs.
Specifically, an incomplete edge space is a stratified space $X$ with a single
singular stratum $Y.$ In keeping with Melrose’s paradigm for analysis on
singular spaces (see e.g., [43, 47]) we resolve $X$ by ‘blowing-up’ $Y$ and
obtain a smooth manifold with boundary $M,$ whose boundary is the total space
of a fibration of smooth manifolds
$Z-\partial M\xrightarrow{\phantom{x}\phi\phantom{x}}Y.$
A (product-type) incomplete edge metric is a metric that, in a collar
neighborhood of the boundary, takes the form
$g=dx^{2}+x^{2}g_{Z}+\phi^{*}g_{Y}$ (1)
with $x$ a defining function for $\partial M,$ $g_{Y}$ a metric on $Y$ and
$g_{Z}$ a family of two-tensors that restrict to a metric on each fiber of
$\phi.$ Thus we see that metrically the fibers of the boundary fibration are
collapsed, as they are in $X.$
We also replace the cotangent bundle of $M$ by a bundle adapted to the
geometry, the ‘incomplete edge cotangent bundle’ $T_{\operatorname{ie}}^{*}M$,
see (1.3) below. This bundle is locally spanned by forms like $dx,$ $x\;dz,$
and $dy,$ and the main difference with the usual cotangent bundle is that the
form $x\;dz$ is a non-vanishing section of $T_{\operatorname{ie}}^{*}M$ all
the way to $\partial M.$
We assume that $(M,g)$ is spin and denote a spin bundle on $M$ by
$\mathcal{S}\longrightarrow M$ and the associated Dirac operator by $\eth.$
The operator $\eth$ does not induce an operator on the boundary in the usual
sense, due to the degeneracy of the metric there, but we do have
$x\eth\big{\rvert}_{\partial M}=c(dx)(\tfrac{1}{2}\dim Z+\eth_{Z})$
where $\eth_{Z}$ is a vertical family of Dirac operators on $\partial M.$ It
turns out, as in the conic case mentioned above, and the analogous study of
the signature operator in [2, 3], that much of the functional analytic
behavior of $\eth$ is tied to that of $\eth_{Z}.$ Indeed, in Section 2.4 below
we prove the following theorem.
###### Theorem 1.
Assume that $\eth$ is a Dirac operator on a spin incomplete edge space
$(M,g),$ satisfying the “geometric Witt-assumption”
$\mathrm{Spec}\;(\eth_{Z})\cap(-1/2,1/2)=\emptyset.$ (2)
Then the unbounded operator $\eth$ on $L^{2}(M;\mathcal{S})$ with core domain
$C^{\infty}_{c}(M;\mathcal{S})$ is essentially self-adjoint. Moreover, letting
$\mathcal{D}$ denote the domain of this self-adjoint extension, the map
$\eth\colon\mathcal{D}\longrightarrow L^{2}(M;\mathcal{S})$
is Fredholm.
The spin bundle admits the standard $\mathbb{Z}/2\mathbb{Z}$ grading into even
and odd spinors
$\mathcal{S}=\mathcal{S}^{+}\oplus\mathcal{S}^{-},$
and thus we have the chirality spaces $\mathcal{D}^{\pm}=\mathcal{D}\cap
L^{2}(M;\mathcal{S}^{\pm})$ and the restriction of the Dirac operator
satisfies
$\eth\colon\mathcal{D}^{+}\longrightarrow L^{2}(M;\mathcal{S}^{-}).$
This map is Fredholm and our main result is an explicit formula for its index.
The metric (1) naturally defines a bundle metric on
$T_{\operatorname{ie}}^{*}M,$ non-degenerate at $\partial M,$ and the Levi-
Civita connection of $g$ naturally defines a connection $\nabla$ on
$T_{\operatorname{ie}}^{*}M.$ Our index formula involves the transgression of
a characteristic class between two related connections. The restriction of
$T_{\operatorname{ie}}M$ to $\partial M$ can be identified with $N_{M}\partial
M\oplus T\partial M/Y\oplus\phi^{*}Y.$ Let
$\mathbf{n}:T_{\operatorname{ie}}M\big{\rvert}_{\partial M}\longrightarrow
N_{M}\partial M,\quad\mathbf{v}:T_{\operatorname{ie}}M\big{\rvert}_{\partial
M}\longrightarrow T\partial M/Y,$
be the orthogonal projections onto the normal bundle of $\partial M$ in $M,$
$N_{M}\partial M=\langle\partial_{x}\rangle,$ and the vertical bundle of
$\phi,$ respectively, and let $\mathbf{v}_{+}=\mathbf{n}\oplus\mathbf{v}.$
Both
$\nabla^{v_{+}}=\mathbf{v}_{+}\circ\nabla\big{\rvert}_{\partial
M}\circ\mathbf{v}_{+},\text{ and
}\nabla^{\operatorname{pt}}=\mathbf{n}\circ\nabla\big{\rvert}_{\partial
M}\circ\mathbf{n}\;\oplus\;\mathbf{v}\circ j_{0}^{*}\nabla\circ\mathbf{v},$
(3)
where $\operatorname{pt}$ stands for ‘product’, are connections on
$N_{M}\partial M\oplus T\partial M/Y\longrightarrow\partial M.$
###### Main Theorem.
Let $X$ be stratified space with a single singular stratum endowed with an
incomplete edge metric $g$ and let $M$ be its resolution. If $\eth$ is a Dirac
operator associated to a spin bundle $\mathcal{S}\longrightarrow M$ and $\eth$
satisfies the geometric Witt condition (2), then
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=\int_{M}\widehat{A}(M)+\int_{Y}\widehat{A}(Y)\left(-\frac{1}{2}\widehat{\eta}(\eth_{Z})+\int_{Z}T\widehat{A}(\nabla^{v_{+}},\nabla^{\operatorname{pt}})\right)$
(4)
where $\widehat{A}$ denotes the $\widehat{A}$-genus,
$T\widehat{A}(\nabla^{v_{+}},\nabla^{\operatorname{pt}})$ denotes the
transgression form of the $\widehat{A}$ genus associated to the connections
(3), and $\widehat{\eta}$ the $\eta$-form of Bismut-Cheeger [11].
The simplest setting of incomplete edge spaces occurs when $Z$ is a sphere, as
then $X$ is a smooth manifold and the singularity at $Y$ is entirely in the
metric. Atiyah and LeBrun have recently studied the case where
$Z=\mathbb{S}^{1}$ and $X$ is four-dimensional, so that $Y$ is an embedded
surface, and the metric $g$ asymptotically has the form
$dx^{2}+x^{2}\beta^{2}d\theta^{2}+\phi^{*}g_{Y}.$
The cone angle $2\pi\beta$ is assumed to be constant along $Y.$ In [6] they
find formulas for the signature and the Euler characteristic of $X$ in terms
of the curvature of this incomplete edge metric. In this setting, assuming
that $g$ is Einstein and self-dual or anti-self-dual, Lock and Viaclovsky [38]
compute the index of the ‘anti-self-dual deformation complex’. Using work of
Dai [25] and Dai-Zhang [27], we recover the Atiyah-LeBrun formula for the
signature (see Theorem 5.2 below) and show that our formula for the index of
the Dirac operator (4) simplifies substantially in this case.
###### Corollary 2.
If $\eth$ is a Dirac operator on a smooth four-dimensional manifold $X,$
associated to an incomplete edge metric with constant cone angle
$2\pi\beta\leq 2\pi$ along an embedded surface $Y,$ then its index is given by
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=-\frac{1}{24}\int_{M}p_{1}(M)+\frac{1}{24}(\beta^{2}-1)[Y]^{2},$
(5)
where $[Y]^{2}$ is the self-intersection number of $Y$ in $X$.
The formulas in (4) and (5), and indeed our proof, are obtained by taking the
limit of the index formula for the Dirac operators on the manifolds with
boundary
$M_{\varepsilon}=\\{x\geq\varepsilon\\},$
so in particular the contribution from the singular stratum $Y$ is the
adiabatic limit [54, 11, 25] of the $\eta$-invariant from the celebrated
classical theorem of Atiyah, Patodi, and Singer [7], which we review in
Section 5. It is important to note that the analogous statement for an
arbitrary Dirac-type operator is false and the general index formula requires
an extra contribution from the singularity. We will return to this in a
subsequent publication. We also point out to the reader that there exist other
derivations of index formulas on manifolds with structured ends in which the
computation is reduced to taking a limit in the Atiyah-Patodi-Singer index
formula; see for example [18] or [36].
One very interesting aspect of the spin Dirac operator is its close relation
to the existence of positive scalar curvature metrics. Most directly, the
Lichnerowicz formula shows that the index of the Dirac operator is an
obstruction to the existence of such a metric. This is still true among
metrics with incomplete edge singularities. Analogously to the results of Chou
for conic singularities [23] we prove the following theorem in §6.
###### Theorem 3.
Let $(M,g)$ be a spin space with an incomplete edge metric.
a) If the scalar curvature of $g$ is non-negative in a neighborhood of
$\partial M$ then the “geometric Witt assumption” (2) holds.
b) If the scalar curvature of $g$ is non-negative on all of $M,$ and positive
somewhere, then $\operatorname{Ind}(\eth)=0.$
Notice that the first part of this theorem, as indeed Theorem 1, show that our
geometric Witt assumption is a natural assumption on $\eth.$
Now let us indicate in more detail how these theorems are proved. For
convenience we work throughout with a product-type incomplete edge metric as
described above, but removing this assumption would only result in slightly
more intricate computations below. The proof of Theorem 1 follows the
arguments employed in [2, 3] to prove the analogous result for the signature
operator. Thus we start with the two canonical closed extensions of $\eth$
from $C^{\infty}_{comp}(M),$ namely
$\begin{split}\mathcal{D}_{max}&:=\left\\{u\in L^{2}(M;\mathcal{S}):\eth u\in
L^{2}(M;\mathcal{S})\right\\},\\\
\mathcal{D}_{min}&:=\left\\{u\in\mathcal{D}_{max}:\exists u_{k}\in
C^{\infty}_{comp}(M)\mbox{ with }u_{k}\to u,\eth u_{k}\to\eth u\mbox{ as
}k\to\infty\right\\},\end{split}$ (6)
where the convergence in the second definition is in $L^{2}(M;\mathcal{S}),$
and we show that under Assumption (2), these domains coincide
$\mathcal{D}_{min}=\mathcal{D}_{max}=\mathcal{D}.$ (7)
Since $\eth$ is a symmetric operator, this shows that it is essentially self-
adjoint.
One difference between the case of isolated conic singularities ($\dim Y=0$)
and the general incomplete edge case is that in the former, even if Assumption
(2) does not hold, $\mathcal{D}_{max}/\mathcal{D}_{min}$ is a finite
dimensional space. In contrast, when $\dim Y>0,$ this space is generally
infinite-dimensional.
We prove (7) by constructing a parametrix $\overline{Q}$ for $\eth$ in Section
2. From the mapping properties of $\overline{Q}$, we deduce both that $\eth$
is essentially self-adjoint, and that it is a Fredholm operator from the
domain of its unique self-adjoint extension to $L^{2}$. The relationship
between the mapping properties of $\overline{Q}$ and the stated conclusions
can be seen largely through (2.31) below, which states that the maximal domain
has ‘extra’ vanishing, i.e. sections in $\mathcal{D}_{max}$ lie in weighted
spaces $x^{\delta}L^{2}$ with weight $\delta$ higher than generically
expected. This shows that the inclusion of the domain into $L^{2}$ is a
compact operator, which in particular gives that the kernel of $\eth$ on the
maximal domain is finite dimensional.
$M$$M_{\varepsilon}$$x$$\partial M$$Z$ Figure 2. $M$ as a smooth manifold with
boundary whose boundary $\partial M$ is a fiber bundle. Here $x$ is a boundary
defining function and the space $M_{\varepsilon}$ are given by
$M_{\varepsilon}=\left\\{x\geq\varepsilon\right\\}.$
Once this is established we give a precise description of the Schwartz kernel
of the generalized inverse $Q$ of $\eth$ using the technology of [39, 41]. In
Section 3, we use $Q$ and standard methods from layer potentials to construct
a family of pseudodifferential projectors
$\mathcal{E}_{\varepsilon}\in\Psi^{0}(\partial M_{\varepsilon};\mathcal{S})$
such that
$\eth\big{\rvert}_{M_{\varepsilon}}\text{ with domain
}\mathcal{D}_{\varepsilon}=\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):(Id-\mathcal{E}_{\varepsilon})(u\rvert_{\partial
M_{\varepsilon}})=0\right\\}.$
is Fredholm and has the same index as $(\eth,\mathcal{D}).$ This domain is
constructed so that the boundary values coincide with boundary values of
‘$\eth-$harmonic’ $L^{2}$-sections over the excised neighborhood of the
singularity, $M-M_{\varepsilon}.$
To compute this index, we consider the operators
$\eth\big{\rvert}_{M_{\varepsilon}}\text{ with domain
}\mathcal{D}_{APS,\varepsilon}=\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):(Id-\pi_{APS,\varepsilon})(u\rvert_{\partial
M_{\varepsilon}})=0\right\\}.$
where $\pi_{APS,\varepsilon},$ is the projection onto the positive spectrum of
$\eth\big{\rvert}_{\partial M_{\varepsilon}}.$ From [7] we know that these are
Fredholm operators and
$\operatorname{Ind}(\eth,\mathcal{D}^{+}_{APS,\varepsilon}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=\int_{M}\widehat{A}(M)-\tfrac{1}{2}\eta(\eth_{\partial
M_{\varepsilon}})+\mathcal{L}_{\varepsilon},$ (8)
where $\mathcal{L}_{\varepsilon}$ is a local integral over $\partial
M_{\varepsilon}$ compensating for the fact that the metric is not of product-
type at $\partial M_{\varepsilon}.$ These domains depend fundamentally on
$\varepsilon.$ Not only does $\mathcal{D}_{APS,\varepsilon}$ vary as
$\varepsilon\to 0$, it does not limit to a fixed subspace of $L^{2}(\partial
M)$ with any natural metric. (More precisely, the boundary value projectors
$\pi_{APS,\varepsilon}$ which define the boundary condition do not converge in
norm.)
Through a semiclassical analysis, which we carry out using the adiabatic
calculus of Mazzeo-Melrose [40], we show that the projections
$\mathcal{E}_{\varepsilon}$ and $\pi_{APS,\varepsilon}$ are homotopic for
small enough $\varepsilon,$ with a homotopy through operators with the same
principal symbol. The adiabatic calculus technology boils this down to an
explicit analysis of modified Bessel functions, which we carry out in the
appendix. Then we can appeal to arguments from Booss-Bavnbek-Wojciechowski
[16] to see that the two boundary value problems have the same index. Having
shown that the index of $(\eth,\mathcal{D})$ is equal to the adiabatic limit
of the index formula (8), the Main Theorem follows as shown in Section 5.
Acknowledgements. P. A. was supported by NSF grant DMS-1104533 and Simons
Foundation grant #317883. The authors are happy to thank Rafe Mazzeo and
Richard Melrose for many useful and interesting discussions.
###### Contents
1. 1 Connection and Dirac operator
1. 1.1 Incomplete edge metrics and their connections
2. 1.2 Clifford bundles and Clifford actions
3. 1.3 The APS boundary projection
2. 2 Mapping properties of $\eth$
1. 2.1 The “geometric Witt condition”
2. 2.2 Review of edge and incomplete edge operators
3. 2.3 Parametrix of $x\eth$ on weighted edge spaces
4. 2.4 Proof of Theorem 1 and the generalized inverse of $\eth$
3. 3 Boundary values and boundary value projectors
1. 3.1 Boundary value projector for $\mathcal{D}_{\varepsilon}$
4. 4 Equivalence of indices
1. 4.1 Review of the adiabatic calculus
2. 4.2 APS projections as an adiabatic family
5. 5 Proof of Main Theorem: limit of the index formula
1. 5.1 Four-dimensions with circle fibers
6. 6 Positive scalar curvature metrics
7. 7 Appendix
## 1\. Connection and Dirac operator
Let $(M,g)$ be an incomplete edge space which is spin,
$\mathcal{S}\longrightarrow M$ the spinor bundle for a fixed spin structure
with connection $\nabla$, and let $\eth$ be the corresponding Dirac operator.
Given an orthonormal frame $e_{i}$ of the tangent bundle of $M$, the Dirac
operator satisfies
$\eth=\sum_{i}c(e_{i})\nabla_{e_{i}},$ (1.1)
where $c(v)$ denotes Clifford multiplication by the vector $v$. See [50, 35]
for background on spinor bundles and Dirac operators. The main goal of this
section is to prove Lemma 1.1 below, where we produce a tractable form of the
Dirac operator on a collar neighborhood of the boundary $\partial M$, or
equivalently of the singular stratum $Y\subset X$.
### 1.1. Incomplete edge metrics and their connections
Let $M$ be the interior of a compact manifold with boundary. Assume that
$\partial M=N$ participates in a fiber bundle
$Z\operatorname{---}N\xrightarrow{\phantom{x}\phi\phantom{x}}Y.$
Let $X$ be the singular space obtained from $M$ by collapsing the fibers of
the fibration $\phi.$ If we want to understand the differential forms on $X$
while working on $M,$ it is natural to restrict our attention to
$\\{\omega\in{\mathcal{C}}^{\infty}(\overline{M};T^{*}\overline{M}):i_{N}^{*}\omega\in\phi^{*}{\mathcal{C}}^{\infty}(Y;T^{*}Y)\\}.$
(1.2)
Following Melrose’s approach to analysis on singular spaces [46] let
$T_{\operatorname{ie}}^{*}M\longrightarrow\overline{M}$ (1.3)
be the vector bundle whose space of sections is (1.2). We call
$T_{\operatorname{ie}}^{*}M$ the ‘incomplete edge cotangent bundle’, and its
dual bundle $T_{\operatorname{ie}}M,$ the ‘incomplete edge tangent bundle’.
(Note that $T_{\operatorname{ie}}M$ is simply a rescaled bundle of the
(complete) ‘edge tangent bundle’ of Mazzeo [39].) The incomplete edge tangent
bundle is, over $M,$ canonically isomorphic to $TM,$ but its extension to
$\overline{M}$ is not canonically isomorphic to $T\overline{M}$ (though they
are of course isomorphic bundles).
Let $x$ be a boundary defining function on $M,$ meaning a smooth non-negative
function $x\in{\mathcal{C}}^{\infty}(\overline{M};[0,\infty))$ such that
$\\{x=0\\}=N$ and $|dx|$ has no zeroes on $N.$ We will typically work in local
coordinates
$x,\quad y,\quad z$ (1.4)
where $y$ are coordinates along $Y$ and $z$ are coordinates along $Z.$ In
local coordinates the sections of $T_{\operatorname{ie}}^{*}M$ are spanned by
$dx,\quad x\;dz,\quad dy$
where $dz$ denotes a $\phi$-vertical one form and $dy$ a $\phi$-horizontal one
form. The crucial fact is that $x\;dz$ vanishes at $N$ as a section of
$T^{*}\overline{M},$ but it does not vanish at $N$ as a section of
$T_{\operatorname{ie}}^{*}M$ because the ‘$x$’ is here part of the basis
element and not a coefficient. Similarly, in local coordinates the sections of
$T_{\operatorname{ie}}M$ are spanned by
$\partial_{x},\quad\tfrac{1}{x}\partial_{z},\quad\partial_{y},$
and, in contrast to $T\overline{M},$ the vector field
$\tfrac{1}{x}\partial_{z}$ is non-degenerate at $N$ as a section of
$T_{\operatorname{ie}}M.$
Next consider a metric on $M$ that reflects the collapse of the fibers of
$\phi.$ Let $\mathscr{C}$ be a collar neighborhood of $N$ in $M$ compatible
with $x,$ $\mathscr{C}\cong[0,1]_{x}\times N.$ Fix a splitting
$T\mathscr{C}=\langle\partial_{x}\rangle\oplus TN/Y\oplus\phi^{*}TY.$
A ‘product-type incomplete edge metric’ is a Riemannian metric on $M$ that,
for some choice of collar neighborhood and splitting, has the form
$g_{\operatorname{ie}}=dx^{2}+x^{2}g_{Z}+\phi^{*}g_{Y}$ (1.5)
where $g_{Z}+\phi^{*}g_{Y}$ is a submersion metric for $\phi$ independent of
$x.$ Note that this metric naturally induces a bundle metric on
$T_{\operatorname{ie}}M$ with the advantage that it extends non-degenerately
to $\overline{M}.$ We will consider this as a metric on
$T_{\operatorname{ie}}M$ from now on. (A general incomplete edge metric is
simply a bundle metric on
$T_{\operatorname{ie}}M\longrightarrow\overline{M}.$)
To describe the asymptotics of the Levi-Civita connection of
$g_{\operatorname{ie}},$ let us start by recalling the behavior of the Levi-
Civita connection of a submersion metric. Endow $N=\partial M$ with a
submersion metric of the form $g_{N}=\phi^{*}g_{Y}+g_{Z}.$ Given a vector
field $U$ on $Y,$ let us denote its horizontal lift to $N$ by $\widetilde{U}.$
Also let us denote the projections onto each summand by
$\mathbf{h}:TN\longrightarrow\phi^{*}TY,\quad\mathbf{v}:TN\longrightarrow
TN/Y.$
The connection $\nabla^{N}$ differs from the connections $\nabla^{Y}$ on the
base and the connections $\nabla^{N/Y}$ on the fibers through two tensors. The
second fundamental form of the fibers is defined by
$\mathcal{S}^{\phi}:TN/Y\times
TN/Y\longrightarrow\phi^{*}TY,\quad\mathcal{S}^{\phi}(V_{1},V_{2})=\mathbf{h}(\nabla^{N/Y}_{V_{1}}V_{2})$
and the curvature of the fibration is defined by
$\mathcal{R}^{\phi}:\phi^{*}TY\times\phi^{*}TY\longrightarrow
TN/Y,\quad\mathcal{R}^{\phi}(\widetilde{U}_{1},\widetilde{U}_{2})=\mathbf{v}([\widetilde{U}_{1},\widetilde{U}_{2}]).$
The behavior of the Levi-Civita connection (cf. [33, Proposition 13]) is then
summed up in the table:
$g_{N}\left(\nabla^{N}_{W_{1}}W_{2},W_{3}\right)$ | $V_{0}$ | $\widetilde{U}_{0}$
---|---|---
$\nabla^{N}_{V_{1}}V_{2}$ | $g_{N/Y}\left(\nabla^{N/Y}_{V_{1}}V_{2},V_{0}\right)$ | $\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V_{1},V_{2}),\widetilde{U}_{0})$
$\nabla^{N}_{\widetilde{U}}V$ | $g_{N/Y}\left([\widetilde{U},V],V_{0}\right)-\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$ | $-\frac{1}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$\nabla^{N}_{V}\widetilde{U}$ | $-\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$ | $\frac{1}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$\nabla^{N}_{\widetilde{U}_{1}}\widetilde{U}_{2}$ | $\frac{1}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U}_{1},\widetilde{U}_{2}),V_{0})$ | $g_{Y}(\nabla^{Y}_{U_{1}}U_{2},U_{0})$
We want a similar description of the Levi-Civita connection of an incomplete
edge metric. The splitting of the tangent bundle of $\mathscr{C}$ induces a
splitting
$T_{\operatorname{ie}}\mathscr{C}=\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y\oplus\phi^{*}TY,$
(1.6)
in terms of which a convenient choice of vector fields is
$\partial_{x},\quad\tfrac{1}{x}V,\quad\widetilde{U}$
where $V$ denotes a vertical vector field at $\\{x=0\\}$ extended trivially to
$\mathscr{C}$ and $\widetilde{U}$ denotes a vector field on $Y,$ lifted to
$\partial M$ and then extended trivially to $\mathscr{C}.$ Note that, with
respect to $g_{\operatorname{ie}},$ these three types of vector fields are
orthogonal, and that their commutators satisfy
$\begin{gathered}\left[\partial_{x},\tfrac{1}{x}V\right]=-\tfrac{1}{x^{2}}V\in
x^{-1}{\mathcal{C}}^{\infty}(\mathscr{C},\tfrac{1}{x}T\partial
M/Y),\quad\left[\partial_{x},\widetilde{U}\right]=0,\\\
\left[\tfrac{1}{x}V_{1},\tfrac{1}{x}V_{2}\right]=\tfrac{1}{x^{2}}\left[V_{1},V_{2}\right]\in
x^{-1}{\mathcal{C}}^{\infty}(\mathscr{C},\tfrac{1}{x}T\partial
M/Y),\quad\left[\tfrac{1}{x}V,\widetilde{U}\right]=\tfrac{1}{x}\left[V,\widetilde{U}\right]\in{\mathcal{C}}^{\infty}(\mathscr{C},\tfrac{1}{x}T\partial
M/Y),\\\ \left[\widetilde{U}_{1},\widetilde{U}_{2}\right]\in
x{\mathcal{C}}^{\infty}(\mathscr{C},\tfrac{1}{x}T\partial
M/Y)+{\mathcal{C}}^{\infty}(\mathscr{C},\phi^{*}TY).\end{gathered}$
We define an operator $\nabla$ on sections of the $\operatorname{ie}$-bundle
through the Koszul formula for the Levi-Civita connection
$2g_{\operatorname{ie}}(\nabla_{W_{0}}W_{1},W_{2})=W_{0}g_{\operatorname{ie}}(W_{1},W_{2})+W_{1}g_{\operatorname{ie}}(W_{0},W_{2})-W_{2}g_{\operatorname{ie}}(W_{0},W_{1})\\\
+g_{\operatorname{ie}}(\left[W_{0},W_{1}\right],W_{2})-g_{\operatorname{ie}}(\left[W_{0},W_{2}\right],W_{1})-g_{\operatorname{ie}}(\left[W_{1},W_{2}\right],W_{0})$
where $W_{1}$ and $W_{2}$ are $\operatorname{ie}$-vector fields.
If $W_{0}\in\\{\partial_{x},V,\widetilde{U}\\}$ and
$W_{1},W_{2}\in\\{\partial_{x},\tfrac{1}{x}V,\widetilde{U}\\}$ then we find
$g_{\operatorname{ie}}(\nabla_{W_{0}}W_{1},W_{2})=0\text{ if
}\partial_{x}\in\\{W_{0},W_{1},W_{2}\\}\\\ \text{ except for
}g_{\operatorname{ie}}(\nabla_{V_{1}}\partial_{x},\tfrac{1}{x}V_{2})=-g_{\operatorname{ie}}(\nabla_{V_{1}}\tfrac{1}{x}V_{2},\partial_{x})=g_{Z}(V_{1},V_{2})$
and otherwise
$g_{\operatorname{ie}}\left(\nabla_{W_{1}}W_{2},W_{3}\right)$ | $\tfrac{1}{x}V_{0}$ | $\widetilde{U}_{0}$
---|---|---
$\nabla_{V_{1}}\tfrac{1}{x}V_{2}$ | $g_{N/Y}\left(\nabla^{N/Y}_{V_{1}}V_{2},V_{0}\right)$ | $x\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V_{1},V_{2}),\widetilde{U}_{0})$
$\nabla_{\widetilde{U}}\tfrac{1}{x}V$ | $g_{N/Y}\left([\widetilde{U},V],V_{0}\right)-\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$ | $-\frac{x}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$\nabla_{V}\widetilde{U}$ | $-x\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$ | $\frac{x^{2}}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$\nabla_{\widetilde{U}_{1}}\widetilde{U}_{2}$ | $\frac{x}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U}_{1},\widetilde{U}_{2}),V_{0})$ | $g_{Y}(\nabla^{Y}_{U_{1}}U_{2},U_{0})$
We point out a few consequences of these computations. First note that the
$\nabla:{\mathcal{C}}^{\infty}(M;T_{\operatorname{ie}}M)\longrightarrow{\mathcal{C}}^{\infty}(M;T^{*}M\otimes
T_{\operatorname{ie}}M)$
defines a connection on the incomplete edge tangent bundle. Also note that
this connection asymptotically preserves the splitting of
$T_{\operatorname{ie}}\mathscr{C}$ into two bundles
$T_{\operatorname{ie}}\mathscr{C}=\left[\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y\right]\oplus\phi^{*}TY$
(1.7)
in that if $W_{1},W_{2}\in\mathcal{V}_{\operatorname{ie}}$ are sections of the
two different summands then
$g_{\operatorname{ie}}(\nabla_{W_{0}}W_{1},W_{2})=\mathcal{O}(x)\text{ for all
}W_{0}\in{\mathcal{C}}^{\infty}(M;TM).$
In fact, let us denote the projections onto each summand of (1.7) by
$\mathbf{v}_{+}:T_{\operatorname{ie}}\mathscr{C}\longrightarrow\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y,\quad\mathbf{h}:T_{\operatorname{ie}}\mathscr{C}\longrightarrow\phi^{*}TY,$
and define connections
$\begin{gathered}\nabla^{v_{+}}=\mathbf{v}_{+}\circ\nabla\circ\mathbf{v}_{+}:{\mathcal{C}}^{\infty}(\mathscr{C};\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y)\longrightarrow{\mathcal{C}}^{\infty}(\mathscr{C};T^{*}\mathscr{C}\otimes\left(\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y\right))\\\
\nabla^{h}=\phi^{*}\nabla^{Y}:{\mathcal{C}}^{\infty}(\mathscr{C};\phi^{*}TY)\longrightarrow{\mathcal{C}}^{\infty}(\mathscr{C};T^{*}\mathscr{C}\otimes\phi^{*}TY).\end{gathered}$
Denote by
$j_{\varepsilon}:\\{x=\varepsilon\\}\hookrightarrow\mathscr{C}$ (1.8)
the inclusion, and identify $\\{x=\varepsilon\\}$ with $N=\\{x=0\\},$ note
that the pull-back connections $j_{\varepsilon}^{*}\nabla^{v_{+}}$ and
$j_{\varepsilon}^{*}\nabla^{h}$ are independent of $\varepsilon$ and
$j_{0}^{*}\nabla=j_{0}^{*}\nabla^{v_{+}}\oplus j_{0}^{*}\nabla^{h}.$ (1.9)
In terms of the local connection one-form $\omega$ and the splitting (1.7), we
have
$P^{V^{+}}\omega=\begin{pmatrix}\omega_{N/Y}&\mathcal{O}(x)\\\
\mathcal{O}(x)&\mathcal{O}(x^{2})\end{pmatrix},P^{H}\omega=\begin{pmatrix}\omega_{\mathcal{S}}&\mathcal{O}(x)\\\
\mathcal{O}(x)&\phi^{*}\omega_{Y}\end{pmatrix},\omega=\begin{pmatrix}\omega_{v_{+}}&\mathcal{O}(x)\\\
\mathcal{O}(x)&\phi^{*}\omega_{Y}+\mathcal{O}(x^{2})\end{pmatrix}$ (1.10)
where $P^{V^{+}}\omega$ is the projection onto the dual bundle of
$\left[\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y\right],$ $P^{H}\omega$
is the projection onto the dual bundle of $\phi^{*}TY,$ and the forms
$\omega_{N/Y},$ $\omega_{\mathcal{S}},$ $\omega_{Y},$ $\omega_{v+}$ are
defined by these equations. Finally, consider the curvature
$R_{\operatorname{ie}}$ of $\nabla.$ If
$W_{1},W_{2}\in\mathcal{V}_{\operatorname{ie}}$ are sections of two different
summands of (1.7) and
$W_{3},W_{4}\in{\mathcal{C}}^{\infty}(\mathscr{C},TN/Y\oplus\phi^{*}TY)$ then
$g_{\operatorname{ie}}(R_{\operatorname{ie}}(W_{3},W_{4})W_{1},W_{2})=\mathcal{O}(x),$
but
$g_{\operatorname{ie}}(R_{\operatorname{ie}}(\partial_{x},W_{4})W_{1},W_{2})=\frac{1}{x}g_{\operatorname{ie}}(\nabla_{W_{4}}W_{1},W_{2})=\mathcal{O}(1).$
We will be interested in the curvature along the level sets of $x.$
Schematically, if $\Omega$ denotes the
$\operatorname{End}(T_{\operatorname{ie}}M)$-valued two-form corresponding to
the curvature of $\nabla,$ then with respect to the splitting (1.7) we have
$\Omega\big{\rvert}_{x=\varepsilon}=\begin{pmatrix}\Omega_{v_{+}}&\mathcal{O}(\varepsilon)\\\
\mathcal{O}(\varepsilon)&\phi^{*}\Omega_{Y}\end{pmatrix}$
where $\Omega_{v_{+}}$ is the tangential curvature associated to
$\omega_{N/Y}+\omega_{\mathcal{S}}$ and $\Omega_{Y}$ is the curvature
associated to $\omega_{Y},$ and analogously to (1.9),
$j_{0}^{*}\Omega=j_{0}^{*}\Omega_{v_{+}}+\phi^{*}\Omega_{Y}.$ (1.11)
Following [12] and [33], it will be convenient to use the block-diagonal
connection $\widetilde{\nabla}$ on $T_{\operatorname{ie}}M$ from the splitting
(1.6). Thus
$\widetilde{\nabla}:{\mathcal{C}}^{\infty}(\mathscr{C},T_{\operatorname{ie}}\mathscr{C})\longrightarrow{\mathcal{C}}^{\infty}(\mathscr{C};T^{*}\mathscr{C}\otimes
T_{\operatorname{ie}}\mathscr{C})$ (1.12)
satisfies
$\begin{gathered}\widetilde{\nabla}\partial_{x}=0,\quad\widetilde{\nabla}_{\partial_{x}}=0,\text{
and }\\\
\begin{tabular}[]{|c||c|c|}\hline\cr$g_{\operatorname{ie}}\left(\widetilde{\nabla}_{W_{1}}W_{2},W_{3}\right)$&$\tfrac{1}{x}V_{0}$&$\widetilde{U}_{0}$\\\
\hline\cr\hline\cr$\widetilde{\nabla}_{V_{1}}\tfrac{1}{x}V_{2}$&$g_{N/Y}\left(\nabla^{N/Y}_{V_{1}}V_{2},V_{0}\right)$&$0$\\\
\hline\cr$\widetilde{\nabla}_{\widetilde{U}}\tfrac{1}{x}V$&$g_{N/Y}\left([\widetilde{U},V],V_{0}\right)-\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$&$0$\\\
\hline\cr$\widetilde{\nabla}_{V}\widetilde{U}$&$0$&$0$\\\
\hline\cr$\widetilde{\nabla}_{\widetilde{U}_{1}}\widetilde{U}_{2}$&$0$&$g_{Y}(\nabla^{Y}_{U_{1}}U_{2},U_{0})$\\\
\hline\cr\end{tabular}\end{gathered}$
The connection $\widetilde{\nabla}$ is a metric connection and preserves the
splitting (1.7).
### 1.2. Clifford bundles and Clifford actions
The incomplete edge Clifford bundle, denoted $Cl_{\operatorname{ie}}(M,g)$, is
the bundle obtained by taking the Clifford algebra of each fiber of
$T_{\operatorname{ie}}M$. Concretely,
$Cl_{\operatorname{ie}}(M,g)=\sum_{k=0}^{\infty}T_{\operatorname{ie}}M^{\otimes
k}/(x\otimes y-y\otimes x=-2\langle x,y\rangle_{g}).$ (1.13)
This is a smooth vector bundle on all of $\overline{M}$.
We assume that $M$ is spin and fix a spin bundle $\mathcal{S}\longrightarrow
M.$ Denote Clifford multiplication by
$c:{\mathcal{C}}^{\infty}(M,T_{\operatorname{ie}}M)\longrightarrow{\mathcal{C}}^{\infty}(M;\operatorname{End}(\mathcal{S})).$
We denote the connection induced on $\mathcal{S}$ by the Levi-Civita
connection $\nabla$ by the same symbol. Let $\eth$ denote the corresponding
Dirac operator.
###### Lemma 1.1.
Let $\mathscr{C}\cong[0,1)_{x}\times N$ be a collar neighborhood of the
boundary. Let
$\partial_{x},\quad\tfrac{1}{x}V_{\alpha},\quad\widetilde{U}_{i}$ (1.14)
denote a local orthonormal frame consistent with the splitting (1.6). In terms
of this frame and the connection $\widetilde{\nabla}$ from (1.12), the Dirac
operator $\eth$ decomposes as
$\begin{split}\eth&=c(\partial_{x})\partial_{x}+\frac{f}{2x}c(\partial_{x})+\frac{1}{x}\sum_{\alpha=1}^{f}c(\frac{1}{x}V_{\alpha})\widetilde{\nabla}_{V_{\alpha}}+\sum_{i=1}^{b}c(\widetilde{U}_{i})\widetilde{\nabla}_{\widetilde{U}_{i}}+B,\end{split}$
(1.15)
where $f=\dim Z,$ $b=\dim Y,$ and
$B\in{\mathcal{C}}^{\infty}(\overline{M},\operatorname{End}(\mathcal{S})),\quad\left\|B\right\|=O(1).$
(1.16)
###### Proof.
Consider the difference of connections (on the tangent bundle)
$A=\nabla-\widetilde{\nabla}\in{\mathcal{C}}^{\infty}(\mathscr{C};T^{*}M\otimes\operatorname{End}({}^{\operatorname{ie}}T\mathscr{C})).$
From [15] we have
$\eth=\sum_{i}c(e_{i})\left(\widetilde{\nabla}_{e_{i}}+\widetilde{A}(e_{i})\right),$
(1.17)
where
$\widetilde{A}(W):=\frac{1}{4}\sum_{jk}g_{\operatorname{ie}}(A(W)e_{j},e_{k})c(e_{j})c(e_{k})$.
From §1.1 we have
$g_{\operatorname{ie}}(A(W_{0})W_{1},W_{2})=0\text{ if
}\partial_{x}\in\\{W_{0},W_{1},W_{2}\\}\\\ \text{ except for
}g_{\operatorname{ie}}(A(\tfrac{1}{x}V_{\alpha})\partial_{x},\tfrac{1}{x}V_{\beta})=-g_{\operatorname{ie}}(\tfrac{1}{x}A(V_{\alpha})\tfrac{1}{x}V_{\beta},\partial_{x})=\frac{1}{x}g_{Z}(V_{\alpha},V_{\beta})$
and otherwise
$g_{\operatorname{ie}}\left(A(W_{1})W_{2},W_{3}\right)$ | $\tfrac{1}{x}V_{0}$ | $\widetilde{U}_{0}$
---|---|---
$A(\tfrac{1}{x}V_{1})\tfrac{1}{x}V_{2}$ | $0$ | $\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V_{1},V_{2}),\widetilde{U}_{0})$
$A(\widetilde{U})\tfrac{1}{x}V$ | $0$ | $-\frac{x}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$A(\tfrac{1}{x}V)\widetilde{U}$ | $-\phi^{*}g_{Y}(\mathcal{S}^{\phi}(V,V_{0}),\widetilde{U})$ | $\frac{x}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U},\widetilde{U}_{0}),V)$
$A(\widetilde{U}_{1})\widetilde{U}_{2}$ | $\frac{x}{2}g_{N/Y}(\mathcal{R}^{\phi}(\widetilde{U}_{1},\widetilde{U}_{2}),V_{0})$ | $0$
Hence
$\tfrac{1}{4}\sum_{s,t,u}g_{\operatorname{ie}}(A(e_{s})e_{t},e_{u})c(e_{s})c(e_{t})c(e_{u})$
has terms of order $\mathcal{O}(\tfrac{1}{x}),$ $\mathcal{O}(1),$ and
$\mathcal{O}(x).$ The terms of order $\tfrac{1}{x}$ are
$\frac{1}{4}\sum_{\alpha}\left(g_{\operatorname{ie}}(A(\tfrac{1}{x}V_{\alpha})\partial_{x},\tfrac{1}{x}V_{\alpha})c(\partial_{x})+g_{\operatorname{ie}}(A(\tfrac{1}{x}V_{\alpha})\tfrac{1}{x}V_{\alpha},\partial_{x})(-c(\partial_{x}))\right)\\\
=\frac{1}{4}\sum_{\alpha}2\frac{c(\partial_{x})}{x}=\frac{f}{2x}c(\partial_{x}),$
which establishes (1.15). ∎
### 1.3. The APS boundary projection
We now define the APS boundary condition discussed in the introduction. We
will make use of a simplified coordinate system near the boundary of $M$,
namely, let $(x,x^{\prime})$ be coordinates near a point on $\partial M$ for
which $x^{\prime}\in\mathbb{R}^{n-1}$ are coordinates on $\partial M$ and $x$
is the same fixed boundary defining function used in (1.4). For the cutoff
manifold $M_{\varepsilon}=\\{x\geq\varepsilon\\},$ consider the differential
operator on sections of $\mathcal{S}$ over $\partial M_{\varepsilon}$ defined
by choosing any orthonormal frame $e_{p}$, $p=1,\dots n-1$ of the distribution
of the tangent bundle orthogonal to $\partial_{x}$ and setting
$\begin{split}\frac{1}{\varepsilon}\widetilde{\eth}_{\varepsilon}&:=\left.-c(\partial_{x})\left(\sum_{p=1}^{n-1}c(e_{p})\widetilde{\nabla}_{e_{p}}\right)\right|_{x=\varepsilon},\end{split}$
where $\widetilde{\nabla}$ is the connection from (1.12). The operator
$\widetilde{\eth}_{\varepsilon}$ is defined independently of the choice of
frame, so we may take frames as in (1.14) to obtain
$\begin{split}\widetilde{\eth}_{\varepsilon}&=\left.-xc(\partial_{x})\left(\sum_{\alpha=1}^{f}c(\frac{1}{x}V_{\alpha})\widetilde{\nabla}_{\frac{1}{x}V_{\alpha}}+\sum_{j=1}^{b}c(\widetilde{U}_{j})\widetilde{\nabla}_{\widetilde{U}_{j}}\right)\right|_{x=\varepsilon}.\end{split}$
(1.18)
We refer to $\widetilde{\eth}_{\varepsilon}$ below as the tangential operator,
since for every $\varepsilon$ it acts tangentially along the boundary
$\partial M_{\varepsilon}$. The operator $\widetilde{\eth}_{\varepsilon}$ is
self-adjoint on $L^{2}(\partial M_{\varepsilon},\mathcal{S}).$
We denote the dual coordinates on $T^{*}_{x,x^{\prime}}M$ by
$(\xi,\xi^{\prime})$. Using the identification of $T^{*}M$ with $TM$ induced
by the metric $g$, the principal symbol of $\eth$ is given by
$\sigma(\eth)(x,x^{\prime})=i\xi
c(\partial_{x})+ic(\xi^{\prime}\cdot\partial_{x^{\prime}}),$ (1.19)
where
$\xi^{\prime}\cdot\partial_{x^{\prime}}=\sum_{i=1}^{n-1}\xi^{\prime}_{j}\partial_{x^{\prime}_{j}}$.
Note that using coordinates as in (1.19), for $x=\varepsilon$,
$\widetilde{\eth}_{\varepsilon}$ has principal symbol
$\sigma(\widetilde{\eth}_{\varepsilon})(x^{\prime},\xi^{\prime})=-i\varepsilon
c(\partial_{x})c(\xi^{\prime}\cdot\partial_{x^{\prime}}).$
Since
$\sigma(\widetilde{\eth}_{\varepsilon})(x^{\prime},\xi^{\prime})^{2}=\varepsilon^{2}\left\lvert(0,\xi^{\prime})\right\rvert_{g}^{2}$,
if we define
$\widehat{\xi}^{\prime}=\xi^{\prime}/\left\lvert(0,\xi^{\prime})\right\rvert_{g}$,
then
$\begin{split}\sigma(\widetilde{\eth}_{\varepsilon})(x^{\prime},\xi^{\prime})&=\left\lvert(0,\xi^{\prime})\right\rvert_{g}\sigma(\widetilde{\eth}_{\varepsilon})(x^{\prime},\widehat{\xi}^{\prime})=\varepsilon\left\lvert(0,\xi^{\prime})\right\rvert_{g}(\pi_{\varepsilon,+,\widehat{\xi}^{\prime}}(x^{\prime})-\pi_{\varepsilon,-,\widehat{\xi}^{\prime}}(x^{\prime}))\end{split}$
(1.20)
where $\pi_{\varepsilon,\pm,\widehat{\xi}^{\prime}}(x^{\prime})$ are
orthogonal projections onto $\pm$ eigenspaces of
$\sigma(\widetilde{\eth}_{\varepsilon})(x^{\prime},\widehat{\xi^{\prime}})$.
We will define a boundary condition for $\eth$ on the cutoff manifolds
$M_{\varepsilon},$
$\begin{split}\pi_{APS,\varepsilon}&:=L^{2}\mbox{ orthogonal projection onto
}V_{-,\varepsilon},\end{split}$ (1.21)
where $V_{-,\varepsilon}$ is the direct sum of eigenspaces of
$\widetilde{\eth}_{\varepsilon}$ with negative eigenvalues. We recall basic
facts about $\pi_{APS,\varepsilon}$.
###### Theorem 1.2 ([7, 51]).
For fixed $\varepsilon$, the operator $\pi_{APS,\varepsilon}$ is a
pseudodifferential operator of order $0$, i.e.
$\pi_{APS,\varepsilon}\in\Psi^{0}(\partial M_{\varepsilon};\mathcal{S})$. Its
principal symbol satisfies
$\sigma(\pi_{APS,\varepsilon})(x^{\prime},\xi^{\prime})=\pi_{\varepsilon,-,\widehat{\xi}^{\prime}}(x^{\prime}),$
where $\pi_{\varepsilon,-,\widehat{\xi^{\prime}}}(x^{\prime})$ is projection
onto the negative eigenspace of
$-ic(\partial_{x})c(\widehat{\xi}^{\prime}\cdot\partial_{x^{\prime}})$ from
(1.20).
Consider the domains for $\eth$ on $L^{2}(M_{\varepsilon};\mathcal{S})$
defined as follows
$\begin{split}\mathcal{D}_{APS,\varepsilon}&:=\left\\{u\in
H^{1}(M_{\varepsilon},\mathcal{S}):(Id-\pi_{APS,\varepsilon})u=0\right\\}\\\
\mathcal{D}^{+}_{APS,\varepsilon}&:=\left\\{u\in\mathcal{D}_{APS,\varepsilon}:\mbox{image}(u)\subset\mathcal{S}^{+}\right\\}\end{split}$
(1.22)
In one of the main results of this paper, we will show that a Dirac operator
satisfying the geometric Witt assumption (2) has a unique closed extension.
Denoting this domain by $\mathcal{D}$ and its restriction to positive spinors
by $\mathcal{D}^{+}$ we will show that, for $\varepsilon>0$ sufficiently
small,
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}_{APS,\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))=\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-})).$
Indeed, this will follow from Theorems 3.1 and 4.1 below.
## 2\. Mapping properties of $\eth$
In this section we will use the results and techniques in [2, 3] to prove
Theorem 1. We proceed by constructing a parametrix for $\eth$ and analyzing
the mapping properties of this parametrix.
Let $\mathcal{D}$ denote the domain of the unique self-adjoint extention of
$\eth$. At the end of this section, we analyze the structure of the
generalized inverse $Q$ for $\eth$, that is, the map
$\begin{split}Q&\colon
L^{2}(M;\mathcal{S})\longrightarrow\mathcal{D}\quad\mbox{ satisfying }\\\ \eth
Q&=Id-\pi_{ker}\mbox{ and }Q=Q^{*},\end{split}$
where $\pi_{ker}$ is $L^{2}-$orthogonal projection onto the kernel of $\eth$.
Here the adjoint $Q^{*}$ is taken with respect to the pairing defined for
sections $\phi,\psi$ by
$\langle\phi,\psi\rangle_{L^{2}}=\int\langle\phi,\psi\rangle_{G}\;dVol_{g},$
where $G$ is the Hermitian inner product on $\mathcal{S}$.
### 2.1. The “geometric Witt condition”
The proof of Theorem 1 relies on an assumption on an induced family of Dirac
operators on the fiber $Z$ which we describe now. By Lemma 1.1, on a collar
neighborhood of the boundary,
$\mathcal{U}\subset M\mbox{ with
}\mathcal{U}\simeq[0,\varepsilon_{0})_{x}\times\partial M$ (2.1)
we can write
$\eth=c(\partial_{x})\left(\partial_{x}+\frac{f}{2x}+\frac{1}{x}\eth^{Z}_{y}-\sum_{i=1}^{b}c(\partial_{x})c(\widetilde{U}_{i})\widetilde{\nabla}_{\widetilde{U}_{i}}\right)+B$
(2.2)
with $\left\|B\right\|=O(1)$ and where, for $y$ in the base $Y$
$\eth^{Z}_{y}=-c(\partial_{x})\cdot\sum_{\alpha=1}^{f}c(\frac{1}{x}V_{\alpha})\cdot\widetilde{\nabla}_{V_{\alpha}}$
(2.3)
The operator $\eth^{Z}_{y}$ defines a self-adjoint operator on the fiber over
$y\in Y$ in the boundary fibration
$N\xrightarrow{\phantom{x}\phi\phantom{x}}Y$ acting on sections of the
restriction of the spin bundle $\mathcal{S}_{y}.$
We will assume the following “geometric Witt condition” discussed in the
introduction.
###### Assumption 2.1.
The fiber operator $\eth^{Z}_{y}$ in (2.3) satisfies
$(-1/2,1/2)\cap\operatorname{spec}(\eth^{Z}_{y})=\varnothing\mbox{ for all
}y.$ (2.4)
### 2.2. Review of edge and incomplete edge operators
A vector field on $\overline{M}$ is an ‘edge vector field’ if its restriction
to $N=\partial M$ is tangent to the fibers of $\phi$ [39]. A differential
operator is an edge differential operator if in every coordinate chart it can
be written as a polynomial in edge vector fields. Thus if $E$ and $F$ are
vector bundles over $M,$ we say that $P^{\prime}$ is an $m^{\text{th}}$ order
edge differential operator between sections of $E$ and $F,$ denoted
$P^{\prime}\in\operatorname{Diff}_{e}^{m}(M;E,F),$ if in local coordinates we
have
$P^{\prime}=\sum_{j+|\alpha|+|\gamma|\leq
m}a_{j,\alpha,\gamma}(x,y,z)(x\partial_{x})^{j}(x\partial_{y})^{\alpha}(\partial_{z})^{\gamma}$
where $\alpha$ denotes a multi-index $(\alpha_{1},\ldots,\alpha_{b})$ with
$|\alpha|=\alpha_{1}+\ldots+\alpha_{b}$ and similarly for
$\gamma=(\gamma_{1},\ldots,\gamma_{f}),$ and each $a_{j,\alpha,\gamma}(x,y,z)$
is a local section of $\hom(E,F).$
A differential operator $P$ is an ‘incomplete edge differential operator’ of
order $m$ if $P^{\prime}=x^{m}P$ is an edge differential operator of order
$m.$ Thus, symbolically,
$\operatorname{Diff}_{\operatorname{ie}}^{m}(M;E,F)=x^{-m}\operatorname{Diff}_{e}^{m}(M;E,F),$
(2.5)
and in local coordinates
$P=x^{-m}\sum_{j+|\alpha|+|\gamma|\leq
m}a_{j,\alpha,\gamma}(x,y,z)(x\partial_{x})^{j}(x\partial_{y})^{\alpha}(\partial_{z})^{\gamma}.$
The (incomplete edge) principal symbol of $P$ is defined on the incomplete
edge cotangent bundle,
$\sigma(P)\in{\mathcal{C}}^{\infty}(T_{\operatorname{ie}}^{*}M;\pi^{*}\hom(E,F)),$
where $\pi:T_{\operatorname{ie}}^{*}M\longrightarrow M$ denotes the bundle
projection. In local coordinates it is given by
$\sigma(P)(x,y,z,\xi,\eta,\zeta):=\sum_{j+|\alpha|+|\gamma|=m}a_{j,\alpha,\gamma}(x,y,z)(\xi)^{j}(\eta)^{\alpha}(\zeta)^{\gamma}.$
We say that $P$ is elliptic if this symbol is invertible whenever
$(\xi,\eta,\zeta)\neq 0.$
###### Remark 2.2.
Clearly ellipticity of the incomplete edge symbol is equivalent to ellipticity
of the usual principal symbol of a differential operator. The advantage of the
former is that it will be uniformly elliptic over $\overline{M},$ whereas the
latter typically will not.
###### Lemma 2.3.
The Dirac operator $\eth$ on an incomplete edge space is an elliptic
incomplete edge differential operator of order $1$, i.e. is an elliptic
element of $\operatorname{Diff}^{1}_{\operatorname{ie}}(M;\mathcal{S})$. In
particular, $x\eth$ is an elliptic element of
$\operatorname{Diff}^{1}_{e}(M;\mathcal{S})$.
###### Proof.
This follows from equation (1.15) in Lemma 1.1. ∎
### 2.3. Parametrix of $x\eth$ on weighted edge spaces
Lemma 2.3 shows that $x\eth$ is an elliptic edge operator. By the theory of
edge operators [39], this implies that $x\eth$ is a bounded operator between
appropriate weighted Sobolev spaces, whose definition we now recall.
Let $\mathcal{D}^{\prime}(M;\mathcal{S})$ denote distributional sections.
Given $k\in\mathbb{N}$, let
$H^{k}_{e}(M;\mathcal{S}):=\left\\{u\in\mathcal{D}^{\prime}(M;\mathcal{S}):A^{1}\cdots
A^{j}u\in L^{2}(M;\mathcal{S})\mbox{ for }j\leq k\mbox{ and
}A^{i}\in\operatorname{Diff}^{1}_{e}(M;\mathcal{S})\right\\}.$
In particular, $u\in H^{1}_{e}(M;\mathcal{S})$ if and only if, for any edge
vector field $V\in C^{\infty}(M;T_{e}M)$, $\nabla_{V}u\in
L^{2}(M;\mathcal{S})$. The weighted edge Sobolev spaces are defined by
$x^{\delta}H^{k}_{e}(M;\mathcal{S}):=\left\\{u:x^{-\delta}u\in
H^{k}_{e}(M;\mathcal{S})\right\\}.$
Thus, the map
$x\eth\colon x^{\delta}H^{k}_{e}(M;\mathcal{S})\longrightarrow
x^{\delta}H^{k-1}(M;\mathcal{S})$ (2.6)
is bounded for all $\delta\in\mathbb{R},k\in\mathbb{N}$, in fact for
$k\in\mathbb{R}$ by interpolation. We will prove the following
###### Proposition 2.4.
Under the Witt assumption (Assumption 2.1), the map (2.6) is Fredholm for
$0<\delta<1$.
###### Remark 2.5.
Recall that the space $L^{2}(M;\mathcal{S})$ used to define the
$x^{\delta}H^{k}(M;\mathcal{S})$ is equipped with the inner product from the
Hermitian metric on $\mathcal{S}$ and the volume form of the incomplete edge
metric $g$.
In fact we will need more than Proposition 2.4; the proof of the Main Theorem
requires a detailed understanding of the structure of parametrices for
$x\eth$. To understand these, we must recall of edge double space $M^{2}_{e}$,
depicted heuristically in Figure 3 below, and edge pseudodifferential
operators, defined in [39] with background material in [44]. The edge double
space $M^{2}_{e}$ is a manifold with corners, obtained by radial blowup of
$M\times M$, namely $M^{2}_{e}:=[M\times M;\operatorname{diag}_{fib}(\partial
M\times\partial M)]$, where the notation is that in [44]. Here
$\operatorname{diag}_{fib}(\partial M\times\partial M)$ denotes the fiber
diagonal
$\operatorname{diag}_{fib}(N\times N)=\left\\{(p,q)\in N\times
N:\phi(p)=\phi(q)\right\\},$ (2.7)
where $\phi:N\longrightarrow Y$ is fiber bundle projection onto $Y.$ Whereas
$M\times M$ is a manifold with corners with two boundary hypersurfaces,
$M^{2}_{e}$ has a third boundary hypersurface introduced by the blowup.
Let $\operatorname{ff}$ be the boundary hypersurface of $M^{2}_{e}$ introduced
by the blowup.
Furthermore we have a blowdown map.
$\beta\colon M^{2}_{e}\longrightarrow M\times M,$
which is a $b-$map in the sense of [44] and a diffeomorphism from the interior
of $M^{2}_{e}$ to that of $M\times M$. The two boundary hypersurfaces of
$M\times M$, $\left\\{x=0\right\\}$ and $\left\\{x^{\prime}=0\right\\}$, lift
to boundary hypersurfaces of $M^{2}_{e}$ which we denote by
$\operatorname{lf}:=\beta^{-1}(\left\\{x=0\right\\}^{int})\mbox{ and
}\operatorname{rf}:=\beta^{-1}(\left\\{x^{\prime}=0\right\\}^{int})$
The edge front face, $\operatorname{ff}$, is the radial compactification of
the total space of a fiber bundle
$Z^{2}\times\mathbb{R}^{b}\times\mathbb{R}_{+}\operatorname{---}\operatorname{ff}\longrightarrow
Y,$
where $b=\dim Y$. This bundle is the fiber product of two copies of $\partial
M$ and the tangent bundle $TY$. Choosing local coordinates $(x,y,z)$ as in
(1.4), and our fixed bdf $x$, and letting
$(x,y,z,x^{\prime},y^{\prime},z^{\prime})$ denote coordinates on $M\times M$,
the functions
$\begin{split}x^{\prime},\quad\sigma=\frac{x}{x^{\prime}},\quad\mathcal{Y}=\frac{y-y^{\prime}}{x^{\prime}},\quad
y^{\prime},\quad z,\quad z^{\prime}.\end{split}$ (2.8)
define coordinates near $\operatorname{ff}$ in the set $0\leq\sigma<\infty$,
and in these coordinates $x^{\prime}$ is a boundary defining function for
$\operatorname{ff}$, meaning that $\left\\{x^{\prime}=0\right\\}$ coincides
with $\operatorname{ff}$ on $0\leq\sigma<\infty$, and $x^{\prime}$ has non-
vanishing differential on $\operatorname{ff}$. When $x^{\prime}=0$, $\sigma$
gives coordinates on the $\mathbb{R}_{+}$ fiber, $\mathcal{Y}$ on the
$\mathbb{R}^{b}$ fiber, and $z,z^{\prime}$ on the $Z^{2}$ fiber. Below we will
also use polar coordinates near $\operatorname{ff}$. These have the advantage
that they are defined on open neighborhoods of sets in $\operatorname{ff}$
which lie over open sets $V$ in the base $Y$. With $(x,y,z)$ as above, let
$\begin{split}\rho&=(x^{2}+(x^{\prime})^{2}+\left\lvert
y-y^{\prime}\right\rvert^{2})^{1/2}\\\
\phi&=\left(\frac{x}{\rho},\frac{x^{\prime}}{\rho},\frac{y-y^{\prime}}{\rho}\right),\end{split}$
so $(\rho,\phi,y^{\prime},z,z^{\prime})$ form polar coordinates (in the sense
that $\left\lvert\phi\right\rvert^{2}=1$) near the lift of $V$ to
$\operatorname{ff}$ and in the domain of validity of $y,z$.
$\operatorname{lf}$$\operatorname{rf}$$\Delta_{e}$$\operatorname{ff}$$M^{2}_{e}$$\operatorname{ff}_{y}$$\mathcal{Y}=0$$Z^{2}$$\mathcal{Y}$$\times[0,\infty]_{\sigma}$
Figure 3.
We now define the calculus of edge pseudodifferential operators with bounds,
which is similar to the large calculus of pseudodifferential edge operators
defined in [39]. Thus, $\Psi^{m}_{e,bnd}(M;\mathcal{S})$ will denote the set
of operators $A$ mapping $C^{\infty}_{comp}(M;\mathcal{S})$ to distributional
sections $\mathcal{D}^{\prime}(M;\mathcal{S})$, whose Schwartz kernels have
the following structure. Let $\operatorname{End}(\mathcal{S})$ denote the
bundle over $M\times M$ whose fiber at $(p,q)$ is
$\operatorname{End}(\mathcal{S}_{q};\mathcal{S}_{p})$. The Schwartz kernel of
$A$, $K_{A}$ is a distributional section of the bundle
$\operatorname{End}(\mathcal{S})$ over $M\times M$ satisfying that for a
section $\phi\in C^{\infty}_{comp}(M;\mathcal{S})$,
$A\phi(w)=\int_{M}K_{A}(w,w^{\prime})\phi(w^{\prime})dVol_{g}(w^{\prime}),$
(2.9)
where $dVol_{g}$ is the volume form of an incomplete edge metric $g$
asymptotically of the form (1.5). Moreover,
$K_{A}\in\mathcal{A}_{a,b,f}I^{m}(M^{2}_{e},\Delta_{e};\beta^{*}\operatorname{End}(\mathcal{S})),$
meaning that $K_{A}=K_{1}+K_{2}$ where $\rho^{f}K_{1}$ is in the Hörmander
conormal space [34, Chap. 18]
$\rho^{f}K_{1}\in
I^{m}(M^{2}_{e},\Delta_{e};\beta^{*}\operatorname{End}(\mathcal{S})),$
$K_{1}$ is supported near $\Delta_{e}$, and $K_{2}\in
C^{\infty}(M^{int};\operatorname{End}(\mathcal{S}))$ satisfies the bounds
$\begin{split}K_{2}(p)&=O(\rho_{\operatorname{lf}}^{a})\mbox{ as
}p\to\operatorname{lf}\\\ K_{2}(p)&=O(\rho_{\operatorname{rf}}^{b})\mbox{ as
}p\to\operatorname{rf}\end{split}$ (2.10)
where $\rho_{\operatorname{lf}},\rho_{\operatorname{rf}},$ and
$\rho_{\operatorname{ff}}$ are boundary defining functions for
$\operatorname{lf},\operatorname{rf},$ and $\operatorname{ff}$ respectively,
and the bound is in the norm on $\operatorname{End}(\mathcal{S})$ over
$M\times M,$ see [39] for details. Since the bounds $a$ and $b$ in (2.10) will
be of some importance, we let
$\Psi^{m}_{e}(M;\mathcal{S};a,b)$ (2.11)
denote the subspace of $\Psi^{m}_{e,bnd}(M;\mathcal{S})$ of pseudodifferential
edge operators whose Schwartz kernels satisfy (2.10) with bounds $a$ and $b$.
The bounds in (2.10) determine the mapping properties of $A$ on weighted
Sobolev spaces. From [39, Theorem 3.25], we have
###### Theorem 2.6.
An element $A\in\Psi^{m}_{e}(M;\mathcal{S};a,b)$ is bounded as a map
$A\colon x^{\delta}H^{k}_{e}(M;\mathcal{S})\longrightarrow
x^{\delta^{\prime}}H^{k-m}(M;\mathcal{S}),$
if and only if
$a>\delta^{\prime}-f/2-1/2\quad\mbox{ and }\quad b>-\delta-f/2-1/2.$
###### Remark 2.7.
In Mazzeo’s paper [39] the convention used to describe the weights (orders of
vanishing) of the Schwartz kernels of elements in $\Psi^{m}_{e}$ is slightly
different from ours. There one chooses a half-density $\mu$ on $M$ which looks
like $\sqrt{dxdydz}$ near $\partial M$. The choice of $\mu$ gives an
isomorphism between the sections of $\mathcal{S}$ and the sections of
$\mathcal{S}\otimes\Omega^{1/2}(M)$ where $\Omega^{1/2}(M)$ is the half-
density bundle of $M$ (simply by multiplying by $\mu$), and the Schwartz
kernel of an edge pseudodifferential operator, $A$, in this context is the
section $\kappa_{A}$ of
$\operatorname{End}(\mathcal{S})\otimes\Omega^{1/2}(M^{2}_{e})$ with the
property that
$A(\psi\mu)=\int_{M}\kappa_{A}\psi\mu.$ (2.12)
One nice feature of (2.12) is that $\kappa_{A}$ is smooth (away from the
diagonal) down to $\operatorname{ff}$. With our convention in (2.9), it is
singular of order $-f$ due to the factors of $x$ in the volume form of $g.$
Given an elliptic edge operator
$\widetilde{P}\in\operatorname{Diff}_{e}^{m}(M;\mathcal{S})$, to construct a
parametrix for $\widetilde{P}$ one must study two models for $\widetilde{P}$,
the indicial family $I_{y}(\widetilde{P},\zeta)$ and the normal operator
$N(\widetilde{P})_{y}$.
First we discuss the indicial operator. For each $y$ in the base $Y$, the
indicial family $I_{y}(\widetilde{P},\zeta)$ is an elliptic operator-valued
function on $\mathbb{C}$ obtained by taking the Mellin transform (see [39,
Sect. 2]) of the leading order part of $\widetilde{P}$ in $x$. By (2.2), the
leading order part of $\widetilde{P}=x\eth$ is
$c(\partial_{x})\left(x\partial_{x}+f/2+\eth^{Z}_{y}\right),$ so taking the
Mellin transform and ignoring the $c(\partial_{x})$ gives
$i\zeta+f/2+\eth^{Z}_{y}.$ (2.13)
The meaningful values of $\zeta$ are the indicial roots, which we define to be
$\Lambda_{y}=\left\\{i\zeta+f/2+1/2:\eqref{eq:mellin}\mbox{ is not
invertible.}\right\\}$ (2.14)
By definition, (2.13) is invertible as long as
$i\zeta+f/2\not\in\sigma(\eth^{Z}_{y})$, so Assumption 2.1 implies that
$\Lambda_{y}\cap[0,1]\subset\left\\{0,1\right\\}\mbox{ for all }y\in Y.$
###### Remark 2.8.
The shift by $f/2+1/2$ in (2.14) comes from the following considerations. We
want to understand the mapping properties of $x\eth$ on $L^{2}(M;\mathcal{S})$
with the natural measure $dVol_{g}$ given by the incomplete edge metric $g$.
On the other hand, the values of $i\zeta$ for which (2.13) fails to be
invertible give information about the mapping properties of $x\eth$ on the
Sobolev spaces defined with respect to $b-$measure
$\mu_{b}:=x^{-f-1}dVol_{g}.$
In particular, the Fredholm property in Proposition 2.4 is equivalent to
$x\eth$ being a Fredholm map from the space
$x^{\delta-f/2-1/2}H^{1}_{e}(M;\mathcal{S};\mu_{b})$ to the space
$x^{\delta-f/2-1/2}L^{2}_{e}(M;\mathcal{S};\mu_{b}))$, where the Sobolev
spaces are now defined with respect to the $b-$measure. Alternatively, as in
[3] we could define $\widetilde{P}^{\prime}=x^{-f/2-1/2}(x\eth)x^{f/2+1/2}$
take the Mellin transform and use the values of $i\zeta$ as the indicial
roots, but we would get the same answer as in (2.14).
Now we discuss the normal operator $N(\widetilde{P})$. Letting $\widetilde{P}$
act on the left on $M\times M$ (i.e., in the coordinates $(x,y,z)$ in (2.8)),
$\widetilde{P}$ restricts to an operator on $\operatorname{ff}$ acting on the
fibers of $\operatorname{ff}$ and parametrized by the base $Y$. That is, for
every $y\in Y$ we have an operator
$N(\widetilde{P})_{y}\mbox{ acting on the fiber }\operatorname{ff}_{y}\mbox{
over }y.$
To obtain an expression for $N(\widetilde{P})_{y}$ in coordinates, write
$\begin{split}\widetilde{P}=\sum_{i+\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq
m}a_{i,\alpha,k}(x,y,z)(x\partial_{x})^{i}(x\partial_{y})^{\alpha}\partial_{z}^{\beta}\quad\mbox{
where }a_{i,\alpha,\beta}\in
C^{\infty}(M;\operatorname{End}\mathcal{S}),\end{split}$
and use the projective coordinates in (2.8) to write
$\begin{split}N(\widetilde{P})_{y}=\sum_{i+\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq
m}a_{i,\alpha,k}(0,y,z)(\sigma\partial_{\sigma})^{i}(\sigma\partial_{\mathcal{Y}})^{\alpha}\partial_{z}^{\beta}.\end{split}$
The mapping properties of $\widetilde{P}$ are deduced from mapping properties
of the $N(\widetilde{P})_{y}$. In particular, to prove Proposition 2.4 we will
need Lemma 2.10 below, which shows that the Fourier transform of
$N_{y}(x\eth)$ is invertible on certain spaces.
Edge pseudodifferential operators also admit normal operators. Given
$A\in\Psi^{m}_{e,bds}(M;\mathcal{S})$, the restriction
$N(A):=\rho_{\operatorname{ff}}^{f}K_{A}\rvert_{\operatorname{ff}}$ is well
defined, and in fact
$N(A)\in\mathcal{A}_{a,b}I^{m}(\operatorname{ff},\Delta_{e}\rvert_{\operatorname{ff}};\beta^{*}\operatorname{End}(\mathcal{S})\rvert_{\operatorname{ff}}),$
meaning that $N(A)=\kappa_{1}+\kappa_{2}$ where $\kappa_{1}$ is a distribution
on $\operatorname{ff}$ conormal to $\Delta_{e}\cap\operatorname{ff}$ of order
$m$ and $\kappa_{2}$ is a smooth function on $\operatorname{ff}^{int}$ with
bounds in (2.10) (with the point $p$ restricted to $\operatorname{ff}$).
Using (2.2) and the projective coordinates in (2.8), and letting $c_{\nu}$
denote the operator induced by $c(\partial_{x})$ on the bundle
$\mathcal{S}_{y},$ the restriction of the spin bundle to the fiber over $y,$
the normal operator of $x\eth$ satisfies
$N(x\eth)=c_{\nu}\cdot\left(\sigma\frac{\partial}{\partial\sigma}+\frac{f}{2}+\eth^{Z}_{y^{\prime}}\right)+\sigma\eth_{\mathcal{Y}},$
(2.15)
where $\eth_{\mathcal{Y}}$ can be written locally in terms of the limiting
base metric $h_{y}=g_{Y}\rvert_{y}$ in (1.5) as
$\eth_{\mathcal{Y}}=\sum_{i,j=1}^{\dim
Y}c(\partial_{\mathcal{Y}_{i}})h_{y}^{ij}\partial_{\mathcal{Y}_{j}}.$
The operator $N(x\eth)$ acts on sections of $\mathcal{S}_{y}$.
The remainder of this subsection consists in establishing the following
theorem.
###### Theorem 2.9.
Let $0<\delta<1$. Under Assumption 2.1, there exist left and right
parametrices $\widetilde{Q}_{i}$, $i=1,2$ for $x\eth$. Precisely, there are
operators $\widetilde{Q}_{i}\in\Psi^{-1}_{e,bnd}(M;\mathcal{S})$, satisfying
$\begin{split}\widetilde{Q}_{1}x\eth&=Id-\Pi_{ker,\delta}\mbox{ and
}x\eth\widetilde{Q}_{2}=Id-\Pi_{coker,\delta}\mbox{ where }\\\
\widetilde{Q}_{i}&\colon x^{\delta}H^{k}_{e}(M;\mathcal{S})\longrightarrow
x^{\delta}H^{k+1}_{e}(M;\mathcal{S})\mbox{ and }\\\
\Pi_{ker,\delta},\Pi_{coker,\delta}&\colon
x^{\delta}H^{k}_{e}(M;\mathcal{S})\longrightarrow
x^{\delta}H^{\infty}_{e}(M;\mathcal{S})\end{split}$ (2.16)
for any $k$. Here $\Pi_{ker,\delta}$ (resp. $\Pi_{coker,\delta},$) is
$x^{\delta}L^{2}(M;\mathcal{S})$ orthogonal projection onto the kernel (resp.
cokernel) of $x\eth$. The Schwartz kernels satisfy the following bounds.
$\begin{split}\widetilde{Q}_{i}&\in\Psi^{-1}_{e}(M;\mathcal{S};a,b),\Pi_{ker,\delta},\Pi_{coker,\delta}\in\Psi^{-\infty}_{e}(M;\mathcal{S};a,b)\\\
\mbox{ where }a&>\delta-f/2-1/2\mbox{ and }b>-\delta-f/2-1/2.\end{split}$
Furthermore, $N(\Pi_{ker,\delta})\equiv 0\equiv N(\Pi_{coker,\delta})$ so we
have
$N(\widetilde{Q_{1}})N(x\eth)=N(Id)=N(x\eth)N(\widetilde{Q}_{2}).$ (2.17)
In particular this establishes that $x\eth\colon
x^{\delta}H^{k+1}_{e}(M;\mathcal{S})\longrightarrow
x^{\delta}H^{k}_{e}(M;\mathcal{S})$ is Fredholm.
We will see that Theorem 2.9 can be deduced from the work of Mazzeo in [39]
and its modifications in [2, 3, 42]. In order to see that the results of those
papers apply, we must prove that the normal operator $N(x\eth)$ is invertible
in a suitable sense. Taking the Fourier transform of $N_{y}(x\eth)$ in (2.15)
in the $\mathcal{Y}$ variable gives
$L(y,\eta):=\widehat{N_{y}(x\eth)}(\sigma,\eta,z)=c_{\nu}\cdot\left(\sigma\frac{\partial}{\partial\sigma}+\frac{f}{2}+\eth^{Z}_{y}\right)+i\sigma
c(\eta)$ (2.18)
and for each $y$, one considers the mapping of weighted edge Sobolev spaces
defined by picking a positive cutoff function
$\phi\colon[0,\infty)_{\sigma}\longrightarrow\mathbb{R}$ that is $1$ near zero
and $0$ near infinity and letting
$\mathcal{H}^{r,\delta,l}:=\left\\{u\in\mathcal{D}^{\prime}([0,\infty)_{\sigma}\times
Z;\mathcal{S}_{y}):\phi
u\in\sigma^{\delta}H^{r},(1-\phi)u\in\sigma^{-l}H^{r}\right\\},$ (2.19)
where, in terms of $k_{y}=g_{N/Y}\big{\rvert}_{y},$
$H^{r}:=H^{r}(\sigma^{f}d\sigma dVol_{k_{y}};\mathcal{S}_{y})$
i.e. it is the standard Sobolev space on $[0,\infty)_{\sigma}\times Z$ whose
sections take values in the bundle $\mathcal{S}$ restricted to the boundary
over the base point $y$. Consider
$L(y,\eta)\colon\mathcal{H}^{r,\delta,l}\longrightarrow\mathcal{H}^{r-1,\delta,l}.$
###### Lemma 2.10.
If the fiber operators $\eth^{Z}_{y}$ satisfy Assumption 2.1 for each $y$,
then
$L(y,\eta)\colon\mathcal{H}^{r,\delta,l}\longrightarrow\mathcal{H}^{r-1,\delta,l}$
is invertible for $0<\delta<1$, where $L(y,\eta)$ and
$\mathcal{H}^{r,\delta,l}$ are defined in (2.18) and (2.19).
###### Proof of Lemma 2.10.
Given $y\in Y$ and $\eta\in T_{y}Y$ with $\eta\neq 0$, writing
$\widehat{\eta}=\eta/\left\lvert\eta\right\rvert$, we have
$(ic(\widehat{\eta}))^{2}=id$. Furthermore,
$\eth^{Z}_{y}ic(\widehat{\eta})=ic(\widehat{\eta})\eth^{Z}_{y},$
so these operators are simultaneously diagonalizable on
$L^{2}(Z;\mathcal{S}_{0,y},k_{0})$. Thus for each $y$ and $\widehat{\eta}$ we
have an orthonormal basis $\left\\{\phi_{i,\pm}\right\\}_{i=1}^{\infty}$ of
$L^{2}(Z;\mathcal{S}_{0,y},k_{y})$ satisfying
$\eth^{Z}_{y}\phi_{i,\pm}=\pm\mu_{i}\phi_{i,\pm},\qquad
ic(\widehat{\eta})\phi_{i,\pm}=\pm\phi_{i,\pm},\qquad
c_{\nu}\phi_{i,\pm}=\pm\phi_{i,\mp}.$ (2.20)
Note that the existence of such an orthonormal basis is automatic from the
existence of any simultaneous diagonalization $\widetilde{\phi}_{i}$. Indeed,
since $c_{\nu}$ is the operator on the bundle $\mathcal{S}_{y}$ induced by
$c(\partial_{x})$, we have
$ic(\widehat{\eta})c_{\nu}\widetilde{\phi}_{i}=-c_{\nu}\widetilde{\phi}_{i}$,
so we can reindex to obtain $\phi_{i,\pm}$ satisfying the two equations on the
right in (2.20). But then since $c_{\nu}\eth^{Z}_{y}=-\eth^{Z}_{y}c_{\nu}$,
the first equation in (2.20) follows automatically. Using the $\phi_{i,\pm}$,
we define subspaces of $\mathcal{H}^{r,\delta,l}$ by
$W_{i}^{r,\delta,l}=\operatorname{span}\left\\{\left(a(\sigma)\phi_{i,+}+b(\sigma)\phi_{i,-}\right):a,b\in\mathcal{H}^{r,\delta,l}(d\sigma)\right\\},$
(2.21)
where $\mathcal{H}^{r,\delta,l}(\sigma^{f}d\sigma)$ is defined as in (2.19) in
the case that $Z$ is a single point. In particular, for all $\eta$ and $i$,
$W_{i}^{r,\delta,l}\subset\mathcal{H}^{r,\delta,l}.$
Note that multiplication by $c_{\nu}$ defines a unitary isomorphism of
$W_{i}^{r,\delta,l}$. We consider the map $L(y,\eta)$ on each space
individually. We claim that
$L(y,\eta)\colon W_{i}^{r,\delta,l}\longrightarrow W_{i}^{r-1,\delta,l}\mbox{
is invertible for $0<\delta<1$.}$ (2.22)
From (2.18), we compute
$\begin{split}-c_{\nu}\cdot
L(y,\eta)a(\sigma)\phi_{i,\pm}&=\left(\sigma\partial_{\sigma}+\frac{f}{2}\pm\mu\right)a\phi_{i,\pm}-\sigma\left\lvert\eta\right\rvert
a\phi_{i,\mp}\end{split}$ (2.23)
Thus, writing elements in $W_{i}^{r,\delta,l}$ as vector valued functions
$(a,b)^{T}=a\phi_{i,+}+b\phi_{i,-}$, we see that $L(y,\eta)$ indeed maps
$W_{i}^{r,\delta,l}$ to $W_{i}^{r-1,\delta,l}$, acting as the matrix
$-c_{\nu}L(y,\eta)\rvert_{W_{i}}=\sigma\partial_{\sigma}+f/2+\left(\begin{array}[]{cc}\mu&-\sigma\left\lvert\eta\right\rvert\\\
-\sigma\left\lvert\eta\right\rvert&-\mu\end{array}\right).$
From this, one checks that that the kernel of $L(y,\eta)$ can be written using
separation of variables as superpositions of sections given, again in terms of
the $\phi_{i,\pm}$ by
$\begin{split}\mathcal{I}_{\mu,\eta}(\sigma)&:=\sigma^{-f/2+1/2}\left(\begin{array}[]{cc}I_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\\\
I_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\end{array}\right)\mbox{
and }\\\
\mathcal{K}_{\mu,\eta}(\sigma)&:=\sigma^{-f/2+1/2}\left(\begin{array}[]{cc}-K_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\\\
K_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\end{array}\right),\end{split}$
where $I_{\nu}(z)$ and $K_{\nu}(z)$ are the modified Bessel functions [1]. By
the asymptotic formulas [1, 9.7], only the sections involving the
$\mathcal{K}_{\mu,\eta}$ are tempered distributions, and since $K_{\nu}(z)\sim
z^{-\nu}$ as $z\to 0$ for $\nu>0$, by Assumption 2.1,
$\mbox{\eqref{eq:L0mapping} is injective if $\delta>0$}.$
On the other hand, the ordinary differential operator in (2.23) admits an
explicit right inverse if $\delta<1$. Specifically, consider the matrix
$\begin{split}\mathcal{M}_{\mu,\left\lvert\eta\right\rvert}(\sigma,\widetilde{\sigma})&=(\sigma\widetilde{\sigma})^{1/2}\left\lvert\eta\right\rvert\left(\begin{array}[]{cc}I_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)&-K_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\\\
I_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)&K_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\sigma)\end{array}\right)\times\\\
&\quad\left(\begin{array}[]{cc}-H(\widetilde{\sigma}-\sigma)K_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\widetilde{\sigma})&-H(\widetilde{\sigma}-\sigma)K_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\widetilde{\sigma})\\\
-H(\sigma-\widetilde{\sigma})I_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert\widetilde{\sigma})&H(\sigma-\widetilde{\sigma})I_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert\widetilde{\sigma})\end{array}\right).\end{split}$
(2.24)
Then the operator $Q_{y,\mu}$ on $W^{r-1,\delta,l}_{i}$ defined by acting on
elements $a(\sigma)\phi_{i,+}+b(\sigma)\phi_{i,-}$ by
$Q_{y,\mu}\left(\begin{array}[]{c}a\\\
b\end{array}\right):=\sigma^{-f/2}\int_{0}^{\infty}\mathcal{M}_{\mu,\left\lvert\eta\right\rvert}(\left\lvert\eta\right\rvert,\sigma,\widetilde{\sigma})\widetilde{\sigma}^{f/2-1}\left(\begin{array}[]{c}-b(\widetilde{\sigma})\\\
a(\widetilde{\sigma})\end{array}\right)d\widetilde{\sigma}$ (2.25)
satisfies
$L(y,\eta)Q_{y,\mu}=id\mbox{ on }W^{r,\delta,l}_{i}.$ (2.26)
(One checks (2.26) using the recurrence relations and Wronskian identity
$\begin{split}I_{\nu}^{\prime}(z)&=I_{\nu-1}(z)-\frac{\nu}{z}I_{\nu}(z)=\frac{\nu}{z}I_{\nu}(z)+I_{\nu+1}(z)\\\
K_{\nu}^{\prime}(z)&=-K_{\nu-1}(z)-\frac{\nu}{z}K_{\nu}(z)=\frac{\nu}{z}K_{\nu}(z)-K_{\nu+1}(z)\\\
1/z&=I_{\nu}(z)K_{\nu+1}(z)+I_{\nu+1}(z)K_{\nu}(z),\end{split}$ (2.27)
which are equations (9.6.15) and (9.6.26) from [1].) That $Q_{y,\mu}\colon
W^{r,\delta,l}_{i}\longrightarrow W^{r-1,\delta,l}_{i}$ is bounded for
$\delta<1$ can be seen using [39], but one can also check it directly using
the density of polyhomogeneous functions. Invertibility on each
$W^{r,\delta,l}_{i}$ gives invertibility on $\mathcal{H}^{r,\delta,l}$. This
proves Lemma 2.10. ∎
Theorem 2.9 then follows from [39] as explained in [3, Sect. 2] using the
invertibility of the normal operator from Lemma 2.10. In the notation of those
papers, one has the numbers
$\begin{split}\overline{\delta}&:=\inf\left\\{\delta:L(y,\eta):\sigma^{\delta}L^{2}(d\sigma
dVol_{z};\mathcal{S}_{y})\longrightarrow L^{2}(d\sigma
dVol_{z};\mathcal{S}_{y})\mbox{ is injective for all $y$.}\right\\}\\\
\underline{\delta}&:=\sup\left\\{\delta:L(y,\eta):\sigma^{\delta}L^{2}(d\sigma
dVol_{z};\mathcal{S}_{y})\longrightarrow L^{2}(d\sigma
dVol_{z};\mathcal{S}_{y})\mbox{ is surjective for all
$y$.}\right\\}\end{split}$
By our work above, $\overline{\delta}\leq 0<1\leq\underline{\delta}$, and thus
for the map $x\eth\colon x^{\delta}H^{k}_{e}\longrightarrow
x^{\delta}H^{k}_{e}$ with $0<\delta<1$, there exist $\widetilde{Q}_{i}$,
$i=1,2$ satisfying (2.16) for $x\eth$. In particular, by (2.17)
$N(x\eth)_{y}N(\widetilde{Q}_{i})_{y}=N(Id)=\delta_{\beta^{*}\Delta\cap\operatorname{ff}},$
where $\beta^{*}\Delta$ is the lift of the interior of the diagonal
$\Delta\subset M\times M$ to the blown up space $M^{2}$. Thus in the
coordinates (2.8),
$\delta_{\beta^{*}\Delta\cap\operatorname{ff}}=\delta_{\sigma=1,\mathcal{Y}=0}$,
so from (2.24) and (2.25), we can write $N(\widetilde{Q}_{i})_{y}$ as follows.
For fixed $\eta$ and the basis $\phi_{i,\pm}$, $i=1,2,\dots,$ from (2.20), let
$\Pi(i,\eta)$ denote $L^{2}$ orthogonal projection onto $\phi_{i,\pm}$ and
define the vectors
$\Pi(\eta,i)=\left(\begin{array}[]{c}\pi(\eta,i,+)\\\
\pi(\eta,i,-)\end{array}\right),$ (2.28)
where $\pi(\eta,i,\pm)$ is orthogonal projection in
$L^{2}(Z,\mathcal{S}_{0,y},k_{y})$ onto $\phi_{i,\pm}$. We thus have
$\begin{split}\Pi(\eta,i)\widehat{N(\widetilde{Q}_{i})_{y}}\Pi^{*}(\eta,i)&=(\widetilde{\sigma}/\sigma)^{f/2}\widetilde{\sigma}^{-1}\mathcal{M}_{\mu_{i},\left\lvert\eta\right\rvert}(\sigma,1),\end{split}$
(2.29)
where $\Pi^{*}(\eta,i)\left(\begin{array}[]{c}a\\\
b\end{array}\right)=a\phi_{i,+}+b\phi_{i,-}.$
### 2.4. Proof of Theorem 1 and the generalized inverse of $\eth$
In this section we will prove Theorem 1 and describe the properties of the
integral kernel of the generalized inverse of $\eth$. We start by recalling
the statement for the convenience of the reader:
###### Theorem 2.11.
Assume that $\eth$ is a Dirac operator on a spin incomplete edge space
$(M,g),$ satisfying Assumption 2.1, then the unbounded operator $\eth$ on
$L^{2}(M;\mathcal{S})$ with core domain $C^{\infty}_{c}(M;\mathcal{S})$ is
essentially self-adjoint. Moreover, letting $\mathcal{D}$ denote the domain of
this self-adjoint extension, the map
$\eth\colon\mathcal{D}\longrightarrow L^{2}(M;\mathcal{S})$
is Fredholm.
###### Proof.
The proof of Theorem 1 will follow from combining various elements of [2, 3].
The first and main step is the construction of a left parametrix for the map
$\eth\colon\mathcal{D}_{max}\longrightarrow L^{2}(M;\mathcal{S})$, where
$\mathcal{D}_{max}$ is the maximal domain defined in (6).
Consider $\widetilde{Q}_{1}$ from (2.16) and set
$\widetilde{Q}_{1}x=\overline{Q}_{1}$. Then by (2.16)
$\begin{split}\overline{Q}_{1}\eth&=Id-\Pi_{ker,\delta},\end{split}$ (2.30)
where both sides of this equation are thought of as maps of
$x^{\delta}L^{2}_{e}(M;\mathcal{S})$. We claim that in fact equation (2.30)
holds not only on $x^{\delta}L^{2}_{e}(M;\mathcal{S})$, but on the maximal
domain $\mathcal{D}_{max}$ defined in (6). This follows from [3, Lemma 2.7] as
follows. In the notation of that paper, $L=\eth$ and $P=x\eth$. Taking (again,
in the notation of that paper) $\mathcal{E}(L)$ to be $\mathcal{D}_{max}$, by
[3, Lemma 2.1], $\mathcal{E}^{(\tau)}(L)=\mathcal{E}(L)$. Furthermore,
$\mathcal{E}_{\tau}(L)=x^{\tau}L^{2}(M;\mathcal{S})\cap\mathcal{D}_{max}$.
Since $\widetilde{Q}_{1}$ maps $x^{\delta}L^{2}(M;\mathcal{S})$ to
$x^{\delta}H^{1}_{e}$, we have $Id-\overline{Q}_{1}\eth$ is bounded on
$\mathcal{E}_{\tau}$. Futhermore, $x\eth$ maps $\mathcal{D}_{max}$ to
$xL^{2}(M;\mathcal{S})\subset x^{\delta}L^{2}(M;\mathcal{S})$, so
$\overline{Q}_{1}\eth=\widetilde{Q}_{1}x\eth$ maps $\mathcal{D}_{max}$ to
$x^{\delta}L^{2}(M;\mathcal{S})$. Thus [3, Lemma 2.7] applies and (2.30) holds
on $\mathcal{D}_{max}$, as advertised.
Thus $Id=\overline{Q}_{1}\eth+\Pi_{ker,\delta}$ on $\mathcal{D}_{max}$, and
since the right hand side is bounded $L^{2}(M;\mathcal{S})$ to
$x^{\delta}L^{2}$, for any $\delta\in(0,1)$, we have
$\mathcal{D}_{max}\subset\bigcap_{\delta<1}x^{\delta}L^{2}(M,\mathcal{S}),$
(2.31)
in particular for any $\delta<1$, $\mathcal{D}_{max}\subset H^{1}_{loc}\cap
x^{\delta}L^{2}(M,\mathcal{S})$ which is a compact subset of
$L^{2}(M;\mathcal{S})$. It then follows from Gil-Mendoza [30] (see [2, Prop.
5.11]) that $\mathcal{D}_{max}\subset\mathcal{D}_{min}$, i.e. that $\eth$ is
essentially self-adjoint. By a standard argument, e.g. [45, Lemma 4.2], the
compactness of $\mathcal{D}_{max}$ implies that $\eth$ has finite dimensional
kernel and closed range. But the self-adjointness of $\eth$ on $\mathcal{D}$
now implies that $\eth$ has finite dimensional cokernel, so $\eth$ is self-
adjoint and Fredholm. ∎
Thus $\eth$ admits a generalized inverse $Q\colon
L^{2}(M;\mathcal{S})\longrightarrow\mathcal{D}$ satisfying
$\eth Q=Id-\pi_{ker}\mbox{ and }Q=Q^{*},$
where $\pi_{ker}$ is $L^{2}$ orthogonal projection onto the kernel of $\eth$
in $\mathcal{D}$ with respect to the pairing induced by the Hermitian inner
product on $\mathcal{S}.$ To be precise, if $\left\\{\phi_{i}\right\\}$,
$i=1,\dots,N$ is an orthonormal basis for the kernel of $\eth$ on
$\mathcal{D}$, then $\pi_{ker}$ has Schwartz kernel
$K_{\pi_{ker}}(w,w^{\prime})=\sum_{i=1}^{N}\phi_{i}(w)\otimes\overline{\phi_{i}(w^{\prime})}.$
From (2.31), we see that $\pi_{ker}\in\Psi^{-\infty}_{e}(M;\mathcal{S};a,b)$.
The properties of the integral kernel of $Q$ can be deduced from those of the
parametrices $\widetilde{Q}_{i}$ in (2.16). Indeed, setting
$\widetilde{Q}=Qx^{-1}$, we see that
$\widetilde{Q}(x\eth)=Id-\pi_{ker}\mbox{ and
}(x\eth)\widetilde{Q}=Id-x\cdot\pi_{ker}\cdot x^{-1}.$
Applying the argument from [39, Sect. 4], specifically equations (4.24) and
(4.25) there, shows that $\widetilde{Q}\in\Psi^{-1}_{e}(M;\mathcal{S};a,b)$
for the same $a,b$ as in (2.16), and in particular that
$N(\widetilde{Q})=N(\widetilde{Q}_{i}$). In particular, by Theorem 2.9, we
have the bounds
$K_{Q}(p)=O(\rho_{\operatorname{lf}}^{a})\mbox{ as
}p\to\operatorname{lf}\mbox{ and
}K_{Q}(p)=O(\rho_{\operatorname{rf}}^{b})\mbox{ as }p\to\operatorname{rf}$
where $a>\delta-f/2-1/2$ and $b>-\delta-f/2+1/2$, $0<\delta<1$, and again the
bounds hold for $K_{Q}$ as a section of $\operatorname{End}(\mathcal{S})$ over
$M\times M$. Finally, by self-adjointness of $Q$, we have that
$K_{Q}(w,w^{\prime})=K_{Q}^{*}(w^{\prime},w)\mbox{ for all }w,w^{\prime}\in
M^{int}.$ (2.32)
By (2.32), the bound at $\operatorname{rf}$, which one approaches in
particular if $w$ remains fixed in the interior of $M$ and $w^{\prime}$ goes
to the boundary, gives a bound at $\operatorname{lf}$. Thus we obtain the
following.
###### Proposition 2.12.
The distributional section $K_{Q}$ of $\operatorname{End}(\mathcal{S})$ over
$M\times M$ with the property that
$Q\phi=\int_{M}K_{Q}(w,w^{\prime})\phi^{\prime}(w^{\prime})dVol_{g}(w^{\prime})$
is conormal at $\Delta_{e}$, and $\rho^{f-1}K_{Q}$ is smoothly conormal up to
$\operatorname{ff}$, where
$\rho^{f}K_{Q}x^{-1}\rvert_{\operatorname{ff}}=\rho^{f}\widetilde{Q}\rvert_{\operatorname{ff}}$
satisfies (2.29). Moreover, for coordinates
$(x,y,z,x^{\prime},y^{\prime},z^{\prime})$ on $M\times M$ as in (1.4),
$K_{Q}(x,y,z,x^{\prime},y^{\prime},z^{\prime})=O(x^{a}),\mbox{ uniformly for
}x^{\prime}\geq c>0,$ (2.33)
where $a>-\delta-f/2+1/2$ for any $\delta>0$ and $c$ is an arbitrary small
positive number.
## 3\. Boundary values and boundary value projectors
Recall that $M_{\varepsilon}=\\{x\geq\varepsilon\\}$ is a smooth manifold with
boundary, and $M-M_{\varepsilon}$ is a tubular neighborhood of the
singularity. Consider the space of harmonic sections over $M-M_{\varepsilon}$
$\mathcal{H}_{loc,\varepsilon}=\left\\{u\in
L^{2}(M-M_{\varepsilon};\mathcal{S}):\eth
u=0,\exists\;\widetilde{u}\in\mathcal{D}\text{ s.t.
}u=\widetilde{u}\rvert_{M-M_{\varepsilon}}\right\\},$
where $\mathcal{D}$ is the domain for $\eth$ from Theorem 1; in particular,
$\mathcal{D}\subset H^{1}_{loc}$. By the standard restriction theorem for
$H^{1}$ sections [52, Prop 4.5, Chap 4], any element
$u\in\mathcal{H}_{loc,\varepsilon}$ has boundary values $u\rvert_{\partial
M_{\varepsilon}}\in H^{1/2}(\partial M_{\varepsilon})$. We define a domain for
$\eth$ on the cutoff manifold $M_{\varepsilon}$ by
$\mathcal{D}_{\varepsilon}:=\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):u\rvert_{\partial
M_{\varepsilon}}=v\rvert_{\partial M_{\varepsilon}}\mbox{ for some
}v\in\mathcal{H}_{loc,\varepsilon}\right\\}\subset
L^{2}(M_{\varepsilon};\mathcal{S}).$ (3.1)
Essentially, $\mathcal{D}_{\varepsilon}$ consists of sections over
$M_{\varepsilon}$ whose boundary values correspond with the boundary values of
an $L^{2}$ harmonic section over $M-M_{\varepsilon}$. We also have the
chirality spaces
$\mathcal{D}_{\varepsilon}^{\pm}=\mathcal{D}_{\varepsilon}\cap
L^{2}(M_{\varepsilon};\mathcal{S}^{\pm})$
where $\mathcal{S}^{\pm}$ are the chirality subbundles of even and odd
spinors. In this section we will prove the following.
###### Theorem 3.1.
For $\varepsilon>0$ sufficiently small and $\mathcal{D}_{\varepsilon}$ as in
(3.1), the map $\eth\colon\mathcal{D}_{\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S})$ is Fredholm, and
$\operatorname{Ind}(\eth\colon\mathcal{D}_{\varepsilon}^{+}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))=\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-})).$
In the process of proving Theorem 3.1, we will construct a family of boundary
value projectors $\pi_{\varepsilon}$ which define $\mathcal{D}_{\varepsilon}$
in the sense of Claim 3.2 below, and whose microlocal structure we will use in
Section 4 to relate the index of $\eth$ on $M_{\varepsilon}$ with domain
$\mathcal{D}_{\varepsilon}$ to the index of $\eth$ on $M_{\varepsilon}$ with
the APS boundary condition, see Theorem 4.1.
### 3.1. Boundary value projector for $\mathcal{D}_{\varepsilon}$
As already mentioned, the main tool for proving Theorem 3.1 and also for
proving Theorem 4.1 below will be to express the boundary condition in the
definition of $\mathcal{D}_{\varepsilon}$ in (3.1) in terms of a
pseudodifferential projection over $\partial M_{\varepsilon}$. We discuss the
construction of this projection now.
First we claim that the invertible double construction of [16, Chapter 9]
holds in this context in the following form: there exists an incomplete edge
manifold $M^{\prime}$ with spinor bundle $\mathcal{S}^{\prime}$ and Dirac
operator $\eth^{\prime}$, together with an isomorphism
$\Phi\colon(M-M_{\varepsilon_{0}},\mathcal{S})\longrightarrow(M^{\prime}-M^{\prime}_{\varepsilon_{0}},\mathcal{S}^{\prime})$
such that, with identifications induced by $\Phi,$ the operators $\eth$ and
$\eth^{\prime}$ are equal over
$M-M_{\varepsilon_{0}}(=\Phi^{-1}(M^{\prime}-M^{\prime}_{\varepsilon_{0}})),$
and finally such that $\eth^{\prime}$ is invertible. In particular, the
inverse $Q^{\prime}$ satisfies
$\begin{split}Q^{\prime}&:\mathcal{D}^{\prime}\longrightarrow
L^{2}(M^{\prime})\\\ \eth Q^{\prime}&=id=Q^{\prime}\eth,\end{split}$ (3.2)
where $\mathcal{D}^{\prime}$ is the unique self-adjoint domain for
$\eth^{\prime}$ on $M^{\prime}$ with core domain
$C_{c}^{\infty}(M^{\prime},\mathcal{S}^{\prime})$. Moreover, $Q^{\prime}$
satisfies all of the properties in Proposition 2.12.
We describe the construction of this “invertible double” for the convenience
of the reader, though it is essentially identical to that in [16, Chapter 9],
the only difference being that we must introduce a product type boundary while
they have one to begin with. Choosing any point $p\in M_{\varepsilon_{0}}$,
let $D_{1},D_{2}$ denote open discs around $p$ with $p\in D_{1}\Subset D_{2}$
and $D_{2}\cap(M-M_{\varepsilon_{0}})=\varnothing$. We can identify the
annulus $D_{2}-D_{1}$ with $[1,2)_{s}\times\mathbb{S}^{d-1}$ by a
diffeomorphism and the metric $g$ is homotopic to a product metric
$ds^{2}+\left\lvert dx\right\rvert^{2}$ where $x$ is the standard coordinate
on $\mathbb{B}^{d-1}$. Furthermore, the connection can be deformed so that the
induced Dirac operator $\eth^{\prime}$ is of product type on the annulus (see
equation 9.4 in [16]). Call the bundle over $N_{1}:=M-D_{1}$ thus obtained
$\widetilde{\mathcal{S}}$. Letting $N_{2}:=-N_{1}$, the same incomplete edge
space with the opposite orientation, let $M^{\prime}=N_{1}\sqcup
N_{2}/\left\\{s=1\right\\}$ and consider the vector bundle
$\mathcal{S}^{\prime}$ over $M^{\prime}$ obtained by taking
$\widetilde{\mathcal{S}}^{+}$ over $N_{1}$ and $\widetilde{\mathcal{S}}^{-}$
over $N_{2}$ and identifying the two bundles over $D_{2}$ using Clifford
multiplication by $\partial_{s}$. The resulting Dirac operator, which we still
denote by $\eth^{\prime}$, is seen to be invertible on $M^{\prime}$ by the
symmetry and unique continuation argument in Lemma 9.2 of [16].
We will now work on a neighborhood in $M-M_{\varepsilon}$ of $\partial M$ (or
equivalently of the singular stratum $Y$), so we drop the distinction between
$M$ and $M^{\prime}$. Using notation as in (3.2), and given $f\in
C^{\infty}(\partial M_{\varepsilon};\mathcal{S})$, define the harmonic
extension
$\begin{split}\operatorname{Ext}_{\varepsilon}f(w)&:=\int_{w^{\prime}\in\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}f(w^{\prime})dVol_{\partial
M_{\varepsilon}},\end{split}$ (3.3)
where $K_{Q^{\prime}}$ is the Schwartz kernel of $Q^{\prime}$ (see (2.9)), and
$c_{\nu}=c(\partial_{x}).$ Since
$\eth^{\prime}K_{Q^{\prime}}(w,w^{\prime})=0\mbox{ away from }w=w^{\prime}$
(3.4)
$\eth^{\prime}\operatorname{Ext}_{\varepsilon}f(w)=0$ for $w\not\in\partial
M_{\varepsilon}$. Recall Green’s formula for Dirac operators; specifically,
for a smoothly bounded region $\Omega$ with normal vector $\partial_{\nu}$,
$\int_{\Omega}\left(\langle\eth u,v\rangle-\langle u,\eth
v\rangle\right)dVol_{\Omega}=\int_{\partial\Omega}\langle
c(\partial_{\nu})u,v\rangle dVol_{\partial\Omega}.$ (3.5)
Green’s formula for sections $u$ satisfying $\eth u\equiv 0$ in
$M-M_{\varepsilon}$ gives that for $u\in\mathcal{H}_{loc,\varepsilon}$,
$\begin{split}u(w)=-\int_{\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}u(w^{\prime})dVol_{\partial
M_{\varepsilon}},\end{split}$ (3.6)
_provided_
$\forall u\in\mathcal{H}_{loc,\varepsilon},w\in
M^{int},\quad\lim_{\widetilde{\varepsilon}\to 0}\int_{\partial
M_{\widetilde{\varepsilon}}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}u(w^{\prime})dVol_{\partial
M_{\widetilde{\varepsilon}}}=0.$ (3.7)
The identity in (3.6) is obtained by integrating by parts in
$\int_{M-M_{\varepsilon}}\eth
K_{Q^{\prime}}(w,w^{\prime})u(w^{\prime})-K_{Q^{\prime}}(w,w^{\prime})\eth
u(w^{\prime})dVol_{w^{\prime}}$
and using (3.4). In fact, as we will see in the proof of Claim 3.2 below,
(3.7), and thus (3.6), hold for all $u\in\mathcal{H}_{loc,\varepsilon}$.
It follows from (3.3) and (3.6) that, for $\eth u=0$ satisfying (3.7),
$\begin{split}u\rvert_{\partial
M_{\varepsilon}}(w)&=\mathcal{E}_{\varepsilon}(u\rvert_{\partial
M_{\varepsilon}})(w)\mbox{ where }\\\
\mathcal{E}_{\varepsilon}(f)(w)&:=\lim_{\begin{subarray}{c}\widetilde{w}\to
w\\\ \widetilde{w}\in
M-M_{\varepsilon}\end{subarray}}\operatorname{Ext}_{\varepsilon}(f)(w)\end{split}$
(3.8)
We will show that the $\mathcal{E}_{\varepsilon}$ define the domains
$\mathcal{D}_{\varepsilon}$ as follows.
###### Claim 3.2.
The operator $\mathcal{E}_{\varepsilon}$ in (3.8) is a projection operator on
$L^{2}(\partial M_{\varepsilon},\mathcal{S})$, and the domain
$\mathcal{D}_{\varepsilon}$ in (3.1) is given by
$\mathcal{D}_{\varepsilon}=\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):(Id-\mathcal{E}_{\varepsilon})(u\rvert_{\partial
M_{\varepsilon}})=0\right\\}.$ (3.9)
Moreover, there exists $B_{\varepsilon}\in\Psi^{0}(\partial
M_{\varepsilon};\mathcal{S})$ such that
$\begin{split}\mathcal{E}_{\varepsilon}&=\frac{1}{2}Id+B_{\varepsilon}\mbox{
and }\\\ B_{\varepsilon}f(w)&=-\int_{\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}f(w^{\prime})dVol_{\partial
M_{\varepsilon}}\mbox{ for }w\in\partial M_{\varepsilon},\end{split}$ (3.10)
and the principal symbol of $\mathcal{E}_{\varepsilon}$ satisfies
$\sigma(\mathcal{E}_{\varepsilon})=\sigma(\pi_{APS,\varepsilon}),$ (3.11)
where $\pi_{APS,\varepsilon}$ is the APS projection defined in (1.21).
Assuming Claim 3.2 for the moment, we prove Theorem 3.1.
###### Proof of Theorem 3.1 assuming Claim 3.2.
The main use of Claim 3.2 in this context (it will be used again in Theorem
4.1 ) is to show that the map
$\eth\colon\mathcal{D}_{\varepsilon}\longrightarrow L^{2}(M_{\varepsilon}).$
(3.12)
is self-adjoint on $L^{2}(M_{\varepsilon};\mathcal{S})$ and Fredholm. The
Fredholm property follows from the principal symbol equality (3.11), since
from [16] any projection in $\Psi^{0}(\partial M_{\varepsilon},\mathcal{S})$
with principal symbol equal to that of the Atiyah-Patodi-Singer boundary
projection defines a Fredholm problem. To see that it is self-adjoint, note
that from (3.5) the adjoint boundary condition is
$\mathcal{D}_{\varepsilon}^{*}=\left\\{\phi:\langle
g,cl_{\nu}\phi\rvert_{\partial M_{\varepsilon}}\rangle_{\partial
M_{\varepsilon}}=0\mbox{ for all }g\mbox{ with
}(Id-\mathcal{E}_{\varepsilon})g=0\right\\}.$
Again by (3.5), for any $v\in\mathcal{H}_{loc,\varepsilon}$,
$v\rvert_{\partial M_{\varepsilon}}\in\mathcal{D}_{\varepsilon}^{*}.$ Thus
$\mathcal{D}_{\varepsilon}\subset\mathcal{D}_{\varepsilon}^{*}$, and it
remains to show that
$\mathcal{D}_{\varepsilon}^{*}\subset\mathcal{D}_{\varepsilon}$. Let
$\phi\in\mathcal{D}_{\varepsilon}^{*}$, and set $f:=\phi\rvert_{\partial
M_{\varepsilon}}$. We want to show that $(I-\mathcal{E}_{\varepsilon})f=0$, or
equivalently
$\begin{split}\langle(I-\mathcal{E}_{\varepsilon})f,g\rangle_{\partial
M_{\varepsilon}}&=0\quad\forall g\\\ \iff\langle
f,(I-\mathcal{E}_{\varepsilon}^{*})g\rangle_{\partial
M_{\varepsilon}}&=0\quad\forall g.\end{split}$ (3.13)
Since $\langle f,c_{\nu}g\rangle=-\langle c_{\nu}f,g\rangle$, by (3.5) we have
$\langle f,c_{\nu}g\rangle=0$ for every $g\in\operatorname{Ran}\mathcal{E}$,
and thus (3.13) will hold if
$(I-\mathcal{E}_{\varepsilon}^{*})g\in\operatorname{Ran}c_{\nu}\mathcal{E}$.
In fact, we claim that
$I-\mathcal{E}_{\varepsilon}^{*}=-c_{\nu}\mathcal{E}c_{\nu}.$
To see that his holds, note that by Claim 3.2 and self-adjointness of
$Q^{\prime}$, specifically (2.32),
$B_{\varepsilon}^{*}=c_{\nu}B_{\varepsilon}c_{\nu}$, so
$I-\mathcal{E}_{\varepsilon}^{*}=I-(\frac{1}{2}+B_{\varepsilon})^{*}=\frac{1}{2}-B_{\varepsilon}^{*}=-c_{\nu}(\frac{1}{2}+B_{\varepsilon})c_{\nu}=-c_{\nu}\mathcal{E}_{\varepsilon}c_{\nu},$
which proves self-adjointness.
Now that we know that (3.12) is self-adjoint, we proceed as follows. We claim
that for $\varepsilon>0$ sufficiently small, the map
$\begin{split}\ker(\eth\colon\mathcal{D}\longrightarrow
L^{2}(M;\mathcal{S}))&\longrightarrow\ker(\eth\colon\mathcal{D}_{\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}))\\\
\widetilde{\phi}&\longmapsto\phi=\widetilde{\phi}\rvert_{M_{\varepsilon}}\end{split}$
is well defined and an isomorphism. It is well defined since by definition any
section $\widetilde{\phi}\in\ker(\eth\colon\mathcal{D}\longrightarrow
L^{2}(M;\mathcal{S}))$ satisfies that
$\phi=\widetilde{\phi}\rvert_{M_{\varepsilon}}\in\mathcal{D}_{\varepsilon}$.
It is injective by unique continuation. For surjectivity, note that for any
element $\phi\in\ker(\eth\colon\mathcal{D}_{\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}))$, by definition there is a
$u\in\mathcal{H}_{loc,\varepsilon}$ such that $u\rvert_{\partial
M_{\varepsilon}}=\phi\rvert_{\partial M_{\varepsilon}}$. It follows that
$\widetilde{\phi}(w):=\left\\{\begin{array}[]{ccc}\phi(w)&\mbox{ for }&w\in
M_{\varepsilon}\\\ u(w)&\mbox{ for }&w\in M-M_{\varepsilon}\end{array}\right.$
is in $H^{1}$ and satisfies $\eth\widetilde{\phi}=0$ on all of $M$, i.e.
$\widetilde{\phi}\in\ker(\eth\colon\mathcal{D}\longrightarrow
L^{2}(M;\mathcal{S})).$ Since the full operator $\eth$ on $\mathcal{D}$ is
self-adjoint, and since the operator in (3.12) is also, the cokernels of both
maps are equal to the respective kernels. Restricting $\eth$ to a map from
sections of $\mathcal{S}^{+}$ to sections of $\mathcal{S}^{-}$ gives the
theorem.
This completes the proof.∎
Thus to prove Theorem 3.1 it remains to prove Claim 3.2, which we proceed to
do
###### Proof of Claim 3.2.
We begin by proving (3.10). It is a standard fact (see [53, Sect. 7.11]) that
$\mathcal{E}_{\varepsilon}\in\Psi^{0}(\partial M_{\varepsilon};\mathcal{S}).$
Obviously,
$\begin{split}\mathcal{E}_{\varepsilon}&=A+B_{\varepsilon}\mbox{ where }\\\
B_{\varepsilon}f(w)&=-\int_{\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}f(w^{\prime})dVol_{\partial
M_{\varepsilon}}\mbox{ for }w\in\partial M_{\varepsilon}\mbox{ and }\\\
\operatorname{supp}A&\subset\operatorname{diag}(\partial
M_{\varepsilon}\times\partial M_{\varepsilon}),\end{split}$ (3.14)
where the last containment refers to the Schwartz kernel of $A$. We claim that
$A=\frac{1}{2}id\quad\mbox{ and }\quad B_{\varepsilon}\in\Psi^{0}(\partial
M_{\varepsilon};\mathcal{S}).$
Using that $Q^{\prime}$ has principal symbol
$\sigma(Q^{\prime})=\sigma(\eth)^{-1}$ we can write $Q^{\prime}$ in local
coordinates $w$ as
$\begin{split}Q^{\prime}&=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{-(w-\widetilde{w})\cdot\xi}a(w,\widetilde{w},\xi)d\xi\mbox{
locally, where }\\\
a(w,\widetilde{w},\xi)&=\left\lvert\xi\right\rvert_{g(w)}^{-2}ic(\xi)+O(1)\mbox{
for }\left\lvert\xi\right\rvert\geq c>0.\end{split}$
Given a bump function $\chi$ supported near $w_{0}\in\partial
M_{\varepsilon}$, let $Q^{\prime}_{\chi}:=\chi Q^{\prime}\chi$ and define the
distributions
$\begin{split}K_{Q^{\prime}_{\chi}}&=K_{1}+K_{2}\\\ \mbox{ where
}K_{1}&=\mathcal{F}_{\xi}^{-1}\left(\left\lvert\xi\right\rvert_{g(w)}^{-2}ic(\xi)\right)\mathcal{F}_{\widetilde{x}}\end{split}$
(3.15)
where, as in (2.9), $K_{Q^{\prime}_{\chi}}$ denotes the Schwartz kernel of
$Q^{\prime}_{\chi}$. The distribution $K_{2}$ is that of a pseudodifferential
operator of order $-2$ on $M$, and it follows from the theory of homogeneous
distributions (see [53, Chapter 7]) that the distribution $K_{2}$ restricts to
$\partial M_{\varepsilon}$ to be the Schwartz kernel of a pseudodifferential
operator of order $-1$. The distribution $K_{1}$ is that of a
pseudodifferential operator on $M$ of order $-1$. It is smooth in
$\widetilde{x}$ with values in homogeneous distributions in $x-\widetilde{x}$
of order $-n+1$, and it follows that the restriction of the Schwartz kernel
$K_{1}(w,w^{\prime})$ to $\partial M_{\varepsilon}$ gives a pseudodifferential
operator of order zero. Letting $B_{\varepsilon}$ in (3.14) be the operator
defined by the restriction of $K_{1}$ to $\partial M_{\varepsilon}$, we have
that $B_{\varepsilon}$ is in $\Psi^{0}(\partial M_{\varepsilon};\mathcal{S})$
and it remains to calculate $A$. Choosing coordinates of the form
$w=(x,x^{\prime})$ and $\widetilde{w}=(\widetilde{x},\widetilde{x}^{\prime})$
of the form in (1.19) and such that at the fixed value
$w_{0}=(\varepsilon,x^{\prime}_{0})\in\partial M_{\varepsilon}$ the metric
satisfies $g(x)=id$, it follows (see [4]) that the Schwartz kernel of $B$ in
(3.15) satisfies
$B(x_{0},\widetilde{x})=-\frac{1}{\omega_{n-1}}\frac{c(x_{0})-c(\widetilde{x})}{\left\lvert
x_{0}-\widetilde{x}\right\rvert^{n}}+O(\left\lvert
x_{0}-\widetilde{x}\right\rvert^{2-n})$
where $\omega_{n-1}$ is the volume of the unit sphere $\mathbb{S}^{n-1}$. If
we let
$\widetilde{B}(x^{\prime},\widetilde{x}^{\prime})=B(0,x^{\prime},0,\widetilde{x}^{\prime})$,
then near $x_{0}$
$\begin{split}\operatorname{Ext}_{\varepsilon}f(\delta,x^{\prime})&=-\frac{1}{\omega_{n-1}}\int\left(\frac{c((\delta,x^{\prime}))-c((0,\widetilde{x}^{\prime}))}{\left\lvert(\delta,x^{\prime})-(0,\widetilde{x}^{\prime})\right\rvert^{n}}\right)c_{\nu}f(\widetilde{x}^{\prime})d\widetilde{x}^{\prime}\\\
&=-\frac{1}{\omega_{n-1}}\int\left(\frac{\delta
c_{\nu}}{\left\lvert(\delta,x^{\prime})-(0,\widetilde{x}^{\prime})\right\rvert^{n}}+\frac{c((0,x^{\prime}))-c((0,\widetilde{x}^{\prime}))}{\left\lvert(\delta,x^{\prime})-(0,\widetilde{x}^{\prime})\right\rvert^{n}}\right)c_{\nu}f(\widetilde{x}^{\prime})d\widetilde{x}^{\prime}.\\\
&\to\frac{1}{2}f(x^{\prime})+\int\widetilde{B}(x^{\prime},y^{\prime})c_{\nu}f(y^{\prime})dy^{\prime}\mbox{
as }\delta\to 0.\end{split}$
This proves that $A=1/2.$
The principal symbol of $\mathcal{E}_{\varepsilon}$ (again see [53, Sect.
7.11]) is given by the integral
$\begin{split}\sigma(\mathcal{E})(x^{\prime},\xi^{\prime})&=\frac{-1}{2\pi}\lim_{x\to\varepsilon^{-}}\int_{\mathbb{R}}e^{i(x-\varepsilon)\xi}\frac{1}{|(\xi,\xi^{\prime})|_{g}^{2}}(ic(\xi\partial_{x})+ic(\xi^{\prime}\cdot\partial_{x^{\prime}}))c(\partial_{x})d\xi\\\
&=\frac{-1}{2\pi}\lim_{x\to\varepsilon^{-}}\left(\int_{\mathbb{R}}e^{i(x-\varepsilon)\xi}\frac{\xi}{|(\xi,\xi^{\prime})|_{g}^{2}}d\xi\right)ic(\partial_{x})^{2}-\frac{1}{2}ic(\widehat{\xi}^{\prime}\cdot\partial_{x^{\prime}})c(\partial_{x})\\\
&=\frac{-1}{2\pi}\left(-\frac{2\pi
i}{2}\right)ic(\partial_{x})^{2}-\frac{1}{2}(-ic(\partial_{x}c(\widehat{\xi}^{\prime}\cdot\partial_{x^{\prime}}))\\\
&=\frac{1}{2}-\frac{1}{2}(-ic(\partial_{x})c(\widehat{\xi}^{\prime}\cdot\partial_{x^{\prime}}))).\end{split}$
where in the third line we used the residue theorem. Now recall that the term
$-ic(\partial_{x})c(\widehat{\xi}^{\prime}\cdot\partial_{x^{\prime}}))$ is
precisely the endomorphism appearing in (1.20), so
$\begin{split}\sigma(\mathcal{E})(x^{\prime},\xi^{\prime})&=\frac{1}{2}-\frac{1}{2}(\pi_{\varepsilon,+,\widehat{\xi}^{\prime}}(x^{\prime})-\pi_{\varepsilon,-,\widehat{\xi}^{\prime}}(x^{\prime}))=\pi_{\varepsilon,-,\widehat{\xi}^{\prime}}(x^{\prime}),\end{split}$
where the projections are those in (1.20). Thus, Theorem 1.2 implies the
desired formula for the principal symbl of $\mathcal{E}_{\varepsilon}$,
(3.11).
To finish the claim, we must show the equivalence of domains in (3.9). We
first show that for any $u\in\mathcal{H}_{loc,\varepsilon}$, the formula in
(3.6) holds. This will show that any $f\in H^{1/2}(\partial
M_{\varepsilon};\mathcal{S})$ with $f=u\rvert_{\partial M_{\varepsilon}}$ for
some $u\in\mathcal{H}_{loc,\varepsilon}$ satisfies
$(Id-\mathcal{E}_{\varepsilon})f=0$, i.e., that
$\mathcal{D}_{\varepsilon}\subset\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):(Id-\mathcal{E}_{\varepsilon})(u\rvert_{\partial
M_{\varepsilon}})=0\right\\}.$
Thus we must show that (3.7) holds for $u\in\mathcal{H}_{loc,\varepsilon}$.
For such $u$, we claim that for some $\delta>0$, as $\varepsilon\to 0$
$\int_{\partial M_{\varepsilon}}\left\|u\right\|^{2}dVol_{\partial
M_{\varepsilon}}=O(\varepsilon^{-f-\delta}).$ (3.16)
To see this, note first that $u\in
x^{1-\delta}H^{1}_{e}(M-M_{\varepsilon},\mathcal{S})$ for every $\delta>0$,
which follows since $u$ has an extension to a section in
$\mathcal{D}_{max}\subset
H^{1}_{loc}\cap_{\delta>0}x^{1-\delta}L^{2}(M;\mathcal{S})$. In particular,
$x^{\delta+f/2}u\in H^{1}(M,dxdydz)$, the standard Sobolev space of order $1$
on the manifold with boundary $M$. Using the restriction theorem [52, Prop
4.5, Chap 4], $x^{\delta+f/2}u=\varepsilon^{\delta+f/2}u\in H^{1/2}(\partial
M)$ uniformly in $\varepsilon$, so (3.16) holds. Thus, for fixed $w\in
M-M_{\varepsilon}$, writing $dVol_{g}=x^{f}adxdydz$ for some $a=a(x,y,z)$ with
$a(0,y,z)\neq 0$, we can use the bound for $K_{Q^{\prime}}$ in (2.33) with
$x^{\prime}$ fixed and $x=\varepsilon$ to conclude
$\begin{split}\left(\int_{\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}u(w^{\prime})dVol_{\partial
M_{\varepsilon}}\right)^{2}&=\left(\int_{\partial
M_{\varepsilon}}K_{Q^{\prime}}(w,w^{\prime})c_{\nu}u(w^{\prime})\varepsilon^{f}ady^{\prime}dz^{\prime}\right)^{2}\\\
&\leq\varepsilon^{2f}(\int_{\partial
M_{\varepsilon}}\left\|K_{Q^{\prime}}(w,w^{\prime})\right\|^{2}ady^{\prime}dz^{\prime})\\\
&\qquad\times(\int_{\partial
M_{\varepsilon}}\left\|u\right\|^{2}ady^{\prime}dz^{\prime})\\\
&=\varepsilon^{2f}o(\varepsilon^{-2\delta-f+1})o(\varepsilon^{-f-2\delta})\mbox{
for all }\delta>0\\\ &=o(\varepsilon^{-4\delta+1})\to 0\mbox{ as
}\varepsilon\to 0.\end{split}$
To prove the other direction of containment in (3.9), we need to know that for
$f\in H^{1/2}(\partial M_{\varepsilon})$ satisfying
$(Id-\mathcal{E}_{\varepsilon})f=0$, the section
$u:=\operatorname{Ext}_{\varepsilon}f\rvert_{M-M_{\varepsilon}}\in\mathcal{H}_{loc,\varepsilon}$,
where $\operatorname{Ext}_{\varepsilon}$ is the extension operator in (3.3).
This is true since for any $H^{1/2}$ section $h$ over $\partial
M_{\varepsilon}$, there is an $H^{1}$ extension $v$ to the manifold
$M^{\prime}$ defined above, that can be taken with support away from the
singular locus. If $1_{M^{\prime}_{\varepsilon}}$ is the indicator function of
$M^{\prime}_{\varepsilon}$, then
$\eth^{\prime}(\operatorname{Ext}_{\varepsilon}f+1_{M^{\prime}_{\varepsilon}}v)=\delta_{\partial
M_{\varepsilon}}(f+h)+1_{M^{\prime}_{\varepsilon}}\eth^{\prime}v$. Taking $h$
to cancel $f$ gives that
$\eth^{\prime}(\operatorname{Ext}_{\varepsilon}f+1_{M^{\prime}_{\varepsilon}}v)\in
L^{2}(M^{\prime};\mathcal{S})$. Since $1_{M^{\prime}_{\varepsilon}}v$ is an
extendible $H^{1}$ distribution on $M_{\varepsilon}^{\prime}$ near $\partial
M_{\varepsilon}$, $\operatorname{Ext}f\rvert_{M-M_{\varepsilon}}$ is an
extendible $H_{loc}^{1}$ distribution on $M-M_{\varepsilon}$ near $\partial
M_{\varepsilon}$. This completes the proof of Claim 3.2.
∎
## 4\. Equivalence of indices
In the previous section we have shown
$\operatorname{Ind}(\eth\colon\mathcal{D}_{\varepsilon}^{+}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))=\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-})),$
in this section we will prove the following.
###### Theorem 4.1.
Let $\pi_{APS,\varepsilon}$ denote the APS projector from (1.22). Then for
$\varepsilon>0$ sufficiently small,
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}_{APS,\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))=\operatorname{Ind}(\eth\colon\mathcal{D}^{+}_{\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-})),$
where $\mathcal{D}_{\varepsilon}$ is the domain in (3.9) and
$\mathcal{D}_{APS,\varepsilon}$ is the domain in (1.22).
The main tool for proving Theorem 4.1 is the following theorem from [16]. We
define the ‘pseudodifferential Grassmanians’
$Gr_{APS,\varepsilon}=\left\\{\pi\in\Psi^{0}(\partial
M_{\varepsilon};\mathcal{S}):\pi^{2}=\pi\mbox{ and
}\sigma(\pi)=\sigma(\pi_{APS,\varepsilon})\right\\}.$ (4.1)
We endow $Gr_{APS,\varepsilon}$ with the norm topology. If $\pi\in
Gr_{APS,\varepsilon}$, then defining the domain
$\mathcal{D}_{\pi,\varepsilon}=\left\\{u\in
H^{1}(M_{\varepsilon};\mathcal{S}):(Id-\pi)(u\rvert_{\partial
M_{\varepsilon}})=0\right\\}$, the map
$\eth\colon\mathcal{D}_{\pi,\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S})$
is Fredholm. The following follows from [16, Theorem 20.8] and [16, Theorem
15.12]
###### Theorem 4.2.
If $\pi_{i}\in Gr_{APS,\varepsilon}$, $i=1,2$ lie in the same connected
component of $Gr_{APS,\varepsilon}$ then the elliptic boundary problems
$\eth\colon\mathcal{D}^{+}_{\pi_{i},\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon},\mathcal{S}^{-})$
have equal indices.
To apply Theorem 4.2 in our case, we will study the two families of boundary
values projectors $\pi_{APS,\varepsilon}$ and $\mathcal{E}_{\varepsilon}$
using the adiabatic calculus of Mazzeo and Melrose, [40].
### 4.1. Review of the adiabatic calculus
Consider a fiber bundle
$Z\hookrightarrow\widetilde{X}\xrightarrow{\phantom{x}\pi\phantom{x}}Y$. The
adiabatic double space $\widetilde{X}^{2}_{ad}$ is formed by radial blow up of
$\widetilde{X}^{2}\times[0,\varepsilon_{0})_{\varepsilon}$ along the fiber
diagonal, $\operatorname{diag}_{fib}(\widetilde{X})$ (see (2.7)) at
$\varepsilon=0$. That is,
$\widetilde{X}^{2}_{ad}=[\widetilde{X}^{2}\times[0,\varepsilon_{0})_{\varepsilon};\operatorname{diag}_{fib}(\widetilde{X})\times\left\\{\varepsilon=0\right\\}].$
Thus, $\widetilde{X}^{2}_{ad}$ is a manifold with corners with two boundary
hypersurfaces: the lift of $\left\\{\varepsilon=0\right\\}$, which we continue
to denote by $\left\\{\varepsilon=0\right\\}$, and the one introduced by the
blowup, which we call $\operatorname{ff}$. Similar to the edge front face
above, $\operatorname{ff}$ is a bundle over $Y$ whose fibers are isomorphic to
$Z^{2}\times\mathbb{R}^{b}$ where $b=\dim Y$, and in fact this is the fiber
product of $\pi^{*}T^{*}Y$ and $\widetilde{X}.$
$\mbox{We define $\operatorname{ff}_{y}$ to be the fiber of
$\operatorname{ff}$ lying above $y$}.$
$\mathcal{Y}=0$$Z^{2}$$\mathcal{Y}$$\Delta_{ad}$$\left\\{\varepsilon=0\right\\}$$\operatorname{ff}$$\operatorname{ff}_{y}$
Figure 4. The adiabatic double space.
The adiabatic vector fields on the fibration $\widetilde{X}$ are families of
vector fields $V_{\varepsilon}$ parametrized smoothly in
$\varepsilon\in[0,\varepsilon)$, such that $V_{0}$ is a vertical vector field,
i.e., a section of $T\widetilde{X}/Y.$ Locally these are
$C^{\infty}(\widetilde{X}\times[0,\varepsilon_{0})_{\varepsilon})$ linear
combinations of the vector fields
$\partial_{z},\quad\varepsilon\partial_{y}.$
Such families of vector fields are in fact sections of a vector bundle
$T_{ad}(\partial M)\longrightarrow\partial
M\times[0,\varepsilon_{0})_{\varepsilon}.$ (4.2)
We will now define adiabatic differential operators on sections of
$\mathcal{S}$. The space of $m^{th}$ order adiabatic differential operators
$\operatorname{Diff}^{m}_{ad}(\widetilde{X};\mathcal{S})$ is the space of
differential operators obtained by taking
$C^{\infty}(\widetilde{X};\operatorname{End}(\mathcal{S}))$ combinations of
powers (up to order $m$) of adiabatic vector fields. An adiabatic differential
operator $P$ admits a normal operator $N(P),$ obtained by letting $P$ act on
$\widetilde{X}\times\widetilde{X}\times[0,\varepsilon_{0})_{\varepsilon}$,
pulling back $P$ to $\widetilde{X}^{2}_{ad}$, and restricing it to
$\operatorname{ff}$. The normal operator acts tangentially along the fibers of
$\operatorname{ff}$ over $Y$, and $N(P)_{y}$ will denote the operator on
sections of $\mathcal{S}$ restricted to over $\operatorname{ff}_{y}$. More
concretely, if $P$ is an adiabatic operator of order $m$, then near a point
$y_{0}$ in $Y$, we can write
$P=\sum_{\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq
m}a_{\alpha,\beta}(z,y,\varepsilon)\partial_{z}^{\alpha}(\varepsilon\partial_{y})^{\beta},$
for $y$ near $y_{0}$, where $a_{\alpha,\beta}(z,y,\varepsilon)$ is a smooth
family of endomorphisms of $\mathcal{S}$. The normal operator is given by
$N(P)_{y_{0}}=\sum_{\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq
m}a_{\alpha,\beta}(z,y_{0},0)\partial_{z}^{\alpha}\partial_{\mathcal{Y}}^{\beta},$
where $\mathcal{Y}$ are coordinates on $\mathbb{R}^{b}$. Thus $N(P)_{y_{0}}$
is a differential operator on $Z_{y_{0}}\times T_{y_{0}}Y$ that is constant
coefficient in the $TY$ direction.
Returning to the case that $\widetilde{X}=\partial M$ for $M$ a compact
manifold with boundary, we take a collar neighborhood
$\mathcal{U}\simeq\partial M\times[0,\varepsilon_{0})_{x}$ as in (2.1), and
treating the boundary defining function, $x$, as the parameter $\varepsilon$
in the previous paragraph, identify the adiabatic double space $(\partial
M)^{2}_{ad}$ with a blow up of
$\left\\{x=x^{\prime}\right\\}\subset\mathcal{U}\times\mathcal{U}$.
###### Lemma 4.3.
The tangential operator $\widetilde{\eth}_{\varepsilon}$ defined in (1.18)
lies in $\operatorname{Diff}^{m}_{ad}(\partial M;\mathcal{S})$. The normal
operator of $\widetilde{\eth}_{\varepsilon}$ satisfies
$N(\widetilde{\eth}_{\varepsilon})_{y}=\eth^{Z}_{y}-c_{\nu}\eth_{\mathcal{Y}},$
where $\eth^{Z}_{y}$ is as in (2.3), and $\eth_{\mathcal{Y}}$ is the standard
Dirac operator on $T_{y}Y$.
###### Proof.
This follows from equation (2.15) above. ∎
The space of adiabatic pseudodifferential operators with bounds on
$\widetilde{X}$ of order $m$ acting on sections of $\mathcal{S}$, denoted
$\Psi^{m}_{ad,bnd}(\widetilde{X};\mathcal{S})$, is the space of families of
pseudodifferential operators
$\left\\{A_{\varepsilon}\right\\}_{0<\varepsilon<\varepsilon_{0}}$, where
$A_{\varepsilon}$ is a (standard) $\Psi$ of order $m$ for each $\varepsilon$,
and whose integral kernel of $A_{\varepsilon}$ is conormal to the lifted
diagonal
$\Delta_{ad}:=\overline{\operatorname{diag}_{\widetilde{X}}\times(0,\varepsilon_{0})_{\varepsilon}}$,
smoothly up to $\operatorname{ff}$. To be precise, the Schwartz kernel of an
operator $A\in\Psi^{m}(\partial M;\mathcal{S})$ is given by a family of
Schwartz kernels $K_{A_{\varepsilon}}=K_{1,\varepsilon}+K_{2,\varepsilon}$
where $K_{1,\varepsilon}$ is conormal of order $m$ at $\Delta_{ad}$ smoothly
down to $\operatorname{ff}$ and supported near $\Delta_{ad}$, and
$K_{2,\varepsilon}$ is smooth on the interior and bounded at the boundary
hypersurfaces.
An adiabatic pseudodifferential operator $A\in\Psi^{m}(\partial
M;\mathcal{S})$ with bounds comes with two crucial pieces of data: a principal
symbol and a normal operator. The principal symbol $\sigma(A)(\varepsilon)$ is
the standard one defined for a conormal distribution, i.e. as a homogeneous
section of $N^{*}(\Delta;\operatorname{End}(\mathcal{S}))\otimes\Omega^{1/2}$,
the conormal bundle to the lifted diagonal (with coefficients in half-
densities). In our case $N^{*}(\Delta)$ is canonically isomorphic to
$T^{*}_{ad}(\partial M)$, the dual bundle to $T_{ad}(\partial M)$ defined in
(4.2); in particular, the symbol of $A$ is a map $\sigma(A)\colon
T^{*}_{ad}(\partial M)\longrightarrow
C^{\infty}(M;\operatorname{End}(\mathcal{S}))$, well defined only to leading
order, and smooth down to $\operatorname{ff}$. The normal operator is the
restriction of the Schwartz kernel of $A$ to the front face
$N(A)=K_{A}\rvert_{\operatorname{ff}}.$
We thus have maps
$\Psi^{m-1}_{ad}(\partial M;\mathcal{S})\hookrightarrow\Psi^{m}_{ad}(\partial
M;\mathcal{S})\xrightarrow{\phantom{x}\sigma\phantom{x}}S^{m}(T^{*}_{ad}(\partial
M;\mathcal{S}))\otimes\Omega^{1/2},$
and
$N\colon\Psi^{m}_{ad}(\partial
M;\mathcal{S})\longrightarrow\Psi^{m}_{\operatorname{ff},ad}(\partial
M;\mathcal{S}).$
For fixed $\eta$, we use the same eigenvectors $\phi_{i,\pm}$ $\eth^{Z}_{y}$
and $ic(\widehat{\eta})$ as in (2.20) above, and consider the spaces
$\begin{split}\mathcal{W}_{i}&=\operatorname{span}\left\\{\phi_{i,+},\phi_{i,-}\right\\}.\end{split}$
(4.3)
We have the following.
###### Lemma 4.4.
The layer potential $\mathcal{E}_{\varepsilon}$ is a zero-th order adiabatic
family (with bounds), i.e. $\mathcal{E}_{\varepsilon}\in\Psi^{0}_{ad}(\partial
M;\mathcal{S})$. Using the vectors $\Pi(\eta,i)$ from (2.28), the normal
symbol of $\mathcal{E}_{\varepsilon}$ satisfies
$\begin{split}\Pi(\eta,i)\widehat{N_{y}(\mathcal{E}_{\varepsilon})}(y,\eta)\Pi^{*}(\eta,i)&=\mathcal{N}_{\mu_{i},\left\lvert\eta\right\rvert},\end{split}$
(4.4)
where
$\begin{split}\mathcal{N}_{\mu,\left\lvert\eta\right\rvert}&=\left\lvert\eta\right\rvert\left(\begin{array}[]{cc}I_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert)K_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert)&I_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert)K_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert)\\\
I_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert)K_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert)&I_{\left\lvert\mu-1/2\right\rvert}(\left\lvert\eta\right\rvert)K_{\left\lvert\mu+1/2\right\rvert}(\left\lvert\eta\right\rvert)\end{array}\right)\end{split}$
(4.5)
and $I_{\cdot},$ $K_{\cdot}$ denote modified Bessel functions.
###### Proof.
That $\mathcal{E}_{\varepsilon}$ is an adiabatic pseudodifferential operator
follows from (3.10). The formula in (4.4)-(4.5) follows from the Fourier
decomposition of the normal operator of the generalized inverse $Q$ in
Proposition 2.12, since by (3.3) the operator $\mathcal{E}_{\varepsilon}$ is
obtained by taking the limit in (2.29) as $\sigma=x/x^{\prime}\uparrow 1$ and
checking that $\lim_{\sigma\uparrow
1}\mathcal{M}_{\mu,\left\lvert\eta\right\rvert}(\sigma,1)=\mathcal{N}_{\mu,\left\lvert\eta\right\rvert}$.
∎
### 4.2. APS projections as an adiabatic family
To study the integral kernel of the projector $\pi_{APS,\varepsilon}$ we will
make use of the fact that the boundary Dirac operator
$\widetilde{\eth}_{\varepsilon}$ from (1.18) is invertible for small
$\varepsilon$. This is a general fact about adiabatic pseudodifferential
operators: invertibility at $\varepsilon=0$ implies invertibility for small
epsilon, or formally
###### Theorem 4.5.
Let $A_{\varepsilon}\in\Psi^{m}_{ad}(M;\mathcal{S})$ and assume that on each
fiber $\operatorname{ff}_{y}$ the Fourier transform of the normal operator
$\widehat{N(A_{\varepsilon})}_{y}$ is invertible on
$L^{2}(Z;\mathcal{S}_{y},k_{y})$, with $\mathcal{S}_{y}$ the restriction of
the spinor bundle to the fiber over $y$ and $k_{y}=g_{N/Y}\big{\rvert}_{y}.$
Then $A_{\varepsilon}$ is invertible for small $\varepsilon$.
It is well known [7] that for each fixed $\varepsilon>0$,
$\pi_{APS,\varepsilon}$ is a pseudodifferential operator of order $0$. As
$\varepsilon$ varies, these operators form an adiabatic family:
###### Lemma 4.6.
The family $\pi_{APS,\varepsilon}$ lies in $\Psi^{0}_{ad}(\partial
M;\mathcal{S})$. Its normal symbol $N(\pi_{APS,\varepsilon})$ satisfies
$\begin{split}N(\pi_{APS,\varepsilon})&=\frac{1}{2}N(\widetilde{\eth}_{\varepsilon})^{-1}\left(N(\widetilde{\eth}_{\varepsilon})-\left\lvert
N(\widetilde{\eth}_{\varepsilon})\right\rvert\right).\end{split}$ (4.6)
###### Proof.
By Assumption 2.1, $\widetilde{\eth}_{\varepsilon}$ is invertible for small
$\varepsilon$. Indeed, by (4.8), $N(\widetilde{\eth}_{\varepsilon})_{y}$ does
not have zero as an eigenvalue. The projectors $\pi_{APS,\varepsilon}$ can be
expressed in terms of functions of the tangential operators
$\widetilde{\eth}_{\varepsilon}$ [7] via the formula
$\pi_{APS,\varepsilon}=\frac{1}{2}\widetilde{\eth}_{\varepsilon}^{-1}\left(\widetilde{\eth}_{\varepsilon}-\left\lvert\widetilde{\eth}_{\varepsilon}\right\rvert\right).$
(4.7)
Following [51], the operator
$\widetilde{\eth}_{\varepsilon}^{-1}|\widetilde{\eth}_{\varepsilon}|$ is in
$\Psi_{ad}^{1}(\partial X;\mathcal{S})$ and has the expected normal operator,
namely the one obtained by applying the appropriate functions to the normal
operator of $N(\widetilde{\eth}_{\varepsilon})_{y}$ and composing them. ∎
We compute that the operator $\widehat{N(\widetilde{\eth}_{\varepsilon})}_{y}$
acts on the spaces $\mathcal{W}_{i}$ from (4.3) by
$\begin{split}\widehat{N(\widetilde{\eth}_{\varepsilon})}_{y}(\eta)\phi_{i,\pm}&=\pm\mu\phi_{i,\pm}-\left\lvert\eta\right\rvert\phi_{i,\mp}.\end{split}$
That is to say, with $\Pi(\eta,i)$ as in (2.28),
$\Pi(\eta,i)\widehat{N(\widetilde{\eth}_{\varepsilon})}_{y}(\eta)\Pi^{*}(\eta,i)=\left(\begin{array}[]{cc}\mu&-\left\lvert\eta\right\rvert\\\
-\left\lvert\eta\right\rvert&-\mu\end{array}\right)$ (4.8)
Thus,
$\begin{split}\Pi(\eta,i)\widehat{N(\widetilde{\eth}_{\varepsilon}^{-1}|\widetilde{\eth}_{\varepsilon}|)}_{y}(\eta)\Pi^{*}(\eta,i)=\frac{1}{(\mu^{2}+\left\lvert\eta\right\rvert^{2})^{1/2}}\left(\begin{array}[]{cc}\mu&-\left\lvert\eta\right\rvert\\\
-\left\lvert\eta\right\rvert&-\mu\end{array}\right)\Pi_{\mu_{i},j}\end{split}$
Using (4.7), we obtain
$\begin{split}\Pi(\eta,i)\widehat{N}(\pi_{APS,\varepsilon})_{y}(\eta)\Pi^{*}(\eta,i)&=\mathcal{N}^{APS}_{\mu,\left\lvert\eta\right\rvert}\mbox{
where }\\\
\mathcal{N}^{APS}_{\mu,\left\lvert\eta\right\rvert}&=\frac{1}{2}\left(Id_{2\times
2}+\frac{1}{(\mu^{2}+\left\lvert\eta\right\rvert^{2})^{1/2}}\left(\begin{array}[]{cc}-\mu&\left\lvert\eta\right\rvert\\\
\left\lvert\eta\right\rvert&\mu\end{array}\right)\right)\end{split}$
###### Theorem 4.7.
There exists a smooth family $\pi_{\varepsilon,t}$, parametrized by
$t\in[0,1]$ satisfying:
1) for fixed $t$, $\pi_{\varepsilon,t}\in Gr_{APS,\varepsilon}$, the
Grassmanians defined in (4.1), and
2) $\pi_{\varepsilon,0}=\mathcal{E}_{\varepsilon}\mbox{ and
}\pi_{\varepsilon,1}=\pi_{APS,\varepsilon}.$
###### Proof.
The proof proceeds in two main steps. First, we construct a homotopy from the
normal operators $N(\mathcal{E}_{\varepsilon})$ to $N(\pi_{APS,\varepsilon})$.
Then we extend this homotopy to a homotopy of the adiabatic families as
claimed in the theorem.
For the homotopy of the normal operators, the main lemma will be the following
###### Claim 4.8.
For each $y\in Y$, the normal operators $N(\mathcal{E}_{\varepsilon})_{y}$ and
$N(\pi_{APS,\varepsilon})$, acting on $L^{2}(Z\times T_{y}Y;\mathcal{S}_{y})$,
satisfy
$\left\|N(\mathcal{E}_{\varepsilon})_{y}-N(\pi_{APS,\varepsilon})\right\|_{L^{2}\longrightarrow
L^{2}}<1-\delta,$
for some $\delta>0$ independent of $y$.
Assuming the claim for the moment, the following argument from [16, Chap 15]
furnishes a homotopy. In general, let $P$ and $Q$ be projections on a
separable Hilbert space. Define $T_{t}=Id+t(Q-P)(2P-Id)$, and note that
$T_{1}P=QT_{1}$. Now assume that $T_{t}$ is invertible for all $t$. Then the
operator
$F_{t}=T_{t}^{-1}PT_{t}$
is a homotopy from $P$ to $Q$, i.e. $F_{0}=P$, $F_{1}=Q$. This holds in
particular if $\left\|P-Q\right\|<1$, in which case $T_{t}$ is invertible by
Neumann series for $t\in[0,1]$.
To apply this in our context, we first take
$P=N(\mathcal{E}_{\varepsilon})_{y}$ and $Q=N(\pi_{APS,\varepsilon})_{y}$, and
see that the corresponding operator $T_{t}$ is invertible by Claim 4.8. Now
taking $P=\mathcal{E}_{\varepsilon}$ and $Q=\pi_{APS,\varepsilon}$ (so $P$,
$Q$, and $T_{t}$ depend on $\varepsilon$) by Theorem 4.5, $T_{t}$ is
invertible for small $\varepsilon$. Thus the homotopy
$F_{t}=F_{t}(\varepsilon)=:\pi_{\varepsilon,t}$ is well defined for small
$\varepsilon$. In fact, $\pi_{\varepsilon,t}$ is a smooth family of adiabatic
pseudodifferential projections with principal symbol equal to that of
$\pi_{APS,\varepsilon}$ for all $\varepsilon$.
Thus it remains to prove Claim 4.8. By the formulas for the normal operators
given in (4.4) and (4.6) and Plancherel, the claim will follow if we can show
that for each $\mu$ with $\left\lvert\mu\right\rvert>1/2$, and all
$\left\lvert\eta\right\rvert$, that
$\left\|\mathcal{N}_{\mu,\left\lvert\eta\right\rvert}-\mathcal{N}^{APS}_{\mu,\left\lvert\eta\right\rvert}\right\|<1-\delta,$
(4.9)
for some $\delta$ independent of $\mu\geq 1/2$ and
$\left\lvert\eta\right\rvert$. Here the norm is as a map of $\mathbb{R}^{2}$
with the standard Euclidean norm. We prove the bound in (4.9) using standard
bounds on modified Bessel functions in the Appendix, §7. ∎
## 5\. Proof of Main Theorem: limit of the index formula
Recall (e.g., [48, §2.14]) that if $E\longrightarrow M$ is a real vector
bundle of rank $k$ connection $\nabla^{E}$ and curvature tensor $R^{E}$ then
every smooth function (or formal power series)
$P:\mathfrak{so}(k)\longrightarrow\mathbb{C},$
that is invariant under the adjoint action of $SO(k),$ determines a closed
differential form $P(R^{E})\in{\mathcal{C}}^{\infty}(M;\Lambda^{*}T^{*}M).$ If
$\nabla^{E}_{1}$ is another connection on $E,$ with curvature tensor
$R^{E}_{1}$ then $P(R^{E})$ and $P(R^{E}_{1})$ differ by an exact form.
Indeed, define a family of connections on $E$ by
$\theta=\nabla^{E}_{1}-\nabla^{E}\in{\mathcal{C}}^{\infty}(M;T^{*}M\otimes\operatorname{Hom}(E)),\quad\nabla_{t}^{E}=(1-t)\nabla^{E}+t\nabla^{E}_{1}=\nabla^{E}+t\theta,$
denote the curvature of $\nabla^{E}_{t}$ by $R^{E}_{t},$ and let
$P^{\prime}(A;B)=\left.\frac{\partial}{\partial s}\right|_{s=0}P(A+sB).$
The differential form
$TP(\nabla^{E},\nabla^{E}_{1})=\int_{0}^{1}P^{\prime}(R^{E}_{t};\theta)\;dt$
satisfies
$dTP(\nabla^{E},\nabla^{E}_{1})=P(R^{E})-P(R^{E}_{1}).$
Now consider for $\varepsilon<1$ the truncated manifold
$M_{\varepsilon}=\\{x\geq\varepsilon\\}$ and the corresponding truncated
collar neighborhood $\mathscr{C}_{\varepsilon}=[\varepsilon,1]\times N.$ Let
$\nabla^{\operatorname{pt}}$ be the Levi-Civita connection of the metric
$g_{\operatorname{pt}}=dx^{2}+\varepsilon^{2}g_{Z}+\phi^{*}g_{Y}.$
The Atiyah-Patodi-Singer index theorem on $M_{\varepsilon}$ has the form [31,
32], cf. [26]
$\int_{M_{\varepsilon}}AS(\nabla)+\int_{\partial
M_{\varepsilon}}TAS(\nabla,\nabla^{\operatorname{pt}})-\tfrac{1}{2}\eta(\partial
M_{\varepsilon})$
where $AS$ is a characteristic form associated to a connection $\nabla$ and
$TAS(\nabla,\nabla^{\operatorname{pt}})$ is its transgression form with
respect to the connection $\nabla^{\operatorname{pt}}.$
The Levi-Civita connection of $g_{\operatorname{pt}}$ induces a connection on
$T_{\operatorname{ie}}M_{\varepsilon},$ which we continue to denote
$\nabla^{\operatorname{pt}}.$ Let
$\theta^{\varepsilon}=\nabla-\nabla^{\operatorname{pt}}.$ Since
$g_{\operatorname{ie}}$ and $g_{\operatorname{pt}}$ coincide on
$\\{x=\varepsilon\\}$ we have
$g_{\operatorname{pt}}(\nabla^{\operatorname{pt}}_{A}B,C)\big{\rvert}_{x=\varepsilon}=g_{\operatorname{ie}}(\nabla^{\operatorname{ie}}_{A}B,C)\big{\rvert}_{x=\varepsilon}\text{
if }A,B,C\in{\mathcal{C}}^{\infty}(\mathscr{C}_{\varepsilon};TN).$
On the other hand, if
$A,B,C\in\\{\partial_{x},\tfrac{1}{x}V,\widetilde{U}\\},$ we have
$g_{\operatorname{pt}}(\nabla^{\operatorname{pt}}_{A}B,C)=0\text{ if
}\partial_{x}\in\\{A,B,C\\}\text{ except for
}g_{\operatorname{pt}}(\nabla^{\operatorname{pt}}_{\partial_{x}}\tfrac{1}{x}V_{1},\tfrac{1}{x}V_{2})=-\frac{\varepsilon^{2}}{x^{3}}g_{Z}(V_{1},V_{2}).$
Note that, analogously to (1.9), we have
$j_{0}^{*}\nabla^{\operatorname{pt}}=j_{0}^{*}\nabla^{v}\oplus
j_{0}^{*}\nabla^{h}$
where, as above, $j_{\varepsilon}:N\hookrightarrow\mathscr{C}$ is the
inclusion of $\\{x=\varepsilon\\},$ and
$\nabla^{v}=\mathbf{v}\circ\nabla\circ\mathbf{v}$ is the restriction of the
Levi-Civita connection to $TN/Y.$
Thus
$\theta^{\varepsilon}_{A}(B)\big{\rvert}_{x=\varepsilon}=0\\\ \text{ except
for
}\theta^{\varepsilon}_{\partial_{x}}(\tfrac{1}{x}V)\big{\rvert}_{x=\varepsilon}=\tfrac{1}{\varepsilon}\tfrac{1}{x}V,\quad\theta^{\varepsilon}_{V}(\partial_{x})\big{\rvert}_{x=\varepsilon}=\tfrac{1}{x}V,\quad\theta^{\varepsilon}_{V_{1}}(\tfrac{1}{x}V_{2})\big{\rvert}_{x=\varepsilon}=-g_{Z}(V_{1},V_{2})\partial_{x}.$
(5.1)
In particular note that $j_{\varepsilon}^{*}\theta^{\varepsilon}$ is
independent of $\varepsilon$ and is equal to
$j_{\varepsilon}^{*}\theta^{\varepsilon}=j_{0}^{*}\nabla^{v_{+}}-j_{0}^{*}\nabla^{v}.$
Next we need to compute the restriction to $x=\varepsilon$ of the curvature
$\Omega_{t}$ of the connection
$(1-t)\nabla+t\nabla^{\operatorname{pt}}=\nabla+t\theta^{\varepsilon}.$
Locally, with $\omega$ the local connection one-form of $\nabla$ (1.10), the
curvature $\Omega_{t}$ is given by
$\Omega_{t}=d(\omega+t\theta^{\varepsilon})+(\omega+t\theta^{\varepsilon})\wedge(\omega+t\theta^{\varepsilon})=\Omega+t(d\theta^{\varepsilon}+[\omega,\theta^{\varepsilon}]_{s})+t^{2}\theta^{\varepsilon}\wedge\theta^{\varepsilon}$
where $[\cdot,\cdot]_{s}$ denotes the supercommutator with respect to form
parity, so that
$[\omega,\theta^{\varepsilon}]_{s}=\omega\wedge\theta^{\varepsilon}+\theta^{\varepsilon}\wedge\omega.$
In terms of the splitting (1.7) we have
$\Omega\big{\rvert}_{x=\varepsilon}=\begin{pmatrix}\Omega_{v_{+}}&\mathcal{O}(\varepsilon)\\\
\mathcal{O}(\varepsilon)&\phi^{*}\Omega_{Y}\end{pmatrix},\quad\omega=\begin{pmatrix}\omega_{v_{+}}&\mathcal{O}(x)\\\
\mathcal{O}(x)&\phi^{*}\omega_{Y}+\mathcal{O}(x^{2})\end{pmatrix},\quad
j_{\varepsilon}^{*}\theta^{\varepsilon}=\begin{pmatrix}\widetilde{\theta}&0\\\
0&0\end{pmatrix}$
and hence
$\Omega_{t}\big{\rvert}_{x=\varepsilon}=\begin{pmatrix}\Omega_{v_{+}}+t(d\widetilde{\theta}+[\omega_{v_{+}},\widetilde{\theta}]_{s})+t^{2}\widetilde{\theta}\wedge\widetilde{\theta}&\mathcal{O}(\varepsilon)\\\
\mathcal{O}(\varepsilon)&\phi^{*}\Omega_{Y}\end{pmatrix}.$
In particular, if we denote $\Omega_{v_{+},t}$ the curvature of the connection
$(1-t)\nabla^{v_{+}}+t\nabla^{v}$ on the bundle
$\langle\partial_{x}\rangle+TN/Y,$ we have
$j_{\varepsilon}^{*}\Omega_{t}=j_{0}^{*}\Omega_{t}+\mathcal{O}(\varepsilon),\text{
with }j_{0}^{*}\Omega_{t}=\begin{pmatrix}\Omega_{v_{+},t}&0\\\
0&\phi^{*}\Omega_{Y}\end{pmatrix}.$
It follows that
$\lim_{\varepsilon\to
0}j_{\varepsilon}^{*}T\widehat{A}(\nabla,\nabla^{\operatorname{pt}})=\int_{0}^{1}\left.\frac{\partial}{\partial
s}\right|_{s=0}j_{0}^{*}\widehat{A}(\Omega_{Y})\widehat{A}(\Omega_{v_{+},t}+s\widetilde{\theta})\;dt=\widehat{A}(Y)\wedge
T\widehat{A}(\nabla^{v_{+}},\nabla^{v})$
and similarly for any multiplicative characteristic class.
We can now prove the main theorem, whose statement we recall for the reader’s
convenience.
###### Theorem 5.1.
Let $X$ be stratified space with a single singular stratum endowed with an
incomplete edge metric $g$ and let $M$ be its resolution. If $\eth$ is a Dirac
operator associated to a spin bundle $\mathcal{S}\longrightarrow M$ and $\eth$
satisfies Assumption 2.1, then
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=\int_{M}\widehat{A}(M)+\int_{Y}\widehat{A}(Y)\left(-\frac{1}{2}\widehat{\eta}(\eth_{Z})+\int_{\partial
M/Y}T\widehat{A}(\nabla^{v_{+}},\nabla^{\operatorname{pt}})\right)$
where $\widehat{A}$ denotes the $\widehat{A}$-genus,
$T\widehat{A}(\nabla^{v_{+}},\nabla^{\operatorname{pt}})$ denotes the
transgression form of the $\widehat{A}$ genus associated to the connections
$\nabla^{v_{+}}$ and $\nabla^{\operatorname{pt}}$ above, and $\widehat{\eta}$
the $\eta$-form of Bismut-Cheeger [11].
###### Proof of Main Theorem.
Combining Theorems 3.1 and 4.1 we know that, for $\varepsilon$ small enough,
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=\operatorname{Ind}(\eth\colon\mathcal{D}_{\varepsilon}^{+}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))\\\
=\operatorname{Ind}(\eth\colon\mathcal{D}^{+}_{APS,\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-})).$
Hence
$\operatorname{Ind}(\eth\colon\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=\lim_{\varepsilon\to
0}\operatorname{Ind}(\eth\colon\mathcal{D}^{+}_{APS,\varepsilon}\longrightarrow
L^{2}(M_{\varepsilon};\mathcal{S}^{-}))\\\ =\lim_{\varepsilon\to
0}\int_{M_{\varepsilon}}\widehat{A}(\nabla)+\int_{\partial
M_{\varepsilon}}T\widehat{A}(\nabla,\nabla^{\operatorname{pt}})-\tfrac{1}{2}\eta(\partial
M_{\varepsilon})\\\ =\int_{M}\widehat{A}(M)+\int_{\partial
M}\widehat{A}(Y)\wedge
T\widehat{A}(\nabla^{v_{+}},\nabla^{v})-\tfrac{1}{2}\int_{Y}\widehat{A}(Y)\widehat{\eta}(\eth_{Z}).$
∎
### 5.1. Four-dimensions with circle fibers
An incomplete edge space whose link is a sphere is topologically a smooth
space. So let us consider a four-dimensional manifold $X$ with a submanifold
$Y$ and a Riemannian metric on $X\setminus Y$ that in a tubular neighborhood
of $Y$ takes the form
$dx^{2}+x^{2}\beta^{2}d\theta^{2}+\phi^{*}g_{Y}.$
Here $\beta$ is a constant and $2\pi\beta$ is the ‘cone angle’ along the edge.
Recall that the circle has two distinct spin structures, and with the round
metric the corresponding Dirac operators have spectra equal to either the even
or odd integer multiples of $\pi.$ The non-trivial spin structure on the
circle is the one that extends to the disk, and so any spin structure on $X$
will induce non-trivial spin structures on its link circles. Thus, cf. [24,
Proposition 2.1], the generalized Witt assumption will be satisfied as long as
$\beta\leq 1.$
In this setting the relevant characteristic class is the first Pontryagin
class: for a two-by-two anti-symmetric matrix $A,$ let
$p_{1}(A)=-c_{2}(A)=-\frac{1}{8\pi^{2}}\operatorname{Tr}(A^{2}).$
Note that $p_{1}^{\prime}(A;B)=-\frac{1}{(2\pi)^{2}}\operatorname{Tr}(AB),$
and so
$Tp_{1}(\nabla,\nabla^{\operatorname{pt}})=-\frac{1}{(2\pi)^{2}}\int_{0}^{1}\operatorname{Tr}j_{0}^{*}(\theta\wedge\Omega_{t})\;dt$
with
$\Omega_{t}=\Omega+t(d\theta^{\varepsilon}+[\omega,\theta^{\varepsilon}]_{s})+t^{2}\theta^{\varepsilon}\wedge\theta^{\varepsilon}.$
We can simplify this formula. Indeed, note that if $\\{V_{i}\\}$ are an
orthonormal frame for $TN/Y$ then
$j_{\varepsilon}^{*}(\theta^{\varepsilon}\wedge\theta^{\varepsilon})=\sum\Theta_{ij}V_{i}^{\flat}\wedge
V_{j}^{\flat}\text{ with
}\Theta_{ij}(\tfrac{1}{x}V_{k})=-\delta_{kj}\tfrac{1}{x}V_{i},$
and so in particular $\dim Z=1$ implies
$j_{\varepsilon}^{*}(\theta^{\varepsilon}\wedge\theta^{\varepsilon})=0.$
Moreover with respect to the splitting (1.6), $\theta$ is off-diagonal and
$\Omega$ is on-diagonal, hence
$\operatorname{Tr}j_{0}^{*}(\theta\wedge\Omega)=0$ and
$Tp_{1}(\nabla,\nabla^{\operatorname{pt}})=-\frac{1}{4\pi^{2}}\int_{0}^{1}t\operatorname{Tr}j_{0}^{*}(\theta\wedge
d\theta)\;dt=-\frac{1}{8\pi^{2}}\operatorname{Tr}j_{0}^{*}(\theta\wedge
d\theta).$
Next let us consider $\theta$ in more detail. From (5.1), with respect to the
splitting (1.6), we have
$j_{0}^{*}\theta=\begin{pmatrix}0&\operatorname{Id}&0\\\
-\operatorname{Id}&0&0\\\ 0&0&0\end{pmatrix}\alpha$
where $\alpha$ is a vertical one-form of $g_{Z}$ length one. This form is
closely related to the ‘global angular form’ described in [17, pg. 70].
Indeed, $\alpha$ restricts to each fiber to be $\beta d\theta$ which
integrates out to $2\pi\beta.$ It follows that $d\alpha=-2\pi\beta\phi^{*}e,$
where $e\in{\mathcal{C}}^{\infty}(Y;T^{*}Y)$ is the Euler class of $Y$ as a
submanifold of $X,$ and hence
$j_{0}^{*}(\theta\wedge d\theta)=\begin{pmatrix}-\operatorname{Id}&0&0\\\
0&-\operatorname{Id}&0\\\
0&0&0\end{pmatrix}\alpha\wedge(-2\pi\beta\phi^{*}e).$
Thus we find
$\int_{\partial
M}Tp_{1}(\nabla,\nabla^{\operatorname{pt}})=-\frac{1}{8\pi^{2}}\int_{\partial
M}\operatorname{Tr}j_{0}^{*}(\theta\wedge
d\theta)=-\frac{1}{8\pi^{2}}\int_{\partial
M}(4\pi\beta\alpha\wedge\phi^{*}e)\\\ =-\beta^{2}\int_{Y}e=-\beta^{2}[Y]^{2}.$
(5.2)
This computation yields a formula for the index of the Dirac operator and,
combined with results of Dai and Dai-Zhang, also a proof of the signature
theorem of Atiyah-LeBrun.
###### Theorem 5.2.
Let $X$ be an oriented four dimensional manifold, $Y$ a smooth compact
oriented embedded surface, and $g$ an incomplete edge metric on $X\setminus Y$
with cone angle $2\pi\beta$ along $Y.$
1)If $X$ is spin and $\beta\in(0,1],$
$\operatorname{Ind}(\eth:\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=-\frac{1}{24}\int_{M}p_{1}(M)+\frac{1}{24}(\beta^{2}-1)[Y]^{2}.$
2)[Atiyah-LeBrun [6]] The signature of $X$ is given by
$\operatorname{sgn}(X)=\frac{1}{12\pi^{2}}\int_{M}(|W_{+}|^{2}-|W_{-}|^{2})\;d\mu+\frac{1-\beta^{2}}{3}[Y]^{2}.$
###### Proof.
1) As mentioned above, the fact that the spin structure extends to all of $X$
and $\beta\in(0,1]$ implies that the generalized Witt assumption for $\eth$ is
satisfied. The degree four term of the $\widehat{A}$ genus is $-p_{1}/24,$ so
applying our index formula (4) and using the derivation of the local boundary
term for $p_{1}$ in (5.2) gives
$\operatorname{Ind}(\eth:\mathcal{D}^{+}\longrightarrow
L^{2}(M;\mathcal{S}^{-}))=-\frac{1}{24}\int_{M}p_{1}(M)-\frac{1}{24}\beta^{2}[Y]^{2}+\int_{Y}\widehat{A}(Y)\left(-\frac{1}{2}\widehat{\eta}(\eth_{Z})\right),$
where the final term on the right is the limit $(1/2)\lim_{\varepsilon\to
0}\eta_{\varepsilon}$ where $\varepsilon_{\varepsilon}$ is the eta-invariants
induced on the boundary of $M_{\varepsilon}$ as $\varepsilon\to 0$. Thus we
claim (and it remains to prove) that the adiabatic limit of the eta-invariant
for the spin Dirac operator is
$\lim_{\varepsilon\to 0}\tfrac{1}{2}\eta_{\varepsilon}=\frac{1}{24}[Y]^{2},$
(5.3)
i.e. the limit of the eta-invariants is the opposite of the local boundary
term when $\beta=1$, which indeed it should be since in that case the metric
is smooth across $x=0$.
Although other derivations of the adiabatic eta invariant exist [27], we
prefer to give on here which we find intuitive and which fits nicely with
arguments above. To this end, we consider $N$, a disc bundle over a smooth
manifold $Y$, and we assume $N$ is spin. We will show below that $N$ admits a
positive scalar curvature metric. Thus, given a spin structure and metric, the
index of $\eth$ vanishes on $N$. If we furthermore note that $N$ is
diffeomorphic to $[0,1)_{x}\times X$ where $X$ is a circle bundle over $Y$,
and let $N^{\varepsilon}=[0,\varepsilon)_{x}\times X$, we may consider metrics
$g=dx^{2}+f^{2}(x)k+h,$ (5.4)
where $h$ is the pullback of a metric on $Y$, $k\in
Sym^{0,2}(N^{\varepsilon})$ $x$ and $dx$-independent and restrics to a
Riemannian metric on the fibers of $X$. We assume $f$ is smooth across $x=0$
with $f(x)=x+O(x^{2})$ which implies that $g$ is smooth on $N$. Using the
computation of the connection above, with respect to the orthonormal basis,
$X_{i},\frac{1}{f}U,\partial_{x}$ the connection one form of $g$ is
$\begin{split}\omega&=\left(\begin{array}[]{c|c|c}\widetilde{\omega}_{\widetilde{h}}-f^{2}\frac{1}{2}g^{\partial}\mathcal{R}&-fg^{\partial}\left(\widehat{\@slowromancap
ii@}+\frac{1}{2}\widehat{\mathcal{R}}\right)&0\\\ \hline\cr
fg^{\partial}\left(\widehat{\@slowromancap
ii@}+\frac{1}{2}\widehat{\mathcal{R}}\right)&0&f^{\prime}U^{\sharp}\\\
\hline\cr 0&-f^{\prime}U^{\sharp}&0\end{array}\right)\end{split}$ (5.5)
where $g^{\partial}=k+h$ is the metric on the circle bundle $X$. We will take
$f(x)=f_{\varepsilon}(x)=x\chi_{\varepsilon}(x),$
where $\chi$ is a smooth positive function that is monotone decreasing with
$\chi(x)=1$ for $x\leq 1/3$ and $\chi(x)=\beta$ for $x\geq 2/3$. Then
$f=f^{\prime}=f^{\prime\prime}=O(1/\varepsilon)$, and using
$\Omega=d\omega+\omega\wedge\omega$, we see that
$\widehat{A}_{g}=FdVol_{g}$
where $F$ is a function that is $O(1/\varepsilon)$. Since
$Vol(N_{\varepsilon})=O(\varepsilon^{2})$,
$\int_{N_{\varepsilon}}\widehat{A}_{\varepsilon}=-\frac{1}{24}\int_{N_{\varepsilon}}p_{1}\to
0\mbox{ as }\varepsilon\to 0.$
Since the index of the Dirac operator vanishes on $N^{\varepsilon}$, applying
the APS formula gives
$\begin{split}0=-\frac{1}{24}\int_{N^{\varepsilon}}p_{1}+\int_{\partial
N^{\varepsilon}}Tp_{1}(\nabla,\nabla^{\operatorname{pt}})-\tfrac{1}{2}\eta(\partial
N^{\varepsilon}),\end{split}$ (5.6)
where $\nabla^{pt}$ is as in (3), and thus the limit of the trangression forms
is exactly as computed above. Thus by taking the $\varepsilon\to 0$ limit we
obtain (5.3).
To prove part 1) it remains to prove the existence of a positive scalar
curvature metric on $N$. To this end we take the metric $g$ as in (5.4) on
$N^{\varepsilon}$ now with
$f(x)=f_{\delta}(x)=\delta\sin(x/\delta).$ (5.7)
Note that $f=O(\varepsilon),f^{\prime}=O(\varepsilon/\delta)$. Then curvature
equals
$\begin{split}\Omega&=d\omega+\omega\wedge\omega\\\
&=\left(\begin{array}[]{c|c|c}\widetilde{\Omega}_{\widetilde{h}}&0&0\\\
\hline\cr 0&0&f^{\prime\prime}dx\wedge U^{\sharp}\\\ \hline\cr
0&-f^{\prime\prime}dx\wedge
U^{\sharp}&0\end{array}\right)+O(\varepsilon)+O(\varepsilon/\delta).\end{split}$
(5.8)
Denoting our orthonormal basis by $e_{i},i=1,\dots,n$ and taking traces gives
$scal_{g}=\delta^{ik}\delta^{jl}\Omega_{ij}(e_{k},e_{l})=scal_{h}+\frac{2}{\delta^{2}}+O(\varepsilon/\delta),$
and thus taking $\varepsilon/\delta=1$ and $\delta$ small gives a positive
scalar curvature metric.
2) Since $X$ is a smooth manifold we can use Novikov additivity of the
signature to decompose the signature as
$\operatorname{sgn}(X)=\operatorname{sgn}(X\setminus
M_{\varepsilon})+\operatorname{sgn}(M_{\varepsilon}).$
Identifying $X\setminus M_{\varepsilon}$ with a disk bundle over $Y$ we have
from [25, Pg. 314] that
$\operatorname{sgn}(X\setminus
M_{\varepsilon})=\operatorname{sgn}\left(\int_{Y}e\right),$
i.e., the signature is the sign of the self-intersection number of $Y$ in $X.$
In fact this is a simple exercise using the Thom isomorphism theorem.
The Atiyah-Patodi-Singer index theorem for the signature of $M_{\varepsilon}$
yields
$\operatorname{sgn}(M_{\varepsilon})=\frac{1}{3}\int_{M_{\varepsilon}}p_{1}(\nabla)+\frac{1}{3}\int_{\partial
M_{\varepsilon}}Tp_{1}(\nabla,\nabla^{\operatorname{pt}})-\eta^{\operatorname{even}}_{\varepsilon}$
where $\eta^{\operatorname{even}}_{\varepsilon}$ is the eta-invariant of the
boundary signature operator restricted to forms of even degree. As
$\varepsilon\to 0,$ the eta invariant is undergoing adiabatic degeneration and
its limit is computed in [27, Theorem 3.2],
$\lim_{\varepsilon\to 0}=-\int_{Y}L(TY)(\coth
e-e^{-1})+\operatorname{sgn}\left(B_{e}\right)$
where $B_{e}$ is the bilinear form on $H^{0}(Y)$ given by $H^{0}(Y)\ni
c,c^{\prime}\mapsto cc^{\prime}\langle e,Y\rangle\in\mathbb{R},$ i.e., it is
again the sign of the self-intersection of $Y.$ (In comparing with [27] note
that the orientation of $\partial M_{\varepsilon}$ is the opposite of the
orientation of the spherical normal bundle of $Y$ in $X,$ and so
$\operatorname{sgn}(B_{e})=-\operatorname{sgn}(X\setminus M_{\varepsilon}).$)
The only term in $L(TY)(\coth e-e^{-1})$ of degree two is $\tfrac{1}{3}e,$ and
hence
$\operatorname{sgn}(X)=\frac{1}{3}\int_{X}p_{1}+\frac{1}{3}[Y]^{2}+\frac{1}{3}\left(-\beta^{2}[Y]^{2}\right)$
as required. (Note that we could also argue as in the Dirac case to compute
the limit of the eta invariants.)
∎
## 6\. Positive scalar curvature metrics
In this short section, we prove Theorem 3 following [23]. We recall the
statement of the theorem for the benefit of the reader:
###### Theorem 6.1.
Let $(M,g)$ be a spin space with an incomplete edge metric.
a) If the scalar curvature of $g$ is non-negative in a neighborhood of
$\partial M$ then the geometric Witt assumption (Assumption 2.1) holds.
b) If the scalar curvature of $g$ is non-negative on all of $M,$ and positive
somewhere, then $\operatorname{Ind}(\eth)=0.$
###### Proof.
a) Taking traces in (1.11), the scalar curvature $R_{g}$ satisfies
$R_{g}=R_{cone}+\mathcal{O}(1),$
where $R_{cone}$ is the scalar curvature of the cone with metric
$dx^{2}+x^{2}g_{N/Y}\rvert_{\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y}$,
as in (1.7). On the other hand, by [23, Sect. 4], the scalar curvature of an
exact cone $C(Z)$ is equal to $x^{-2}(R_{Z}-\dim(Z)(\dim(Z)-1))$, where
$R_{Z}$ is the scalar curvature of $Z$. Thus $R_{g}\geq 0$ implies that
$R_{Z}\geq 0$, which by [23, Lemma 3.5] shows that Assumption 2.1 holds.
b) First off, by Theorem 1, $\eth$ is essentially self-adjoint. That is, the
graph closure of $\eth$ on $C^{\infty}_{comp}(M)$ is self-adjoint, with domain
$\mathcal{D}$ from Theorem 1, and furthermore by the Main Theorem its index
satisfies (4).
From the Lichnerowicz formula [10],
$\eth^{*}\eth=\nabla^{*}\nabla+R/4$
where $R$ is the scalar curvature. Thus, for every $\phi\in
C^{\infty}_{comp}(M)$,
$\left\|\eth\phi\right\|_{L^{2}}=\left\|\nabla\phi\right\|_{L^{2}}+\langle
R\phi,\phi\rangle_{L^{2}}.$ We conclude that for all $\phi\in
C^{\infty}_{comp}(M),$
$\left\|\eth\phi\right\|_{L^{2}}\geq\left\|\eth\phi\right\|_{L^{2}}-\langle
R\phi,\phi\rangle_{L^{2}}\geq\left\|\nabla\phi\right\|_{L^{2}}\geq 0.$ (6.1)
This implies in particular that
$\mathcal{D}_{min}(\eth)=\mathcal{D}\subset\mathcal{D}_{min}(\nabla)$, where
we recall that $\mathcal{D}_{min}(P)$ refers to the graph closure of the
operator $P$ with domain $C^{\infty}_{comp}(M)$. We claim that the index of
the operator
$\eth\colon\mathcal{D}^{+}\longrightarrow L^{2}(M;\mathcal{S}^{-}).$
vanishes, so by formula (4), Theorem 3(b) holds. In fact, the kernel of $\eth$
on $\mathcal{D}$ consists only of the zero vector, since if
$\phi\in\mathcal{D}$ has $\eth\phi=0$, then since (6.1) holds on
$\mathcal{D}$, $\nabla\phi=0$ also. By the Lichnerowicz formula again,
$R\phi=0$, but since by assumption $R$ is not identically zero, $\phi$ must
vanish somewhere and by virtue of its being parallel, $\phi\equiv 0$. ∎
## 7\. Appendix
In this appendix we prove Claim 4.8 by using standard bounds on modified
Bessel functions to prove the sup norm bound (4.9): for each $\mu$ with
$\left\lvert\mu\right\rvert>1/2$, and all $\left\lvert\eta\right\rvert$,
$\left\|\mathcal{N}_{\mu,\left\lvert\eta\right\rvert}-\mathcal{N}^{APS}_{\mu,\left\lvert\eta\right\rvert}\right\|<1-\delta,$
Among references for modified Bessel functions we recall [5, 8, 9, 49].
To begin with, using the Wronskian equation (2.27), note that
$\operatorname{Tr}\mathcal{N}_{\mu,z}=\operatorname{Tr}\mathcal{N}^{APS}_{\mu,z}=1.$
Thus the difference $\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z}$ has two
equal eigenvalues and hence its norm is the square root of the determinant. We
now assume that $\mu\geq 1/2$, since the $\mu\leq-1/2$ case is treated the
same way. Using (2.27) again, we see that
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(\mu(I_{\mu-1/2}K_{\mu+1/2}-I_{\mu+1/2}K_{\mu-1/2})\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right),\end{split}$
(7.1)
and we want to show that for some $\delta>0$ independent of $\mu\geq 1/2$ and
$z\geq 0$,
$\begin{split}-1+\delta\leq\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})\leq
1-\delta.\end{split}$ (7.2)
To begin with, we prove that
$0\leq zI_{\nu}(z)K_{\nu}(z)\leq 1/2\quad\mbox{ for }\quad\nu\geq 1/2,z\geq
0.$ (7.3)
In fact, we claim that for $\nu\geq 1/2$, $zK_{\nu}(z)I_{\nu}(z)$ is monotone.
To see that this holds, differentiate
$\begin{split}(zK_{\nu}(z)I_{\nu}(z))^{\prime}&=K_{\nu}I_{\nu}+z(K^{\prime}_{\nu}I_{\nu}+K_{\nu}I^{\prime}_{\nu})=K_{\nu}I_{\nu}(1+\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}).\end{split}$
Thus we want to show that
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)})\geq-1$.
Using [9, Eqn. 5.1], for $\nu\geq 1/2$
$\left(\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}\right)^{\prime}+\left(\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}\right)^{\prime}\leq
0,$
so the quantity
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}$
is monotone decreasing. In fact, we claim that
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}\left\\{\begin{array}[]{cc}\to
0&\mbox{ as }z\to 0\\\ \to-1&\mbox{ as }z\to\infty.\end{array}\right.$
The limit as $z\to\infty$ can be seen using the large argument asymptotic
formulas from [1, Sect. 9.7], while the limit as $z\to 0$ follows from the
recurrence relations (2.27) and the small argument asymptotics in [1, Sect.
9.6]. Thus $zK_{\nu}(z)I_{\nu}(z)$ is monotone on the region under
consideration. Using the asymptotic formulas again shows that
$zK_{\nu}(z)I_{\nu}(z)\left\\{\begin{array}[]{cc}\to 0&\mbox{ as }z\to 0\\\
\to 1/2&\mbox{ as }z\to\infty.\end{array}\right.$
so (7.3) holds.
We can now show the upper bound in (7.2). Using the Wronskian relation in
(2.27), we write
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}+\frac{1}{2}\frac{\mu}{(\mu^{2}+z^{2})^{1/2}}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(-2\mu
I_{\mu+1/2}K_{\mu-1/2}\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right)\\\
&\leq\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right)\end{split}$
(7.4)
Now, if $\mu\geq 1$, by (7.3), the right hand side in the final inequality is
bounded by $1/2$, establishing the upper bound in (7.1) in this case (with
$\delta=1/2$). If $\mu\in[1/2,1]$, we use the following inequalities of Barciz
[9, Equations 2.3, 2.4]
$\frac{zI_{\nu}^{\prime}(z)}{I_{\nu}(z)}<\sqrt{z^{2}+\nu^{2}}\qquad\mbox{ and
}\qquad\frac{zK_{\nu}^{\prime}(z)}{K_{\nu}(z)}<-\sqrt{z^{2}+\nu^{2}},$
for $\nu\geq 0,z\geq 0$. Using these inequalities and the recurrence relation
(2.27) gives
$\frac{I_{\mu-1/2}}{I_{\mu+1/2}}<\frac{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}{z},\quad\frac{K_{\mu-1/2}}{K_{\mu+1/2}}<\frac{z}{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2},$
so continuing the inequality (7.4) gives
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&\leq\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(zI_{\mu+1/2}K_{\mu+1/2}\right)\times\\\
&\qquad\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).\end{split}$
(7.5)
One checks that for $1/2\leq\mu$, the fraction in the second line is monotone
decreasing in $z$, and thus by (7.3), for $z\geq 1$ the determinant is bounded
by
$\frac{1}{4}\left(1+\frac{\sqrt{1+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{1+(\mu-1/2)^{2}}+\mu-1/2}\right)\leq\frac{1}{4}(1+(1+\sqrt{2}))\leq
1-\delta$ (7.6)
where the middle bound is obtained by checking that the fraction on the left
is monotone decreasing in $\mu$ for $\mu\geq 1/2$ and equal to $1+\sqrt{2}$ at
$\mu=1/2$. Thus, we have established the upper bound in (7.2) in the region
$z\geq 1$. For $z\leq 1$, rewrite the bound in (7.5) as
$\begin{split}\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(I_{\mu+1/2}K_{\mu+1/2}\right)\times
z\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).\end{split}$
For $\mu\geq 1/2$, by [49], the function $I_{\mu+1/2}(z)K_{\mu+1/2}(z)$ is
monotone decreasing, and by the asymptotic formulas it is goes to $1/2$ as
$z\to 0$. Thus in $0\leq z\leq 1$ the determinant is bounded about by
$\frac{1}{4}\times
z\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).$
This function is monotone increasing in $z$ for $\mu\in[1/2,1]$, so the max is
obtained at $z=1$, i.e. it is bounded by the left hand side of (7.6), in
particular by $1-\delta$ for the same $\delta$. This establishes the upper
bound in (7.2)
Finally we establish the lower bound. First, we rewrite the determinant again,
this time using the Wronskian relation in the opposite direction to obtain
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}-\frac{1}{2}\frac{\mu}{(\mu^{2}+z^{2})^{1/2}}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(2\mu
I_{\mu-1/2}K_{\mu+1/2}\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right).\end{split}$
(7.7)
Now, recalling that $zI_{\mu+1/2}(1)K_{\mu+1/2}(1)$ is monotone increasing,
using the asymptotic formulas [1, 9.7.7, 9.7.8] we see that
$I_{\mu+1/2}(1)K_{\mu+1/2}(1)\to\frac{1}{2(\mu+1/2)}$
as $\mu\to\infty$, we use the inequality [5, Eqn. 11], namely
$I_{\mu-1/2}(z)\geq\frac{\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}}{z}I_{\mu+1/2}(z).$
On the region $z\in[0,1]$,
$\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}\geq\delta_{0}>0$. Dropping the terms with
equal order in (7.7) then gives
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&>-1+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}2\mu
I_{\mu-1/2}K_{\mu+1/2}\\\
&\geq-1+\frac{1}{2}\frac{2\mu}{(\mu^{2}+z^{2})^{1/2}}I_{\mu+1/2}K_{\mu+1/2}(\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2})\\\
&\geq-1+\delta_{0}\frac{1}{2}\frac{\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}}{(\mu^{2}+z^{2})^{1/2}}\\\
&\geq-1+\delta.\end{split}$
This completes the proof of (7.2).
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|
arxiv-papers
| 2013-12-16T03:54:25 |
2024-09-04T02:49:55.461947
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Pierre Albin, Jesse Gell-Redman",
"submitter": "Jesse Gell-Redman",
"url": "https://arxiv.org/abs/1312.4241"
}
|
1312.4270
|
# Spin-polarized hydrogen adsorbed on the surface of superfluid 4He
J. M. Marína, L. Vranješ Markićb and J. Boronata a Departament de Física i
Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5,
E-08034, Barcelona, Spain
b Faculty of Science, University of Split, HR-21000 Split, Croatia
###### Abstract
The experimental realization of a thin layer of spin-polarized hydrogen
H$\downarrow$ adsorbed on top of the surface of superfluid 4He provides one of
the best examples of a stable nearly two-dimensional quantum Bose gas. We
report a theoretical study of this system using quantum Monte Carlo methods in
the limit of zero temperature. Using the full Hamiltonian of the system,
composed of a superfluid 4He slab and the adsorbed H$\downarrow$ layer, we
calculate the main properties of its ground state using accurate models for
the pair interatomic potentials. Comparing the results for the layer with the
ones obtained for a strictly two-dimensional (2D) setup, we analyze the
departure from the 2D character when the density increases. Only when the
coverage is rather small the use of a purely 2D model is justified. The
condensate fraction of the layer is significantly larger than in 2D at the
same surface density, being as large as 60% at the largest coverage studied.
###### pacs:
67.65.+z,02.70.Ss,67.63.Gh
## I Introduction
Electron-spin-polarized hydrogen (H$\downarrow$) was proposed long time ago as
the system in which a Bose-Einstein condensate state (BEC) could be obtained.
stwaley ; miller Intensive theoretical and experimental work was made in the
eighties and nineties of the past century to devise experimental setups able
to reach the predicted density and temperature regimes for BEC. silvera1 ;
greytak ; silvera2 The high recombination rate in the walls of the containers
hindered this achievement for a long time, and only after working with a wall-
free confinement, Fried et al. fried were able to realize its BEC in 1998.
However, this was not the first BEC because three years before the BEC state
was impressively obtained working with cold metastable alkali gases. becgas
The same year BEC of H$\downarrow$ was obtained, Safonov et al. safonov
observed for the first time a quasi-condensate of nearly two-dimensional
H$\downarrow$ adsorbed on the surface of superfluid 4He.
In spite of hydrogen losing the race against alkali gases to be the first BEC
system, it still deserves interest for both theory and experiment. Hydrogen is
the lightest and most abundant element of the Universe and, when it is spin
polarized with the use of a proper magnetic field, it is the only system that
remains in the gas state down to the limit of zero temperature. H$\downarrow$
is therefore extremely quantum matter. A standard measure of the quantum
nature of a system is the de Boer parameter miller
$\eta=\frac{\hbar^{2}}{m\epsilon\sigma^{2}}\ ,$ (1)
with $\epsilon$ and $\sigma$ the well depth and core radius of the pair
interaction, respectively. According to this definition, $\eta=0.5$ for
H$\downarrow$ which is the largest value for $\eta$ among all the quantum
fluids (for instance, $\eta=0.2$ for 4He). This large value for $\eta$ results
from the shallow minimum ($\sim 6$ K) of the triplet potential $b$
${}^{3}\Sigma_{u}^{+}$ between spin-polarized hydrogen atoms and their small
mass. kolos
Adsorption of H$\downarrow$ on the surface of liquid 4He has been extensively
used because of its optimal properties. walraven ; berkhout ; mosk ; ahokas ;
ahokas2 ; jarvinen On one hand, the interaction of any adsorbant with the 4He
surface is the smallest known, and on the other, at temperatures $T<300$ mK
the 4He vapor pressure is negligible and thereby above the free surface one
can reasonably assume vacuum. In fact, liquid 4He was also extensively used in
the search of the three-dimensional H$\downarrow$ BEC state when the cells
were coated with helium films to avoid adsorption of H$\downarrow$ on the
walls and the subsequent recombination to form molecular hydrogen H2. silvera1
; greytak ; silvera2 Helium is chemically inert and only a small fraction of
3He ($6.6$ %) is soluble in bulk 4He; spin-polarized hydrogen, and its
isotopes deuterium and tritium, are expelled to the surface where they have a
single bound state. For instance, in the case of H$\downarrow$, the chemical
potential of a single atom in bulk 4He is marin1 $36$ K to be compared with
the negative value on the surface, $-1.14$ K. safonov2 ; mantz ; silvera3
The quantum degeneracy of H$\downarrow$ adsorbed on 4He is quantified by
defining the quantum parameter $\sigma\Lambda^{2}$, with $\sigma$ the surface
density and $\Lambda$ the thermal de Broglie wave length. jarvinen
Experiments try to increase this parameter as much as possible by increasing
the surface density and lowering the temperature of the film. To this end, two
methods for local compression have been used. The first one, that relies on
the application of a high magnetic field, is able to attain large quantum
parameter values, $\sigma\Lambda^{2}\simeq 9$. safonov ; mosk However, to
measure the main properties of the quasi-two dimensional gas becomes difficult
due to the large magnetic field. jarvinen2 An alternative to this method is
to work with thermal compression, in which a small spot on the sample cell is
cooled down to a temperature below the one of the cell. matsubara ; vasyliev
This second method achieves lower values for quantum degeneracy
$\sigma\Lambda^{2}\simeq 1.5$ but allows for direct observation of the sample.
Up to now, it has not been possible to arrive to the value
$\sigma\Lambda^{2}\simeq 4$ where the Berezinskii-Kosterlitz-Thouless
superfluid transition is expected to set in. Nevertheless, the quantum
degeneracy of the gas has been observed as a decrease of the three-body
recombination rate at temperatures $T=120$-$200$ mK and densities
$\sigma\simeq 4\times 10^{12}$ cm-2. jarvinen2
The zero-temperature equations of state of bulk gas leandra1 H$\downarrow$
and liquid leandra2 T$\downarrow$ have been recently calculated using
accurate quantum Monte Carlo methods. Properties like the condensate fraction,
distribution functions and localization of the gas(liquid)-solid phase
transitions have been established with the help of the ab initio
H$\downarrow$-H$\downarrow$ interatomic potential. kolos ; jamieson ; yan
From the theoretical side, much less is known about the ground-state
properties of two-dimensional H$\downarrow$ or H$\downarrow$ adsorbed on a
free 4He surface. In a pioneering work, Mantz and Edwards mantz used the
variational Feynman-Lekner approximation to calculate the effective potential
felt by a hydrogen atom on the 4He surface. Solving the Schrödinger equation
for the atom in this effective potential they concluded that H$\downarrow$,
D$\downarrow$, and T$\downarrow$ have a single bound state and calculated the
respective binding energies. The main drawback of this treatment is that the
adsorbent is substituted by an effective field representing a static and
undisturbed surface. In fact, a quantitatively accurate approach to this
problem requires a good model for the 4He surface. krotscheck The use of
accurate He-He potentials and ground-state quantum Monte Carlo methods has
proved to be able to reproduce experimental data directly related to the
surface, like the surface tension and the surface width. marin2 In the
present work, we rely on a similar methodology to the one previously used in
the study of the free 4He surface marin2 in order to microscopically
characterize the ground-state of H$\downarrow$ adsorbed on its surface. Our
study is complemented by a purely two-dimensional simulation of H$\downarrow$
in order to establish the degree of two-dimensionality of the adsorbed film.
The rest of the paper is organized as follows. The quantum Monte Carlo method
used for this study is described in Sec. II. The results obtained for
H$\downarrow$ adsorbed on the 4He surface within a slab geometry are presented
in Sec. III together with the comparison with the strictly two-dimensional
case. Finally, Sec. IV comprises a brief summary and the main conclusions of
the work.
## II Quantum Monte Carlo method
We have studied the ground-state (zero temperature) properties of a thin layer
of H$\downarrow$ adsorbed on the free surface of a 4He slab and also the
limiting case of a strictly two-dimensional (2D) H$\downarrow$ gas. Focusing
first on the slab geometry, the Hamiltonian of the system composed by $N_{\rm
He}$ 4He and $N_{\rm H}$ H$\downarrow$ atoms is
$\displaystyle H$ $\displaystyle=$ $\displaystyle-\frac{\hbar^{2}}{2m_{\rm
He}}\sum_{I=1}^{N_{\rm He}}{\bm{\nabla}}_{I}^{2}-\frac{\hbar^{2}}{2m_{\rm
H}}\sum_{i=1}^{N_{\rm H}}{\bm{\nabla}}_{i}^{2}+\sum_{1=I<J}^{N_{\rm He}}V_{\rm
He-He}(r_{IJ})$ (2) $\displaystyle+\sum_{1=i<j}^{N_{\rm H}}V_{\rm
H-H}(r_{ij})+\sum_{1=I,i}^{N_{\rm He},N_{\rm H}}V_{\rm He-H}(r_{Ii})\ ,$
with capital and normal indices standing for 4He and H$\downarrow$ atoms,
respectively. The pair potential between He atoms is the Aziz HFD-B(HE) model
aziz used extensively in microscopic studies of liquid and solid helium. The
H$\downarrow$-H$\downarrow$ interaction ($b~{}^{3}\Sigma_{u}^{+}$ triplet
potential) was calculated with high accuracy by Kolos and Wolniewicz (KW).
kolos More recently, this potential has been recalculated up to larger
interatomic distances by Jamieson, Dalgarno, and Wolniewicz (JDW). jamieson
We have used the JDW data smoothly connected with the long-range behavior of
the H$\downarrow$-H$\downarrow$ potential as calculated by Yan et al. yan The
JDW potential has a core diameter of $3.67$ Å and a minimum $\epsilon=-6.49$ K
(slightly deeper than KW) at a distance $r_{\text{m}}=4.14$ Å. Finally, we
take the H-He pair potential from Das et al.; das this model has been used in
the past in the study of a single H$\downarrow$ impurity marin1 in liquid 4He
and in mixed T$\downarrow$-4He clusters. petar The Das potential das has a
minimum $\epsilon=-6.53$ K at a distance $r_{\text{m}}=3.60$ Å.
The quantum $N$-body problem is solved stochastically using the diffusion
Monte Carlo (DMC) method. hammond DMC is nowadays one of the most accurate
tools for the study of quantum fluids and gases, providing exact results for
boson systems within some statistical errors. In brief, DMC solves the
imaginary-time ($\tau$) $N$-body Schrödinger equation for the function
$f({\bm{R}},\tau)=\psi({\bm{R}})\Psi({\bm{R}},\tau)$, with
$\Psi_{0}({\bm{R}})=\lim_{\tau\to\infty}\Psi({\bm{R}},\tau)$ the exact ground-
state wave function. The auxiliary wave function $\psi({\bm{R}})$ acts as a
guiding wave function in the diffusion process towards the ground state when
$\tau\to\infty$. The direct statistical sampling with $f({\bm{R}},\tau)$,
called mixed estimator, is unbiased for the energy but not completely for
operators which do not commute with the Hamiltonian. In these cases, we rely
on the use of pure estimators based on the forward walking strategy. pures
The influence of the finite time step used in the iterative process is reduced
by working with a second-order expansion for the imaginary-time Green’s
function. dmccasu The last systematic error that one has to deal with is the
finite number of walkers ${\bm{R}}_{i}$ which represent the wave function
$\Psi({\bm{R}},\tau)$. As usual, we analyze which is the number of walkers
required to reduce any bias coming from it to the level of the statistical
uncertainties.
The 4He surface is simulated using a slab which grows symmetrically in the $z$
direction and with periodic boundary conditions in the $x-y$ plane. marin2
The guiding wave function is then the product of two terms
$\psi({\bm{R}})=\psi_{J}({\bm{R}})\,\phi({\bm{R}})\ ,$ (3)
the first one accounting for dynamical correlations induced by the interatomic
potentials and the second for the finite size of the liquid in the $z$
direction. Explicitly, $\psi_{J}({\bm{R}})$ is built as a product of two-body
Jastrow factors between the different particles,
$\psi_{J}({\bm{R}})=\prod_{1=I<J}^{N_{\rm He}}f_{\rm
He}(r_{IJ})\prod_{1=i<j}^{N_{\rm H}}f_{\rm H}(r_{ij})\prod_{1=I,i}^{N_{\rm
He},N_{\rm H}}f_{\rm He-H}(r_{Ii})\ .$ (4)
The one-body correlations that confine the system to a slab geometry are
introduced in $\phi({\bm{R}})$,
$\phi({\bm{R}})=\prod_{I=1}^{N_{\rm He}}h_{\rm He}(z_{I})\prod_{i=1}^{N_{\rm
H}}h_{\rm H}(z_{i})\ .$ (5)
The 4He-4He ($f_{\rm He}(r)$) and 4He-H$\downarrow$ ($f_{\rm He-H}(r)$) two-
body correlation factors (4) are chosen of Schiff-Verlet type,
$f(r)=\exp\left[-\frac{1}{2}\left(\frac{c}{r}\right)^{5}\right]\ ,$ (6)
whereas the H$\downarrow$-H$\downarrow$ one is taken as
$f_{\rm H}(r)=\exp[-b_{1}\exp(-b_{2}r)]\ ,$ (7)
because it has been shown to be variationally better for describing the
hydrogen correlations. leandra1 The parameters entering Eqs. (6,7) have been
optimized using the variational Monte Carlo method. We have used $c_{\rm
He}=c_{{\rm He-H}}=3.07$ Å, $b_{1}=101$, and $b_{2}=1.30$ Å-1, neglecting
their slight dependence on density. The one-body functions in Eq. (5) are of
Fermi type,
$h(z)=\left\\{1+\exp[\,k(\,|z-z_{\text{cm}}|-z_{0})]\right\\}^{-1}\ ,$ (8)
with variational parameters $k$ and $z_{0}$ related to the width and location
of the interface, respectively. The main goal of these one-body terms is to
avoid eventual evaporation of particles by introducing a restoring drift force
only when particles want to escape to unreasonable distances. Any spurious
kinetic energy contribution due to the movement of the center of mass of the
full system (4He+H$\downarrow$) is removed by subtracting $z_{\text{cm}}$ from
each particle coordinate $z$, either of 4He or H$\downarrow$, in Eq. (8). The
optimal values used in the DMC simulations are $z_{0}$(4He)$=22.10$ Å,
$z_{0}$(H$\downarrow$)$=37.06$ Å, and $k$(4He)$=k$(H$\downarrow$)$=1$ Å-1.
Our study of the thin layer of H$\downarrow$ adsorbed on 4He is complemented
with some calculations of a strictly 2D H$\downarrow$ gas with the Hamiltonian
$H_{\text{2D}}=-\frac{\hbar^{2}}{2m_{\rm H}}\sum_{i=1}^{N_{\rm
H}}{\bm{\nabla}}_{i}^{2}+\sum_{1=i<j}^{N_{\rm H}}V_{\rm H-H}(r_{ij})\ ,$ (9)
using as a guiding wave function a Jastrow factor with the same two-body
correlation factors as in the slab (7). h2d
## III Results
The 4He surface where H$\downarrow$ is adsorbed is simulated with the DMC
method using a slab geometry. We use a square cell in the $x-y$ plane that is
made continuous by considering periodic boundary conditions in both
directions. In the transverse direction $z$ the system is finite, with two
symmetric free surfaces at the same distance from the center $z=0$. The
surface of the basic simulation cell is $A=290.30$ Å2 and $N_{\rm He}=324$.
With these conditions we guarantee an accurate model for the free surface of
4He, as shown in Ref. marin2, .
Figure 1: (Color online) Density profile of the 4He slab (dashed line) and of
the H$\downarrow$ adsorbed gas (solid line) corresponding to a surface density
$\sigma=9.57\times 10^{-3}$ Å-2.
On top of one of the slab surfaces we introduce a variable number $N_{\rm H}$
of H$\downarrow$ atoms that form a thin layer of surface densities
$\sigma=N_{\rm H}/A$. In order to reach lower densities than $\sigma=1/A$ we
have replicated the basic slab cell the required number of times. In Fig. 1,
we show the density profiles of the 4He slab and of the H$\downarrow$ layer
for a surface density $\sigma=9.57\times 10^{-3}$ Å-2. This layer has an
approximate width of 8 Å and virtually floats on the helium surface: the
center of the H$\downarrow$ layer is located out of the surface, where the 4He
density is extremely small. The picture is similar to the one obtained
previously by Mantz and Edwards mantz in a variational description of the
adsorption of a single H$\downarrow$ atom. However, contrarily to the
exponential tail of the density profile derived by Krotschek and Zillich
krotscheck in a thorough description of the impurity problem, we observe a
faster decay to zero and a rather isotropic profile. We attribute this
difference to the residual bias of the one-body factor $h(z)$ (8) used to
avoid spurious evaporation of particles. On the other hand, the more well
studied case of 3He adsorbed on the 4He surface shows a similar density
profile, guardiola located on the surface, but in this case centered not so
far from the bulk.
Figure 2: (Color online) Energy per particle of H$\downarrow$ on top of the
4He surface (points with error bars). The energy at the zero-dilution limit is
subtracted in such a way that the energy is zero in the limit
$\sigma\rightarrow 0$. The line on top of the DMC data corresponds to the
polynomial fit of Eq. (10).
Figure 3: (Color online) Comparison between the energy per particle of
H$\downarrow$ adsorbed on the 4He slab (full circles) and the energy of purely
two-dimensional H$\downarrow$ (open squares). The solid line is the polynomial
fit (10) and the dashed line is a fit of the 2D energies (11).
One of the most relevant magnitudes that characterize the H$\downarrow$ film
is its energy per particle at different coverages. In Fig. 2, we plot the DMC
energy per particle of H$\downarrow$ as a function of the surface density
$\sigma$. In order to better visualize the energy of the adsorbed gas, we have
subtracted from the computed energies the energy in the infinite dilution
limit $\sigma\rightarrow 0$. The energy increases monotonously with the
density and its behavior is well accounted for by the simple polynomial law
$E/N(\sigma)=B\sigma+C\sigma^{2}\ ,$ (10)
with optimal parameters $B=48(2)$ KÅ2 and $C=5.6(9)\times 10^{2}$ KÅ4, the
figures in parenthesis being the statistical uncertainties.
H$\downarrow$ floating on top of the 4He free surface has been currently
considered as a nice representation of a quasi-two-dimensional quantum gas. In
order to be quantitatively accurate in this comparison, we have carried out
DMC simulations of strictly 2D H$\downarrow$ gas without any adsorbing
surface. h2d The results obtained for the energy per particle of the 2D gas
at different densities are shown in Fig. 3. The energies are well reproduced
by a polynomial law
$E/N(\sigma)=B_{\rm{2D}}\sigma+C_{\rm{2D}}\sigma^{2}\ ,$ (11)
with $B_{\rm{2D}}=35(3)$ KÅ2 and $C_{\rm{2D}}=6.4(1)\times 10^{4}$ KÅ4. In the
same figure, we plot the energies for the adsorbed gas at the same coverage.
As one can see, the agreement between the strictly 2D gas and the film is good
for densities $\sigma\lesssim 5\times 10^{-3}$ Å-2. At higher densities, the
additional degree of freedom in the $z$ direction makes the growth of the
energy with the surface density in the layer nearly linear up to the shown
density, in contrast with the significant quadratic increase observed in the
2D gas ($C<<C_{\rm{2D}}$).
Figure 4: (Color online) Comparison between the energy per particle of
H$\downarrow$ adsorbed on the 4He slab and the energy of bulk H$\downarrow$
(solid line) from Ref. leandra1, . Full squares, full circles, and full
diamonds correspond to the layer where we have considered a width in $z$ of 9,
8, and 7 Å, respectively.
A possible scenario when the density increases and the equation of state of
the layer departs from the 2D law is the existence of a nearly three-
dimensional (3D) gas. We have analyzed this possibility by considering a width
in $z$ given by the density profile (Fig. 1) and by estimating the 3D density
of the adsorbed gas as the coverage divided by the layer width. In Fig. 4, we
show the energy per particle of adsorbed H$\downarrow$ as a function of the
density considering our best estimation for the layer width, $z=8$ Å, and also
$z=9$ and 7 Å. The possible 3D behavior of the energy is analyzed by comparing
the results of the layer with the ones of the bulk 3D gas. At low densities,
the energies of the adsorbed phase are higher than the 3D gas and, when the
density increases, both results tend to cross. As one can see, the energies of
adsorbed H$\downarrow$ are not well described by a 3D equation of state at any
density within the regime studied.
Figure 5: (Color online) Two-body distribution function $g(z,r)$ of
H$\downarrow$ adsorbed on 4He, with $r=\sqrt{x^{2}+y^{2}}$, at surface density
$\sigma=0.0215$ Å-2.
The structure and the distribution functions of H$\downarrow$ atoms in the
layer can be studied by doing slices of small width ($\Delta z=1$ Å) and,
within a given slice, as a function of the radial distance between particles
in the plane $r=\sqrt{x^{2}+y^{2}}$. In Fig. 5, we report results of the two-
body radial distribution function $g(z,r)$ where $z$ is the distance to the
center of the 4He slab at a coverage $\sigma=0.0215$ Å-2. Around the center of
the H$\downarrow$ density profile, $g(r)$ is nearly independent of $z$ with a
main peak of a height smaller than 1.2. In the wings of $\rho_{\text{H}}(z)$,
where the local density is smaller, $g(r)$ shows less structure and the noise
of the DMC data also increases due to low statistics.
Figure 6: (Color online) Comparison between the two-body distribution function
in the center of the slab, corresponding to the density $\sigma=0.0095$ Å-2
(solid line) with the one corresponding to a purely 2D H$\downarrow$ gas at
the same surface density (dotted line).
It is interesting to know if the spatial structure of H$\downarrow$ atoms on
the 4He surface is similar to the one in a strictly 2D geometry. To this end,
we show in Fig. 6 results of the radial distribution function for both systems
at the same surface density ($\sigma=0.0095$ Å-2). The result corresponding to
the layer is taken from a slice $\Delta z$ in the center of the density
profile. As one can see, both functions do not show any significant peak
because the density is rather small. However, the behavior at small
interparticle distances is appreciably different. In the layer, atoms can be
closer (in the in-plane distance $r=\sqrt{x^{2}+y^{2}}$ ) than in 2D because
of the small but nonzero width of the slice used for its calculation. In fact,
we have shown previously in Fig. 2 that, at the density $\sigma=0.0095$ Å-2
used in Fig. 6, the energies per particle of the layer and the strictly 2D gas
start to be significantly different, in agreement with the differences
observed here in the distribution function $g(r)$.
Figure 7: (Color online) One-body distribution function $\rho_{1}(z,r)$ of
H$\downarrow$ adsorbed on 4He, with $r=\sqrt{x^{2}+y^{2}}$, at surface density
$\sigma=0.0215$ Å-2.
A key magnitude in the study of any quantum Bose gas is the one-body
distribution function $\rho_{1}(r)$ since it furnishes evidence of the
presence of off-diagonal long-range order in the system. As it is well known,
its asymptotic behavior in a homogeneous system
$\lim_{r\rightarrow\infty}\rho_{1}(r)=n_{0}$ gives the fraction of particles
occupying the zero-momentum state, that is the condensate fraction $n_{0}$. In
Fig. 7, we show a surface plot containing results of $\rho_{1}(z,r)$ at
density $\sigma=0.0215$ Å-2 obtained following the same method as in the grid
of $g(z,r)$ shown in Fig. 5. In the outer part of the density profile the
condensate fraction approaches one because the density is very small. When $z$
decreases the condensate fraction also decreases and reaches a plateau in the
central part of $\rho_{\text{H}}(r)$. If $z$ is reduced even more and
$\rho_{\text{He}}(r)$ starts to increase, the H$\downarrow$ condensate
fraction decreases again due to the small but nonzero 4He density; the low
statistics in this part makes the signal very noisy and therefore we do not
plot data for $z<27$ Å in Fig. 7.
Figure 8: (Color online) Comparison between the one-body distribution function
in the center of the slab, corresponding to a density $\sigma=0.0095$ Å-2
(solid line) with the one corresponding to a purely 2D H$\downarrow$ gas at
the same surface density (dotted line).
A relevant issue in the study of the off-diagonal long-range order in the
adsorbed gas is the dimensionality of the results achieved. As we have made
before for the two-body distribution functions, we compare $\rho_{1}(r)$ for a
2D gas and for a slice in the center of the adsorbed layer at the same density
in Fig. 8. The results show that in this case the behavior in the layer is
significantly different from the one observed in strictly 2D. The difference
is larger than the one we have observed at the same density for $g(r)$ (Fig.
6), with values for the condensate fraction that differs in $\sim 30$ %. The
condensate fraction of the 2D gas is clearly smaller than the one of the layer
due to the transverse degree of freedom $z$ that translates into an effective
surface density smaller than the one of the full layer.
Figure 9: (Color online) Condensate fraction as a function of the surface
density $\sigma$. Solid circles correspond to H$\downarrow$ on 4He and open
squares to a 2D gas. The lines on top of the DMC data are fits to guide the
eye.
The density dependence of the condensate fraction of adsorbed H$\downarrow$ is
shown in Fig. 9. The values reported have been obtained from the asymptotic
value of the one-body distribution function in the central part of the density
profile. As expected, the condensate fraction is nearly 1 at very low
densities and then decreases when $\sigma$ increases. However, the decrease is
quite slow in such a way that even at densities as large as $\sigma=0.02$ Å-2
the condensate fraction is still $n_{0}\simeq 0.6$. At the same density, the
condensate fraction of the 2D gas is half this value, $n_{0}\simeq 0.3$. The
dependence of $n_{0}$ with the density for the 2D geometry, shown in Fig. 9
for comparison, is significantly stronger with a larger depletion of the
condensate fraction for all densities.
## IV Summary and Conclusions
The experimental realization of an extremely thin layer of H$\downarrow$
adsorbed on the surface of superfluid 4He provides a unique opportunity for
the study of nearly two-dimensional quantum gases. The system is stable and
the influence of the liquid substrate is nearly negligible, without the
corrugation effects that a solid surface like graphite provides. Moreover,
spin-polarized hydrogen is a specially appealing system from the theoretical
side because it is the best example of quantum matter (it remains gas even in
the zero temperature limit) and its interatomic interaction is known with high
accuracy. In the present work, we have addressed its study from a microscopic
approach relying on the use of quantum Monte Carlo methods by means of a
simulation that incorporates the full Hamiltonian of the system, composed by a
realistic 4He surface and the layer of H$\downarrow$ adsorbed on it.
From very low coverages up to relatively high surface densities, we have
reported results of the main properties of adsorbed H$\downarrow$: energy,
density profile, two- and one-body distribution functions, and the condensate
fraction. Our results point to a $\sim 8$ Å thick layer that virtually floats
on top of 4He. We have calculated the energy as a function of the surface
density $\sigma$ and compared these energies with the results obtained in a
purely 2D H$\downarrow$ gas in order to establish the degree of two-
dimensionality of the layer. The agreement between both simulations is only
satisfactory for small densities $\sigma\lesssim 5\times 10^{-3}$ Å-2 and,
from then on, the additional degree of freedom in the $z$ direction of the
layer causes its energy to grow slower than in strictly 2D. Significant
departures of strictly 2D behavior are also observed in the two-body radial
distribution function and mainly in the condensate fraction values. Our DMC
results show that the condensate fraction for the layer is appreciably higher
than in 2D, with values as large as $n_{0}=0.6$ at the largest coverages
studied. If we convert this coverage to volume density by using the layer
width of 8 Å, we see that the condensate fraction is quite close to published
3D values in Ref. leandra1, . From these results we can be certain that a BKT
phase transition would be a realistic scenario at low surface densities. For
higher densities, further study using intensive path-integral Monte Carlo
simulations at finite temperatures would be needed.
###### Acknowledgements.
The authors acknowledge partial financial support from the DGI (Spain) Grant
No. FIS2011-25275, Generalitat de Catalunya Grant No. 2009SGR-1003, Qatar
National Research Fund NPRP 5-674-1-114 as well as the support from MSES
(Croatia) under Grant No. 177-1770508-0493.
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* (35) A study of the phase diagram of strictly two-dimensional H$\downarrow$ gas can be found in L. Vranješ Markić and J. Boronat, J. Low. Temp. Phys. 171, 685 (2013).
* (36) R. Guardiola and J. Navarro, Phys. Rev. Lett. 89, 193401 (2002).
|
arxiv-papers
| 2013-12-16T09:11:40 |
2024-09-04T02:49:55.481869
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. M. Marin, L. Vranjes Markic, and J. Boronat",
"submitter": "Jordi Boronat",
"url": "https://arxiv.org/abs/1312.4270"
}
|
1312.4362
|
# Hot Spin Polarized Strange Quark Stars in the Presence of Magnetic Field
using a density dependent bag constant
G. H. Bordbar1,2 111Corresponding author. E-mail: [email protected]
and Z. Alizade 1Department of Physics, Shiraz University, Shiraz 71454, Iran
and
2Center for Excellence in Astronomy and Astrophysics (CEAA-RIAAM)-Maragha,
P.O. Box 55134-441, Maragha 55177-36698, Iran
###### Abstract
The effect of magnetic field on the structure properties of hot spin polarized
strange quark stars has been investigated. For this purpose, we use the MIT
bag model with a density dependent bag constant to calculate the thermodynamic
properties of spin polarized strange quark matter such as energy and equation
of state. We see that the energy and equation of state of strange quark matter
changes significantly in a strong magnetic field. Finally, using our equation
of state, we compute the structure of spin polarized strange quark star at
different temperatures and magnetic fields.
## I Introduction
A strange quark star is a hypothetical type of exotic star composed of strange
quark matter. This is an ultra-dense phase of degenerate matter theorized to
form inside particularly massive neutron stars. It is theorized that when the
degenerate neutron matter which makes up a neutron star is put under
sufficient pressure due to the star’s gravity, neutrons break down into their
constituent up and down quarks. Some of these quarks may then become strange
quarks and form strange matter, and hence a strange quark star, similar to a
single gigantic hadron (but bound by gravity rather than the strong force).
Actually, until recently, astrophysicists were not sure there was a gray area
between neutron stars and black holes, stellar remnants from a massive star’s
death had to be one or the other. Now, it is thought there is another bizarre
creature out there, more massive than a neutron star, yet too small to
collapse in on itself to form a black hole. Although they have yet to be
observed, strange quark stars should exist, and scientists are only just
beginning to realize how strange these things are. Neutron stars, strange
quark stars and black holes are all born via the same mechanism: a supernova
collapse. But each of them are progressively more massive, so they originate
from supernovae produced by progressively more massive stars. The collapsing
supernova will turn into a neutron star only if its mass is about
$1.4-3M_{sun}$. In a neutron star, if density of the core is high enough
($10^{15}\frac{gr}{cm^{3}}$) the nucleons dissolve to their components,
quarks, and a hybride star (neutron star with a core of strange quark matter
(SQM)) is formed. If after the explosion of the supernova density high enough
($10^{15}\frac{gr}{cm^{3}}$), the pure strange quark star (SQS) may be formed
directly. The composition of SQS was first proposed by Itoh rk1 with
formulation of Quantum Charmo Dynamics (QCD).
One of the most important characteristics of a compact star is its magnetic
field which is about $10^{15}-10^{19}\ G$ for pulsars, magnetars, neutron
stars and SQS rk1017 ; rk1018 . This strong magnetic field has an important
influence on compact stars. Therefore, investigating the effect of an strong
magnetic field on strange quark matter (SQM) properties is important in
astrophysics. In recent years much interesting work has been done on the
properties of dense astrophysical matter in the presence of a strong magnetic
field rk2 ; rk3 . The effect of the strong magnetic field on SQM has been
investigated using the MIT bag model as well as the D3QM model of confinement
rk4 ; rk5 . We have studied the effects of strong magnetic fields on the
neutron star structure employing the lowest order constrained variational
technique rk5-1 . Recently, we have also calculated the structure of polarized
SQS at zero temperature rk6 , the structure of unpolarized SQS at finite
temperature rk7 , structure of the neutron star with the quark core at zero
temperature rk8 and finite temperature rk9 ; rk9p , structure of spin
polarized SQS in the presence of magnetic field at zero temperature using
density dependent bag constant rk1015 and at finite temperature using a fixed
bag constant rk1016 . The aim of the present work is calculating some
properties of polarized SQS at finite temperature in the presence of a strong
magnetic field using the MIT bag model with a density dependent bag constant.
To this aim, in section II, we calculate the energy and equation of state of
SQM in the presence of magnetic field at finite temperatures by MIT bag model
using a density dependent bag constant. Finally in section III, we solve the
TOV equation, and calculate structure of SQS.
## II Calculation of energy and equation of state of strange quark matter
We study the properties of strange quark matter and resulting equation of
state. The equation of state plays an important role in obtaining the
structure of a star. From a basic point of view, the equation of state for SQM
should be calculated by Quantum chromodynamics (QCD). Previous researchers
have investigated the properties of the strange stars using diffrent equations
of state with interesting results rk21 ; rk22 ; rk23 . There are many
different models for deriving the equation of state of strange quark matter
(SQM) such as MIT bag model rk10 ; rk11 , NJL model rk12 ; rk13 and
perturbation QCD model rk14 ; rk15 . Here, we use MIT bag model using a
density dependent bag constant to calculate the equation of state of SQM in
the presence of a strong magnetic field.
The MIT bag model confines three non-interacting quarks to a spherical cavity,
with the boundary condition that the quark vector current vanishes on the
boundary. The non-interacting treatment of the quarks is justified by
appealing to the idea of asymptotic freedom, whereas the hard boundary
condition is justified by quark confinement. This model developed in 1947 at
”Massachusetts Institute of Technology”. In this model quarks are forced by a
fixed external pressure to move only inside a given spatial region and occupy
single particle orbital. The shape of the bag is spherical if all the quarks
are in ground state. Inside the bag, quarks are allowed to move quasi-free. It
is an appropriate boundary condition at the bag surface that guarantees that
no quark can leave the bag. This implies that there are no quarks outside the
bag rk1020 .
### II.1 Density dependent bag constant
In the MIT bag model, the energy per volume for the strange quark matter is
equal to the kinetic energy of the free quarks plus a bag constant $({\cal
B}_{bag})$ rk10 , which is the difference between energy densities of the
noninteracting quarks and interacting quarks. There are two cases for the bag
constant, a fixed value, and a density dependent value. In the initial MIT bag
model, two different values such as $55$ and $90\ MeV/fm^{3}$ were considered
for the bag constant. Since the density of strange quark matter increases from
the surface to the core of a strange quark star, it is more realistic that we
use a density dependent bag constant rk1000 ; rk1001 ; rk1002 ; rk1003 . By
considering the experimental date received at CERN, the quark-hadron
transition occurs at a density about seven times the normal nuclear matter
energy density $(156\ MeV/fm^{3})$ rk15 ; rk1004 . By supposing that
transition of quark-gluon plasma is only defined by the value of the energy
density, the density dependence of ${\cal B}_{bag}$ has been considered to
have a Gaussian form,
${{\cal B}_{bag}}(n)={{\cal B}}_{\infty}+({\cal B}_{0}-{{\cal
B}}_{\infty})e^{-\gamma(\frac{n}{n_{0}})^{2}},$ (1)
where ${\cal B}_{0}$ parameter is equal to ${\cal B}(n=0)$, and it has fixed
value ${\cal B}_{0}=400\ MeV/fm^{3}$. $\gamma$ is a numerical parameter, and
usually equal to $0.17$, the normal nuclear matter density rk1003 . ${\cal
B}_{\infty}$ depends only on the free parameter ${\cal B}_{0}$.
For obtaining ${\cal B}_{\infty}$, we use the equation of state of the
asymmetric nuclear matter, which should agree with empirical data. For
computing the equation of state of asymmetric nuclear matter, we apply the
lowest order constrained variational (LOCV) many-body procedure as follows
rk1005 ; rk1006 ; rk1007 ; rk1008 ; rk1009 ; rk1010 ; rk1011 ; rk1012 ; rk1013
.
The asymmetric nuclear matter is defined as a system consisting of $Z$ protons
$(pt)$ and $N$ neutrons $(nt)$ with the total number density $n=n_{pt}+n_{nt}$
and proton fraction $x_{pt}=\frac{n_{pt}}{n}$, where $n_{pt}$ and $n_{nt}$ are
the number densities of protons and neutrons, respectively. For this system,
we consider a trial wave function as follows:
$\psi=F\phi,$ (2)
where $\phi$ is the Slater determination of the single-particle wave function
and F is the A-body correlation operator $(A\ =\ Z\ +\ N)$, which is taken to
be
$F=\textrm{S}\prod f(ij),$ (3)
and S is a symmetrizing operator. For the asymmetric nuclear matter, the
energy per nucleon up to the two-body term in the cluster expansion is
$E([f])=\frac{1}{A}\frac{<\psi|H|\psi>}{<\psi|\psi>}=E_{1}+E_{2}.$ (4)
The one-body energy, $E_{1}$, is
$E_{1}=\sum\sum\frac{\hbar^{2}k_{i}^{2}}{2m},$ (5)
where labels $1$ and $2$ are used for the proton and neutron respectively, and
$k_{i}$ is the momentum of particle $i$. The two-body energy, $E_{2}$, is
$E_{2}=\frac{1}{2A}\sum<ij|v(12)|ij-ji>,$ (6)
where
$v(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]+f(12)V(12)f(12).$
(7)
$f(12)$ and $V(12)$ are the two-body correlation and nucleon-nucleon
potential, respectively. In our calculations, we use $UV_{14}+TNI$ nucleon-
nucleon potential rk1014 . The procedure of these calculations has been
studied in rk1006 . According to this discussion, we minimize the two-body
energy with relation to the variations in the correlation function subject to
the normalization constraint. From minimization of the two-body energy, we get
a set of differential equations. We can compute the correlation function by
numerically solving these differential equations. Finally, we get the two-body
energy, and then the energy of asymmetric nuclear matter.
The empirical consequence at CERN acknowledge a proton fraction $x_{pt}=0.4$
(data are from probation accelerated nuclei) rk1003 ; rk1004 .Therefore to
calculate ${\cal B}_{\infty}$, we use our results of the above formalism for
the asymmetric nuclear matter characterized by a proton fraction $x_{pt}=0.4$.
According to the following method, the assumptions of the hadron-quark
transition takes place at energy density equal to $1100\ MeV/fm^{3}$ rk1003 ;
rk1004 . We find that the baryonic density of the nuclear matter is
$n_{0}=0.98\ fm^{-3}$ (transition density). At densities lower than this
value, the energy density of the quark matter is higher than that of the
nuclear matter. By increasing the baryonic density, these two energy densities
become equal at the transition density, and above this value the nuclear
matter energy density remains always higher. Also, we determine ${\cal
B}_{\infty}=8.99\ MeV/fm^{3}$ by putting the energy density of the quark
matter and that of the nuclear matter equal to each other.
### II.2 Energy of spin polarized strange quark matter at finite temperature
in the presence of magnetic field
In this section, we derive the EOS of SQM in the presence of magnetic field.
First, we calculate the energy of SQM. For this, we should find the quark
densities in term of baryonic number density ($n_{B}$). By imposing charge
neutrality and chemical equilibrium (we suppose that neutrinons leave the
system freely), we get the following relations rk15 ,
$\mu_{d}=\mu_{u}+\mu_{e},$ (8) $\mu_{s}=\mu_{u}+\mu_{e},$ (9)
$\mu_{s}=\mu_{d},$ (10) $2/3n_{u}-1/3n_{s}-1/3n_{d}-n_{e}=0,$ (11)
where $\mu_{i}$ is the chemical potential and $n_{i}$ is the number density of
quark $i$. We can ignore the electrons ($n_{e}=0$) rk16 ; rk17 ; rk18 , and
consider the strange quark matter (SQM) including u, d and s quarks.
Therefore, we have
$n_{u}=1/2(n_{s}+n_{d}).$ (12)
In the presence of the magnetic field, we have the spin polarized SQM
including spin-up and spin-down u, d and s quarks. Now, we introduce the
polarization parameter as follows,
$\zeta_{i}=\frac{n_{i}^{+}-n_{i}^{-}}{n_{i}}.$ (13)
In the above equation, $n_{i}^{+}$ is the number density of spin-up quark $i$
and $n_{i}^{-}$ is the number density of spin-down quark $i$, where
$0\leq\zeta_{i}\leq 1$ and $n_{i}=n_{i}^{+}+n_{i}^{-}$.
The chemical potential $\mu_{i}$ for any value of the temperature ($T$) and
number density ($n_{i}$) is obtained using the following constraint,
$n_{i}=\sum_{p=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}f(n_{i}^{(p)},k,T)k^{2}dk,$
(14)
where $g$ is degeneracy number of the system and
$f(n_{i}^{(p)},k,T)=\frac{1}{exp\left(\beta((m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}-\mu_{i}(n_{i}^{(p)},T))\right)+1}$
(15)
is the Fermi-Dirac distribution function. In the above equation
$\beta=1/k_{B}T$ and $m_{i}$ is the mass of quark $i$. It should be noted that
in our calculations, we ignore the masses of u and d quarks, and we consider
$m_{s}=150\ MeV$.
The energy of spin polarized SQM in the presence of the magnetic field within
the MIT bag model is as follows,
$\varepsilon_{tot}=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}+\varepsilon_{M}+{\cal
B}_{bag},$ (16)
where
$\varepsilon_{i}=\sum_{p=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}(m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}f(n_{i}^{(p)},k,T)k^{2}dk.$
(17)
In our calculations, we suppose that $\zeta=\zeta_{u}=\zeta_{d}=\zeta_{s}$. In
Eq. (16), ${\cal B}_{bag}$ is the bag constant with a density-dependent value
which has been introduced in Eq. (1), and $\varepsilon_{M}=\frac{E_{M}}{V}$ is
the magnetic energy density of SQM, where $E_{M}=-M.B$ is the magnetic energy.
If we consider the uniform magnetic field along $z$ direction, the
contribution of magnetic energy of the spin polarized SQM is given by
$E_{M}=-\sum_{i=u,d,s}M_{z}^{(i)}B,$ (18)
where $M_{z}^{(i)}$ is the magnetization of the system corresponding to
particle $i$ which is given by
$M_{z}^{(i)}=N_{i}{\mu_{i}}\zeta_{i}.$ (19)
In the above equation, $N_{i}$ and ${\mu_{i}}$ are the number and magnetic
moment of particle $i$, respectively (${\mu_{s}=-0.581\mu_{N}}$,
$\mu_{u}=1.852\mu_{N}$ and $\mu_{d}=-0.972\mu_{N}$, where $\mu_{N}=5.05\times
10^{-27}\ J/T$ is the nuclear magnetic moment rk0019 ). Finally, the magnetic
energy density of spin polarized SQM can be obtained using the following
relation,
$\varepsilon_{M}=-\sum_{i}n_{i}\mu_{i}\zeta_{i}B.$ (20)
We obtain the thermodynamic properties of the system using the Helmholtz free
energy,
$F=\varepsilon_{tot}-TS_{tot},$ (21)
where $S_{tot}$ is the total entropy of SQM,
$S_{tot}=S_{u}+S_{d}+S_{s}.$ (22)
In Eq. (22), $S_{i}$ is entropy of particle $i$,
$\displaystyle S_{i}(n_{i},T)$ $\displaystyle=$
$\displaystyle-\sum_{p=\pm}\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}[f(n_{i}^{(p)},k,T)ln(f(n_{i}^{(p)},k,T))$
$\displaystyle+$
$\displaystyle(1-f(n_{i}^{(p)},k,T))ln(1-f(n_{i}^{(p)},k,T))]k^{2}dk.$
### II.3 Equation of state of spin polarized strange quark matter
Equation of state of strange quark matter plays an important role in
investigating the structure of strange quark star rk19 ; rk8 ; rk20 . We can
use the free energy to derive the equation of SQM in the presence of the
magnetic field with a density dependent bag constant, by the following
relation,
$P=\sum_{i}(n_{i}\frac{\partial F_{i}}{\partial n_{i}}-F_{i}),$ (24)
where $P$ is the pressure of system and $F_{i}$ is the free energy of particle
$i$ .
## III Results and discussion
### III.1 Thermodynamic properties of spin polarized strange quark matter
In Fig. 1, we have plotted the polarization parameter versus the baryonic
density in the presence of magnetic field ($B=5\times 10^{18}\ G$) at
different temperatures. From this figure, we can see that the polarization
parameter decreases by increasing the baryonic density. However, at high
densities, the polarization parameter gets a constant value. In Fig.1, we have
also shown the influence of increasing the temperature on the polarization of
SQM. We see that at a fixed density, the polarization parameter decreases by
increasing the temperature. In fact, at high temperatures, the kinetic energy
of quarks increases, and the contribution of magnetic energy is therefore
lower. We have also shown the polarization parameter versus the baryonic
density at a fixed temperature ($T=30\ MeV$) in different magnetic fields in
Fig. 2. This indicates that by increasing the baryonic density, the
polarization parameter decreases. We see that at high densities, this
parameter gets a constant value, and it increases by increasing the magnetic
field. Fig. 2 shows that at high densities, for the magnetic fields lower than
$B=5\times 10^{17}\ G$, the polarization parameter becomes nearly zero. In the
other words, at high densities for low magnetic fields, the SQM becomes nearly
unpolarized.
We have presented the total free energy per volume of the spin polarized SQM
as a function of the baryonic density in Fig. 3 for the magnetic field
$B=5\times 10^{18}\ G$ at different temperatures. We can see that the free
energy of spin polarized SQM increases by increasing the baryonic density, and
at high densities, the increasing of free energy is faster than at low
densities. At any density, the free energy decreases by increasing the
temperature. This is due to the fact that the magnitude of second term of Eq.
(21) ($TS_{tot}$) increases as the temperature increases. In Fig. 4, we have
seen that at a fixed temperature ($T=30MeV$), the free energy of the spin
polarized SQM decreases as the magnetic field increases. In fact, the presence
of magnetic field helps the orientation of quarks to a more regular and stable
system with the lower energy.
In Fig. 5, we have shown the pressure of spin polarized SQM versus density in
the presence of magnetic field ($B=5\times 10^{18}\ G$) at different
temperatures. From this figure, we have found that at each density, by
increasing the temperature, the pressure increases. In the other word, the
equation of state of spin polarized SQM becomes stiffer by increasing the
temperature. In Fig. 6, the equation of state of spin polarized SQM at fixed
temperature ($T=30MeV$) for different magnetic fields has been plotted. This
figure indicates that the presence of magnetic field leads to the stiffer
equation of state for the spin polarized SQM. As can be seen from Figs. 3 and
4, by increasing both temperature and magnetic field, increasing the free
energy versus density takes place with the higher slope. This leads to higher
pressure at higher temperatures and magnetic fields. The equation of state of
system for the density dependent bag constant at $T=30\ MeV$ and $B=5\times
10^{18}\ G$ has been plotted in Fig. 7. In this figure, we have also given the
results for the case of fixed bag constant (${\cal B}_{bag}=90\
\frac{MeV}{fm^{3}}$) rk1016 for comparison. Fig. 7 indicates that with the
density dependent bag constant, the equation of state of spin polarized SQM is
stiffer than that with the fixed bag constant.
### III.2 Structure of spin polarized strange quark star
Mass and radius are the important macroscopic parameters for a compact star
playing crucial roles in investigation of its structure. Since strange quark
stars are relativistic systems, for calculating the structure properties of
these systems, we use general relativity. We assume the strange quark star to
be spherically symmetric, the structure of this star is determined by
numerically integrating the Tolman-Oppenheimer-Volkoff equations rk24 ; rk25 ;
rk26 using the equation of state of the system,
$\frac{dP}{dr}=-\frac{G\left[\varepsilon(r)+\frac{P(r)}{c^{2}}\right]\left[m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}\right]}{r^{2}\left[1-\frac{2Gm(r)}{rc^{2}}\right]},$ (25)
$\frac{dm}{dr}=4\pi r^{2}\varepsilon(r),$ (26)
where $G=6.707\times 10^{-45}\ MeV^{-2}$ is the gravitational constant, $r$ is
the distance from the center of the star, $\varepsilon(r)$ is the energy
density, $m(r)=m$ is the mass within the radius $r$, and $P=P(r)$ is the
pressure. The boundary condition is $P(r=0)\equiv P_{c}=P(\varepsilon_{c})$,
where $\varepsilon_{c}$ denotes the energy density at the star’s center. For
all pressure, we have $P<P_{c}$.
In Fig. 8, we have presented the gravitational mass of spin polarized SQS
versus the central energy density at different temperatures for the magnetic
field $B=5\times 10^{18}\ G$. In this figure, we have also given the results
at $T=0\ MeV$ and $B=5\times 10^{18}\ G$ for comparison rk1015 . We can see
that for all temperatures, the gravitational mass increases rapidly by
increasing the central energy density, and finally gets a limiting value
(maximum gravitational mass). This limiting value decreases by increasing the
temperature. The effect of magnetic field on the gravitational mass of spin
polarized SQS at a fixed temperature $T=30\ MeV$ has been shown in Fig. 9. We
see that by increasing the magnetic field, the gravitational mass decreases.
In Table 1, we have given the maximum mass and the corresponding radius of
spin polarized SQS at different temperatures for $B=5\times 10^{18}\ G$. It is
shown that as the temperature increases, the maximum mass and corresponding
radius of spin polarized SQS decreases. We have also presented the maximum
mass and the corresponding radius of spin polarized SQS for different magnetic
fields at fixed temperature $T=30\ MeV$ in Table 2. We see that the maximum
mass and corresponding radius of the spin polarized SQS decreases by
increasing the magnetic field. The above results indicate that at higher
temperatures and magnetic fields, the spin polarized SQS with the lower
gravitational mass can be stable. From Figs. 5 and 6, we see that by
increasing the temperature and magnetic field, the equation of state of system
becomes stiffer. Here, we can conclude that the stiffer equation of state for
spin polarized SQS leads to the lower values for its gravitational mass. In
Fig. 10, We have compared our results for two cases of density dependent and
density independent bag constant (${\cal B}_{bag}=90\ \frac{MeV}{fm^{3}}$)
rk1016 at $T=30\ MeV$ and $B=5\times 10^{18}\ G$. We can see that in the case
of density dependent bag constant, the gravitational mass of spin polarized
SQS is lower than that in the case of fixed bag constant. This corresponds to
the result of Fig. 7 in which we have shown that the equation of state with
the density dependent bag constant is stiffer than with the density
independent bag constant. In Table 3, at $T=30\ MeV$ for $B=5\times 10^{18}\
G$, our results for the maximum mass and corresponding radius of spin
polarized SQS has been compared with the results of density independent bag
constant rk1016 . We can see that the maximum mass for the density dependent
${\cal B}_{bag}$ is less than that for the fixed ${\cal B}_{bag}$.
## IV Summary and conclusions
In this article, we have studied the properties of a hot spin polarized
strange quark matter (SQM) in the presence of the strong magnetic field by the
MIT bag model using a density dependent bag constant. We have shown that by
increasing both magnetic field and temperature, the polarization parameter
decreases. We have calculated the energy density and the equation of state of
spin polarized SQM at different temperatures and magnetic fields. Our results
show that by increasing both temperature and magnetic field, the energy
density decreases. It is seen that the equation of state of spin polarized SQM
becomes stiffer by increasing both temperature and magnetic field. We have
used TOV equations to calculate the structure properties of spin polarized
SQS. Our results show that the gravitational mass increases by increasing the
central energy density and reaches a maximum value. This maximum value
decreases by increasing both temperature and magnetic field. From these
results, we have concluded that at higher temperatures and magnetic fields,
the SQS with lower gravitational mass can be stable. We have compared our
results of the density dependent bag constant with results of a fixed bag
constant. It is shown that the maximum mass with the density dependent bag
constant is lower than that with a fixed bag constant.
## Acknowledgements
This work has been supported financially by the Center for Excellence in
Astronomy and Astrophysics (CEAA-RIAAM). We wish to thank the Shiraz
University Research Council.
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Table 1: Maximum mass and the corresponding radius of spin polarized SQS for $B=5\times 10^{18}\ G$ at different temperatures. The results of $T=0\ MeV$ have been also given for comparison rk1015 . $T\ (MeV)$ | | $M_{max}\ (M_{\odot})$ | | $R\ (km)$
---|---|---|---|---
$0$ | | 1.62 | | 8.36
$30$ | | 1.15 | | 7.1
$70$ | | 0.77 | | 6.89
Table 2: Maximum mass and the corresponding radius of spin polarized SQS for different magnetic fields at $T=30\ MeV$. $B\ (G)$ | | $M_{max}\ (M_{\odot})$ | | $R\ (km)$
---|---|---|---|---
$0$ | | 1.39 | | 8.5
$5\times 10^{18}$ | | 1.15 | | 7.1
$5\times 10^{19}$ | | 0.99 | | 7.09
Table 3: Maximum mass and the corresponding radius of spin polarized SQS for $B=5\times 10^{18}\ G$ at $T=30\ MeV$. The results of $T=30\ MeV$ by a fixed bag constant have been also given for comparison rk1016 . ${\cal B}_{bag}\ (MeV/fm^{3})$ | | $M_{max}\ (M_{\odot})$ | | $R\ (km)$
---|---|---|---|---
density dependent | | 1.15 | | 7.1
90 | | 1.17 | | 7.37
Figure 1: The polarization parameter versus baryonic density for $B=5\times
10^{18}\ G$ at different temperatures $(T)$.
Figure 2: The polarization parameter versus baryonic density at $T=30\ MeV$
for different magnetic fields $(B)$.
Figure 3: The total free energy per volume of the spin polarized SQM as a
function of the baryonic density for $B=5\times 10^{18}\ G$ at different
temperatures $(T)$. Figure 4: The total free energy per volume of the spin
polarized SQM as a function of the baryonic density at $T=30\ MeV$ for
different magnetic fields $(B)$.
Figure 5: The pressure of the spin polarized SQM versus the baryonic density
for $B=5\times 10^{18}\ G$ at different temperatures $(T)$. Figure 6: The
pressure of the spin polarized SQM the baryonic density at $T=30\ MeV$ for
different magnetic fields $(B)$.
Figure 7: The pressure of the spin polarized SQM the baryonic density at
$T=30\ MeV$ and for $B=5\times 10^{18}\ G$ calculated by a density dependent
bag constant (solid curve). The results for ${\cal B}_{bag}=90\ MeVfm^{-3}$
(dashed curve) have also been given for comparison.
Figure 8: The gravitational mass of spin polarized SQS versus the central
energy density in $B=5\times 10^{18}\ G$ at different temperatures $(T)$. The
results at $T=0\ MeV$ (dashed dotted curve) have also been given for
comparison.
Figure 9: The gravitational mass of spin polarized SQS versus the central
energy density at $T=30\ MeV$ for different magnetic fields $(B)$.
Figure 10: The gravitational mass of spin polarized SQS versus the central
energy density at $T=30\ MeV$ for $B=5\times 10^{18}\ G$ calculated by a
density dependent bag constant (solid curve).The results for ${\cal
B}_{bag}=90\ MeVfm^{-3}$ (dashed curve) have also been given for comparison.
|
arxiv-papers
| 2013-12-16T14:03:38 |
2024-09-04T02:49:55.492024
|
{
"license": "Public Domain",
"authors": "G. H. Bordbar and Z. Alizade",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1312.4362"
}
|
1312.4382
|
# Single-trial estimation of stimulus and spike-history effects on time-
varying ensemble spiking activity of multiple neurons: a simulation study
Hideaki Shimazaki RIKEN Brain Science Institute, Wako, Saitama, Japan
[email protected]
###### Abstract
Neurons in cortical circuits exhibit coordinated spiking activity, and can
produce correlated synchronous spikes during behavior and cognition. We
recently developed a method for estimating the dynamics of correlated ensemble
activity by combining a model of simultaneous neuronal interactions (e.g., a
spin-glass model) with a state-space method (Shimazaki et al. 2012 PLoS Comput
Biol 8 e1002385). This method allows us to estimate stimulus-evoked dynamics
of neuronal interactions which is reproducible in repeated trials under
identical experimental conditions. However, the method may not be suitable for
detecting stimulus responses if the neuronal dynamics exhibits significant
variability across trials. In addition, the previous model does not include
effects of past spiking activity of the neurons on the current state of
ensemble activity. In this study, we develop a parametric method for
simultaneously estimating the stimulus and spike-history effects on the
ensemble activity from single-trial data even if the neurons exhibit dynamics
that is largely unrelated to these effects. For this goal, we model ensemble
neuronal activity as a latent process and include the stimulus and spike-
history effects as exogenous inputs to the latent process. We develop an
expectation-maximization algorithm that simultaneously achieves estimation of
the latent process, stimulus responses, and spike-history effects. The
proposed method is useful to analyze an interaction of internal cortical
states and sensory evoked activity.
## 1 Introduction
Neurons in the brain make synaptic contacts to each other and form specific
signaling networks. A typical cortical neuron receives synaptic inputs from
$3000-10000$ other neurons, and makes synaptic contacts to several thousands
of other neurons. They send and receive signals using pulsed electrical
discharges known as action potentials, or spikes. Therefore, individual
neurons in a circuit can be activated in a coordinated manner when relevant
information is processed. In particular, nearly simultaneous spiking activity
of multiple neurons (synchronous spikes) occurs dynamically in relation to a
stimulus presented to an animal, the animal’s behavior, and the internal state
of the brain (attention and expectation) [1, 2, 3, 4, 5].
Recently, it was reported that a model of synchronous spiking activity that
accounts for spike rates of individual neurons and interactions between pairs
of neurons can explain $\sim 90$% of the synchronous spiking activity of a
small subset ($\sim 10$) of retinal ganglion cells [6, 7] and cortical neurons
[8] in vitro. This model is known as a maximum entropy model or an Ising/spin-
glass model in statistical physics. However, since the model assumes
stationary data, it is not directly applicable to non-stationary data recorded
from awake behaving animals. In these data sets, spike-rates of individual
neurons and even interactions among them may vary across time.
In order to analyze time-dependent synchronous activity of neurons, we
recently developed a method for estimating the dynamics of correlations
between neurons by combining the model of neuronal interactions (e.g., the
Ising/spin-glass model) with a state-space method [9, 10]. In classical
neurophysiological experiments, neuronal activity is repeatedly recorded under
identical experimental conditions in order to obtain reproducible features in
the spiking activity across the ‘trials’. Typically, neurophysiologists
estimate average time-varying firing rates of individual neurons in response
to a stimulus from the repeated trials [11, 12]. In the same fashion, the
state-space method in [9, 10] aims to estimate the dynamics of the neuronal
interactions, including higher-order interactions, that occurs repeatedly upon
the onset of externally triggered events. When this method is applied to three
neurons recorded simultaneously from the primary motor cortex of a monkey
engaged in a delayed motor task (data from [2]), it was revealed that these
neurons dynamically organized into a group characterized by the presence of a
higher-order (triple-wise) interaction, depending on the behavioral demands to
the monkey [12].
However, neurophysiological studies in the past decades revealed that spiking
activity of individual neurons is subject to large variability across trials
due to structured ongoing activity of the networks that arises internally to
the brain [13, 14, 15]. In these conditions, the method developed in [12]
would not efficiently detect the stimulus responses because a signal-to-noise
ratio may be small even in the trial-averaged activity. Although statistical
methods for detecting responses of individual neurons from single-trial data
have been investigated [16, 17, 18, 19], no methods are available for
estimating synchronous responses of multiple neurons to a stimulus in a single
trial when these neurons are subject to the activity that is largely unrelated
to the stimulus.
In the analysis of single-trial data, it is critical to consider dependency of
the current activity of neurons on the past history of their activity. A
neuron undergoes an inactivation period known as a refractory period after it
generates an action potential. Therefore, a model neuron significantly
improves its goodness-of-fit to data if it captures this biophysical property
[20, 21]. In addition, estimating the dependency of the current activity level
of a neuron on past spiking history of another neuron allows us to construct
effective connectivity of the network within an observed set of neurons [22,
23]. Including the spike-history effects in the models of synchronous ensemble
activity is thus an important topic, and investigated also in [24] in the
framework of a continuous-time point process theory.
In this study we construct a method for simultaneously estimating the stimulus
and spike-history effects on ensemble spiking activity when the activity of
these neurons is dominated by ongoing activity. For this goal, we extend the
previously developed state-space model of neuronal interactions: We model the
ongoing activity, i.e., time-varying spike rates and interactions, of neurons
as a latent process, and include the stimulus and spike-history effects on the
activity as exogenous inputs to the latent process. We develop an expectation-
maximization (EM) algorithm for this model, which efficiently combines
construction of a posterior density of the latent process and estimation of
the parameters for stimulus and spike-history effects. The method is tested
using simulated spiking activity of 3 neurons with known underlying
architecture. We provide an approximation method for determining inclusion of
these exogenous inputs in the model and a surrogate method to test
significance of the estimated parameters.
## 2 Methods
In this study, we analyze spike sequences simultaneously obtained from $N$
neurons. From these spike sequences, we construct binary spike patterns at
discrete time steps by dividing the sequences into disjoint time bins with an
equal width of $\Delta$ ms (in total, $T$ bins). The width $\Delta$ determines
a permissible range of synchronous activity of neurons in this analysis. We
let $X_{i}^{t}$ be a binary variable of the $i$-th neuron ($i=1,2,\ldots,N$)
in the $t$-th time bin ($t=1,2,\ldots,T$). Here a time bin containing ‘$1$’
indicates that one or more spikes exist in the time bin whereas ‘$0$’
indicates that no spike exists in the time bin. The binary pattern of $N$
neurons at time bin $t$ is denoted as
$\mathbf{X}_{t}=[X_{1}^{t},X_{2}^{t},\ldots,X_{N}^{t}]^{\prime}$. The prime
indicates the transposition operation to the vector. The entire observation of
the discretized ensemble spiking activity is represented as
$\mathbf{X}_{1:T}=[\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{T}]$.
### 2.1 The model of time-varying simultaneous interactions of neurons
We analyze the ensemble spike patterns using time-dependent formulation of a
joint probability mass function for binary random variables. Let $x_{i}$ be a
binary variable, namely $x_{i}=\left\\{0,1\right\\}$. The joint probability
mass function of $N$-tuple binary variables,
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{N}]$, at time bin $t$ ($t=1,2,\ldots,T$)
can be written in an exponential form as
$\displaystyle p(\mathbf{x}|\boldsymbol{\theta}_{t})$
$\displaystyle=\exp\left[\sum_{i}\theta_{i}^{t}x_{i}+\sum_{i<j}\theta_{ij}^{t}x_{i}x_{j}+\cdots+\theta_{1\cdots
N}^{t}x_{1}\cdots x_{N}-\psi(\boldsymbol{\theta}_{t})\right].$ (1)
Here
$\boldsymbol{\theta}_{t}=[\theta_{1}^{t},\theta_{2}^{t},\ldots,\theta_{12}^{t},\theta_{13}^{t},\ldots,\theta_{1\cdots
N}^{t}]^{\prime}$ summarizes the time-dependent canonical parameters of the
exponential family distribution. The canonical parameters for the interaction
terms, e.g., $\theta_{ij}^{t}$ ($i,j=1,\ldots,N$), represent time-dependent
instantaneous interactions at time bin $t$ among the neurons denoted in its
subscript. $\psi(\boldsymbol{\theta}_{t})$ is a log normalization parameter to
satisfy $\sum p(\mathbf{x}|\boldsymbol{\theta}_{t})=1$.
Using a feature vector that captures simultaneous spiking activities of
subsets of the neurons,
$\mathbf{f}=[f_{1},f_{2},\ldots,f_{12},f_{13},\ldots,f_{1\cdots N}]^{\prime}$,
where
$\begin{array}[]{cc}f_{i}\left(\mathbf{x}\right)=x_{i},&i=1,\cdots,N\\\
f_{ij}\left(\mathbf{x}\right)=x_{i}x_{j},&i<j\\\ \vdots&\mbox{}\\\ f_{1\cdots
N}\left(\mathbf{x}\right)=x_{1}\cdots x_{N},&\mbox{}\end{array}$
the probability mass function (Eq. 1) is compactly written as
$p({\mathbf{x}}|\boldsymbol{\theta}_{t})=\exp\left[\boldsymbol{\theta}_{t}^{\prime}\mathbf{f}\left(\mathbf{x}\right)-\psi(\boldsymbol{\theta}_{t})\right]$.
The expected occurrence rates of simultaneous spikes of multiple neurons is
given by a vector
$\boldsymbol{\eta}_{t}=E\left[\mathbf{f}\left(\mathbf{x}\right)|\boldsymbol{\theta}_{t}\right]$,
where expectation is performed using
$p({\mathbf{x}}|\boldsymbol{\theta}_{t})$.
Eq. 1 specifies the probabilities of all $2^{N}$ spike patterns by using
$2^{N}-1$ parameters. One reasonable approach to reduce the number of
parameters is to select and fix interesting features in the spiking activity,
and construct a probability model that maximizes entropy. For example,
maximization of entropy of the spike patterns given the low-order features,
$\mathbf{f}=[f_{1},f_{2},\ldots,f_{N},f_{12},f_{13},\ldots,f_{N-1,N}]^{\prime}$,
yields a spin-glass model that is similar to Eq. 1, but does not include
interactions higher than the second order. While it is important to explore a
characteristic feature vector to neuronal ensembles, here we note that the
method developed in this study does not depend on the choice of the vector,
$\mathbf{f}$. Below, we denote $d$ as the number of elements in the vector,
$\mathbf{f}$.
Given the observed ensemble spiking activity $\mathbf{X}_{1:T}$, the
likelihood function of
$\boldsymbol{\theta}_{1:T}=[\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2},\ldots,\boldsymbol{\theta}_{T}]$
is given as
$\displaystyle p\left(\mathbf{X}_{1:T}|\boldsymbol{\theta}_{1:T}\right)$
$\displaystyle=\prod\limits_{t=1}^{T}\exp[\boldsymbol{\theta}_{t}^{\prime}\mathbf{f}\left(\mathbf{X}_{t}\right)-\psi(\boldsymbol{\theta}_{t})],$
(2)
assuming conditional independence across the time bins. Eq.2 constitutes an
observation equation of our state-space model.
### 2.2 Inclusion of stimulus and spike-history effects in the state model
The main focus of attention in this study is modeling of a process for the
time-dependent canonical parameters, $\boldsymbol{\theta}_{t}$, in Eq. 1. We
model their evolution as a first-order auto-regressive (AR) model. The effects
of the stimulus and spike history are included as exogenous inputs to the AR
model (an ARX model). In its full expression, the state model is written as
$\boldsymbol{\theta}_{t}=\mathbf{F}\boldsymbol{\theta}_{t-1}+\mathbf{G}\mathbf{S}_{t}+\sum_{i=1}^{p}\mathbf{H}_{i}\mathbf{X}_{t-i}+\boldsymbol{\xi}_{t},$
(3)
for $t=2,\ldots,T$. Here the matrix $\mathbf{F}$ ($d\times d$ matrix) is the
first order auto-regressive parameter. $\boldsymbol{\xi}_{t}$ ($d\times 1$
matrix) is a random vector independently drawn from a zero-mean multivariate
normal distribution with covariance matrix $\mathbf{Q}$ ($d\times d$ matrix)
at each time bin. The state process starts with an initial value
$\boldsymbol{\theta}_{1}$ that follows a normal distribution with mean
$\boldsymbol{\mu}$ ($d\times 1$ matrix) and covariance matrix
$\boldsymbol{\Sigma}$ ($d\times d$ matrix), namely
$\boldsymbol{\theta}_{1}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
Below, we describe details of the exogenous terms.
The second term represents responses to external signals, or stimuli,
$\mathbf{S}_{t}$, which are observed concurrently with the spike sequences.
The vector $\mathbf{S}_{t}$ is a column vector of $n_{s}$ external signals at
time bin $t$. The each element is the value of an external signal at time bin
$t$. If an external signal is represented as a sequence of discrete events, we
denote the corresponding element of $\mathbf{S}_{t}$ by ‘1’ if an event
occurred within time bin $t$ and ‘0’ otherwise. Multiplying $\mathbf{S}_{t}$
by the matrix $\mathbf{G}$ ($d\times n_{s}$ matrix) produces weighted linear
combinations of the external signals at time bin $t$.
The third term represents the effects of spiking activity during the previous
$p$ time bins, $\mathbf{X}_{t-i}$ $(i=1,\ldots,p)$, on the current activity.
The matrix $\mathbf{H}_{i}$ ($d\times N$ matrix) represents the spike-history
effects of spiking activity in the previous time bin $t-i$ on the state at
time bin $t$. The spike-history effects are collectively denoted as
$\mathbf{H}\equiv[\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{p}]$
($d\times Np$ matrix).
Eq. 3 constitutes a prior density of the latent process in our state-space
model. We denote the set of parameters in the prior distribution, called
hyper-parameters, as
$\mathbf{w}\equiv\left[\mathbf{F},\mathbf{G},\mathbf{H},\mathbf{Q},\boldsymbol{\mu},\boldsymbol{\Sigma}\right]$.
In this study, we refer to $\mathbf{w}$ as a parameter. In addition, we
simplify Eq. 3 as
$\boldsymbol{\theta}_{t}=\mathbf{F}\boldsymbol{\theta}_{t-1}+\mathbf{U}\mathbf{u}_{t}+\boldsymbol{\xi}_{t},$
where $\mathbf{u}_{t}$ is a single column vector constructed by stacking the
stimulus vector and spike-history vectors at time bin $t$ in a row, i.e.,
$\mathbf{u}_{t}=[\mathbf{S}_{t};\mathbf{X}_{t-1};\mathbf{X}_{t-2};\ldots;\mathbf{X}_{t-p}]$.
Similarly, we define a matrix $\mathbf{U}$ as
$\mathbf{U}=[\mathbf{G},\mathbf{H}]$. With this simplification, the prior
density defined in Eq. 3 is written as
$p(\boldsymbol{\theta}_{1:T}|\mathbf{w})=p(\boldsymbol{\theta}_{1}|\boldsymbol{\mu},\boldsymbol{\Sigma})\prod_{t=2}^{T}p(\boldsymbol{\theta}_{t}|\boldsymbol{\theta}_{t-1},\mathbf{F},\mathbf{U},\mathbf{Q})$,
where the transition probability,
$p(\boldsymbol{\theta}_{t}|\boldsymbol{\theta}_{t-1},\mathbf{F},\mathbf{U},\mathbf{Q})$,
is given as a normal distribution with mean
$\mathbf{F}\boldsymbol{\theta}_{t-1}+\mathbf{U}\mathbf{u}_{t}$ and covariance
matrix $\mathbf{Q}$.
## 3 Estimation of stimulus responses and spike-history effects
We estimate the parameter, $\mathbf{w}$, based on the principle of maximizing
a (log) marginal likelihood function. Namely, we select the parameter that
maximizes
$l\left(\mathbf{w}\right)=\log\int
p\left(\mathbf{X}_{1:T},\boldsymbol{\theta}_{1:T}|\mathbf{w}\right)d\boldsymbol{\theta}_{1:T}.$
(4)
For this goal, we use the expectation-maximization (EM) algorithm [25, 26,
27]. In this method, we iteratively obtain the optimal parameter
$\mathbf{\mathbf{w}^{\ast}}$ that maximizes the lower bound of the above log
marginal likelihood. This alternative function, known as the expected complete
data log-likelihood (a.k.a., $q$-function), is computed as
$\displaystyle q\left(\mathbf{\mathbf{w}^{\ast}}|\mathbf{w}\right)$
$\displaystyle\equiv E\left[\log
p\left(\mathbf{X}_{1:T},\boldsymbol{\mathbf{\theta}}_{1:T}|\mathbf{\mathbf{w}^{\ast}}\right)|\mathbf{X}_{1:T},\mathbf{w}\right]$
$\displaystyle=\sum\limits_{t=1}^{T}\left(E\boldsymbol{\theta}^{\prime}_{t}\mathbf{f}\left(\mathbf{X}_{t}\right)-E\psi\left(\boldsymbol{\theta}_{t}\right)\right)-\frac{d}{2}\log{2\pi}-\frac{1}{2}\log{\det\boldsymbol{\Sigma}^{\ast}}$
$\displaystyle-\frac{1}{2}E[\left(\boldsymbol{\theta}_{1}-\boldsymbol{\mu}^{\ast}\right)^{\prime}\boldsymbol{\Sigma}^{\ast-1}\left(\boldsymbol{\theta}_{1}-\boldsymbol{\mu}^{\ast}\right)]-\frac{\left(T-1\right)d}{2}\log{2\pi}-\frac{\left(T-1\right)}{2}\log{\det\mathbf{Q}^{\ast}}$
$\displaystyle-\frac{1}{2}\sum\limits_{t=2}^{T}E[\left(\boldsymbol{\theta}_{t}-\mathbf{F}^{\ast}\boldsymbol{\theta}_{t-1}-\mathbf{U}^{\ast}\mathbf{u}_{t}\right)^{\prime}\mathbf{Q}^{\ast-1}\left(\boldsymbol{\theta}_{t}-\mathbf{F}^{\ast}\boldsymbol{\theta}_{t-1}-\mathbf{U}^{\ast}\mathbf{u}_{t}\right)].$
(5)
The expectation, $E[\centerdot|\mathbf{X}_{1:T},\mathbf{w}]$, in Eq. 5 is
performed using the smoother posterior density of the state obtained by a
nominal parameter, $\mathbf{w}$, namely
$p\left(\boldsymbol{\theta}_{1:T}|\mathbf{X}_{1:T},\mathbf{w}\right)=\frac{p\left(\mathbf{X}_{1:T}|\boldsymbol{\theta}_{1:T}\right)p\left(\boldsymbol{\theta}_{1:T}|\mathbf{w}\right)}{p\left(\mathbf{X}_{1:T}|\mathbf{w}\right)}.$
(6)
In particular, Eq. 5 can be computed using the following expected values by
the posterior density: The smoother mean
$\boldsymbol{\theta}_{t|T}=E\left[\boldsymbol{\theta}_{t}|\mathbf{X}_{1:T},\mathbf{w}\right]$,
the smoother covariance matrix
$W_{t|T}=E[(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|T})(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|T})^{\prime}|\mathbf{X}_{1:T},\mathbf{w}]$,
and the lag-one covariance matrix,
$W_{t,t-1|T}=E[(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|T})(\boldsymbol{\theta}_{t-1}-\boldsymbol{\theta}_{t-1|T})^{\prime}|\mathbf{X}_{1:T},\mathbf{w}]$.
These values are obtained using the approximate recursive Bayesian
filtering/smoothing algorithm developed in [9, 10] (See Appendix A and Eqs.
22, 23, and 24 therein).
In the EM-algorithm, we obtain the parameter that maximizes the $q$-function
by alternating the expectation (E) and maximization (M) steps. In the E-step,
we obtain the above expected values in Eq. 5 by the approximate recursive
Bayesian filtering/smoothing algorithm using a fixed $\mathbf{w}$ (Appendix
A). In the M-step, we obtain the parameter, $\mathbf{\mathbf{w}^{\ast}}$, that
maximizes Eq. 5. The resulting $\mathbf{\mathbf{w}^{\ast}}$ is then used in
the next E-step. Below, we derive an algorithm for optimizing the parameter at
the M-step.
For the state model that includes the auto-regressive parameter and stimulus
and/or spike-history effects, these parameters are estimated simultaneously.
From
$\frac{\partial}{\partial\mathbf{F}^{\ast}}q\left(\mathbf{w}^{\ast}|\mathbf{w}\right)=\mathbf{0}$,
we obtain
$\mathbf{F}^{\ast}\sum_{t=2}^{T}\left(\mathbf{W}_{t-1,t|T}+\boldsymbol{\theta}_{t-1|T}\boldsymbol{\theta}_{t-1|T}^{\prime}\right)+\mathbf{U}^{\ast}\sum_{t=2}^{T}\boldsymbol{u}_{t}\boldsymbol{\theta}_{t-1|T}^{\prime}=\sum_{t=2}^{T}\left(\mathbf{W}_{t-1,t|T}+\boldsymbol{\theta}_{t|T}\boldsymbol{\theta}_{t-1|T}^{\prime}\right).$
(7)
Here, $\boldsymbol{\theta}_{t|T}$, $\mathbf{W}_{t|T}$, and
$\mathbf{W}_{t-1,t|T}$ are the smoother mean and covariance, and the lag-one
covariance matrix given by Eqs. 22, 23, and 24, respectively. Similarly, from
$\frac{\partial}{\partial\mathbf{U}^{\ast}}q\left(\mathbf{w}^{\ast}|\mathbf{w}\right)=\mathbf{0}$,
we obtain
$\mathbf{F}^{\ast}\sum_{t=2}^{T}\boldsymbol{\theta}_{t-1|T}\boldsymbol{u}_{t}^{\prime}+\mathbf{U}^{\ast}\sum_{t=2}^{T}\boldsymbol{u}_{t}\boldsymbol{u}_{t}^{\prime}=\sum_{t=2}^{T}\boldsymbol{\theta}_{t|T}\boldsymbol{u}_{t}^{\prime}.$
(8)
Hence, the simultaneous update rule for $\mathbf{F}^{\ast}$ and
$\mathbf{U}^{\ast}$ is given as
$\displaystyle\left[\begin{array}[]{cc}\mathbf{F}^{\ast}&\mathbf{U}^{\ast}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}\sum_{t=2}^{T}\left(\mathbf{W}_{t-1,t|T}+\boldsymbol{\theta}_{t|T}\boldsymbol{\theta}_{t-1|T}^{\prime}\right)&\sum_{t=2}^{T}\boldsymbol{\theta}_{t|T}\boldsymbol{u}_{t}^{\prime}\end{array}\right]$
(11)
$\displaystyle\left[\begin{array}[]{cc}\sum_{t=2}^{T}\left(\mathbf{W}_{t-1,t|T}+\boldsymbol{\theta}_{t-1|T}\boldsymbol{\theta}_{t-1|T}^{\prime}\right)&\sum_{t=2}^{T}\boldsymbol{\theta}_{t-1|T}\boldsymbol{u}_{t}^{\prime}\\\
\sum_{t=2}^{T}\boldsymbol{u}_{t}\boldsymbol{\theta}_{t-1|T}^{\prime}&\sum_{t=2}^{T}\boldsymbol{u}_{t}\boldsymbol{u}_{t}^{\prime}\end{array}\right]^{-1}.$
(14)
Here the inverse matrix on the r.h.s. is obtained by using the blockwise
inversion formula:
$\left[\begin{array}[]{cc}A&B\\\
C&D\end{array}\right]^{-1}=\left[\begin{array}[]{cc}A^{-1}+A^{-1}B\left(D-CA^{-1}B\right)^{-1}CA^{-1}&-A^{-1}B\left(D-CA^{-1}B\right)^{-1}\\\
-\left(D-CA^{-1}B\right)^{-1}CA^{-1}&\left(D-CA^{-1}B\right)^{-1}\end{array}\right].$
The covariance matrix, $\mathbf{Q}$, can be optimized separately. From
$\frac{\partial}{\partial\mathbf{Q}^{\ast}}q\left(\mathbf{w}^{\ast}|\mathbf{w}\right)=\mathbf{0}$,
the update rule of $\mathbf{Q}$ is obtained as
$\displaystyle\mathbf{Q}^{\ast}=$
$\displaystyle\frac{1}{T-1}\sum_{t=2}^{T}[\mathbf{W}_{t|T}-\mathbf{W}_{t-1,t|T}\mathbf{F}^{\prime}-\mathbf{FW}_{t-1,t|T}^{\prime}+\mathbf{FW}_{t-1|T}\mathbf{F}^{\prime}]$
$\displaystyle+$
$\displaystyle\frac{1}{T-1}\sum_{t=2}^{T}\left(\boldsymbol{\theta}_{t|T}-\mathbf{F}\boldsymbol{\theta}_{t-1|T}-\mathbf{U}\mathbf{u}_{t}\right)\left(\boldsymbol{\theta}_{t|T}-\mathbf{F}\boldsymbol{\theta}_{t-1|T}-\mathbf{U}\mathbf{u}_{t}\right)^{\prime}.$
(15)
Finally, the mean of the initial distribution is updated with
$\boldsymbol{\mu}^{\ast}=\boldsymbol{\theta}_{1|T}$ from
$\frac{\partial}{\partial\boldsymbol{\mu}^{\ast}}q\left(\mathbf{w}^{\ast}|\mathbf{w}\right)=\mathbf{0}$.
The covariance matrix $\boldsymbol{\Sigma}$ for the initial parameters is
fixed in this optimization.
## 4 Results
### 4.1 Simulation of a network of 3 neurons
Figure 1: (A) Schematic diagram of a simulated network of 3 neurons. Neuron 1
makes excitatory synaptic contacts to Neuron 2 and 3. Stimulus 1 excites
Neuron 1 whereas Stimulus 2 excites Neuron 2 and 3 simultaneously. In
addition, all neurons receive sinusoidal rate modulation. (B) Simulated
spiking activity of the network. A short period (1 s) of the total 30 s length
is shown. The magenta and cyan triangles represent occurrence times of
Stimulus 1 and 2, respectively. The gray bar highlights simultaneous spikes of
Neuron 2 and 3 that are causally induced 5 ms after a spike occurs in Neuron
1. (C) Instantaneous spike rates. (Top) The red trace is the instantaneous
spike rate of Neuron 1 simulated as an inhomogeneous renewal point process
whos instantaneous inter-spike interval is given by the inverse Gaussian
distribution ($f\left(t;\kappa\right)=\sqrt{\frac{\kappa}{2\pi
t^{3}}}\exp\left[-\frac{\kappa}{2t}\left(t-1\right)^{2}\right]\text{for
}x>0\text{, }0\text{ for }x<0$, $\kappa=1.8$ for all neurons). The
inhomogeneous rate is modulated by the sinusoidal function (black solid line,
frequency: 1 Hz; mean rate and amplitude: 30 spikes/s). (Bottom) Instantaneous
spike rates of Neuron 2 and 3 (solid green line and dashed blue line,
respectively).
In order to test the method, we simulate spiking activity of 3 neurons that
possess specific characteristics in spike generation and connectivity as
follows (See Fig. 1A). (1) The instantaneous firing rate of each neuron model
depends on its own spike history in order to reproduce refractoriness in
neuronal spiking activity. To achieve this, we adopt a renewal point process
model whose instantaneous inter-spike interval (ISI) distribution is given by
an inverse Gaussian distribution as a model of the stochastic spiking
activity. (2) The firing rate of each neuron model varies across time in order
to reproduce the ongoing activity. For that purpose, spike times of each
neuron are generated from the renewal process by adding inhomogeneity to the
underlying rate using the time-rescaling method described in [20]. The
underlying rate of the inhomogeneous renewal point process model is modulated
using a sinusoidal function (frequency: 1 Hz; mean and amplitude: 30
spikes/s). This rate modulation is common to the 3 neurons. (3) The neurons
are activated by externally triggered stimulus inputs. To realize the stimulus
responses, we deterministically induce spikes at predetermined timings of the
stimuli. We consider two stimuli, one (Stimulus 1) that induces a spike in
Neuron 1, and the other (Stimulus 2) that induces synchronous spikes in Neuron
2 and Neuron 3. The timings of external stimuli are not related to the
sinusoidal time-varying rate, but randomly selected in the observation period
(On average each stimulus happens once in 1 second). (4) There is feedforward
circuitry in the network. We assume that Neuron 1 makes excitatory synaptic
contacts to Neuron 2 and Neuron 3. To realize this, 5ms after a spike occurs
in Neuron 1, we induce simultaneous spikes in Neuron 2 and Neuron 3 with a
probability 0.5.
We simulate spike sequences with a length of 30 seconds using 1 ms resolution
for numerical time steps (An example of a short period (1 s) is shown in Fig.
1B). Figure 1C displays the instantaneous spike-rates (conditional intensity
functions of point processes) of Neuron 1 (Top, red line) and Neuron 2 & 3
(Bottom, green and blue lines) underlying the spiking activity in Fig. 1B. The
black lines indicate sinusoidal rate modulation common to all neurons. In
addition, spikes are induced in Neuron 1 at the onsets of Stimulus 1 (magenta
triangles). Similarly, simultaneous spikes of Neuron 2 and Neuron 3 are
generated at the onsets of Stimulus 2 (cyan triangles). In the traces of
instantaneous spike-rates in Fig. 1C, instantaneous increases caused by the
stimuli and synaptic inputs are not displayed. The instantaneous spike-rate of
a neuron is reset to zero whenever a spike is induced in that neuron.
### 4.2 Selection of a state model
Figure 2: Comparison of state models by the Akaike Information Criterion
(AIC). The state-space models with the following five different state models
are comapared: [$\mathbb{Q}$], [$\mathbb{Q}$,
$\mathbb{F}$],[$\mathbb{Q}$,$\mathbb{F}$,$\mathbb{G}$],
[$\mathbb{Q}$,$\mathbb{F}$, $\mathbb{G}$, $\mathbb{H}6$], and
[$\mathbb{Q}$,$\mathbb{F}$, $\mathbb{G}$, $\mathbb{H}12$] (See details of the
models for main text). The reduction of the AIC of the last four models from
the AIC of the model [$\mathbb{Q}$] ($\Delta$AIC) was repeatedly computed for
10 times. The height of the bar indicates the average $\Delta$AIC. The error
bar indicates $\pm$ 2 S.E. The numbers marked on each bar are dimensions of
the models (The number of free parameters in the state model).
We analyze the simulated ensemble activity by the proposed state-space model.
For this goal, we first construct binary spike patterns, $\mathbf{X}_{1:T}$,
from the simulated spike sequences of 30 seconds (Note: spike times are
recorded in 1 ms resolution) by discretizing them using disjoint time bins
with 2 ms width. We then apply state-space models to the binary data. The
observation model used here contains interactions up to the second order (a
pairwise interaction model):
$\displaystyle p(\mathbf{x}|\boldsymbol{\theta}_{t})$
$\displaystyle=\exp\left[\theta_{1}^{t}x_{1}+\theta_{2}^{t}x_{2}+\theta_{3}^{t}x_{3}+\theta_{12}^{t}x_{1}x_{2}+\theta_{13}^{t}x_{1}x_{3}+\theta_{23}^{t}x_{2}x_{3}-\psi(\boldsymbol{\theta}_{t})\right].$
(16)
For the state model, we consider 5 different models that include a set of
different components in Eq. 3. We select a model based on the framework of
model selection in order to avoid over-fitting of a model to the data. Details
of each state model are described as follows.
The first state model assumes $\mathbf{F}=\mathbf{I}$, where $\mathbf{I}$ is
an identity matrix, and does not include any of exogenous inputs. In this
model, we optimize only the covariance matrix $\mathbf{Q}$. The first model is
denoted as $[\mathbf{Q}]$. The second state model, denoted as
$[\mathbf{Q},\mathbf{F}]$, is the first-order auto-regressive model. In this
model, we optimize both the covariance matrix $\mathbf{Q}$ and the auto-
regressive parameter $\mathbf{F}$. The third model, denoted as
$[\mathbf{Q},\mathbf{F},\mathbf{G}]$, additionally includes the stimulus term
as exogenous inputs (Stimulus 1 and Stimulus 2). Both the matrix $\mathbf{F}$
and $\mathbf{G}$ are optimized simultaneously in addition to $\mathbf{Q}$. The
fourth model includes both stimulus and spike-history terms. In this model,
the state model includes the history of spiking activity up to the last 6 time
bins ($p=6$). All parameters $\mathbf{F}$, $\mathbf{G}$, and $\mathbf{H}$ are
optimized simultaneously in addition to $\mathbf{Q}$. This model is denoted as
$[\mathbf{Q},\mathbf{F},\mathbf{G},\mathbf{H}6]$. The structure of the fifth
model is the same as the fourth model, but contains the history of spiking
activity up to the last 12 time bins ($p=12$). The last model is denoted as
$[\mathbf{Q},\mathbf{F},\mathbf{G},\mathbf{H}12]$.
In order to select the most predictive model among them, we select the state-
space model that minimizes the Akaike (Bayesian) information criterion (AIC)
[28],
$\textrm{AIC}=-2l\left(\mathbf{w}^{\ast}\right)+2\textrm{dim}\,\mathbf{w}^{\ast},$
(17)
where $\mathbf{w}^{\ast}$ is the optimized parameter in the Methods section.
The (marginal) likelihood function in Eq. 17 is obtained by a log-quadratic
approximation, i.e, the Laplace method [10] (See Appendix B for the complete
equation). Figure 2B displays decreases in AICs ($\Delta\textrm{AIC}$) of the
last four models from the AIC of the first model, $[\mathbf{Q}]$. The larger
the $\Delta\textrm{AIC}$ is, the better the state-space model is expected to
predict unseen data. For these data sets, inclusion of exogenous inputs, in
particular the spike history, significantly decreases the AIC. From this
result, we select the state model that includes the stimulus response term and
the spike-history terms up to the previous 6 time bins.
### 4.3 Parameter estimation
Figure 3: Parameter estimation of the state-space model. (A) Effects of
Stimulus 1 and 2 on the canonical paramters (the first and second column of
$\mathbb{G}$). The vertical ticks on abscissa indicate the 95% confidence
bounds for each parameters obtained by the surrogate method. (B) Summed spike-
history effects. The matrices of spike-history effects, $\mathbb{H}_{p}$, are
summed over the time-lags and shown using color. (C) The effect of a spike
occurrence in Neuron $i$ at $p$ time bins before the $t$th bin on
$\theta^{(t)}_{i}$ ($i=1,2,3$). (D) The effect of a spike occurence in Neuron
$i$ ($i=1,2,3$) on the interaction parameter $\theta^{(t)}_{23}$.
We now look at the estimated parameters of the model selected by the AIC,
namely $[\mathbf{Q},\mathbf{F},\mathbf{G},\mathbf{H}6]$. Due to the limitation
in the space, we do not display the estimated dynamics of the canonical
parameters, $\boldsymbol{\theta}_{t}$, by the recursive Bayesian method (See
[9, 10] for the detailed analysis on dynamics of $\boldsymbol{\theta}_{t}$ by
this method). The estimated parameters, $\mathbf{G}$ and $\mathbf{H}$, are
summarized in Fig. 3.
Here, in order to test the significance of the estimated parameters, we
construct confidence bounds of the estimates, using a surrogate method. In
this approach, we apply the same state-space model,
$[\mathbf{Q},\mathbf{F},\mathbf{G},\mathbf{H}6]$, to the surrogate data set
for the exogenous inputs. In the surrogate data set, the onset times of
external signals (Stimulus 1 and Stimulus 2) are randomized in the observation
period. Similarly, we randomly select $p=6$ bins from the past spiking
activity to obtain surrogate spike history, instead of selecting the last
consecutive 6 bins from time bin $t$. Thus the estimated parameters,
$\mathbf{G}$ and $\mathbf{H}$, are not related to the structure specified in
the Section 4.1. We repeatedly applied the state-space model to the surrogate
data (1000 times) to obtain the 95% confidence bound for the parameter
estimation (vertical ticks in Fig. 3A, C and D).
Figure 3A displays effects of the two stimuli, $\mathbf{G}$, on the respective
elements in $\boldsymbol{\theta}_{t}$. First, Stimulus 1 significantly
increases $\theta_{1}^{(t)}$ whereas changes in the pairwise interactions by
Stimulus 1 are relatively small, indicating that Stimulus 1 induces spikes in
Neuron 1. On the contrary, Stimulus 2 increases to $\theta_{2}^{(t)}$,
$\theta_{3}^{(t)}$, and $\theta_{23}^{(t)}$. In particular, the increase in
the interaction parameter $\theta_{23}^{(t)}$ by Stimulus 2 indicates that the
presence of Stimulus 2 induces excess simultaneous spikes in Neuron 2 and
Neuron 3 more often than the chance coincidence expected for the two neurons.
The spike-history effects are summarized as a summed matrix,
$\sum_{i=1}^{p}\mathbf{H}_{i}$, shown in Fig. 3B. Two major effects are
observed. First, the spike history of Neuron $i$ significantly decreases
$\theta_{i}^{(t)}$ ($i=1,2,3$) (See diagonal of the first $3\times 3$ matrix
in Fig. 3B). Figure 3C displays the contribution of a spike in Neuron $i$
during the previous $p$ time bins to the parameter $\theta_{i}^{(t)}$. These
components primarily, albeit not exclusively, capture the renewal property of
the simulated neuron models. Second, the spike history of Neuron 1 increases
$\theta_{2}^{(t)}$, $\theta_{3}^{(t)}$, and $\theta_{23}^{(t)}$ (See the first
column in Fig. 3B), indicating that spike interactions from Neuron 1 to Neuron
2 & 3\. In particular, the increase in $\theta_{23}^{(t)}$ due to the spikes
in Neuron 1 during previous 1-3 time bins (Fig. 3D) indicates that the inputs
from Neuron 1 induces excess synchronous spikes in the other two neurons with
approximately 2-6 ms delay.
## 5 Conclusion
We developed a parametric method for estimating stimulus responses and spike-
history effects on the simultaneous spiking activity of multiple neurons when
the ensemble themselves exhibit ongoing activity. The method was tested by
simulated multiple neuronal spiking activity with known underlying
architecture. We provided two methods to corroborate the fitted models. First,
based on the result in the preceding paper, we provided an approximate
equation for the log marginal likelihood (see Appendix B), which was used to
select the most predictive state-space model. Second, we provided a method for
obtaining confidence bounds of the estimated parameters based on a surrogate
approach.
Example spike sequences simulated in this study are overly simplified.
Therefore, the method needs be tested using real neuronal spike data, e.g.,
from cultured neurons whose underlying circuit is identified by
electrophysiological studies. In practical applications, it is recommended to
utilize basis functions such as raised cosine bumps used in [23] in the
exogenous terms in order to capture the stimulus and spike-history effects
with a fewer parameters. In addition, an appropriate bin size must be selected
in order to obtain a meaningful result in the analysis of real data. Since the
bin size determines a permissible range of synchronous activity, a
physiological interpretation of the result depends on the choice of the bin
size. It is thus recommended to present results based on multiple different
bin sizes in order to confirm a specific hypothesis in a study as shown in
[29, 10]. Methods to overcome an artifact due to the disjoint binning are
discussed in [30, 31, 32, 33, 34]. In future, inclusion of such advanced
methods will allow us to detect near-synchronous responses without sacrificing
temporal resolution of the analysis.
Given that applicability of the method is confirmed in real data, the proposed
method is useful to investigate how ensemble activity of multiple neurons in a
local circuit changes configurations of their simultaneous responses
(synchronous responses) to different stimuli applied to an animal. Further, it
would be interesting to see different effects of the same stimulus on the
ensemble activity when an animal undergoes different cortical states.
The present study is based on the modeling framework developed in [9, 10]. The
author acknowledges Prof. Shun-ichi Amari, Prof. Emery N. Brown, and Prof.
Sonja Grün for their support in construction of the original model. The author
also thanks to Dr. Christopher L. Buckley and Dr. Erin Munro for critical
reading of the manuscript.
## Appendix A Construction of a posterior density by the recursive Bayesian
filtering/smoothing algorithm
A posterior density of the time-varying $\boldsymbol{\theta}_{t}$, which
specifies the joint probability mass function of spike patterns at time bin
$t$, are obtained by a non-linear recursive Bayesian estimation method
developed in [9, 10]. The method allows us to find a maximum a posteriori
(MAP) estimate of $\boldsymbol{\theta}_{t}$ and its uncertainty, namely the
most probable paths of time-varying canonical parameters
$\boldsymbol{\theta}_{t}$ and their confidence bounds given the observed
simultaneous activity of multiple neurons. The estimation procedure completes
by a forward recursion to construct a filter posterior density and then by a
backward recursion to construct a smoother posterior density. In this
approach, the posterior densities are approximated as a multivariate normal
probability density function.
In the forward filtering step, we first compute mean and covariance of one
step prediction density:
$\displaystyle\boldsymbol{\theta}_{t|t-1}$
$\displaystyle=\mathbf{F}\boldsymbol{\theta}_{t-1|t-1}+\mathbf{U}\mathbf{u}_{t},$
(18) $\displaystyle\mathbf{W}_{t|t-1}$
$\displaystyle=\mathbf{F}W_{t-1|t-1}\mathbf{F}^{\prime}+\mathbf{Q}.$ (19)
Then, a mean vector and covariance matrix of the filter posterior density,
which is approximated as a normal density, is given as
$\displaystyle\boldsymbol{\theta}_{t|t}$
$\displaystyle=\boldsymbol{\theta}_{t|t-1}+n\mathbf{W}_{t|t-1}(\mathbf{y}_{t}-\boldsymbol{\eta}_{t|t}),$
(20) $\displaystyle\mathbf{W}_{t|t}^{-1}$
$\displaystyle=\mathbf{W}_{t|t-1}^{-1}+n\mathbf{J}_{t|t},$ (21)
where
$\boldsymbol{\eta}_{t|t}=E\left[\mathbf{f}\left(\mathbf{x}\right)|\mathbf{X}_{1:t},\mathbf{w}\right]$
is the simultaneous spike rates at time bin $t$ expected from the joint
probability mass function, Eq. 1, specified by $\boldsymbol{\theta}_{t|t}$.
Thus Eq. 20 is a non-linear equation. We solve Eq. 20 by a Newton-Raphson
method. It can be shown that the solution is unique. The matrix
$\mathbf{J}_{t|t}$ is a Fisher information matrix of Eq. 1 evaluated at
$\boldsymbol{\theta}_{t|t}$.
Finally, we compute mean and covariance of a smoother posterior density as
$\displaystyle\boldsymbol{\theta}_{t|T}$
$\displaystyle=\boldsymbol{\theta}_{t|t}+\mathbf{A}_{t}\left(\boldsymbol{\theta}_{t+1|T}-\boldsymbol{\theta}_{t+1|t}\right),$
(22) $\displaystyle\mathbf{W}_{t|T}$
$\displaystyle=\mathbf{W}_{t|t}+\mathbf{A}_{t}\left(\mathbf{W}_{t+1|T}-\mathbf{W}_{t+1|t}\right)\mathbf{A}_{t}^{\prime}.$
(23)
with
$\mathbf{A}_{t}=\mathbf{W}_{t|t}\mathbf{F}^{\prime}\mathbf{W}_{t+1|t}^{-1}$
for $t=T,T-1,\ldots,2,1$. Namely, we start computing Eqs. 22 and 23 in a
backward manner, using $\boldsymbol{\theta}_{T|T}$ and $\mathbf{W}_{T|T}$
obtained in the filtering method at the initial step. The lag-one covariance
smoother, $W_{t-1,t|T}$, is obtained using the method of De Jong and Mackinnon
[35]:
$\displaystyle\mathbf{W}_{t-1,t|T}$ $\displaystyle\equiv
E[\left.(\boldsymbol{\theta}_{t-1}-\boldsymbol{\theta}_{t-1|T})(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|T})^{\prime}\right|y_{1:T}]=\mathbf{A}_{t-1}\mathbf{W}_{t|T}.$
(24)
## Appendix B Approximate marginal likelihood function
The approximated formula of the log marginal likelihood (Eq. 4) was obtained
in [10] as
$\displaystyle l(\mathbf{w})\approx$
$\displaystyle\sum_{t=1}^{T}n\left(\mathbf{y}_{t}^{\prime}\boldsymbol{\theta}_{t|t}-\psi\left(\boldsymbol{\theta}_{t|t}\right)\right)+\frac{1}{2}\sum_{t=1}^{T}\left(\log{\det
W_{t|t}}-\log{\det W_{t|t-1}}\right)$
$\displaystyle-\frac{1}{2}\sum_{t=1}^{T}tr\left[\mathbf{W}_{t|t-1}^{-1}\left(\boldsymbol{\theta}_{t|t}-\boldsymbol{\theta}_{t|t-1}\right)\left(\boldsymbol{\theta}_{t|t}-\boldsymbol{\theta}_{t|t-1}\right)^{\prime}\right].$
(25)
Here we briefly provide the derivation (See [10] for details). The log
marginal likelihood is written as
$\displaystyle l(\mathbf{w})$ $\displaystyle=\sum_{t=1}^{T}\log
p(\mathbf{y}_{t}|\mathbf{y}_{1:t-1},\mathbf{w})=\sum_{t=1}^{T}\log\int
p(\mathbf{y}_{t}|\boldsymbol{\theta}_{t})p(\boldsymbol{\theta}_{t}|\mathbf{y}_{1:t-1},\mathbf{w})d\boldsymbol{\theta}_{t}.$
(26)
The integral in the above equation is approximated as
$\displaystyle\int
p(\mathbf{y}_{t}|\boldsymbol{\theta}_{t})p(\boldsymbol{\theta}_{t}|\mathbf{y}_{1:t-1},\mathbf{w})d\boldsymbol{\theta}_{t}$
$\displaystyle=\frac{1}{\sqrt{(2\pi)^{d}|W_{t|t-1}|}}\int\exp\left[q(\boldsymbol{\theta}_{t})\right]d\boldsymbol{\theta}_{t}\approx\frac{\sqrt{(2\pi)^{d}|W_{t|t}|}}{\sqrt{(2\pi)^{d}|W_{t|t-1}|}}\exp\left[q\left(\boldsymbol{\theta}_{t|t}\right)\right],$
(27)
where
$q(\boldsymbol{\theta}_{t})=n\left(\mathbf{y}_{t}^{\prime}\boldsymbol{\theta}_{t}-\psi\left(\boldsymbol{\theta}_{t}\right)\right)-\frac{1}{2}\left(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|t-1}\right)^{\prime}\mathbf{W}_{t|t-1}^{-1}\left(\boldsymbol{\theta}_{t}-\boldsymbol{\theta}_{t|t-1}\right).$
To obtain the second approximate equality, we used the Laplace approximation:
the integral in Eq. 27 is given as
$\int\exp\left[q\left(\boldsymbol{\theta}_{t}\right)\right]d\boldsymbol{\theta}_{t}\approx\sqrt{(2\pi)^{d}|W_{t|t}|}\exp\left[q\left(\boldsymbol{\theta}_{t|t}\right)\right]$.
Here we note that a solution of $q(\boldsymbol{\theta}_{t})=0$ is equivalent
to the filter MAP estimate, $\boldsymbol{\theta}_{t|t}$. By applying Eq. 27 to
Eq. 26, we obtain Eq. 25.
## References
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|
arxiv-papers
| 2013-12-16T14:50:26 |
2024-09-04T02:49:55.500226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hideaki Shimazaki",
"submitter": "Hideaki Shimazaki",
"url": "https://arxiv.org/abs/1312.4382"
}
|
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